fitdist: Fit distributions to elicited probabilities

In SHELF: Tools to Support the Sheffield Elicitation Framework

Fit distributions to elicited probabilities

Description

Takes elicited probabilities as inputs, and fits parametric distributions using least squares on the cumulative distribution function. If separate judgements from multiple experts are specified, the function will fit one set of distributions per expert.

Usage

fitdist(
  vals,
  probs,
  lower = -Inf,
  upper = Inf,
  weights = 1,
  tdf = 3,
  expertnames = NULL,
  excludelogt = FALSE
)

Arguments

vals	A vector of elicited values for one expert, or a matrix of elicited values for multiple experts (one column per expert). Note that the an elicited judgement about X should be of the form P(X<= vals[i,j]) = probs[i,j]
probs	A vector of elicited probabilies for one expert, or a matrix of elicited values for multiple experts (one column per expert). A single vector can be used if the probabilities are the same for each expert. For each expert, there should be at least one non-zero probability less than 0.4, and at least one elicited probability less and 1 and greater than 0.6. Exponential distributions can be fitted by specifying one limit (lower or upper) and one probability between 0 and 1.
lower	A single lower limit for the uncertain quantity X, or a vector of different lower limits for each expert. Specifying a lower limit will allow the fitting of distributions bounded below.
upper	A single upper limit for the uncertain quantity X, or a vector of different lower limits for each expert. Specifying both a lower limit and an upper limit will allow the fitting of a Beta distribution.
weights	A vector or matrix of weights corresponding to vals if weighted least squares is to be used in the parameter fitting.
tdf	The number of degrees of freedom to be used when fitting a t-distribution.
expertnames	Vector of names to use for each expert.
excludelogt	Set to TRUE to exclude log-t and mirror log-t when identifying best fitting distribution.

Value

An object of class elicitation. This is a list containing the elements

Normal	Parameters of the fitted normal distributions.
Student.t	Parameters of the fitted t distributions. Note that (X - location) / scale has a standard t distribution. The degrees of freedom is not fitted; it is specified as an argument to fitdist.
Skewnormal	Parameters of the fitted skew-normal distribution. The skew-normal distribution is implemented using the sn package. See sn::dsn for details. This distribution requires at least three elicited probabilities, including at least one in each interval (0, 0.4) and (0.6, 1).
Gamma	Parameters of the fitted gamma distributions. Note that E(X - lower) = shape / rate.
Log.normal	Parameters of the fitted log normal distributions: the mean and standard deviation of log (X - lower).
Log.Student.t	Parameters of the fitted log student t distributions. Note that (log(X- lower) - location) / scale has a standard t distribution. The degrees of freedom is not fitted; it is specified as an argument to fitdist.
Beta	Parameters of the fitted beta distributions. X is scaled to the interval [0,1] via Y = (X - lower)/(upper - lower), and E(Y) = shape1 / (shape1 + shape2).
mirrorgamma	Parameters of ('mirror') gamma distributions fitted to Y = upper - X. Note that E(Y) = shape / rate.
mirrorlognormal	Parameters of ('mirror') log normal distributions fitted to Y = upper - X.
mirrorlogt	Parameters of ('mirror') log Student-t distributions fitted to Y = upper - X. Note that (log(Y) - location) / scale has a standard t distribution. The degrees of freedom is not fitted; it is specified as an argument to fitdist.
ssq	Sum of squared errors for each fitted distribution and expert. Each error is the difference between an elicited cumulative probability and the corresponding fitted cumulative probability.
best.fitting	The best fitting distribution for each expert, determined by the smallest sum of squared errors. Note that with three judgements only, this is likely to be the skew-normal, as this is a three parameter distribution.
vals	The elicited values used to fit the distributions.
probs	The elicited probabilities used to fit the distributions.
limits	The lower and upper limits specified by each expert (+/- Inf if not specified).

Note

The least squares parameter values are found numerically using the optim command. Starting values for the distribution parameters are chosen based on a simple normal approximation: linear interpolation is used to estimate the 0.4, 0.5 and 0.6 quantiles, and starting parameter values are chosen by setting E(X) equal to the 0.5th quantile, and Var(X) = (0.6 quantile - 0.4 quantile)^2 / 0.25. Note that the arguments lower and upper are not included as elicited values on the cumulative distribution function. To include a judgement such as P(X<=a)=0, the values a and 0 must be included in vals and probs respectively.

Author(s)

Jeremy Oakley <[e-mail]>

Examples

## Not run: 
# One expert, with elicited probabilities
# P(X<20)=0.25, P(X<30)=0.5, P(X<50)=0.75
# and X>0.
v <- c(20,30,50)
p <- c(0.25,0.5,0.75)
fitdist(vals=v, probs=p, lower=0)

# Now add a second expert, with elicited probabilities
# P(X<55)=0.25, P(X<60=0.5), P(X<70)=0.75
v <- matrix(c(20,30,50,55,60,70),3,2)
p <- c(0.25,0.5,0.75)
fitdist(vals=v, probs=p, lower=0)

# Two experts, different elicited quantiles and limits.
# Expert A: P(X<50)=0.25, P(X<60=0.5), P(X<65)=0.75, and provides bounds 10<X<100
# Expert B: P(X<40)=0.33, P(X<50=0.5), P(X<60)=0.66, and provides bounds 0<X
v <- matrix(c(50,60,65,40,50,60),3,2)
p <- matrix(c(.25,.5,.75,.33,.5,.66),3,2)
l <- c(10,0)
u <- c(100, Inf)
fitdist(vals=v, probs=p, lower=l, upper=u)

## End(Not run)

SHELF documentation built on Sept. 11, 2024, 6:54 p.m.