Moving Average
- 1. Moving Average Methods Edward L. Boone Department of Statistical Sciences and Operations Research Virginia Commonwealth University November 11, 2013 Edward L. Boone
- 2. Simple Moving Average We are considering time series data xt where t = 1, 2, ..., T . The order of the observations matter. A simple moving average attempts to find a local mean. This can be done simply by taking the average of the points around the time of interest. For example if we are interested in a window of width k we simply take xt , xt−1 , xt+1 ,...,xt+k ,xt−k and compute their average. Edward L. Boone
- 3. Example Consider the following example: x1 1.2 x2 1.3 x3 1.1 x4 1.2 x5 1.4 x6 1.7 x7 1.6 x8 1.8 x9 1.5 x10 1.6 If we want the moving average at time t = 3 with window 2. ¯ x3,2 = x1 + x2 + x3 + x4 + x5 1.2 + 1.3 + 1.1 + 1.2 + 1.4 = = 1.24 5 5 If we want the moving average at time t = 7 with window 2. ¯ x7,2 = x5 + x6 + x7 + x8 + x9 1.4 + 1.7 + 1.6 + 1.8 + 1.5 = = 1.6 5 5 Notice that the “local” means are not similar. Edward L. Boone
- 4. Trailing Moving Average The problem with a standard moving average is that for the mean at time t we need to know t + 1, t + 2,...,t + k , which is in the future. In many useful cases we don’t know the future. We want to just use past values. This leads to the idea of the trailing moving average. Only take the average of xt−k , xt−k +1 ,...x1 , xt . ¯ xt,k = Edward L. Boone 1 k t xt i=t−k
- 5. Example Again consider the following example. x1 1.2 x2 1.3 x3 1.1 x4 1.2 x5 1.4 x6 1.7 x7 1.6 x8 1.8 x9 1.5 Trailing moving average with window k = 2 ¯ x3,2 = 1.3 + 1.1 + 1.2 x2 + x3 + x4 = = 1.2 3 3 . . . x5 + x6 + x7 1.4 + 1.7 + 1.6 ¯ = = = 1.56 3 3 ¯ x4,2 = ¯ x7,2 Edward L. Boone x1 + x2 + x3 1.2 + 1.3 + 1.1 = = 1.2 3 3 x10 1.6
- 6. Simple vs. Trailing Moving Average There are some issues that we will have to confront with all time series methods. How to handle the starting values? Outliers? Gaps? Prediction? Some of these are easier to deal with than others. Edward L. Boone
- 7. Simple vs. Trailing Moving Average 23 Consider the example to the right. Edward L. Boone 22 x 20 19 18 17 Notice that the red line is “smoother” than the blue line. 21 Red is centered moving average. Blue is trailing moving average. True Center Trail 0 20 40 60 t 80 100
- 8. Issues with Moving Averages Problems with simple moving average techniques. In the previous methods, all observations in window get the same weight. We may wish to downweight observations as they get farther in the past and always use all observations. Gaps? Prediction? Some of these are easier to deal with than others. Edward L. Boone
- 9. Exponentially Weighted Moving Average A “simple” way to address the weighting problem is using a weighted moving average. There are several versions of these. Each attempts to model the components of a time series dataset. If we just want to model the level then the Exponentially Weighted Moving Average may be reasonable. S1 = x1 St = αxt + (1 − α)St−1 These downweight the previous observations but still use all observations. Edward L. Boone
- 10. Example Again consider the following example using α = 0.3. x1 1.2 S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 Edward L. Boone = = = = = = = = = = x2 1.3 x3 1.1 x4 1.2 x1 = 1.2 αx2 + (1 − α)S1 αx3 + (1 − α)S2 αx4 + (1 − α)S3 αx5 + (1 − α)S4 αx6 + (1 − α)S5 αx7 + (1 − α)S6 αx8 + (1 − α)S7 αx9 + (1 − α)S8 ?? x5 1.4 x6 1.7 x7 1.6 x8 1.8 x9 1.5 x10 1.6 = 0.3(1.3) + 0.7(1.2) = 1.23 = 0.3(1.1) + 0.7(1.23) = 1.191 = 0.3(1.2) + 0.7(1.191) = 1.1937 = 0.3(1.4) + 0.7(1.1937) = 1.2559 = 0.3(1.7) + 0.7(1.2559) = 1.3891 = 0.3(1.6) + 0.7(1.3891) = 1.4523 = 0.3(1.8) + 0.7(1.4523) = 1.5566 = 0.3(1.5) + 0.7(1.5566) = 1.5396
- 11. Example 1.1 1.2 1.3 1.4 x 1.5 1.6 1.7 1.8 A picture of what the calculations give. 2 4 6 time Edward L. Boone 8 10
- 12. Exponentially Weighted Moving Average What we have looked at so far is only concerned in estimating a level (mean). Only models the level. We want to model the trend. We want to model the Seasonality as well. In order to do this we will need to build the model using these basic components. Edward L. Boone
- 13. Double Exponential Smoothing Now we can add in a trend term bt . Starting values: S1 = x1 b1 = x2 − x1 Process smoothing: St bt Edward L. Boone = αxt + (1 − α)(St−1 + bt−1 ) = β(St − St−1 ) + (1 − β)bt−1
- 14. Example Again consider the following example using α = 0.3 and β = 0.2. x1 1.2 x2 1.3 x3 1.1 x4 1.2 x5 1.4 x6 1.7 x7 1.6 x8 1.8 x9 1.5 S1 = x1 = 1.2 b1 = x2 − x1 = 1.3 − 1.2 = 0.1 S2 = αx2 + (1 − α)(S1 + b1 ) = 0.3(1.3) + 0.7(1.2 + 0.1) = 1.3 b2 = β(S1 − S2 ) + (1 − β)(b1 ) = 0.2(1.3 − 1.2) + 0.8(0.1) = 0.1 . . . S10 = 1.7343 b10 = 0.0576 Edward L. Boone x10 1.6
- 15. Example Again consider the following example using α = 0.3 and β = 0.2. x1 1.2 x2 1.3 x3 1.1 x4 1.2 x5 1.4 x6 1.7 x7 1.6 x8 1.8 Predict x11 and x12 . S10 = 1.7343 b10 = 0.0576 x11 = S10 + b10 = 1.7343 + 0.0567 = 1.7919 x12 = S10 + 2b10 = 1.7343 + 2(0.0567) = 1.8496 Edward L. Boone x9 1.5 x10 1.6
- 16. Triple Exponential Smoothing To have a level, trend and season can get a bit complicated. We need to know what period the seasonality manifests. Quarterly data the seasonal “lag" L may be 4. Monthly data the seasonal “lag" L may be 12. Weekly data the seasonal “lag" L may be 52. Daily data the seasonal “lag" L may be 365. Think how complicated hourly data would be. For simplicity we will consider Quarterly data with L = 4. Edward L. Boone
- 17. Triple Exponential Smoothing This is also known as the Holt-Winters method. Process smoothing: xt + (1 − α)(St−1 + bt−1 ) St = α Ct−L bt = β(St − St−1 ) + (1 − β)bt−1 xt Ct = γ + (1 − γ)Ct−L st Starting values: These are more difficult to get. Some use the first few cycles and get means. Some use regression to get S0 and b0 , then use those to get the initial C’s. We will let R do this for us so we don’t have to worry about it. Edward L. Boone
- 18. Example 4000 2000 3000 Sales 5000 6000 Consider the following data: SeasonalSales.csv 0 10 20 30 Time Edward L. Boone 40
- 19. HoltWinters Function in R Using the HoltWinters function in R we can estimate: smoothing parameters fitted values the fitted values also contain the level, trend and seasonal components Edward L. Boone
- 20. Example 4000 2000 3000 Sales 5000 6000 Consider the following data: SeasonalSales.csv 2 4 6 8 Time Edward L. Boone 10 12
- 21. Example 4000 2000 3000 Sales 5000 6000 A closer look: 8.0 8.5 9.0 Time Edward L. Boone 9.5 10.0
- 22. Example Sales 2000 3000 4000 5000 6000 7000 Predict into the future. 2 4 6 8 Time Edward L. Boone 10 12 14
- 23. Example Sales 2000 3000 4000 5000 6000 7000 A closer look. 12.0 12.5 13.0 13.5 Time Edward L. Boone 14.0 14.5 15.0
- 24. Example 2350 2400 2450 Sales 2500 2550 2600 An even closer look. 12.0 12.5 13.0 13.5 Time Edward L. Boone 14.0 14.5 15.0
- 25. Conclusion Moving average and “smoothing" methods are ad hoc methods for analyzing time series data. MA methods require user input k . Smoothing methods downweight past observations. These methods can directly model the level, trend and season components. Since they are ad hoc they can produce odd results at times. While these methods are useful we need to be careful because they have no clear theory to back them up. Edward L. Boone