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.

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 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.
The first step is obviously to establish the degree of the predicting power between the two
variables 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 “WK1
model” will calculate the trend line in the usual way.
Finally, the results of the first two steps are combined with the calculation of the different
prediction intervals (e.g. 60% probability, 75%, 90%, 95%, 99%, 99.5%) to provide the
decision-maker a forecast supplemented with its prediction intervals (outcome
probability), instead of a single point forecast. These ranges are based on the trend line
value, 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) in
Brussels, 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 be
available separately (spreadsheet template) and included in the predictive analytics module of
Consolite.
Number of pages in PDF: 10
Keywords: predictive analytics, prediction intervals, correlation, covariance, trends, future values,
estimates, WK1 model, Martin van Wunnik, Consolite, ARSIMA, ARSIMA Projects
JEL Classifications: C53, C13, C49, D89, G10
Martin van Wunnik © 2011 ARSIMA Projects [Consolite] 1 - 10

2. Introduction
In the words of Dr. Chris Chatfield: “Predictions are often given as point forecasts with no
guidance as to their likely accuracy (and perhaps even with an unreasonable high
number of significant digits implying spurious accuracy!)” 1. We set out to improve these
usual 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 and
decide accordingly what forecast to be used.
We have called this approach the “WK1 model” 2 (www.WK1model.com), and it is in theory
applicable in every area where future values are predicted from a trend line distillated from
available historical data.
However, the reader must note that our prediction intervals come from an assessment of the
existing historical data: Were the two variables related in the past? If so, the “WK1 model” uses
this historical relation for its future predictions calculations. “Unexpected” new events (i.e. not
happened 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 data
From the available, historical data, our very first step is to assess the pertinence and predicting
power of one variable for another.
Although in statistics, it is said that every thing can be correlated to everything, as long as you
search for it long enough, our main purpose here is to define which two variables the decision
maker 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 historical
data, 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 121
2
The name WK1 comes from the first development of this approach in 1991 by the Author with two fellow students for the
requested production planning of the Belgian subsidiary of BMW motorcycles (with the sport-touring 4-cylinder BMW K1
motorcycle), on a Lotus 123 spreadsheet ( standard filename extension *.wk1).
3
http://en.wikipedia.org/wiki/Knightian_uncertainty
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3. We will illustrate the theories and algorithms stated in this paper with a simple business case
example: How to estimate the yearly sales of a current year, based on the cumulative sales up to a
given month (e.g. June). We are going to start with the yearly sales of the last eight years, broken
down 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 Month
We now simply add up these monthly sales into cumulative ones, so that the new aggregated
matrix (see Illustration 2) shows cumulative sales up until any given month. In addition, we also
calculate 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 Month
Illustration 3 represents these monthly sales and the cumulative monthly sales graphically for each
historical year.
Illustration 3: Yearly Historical Sales, monthly sales and cumulative monthly sales graphics
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4. Now assume that for the current year, we have a monthly sales breakdown until the month of
June, 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 Month
Before we can proceed, we absolutely need to assess the predicting “power” of cumulative June
sales figures for the predicted sales for the whole year. In other words, do we over the years have
a 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 deducting
the average from the historical value, and divide by the historical standard deviation, to obtain the
so-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 figures
Illustration 6: Non-standardized and related Standardized cumulative monthly sales figures graphs
The purpose of this standardization process is to obtain a comparable view, expressed in the same
unit, of two variables. Illustration 7 shows an example of the June cumulative sales versus the
yearly 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.
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5. Illustration 7: Additional Non-standardized and related Standardized Yearly Sales
On these standardized figures, we will now check with the statistical covariance the predicting
power of the cumulative monthly sales for estimated yearly sales. To be significant, meaning that
the two variables are correlated (i.e. we can predict one variable from the other variable), this
covariance must be above the 70% limit.
In our example, for each year, we multiply the cumulative June U-value with the respective yearly
sales (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 against
the 70% limit.
The Illustration 8 shows us that the covariance is not high enough until the month of May. Only
from June onwards are we above the 70% limit. Obviously the covariance is increasing with the
number of months taken into account, and, always a good check of any model, the cumulative
December figures are obviously correlated for 100% with the yearly sales figures.
Illustration 8: Covariance per Month
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6. Calculate and define the trend line
Where the cumulative sales figures are more than 70% correlated with the yearly sales figures, we
can now define a linear or non-linear trend line, being a straight or curved line to show the general
pattern 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 the
most commonly used method to define a straight (trend) line through a set of points on a
scattered plot (see Illustration 9).
Illustration 9: Linear trend line through scattered plot
For “a”, we multiply the covariance with the yearly sales standard deviation, and divide this
product by the standard deviation of the month considered. Deducting “a” times the monthly
cumulative 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 results
for the month of June.
Illustration 10: Linear function for trend line Illustration 11: Trend line values for historical data
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7. Calculate and define the prediction intervals
Because “A point estimate of a parameter is not very meaningful without some measure of the
possible error in the estimate” 4, the essence of the “WK1 model” is to add prediction
intervals to these point forecasts from the trend line.
For this purpose we can use a standard normal distribution or a Student t-distribution when the
number of occurrences is low (less than 30). In our example (where we have only 8 historical
occurrences), 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 the
following 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 to
six are represented in Illustration 14.
4
Mood, Alexander M., Graybill, Franklin A., “Introduction to the theory of statistics – second edition”, McGraw-Hill Book
Company Inc., New York, 1963, pp 248
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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)^2
Illustration 14: Calculate width of prediction interval – steps 4 to 6
With all these statistical foundations used and calculated until now, we are able to define the width
of the different prediction intervals, whereby half of it needs to be added to the obtained discrete
trend value (i.e. point forecast) for the upper limit, and half of it needs to be deducted from the
obtained discrete trend value for the lower limit, as shown in Illustration 15 and graphically in
Illustration 16.
Illustration 15: Top and Bottom limits of Prediction intervals
Martin van Wunnik © 2011 ARSIMA Projects [Consolite] 8 - 10

9. Illustration 16: WK1 model
We started our example with 96 discrete historical figures and actual sales figures for the on-going
year until June. Based hereupon with the “WK1 model”, we have at the finish line the different
prediction intervals (upper and lower limits for each probability) for the forecasted yearly sales
from the trend line.
Conclusion
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).
We demonstrated that 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 (e.g. 60%, 75%, 90%, 95%, 99%, 99.5%)
VAN WUNNIK, Martin
November 6 th 2011
Brussels & Lede - Belgium
Available at SSRN: http://ssrn.com/abstract=1955450
Martin van Wunnik © 2011 ARSIMA Projects [Consolite] 9 - 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 121-135
Luan, Jiahui, Tang, Jian, Lu, Chen, “Prediction Interval on Spacecraft Telemetry Data Based on Modified
Block Bootstrap Method” in “Artificial Intelligence and Computional Intelligence”, International Conference,
AICI 2010, Sanya, China, October 2010, Proceedings, Part II, Springer-Verlag Berlin-Heidelberg, 2010
Mimmack, Gillian M., Manas, Gary J., Meyer, Denny H., “Introductory statistics for business: the analysis of
business data”, Pearson Education South Africa, 2001
Montgomery, Douglas C., Runger, George C., “Applied Statistics and Probability for Engineers”, John Wiley
& Sons Inc., 2011
Mood, Alexander M., and Graybill, Franklin A., “Introduction to the theory of statistics – second edition”,
McGraw-Hill Book Company Inc., 1963
Ryan, Thomas P., “Statistical Methods for Quality Improvement – Third edition”, John Wiley & Sons Inc.,
2011
Watts, S. Humphrey, “PSP(sm): A Self-Improvement Process for Software Engineers”, Pearson Education
Inc., 2005
http://en.wikipedia.org/wiki/Prediction_interval
Martin van Wunnik © 2011 ARSIMA Projects [Consolite] 10 - 10