Post on 11-Jan-2016
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
TES
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Data
SLR w SI
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
0
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025
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0M
ore
Forecast Error
Fre
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Approximate with Normal Distribution
“Standard Error” of the forecast errors
Errors
02468
101214161820
Forecast Error
Freq
uenc
y
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
-
5 ,0 0 0
1 0 ,0 0 0
1 5 ,0 0 0
2 0 ,0 0 0
2 5 ,0 0 0
3 0 ,0 0 0
3 5 ,0 0 0
4 0 ,0 0 0
1 9 9 2 - 2 0 0 4
Sa
les
in $
mill
ion
s
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
1 5 ,0 0 0
2 0 ,0 0 0
2 5 ,0 0 0
3 0 ,0 0 0
3 5 ,0 0 0
4 0 ,0 0 0
JAN
FE
BM
AR
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RM
AY
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EP
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NO
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2 0 0 1 2 0 0 2 2 0 0 3 2 0 0 4
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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Number of calls / month
Weekly Seasonality
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Avg. # of calls / hr., 2004