betaPERT | R Documentation |
The Beta-PERT methodology allows to parametrize a generalized Beta distribution based on expert opinion regarding
a pessimistic estimate (minimum value), a most likely estimate (mode),
and an optimistic estimate (maximum value). The betaPERT
function incorporates two methods of
calculating the parameters of a Beta-PERT distribution, designated "classic"
and "vose"
.
betaPERT(a, m, b, k = 4, method = c("classic", "vose")) ## S3 method for class 'betaPERT' print(x, conf.level = .95, ...) ## S3 method for class 'betaPERT' plot(x, y, ...)
a |
Pessimistic estimate (Minimum value) |
m |
Most likely estimate (Mode) |
b |
Optimistic estimate (Maximum value) |
k |
Scale parameter |
method |
|
x |
Object of class |
y |
Currently ignored |
conf.level |
Confidence level used in printing quantiles of resulting Beta-PERT distribution |
... |
Other arguments to pass to function |
The Beta-PERT methodology was developed in the context of Program Evaluation and Review Technique (PERT). Based on a pessimistic estimate (minimum value), a most likely estimate (mode), and an optimistic estimate (maximum value), typically derived through expert elicitation, the parameters of a Beta distribution can be calculated. The Beta-PERT distribution is used in stochastic modeling and risk assessment studies to reflect uncertainty regarding specific parameters.
Different methods exist in literature for defining the parameters of a Beta distribution based on PERT. The two most common methods are included in the BetaPERT
function:
The standard formulas for mean, standard deviation, α and β, are as follows:
mean = (a + k*m + b) / (k + 2)
sd = (b - a) / (k + 2)
α = { (mean - a) / (b - a) } * { (mean - a) * (b - mean) / sd^{2} - 1 }
β = α * (b - mean) / (mean - a)
The resulting distribution is a 4-parameter Beta distribution: Beta(α, β, a, b).
Vose (2000) describes a different formula for α:
(mean - a) * (2 * m - a - b) / { (m - mean) * (b - a) }
Mean and β are calculated using the standard formulas; as for the classical PERT,
the resulting distribution is a 4-parameter Beta distribution: Beta(α, β, a, b).
Note: If m = mean, α is calculated as 1 + k/2, in accordance with the mc2d package (see 'Note').
A list of class "betaPERT"
:
alpha |
Parameter α (shape1) of the Beta distribution |
beta |
Parameter β (shape2) of the Beta distribution |
a |
Pessimistic estimate (Minimum value) |
m |
Most likely estimate (Mode) |
b |
Optimistic estimate (Maximum value) |
method |
Applied method |
Available generic functions for class "betaPERT"
are print
and plot
.
The mc2d package provides
the probability density function, cumulative distribution function, quantile function and random number generation function
for the PERT distribution, parametrized by the "vose"
method.
Brecht Devleesschauwer <brechtdv@gmail.com>
Malcolm DG, Roseboom JH, Clark CE, Fazar W (1959) Application of a technique for research and development program evaluation. Oper Res 7(5):646-669.
David Vose. Risk analysis, a quantitative guide, 2nd edition. Wiley and Sons, 2000.
PERT distribution in ModelRisk (Vose software)
betaExpert
, for modelling a standard Beta distribution based on expert opinion
## The value of a parameter of interest is believed to lie between 0 and 50 ## The most likely value is believed to be 10 # Classical PERT betaPERT(a = 0, m = 10, b = 50, method = "classic") # Vose parametrization betaPERT(a = 0, m = 10, b = 50, method = "vose")
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