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

Slide 2

Trend

Represents long-term movement

Estimated using linear regression, independentvariable is the time t

Slide 3

Trend : Linear

Suitable when there is no obvious curvature in the time series plot

0 1tT = b bt

Slide 4

Trend : Quadratic

Suitable when the time series plot displays obvious curvature

20 1 2tT = b bt b t

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 ( , )

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

Slide 7

SI : One Year Quarterly Data

Slide 8

SI : Two Years Quarterly Data

Slide 9

SI : Two Years Monthly Data

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

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

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

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

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