Forecasting by jizhen1947

VIEWS: 5 PAGES: 40

									                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


                                               1
      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.
                                                                2
        Time Series Methods
   Three time series methods are:
     smoothing
     trend projection




                                     3
    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.




                                                        4
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
    advance.




                                                            5
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.
                                                            6
           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.




                                                        8
   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?




                                                    9
       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




                                         10
        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.
                                                    12
        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
                                                   13
              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.
                                                   14
          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.


                                                    15
           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.


                                                                     16
             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.

                                                          17
                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
                                                                   18
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




                                                         19
    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


                                                        20
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

                                                             21
     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
                                                     22
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


                                                          23
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
                                                      24
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
                                      25
             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.
                                                              26
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
                                                     27
             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.




                                                                28
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
                                                             29
        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.

                                               30
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




                                                31
    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
                                            32
    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

                                                                      33
    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
                                             34
    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 . . . . . . .

                                                               35
    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.




                                                     36
    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
                                                    37
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.




                                                      38
    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)

                                                                     39
    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.
                                                    40

								
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