TimeSeries Analysis

1
Time-Series Analysis

• A time series is numerical sequence of
• values generated over regular time intervals.
• The classical time-series model involves
• four components
• Secular trend (Tt).
• Cyclical movement (Ct).
• Seasonal fluctuation (St).
• Irregular variation (It).
• The multiplicative model determine the
• level of the forecast variable Yt
• Yt Tt Ct St It

2
Classical Time-Series Model
3
Exponential Smoothing

• Finding the components is difficult.
• A direct approach averages past Yt values by
  exponential smoothing.
• The forecast value is computed from
• Ft1 aYt (1- a)Ft
• The above involves a single parameter, the
  smoothing constant (a) alpha.
• All previous time periods are reflected in the
  Fs, and greater weight is given to the more
  recent.

4
Forecasts with Single-Parameter Exponential
Smoothing
5
Single-Parameter Forecasts

• The preceding slide shows single-parameter
  forecasts of Blitz Beer sales. These were
  generated by computer.
• The level for a was .20. A greater a will assign
  more weight to the present.
• Quality of forecasts may be measured. Most
  common is the mean squared error
• which averages errors over all forecasts made.
• Other measures are the mean absolute deviation
  (MAD) and mean absolute percent error (MAPE).

6
Two-Parameter Exponential Smoothing

• The smoothing constant can be tuned to the past,
  possibly providing better forecasts.
• But single-parameter forecasts may still lead or
  lag actuals, as seen for Blitz Beer, because the
  impact of trends is delayed.
• Trend Tt can be incorporated with a second trend
  smoothing constant g (gamma)
• Tt aYt (1 a)(Tt 1 bt 1)
• bt g(Tt Tt 1) (1 g)bt 1
• Ft1 Tt bt
• That greatly reduces Blitz Beers MSE.

7
Forecasts with Two-Parameters
8
Seasonal Exponential Smoothing with Three
Parameters

• Many time series have regular seasonal patterns
  to be incorporated into forecasts.
• The three-parameter model incorporates a seasonal
  smoothing constant b (beta)
• Tt a(Yt /St p) (1 a)(Tt 1 bt 1)
• bt g(Tt Tt 1) (1 g)bt 1
• St b(Yt /Tt) (1 b)St p
• Ft1 (Tt bt) St p1

9
Forecasting withThree Parameters
10
Forecasting withThree Parameters

• The above works for p 4 quarters or p 12
  months.
• The preceding slide needs 6 quarters to generate
  the first (very bad) forecast.
• The process settles quickly, providing good
  forecasts p periods into the future.