1. CHAPTER 5
TIME SERIES AND
THEIR COMPONENTS
(Page 165)

2. DECOMPOSITION
An attempt to identify the component factors that influence each value in a time series. Each component is identified separately. Projections of each of the components can then be combined to produce forecasts.

3. A model that treats the time series values as a sum of the components is called an additive components model.
A model that treats the time series values as the product of the components is called a multiplicative components model.
Both models are sometimes referred to as unobserved components models.

4. The additive components model The multiplicative components model
(5.1)
The multiplicative components model
(5.2)
Where:
Observed value,
Trend component,
Seasonal component,
Irregular component.

5. The additive components model works best when the time series has roughly the same variability throughout the length of the series.
The multiplicative components model works best when the time series increases with the level.
Figure (page 168)

6. Trend is the predicted value for the trend at time t.
If the trend appears to be roughly linear, then it is represented by the equation:
(5.3)
is the predicted value for the trend at time t.
t represents time, the independent variable, and assumes integer values 1, 2, 3, … corresponding to consecutive time periods. The slope coefficient, , is the average increase or decrease in T for each one-period increase in time.

7. The values of the coefficients and in the trend equation are calculated using the method of least squares so that the estimated trend values ( ) are close to the actual values ( ) as measured by the sum of squared errors criterion.
(5.4)

8. Example (page 169)
Data is in Table 5-1, and plotted in Figure 5-2.
The fitted trend line has the equation:
Figure 5-3 shows the straight-line trend fitted to the actual data. It shows also forecasts of new car registrations for 2 more years (t = 34 and t = 35) obtained by extrapolating the trend line. The errors
are used to compute the measures of fit, the MAD, MSD, and MAPE. “ see the Minitab Application section at the end of this chapter”

9. Additional Trend Curves
A simple function that allows for curvature is the Quadratic Trend
(5.5)
Figure 5-5

10. Exponential Trend
(5.6)
Exponential Trend can be fitted when a time series starts slowly and then appears to be increasing at an increasing rate such that the percentage difference from observation to observation is constant.
It is given by:
The coefficient is related to the growth rate. If the exponential trend is fit to annual data, the growth rate is estimated to be 100( -1)%

11. Example
Given the number of mutual fund salespeople employed by a particular company for several consecutive years. The increase in the number of salespeople is not constant. It appears as if increasing larger numbers of people are being added in the later years.
An exponential trend curve fit to the salespeople data has the equation: implying an annual growth rate of about 31%.

12. Salespeople
Year 13
2000 18
2001 22
2002 29
2003 41
2004 50
2005 68
2006

13. Growth Curves
Gompertz Trend Curve, and Logistic (Pearl-Reed) Trend Curve.
Logistic (Pearl-Reed) Trend Curve is called:
S-Curve (Pearl-Reed Logistic) in Minitab.

14. Forecasting Trend For the Linear Trend model:
n = end of time series, the forecasting origin.
p = lead time, the p steps to forecast

15. Trend curve models are based on the following assumptions:
1- The correct trend curve has been selected.
2- The curve that fits the past is indicative of the future.

16. Method Pattern of Data Time Horizon Type of Model
Minimal Data Requirements
Nonseasonal
Seasonal
Exponential trend models T
I , L TS 10
S-curve fitting
Gompertz models
Growth curves
Pattern of data: ST, stationary; T, trended; S, seasonal; C, cyclical Time horizon: S, short term (less than three months); I, intermediate; L, long term Type of model: TS, time series; C, causal. Seasonal: s, length of seasonality. of Variable: V, number variables.