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inversegaussian.jl
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inversegaussian.jl
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"""
InverseGaussian(μ,λ)
The *inverse Gaussian distribution* with mean `μ` and shape `λ` has probability density function
```math
f(x; \\mu, \\lambda) = \\sqrt{\\frac{\\lambda}{2\\pi x^3}}
\\exp\\!\\left(\\frac{-\\lambda(x-\\mu)^2}{2\\mu^2x}\\right), \\quad x > 0
```
```julia
InverseGaussian() # Inverse Gaussian distribution with unit mean and unit shape, i.e. InverseGaussian(1, 1)
InverseGaussian(μ), # Inverse Gaussian distribution with mean μ and unit shape, i.e. InverseGaussian(μ, 1)
InverseGaussian(μ, λ) # Inverse Gaussian distribution with mean μ and shape λ
params(d) # Get the parameters, i.e. (μ, λ)
mean(d) # Get the mean parameter, i.e. μ
shape(d) # Get the shape parameter, i.e. λ
```
External links
* [Inverse Gaussian distribution on Wikipedia](http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution)
"""
struct InverseGaussian{T<:Real} <: ContinuousUnivariateDistribution
μ::T
λ::T
InverseGaussian{T}(μ::T, λ::T) where {T<:Real} = new{T}(μ, λ)
end
function InverseGaussian(μ::T, λ::T; check_args::Bool=true) where {T<:Real}
check_args && @check_args(InverseGaussian, μ > zero(μ) && λ > zero(λ))
return InverseGaussian{T}(μ, λ)
end
InverseGaussian(μ::Real, λ::Real; check_args::Bool=true) = InverseGaussian(promote(μ, λ)...; check_args=check_args)
InverseGaussian(μ::Integer, λ::Integer; check_args::Bool=true) = InverseGaussian(float(μ), float(λ); check_args=check_args)
InverseGaussian(μ::Real; check_args::Bool=true) = InverseGaussian(μ, one(μ); check_args=check_args)
InverseGaussian() = InverseGaussian{Float64}(1.0, 1.0)
@distr_support InverseGaussian 0.0 Inf
#### Conversions
function convert(::Type{InverseGaussian{T}}, μ::S, λ::S) where {T <: Real, S <: Real}
InverseGaussian(T(μ), T(λ))
end
function convert(::Type{InverseGaussian{T}}, d::InverseGaussian{S}) where {T <: Real, S <: Real}
InverseGaussian(T(d.μ), T(d.λ), check_args=false)
end
#### Parameters
shape(d::InverseGaussian) = d.λ
params(d::InverseGaussian) = (d.μ, d.λ)
partype(::InverseGaussian{T}) where {T} = T
#### Statistics
mean(d::InverseGaussian) = d.μ
var(d::InverseGaussian) = d.μ^3 / d.λ
skewness(d::InverseGaussian) = 3sqrt(d.μ / d.λ)
kurtosis(d::InverseGaussian) = 15d.μ / d.λ
function mode(d::InverseGaussian)
μ, λ = params(d)
r = μ / λ
μ * (sqrt(1 + (3r/2)^2) - (3r/2))
end
#### Evaluation
function pdf(d::InverseGaussian{T}, x::Real) where T<:Real
if x > 0
μ, λ = params(d)
return sqrt(λ / (twoπ * x^3)) * exp(-λ * (x - μ)^2 / (2μ^2 * x))
else
return zero(T)
end
end
function logpdf(d::InverseGaussian{T}, x::Real) where T<:Real
if x > 0
μ, λ = params(d)
return (log(λ) - (log2π + 3log(x)) - λ * (x - μ)^2 / (μ^2 * x))/2
else
return -T(Inf)
end
end
function cdf(d::InverseGaussian, x::Real)
μ, λ = params(d)
y = max(x, 0)
u = sqrt(λ / y)
v = y / μ
z = normcdf(u * (v - 1)) + exp(2λ / μ) * normcdf(-u * (v + 1))
# otherwise `NaN` is returned for `+Inf`
return isinf(x) && x > 0 ? one(z) : z
end
function ccdf(d::InverseGaussian, x::Real)
μ, λ = params(d)
y = max(x, 0)
u = sqrt(λ / y)
v = y / μ
z = normccdf(u * (v - 1)) - exp(2λ / μ) * normcdf(-u * (v + 1))
# otherwise `NaN` is returned for `+Inf`
return isinf(x) && x > 0 ? zero(z) : z
end
function logcdf(d::InverseGaussian, x::Real)
μ, λ = params(d)
y = max(x, 0)
u = sqrt(λ / y)
v = y / μ
a = normlogcdf(u * (v - 1))
b = 2λ / μ + normlogcdf(-u * (v + 1))
z = logaddexp(a, b)
# otherwise `NaN` is returned for `+Inf`
return isinf(x) && x > 0 ? zero(z) : z
end
function logccdf(d::InverseGaussian, x::Real)
μ, λ = params(d)
y = max(x, 0)
u = sqrt(λ / y)
v = y / μ
a = normlogccdf(u * (v - 1))
b = 2λ / μ + normlogcdf(-u * (v + 1))
z = logsubexp(a, b)
# otherwise `NaN` is returned for `+Inf`
return isinf(x) && x > 0 ? oftype(z, -Inf) : z
end
@quantile_newton InverseGaussian
#### Sampling
# rand method from:
# John R. Michael, William R. Schucany and Roy W. Haas (1976)
# Generating Random Variates Using Transformations with Multiple Roots
# The American Statistician , Vol. 30, No. 2, pp. 88-90
function rand(rng::AbstractRNG, d::InverseGaussian)
μ, λ = params(d)
z = randn(rng)
v = z * z
w = μ * v
x1 = μ + μ / (2λ) * (w - sqrt(w * (4λ + w)))
p1 = μ / (μ + x1)
u = rand(rng)
u >= p1 ? μ^2 / x1 : x1
end
#### Fit model
"""
Sufficient statistics for `InverseGaussian`, containing the weighted
sum of observations, the weighted sum of inverse points and sum of weights.
"""
struct InverseGaussianStats <: SufficientStats
sx::Float64 # (weighted) sum of x
sinvx::Float64 # (weighted) sum of 1/x
sw::Float64 # sum of sample weight
end
function suffstats(::Type{<:InverseGaussian}, x::AbstractVector{<:Real})
sx = sum(x)
sinvx = sum(inv, x)
InverseGaussianStats(sx, sinvx, length(x))
end
function suffstats(::Type{<:InverseGaussian}, x::AbstractVector{<:Real}, w::AbstractVector{<:Real})
n = length(x)
if length(w) != n
throw(DimensionMismatch("Inconsistent argument dimensions."))
end
T = promote_type(eltype(x), eltype(w))
sx = zero(T)
sinvx = zero(T)
sw = zero(T)
@inbounds @simd for i in eachindex(x)
sx += w[i]*x[i]
sinvx += w[i]/x[i]
sw += w[i]
end
InverseGaussianStats(sx, sinvx, sw)
end
function fit_mle(::Type{<:InverseGaussian}, ss::InverseGaussianStats)
mu = ss.sx / ss.sw
invlambda = ss.sinvx / ss.sw - inv(mu)
InverseGaussian(mu, inv(invlambda))
end