Forecasting using simple models

1. Forecasting using simple models

2. Outline • Basic forecasting models • The basic ideas behind each model • When each model may be appropriate • Illustrate with examples • Forecast error measures • Automatic model selection • Adaptive smoothing methods • (automatic alpha adaptation) • Ideas in model based forecasting techniques • Regression • Autocorrelation • Prediction intervals

3. Basic Forecasting Models • Moving average and weighted moving average • First order exponential smoothing • Second order exponential smoothing • First order exponential smoothing with trends and/or seasonal patterns • Croston’s method

4. M-Period Moving Average • i.e. the average of the last M data points • Basically assumes a stable (trend free) series • How should we choose M? • Advantages of large M? • Advantages of large M? • Average age of data = M/2

5. Weighted Moving Averages • The Wi are weights attached to each historical data point • Essentially all known (univariate) forecasting schemes are weighted moving averages • Thus, don’t screw around with the general versions unless you are an expert

6. Simple Exponential Smoothing • Pt+1(t) = Forecast for time t+1 made at time t • Vt = Actual outcome at time t • 0<<1 is the “smoothing parameter”

7. Two Views of Same Equation • Pt+1(t) = Pt(t-1) + [Vt – Pt(t-1)] • Adjust forecast based on last forecast error OR • Pt+1(t) = (1- )Pt(t-1) + Vt • Weighted average of last forecast and last Actual

8. Simple Exponential Smoothing • Is appropriate when the underlying time series behaves like a constant + Noise • Xt =  + Nt • Or when the mean  is wandering around • That is, for a quite stable process • Not appropriate when trends or seasonality present

9. ES would work well here

10. Simple Exponential Smoothing • We can show by recursive substitution that ES can also be written as: • Pt+1(t) = Vt + (1-)Vt-1 + (1-)2Vt-2 + (1-)3Vt-3 +….. • Is a weighted average of past observations • Weights decay geometrically as we go backwards in time

12. Simple Exponential Smoothing • Ft+1(t) = At + (1-)At-1 + (1-)2At-2 + (1-)3At-3 +….. • Large  adjusts more quickly to changes • Smaller  provides more “averaging” and thus lower variance when things are stable • Exponential smoothing is intuitively more appealing than moving averages

13. Exponential Smoothing Examples

14. Zero Mean White Noise

17. Shifting Mean + Zero Mean White Noise

20. Automatic selection of  • Using historical data • Apply a range of  values • For each, calculate the error in one-step-ahead forecasts • e.g. the root mean squared error (RMSE) • Select the  that minimizes RMSE

21. RMSE vs Alpha 1.45 1.4 1.35 RMSE 1.3 1.25 1.2 1.15 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Alpha

22. Recommended Alpha • Typically alpha should be in the range 0.05 to 0.3 • If RMSE analysis indicates larger alpha, exponential smoothing may not be appropriate

25. Might look good, but is it?

28. Series and Forecast using Alpha=0.9 2 1.5 1 Forecast 0.5 0 -0.5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Period

29. Forecast RMSE vs Alpha 0.67 0.66 0.65 0.64 0.63 Forecast RMSE 0.62 Series1 0.61 0.6 0.59 0.58 0.57 0 0.2 0.4 0.6 0.8 1 Alpha

32. Forecast RMSE vs Alpha for Lake Huron Data 1.1 1.05 1 0.95 0.9 RMSE 0.85 0.8 0.75 0.7 0.65 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Alpha

35. Forecast RMSE vs Alpha for Monthly Furniture Demand Data 45.6 40.6 35.6 30.6 25.6 RMSE 20.6 15.6 10.6 5.6 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Alpha

36. Exponential smoothing will lag behind a trend • Suppose Xt=b0+ b1t • And St= (1- )St-1 + Xt • Can show that

38. Double Exponential Smoothing • Modifies exponential smoothing for following a linear trend • i.e. Smooth the smoothed value

39. St Lags St[2] Lags even more

40. 2St -St[2] doesn’t lag

44. Example

45. =0.2

46. Single Lags a trend

47. 6 5 4 Double Over-shoots a change (must “re-learn” the slope) 3 Trend 2 Series Data Single Smoothing Double smoothing 1 0 -1 1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101

48. Holt-Winters Trend and Seasonal Methods • “Exponential smoothing for data with trend and/or seasonality” • Two models, Multiplicative and Additive • Models contain estimates of trend and seasonal components • Models “smooth”, i.e. place greater weight on more recent data

49. Winters Multiplicative Model • Xt = (b1+b2t)ct + t • Where ct are seasonal terms and • Note that the amplitude depends on the level of the series • Once we start smoothing, the seasonal components may not add to L

50. Holt-Winters Trend Model • Xt = (b1+b2t) + t • Same except no seasonal effect • Works the same as the trend + season model except simpler