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//! Probabilistic distributions //! //! ## Probability Distribution //! //! * There are some famous pdf in Peroxide (not checked pdfs will be implemented soon) //! * Bernoulli //! * Binomial //! * Beta //! * Dirichlet //! * Gamma //! * Normal //! * Student's t //! * Uniform //! * There are two enums to represent probability distribution //! * `OPDist<T>` : One parameter distribution (Bernoulli) //! * `TPDist<T>` : Two parameter distribution (Uniform, Normal, Beta, Gamma) //! * `T: PartialOrd + SampleUniform + Copy + Into<f64>` //! * There are some traits for pdf //! * `RNG` trait - extract sample & calculate pdf //! * `Statistics` trait - already shown above //! //! ### `RNG` trait //! //! * `RNG` trait is composed of two fields //! * `sample`: Extract samples //! * `pdf` : Calculate pdf value at specific point //! ```rust //! extern crate rand; //! use rand::distributions::uniform::SampleUniform; //! //! pub trait RNG { //! /// Extract samples of distributions //! fn sample(&self, n: usize) -> Vec<f64>; //! //! /// Probability Distribution Function //! /// //! /// # Type //! /// `f64 -> f64` //! fn pdf<S: PartialOrd + SampleUniform + Copy + Into<f64>>(&self, x: S) -> f64; //! } //! ``` //! //! ### Bernoulli Distribution //! //! * Definition //! $$ \text{Bern}(x | \mu) = \mu^x (1-\mu)^{1-x} $$ //! * Representative value //! * Mean: $\mu$ //! * Var : $\mu(1 - \mu)$ //! * In peroxide, to generate $\text{Bern}(x | \mu)$, use simple traits //! 1. Generate $U \sim \text{Unif}(0, 1)$ //! 2. If $U \leq \mu$, then $X = 1$ else $X = 0$ //! * Usage is very simple //! //! ```rust //! extern crate peroxide; //! use peroxide::fuga::*; //! //! fn main() { //! let b = Bernoulli(0.1); // Bern(x | 0.1) //! b.sample(100).print(); // Generate 100 samples //! b.pdf(0).print(); // 0.9 //! b.mean().print(); // 0.1 //! b.var().print(); // 0.09 (approximately) //! b.sd().print(); // 0.3 (approximately) //! } //! ``` //! //! ### Uniform Distribution //! //! * Definition //! $$\text{Unif}(x | a, b) = \begin{cases} \frac{1}{b - a} & x \in [a,b]\\\ 0 & \text{otherwise} \end{cases}$$ //! * Representative value //! * Mean: $\frac{a + b}{2}$ //! * Var : $\frac{1}{12}(b-a)^2$ //! * To generate uniform random number, Peroxide uses `rand` crate //! * **Caution**: `Uniform(T, T)` generates `T` type samples (only for `Uniform`) //! //! ```rust //! extern crate peroxide; //! use peroxide::fuga::*; //! //! fn main() { //! // Uniform(start, end) //! let a = Uniform(0f64, 1f64); // It will generate `f64` samples. //! a.sample(100).print(); //! a.pdf(0.2).print(); //! a.mean().print(); //! a.var().print(); //! a.sd().print(); //! } //! ``` //! //! ### Normal Distribution //! //! * Definition //! $$\mathcal{N}(x | \mu, \sigma^2) = \frac{1}{\sqrt{2\pi \sigma^2}} \exp{\left( - \frac{(x - \mu)^2}{2\sigma^2}\right)}$$ //! * Representative value //! * Mean: $\mu$ //! * Var: $\sigma^2$ //! * To generate normal random number, there are two famous algorithms //! * Marsaglia-Polar method //! * Ziggurat traits //! * In peroxide (after ver 0.19.1), use `rand_distr` to generate random normal samples. //! * <del>In peroxide, main traits is Ziggurat - most efficient traits to generate random normal samples.</del> //! * <del>Code is based on a [C implementation](https://www.seehuhn.de/pages/ziggurat.html) by Jochen Voss.</del> //! ```rust //! extern crate peroxide; //! use peroxide::fuga::*; //! //! fn main() { //! // Normal(mean, std) //! let a = Normal(0, 1); // Standard normal //! a.sample(100).print(); //! a.pdf(0).print(); // Maximum probability //! a.mean().print(); //! a.var().print(); //! a.sd().print(); //! } //! ``` //! //! ### Beta Distribution //! //! ### Gamma Distribution //! //! ### Binomial Distribution //! extern crate rand; extern crate rand_distr; use self::rand::distributions::uniform::SampleUniform; use self::rand::prelude::*; use self::rand_distr::Distribution; pub use self::OPDist::*; pub use self::TPDist::*; use crate::special::function::*; //use statistics::rand::ziggurat; use crate::statistics::{ops::C, stat::Statistics}; use std::convert::Into; use std::f64::consts::E; /// One parameter distribution /// /// # Distributions /// * `Bernoulli(prob)`: Bernoulli distribution #[derive(Debug, Clone)] pub enum OPDist<T: PartialOrd + SampleUniform + Copy + Into<f64>> { Bernoulli(T), StudentT(T), } /// Two parameter distribution /// /// # Distributions /// * `Uniform(start, end)`: Uniform distribution /// * `Normal(mean, std)`: Normal distribution #[derive(Debug, Clone)] pub enum TPDist<T: PartialOrd + SampleUniform + Copy + Into<f64>> { Uniform(T, T), Binomial(usize, T), Normal(T, T), Beta(T, T), Gamma(T, T), } /// Extract parameter pub trait ParametricDist { type Parameter; fn params(&self) -> Self::Parameter; } impl<T: PartialOrd + SampleUniform + Copy + Into<f64>> ParametricDist for OPDist<T> { type Parameter = f64; fn params(&self) -> Self::Parameter { match self { Bernoulli(mu) => (*mu).into(), StudentT(nu) => (*nu).into(), } } } impl<T: PartialOrd + SampleUniform + Copy + Into<f64>> ParametricDist for TPDist<T> { type Parameter = (f64, f64); fn params(&self) -> Self::Parameter { match self { Uniform(a, b) => ((*a).into(), (*b).into()), Binomial(a, b) => (*a as f64, (*b).into()), Normal(mu, sigma) => ((*mu).into(), (*sigma).into()), Beta(a, b) => ((*a).into(), (*b).into()), Gamma(a, b) => ((*a).into(), (*b).into()), } } } /// Random Number Generator trait /// /// # Methods /// * `sample`: extract samples pub trait RNG { /// Extract samples of distributions fn sample(&self, n: usize) -> Vec<f64>; /// Probability Distribution Function /// /// # Type /// `f64 -> f64` fn pdf<S: PartialOrd + SampleUniform + Copy + Into<f64>>(&self, x: S) -> f64; /// Cumulative Distribution Function /// /// # Type /// `f64` -> `f64` fn cdf<S: PartialOrd + SampleUniform + Copy + Into<f64>>(&self, x: S) -> f64; } /// RNG for OPDist impl<T: PartialOrd + SampleUniform + Copy + Into<f64>> RNG for OPDist<T> { fn sample(&self, n: usize) -> Vec<f64> { match self { Bernoulli(prob) => { assert!( (*prob).into() <= 1f64, "Probability should be smaller than 1" ); let mut rng = thread_rng(); let mut v = vec![0f64; n]; for i in 0..n { let uniform = rng.gen_range(0f64, 1f64); if uniform <= (*prob).into() { v[i] = 1f64; } else { v[i] = 0f64; } } v } StudentT(nu) => { let mut rng = thread_rng(); let stud = rand_distr::StudentT::<f64>::new((*nu).into()).unwrap(); stud.sample_iter(&mut rng).take(n).collect() } } } fn pdf<S: PartialOrd + SampleUniform + Copy + Into<f64>>(&self, x: S) -> f64 { match self { Bernoulli(prob) => { if x.into() == 1f64 { (*prob).into() } else { 1f64 - (*prob).into() } } StudentT(nu) => { let dof = (*nu).into(); let t = x.into(); 1f64 / (dof.sqrt() * beta(0.5f64, dof / 2f64)) * (1f64 + t.powi(2) / dof).powf(-(dof + 1f64) / 2f64) } } } fn cdf<S: PartialOrd + SampleUniform + Copy + Into<f64>>(&self, x: S) -> f64 { match self { Bernoulli(prob) => { let k: f64 = x.into(); if k < 0f64 { 0f64 } else if k < 1f64 { 1f64 - (*prob).into() } else { 1f64 } } StudentT(nu) => { let x: f64 = x.into(); let nu: f64 = (*nu).into(); let _odd_nu = (nu + 1f64) / 2f64; let even_nu = nu / 2f64; if x > 0f64 { let x_t = nu / (x.powi(2) + nu); 1f64 - 0.5 * inc_beta(even_nu, 0.5, x_t) } else if x < 0f64 { self.cdf(-x) - 0.5 } else { 0.5 } // 0.5f64 + x * gamma(odd_nu) * hyp2f1(0.5, odd_nu, 1.5, -x.powi(2) / (*nu).into()) / (PI * (*nu).into()).sqrt() * gamma(even_nu) } } } } /// RNG for TPDist impl<T: PartialOrd + SampleUniform + Copy + Into<f64>> RNG for TPDist<T> { fn sample(&self, n: usize) -> Vec<f64> { match self { Uniform(start, end) => { let mut rng = thread_rng(); let mut v = vec![0f64; n]; for i in 0..n { v[i] = rng.gen_range(*start, *end).into(); } v } Binomial(num, mu) => { let mut rng = thread_rng(); let binom = rand_distr::Binomial::new(*num as u64, (*mu).into()).unwrap(); binom .sample_iter(&mut rng) .take(n) .map(|t| t as f64) .collect() } Normal(m, s) => { let mut rng = thread_rng(); let normal = rand_distr::Normal::<f64>::new((*m).into(), (*s).into()).unwrap(); normal.sample_iter(&mut rng).take(n).collect() } // Normal(m, s) => { // let mut rng = thread_rng(); // let mut v = vec![0f64; n]; // // for i in 0..n { // v[i] = ziggurat(&mut rng, (*s).into()) + (*m).into(); // } // v // } Beta(a, b) => { let mut rng = thread_rng(); let beta = rand_distr::Beta::<f64>::new((*a).into(), (*b).into()).unwrap(); beta.sample_iter(&mut rng).take(n).collect() } // Beta(a, b) => { // let mut rng1 = thread_rng(); // let mut rng2 = thread_rng(); // let mut v = vec![0f64; n]; // // let a_f64 = (*a).into(); // let b_f64 = (*b).into(); // // // For acceptance-rejection method // let c_x = (a_f64 - 1f64) / (a_f64 + b_f64 - 2f64); // let c = self.pdf(c_x); // Beta(mode(x) | a, b) // // let mut iter_num = 0usize; // // while iter_num < n { // let u1 = rng1.gen_range(0f64, 1f64); // let u2 = rng2.gen_range(0f64, 1f64); // // if u2 <= 1f64 / c * self.pdf(u1) { // v[iter_num] = u1; // iter_num += 1; // } // } // v // } Gamma(shape, scale) => { let mut rng = thread_rng(); let gamma = rand_distr::Gamma::<f64>::new((*shape).into(), (*scale).into()).unwrap(); gamma.sample_iter(&mut rng).take(n).collect() } // Gamma(a, b) => { // let a_f64 = (*a).into(); // let b_f64 = (*b).into(); // // // for Marsaglia & Tsang's Method // let d = a_f64 - 1f64 / 3f64; // let c = 1f64 / (9f64 * d).sqrt(); // // let mut rng1 = thread_rng(); // let mut rng2 = thread_rng(); // // let mut v = vec![0f64; n]; // let mut iter_num = 0usize; // // while iter_num < n { // let u = rng1.gen_range(0f64, 1f64); // let z = ziggurat(&mut rng2, 1f64); // let w = (1f64 + c * z).powi(3); // // if z >= -1f64 / c && u.ln() < 0.5 * z.powi(2) + d - d * w + d * w.ln() { // v[iter_num] = d * w / b_f64; // iter_num += 1; // } // } // v // } } } fn pdf<S: PartialOrd + SampleUniform + Copy + Into<f64>>(&self, x: S) -> f64 { match self { Uniform(a, b) => { let val = x.into(); let a_f64 = (*a).into(); let b_f64 = (*b).into(); if val >= a_f64 && val <= b_f64 { let length = b_f64 - a_f64; 1f64 / length } else { 0f64 } } Binomial(n, mu) => { let n = *n; let mu = (*mu).into(); let m = x.into() as usize; (C(n, m) as f64) * mu.powi(m as i32) * (1f64 - mu).powi((n - m) as i32) } Normal(m, s) => { let mean = (*m).into(); let std = (*s).into(); gaussian(x.into(), mean, std) } Beta(a, b) => { let a_f64 = (*a).into(); let b_f64 = (*b).into(); 1f64 / beta(a_f64, b_f64) * x.into().powf(a_f64 - 1f64) * (1f64 - x.into()).powf(b_f64 - 1f64) } Gamma(a, b) => { let a_f64 = (*a).into(); let b_f64 = (*b).into(); 1f64 / gamma(a_f64) * b_f64.powf(a_f64) * x.into().powf(a_f64 - 1f64) * E.powf(-b_f64 * x.into()) } } } fn cdf<S: PartialOrd + SampleUniform + Copy + Into<f64>>(&self, x: S) -> f64 { let x: f64 = x.into(); match self { Uniform(a, b) => { let a: f64 = (*a).into(); let b: f64 = (*b).into(); if x < a { 0f64 } else if x <= b { (x - a) / (b - a) } else { 1f64 } } Binomial(n, mu) => { let n = *n; let p = (*mu).into(); let q = 1f64 - p; let k: f64 = x.into(); inc_beta(n as f64 - k, k + 1f64, q) } Normal(m, s) => phi((x - (*m).into()) / (*s).into()), Beta(a, b) => { let a: f64 = (*a).into(); let b: f64 = (*b).into(); inc_beta(a, b, x) } Gamma(a, b) => { let a: f64 = (*a).into(); let b: f64 = (*b).into(); inc_gamma(a, b * x) } } } } impl<T: PartialOrd + SampleUniform + Copy + Into<f64>> Statistics for OPDist<T> { type Array = Vec<f64>; type Value = f64; fn mean(&self) -> Self::Value { match self { Bernoulli(mu) => (*mu).into(), StudentT(_) => 0f64, } } fn var(&self) -> Self::Value { match self { Bernoulli(mu) => { let mu_f64 = (*mu).into(); mu_f64 * (1f64 - mu_f64) } StudentT(nu) => { let nu_f64 = (*nu).into(); nu_f64 / (nu_f64 - 2f64) } } } fn sd(&self) -> Self::Value { match self { Bernoulli(_mu) => self.var().sqrt(), StudentT(_nu) => self.var().sqrt(), } } fn cov(&self) -> Self::Array { unimplemented!() } fn cor(&self) -> Self::Array { unimplemented!() } } impl<T: PartialOrd + SampleUniform + Copy + Into<f64>> Statistics for TPDist<T> { type Array = Vec<f64>; type Value = f64; fn mean(&self) -> Self::Value { match self { Uniform(a, b) => ((*a).into() + (*b).into()) / 2f64, Binomial(n, mu) => (*n as f64) * (*mu).into(), Normal(m, _s) => (*m).into(), Beta(a, b) => (*a).into() / ((*a).into() + (*b).into()), Gamma(a, b) => (*a).into() / (*b).into(), } } fn var(&self) -> Self::Value { match self { Uniform(a, b) => ((*b).into() - (*a).into()).powi(2) / 12f64, Binomial(n, mu) => (*n as f64) * (*mu).into() * (1f64 - (*mu).into()), Normal(_m, s) => (*s).into().powi(2), Beta(a, b) => { let a_f64 = (*a).into(); let b_f64 = (*b).into(); a_f64 * b_f64 / ((a_f64 + b_f64).powi(2) * (a_f64 + b_f64 + 1f64)) } Gamma(a, b) => (*a).into() / (*b).into().powi(2), } } fn sd(&self) -> Self::Value { match self { Uniform(_a, _b) => self.var().sqrt(), Binomial(_n, _mu) => self.var().sqrt(), Normal(_m, s) => (*s).into(), Beta(_a, _b) => self.var().sqrt(), Gamma(_a, _b) => self.var().sqrt(), } } fn cov(&self) -> Self::Array { unimplemented!() } fn cor(&self) -> Self::Array { unimplemented!() } }