Chap11.ppt

RMSE MSE We will focus on the MSE.
Extrapolation Models
Extrapolation models try to account for the past behavior of a time series variable in an effort to predict the future behavior of the variable.
MAD =

i 1
n
Yi Yi n
100 – mean absolute percent error, MAPE = n i 1
n
n
ˆ Yi Yi Yi
– the mean square error,
– root mean square error.
2 Yi Yi MSE = n i 1
Forecasts for time periods 25 and 26 at time period 24:
ˆ ˆ ˆ Y25 Y24 (Y24 Y24 ) 35.74 0.268(36 35.74) 35.81
ˆ ˆ ˆ ˆ ˆ ˆ ˆ Y26 Y25 (Y25 Y25 ) Y25 (Y25 Y25 ) Y25 35.81
Forecasting With The Moving Average Model
Forecasts for time periods 25 and 26 at time period 24:
Y24 Y23 36 + 35 355 Y25 . 2 2
Y25 Y24 35.5 + 36 Y26 35.75 2 2
A time-series is a set of observations on a quantitative variable collected over time. Examples Dow Jones Industrial Averages Historical data on sales, inventory, customer counts, interest rates, costs, etc Businesses are often very interested in forecasting time series variables. Often, independent variables are not available to build a regression model of a time series variable. In time series analysis, we analyze the past behavior of a variable in order to predict its future behavior.
Forecasts for time periods 25 and 26 at time period 24:
ˆ Y25 w1Y24 w2 Y23 0.291 36 0.709 35 35.29 ˆ ˆ Y26 w1Y25 w2 Y24 0.291 35.29 0.709 36 35.79
Weighted Moving Average
The moving average technique assigns equal weight to all previous observations 1 Y 1 Y 1 Y Yt 1 k t k t-1 k t- k -1 The weighted moving average technique allows for different weights to be assigned to previous observations. Yt 1 w1Yt w2 Yt-1 wk Yt-k -1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
T im e P e r io d
Stationary Data With Additive Seasonal Effects
where
ˆ Yt n E t St n p
Et (Yt - St-p ) (1- )Et 1
Note that,
Yt 3581, for t = 25, 26, 27, .
Seasonality
Seasonality is a regular, repeating pattern in time series data. May be additive or multiplicative in nature...
Moving Averages
Yt Yt-1 Yt- k +1 Yt 1 k
No general method exists for determining k. We must try out several k values to see what works best.
Spreadsheet Modeling & Decision Analysis
A Practical Introduction to Management Science
5th edition
Biblioteka Baidu
Cliff T. Ragsdale
Chapter 11
Time Series Forecasting
Introduction to Time Series Analysis
St (Yt - Et ) (1- )St p 0 1 0 1
Implementing the Model
See file Fig11-2.xls
A Comment on Comparing MSE Values
Care should be taken when comparing MSE values of two different forecasting techniques. The lowest MSE may result from a technique that fits older values very well but fits recent values poorly. It is sometimes wise to compute the MSE using only the most recent values.
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Time Period
Implementing the Model
See file Fig11-8.xls
Forecasting With The Exponential Smoothing Model
Measuring Accuracy
We need a way to compare different time series techniques for a given data set. Four common techniques are the:
– mean absolute deviation,
Exponential Smoothing
Yt 1 Yt (Yt Yt )
where 0 1
It can be shown that the above equation is equivalent to:
Yt 1 Yt (1 )Yt 1 (1 ) 2 Yt 2 (1 ) n Yt n
Yt 1 f Yt , Yt 1 , Yt 2 ,
We’ll first talk about several extrapolation techniques that are appropriate for stationary data.
An Example
Electra-City is a retail store that sells audio and video equipment for the home and car. Each month the manager of the store must order merchandise from a distant warehouse. Currently, the manager is trying to estimate how many VCRs the store is likely to sell in the next month. He has collected 24 months of data. See file Fig11-1.xls
Some Time Series Terms
Stationary Data - a time series variable exhibiting no significant upward or downward trend over time. Nonstationary Data - a time series variable exhibiting a significant upward or downward trend over time. Seasonal Data - a time series variable exhibiting a repeating patterns at regular intervals over time.
Examples of Two Exponential Smoothing Functions
42 40 38
Units Sold
36 34 32
Number of VCRs Sold
30 28 1 2 3 4 5 6 7 8
Exp. Smoothing alpha=0.1 Exp. Smoothing alpha=0.9
Approaching Time Series Analysis
There are many, many different time series techniques. It is usually impossible to know which technique will be best for a particular data set. It is customary to try out several different techniques and select the one that seems to work best. To be an effective time series modeler, you need to keep several time series techniques in your “tool box.”
Stationary Seasonal Effects
A d d itiv e S e a s o n a l E ffe c ts
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
T im e P e r io d
M u ltip lic a tiv e S e a s o n a l E ffe c ts
where 0 wi 1 and wi 1
We must determine values for k and the wi
Implementing the Model
See file Fig11-4.xls
Forecasting With The Weighted Moving Average Model