UncertainLMH distribution

UncertainLMH( xLow, xMedian, xHigh, pLow, lb, ub)

A simple and convenient way to specify a smooth probability distribution with three points, «xLow», «xMedian» and «xHigh». By default it assumes these parameters are the 10th, 50th, and 90th percentiles. Or you can specify the optional «pLow» probability percentile for «xLow», and it uses 1- «pLow» for «xHigh». For example,

UncertainLMH(10, 20, 30, pLow: 25%)

Treats parameters 10, 20, 30 as the 25th, 50th, and 75th percentiles.

0.25 treats «xLow»-«xMedian»-«xHigh» as 25-50-75 percentile estimates. «xMedian» is always the median. You can set bounds low, above, or both: The distribution is unbounded below unless you specify a value for «lb» (lower bound). Similarly it is unbounded above unless you specify a value for «ub» (upper bound). The result is a sample from a 3-term Keelin distribution, also known as a MetaLog, with a Symmetric Percentile Triplet (SPT), introduced in:

Thomas W. Keelin (Nov. 2016), "The Metalog Distribution", Decision Analysis, 13(4):243-277,

This SPT Keelin distribution is a convenient special case of the full Keelin distribution.

See #Analytic distribution functions below for functions that give exact values for the cumulative, inverse cumulative, and density functions:

CumUncertainLMH( x, xLow, xMedian, xHigh, pLow, lb, ub)

CumUncertainLMHInv( p, xLow, xMedian, xHigh, pLow, lb, ub)

DensUncertainLMH( x, xLow, xMedian, xHigh, pLow, lb, ub)

Feasibility

The parameters must be ordered as «lb» < «xLow» < «xMedian» < «xHigh» < «ub». But, not all ordered parameters result in a valid 3-term Keelin distribution. Those combinations that are valid are called feasible, and parameter combinations that cannot be fit exactly are called infeasible. Infeasible combinations are typically very extreme. If the parameters are infeasible, an error message identifies the range of possible values for «xMedian» that would be feasible given the other parameters.

For the unbounded case, the parameter combination is feasible when (Keelin 2016, Proposition 2)

[math]\displaystyle{ k \lt r \lt 1-k }[/math]

where

[math]\displaystyle{ k = {1\over 2} ( 1 - 1.66711 ({1\over 2} - p_{low} ) }[/math]

[math]\displaystyle{ r = {{x_{median} - x_{low}} \over { x_{high} - x_{low} } } }[/math]

k = 0.16658 in the 10-50-90 case.

Examples

An unbounded smooth continuous distribution with a 10% probability of being <= 8, a 50% probability of being <= 29, and a 90% probability of being less that 44. In other words, the 10-50-90 estimates are 8,29,44:

UncertainLMH(8, 29, 44) →

A positive-only (semi-bounded) distribution with 10-50-90 percentiles of 8, 29 and 44:

UncertainLMH(8, 29, 44, lb:0)→

A semi-bounded from above distribution with 10-50-90 percentiles of 8, 29, 44 and an upper bound of 60

UncertainLMH(8, 29, 44, ub:60)→

A full bounded (between 0 and 60) distribution with 10-50-90 percentiles of 8, 29 and 44.

UncertainLMH(8, 29, 44, lb:0 ub:60)→

Comparison of three unbounded distributions with different percentile levels. 5-50-95, 10-50-90 and 25-50-75. Notice that the CDF's value at X=8 is equal to the percentile level, that all three CDF curves pass through (29, 0.5), and that the CDF at x=44 is 1 minus the percentile level.

Index percentile := [5%, 10%, 25%] Do UncertainLMH(8, 29, 44, percentile)→

Analytic distribution functions

The analytic distribution functions compute the exact metric at a point on a distribution without any sampling error.

CumUncertainLMH( x, xLow, xMedian, xHigh, pLow, lb, ub)

The cumulative probability function, computes the probability that the true value (or a value sampled from the distribution) is less than or equal to «x»

CumUncertainLMHInv( p, xLow, xMedian, xHigh, pLow, lb, ub)

The inverse cumulative probability function, aka called the quantile function. Returns the value x where there is a «p» probability of the true value (or a value sampled randomly from the distribution) being less than or equal to x.

DensUncertainLMH( x, xLow, xMedian, xHigh, pLow, lb, ub)

The probability density function. Returns the probability density at «x».