1 Forecasting Forecasting Terminology Simple Moving Average Weighted Moving AverageExponential SmoothingSimple Linear Regression ModelHolt’s Trend ModelSeasonal Model (No Trend)Winter’s Model for Data with Trend and Seasonal Components
2 Evaluating Forecasts Visual Review Errors Errors Measure MPE and MAPE Tracking Signal
4 Forecasting Terminology “We are now looking at a future from here, and the future we were looking at in February now includes some of our past, and we can incorporate the past into our forecast , the first half, which is now the past and was the future when we issued our first forecast, is now over”Laura D’Andrea Tyson, Head of the President’s Council of Economic Advisors, quoted in November of 1993 in the Chicago Tribune, explaining why the Administration reduced its projections of economic growth to 2 percent from the 3.1percent it predicted in February.
5 Forecasting ProblemSuppose your fraternity/sorority house consumed the following number of cases of beer for the last 6 weekends: 8, 5, 7, 3, 6, 9How many cases do you think your fraternity / sorority will consume this weekend?
6 Forecasting: Simple Moving Average Method Using a three period moving average, we would get the following forecast:12345678910WeekCases
7 Forecasting: Simple Moving Average Method What if we used a two period moving average?12345678910WeekCases
8 Forecasting: Simple Moving Average Method The number of periods used in the moving average forecast affects the “responsiveness” of the forecasting method:1 Period12345678910WeekCases2 Periods3 Periods
9 Forecasting Terminology Applying this terminology to our problem using the Moving Average forecast:ModelEvaluationInitializationExPostForecastForecast
10 Forecasting: Weighted Moving Average Method Rather than equal weights, it might make sense to use weights which favor more recent consumption values.With the Weighted Moving Average, we have to select weights that are individually greater than zero and less than 1, and as a group sum to 1:Valid Weights: (.5, .3, .2) , (.6,.3,.1), (1/2, 1/3, 1/6)Invalid Weights: (.5, .2, .1), (.6, -.1, .5), (.5,.4,.3,.2)
11 Forecasting: Weighted Moving Average Method A Weighted Moving Average forecast with weights of (1/6, 1/3, 1/2), is performed as follows:How do you make the Weighted Moving Average forecast more responsive?
12 Forecasting: Exponential Smoothing Exponential Smoothing is designed to give the benefits of the Weighted Moving Average forecast without the cumbersome problem of specifying weights. In Exponential Smoothing, there is only one parameter ():= smoothing constant (between 0 and 1)
20 Forecasting: Simple Linear Regression Model Simple linear regression can be used to forecast data with trendsDIbaD is the regressed forecast value or dependent variable in the model, a is the intercept value of the regression line, and b is the slope of the regression line.35
21 Forecasting: Simple Linear Regression Model ErrorIn linear regression, thesquared errors are minimized
23 Limitations in Linear Regression Model 50100150200250246810121416As with the simple moving average model, all data pointscount equally with simple linear regression.
24 Forecasting: Holt’s Trend Model To forecast data with trends, we can use an exponential smoothing model with trend, frequently known as Holt’s model:L(t) = aA(t) + (1- a) F(t)T(t) = [L(t) - L(t-1) ] + (1- ) T(t-1)F(t+1) = L(t) + T(t)We could use linear regression to initialize the model
31 Seasonal Model Formulas L(t) = aA(t) / S(t-p) + (1- a) L(t-1)S(t) = g [A(t) / L(t)] + (1- g) S(t-p)F(t+1) = L(t) * S(t+1-p)p is the number of periods in a seasonQuarterly data: p = 4Monthly data: p = 12
33 Seasonal Model Forecasting A(t)L(t)SeasonalFactorS(t)F(t)2004Spring160.60Summer261.00Fall431.55Winter2326.500.85g = 0.3= 0.42005Spring14Summer29Fall41Winter2225.180.5916.0026.711.0325.1826.621.5541.3226.340.8422.602006SpringSummerFallWinter15.5327.0240.6922.25
34 Seasonal Model Forecasting 51015202530354045502468121416
35 Forecasting: Winter’s Model for Data with Trend and Seasonal Components L(t) = aA(t) / S(t-p) + (1- a)[L(t-1)+T(t-1)]T(t) = b [L(t) - L(t-1)] + (1- b) T(t-1)S(t) = g [A(t) / L(t)] + (1- g) S(t-p)F(t+1) = [L(t) + T(t)] S(t+1-p)
36 Seasonal-Trend Model Decomposition To initialize Winter’s Model, we will use Decomposition Forecasting, which itself can be used to make forecasts.
37 Decomposition Forecasting There are two ways to decompose forecast data with trend and seasonal components:Use regression to get the trend, use the trend line to get seasonal factorsUse averaging to get seasonal factors, “de-seasonalize” the data, then use regression to get the trend.
38 Decomposition Forecasting The following data contains trend and seasonal components:
39 Decomposition Forecasting The seasonal factors are obtained by the same method used for the Seasonal Model forecast:Seas.Factor0.801.351.050.791.00PeriodQuarterSales1Spring902Summer1573Fall1234Winter935128621171638122Qtr. Ave.109184143107.5Average to 1Average =135.9
40 Decomposition Forecasting With the seasonal factors, the data can be de-seasonalized by dividing the data by the seasonal factors:Regression on the De-seasonalized data will give the trend
42 Decomposition Forecast Regression on the de-seasonalized data produces the following results:Slope (m) = 7.71Intercept (b) = 101.2Forecasts can be performed using the following equation[mx + b](seasonal factor)
44 Winter’s Model Initialization We can use the decomposition forecast to define the following Winter’s Model parameters:L(n) = b + m (n)T(n) = mS(j) = S(j-p)So from our previous model, we haveL(8) = (7.71) =T(8) = 7.71S(5) = 0.80S(6) = 1.35S(7) = 1.05S(8) = 0.79
45 Winter’s Model Example = 0.3b= 0.4g= 0.2PeriodQuarterSalesL(t)T(t)S(t)F(t)1Spring902Summer1573Fall1234Winter9351280.862111.3571621.058122162.887.710.799Spring15210Summer30311Fall23212Winter171176.4110.040.81136.47197.8514.601.39251.71215.0015.621.06223.07226.3713.920.78182.1913Spring14Summer15Fall16Winter195.19352.41283.09220.87
46 Winter’s Model Example 5010015020025030035040012345678910111213141516
47 Evaluating Forecasts “Trust, but Verify” Ronald W. ReaganComputer software gives us the ability to mess up more data on a greater scale more efficientlyWhile software like SAP can automatically select models and model parameters for a set of data, and usually does so correctly, when the data is important, a human should review the model resultsOne of the best tools is the human eye
48 Visual Review How would you evaluate this forecast? 10 20 30 40 50 60 1020304050601234567891112131415
49 Forecast Evaluation Where Forecast is Evaluated ExPost Initialization 5010015020025030035040010203040Do not includeinitialization datain evaluationExPostForecastInitializationForecast
50 Errors501001502002503003504002025303540All error measures compare the forecast model to the actual data for the ExPost Forecast region
51 Errors MeasureAll error measures are based on the comparison of forecast values to actual values in the ExPost Forecast region—do not include data from initialization.
53 Bias and MADBias tells us whether we have a tendency to over- or under-forecast. If our forecasts are “in the middle” of the data, then the errors should be equally positive and negative, and should sum to 0.MAD (Mean Absolute Deviation) is the average error, ignoring whether the error is positive or negative.Errors are bad, and the closer to zero an error is, the better the forecast is likely to be.Error measures tell how well the method worked in the ExPost forecast region. How well the forecast will work in the future is uncertain.
54 Absolute vs. Relative Measures Forecasts were made for two sets of data. Which forecast was better?Data Set 1Bias = 18.72MAD = 43.99Data Set 2Bias = 182MAD = 912.5Data Set 1Data Set 2
55 MPE and MAPEWhen the numbers in a data set are larger in magnitude, then the error measures are likely to be large as well, even though the fit might not be as “good”.Mean Percentage Error (MPE) and Mean Absolute Percentage Error (MAPE) are relative forms of the Bias and MAD, respectively.MPE and MAPE can be used to compare forecasts for different sets of data.
56 MPE and MAPE Mean Percentage Error (MPE) Mean Absolute Percentage Error (MAPE)
60 Tracking SignalWhat’s happened in this situation? How could we detect this in an automatic forecasting environment?1020304050601234567891112131415
61 Tracking SignalThe tracking signal can be calculated after each actual sales value is recorded. The tracking signal is calculated as:The tracking signal is a relative measure, like MPE and MAPE, so it can be compared to a set value (typically 4 or 5) to identify when forecasting parameters and/or models need to be changed.
62 Tracking Signal t A(t) F(t) F(t) - A(t) RSFE | F(t) - A(t) | S MAD TS 115.1216.815.9311.422.214.171.124.00418.715.8-126.96.36.199.13.050.105188.8.131.52.92.971.04617.215.4-184.108.40.2060.72.680.49712.91.73.012.42.481.21822.917.1-5.8-2.85.818.23.03-0.92924.019.2-4.8-7.64.8233.29-2.311032.623.2-9.4-17.09.432.44.05-4.201138.527.8-10.7-27.743.14.79-5.781236.630.4-6.2-220.127.116.11.93-6.881340.633.5-7.1-41.07.156.45.13-8.001451.038.7-12.3-53.312.368.75.73-9.311551.942.7-9.2-62.59.277.95.99-10.43