Determining the joint uncertainty distribution for parameters of a distribution

See also: Fitting distributions to data, Fitting in ModelRisk, Analyzing and using data introduction

This topic explains the technique behind determining the joint distribution of uncertainty for parameters of a fitted distribution.

With ModelRisk this is done using the VoseDistributionFitP function with the Uncertainty parameter set to TRUE.

We explain the technique using a Weibull(alpha,beta) distribution as example, but it can be extended to any distribution.

1. A prior distribution is chosen for alpha and beta (E.g. Uniform distributions);

2. A range of the parameters alpha and beta is set;

3. A table calculates the likelihood function for the observed data using each combination of alpha and beta;

4. The marginal distribution for alpha is determined by summing over all likelihoods for the tested values of beta, and multiplied by the prior density.

5. The marginal distributions are graphed, normalizing to set their peaks to unity:

6. To simulate a value for alpha, a Relative distribution is constructed from the calculated marginal densities for alpha. The resulting value is plotted as a red diamond.

7. The conditional distribution for beta is determined by looking up the generated alpha value within the table and interpolating to get the necessary beta density. The plot is shown with a dashed blue line.

8. To simulate a value for beta, a Relative distribution is constructed from the calculated marginal densities for beta. The resulting value is plotted as a blue diamond.

Correlation of a and b

The mean of a Weibull distribution is given by:

which means that if we roughly know the mean (which is the first moment that one can estimate with accuracy as one acquires more data) then for a high value of alpha we must also have a high value for beta to maintain the same mean. Thus, you will notice from the model that if the generated value for alpha is in the high end of its uncertainty distribution, the conditional distribution (dashed blue line) for beta will also be high relative to the marginal distribution (full blue line).

Scatter plots of alpha and beta values generated from the VoseWeibullFitP functions show a strong correlation pattern.

 

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