# VoseTimeSMESaturation

VoseTimeSMESaturation({Probabilities}, Initial Population, Conditional)

Time series function modeling a variable estimated for each period by minimum, most likely and maximum values.

• {Probabilities} is an array of probabilities per period that an individual from the InitialPopulation will ‘convert’.

• InitialPopulation is the size of the population at time zero that might ‘convert’.

• Conditional is a Boolean variable. If TRUE then {Probabilities} define the probability of ‘converting’ in each period given that the individual has not yet ‘converted’. If FALSE then {Probabilities} define the probability of ‘converting’ in each period, and the sum of {Probabilities} may not them exceed 1.

Explanation and Uses

The SMESaturation time series function allows one to model ‘conversion’ of a population over time, where each conversion is assumed to occur independently of all others. For example, one might be interested in modeling how many of a population of potential clients will make a purchase, or how many people will get vaccinated, etc. The principle behind the model is that a ‘conversion’ occurs just once so, in terms of a sale for example, one would only expect a client to make a single purchase or none at all.

The function operates in two modes according to the Conditional parameter. If this parameter is set to FALSE, then the {Probabilities} define the chance that any individual will convert in each given period. So, for example, consider the following parameter set:

{Probabilities} = {0.2, 0.15, 0.1, 0.05}

InitialPopulation = 1000

Conditional = FALSE

The number of conversions in each year will then be:

{X1:X4} = Multinomial(1000, {0.2, 0.15, 0.1, 0.05})

The sum of probabilities must not exceed 1 (0.2 + 0.15 + 0.1 + 0.05 = 0.5) since these are the probabilities for an individual converting in each year, and they may do so only once.

The second mode for this function is when the Conditional parameter is set to TRUE, in which case the {Probabilities} define the chance that any individual will convert in each given period given that the individual has not yet converted. The function models this as a set of nested Binomial distributions. So, for example, consider the following parameter set:

{Probabilities} = {0.4, 0.5, 0.3, 0.2}

InitialPopulation = 1000

Conditional = TRUE

The number of conversions in each year will then be:

X1 = Binomial(1000,0.4)

X2 = Binomial(1000 – X1, 0.5)

X3 = Binomial(1000 – X1 – X2, 0.3)

X4 = Binomial(1000 – X1 – X2 – X3, 0.2)

In other words, in each year the size of the population that has not yet converted up to that period is calculated and the probability that those remaining convert in that period is defined by the {Probabilities} parameter.

Converting between the two modes

Let be individual probabilities when the Conditional parameter is FALSE.

Let be individual probabilities when the Conditional parameter is TRUE.

The models are equivalent when:

Thus:

Behavior of the model

When the Conditional parameter is set to FALSE the mean value for each period is just the appropriate value from the {Probabilities} array multiplied by the InitialPopulation. Thus one will tend to see the same up and down pattern in {Probabilities} repeated in the series itself.

When the Conditional parameter is set to FALSE, the observed pattern will be quite different from the {Probabilities} array because we are modeling only the remaining population at each stage, not the entire population. Thus, for example, if Conditional = FALSE and {Probabilities} = {0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1} – i.e. the probability of converting each period given one hasn’t yet converted is independent of how much time has already passed – we get a decaying pattern of conversions because the number remaining decreases each year (in this graphed example, InitialPopulation = 1000):