# VoseTimeSMEPoisson

VoseTimeSMEPoisson({Mean Values}, Spread Multiplier, Gamma Correlation)

Time series function modeling a variable that occurs randomly in time.

• {Mean Values} is an array of values of the expected number of counts in each period of the forecast. Each MeanValue must be greater than zero.

• Spread Multiplier increases the spread of the series. Must lie on [1,10]. It is an optional, and is set to 1 if omitted.

• Gamma Correlation applies a positive correlation  factor described below. GammaCorrelation must lie on [0,1]. It is an optional parameter, and is set to zero if omitted.

Explanation and Uses

The SMEPoisson time series is based on the Poisson distribution which is commonly used to model a discrete counting variable that occurs independently and randomly in time. For example, one might consider that telephone exchange failures, the number of sales of large items in a store, or the number of insured accidents occur randomly in time. A Poisson distribution takes a single input parameter: the expected (average) number of events that will occur and applies a distribution around that value to reflect the variation that might occur. For example, Poisson(9.5) is centered on an expected rate of 9.5 counts, but has the following spread around that value:

The Poisson distribution is widely used in many fields, but has often been observed to underestimate the amount of spread around a mean value compared with what is observed in the real world. The reason is generally because the expected value is itself a random variable: in some weeks (or months, years, etc. depending on the time increments used), for example, during winter one might have very cold frosts resulting in more car accidents and in other weeks no frost at all resulting in fewer accidents: although within each week one might consider each accident to occur independently, the expectation of accidents occurring changes.

In probability modeling the usual approach to dealing with random variation in the expected rate of occurrence is to model the rate using a Gamma distribution . The main reason for choosing a Gamma is convenience: it turns out that a Poisson(Gamma(a,b)) follows a Pólya distribution which has a fairly convenient mathematical form. Other reasons are that the Gamma distribution is always greater than zero (which is of course a requirement) and that it can take a variety of shapes from very right skewed to essentially normally distributed.

In the SMEPoisson function, the Pólya distribution comes into play if one selects a SpreadMultiplier greater than 1. For example, if one chooses a SpreadMultiplier of 2, the function determines the parameters of Polya distributions that would give the defined mean values but also give twice the spread (standard deviation) that a Poisson distribution would produce. The following screen shots illustrate the principle:

{=VoseTimeSMEPoisson({5,6,7,8,9,10},1,0)}

{=VoseTimeSMEPoisson({5,6,7,8,9,10},2,0)}

Note: the Spread Multiplier value is limited to a maximum of 10 because this is an extremely high multiplier for a modification to a Poisson process, and you should probably consider one of the other SME time series functions instead.

The Gamma Correlation parameter allows one to apply a positive correlation to the Gamma distributions that are used (i.e. when the Spread Multiplier is greater than 1). The effect is most visible when the {Mean Values} are relatively large (say >100) because the Gamma distributions are then more dominant than the Poisson distributions they sit within in terms of their contribution to randomness. The following screen shots illustrate the idea where, in each plot, two possible pathways have been drawn (in black). The Gamma Correlation parameter controls how much a simulated pathway will stay at a high value if it starts off high, and vice versa:

{=VoseTimeSMEPoisson({5,6,7,8,9,10},2,0.0)} v {=VoseTimeSMEPoisson({5,6,7,8,9,10},2,0.9)}

{=VoseTimeSMEPoisson({50,60,70,80,90,100},2,0.0)} v

{=VoseTimeSMEPoisson({50,60,70,80,90,100},2,0.9)}

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