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Unformatted text preview: 540:453 Production Control Lecture 4: Forecasting (Ch. 2) Prof. T. Boucher 1 Winter’s Method: Exponential Smoothing
w/ Trend and Seasonality • Basic Data Pattern 2 Winter’s Method: Exponential Smoothing
w/ Trend and Seasonality
• Data Generating Process
X t ,t (at bt )C t t • The Model , let N be the number of seasons
ˆ
ˆˆˆ
(a b )C
F
X
t ,t t ,t t t N • Steps ˆ
– Initialize the model parameters: at bt
ˆ ˆ Ct
– Forecast
– Revise
ˆˆˆ
a b C based on new data
t t t 3 Winter’s Method: Procedure
• Obtain initial estimates of the slope, and the
seasonal factors (a minimum of two seasons of
data)
• Assume we have two seasons of data, each with
N data points
1. Calculate the sample means (V1 and V2) for the two
separate seasons
2. Define the initial trend b0 = (V2 – V1)/N
3. Estimate the initial interception a0 = V2 + b0[(N1)/2]
4. Calculate the initial seasonal factors 4 Example
• quarterly sales data • Initialization
S1 30 48 60 35
4 43.25 54.5 43.25
4 ˆ
b
ˆ
a8 S2 S2 ˆ n 1)
b(
2 42 58 74 44
4 2 .8 3
54.5 2.8( )
2 58.7 54.5 Example
• quarterly sales data • Initialization Ct Xt
ˆ
a0 ˆ
bt Forecasting from period 8 ˆ
X 8, 9 [58.7 2.8(1)]0.8 49.4 ˆ
X 8,10 [58.7 2.8(2)]1.12 72.0 ˆ
X 8,11 [58.7 2.8(3)]1.33 89.2 ˆ
X 8,12 [58.7 2.8(4)]0.75 52.4 7 Updating Model Parameters
• Assume the following forecast and actual for period t = 9 ˆ
X 8, 9 [58.7 2.8(1)]0.8 X9 49.4 48.0 • Update at by smoothing two adjustments to the new level
ˆ
at X
ˆ
[ t ] (1 a)[at
Ct N 1 ˆ
bt 1 ] ˆ
a9 0.2[ 48 (0.8)[58.7 2.8]
0 .8 61.2 • Update Ct by smoothing two adjustments to the new level Xt
[ ] (1
ˆ
at ˆ
Ct ˆ
)Ct ˆ
C1 N 0.2[ 48 (0.8)(0.8)
61.2 0.8 • Update bt by smoothing two adjustments to the new level
ˆ
bt ˆ
[at ˆ
at 1 ] (1 ˆ
)bt 1 ˆ
b9 0.2[61.2 58.7] (0.8)2.8 2.74 8 Updating the forecast for period 10
• Prior Forecast ˆ
X 8,10 [58.7 2.8(2)]1.12 72.0 • Updated Forecast ˆ
X 9,10 [61.2 2.74(1)]1.12 71.6 Confidence Interval of Forecast 10 Time Series Models and Tracking Signals
• • Because time series models are not causal, but depend on the
same variable moving through time, it is possible that the model
parameters will change.
A tracking signal is used to monitor the performance of the model
and will signal when the model should be reevaluated. Tt t
t
i0 Linear Regression
• Used in causal models in which there is a structural
relationship between variables
– amount of advertising and subsequent sales
– amount of fertilizer used per acre and crop yield • Fundamentally different for time series in which there
is no cause and effect relationship necessary Linear Regression
• Data Generating Process: Yi a bX i
e ~ NID( ei
0, e ) • Model: need to estimate the parameters a and b Yi a bX i Linear Regression
• If we have data with paired observations of X and Y, we can
empirically derive the parameters of the model by finding the
linear function that minimizes the sum of squared errors. ei
L n ei 2 i1 2 n
i1 (Yi
(Yi ˆ
Yi ) 2 ˆ
Yi ) 2
n (Yi ˆˆ
a bX i ) 2 i1 • Transform the data set and the model by moving the reference
point to
and
Y
X L n
i1 {(Yi ˆ
Y ) b( X i X )}2 Linear Regression
• The regression line goes through
the mean of X and Y. Yi ˆ
a
L n ei n 2 i1 L i1 n
i1 {(Yi (Yi ˆ
Yi ) n 2 ˆ
b( X i Y Y
(Yi ˆˆ
a bX i ) 2 i1 ˆ
Y ) b( X i ˆ
bX X )}2 X) Linear Regression
n L ˆ
Y ) b( X i {(Yi X )}2 i1 • Taking partial derivatives: L
Y n 2 [Yi Y ˆ
b( X i X )] 0 i1 L
ˆ
b 2 n [Yi Y ˆ
b( X i X )]( X i X) 0 i1 • Which Yields:
n ˆ
b • Also: Yi ( X i X) i1
n (X i
i1 X) 2 ˆ
a Y ˆ
bX Linear Regression
• Example: X 400 Y 60 • Model: • Computations:
n ˆ
b i1
n Yi ( X i
(X i X)
X )2 19000
280000 ˆ
Yi
0.068 32.8 0.068 X i
• Estimation: i1 ˆ
a Y ˆ
bX 60 0.068(400) 32.8 ˆ
Yi 32.8 0.068(500) 66.8 Linear Regression
Prediction Interval: it can be shown that the 95% confidence interval
around a forecast is given by s s2 2 1
n2 1
(35.72)
5 [Yi ˆ
Yi ] 7.144 2 CI CI ˆ
Yk ˆ
Yk 1
t0.025 s
n 1
t 0.025 s
n
( X k X )2
ˆ
1; Y700
( X i X )2 X k2
X i2 1 2.571(2.67) 1
7 90000
1 8.31
280000 Linear Regression
• Coefficient of Multiple Determination, R2: • If you had no function, Y would be the “best guess” at any given value of X. ˆ
• b( X i X ) is the amount of variation “explained” by the function.
ˆ
[b( X i R 2 1 X )] 2 i (Yi
i Y) 2 0 R2 1 Linear Regression
• Example: ˆ
[b( X i
R2 1 X )] 2 i (Yi
i Y) 2 1 1294.72
1325 0.98 Quantitative Forecasting Applications
Small and Large Firms
Low
Sales
Technique
Moving average
Straight line projection
Naive
Exponential smoothing
Regression
Simulation
Classical decomposition
BoxJenkins
Number of Firms High
Sales < $100M > $500M 29.6%
14.8%
18.5%
14.8%
22.2%
3.7%
3.7%
3.7%
27 29.2%
14.6%
14.6%
20.8%
27.1%
10.4%
8.3%
6.3%
48 Source: Nada Sanders and Karl Mandrodt (1994) “Practitioners Continue to Rely on Judgmental Forecasting
Methods Instead of Quantitative Methods,” Interfaces, vol. 24, no. 2, pp. 92100. 21 Time Series Forecasting using Excel
• Excel can be used to develop forecasts:
– Moving average
– Exponential smoothing
– Linear trend line 1222 ...
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