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Implement Wishart(v, 𝗩) matrix-variate distribution #106

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284 changes: 280 additions & 4 deletions src/statistics/dist.rs
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Original file line number Diff line number Diff line change
Expand Up @@ -241,6 +241,7 @@
//! * Mean: $e^{\mu + \frac{\sigma^2}{2}}$
//! * Var: $(e^{\sigma^2} - 1)e^{2\mu + \sigma^2}$
//! * To generate log-normal random samples, Peroxide uses the `rand_distr::LogNormal` distribution from the `rand_distr` crate.
//!
//! ### `MVRNG` trait
//!
//! * `MVRNG` trait is composed of four fields
Expand Down Expand Up @@ -270,8 +271,8 @@
//! ### Dirichlet Distribution
//!
//! * Definition
//! $$ \text{Dir}(\mathbf{x} | \boldsymbol{\alpha}) = \frac{1}{\text{B}(\boldsymbol{\alpha})} \prod_{i=1}^K x_i^{\alpha_i - 1} $$
//! where $\text{B}(\boldsymbol{\alpha}) = \frac{\prod_{i=1}^K \Gamma(\alpha_i)}{\Gamma(\sum_{i=1}^K \alpha_i)}$
//! $$ \text{Dir}(\mathbf{x} | \boldsymbol{\alpha}) = \frac{1}{\text{B}(\boldsymbol{\alpha})} \prod_{i=1}^K x_i^{\alpha_i - 1} $$
//! where $\text{B}(\boldsymbol{\alpha}) = \frac{\prod_{i=1}^K \Gamma(\alpha_i)}{\Gamma(\sum_{i=1}^K \alpha_i)}$
//! * Representative value
//! * Mean: $\frac{\alpha_i}{\alpha_0}$
//! * Var : $\frac{\alpha_i(\alpha_0 - \alpha_i)}{\alpha_0^2(\alpha_0 + 1)}$
Expand All @@ -292,6 +293,68 @@
//! a.cov().print(); // Covariance matrix
//! }
//! ```
//!
//! ### `MatRNG` trait
//!
//! * `MatRNG` trait is composed of four fields for Matrix-Variate distributions
//! * `sample`: Extract samples
//! * `sample_with_rng`: Extract samples with specific rng
//! * `pdf` : Calculate pdf value at specific point
//! * `ln_pdf` : Calculate log-pdf value at specific point
//! ```no_run
//! use rand::Rng;
//! use peroxide::fuga::*;
//!
//! pub trait MatRNG {
//! /// Extract samples of matrix-variate distributions
//! fn sample(&self, n: usize) -> Vec<Matrix>;
//!
//! /// Extract samples of distributions with specific rng
//! fn sample_with_rng<R: Clone + Rng>(&self, rng: &mut R, n: usize) -> Vec<Matrix>;
//!
//! /// Probability Density Function
//! fn pdf(&self, x: &Matrix) -> f64;
//!
//! /// Log Probability Density Function
//! fn ln_pdf(&self, x: &Matrix) -> f64;
//! }
//! ```
//!
//! ### Wishart Distribution
//!
//! * Definition
//! $$ \mathcal{W}_p(\nu, \mathbf{V}) = \frac{|\mathbf{X}|^{(\nu - p - 1)/2} \exp\left(-\frac{1}{2}\text{tr}(\mathbf{V}^{-1}\mathbf{X})\right)}{2^{\nu p / 2} |\mathbf{V}|^{\nu/2} \Gamma_p\left(\frac{\nu}{2}\right)} $$
//! where $\nu$ is the degrees of freedom (`df`), $\mathbf{V}$ is the $p \times p$ scale matrix, and $\Gamma_p$ is the multivariate Gamma function.
//! * Representative value
//! * Mean: $\nu \mathbf{V}$
//! * Variance: $\nu(V_{ij}^2 + V_{ii}V_{jj})$ (Element-wise)
//! * Covariance: $\text{Cov}(X_{ij}, X_{kl}) = \nu(V_{ik}V_{jl} + V_{il}V_{jk})$ (Vectorized to $p^2 \times p^2$)
//! * To generate Wishart random samples, Peroxide uses the Bartlett Decomposition: $\mathbf{X} = \mathbf{L} \mathbf{A} \mathbf{A}^T \mathbf{L}^T$.
//! * **Caution**: `MatDist` utilizes the existing `Statistics` trait. Covariance and correlation are returned as flattened $p^2 \times p^2$ matrices.
//!
//! ```rust
//! use peroxide::fuga::*;
//!
//! fn main() {
//! // Ensure O3 feature is enabled for LAPACK operations
//! #[cfg(feature = "O3")]
//! {
//! let mut rng = smallrng_from_seed(42);
//! let scale = matrix(vec![1.0, 0.5, 0.5, 1.0], 2, 2, Row);
//! let w = MatDist::Wishart(5.0, scale); // Wishart(df, V)
//!
//! w.sample(10)[0].print(); // Print first sample
//! w.sample_with_rng(&mut rng, 10)[0].print(); // Generate with specific rng
//!
//! let test_x = matrix(vec![2.0, 0.0, 0.0, 2.0], 2, 2, Row);
//! w.pdf(&test_x).print(); // Probability density
//!
//! w.mean().print(); // Mean matrix
//! w.var().print(); // Element-wise variance matrix
//! w.cov().print(); // Vectorized p^2 x p^2 covariance
//! }
//! }
//! ```

extern crate rand;
extern crate rand_distr;
Expand All @@ -303,16 +366,19 @@ pub use self::MVDist::*;
pub use self::OPDist::*;
pub use self::TPDist::*;
use crate::special::function::*;
use crate::traits::fp::FPVector;
use crate::traits::fp::{FPMatrix, FPVector};
//use statistics::rand::ziggurat;
use self::WeightedUniformError::*;
use crate::statistics::{ops::C, stat::Statistics};
use crate::structure::matrix::{matrix, Matrix, Row};
use crate::util::non_macro::{linspace, seq};
use crate::util::non_macro::{linspace, seq, zeros};
use crate::util::useful::{auto_zip, find_interval};
use anyhow::{bail, Result};
use std::f64::consts::E;

#[cfg(feature = "O3")]
use crate::traits::matrix::{LinearAlgebra, MatrixTrait, UPLO};

/// One parameter distribution
///
/// # Distributions
Expand Down Expand Up @@ -347,6 +413,15 @@ pub enum MVDist<T: PartialOrd + SampleUniform + Copy + Into<f64>> {
Dirichlet(Vec<T>),
}

/// Matrix-Variate Distribution
///
/// # Distributions
/// * `Wishart(df, scale)`: Wishart distribution
#[derive(Debug, Clone)]
pub enum MatDist {
Wishart(f64, Matrix),
}

pub struct WeightedUniform<T: PartialOrd + SampleUniform + Copy + Into<f64>> {
weights: Vec<T>,
sum: T,
Expand Down Expand Up @@ -1215,3 +1290,204 @@ impl<T: PartialOrd + SampleUniform + Copy + Into<f64>> MVRNG for MVDist<T> {
}
}
}

/// Matrix-Variate Random Number Generator Trait
pub trait MatRNG {
/// Extract samples of matrix-variate distributions
fn sample(&self, n: usize) -> Vec<Matrix> {
let mut rng = rand::rng();
self.sample_with_rng(&mut rng, n)
}

/// Extract samples of distributions with specific rng
fn sample_with_rng<R: Rng + Clone>(&self, rng: &mut R, n: usize) -> Vec<Matrix>;

/// Probability Density Function
fn pdf(&self, x: &Matrix) -> f64 {
self.ln_pdf(x).exp()
}

/// Log Probability Density Function
fn ln_pdf(&self, x: &Matrix) -> f64;
}

impl Statistics for MatDist {
type Array = Matrix;
type Value = Matrix;

fn mean(&self) -> Self::Value {
match self {
MatDist::Wishart(df, scale) => scale.clone() * *df,
}
}

fn var(&self) -> Self::Value {
match self {
MatDist::Wishart(df, scale) => {
let p = scale.row;
let mut v = matrix(vec![0f64; p * p], p, p, Row);
for i in 0..p {
for j in 0..p {
v[(i, j)] = *df * (scale[(i, j)].powi(2) + scale[(i, i)] * scale[(j, j)]);
}
}
v
}
}
}

fn sd(&self) -> Self::Value {
self.var().fmap(|x| x.sqrt())
}

fn cov(&self) -> Self::Array {
match self {
MatDist::Wishart(df, scale) => {
let p = scale.row;
let p_sq = p * p;

// Allocate a flattened p^2 x p^2 covariance matrix
let mut cov_mat = zeros(p_sq, p_sq);

for i in 0..p {
for j in 0..p {
for k in 0..p {
for l in 0..p {
let row_idx = i * p + j;
let col_idx = k * p + l;

cov_mat[(row_idx, col_idx)] = *df
* (scale[(i, k)] * scale[(j, l)]
+ scale[(i, l)] * scale[(j, k)]);
}
}
}
}
cov_mat
}
}
}

fn cor(&self) -> Self::Array {
let cov_mat = self.cov();
let sd_mat = self.sd();
let p = sd_mat.row;
let p_sq = p * p;

let mut cor_mat = zeros(p_sq, p_sq);

for i in 0..p {
for j in 0..p {
for k in 0..p {
for l in 0..p {
let row_idx = i * p + j;
let col_idx = k * p + l;

// Scale the covariance element by the standard deviations
cor_mat[(row_idx, col_idx)] =
cov_mat[(row_idx, col_idx)] / (sd_mat[(i, j)] * sd_mat[(k, l)]);
}
}
}
}
cor_mat
}
}

impl MatRNG for MatDist {
#[cfg_attr(not(feature = "O3"), allow(unused))]
fn sample_with_rng<R: Rng + Clone>(&self, rng: &mut R, n: usize) -> Vec<Matrix> {
match self {
MatDist::Wishart(df, scale) => {
let p = scale.row;

#[cfg(feature = "O3")]
{
// Bartlett Decomposition Requires Cholesky on the Scale Matrix
let l_mat = scale.cholesky(UPLO::Lower);
let mut samples = Vec::with_capacity(n);
let normal = rand_distr::StandardNormal;

for _ in 0..n {
let mut a_mat = matrix(vec![0f64; p * p], p, p, Row);

// Populate the Bartlett Matrix A
for i in 0..p {
for j in 0..=i {
if i == j {
let chi = rand_distr::ChiSquared::new(*df - i as f64).unwrap();
a_mat[(i, i)] = chi.sample(rng).sqrt();
} else {
a_mat[(i, j)] = rng.sample(normal);
}
}
}

// Compute X = L * A * A^T * L^T
let la = &l_mat * &a_mat;
let x = &la * &la.t();

samples.push(x);
}

samples
}
#[cfg(not(feature = "O3"))]
{
unimplemented!(
"Wishart sampling requires the 'O3' feature for Cholesky decomposition."
);
}
}
}
}

#[cfg_attr(not(feature = "O3"), allow(unused))]
fn ln_pdf(&self, x: &Matrix) -> f64 {
match self {
MatDist::Wishart(df, scale) => {
#[cfg(feature = "O3")]
{
let p_usize = scale.row;
let p = p_usize as f64;

let det_x = x.det();
let det_v = scale.det();
let inv_v = scale.inv();

// Trace of (V^-1 * X)
let v_inv_x = &inv_v * x;
let tr_v_inv_x = v_inv_x.diag().iter().sum::<f64>();

let term1 = (*df - p - 1.0) / 2.0 * det_x.ln();
let term2 = -0.5 * tr_v_inv_x;
let term3 = -(*df * p) / 2.0 * 2f64.ln();
let term4 = -(*df / 2.0) * det_v.ln();
let term5 = -mv_ln_gamma(p_usize, *df / 2.0);

term1 + term2 + term3 + term4 + term5
}
#[cfg(not(feature = "O3"))]
{
unimplemented!("Wishart PDF calculation requires the 'O3' feature for matrix inversion and determinants.");
}
}
}
}
}

#[cfg_attr(not(feature = "O3"), allow(unused))]
fn mv_ln_gamma(p: usize, a: f64) -> f64 {
let p_f64 = p as f64;

if a <= (p_f64 - 1.0) / 2.0 {
return f64::NAN;
}

let mut sum = p_f64 * (p_f64 - 1.0) / 4.0 * std::f64::consts::PI.ln();
for j in 1..=p {
sum += ln_gamma(a + (1.0 - j as f64) / 2.0);
}

sum
}
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