商务与经济统计ppt第18章B
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Least squares, also used in regression analysis, determines the unique trend line forecast which minimizes the mean square error between the trend line forecasts and the actual observed values for the time series.
Y = average values of the time series
t = average value of t
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 11
Holt’s Linear Exponential Smoothing
Charles Holt developed a version of exponential smoothing that can be used to forecast a time series with a linear trend.
Slide 9
Trend Projection
Three-Month Weighted Moving Average The forecast for December will be the weighted
average of the preceding three months: September, October, and November.
Slide 4
Linear Trend Regression
For the trend projection equation Tt = b0 + b1t
n
(t t )(Yt Y)
b1 t1 n
(t t )2
t1
b0 Y b1t
where: Yt = value of the time series in period t n = number of time periods (observations)
134.68 -0.99 89.34 12.67
(July) 5 0 0 396 9.33 (Aug.) 6 1 1 409 22.33 (Sep.) 7 2 4 399 12.33 (Oct.) 8 3 9 412 25.33 (Nov.) 9 4 16 408 21.33
0 22.33 24.66 75.99 85.32
Statistics for Business and Economics
Anderson Sweeney Williams
Slides by
John Loucks
St. Edward’s University
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
F10 = .1YSep. + .3YOct. + .6YNov. = .1(399) + .3(412) + .6(408) = 408.3
Trend Projection F10 = 422.27 (from earlier slide)
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 7
Linear Trend Regression
t 45/9 5 Y 3480/9 386.667
n
b1
(t t )(Yt Y)
t1 n
(t t )2
3480 60
7.12
t1
b0 Y b1t 386.667 7.12(5) 351.07
The independent variable is the time period and the dependent variable is the actual observed value in the time series.
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 2
Trend Pቤተ መጻሕፍቲ ባይዱojection
If a time series exhibits a linear trend, the method of least squares may be used to determine a trend line (projection) for future forecasts.
Slide 10
Trend Projection
Conclusion Due to the positive trend component in the time
series, the trend projection produced a forecast that is more in line with the trend that exists. The weighted moving average, even with heavy (.6) weight placed on the current period, produced a forecast that is lagging behind the changing data.
Slide 8
Trend Projection
Example: Auger’s Plumbing Service
Forecast for December (Month 10) using a
three-period (k = 3) weighted moving average with
weights of .6, .3, and .1 for the newest to oldest data, respectively. Then, compare this Month 10 weighted moving average forecast with the Month 10 trend projection forecast.
Sum 45
60 3480
444.00
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 5
Linear Trend Regression
Example: Auger’s Plumbing Service
The number of plumbing repair jobs performed by
Auger's Plumbing Service in the last nine months is
T10 = 351.07 + (7.12)(10) = 422.27
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 3
Linear Trend Regression
Using the method of least squares, the formula for the trend projection is:
Tt = b0 + b1t
where:
Tt = linear trend forecast in period t b0 = intercept of the linear trend line b1 = slope of the linear trend line t = time period
Month Jobs August 409 September 399 October 412 November 408
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
listed on the right. Forecast the number of repair jobs Auger's will perform in December using the least squares method.
Month Jobs March 353 April 387 May 342 June 374 July 396
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 1
Chapter 18, Part B Forecasting
Trend Projection Seasonality and Trend Time Series Decomposition
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 6
Linear Trend Regression
(month) t t t (t t )2 Yt (Yt Y) (t t )(Yt Y)
(Mar.) 1 -4 16 353 -33.67 (Apr.) 2 -3 9 387 0.33 (May) 3 -2 4 342 -44.67 (June) 4 -1 1 374 -12.67
Month Jobs March 353 April 387 May 342 June 374 July 396
Month Jobs August 409 Septem. 399 October 412 Novem. 408
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Y = average values of the time series
t = average value of t
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 11
Holt’s Linear Exponential Smoothing
Charles Holt developed a version of exponential smoothing that can be used to forecast a time series with a linear trend.
Slide 9
Trend Projection
Three-Month Weighted Moving Average The forecast for December will be the weighted
average of the preceding three months: September, October, and November.
Slide 4
Linear Trend Regression
For the trend projection equation Tt = b0 + b1t
n
(t t )(Yt Y)
b1 t1 n
(t t )2
t1
b0 Y b1t
where: Yt = value of the time series in period t n = number of time periods (observations)
134.68 -0.99 89.34 12.67
(July) 5 0 0 396 9.33 (Aug.) 6 1 1 409 22.33 (Sep.) 7 2 4 399 12.33 (Oct.) 8 3 9 412 25.33 (Nov.) 9 4 16 408 21.33
0 22.33 24.66 75.99 85.32
Statistics for Business and Economics
Anderson Sweeney Williams
Slides by
John Loucks
St. Edward’s University
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
F10 = .1YSep. + .3YOct. + .6YNov. = .1(399) + .3(412) + .6(408) = 408.3
Trend Projection F10 = 422.27 (from earlier slide)
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 7
Linear Trend Regression
t 45/9 5 Y 3480/9 386.667
n
b1
(t t )(Yt Y)
t1 n
(t t )2
3480 60
7.12
t1
b0 Y b1t 386.667 7.12(5) 351.07
The independent variable is the time period and the dependent variable is the actual observed value in the time series.
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 2
Trend Pቤተ መጻሕፍቲ ባይዱojection
If a time series exhibits a linear trend, the method of least squares may be used to determine a trend line (projection) for future forecasts.
Slide 10
Trend Projection
Conclusion Due to the positive trend component in the time
series, the trend projection produced a forecast that is more in line with the trend that exists. The weighted moving average, even with heavy (.6) weight placed on the current period, produced a forecast that is lagging behind the changing data.
Slide 8
Trend Projection
Example: Auger’s Plumbing Service
Forecast for December (Month 10) using a
three-period (k = 3) weighted moving average with
weights of .6, .3, and .1 for the newest to oldest data, respectively. Then, compare this Month 10 weighted moving average forecast with the Month 10 trend projection forecast.
Sum 45
60 3480
444.00
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 5
Linear Trend Regression
Example: Auger’s Plumbing Service
The number of plumbing repair jobs performed by
Auger's Plumbing Service in the last nine months is
T10 = 351.07 + (7.12)(10) = 422.27
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 3
Linear Trend Regression
Using the method of least squares, the formula for the trend projection is:
Tt = b0 + b1t
where:
Tt = linear trend forecast in period t b0 = intercept of the linear trend line b1 = slope of the linear trend line t = time period
Month Jobs August 409 September 399 October 412 November 408
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
listed on the right. Forecast the number of repair jobs Auger's will perform in December using the least squares method.
Month Jobs March 353 April 387 May 342 June 374 July 396
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 1
Chapter 18, Part B Forecasting
Trend Projection Seasonality and Trend Time Series Decomposition
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 6
Linear Trend Regression
(month) t t t (t t )2 Yt (Yt Y) (t t )(Yt Y)
(Mar.) 1 -4 16 353 -33.67 (Apr.) 2 -3 9 387 0.33 (May) 3 -2 4 342 -44.67 (June) 4 -1 1 374 -12.67
Month Jobs March 353 April 387 May 342 June 374 July 396
Month Jobs August 409 Septem. 399 October 412 Novem. 408
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.