Advanced Forecasting Copyright (c) 2008 by The McGraw-Hill Companies. This material is intended solely for educational use by licensed users of LearningStats. It may not be copied or resold for profit. Copyright Notice Portions of MINITAB Statistical Software input and output contained in this document are printed with permission of Minitab, Inc. MINITABTM is a trademark of Minitab Inc. in the United States and other countries and is used herein with the owner's permission. Trendless Data For an observed time series y1, y2, ..., yt with no consistent trend, here are three common ways to forecast one period ahead (Ft is the forecast and yt is the actual). Forecast Method Characteristics Ft+1 = yt Same as Last Period Very simple, but accuracy depends solely on the most recent actual data point. Ft+1 = (yt + yt-1 + ... + yt-n)/n Average of Past n Periods Simple, but each past data point has equal weight. Ft+1 = a yt + (1-a) Ft Exponential Smoothing Assigns a given weight to the most recent actual data point, and its reciprocal weight to the most recent forecast. Single Smoothing The single-smoothing updating model is Ft+1 = a yt + (1-a) Ft where yt = actual data in period t. Ft = forecast for period t. a = weight given to the current data (0 a 1) But how do we “seed” the initial forecast for period 1? Minitab sets F1 to the average of the first six values of yt. Excel's Tools > Data Analysis > Exponential Smoothing sets F1 to y1. Unless n is small, it doesn’t make very much difference how we seed F1. Trended Data For an observed time series y1, y2, ..., yt with consistent trend, here are three common ways to forecast one period ahead (Ft is the forecast and yt is the actual for period t). Forecast Method Characteristics Graph extrapolation Eyeball Method Visually fitting a trend and projecting it. Simple, but imprecise. Ft+1 = yt + (yt yt-n)/n Same Average Change Simple, but assumes linearity and does not consider variation in trend. Ft+1 = Lt + Tt Double Smoothing Uses an updating mechanism to calculate level (Lt) and trend (Tt) components for period t. Double Smoothing The double-smoothing updating model is: Lt = a yt + (1a) Ft Tt = b (Lt Lt-1) + (1 b) Tt-1 Ft+1 = Lt + Tt where yt is the actual data value in period t Lt is the level component in period t Tt is the trend component in period t Ft is the forecast for period t a and b are constants between 0 and 1. Double Smoothing: Example Living donor transplants in California have a clear upward trend, so double exponential smoothing is appropriate. This example uses a = 0.20 and b = 0.20. MINITAB’s smoothed (“Fits”) forecasts (shown in red) track the data well for 1988-2002 . The one-year forecast for 2003 (shown in green) is believable, ceteris paribus. Note MINITAB’s addition of 95% confidence limits. Year 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 Transplants 12,786 13,471 15,462 15,687 16,043 17,533 18,170 19,218 19,518 20,052 21,223 21,594 22,773 24,076 24,851 http://www.gsds.org Seasonal Data For an observed time series y1, y2, ..., yt with both trend and seasonal periodicity p, here are three common ways to forecast one period ahead (Ft is forecast, yt is actual,). Forecast Method Characteristics Ft+1 = yt-p+1 Same as p periods earlier For example, forecast next July the same as last July. Easy, but ignores trend. Extrapolate trend by month (or quarter or whatever) p-Trends For example, project fitted trend for all past Julys. Simple, attractive way to handle seasonality. Ft+m = (Ft + mTt) St Winters’ Method Forecast m periods ahead based on smoothed forecasts (Ft), trend (Tt), and seasonality (St). Widely used, but hard to explain. Winters’ Method To obtain a forecast Ft+m for m periods ahead with seasonal data, the updating equations are: Ft+m = (Ft + mTt)St Ft = a yt/St-p + (1a)(Ft-1 + Tt-1) St = d yt/Ft + (1d)St-p Tt = g(FtFt-1) + (1g)Tt-1 where Ft = smoothed series in period t yt = actual value in period t Tt = trend estimate in period t St = seasonality estimate in period t a = smoothing constant for level g = smoothing constant for trend d = smoothing constant for seasonal Note Smoothing constants follow MINITAB’s notation. See J. Holton Wilson and Barry Keating, Business Forecasting 2/e, Irwin, 1994), p. 116. Winters’ Method: Example U.S. oil imports are recorded per 4-week period (13 “months” per year). Using periodicity 13 with data from 1996-2003, MINITAB uses Winters’ method to forecast 2004 (shown in green with 95% confidence limits in blue). Month Imports 1996-01 9364 1996-02 8390 1996-03 9092 1996-04 9429 1996-05 10007 1996-06 9938 1996-07 9820 1996-08 9986 1996-09 9142 1996-10 9837 1996-11 9244 1996-12 9417 1996-13 9479 1997-01 9763 1997-02 9561 1997-03 9833 1997-04 10114 1997-05 10818 1997-06 10737 1997-07 10008 1997-08 10465 1997-09 10537 1997-10 10792 1997-11 9948 1997-12 9328 1997-13 10162 1998-01 10127 1998-02 9991 1998-03 10034 1998-04 11105 1998-05 11104 1998-06 10926 1998-07 11649 1998-08 11032 1998-09 10499 1998-10 10861 1998-11 10860 1998-12 10258 1998-13 10708 Month Imports 1999-01 10424 1999-02 10650 1999-03 10658 1999-04 11618 1999-05 11511 1999-06 11160 1999-07 11697 1999-08 11142 1999-09 10657 1999-10 10595 1999-11 10033 1999-12 10065 1999-13 10852 2000-01 10140 2000-02 11003 2000-03 11052 2000-04 11558 2000-05 11416 2000-06 12032 2000-07 11588 2000-08 12173 2000-09 11900 2000-10 11290 2000-11 11309 2000-12 12053 2000-13 11459 2001-01 12555 2001-02 11643 2001-03 12132 2001-04 12653 2001-05 12529 2001-06 11732 2001-07 11760 2001-08 11622 2001-09 11818 2001-10 11379 2001-11 11629 2001-12 10994 2001-13 11871 Month Imports 2002-01 11088 2002-02 10904 2002-03 11198 2002-04 11765 2002-05 11769 2002-06 11753 2002-07 11624 2002-08 11890 2002-09 11075 2002-10 11893 2002-11 12268 2002-12 11100 2002-13 11530 2003-01 11008 2003-02 10764 2003-03 11857 2003-04 12446 2003-05 12814 2003-06 12941 2003-07 12788 2003-08 12904 2003-09 13042 2003-10 12526 2003-11 11846 2003-12 12011 2003-13 12254 2004-01 12370 2004-02 12205 2004-03 12716 2004-04 13430 2004-05 13606 2004-06 13539 2004-07 13508 2004-08 13599 2004-09 13280 2004-10 13322 2004-11 13108 2004-12 12830 2004-13 13327 Source: http://tonto.eia.doe.gov Decomposition Time series decomposition seeks to separate a time series Y into four components: trend (T), cycle (C), seasonality (S), and irregular (I). These components are assumed to follow either an additive or a multiplicative model: Model Components Used For Additive Y=T+C+S+I Data of similar magnitude (short run or trend-free data) with constant absolute growth or decline. Multiplicative Y=T×C×S×I Data of increasing or decreasing magnitude (long run or trended data) with constant percent growth or decline. Decomposition We ignore cycles on grounds that there is no accepted theory of cycles. To decompose a monthly* time series Y: Fit the trend T (assumed linear) Estimate the seasonal factor Sj for each month: If additive, average YT for each month If multiplicative, average Y/T for each month If forecasts are desired: extrapolate the trend add (or multiply by) the seasonal factor for each month *Quarters or other sub-periods follow similar steps Decomposition: Example U.S. oil imports are recorded per 4-week period (13 “months” per year). Here is MINITAB’s decomposition using periodicity 13 with data from 1996-2003, and forecasts for 13 “months” in 2004: Month Imports 1996-01 9364 1996-02 8390 1996-03 9092 1996-04 9429 1996-05 10007 1996-06 9938 1996-07 9820 1996-08 9986 1996-09 9142 1996-10 9837 1996-11 9244 1996-12 9417 1996-13 9479 1997-01 9763 1997-02 9561 1997-03 9833 1997-04 10114 1997-05 10818 1997-06 10737 1997-07 10008 1997-08 10465 1997-09 10537 1997-10 10792 1997-11 9948 1997-12 9328 1997-13 10162 1998-01 10127 1998-02 9991 1998-03 10034 1998-04 11105 1998-05 11104 1998-06 10926 1998-07 11649 1998-08 11032 1998-09 10499 1998-10 10861 1998-11 10860 1998-12 10258 1998-13 10708 Month Imports 1999-01 10424 1999-02 10650 1999-03 10658 1999-04 11618 1999-05 11511 1999-06 11160 1999-07 11697 1999-08 11142 1999-09 10657 1999-10 10595 1999-11 10033 1999-12 10065 1999-13 10852 2000-01 10140 2000-02 11003 2000-03 11052 2000-04 11558 2000-05 11416 2000-06 12032 2000-07 11588 2000-08 12173 2000-09 11900 2000-10 11290 2000-11 11309 2000-12 12053 2000-13 11459 2001-01 12555 2001-02 11643 2001-03 12132 2001-04 12653 2001-05 12529 2001-06 11732 2001-07 11760 2001-08 11622 2001-09 11818 2001-10 11379 2001-11 11629 2001-12 10994 2001-13 11871 Month Imports 2002-01 11088 2002-02 10904 2002-03 11198 2002-04 11765 2002-05 11769 2002-06 11753 2002-07 11624 2002-08 11890 2002-09 11075 2002-10 11893 2002-11 12268 2002-12 11100 2002-13 11530 2003-01 11008 2003-02 10764 2003-03 11857 2003-04 12446 2003-05 12814 2003-06 12941 2003-07 12788 2003-08 12904 2003-09 13042 2003-10 12526 2003-11 11846 2003-12 12011 2003-13 12254 2004-01 12103 2004-02 12169 2004-03 12344 2004-04 13078 2004-05 13304 2004-06 13060 2004-07 13018 2004-08 13110 2004-09 12535 2004-10 12840 2004-11 12374 2004-12 12201 2004-13 12638 Source: http://tonto.eia.doe.gov Decomposition: Other Exhibits Here are some additional MINITAB graphs from decomposition of the oil import data. ARIMA Models Basic Model Notation Original series Differenced series yt = y1, y2, ... , yn zt = yt yt-1 = z1, z2, ... , zn-d AR(1) model (autoregressive of order 1): AR(p) model (autoregressive of order q): n observations nd observations zt = d + f1zt-1 + et zt = d + f1zt-1+ f2zt-2+ ... + ft-pzt-p + et MA(1) model (moving average of order 1): zt = d + et q1et-1 MA(q) model (moving average of order q): zt = d + et q1et-1 q2et-2 ... qt-qet-q ARMA (p,q) models: zt = d + f1zt-1+ f2zt-2+ ... + ft-pzt-p + et - q1et-1 q2et-2 ... qt-qet-q ARIMA(p,d,q) models (d is number of differences in working series) See J. Holton Wilson and Barry Keating, Business Forecasting 2/e, Irwin, 1994). ARIMA Model identification depends on the data. It is rarely necessary to go beyond AR(1) or MA(1), although seasonal data will require fancier models. Criteria for model adequacy can be found in any forecasting textbook. Time series modeling using ARIMA is an advanced topic that requires additional study*. * See J. Holton Wilson and Barry Keating, Business Forecasting 2/e, Irwin, 1994). AR(1) Patterns See J. Holton Wilson and Barry Keating, Business Forecasting 2/e, Irwin, 1994). MA(1) Patterns See J. Holton Wilson and Barry Keating, Business Forecasting 2/e, Irwin, 1994). ARIMA: Example Here is MINITAB’s ARIMA for U.S. Oil Imports using an ARIMA model with one seasonal difference on data from 1996-2003 with forecasts for 13 “months” in 2004. The forecasts have very wide confidence bands. Month Imports 1996-01 9364 1996-02 8390 1996-03 9092 1996-04 9429 1996-05 10007 1996-06 9938 1996-07 9820 1996-08 9986 1996-09 9142 1996-10 9837 1996-11 9244 1996-12 9417 1996-13 9479 1997-01 9763 1997-02 9561 1997-03 9833 1997-04 10114 1997-05 10818 1997-06 10737 1997-07 10008 1997-08 10465 1997-09 10537 1997-10 10792 1997-11 9948 1997-12 9328 1997-13 10162 1998-01 10127 1998-02 9991 1998-03 10034 1998-04 11105 1998-05 11104 1998-06 10926 1998-07 11649 1998-08 11032 1998-09 10499 1998-10 10861 1998-11 10860 1998-12 10258 1998-13 10708 Month Imports 1999-01 10424 1999-02 10650 1999-03 10658 1999-04 11618 1999-05 11511 1999-06 11160 1999-07 11697 1999-08 11142 1999-09 10657 1999-10 10595 1999-11 10033 1999-12 10065 1999-13 10852 2000-01 10140 2000-02 11003 2000-03 11052 2000-04 11558 2000-05 11416 2000-06 12032 2000-07 11588 2000-08 12173 2000-09 11900 2000-10 11290 2000-11 11309 2000-12 12053 2000-13 11459 2001-01 12555 2001-02 11643 2001-03 12132 2001-04 12653 2001-05 12529 2001-06 11732 2001-07 11760 2001-08 11622 2001-09 11818 2001-10 11379 2001-11 11629 2001-12 10994 2001-13 11871 Month Imports 2002-01 11088 2002-02 10904 2002-03 11198 2002-04 11765 2002-05 11769 2002-06 11753 2002-07 11624 2002-08 11890 2002-09 11075 2002-10 11893 2002-11 12268 2002-12 11100 2002-13 11530 2003-01 11008 2003-02 10764 2003-03 11857 2003-04 12446 2003-05 12814 2003-06 12941 2003-07 12788 2003-08 12904 2003-09 13042 2003-10 12526 2003-11 11846 2003-12 12011 2003-13 12254 2004-01 11612 2004-02 11345 2004-03 12419 2004-04 12991 2004-05 13344 2004-06 13457 2004-07 13294 2004-08 13400 2004-09 13529 2004-10 13006 2004-11 12318 2004-12 12478 2004-13 12716 ARIMA: Model Fit MINITAB’s results show that the model needs refinement (p-values for B-P statistics are too small) and the ACF and PACF plots (next screen) suggest possible need for seasonal terms. Type AR 1 MA 1 Constant Coef 0.8722 0.5449 54.61 SE Coef 0.0850 0.1445 27.66 T 10.27 3.77 1.97 P 0.000 0.000 0.051 Differencing: 0 regular, 1 seasonal of order 13 Number of observations: Original series 104, after differencing 91 Residuals: SS = MS = 29397038 (backforecasts excluded) 334057 DF = 88 Modified Box-Pierce (Ljung-Box) Chi-Square statistic Lag 12 24 36 48 Chi-Square 17.8 55.5 66.1 75.5 DF 9 21 33 45 P-Value 0.037 0.000 0.001 0.003 ARIMA: ACF and PACF Residual Plots Spikes at 7 and 13 suggest model revision. Other Methods Seasonal binaries in regression Moving average to remove trend (instead of linear OLS trend) then decomposition on residuals Ad hoc methods Final Advice Occam's Razor Given two sufficient explanations, we prefer the simpler. Named for the English philosopher William of Occam (d. 1349). Remember Occam’s Razor Don’t be dazzled by equations Think about underlying causes Are your forecasts credible? Clean, simple graphs help
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