Chapter 16
Time-series Forecasting

Time series consists of 4 components: Trend, Seaonal, Cyclical, and Irregular(error).
Trend- Long run increase or decrease overtime. (over long period of time)
- Can be upward or downward
- Can be linear or non-linear
Seasonal Component –
- short term regular wave-like patterns,
- Observed within 1 year, often monthly or quartery.
Cyclical Component – Long term wave like patterns
- Regularly occur but may vary in length
- Often measured from peak to peak or trough to trough
Irregular Component – Unpredictable, random, residuals fluctuations
- Due to random variations of nature or accidents
Multiplicative time series
Used primarily for forecasting, observed value in time series is the product of components
Ysubi= Tsubi *Csubi *Isubi
Tsubi = trend value at year i
Csubi = cyclical value at year i
Isubi= irregular value at year i
Seaonsal component of multiplicative time series
Ysubi =Tsubi * Ssubi *Csubi *Isubi
**Ssubi = seasonal value at time i
Moving average: averages of consecutive time series for a chosen time period.
- Used for smoothing
- Arthimetic means over time
- Result depend upon choice of L
- Refer to example to see**
Exponential Smoothing – Used for smoothing and short term forecasting(often one period into the future)
- Weight declines exponentially
- Most recent observation weighted most
The weight (smoothing coefficient) is W
- Range from 0-1
- Smaller W gives more smoothing, Larger W gives less smoothing.
The weight is
- Close to 0 for smoothing out unwanted cyclical and irregular components
- Close to 1 for forecasting
Exponential Smoothing Model
Esub1 = Ysub1
Esubi = WYsubi + (1-W)Esubi-1
Esubi = exponentially smoother value for period i
Ei-1 = exponentially smoother value already computed for period i-1
Ysubi = observed value in period i
W= weight (smoothing coefficient), 0