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Cal Poly STAT 252 - Smoothing and Forecasting Time Series

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1 STAT 252 Handout 23 Winter 2010 Smoothing and Forecasting Time Series A common goal with analyzing time series data is to reduce the random variation by smoothing the data. The smoothed data can then be used for making predictions or forecasts. We will study two techniques for smoothing time series: moving averages and exponential smoothing. • A moving average of length m simply takes the average of the current value and the previous (m-1) values. o Smaller values of m put more weight on recent data. o Larger values of m put more weight on past data. Example 1: Super Bowls The following table displays the total points scored in (an excerpt from) the 43 Super Bowl football games that have been played: Number Total Points Smoothed (MA2) Smoothed (MA4) 1 45 * * 2 47 * 3 23 * 4 30 5 29 6 27 … 40 31 41 46 42 31 43 50 44 48 a) Calculate (by hand) the moving averages of length 2 for game #2. Record this in the table. b) Calculate (by hand) the moving averages of length 2 for games 3-6. Record these in the table.2 c) Explain what additional information you need to calculate this moving average for game #40. d) Calculate (by hand) the moving averages of length 2 for games #41-44. Record these in the table. e) Now consider moving averages of length 4. Calculate these moving averages for all games for which you have enough information. Record these in the table. f) Use Minitab (SuperBowls.mtw) to produce the smoothed values for a moving average of length 2 (Stat> Time Series> Moving Average …; click on Storage and store the moving averages). Check your by-hand answers above, and also fill in the missing values that you could not compute earlier. g) Repeat f) for a moving average of length 4. h) Comment on how the two smoothed series compare. i) Describe what a moving average of size 1 would look like.3 Example 2: Super Bowls (cont.) Reconsider the Super Bowl data, a portion of which is reproduced below: Number Total Points Smoothed (Expo w = .2) Smoothed (Expo w = .5) 1 45 45 45 2 47 3 23 4 30 5 29 6 27 … 40 31 41 46 42 31 43 50 44 48 • Exponential smoothing takes a weighted average of the current value and the smoothed value for the previous time period. o Let w represent the weight assigned to the current value (0 < w < 1).  Smaller values of the weight w put more weight on past data.  Larger values of the weight w put more weight on recent data. o If we let St represent the smoothed value at time t, exponential smoothing uses the formula St = w × yt + (1-w) × St-1, where yt is the actual value at time t. o We typically let the smoothed value for the first time period be equal to its actual value. a) Take the weight to be w = 0.2. Calculate (by hand) the exponentially smoothed value for game #2. Record this in the table. b) Calculate the exponentially smoothed values for games #3-6. c) Explain why you cannot calculate the exponentially smoothed values for games #40-44 from the information presented in the table.4 d) Use Minitab to determine the exponentially smoothed values for all 44 games, still using a weight of w = 0.2 (Stat> Time Series> Single Exp Smoothing …; choose options and specify 1 for the initial smoothed value; choose storage and store the smoothed values). Check your by-hand answers above, and also fill in the missing values that you could not compute earlier. e) Change the smoothing parameter (weight) to be w = 0.5, and recalculate the smoothed values for games #2 and #3 by hand. Also describe how these differ from the earlier smoothed values, and explain why this makes sense. f) Use Minitab to determine the exponentially smoothed values for all 43 games, using a weight of w = 0.5. g) Comment on how the two smoothed series compare. h) Describe what an exponentially smoothed series with weight w = 1 would look like.5 • A primary application of smoothing time series is forecasting. • The smoothed value for time t is used as the forecast (prediction) for time (t + 1). • The accuracy of forecasts is evaluated with three measures: o MAPE (mean absolute percentage error) = nyFyttt∑−100 o MAD (mean absolute deviation) = nFytt∑− o MSD (mean squared deviation) = ()nFytt2∑− Example 3: Super Bowls (cont.) Reconsider again the Super Bowl data, a portion of which is reproduced below along with moving averages of size 2. Number Total Points Smoothed (MA2) Forecast value Abs pct error Abs deviation Sq deviation 1 45 * * * * * 2 47 46.0 * * * * 3 23 35.0 4 30 26.5 5 29 29.5 6 27 28.0 … … … … … … … a) Fill in the forecast (predicted) values for games #3-6. b) For games #3-6, calculate the values of absolute percentage error, absolute deviation, and squared deviation. Record these in the table.6 c) Use Minitab to implement a moving average of length 2. Record the values of these three accuracy measures in the first column below: Accuracy Measure MA(2) MA(4) Expo(0.2) Expo(0.5) Previous value MAPE MAD MSD d) Repeat (c) for a moving average of length 4, and then for exponentially smoothed series with weights 0.2 and 0.5. e) Repeat (c) using a smoothing procedure that forecasts each new value to be exactly equal to the previous value. f) For each of the three accuracy measures, identify which of these 5 smoothing procedures provides the best forecasts. MAPE: MAD: MSD: g) With whichever procedure(s) you identified in f), predict the total points that will be scored in next year’s Super Bowl. h) Use trial-and-error (and Minitab) to try to find a length of moving average that gives better forecasts than either the MA(2) or MA(4) models. If you find a better model, report it and the values of the accuracy measures. i) Use trial-and-error (and Minitab) to try to find a weight for exponential that gives better forecasts than either the w = 0.2 or w = 0.5 weights. If you find a better model, report it and the values of the accuracy


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Cal Poly STAT 252 - Smoothing and Forecasting Time Series

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