Forecasting by jizhen1947

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```									                Forecasting
   Quantitative Approaches to Forecasting
   The Components of a Time Series
   Measures of Forecast Accuracy
   Using Smoothing Methods in Forecasting
   Using Trend Projection in Forecasting
   Using Regression Analysis in Forecasting
   Qualitative Approaches to Forecasting

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Quantitative Approaches to
Forecasting
   Quantitative methods are based on an analysis of historical
data concerning one or more time series.
   A time series is a set of observations measured at
successive points in time or over successive periods of
time.
   If the historical data used are restricted to past values of
the series that we are trying to forecast, the procedure is
called a time series method.
   If the historical data used involve other time series that are
believed to be related to the time series that we are trying
to forecast, the procedure is called a causal method.
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Time Series Methods
   Three time series methods are:
 smoothing
 trend projection

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Components of a Time Series
   The trend component accounts for the gradual
shifting of the time series over a long period of
time.
   Any regular pattern of sequences of values
above and below the trend line is attributable to
the cyclical component of the series.

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Components of a Time Series
   The seasonal component of the series accounts for
regular patterns of variability within certain time
periods, such as over a year.
   The irregular component of the series is caused by
short-term, unanticipated and non-recurring factors
that affect the values of the time series. One cannot
attempt to predict its impact on the time series in

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Measures of Forecast Accuracy
   Mean Squared Error
The average of the squared forecast errors for
the historical data is calculated. The forecasting
method or parameter(s) which minimize this mean
squared error is then selected.
   Mean Absolute Deviation
The mean of the absolute values of all forecast
errors is calculated, and the forecasting method or
parameter(s) which minimize this measure is selected.
The mean absolute deviation measure is less sensitive
to individual large forecast errors than the mean
squared error measure.
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Smoothing Methods
   In cases in which the time series is fairly stable
and has no significant trend, seasonal, or
cyclical effects, one can use smoothing
methods to average out the irregular
components of the time series.
   Four common smoothing methods are:
 Moving averages
 Centered moving averages

 Weighted moving averages

 Exponential smoothing                              7
Smoothing Methods
   Moving Average Method
The moving average method consists of
computing an average of the most recent n
data values for the series and using this average
for forecasting the value of the time series for
the next period.

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Example: Rosco Drugs
Sales of Comfort brand headache medicine for
the past ten weeks at Rosco Drugs
are shown on the next slide. If
Rosco Drugs uses a 3-period
moving average to forecast sales,
what is the forecast for Week 11?

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Example: Rosco Drugs
   Past Sales

Week   Sales   Week   Sales
1     110      6     120
2     115      7     130
3     125      8     115
4     120      9     110
5     125     10     130

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Example: Rosco Drugs
   Excel Spreadsheet Showing Input
Data           A      B      C
1    Robert's Drugs
2
3    Week (t )   Salest   Forect+1
4      1         110
5      2         115
6      3         125
7      4         120
8      5         125
9      6         120
10      7         130
11      8         115
12      9         110
11
13      10        130
Example: Rosco Drugs
   Steps to Moving Average Using Excel
Step 1: Select the Tools pull-down menu.
Step 2: Select the Data Analysis option.
Step 3: When the Data Analysis Tools dialog
appears, choose Moving Average.
Step 4: When the Moving Average dialog box
appears:
Enter B4:B13 in the Input Range box.
Enter 3 in the Interval box.
Enter C4 in the Output Range box.
Select OK.
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Example: Rosco Drugs
   Spreadsheet Showing Results Using n =
3              A      B       C
1    Robert's Drugs
2
3    Week (t )   Salest   Forect+1
4      1         110      #N/A
5      2         115      #N/A
6      3         125      116.7
7      4         120      120.0
8      5         125      123.3
9      6         120      121.7
10      7         130      125.0
11      8         115      121.7
12      9         110      118.3
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13      10        130      118.3
Smoothing Methods
   Centered Moving Average Method
The centered moving average method
consists of computing an average of n periods'
data and associating it with the midpoint of the
periods. For example, the average for periods 5,
6, and 7 is associated with period 6. This
methodology is useful in the process of
computing season indexes.
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Smoothing Methods
   Weighted Moving Average Method
In the weighted moving average method for
computing the average of the most recent n
periods, the more recent observations are
typically given more weight than older
observations. For convenience, the weights
usually sum to 1.

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Smoothing Methods
   Exponential Smoothing
 Using exponential smoothing, the forecast for the next
period is equal to the forecast for the current period plus
a proportion () of the forecast error in the current
period.
 Using exponential smoothing, the forecast is calculated
by:
[the actual value for the current period] +
(1- )[the forecasted value for the current period],
where the smoothing constant,  , is a number between
0 and 1.

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Trend Projection
   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.
   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.
   The independent variable is the time period and the
dependent variable is the actual observed value in the
time series.

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Trend Projection
   Using the method of least squares, the formula for the trend
projection is: Tt = b0 + b1t.
where: Tt = trend forecast for time period t
b1 = slope of the trend line
b0 = trend line projection for time 0

b1 = ntYt - t Yt      b0  Y  b1 t
nt 2 - (t )2
where: Yt = observed value of the time series at time
period t
Y= average of the observed values for Yt
t= average time period for the n observations
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Example: Rosco Drugs (B)
If Rosco Drugs uses exponential smoothing to
forecast sales, which value for the smoothing constant
, .1 or .8, gives better forecasts?

Week    Sales       Week     Sales
1       110          6        120
2       115          7        130
3       125          8         115
4       120          9        110
5       125         10        130

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Example: Rosco Drugs (B)
   Exponential Smoothing
To evaluate the two smoothing constants,
determine how the forecasted values would compare
with the actual historical values in each case.
Let: Yt = actual sales in week t
Ft = forecasted sales in week t
F1 = Y1 = 110
For other weeks, Ft+1 = .1Yt + .9Ft

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Example: Rosco Drugs (B)
   Exponential Smoothing ( = .1, 1 -  = .9)

F1                                         = 110
F2 = .1Y1 + .9F1 = .1(110) + .9(110)      = 110
F3 = .1Y2 + .9F2 = .1(115) + .9(110)      = 110.5
F4 = .1Y3 + .9F3 = .1(125) + .9(110.5)    = 111.95
F5 = .1Y4 + .9F4 = .1(120) + .9(111.95)   = 112.76
F6 = .1Y5 + .9F5 = .1(125) + .9(112.76)   = 113.98
F7 = .1Y6 + .9F6 = .1(120) + .9(113.98)   = 114.58
F8 = .1Y7 + .9F7 = .1(130) + .9(114.58)   = 116.12
F9 = .1Y8 + .9F8 = .1(115) + .9(116.12)   = 116.01
F10= .1Y9 + .9F9 = .1(110) + .9(116.01)   = 115.41

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Example: Rosco Drugs (B)
   Exponential Smoothing ( = .8, 1 -  = .2)

F1                         = 110
F2 = .8(110) + .2(110) = 110
F3 = .8(115) + .2(110)    = 114
F4 = .8(125) + .2(114)    = 122.80
F5 = .8(120) + .2(122.80) = 120.56
F6 = .8(125) + .2(120.56) = 124.11
F7 = .8(120) + .2(124.11) = 120.82
F8 = .8(130) + .2(120.82) = 128.16
F9 = .8(115) + .2(128.16) = 117.63
F10= .8(110) + .2(117.63) = 111.53
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Example: Rosco Drugs (B)

   Mean Squared Error
In order to determine which smoothing
constant gives the better performance, calculate,
for each, the mean squared error for the nine
weeks of forecasts, weeks 2 through 10 by:

[(Y2-F2)2 + (Y3-F3)2 + (Y4-F4)2 + . . . + (Y10-
F10)2]/9

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Example: Rosco Drugs (B)
α = .1             α = .8
Week    Yt    Ft      (Yt - Ft)2    Ft   (Yt - Ft)2

1    110
2    115   110.00    25.00       110.00 25.00
3    125   110.50   210.25       114.00 121.00
4    120   111.95    64.80       122.80   7.84
5    125   112.76   149.94       120.56 19.71
6    120   113.98    36.25       124.11 16.91
7    130   114.58   237.73       120.82 84.23
8    115   116.12     1.26       128.16 173.30
9    110   116.01    36.12       117.63 58.26
10    130   115.41   212.87       111.53 341.27

Sum 974.22            Sum 847.52
MSE   Sum/9 108.25          Sum/9
94.17
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Example: Rosco Drugs (B)
   Excel Spreadsheet Showing Input
Data           A      B      C
1   Robert's Drugs
2
3    Week      Sales
4     1         110
5     2         115
6     3         125
7     4         120
8     5         125
9     6         120
10     7         130
11     8         115
12     9         110
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13     10        130
Example: Rosco Drugs (B)
   Steps to Exponential Smoothing Using Excel
Step 1: Select the Tools pull-down menu.
Step 2: Select the Data Analysis option.
Step 3: When the Data Analysis Tools dialog
appears, choose Exponential Smoothing.
Step 4: When the Exponential Smoothing dialog box
appears:
Enter B4:B13 in the Input Range box.
Enter 0.9 (for  = 0.1) in Damping Factor box.
Enter C4 in the Output Range box.
Select OK.
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Example: Rosco Drugs (B)
   Spreadsheet Showing Results Using 
= 0.1
A         B            C
1    Robert's Drugs
2                           = 0.1
3    Week (t )   Sales t   Forec t +1
4      1         110       #N/A
5      2         115       110.0
6      3         125       110.5
7      4         120       112.0
8      5         125       112.8
9      6         120       114.0
10      7         130       114.6
11      8         115       116.1
12      9         110       116.0
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13      10        130       115.4
Example: Rosco Drugs (B)
   Repeating the Process for  = 0.8
   Step 4: When the Exponential Smoothing dialog
box          appears:
Enter B4:B13 in the Input Range box.
Enter 0.2 (for  = 0.8) in Damping Factor box.
Enter D4 in the Output Range box.
Select OK.

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Example: Rosco Drugs (B)
   Spreadsheet Results for  = 0.1 and 
= 0.8
A         B            C            D
1   Robert's Drugs
2                          = 0.1       = 0.8
3   Week (t )   Sales t   Forec t +1   Forec t +1
4      1         110       #N/A         #N/A
5      2         115       110.0        110.0
6      3         125       110.5        114.0
7      4         120       112.0        122.8
8      5         125       112.8        120.6
9      6         120       114.0        124.1
10      7         130       114.6        120.8
11      8         115       116.1        128.2
12      9         110       116.0        117.6
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13      10        130       115.4        111.5
Example: Auger’s Plumbing
Service
The number of plumbing repair jobs
performed by
Auger's Plumbing Service in each of the last
nine
months is listed on the next slide. Forecast
the number of repair jobs Auger's will
perform in December using the least
squares method.

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Example: Auger’s Plumbing
Service
Month Jobs   Month    Jobs   Month     Jobs
March 353    June      374   September 399
April  387   July      396   October   412
May   342    August    409   November 408

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Example: Auger’s Plumbing
Service
   Trend Projection

(month) t   Yt    tYt   t2

(Mar.) 1     353   353   1
(Apr.) 2     387   774   4
(May) 3      342 1026    9
(June) 4     374 1496 16
(July) 5     396 1980 25
(Aug.) 6    409 2454 36
(Sep.) 7     399 2793 49
(Oct.) 8      412 3296 64
(Nov.) 9     408 3672 81
Sum 45      3480 17844 285
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Example: Auger’s Plumbing
Service
   Trend Projection (continued)

t = 45/9 = 5     Y = 3480/9 = 386.667

ntYt - t Yt       (9)(17844) - (45)(3480)
b1 =                    =                             = 7.4
nt 2 - (t)2           (9)(285) - (45)2

b0  Y  b1 t = 386.667 - 7.4(5) = 349.667

T10 = 349.667 + (7.4)(10) = 423.667

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Example: Auger’s Plumbing
Service
   Excel Spreadsheet Showing Input Data
A         B        C
1   Auger's Plumbing Service
2
3   Month     Calls
4    1        353
5    2        387
6    3        342
7    4        374
8    5        396
9    6        409
10    7        399
11    8        412
12    9        408
13
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Example: Auger’s Plumbing
Service
   Steps to Trend Projection Using Excel
Step 1:   Select an empty cell (B13) in the worksheet.
Step 2:   Select the Insert pull-down menu.
Step 3:   Choose the Function option.
Step 4:   When the Paste Function dialog box appears:
Choose Statistical in Function Category box.
Choose Forecast in the Function Name box.
Select OK.
more . . . . . . .

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Example: Auger’s Plumbing
Service
   Steps to Trend Projecting Using Excel
(continued)
Step 5: When the Forecast dialog box appears:
Enter 10 in the x box (for month 10).
Enter B4:B12 in the Known y’s box.
Enter A4:A12 in the Known x’s box.
Select OK.

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Example: Auger’s Plumbing
Service
   Spreadsheet with Trend Projection for
Month 10          A      B       C
1   Auger's Plumbing Service
2
3   Month     Calls
4    1         353
5    2         387
6    3         342
7    4         374
8    5         396
9    6         409
10    7         399
11    8         412
12    9         408
13    10      423.667   Projected
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Example: Auger’s Plumbing
Service (B)
Forecast for December (Month 10) using a
three-period (n = 3) weighted moving average with
weights of .6, .3, and .1.
Then, compare this Month 10 weighted moving
average forecast with the Month 10 trend projection
forecast.

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Example: Auger’s Plumbing
Service (B)
   Three-Month Weighted Moving Average
The forecast for December will be the weighted average of
the preceding three months: September, October, and
November.
F10 = .1YSep. + .3YOct. + .6YNov.
= .1(399) + .3(412) + .6(408)
= 408.3

   Trend Projection
F10 = 423.7 (from earlier slide)

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Example: Auger’s Plumbing
Service (B)
   Conclusion
Due to the positive trend component in the
time series, the trend projection produced a
forecast that is more in tune with the trend that
exists. The weighted moving average, even with
heavy (.6) placed on the current period,
produced a forecast that is lagging behind the
changing data.
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