decomposition of time series
DESCRIPTION
This document describes how a time series can be decomposed into three parts - trend, seasonal component and noise. The ultimate goal is in forecasting future values of the series.TRANSCRIPT
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Slide 1
Decomposition of Time Series
Decomposition is the breaking down of a time series (Y) into Trend (T), Seasonal component (S) and Irregular component (I) so that
Yt = Tt St It Estimates of the trend and seasonal
component are multiplied together to produce a forecast of the original series
t t tY T S
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Slide 2
Trend
Represents long-term movement
Estimated using linear regression, independentvariable is the time t
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Slide 3
Trend : Linear
Suitable when there is no obvious curvature in the time series plot
0 1tT = b bt
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Slide 4
Trend : Quadratic
Suitable when the time series plot displays obvious curvature
20 1 2tT = b bt b t
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Slide 5
Trend : Exponential
Useful when time series plot a. rises at increasing rate
b. drops at decreasing rate
0 1 0 1 0ttT = b b b b ( , )
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Slide 6
Seasonal Component
The seasonal component is represented by a collection of seasonal indices (SI)
SI are usually extracted by the average-all-data(AAD) method
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Slide 7
SI : One Year Quarterly Data
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Slide 8
SI : Two Years Quarterly Data
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Slide 9
SI : Two Years Monthly Data
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Slide 10
SI : Interpretation
A seasonal index of 1.00 for a particular period means that the average of that period is equal to the annual mean
A seasonal index of 1.25 means that the average of that period is 25% higher than the annual mean
A seasonal index of 0.70 indicates that the average of that period is 30% lower than the annual mean
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Slide 11
Deseasonalized Data
Deseasonalizing (Y/S) means removing the seasonal component from the data
After deseasonalizing, only (T I) remains Also known as seasonally adjusted data
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Slide 12
Sales Data : Before & After Adjustedt Year Q Sales Sales/Average SI SeasonallyAdj.Sales1 1 1 232.7 0.8743 0.7903 294.52 2 309.2 1.1618 1.0111 305.83 3 310.7 1.1674 1.1129 279.24 4 293 1.1009 1.0857 269.95 2 1 205.1 0.7706 259.56 2 234.4 0.8807 231.87 3 285.4 1.0723 256.48 4 258.7 0.9720 238.39 3 1 193.2 0.7259 244.510 2 263.7 0.9908 260.811 3 292.5 1.0990 262.812 4 315.2 1.1843 290.3
Average 266.15
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Slide 13
Forecasting Future Values
Fit a trend model to deseasonalized data
Use the trend model to forecast the trend value for a given time period
Multiply the forecast trend value by the seasonal index of that period to get the overall forecast
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Slide 14
Forecasts for Next Year Estimated trend value for Q1 :
282.675 2.542 13 = 249.629
SI for Q1 = 0.7903
Forecast for Q1 the following year : 249.629 0.7903 = 197.282
Similarly, forecast for Q2 = (282.675 2.542 14) 1.0111 = 249.830Q3 = (282.675 2.542 15) 1.1129 = 272.154Q4 = (282.675 2.542 16) 1.0857 = 262.743