Chapter TopicsChapter Topics • Introduction to exponential smoothing • Simple Exponential Smoothing • Holt’s Trend Corrected Exponential Smoothing • Holt-Winters Methods – Multiplicative Holt-Winters method – Additive Holt-Winters method Slide 2
Motivation of Exponential SmoothingMotivation of Exponential Smoothing • Simple moving average method assigns equal weights (1/k) to all k data points. • Arguably, recent observations provide more relevant information than do observations in the past. • So we want a weighting scheme that assigns decreasing weights to the more distant observations. Slide 3
Exponential SmoothingExponential Smoothing • Exponential smoothing methods give larger weights to more recent observations, and the weights decrease exponentially as the observations become more distant. • These methods are most effective when the parameters describing the time series are changing SLOWLY over time. Slide 4
Data vs MethodsData vs Methods No trend or N Linear trend N Both trend N seasonal and no seasonal and seasonal pattern? pattern? pattern? Y Y Y Holt’s Trend Single Corrected Exponential Holt-Winters Use Other Exponential Smoothing Methods Methods Smoothing Method Method Slide 5
Simple Exponential SmoothingSimple Exponential Smoothing • The Simple Exponential Smoothing method is used for forecasting a time series when there is no trend or seasonal pattern, but the mean (or level) of the time series y is t slowly changing over time. • NO TREND model y t o t Slide 6
Procedures of Simple ExponentialProcedures of Simple Exponential Smoothing MethodSmoothing Method • Step 1: Compute the initial estimate of the mean (or level) of the series at time period t = 0 n y t y t1 0 n • Step 2: Compute the updated estimate by using the smoothing equation y (1 ) T T T1 where is a smoothing constant between 0 and 1. Slide 7
Procedures of Simple ExponentialProcedures of Simple Exponential Smoothing MethodSmoothing Method Note that y (1 ) T T T 1 y (1 )[y (1 ) ] T T1 T 2 y (1 ) y (1 )2 T T1 T 2 y (1 ) y (1 )2y ...(1 )T1y (1 )T T T1 T 2 1 0 The coefficients measuring the contributions of the observations decrease exponentially over time. Slide 8
Simple Exponential SmoothingSimple Exponential Smoothing • Point forecast made at time T for y T+p yˆ (T) (p 1, 2,3,...) T p T • SSE, MSE, and the standard errors at time T T SSE [y yˆ (t 1)]2 t t t1 SSE MSE , s MSE T 1 Note: There is no theoretical justification for dividing SSE by (T – number of smoothing constants). However, we use this divisor because it agrees to the computation of s in Box- Jenkins models introduced later. Slide 9
Example: Cod CatchExample: Cod Catch • The Bay City Seafood Company recorded the monthly cod catch for the previous two years, as given below. Cod Catch (In Tons) Month Year 1 Year 2 January 362 276 February 381 334 March 317 394 April 297 334 May 399 384 June 402 314 July 375 344 August 349 337 September 386 345 October 328 362 November 389 314 December 343 365 Slide 10
Example: Cod CatchExample: Cod Catch • The plot of these data suggests that there is no trend or seasonal pattern. Therefore, a NO TREND model is suggested: y t o t It is also possible that the mean (or level) is slowly changing over time. Monthly Cod Catch 410 390 370 350 330 310 Cod Catch (in tons)290 270 250 0 5 10 15 20 25 Month Slide 11
Example: Cod CatchExample: Cod Catch • Step 1: Compute ℓ by averaging the first twelve 0 time series values. 12 y t 362381...343 t1 360.6667 0 12 12 Though there is no theoretical justification, it is a common practice to calculate initial estimates of exponential smoothing procedures by using HALF of the historical data. Slide 12
Example: Cod CatchExample: Cod Catch • Step 2: Begin with the initial estimate ℓ = 360.6667 0 and update it by applying the smoothing equation to the 24 observed cod catches. Set = 0.1 arbitrarily and judge the appropriateness of this choice of by the model’s in-sample fit. y (1 ) 0.1(362)0.9(360.6667) 360.8000 1 1 0 y (1 ) 0.1(381)0.9(360.8000) 362.8200 2 2 1 Slide 13
One-period-ahead ForecastingOne-period-ahead Forecasting n alpha SSE MSE s 24 0.1 28735.1092 1249.3526 35.3462 Time Smoothed Estimate Forecast Made Forecast Squared Forecast Period y for Level Last Period Error Error 0 360.6667 1 362 360.8000 360.6667 1.3333 1.7777 2 381 362.8200 360.8000 20.2000 408.0388 3 317 358.2380 362.8200 -45.8200 2099.4749 4 297 352.1142 358.2380 -61.2380 3750.0956 5 399 356.8028 352.1142 46.8858 2198.2762 6 402 361.3225 356.8028 45.1972 2042.7869 7 375 362.6903 361.3225 13.6775 187.0735 8 349 361.3212 362.6903 -13.6903 187.4234 9 386 363.7891 361.3212 24.6788 609.0411 10 328 360.2102 363.7891 -35.7891 1280.8609 11 389 363.0892 360.2102 28.7898 828.8523 12 343 361.0803 363.0892 -20.0892 403.5753 13 276 352.5722 361.0803 -85.0803 7238.6517 14 334 350.7150 352.5722 -18.5722 344.9281 15 394 355.0435 350.7150 43.2850 1873.5899 16 334 352.9392 355.0435 -21.0435 442.8295 17 384 356.0452 352.9392 31.0608 964.7756 18 314 351.8407 356.0452 -42.0452 1767.8027 19 344 351.0566 351.8407 -7.8407 61.4769 20 337 349.6510 351.0566 -14.0566 197.5894 21 345 349.1859 349.6510 -4.6510 21.6317 22 362 350.4673 349.1859 12.8141 164.2015 23 314 346.8206 350.4673 -36.4673 1329.8638 Slide 14 24 365 348.6385 346.8206 18.1794 330.4918
Example: Cod CatchExample: Cod Catch • Results associated with different values of Smoothing Sum of Squared Constant Errors 0.1 28735.11 0.2 30771.73 0.3 33155.54 0.4 35687.69 0.5 38364.24 0.6 41224.69 0.7 44324.09 0.8 47734.09 Slide 15
Example: Cod CatchExample: Cod Catch • Step 3: Find a good value of that provides the minimum value for MSE (or SSE). – Use Solver in Excel as an illustration SSE alpha Slide 16
Holt’s Trend Corrected ExponentialHolt’s Trend Corrected Exponential SmoothingSmoothing • If a time series is increasing or decreasing approximately at a fixed rate, then it may be described by the LINEAR TREND model y t t 0 1 t If the values of the parameters β and β are slowly changing over time, Holt’s trend corrected exponential0 1 smoothing method can be applied to the time series observations. Note: When neither β nor β is changing over time, 0 1 regression can be used to forecast future values of y . t • Level (or mean) at time T: β + β T 0 1 Growth rate (or trend): β 1 Slide 18
Holt’s Trend Corrected ExponentialHolt’s Trend Corrected Exponential SmoothingSmoothing • A smoothing approach for forecasting such a time series that employs two smoothing constants, denoted by and . • There are two estimates ℓ and b . T-1 T-1 – ℓ is the estimate of the level of the time T-1 series constructed in time period T–1 (This is usually called the permanent component). – b is the estimate of the growth rate of the T-1 time series constructed in time period T–1 (This is usually called the trend component). Slide 19
Holt’s Trend Corrected ExponentialHolt’s Trend Corrected Exponential SmoothingSmoothing • Level estimate y (1 )( b ) T T T 1 T 1 • Trend estimate b ( )(1 )b T T T 1 T 1 where = smoothing constant for the level (0 ≤ ≤ 1) = smoothing constant for the trend (0 ≤ ≤ 1) Slide 20
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