Forecasting

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Chapter 2

• Forecasting

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Introduction to Forecasting

• What is forecasting?
• Primary Function is to Predict the Future
• Why are we interested?
• Affects the decisions we make today
• Examples who uses forecasting in their jobs?
• forecast demand for products and services
• forecast availability of manpower for
  manufacturing or services.
• forecast inventory and materiel needs daily for
  mfg or services

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Characteristics of Forecasts

• They are usually wrong!
• A good forecast is more than a single number
• mean and standard deviation
• range (high and low)
• Aggregate forecasts are usually more accurate
• Accuracy erodes as we go further into the future.
  
• Forecasts should not be used to the exclusion of
  known information

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What Makes a Good Forecast

• It should be timely
• It should be as accurate as possible
• It should be reliable
• It should be in meaningful units
• It should be presented in writing
• The method should be easy to use and understand
  in most cases.

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Forecast Horizons in Operation Planning
Figure 2.1
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Subjective Forecasting Methods

• Sales Force Composites
• Aggregation of sales personnel estimates
• Customer Surveys
• Jury of Executive Opinion
• The Delphi Method
• Individual opinions are compiled and
  reconsidered. Repeat until and overall group
  consensus is (hopefully) reached.

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Judgmental Forecasts

• There may not be enough time to gather data and
  analyze quantitative data or no data at all.
• Expert Judgment managers(marketing,operations,fi
  nance,etc.)
• Be careful about who you call an expert
• Sales force composite
• Recent experience may influence their perceptions
• Consumer surveys
• Requires considerable amount of knowledge and
  skill
• Opinions of managers and staff
• Delphi method a series of questionnaire,
  responses are kept anonymous, new questionnaires
  are developed based on earlier results Rand
  corporation (1948)

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Objective Forecasting Methods

• Two primary methods causal models and time
  series methods
• Causal Models
• Let Y be the quantity to be forecasted and
  (X1, X2, . . . , Xn) be n variables that have
  predictive power for Y.
• A causal model is Y f (X1, X2, . . . , Xn).
• A typical relationship is a linear one. That
  is,
• Y a0 a1X1 . . . an Xn.

What might be such variables for average income
for Turkey for 2007?
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Time Series Methods

• A time series is just collection of past values
  of the variable being predicted. Also known as
  naïve methods. Goal is to isolate patterns in
  past data. (See Figures on following pages)
• Trend
• Seasonality
• Cycles
• Randomness

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Time Series Model Building

• A time-series is a time ordered sequence of
  observations taken at regular intervals over a
  period of time.
• The data may be demand, earnings, profit,
  accidents, consumer price index,etc.
• The assumption is future values of the series can
  be estimated from past values
• One need to identify the underlying behavior of
  the series - pattern of the data

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Some Behaviors Typically Observed

• Trend
• E.g., population shifts, change in income.
  Usually a long-term movement in data
• Seasonality
• Fairly regular variations, e.g., Friday nights in
  restaurants, new year in shopping malls, rush
  hour traffic., etc.
• Cycles
• Wavelike variations lasting more than a year,
  e.g. economic recessions, etc.
• Irregular variations
• Caused by unusual circumstances, e.g., strikes,
  weather conditions, etc.
• Random variations
• Residual variations after all other behaviors are
  accounted for. Caused by chance

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Forecast Variations
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Types of Time Series Models

• We will cover the following techniques in this
  section
• Naïve
• Techniques for averaging
• Moving average
• Weighted moving average
• Exponential smoothing
• Techniques for trend
• Linear equations
• Trend adjusted exponential smoothing
• Techniques for seasonality
• Techniques for Cycles

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Notation Conventions

• Let D1, D2, . . . Dn, . . . be the past values of
  the series to be predicted (demand). If we are
  making a forecast in period t, assume we have
  observed Dt , Dt-1 etc.
• Let Ft, t t forecast made in period t for the
  demand in period t t where t 1, 2, 3,
• Then Ft -1, t is the forecast made in t-1 for t
  and Ft, t1 is the forecast made in t for
  t1. (one step ahead) Use shorthand notation Ft
  Ft - 1, t .

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Evaluation of Forecasts

• The forecast error in period t, et, is the
  difference between the forecast for demand in
  period t and the actual value of demand in t.
• For a multiple step ahead forecast et Ft - t,
  t - Dt.
• For one step ahead forecast et Ft - Dt.
• e1, e2, .. , en forecast errors over n periods
• Mean Absolute Deviation MAD
  (1/n) S e i
• Mean Absolute Percentage Error MAPE
  (1/n) S e i /Di 100
• Mean Square Error MSE (1/n) S ei 2

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Measures of Forecast Accuracy

• Error - difference between actual value and
  predicted value
• Mean absolute deviation (MAD)
• Average absolute error
• Mean squared error (MSE)
• Average of squared error
• Tracking signal
• Ratio of cumulative error and MAD

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MAD MSE
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Tracking Signal
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Biases in Forecasts

• A bias occurs when the average value of a
  forecast error tends to be positive or negative
  (ie, deviation from truth).
• Mathematically an unbiased forecast is one in
  which E (e i ) 0. See Figure 2.3 (next slide).
• S e i 0

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Forecast Errors Over Time Figure
2.3
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Ex. 2.1
week F1 D1 E1 E1/D1 F2 D2 E2 E2/D2
1 92 88 4 0.0455 96 91 5 0.0549
2 87 88 1 0.0114 89 89 0 0.0000
3 95 97 2 0.0206 92 90 2 0.0222
4 90 83 7 0.0843 93 90 3 0.0333
5 88 91 3 0.0330 90 86 4 0.0465
6 93 93 0 0.0000 85 89 4 0.0449
Which forecast is a better forecast? MAD, MAPE
and MSE
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• MAD1 17/6 2.83 better
• MAD2 18/6 3.00
• MSE1 79/6 13.17
• MSE2 70/6 11.67 better
• MAPE1 0.0325 better
• MAPE2 0.0333

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Forecasting for Stationary Series

• A stationary time series has the form
• Dt m e t where m is a constant (mean of the
  series) and e t is a random variable with mean 0
  and var s2 .
• Two common methods for forecasting stationary
  series are moving averages and exponential
  smoothing.

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Moving Averages

• In words the arithmetic average of the N most
  recent observations. For a one-step-ahead
  forecast
• Ft (1/N) (Dt - 1 Dt - 2 . . . Dt - N )

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Summary of Moving Averages

• Advantages of Moving Average Method
• Easily understood
• Easily computed
• Provides stable forecasts
• Disadvantages of Moving Average Method
• Requires saving all past N data points
• Lags behind a trend
• Ignores complex relationships in data

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Moving Average Lags a Trend
Figure 2.4
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Exponential Smoothing Method

• A type of weighted moving average that applies
  declining weights to past data.
• Based on the idea More recent data is more
  relevant
• 1. New Forecast a (most recent observation)
• (1 - a) (last forecast)
• or
• 2. New Forecast last forecast - a (last
  forecast error)
• where 0 lt a lt 1 and generally is small for
  stability of forecasts ( around .1 to .2)

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Exponential Smoothing (cont.)

• In symbols
• Ft1 a Dt (1 - a ) Ft
• a Dt (1 - a ) (a Dt-1 (1 - a ) Ft-1)
• a Dt (1 - a )(a )Dt-1 (1 - a)2 (a )Dt - 2
  . . .
• Hence the method applies a set of
  exponentially declining weights to past data. It
  is easy to show that the sum of the weights is
  exactly one.
• (Or Ft 1 Ft - a (Ft -
  Dt) )

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Weights in Exponential Smoothing Fig. 2-5
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Comparison of ES and MA

• Similarities
• Both methods are appropriate for stationary
  series
• Both methods depend on a single parameter
• Both methods lag behind a trend
• One can achieve the same distribution of forecast
  error by setting a 2/ ( N 1).
• Differences
• ES carries all past history. MA eliminates bad
  data after N periods
• MA requires all N past data points while ES only
  requires last forecast and last observation.

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Exponential Smoothing for different values of
alpha
So how does alpha effect forecast?
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Example of Exponential Smoothing
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Picking a Smoothing Constant
Lower values of ??are preferred when the
underlying trend is stable and higher values of
??are preferred when it is susceptible to change.
Note that if ??is low your next forecast highly
depends on your previous ones and feedback is
less effective.
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Using Regression for Times Series Forecasting

• Regression Methods Can be Used When Trend is
  Present.
• Model Dt a bt.
• If t is scaled to 1, 2, 3, . . . , then the least
  squares estimates for a and b can be computed as
  follows
• Set Sxx n2 (n1)(2n1)/6 - n(n1)/22
• Set Sxy n S i Di - n(n 1)/2 S Di
  
• _
  
• Let b Sxy / Sxx and a D - b (n1)/2
• These values of a and b provide the best fit
  of the data in a least squares sense.

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An Example of a Regression Line
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Linear Trend Equation - Notation
A linear trend equation has the form Yt a
bt

• b is similar to the slope. However, since it is
  calculated with the variability of the data in
  mind, its formulation is not as straight-forward
  as our usual notion of slope.

yt Forecast for period t, a value of yt at t0
and b is the slope of the line.
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Insights For Calculating a and b

• Suppose that you think that there is a linear
  relation between the height (ft.) and weight
  (pounds) of humans. You collected data and want
  to fit a linear line to this data.
• Weight a b Height
• How do you estimate a and b?

For further information refer to http//www.stat.p
su.edu/bart/0515.doc or any statistics book!
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More Insights For Calculating a and b

• Demand observed for the past 11 weeks are given.
• We want to fit a linear line (DabT) and
  determine a and b that minimizes the sum of the
  squared deviations. (Why squared?)

A little bit calculus, take the partial
derivatives and set it equal to 0 and solve for a
and b!
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Linear Trend Equation Example
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Linear Trend Calculation
If we fit a line to the observed sales of the
last five months,
Question is forecasting the sales for the 6th
period. What do you think it will be?
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Linear Trend Calculation
812
-
6.3(15)
a




143.5

5
y 143.5 6.3t
y 143.5 6.36 181.5
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Other Methods When Trend is Present
Double exponential smoothing, of which Holts
method is only one example, can also be used to
forecast when there is a linear trend present in
the data. The method requires separate smoothing
constants for slope and intercept.
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Trends Adjusted Exponential Smoothing

• A variation of simple Exponential Smoothing can
  be used when trend is observed in historical
  data.
• It is also referred as double smoothing.
• Note that if a series has a trend and simple
  smoothing is used the forecasts will all lag the
  trend. If data are increasing each forecast will
  be low! When trend exists we may improve the
  model by adjusting for this trend. (C.C. Holt)
• Trend Adjusted Forecasts (TAF) is composed of two
  elements a smoothed error and a trend factor
• TAFt1 St Tt where
• St smoothed forecast TAFt ?(At TAFt)
• Tt current trend estimate Tt-1 b(TAFt
  TAFt-1 Tt-1)

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Insights TAES

• TAFt1 St Tt where
• St smoothed forecast TAFt ?(At TAFt)
• Tt current trend estimate Tt-1 b(TAFt TAFt-1
  Tt-1) (1-b) Tt-1 b(TAFt TAFt-1 )
  Weighted average of last trend and last forecast
  error.
• ? and b are smoothing constants to be selected
  by the modeler.
• St is same with original ES feedback for the
  forecast error is added to previous forecast with
  a percentage of ?
• If there is trend ES will have a lag. We must
  also include this lag to our model. Hence Tt is
  added where
• Tt is the trend and updated each period.

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Forecasting For Seasonal Series

• Seasonality corresponds to a pattern in the data
  that repeats at regular intervals. (See figure
  next slide)
• Multiplicative seasonal factors c1 , c2 , . . .
  , cN where i 1 is first period of season, i 2
  is second period of the season, etc..
• S ci N.
• ci 1.25 implies 25 higher than the
  baseline on avg.
• ci 0.75 implies 25 lower than the
  baseline on avg.

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A Seasonal Demand Series
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Quick and Dirty Method of Estimating Seasonal
Factors

• Compute the sample mean of the entire data set
  (should be at least several seasons of data).
• Divide each observation by the sample mean. (This
  gives a factor for each observation.)
• Average the factors for like periods in a season.
• The resulting N numbers will exactly add to N and
  correspond to the N seasonal factors.

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Deseasonalizing a Series

• To remove seasonality from a series, simply
  divide each observation in the series by the
  appropriate seasonal factor. The resulting series
  will have no seasonality and may then be
  predicted using an appropriate method. Once a
  forecast is made on the deseasonalized series,
  one then multiplies that forecast by the
  appropriate seasonal factor to obtain a forecast
  for the original series.

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Seasonal series with increasing trend Fig 2-10
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Initialization for Winterss Method
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Practical Considerations

• Overly sophisticated forecasting methods can be
  problematic, especially for long term
  forecasting. (Refer to Figure on the next slide.)
• Tracking signals may be useful for indicating
  forecast bias.
• Box-Jenkins methods require substantial data
  history, use the correlation structure of the
  data, and can provide significantly improved
  forecasts under some circumstances.

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The Difficulty with Long-Term Forecasts
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Tracking the Mean When Lost Sales are
Present Fig. 2-13
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Tracking the Standard Deviation When Lost Sales
are Present Fig. 2-14
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Case Study Sport Obermeyer Saves Money Using
Sophisticated Forecasting Methods

• Problem Company had to commit at least half of
  production based on forecasts, which were often
  very wrong. Standard jury of executive opinion
  method of forecasting was replaced by a type of
  Delphi Method which could itself predict forecast
  accuracy by the dispersion in the forecasts
  received. Firm could commit early to items that
  had forecasts more likely to be accurate and hold
  off on items in which forecasts were probably
  off. Use of early information from retailers
  improved forecasting on difficult items.
• Consensus forecasting in this case was not the
  best method.