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Forecasting

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Forecasting Forecasting Terminology Simple Moving Average Weighted Moving Average Exponential Smoothing Simple Linear Regression Model Holt s Trend Model – PowerPoint PPT presentation

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Title: Forecasting


1
Forecasting
  • Forecasting Terminology
  • Simple Moving Average
  • Weighted Moving Average
  • Exponential Smoothing
  • Simple Linear Regression Model
  • Holts Trend Model
  • Seasonal Model (No Trend)
  • Winters Model for Data with Trend and Seasonal
    Components

2
Evaluating Forecasts
  • Visual Review
  • Errors
  • Errors Measure
  • MPE and MAPE
  • Tracking Signal

3
Forecasting Terminology
Historical Data
ExPost Forecast
Initialization
Forecast
4
Forecasting Terminology
  • We are now looking at a future from here, and
    the future we were looking at in February now
    includes some of our past, and we can incorporate
    the past into our forecast. 1993, the first
    half, which is now the past and was the future
    when we issued our first forecast, is now over
  • Laura DAndrea Tyson, Head of the Presidents
    Council of Economic Advisors, quoted in November
    of 1993 in the Chicago Tribune, explaining why
    the Administration reduced its projections of
    economic growth to 2 percent from the 3.1percent
    it predicted in February.

5
Forecasting Problem
  • Suppose your fraternity/sorority house consumed
    the following number of cases of beer for the
    last 6 weekends 8, 5, 7, 3, 6, 9
  • How many cases do you think your fraternity /
    sorority will consume this weekend?

6
ForecastingSimple Moving Average Method
  • Using a three period moving average, we would get
    the following forecast

7
ForecastingSimple Moving Average Method
  • What if we used a two period moving average?

8
ForecastingSimple Moving Average Method
  • The number of periods used in the moving average
    forecast affects the responsiveness of the
    forecasting method

9
Forecasting Terminology
  • Applying this terminology to our problem using
    the Moving Average forecast

Model Evaluation
Initialization
ExPost Forecast
Forecast
10
ForecastingWeighted Moving Average Method
  • Rather than equal weights, it might make sense to
    use weights which favor more recent consumption
    values.
  • With the Weighted Moving Average, we have to
    select weights that are individually greater than
    zero and less than 1, and as a group sum to 1
  • Valid Weights (.5, .3, .2) , (.6,.3,.1), (1/2,
    1/3, 1/6)
  • Invalid Weights (.5, .2, .1), (.6, -.1, .5),
    (.5,.4,.3,.2)

11
ForecastingWeighted Moving Average Method
  • A Weighted Moving Average forecast with weights
    of (1/6, 1/3, 1/2), is performed as follows
  • How do you make the Weighted Moving Average
    forecast more responsive?

12
ForecastingExponential Smoothing
  • Exponential Smoothing is designed to give the
    benefits of the Weighted Moving Average forecast
    without the cumbersome problem of specifying
    weights. In Exponential Smoothing, there is only
    one parameter (?)

? smoothing constant (between 0 and 1)
13
ForecastingExponential Smoothing
  • Initialization

14
ForecastingExponential Smoothing
  • Using a 0.4,

t A(t) F(t)
1 8  
2 5 6.5
3 7 5.9
4 3 6.34
5 6 5
6 9 5.4
7   6.84
8   6.84
9   6.84
10   6.84
Initialization
ExPost Forecast
Forecast
15
ForecastingExponential Smoothing
16
ForecastingExponential Smoothing
17
Outliers (eloping point)
18
Data with Trends
19
Data with Trends
20
ForecastingSimple Linear Regression Model
Simple linear regression can be used to forecast
data with trends
a
D is the regressed forecast value or dependent
variable in the model, a is the intercept value
of the regression line, and b is the slope of the
regression line.
21
ForecastingSimple Linear Regression Model
In linear regression, the squared errors are
minimized
22
ForecastingSimple Linear Regression Model
23
Limitations in Linear Regression Model
As with the simple moving average model, all data
points count equally with simple linear
regression.
24
ForecastingHolts Trend Model
  • To forecast data with trends, we can use an
    exponential smoothing model with trend,
    frequently known as Holts model

L(t) aA(t) (1- a) F(t)
T(t) ? L(t) - L(t-1) (1- ?) T(t-1)
F(t1) L(t) T(t)
  • We could use linear regression to initialize the
    model

25
Holts Trend ModelInitialization
First, well initialize the model
L(4) 20.54(9.9)60.1 T(4) 9.9
26
Holts Trend ModelUpdating
  • 0.3
  • b 0.4

52
64.6
7.74
L(t) aA(t) (1- a) F(t)
L(5) 0.3 (52) 0.7 (70)64.6
T(t) ? L(t) - L(t-1) (1- ?) T(t-1)
T(5) 0.4 64.6 60.1 0.6 (9.9) 7.74
F(t1) L(t) T(t)
F(6) 64.6 7.74 72.34
27
Holts Trend Model Updating
  • 0.3
  • b 0.4

63
69.54
6.62
72
L(6) 0.3 (63) 0.7 (72.34)69.54
T(6) 0.4 69.54 64.60 0.6 (7.74) 6.62
F(7) 69.54 6.62 76.16
28
Holts Model Results
Initialization
ExPost Forecast
Forecast
29
Holts Model Results
Initialization
ExPost Forecast
Forecast
30
Forecasting Seasonal Model (No Trend)
31
Seasonal Model Formulas
L(t) aA(t) / S(t-p) (1- a) L(t-1)
S(t) g A(t) / L(t) (1- g) S(t-p)
F(t1) L(t) S(t1-p)
p is the number of periods in a season Quarterly
data p 4 Monthly data p 12
32
Seasonal Model Initialization
S(5) 0.60 S(6) 1.00 S(7) 1.55 S(8)
0.85 L(8) 26.5
33
Seasonal Model Forecasting
  • g 0.3
  • 0.4

34
Seasonal Model Forecasting
35
Forecasting Winters Model for Data with Trend
and Seasonal Components
L(t) aA(t) / S(t-p) (1- a)L(t-1)T(t-1)
T(t) b L(t) - L(t-1) (1- b) T(t-1)
S(t) g A(t) / L(t) (1- g) S(t-p)
F(t1) L(t) T(t) S(t1-p)
36
Seasonal-Trend Model Decomposition
  • To initialize Winters Model, we will use
    Decomposition Forecasting, which itself can be
    used to make forecasts.

37
Decomposition Forecasting
  • There are two ways to decompose forecast data
    with trend and seasonal components
  • Use regression to get the trend, use the trend
    line to get seasonal factors
  • Use averaging to get seasonal factors,
    de-seasonalize the data, then use regression to
    get the trend.

38
Decomposition Forecasting
  • The following data contains trend and seasonal
    components

39
Decomposition Forecasting
  • The seasonal factors are obtained by the same
    method used for the Seasonal Model forecast

Average to 1
40
Decomposition Forecasting
  • With the seasonal factors, the data can be
    de-seasonalized by dividing the data by the
    seasonal factors

Regression on the De-seasonalized data will give
the trend
41
Decomposition Forecasting Regression Results
42
Decomposition Forecast
  • Regression on the de-seasonalized data produces
    the following results
  • Slope (m) 7.71
  • Intercept (b) 101.2
  • Forecasts can be performed using the following
    equation
  • mx b(seasonal factor)

43
Decomposition Forecasting
44
Winters Model Initialization
  • We can use the decomposition forecast to define
    the following Winters Model parameters

L(n) b m (n) T(n) m S(j) S(j-p)
So from our previous model, we have
L(8) 101.2 8 (7.71) 162.88 T(8) 7.71 S(5)
0.80 S(6) 1.35 S(7) 1.05 S(8) 0.79
45
Winters Model Example
46
Winters Model Example
47
Evaluating Forecasts
Trust, but Verify Ronald W. Reagan
  • Computer software gives us the ability to mess up
    more data on a greater scale more efficiently
  • While software like SAP can automatically select
    models and model parameters for a set of data,
    and usually does so correctly, when the data is
    important, a human should review the model
    results
  • One of the best tools is the human eye

48
Visual Review
  • How would you evaluate this forecast?

49
Forecast Evaluation
Where Forecast is Evaluated
Do not include initialization data in evaluation
ExPost Forecast
Initialization
Forecast
50
Errors
All error measures compare the forecast model to
the actual data for the ExPost Forecast region
51
Errors Measure
All error measures are based on the comparison of
forecast values to actual values in the ExPost
Forecast regiondo not include data from
initialization.
52
Bias and MAD
53
Bias and MAD
  • Bias tells us whether we have a tendency to over-
    or under-forecast. If our forecasts are in the
    middle of the data, then the errors should be
    equally positive and negative, and should sum to
    0.
  • MAD (Mean Absolute Deviation) is the average
    error, ignoring whether the error is positive or
    negative.
  • Errors are bad, and the closer to zero an error
    is, the better the forecast is likely to be.
  • Error measures tell how well the method worked in
    the ExPost forecast region. How well the
    forecast will work in the future is uncertain.

54
Absolute vs. Relative Measures
  • Forecasts were made for two sets of data. Which
    forecast was better?

Data Set 1 Bias 18.72 MAD 43.99
Data Set 2 Bias 182 MAD 912.5
55
MPE and MAPE
  • When the numbers in a data set are larger in
    magnitude, then the error measures are likely to
    be large as well, even though the fit might not
    be as good.
  • Mean Percentage Error (MPE) and Mean Absolute
    Percentage Error (MAPE) are relative forms of the
    Bias and MAD, respectively.
  • MPE and MAPE can be used to compare forecasts for
    different sets of data.

56
MPE and MAPE
  • Mean Percentage Error (MPE)
  • Mean Absolute Percentage Error (MAPE)

57
MPE and MAPE
Data Set 1
58
MPE and MAPE
Data Set 2
59
MPE and MAPE
Data Set 2
Data Set 1
60
Tracking Signal
  • Whats happened in this situation? How could we
    detect this in an automatic forecasting
    environment?

61
Tracking Signal
  • The tracking signal can be calculated after each
    actual sales value is recorded. The tracking
    signal is calculated as
  • The tracking signal is a relative measure, like
    MPE and MAPE, so it can be compared to a set
    value (typically 4 or 5) to identify when
    forecasting parameters and/or models need to be
    changed.

62
Tracking Signal
63
Tracking Signal
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