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8/3/2019 Estimation of Seasonality Index 15OCT2011
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Time Series with Cyclical and Seasonal
Variations
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Seasonal Effect
Seasonal effect is defined as the repetitive and predictable
pattern data behaviour in a time-series around the trend line.
To measure the seasonal effect the time period must be less thanone year, such as, days, weeks, months or quarters.
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Seasonal effect
Description of seasonal effect provides better understanding of
the seasonal component.
Seasonal effect can be eliminated from the time-series. This
process is called deseasonalizing or seasonal adjusting.
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Seasonal Adjusting
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modeladditiveinadjustmentSeasonal
100ECTECT
ESCTeffectSeasonal
modeltivemultiplicainadjustmentSeasonal
ESS
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Seasonal Index
Method of simple averages
Ratio-to-moving average method
Ratio to Trend method
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Method of simple averages
Average the unadjusted data by months (or quarters, if quarterlydata is given).
Add the data for each month (or quarter) and calculate theaverage by diving the monthly (quarterly) totals by number of
years. Calculate the average of monthly averages.
Seasonal index for month i is the ratio of monthly average ofmonth i to the average of monthly averages times 100.
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Example: Use the method of simple averages to calculate seasonal index and findthe forecast for October 2005
Month 2006 2007 2008
Jan 15 23 25
Feb 16 22 25
Mar 18 28 35
Apr 18 27 36
May 23 31 36
June 23 28 30
July 20 22 30
Aug 28 28 34
Sep 29 32 38
Oct 33 37 47
Nov 33 34 41
Dec 38 44 53
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Month 2006 2007 2008 Monthly Total Monthly Average Percentage Average
of Monthly
Averages
Jan 15 23 25 6321 70
Feb 16 22 25 63
21 70Mar 18 28 35 81
27 90
Apr 18 27 36 8127 90
May 23 31 36 9030 100
June 23 28 30 81 27 90
July 20 22 30 7224 80
Aug 28 28 34 9030 100
Sep 29 32 38 9933 110
Oct 33 37 47 11739 130
Nov 33 34 41 10836 120
Dec 38 44 53 13545 150
90 30 1200
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Month 2006 2006
deseasonalized
data
2007 2007 deseasonalized
data
2008 2008
deseasonalized
data
Jan 1521.428571
2332.857143
2535.714286
Feb 16 22.857143 22 31.428571 25 35.714286
Mar 1820
2831.111111
3538.888889
Apr 1820
2730
3640
May 2323
3131
3636
June 2325.555556
2831.111111
3033.333333
July 2025
2227.5
3037.5
Aug 2828
2828
3434
Sep 2926.363636
3229.090909
3834.545455
Oct 3325.384615
3728.461538
4736.153846
Nov 3327.5
3428.333333
4134.166667
Dec 3825.333333
4429.333333
5335.333333
Deseasonalized data = actual data / seasonality index
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Trend
Trend is calculated using regression on
deseasonalized data.
Deseasonalized data is obtained by dividing theactual data with its seasonality index.
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Forecasting using method of averages in the presence of seasonality
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Forecast using Method of Averages
Let us assume that we want to forecast value for
2009 October ( t = 46).
Trend Component = 21.94 + 0.4352 x 46 = 41.97
Seasonality Index for October = 130
Forecasted value for October 2005 = 41.97 x 1.3 =
54.56
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Ratio to Moving Average Method
A moving average smoothes the data of their
variations.
In a multiplicative time series model, the ratio to
moving average results in Seasonal and random
error component.
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The ratio to moving average
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t EST
EST
MA
EST
Y
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Ratio to Moving Average - Steps
Compute a moving average (based on the number of seasons,
that is n is the equal to the number of seasons).
Center the moving averages by averaging every consecutive pair.
For each data point, divide the original series value by thecorresponding moving average and multiply by 100. This gives
ratio to moving average.
For each season average all data corresponding to the season.
This will result in seasonal index. The seasonal indexes are
adjusted so that the mean is 100.
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Forecasting using Ratio to moving
average
Approach 1: Use moving average values to
get the trend equation using regression.
Approach 2: Deseasonalize the data by
dividing the actual data with seasonality
index. Derive the trend equation usingdeseasonalized data (using regression)
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Year Quarter1 2 3 4
2005 75 60 54 59
2006 86 65 63 80
2007 90 72 66 85
2008 100 78 72 93
Forecast the value for Q3 2009
Ratio to Moving Average Example
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Year Quarter Value 4Q MA Centered MA Ratio to MA
Seasonality
Index
Deseasonalized
Data
2005 Q1 75
Q2 60
Q3 54 62 63.375 85.2071006 84.68 63.76949
Q4 59 64.75 65.375 90.248566 100.49 58.71231
2006 Q1 86 66 67.125 128.119181 122.34 70.2959Q2 65 68.25 70.875 91.7107584 92.47 70.29307
Q3 63 73.5 74 85.1351351 84.68 74.39773
Q4 80 74.5 75.375 106.135987 100.49 79.60991
2007 Q1 90 76.25 76.625 117.455139 122.34 73.56547
Q2 72 77 77.625 92.7536232 92.47 77.86309
Q3 66 78.25 79.5 83.0188679 84.68 77.94048
Q4 85 80.75 81.5 104.294479 100.49 84.58553
2008 Q1 100 82.25 83 120.481928 122.34 81.73941
Q2 78 83.75
Q3 72
Q4 93
Ratio to Moving Average
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Forecast for 2009 Q3
t = 19
Trend = 57.12 + 2.10 x 19 = 97.02
Forecast = 97.02 x 84.68 / 100 = 82.15
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Recommended Readings
Amir D Aczel and J Sounderpandian, Complete Business
Statistics, The McGraw Hill, 2009.