# Ogive distribution

Format: Ogive(min, max, {data}, U) and VoseOgiveU(min, max, {data})

An Ogive distribution is the result of the fitting of a continuous non-parametric first-order distribution to data.

It is used to convert a set of data values into an empirical distribution, using the following steps:

• The minimum (min) and maximum (max) for the empirical distribution are subjectively determined based on the analyst's knowledge of the variable. For a continuous variable, these values will generally be outside the observed range of the data.

• The data points are ranked in ascending order between the min and max values.

• The cumulative probability F(xi) for each xi-value is calculated as follows:

where n is the number of data points. This formula maximizes the chance of replicating the true distribution, so it is a "best guess".

• The two arrays, {xi} and {F(xi)}, along with the minimum and maximum values, can then be used as direct inputs into a Cumulative ascending distribution CumulA(min, max, {xi}, {F(xi)}).

To generate an array of just the F(xi) values use the VoseOgive1() function. Note that this function does not take any parameters, as the values generated are only based on n, i.e. the length of the output array.

## VoseOgiveU - the second-order non-parametric distribution

The technique explained above provides us with a "best guess" for the non-parametric distribution. However, the smaller the dataset, the greater the uncertainty about the non-parametric distribution will be. We can use the Bayesian technique explained here to take this uncertainty into account.

The VoseOgiveU(min, max, {data}, U) function in ModelRisk constructs the non-parametric distribution with this uncertainty about the distribution itself (i.e. second-order) taken into account.

To generate an array of just the (uncertain) F(xi) values use ModelRisk's VoseOgive2() function. Note that this function does not take any parameters, as the values generated are only based on n, i.e. the length of the output array. As opposed to VoseOgive1, VoseOgive2 will generate a new array of values on each recalculation.

## ModelRisk functions added to Microsoft Excel for the Ogive distribution

VoseOgive generates random values from this distribution for Monte Carlo simulation, or calculates a percentile if used with a U parameter.

VoseOgiveU generates values from this distribution (with second-order uncertainty included) or calculates a percentile.

VoseOgiveObject constructs a distribution object for this distribution.

VoseOgiveProb returns the probability density or cumulative distribution function for this distribution.

VoseOgiveProb10 returns the log10 of the probability density or cumulative distribution function.

VoseOgive1 generates an array of best guess F(x) values for an Ogive distribution, as explained above.

VoseOgive2: generates an array of F(x) values with uncertainty included for an Ogive distribution, as explained above.