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Chapter 4: Seasonal Series: Forecasting and
Decomposition
4.1 Components of a Time Series
4.2 Forecasting Purely Seasonal Series
4.3 Forecasting Using a Seasonal Decomposition
4.4 Pure Decomposition
4.5 The Census X-12 Decomposition
4.6 The Holt-Winters Seasonal Smoothing Methods
4.7 The Multiplicative Holt-Winters Method
4.8 Weekly Data
4.9 Prediction Intervals
4.10 Principles
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4.1: Components of a Time Series
• A time series is said to have a seasonal component if it
displays a recurrent pattern with a fixed and known
duration [e.g. months of the year, days of the week].
• A time series is said to have a cyclical component if it
displays somewhat regular fluctuations about the trend
but those fluctuations have a periodicity of variable and
unknown duration [e.g. a business cycle].
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4.1: Components of a Time Series
• Define T = Trend, S = Seasonal, E = Error
• Additive model: Y  T  S  E
• Multiplicative model: Y  TSE
• Mixed additive-multiplicative model: Y  TS  E
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4.1: Components of a Time Series
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4.2: Forecasting Purely Seasonal Series
• Purely seasonal series usually occur only because the series
has been decomposed into separate trend and seasonal
components.
• Use an update from the previous seasonal value (e.g. July to
July or Monday to Monday).
• If m denotes the number of seasons [m=12 for monthly, etc.],
the forecasts function is:
Ft m (m)  Ft (m)   (Yt  Ft (m))
• Essentially similar to SES, save that we update one sub-series
m time periods; also we use a common smoothing parameter.
• Note that m starting values are required.
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Figure 4.2: Monthly Temperatures in Boulder
Monthly Average Temperatures in Boulder , CO, 1991 - 2007
80
Temperature
70
60
50
40
30
20
Month Jan
Year 1991
Jan
1993
Jan
1995
Jan
1997
Jan
1999
Jan
2001
Jan
2003
Jan
2005
Jan
2007
Source: Earth System Research Laboratory, Physical Sciences Division, National Oceanic and Atmospheric
Administration (www.cdc.noaa.gov/Boulder/Boulder.mm.html). Data shown is from file Boulder.xlsx.
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Figure 4.3: Plot of Seasonal Patterns for Boulder Data
Scatterplot of Temperature against Month, by Year
Data for Boulder, CO, 1991 - 2007
80
Year
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
Temperature
70
60
50
40
30
20
0
©Cengage
2
4
6
Month
8
10
12
Learning 2013.
8
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Table 4.2: Summary Results for
Boulder Temperature Data
• Optimal γ=0.177
• Also consider γ=0 [long-run average] and γ=1 [last
year]
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4.3: Forecasting Using a Seasonal Decomposition
• The basic steps in the process are:
1. Generate estimates of the trend by averaging out the
seasonal component.
2. Estimate the seasonal component.
3. Create a deseasonalized series.
4. Forecast the trend and the seasonal pattern separately.
5. Recombine the trend forecast with the seasonal
component to produce a forecast for the original series.
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4.3: Forecasting Using a Seasonal Decomposition
• Pure Decomposition - use all n data points to
estimate the fitted value at each time t (two-sided
decomposition)
• Forecasting Decomposition - use only the data up to
and including time t to predict value at time t+1 (onesided decomposition)
Q: Under what circumstances is one approach preferable to the other?
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4.3: Forecasting Using a Seasonal Decomposition
Moving Average
• A (simple) moving average of order K, denoted by
MA(K), is the average of K successive terms in a
time series, taken over successive sets of K, so
that the first average is (Y1  Y2  ...  YK ) / K , the
second is (Y2  Y3  ...  YK 1 ) / K , and so on.
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4.3: Forecasting Using a Seasonal Decomposition
Centered Moving Average
• A centered moving average (CMA) of order K, denoted by CM(t|K),
is defined by taking the average of successive pairs of simple
moving averages. Thus, denote the first average of K terms by
M (t  1| K )  (Yt 1  Yt 2 
and the second one by
M (t | K )  (Yt  Yt 1 
 Yt  K ) / K
 Yt  K 1 ) / K
then define the centered MA as
CM (t | K )  [ M (t  1| K )  M (t | K )] / 2.
•
When K is odd, it is usually not necessary to calculate a CMA.
However, when K is even, K=2J say, the first average corresponds
to an “average time” of t-J-0.5 and the second to t-J+0.5 so the
centered MA corresponds to time t-J as desired.
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Table 4.3: Forecasting Using a Seasonal Decomposition
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Table 4.3: Forecasting Using a Seasonal Decomposition
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4.4: Pure Decomposition
1. Calculate the 4-term MA and then the centered MA. The first
CMA term corresponds to period 3 and the last one to period
(t-2); K = 4 so we “lose” two values at each end of the series.
2. Divide [subtract] observations 3, …, (t-2) by [from] their
corresponding CMA to obtain a detrended series.
3. Calculate the average value (across years) of the detrended
series for each quarter j (j = 1, 2, 3, 4) to produce the initial
seasonal factors.
4. Standardize the seasonal factors by computing their average
and then setting the final seasonal factor equal to the initial
value divided by [minus] the overall average.
5. Estimate the error term by dividing the detrended series by the
final seasonal factor [subtracting the final seasonal factor from
the detrended series].
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4.4: Pure Decomposition
Table 4.4:
Multiplicative
Decomposition
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4.5: The Census X-12 Decomposition
Brief outline of steps:
1. Identify anomalous observations and make any
necessary adjustments.
2. Use a (usually multiplicative) decomposition to estimate
preliminary seasonal factors.
3. Apply an initial set of seasonal adjustments using [more
complex] moving averages.
4. Extend the series using ARIMA so that moving averages
can be calculated up to the end of the series.
5. Apply a final round of detrending and then estimate the
components.
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4.6: The Holt-Winters Seasonal Smoothing Methods
• The forecast function includes both local trend and
seasonal components. The additive form is:
F (h)  level (t )  h * trend (t )  seasonal (t  m  h)
t h
 Lt  hTt  S
t mh
o where
 Lt is local estimate of level at time t.
 Tt is local estimate of trend at time t.
 St m h is the local estimate of the seasonal effect
[from same ‘month’ last year].
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4.6.1: The Additive Holt-Winters Method
• When a new observation is recorded, the complete set
of updating equations takes the form:
o Observed error:
et  Yt  Lt 1  Tt 1  St m
o 1-step ahead forecast:
Ft 1 1  Ft 1  Lt  Tt  St m1
o Updating relationships:
Lt  Lt 1  Tt 1   [Yt  Lt 1  Tt 1  St m ]
Tt  Tt 1   [ Lt  Lt 1  Tt 1 ]
St  St m   [Yt  Lt 1  Tt 1  St m ]
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4.6.1: The Additive Holt-Winters Method
• The updating equations may be expressed in error correction
form:
Lt  L  T   et
t 1 t 1
Tt  T   et
t 1
St  St m   et
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4.6.1: The Additive Holt-Winters Method
A variety of special cases exists:
o Fixed seasonal pattern: γ = 0 (no seasonal updating)
o No seasonal pattern: γ = 0 and all initial S values are set
equal to zero
o Fixed trend: β = 0
o Zero trend: β = 0 and T0 = 0
o All fixed components: α = β = γ = 0 [leading to the regression
model in Section 9.1.1]
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4.6: The Holt-Winters Seasonal Smoothing Methods
•Starting Values
o Can use the Method of Least Squares to estimate {, ,
} BUT
o Need m+2 starting values [=14 for monthly data];
problematic for short series, so various heuristic methods
are used for the starting values.
Q: Suppose you has three years of monthly data.
How would you specify starting values for the level,
trend and seasonal components?
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Example 4.4 [Table 4.5]
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Example 4.5: U.S. Auto Sales
Original series
Components
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4.6: The Holt-Winters Seasonal Smoothing Methods
Q: Which method would you use?
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4.7: The Multiplicative Holt-Winters Method
• The forecast function includes the local trend and
seasonal components as before, but the seasonal effect
is now multiplicative:
Ft h (h)  [level (t )  h * slope(t )]* seasonal (t  m  h)
 [ Lt  hTt ]St mh
• The error correction updating relationships are:
Lt  Lt 1  Tt 1   (et / St  m )
Tt  Tt 1   (et / St m )
St  St m   (et / ( Lt 1  Tt 1 ))
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4.8: Weekly Data
• Often working with one or
two years of data, yet need
52 “seasonal values”.
• Proceed by estimating
common (multiplicative
seasonal values for similar
products.
• Deseasonalize the series
and apply SES to each
series.
Plot suggests similar multiplicative seasonal factors.
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4.9: Prediction Intervals
• As before, we can define a prediction interval using the
normal distribution as:
Forecast  z 2 *( RMSE )
Q: These intervals are very wide. Why?
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4.10: Principles
• Examine carefully the structure of the seasonal pattern.
• When there are limited data from which to calculate
seasonal components, consider using the seasonals
from elated data series, such as the aggregate.
Alternatively, average the estimates across similar
products.
30
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Take-Aways
• Seasonal series need to be decomposed into trend and
seasonal components.
• The type of decomposition used depends upon whether
we are interested in pure forecasting or smoothing.
• The seasonal component may have either an additive or
a multiplicative interaction with the trend.
• Short series mean that special methods involving the
assumption of common seasonal patterns must be used.
31
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