The KIV model - nonlinear spatio-temporal dynamics of the primordial vertebrate forebrain.

2097-3012(2023)01-0041-08 Journal of Spatio-temporal Information 时空信息学报收稿日期: 2023-01-03；修订日期: 2023-04-14 基金项目: 国家自然科学基金项目（42201474）作者简介: 刘欣怡，研究方向为点云影像联合处理、实景三维建模等。

E-mail:*****************.cn无人机倾斜摄影三维建模技术研究现状及展望刘欣怡，张永军，范伟伟，王森援，岳冬冬，刘梓航，贾琛，景慧莹，钟佳辰武汉大学 遥感信息工程学院，武汉 430079摘 要：无人机倾斜摄影从航空平台多视角同步采集影像，可通过倾斜摄影测量，三维建模等技术生成真实的三维模型，是目前地形级到城市级实景三维模型重建最主要的技术手段之一。

本文总结归纳了目前无人机倾斜摄影三维建模过程中的关键技术，重点介绍了三维场景重建和倾斜摄影三角网模型的语义提取、单体化、实体化处理等技术方法及研究现状，指出了目前无人机倾斜摄影三维建模技术仍存在的问题，并从无人机路径规划、数据获取、建模技术、模型表达四方面对该领域潜在的发展趋势和研究方向进行了分析与展望。

关键词：无人机；倾斜摄影测量；实景三维中国；三维建模；实体化引用格式：刘欣怡, 张永军, 范伟伟, 王森援, 岳冬冬, 刘梓航, 贾琛, 景慧莹, 钟佳辰. 2023. 无人机倾斜摄影三维建模技术研究现状及展望. 时空信息学报, 30(1): 41-48Liu X Y, Zhang Y J, Fan W W, Wang S Y, Yue D D, Liu Z H, Jia C, Jing H Y, Zhong J C. 2023. 3D modeling based on UAV oblique photogrammetry: Research status and prospect. Journal of Spatio-temporal Information, 30(1): 41-48, doi: 10.20117/j.jsti.2023010061 引 言实景三维是对人类生产、生活、生态空间进行真实、立体、时序化反映和表达的数字虚拟空间，是国家新型基础设施建设的重要组成部分。

隐马尔可夫模型状态空间模型
隐马尔可夫模型（Hidden Markov Model，HMM）和状态空间模型都是用于描述时间序列数据的统计模型。

隐马尔可夫模型是一种基于概率的图模型，用于描述一个序列的状态随时间变化的过程。

其中，观测序列代表着我们观察到的数据序列，而状态序列则是指导着这些数据生成的隐藏状态序列。

HMM的核心是建立起一个概率转移矩阵，描述了当前状态之间的转移概率；以及一个观测概率矩阵，描述了当前状态下生成观测序列的概率。

HMM常用于语音识别、自然语言处理、音乐分析、生物信息学等领域。

状态空间模型（State Space Model，SSM）也是一种描述时间序列数据的统计模型。

状态空间模型通常由两个部分组成：状态方程和观测方程。

状态方程描述了系统的状态如何随着时间推移而变化，而观测方程则描述了如何从这个状态产生观测值。

SSM也可以看作是一个概率图模型，其中状态变量是在时间上链接的随机变量，不可被直接观测到；观测变量是其生成的可观测结果。

SSM常用于时间序列分析、金融预测、天气预报等领域。

基于帕累托前沿面曲率预估的超多目标进化算法基于帕累托前沿面曲率预估的超多目标进化算法序言：超多目标优化问题在现实世界中非常常见，涉及到多个冲突的目标。

为了解决这类问题，进化算法被广泛采用。

然而，当目标超过三个时，直接应用进化算法面临挑战，其中之一是如何有效地选择适当的解集。

对于这个问题，一种新的方法——基于帕累托前沿面曲率预估的超多目标进化算法应运而生。

介绍：帕累托前沿面曲率预估是一种通过分析帕累托前沿面的曲率特征来预测解的优劣的方法。

在超多目标进化算法中，该方法可以用于帮助选择最优解集。

在本文中，我将深入探讨基于帕累托前沿面曲率预估的超多目标进化算法的原理、优势、应用以及我的个人观点和理解。

一、基本原理1.1 帕累托前沿面曲率预估的概念帕累托前沿面曲率预估是基于帕累托前沿面的曲率进行预测的方法。

帕累托前沿面是一组最优解的集合，其中任何解的改进都会导致至少一个目标的恶化。

曲率被认为是评估前沿面的弯曲程度的一种方式。

通过分析前沿面上的点的曲率，可以得出一些关于全局优化的启示。

1.2 算法流程基于帕累托前沿面曲率预估的超多目标进化算法的流程如下：1) 初始化种群；2) 计算种群中每个个体的目标函数值，并按照帕累托支配关系将个体分为不同的支配层次；3) 对于每个支配层次，计算该层次上每个个体在前沿面上的曲率；4) 根据曲率预估，选择某个阈值，将曲率小于该阈值的个体加入解集；5) 将其他个体作为种群重新进行进化操作；6) 重复步骤2至5，直到满足停止条件。

二、优势与应用2.1 优势基于帕累托前沿面曲率预估的超多目标进化算法具有以下优势：- 可以预测解的优劣，帮助选择最优解集；- 通过曲率分析，能够发现前沿面上的局部极值点；- 可以加速算法的收敛过程，提高求解效率；- 在处理带有冲突目标的问题时，表现出较好的性能。

2.2 应用基于帕累托前沿面曲率预估的超多目标进化算法已经在多个领域得到了成功应用，比如：- 交通规划中的路网设计优化；- 供应链管理中的供应商选择问题；- 机器学习中的特征选择与神经网络设计；- 网络安全领域的漏洞修复策略制定等。

Modeling the Spatial Dynamics of Regional Land Use:The CLUE-S ModelPETER H.VERBURG*Department of Environmental Sciences Wageningen UniversityP.O.Box376700AA Wageningen,The NetherlandsandFaculty of Geographical SciencesUtrecht UniversityP.O.Box801153508TC Utrecht,The NetherlandsWELMOED SOEPBOERA.VELDKAMPDepartment of Environmental Sciences Wageningen UniversityP.O.Box376700AA Wageningen,The NetherlandsRAMIL LIMPIADAVICTORIA ESPALDONSchool of Environmental Science and Management University of the Philippines Los Ban˜osCollege,Laguna4031,Philippines SHARIFAH S.A.MASTURADepartment of GeographyUniversiti Kebangsaan Malaysia43600BangiSelangor,MalaysiaABSTRACT/Land-use change models are important tools for integrated environmental management.Through scenario analysis they can help to identify near-future critical locations in the face of environmental change.A dynamic,spatially ex-plicit,land-use change model is presented for the regional scale:CLUE-S.The model is specifically developed for the analysis of land use in small regions(e.g.,a watershed or province)at afine spatial resolution.The model structure is based on systems theory to allow the integrated analysis of land-use change in relation to socio-economic and biophysi-cal driving factors.The model explicitly addresses the hierar-chical organization of land use systems,spatial connectivity between locations and stability.Stability is incorporated by a set of variables that define the relative elasticity of the actual land-use type to conversion.The user can specify these set-tings based on expert knowledge or survey data.Two appli-cations of the model in the Philippines and Malaysia are used to illustrate the functioning of the model and its validation.Land-use change is central to environmental man-agement through its influence on biodiversity,water and radiation budgets,trace gas emissions,carbon cy-cling,and livelihoods(Lambin and others2000a, Turner1994).Land-use planning attempts to influence the land-use change dynamics so that land-use config-urations are achieved that balance environmental and stakeholder needs.Environmental management and land-use planning therefore need information about the dynamics of land use.Models can help to understand these dynamics and project near future land-use trajectories in order to target management decisions(Schoonenboom1995).Environmental management,and land-use planning specifically,take place at different spatial and organisa-tional levels,often corresponding with either eco-re-gional or administrative units,such as the national or provincial level.The information needed and the man-agement decisions made are different for the different levels of analysis.At the national level it is often suffi-cient to identify regions that qualify as“hot-spots”of land-use change,i.e.,areas that are likely to be faced with rapid land use conversions.Once these hot-spots are identified a more detailed land use change analysis is often needed at the regional level.At the regional level,the effects of land-use change on natural resources can be determined by a combina-tion of land use change analysis and specific models to assess the impact on natural resources.Examples of this type of model are water balance models(Schulze 2000),nutrient balance models(Priess and Koning 2001,Smaling and Fresco1993)and erosion/sedimen-tation models(Schoorl and Veldkamp2000).Most of-KEY WORDS:Land-use change;Modeling;Systems approach;Sce-nario analysis;Natural resources management*Author to whom correspondence should be addressed;email:pverburg@gissrv.iend.wau.nlDOI:10.1007/s00267-002-2630-x Environmental Management Vol.30,No.3,pp.391–405©2002Springer-Verlag New York Inc.ten these models need high-resolution data for land use to appropriately simulate the processes involved.Land-Use Change ModelsThe rising awareness of the need for spatially-ex-plicit land-use models within the Land-Use and Land-Cover Change research community(LUCC;Lambin and others2000a,Turner and others1995)has led to the development of a wide range of land-use change models.Whereas most models were originally devel-oped for deforestation(reviews by Kaimowitz and An-gelsen1998,Lambin1997)more recent efforts also address other land use conversions such as urbaniza-tion and agricultural intensification(Brown and others 2000,Engelen and others1995,Hilferink and Rietveld 1999,Lambin and others2000b).Spatially explicit ap-proaches are often based on cellular automata that simulate land use change as a function of land use in the neighborhood and a set of user-specified relations with driving factors(Balzter and others1998,Candau 2000,Engelen and others1995,Wu1998).The speci-fication of the neighborhood functions and transition rules is done either based on the user’s expert knowl-edge,which can be a problematic process due to a lack of quantitative understanding,or on empirical rela-tions between land use and driving factors(e.g.,Pi-janowski and others2000,Pontius and others2000).A probability surface,based on either logistic regression or neural network analysis of historic conversions,is made for future conversions.Projections of change are based on applying a cut-off value to this probability sur-face.Although appropriate for short-term projections,if the trend in land-use change continues,this methodology is incapable of projecting changes when the demands for different land-use types change,leading to a discontinua-tion of the trends.Moreover,these models are usually capable of simulating the conversion of one land-use type only(e.g.deforestation)because they do not address competition between land-use types explicitly.The CLUE Modeling FrameworkThe Conversion of Land Use and its Effects(CLUE) modeling framework(Veldkamp and Fresco1996,Ver-burg and others1999a)was developed to simulate land-use change using empirically quantified relations be-tween land use and its driving factors in combination with dynamic modeling.In contrast to most empirical models,it is possible to simulate multiple land-use types simultaneously through the dynamic simulation of competition between land-use types.This model was developed for the national and con-tinental level,applications are available for Central America(Kok and Winograd2001),Ecuador(de Kon-ing and others1999),China(Verburg and others 2000),and Java,Indonesia(Verburg and others 1999b).For study areas with such a large extent the spatial resolution of analysis was coarse(pixel size vary-ing between7ϫ7and32ϫ32km).This is a conse-quence of the impossibility to acquire data for land use and all driving factors atfiner spatial resolutions.A coarse spatial resolution requires a different data rep-resentation than the common representation for data with afine spatial resolution.Infine resolution grid-based approaches land use is defined by the most dom-inant land-use type within the pixel.However,such a data representation would lead to large biases in the land-use distribution as some class proportions will di-minish and other will increase with scale depending on the spatial and probability distributions of the cover types(Moody and Woodcock1994).In the applications of the CLUE model at the national or continental level we have,therefore,represented land use by designating the relative cover of each land-use type in each pixel, e.g.a pixel can contain30%cultivated land,40%grass-land,and30%forest.This data representation is di-rectly related to the information contained in the cen-sus data that underlie the applications.For each administrative unit,census data denote the number of hectares devoted to different land-use types.When studying areas with a relatively small spatial ex-tent,we often base our land-use data on land-use maps or remote sensing images that denote land-use types respec-tively by homogeneous polygons or classified pixels. When converted to a raster format this results in only one, dominant,land-use type occupying one unit of analysis. The validity of this data representation depends on the patchiness of the landscape and the pixel size chosen. Most sub-national land use studies use this representation of land use with pixel sizes varying between a few meters up to about1ϫ1km.The two different data represen-tations are shown in Figure1.Because of the differences in data representation and other features that are typical for regional appli-cations,the CLUE model can not directly be applied at the regional scale.This paper describes the mod-ified modeling approach for regional applications of the model,now called CLUE-S(the Conversion of Land Use and its Effects at Small regional extent). The next section describes the theories underlying the development of the model after which it is de-scribed how these concepts are incorporated in the simulation model.The functioning of the model is illustrated for two case-studies and is followed by a general discussion.392P.H.Verburg and othersCharacteristics of Land-Use SystemsThis section lists the main concepts and theories that are prevalent for describing the dynamics of land-use change being relevant for the development of land-use change models.Land-use systems are complex and operate at the interface of multiple social and ecological systems.The similarities between land use,social,and ecological systems allow us to use concepts that have proven to be useful for studying and simulating ecological systems in our analysis of land-use change (Loucks 1977,Adger 1999,Holling and Sanderson 1996).Among those con-cepts,connectivity is important.The concept of con-nectivity acknowledges that locations that are at a cer-tain distance are related to each other (Green 1994).Connectivity can be a direct result of biophysical pro-cesses,e.g.,sedimentation in the lowlands is a direct result of erosion in the uplands,but more often it is due to the movement of species or humans through the nd degradation at a certain location will trigger farmers to clear land at a new location.Thus,changes in land use at this new location are related to the land-use conditions in the other location.In other instances more complex relations exist that are rooted in the social and economic organization of the system.The hierarchical structure of social organization causes some lower level processes to be constrained by higher level dynamics,e.g.,the establishments of a new fruit-tree plantation in an area near to the market might in fluence prices in such a way that it is no longer pro fitable for farmers to produce fruits in more distant areas.For studying this situation an-other concept from ecology,hierarchy theory,is use-ful (Allen and Starr 1982,O ’Neill and others 1986).This theory states that higher level processes con-strain lower level processes whereas the higher level processes might emerge from lower level dynamics.This makes the analysis of the land-use system at different levels of analysis necessary.Connectivity implies that we cannot understand land use at a certain location by solely studying the site characteristics of that location.The situation atneigh-Figure 1.Data representation and land-use model used for respectively case-studies with a national/continental extent and local/regional extent.Modeling Regional Land-Use Change393boring or even more distant locations can be as impor-tant as the conditions at the location itself.Land-use and land-cover change are the result of many interacting processes.Each of these processes operates over a range of scales in space and time.These processes are driven by one or more of these variables that influence the actions of the agents of land-use and cover change involved.These variables are often re-ferred to as underlying driving forces which underpin the proximate causes of land-use change,such as wood extraction or agricultural expansion(Geist and Lambin 2001).These driving factors include demographic fac-tors(e.g.,population pressure),economic factors(e.g., economic growth),technological factors,policy and institutional factors,cultural factors,and biophysical factors(Turner and others1995,Kaimowitz and An-gelsen1998).These factors influence land-use change in different ways.Some of these factors directly influ-ence the rate and quantity of land-use change,e.g.the amount of forest cleared by new incoming migrants. Other factors determine the location of land-use change,e.g.the suitability of the soils for agricultural land use.Especially the biophysical factors do pose constraints to land-use change at certain locations, leading to spatially differentiated pathways of change.It is not possible to classify all factors in groups that either influence the rate or location of land-use change.In some cases the same driving factor has both an influ-ence on the quantity of land-use change as well as on the location of land-use change.Population pressure is often an important driving factor of land-use conver-sions(Rudel and Roper1997).At the same time it is the relative population pressure that determines which land-use changes are taking place at a certain location. Intensively cultivated arable lands are commonly situ-ated at a limited distance from the villages while more extensively managed grasslands are often found at a larger distance from population concentrations,a rela-tion that can be explained by labor intensity,transport costs,and the quality of the products(Von Thu¨nen 1966).The determination of the driving factors of land use changes is often problematic and an issue of dis-cussion(Lambin and others2001).There is no unify-ing theory that includes all processes relevant to land-use change.Reviews of case studies show that it is not possible to simply relate land-use change to population growth,poverty,and infrastructure.Rather,the inter-play of several proximate as well as underlying factors drive land-use change in a synergetic way with large variations caused by location specific conditions (Lambin and others2001,Geist and Lambin2001).In regional modeling we often need to rely on poor data describing this complexity.Instead of using the under-lying driving factors it is needed to use proximate vari-ables that can represent the underlying driving factors. Especially for factors that are important in determining the location of change it is essential that the factor can be mapped quantitatively,representing its spatial vari-ation.The causality between the underlying driving factors and the(proximate)factors used in modeling (in this paper,also referred to as“driving factors”) should be certified.Other system properties that are relevant for land-use systems are stability and resilience,concepts often used to describe ecological systems and,to some extent, social systems(Adger2000,Holling1973,Levin and others1998).Resilience refers to the buffer capacity or the ability of the ecosystem or society to absorb pertur-bations,or the magnitude of disturbance that can be absorbed before a system changes its structure by changing the variables and processes that control be-havior(Holling1992).Stability and resilience are con-cepts that can also be used to describe the dynamics of land-use systems,that inherit these characteristics from both ecological and social systems.Due to stability and resilience of the system disturbances and external in-fluences will,mostly,not directly change the landscape structure(Conway1985).After a natural disaster lands might be abandoned and the population might tempo-rally migrate.However,people will in most cases return after some time and continue land-use management practices as before,recovering the land-use structure (Kok and others2002).Stability in the land-use struc-ture is also a result of the social,economic,and insti-tutional structure.Instead of a direct change in the land-use structure upon a fall in prices of a certain product,farmers will wait a few years,depending on the investments made,before they change their cropping system.These characteristics of land-use systems provide a number requirements for the modelling of land-use change that have been used in the development of the CLUE-S model,including:●Models should not analyze land use at a single scale,but rather include multiple,interconnected spatial scales because of the hierarchical organization of land-use systems.●Special attention should be given to the drivingfactors of land-use change,distinguishing drivers that determine the quantity of change from drivers of the location of change.●Sudden changes in driving factors should not di-rectly change the structure of the land-use system asa consequence of the resilience and stability of theland-use system.394P.H.Verburg and others●The model structure should allow spatial interac-tions between locations and feedbacks from higher levels of organization.Model DescriptionModel StructureThe model is sub-divided into two distinct modules,namely a non-spatial demand module and a spatially explicit allocation procedure (Figure 2).The non-spa-tial module calculates the area change for all land-use types at the aggregate level.Within the second part of the model these demands are translated into land-use changes at different locations within the study region using a raster-based system.For the land-use demand module,different alterna-tive model speci fications are possible,ranging from simple trend extrapolations to complex economic mod-els.The choice for a speci fic model is very much de-pendent on the nature of the most important land-use conversions taking place within the study area and the scenarios that need to be considered.Therefore,the demand calculations will differ between applications and scenarios and need to be decided by the user for the speci fic situation.The results from the demandmodule need to specify,on a yearly basis,the area covered by the different land-use types,which is a direct input for the allocation module.The rest of this paper focuses on the procedure to allocate these demands to land-use conversions at speci fic locations within the study area.The allocation is based upon a combination of em-pirical,spatial analysis,and dynamic modelling.Figure 3gives an overview of the procedure.The empirical analysis unravels the relations between the spatial dis-tribution of land use and a series of factors that are drivers and constraints of land use.The results of this empirical analysis are used within the model when sim-ulating the competition between land-use types for a speci fic location.In addition,a set of decision rules is speci fied by the user to restrict the conversions that can take place based on the actual land-use pattern.The different components of the procedure are now dis-cussed in more detail.Spatial AnalysisThe pattern of land use,as it can be observed from an airplane window or through remotely sensed im-ages,reveals the spatial organization of land use in relation to the underlying biophysical andsocio-eco-Figure 2.Overview of the modelingprocedure.Figure 3.Schematic represen-tation of the procedure to allo-cate changes in land use to a raster based map.Modeling Regional Land-Use Change395nomic conditions.These observations can be formal-ized by overlaying this land-use pattern with maps de-picting the variability in biophysical and socio-economic conditions.Geographical Information Systems(GIS)are used to process all spatial data and convert these into a regular grid.Apart from land use, data are gathered that represent the assumed driving forces of land use in the study area.The list of assumed driving forces is based on prevalent theories on driving factors of land-use change(Lambin and others2001, Kaimowitz and Angelsen1998,Turner and others 1993)and knowledge of the conditions in the study area.Data can originate from remote sensing(e.g., land use),secondary statistics(e.g.,population distri-bution),maps(e.g.,soil),and other sources.To allow a straightforward analysis,the data are converted into a grid based system with a cell size that depends on the resolution of the available data.This often involves the aggregation of one or more layers of thematic data,e.g. it does not make sense to use a30-m resolution if that is available for land-use data only,while the digital elevation model has a resolution of500m.Therefore, all data are aggregated to the same resolution that best represents the quality and resolution of the data.The relations between land use and its driving fac-tors are thereafter evaluated using stepwise logistic re-gression.Logistic regression is an often used method-ology in land-use change research(Geoghegan and others2001,Serneels and Lambin2001).In this study we use logistic regression to indicate the probability of a certain grid cell to be devoted to a land-use type given a set of driving factors following:LogͩP i1ϪP i ͪϭ␤0ϩ␤1X1,iϩ␤2X2,i......ϩ␤n X n,iwhere P i is the probability of a grid cell for the occur-rence of the considered land-use type and the X’s are the driving factors.The stepwise procedure is used to help us select the relevant driving factors from a larger set of factors that are assumed to influence the land-use pattern.Variables that have no significant contribution to the explanation of the land-use pattern are excluded from thefinal regression equation.Where in ordinal least squares regression the R2 gives a measure of modelfit,there is no equivalent for logistic regression.Instead,the goodness offit can be evaluated with the ROC method(Pontius and Schnei-der2000,Swets1986)which evaluates the predicted probabilities by comparing them with the observed val-ues over the whole domain of predicted probabilities instead of only evaluating the percentage of correctly classified observations at afixed cut-off value.This is an appropriate methodology for our application,because we will use a wide range of probabilities within the model calculations.The influence of spatial autocorrelation on the re-gression results can be minimized by only performing the regression on a random sample of pixels at a certain minimum distance from one another.Such a selection method is adopted in order to maximize the distance between the selected pixels to attenuate the problem associated with spatial autocorrelation.For case-studies where autocorrelation has an important influence on the land-use structure it is possible to further exploit it by incorporating an autoregressive term in the regres-sion equation(Overmars and others2002).Based upon the regression results a probability map can be calculated for each land-use type.A new probabil-ity map is calculated every year with updated values for the driving factors that are projected to change in time,such as the population distribution or accessibility.Decision RulesLand-use type or location specific decision rules can be specified by the user.Location specific decision rules include the delineation of protected areas such as nature reserves.If a protected area is specified,no changes are allowed within this area.For each land-use type decision rules determine the conditions under which the land-use type is allowed to change in the next time step.These decision rules are implemented to give certain land-use types a certain resistance to change in order to generate the stability in the land-use structure that is typical for many landscapes.Three different situations can be distinguished and for each land-use type the user should specify which situation is most relevant for that land-use type:1.For some land-use types it is very unlikely that theyare converted into another land-use type after their first conversion;as soon as an agricultural area is urbanized it is not expected to return to agriculture or to be converted into forest cover.Unless a de-crease in area demand for this land-use type occurs the locations covered by this land use are no longer evaluated for potential land-use changes.If this situation is selected it also holds that if the demand for this land-use type decreases,there is no possi-bility for expansion in other areas.In other words, when this setting is applied to forest cover and deforestation needs to be allocated,it is impossible to reforest other areas at the same time.2.Other land-use types are converted more easily.Aswidden agriculture system is most likely to be con-verted into another land-use type soon after its396P.H.Verburg and othersinitial conversion.When this situation is selected for a land-use type no restrictions to change are considered in the allocation module.3.There is also a number of land-use types that oper-ate in between these two extremes.Permanent ag-riculture and plantations require an investment for their establishment.It is therefore not very likely that they will be converted very soon after into another land-use type.However,in the end,when another land-use type becomes more pro fitable,a conversion is possible.This situation is dealt with by de fining the relative elasticity for change (ELAS u )for the land-use type into any other land use type.The relative elasticity ranges between 0(similar to Situation 2)and 1(similar to Situation 1).The higher the de fined elasticity,the more dif ficult it gets to convert this land-use type.The elasticity should be de fined based on the user ’s knowledge of the situation,but can also be tuned during the calibration of the petition and Actual Allocation of Change Allocation of land-use change is made in an iterative procedure given the probability maps,the decision rules in combination with the actual land-use map,and the demand for the different land-use types (Figure 4).The following steps are followed in the calculation:1.The first step includes the determination of all grid cells that are allowed to change.Grid cells that are either part of a protected area or under a land-use type that is not allowed to change (Situation 1,above)are excluded from further calculation.2.For each grid cell i the total probability (TPROP i,u )is calculated for each of the land-use types u accord-ing to:TPROP i,u ϭP i,u ϩELAS u ϩITER u ,where ITER u is an iteration variable that is speci fic to the land use.ELAS u is the relative elasticity for change speci fied in the decision rules (Situation 3de-scribed above)and is only given a value if grid-cell i is already under land use type u in the year considered.ELAS u equals zero if all changes are allowed (Situation 2).3.A preliminary allocation is made with an equalvalue of the iteration variable (ITER u )for all land-use types by allocating the land-use type with the highest total probability for the considered grid cell.This will cause a number of grid cells to change land use.4.The total allocated area of each land use is nowcompared to the demand.For land-use types where the allocated area is smaller than the demanded area the value of the iteration variable is increased.For land-use types for which too much is allocated the value is decreased.5.Steps 2to 4are repeated as long as the demandsare not correctly allocated.When allocation equals demand the final map is saved and the calculations can continue for the next yearly timestep.Figure 5shows the development of the iteration parameter ITER u for different land-use types during asimulation.Figure 4.Representation of the iterative procedure for land-use changeallocation.Figure 5.Change in the iteration parameter (ITER u )during the simulation within one time-step.The different lines rep-resent the iteration parameter for different land-use types.The parameter is changed for all land-use types synchronously until the allocated land use equals the demand.Modeling Regional Land-Use Change397Multi-Scale CharacteristicsOne of the requirements for land-use change mod-els are multi-scale characteristics.The above described model structure incorporates different types of scale interactions.Within the iterative procedure there is a continuous interaction between macro-scale demands and local land-use suitability as determined by the re-gression equations.When the demand changes,the iterative procedure will cause the land-use types for which demand increased to have a higher competitive capacity (higher value for ITER u )to ensure enough allocation of this land-use type.Instead of only being determined by the local conditions,captured by the logistic regressions,it is also the regional demand that affects the actually allocated changes.This allows the model to “overrule ”the local suitability,it is not always the land-use type with the highest probability according to the logistic regression equation (P i,u )that the grid cell is allocated to.Apart from these two distinct levels of analysis there are also driving forces that operate over a certain dis-tance instead of being locally important.Applying a neighborhood function that is able to represent the regional in fluence of the data incorporates this type of variable.Population pressure is an example of such a variable:often the in fluence of population acts over a certain distance.Therefore,it is not the exact location of peoples houses that determines the land-use pattern.The average population density over a larger area is often a more appropriate variable.Such a population density surface can be created by a neighborhood func-tion using detailed spatial data.The data generated this way can be included in the spatial analysis as anotherindependent factor.In the application of the model in the Philippines,described hereafter,we applied a 5ϫ5focal filter to the population map to generate a map representing the general population pressure.Instead of using these variables,generated by neighborhood analysis,it is also possible to use the more advanced technique of multi-level statistics (Goldstein 1995),which enable a model to include higher-level variables in a straightforward manner within the regression equa-tion (Polsky and Easterling 2001).Application of the ModelIn this paper,two examples of applications of the model are provided to illustrate its function.TheseTable nd-use classes and driving factors evaluated for Sibuyan IslandLand-use classes Driving factors (location)Forest Altitude (m)GrasslandSlope Coconut plantation AspectRice fieldsDistance to town Others (incl.mangrove and settlements)Distance to stream Distance to road Distance to coast Distance to port Erosion vulnerability GeologyPopulation density(neighborhood 5ϫ5)Figure 6.Location of the case-study areas.398P.H.Verburg and others。

第 41 卷 ，第 2 期 2024 年4 月15 日国土资源科技管理Vol. 41，No.2Apr. 15，2024 Scientific and Technological Management of Land and Resourcesdoi：10.3969/j.issn.1009-4210.2024.02.005河南省耕地“非农化”时空演变特征分析余庆年，王虎威（河海大学　公共管理学院，江苏　南京　211100）摘　要：深入理解和掌握耕地“非农化”的时空演变特征及其原因对保护耕地并确保粮食安全具有重要意义。

本文针对中国产粮第二大省河南，以县域为单位，基于河南省1980—2020年土地利用长时序空间数据，采用重心转移模型和空间自相关分析等方法，定量揭示全省158个县级评价单元1980—2020年来耕地“非农化”的空间分布特征、时空迁移路径和集聚特征，以期为河南省耕地资源的保护与可持续利用提供依据。

结果表明：（1）河南省耕地资源东西分布差异大，集中连片耕地主要集中在东部，耕地总面积随时间推移呈波动减少。

（2）耕地“非农化”等级时空差异较大，豫西地区耕地“非农化”较为缓和，中部和东部地区呈先快速上升后缓慢下降的态势。

（3）河南省耕地“非农化”空间不均衡性增强，空间格局小幅波动，耕地“非农化”重心以先向东南和西南后向东北的路径迁移。

（4）河南省耕地“非农化”空间分布格局在4个时期均呈现出集聚特性，空间集聚程度先增大后减小，高—高和低—低聚类主导格局变化。

本研究揭示了1980—2020年河南省耕地“非农化”的时空演变格局，为政府管控耕地“非农化”现象、实现耕地资源的可持续发展提供参考和借鉴。

关键词：耕地“非农化”；时空演变特征；重心模型；空间自相关中图分类号：F323.21 文献标志码：Ａ 文章编号：1009-4210-（2024）02-50-12Analysis on Temporal and Spatial Evolution Characteristics of FarmlandConversion in Henan ProvinceYU Qingnian，WANG Huwei（School of Public Administration，Hohai university，Nanjing 211100，Jiangsu，China）Abstract: Understanding and mastering the characteristics and underlying reasons for the spatial-temporal evolution of cultivated land “non-agriculturalization” is of paramount importance to ensure food security. Based on the long-term spatial data of land use in Henan from 1980 to 2020，the second largest grain-producing province in China，this paper adopts the methods of gravity shift model and spatial autocorrelation analysis. To provide a solid foundation for the protection and sustainable use of cultivated land resources in Henan Province，this study quantitatively examines the spatial distribution patterns，temporal-spatial migration routes，and agglomeration characteristics of cultivated land “non-agriculturalization” across 158 county-level units from 1980 through to 2020. The results show that：（1）the distribution of cultivated land resources in Henan Province varies greatly from east to west. The收稿日期：2023-10-31作者简介：余庆年（1977—），女，博士，副教授，硕士生导师，从事土地政策与制度研究。

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Bipedal Walking on Rough Terrain Using Manifold ControlTom Erez and William D.SmartMedia and Machines Lab,Department of Computer Science and EngineeringWashington University in St.Louis,MOetom,wds@Abstract—This paper presents an algorithm for adapting periodic behavior to gradual shifts in task parameters.Since learning optimal control in high dimensional domains is subject to the’curse of dimensionality’,we parametrize the policy only along the limit cycle traversed by the gait,and thus focus the computational effort on a closed one-dimensional manifold,embedded in the high-dimensional state space.We take an initial gait as a departure point,and iterate between modifying the task slightly,and adapting the gait to this modification.This creates a sequence of gaits,each optimized for a different variant of the task.Since every two gaits in this sequence are very similar,the whole sequence spans a two-dimensional manifold,and combining all policies in this 2-manifold provides additional robustness to the system.We demonstrate our approach on two simulations of bipedal robots —the compass gait walker,which is a four-dimensional system, and RABBIT,which is ten-dimensional.The walkers’gaits are adapted to a sequence of changes in the ground slope,and when all policies in the sequence are combined,the walkers can safely traverse a rough terrain,where the incline changes at every step.I.INTRODUCTIONThis paper deals with the general task of augmenting the capacities of legged robots by using reinforcement learn-ing1.The standard paradigm in Control Theory,whereby an optimized reference trajectory is foundfirst,and then a stabilizing controller is designed,can become laborious when a whole range of task variants are considered.Standard algo-rithms of reinforcement learning cannot yet offer compelling alternatives to the control theory paradigm,mostly because of the prohibitive effect of the curse of dimensionality. Legged robots often constitute a high-dimensional system, and standard reinforcement learning methods,with their focus on Markov Decision Processes models,usually cannot overcome the exponential growth in state space volume. Most previous work in machine learning for gait domains required either an exhaustive study of the state space[1], or the use of non-specific optimization techniques,such as genetic algorithms[2].In this paper,we wish to take a first step towards efficient reinforcement learning in high-dimensional domains by focusing on periodic tasks.We make the observation that while legged robots have a high-dimensional state space,not every point in the state space represents a viable pose.By definition,a proper gait would always converge to a stable limit cycle,which traces a closed one-dimensional manifold embedded in the 1The interested reader is encouraged to follow the links mentioned in the footnotes to section IV to see movies of our simulations high-dimensional state space.This is true for any system performing a periodic task,regardless of the size of its state space(see also[3],section3.1,and[4],figure19,for a validation of this point in the model discussed below).This observation holds a great promise:a controller that can keep the system close to one particular limit cycle despite minor perturbations(i.e.has a non-trivial basin of attraction)is free to safely ignore the entire volume of the state space. Finding such a stable controller is far from trivial,and amounts to creating a stable gait.However,for our purpose, such a controller can be suboptimal,and may be supplied by a human tele-operating the system,by leveraging on passive dynamic properties of the system(as in section IV-A),or by applying control theory tools(as in section IV-B).In all cases,the one-dimensional manifold traced by the gait of a stable controller can be identified in one cycle of the gait, simply by recording the state of the system at every time step. Furthermore,by querying the controller,we can identify the local policy on and around that manifold.With these two provided,we can create a local representation of the policy which generated the gait by approximating the policy only on and around that manifold,like a ring embedded in state space,and this holds true regardless of the dimensionality of the state space.By representing the original control function in a compact way we may focus our computational effort on the relevant manifold alone,and circumvent the curse of dimensionality as such a parametrization does not scale exponentially with the number of dimensions.This opens a door for an array of reinforcement learning methods(such as policy gradient)which may be used to adapt the initial controller to different conditions,and thus augment the capacities of the system.In this article we report two experiments.Thefirst studies the Compass-Gait walker([9],[10],[11],a system known for its capacity to walk without actuation on a small range of downward slopes.The second experiment uses a simulation of the robot RABBIT[3],a biped robot with knees and a torso,but no feet,which has been studied before by the control theory community[5],[4],[6].Thefirst model has a four-dimensional state space,and the second model has10 state dimensions and4action dimensions.By composing together several controllers,each adapted to a different incline,we are able to create a composite controller that can stably traverse a rough terrain going downhill.The same algorithm was successfully applied to the second system too, although the size of that problem would be prohibitive formost reinforcement learning algorithms.In the following wefirst give a short review of previous work in machine learning,and then explain the technical aspects of constructing a manifold controller,as well as the learning algorithms used.We then demonstrate the effec-tiveness of Manifold Control by showing how it is used to augment the capacities of existing controllers in two different models of bipedal walk.We conclude by discussing the potential of our approach,and offer directions for future work.II.P REVIOUS W ORKThe generalfield of gait design has been at the focus of mechanical engineering for many years,and recent years saw an increase in the contribution from the domain of machine learning.For example,Stilman et al.[7]studied an eight-dimensional system of a biped robot with knees, similar to the one studied below.They showed that in their particular case the dimensionality can be reduced through some specific approximations during different phases.Then, they partitioned the entire volume of the reduced state space into a grid,and performed Q-learning using a simulation model of the system’s dynamics.The result was a robot walker that can handle a range of slopes around the level horizontal plane.In addition,there is a growing interest in recent years in gaits that can effectively take advantage of the passive dynamics(see the review by Collins et al.[8]for a thorough coverage).Tedrake[9]discusses several versions the com-pass gait walker which were built and analyzed.Controllers for the compass gait based on an analytical treatment of the system equations wasfirst suggested by Goswami et al.[10],who used both hip and ankle actuation.Further results were described by Spong and Bhatia[11],where the case of uneven terrain was also discussed.Ramamoorthy and Kuipers[12]suggested hybrid control of walking over irregular terrain by seeking inspiration from human walking.III.M ANIFOLD C ONTROLA.The Basic Control SchemeThe basic idea in manifold control is to focus the com-putational effort on the limit cycle.Therefore,the policy is approximated using locally activated processing elements (PEs),positioned along the manifold spanned by the limit cycle.Each PE defines a local policy,linear in the position of the state relative to that PE.When the policy is queried with a given state x,the local policy of each PE is calculated as:µi(x)=[1(x−c i)T M T]G i(1) where c i is the location of element i,M is a diagonal matrix which determines the scale of each dimension,and G i is an (n+1)-by-m matrix,where m is the action dimension and n is the number of state space dimensions.G i is made of m columns,one for each action dimension,and each column is an(n+1)-sized gain vector.Thefinal policy u(x)is calculated by mixing the local policies of each PEaccordingFig.1.Illustrations of the models used.On the left,the compass-gait walker:the system’s state is defined by the two legs’angles from the vertical direction and the associated angular velocities,for a total of four dimensions. Thisfigure also depicts the incline of the sloped ground.On the right, RABBIT:the system’s state is defined bt the angle of the torso form the vertical direction,the angles between the thighs and the torso,and the knee angles between the shank and the thigh.This model of RABBIT has ten state dimensions,where at every moment one leg isfixed to the ground, and the other leg is free to swing.to a normalized Gaussian activation function,usingσas a scalar bandwidth term:w i=exp(−(x−c i)T M TσM(x−c i)),(2)u(x)= n i=1w iµitraverse a path of higher value(i.e.collect more rewards,or less cost)along its modified limit cycle.1)Defining the Value Function:In the present work we consider a standard nondiscounted reinforcement learning formulation with afinite time horizon and no terminal costs. More accurately,we define the Value Vπ(x0)of a given state x0under afixed policyπ(x)as:Vπ(x0)= T0r(x t,π(x t))dt(4) where r(x,u)is the reward determined by the current state and the selected action,T is the time horizon,and x t is the solution of the time invariant ordinary differential equation ˙x=f(x,π(x))with the initial condition x=x0,so thatx t= t0f(xτ,π(xτ))dτ.(5) 2)Approximating the Policy Gradient:With C being the locations of the processing elements,and G being the set of their local gains,we make use of a method,due to[14],of piecewise estimation of the gradient of the value function at a given initial state x0with respect to the parameter set G. As Munos showed in[14],from(4)we can write∂V∂G,(6) and for the general form r=r(x,u)we can decompose∂r/∂G as∂r∂u ∂u∂x∂xFig.2.The path used to test the compass gait walker,and an overlay of the walker traversing this path.Note how the step length adapts to the changing incline.the forward swing,but will undergo an inelastic collision with thefloor during the backward swing.At this point it will become the stance leg,and the other leg will be set free to swing.The entire system is placed on a plane that is inclined at a variable angleφfrom the horizon.In the interests of brevity,we omit a complete description of the system dynamics here,referring the interested reader to the relevant literature[15],[10],[11].Although previous work considered actuation both at the hip and the ankle,we chose to study the case of actuation at the hip only.The learning phase in this system was done using simple stochastic gradient ascent,rather than the elaborate policy gradient estimation described in section III-B.2.The initial manifold was sampled at an incline ofφ=−0.05(the initial policy is the zero policy,so there were no approximation errors involved).One shaping iteration consisted of the following:first,G was modified to G tent=G+ηδG, withη=0.1andδG drawn at random from a multinormal distribution with unit covariance.The new policy’s perfor-mance was measured as a sum of all the rewards along20 steps.If the value of this new policy was bigger than the present one,it was adopted,otherwise it was rejected.Then, a newδG was drawn,and the process repeated itself.After 3successful adoptions,the shaping iterations step concluded with a resampling of the new controller,and the incline was decreased by0.01.After10shaping iteration steps,we had controllers that could handle inclines up toφ=−0.14.After another10 iteration steps with the incline increasing by0.005,we had controllers that could handle inclines up toφ=0.0025(a slight uphill).This range is approxmately double the limit of the passive walker[15].Finally,we combined the various controllers into one composite controller.This new controller used1500charts to span a two-dimensional manifold embedded in the four-dimensional state space.The performance of the composite controller was tested on an uneven terrain where the incline was gradually changed fromφ=0toφ=0.15radians, made of“tiles”of variable length whose inclines were0.01 radians apart.figure III-B.2shows an overlay of the walker’s downhill path.A movie of this march is available online.2B.The Uphill-Walking RABBIT RobotWe applied manifold control also to simulations of the legged robot RABBIT,using code from Prof.Jessy Grizzle that is freely available online[16].RABBIT is a biped robot with a torso,two knees and no feet(seefigure1b.),and is actuated at four places:both hip joins(where thighs are actuated against the torso),and both knees(where shanks are actuated against the thighs).The simulation assumes a stance leg with no slippage,and a swing leg that is free to move at all directions until it collides inelastically with the floor,and becomes the stance leg,freeing the other leg to swing.This robot too is modeled as a nonlinear system with impulse effects.Again,we are forced to omit a complete reconstruction of the model’s details,and refer the interested reader to[4],equation8.This model was studied extensively by the control theory community.In particular,an optimal desired signal was derived in[6],and a controller that successfully realizes this signal was presented in[4].However,all the efforts were focused on RABBIT walking on even terrain.We sought a way to augment the capacities of the RABBIT model,and allow it to traverse a rough,uneven terrain.We found that the controller suggested by[4]can easily handle negative (downhill)incline of0.2radiand and more,but cannot handle positive(uphill)inclines.3.Learning started by approximating the policy from[4]as a manifold controller,using400processing elements with a mean distance of about0.03state space length units.The performance of the manifold controller was indistinguishable to the naked eye from the original controller,and perfor-mance,as measured by the performance criterion C3in[6] (the same used by[4]),was only1%worse,probably due to minor approximation errors.The policy gradient was estimated using(6),according to a simple reward model:r(x,u)=10v x hip−1Fig.3.The rough terrain traversed by RABBIT.Since this model has knees,it can walk both uphill and downhill.Note how the step length adapts to the changing terrain.The movie of this parade can be seen at tt http:/2b8sdm,which is a shortcut to the YouTube website.where v xhip is the velocity of the hip joint(where the thighand the torso meet)in the positive direction of the X-axis, and u max is a scalar parameter(in our case,chosen to be 120)that tunes the nonlinear action penalty and promotes energetic efficiency.After the initial manifold controller was created,the system followed a fully automated shaping protocol for 20iterations:at every iteration,∂V/∂G was estimated, andηwasfixed to0.1%of|G|.This small learning rate ensured that we don’t modify the policy too much and lose stability.The modified policy,assumed to be slightly better, was then tested on a slightly bigger incline(the veryfirst manifold controller was tried on an incline of0rad.,and in every iteration we increased the incline in0.003rad.).This small modification to the model parameters ensured that the controller can still walk stably on the incline.If stability was not lost(as was the case in all our iterations),we resampled u(·;C,G new)so that C adj overlapped the limit cycle of the modified system(with the new policy and new incline),and the whole process repeated.This procedure allowed a gradual increase in the system’s maximal stable incline.Figure4depicts the evolution of the stability margins of every ring along the shaping iteration:for every iteration we present an upper(and lower)bound on the incline for which the controller can maintain stability.Thiswas tested by setting a test incline,and allowing the system to run for10 seconds.If no collapse happened by this time,the test incline was raised(lowered),until an incline was found for which the system can no longer maintain stability.As this picture shows,our automated shaping protocol does not maintain a tight control on the stability margins-for most iterations,a modest improvement is recorded.The system’s nonlinearity is well illustrated by the curious case of iteration9,where the same magnitude ofδG causes a massive improvement, despite the fact that the control manifold itself didn’t change dramatically(seefigure5).The converse is also true for some iterations(such as17and18)there is a decrease in the stability margins,but this is not harming the overall effectiveness,since these iterations are using training data obtained at an incline that is very far from the stability Fig.4.Thisfigure shows the inclines for which each iteration could maintain a stable gait on the RABBIT model.The diagonal line shows the incline for which each iteration was trained.Iteration0is the original controller.The initial manifold control approximation degrades most of the stability margin of the original control,but this is quickly regained through adaptation.Note that both the learning rate and the incline change rate were held constant through the entire process.The big jump in iteration 9exemplifies the nonlinearity of the system,as small changes may have unpredictable results,in this case,for the best.margin.Finally,three iterations were composed together,and the resulting controller successfully traversed a rough terrain that included inclines from−0.05to0.15radians.Figure3 shows an overlay image of the rough path.V.C ONCLUSION AND F UTURE W ORKIn this paper we present a compact representation of the policy for periodic tasks,and apply a trajectory-based policy gradient algorithm to it.Most importantly,the methods we present do not scale exponentially with the number of dimensions,and hence allow us to circumvent the curse of dimensionality in the particular case of periodic tasks.By following a gradual shaping process,we are able to create robust controllers that augment the capacities of existing systems in a consistent way.33.54−1.5−0.2anglea n g . v e l.anglea n g . v el.anglea n g . v e l.anglea n g . v el.anglea n g . v e l.anglea n g . v el.anglea n g . v e l.anglea n g . v el.anglea n g . v e l.anglea n g . v el.anglea n g . v e l.anglea n g . v el.anglea n g . v e l.anglea n g . v e l.Fig.5.A projection of the manifold of several stages of the shaping process for the RABBIT model.The top row shows the angle and angular velocity between the torso and the stance thigh,and the bottom row shows the angle and angular velocity of the knee of the swing leg.Every two consecutive iterations are only slightly different from each other.Throughout the entire shaping process,changes accumulate,and new solutions emerge.Manifold control may also be used when the initial con-troller is profoundly suboptimal 4.It is also important to note that the rough terrain was traversed without informing the walker of the current terrain.We may say that the walkers walked blindly on their rough path.This demonstrates how stable a composite manifold controller can be.However,in some practical applications it could be beneficial to represent this important piece of information explicitly,and select the most appropriate ring at every step.We believe that the combination of local learning and careful shaping holds a great promise to many applications of periodic tasks,and hope to demonstrate it through future work on even higher-dimensional systems.Future research directions could include methods that allow second-order convergence,and learning a model of the plant.R EFERENCES[1]M.Stilman,C.G.Atkeson,J.J.Kuffner,and G.Zeglin,“Dynamicprogramming in reduced dimensional spaces:Dynamic planning for robust biped locomotion,”in Proceedings of the 2005IEEE Interna-tional Conference on Robotics and Automation (ICRA 2005),2005,pp.2399–2404.[2]J.Buchli, F.Iida,and A.Ijspeert,“Finding resonance:Adaptivefrequency oscillators for dynamic legged locomotion,”in Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS).IEEE,2006,pp.3903–3909.[3] C.Chevallereau and P.Sardain,“Design and actuation optimization ofa 4-axes biped robot for walking and running,”in Proceedings of the IEEE International Conference on Robotics and Automation (ICRA),2000.[4] F.Plestan,J.W.Grizzle,E.Westervelt,and G.Abba,“Stable walkingof a 7-dof biped robot,”IEEE Trans.Robot.Automat.,vol.19,no.4,pp.653–668,Aug.2003.4theinterested reader is welcome to see other results of manifold learning on a 14-dimensional system at /2h3qny and /2462j7.[5] C.Sabourin,O.Bruneau,and G.Buche,“Experimental validation ofa robust control strategy for the robot rabbit,”in Proceedings of the IEEE International Conference on Robotics and Automation (ICRA),2005.[6] C.Chevallereau and Y .Aoustin,“Optimal reference trajectories forwalking and running of a biped robot,”Robotica ,vol.19,no.5,pp.557–569,2001.[7]M.Stilman,C.G.Atkeson,J.J.Kuffner,and G.Zeglin,“Dynamicprogramming in reduced dimensional spaces:Dynamic planning for robust biped locomotion,”in Proceedings of the 2005IEEE Interna-tional Conference on Robotics and Automation (ICRA 2005),2005,pp.2399–2404.[8]S.H.Collins,A.Ruina,R.Tedrake,,and M.Wisse,“Efficient bipedalrobots based on passive-dynamic walkers,”Science ,pp.307:1082–1085,February 2005.[9]R.L.Tedrake,“Applied optimal control for dynamically stable leggedlocomotion,”Ph.D.dissertation,Massachusetts Institute of Technol-ogy,August 2004.[10] A.Goswami, B.Espiau,and A.Keramane,“Limit cycles in apassive compass gait biped and passivity-mimicking control laws,”Autonomous Robots ,vol.4,no.3,pp.273–286,1997.[11]M.W.Spong and G.Bhatia,“Further results on the control of the com-pass gait biped,”in Proceedings of the 2003IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2003),vol.2,2003,pp.1933–1938.[12]S.Ramamoorthy and B.Kuipers,“Qualitative hybrid control ofdynamic bipedal walking,”in Robotics :Science and Systems II ,G.S.Sukhatme,S.Schaal,W.Burgard,and D.Fox,Eds.MIT Press,2007.[13]S.Schaal and C.Atkeson,“constructive incremental learning fromonly local information,”neural computation ,no.8,pp.2047–2084,1998.[14]R.Munos,“Policy gradient in continuous time,”Journal of MachineLearning Research ,vol.7,pp.771–791,2006.[15] A.Goswami,B.Thuilot,and B.Espiau,“Compass-like biped robotpart i:Stability and bifurcation of passive gaits,”INRIA,Tech.Rep.2996,October 1996.[16] E.Westervelt, B.Morris,and J.Grizzle.(2003)Five link walker.IEEE-CDC Workshop:Feedback Control of Biped Walking Robots.[Online].Available:/2znlz2。

a r X i v :n l i n /0212002v 2 [n l i n .S I ] 21 M a r 2003Nonlinear superposition formula for N =1supersymmetric KdV Equation Q.P.Liu and Y.F.Xie Department of Mathematics,China University of Mining and Technology,Beijing 100083,China.Abstract In this paper,we derive a B¨a cklund transformation for the supersymmetric Korteweg-de Vries equation.We also construct a nonlinear superposition formula,which allows us to rebuild systematically for the supersymmetric KdV equation the soliton solutions of Carstea,Ramani and Grammaticos.The celebrated Korteweg-de Vries (KdV)equation was extended into super frame-work by Kupershmidt [3]in 1984.Shortly afterwards,Manin and Radul [7]proposed another super KdV system which is a particular reduction of their general supersymmet-ric Kadomstev-Petviashvili hierarchy.In [8],Mathieu pointed out that the super version of Manin and Radul for the KdV equation is indeed invariant under a space supersymmetric transformation,while Kupershmidt’s version does not.Thus,the Manin-Radul’s super KdV is referred to the supersymmetric KdV equation.We notice that the supersymmetric KdV equation has been studied extensively in lit-erature and a number of interesting properties has been established.We mention here the infinite conservation laws [8],bi-Hamiltonian structures [10],bilinear form [9][2],Darboux transformation [6].By the constructed Darboux transformation,Ma˜n as and one of us calculated the soli-ton solutions for the supersymmetric KdV system.This sort of solutions was also obtained by Carstea in the framework of bilinear formalism [1].However,these solutions are char-acterized by the presentation of some constraint on soliton parameters.Recently,using super-bilinear operators,Carstea,Ramani and Grammaticos [2]constructed explicitly new two-and three-solitons for the supersymmetric KdV equation.These soliton solutions are interesting since they are free of any constraint on soliton parameters.Furthermore,the fermionic part of these solutions is dressed through the interactions.In addition to the bilinear form approach,B¨a cklund transformation (BT)is also a powerful method to construct solutions.Therefore,it is interesting to see if the soliton solutions of Carstea-Ramani-Grammaticos can be constructed by BT approach.In this paper,we first construct a BT for the supersymmetric KdV equation.Then,we derive a nonlinear superposition formula.In this way,the soliton solutions can be producedsystematically.We explicitly show that the two-soliton solution of Carstea,Ramani and Grammaticos appears naturally in the framework of BT.To introduce the supersymmetric extension for the KdV equation,we recall some terminology and notations.The classical spacetime is(x,t)and we extend it to a super-spacetime(x,t,θ),whereθis a Grassmann odd variable.The dependent variable u(x,t) in the KdV equation is replaced by a fermionic variableΦ=Φ(x,t,θ).Now the super-symmetric KdV equation reads asΦt−3(ΦDΦ)x+Φxxx=0,(1)where D=∂∂x is the superderivative.Mathieu found the following supersymmetricversion of Gardner type mapΦ=χ+ǫχx+ǫ2χ(Dχ),(2) whereǫis an ordinary(bosonic)parameter.It is easy to show thatχsatisfies the following supersymmetric Gardner equationχt−3(χDχ)x−3ǫ2(Dχ)(χDχ)x+χxxx=0.(3) This map was used in[8]to prove that there exists an infinite number of conservation laws for the supersymmetric KdV equation(1).In the classical case,Gardner type of map was studied extensively by Kupershmidt[4].It is well known that such map may be used to construct interesting BT.We will show that it is also the case for the supersymmetric KdV equation.We notice that the supersymmetric Gardner equation(3)is invariant underǫ→−ǫ. The new solution of the supersymmetric KdV equation corresponding to with−ǫis denoted as˜Φ.Thus we have˜Φ=χ−ǫχx+ǫ2χ(Dχ).(4) From above relations(3-4),wefindΦ−˜Φ=2ǫχx,(5)Φ+˜Φ=2χ+2ǫχ(Dχ).(6) Let us introduce the potentials as followsΦ=Ψx,˜Φ=˜Ψx,thus,the equation(5)provides usχ=12(Ψ−˜Ψ)(DΨ−D˜Ψ),(8)whereλ=1/ǫis the B¨a cklund parameter.The transformation(8)is in fact the spatial part of BT.Its temporal counterpart can be easily worked out.We also remark here that the BT is reduced to the well known BT for the classical KdV equation ifη=0,as it should be.A BT can be used to generate special solutions.If we start with the trivial solution ˜Ψ=0,we obtain2a(ζ+θλ)eλx−λ3tΨ=−(Ψ1−Ψ0)(DΨ1−DΨ0),(10)2and1(Ψ0+Ψ2)x=λ2(Ψ2−Ψ0)+(Ψ3−Ψ1)(DΨ3−DΨ1),(12)2and1(Ψ2+Ψ3)x=λ1(Ψ3−Ψ2)+Subtraction (10)form (11),we have(Ψ2−Ψ1)x =λ2Ψ2−λ1Ψ1+(λ1−λ2)Ψ0+12Ψ2(D Ψ0)−12Ψ1(D Ψ1)+12Ψ0(D Ψ1),(14)similarly,from (12)and (13)we have(Ψ2−Ψ1)x = λ1−λ2+12(D Ψ2) Ψ3+12Ψ2(D Ψ2)−12(D Ψ1)−12(Ψ1−Ψ2)[(D Ψ3)+2λ1+2λ2−(D Ψ0)]=0.(16)The equation (16)is a differential equation for Ψ3.Solving it we obtainΨ3=Ψ0−(λ1+λ2)(Ψ1−Ψ2)λ1−λ2+v 1−v 2−(λ1+λ2)(η1−η2)(η1,x −η2,x )(λ1−λ2+v 1−v 2).It is clear that our nonlinear superposition formula reduces to the well-known superposition formula for the KdV as it should be.The advantage to have a superposition formula is that it is an algebraic one and can be used easily to find solutions.Using the solutions (9)as our seeds,we may construct a 2-soliton solution of (1)by means of our superposition formula.Indeed,letΨ0=0,Ψ1=−2a 1(ζ1+θλ1)e λ1x −λ31t1+a 2e λ2x −λ32tthen from our nonlinear superposition formula (17),we obtainΨ3=2 ζ1a 1e δ1−ζ2a 2e δ2+(ζ1−ζ2)a 1a 2e δ1+δ2+θ(λ1a 1e δ1−λ2a 2e δ2+(λ1−λ2)a 1a 2e δ1+δ2)whereδi=λi x−λ3i t+θζi(i=1,2).Now,takingλ1>λ2,a1=−λ1−λ2λ1+λ2we recover the2-soliton solution foundfirst by Carstea,Ramani and Grammaticos[2].As in the classical KdV case[11],we generate this2-soliton from a regular solution and a singular solution.We could continue this process to build the higher soliton solutions and the calculation will be tedious but straightforward.Acknowledgment The work is supported in part by National Natural Scientific Foun-dation of China(grant number10231050)and Ministry of Education of China. References[1]Carstea A S2000Extension of the bilinear formalism to supersymmetric KdV-typeequations Nonlinearity131645.[2]Carstea A S,Ramani A and Grammaticos B2001Constructing the soliton solutionsfor the N=1supersymmetric KdV hierarchy Nonlinearity141419.[3]Kupershmidt B A1984A super Korteweg-de Vries equation Phys.Lett.102A213.[4]Kupershmidt B A1981On the nature of Gardner transformation J.Math.Phys.22449;Kupershmidt B A1983Deformations of integrable systems Proc.R.Soc.Irish 83A45.[5]Liu Q P1995Darboux transformation for the supersymmetric KdV equations Lett.Math.Phys.35115.[6]Liu Q P and Ma˜n as M1997Darboux transformation for the Mann-Radul supersym-metric KdV equation Phys.Lett.B394337;Liu Q P and Ma˜n as M in Supersymmetry and Integrable Systems eds.H.Aratyn et al Lecture Notes in Physics502(Springer).[7]Manin Yu I and Radul A O1985A supersymmetric extension of the Kadomtzev-Petviashvili hierarchy Commun.Math.Phys.9865.[8]Mathieu P1988Supersymmetric extension of the Korteweg-de Vries equation J.Math.Phys.292499.[9]McArthur I N and Yung C M1993Hirota bilinearfform for the super KdV hierarchyMod.Phys.Lett.8A1739.[10]Oevel W and Popowicz Z1991The bi-Hamiltonian structure of fully supersymmetricKorteweg-de Vries systems Commun.Math.Phys.139441;Figueroa-O’Farril J,Mas J and Ramos E1991Integrability and biHamiltonian structure of the even order sKdV hierarchies Rev.Math.Phys.3479.[11]Wahlquist H D and Estabrook F B1974B¨a cklund transformation for solutions of theKorteweg-de Vries equation Phys.Rev.Lett.311386.。

904N.E.Huang and others10.Discussion98711.Conclusions991References993 A new method for analysing has been devel-oped.The key part of the methodany complicated data set can be decomposed intoof‘intrinsic mode functions’Hilbert trans-This decomposition method is adaptive,and,highly efficient.Sinceapplicable to nonlinear and non-stationary processes.With the Hilbert transform,Examplesthe classical nonlinear equation systems and dataare given to demonstrate the power new method.data are especially interesting,for serve to illustrate the roles thenonlinear and non-stationary effects in the energy–frequency–time distribution.Keywords:non-stationary time series;nonlinear differential equations;frequency–time spectrum;Hilbert spectral analysis;intrinsic time scale;empirical mode decomposition1.Introductionsensed by us;data analysis serves two purposes:determine the parameters needed to construct the necessary model,and to confirm the model we constructed to represent the phe-nomenon.Unfortunately,the data,whether from physical measurements or numerical modelling,most likely will have one or more of the following problems:(a)the total data span is too short;(b)the data are non-stationary;and(c)the data represent nonlinear processes.Although each of the above problems can be real by itself,the first two are related,for a data section shorter than the longest time scale of a sta-tionary process can appear to be non-stationary.Facing such data,we have limited options to use in the analysis.Historically,Fourier spectral analysis has provided a general method for examin-the data analysis has been applied to all kinds of data.Although the Fourier transform is valid under extremely general conditions(see,for example,Titchmarsh1948),there are some crucial restrictions of Proc.R.Soc.Lond.A(1998)Nonlinear and non-stationary time series analysis905the Fourier spectral analysis:the system must be linear;and the data must be strict-ly periodic or stationary;otherwise,the resulting spectrum will make little physicalsense.to the Fourier spectral analysis methods.Therefore,behoves us review the definitions of stationarity here.According to the traditional definition,a time series,X (t ),is stationary in the wide sense,if,for all t ,E (|X (t )2|)<∞,E (X (t))=m,C (X (t 1),X (t 2))=C (X (t 1+τ),X (t 2+τ))=C (t 1−t 2),(1.1)in whichE (·)is the expected value defined as the ensemble average of the quantity,and C (·)is the covariance function.Stationarity in the wide sense is also known as weak stationarity,covariance stationarity or second-order stationarity (see,forexample,Brockwell &Davis 1991).A time series,X (t ),is strictly stationary,if the joint distribution of [X (t 1),X (t 2),...,X (t n )]and [X (t 1+τ),X (t 2+τ),...,X (t n +τ)](1.2)are the same for all t i and τ.Thus,a strictly stationaryprocess with finite second moments is alsoweakly stationary,but the inverse is not true.Both definitions arerigorous but idealized.Other less rigorous definitions have also beenused;for example,that is stationary within a limited timespan,asymptotically stationary is for any random variableis stationary when τin equations (1.1)or (1.2)approaches infinity.In practice,we can only have data for finite time spans;these defini-tions,we haveto makeapproximations.Few of the data sets,from either natural phenomena or artificial sources,can satisfy these definitions.It may be argued thatthe difficulty of invoking stationarity as well as ergodicity is not on principlebut on practicality:we just cannot have enough data to cover all possible points in thephase plane;therefore,most of the cases facing us are transient in nature.This is the reality;we are forced to face it.Fourier spectral analysis also requires linearity.can be approximated by linear systems,the tendency tobe nonlinear whenever their variations become finite Compounding these complications is the imperfection of or numerical schemes;theinteractionsof the imperfect probes even with a perfect linear systemcan make the final data nonlinear.For the above the available data are ally of finite duration,non-stationary and from systems that are frequently nonlinear,either intrinsicallyor through interactions with the imperfect probes or numerical schemes.Under these conditions,Fourier spectral analysis is of limited use.For lack of alternatives,however,Fourier spectral analysis is still used to process such data.The uncritical use of Fourier spectral analysis the insouciant adoption of the stationary and linear assumptions may give cy range.a delta function will giveProc.R.Soc.Lond.A (1998)906N.E.Huang and othersa phase-locked wide white Fourier spectrum.Here,added to the data in the time domain,Constrained bythese spurious harmonics the wide frequency spectrum cannot faithfully represent the true energy density in the frequency space.More seri-ously,the Fourier representation also requires the existence of negative light intensity so that the components can cancel out one another to give thefinal delta function. Thus,the Fourier components might make mathematical sense,but do not really make physical sense at all.Although no physical process can be represented exactly by a delta function,some data such as the near-field strong earthquake records areFourier spectra.Second,tions;wave-profiles.Such deformations,later,are the direct consequence of nonlinear effects.Whenever the form of the data deviates from a pure sine or cosine function,the Fourier spectrum will contain harmonics.As explained above, both non-stationarity and nonlinearity can induce spurious harmonic components that cause energy spreading.The consequence is the misleading energy–frequency distribution forIn this paper,modemode functions The decomposition is based on the direct extraction of theevent on the time the frequency The decomposition be viewed as an expansion of the data in terms of the IMFs.Then,based on and derived from the data,can serve as the basis of that expansion linear or nonlinear as dictated by the data,Most important of all,it is adaptive.As will locality and adaptivity are the necessary conditions for the basis for expanding nonlinear and non-stationary time orthogonality is not a necessary criterionselection for a nonlinearon the physical time scaleslocal energy and the instantaneous frequencyHilbert transform can give us a full energy–frequency–time distribution of the data. Such a representation is designated as the Hilbert spectrum;it would be ideal for nonlinear and non-stationary data analysis.We have obtained good results and new insights by applying the combination of the EMD and Hilbert spectral analysis methods to various data:from the numerical results of the classical nonlinear equation systems to data representing natural phe-nomena.The classical nonlinear systems serve to illustrate the roles played by the nonlinear effects in the energy–frequency–time distribution.With the low degrees of freedom,they can train our eyes for more complicated cases.Some limitations of this method will also be discussed and the conclusions presented.Before introducing the new method,we willfirst review the present available data analysis methods for non-stationary processes.Proc.R.Soc.Lond.A(1998)Nonlinear and non-stationary time series analysis9072.Review of non-stationary data processing methodsWe willfirstgivea brief survey of themethodsstationary data.are limited to linear systems any method is almost strictly determined according to the special field in which the application is made.The available methods are reviewed as follows.(a )The spectrogramnothing but a limited time window-width Fourier spectral analysis.the a distribution.Since it relies on the tradition-al Fourier spectral analysis,one has to assume the data to be piecewise stationary.This assumption is not always justified in non-stationary data.Even if the data are piecewise stationary how can we guarantee that the window size adopted always coincides with the stationary time scales?What can we learn about the variations longer than the local stationary time scale?Will the collection of the locally station-ary pieces constitute some longer period phenomena?Furthermore,there are also practical difficulties in applying the method:in order to localize an event in time,the window width must be narrow,but,on the other hand,the frequency resolu-tion requires longer time series.These conflicting requirements render this method of limited usage.It is,however,extremely easy to implement with the fast Fourier transform;thus,ithas attracted a wide following.Most applications of this methodare for qualitative display of speech pattern analysis (see,for example,Oppenheim &Schafer 1989).(b )The wavelet analysisThe wavelet approach is essentially an adjustable window Fourier spectral analysiswith the following general definition:W (a,b ;X,ψ)=|a |−1/2∞−∞X (t )ψ∗ t −b ad t,(2.1)in whichψ∗(·)is the basic wavelet function that satisfies certain very general condi-tions,a is the dilation factor and b is the translationof theorigin.Although time andfrequency do not appear explicitly in the transformed result,the variable 1/a givesthe frequency scale and b ,the temporal location of an event.An intuitive physical explanation of equation (2.1)is very simple:W (a,b ;X,ψ)is the ‘energy’of X ofscale a at t =b .Because of this basic form of at +b involvedin thetransformation,it is also knownas affinewavelet analysis.For specific applications,the basic wavelet function,ψ∗(·),can be modified according to special needs,but the form has to be given before the analysis.In most common applications,however,the Morlet wavelet is defined as Gaussian enveloped sine and cosine wave groups with 5.5waves (see,for example,Chan 1995).Generally,ψ∗(·)is not orthogonalfordifferent a for continuous wavelets.Although one can make the wavelet orthogonal by selecting a discrete set of a ,thisdiscrete wavelet analysis will miss physical signals having scale different from theselected discrete set of a .Continuous or discrete,the wavelet analysis is basically a linear analysis.A very appealing feature of the wavelet analysis is that it provides aProc.R.Soc.Lond.A (1998)908N.E.Huang and othersuniform resolution for all the scales.Limited by the size of thebasic wavelet function,the downside of the uniform resolution is uniformly poor resolution.Although wavelet analysis has been available only in the last ten years or so,it hasbecome extremelypopular.Indeed,it is very useful in analysing data with gradualfrequency changes.Since it has an analytic form for the result,it has attracted extensive attention of the applied mathematicians.Most of its applications have been in edge detection and image compression.Limited applications have also been made to the time–frequency distribution in time series (see,for example,Farge 1992;Long et al .1993)andtwo-dimensionalimages (Spedding et al .1993).Versatile as the wavelet analysis is,the problem with the most commonly usedMorlet wavelet is its leakage generated by the limited length of the basic wavelet function,whichmakesthe quantitativedefinitionof the energy–frequency–time dis-tribution difficult.Sometimes,the interpretation of the wavelet can also be counter-intuitive.For example,to define a change occurring locally,one must look for theresult in the high-frequencyrange,for the higher the frequency the more localized thebasic wavelet will be.If a local event occurs only in the low-frequency range,one willstill be forced to look for its effects inthe high-frequencyrange.Such interpretationwill be difficultif it is possible at all (see,for example,Huang et al .1996).Another difficulty of the wavelet analysis is its non-adaptive nature.Once the basic waveletis selected,one will have to use it to analyse all the data.Since the most commonlyused Morlet wavelet is Fourier based,it also suffers the many shortcomings of Fouri-er spectral analysis:it can only give a physically meaningful interpretation to linear phenomena;it can resolve the interwave frequency modulation provided the frequen-cy variationis gradual,but it cannot resolve the intrawave frequency modulation because the basic wavelet has a length of 5.5waves.Inspite of all these problems,wavelet analysisisstillthe bestavailable non-stationary data analysis method so far;therefore,we will use it in this paper as a reference to establish the validity and thecalibration of the Hilbert spectrum.(c )The Wigner–Ville distributionThe Wigner–Ville distribution is sometimes alsoreferred toas the Heisenberg wavelet.By definition,it is the Fourier transform of the central covariance function.For any time series,X (t ),we can define the central variance as C c (τ,t )=X (t −12τ)X ∗(t +12τ).(2.2)Then the Wigner–Ville distribution is V (ω,t )=∞−∞C c (τ,t )e −i ωτd τ.(2.3)This transform has been treated extensively by Claasen &Mecklenbr¨a uker (1980a ,b,c )and by Cohen (1995).It has been extremely popular with the electrical engi-neering community.The difficulty with this method is the severe cross terms as indicated by the exis-tence of negativepowerfor some frequency ranges.Although this shortcoming canbe eliminated by using the Kernel method (see,for example,Cohen 1995),the resultis,then,basically that of a windowed Fourier analysis;therefore,itsuffers all thelim-itations of the Fourier analysis.An extension of this method has been made by Yen(1994),who used the Wigner–Ville distribution to define wave packets that reduce Proc.R.Soc.Lond.A (1998)Nonlinear and non-stationary time series analysis909 a complicated data set to afinite number of simple components.This extension is very powerful and can be applied to a variety of problems.The applications to complicated data,however,require a great amount of judgement.(d)Evolutionary spectrumThe evolutionary spectrum wasfirst proposed by Priestley(1965).The basic idea is to extend the classic Fourier spectral analysis to a more generalized basis:from sine or cosine to a family of orthogonal functions{φ(ω,t)}indexed by time,t,and defined for all realω,the frequency.Then,any real random variable,X(t),can beexpressed asX(t)= ∞−∞φ(ω,t)d A(ω,t),(2.4)in which d A(ω,t),the Stieltjes function for the amplitude,is related to the spectrum asE(|d A(ω,t)|2)=dµ(ω,t)=S(ω,t)dω,(2.5) whereµ(ω,t)is the spectrum,and S(ω,t)is the spectral density at a specific time t,also designated as the evolutionary spectrum.If for eachfixedω,φ(ω,t)has a Fourier transformφ(ω,t)=a(ω,t)e iΩ(ω)t,(2.6) then the function of a(ω,t)is the envelope ofφ(ω,t),andΩ(ω)is the frequency.If, further,we can treatΩ(ω)as a single valued function ofω,thenφ(ω,t)=α(ω,t)e iωt.(2.7) Thus,the original data can be expanded in a family of amplitude modulated trigono-metric functions.The evolutionary spectral analysis is very popular in the earthquake communi-ty(see,for example,Liu1970,1971,1973;Lin&Cai1995).The difficulty of its application is tofind a method to define the basis,{φ(ω,t)}.In principle,for this method to work,the basis has to be defined a posteriori.So far,no systematic way has been offered;therefore,constructing an evolutionary spectrum from the given data is impossible.As a result,in the earthquake community,the applications of this method have changed the problem from data analysis to data simulation:an evo-lutionary spectrum will be assumed,then the signal will be reconstituted based on the assumed spectrum.Although there is some general resemblance to the simulated earthquake signal with the real data,it is not the data that generated the spectrum. Consequently,evolutionary spectrum analysis has never been very useful.As will be shown,the EMD can replace the evolutionary spectrum with a truly adaptive representation for the non-stationary processes.(e)The empirical orthogonal function expansion(EOF)The empirical orthogonal function expansion(EOF)is also known as the principal component analysis,or singular value decomposition method.The essence of EOF is briefly summarized as follows:for any real z(x,t),the EOF will reduce it toz(x,t)=n1a k(t)f k(x),(2.8)Proc.R.Soc.Lond.A(1998)910N.E.Huang and othersin whichf j·f k=δjk.(2.9)The orthonormal basis,{f k},is the collection of the empirical eigenfunctions defined byC·f k=λk f k,(2.10)where C is the sum of the inner products of the variable.EOF represents a radical departure from all the above methods,for the expansion basis is derived from the data;therefore,it is a posteriori,and highly efficient.The criticalflaw of EOF is that it only gives a distribution of the variance in the modes defined by{f k},but this distribution by itself does not suggest scales or frequency content of the signal.Although it is tempting to interpret each mode as indepen-dent variations,this interpretation should be viewed with great care,for the EOF decomposition is not unique.A single component out of a non-unique decomposition, even if the basis is orthogonal,does not usually contain physical meaning.Recently, Vautard&Ghil(1989)proposed the singular spectral analysis method,which is the Fourier transform of the EOF.Here again,we have to be sure that each EOF com-ponent is stationary,otherwise the Fourier spectral analysis will make little sense on the EOF components.Unfortunately,there is no guarantee that EOF compo-nents from a nonlinear and non-stationary data set will all be linear and stationary. Consequently,singular spectral analysis is not a real improvement.Because of its adaptive nature,however,the EOF method has been very popular,especially in the oceanography and meteorology communities(see,for example,Simpson1991).(f)Other miscellaneous methodsOther than the above methods,there are also some miscellaneous methods such as least square estimation of the trend,smoothing by moving averaging,and differencing to generate stationary data.Methods like these,though useful,are too specialized to be of general use.They will not be discussed any further here.Additional details can be found in many standard data processing books(see,for example,Brockwell &Davis1991).All the above methods are designed to modify the global representation of the Fourier analysis,but they all failed in one way or the other.Having reviewed the methods,we can summarize the necessary conditions for the basis to represent a nonlinear and non-stationary time series:(a)complete;(b)orthogonal;(c)local;and (d)adaptive.Thefirst condition guarantees the degree of precision of the expansion;the second condition guarantees positivity of energy and avoids leakage.They are the standard requirements for all the linear expansion methods.For nonlinear expansions,the orthogonality condition needs to be modified.The details will be discussed later.But even these basic conditions are not satisfied by some of the above mentioned meth-ods.The additional conditions are particular to the nonlinear and non-stationary data.The requirement for locality is the most crucial for non-stationarity,for in such data there is no time scale;therefore,all events have to be identified by the time of their occurences.Consequently,we require both the amplitude(or energy) and the frequency to be functions of time.The requirement for adaptivity is also crucial for both nonlinear and non-stationary data,for only by adapting to the local variations of the data can the decomposition fully account for the underlying physics Proc.R.Soc.Lond.A(1998)Nonlinear and non-stationary time series analysis911of the processes and not just to fulfil the mathematical requirements for fitting the data.This is especially important for the nonlinear phenomena,for a manifestation of nonlinearity is the ‘harmonic distortion’in the Fourier analysis.The degree of distortion depends on the severity of nonlinearity;therefore,one cannot expect a predetermined basis to fit all the phenomena.An easy way to generate the necessary adaptive basis is to derive the basis from the data.In this paper,we will introduce a general method which requires two steps in analysing the data as follows.The first step is to preprocess the data by the empirical mode decomposition method,with which the data are decomposed into a number of intrinsic mode function components.Thus,we will expand the data in a basis derived from the data.The second step is to apply the Hilbert transform to the decomposed IMFs and construct the energy–frequency–time distribution,designated as the Hilbert spectrum,from which the time localities of events will be preserved.In other words,weneed the instantaneous frequency and energy rather than the global frequency and energy defined by the Fourier spectral analysis.Therefore,before goingany further,we have to clarify the definition of the instantaneous frequency.3.Instantaneous frequencyis to accepting it only for special ‘monocomponent’signals 1992;Cohen 1995).Thereare two basicdifficulties with accepting the idea of an instantaneous fre-quency as follows.The first one arises from the influence of theFourier spectral analysis.In the traditional Fourier analysis,the frequency is defined for thesineor cosine function spanning the whole data length with constant ampli-tude.As an extension of this definition,the instantaneous frequencies also have torelate to either a sine or a cosine function.Thus,we need at least one full oscillationof a sineor a cosine wave to define the local frequency value.According to this logic,nothing full wave will do.Such a definition would not make sense forThe secondarises from the non-unique way in defining the instantaneousfrequency.Nevertheless,this difficulty is no longer serious since the introduction ofthe meanstomakethedata analyticalthrough the Hilbert transform.Difficulties,however,still exist as ‘paradoxes’discussed by Cohen (1995).For an arbitrary timeseries,X (t ),we can always have its Hilbert Transform,Y (t ),as Y (t )=1πP∞−∞X (t )t −t d t,(3.1)where P indicates the Cauchy principal value.This transformexists forallfunctionsof class L p(see,for example,Titchmarsh 1948).With this definition,X (t )and Y (t )form the complex conjugate pair,so we can have an analytic signal,Z (t ),as Z (t )=X (t )+i Y (t )=a (t )e i θ(t ),(3.2)in which a (t )=[X 2(t )+Y 2(t )]1/2,θ(t )=arctanY (t )X (t ).(3.3)Proc.R.Soc.Lond.A (1998)912N.E.Huang andothers Theoretically,there are infinitely many ways of defining the imaginary part,but the Hilbert transform provides a unique way of defining the imaginary part so that the result is ananalyticfunction.A brief tutorial on the Hilbert transform with theemphasis on its physical interpretation can be found in Bendat &Piersol is the bestlocal fitan amplitude and phase varying trigonometric function to X (t ).Even with the Hilbert transform,there is still controversy in defining the instantaneous frequency as ω=d θ(t )d t .(3.4)This leads Cohen (1995)to introduce the term,‘monocomponent function’.In prin-ciple,some limitations on the data are necessary,forthe instantaneous frequencygiven in equation (3.4)is a single value function of time.At any given time,thereis only one frequency value;therefore,it can only represent one component,hence ‘monocomponent’.Unfortunately,no cleardefinition of the ‘monocomponent’signalwas given to judge whether a function is or is not ‘monocomponent’.For lack ofa precise definition,‘narrow band’was adopted a on the data for the instantaneous frequency to make sense (Schwartz et al .1966).There are two definitions for bandwidth.The first one is used in the study of the probability properties of the signalsand waves,wherethe processes are assumed tobe stationary and Gaussian.Then,the bandwidth can be defined in spectral moments The expected number of zero crossings per unit time is given byN 0=1π m 2m 0 1/2,(3.5)while the expected number of extrema per unit time is given byN 1=1π m 4m 2 1/2,(3.6)in which m i is the i th moment of the spectrum.Therefore,the parameter,ν,definedas N 21−N 20=1π2m 4m 0−m 22m 2m 0=1π2ν2,(3.7)offers a standard bandwidth measure (see,for example,Rice 1944a,b ,1945a,b ;Longuet-Higgins 1957).For a narrow band signal ν=0,the expected numbers extrema and zero crossings have to equal.the spectrum,but in a different way.coordinates as z (t )=a (t )e i θ(t ),(3.8)with both a (t )and θ(t )being functions of time.If this function has a spectrum,S (ω),then the mean frequency is given byω = ω|S (ω)|2d ω,(3.9)Proc.R.Soc.Lond.A (1998)Nonlinear and non-stationary time series analysis913which can be expressed in another way asω =z ∗(t )1i dd tz (t )d t=˙θ(t )−i ˙a (t )a (t )a 2(t )d t =˙θ(t )a 2(t )d t.(3.10)Based on this expression,Cohen (1995)suggested that ˙θbe treated as the instanta-neous frequency.With these notations,the bandwidth can be defined asν2=(ω− ω )2 ω 2=1 ω 2(ω− ω )2|S (ω)|2d ω=1 ω 2z ∗(t ) 1i d d t− ω 2z (t )d t =1 ω 2 ˙a 2(t )d t +(˙θ(t )− ω )2a 2(t )d t .(3.11)For a narrow band signal,this value has to be small,then both a and θhave to begradually varying functions.Unfortunately,both equations (3.7)and (3.11)defined the bandwidth in the global sense;they are both overly restrictive and lack preci-sion at the same time.Consequently,the bandwidth limitation on the Hilbert trans-form to give a meaningful instantaneous frequency has never been firmly established.For example,Melville (1983)had faithfully filtered the data within the bandwidth requirement,but he still obtained many non-physical negative frequency values.It should be mentioned here that using filtering to obtain a narrow band signal is unsat-isfactory for another reason:the filtered data have already been contaminated by the spurious harmonics caused by the nonlinearity and non-stationarity as discussed in the introduction.In order to obtain meaningful instantaneous frequency,restrictive conditions have to be imposed on the data as discussed by Gabor (1946),Bedrosian (1963)and,more recently,Boashash (1992):for any function to have a meaningful instantaneous frequency,the real part of its Fourier transform has to have only positive frequency.This restriction can be proven mathematically as shown in Titchmarsh (1948)but it is still global.For data analysis,we have to translate this requirement into physically implementable steps to develop a simple method for applications.For this purpose,we have to modify the restriction condition from a global one to a local one,and the basis has to satisfy the necessary conditions listed in the last section.Let us consider some simple examples to illustrate these restrictions physically,by examining the function,x (t )=sin t.(3.12)Its Hilbert transform is simply cos t .The phase plot of x –y is a simple circle of unit radius as in figure 1a .The phase function is a straight line as shown in figure 1b and the instantaneous frequency,shown in figure 1c ,is a constant as expected.If we move the mean offby an amount α,say,then,x (t )=α+sin t.(3.13)Proc.R.Soc.Lond.A (1998)。

s Data mining and knowledge discovery in databases have been attracting a significant amount of research, industry, and media atten-tion of late. What is all the excitement about?This article provides an overview of this emerging field, clarifying how data mining and knowledge discovery in databases are related both to each other and to related fields, such as machine learning, statistics, and databases. The article mentions particular real-world applications, specific data-mining techniques, challenges in-volved in real-world applications of knowledge discovery, and current and future research direc-tions in the field.A cross a wide variety of fields, data arebeing collected and accumulated at adramatic pace. There is an urgent need for a new generation of computational theo-ries and tools to assist humans in extracting useful information (knowledge) from the rapidly growing volumes of digital data. These theories and tools are the subject of the emerging field of knowledge discovery in databases (KDD).At an abstract level, the KDD field is con-cerned with the development of methods and techniques for making sense of data. The basic problem addressed by the KDD process is one of mapping low-level data (which are typically too voluminous to understand and digest easi-ly) into other forms that might be more com-pact (for example, a short report), more ab-stract (for example, a descriptive approximation or model of the process that generated the data), or more useful (for exam-ple, a predictive model for estimating the val-ue of future cases). At the core of the process is the application of specific data-mining meth-ods for pattern discovery and extraction.1This article begins by discussing the histori-cal context of KDD and data mining and theirintersection with other related fields. A briefsummary of recent KDD real-world applica-tions is provided. Definitions of KDD and da-ta mining are provided, and the general mul-tistep KDD process is outlined. This multistepprocess has the application of data-mining al-gorithms as one particular step in the process.The data-mining step is discussed in more de-tail in the context of specific data-mining al-gorithms and their application. Real-worldpractical application issues are also outlined.Finally, the article enumerates challenges forfuture research and development and in par-ticular discusses potential opportunities for AItechnology in KDD systems.Why Do We Need KDD?The traditional method of turning data intoknowledge relies on manual analysis and in-terpretation. For example, in the health-careindustry, it is common for specialists to peri-odically analyze current trends and changesin health-care data, say, on a quarterly basis.The specialists then provide a report detailingthe analysis to the sponsoring health-care or-ganization; this report becomes the basis forfuture decision making and planning forhealth-care management. In a totally differ-ent type of application, planetary geologistssift through remotely sensed images of plan-ets and asteroids, carefully locating and cata-loging such geologic objects of interest as im-pact craters. Be it science, marketing, finance,health care, retail, or any other field, the clas-sical approach to data analysis relies funda-mentally on one or more analysts becomingArticlesFALL 1996 37From Data Mining to Knowledge Discovery inDatabasesUsama Fayyad, Gregory Piatetsky-Shapiro, and Padhraic Smyth Copyright © 1996, American Association for Artificial Intelligence. All rights reserved. 0738-4602-1996 / $2.00areas is astronomy. Here, a notable success was achieved by SKICAT ,a system used by as-tronomers to perform image analysis,classification, and cataloging of sky objects from sky-survey images (Fayyad, Djorgovski,and Weir 1996). In its first application, the system was used to process the 3 terabytes (1012bytes) of image data resulting from the Second Palomar Observatory Sky Survey,where it is estimated that on the order of 109sky objects are detectable. SKICAT can outper-form humans and traditional computational techniques in classifying faint sky objects. See Fayyad, Haussler, and Stolorz (1996) for a sur-vey of scientific applications.In business, main KDD application areas includes marketing, finance (especially in-vestment), fraud detection, manufacturing,telecommunications, and Internet agents.Marketing:In marketing, the primary ap-plication is database marketing systems,which analyze customer databases to identify different customer groups and forecast their behavior. Business Week (Berry 1994) estimat-ed that over half of all retailers are using or planning to use database marketing, and those who do use it have good results; for ex-ample, American Express reports a 10- to 15-percent increase in credit-card use. Another notable marketing application is market-bas-ket analysis (Agrawal et al. 1996) systems,which find patterns such as, “If customer bought X, he/she is also likely to buy Y and Z.” Such patterns are valuable to retailers.Investment: Numerous companies use da-ta mining for investment, but most do not describe their systems. One exception is LBS Capital Management. Its system uses expert systems, neural nets, and genetic algorithms to manage portfolios totaling $600 million;since its start in 1993, the system has outper-formed the broad stock market (Hall, Mani,and Barr 1996).Fraud detection: HNC Falcon and Nestor PRISM systems are used for monitoring credit-card fraud, watching over millions of ac-counts. The FAIS system (Senator et al. 1995),from the U.S. Treasury Financial Crimes En-forcement Network, is used to identify finan-cial transactions that might indicate money-laundering activity.Manufacturing: The CASSIOPEE trou-bleshooting system, developed as part of a joint venture between General Electric and SNECMA, was applied by three major Euro-pean airlines to diagnose and predict prob-lems for the Boeing 737. To derive families of faults, clustering methods are used. CASSIOPEE received the European first prize for innova-intimately familiar with the data and serving as an interface between the data and the users and products.For these (and many other) applications,this form of manual probing of a data set is slow, expensive, and highly subjective. In fact, as data volumes grow dramatically, this type of manual data analysis is becoming completely impractical in many domains.Databases are increasing in size in two ways:(1) the number N of records or objects in the database and (2) the number d of fields or at-tributes to an object. Databases containing on the order of N = 109objects are becoming in-creasingly common, for example, in the as-tronomical sciences. Similarly, the number of fields d can easily be on the order of 102or even 103, for example, in medical diagnostic applications. Who could be expected to di-gest millions of records, each having tens or hundreds of fields? We believe that this job is certainly not one for humans; hence, analysis work needs to be automated, at least partially.The need to scale up human analysis capa-bilities to handling the large number of bytes that we can collect is both economic and sci-entific. Businesses use data to gain competi-tive advantage, increase efficiency, and pro-vide more valuable services to customers.Data we capture about our environment are the basic evidence we use to build theories and models of the universe we live in. Be-cause computers have enabled humans to gather more data than we can digest, it is on-ly natural to turn to computational tech-niques to help us unearth meaningful pat-terns and structures from the massive volumes of data. Hence, KDD is an attempt to address a problem that the digital informa-tion era made a fact of life for all of us: data overload.Data Mining and Knowledge Discovery in the Real WorldA large degree of the current interest in KDD is the result of the media interest surrounding successful KDD applications, for example, the focus articles within the last two years in Business Week , Newsweek , Byte , PC Week , and other large-circulation periodicals. Unfortu-nately, it is not always easy to separate fact from media hype. Nonetheless, several well-documented examples of successful systems can rightly be referred to as KDD applications and have been deployed in operational use on large-scale real-world problems in science and in business.In science, one of the primary applicationThere is an urgent need for a new generation of computation-al theories and tools toassist humans in extractinguseful information (knowledge)from the rapidly growing volumes ofdigital data.Articles38AI MAGAZINEtive applications (Manago and Auriol 1996).Telecommunications: The telecommuni-cations alarm-sequence analyzer (TASA) wasbuilt in cooperation with a manufacturer oftelecommunications equipment and threetelephone networks (Mannila, Toivonen, andVerkamo 1995). The system uses a novelframework for locating frequently occurringalarm episodes from the alarm stream andpresenting them as rules. Large sets of discov-ered rules can be explored with flexible infor-mation-retrieval tools supporting interactivityand iteration. In this way, TASA offers pruning,grouping, and ordering tools to refine the re-sults of a basic brute-force search for rules.Data cleaning: The MERGE-PURGE systemwas applied to the identification of duplicatewelfare claims (Hernandez and Stolfo 1995).It was used successfully on data from the Wel-fare Department of the State of Washington.In other areas, a well-publicized system isIBM’s ADVANCED SCOUT,a specialized data-min-ing system that helps National Basketball As-sociation (NBA) coaches organize and inter-pret data from NBA games (U.S. News 1995). ADVANCED SCOUT was used by several of the NBA teams in 1996, including the Seattle Su-personics, which reached the NBA finals.Finally, a novel and increasingly importanttype of discovery is one based on the use of in-telligent agents to navigate through an infor-mation-rich environment. Although the ideaof active triggers has long been analyzed in thedatabase field, really successful applications ofthis idea appeared only with the advent of theInternet. These systems ask the user to specifya profile of interest and search for related in-formation among a wide variety of public-do-main and proprietary sources. For example, FIREFLY is a personal music-recommendation agent: It asks a user his/her opinion of several music pieces and then suggests other music that the user might like (<http:// www.ffl/>). CRAYON(/>) allows users to create their own free newspaper (supported by ads); NEWSHOUND(<http://www. /hound/>) from the San Jose Mercury News and FARCAST(</> automatically search information from a wide variety of sources, including newspapers and wire services, and e-mail rele-vant documents directly to the user.These are just a few of the numerous suchsystems that use KDD techniques to automat-ically produce useful information from largemasses of raw data. See Piatetsky-Shapiro etal. (1996) for an overview of issues in devel-oping industrial KDD applications.Data Mining and KDDHistorically, the notion of finding useful pat-terns in data has been given a variety ofnames, including data mining, knowledge ex-traction, information discovery, informationharvesting, data archaeology, and data patternprocessing. The term data mining has mostlybeen used by statisticians, data analysts, andthe management information systems (MIS)communities. It has also gained popularity inthe database field. The phrase knowledge dis-covery in databases was coined at the first KDDworkshop in 1989 (Piatetsky-Shapiro 1991) toemphasize that knowledge is the end productof a data-driven discovery. It has been popular-ized in the AI and machine-learning fields.In our view, KDD refers to the overall pro-cess of discovering useful knowledge from da-ta, and data mining refers to a particular stepin this process. Data mining is the applicationof specific algorithms for extracting patternsfrom data. The distinction between the KDDprocess and the data-mining step (within theprocess) is a central point of this article. Theadditional steps in the KDD process, such asdata preparation, data selection, data cleaning,incorporation of appropriate prior knowledge,and proper interpretation of the results ofmining, are essential to ensure that usefulknowledge is derived from the data. Blind ap-plication of data-mining methods (rightly crit-icized as data dredging in the statistical litera-ture) can be a dangerous activity, easilyleading to the discovery of meaningless andinvalid patterns.The Interdisciplinary Nature of KDDKDD has evolved, and continues to evolve,from the intersection of research fields such asmachine learning, pattern recognition,databases, statistics, AI, knowledge acquisitionfor expert systems, data visualization, andhigh-performance computing. The unifyinggoal is extracting high-level knowledge fromlow-level data in the context of large data sets.The data-mining component of KDD cur-rently relies heavily on known techniquesfrom machine learning, pattern recognition,and statistics to find patterns from data in thedata-mining step of the KDD process. A natu-ral question is, How is KDD different from pat-tern recognition or machine learning (and re-lated fields)? The answer is that these fieldsprovide some of the data-mining methodsthat are used in the data-mining step of theKDD process. KDD focuses on the overall pro-cess of knowledge discovery from data, includ-ing how the data are stored and accessed, howalgorithms can be scaled to massive data setsThe basicproblemaddressed bythe KDDprocess isone ofmappinglow-leveldata intoother formsthat might bemorecompact,moreabstract,or moreuseful.ArticlesFALL 1996 39A driving force behind KDD is the database field (the second D in KDD). Indeed, the problem of effective data manipulation when data cannot fit in the main memory is of fun-damental importance to KDD. Database tech-niques for gaining efficient data access,grouping and ordering operations when ac-cessing data, and optimizing queries consti-tute the basics for scaling algorithms to larger data sets. Most data-mining algorithms from statistics, pattern recognition, and machine learning assume data are in the main memo-ry and pay no attention to how the algorithm breaks down if only limited views of the data are possible.A related field evolving from databases is data warehousing,which refers to the popular business trend of collecting and cleaning transactional data to make them available for online analysis and decision support. Data warehousing helps set the stage for KDD in two important ways: (1) data cleaning and (2)data access.Data cleaning: As organizations are forced to think about a unified logical view of the wide variety of data and databases they pos-sess, they have to address the issues of map-ping data to a single naming convention,uniformly representing and handling missing data, and handling noise and errors when possible.Data access: Uniform and well-defined methods must be created for accessing the da-ta and providing access paths to data that were historically difficult to get to (for exam-ple, stored offline).Once organizations and individuals have solved the problem of how to store and ac-cess their data, the natural next step is the question, What else do we do with all the da-ta? This is where opportunities for KDD natu-rally arise.A popular approach for analysis of data warehouses is called online analytical processing (OLAP), named for a set of principles pro-posed by Codd (1993). OLAP tools focus on providing multidimensional data analysis,which is superior to SQL in computing sum-maries and breakdowns along many dimen-sions. OLAP tools are targeted toward simpli-fying and supporting interactive data analysis,but the goal of KDD tools is to automate as much of the process as possible. Thus, KDD is a step beyond what is currently supported by most standard database systems.Basic DefinitionsKDD is the nontrivial process of identifying valid, novel, potentially useful, and ultimate-and still run efficiently, how results can be in-terpreted and visualized, and how the overall man-machine interaction can usefully be modeled and supported. The KDD process can be viewed as a multidisciplinary activity that encompasses techniques beyond the scope of any one particular discipline such as machine learning. In this context, there are clear opportunities for other fields of AI (be-sides machine learning) to contribute to KDD. KDD places a special emphasis on find-ing understandable patterns that can be inter-preted as useful or interesting knowledge.Thus, for example, neural networks, although a powerful modeling tool, are relatively difficult to understand compared to decision trees. KDD also emphasizes scaling and ro-bustness properties of modeling algorithms for large noisy data sets.Related AI research fields include machine discovery, which targets the discovery of em-pirical laws from observation and experimen-tation (Shrager and Langley 1990) (see Kloes-gen and Zytkow [1996] for a glossary of terms common to KDD and machine discovery),and causal modeling for the inference of causal models from data (Spirtes, Glymour,and Scheines 1993). Statistics in particular has much in common with KDD (see Elder and Pregibon [1996] and Glymour et al.[1996] for a more detailed discussion of this synergy). Knowledge discovery from data is fundamentally a statistical endeavor. Statistics provides a language and framework for quan-tifying the uncertainty that results when one tries to infer general patterns from a particu-lar sample of an overall population. As men-tioned earlier, the term data mining has had negative connotations in statistics since the 1960s when computer-based data analysis techniques were first introduced. The concern arose because if one searches long enough in any data set (even randomly generated data),one can find patterns that appear to be statis-tically significant but, in fact, are not. Clearly,this issue is of fundamental importance to KDD. Substantial progress has been made in recent years in understanding such issues in statistics. Much of this work is of direct rele-vance to KDD. Thus, data mining is a legiti-mate activity as long as one understands how to do it correctly; data mining carried out poorly (without regard to the statistical as-pects of the problem) is to be avoided. KDD can also be viewed as encompassing a broader view of modeling than statistics. KDD aims to provide tools to automate (to the degree pos-sible) the entire process of data analysis and the statistician’s “art” of hypothesis selection.Data mining is a step in the KDD process that consists of ap-plying data analysis and discovery al-gorithms that produce a par-ticular enu-meration ofpatterns (or models)over the data.Articles40AI MAGAZINEly understandable patterns in data (Fayyad, Piatetsky-Shapiro, and Smyth 1996).Here, data are a set of facts (for example, cases in a database), and pattern is an expres-sion in some language describing a subset of the data or a model applicable to the subset. Hence, in our usage here, extracting a pattern also designates fitting a model to data; find-ing structure from data; or, in general, mak-ing any high-level description of a set of data. The term process implies that KDD comprises many steps, which involve data preparation, search for patterns, knowledge evaluation, and refinement, all repeated in multiple itera-tions. By nontrivial, we mean that some search or inference is involved; that is, it is not a straightforward computation of predefined quantities like computing the av-erage value of a set of numbers.The discovered patterns should be valid on new data with some degree of certainty. We also want patterns to be novel (at least to the system and preferably to the user) and poten-tially useful, that is, lead to some benefit to the user or task. Finally, the patterns should be understandable, if not immediately then after some postprocessing.The previous discussion implies that we can define quantitative measures for evaluating extracted patterns. In many cases, it is possi-ble to define measures of certainty (for exam-ple, estimated prediction accuracy on new data) or utility (for example, gain, perhaps indollars saved because of better predictions orspeedup in response time of a system). No-tions such as novelty and understandabilityare much more subjective. In certain contexts,understandability can be estimated by sim-plicity (for example, the number of bits to de-scribe a pattern). An important notion, calledinterestingness(for example, see Silberschatzand Tuzhilin [1995] and Piatetsky-Shapiro andMatheus [1994]), is usually taken as an overallmeasure of pattern value, combining validity,novelty, usefulness, and simplicity. Interest-ingness functions can be defined explicitly orcan be manifested implicitly through an or-dering placed by the KDD system on the dis-covered patterns or models.Given these notions, we can consider apattern to be knowledge if it exceeds some in-terestingness threshold, which is by nomeans an attempt to define knowledge in thephilosophical or even the popular view. As amatter of fact, knowledge in this definition ispurely user oriented and domain specific andis determined by whatever functions andthresholds the user chooses.Data mining is a step in the KDD processthat consists of applying data analysis anddiscovery algorithms that, under acceptablecomputational efficiency limitations, pro-duce a particular enumeration of patterns (ormodels) over the data. Note that the space ofArticlesFALL 1996 41Figure 1. An Overview of the Steps That Compose the KDD Process.methods, the effective number of variables under consideration can be reduced, or in-variant representations for the data can be found.Fifth is matching the goals of the KDD pro-cess (step 1) to a particular data-mining method. For example, summarization, clas-sification, regression, clustering, and so on,are described later as well as in Fayyad, Piatet-sky-Shapiro, and Smyth (1996).Sixth is exploratory analysis and model and hypothesis selection: choosing the data-mining algorithm(s) and selecting method(s)to be used for searching for data patterns.This process includes deciding which models and parameters might be appropriate (for ex-ample, models of categorical data are differ-ent than models of vectors over the reals) and matching a particular data-mining method with the overall criteria of the KDD process (for example, the end user might be more in-terested in understanding the model than its predictive capabilities).Seventh is data mining: searching for pat-terns of interest in a particular representa-tional form or a set of such representations,including classification rules or trees, regres-sion, and clustering. The user can significant-ly aid the data-mining method by correctly performing the preceding steps.Eighth is interpreting mined patterns, pos-sibly returning to any of steps 1 through 7 for further iteration. This step can also involve visualization of the extracted patterns and models or visualization of the data given the extracted models.Ninth is acting on the discovered knowl-edge: using the knowledge directly, incorpo-rating the knowledge into another system for further action, or simply documenting it and reporting it to interested parties. This process also includes checking for and resolving po-tential conflicts with previously believed (or extracted) knowledge.The KDD process can involve significant iteration and can contain loops between any two steps. The basic flow of steps (al-though not the potential multitude of itera-tions and loops) is illustrated in figure 1.Most previous work on KDD has focused on step 7, the data mining. However, the other steps are as important (and probably more so) for the successful application of KDD in practice. Having defined the basic notions and introduced the KDD process, we now focus on the data-mining component,which has, by far, received the most atten-tion in the literature.patterns is often infinite, and the enumera-tion of patterns involves some form of search in this space. Practical computational constraints place severe limits on the sub-space that can be explored by a data-mining algorithm.The KDD process involves using the database along with any required selection,preprocessing, subsampling, and transforma-tions of it; applying data-mining methods (algorithms) to enumerate patterns from it;and evaluating the products of data mining to identify the subset of the enumerated pat-terns deemed knowledge. The data-mining component of the KDD process is concerned with the algorithmic means by which pat-terns are extracted and enumerated from da-ta. The overall KDD process (figure 1) in-cludes the evaluation and possible interpretation of the mined patterns to de-termine which patterns can be considered new knowledge. The KDD process also in-cludes all the additional steps described in the next section.The notion of an overall user-driven pro-cess is not unique to KDD: analogous propos-als have been put forward both in statistics (Hand 1994) and in machine learning (Brod-ley and Smyth 1996).The KDD ProcessThe KDD process is interactive and iterative,involving numerous steps with many deci-sions made by the user. Brachman and Anand (1996) give a practical view of the KDD pro-cess, emphasizing the interactive nature of the process. Here, we broadly outline some of its basic steps:First is developing an understanding of the application domain and the relevant prior knowledge and identifying the goal of the KDD process from the customer’s viewpoint.Second is creating a target data set: select-ing a data set, or focusing on a subset of vari-ables or data samples, on which discovery is to be performed.Third is data cleaning and preprocessing.Basic operations include removing noise if appropriate, collecting the necessary informa-tion to model or account for noise, deciding on strategies for handling missing data fields,and accounting for time-sequence informa-tion and known changes.Fourth is data reduction and projection:finding useful features to represent the data depending on the goal of the task. With di-mensionality reduction or transformationArticles42AI MAGAZINEThe Data-Mining Stepof the KDD ProcessThe data-mining component of the KDD pro-cess often involves repeated iterative applica-tion of particular data-mining methods. This section presents an overview of the primary goals of data mining, a description of the methods used to address these goals, and a brief description of the data-mining algo-rithms that incorporate these methods.The knowledge discovery goals are defined by the intended use of the system. We can distinguish two types of goals: (1) verification and (2) discovery. With verification,the sys-tem is limited to verifying the user’s hypothe-sis. With discovery,the system autonomously finds new patterns. We further subdivide the discovery goal into prediction,where the sys-tem finds patterns for predicting the future behavior of some entities, and description, where the system finds patterns for presenta-tion to a user in a human-understandableform. In this article, we are primarily con-cerned with discovery-oriented data mining.Data mining involves fitting models to, or determining patterns from, observed data. The fitted models play the role of inferred knowledge: Whether the models reflect useful or interesting knowledge is part of the over-all, interactive KDD process where subjective human judgment is typically required. Two primary mathematical formalisms are used in model fitting: (1) statistical and (2) logical. The statistical approach allows for nondeter-ministic effects in the model, whereas a logi-cal model is purely deterministic. We focus primarily on the statistical approach to data mining, which tends to be the most widely used basis for practical data-mining applica-tions given the typical presence of uncertain-ty in real-world data-generating processes.Most data-mining methods are based on tried and tested techniques from machine learning, pattern recognition, and statistics: classification, clustering, regression, and so on. The array of different algorithms under each of these headings can often be bewilder-ing to both the novice and the experienced data analyst. It should be emphasized that of the many data-mining methods advertised in the literature, there are really only a few fun-damental techniques. The actual underlying model representation being used by a particu-lar method typically comes from a composi-tion of a small number of well-known op-tions: polynomials, splines, kernel and basis functions, threshold-Boolean functions, and so on. Thus, algorithms tend to differ primar-ily in the goodness-of-fit criterion used toevaluate model fit or in the search methodused to find a good fit.In our brief overview of data-mining meth-ods, we try in particular to convey the notionthat most (if not all) methods can be viewedas extensions or hybrids of a few basic tech-niques and principles. We first discuss the pri-mary methods of data mining and then showthat the data- mining methods can be viewedas consisting of three primary algorithmiccomponents: (1) model representation, (2)model evaluation, and (3) search. In the dis-cussion of KDD and data-mining methods,we use a simple example to make some of thenotions more concrete. Figure 2 shows a sim-ple two-dimensional artificial data set consist-ing of 23 cases. Each point on the graph rep-resents a person who has been given a loanby a particular bank at some time in the past.The horizontal axis represents the income ofthe person; the vertical axis represents the to-tal personal debt of the person (mortgage, carpayments, and so on). The data have beenclassified into two classes: (1) the x’s repre-sent persons who have defaulted on theirloans and (2) the o’s represent persons whoseloans are in good status with the bank. Thus,this simple artificial data set could represent ahistorical data set that can contain usefulknowledge from the point of view of thebank making the loans. Note that in actualKDD applications, there are typically manymore dimensions (as many as several hun-dreds) and many more data points (manythousands or even millions).ArticlesFALL 1996 43Figure 2. A Simple Data Set with Two Classes Used for Illustrative Purposes.。

Feb. 2021Vol. 49 No. 32021 年 2 月第 49 卷 第 3 期机床与液压MACHINE TOOL & HYDRAULICSDOI ： 10.3969/j. issn. 1001 — 3881. 2021. 03. 011本文引用格式：吴晓燕，虞启凯,韩江义.关节空间内工业机器人抗干扰轨迹跟踪控制[J].机床与液压,2021,49(3):52-57.WU Xiaoyan ，YU Qikai ，HAN Jiangyi. Trajectory tracking control for an industrial robot in joint space with disturbancerejection technique [J]. Machine Tool & Hydraulics , 2021,49(3) ：52-57.关节空间内工业机器人抗干扰轨迹跟踪控制吴晓燕1，虞启凯1，韩江义2(1.南京科技职业学院智能制造学院，江苏南京210048；2.江苏大学汽车与交通工程学院，江苏镇江212013)摘要：针对工业机器人在关节空间内轨迹跟踪精度差和易受集总干扰影响等问题，提出一种基于非线性扰动观测器的 快速连续非奇异终端滑模控制策略。

根据拉格朗日方程推导出四轴工业机器人的动力学模型，获得系统的输入输出关系。

引入非线性扰动观测器对集总干扰进行估计与补偿，设计快速连续非奇异终端滑模控制器来加快系统状态量的收敛速率, 提高关节空间内轨迹跟踪的精度。

由李雅普诺夫稳定性理论证明了此控制器的全局稳定性。

通过仿真案例与试验研究验证 了所设计控制策略的有效性，结果表明：该控制器能有效抑制集总干扰影响，保证工业机器人轨迹跟踪的精度，具有一定的工程参考价值。

关键词：工业机器人；轨迹跟踪控制；集总干扰；非线性扰动观测器；终端滑模中图分类号： TP242Trajectory Tracking Control for an Industrial Robot in JointSpace with Disturbance Rejection TechniqueWU Xiaoyan 1 , YU Qikai 1 , HAN Jiangyi 2(1. School of Intelligent Manufacturing , Nanjing Polytechnic Institute , Nanjing Jiangsu 210048 , China ；2. School of Automotive and Traffic Engineering , Jiangsu University , Zhenjiang Jiangsu 212013, China)Abstract : Aiming at the difficulty of poor trajectory tracking precision and susceptible to lumped disturbances in joint space con ­trol for an industrial robot ， a fast-continuous nonsingular terminal sliding mode control strategy was proposed based on nonlinear dis ­turbance observer. The dynamical model of a 4-DOF robot was modeled by using Lagrangian equation ， by which the inputs and outputsof the system were obtained. A nonlinear disturbance observer was introduced to estimate and compensate the lumped disturbances. Anda fast-continuous nonsingular terminal sliding mode controller was designed to accelerate the convergence of the system state variables ，which could improve the trajectory tracking precision in joint space. Meanwhile ， the global stability of the controller was proved byusing Lyapunov theory. Some simulation cases and experiments were conducted to test the efficiency of the controller. The results showthat the proposed controller can be used to retrain the lumped disturbances and to ensure the trajectory tracking precision ， which has aproject application value.Keywords ： Industrial robot ； Trajectory tracking control ； Lumped disturbances ； Nonlinear disturbance observer ； Terminal slidingmode0 前言智能制造是现代工业升级的基础，随着信息技术、机器人技术、新能源、人工智能等重要领域和前 沿方向的革命性突破和交叉融合，正在引发新一轮的 产业变革°工业机器人是智能制造中的重要载体，支撑着产业发展°作为工业机器人的关键性问题之一， 关节空间内高精度轨迹跟踪控制一直是国内外学者们的研究热点与重点[1-2]°然而，机器人系统是一个强耦合、高非线性、多变量的系统，要获得高精度的轨 迹跟踪控制难度颇大°另外，系统的未建模特性、关 节摩擦间隙、外界干扰和末端未知负载等因素构成的 集总干扰也会进一步加剧设计轨迹跟踪控制器的难 度°上述问题导致机器人关节空间内抗干扰轨迹跟踪控制极具挑战性°收稿日期：2019-11-29基金项目：南京科技职业学院2019年院级科研项目(NHKY-2019-12)作者简介：吴晓燕(1979—),女，硕士，讲师，主要从事智能制造研究。

More informationNONLINEAR TIME SERIES ANALYSISThis book represents a modern approach to time series analysis which is based onthe theory of dynamical systems.It starts from a sound outline of the underlyingtheory to arrive at very practical issues,which are illustrated using a large number ofempirical data sets taken from variousfields.This book will hence be highly usefulfor scientists and engineers from all disciplines who study time variable signals,including the earth,life and social sciences.The paradigm of deterministic chaos has influenced thinking in manyfields ofscience.Chaotic systems show rich and surprising mathematical structures.In theapplied sciences,deterministic chaos provides a striking explanation for irregulartemporal behaviour and anomalies in systems which do not seem to be inherentlystochastic.The most direct link between chaos theory and the real world is the anal-ysis of time series from real systems in terms of nonlinear dynamics.Experimentaltechnique and data analysis have seen such dramatic progress that,by now,mostfundamental properties of nonlinear dynamical systems have been observed in thelaboratory.Great efforts are being made to exploit ideas from chaos theory where-ver the data display more structure than can be captured by traditional methods.Problems of this kind are typical in biology and physiology but also in geophysics,economics and many other sciences.This revised edition has been significantly rewritten an expanded,includingseveral new chapters.In view of applications,the most relevant novelties will be thetreatment of non-stationary data sets and of nonlinear stochastic processes insidethe framework of a state space reconstruction by the method of delays.Hence,non-linear time series analysis has left the rather narrow niche of strictly deterministicsystems.Moreover,the analysis of multivariate data sets has gained more atten-tion.For a direct application of the methods of this book to the reader’s own datasets,this book closely refers to the publicly available software package TISEAN.The availability of this software will facilitate the solution of the exercises,so thatreaders now can easily gain their own experience with the analysis of data sets.Holger Kantz,born in November1960,received his diploma in physics fromthe University of Wuppertal in January1986with a thesis on transient chaos.InJanuary1989he obtained his Ph.D.in theoretical physics from the same place,having worked under the supervision of Peter Grassberger on Hamiltonian many-particle dynamics.During his postdoctoral time,he spent one year on a Marie Curiefellowship of the European Union at the physics department of the University ofMore informationFlorence in Italy.In January1995he took up an appointment at the newly foundedMax Planck Institute for the Physics of Complex Systems in Dresden,where heestablished the research group‘Nonlinear Dynamics and Time Series Analysis’.In1996he received his venia legendi and in2002he became adjunct professorin theoretical physics at Wuppertal University.In addition to time series analysis,he works on low-and high-dimensional nonlinear dynamics and its applications.More recently,he has been trying to bridge the gap between dynamics and statis-tical physics.He has(co-)authored more than75peer-reviewed articles in scien-tific journals and holds two international patents.For up-to-date information seehttp://www.mpipks-dresden.mpg.de/mpi-doc/kantzgruppe.html.Thomas Schreiber,born1963,did his diploma work with Peter Grassberger atWuppertal University on phase transitions and information transport in spatio-temporal chaos.He joined the chaos group of Predrag Cvitanovi´c at the Niels BohrInstitute in Copenhagen to study periodic orbit theory of diffusion and anomaloustransport.There he also developed a strong interest in real-world applications ofchaos theory,leading to his Ph.D.thesis on nonlinear time series analysis(Univer-sity of Wuppertal,1994).As a research assistant at Wuppertal University and duringseveral extended appointments at the Max Planck Institute for the Physics of Com-plex Systems in Dresden he published numerous research articles on time seriesmethods and applications ranging from physiology to the stock market.His habil-itation thesis(University of Wuppertal)appeared as a review in Physics Reportsin1999.Thomas Schreiber has extensive experience teaching nonlinear dynamicsto students and experts from variousfields and at all levels.Recently,he has leftacademia to undertake industrial research.NONLINEAR TIME SERIES ANALYSIS HOLGER KANTZ AND THOMAS SCHREIBERMax Planck Institute for the Physics of Complex Systems,DresdenMore informationMore informationpublished by the press syndicate of the university of cambridgeThe Pitt Building,Trumpington Street,Cambridge,United Kingdomcambridge university pressThe Edinburgh Building,Cambridge CB22RU,UK40West20th Street,New York,NY10011–4211,USA477Williamstown Road,Port Melbourne,VIC3207,AustraliaRuiz de Alarc´o n13,28014Madrid,SpainDock House,The Waterfront,Cape Town8001,South AfricaC Holger Kantz and Thomas Schreiber,2000,2003This book is in copyright.Subject to statutory exceptionand to the provisions of relevant collective licensing agreements,no reproduction of any part may take place withoutthe written permission of Cambridge University Press.First published2000Second edition published2003Printed in the United Kingdom at the University Press,CambridgeTypeface Times11/14pt.System L A T E X2ε[tb]A catalogue record for this book is available from the British LibraryLibrary of Congress Cataloguing in Publication dataKantz,Holger,1960–Nonlinear time series analysis/Holger Kantz and Thomas Schreiber.–[2nd ed.].p.cm.Includes bibliographical references and index.ISBN0521821509–ISBN0521529026(paperback)1.Time-series analysis.2.Nonlinear theories.I.Schreiber,Thomas,1963–II.TitleQA280.K3552003519.5 5–dc212003044031ISBN0521821509hardbackISBN0521529026paperbackThe publisher has used its best endeavours to ensure that the URLs for external websites referred to in this bookare correct and active at the time of going to press.However,the publisher has no responsibility for the websites and can make no guarantee that a site will remain live or that the content is or will remain appropriate.More informationContentsPreface to thefirst edition page xiPreface to the second edition xiiiAcknowledgements xvI Basic topics11Introduction:why nonlinear methods?32Linear tools and general considerations132.1Stationarity and sampling132.2Testing for stationarity152.3Linear correlations and the power spectrum182.3.1Stationarity and the low-frequency component in thepower spectrum232.4Linearfilters242.5Linear predictions273Phase space methods303.1Determinism:uniqueness in phase space303.2Delay reconstruction353.3Finding a good embedding363.3.1False neighbours373.3.2The time lag393.4Visual inspection of data393.5Poincar´e surface of section413.6Recurrence plots434Determinism and predictability484.1Sources of predictability484.2Simple nonlinear prediction algorithm504.3Verification of successful prediction534.4Cross-prediction errors:probing stationarity564.5Simple nonlinear noise reduction58vMore informationvi Contents5Instability:Lyapunov exponents655.1Sensitive dependence on initial conditions655.2Exponential divergence665.3Measuring the maximal exponent from data696Self-similarity:dimensions756.1Attractor geometry and fractals756.2Correlation dimension776.3Correlation sum from a time series786.4Interpretation and pitfalls826.5Temporal correlations,non-stationarity,and space timeseparation plots876.6Practical considerations916.7A useful application:determination of the noise level using thecorrelation integral926.8Multi-scale or self-similar signals956.8.1Scaling laws966.8.2Detrendedfluctuation analysis1007Using nonlinear methods when determinism is weak1057.1Testing for nonlinearity with surrogate data1077.1.1The null hypothesis1097.1.2How to make surrogate data sets1107.1.3Which statistics to use1137.1.4What can go wrong1157.1.5What we have learned1177.2Nonlinear statistics for system discrimination1187.3Extracting qualitative information from a time series1218Selected nonlinear phenomena1268.1Robustness and limit cycles1268.2Coexistence of attractors1288.3Transients1288.4Intermittency1298.5Structural stability1338.6Bifurcations1358.7Quasi-periodicity139II Advanced topics1419Advanced embedding methods1439.1Embedding theorems1439.1.1Whitney’s embedding theorem1449.1.2Takens’s delay embedding theorem1469.2The time lag148More informationContents vii9.3Filtered delay embeddings1529.3.1Derivative coordinates1529.3.2Principal component analysis1549.4Fluctuating time intervals1589.5Multichannel measurements1599.5.1Equivalent variables at different positions1609.5.2Variables with different physical meanings1619.5.3Distributed systems1619.6Embedding of interspike intervals1629.7High dimensional chaos and the limitations of the time delayembedding1659.8Embedding for systems with time delayed feedback17110Chaotic data and noise17410.1Measurement noise and dynamical noise17410.2Effects of noise17510.3Nonlinear noise reduction17810.3.1Noise reduction by gradient descent17910.3.2Local projective noise reduction18010.3.3Implementation of locally projective noise reduction18310.3.4How much noise is taken out?18610.3.5Consistency tests19110.4An application:foetal ECG extraction19311More about invariant quantities19711.1Ergodicity and strange attractors19711.2Lyapunov exponents II19911.2.1The spectrum of Lyapunov exponents and invariantmanifolds20011.2.2Flows versus maps20211.2.3Tangent space method20311.2.4Spurious exponents20511.2.5Almost two dimensionalflows21111.3Dimensions II21211.3.1Generalised dimensions,multi-fractals21311.3.2Information dimension from a time series21511.4Entropies21711.4.1Chaos and theflow of information21711.4.2Entropies of a static distribution21811.4.3The Kolmogorov–Sinai entropy22011.4.4The -entropy per unit time22211.4.5Entropies from time series data226More informationviii Contents11.5How things are related22911.5.1Pesin’s identity22911.5.2Kaplan–Yorke conjecture23112Modelling and forecasting23412.1Linear stochastic models andfilters23612.1.1Linearfilters23712.1.2Nonlinearfilters23912.2Deterministic dynamics24012.3Local methods in phase space24112.3.1Almost model free methods24112.3.2Local linearfits24212.4Global nonlinear models24412.4.1Polynomials24412.4.2Radial basis functions24512.4.3Neural networks24612.4.4What to do in practice24812.5Improved cost functions24912.5.1Overfitting and model costs24912.5.2The errors-in-variables problem25112.5.3Modelling versus prediction25312.6Model verification25312.7Nonlinear stochastic processes from data25612.7.1Fokker–Planck equations from data25712.7.2Markov chains in embedding space25912.7.3No embedding theorem for Markov chains26012.7.4Predictions for Markov chain data26112.7.5Modelling Markov chain data26212.7.6Choosing embedding parameters for Markov chains26312.7.7Application:prediction of surface wind velocities26412.8Predicting prediction errors26712.8.1Predictability map26712.8.2Individual error prediction26812.9Multi-step predictions versus iterated one-step predictions27113Non-stationary signals27513.1Detecting non-stationarity27613.1.1Making non-stationary data stationary27913.2Over-embedding28013.2.1Deterministic systems with parameter drift28013.2.2Markov chain with parameter drift28113.2.3Data analysis in over-embedding spaces283More informationContents ix13.2.4Application:noise reduction for human voice28613.3Parameter spaces from data28814Coupling and synchronisation of nonlinear systems29214.1Measures for interdependence29214.2Transfer entropy29714.3Synchronisation29915Chaos control30415.1Unstable periodic orbits and their invariant manifolds30615.1.1Locating periodic orbits30615.1.2Stable/unstable manifolds from data31215.2OGY-control and derivates31315.3Variants of OGY-control31615.4Delayed feedback31715.5Tracking31815.6Related aspects319A Using the TISEAN programs321A.1Information relevant to most of the routines322A.1.1Efficient neighbour searching322A.1.2Re-occurring command options325A.2Second-order statistics and linear models326A.3Phase space tools327A.4Prediction and modelling329A.4.1Locally constant predictor329A.4.2Locally linear prediction329A.4.3Global nonlinear models330A.5Lyapunov exponents331A.6Dimensions and entropies331A.6.1The correlation sum331A.6.2Information dimension,fixed mass algorithm332A.6.3Entropies333A.7Surrogate data and test statistics334A.8Noise reduction335A.9Finding unstable periodic orbits336A.10Multivariate data336B Description of the experimental data sets338B.1Lorenz-like chaos in an NH3laser338B.2Chaos in a periodically modulated NMR laser340B.3Vibrating string342B.4Taylor–Couetteflow342B.5Multichannel physiological data343More informationx ContentsB.6Heart rate during atrialfibrillation343B.7Human electrocardiogram(ECG)344B.8Phonation data345B.9Postural control data345B.10Autonomous CO2laser with feedback345B.11Nonlinear electric resonance circuit346B.12Frequency doubling solid state laser348B.13Surface wind velocities349References350Index365More informationPreface to thefirst editionThe paradigm of deterministic chaos has influenced thinking in manyfields of sci-ence.As mathematical objects,chaotic systems show rich and surprising structures.Most appealing for researchers in the applied sciences is the fact that determinis-tic chaos provides a striking explanation for irregular behaviour and anomalies insystems which do not seem to be inherently stochastic.The most direct link between chaos theory and the real world is the analysis oftime series from real systems in terms of nonlinear dynamics.On the one hand,experimental technique and data analysis have seen such dramatic progress that,by now,most fundamental properties of nonlinear dynamical systems have beenobserved in the laboratory.On the other hand,great efforts are being made to exploitideas from chaos theory in cases where the system is not necessarily deterministicbut the data displays more structure than can be captured by traditional methods.Problems of this kind are typical in biology and physiology but also in geophysics,economics,and many other sciences.In all thesefields,even simple models,be they microscopic or phenomenological,can create extremely complicated dynamics.How can one verify that one’s model isa good counterpart to the equally complicated signal that one receives from nature?Very often,good models are lacking and one has to study the system just from theobservations made in a single time series,which is the case for most non-laboratorysystems in particular.The theory of nonlinear dynamical systems provides new toolsand quantities for the characterisation of irregular time series data.The scope ofthese methods ranges from invariants such as Lyapunov exponents and dimensionswhich yield an accurate description of the structure of a system(provided thedata are of high quality)to statistical techniques which allow for classification anddiagnosis even in situations where determinism is almost lacking.This book provides the experimental researcher in nonlinear dynamics with meth-ods for processing,enhancing,and analysing the measured signals.The theorist willbe offered discussions about the practical applicability of mathematical results.ThexiMore informationxii Preface to thefirst editiontime series analyst in economics,meteorology,and otherfields willfind inspira-tion for the development of new prediction algorithms.Some of the techniquespresented here have also been considered as possible diagnostic tools in clinical re-search.We will adopt a critical but constructive point of view,pointing out ways ofobtaining more meaningful results with limited data.We hope that everybody whohas a time series problem which cannot be solved by traditional,linear methodswillfind inspiring material in this book.Dresden and WuppertalNovember1996More informationPreface to the second editionIn afield as dynamic as nonlinear science,new ideas,methods and experimentsemerge constantly and the focus of interest shifts accordingly.There is a continuousstream of new results,and existing knowledge is seen from a different angle aftervery few years.Five years after thefirst edition of“Nonlinear Time Series Analysis”we feel that thefield has matured in a way that deserves being reflected in a secondedition.The modification that is most immediately visible is that the program listingshave been be replaced by a thorough discussion of the publicly available softwareTISEAN.Already a few months after thefirst edition appeared,it became clearthat most users would need something more convenient to use than the bare libraryroutines printed in the book.Thus,together with Rainer Hegger we prepared stand-alone routines based on the book but with input/output functionality and advancedfeatures.Thefirst public release was made available in1998and subsequent releasesare in widespread use now.Today,TISEAN is a mature piece of software thatcovers much more than the programs we gave in thefirst edition.Now,readerscan immediately apply most methods studied in the book on their own data usingTISEAN programs.By replacing the somewhat terse program listings by minuteinstructions of the proper use of the TISEAN routines,the link between book andsoftware is strengthened,supposedly to the benefit of the readers and users.Hencewe recommend a download and installation of the package,such that the exercisescan be readily done by help of these ready-to-use routines.The current edition has be extended in view of enlarging the class of data sets to betreated.The core idea of phase space reconstruction was inspired by the analysis ofdeterministic chaotic data.In contrast to many expectations,purely deterministicand low-dimensional data are rare,and most data fromfield measurements areevidently of different nature.Hence,it was an effort of our scientific work over thepast years,and it was a guiding concept for the revision of this book,to explore thepossibilities to treat other than purely deterministic data sets.xiiiMore informationxiv Preface to the second editionThere is a whole new chapter on non-stationary time series.While detectingnon-stationarity is still briefly discussed early on in the book,methods to deal withmanifestly non-stationary sequences are described in some detail in the secondpart.As an illustration,a data source of lasting interest,human speech,is used.Also,a new chapter deals with concepts of synchrony between systems,linear andnonlinear correlations,information transfer,and phase synchronisation.Recent attempts on modelling nonlinear stochastic processes are discussed inChapter12.The theoretical framework forfitting Fokker–Planck equations to datawill be reviewed and evaluated.While Chapter9presents some progress that hasbeen made in modelling input–output systems with stochastic but observed inputand on the embedding of time delayed feedback systems,the chapter on mod-elling considers a data driven phase space approach towards Markov chains.Windspeed measurements are used as data which are best considered to be of nonlinearstochastic nature despite the fact that a physically adequate mathematical model isthe deterministic Navier–Stokes equation.In the chapter on invariant quantities,new material on entropy has been included,mainly on the -and continuous entropies.Estimation problems for stochastic ver-sus deterministic data and data with multiple length and time scales are discussed.Since more and more experiments now yield good multivariate data,alternativesto time delay embedding using multiple probe measurements are considered at var-ious places in the text.This new development is also reflected in the functionalityof the TISEAN programs.A new multivariate data set from a nonlinear semicon-ductor electronic circuit is introduced and used in several places.In particular,adifferential equation has been successfully established for this system by analysingthe data set.Among other smaller rearrangements,the material from the former chapter“Other selected topics”,has been relocated to places in the text where a connectioncan be made more naturally.High dimensional and spatio-temporal data is now dis-cussed in the context of embedding.We discuss multi-scale and self-similar signalsnow in a more appropriate way right after fractal sets,and include recent techniquesto analyse power law correlations,for example detrendedfluctuation analysis.Of course,many new publications have appeared since1997which are potentiallyrelevant to the scope of this book.At least two new monographs are concerned withthe same topic and a number of review articles.The bibliography has been updatedbut remains a selection not unaffected by personal preferences.We hope that the extended book will prove its usefulness in many applicationsof the methods and further stimulate thefield of time series analysis.DresdenDecember2002More informationAcknowledgementsIf there is any feature of this book that we are proud of,it is the fact that almost allthe methods are illustrated with real,experimental data.However,this is anythingbut our own achievement–we exploited other people’s work.Thus we are deeplyindebted to the experimental groups who supplied data sets and granted permissionto use them in this book.The production of every one of these data sets requiredskills,experience,and equipment that we ourselves do not have,not forgetting thehours and hours of work spent in the laboratory.We appreciate the generosity ofthe following experimental groups:NMR laser.Our contact persons at the Institute for Physics at Z¨u rich University were Leci Flepp and Joe Simonet;the head of the experimental group is E.Brun.(See AppendixB.2.)Vibrating string.Data were provided by Tim Molteno and Nick Tufillaro,Otago University, Dunedin,New Zealand.(See Appendix B.3.)Taylor–Couetteflow.The experiment was carried out at the Institute for Applied Physics at Kiel University by Thorsten Buzug and Gerd Pfister.(See Appendix B.4.) Atrialfibrillation.This data set is taken from the MIT-BIH Arrhythmia Database,collected by G.B.Moody and R.Mark at Beth Israel Hospital in Boston.(See Appendix B.6.) Human ECG.The ECG recordings we used were taken by Petr Saparin at Saratov State University.(See Appendix B.7.)Foetal ECG.We used noninvasively recorded(human)foetal ECGs taken by John F.Hofmeister as the Department of Obstetrics and Gynecology,University of Colorado,Denver CO.(See Appendix B.7.)Phonation data.This data set was made available by Hanspeter Herzel at the Technical University in Berlin.(See Appendix B.8.)Human posture data.The time series was provided by Steven Boker and Bennett Bertenthal at the Department of Psychology,University of Virginia,Charlottesville V A.(SeeAppendix B.9.)xvMore informationxvi AcknowledgementsAutonomous CO2laser with feedback.The data were taken by Riccardo Meucci and Marco Ciofini at the INO in Firenze,Italy.(See Appendix B.10.)Nonlinear electric resonance circuit.The experiment was designed and operated by M.Diestelhorst at the University of Halle,Germany.(See Appendix B.11.)Nd:YAG laser.The data we use were recorded in the University of Oldenburg,where we wish to thank Achim Kittel,Falk Lange,Tobias Letz,and J¨u rgen Parisi.(See AppendixB.12.)We used the following data sets published for the Santa Fe Institute Time SeriesContest,which was organised by Neil Gershenfeld and Andreas Weigend in1991:NH3laser.We used data set A and its continuation,which was published after the contest was closed.The data was supplied by U.H¨u bner,N.B.Abraham,and C.O.Weiss.(SeeAppendix B.1.)Human breath rate.The data we used is part of data set B of the contest.It was submitted by Ari Goldberger and coworkers.(See Appendix B.5.)During the composition of the text we asked various people to read all or part of themanuscript.The responses ranged from general encouragement to detailed technicalcomments.In particular we thank Peter Grassberger,James Theiler,Daniel Kaplan,Ulrich Parlitz,and Martin Wiesenfeld for their helpful remarks.Members of ourresearch groups who either contributed by joint work to our experience and knowl-edge or who volunteered to check the correctness of the text are Rainer Hegger,Andreas Schmitz,Marcus Richter,Mario Ragwitz,Frank Schm¨u ser,RathinaswamyBhavanan Govindan,and Sharon Sessions.We have also considerably profited fromcomments and remarks of the readers of thefirst edition of the book.Their effortin writing to us is gratefully appreciated.Last but not least we acknowledge the encouragement and support by SimonCapelin from Cambridge University Press and the excellent help in questions ofstyle and English grammar by Sheila Shepherd.。

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基于深度学习的多模态时空动作识别①吴　敏, 王　敏(河海大学 计算机与信息学院, 南京 211100)通讯作者: 吴　敏摘　要: 针对视频理解中的时序难点以及传统方法计算量大的困难, 提出了一种带有时空模块的方法用于动作识别. 该方法采用残差网络作为框架, 加入时空模块提取图像以及时序信息, 并且加入RGB 差值信息增强数据, 采用NetVLAD 方法聚合所有的特征信息, 最后实现行为动作的分类. 实验结果表明, 基于时空模块的多模态方法具有较好的识别精度.关键词: 时空模型; 多模态; 动作识别引用格式: 吴敏,王敏.基于深度学习的多模态时空动作识别.计算机系统应用,2021,30(3):272–275. /1003-3254/7840.htmlMulti-Modal Spatiotemporal Action Recognition Based on Deep LearningWU Min, WANG Min(College of Computer and Information, Hohai University, Nanjing 211100, China)Abstract : In view of the time-series difficulty in video understanding and a large amount of calculation in traditional methods, we propose a method with spatio-temporal module for action recognition. With a residual network as the framework, this method adds spatio-temporal module to extract images and time series, adds RGB difference to enhance data, and finally uses the NetVLAD method to aggregate all feature information. In this way, actions are classified. The experimental results show that the multimodal method based on spatio-temporal module has better recognition accuracy.Key words : spatio-temporal model; multi-modal; action recognition1 引言由于互联网的快速发展, 传播媒介的日渐丰富, 网络视频的数量以指数级的速度大量增长. 如何理解视频内容成为一个亟需解决的问题. 动作识别作为计算机视觉中的一个热门领域受到了广泛的关注, 在监控分析、人机交互、体育视频解读等领域内有着广阔的应用前景.在动作识别中, 有两个关键的有效信息: 空间信息和运动信息. 一个识别系统的性能在很大程度上取决于它能否从中提取和利用相关信息. 然而, 由于许多复杂的因素, 例如比例变化、视角变化和相机运动等, 提取这些信息是非常困难的. 因此, 设计有效的特征表示和模型方法来处理这些挑战, 同时保留动作的有效分类信息就变的至关重要. 随着深度学习在图像、文本等领域内取得了成功后, 该方法在动作识别领域内也得到了广泛的应用, 由早期的手工特征的方法, 转变为基于深度学习的方法. 卷积神经网络有着强大的建模能力, 近年来, 机器设备计算能力的提升和大型数据集的出现, 使得基于深度学习的方法成为视频分析动作识别的参考标准.动作识别作为动作预测领域和人体姿态分析的基础, 其主要目标就是对视频中的人物所做的行为动作进行理解分类, 那么如何有效利用视频中的各部分有效信息进行识别是首要问题. 视频识别和图像识别中最大的区分点就是时序信息的使用和建模. 早期采用时空描述符用于特征的提取和分类, Wang 等[1]提出采计算机系统应用 ISSN 1003-3254, CODEN CSAOBNE-mail: Computer Systems & Applications,2021,30(3):272−275 [doi: 10.15888/ki.csa.007840] ©中国科学院软件研究所版权所有.Tel: +86-10-62661041① 收稿时间: 2020-07-19; 修改时间: 2020-08-28; 采用时间: 2020-09-01; csa 在线出版时间: 2021-03-03272用Fisher向量对密集运动轨迹DT进行编码表示. 基于此, Wang等[2]提出改进后的IDT算法, 改进特征正则化方式和特征编码方式, 在动作识别上取得了显著的成果. 深度学习的出现使得能够更好地进行特征的提取和学习. 2D卷积建模用于视频理解主要是对单帧视频进行特征的提取, 不能够很好地对时序信息进行建模. Simonyan等[3]提出了将基于外观的信息与运动相关的信息分离出来, 使用两个并行的卷积网络处理RGB和光流输入, 基于空间流和光流图的双流卷积网络方法用于动作识别, 识别率高. Wang等[4]基于双流网络提出时间片段网络, 将整段视频分割成连续的视频片段, 将每段视频分别输入到网络中, 它将稀疏时间采样策略和基于视频的监督相结合, 使用整个视频有效的学习. 3D卷积[5]能够对时空信息进行更好的捕捉,但是所需的计算成本太大. Ji等[6]首先提出了扩展时间信息后的3D卷积网络用于动作识别, 使用三维核从空间和时间维度中提取特征. Tran等[7]使用三维卷积和三维池化进一步改进3D卷积网络并命名为C3D.近年来对于识别的实时性要求不断提高, 网络架构转向采用轻量级的模块来替代传统的光流方法来减少计算量. Lee等[8]提出了包含运动模块的运动特征网络MFNet, 该运动块可以在端到端训练的统一网络中的相邻帧之间编码时空信息. Jiang等[9]将2D网络作为主干架构, 提出一个简单高效的STM模块用于编码空间和运动信息. Feichtenhofer等[10]提出了快慢结合的模型, 使用了一个慢速高分辨率CNN(Slow通道)来分析视频中的静态内容, 同时使用一个快速低分辨率CNN(Fast通道)来分析视频中的动态内容, 对同一个视频片段应用两个平行的卷积神经网络, 取得了显著的效果. Zhao等[11]将RGB和光流嵌入到一个具有新层的二合一流网络中, 在运动条件层从流图像中提取运动信息, 在运动调制层利用这些信息生成用于调制低层RGB特征的变换参数, 进行端到端的训练, 利用运动条件对RGB特征进行调制可以提高检测精度.本文采用时空模块提取图像以及时序信息, 使用平移部分通道的方法来实现时空信息的融合, 减少计算量, 同时加入RGB差值信息增强数据, 最后采用NetVLAD聚合所有的特征信息实现行为动作的分类.2 基于深度学习的多模态时空动作识别2.1 移位时空模块该模块的主要功能是对不同时间点提取的视频帧特征图进行信息交换, 从而实现时序特征的提取. 地址的移位用于图像识别取得了较好的效果. 根据文献[12]中的模型, 不同于卷积操作, 移位操作本身不需要参数或浮点运算, 相反移位操作包含一系列的记忆性操作,可以通过移位操作融合1×1卷积来提取聚合特征信息,从而减少计算量. 以普通一维卷积举例来说, 预测值表示为对不同输入进行加权求和的结果值, 如式(1)所示. 换一个角度如果将输入值看成是当前时间点和相邻时间点的输入, 也就是输入值看成移位后的−1, 0, 1三个时间点的输入值后, 如式(2)所示, 再进行乘性相加, 如式(3)所示. 由此移位卷积可以概括为移位和乘性相加两个过程的结果.将T帧图片C通道的输入进行排列后的张量, 如图1所示.图1 移位模块的特征示意图每行的各图片通道都表示的是不同时间点获取的图片帧特征值. 对于不同时间点下的同一通道的特征值沿着时间维度进行平移, 部分通道值向下平移一格,部分通道值向上平移一格, 移位后空缺的部分补0, 多出的特征图通道值移出, 从而实现双向平移, 相邻帧的特征信息在移动后与当前帧混合. 但并不是平移越多,交换的信息也越多. 如果移位的比例太小, 时间建模的能力可能不足以处理复杂的时间关系; 如果移位的比例太大, 空间特征学习能力可能会降低过多. 为了进行有效的时空信息融合, 避免移动过多的通道而导致空间建模能力下降, 只移动部分通道, 从而达到平衡空间特征学习和时间特征学习的模型能力.本文将该移位模块加入到残差网络的每个分支残差块中, 在卷积操作前进行移位操作, 不增加3D计算2021 年 第 30 卷 第 3 期计算机系统应用273量的情况下实现时空信息的融合, 对于每个插入的移位时空模块, 时间感受野被放大2倍, 由此进行复杂的时间建模.2.2 多模态除了充分利用时空信息之外, 本文还加入叠加的RGB 差值进行多模态的输入, 实现信息增强的效果. 常用的提取光流图来表征运动信息的方法计算量大, 在光流图计算过程中的关键步骤是将像素值沿时间方向求偏导, 所以本文将其简化成RGB 差值来作为输入,来表示外观变化和显著运动的区域, 从中训练学习运动信息, 从而大大节省了光流提取的时间. 得出的预估分数与时空特征得出的分数进行相加平均用于识别结果.2.3 NetVLAD 方法VLAD 方法[13]在图像检索领域中作为局部聚合描述符向量, 对提取的图像特征进行后处理编码用于图像的表示, 近年来开始应用到端到端的卷积神经网络中用来表示图像特征. 本文采用NetVLAD 方法[14]来作为池化层加入到卷积层的最后, 作为池化层来聚合特征信息.i ∈{1···N }x i ∈R D K c k R D x i x i −c k 对于一张特征图x , 需要从空间位置获取D 维的特征向量来表示该特征图. 首先给定个聚类中心将特征空间划分成K 个单元. 每一个特征向量都对应着一个单元, 并用残差向量表示特征向量和聚类中心的差值, 由此得到的差分向量表示为:x i (j )c k (j )x i c k αv ∈R KD 其中, 和表示特征向量和聚类中心的第j 个分量, 是可训练的超参数. 输出矩阵V 中的第K 列表示的是第K 个单元中聚合的特征向量, 接着将矩阵按列进行归一化, 以及L 2-归一化后化为一维向量表征特征图. 最后将输出值送入到全连接层用于分类.3 实验分析3.1 实验环境与实验数据实验硬件配置为GTX 1080 Ti, 编程语言为Python,基于PyTorch 框架. 数据集是UCF101[15]和HMDB51[16].UCF101数据集包含101个动作类别, 共13 320个视频片段. HMDB51数据集是一个包含电影、网络视频等多个来源的真实动作视频的集合, 共51个类别, 6766个视频片段. 数据集都提供了相应的训练集和测试集的划分. 调整视频帧为224×224作为网络的输入.训练参数为: 50个epoch, 初始学习率为0. 01, 权值衰减率为1e–4, 批处理大小为16, dropout 值为0. 5.本文使用从Kinetics 数据集[17]预先训练的权重进行微调,并冻结批处理规范化层. 对于残差移位模块, 根据文献[18]中的研究结果, 当部分移位信道1/4 (双向移位每个方向1/8)时, 性能达到峰值.时空模块的部分采用的是ResNet50框架, 将时空模块加入到残差网络的分支残差块中, 获得更好的空间特征学习能力.3.2 实验结果分析从表1可以看出, 本文中加入时空模块以及多模态的方法确实能够对识别精度有一定程度的提升, 对比C3D 、ArtNet 方法, 在预训练数据集相同, 浮点运算的数量级相同的条件下, 对两个数据集的识别精度分别达到了不同程度的提升. 对比TSN 方法, 识别精度得到了很大的提升, 在两个数据集上分别提升了8.8和19.4个百分点, 也能够看出使用大型动作数据集进行预训练得出的参数优化能够使实验结果精度得到更大的提升. 在计算资源充足的条件下, 预训练能够对识别的精度起到较大的提升影响. 同时对比I3D 方法, 本文方法在基于2D 模型下的浮点计算量, 能够达到与之相匹敌的识别精度, 实现了计算量和识别精度两方面的平衡.表1 实验结果方法预训练数据集基础架构FLOPs (GB)UCF101HMDB51TSN [19]ImageNet+Kinetics Inception V21686. 453. 7C3D [7]Kinetics ResNet-182089. 862. 1ArtNet[20]KineticsResNet-182494. 370. 9I3D [21]ImageNet+Kinetics 3DResNet-5015395. 674. 8本文Kinetics ResNet-503395. 273. 14 结论与展望本文提出了一种带时空模块的多模态方法. 该方法将时空模块引入到2D 卷积网络中, 实现时空信息的提取融合, 不增加浮点运算, 同时加入RGB 差值进行信息增强, 并采用NetVLAD 方法聚合所有的特征信息,最后实现行为动作的分类, 在数据集UCF101和HMDB51计算机系统应用2021 年 第 30 卷 第 3 期274上达到了比较理想的识别精度, 且与3D 方法的计算量相比较, 较好地实现了计算量和识别精度的平衡.参考文献Wang H, Klaser A, Schmid C, et al . 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第23卷第4期2023年8月交　通　工　程Vol.23No.4Aug.2023DOI:10.13986/ki.jote.2023.04.004基于信息熵与高风险行为的驾驶行为风险评估方法孙宫昊1,常　鑫2,高亚聪3,陈桂华1,毋　超1(1.国汽(北京)智能网联汽车研究院有限公司,北京　100176;2.中国民航大学,天津　300300;3.北京工业大学,北京　100124)摘　要:为客观评估驾驶人的驾驶安全性,提出以信息熵与高风险行为作为风险指标的驾驶行为风险评估方法.基于驾驶模拟实验获得个体驾驶人行为数据,根据个体驾驶行为特征,通过专家评估方式,获取驾驶行为风险评估比对标签;基于信息熵及高风险行为事件提取关键驾驶风险特征;利用随机森林算法进行驾驶行为风险分类.通过与驾驶行为风险评估标签进行比对验证,结果表明,该方法的驾驶行为风险总体辨识精度达到80%,基于信息熵与高风险行为的驾驶行为风险特征指标选择,能客观描述驾驶行为数据的分布差异,精确分析个体驾驶行为风险特性,可为个性化设计车辆安全辅助系统提供依据.关键词:驾驶行为风险;驾驶特征;信息熵;随机森林;分类模型中图分类号:U 463.6文献标志码:A文章编号:2096⁃3432(2023)04⁃022⁃07收稿日期:2022⁃06⁃30.作者简介:孙宫昊(1996 ),男,硕士,研究方向为智能网联汽车驾驶安全㊁技术标准.E⁃mail:sungonghao@china⁃.通讯作者:常鑫(1991 ),男,讲师,博士,研究方向为智能交通技术.E⁃mail:xchang@.Evaluation of Driving Behavior Risk Based on Information Entropy and High⁃Risk Driving BehaviorSUN Gonghao 1,CHANG Xin 2,GAO Yacong 3,CHEN Guihua 1,WU Chao 1(1.China Intelligent and Connected Vehicles (Beijing)Research Institute Co.,Ltd.,Beijing 100176,China;2.Civil Aviation University of China,Tianjin 300300,China;3.Beijing University of Technology,Beijing 100124,China)Abstract :In order to objectively evaluate the driving behavior risks of drivers,this paper proposes a risk assessment method based on information entropy and high⁃risk behaviors.Fine⁃grained driving behavior data of individual drivers are obtained based on driving simulation experiments.According to the characteristics of individual driving behavior,and through expert evaluation,the risk assessment comparison label of driving behavior is obtained.Key driving risk characteristics are extracted based oninformation entropy and high⁃risk behavior events.The random forest algorithm is used to classify driving behavior risk.Through comparison and verification with the driving behavior risk assessment label,the results show that the overall identification accuracy of the driving risk assessment model reaches 80%.Based on information entropy and high⁃risk behaviors,the selection of driving behavior risk characteristic index is proposed.This method can objectively describe the overall distribution of driving behavior data and accurately analyze the individual driving behavior risk characteristics.The evaluation result of this method is more accurate.The analysis results can provide a basis for the customized design of vehicle safety assistance system.Key words :driving behavior risk;feature of driving behavior;information entropy;random forest;disaggregated model　第4期孙宫昊,等:基于信息熵与高风险行为的驾驶行为风险评估方法0　引言世卫组织研究报告显示,全球每年因交通事故造成的受伤人数可达2000~5000万人[1].在我国,随着汽车技术的发展,在人们出行方便的同时,也带来了一系列交通安全问题.根据交管局公开数据显示,2016年交通事故总数相比于2015年增加25065起,造成交通事故的众多因素中,人因是引发事故的重要原因,2016年由驾驶人引发的交通事故数占事故总数的91.23%[2].因此,对驾驶人的行为风险进行预测评估,是提升道路交通安全的有效手段,在车辆安全辅助系统的个性化设计等应用方面具有极大潜力.以往的研究表明,获取驾驶行为特性数据能实现辨识危险驾驶行为㊁预测交通事故几率㊁提出交通事故预防措施等目的[3],并且驾驶行为数据分析法能客观地描述驾驶人的风险特性.目前,基于客观驾驶数据的驾驶行为风险评估研究方法主要分为2种:①以驾驶数据相应阈值识别高风险驾驶行为事件,评估驾驶风险;②以驾驶特征作为评估参数,通过机器学习等算法进行评估.其中,基于阈值识别高风险驾驶行为事件的驾驶行为风险研究, Toledo等[4]通过驾驶行为数据识别了20种驾驶行为事件,包括加速㊁减速㊁换道等事件,结合驾驶事件的风险程度与频率,构建综合指标评估体系,将驾驶风格分为3类.吴振昕等[5]从数据库中提取7种典型驾驶工况,用k⁃means和D⁃S证据理论的方法进行聚类,将驾驶风格分为3类.Eren等[6]用手机传感器获取速度㊁加速度等数据,通过动态时间规整算法识别高风险行为事件,实现对驾驶人驾驶行为安全性的评估.在基于驾驶特征参数建模的驾驶行为风险的研究中,朱冰等[7]用跟车过程中的驾驶数据作为特征指标,以层次聚类方法获得驾驶行为标签,建立了基于随机森林的驾驶人驾驶习性辨识模型.李经纬等[8]采集商用车和乘用车的驾驶行为数据,通过主成分分析法实现特征指标降维,利用k⁃means法进行驾驶风格的识别.Van LY等[9]获取速度㊁加速度㊁制动踏板压力等数据,利用支持向量机和K均值聚类算法对驾驶人进行分类.综上所述,以往研究中,基于识别高风险事件的驾驶行为风险评估研究,对风险事件的阈值判别标准不一致;基于特征参数建模的驾驶行为风险研究,集中于通过统计分析及建模实现驾驶风险的预测.为此,本研究引入信息熵指标客观描述特征参数,同时识别驾驶人的高风险驾驶行为事件,分析驾驶人个体风险行为特性,基于随机森林算法对驾驶行为风险等级进行识别,建立驾驶行为风险评估模型.提出了1种基于信息熵与高风险行为的驾驶行为风险评估方法.1　驾驶模拟实验1.1　实验设备及场景研究所需的驾驶行为数据借助驾驶模拟实验平台获取,实验选取双向四车道的高速公路路段为实验模拟路段,每条车道路宽为3.75m,实验路段长度为5.6km,限速120km/h,本实验主要针对非违法状态下的高速工况展开研究.实验过程中驾驶人仅需根据其日常驾驶习惯完成模拟路段的驾驶即可.借助驾驶模拟实验平台可收集驾驶模拟器产生实验车自身及周边车辆的位置㊁距离等数据信息[10].驾驶模拟实验平台如图1所示,获取的驾驶行为数据包括方向盘转角㊁刹车踏板深度㊁油门踏板深度㊁时间㊁车辆坐标㊁速度㊁加速度㊁侧位移㊁与前车在250m内的距离㊁前车速度等驾驶操作参数.图1　驾驶模拟器1.2　实验人员本次实验共招募35名驾驶经验丰富的驾驶人, 35名驾驶人的个体属性分布如表1所示.每位驾驶人均拥有C级机动车驾驶证,同时,为了避免其他身体因素影响驾驶实验,在实验前,要求驾驶人保证充足睡眠并且避免大量进食,确保身体状况良好.表1　驾驶人个体属性统计分布性别年龄/周岁驾龄/a 男女>45<45>10<10 201592618172　驾驶行为风险比对标签基于驾驶人实验数据,由来自相关企业㊁高效㊁32交　通　工　程2023年科研机构的17名交通领域经验丰富的专家,对驾驶人的驾驶行为风险等级进行评价,得到的评估结果作为所提出算法的输入值,并与算法的预测结果进行比对.驾驶模式是由一段时间内一系列的驾驶操作所构成的,具有时间序列的连续性,通过驾驶模式的变化特征,可直观反映驾驶人的决策偏好.因此,以每位驾驶人的速度㊁加速度㊁车头时距及在实验路段上的行驶状态变化等数据为辨识特征参数,根据以往研究总结得到的高速工况驾驶模式分解模型,驾驶人的驾驶模式变化可有效反映驾驶人的决策偏好[11].将驾驶操作数据分解为9种模式:自由直行㊁迫近㊁远距离跟驰㊁中距离跟驰㊁近距离跟驰㊁渐远㊁受限换道㊁自由换道㊁紧急制动,绘制驾驶人在行程中的驾驶行为模式图,可视化描述驾驶人在路段上的驾驶行为,根据驾驶行为模式图可直观表现驾驶人的风险等级,安全型驾驶人较多保持单一直行模式,危险型驾驶人则有较大概率采用换道模式行驶.驾驶风险越低,驾驶模式的转移形式越单一,驾驶风险越高,不同模式间的转移越频繁,驾驶人更倾向于通过多种驾驶模式之间的组合行驶,以达到缩短行程时间的目的[12].专家根据每位驾驶人的驾驶行程图表,采用投票法,综合所有专家的评分,得到票数最多的等级作为驾驶人的驾驶行为风险等级标签对驾驶人的驾驶行为风险,评价等级分安全㊁一般㊁危险3个等级.驾驶行为模式图如图2所示,图中横轴为行程距离,纵轴为驾驶模式,图像描述了该驾驶人共进行了2次紧急制动,2次自由换道以及1次迫近,驾驶人在每次紧急制动后,会进行1次自由换道.可明显看出,在无前车干扰的情况下,该驾驶人也会通过频繁换道以达到在期望车道行驶的目的,驾驶风险程度偏高.图2　驾驶人的驾驶转移模式部分驾驶人的专家评分统计如表2所示.表2　专家反馈结果驾驶人ID 专家答题百分比/%描述项安全一般危险均值中位数众数标准差129.435.335.32.062.0020.83258.823.517.61.591.0010.80358.823.517.61.591.0010.80458.823.517.61.591.0010.80547.123.529.41.822.0010.88652.929.417.61.651.0010.79723.547.129.42.062.0020.75 采用克朗巴赫系数检验评价结果的一致性,系数达到0.6以上认为内在信度的一致性可接受[13].结果显示,17名专家评价结果的克朗巴赫系数为0.972,大于0.6,评价结果显示出良好的一致性.根据专家评估的结果,获得驾驶人的驾驶行为风险等级比对标准,得到安全型驾驶人24位,一般型驾驶人6位,危险型驾驶人5位.3　驾驶行为风险评估模型3.1　基于信息熵与高风险行为的特征选取本研究共选择8个参数对驾驶人的驾驶行为进行辨识,包括速度㊁纵向加速度㊁纵向减速度㊁横向加速度㊁横向减速度㊁车头时距㊁车头间距7个车辆运行参数的信息熵,以及辨识高风险驾驶事件的累积紧急制动次数.在正常驾驶过程中,频繁地发生急刹车㊁急加速等高风险驾驶行为,能表征驾驶人驾驶行为风险较高.Wen 等[14]研究发现,如果驾驶人每小时多做1次急刹车行为,交通事故发生的增长率将增长12.5%.Wang 等[15]通过最大减速度㊁平均减速度以及车辆动能减少比率作为参数,采用k⁃means 聚类的方法实现对驾驶风险程度的分类.Guo 等[16]提出logistic 预测模型预测驾驶风格,发现频繁的加速度/减速度行为可有效预测高风险驾驶人.因此,为实现具体高风险行为的辨析,引入累积紧急制动次数指标对驾驶人的驾驶行为风险进行辨识.其中,紧急制动行为定义为减速度小于3m /s 2的状态[3].信息论是应用概率论㊁随机过程㊁数理统计以及近世代数㊁矩阵理论等方法研究信息本质和传输规律的科学理论,熵最初是热力学中的1个常用概念,42　第4期孙宫昊,等:基于信息熵与高风险行为的驾驶行为风险评估方法Shannon [17]将热力学中熵的概念引入到信息论,即Shannon 信息熵.信息熵是信息论中用于度量信息量的1个概念,变量的不确定性越大,熵也就越大,1个系统越是有序,信息熵就越低,反之,1个系统越是混乱,信息熵就越高[18].Shannon [19]提出了信息熵的计算公式:H (X )=-C ∑x ∈Xp (x )log p (x )(1)式中,x 为离散型随机变量X 的可能取值;c 为常数,一般将其归化为1.驾驶员反应主要有3种操作:加速㊁减速和转向[20],体现在车辆运行参数上可通过速度㊁横向加速度㊁纵向加速度及车头间距等特征参数表示.驾驶人纵向加速度数据分布情况如图3所示,驾驶人的横向加速度集中分布在(-2~2)m /s 2,对于急加速与急减速的极端驾驶行为出现概率较低.信息熵可客观地描述驾驶人的极端驾驶行为分布情况.当驾驶人较少出现极端驾驶行为时,加速度数据分布集中,较少出现极端驾驶行为,熵值较低,当驾驶人出现极端驾驶行为增多时,数据波动频繁,熵值较高.因此,本研究选择纵向加速度㊁纵向减速度㊁横向加速度㊁横向减速度㊁车头时距㊁车头间距的信息熵来描述车辆在速度㊁方向和距离上的变化,并根据信息熵的累加性特征,将纵向加速度㊁纵向减速度㊁横向加速度㊁横向减速度的熵值合并为总加速度熵,将车头时距㊁车头间距的熵值合并为总车头距离熵.图3　驾驶人加速度分布图由于速度㊁加速度㊁车头间距㊁车头时距数据为连续变量.在计算信息熵时,需要先将数据离散化,对数据进行区间划分,4种加速度分别划分为10个区间,区间为:当a long ,a lat >0时:int =[0~0.5,0.5~1,1~1.5,1.5~2,2~2.5,2.5~3,3~3.5,3.5~4,4~4.5,4.5~∞]当a long ,a lat <0时:int =[-∞-4.5,-4.5~-4,-4~-3.5,-3.5~-3,-3~-2.5,-2.5~-2,-2~-1.5,-1.5~-1,-1~-0.5,-0.5~0]车头间距划分为17个区间,区间为:int =[0~15,15~30,30~45,45~60,60~75,75~90,90~105,105~120,120~135,135~150,150~165,165~180,180~195,195~210,210~225,225~240,240~250]车头时距划分为10个区间,区间为:int =[0~0.5,0.5~1,1~1.5,1.5~2,2~2.5,2.5~3,3~3.5,3.5~4,4~4.5,4.5~∞]统计各驾驶人在各参数区间内的分布比例p i ,分别计算6个参数的信息熵H k :H k =∑Ni =1-p i ×log 10(p i ),i =1,2,3, ,N (2)式中,H k 为6个参数的熵值,k =1,2,3,4,5,6,分别为纵向加速度㊁纵向减速度㊁横向加速度㊁横向减速度㊁车头间距㊁车头时距6个参数的熵值;N 为参数区间划分总数.3.2　基于随机森林的驾驶行为风险评估随机森林是1种集成学习算法,由多个决策树组成.Nadezda 等[21]使用K 均值聚类㊁神经网络㊁决策树㊁随机森林等算法对驾驶人的驾驶风格进行分类辨识,结果显示随机森林算法拥有分类性能好,分类速度快,在实际应用过程中能得到较好的应用的特点.本研究样本量较少,留一法交叉验证具有适用于小样本情形,可充分利用数据的优点[22].因此本研究的随机森林分类器使用留一法交叉验证,将35个样本的数据集分为35组,34组的数据作为训练集,剩余1组作为测试集,共进行35次循环实验,方法流程如图4所示.使用35位驾驶人的数据进行随机森林的留一法交叉验证,模型辨识效果的ROC 曲线如图5所示.ROC 曲线是用来反映敏感性和特异性连续变量的相互关系,曲线下面积(Area Under roc Curve ,AUC)越大,代表着诊断准确性越高[23].AUC 大于等于0.75,可认为该判别指标或检测方法具有较高的准确性[24].模型的AUC 面积约为0.95,说明提出的模型具有较高的检测真实性.各风险等级驾驶人的车头距离熵㊁总加速度熵分布如图6所示,对比不同风险等级驾驶人的车头距离熵值发现,安全型驾驶人熵值最低且数据分布52交　通　工　程2023年图4　方法流程图5　模型辨识效果的ROC曲线图集中,说明安全型驾驶人会与前车保持较为平稳的距离行驶,与前车长期维持安全距离行驶,一般型驾驶人的熵值明显较高,且数据分布集中,说明一般型驾驶人车头距离变化较频繁,危险型驾驶人则熵值数据更为分散且数值较高,说明其数据不确定性高,距离变化频繁,部分危险型驾驶人为在期望车道行驶,会通过接近前车以及频繁换道等行为以获得期望的行驶速度.对比总加速度熵值发现,随着风险等级的上升,总加速度熵值呈上升趋势,说明危险型驾驶人会频繁出现急加速或急减速行为,极端驾驶行为出现次数较多,与安全型及一般型驾驶人危险程度差异明显.各风险等级驾驶人的平均紧急制动次数如图7所示,随着风险等级的上升,累积紧急制动持续时图6　信息熵指标箱线图间呈上升趋势,一般型驾驶人与危险型驾驶人的平均紧急制动次数较为接近,相比于安全型驾驶人制动次数有着明显的增多,说明一般型与危险型驾驶人均有着频繁的紧急制动行为,风险程度较高.图7　平均紧急制动次数分布图表3为随机森林分类器的分类结果与专家评估标签的对比,安全型驾驶人有1人被辨识为一般型,其余均被辨识为安全型,在一般型驾驶人中,有2人被辨识为安全型,其余辨识为一般型,危险型驾驶人中,有3人被辨识为安全型,1人辨识为一般型,1人辨识为危险型,模型的总体辨识精度为80%.由于危险型驾驶人与一般型驾驶人的平均紧急制动次数相似,同时,本研究所用数据有限.因此,相比于其他两种风险类型的识别精度,模型在危险型驾驶人的分类上出现轻微偏差.利用信息熵与高风险驾驶行为描述驾驶行为特征,并基于随机森林算法建立驾驶风险识别模型,模型在危险型驾驶人的分类上出现轻微偏差,但总体辨识精度较高,识别精度达到80%,可满足驾驶风险的识别需要.表3摇随机森林分类器的辨识结果专家评估安全一般危险安全2310一般240危险31162　第4期孙宫昊,等:基于信息熵与高风险行为的驾驶行为风险评估方法3.3　驾驶行为风险评估模型验证本文所提出的基于信息熵与高风险行为的驾驶行为风险评估方法,具有客观描述驾驶行为数据的总体分布,针对性描述驾驶人的个体高风险行为特征的特性,同时,利用有监督机器学习中的随机森林算法建立分类模型,对于异常数据值具有较好的稳健性,并且泛化能力较强.在以往很多驾驶行为风险评估研究中,也会利用非监督机器学习中聚类分析的方法建立分类模型实现对驾驶风险的评估,虽然聚类分析的方法存在对数据中的异常值较为敏感等问题,但其具有实现方式简便㊁效率高㊁能发现数据中的内在结构等特性,因此,在驾驶风险评估方法上,仍能在一定程度上反映驾驶人的行驶风险情况.为验证本文所提出方法的合理性,引用以往研究中1种基于非监督机器学习的k⁃means 聚类方法[25],建立分类模型,对比2种方法的评估结果.该方法中,选取车速超过限速80%的时间比例㊁车速平均值㊁车速标准差㊁总加速度标准差㊁加速度平均值㊁加速度标准差㊁减速度平均值㊁减速度标准差,共8个参数作为驾驶行为风险分类的特征指标;利用因子分析的方法实现特征指标的降维,因子分数的计算公式如式(3)所示,得分越高,驾驶人的驾驶行为风险越高;最后,选定因子得分为指标展开系统聚类,得到驾驶风险聚类结果,见式(3).^F kj =b j 1^x k 1+b j 2^x k 2+ +b j 8^x k 8(3)式中,^Fkj为第k 个驾驶人的第j 个因子分数的估计值,b j 1,b j 2, ,b j 8为因子分数系数,^x k 1,^x k 2, ,^x k 8为特征变量标准化后的数值,见式(4).^xik =x ik -x iσxi(4)式中,x i 和σxi 分别为驾驶人第i 个特征指标的均值和标准差.各特征指标变量在规范化后均满足均值为0,标准差为1.用该方法对本文实验数据进行风险分析,因子分析的特征值及方差贡献率如表4所示,前3个因子的累积方差贡献率为77.208%,可解释77.208%的信息量,因子分析效果较好,因此,将8个特征参表4　特征值及方差贡献率%因子因子分析总特征值方差贡献率累计贡献率13.02337.78337.78321.65720.71858.50131.49718.70777.208数降维为3个因子.以3个因子的分数作为变量,利用系统聚类的方法将驾驶行为风险聚为3类,系统聚类谱系图如图8所示.图8　系统聚类谱系图表5为基于以往文献中的方法与本文方法评估结果的识别准确率对比.表5　评估结果对比%本文方法基于以往文献的方法识别结果相同样本数占比识别准确率806071.43 从表5可见,基于以往文献中的方法识别准确率为60%,与本文所提方法的评估结果相比,共有71.43%的样本识别结果相同,表明本文提出的方法与以往研究中使用的方法具有一定的一致性,同时,基于信息熵与高风险行为的驾驶行为风险评估方法,能客观描述驾驶行为数据的分布差异,精确化描述驾驶人的驾驶行为风险特征,评估结果更具有准确性,能用于驾驶行为风险评估.72交　通　工　程2023年4摇结束语本文通过引入信息熵描述驾驶行为特征并结合高风险行为事件的辨识,基于随机森林分类器建立驾驶行为风险评估模型,得到以下结论.1)信息熵能有效描述操作数据的分布情况,相比于安全型驾驶人,高风险驾驶人会更为频繁出现极端驾驶行为,信息熵值较高.2)不同等级驾驶人在发生紧急制动行为的频率上差异性明显,一般型驾驶人与危险型驾驶人会更为频繁地发生紧急制动行为.3)信息熵可客观度量数据信息的不确定性,结合信息熵与高风险行为描述驾驶行为特征,能客观描述速度㊁加速度等参数的分布差异,精确化描述驾驶人的驾驶行为风险特征,模型总体辨识精度为80%,辨识精度较高,本研究可根据驾驶人的不同驾驶风险情况,用于个性化调整车辆安全辅助系统的参数,以更好适应驾驶人的行车情况,并对提升行车安全,降低车辆能耗具有重要意义.随着车联网技术的发展,基于智能网联环境下驾驶人的驾驶行为数据,分析多场景下驾驶人的驾驶风险情况,解析风险驾驶原因,提高智能网联汽车的驾驶安全,将是未来的重要研究方向.参考文献:[1]World Health Organization.Global status report on road safety2018[R].Geneva:World Health Organization, 2018.[2]公安部交通管理局.中华人民共和国道路交通事故统计年报(2016年度)[R].北京:公安部交通管理局, 2017.[3]杨海飞.驾驶行为特性数据的实验观测方法研究[J].山西建筑,2018,44(31):126⁃128.[4]Toledo T,Musicant O,Lotan T.In⁃vehicle data recorders for monitoring and feedback on drivers’behavior[J]. 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人工势场法 python人工势场法是一种智能控制方法，也是机器人领域中常用的路径规划方法之一。

本文将介绍如何使用Python编写人工势场法算法。

首先，我们需要了解什么是人工势场法。

人工势场法是一种基于能量势场的机器人路径规划方法，它通过构建障碍物的势场，使机器人在移动时受到斥力，并在目标点处设置一个吸引力场，使机器人向该点移动。

该方法具有简单、实用的特点，适用于机器人的路径规划。

在实现算法之前，我们需要定义机器人的模型。

假设机器人是一个点，那么我们可以定义机器人和障碍物的距离作为机器人受到的斥力，机器人和目标点的距离作为机器人受到的吸引力。

接下来，我们可以用Python实现人工势场法算法。

首先，我们需要导入必要的库：```pythonimport numpy as npimport matplotlib.pyplot as plt```然后，我们可以定义机器人和障碍物的初始位置和速度：```pythonrobot_pos = np.array([0, 0]) #机器人初始位置robot_vel = np.array([0, 0]) #机器人初始速度obstacle_pos = np.array([3, 3]) #障碍物位置obstacle_radius = 1 #障碍物半径goal_pos = np.array([5, 5]) #目标点位置```接着，我们可以定义斥力和吸引力的函数：```pythondef repulsive_force(robot_pos, obstacle_pos,obstacle_radius):distance = np.linalg.norm(robot_pos - obstacle_pos) #机器人和障碍物的距离if distance < obstacle_radius:return (robot_pos - obstacle_pos) / distanceelse:return np.array([0, 0])def attractive_force(robot_pos, goal_pos):return goal_pos - robot_pos```在计算总力之前，我们需要定义一些参数，如斥力和吸引力的权重、最大速度和最大加速度等：```pythonk_repulsive = 5 #斥力权重k_attractive = 1 #吸引力权重max_speed = 1 #最大速度max_acceleration = 0.5 #最大加速度```接下来，我们可以计算机器人受到的总力，并根据总力计算机器人的速度和位置：```pythonrepulsive_force_sum = np.zeros(2) #斥力总和for obstacle in obstacles:repulsive_force_sum += repulsive_force(robot_pos, obstacle_pos, obstacle_radius)attractive_force_sum = attractive_force(robot_pos,goal_pos) #吸引力总和total_force = k_repulsive * repulsive_force_sum +k_attractive * attractive_force_sum #总力acceleration = total_force / np.linalg.norm(total_force) * max_acceleration #加速度robot_vel += acceleration #更新速度robot_vel = np.clip(robot_vel, -max_speed, max_speed) #限制速度robot_pos += robot_vel #更新位置```最后，我们可以将机器人、障碍物和目标点绘制在二维平面上： ```pythonplt.scatter(robot_pos[0], robot_pos[1], color='blue', marker='o') #绘制机器人plt.scatter(obstacle_pos[0], obstacle_pos[1],color='red', marker='o') #绘制障碍物plt.scatter(goal_pos[0], goal_pos[1], color='green', marker='o') #绘制目标点plt.xlim((-1, 7)) #设置x轴范围plt.ylim((-1, 7)) #设置y轴范围plt.show() #显示图像```以上就是使用Python实现人工势场法算法的全部内容。

在人工智能领域中，马尔科夫决策过程（MDP）一直是一个研究热点。

MDP是用于描述一个决策问题的数学框架，其核心思想是在不确定的环境中做出最优的决策。

然而，在实际应用中，MDP模型也存在一些常见的问题，本文将针对这些问题进行讨论，并提出解决方法。

问题一：稀疏奖励在实际问题中，很多状态下可能都没有立即的奖励反馈，这就导致了MDP模型的稀疏奖励问题。

换句话说，智能体在某些状态下无法获得奖励，这会导致学习变得非常困难。

解决方法：一种解决稀疏奖励的方法是引入一些额外的奖励机制，比如使用一些奖励函数进行辅助，或者采用一些技巧来平滑奖励信号，使得智能体能够更好地进行学习。

问题二：状态空间过大在一些复杂的环境中，状态空间可能会非常庞大，这就会导致MDP模型的计算复杂度急剧增加，影响模型的有效性和可行性。

解决方法：为了应对状态空间过大的问题，可以采用一些近似方法，比如使用函数逼近技术来减少状态空间的维度，或者采用一些采样方法来对状态空间进行抽样，从而降低计算复杂度。

问题三：非马尔科夫性在实际问题中，很多环境并不满足马尔科夫性，即未来的状态和决策可能受当前状态之外的因素影响，这就会导致MDP模型的失效。

解决方法：针对非马尔科夫性问题，可以采用一些延迟奖励的技巧来考虑未来状态的影响，或者使用一些更复杂的模型来描述环境的动态变化，从而更好地适应非马尔科夫性的情况。

问题四：策略学习困难在MDP模型中，有效的策略学习是非常困难的，尤其是在状态空间和动作空间非常大的情况下，这就需要找到一种高效的策略学习方法。

解决方法：为了解决策略学习困难的问题，可以采用一些强化学习算法，比如Q学习、深度强化学习等，这些算法可以帮助智能体更快地找到最优的策略，提高模型的学习效率和效果。

问题五：探索与利用的平衡在MDP模型中，探索与利用的平衡是一个非常重要的问题，因为智能体需要不断地探索新的状态和动作，同时又需要利用已有的知识来做出最优的决策。

Modelling Temporal Thematic Map ContentsAlberto d’OnofrioEuropean Institute of Oncology,Division of Epidemiology and Biostatistics Via Ripamonti,435I-20141Milano,Italy,alberto.d’onofrio@ieo.itElaheh PourabbasIstituto di Analisi dei Sistemi ed Informatica”Antonio Ruberti”-CNR Viale Manzoni,30I-00185Roma,Italy,pourabbas@r.it1IntroductionIn the last decade the database research community gave valuable results in modelling and retrieving spa-tial objects,in a temporal framework,e.g.,[8][9].It is recognized that the representation and the manage-ment of GIS play a central part in data manipulation and querying.In fact,the incorporation of time in them may lead to consider new types of information for modelling those highly time dependent spatial data(e.g.,temperatures,land use coverage,epidemic information related to a given area,etc.)and moving objects(e.g.,entities that change the relative spatial position).The trend of temporal data modelling in GIS is moving from time-stamping layer(the snap-shot models,[2]),attributes(space-time composites, [5]),and spatial objects to time-stamping events or processes that are mainly based on the concepts of time sequences[8].Generally,in GIS the properties and the relations of geographic features are captured visually by maps.These maps are processed at one time and for one purpose.The class of maps that are developed to aid in decisions are called thematic maps.A thematic map is a map depicting selected kinds of information relating to one,or more specific themes,such as soil type,the incidence of a disease or land classification.Therefore,the manipulation of thematic maps for spatial query languages is an important task.In this paper,we will focus on tem-poral evolution of thematic maps in order to answer queries that refer to the time varying elements char-acterizing a thematic map,i.e.,the geometric com-ponent of the map and the associated alphanumeric information.In this context,wefirst define through a functional approach(functional in the sense that it uses the function as mathematical structure)the notions of basic maps,thematic maps,and a set of operators for their manipulation.Then,we define a space-time composite functional approach by associ-ating time to the above mentioned notions in order to model the temporal evolution of maps information, and we propose some new temporal operators.In our model,time can be conceptualized as instances (point-based)or intervals[1][4].To motivate our proposal,let us consider the query as follows:”Show distribution of the number of viral hepatitis-B cases in children below10years old in the Italian regions in the last century”.For answering this query,the map model besides the spatial”where”,and thematic ”what”,must support”when”.Although many stud-ies have been made in the context of spatio-temporal databases,current databases and systems related to them lack the ability to handle the temporal evo-lution of information in the thematic maps.In[7], for instance,efforts have been made to provide the designer of geographic databases with a query lan-guage extensible by a set of operations for static the-matic map manipulation.In this work,the thematic map model is based on an object model and empha-sizes the separation between map level and geometric level.The main idea is the association of the geomet-ric operations with the data model through generalconstructs.However,in the proposed model the set constructor is used intensively inside ADT manipu-lation,which leads to a model that is difficult to un-derstand.In [3],several operators on map which are based on a formal model of spatial partitions are in-troduced.Although,this work provides a theoretical framework for studying static maps,but the legend is described simply by mapping to a spatial partition a ”label”type.Our approach overcomes the limita-tions of the above mentioned proposals by integrat-ing both map,and geometric model into a unique functional model,and in addition defines by suitable functions the temporal variations of map contents.Furthermore,our approach is compatible with DBV version mechanism discussed in [6],which keeps track of different versions of the same geographic entity through time.The paper is structured as follows:Section 2models general thematic maps called static thematic maps by a functional approach.Section 3proposes a set of operators for manipulating the the-matic maps depending on the number of operands in input,and user applications.Section 4describes the definitions of temporal thematic map.Section 5ex-tends the operators defined in section 3with time.Section 6illustrates the proposed operators by some query examples,and finally,Section 7concludes.2Functional ModellingIn this work,the geometry of the geographic features that we will refer to for describing all the follow-ing definitions and operators is region that is de-noted by r .It is a subset of R 2and it is suchthat ∀ >0,∀x ∈r,∀p ∈r =(R 2−r )it holds:µ(Ball (x,ε) r )>0∧µ(Ball (p,ε) r )>0,where µis the Riemann’s measure.Before introducing the concept of thematic map,we will define some basic notions including first of all the notion of basic maps.A basic map is a set of topographic data displayed in map form providing a frame of reference (i.e.the location data or position data)or contextual infor-mation to the user.Therefore,we give the following definitions for modelling the basic maps.Definition 1A basic map Mis a set of regions M = r 1,...,r n with µ r i r j =0for i =j ,andi,j =1,...,n .From the geometrical point of view,the above def-inition means that two different regions on a given basic map can be disjoint or they can intersect each other in a set of points or polyline.Definition 2Let f :r 1∪...∪r n →R be a func-tion.Let I i = min i ,max i be a range of value where min i and max i represent,respectively,the relative minimum and maximum value.The function f maps each point x of a generic region r i of a basic map to a value belonging to I i .Formally,we have that ∀x ∈r i ,f (x )∈I i .This function satisfies the following con-dition:∀i,j (I i ∩I j =∅)∨(I i =I j )Note that the above mentioned condition specifies that the function f is defined on a set of N p disjoint intervals,where N p ≤N .For sake of simplicity,as-sume that all intervals be consecutive.Then we have: N p q =1min q ;max q=[a,b )where a =min 1,b =max N p .Definition 3Let M and I ={I 1,...,I N p }be,respectively,a basic map and a set of ranges.We define the function ϕ:M →I that map each region to a range.Note that generally,the function ϕis not invertible,this means that to a unique range more regions can be associated.Definition 4Let ψ:I →C be a invertible func-tion that map the ranges to the elements belonging to a set C = c 1,...,c k .This set can represent:a)The set of RGB colours;b)A finite set of pat-tern (bitmap)that are used to fill different areas;c)a finite set of strings.Then,we define the composed function Γ=ψ◦ϕthat maps each region of basic map to an element belonging to C .We will call it correspondence function.Definition 5The legend is defined as the following pair:λ= name,ψ−1 ,where name is a string which indicates the name of the legend.For instance,the name could be ”incidence of cholera”.By using the above definitions,we can for-malize thematic map as follows:Definition 6A thematic map (called θ)is a quadruple:θ= Name,λ,M,Γ ,where Name is a string that represents the name of thematic map.Note that between the name of thematic map andthe name of the associated legend often a relationship exists;for instance,Name =”incidence of cholera in Africa”and λ.name =”incidence of cholera”.The thematic maps will be also denoted by a shortened form as follows: Name,λ, {r 1,c 1},...,{r k ,c k } .For sake of notational simplicity,we will omit the name of the thematic maps where it will not be es-sential in the definitions which follow.We will point out that the concept of thematic map can be general-ized by allowing that the set I is defined as a generic finite set,e.g.,as a set of strings.For instance,a ge-ographic administrative map is also a thematic map assuming that I is a numerable set strings (the re-gions names)and C =I .All the definitions and the set of operators which we will give in the following are valid for the generalization of thematic maps,ex-cept for some operators which will use the properties of the real numbers.Note that,if in the definition of thematic map we assume that M is a set of polylines and µrepresents the length function,then we can ap-ply formally the above definitions and the following operators.In this way,we obtain another kind of the-matic map:the linear thematic map.This type of map could be useful for the description,for example,of the pollution of a river subdivided in a set of lines,for each of which the percentage of the pollution is given.3Set of OperatorsIn the following subsections,we propose a set of oper-ators that can have one,two or more thematic maps as operands in input and output.Furthermore,a set of operators for user-oriented applications is pro-posed.They will refer to the generic set of thematic maps that is represented by the symbol Θ.3.1One operandIn It is related to testing if a given region r is present in a thematic map θ∈Θ.The boolean operator In gives in output the result true if the region r is present in the basic map of θ= Name,λ, {r 1,c 1},...,{r k ,c k } ,false otherwise.Formally:In (r,θ)=∨k i =1(r =r i ).AtLeast Let θ∈Θbe a thematic map and let m be a real number.The operation AtLeast (m,θ)gives as result the set of regions in the thematic map for which the associated thematic value is greater than or equal to m .Formally we have:AtLeast (m,θ)= r i (ψ−1→[p,q ))∧(p ≥m )).AtMax This operator is the dual of AtLeast operator.It gives the set of regions for which the associated thematic value is less than or equal to a real value m :AtMax (m,θ)= r i (ψ−1→[p,q ))∧(q ≤m )).Between Let θ∈Θbe a thematic map and let [a,b ]be a range.This operator gives the set of regions in θfor which the associated thematic value belongs to range [a,b ],then formally:Between ([a,b ],θ)=AtLeast (a,θ)∩AtMax (b,θ).Selection Let θ= Name,λ, {r 1,c 1},...,{r k ,c k }be a thematic map and let x =[a,b )be an interval on real axis.This operation gives the regions of map whose thematic values coincide with interval x :Selection (x,θ)= r ix =ψ−1(c i ) .Fusion This operator allows to obtain a new thematic map realizing the geometric union of the regions of a map which have the same the-matic value.It is defined as follows:Fusion(θ)= Name,λ, ,c | =∪Selection (ψ−1(c ),θ)∧( =∅)∧(c ∈C ).The resulting map is defined by a set of pairs ,c ,where is the geometric union of the regions which have a common thematic value,and c belongs to the set defined in Definition 4.3.2More operandsIntersection between a thematic map and a region Let θ= λ, {r 1,c 1},...,{r k ,c k }be a thematic map and let R be a region.The intersection yields a map defined by the same legend of θand a set of regions that is the result of the intersection of the regionsbelonging to θwith R .Formally:Intersection (θ,R )= λ, r 1∩R,c 1,...,r k ∩R,c k .Union of thematic maps Let θ1= λ1,Γ1,M 1 and θ2= λ2,Γ2,M 2 be thematic maps,thus their basic maps are:M 1and M 2.Two conditions are needed for applying the union operator on them.The first condition is that the thematic maps must share the same legend.The second condition is that the basic maps must be disjoint or,otherwise,the regions belonging to M 1∩M 2have the same thematic values.Summarizing,in order to apply this operator the following condition must hold:(λ1=λ2)∧ M 1∩M 2=∅ ∨∀u ∈M 1∩M 2,Γ1(u )=Γ2(u ).If this condition is satisfied,then by defining the following function:Γ(x )=Γ1(x )if x ∈M 1,Γ2(x )if x ∈M 2,which we will call correspondence function ,the union of θ1,θ2is defined as follows:Union (θ1,θ2)= λ,Γ,M 1∪M 2 .Intersection between a thematic map and a geographic map Let M ={R }be a geographic map defined by a unique region.Let θ= λ, {r 1,c 1},...,{r k ,c k }be a thematic map.The intersection be-tween these maps is:Intersection (θ,{R })= λ, {r 1∩R,c 1},...,{r k ∩R,c k } .When M ={r 1,...,r n },the intersection oper-ation is given by:Intersection (θ,M )=Union Intersection (θ,{r 1}),...,Intersection (θ,{r n }) .In the Figure 1,an example of theapplication of this operation is shown.3.3User-Oriented ApplicationsIn this section,we define a new operator,which we call P icking ,for the interaction with a Thematic Map and we formalize with our functional approach two other operators,windowing and clipping,which were defined previously in [7].Picking Let θ= λ,M,Γ be a thematic map and let W be a selected rectangular or circulararea Figure 1:Illustration of Intersection between a the-matic map and a geographic map.on the screen selected through the mouse.The appli-cation of this operation produces a new thematic map containing all regions of θwhich are totally included in W .The semantic of this operation is defined as follows:P ick (θ,W )= Name,λ,,c |( ∈M )∧( ∩W = )∧(c =Γ( )),where,in the pairs ,c , is a region of the thematic map,whichbelongs totally to W ,and c ∈C .Windowing Let θ= λ,M,Γ be a thematic map and let W be a selected rectangular or circulararea on the screen selected through the mouse.The application of this operation produces a new thematic map containing all regions of θwhich have a not null bi-dimensional intersection withW .The semantic of this operation is defined as follows:W ind (θ,W )= Name,λ,,c |( ∈M )∧( ◦∩W =∅)∧(c =Γ( )) ,where,in the pairs ,c , is a region of the thematic map,whose interior ( ◦)has a not null intersection with the selected window,and c ∈C .Clipping Let θ∈Θbe a thematic map and letW be a selected rectangular or circular area on the screen,that is selected through the mouse.This operation gives as result the regions that are enclosed in the selected area.It is a particular case of the intersection operation between a thematic map anda region:Clip (θ,W )=Intersection (θ,{W }).InFigure 2,an example of application of Pick,Wind and Clip operators is shown.Figure2:Results of application of windowing(A), clipping(B)and picking(C)operators.4Temporal Thematic Maps Considering the definitions given in the previous sec-tions,the concept of time becomes meaningfully im-portant in the context of thematic maps.In fact,in a thematic map either the geographic part or the the-matic part need some temporal elaboration.For in-stance,let us consider a thematic map that represents the”distribution of annual cholera cases in Europe”and the temporal interval[1/1/1989,1/1/1999).In this interval of time,we can consider that the tempo-ral evolution of theme of this map is constant but in the context of geographic map and then the underly-ing subdivision of the boundary of countries there has been some well known variations.Now,let us con-sider the thematic map that represents”distribution of the number of viral hepatitis B cases in children below10years old in the Italian regions”and let the temporal interval be[1/1/1985,1/1/1995).In this period,thanks to the HBV vaccination policy,the annual number of hepatitis B cases(theme)has sig-nificantly decreased,but the geographic part(Italian regions)did not undergo variations.If we consider the thematic map of the distribution of this disease in the period[1/1/1985,1/1/1995)in Europe,both thematic and geographic parts have assumed tempo-ral variations.Generally,the temporal dynamics of theme and geo-graphic one are independent,and for this reason their evolutions are considered to be asynchronous.Start-ing from the formalization of thematic maps intro-duced in the section2,and considering the following assumptions:a)the legend and the name of map do not change in time;b)the basic map changes;c)the functionϕand thenΓchange in time and d)the variations are constant in time intervals(this means that in the considered interval there is no continu-ous variations),we give the definition of temporal(or temporized)thematic map.Note that,in the follow-ing we will use the terms temporal and temporized interchangeably.Definition7A temporized thematic map is defined by the following triple: Name,λ,{(τ1, M1,Γ1 ),...,(τω, Mω,Γω )} ,whereτ1,...,τωrepresents a series of time intervals,in each of which we consider that the evolution of the thematic and geographic parts of the given map are constant.In a generic intervalτi,the considered constant thematic map is defined by basic map M i={r i1,...,r iN i}and the functionΓ{r i1,...,r iN i}={c i1,...,c iN i}. When the indication of the legend and the Name is not essential,a temporal thematic map can be simply represented by a set of couples { τ1,θ1 ,..., τω,θω },each of which represents time interval and a constant thematic map.All these thematic maps share the same legend,and the time ranges are disjoint and ordered.For sake of brevity, we will indicate the set of all Temporized Thematic Maps by the symbol: Θ.Note that a static the-matic map Name,λ,M,Γ can be considered as spe-cial case of a temporized thematic map and can be represented as follows: Name,λ,([t o,now], M,Γ ) , where t o is the initial time which is considered in the system and now refers to the present time.5OperatorsFor the Temporized Thematic Maps,we extend the set of operators defined in section3,and we define new others for modelling some basic interactions with time or time intervals.In the following,we will refer to θfor indicating a generic element of Θ.The temporized unary operators are defined in terms of the static unary operators.They are listed in the Table1,and for each of them the formalization and description are given.We define the following operators with two or more operands.Union of temporized thematic maps The application of this operator,as in the static case,will be possibleOp.s Formalization DescriptionIn In , θ=(∃i,In ( ,θi ))It yields true value if at least for oneelement of temporized thematic maps,the result of the ”static”In operator is True.Selection Selection x, θ = ωj =1 τi ,Selection (x,θi )It applies the static version of Selection to each θ1,...,θωand the result is a temporized thematic map.F usion F usion ( θ)= ωj =1 τi ,F usion (θi )See the description of Selection .AtLeast AtLeast (m, θ)= ωj =1 τi ,AtLeast (m,θi )See the description of Selection .AtMax AtMax (m, θ)= ωj =1 τi ,AtMax (m,θi )See the description of Selection .Between Between [m 1,m 2], θ = ωj =1 τi ,Between ([m 1,m 2],θi )See the description of Selection .Table 1:Operators with one temporized thematic map operand.if and only if there is unique legend shared by all theoperands.First of all,we will consider the simplecase of the union of two very simple elements of Θ:θa ={τa ,θa }, θb ={τb ,θb }.If τ:=τa ∩τb =∅,then their union is defined as:Union ( θa , θb )={ τa ,θa , τb ,θb }={ τa ,θa }∩{ τb ,θb }.If τ=∅,this operator will be defined as follows:Union ( θa , θb )={ τa −τ,θa , τ,Union (θa ,θb ) , τb −τ,θb }={ τa ,θa }∩{ τb ,θb }.For generalizing the above definition,letθ={ τ1,θ1 , τP ,θP }and θ∗={ τ∗1,θ∗1 , τ∗Q ,θ∗Q}be elements of Θ.Their union is:Union( θ, θ∗)= i =P,j =Qi =1,j =1Union ({ τi ,θi },{ τ∗j ,θ∗j }).Intersection of a temporized thematic map with a region This operator is an extension of the previ-ous one defined for static thematic Maps.It will be applied to obtain not only the intersection be-tween a temporized thematic map and a region or map,but also a more time-dependent entities which are temporized region and temporized map.The result of this operator will be a temporized the-matic map.In this paper,we will give a first rough (but complete,for the definition of these op-erators)definition of these entities,which are part of an our work in progress on a general model for temporized geographic objects.The intersec-tion of a temporized thematic map with a regionR can be defined as follows:Intersection( θ,R )=ωi =1Union ({ τi ,Intersection (θi ,R ) }).In this definition R represents a generic region without any temporal variation.Since,the spatial configuration of a region can change in time,we define a new entity which we call temporized region .It consists of a set of pairs formed by a time inter-val and a static region: = σ1,ρ1 ,..., σn ,ρn where σ1,...,σn are consecutive disjoint tempo-ral intervals and ρ1,...,ρn represents different spatial configurations of a given region in each interval.Many examples of temporized regions can be mentioned in various fields such as ecology,history (e.g.Italy became an independent country in 1861and along time the relative boundary is changed,respectively,in 1866,1870,1890,1912,1918,1924,1939,1945,1947and 1954),etc.Then,the intersection of a temporized region with a tem-porized thematic map will be :Intersection ( θ, )=(i,j )∈{(1,1),...,(ω,n )}Union ({ τi ∩σj ,Intersection (θi ,ρj ) }).In the same way by defining a tem-porized map as M = σ1,M 1 ,..., σn ,M n it is possible to define the intersection of a temporized thematic map with a tempo-rized map as follows:Intersection ( θ,M )=(i,j )∈{(1,1),...,(ω,n )}Union ({ τi ∩σj ,Intersection(θi ,M j ) }).Example 1Let us consider a temporized the-matic map whose name is ”Diffusion of flu in Europe between the years 1900and 1999”.A query on this map that invoke the above operator is ”Get the map of the flu in Mediterranean countries of Europe in the period [1/1/1991,1/1/1992)”.TSlice Let θan element of Θand let t an in-stant of time.This binary operator gives the static thematic map belonging to Θat the time t :TSlice( θ,t )=θi ,if ∃i |t ∈τi ,∅otherwise .ExtendedT SliceLet θbe an element of Θand let τbe a time interval.This binary operator gives the temporal thematic map which contains all the the-matic maps belonging to Θduring the time interval τ:ExtendedT Slice ( θ,τ)= ωj =1 τi ∩τ,θi ).For instance,given the temporized thematic map ofExample 1,”get the map of the flu in Europe in the period [1/1/1919,1/1/1920)”.Before defining the operators which model the in-teraction of an user with the elements of Θ,it is,of course,mandatory to define how to visualize a tem-porized thematic map.We identified two different vi-sualization modalities:a)visualization of an unique legend and a series of static thematic maps with the relative temporal intervals and b)visualization of the legend and the representation of a sequence of static thematic map through animation that yields the evo-lution of thematic map objects during each tempo-ral interval.Let θ={ τ1,θ1 ,..., τω,θω }be the element of Θon which the user will interact.At first,we shall examine the user interaction when the first way of visualization is chosen.For each of the θ1,...,θωvisible on the screen,the user will have to select a region trough the mouse.So,at the end of this process,a set of regions w 1,...,w ωis selected.Since a temporal interval is linked to each of the θ1,...,θω,we define the following temporized region which is the implicit result of the selections made by the user on the screen:W ={ τ1,w 1 ,..., τω,w ω }.If in some of the component thematic maps the user does not perform a selection,then we assume the values of the corresponding regions be equal to the empty set.If the chosen visualization mode is theanimation,the user can perform only one region se-lection and this selection has to be performed before the starting of the animation.Also in this case we may define an implicitly selected temporized region W ={ τ1,w 1 ,..., τω,w ω },where this constraint holds:w 1=w 2=···=w ω.Therefore,in boththe above visualization modalities,we can define thethree interaction operators P ick ,W ind and Clip :Pick( θ,W )= ωi =1Union ({ τi ,P ick (θi ,w i ) });Wind( θ,W )= ωi =1Union ({ τi ,W ind (θi ,w i ) });Clip( θ,W )= ωi =1Union ({ τi ,Clip (θi ,w i ) }).6Thematic map querying In this section,we illustrate the applicability ofthe proposed set of operators defined for static and temporal thematic maps to data queries by meansof the following query examples.The queries areexpressed by an OQL-like language.Query 1:”Find all thematic maps to which the nation Italy belongs”.This query can be expressed as follows:SELECT tm FROM tm IN ThematicMaps,n IN NATIONS WHERE =”Italy”AND In(n,tm)Note that in this example both IN operator of OQL language and In operator defined for thematic mapsare used with different semantics.Query 2:”Find the EU nations in which the annual incidence of asthma is greater than 500cases per millions”.SELECT FROM n IN NA-TION,tm IN ThematicMaps WHERE =”annual inciden ce of asthma in EU”AND n in AtLeast(500,tm)Query 3:Find the Italian regions in which the percent incidence of asthma is between 5%and 10%of the populations”.SELECT r FROM r IN REGIONS,tm IN ThematicMaps WHEREFigure3:Application of Clip operation:selection of4rectangular windows(left),and results(right).=”Distribution of asthma in Italian regions”AND r in Between([5,10],tm)In this query REGIONS indicates the italian admin-istrative subdivision.The above queries refer to the static thematic maps.In the following,we will give some query examples in which temporal thematic maps(TTMaps)are involved.Query4:”Get the map of theflu in South Europe in 1900-1999”.SELECT Intersection(ttm,seu)FROM ttm IN TTMaps,seu IN TemporizedMaps WHERE =”Diffusion offlu in Europe in 1900-1999”AND =”South Europe”Query5:”Get the map of theflu in EU in the period [1/1/1919,1/1/1920)”.SELECT ExtendedTSlice(ttm,[1/1/1919, 1/1/1920))FROM ttm IN TTMaps WHERE =”Diffusion offlu in EU in1900-1999”7ConclusionsIn this paper,we have presented a 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