Forecasting

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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

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Evaluating Forecasts

• Visual Review
• Errors
• Errors Measure
• MPE and MAPE
• Tracking Signal

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Forecasting Terminology
Historical Data
ExPost Forecast
Initialization
Forecast
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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.

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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?

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ForecastingSimple Moving Average Method

• Using a three period moving average, we would get
  the following forecast

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ForecastingSimple Moving Average Method

• What if we used a two period moving average?

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ForecastingSimple Moving Average Method

• The number of periods used in the moving average
  forecast affects the responsiveness of the
  forecasting method

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Forecasting Terminology

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

Model Evaluation
Initialization
ExPost Forecast
Forecast
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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)

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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?

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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)
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ForecastingExponential Smoothing

• Initialization

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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
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ForecastingExponential Smoothing
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ForecastingExponential Smoothing
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Outliers (eloping point)
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Data with Trends
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Data with Trends
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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.
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ForecastingSimple Linear Regression Model
In linear regression, the squared errors are
minimized
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ForecastingSimple Linear Regression Model
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Limitations in Linear Regression Model
As with the simple moving average model, all data
points count equally with simple linear
regression.
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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

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Holts Trend ModelInitialization
First, well initialize the model
L(4) 20.54(9.9)60.1 T(4) 9.9
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Holts Trend ModelUpdating

• 0.3
• b 0.4

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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
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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
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Holts Model Results
Initialization
ExPost Forecast
Forecast
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Holts Model Results
Initialization
ExPost Forecast
Forecast
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Forecasting Seasonal Model (No Trend)
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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
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Seasonal Model Initialization
S(5) 0.60 S(6) 1.00 S(7) 1.55 S(8)
0.85 L(8) 26.5
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Seasonal Model Forecasting

• g 0.3
• 0.4

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Seasonal Model Forecasting
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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)
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Seasonal-Trend Model Decomposition

• To initialize Winters Model, we will use
  Decomposition Forecasting, which itself can be
  used to make forecasts.

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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.

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Decomposition Forecasting

• The following data contains trend and seasonal
  components

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Decomposition Forecasting

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

Average to 1
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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
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Decomposition Forecasting Regression Results
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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)

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Decomposition Forecasting
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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
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Winters Model Example
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Winters Model Example
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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

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Visual Review

• How would you evaluate this forecast?

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Forecast Evaluation
Where Forecast is Evaluated
Do not include initialization data in evaluation
ExPost Forecast
Initialization
Forecast
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Errors
All error measures compare the forecast model to
the actual data for the ExPost Forecast region
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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.
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Bias and MAD
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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.

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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
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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.

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MPE and MAPE

• Mean Percentage Error (MPE)

• Mean Absolute Percentage Error (MAPE)

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MPE and MAPE
Data Set 1
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MPE and MAPE
Data Set 2
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MPE and MAPE
Data Set 2
Data Set 1
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Tracking Signal

• Whats happened in this situation? How could we
  detect this in an automatic forecasting
  environment?

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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.

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Tracking Signal
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Tracking Signal