Prediction intervals for your forecasts (WK1 model)

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This paper sets forth a synergy of existing statistical theories to obtain a clear-cut model for calculating forecasts with prediction intervals, named the “WK1 model”.


Many predictive models calculate a linear or non-linear trend from the historical data and generate a single, discrete forecast value, being a single dot on this defined trend line (i.e. point forecast).

Our “WK1 model” increases the power of such a single discrete point forecast by adding its probable accuracy with top and bottom limits. The decision-maker obtains thus different ranges of values, each within several pre-defined prediction intervals to assess for that specific outcome probability.

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Prediction intervals for your forecasts (WK1 model)

  1. 1. WK1 model: Prediction intervals for your forecasts Martin van Wunnik ARSIMA Projects (Consolite)Abstract :This paper sets forth a synergy of existing statistical theories to obtain a clear-cut model forcalculating forecasts with prediction intervals, named the “WK1 model”.Many predictive models calculate a linear or non-linear trend from the historical data and generatea single, discrete forecast value, being a single dot on this defined trend line (i.e. point forecast).Our “WK1 model” increases the power of such a single discrete point forecast by adding itsprobable accuracy with top and bottom limits. The decision-maker obtains thus different ranges ofvalues, each within several pre-defined prediction intervals to assess for that specific outcomeprobability.The first step is obviously to establish the degree of the predicting power between the twovariables that will be used, based on the historical data and their statistical fundamentals(covariance and correlation).Once the predicting power of one variable for another one is proven, the second step of the “WK1model” will calculate the trend line in the usual way.Finally, the results of the first two steps are combined with the calculation of the differentprediction intervals (e.g. 60% probability, 75%, 90%, 95%, 99%, 99.5%) to provide thedecision-maker a forecast supplemented with its prediction intervals (outcomeprobability), instead of a single point forecast. These ranges are based on the trend linevalue, but supplemented with calculated probability margins above and below. By doing so, the“WK1 model” thus includes accuracy and reliability to the point values from the trend line.Martin van Wunnik is owner/manager of ARSIMA Projects bvba/sprl (www.arsimaprojects.eu) inBrussels, Belgium, and is currently developing with partners a consolidation software solution,named Consolite (www.consolite.eu.com). The WK1 model as described in this paper shall beavailable separately (spreadsheet template) and included in the predictive analytics module ofConsolite.Number of pages in PDF: 10Keywords: predictive analytics, prediction intervals, correlation, covariance, trends, future values, estimates, WK1 model, Martin van Wunnik, Consolite, ARSIMA, ARSIMA ProjectsJEL Classifications: C53, C13, C49, D89, G10 Martin van Wunnik © 2011 ARSIMA Projects [Consolite] 1 - 10
  2. 2. IntroductionIn the words of Dr. Chris Chatfield: “Predictions are often given as point forecasts with noguidance as to their likely accuracy (and perhaps even with an unreasonable highnumber of significant digits implying spurious accuracy!)” 1. We set out to improve theseusual single, discrete forecasted values from a calculated trend line by adding prediction intervals.Such intervals help the decision maker to define his/her own acceptable probability margin anddecide accordingly what forecast to be used.We have called this approach the “WK1 model” 2 (www.WK1model.com), and it is in theoryapplicable in every area where future values are predicted from a trend line distillated fromavailable historical data.However, the reader must note that our prediction intervals come from an assessment of theexisting historical data: Were the two variables related in the past? If so, the “WK1 model” usesthis historical relation for its future predictions calculations. “Unexpected” new events (i.e. nothappened in the past and thus missing from our calculations) and so-called Knightian uncertainty 3,by definition immeasurable and impossible to calculate, are thus unaccounted for. Therefore, our“WK1 model”, like many others prediction methods, does obviously not offer absolute guarantees.Establish the degree of the predicting power of the historical dataFrom the available, historical data, our very first step is to assess the pertinence and predictingpower of one variable for another.Although in statistics, it is said that every thing can be correlated to everything, as long as yousearch for it long enough, our main purpose here is to define which two variables the decisionmaker can use to predict one (unknown) variable from a (known) variable.The number of historical occurrences (e.g. years) available here is called “n”. Out of this historicaldata, we are going to search for a predicting variable and use that one for the present moment.1 Chatfield, Chris,“Calculating Interval Forecasts”, Journal of Business & Economic Statistics, Vol.11, No. 2 (April 1993), pp 1212 The name WK1 comes from the first development of this approach in 1991 by the Author with two fellow students for therequested production planning of the Belgian subsidiary of BMW motorcycles (with the sport-touring 4-cylinder BMW K1motorcycle), on a Lotus 123 spreadsheet ( standard filename extension *.wk1).3 http://en.wikipedia.org/wiki/Knightian_uncertainty Martin van Wunnik © 2011 ARSIMA Projects [Consolite] 2 - 10
  3. 3. We will illustrate the theories and algorithms stated in this paper with a simple business caseexample: How to estimate the yearly sales of a current year, based on the cumulative sales up to agiven month (e.g. June). We are going to start with the yearly sales of the last eight years, brokendown per month. This thus gives us in Illustration 1 a matrix of 8 (years) x 12 (months) = [8:12]= 96 discrete values to start off.Illustration 1: Yearly Historical Sales, split by MonthWe now simply add up these monthly sales into cumulative ones, so that the new aggregatedmatrix (see Illustration 2) shows cumulative sales up until any given month. In addition, we alsocalculate the average, being the sum of all yearly totals divided by the number of occurrences (i.e.8 years).Illustration 2:Yearly Historical Sales, cumulative per MonthIllustration 3 represents these monthly sales and the cumulative monthly sales graphically for eachhistorical year.Illustration 3: Yearly Historical Sales, monthly sales and cumulative monthly sales graphics Martin van Wunnik © 2011 ARSIMA Projects [Consolite] 3 - 10
  4. 4. Now assume that for the current year, we have a monthly sales breakdown until the month ofJune, whereby this [1:6] matrix is also defined on a cumulated level (see Illustration 4).Illustration 4: Current Sales until June, split by Month & Cumulative per MonthBefore we can proceed, we absolutely need to assess the predicting “power” of cumulative Junesales figures for the predicted sales for the whole year. In other words, do we over the years havea recurrent and stable evolution between the half year figures and the full year figures?For this assessment, we will first standardize the cumulative monthly sales figures, by deductingthe average from the historical value, and divide by the historical standard deviation, to obtain theso-called U values. The average of standardized values is by definition always equal to zero.Illustrations 5 and 6 show this step for our example.Illustration 5: Standardization cumulative monthly sales figuresIllustration 6: Non-standardized and related Standardized cumulative monthly sales figures graphsThe purpose of this standardization process is to obtain a comparable view, expressed in the sameunit, of two variables. Illustration 7 shows an example of the June cumulative sales versus theyearly sales, first non-standardized and then standardized. Any positive (if one variable goes up,the other one also goes up) or negative (if one variable goes up, the other one goes down)relation between the two variables is visually easier with standardized figures. Martin van Wunnik © 2011 ARSIMA Projects [Consolite] 4 - 10
  5. 5. Illustration 7: Additional Non-standardized and related Standardized Yearly SalesOn these standardized figures, we will now check with the statistical covariance the predictingpower of the cumulative monthly sales for estimated yearly sales. To be significant, meaning thatthe two variables are correlated (i.e. we can predict one variable from the other variable), thiscovariance must be above the 70% limit.In our example, for each year, we multiply the cumulative June U-value with the respective yearlysales (also expressed in U value), after which we summarize these calculations over the (historical)years. This sum is then divided by the number of years (8 in our example), and checked againstthe 70% limit.The Illustration 8 shows us that the covariance is not high enough until the month of May. Onlyfrom June onwards are we above the 70% limit. Obviously the covariance is increasing with thenumber of months taken into account, and, always a good check of any model, the cumulativeDecember figures are obviously correlated for 100% with the yearly sales figures.Illustration 8: Covariance per Month Martin van Wunnik © 2011 ARSIMA Projects [Consolite] 5 - 10
  6. 6. Calculate and define the trend lineWhere the cumulative sales figures are more than 70% correlated with the yearly sales figures, wecan now define a linear or non-linear trend line, being a straight or curved line to show the generalpattern or direction of the historical data.Avoiding the complex calculations of non-linear functions, we will use a simple linear function y = ax+ b further down in our paper and in our example. The reader should note that exactly the same“WK1 model” principles for prediction intervals apply for point forecasts from a non-linear function.The definition of our linear trend line is realized with the method of least squares, which is themost commonly used method to define a straight (trend) line through a set of points on ascattered plot (see Illustration 9).Illustration 9: Linear trend line through scattered plotFor “a”, we multiply the covariance with the yearly sales standard deviation, and divide thisproduct by the standard deviation of the month considered. Deducting “a” times the monthlycumulative average from the average cumulative yearly sales will give us “b”.Applying this y = ax + b function on the historical data gives us the estimated yearly sales values,based on the cumulative sales up to that specific month. Illustrations 10 and 11 show these resultsfor the month of June.Illustration 10: Linear function for trend line Illustration 11: Trend line values for historical data Martin van Wunnik © 2011 ARSIMA Projects [Consolite] 6 - 10
  7. 7. Calculate and define the prediction intervalsBecause “A point estimate of a parameter is not very meaningful without some measure of thepossible error in the estimate” 4, the essence of the “WK1 model” is to add predictionintervals to these point forecasts from the trend line.For this purpose we can use a standard normal distribution or a Student t-distribution when thenumber of occurrences is low (less than 30). In our example (where we have only 8 historicaloccurrences), we will thus use the Student t-distribution table at n-2 degrees of freedom (e.g. 8 -2= 6) for the distinct probability “confidences”, as shown in Illustration 12.Illustration 12: Student t-distribution (n-2)The values from this table are used to define the width of the prediction interval for its given“chance to occur/not occur”. In order to obtain this width information, we need to perform thefollowing separate steps: 1) Quadrate the obtained y value from the trend line for every historical value 2) Calculate (actual yearly historical sales - actual yearly historical sales average)^2 3) Calculate (actual monthly cumulative historical sales - actual monthly cumulative historical sales average)^2 4) Calculate “s2” as s2 = { Σ(Step 2) - ( (a^2) * Σ(Step 3) ) } / (n-2) 5) Calculate “V3”, being the variance of Step 3: V3 = s2 * { (1/n) + ( ( ( actual monthly cumulative current year sales - actual monthly cumulative sales average)^2)/ Σ(Step 3) ) } 6) Calculate square root of V3 from Step 5: sqrt[V3]=3^(1/2)Illustration 13 shows the first three steps for the June figures in our example, while steps four tosix are represented in Illustration 14.4 Mood, Alexander M., Graybill, Franklin A., “Introduction to the theory of statistics – second edition”, McGraw-Hill BookCompany Inc., New York, 1963, pp 248 Martin van Wunnik © 2011 ARSIMA Projects [Consolite] 7 - 10
  8. 8. Illustration 13: Calculate width of prediction interval – steps 1 to 3 Quadrate y^ (actual.dec - actual.dec.avg)^2 (actual.cum.month - actual.cum.month.avg)^2Illustration 14: Calculate width of prediction interval – steps 4 to 6With all these statistical foundations used and calculated until now, we are able to define the widthof the different prediction intervals, whereby half of it needs to be added to the obtained discretetrend value (i.e. point forecast) for the upper limit, and half of it needs to be deducted from theobtained discrete trend value for the lower limit, as shown in Illustration 15 and graphically inIllustration 16.Illustration 15: Top and Bottom limits of Prediction intervals Martin van Wunnik © 2011 ARSIMA Projects [Consolite] 8 - 10
  9. 9. Illustration 16: WK1 modelWe started our example with 96 discrete historical figures and actual sales figures for the on-goingyear until June. Based hereupon with the “WK1 model”, we have at the finish line the differentprediction intervals (upper and lower limits for each probability) for the forecasted yearly salesfrom the trend line.ConclusionMany predictive models calculate a linear or non-linear trend from the historical data and generatea single, discrete forecast value, being a single dot on this defined trend line (i.e. point forecast).We demonstrated that our “WK1 model” increases the power of such a single discrete pointforecast by adding its probable accuracy with top and bottom limits. The decision-maker obtainsthus different ranges of values, each within several pre-defined prediction intervals to assess forthat specific outcome probability (e.g. 60%, 75%, 90%, 95%, 99%, 99.5%)VAN WUNNIK, MartinNovember 6 th 2011Brussels & Lede - BelgiumAvailable at SSRN: http://ssrn.com/abstract=1955450 Martin van Wunnik © 2011 ARSIMA Projects [Consolite] 9 - 10
  10. 10. Bibliography:Van Wunnik, Martin, Lakay, Peter, and Meerschaert, Nicolas, “Voorspelling van de verkoop van BMW-motorfietsen, met behulp van LOTUS 123”, Managementbeslissingen m.b.v. de microcomputer (Prof.Plastria, Ass. A. Thys), March 1st 1991.Chatfield, Chris, “Calculating Interval Forecasts”, Journal of Business & Economic Statistics, Vol.11, No. 2(April 1993), pp 121-135Luan, Jiahui, Tang, Jian, Lu, Chen, “Prediction Interval on Spacecraft Telemetry Data Based on ModifiedBlock Bootstrap Method” in “Artificial Intelligence and Computional Intelligence”, International Conference,AICI 2010, Sanya, China, October 2010, Proceedings, Part II, Springer-Verlag Berlin-Heidelberg, 2010Mimmack, Gillian M., Manas, Gary J., Meyer, Denny H., “Introductory statistics for business: the analysis ofbusiness data”, Pearson Education South Africa, 2001Montgomery, Douglas C., Runger, George C., “Applied Statistics and Probability for Engineers”, John Wiley& Sons Inc., 2011Mood, Alexander M., and Graybill, Franklin A., “Introduction to the theory of statistics – second edition”,McGraw-Hill Book Company Inc., 1963Ryan, Thomas P., “Statistical Methods for Quality Improvement – Third edition”, John Wiley & Sons Inc.,2011Watts, S. Humphrey, “PSP(sm): A Self-Improvement Process for Software Engineers”, Pearson EducationInc., 2005http://en.wikipedia.org/wiki/Prediction_interval Martin van Wunnik © 2011 ARSIMA Projects [Consolite] 10 - 10

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