API reference

bin(A::Vector{<:Vector}; number = fill(10,length(first(A))))

This function creates a data for multidimensional histogram for testing.

Examples

using Random
A = [randn(2) for i in 1:10000]

using MetropolisAlgorithm
b = MetropolisAlgorithm.bin(A, number=[10,10])

using CairoMakie
X = b.center[1]
Y = b.center[2]
Z = b.counter
heatmap(X, Y, Z)

count(center::Vector, width::Vector, A::Vector{<:Vector})

This function counts the number of points in A that fall within the bin defined by center and width.

Arguments

• center::Vector: Center of the bin. A vector of coordinates.
• width::Vector: Width of the bin (rectangle, hypercube). A vector of coordinates.
• A::Vector{<:Vector}: Vector of vectors (points). Each child vector is a point.

Examples

julia> count([0], [2], [randn(1) for i in 1:100000]) / 100 # ≈ 68.3% ()
68.395

julia> count([0], [0.1], [randn(1) for i in 1:100000]) / 100000 / 0.1
0.3988

julia> exp(0) / sqrt(2*π)
0.3989422804014327

metropolis!(f::Function, R::Vector{<:Vector}, r_ini::Vector{<:Real}; type=typeof(first(r_ini)), d::Real=one(type))

This function calculates many-steps of one-walker and overwrites the second argument R. Each child vector in R is a point of the trajectory.

Arguments

• f::Function: Distribution function. It does not need to be normalized.
• r_ini::Vector{<:Real}: Initial value vector. Even in the one-dimensional case, the initial value must be defined as a vector. Each child vector (point) has the same size as r_ini.
• R::Vector{<:Vector}: Vector of vectors (points). Each child vector is a point of the walker. The first element of R is same as r_ini.
• n_steps::Int=10^5: Number of steps. It is same as the length of the output parent vector.
• type::Type=typeof(first(r_ini)): Type of trajectory points. e.g., Float32, Float64, etc..
• d::Real=one(type): Maximum step size. Default value is 1.

metropolis!(f::Function, R::Vector{<:Vector}; type=typeof(first(first(R))), d::Real=one(type))

This function calculates one-step of many-walkers and overwrites the second argument R. Each child vector in R is a point of the walker (not a trajectory).

Arguments

• f::Function: Distribution function. It does not need to be normalized.
• R::Vector{<:Vector}: Vector of vectors (points). Each child vector is a point of the walker.
• type::Type=typeof(first(first(R))): Type of trajectory points. e.g., Float32, Float64, etc..
• d::Real=one(type): Maximum step size.

metropolis(f::Function, r_ini::Vector{<:Real}; n_steps::Int=10^5, type=typeof(first(r_ini)), d::Real=one(type))

This function calculates many-steps of one-walker using metropolis!(f, R, r_ini) and returns the trajectory R as a vector of vectors (points) with memory allocation.

pdf(center::Vector, width::Vector, A::Vector{<:Vector})

This function approximates the probability density function (PDF) with normalizing count(center, width, A). For the Metropolis algorithm, this function is not needed since the distribution function is known. It is used for testing the algorithm.

Arguments

• center::Vector: Center of the bin. A vector of coordinates.
• width::Vector: Width of the bin. A vector of coordinates.
• A::Vector{<:Vector}: Vector of vectors (points). Each child vector is a point.

Examples

julia> pdf([0.0], [0.2], [randn(1) for i in 1:1000000])
0.39926999999999996

julia> exp(0) / sqrt(2*π)
0.3989422804014327

julia> pdf([0.0, 0.0], [0.2, 0.2], [randn(2) for i in 1:1000000])
0.15389999999999998
s
julia> exp(0) / sqrt(2*π)^2
0.15915494309189537

julia> pdf([0.0, 0.0, 0.0], [0.2, 0.2, 0.2], [randn(3) for i in 1:1000000])
0.06162499999999998

julia> exp(0) / sqrt(2*π)^3
0.06349363593424098