statistics.jl

softmax(χ::Vector{T}) where T<:Real

Given a real vector of $k$ non-negative scores $χ=c_1,...,c_k$, return the vector $π=p_1,...,p_k$ of their softmax probabilities, as per

$p_i=\frac{\textrm{e}^{c_i}}{\sum_{i=1}^{k}\textrm{e}^{c_i}}$.

Examples

χ=[1.0, 2.3, 0.4, 5.0]
π=softmax(χ)

std(metric::Metric, ν::Vector{T};
    corrected::Bool=true,
    mean=nothing) where T<:RealOrComplex

Standard deviation of $k$ real or complex scalars, using the specified metric of type Metric::Enumerated type and the specified mean if provided.

Only the Euclidean and Fisher metric are supported by this function. Using the Euclidean metric return the output of standard Julia std function. Using the Fisher metric return the scalar geometric standard deviation, which is defined such as,

$\sigma=\text{exp}\Big(\sqrt{k^{-1}\sum_{i=1}^{k}\text{ln}^2(v_i/\mu})\Big)$.

If corrected is true, then the sum is scaled with $k-1$, whereas if it is false the sum is scaled with $k$.

Examples

using PosDefManifold
# Generate 10 random numbers distributed as a chi-square with 2 df.
ν=[randχ²(2) for i=1:10]
arithmetic_sd=std(Euclidean, ν) # mean not provided
geometric_mean=mean(Fisher, ν)
geometric_sd=std(Fisher, ν, mean=geometric_mean) # mean provided