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Time Series
Analysis
and Forecasting
                  1
2
Introduction

   Any variable that is measured over time in
   sequential order is called a time series.
   We analyze time series to detect patterns.
   The patterns help in forecasting future
   values of the time series.

                            Predicted value
The time series exhibit a
downward trend pattern.
                                              3
20.1 Components of a Time
Series

 A time series can consist of four
 components.
    Long - term trend (T).
    Cyclical effect (C).
    Seasonal effect (S).
    Random variation (R).

 A trend is a long term relatively
 smooth pattern or direction, that
 persists usually for more than one year.
                                            4
Components of a Time Series

  A time series can consists of four
  components.
     Long - term trend (T)
     Cyclical variation (C)
     Seasonal variation (S)
     Random variation (R)
                                             6/90 6/93 6/96 6/99 6/02
  A cycle is a wavelike pattern describing
                                             Cycles are seldom regular, and
  a long term behavior (for more than one
                                             often appear in combination with
  year).
                                             other components.
                                                                          5
Components of a Time Series

    A time series can consists of four
    components.
        Long - term trend (T).
        Cyclical effect (C).
        Seasonal effect (S).
        Random variation (R).
The seasonal component of the time series    6/97 12/97 6/98 12/98 6/99
exhibits a short term (less than one year)
calendar repetitive behavior.
                                                                      6
Components of a Time Series

     A time series can consists of four
     components.
          Long - term trend (T).
          Cyclical effect (C).
          Seasonal effect (S).
          Random variation (R).
                                                We try to remove random variation
Random variation comprises the irregular        thereby, identify the other components.
unpredictable changes in the time series.
It tends to hide the other (more predictable)
components.
                                                                                 7
20.2 Smoothing
Techniques

 To produce a better forecast we need to
 determine which components are present
 in a time series.
 To identify the components present in the
 time series, we need first to remove the
 random variation.
 This can be done by smoothing
 techniques.
                                             8
Moving Averages

   A k-period moving average for time
   period t is the arithmetic average of the
   time series values around period t.
 – For example: A 3-period moving average at period t
   is calculated by (yt-1 + yt + yt+1)/3




                                                        9
Moving Average: Example
Example 20.1
 To forecast future gasoline sales, the last four
 years quarterly sales were recorded.
 Calculate the three-quarter and five-quarter
 moving average. Show the relevant graphs.
 Xm20-01 Period Year/Quarter Gas Sales
                 1     1       1    39
                 2             2    37
                 3             3    61
                 4             4    58
                 5     2       1    18
                 6             2    56          10
Moving Average: Example
 Solution        Period    Gas            3-period    5-period
                           Sales          moving Avg. moving Avg.
   Solving by hand        1          39
                                              *           *
                   (39+37+61)/3=
                          2          37      45.6667      *
                                                         42.6000
                   (37+61+58)/3=
                          3
               (39+37+61+58+18)/5=   61      52.0000
                          4          58      45.6667     46.0000
                          5          18      44.0000     55.0000
                          6          56      52.0000     48.2000
                          7          82      55.0000     44.8000
                          8          27      50.0000     55.0000
                          9          41      45.6667     53.6000
                         10          69      53.0000     50.4000
                         11          49      61.3333     55.8000
                         12          66      56.3333     56.0000
                         13          54      54.0000     60.2000
                         14          42      62.0000     63.6000
                         15          90      66.0000      *
                         16          66
                                              *           *        11
Moving Average: Smoothing
effect

                        Moving Average
                  3-period moving average
          100
  Value




          50

           0                              Comment: We can identify
                1 2 3 4 5 6 7 8 9 10 11 12a 13 14component .
                                            trend 15 16
                                Data Point


  Notice how the averaging process removes some of
  the random variation.
                                                                     12
Moving Average: Smoothing
   effect

                                    3-period moving average
                     100
                      80
                      60
                      40
The 5-period          20
                       0
moving average
                           1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
removes more
variation than the                   5-period moving average
3-period moving      100
                      80
average.              60
                      40
                      20
                       0
                           1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16   13
Centered Moving Average

With an even number of observations
included in the moving average, the average
is placed between the two periods in the
middle.
To place the moving average in an actual
time period, we need to center it.
Two consecutive moving averages are
centered by taking their average, and placing
it in the middle between them. (This is used to
graph the points.)
                                             14
Centered Moving Average:
Example

Calculate the 4-period moving average and
center it, for the data given below:

 Period Time series Moving Avg.       Centerd
  Mov.Avg.
(2.5)1   15         19.0
(3.5)
     2   27                       20.25
                    21.5
(4.5)3   20
                    17.5
                                  19.50
     4   14
     5   25
                                            15
     6   11
Exponential Smoothing


 The exponential smoothing     – When smoothing the
 method provides smoothed        time series at time t,
 values for all the time          > the exponential
 periods observed.                   smoothing method
                                     considers all the data
                                     available at t
 – The moving average
   method does not                the(yt, yt-1,…)
                                      moving average method
                                  considers only the
   provide smoothed
                                  observations included in the
   values (moving                 calculation of the average
   average values) for            value, and “forgets” the
   the first and last set of      rest.
   periods.                                                      16
Exponentially Smoothed Time
Series

             S = wy + (1-w)St-1
             Stt = wytt + (1-w)St-1

   St = exponentially smoothed time series at
   time t.
   yt = time series at time t.
  St-1 = exponentially smoothed time series at
   time t-1.
   w = smoothing constant, where 0 ≤ w ≤ 1.
                                                 17
The Exponentially Smoothed
   Process
          Example 20.2 (Xm20-01)
          Calculate the gasoline sale smoothed time series
          using exponential smoothing with w = .2, and w = .7.
                         Period   Gas
                                  Sales
Set S1 = y1                  1        39   S1 = 39
    S2 = wy2 + (1-w)S1       2        37   S2 = (.2)(37) + (1-.2)(39) = 38.6
    S3 = wy3 + (1-w)S2       3        61   S3 = (.2)(61) + (1-.2)(38.6) = 43.1
                             4        58
                             5        18
                             6        56
                             .         .
                             .         .

                                                                                 18
The Exponentially Smoothed
Process
  Example 20.2-continued
                Exponential Smoothing, w=.2                      Small ‘w’ provides
          100                                                    a lot of smoothing
  Value




           50                                                    Neither value of ‘w’
            0                                                    reveals the seasonality.
                1

                    3

                        5

                            7

                                    9

                                              11

                                                   13

                                                        15
                                               Exponential Smoothing , w=.7

                                    100
                            Value




                                     50

                                      0
                                          1        3    5    7   9   11       13   15
                                                                                        19
The Exponentially Smoothed
Process

 We will let Minitab do our exponential
 smoothing
 Stat>Time Series > Single Exp Smoothing
 This will give us the smoothed graph




                                       20
20.3 Trend and Seasonal Effects
Trend Analysis

  The trend component of a time series
  can be linear or non-linear.
  It is easy to isolate the trend component
  using linear regression.
    For linear trend use the model y = β0 + β1t +
    ε.
    For non-linear trend with one (major) change
    in slope use the quadratic model y = β0 + β1t
    + β2t2 + ε                                   21
Seasonal Analysis

 Seasonal variation may occur within a
 year or within a shorter period (month,
 week)
 To measure the seasonal effects we
 construct seasonal indexes.
 Seasonal indexes express the degree to
 which the seasons differ from the average
 time series value across all seasons.
                                         22
Computing Seasonal Indexes
   Remove the effects of the seasonal and
   random variations by regression analysis
                yt = b 0 + b 1 t
                ˆ

• For each time period compute the ratio
                    yt/yt   >  This is based on the Multiplicative Model.

  which removes most of the trend variation




                                                        >
• For each season calculate the average of yt/yt
  which provides the measure of seasonality.
• Adjust the average above so that the sum of averages
  of all seasons is 1 (if necessary)
                                                                       23
The additive and multiplicative models


There are two mostly used general models to
describe the composition of a time series
  (Assuming no cyclical effects).
The multiplicative model             y t = Tt × S t × R t

The additive model                  yt = Tt + St + Rt

In the seasonal index analysis performed here we use
the multiplicative model because it is mathematically
easier to handle.
                                                            24
Seasonal indexes analysis
 The multiplicative model
        y t Tt × S t × R t
           =               = St × R t
        ˆ
        yt       Tt

    The regression line represents trend.
                        yt
 • We see that the ratio represents the seasonality and
                        ˆ
                        yt
   the random effects.
 • Averaging these ratios for each season type removes
   the random effects and leaves the seasonality.

                                                         25
Seasonal indexes analysis
 The multiplicative model
       y t Tt × S t × R t
          =               = St × R t
       ˆ
       yt       Tt
                         Rate/Predicted rate


       1.5
                                                     Note how no trend is observed, but
        1                                            seasonality and randomness
       0.5
                                                     still exist.

        0
             1   3   5    7     9    11    13   15   17   19

                       yt
 • Averaging the ratios for each season type removes
                       ˆ
                       yt
   the random effects               and leaves the seasonality.                  26
Computing Seasonal Indexes
  Example 20.3 (Xm20-03)
      Calculate the quarterly seasonal indexes
      for hotel occupancy rate in order to
      measure seasonal variation.
      Data:
  Year   Quarter    Rate      Year   Quarter    Rate      Year   Quarter    Rate
    1996        1    0.561      1998        1    0.594      2000        1    0.665
                2    0.702                  2    0.738                  2    0.835
                3       0.8                 3    0.729                  3    0.873
                4    0.568                  4       0.6                 4     0.67
    1997        1    0.575      1999        1    0.622
                2    0.738                  2    0.708
                3    0.868                  3    0.806
                4    0.605                  4    0.632


                                                                             27
Rationale for seasonal indexes

 Computing Seasonal Indexes

      Perform regression analysis for the model
      y = β0 + β1t + ε where t represents the time,
      and y represents the occupancy rate.
       Time (t) Rate                ˆ
                                    y = .639368 + .005246t
        1       0.561
        2       0.702
        3       0.800
        4       0.568
                             Rate




        5       0.575
        6       0.738
        7       0.868
        8       0.605               0         5      10      15      20         25
        .        .                      The regression line represents trend.
                                                          t
        .        .
                                                                                28
yt / yt




                                                                   >
                  The Ratios


t     yt    ˆ
            yt       Ratio
1   .561 .645    .561/.645=.870
2   .702 .650    .702/.650=1.08
3   ………………………………………………….
              =.639368+.005245(1)
                                                        No trend is observed, but
                   yt                                   seasonality and randomness
                      Rate/Predicted rate
                   ˆt
                   y                                    still exist.
    1.5

     1

    0.5

     0
                                         13




                                                   17
          1

              3

                  5

                       7

                            9

                                    11




                                              15




                                                         19
                                                                                29
The Average Ratios by Seasons
Rate/Predicted rate
       0.870
       1.080
                         • To remove most of the random variation
       1.221               but leave the seasonal effects,average
       0.860                               ˆ
                           the terms yt / yt for each season.
       0.864
       1.100
       1.284                                     Rate/Predicted rate
       0.888
       0.865                 1.5
       1.067
       1.046                  1
       0.854
                             0.5
       0.879
       0.993                  0
       1.122
                                   1   3     5    7     9    11    13   15   17   19
       0.874          Average ratio for quarter 1: (.870 + .864 + .865 + .879 + .913)/5 = .878
       0.913
       1.138
                      Average ratio for quarter 2: (1.080+1.100+1.067+.993+1.138)/5 = 1.076
       1.181          Average ratio for quarter 3: (1.221+1.284+1.046+1.122+1.181)/5 = 1.171
       0.900                                                                               30
                      Average ratio for quarter 4: (.860 +.888 + .854 + .874 + .900)/ 5 = .875
Adjusting the Average
Ratios
   In this example the sum of all the averaged ratios
   must be 4, such that the average ratio per season is
   equal to 1.
   If the sum of all the ratios is not 4, we need to adjust
   them proportionately.
       Suppose the(Seasonal averaged ratio) (number of seasons)
                    sum of ratios is equal to 4.1. Then each
Seasonal index be multipliedSum of averaged ratios
       ratio will =         by 4/4.1.

   In our problem the sum of all the averaged ratios is equal to 4:
   .878 + 1.076 + 1.171 + .875 = 4.0.
   No normalization is needed. These ratios become the
   seasonal indexes.
                                                                      31
Interpreting the Seasonal
 Indexes
        The seasonal indexes tell us what is the ratio
        between the time series value at a certain
        season, and the overall seasonal average.
                              17.1% above the
        In our problem:       annual average
                                         117.1%
                   7.6% above the                       12.5% below the
                   annual average                       annual average
Annual average
occupancy (100%)                107.6%
                        87.8%                       87.5%
   12.2% below the
   annual average

                     Quarter 1 Quarter 2 Quarter 3 Quarter 4 Quarter 1 Quarter 2 Quarter 3 Quarter 4


                                                                                            32
The Smoothed Time Series

  The trend component and the seasonality
  component are recomposed using the
                  y = ˆ ˆ            t
                                                 ˆ
  multiplicative model. T × S = (.639 + .0052t ) S
                   ˆ                              t        t                                t

    In period #1 ( quarter 1):           ˆ     ˆ ˆ
                                         y1 = T1 × S1 = (.639 + .0052(1))(.878) = .566
                                         ˆ     ˆ ˆ
                                         y 2 = T2 × S 2 = (.639 + .0052(2))(1.076) = .699
    In period #2 ( quarter 2):

             0.9
                   Actual series             Smoothed series
             0.8

             0.7

             0.6

             0.5                         The linear trend (regression) line
                   1     3       5       7    9       11   13   15   17   19
                                                                                    33
Deseasonalized Time Series



Seasonally adjusted time series = Actual time series
                                   Seasonal index

By removing the seasonality, we can
identify changes in the other components of
the time series, that might have occurred
over time.


                                                       34
Deseasonalized Time Series

 In period #1 ( quarter 1): y 1 / SI1 = .561/ .870 = .639
 In period #2 ( quarter 2): y 2 / SI2 = .708 1.076 = .652
 In period #5 ( quarter 1):  y 5 / SI1 = .575 .870 = .661
     There was a gradual increase in occupancy rate
            1
           0.8

           0.6
           0.4

           0.2
            0
                 0   5        10   15   20    25


                                                            35
20.4 Introduction to
Forecasting

 There are many forecasting models available


            o o o
              *
                *
                            ?              *
                                           o o
                                             *
        o   *                      o     o
                                         *
        *   Model 1                *     Model 2



                      Which model
                      performs better?

                                                   36
20.4 Introduction to
Forecasting
  A forecasting method can be selected
  by evaluating its forecast accuracy
  using the actual time series.
The two most commonly used measures of forecast accuracy
are:                                         n
    Mean Absolute Deviation                    ∑ y t − Ft
                                       MAD = t =1
   Sum of Squares for Forecast Error                    n
                                                n
                                       SSE =   ∑      ( y t − Ft )2
                                               t =1                   37
Measures of Forecast
Accuracy
  Choose SSE if it is important to avoid
  (even a few) large errors. Otherwise,
  use MAD.
A useful procedure for model selection.
   Use some of the observations to develop several
   competing forecasting models.
   Run the models on the rest of the observations.
   Calculate the accuracy of each model (by comparing to
   actual data at a later time period).
   Select the model with the best (lowest) accuracy measure.


                                                               38
Selecting a Forecasting
Model

 Annual data from 1970 to 1996 were used to
 develop three forecasting models.
 Use MAD and SSE to determine which model
 performed best for 1997, 1998, 1999, and
 2000.
                           Forecast value
     Year     Actual y Model 1 Model 2 Model 3
       1997      129     136      118     130
       1998      142     148      141     146
       1999      156     150      158     170
       2000      183     175      163     180


                                                 39
Solution
   For model
                        Actual y   Forecast for y
   1                    in 1991    in 1991

      129 −136 + 142 −148 + 156 −150 + 183 −175
MAD =                                           = 6.75
                          4
SSE = (129 − 136) + (142 − 148) + (156 − 160) + (183 − 175) 2 = 185
                 2               2              2

Summary of results
                      Model 1 Model 2 Model 3
            MAD        6.75     8.5     5.5
            SSE        185     526     222



                                                                      40
Selecting a Forecasting
Model

 In this case, Model 2 is inferior to both Models 1
 and 3
 Using MAD Model 3 is best
 Using SSE Model 1 is most accurate
 Choice – prefer model that consistently
 produces moderately accurate forecasts (Model
 3) or one whose forecasts come quite close
 usually but misses badly in a small number of
 time periods (Model 1)
                                                  41
20.5 Forecasting Models

 The choice of a forecasting technique
 depends on the components identified in
 the time series.
 The techniques discussed next are:
   Exponential smoothing
   Seasonal indexes
   Autoregressive models (a brief discussion)


                                                42
Forecasting with Exponential
Smoothing

  The exponential smoothing model can
  be used to produce forecasts when the
  time series…
    exhibits gradual(not a sharp) trend
    no cyclical effects
    no seasonal effects
  Forecast for period t+k is computed by
                Ft+k = St
  t is the current period; St = ωyt + (1-ω)St-1   43
Forecasting with Exponential
Smoothing - Example

 The quarterly sales (in $ millions) of a
 department store chain were recorded for
 the years 1997 – 2000




                                        44
Forecasting with Exponential
Smoothing - Example

Quarter   1997   1998   1999   2000

1         18     33     25     41

2         22     20     36     33

3         27     38     44     52

4         31     26     29     45


                                      45
Forecasting with Exponential
Smoothing - Example

                    Si ngl e Ex ponent i al Smoot hi ng Pl ot f or Sal es
           55
                                                                                                  Y16 = 45
                                                                                  Variable
                                                                                  Actual
           50                                                                     Fits

                                                                            Smoothing Constant
           45                                                                  Alpha 0.4


           40
                                                                            Accuracy Measures
                                                                             MAPE    22.4486
                                                                                                 S16 =
                                                                             MAD      7.2311
                                                                                                 41.8
   Sales




                                                                             MSD     69.2590
           35

           30

           25

           20


                2     4       6       8     10      12      14      16
                                     I ndex




                                                                                                     46
Forecasting with Exponential
Smoothing - Example

 Forecast for period t+k is computed by
  (t = 16)               Ft+k = St
 t is the current period;   St = ωyt + (1-ω)St-1
 F17= S16 = 41.8
 F18= S16 = 41.8
 F19= S16 = 41.8
 F20= S16 = 41.8
                                                   47
Computing Seasonal Indexes
  Example 20.3 (Xm20-03)
      Calculate the quarterly seasonal indexes
      for hotel occupancy rate in order to
      measure seasonal variation.
      Data:
                        ˆ ˆ                        ˆ
                  yt = Tt × S t = (.639 + .0052t ) S t
                   ˆ
  Year   Quarter    Rate      Year   Quarter    Rate      Year   Quarter    Rate
    1996        1    0.561      1998        1    0.594      2000        1    0.665
                2    0.702                  2    0.738                  2    0.835
                3       0.8                 3    0.729                  3    0.873
                4    0.568                  4       0.6                 4     0.67
    1997        1    0.575      1999        1    0.622
                2    0.738                  2    0.708
                3    0.868                  3    0.806
                4    0.605                  4    0.632


                                                                             48
Forecasting with Seasonal
Indexes; Example

  Example 20.5
      Use the seasonal indexes calculated in
      Example 21.3 along with the trend line, to
      forecast each quarter occupancy rate in 2001.
The solution procedure
   Use simple linear regression to find the trend line.
   Use the trend line to calculate the trend values.
   To calculate Ft multiply the trend value for period t by the
   seasonal index of period t.


                                                              49
Forecasting with Seasonal
Indexes; Example

 The solution procedure
 1. Use simple linear regression to find the trend line.


 From Minitab (Using decomposition )
 Yt = .639 + .00525t




                                                           50
Forecasting with Seasonal
Indexes; Example

 The solution procedure:
   2. Use the trend line to calculate the trend values
                Yt = .639 + .00525t
 We previously used 20 time periods
 Quarter 1: 0.639 + 0.00525(21) = 0.749
 Quarter 2: 0.639 + 0.00525(22) = 0.755
 Quarter 3: 0.639 + 0.00525(23) = 0.760
 Quarter 4: 0.639 + 0.00525(24) = 0.765

                                                         51
Forecasting with Seasonal
Indexes; Example

  The solution procedure
  3. To calculate Ft multiply the trend value for period t by
    the seasonal index of period t.
Quarter 1: 0.749 X 0.878 = 0.658
Quarter 2: 0.755 X 1.076 = 0.812
Quarter 3: 0.760 X 1.171 = 0.890
Quarter 4: 0.765 X 0.875 = 0.670
We forecast that the quarterly occupancy rates for
 the next year will be 65.8%, 81.2%, 89% and
 67%.
                                                           52
Forecasting with Seasonal
    Indexes: Example (Summary)

      Solution
          The trend line was obtained from the
          regression analysis. y t = .639 + .00525t
                               ˆ
          – For the year 2001 we have:
y21 = .639 + .00525 (21) = .749
ˆ                                                         F21 = .749(.878)
                             t Trend value   Quarter SI Forecast
                             21 .749           1      .878 .658
                             22 .755           2     1.076 .812
                             23 .760           3     1.171 .890
                             24 .765           4      .875 .670


                                                                             53
Problem 20-47

 The revenues (in $millions) of a chain of
 ice-cream stores are listed for each
 quarter during the previous 5 years.
 Use the handout with the plots etc. to
 answer the questions.




                                             54
Quarter   1996   1997   1998   1999   2000

  1        16     14     17     18     21

  2        25     27     31     29     30

  3        31     32     40     45     52

  4        24     23     27     24     32



                                             55
20-47

 1. Plot the time series. Is there a
 pattern? Does there appear to be
   A long term trend
   A cyclic effect
   A seasonal effect
 2. Plot the 3-year moving balance. Does
 this change any of your answers from #
 1?

                                       56
20-47 - Answers

 Seems to be a long term trend – going
 upward.
 Does not seem to be a cyclic effect.
 Because we know the data is in quarter
 years, this seems to be a strong seasonal
 effect.
 The three year moving average plot
 strengthens these observations.
                                         57
20-47

 Why is exponential smoothing not
 recommended as a forecasting tool in this
 problem?
 Look at the exponential smoothing plot (w
 = 0.3) Does this confirm your answer
 above?



                                         58
20-47 - Answers

 Exponential smoothing is not
 recommended as a forecasting tool for
 this problem because of the strong
 seasonal effect.
 The 0.3 gives a large smoothing effect,
 and the seasonality is no larger as
 evident.


                                           59
20-47

 Look at the trend analysis plots. Does the
 Linear or the Quadratic trend model seem
 to be better? Why? What is the equation
 of the trend line?




                                          60
20-47 - Answers

 The linear model seems to fit well. The
 quadratic model doesn’t show much of a
 curve. Checking the MAD for both models
 shows 6.8716 for the linear model and
 6.8784 for the quadratic model. Since the
 linear model has a slightly smaller MAD
 we will use it.
 The equation of the trend line, then, is
  Y = 20.2105 + 0.732331t
                                         61
20.47

 Predict the sales for the next 4 quarters
 using the trend analysis line.




                                             62
20-47 - Answer

 The next 4 quarters are numbers 21, 22,
 23, 24
 Y = 20.21 + 0.732 t
 Y21 = [20.21 + 0.732 (21)] = 35.58
 Y22 = [20.21 + 0.732 (22)] = 36.31
 Y23 = [20.21 + 0.732 (23)] = 37.05
 Y24 = [20.21 + 0.732 (24)] = 37.78
                                           63
20-47

 Using the seasonal indexes and the trend
 line, forecast revenues for the next four
 quarters.




                                         64
20-47 - Answer
 The Fitted trend equation for the
 decomposition plot is Y = 21.7511 +
 0.567732 t
 The next time period is number 21.
 Y21 = [21.75 + 0.568 (21)] * 0.63581 =
 21.41
 Y22 = [21.75 + 0.568 (22)] * 1.05732 =
 36.21
 Y23 = [21.75 + 0.568 (23)] * 1.36851 =   65
20-47 (continued)

 If the actual sales numbers for the next
 four quarters were 24, 40, 48, 36.
 Calculate the MAD for the predictions
 from the trend line and seasonal indexes.




                                             66
20-47 - Answer

 MAD = |21.41 - 24| + |36.21 – 40| + |47.46
 – 48| + |33.2 – 36| / 4
 = (2.59 + 3.79 + .54 + .2.8 ) / 4
 = 9.72/4 = 2.43




                                          67
20-47

 Go back to the linear trend equation
 forecasts. If the actual sales numbers for
 the next four quarters were 24, 40, 48, 36.
  Calculate the MAD.
 Which method of forecasting did a better
 job in this case?



                                           68
20-47 Answers

 MAD = |35.58 - 24| + |36.31 – 40| + |37.05 – 48|
 + |37.78 – 36| / 4
 = (11.58 + 3.69 + 10.95 + 1.78 ) / 4
 = 28/4 = 7 This is the MAD for the trend
 analysis.
 The MAD for the seasonal trend analysis was
 2.43.
 The seasonal trend analysis did the better job of
 forecasting.
                                                 69

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Forcasting Techniques

  • 2. 2
  • 3. Introduction Any variable that is measured over time in sequential order is called a time series. We analyze time series to detect patterns. The patterns help in forecasting future values of the time series. Predicted value The time series exhibit a downward trend pattern. 3
  • 4. 20.1 Components of a Time Series A time series can consist of four components. Long - term trend (T). Cyclical effect (C). Seasonal effect (S). Random variation (R). A trend is a long term relatively smooth pattern or direction, that persists usually for more than one year. 4
  • 5. Components of a Time Series A time series can consists of four components. Long - term trend (T) Cyclical variation (C) Seasonal variation (S) Random variation (R) 6/90 6/93 6/96 6/99 6/02 A cycle is a wavelike pattern describing Cycles are seldom regular, and a long term behavior (for more than one often appear in combination with year). other components. 5
  • 6. Components of a Time Series A time series can consists of four components. Long - term trend (T). Cyclical effect (C). Seasonal effect (S). Random variation (R). The seasonal component of the time series 6/97 12/97 6/98 12/98 6/99 exhibits a short term (less than one year) calendar repetitive behavior. 6
  • 7. Components of a Time Series A time series can consists of four components. Long - term trend (T). Cyclical effect (C). Seasonal effect (S). Random variation (R). We try to remove random variation Random variation comprises the irregular thereby, identify the other components. unpredictable changes in the time series. It tends to hide the other (more predictable) components. 7
  • 8. 20.2 Smoothing Techniques To produce a better forecast we need to determine which components are present in a time series. To identify the components present in the time series, we need first to remove the random variation. This can be done by smoothing techniques. 8
  • 9. Moving Averages A k-period moving average for time period t is the arithmetic average of the time series values around period t. – For example: A 3-period moving average at period t is calculated by (yt-1 + yt + yt+1)/3 9
  • 10. Moving Average: Example Example 20.1 To forecast future gasoline sales, the last four years quarterly sales were recorded. Calculate the three-quarter and five-quarter moving average. Show the relevant graphs. Xm20-01 Period Year/Quarter Gas Sales 1 1 1 39 2 2 37 3 3 61 4 4 58 5 2 1 18 6 2 56 10
  • 11. Moving Average: Example Solution Period Gas 3-period 5-period Sales moving Avg. moving Avg. Solving by hand 1 39 * * (39+37+61)/3= 2 37 45.6667 * 42.6000 (37+61+58)/3= 3 (39+37+61+58+18)/5= 61 52.0000 4 58 45.6667 46.0000 5 18 44.0000 55.0000 6 56 52.0000 48.2000 7 82 55.0000 44.8000 8 27 50.0000 55.0000 9 41 45.6667 53.6000 10 69 53.0000 50.4000 11 49 61.3333 55.8000 12 66 56.3333 56.0000 13 54 54.0000 60.2000 14 42 62.0000 63.6000 15 90 66.0000 * 16 66 * * 11
  • 12. Moving Average: Smoothing effect Moving Average 3-period moving average 100 Value 50 0 Comment: We can identify 1 2 3 4 5 6 7 8 9 10 11 12a 13 14component . trend 15 16 Data Point Notice how the averaging process removes some of the random variation. 12
  • 13. Moving Average: Smoothing effect 3-period moving average 100 80 60 40 The 5-period 20 0 moving average 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 removes more variation than the 5-period moving average 3-period moving 100 80 average. 60 40 20 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 13
  • 14. Centered Moving Average With an even number of observations included in the moving average, the average is placed between the two periods in the middle. To place the moving average in an actual time period, we need to center it. Two consecutive moving averages are centered by taking their average, and placing it in the middle between them. (This is used to graph the points.) 14
  • 15. Centered Moving Average: Example Calculate the 4-period moving average and center it, for the data given below: Period Time series Moving Avg. Centerd Mov.Avg. (2.5)1 15 19.0 (3.5) 2 27 20.25 21.5 (4.5)3 20 17.5 19.50 4 14 5 25 15 6 11
  • 16. Exponential Smoothing The exponential smoothing – When smoothing the method provides smoothed time series at time t, values for all the time > the exponential periods observed. smoothing method considers all the data available at t – The moving average method does not the(yt, yt-1,…) moving average method considers only the provide smoothed observations included in the values (moving calculation of the average average values) for value, and “forgets” the the first and last set of rest. periods. 16
  • 17. Exponentially Smoothed Time Series S = wy + (1-w)St-1 Stt = wytt + (1-w)St-1 St = exponentially smoothed time series at time t. yt = time series at time t. St-1 = exponentially smoothed time series at time t-1. w = smoothing constant, where 0 ≤ w ≤ 1. 17
  • 18. The Exponentially Smoothed Process Example 20.2 (Xm20-01) Calculate the gasoline sale smoothed time series using exponential smoothing with w = .2, and w = .7. Period Gas Sales Set S1 = y1 1 39 S1 = 39 S2 = wy2 + (1-w)S1 2 37 S2 = (.2)(37) + (1-.2)(39) = 38.6 S3 = wy3 + (1-w)S2 3 61 S3 = (.2)(61) + (1-.2)(38.6) = 43.1 4 58 5 18 6 56 . . . . 18
  • 19. The Exponentially Smoothed Process Example 20.2-continued Exponential Smoothing, w=.2 Small ‘w’ provides 100 a lot of smoothing Value 50 Neither value of ‘w’ 0 reveals the seasonality. 1 3 5 7 9 11 13 15 Exponential Smoothing , w=.7 100 Value 50 0 1 3 5 7 9 11 13 15 19
  • 20. The Exponentially Smoothed Process We will let Minitab do our exponential smoothing Stat>Time Series > Single Exp Smoothing This will give us the smoothed graph 20
  • 21. 20.3 Trend and Seasonal Effects Trend Analysis The trend component of a time series can be linear or non-linear. It is easy to isolate the trend component using linear regression. For linear trend use the model y = β0 + β1t + ε. For non-linear trend with one (major) change in slope use the quadratic model y = β0 + β1t + β2t2 + ε 21
  • 22. Seasonal Analysis Seasonal variation may occur within a year or within a shorter period (month, week) To measure the seasonal effects we construct seasonal indexes. Seasonal indexes express the degree to which the seasons differ from the average time series value across all seasons. 22
  • 23. Computing Seasonal Indexes Remove the effects of the seasonal and random variations by regression analysis yt = b 0 + b 1 t ˆ • For each time period compute the ratio yt/yt > This is based on the Multiplicative Model. which removes most of the trend variation > • For each season calculate the average of yt/yt which provides the measure of seasonality. • Adjust the average above so that the sum of averages of all seasons is 1 (if necessary) 23
  • 24. The additive and multiplicative models There are two mostly used general models to describe the composition of a time series (Assuming no cyclical effects). The multiplicative model y t = Tt × S t × R t The additive model yt = Tt + St + Rt In the seasonal index analysis performed here we use the multiplicative model because it is mathematically easier to handle. 24
  • 25. Seasonal indexes analysis The multiplicative model y t Tt × S t × R t = = St × R t ˆ yt Tt The regression line represents trend. yt • We see that the ratio represents the seasonality and ˆ yt the random effects. • Averaging these ratios for each season type removes the random effects and leaves the seasonality. 25
  • 26. Seasonal indexes analysis The multiplicative model y t Tt × S t × R t = = St × R t ˆ yt Tt Rate/Predicted rate 1.5 Note how no trend is observed, but 1 seasonality and randomness 0.5 still exist. 0 1 3 5 7 9 11 13 15 17 19 yt • Averaging the ratios for each season type removes ˆ yt the random effects and leaves the seasonality. 26
  • 27. Computing Seasonal Indexes Example 20.3 (Xm20-03) Calculate the quarterly seasonal indexes for hotel occupancy rate in order to measure seasonal variation. Data: Year Quarter Rate Year Quarter Rate Year Quarter Rate 1996 1 0.561 1998 1 0.594 2000 1 0.665 2 0.702 2 0.738 2 0.835 3 0.8 3 0.729 3 0.873 4 0.568 4 0.6 4 0.67 1997 1 0.575 1999 1 0.622 2 0.738 2 0.708 3 0.868 3 0.806 4 0.605 4 0.632 27
  • 28. Rationale for seasonal indexes Computing Seasonal Indexes Perform regression analysis for the model y = β0 + β1t + ε where t represents the time, and y represents the occupancy rate. Time (t) Rate ˆ y = .639368 + .005246t 1 0.561 2 0.702 3 0.800 4 0.568 Rate 5 0.575 6 0.738 7 0.868 8 0.605 0 5 10 15 20 25 . . The regression line represents trend. t . . 28
  • 29. yt / yt > The Ratios t yt ˆ yt Ratio 1 .561 .645 .561/.645=.870 2 .702 .650 .702/.650=1.08 3 …………………………………………………. =.639368+.005245(1) No trend is observed, but yt seasonality and randomness Rate/Predicted rate ˆt y still exist. 1.5 1 0.5 0 13 17 1 3 5 7 9 11 15 19 29
  • 30. The Average Ratios by Seasons Rate/Predicted rate 0.870 1.080 • To remove most of the random variation 1.221 but leave the seasonal effects,average 0.860 ˆ the terms yt / yt for each season. 0.864 1.100 1.284 Rate/Predicted rate 0.888 0.865 1.5 1.067 1.046 1 0.854 0.5 0.879 0.993 0 1.122 1 3 5 7 9 11 13 15 17 19 0.874 Average ratio for quarter 1: (.870 + .864 + .865 + .879 + .913)/5 = .878 0.913 1.138 Average ratio for quarter 2: (1.080+1.100+1.067+.993+1.138)/5 = 1.076 1.181 Average ratio for quarter 3: (1.221+1.284+1.046+1.122+1.181)/5 = 1.171 0.900 30 Average ratio for quarter 4: (.860 +.888 + .854 + .874 + .900)/ 5 = .875
  • 31. Adjusting the Average Ratios In this example the sum of all the averaged ratios must be 4, such that the average ratio per season is equal to 1. If the sum of all the ratios is not 4, we need to adjust them proportionately. Suppose the(Seasonal averaged ratio) (number of seasons) sum of ratios is equal to 4.1. Then each Seasonal index be multipliedSum of averaged ratios ratio will = by 4/4.1. In our problem the sum of all the averaged ratios is equal to 4: .878 + 1.076 + 1.171 + .875 = 4.0. No normalization is needed. These ratios become the seasonal indexes. 31
  • 32. Interpreting the Seasonal Indexes The seasonal indexes tell us what is the ratio between the time series value at a certain season, and the overall seasonal average. 17.1% above the In our problem: annual average 117.1% 7.6% above the 12.5% below the annual average annual average Annual average occupancy (100%) 107.6% 87.8% 87.5% 12.2% below the annual average Quarter 1 Quarter 2 Quarter 3 Quarter 4 Quarter 1 Quarter 2 Quarter 3 Quarter 4 32
  • 33. The Smoothed Time Series The trend component and the seasonality component are recomposed using the y = ˆ ˆ t ˆ multiplicative model. T × S = (.639 + .0052t ) S ˆ t t t In period #1 ( quarter 1): ˆ ˆ ˆ y1 = T1 × S1 = (.639 + .0052(1))(.878) = .566 ˆ ˆ ˆ y 2 = T2 × S 2 = (.639 + .0052(2))(1.076) = .699 In period #2 ( quarter 2): 0.9 Actual series Smoothed series 0.8 0.7 0.6 0.5 The linear trend (regression) line 1 3 5 7 9 11 13 15 17 19 33
  • 34. Deseasonalized Time Series Seasonally adjusted time series = Actual time series Seasonal index By removing the seasonality, we can identify changes in the other components of the time series, that might have occurred over time. 34
  • 35. Deseasonalized Time Series In period #1 ( quarter 1): y 1 / SI1 = .561/ .870 = .639 In period #2 ( quarter 2): y 2 / SI2 = .708 1.076 = .652 In period #5 ( quarter 1): y 5 / SI1 = .575 .870 = .661 There was a gradual increase in occupancy rate 1 0.8 0.6 0.4 0.2 0 0 5 10 15 20 25 35
  • 36. 20.4 Introduction to Forecasting There are many forecasting models available o o o * * ? * o o * o * o o * * Model 1 * Model 2 Which model performs better? 36
  • 37. 20.4 Introduction to Forecasting A forecasting method can be selected by evaluating its forecast accuracy using the actual time series. The two most commonly used measures of forecast accuracy are: n Mean Absolute Deviation ∑ y t − Ft MAD = t =1 Sum of Squares for Forecast Error n n SSE = ∑ ( y t − Ft )2 t =1 37
  • 38. Measures of Forecast Accuracy Choose SSE if it is important to avoid (even a few) large errors. Otherwise, use MAD. A useful procedure for model selection. Use some of the observations to develop several competing forecasting models. Run the models on the rest of the observations. Calculate the accuracy of each model (by comparing to actual data at a later time period). Select the model with the best (lowest) accuracy measure. 38
  • 39. Selecting a Forecasting Model Annual data from 1970 to 1996 were used to develop three forecasting models. Use MAD and SSE to determine which model performed best for 1997, 1998, 1999, and 2000. Forecast value Year Actual y Model 1 Model 2 Model 3 1997 129 136 118 130 1998 142 148 141 146 1999 156 150 158 170 2000 183 175 163 180 39
  • 40. Solution For model Actual y Forecast for y 1 in 1991 in 1991 129 −136 + 142 −148 + 156 −150 + 183 −175 MAD = = 6.75 4 SSE = (129 − 136) + (142 − 148) + (156 − 160) + (183 − 175) 2 = 185 2 2 2 Summary of results Model 1 Model 2 Model 3 MAD 6.75 8.5 5.5 SSE 185 526 222 40
  • 41. Selecting a Forecasting Model In this case, Model 2 is inferior to both Models 1 and 3 Using MAD Model 3 is best Using SSE Model 1 is most accurate Choice – prefer model that consistently produces moderately accurate forecasts (Model 3) or one whose forecasts come quite close usually but misses badly in a small number of time periods (Model 1) 41
  • 42. 20.5 Forecasting Models The choice of a forecasting technique depends on the components identified in the time series. The techniques discussed next are: Exponential smoothing Seasonal indexes Autoregressive models (a brief discussion) 42
  • 43. Forecasting with Exponential Smoothing The exponential smoothing model can be used to produce forecasts when the time series… exhibits gradual(not a sharp) trend no cyclical effects no seasonal effects Forecast for period t+k is computed by Ft+k = St t is the current period; St = ωyt + (1-ω)St-1 43
  • 44. Forecasting with Exponential Smoothing - Example The quarterly sales (in $ millions) of a department store chain were recorded for the years 1997 – 2000 44
  • 45. Forecasting with Exponential Smoothing - Example Quarter 1997 1998 1999 2000 1 18 33 25 41 2 22 20 36 33 3 27 38 44 52 4 31 26 29 45 45
  • 46. Forecasting with Exponential Smoothing - Example Si ngl e Ex ponent i al Smoot hi ng Pl ot f or Sal es 55 Y16 = 45 Variable Actual 50 Fits Smoothing Constant 45 Alpha 0.4 40 Accuracy Measures MAPE 22.4486 S16 = MAD 7.2311 41.8 Sales MSD 69.2590 35 30 25 20 2 4 6 8 10 12 14 16 I ndex 46
  • 47. Forecasting with Exponential Smoothing - Example Forecast for period t+k is computed by (t = 16) Ft+k = St t is the current period; St = ωyt + (1-ω)St-1 F17= S16 = 41.8 F18= S16 = 41.8 F19= S16 = 41.8 F20= S16 = 41.8 47
  • 48. Computing Seasonal Indexes Example 20.3 (Xm20-03) Calculate the quarterly seasonal indexes for hotel occupancy rate in order to measure seasonal variation. Data: ˆ ˆ ˆ yt = Tt × S t = (.639 + .0052t ) S t ˆ Year Quarter Rate Year Quarter Rate Year Quarter Rate 1996 1 0.561 1998 1 0.594 2000 1 0.665 2 0.702 2 0.738 2 0.835 3 0.8 3 0.729 3 0.873 4 0.568 4 0.6 4 0.67 1997 1 0.575 1999 1 0.622 2 0.738 2 0.708 3 0.868 3 0.806 4 0.605 4 0.632 48
  • 49. Forecasting with Seasonal Indexes; Example Example 20.5 Use the seasonal indexes calculated in Example 21.3 along with the trend line, to forecast each quarter occupancy rate in 2001. The solution procedure Use simple linear regression to find the trend line. Use the trend line to calculate the trend values. To calculate Ft multiply the trend value for period t by the seasonal index of period t. 49
  • 50. Forecasting with Seasonal Indexes; Example The solution procedure 1. Use simple linear regression to find the trend line. From Minitab (Using decomposition ) Yt = .639 + .00525t 50
  • 51. Forecasting with Seasonal Indexes; Example The solution procedure: 2. Use the trend line to calculate the trend values Yt = .639 + .00525t We previously used 20 time periods Quarter 1: 0.639 + 0.00525(21) = 0.749 Quarter 2: 0.639 + 0.00525(22) = 0.755 Quarter 3: 0.639 + 0.00525(23) = 0.760 Quarter 4: 0.639 + 0.00525(24) = 0.765 51
  • 52. Forecasting with Seasonal Indexes; Example The solution procedure 3. To calculate Ft multiply the trend value for period t by the seasonal index of period t. Quarter 1: 0.749 X 0.878 = 0.658 Quarter 2: 0.755 X 1.076 = 0.812 Quarter 3: 0.760 X 1.171 = 0.890 Quarter 4: 0.765 X 0.875 = 0.670 We forecast that the quarterly occupancy rates for the next year will be 65.8%, 81.2%, 89% and 67%. 52
  • 53. Forecasting with Seasonal Indexes: Example (Summary) Solution The trend line was obtained from the regression analysis. y t = .639 + .00525t ˆ – For the year 2001 we have: y21 = .639 + .00525 (21) = .749 ˆ F21 = .749(.878) t Trend value Quarter SI Forecast 21 .749 1 .878 .658 22 .755 2 1.076 .812 23 .760 3 1.171 .890 24 .765 4 .875 .670 53
  • 54. Problem 20-47 The revenues (in $millions) of a chain of ice-cream stores are listed for each quarter during the previous 5 years. Use the handout with the plots etc. to answer the questions. 54
  • 55. Quarter 1996 1997 1998 1999 2000 1 16 14 17 18 21 2 25 27 31 29 30 3 31 32 40 45 52 4 24 23 27 24 32 55
  • 56. 20-47 1. Plot the time series. Is there a pattern? Does there appear to be A long term trend A cyclic effect A seasonal effect 2. Plot the 3-year moving balance. Does this change any of your answers from # 1? 56
  • 57. 20-47 - Answers Seems to be a long term trend – going upward. Does not seem to be a cyclic effect. Because we know the data is in quarter years, this seems to be a strong seasonal effect. The three year moving average plot strengthens these observations. 57
  • 58. 20-47 Why is exponential smoothing not recommended as a forecasting tool in this problem? Look at the exponential smoothing plot (w = 0.3) Does this confirm your answer above? 58
  • 59. 20-47 - Answers Exponential smoothing is not recommended as a forecasting tool for this problem because of the strong seasonal effect. The 0.3 gives a large smoothing effect, and the seasonality is no larger as evident. 59
  • 60. 20-47 Look at the trend analysis plots. Does the Linear or the Quadratic trend model seem to be better? Why? What is the equation of the trend line? 60
  • 61. 20-47 - Answers The linear model seems to fit well. The quadratic model doesn’t show much of a curve. Checking the MAD for both models shows 6.8716 for the linear model and 6.8784 for the quadratic model. Since the linear model has a slightly smaller MAD we will use it. The equation of the trend line, then, is Y = 20.2105 + 0.732331t 61
  • 62. 20.47 Predict the sales for the next 4 quarters using the trend analysis line. 62
  • 63. 20-47 - Answer The next 4 quarters are numbers 21, 22, 23, 24 Y = 20.21 + 0.732 t Y21 = [20.21 + 0.732 (21)] = 35.58 Y22 = [20.21 + 0.732 (22)] = 36.31 Y23 = [20.21 + 0.732 (23)] = 37.05 Y24 = [20.21 + 0.732 (24)] = 37.78 63
  • 64. 20-47 Using the seasonal indexes and the trend line, forecast revenues for the next four quarters. 64
  • 65. 20-47 - Answer The Fitted trend equation for the decomposition plot is Y = 21.7511 + 0.567732 t The next time period is number 21. Y21 = [21.75 + 0.568 (21)] * 0.63581 = 21.41 Y22 = [21.75 + 0.568 (22)] * 1.05732 = 36.21 Y23 = [21.75 + 0.568 (23)] * 1.36851 = 65
  • 66. 20-47 (continued) If the actual sales numbers for the next four quarters were 24, 40, 48, 36. Calculate the MAD for the predictions from the trend line and seasonal indexes. 66
  • 67. 20-47 - Answer MAD = |21.41 - 24| + |36.21 – 40| + |47.46 – 48| + |33.2 – 36| / 4 = (2.59 + 3.79 + .54 + .2.8 ) / 4 = 9.72/4 = 2.43 67
  • 68. 20-47 Go back to the linear trend equation forecasts. If the actual sales numbers for the next four quarters were 24, 40, 48, 36. Calculate the MAD. Which method of forecasting did a better job in this case? 68
  • 69. 20-47 Answers MAD = |35.58 - 24| + |36.31 – 40| + |37.05 – 48| + |37.78 – 36| / 4 = (11.58 + 3.69 + 10.95 + 1.78 ) / 4 = 28/4 = 7 This is the MAD for the trend analysis. The MAD for the seasonal trend analysis was 2.43. The seasonal trend analysis did the better job of forecasting. 69