# DistributionFits

Package DistributionFits allows fitting a distribution to a given set of aggregate statistics.

• to specified moments
d = fit(LogNormal, Moments(3.0,4.0))
(mean(d), var(d)) .≈ (3.0, 4.0)
• mean and upper quantile point
d = fit(LogNormal, 3.0, @qp_uu(8))
(mean(d), quantile(d, 0.975)) .≈ (3.0, 8.0)
• to mode and upper quantile point
d = fit(LogNormal, 3.0, @qp_uu(8), Val(:mode))
(mode(d), quantile(d, 0.975)) .≈ (3.0, 8.0)
• to median and upper quantile point
d = fit(LogitNormal, 0.3, @qp_u(0.8), Val(:median))
(median(d), quantile(d, 0.95)) .≈ (0.3, 0.8)
• to two quantiles, i.e confidence range
d = fit(Normal, @qp_ll(1.0), @qp_uu(8))
(quantile(d, 0.025), quantile(d, 0.975)) .≈ (1.0, 8.0)
• approximate a different distribution by matching moments
dn = Normal(3,2)
d = fit(LogNormal, moments(dn))
(mean(d), var(d)) .≈ (3.0, 4.0)

## Fit to statistical moments

StatsAPI.fitMethod
fit(D, m)

Fit a statistical distribution of type D to given moments m.

Arguments

• D: The type of distribution to fit
• m: The moments of the distribution

Notes

This can be used to approximate one distribution by another.

Examples

d = fit(LogNormal, Moments(3.2,4.6));
(mean(d), var(d)) .≈ (3.2,4.6)
d = fit(LogNormal, moments(Normal(3,1.2)));
(mean(d), std(d)) .≈ (3,1.2)
# using StatsPlots
plot(d, label = "lognormal", ylab="probability density")
plot!(Normal(3,1.2), label = "normal")

## Fit to several quantile points

StatsAPI.fitMethod
fit(D, lower::QuantilePoint, upper::QuantilePoint)

Fit a statistical distribution to a set of quantiles

Arguments

• D: The type of the distribution to fit
• lower: lower QuantilePoint (p,q)
• upper: upper QuantilePoint (p,q)

Examples

d = fit(LogNormal, @qp_m(3), @qp_uu(5));
quantile.(d, [0.5, 0.975]) ≈ [3,5]

## Fit to mean, mode, median and a quantile point

StatsAPI.fitMethod
fit(D, val, qp, ::Val{stats} = Val(:mean))

Fit a statistical distribution to a quantile and given statistics

Arguments

• D: The type of distribution to fit
• val: The value of statistics
• qp: QuantilePoint(q,p)
• stats Which statistics to fit: defaults to Val(:mean). Alternatives are: Val(:mode), Val(:median)

Examples

d = fit(LogNormal, 5.0, @qp_uu(14));
(mean(d),quantile(d, 0.975)) .≈ (5,14)
d = fit(LogNormal, 5.0, @qp_uu(14), Val(:mode));
(mode(d),quantile(d, 0.975)) .≈ (5,14)

## Fit to mean and uncertainty parameter

For bayesian inversion it is often required to specify a distribution given the expected value (the predction of the population value) and a description of uncertainty of an observation.

DistributionFits.fit_mean_ΣFunction
fit_mean_Σ(::Type{<:Distribution}, mean, Σ)

Fit a Distribution to mean and uncertainty quantificator Σ.

The meaning of Σ depends on the type of distribution:

• MvLogNormal, MvNormal: the Covariancematrix of the associated normal distribution
• LogNormal, Normal: the scale parameter, i.e. the standard deviation at log-scale, σ

## Currently supported distributions

Univariate continuous

## Implementing support for another distribution

In order to use the fitting framework for a distribution MyDist, one needs to implement the following four methods.

fit(::Type{MyDist}, m::AbstractMoments)

fit_mean_quantile(::Type{MyDist}, mean, qp::QuantilePoint)

fit_mode_quantile(::Type{MyDist}, mode, qp::QuantilePoint)

fit(::Type{MyDist}, lower::QuantilePoint, upper::QuantilePoint)

The default method for fit with stats = :median already works based on the methods for two quantile points. If the general method on two quantile points cannot be specified, one can alternatively implement method:

fit_median_quantile(::Type{MyDist}, median, qp::QuantilePoint)