MGTSC 352Lecture 5: Forecasting

Choosing LS, TS, and SS

SLR w SI = Simple Linear Regression with Seasonality Indices

Range estimates

Choosing Weights

• Find the values for LS, TS and SS that minimize* some performance measure.

* Exception?

• Two methods:– Table – If you want to use more than one

performance measure– Solver – If you want to ‘optimize’ against one

performance measure only

What’s This Solver Thing?

• In Excel: Tools Solver, to bring up:Optimize something (maximize profit, minimize cost, etc.)

By varying some decision variables (“changing cells”)

Keeping in mind any restrictions (“constraints”) on the decision variables

Using Solver to Choose LS, TS, SS

• What to optimize: minimize SE– Could minimize MAD or MAPE, but solver

works more reliably with SE• For the geeks: because SE is a smooth function

• Decision variables: LS, TS, SS

• Constraints:LS

TS

SS

≤≤Something a bit bigger than zero

(f. ex.: 0.01, 0.05)

Something a bit smaller than one

(f. ex.: 0.99, 0.95)

Let’s try it out …

Pg. 33

Why Solver Doesn’t Always Give the Same Solution

Everywhere I look is uphill! I must have reached the lowest

point.

local optimum

global optimum

SLR w SI = Simple Linear Regression with Seasonality Indices

• Captures level, trend, seasonality, like TES

• Details are different• SLR Forecast

– Ft+k = (intercept + [(t + k) slope]) SI

Excel

Pg. 34

TES vs SLRwSI

• TES

Ft+k = (Lt + k Tt) St+k-p

• SLRwSI

Ft+k = (intercept + (t + k) slope) SI

additive trend multiplicative seasonality

TES vs SLRwSI

• Both estimate Level, Trend, Seasonality

• Data points are weighted differently

– TES: weights decline as data age

– SLR w SI: same weight for all points

• TES adapts, SLR w SI does not

Which Method Would Work Well for This Data?

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Data

Patterns in the Data?

• Trend:– Yes, but it is not constant– Zero, then positive, then zero again

• Seasonality?– Yes, cycle of length four

Comparison

• TES: SE = 24.7

• TES trend is adaptive

• SLRwSI: SE = 32.6

• SLR uses constant trend

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Data

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How Good are the Forecasts?

• TES (optimized): Year 5, Quarter 1 sales = 1458.67– Are you willing to bet on it?

• Forecasts are always wrong– How wrong will it be?

• Put limits around a “point forecast”– “Prediction interval”– 95%* sure sales will be between low and high– How do we compute low and high?* (give or take)

Pg. 38

Forecast Error Distribution

Errors

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Approximate with Normal Distribution

“Standard Error” of the forecast errors

Errors

02468

101214161820

Forecast Error

Freq

uenc

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Average Error = .3

Standard Error = 127

95% Prediction Interval

• 1-step Point forecast + bias 2 StdError

• 9 Jan TSX = 12654 + .3 2 127= 12654 254=[12400, 12908]=[low, high]

• Actual 12,467.99

Are TES and SLR w SI it?

• Certainly not– Additive seasonality models

• TES’ or SLR w SD

– Multiplicative trend models• TES’’ or Nonlinear Regression (Dt+1 = 1.1Dt)

Steps in a Forecasting Project

-1: Collect data0: Plot the data (helps detect patterns)1: Decide which models to use

– level – SA, SMA, WMA, ES– level + trend – SLR, DES– level + trend + seas. – TES, SLR w SI, ...

2: Use models3: Compare and select (one or more)4: Generate forecast and range (prediction interval)

More on selection

Pg. 39

How to select a model?

• Look at performance measures– BIAS, MAD, MAPE, MSE

• Use holdout strategy• Example: 4 years of data• Use first 3 years to fit model(s)• Forecast for Year 4 and check the fit(s)• Select model(s)• Refit model(s) adding Year 4 data

• If you have more than one good model...

COMBINE FORECASTS

Pg. 41

Appropriate model...

linearNonlinear (ex. power)

S-curve (ex. any CDF)

DATAB u ild in g M a t e r ia l , G a r d e n E q u ip m e n t a n d S u p p ly D e a le r s

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in $

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TES vs. SLR w/ SI

Which method would you choose?

BIAS 127 BIAS 6MAD 628 MAD 713

MAPE 2.86% MAPE 3.32%MSE 711,039 MSE 1,002,189

TES SLR w/ SI

Holdout Strategy

1. Ignore part of the data (the “holdout data”)

2.Build models using the rest of the data

3.Optimize parameters

4.Forecast for the holdout data

5.Calculate perf. measures for holdout data

6.Choose model that performs best on holdout data

7.Refit parameters of best model, using all data

TES vs. SLR w/ SI…in holdout period

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S a le s T E S S L R w / S I

holdoutperiod

TES vs. SLR w/ SI…in holdout period

Now which method would you choose?

BIAS 1,025 BIAS 2,995MAD 1,319 MAD 2,995

MAPE 4.29% MAPE 9.41%MSE 2,530,775 MSE 11,566,373

TES SLR w/ SI

Calgary EMS Data

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Trend?

Seasonality?

Number of calls / month

Checking for (Yearly) Seasonality

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Weekly Seasonality

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