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| </pre><pre class="rust"><code><span class="comment">// Copyright 2018 Developers of the Rand project. |
| // Copyright 2013 The Rust Project Developers. |
| // |
| // Licensed under the Apache License, Version 2.0 <LICENSE-APACHE or |
| // https://www.apache.org/licenses/LICENSE-2.0> or the MIT license |
| // <LICENSE-MIT or https://opensource.org/licenses/MIT>, at your |
| // option. This file may not be copied, modified, or distributed |
| // except according to those terms. |
| |
| </span><span class="doccomment">//! The normal and derived distributions. |
| |
| </span><span class="kw">use </span><span class="kw">crate</span>::utils::ziggurat; |
| <span class="kw">use </span>num_traits::Float; |
| <span class="kw">use crate</span>::{ziggurat_tables, Distribution, Open01}; |
| <span class="kw">use </span>rand::Rng; |
| <span class="kw">use </span>core::fmt; |
| |
| <span class="doccomment">/// Samples floating-point numbers according to the normal distribution |
| /// `N(0, 1)` (a.k.a. a standard normal, or Gaussian). This is equivalent to |
| /// `Normal::new(0.0, 1.0)` but faster. |
| /// |
| /// See `Normal` for the general normal distribution. |
| /// |
| /// Implemented via the ZIGNOR variant[^1] of the Ziggurat method. |
| /// |
| /// [^1]: Jurgen A. Doornik (2005). [*An Improved Ziggurat Method to |
| /// Generate Normal Random Samples*]( |
| /// https://www.doornik.com/research/ziggurat.pdf). |
| /// Nuffield College, Oxford |
| /// |
| /// # Example |
| /// ``` |
| /// use rand::prelude::*; |
| /// use rand_distr::StandardNormal; |
| /// |
| /// let val: f64 = thread_rng().sample(StandardNormal); |
| /// println!("{}", val); |
| /// ``` |
| </span><span class="attribute">#[derive(Clone, Copy, Debug)] |
| #[cfg_attr(feature = <span class="string">"serde1"</span>, derive(serde::Serialize, serde::Deserialize))] |
| </span><span class="kw">pub struct </span>StandardNormal; |
| |
| <span class="kw">impl </span>Distribution<f32> <span class="kw">for </span>StandardNormal { |
| <span class="attribute">#[inline] |
| </span><span class="kw">fn </span>sample<R: Rng + <span class="question-mark">?</span>Sized>(<span class="kw-2">&</span><span class="self">self</span>, rng: <span class="kw-2">&mut </span>R) -> f32 { |
| <span class="comment">// TODO: use optimal 32-bit implementation |
| </span><span class="kw">let </span>x: f64 = <span class="self">self</span>.sample(rng); |
| x <span class="kw">as </span>f32 |
| } |
| } |
| |
| <span class="kw">impl </span>Distribution<f64> <span class="kw">for </span>StandardNormal { |
| <span class="kw">fn </span>sample<R: Rng + <span class="question-mark">?</span>Sized>(<span class="kw-2">&</span><span class="self">self</span>, rng: <span class="kw-2">&mut </span>R) -> f64 { |
| <span class="attribute">#[inline] |
| </span><span class="kw">fn </span>pdf(x: f64) -> f64 { |
| (-x * x / <span class="number">2.0</span>).exp() |
| } |
| <span class="attribute">#[inline] |
| </span><span class="kw">fn </span>zero_case<R: Rng + <span class="question-mark">?</span>Sized>(rng: <span class="kw-2">&mut </span>R, u: f64) -> f64 { |
| <span class="comment">// compute a random number in the tail by hand |
| |
| // strange initial conditions, because the loop is not |
| // do-while, so the condition should be true on the first |
| // run, they get overwritten anyway (0 < 1, so these are |
| // good). |
| </span><span class="kw">let </span><span class="kw-2">mut </span>x = <span class="number">1.0f64</span>; |
| <span class="kw">let </span><span class="kw-2">mut </span>y = <span class="number">0.0f64</span>; |
| |
| <span class="kw">while </span>-<span class="number">2.0 </span>* y < x * x { |
| <span class="kw">let </span>x_: f64 = rng.sample(Open01); |
| <span class="kw">let </span>y_: f64 = rng.sample(Open01); |
| |
| x = x_.ln() / ziggurat_tables::ZIG_NORM_R; |
| y = y_.ln(); |
| } |
| |
| <span class="kw">if </span>u < <span class="number">0.0 </span>{ |
| x - ziggurat_tables::ZIG_NORM_R |
| } <span class="kw">else </span>{ |
| ziggurat_tables::ZIG_NORM_R - x |
| } |
| } |
| |
| ziggurat( |
| rng, |
| <span class="bool-val">true</span>, <span class="comment">// this is symmetric |
| </span><span class="kw-2">&</span>ziggurat_tables::ZIG_NORM_X, |
| <span class="kw-2">&</span>ziggurat_tables::ZIG_NORM_F, |
| pdf, |
| zero_case, |
| ) |
| } |
| } |
| |
| <span class="doccomment">/// The normal distribution `N(mean, std_dev**2)`. |
| /// |
| /// This uses the ZIGNOR variant of the Ziggurat method, see [`StandardNormal`] |
| /// for more details. |
| /// |
| /// Note that [`StandardNormal`] is an optimised implementation for mean 0, and |
| /// standard deviation 1. |
| /// |
| /// # Example |
| /// |
| /// ``` |
| /// use rand_distr::{Normal, Distribution}; |
| /// |
| /// // mean 2, standard deviation 3 |
| /// let normal = Normal::new(2.0, 3.0).unwrap(); |
| /// let v = normal.sample(&mut rand::thread_rng()); |
| /// println!("{} is from a N(2, 9) distribution", v) |
| /// ``` |
| /// |
| /// [`StandardNormal`]: crate::StandardNormal |
| </span><span class="attribute">#[derive(Clone, Copy, Debug)] |
| #[cfg_attr(feature = <span class="string">"serde1"</span>, derive(serde::Serialize, serde::Deserialize))] |
| </span><span class="kw">pub struct </span>Normal<F> |
| <span class="kw">where </span>F: Float, StandardNormal: Distribution<F> |
| { |
| mean: F, |
| std_dev: F, |
| } |
| |
| <span class="doccomment">/// Error type returned from `Normal::new` and `LogNormal::new`. |
| </span><span class="attribute">#[derive(Clone, Copy, Debug, PartialEq, Eq)] |
| </span><span class="kw">pub enum </span>Error { |
| <span class="doccomment">/// The mean value is too small (log-normal samples must be positive) |
| </span>MeanTooSmall, |
| <span class="doccomment">/// The standard deviation or other dispersion parameter is not finite. |
| </span>BadVariance, |
| } |
| |
| <span class="kw">impl </span>fmt::Display <span class="kw">for </span>Error { |
| <span class="kw">fn </span>fmt(<span class="kw-2">&</span><span class="self">self</span>, f: <span class="kw-2">&mut </span>fmt::Formatter<<span class="lifetime">'_</span>>) -> fmt::Result { |
| f.write_str(<span class="kw">match </span><span class="self">self </span>{ |
| Error::MeanTooSmall => <span class="string">"mean < 0 or NaN in log-normal distribution"</span>, |
| Error::BadVariance => <span class="string">"variation parameter is non-finite in (log)normal distribution"</span>, |
| }) |
| } |
| } |
| |
| <span class="attribute">#[cfg(feature = <span class="string">"std"</span>)] |
| #[cfg_attr(doc_cfg, doc(cfg(feature = <span class="string">"std"</span>)))] |
| </span><span class="kw">impl </span>std::error::Error <span class="kw">for </span>Error {} |
| |
| <span class="kw">impl</span><F> Normal<F> |
| <span class="kw">where </span>F: Float, StandardNormal: Distribution<F> |
| { |
| <span class="doccomment">/// Construct, from mean and standard deviation |
| /// |
| /// Parameters: |
| /// |
| /// - mean (`μ`, unrestricted) |
| /// - standard deviation (`σ`, must be finite) |
| </span><span class="attribute">#[inline] |
| </span><span class="kw">pub fn </span>new(mean: F, std_dev: F) -> <span class="prelude-ty">Result</span><Normal<F>, Error> { |
| <span class="kw">if </span>!std_dev.is_finite() { |
| <span class="kw">return </span><span class="prelude-val">Err</span>(Error::BadVariance); |
| } |
| <span class="prelude-val">Ok</span>(Normal { mean, std_dev }) |
| } |
| |
| <span class="doccomment">/// Construct, from mean and coefficient of variation |
| /// |
| /// Parameters: |
| /// |
| /// - mean (`μ`, unrestricted) |
| /// - coefficient of variation (`cv = abs(σ / μ)`) |
| </span><span class="attribute">#[inline] |
| </span><span class="kw">pub fn </span>from_mean_cv(mean: F, cv: F) -> <span class="prelude-ty">Result</span><Normal<F>, Error> { |
| <span class="kw">if </span>!cv.is_finite() || cv < F::zero() { |
| <span class="kw">return </span><span class="prelude-val">Err</span>(Error::BadVariance); |
| } |
| <span class="kw">let </span>std_dev = cv * mean; |
| <span class="prelude-val">Ok</span>(Normal { mean, std_dev }) |
| } |
| |
| <span class="doccomment">/// Sample from a z-score |
| /// |
| /// This may be useful for generating correlated samples `x1` and `x2` |
| /// from two different distributions, as follows. |
| /// ``` |
| /// # use rand::prelude::*; |
| /// # use rand_distr::{Normal, StandardNormal}; |
| /// let mut rng = thread_rng(); |
| /// let z = StandardNormal.sample(&mut rng); |
| /// let x1 = Normal::new(0.0, 1.0).unwrap().from_zscore(z); |
| /// let x2 = Normal::new(2.0, -3.0).unwrap().from_zscore(z); |
| /// ``` |
| </span><span class="attribute">#[inline] |
| </span><span class="kw">pub fn </span>from_zscore(<span class="kw-2">&</span><span class="self">self</span>, zscore: F) -> F { |
| <span class="self">self</span>.mean + <span class="self">self</span>.std_dev * zscore |
| } |
| |
| <span class="doccomment">/// Returns the mean (`μ`) of the distribution. |
| </span><span class="kw">pub fn </span>mean(<span class="kw-2">&</span><span class="self">self</span>) -> F { |
| <span class="self">self</span>.mean |
| } |
| |
| <span class="doccomment">/// Returns the standard deviation (`σ`) of the distribution. |
| </span><span class="kw">pub fn </span>std_dev(<span class="kw-2">&</span><span class="self">self</span>) -> F { |
| <span class="self">self</span>.std_dev |
| } |
| } |
| |
| <span class="kw">impl</span><F> Distribution<F> <span class="kw">for </span>Normal<F> |
| <span class="kw">where </span>F: Float, StandardNormal: Distribution<F> |
| { |
| <span class="kw">fn </span>sample<R: Rng + <span class="question-mark">?</span>Sized>(<span class="kw-2">&</span><span class="self">self</span>, rng: <span class="kw-2">&mut </span>R) -> F { |
| <span class="self">self</span>.from_zscore(rng.sample(StandardNormal)) |
| } |
| } |
| |
| |
| <span class="doccomment">/// The log-normal distribution `ln N(mean, std_dev**2)`. |
| /// |
| /// If `X` is log-normal distributed, then `ln(X)` is `N(mean, std_dev**2)` |
| /// distributed. |
| /// |
| /// # Example |
| /// |
| /// ``` |
| /// use rand_distr::{LogNormal, Distribution}; |
| /// |
| /// // mean 2, standard deviation 3 |
| /// let log_normal = LogNormal::new(2.0, 3.0).unwrap(); |
| /// let v = log_normal.sample(&mut rand::thread_rng()); |
| /// println!("{} is from an ln N(2, 9) distribution", v) |
| /// ``` |
| </span><span class="attribute">#[derive(Clone, Copy, Debug)] |
| #[cfg_attr(feature = <span class="string">"serde1"</span>, derive(serde::Serialize, serde::Deserialize))] |
| </span><span class="kw">pub struct </span>LogNormal<F> |
| <span class="kw">where </span>F: Float, StandardNormal: Distribution<F> |
| { |
| norm: Normal<F>, |
| } |
| |
| <span class="kw">impl</span><F> LogNormal<F> |
| <span class="kw">where </span>F: Float, StandardNormal: Distribution<F> |
| { |
| <span class="doccomment">/// Construct, from (log-space) mean and standard deviation |
| /// |
| /// Parameters are the "standard" log-space measures (these are the mean |
| /// and standard deviation of the logarithm of samples): |
| /// |
| /// - `mu` (`μ`, unrestricted) is the mean of the underlying distribution |
| /// - `sigma` (`σ`, must be finite) is the standard deviation of the |
| /// underlying Normal distribution |
| </span><span class="attribute">#[inline] |
| </span><span class="kw">pub fn </span>new(mu: F, sigma: F) -> <span class="prelude-ty">Result</span><LogNormal<F>, Error> { |
| <span class="kw">let </span>norm = Normal::new(mu, sigma)<span class="question-mark">?</span>; |
| <span class="prelude-val">Ok</span>(LogNormal { norm }) |
| } |
| |
| <span class="doccomment">/// Construct, from (linear-space) mean and coefficient of variation |
| /// |
| /// Parameters are linear-space measures: |
| /// |
| /// - mean (`μ > 0`) is the (real) mean of the distribution |
| /// - coefficient of variation (`cv = σ / μ`, requiring `cv ≥ 0`) is a |
| /// standardized measure of dispersion |
| /// |
| /// As a special exception, `μ = 0, cv = 0` is allowed (samples are `-inf`). |
| </span><span class="attribute">#[inline] |
| </span><span class="kw">pub fn </span>from_mean_cv(mean: F, cv: F) -> <span class="prelude-ty">Result</span><LogNormal<F>, Error> { |
| <span class="kw">if </span>cv == F::zero() { |
| <span class="kw">let </span>mu = mean.ln(); |
| <span class="kw">let </span>norm = Normal::new(mu, F::zero()).unwrap(); |
| <span class="kw">return </span><span class="prelude-val">Ok</span>(LogNormal { norm }); |
| } |
| <span class="kw">if </span>!(mean > F::zero()) { |
| <span class="kw">return </span><span class="prelude-val">Err</span>(Error::MeanTooSmall); |
| } |
| <span class="kw">if </span>!(cv >= F::zero()) { |
| <span class="kw">return </span><span class="prelude-val">Err</span>(Error::BadVariance); |
| } |
| |
| <span class="comment">// Using X ~ lognormal(μ, σ), CV² = Var(X) / E(X)² |
| // E(X) = exp(μ + σ² / 2) = exp(μ) × exp(σ² / 2) |
| // Var(X) = exp(2μ + σ²)(exp(σ²) - 1) = E(X)² × (exp(σ²) - 1) |
| // but Var(X) = (CV × E(X))² so CV² = exp(σ²) - 1 |
| // thus σ² = log(CV² + 1) |
| // and exp(μ) = E(X) / exp(σ² / 2) = E(X) / sqrt(CV² + 1) |
| </span><span class="kw">let </span>a = F::one() + cv * cv; <span class="comment">// e |
| </span><span class="kw">let </span>mu = F::from(<span class="number">0.5</span>).unwrap() * (mean * mean / a).ln(); |
| <span class="kw">let </span>sigma = a.ln().sqrt(); |
| <span class="kw">let </span>norm = Normal::new(mu, sigma)<span class="question-mark">?</span>; |
| <span class="prelude-val">Ok</span>(LogNormal { norm }) |
| } |
| |
| <span class="doccomment">/// Sample from a z-score |
| /// |
| /// This may be useful for generating correlated samples `x1` and `x2` |
| /// from two different distributions, as follows. |
| /// ``` |
| /// # use rand::prelude::*; |
| /// # use rand_distr::{LogNormal, StandardNormal}; |
| /// let mut rng = thread_rng(); |
| /// let z = StandardNormal.sample(&mut rng); |
| /// let x1 = LogNormal::from_mean_cv(3.0, 1.0).unwrap().from_zscore(z); |
| /// let x2 = LogNormal::from_mean_cv(2.0, 4.0).unwrap().from_zscore(z); |
| /// ``` |
| </span><span class="attribute">#[inline] |
| </span><span class="kw">pub fn </span>from_zscore(<span class="kw-2">&</span><span class="self">self</span>, zscore: F) -> F { |
| <span class="self">self</span>.norm.from_zscore(zscore).exp() |
| } |
| } |
| |
| <span class="kw">impl</span><F> Distribution<F> <span class="kw">for </span>LogNormal<F> |
| <span class="kw">where </span>F: Float, StandardNormal: Distribution<F> |
| { |
| <span class="attribute">#[inline] |
| </span><span class="kw">fn </span>sample<R: Rng + <span class="question-mark">?</span>Sized>(<span class="kw-2">&</span><span class="self">self</span>, rng: <span class="kw-2">&mut </span>R) -> F { |
| <span class="self">self</span>.norm.sample(rng).exp() |
| } |
| } |
| |
| <span class="attribute">#[cfg(test)] |
| </span><span class="kw">mod </span>tests { |
| <span class="kw">use super</span>::<span class="kw-2">*</span>; |
| |
| <span class="attribute">#[test] |
| </span><span class="kw">fn </span>test_normal() { |
| <span class="kw">let </span>norm = Normal::new(<span class="number">10.0</span>, <span class="number">10.0</span>).unwrap(); |
| <span class="kw">let </span><span class="kw-2">mut </span>rng = <span class="kw">crate</span>::test::rng(<span class="number">210</span>); |
| <span class="kw">for _ in </span><span class="number">0</span>..<span class="number">1000 </span>{ |
| norm.sample(<span class="kw-2">&mut </span>rng); |
| } |
| } |
| <span class="attribute">#[test] |
| </span><span class="kw">fn </span>test_normal_cv() { |
| <span class="kw">let </span>norm = Normal::from_mean_cv(<span class="number">1024.0</span>, <span class="number">1.0 </span>/ <span class="number">256.0</span>).unwrap(); |
| <span class="macro">assert_eq!</span>((norm.mean, norm.std_dev), (<span class="number">1024.0</span>, <span class="number">4.0</span>)); |
| } |
| <span class="attribute">#[test] |
| </span><span class="kw">fn </span>test_normal_invalid_sd() { |
| <span class="macro">assert!</span>(Normal::from_mean_cv(<span class="number">10.0</span>, -<span class="number">1.0</span>).is_err()); |
| } |
| |
| <span class="attribute">#[test] |
| </span><span class="kw">fn </span>test_log_normal() { |
| <span class="kw">let </span>lnorm = LogNormal::new(<span class="number">10.0</span>, <span class="number">10.0</span>).unwrap(); |
| <span class="kw">let </span><span class="kw-2">mut </span>rng = <span class="kw">crate</span>::test::rng(<span class="number">211</span>); |
| <span class="kw">for _ in </span><span class="number">0</span>..<span class="number">1000 </span>{ |
| lnorm.sample(<span class="kw-2">&mut </span>rng); |
| } |
| } |
| <span class="attribute">#[test] |
| </span><span class="kw">fn </span>test_log_normal_cv() { |
| <span class="kw">let </span>lnorm = LogNormal::from_mean_cv(<span class="number">0.0</span>, <span class="number">0.0</span>).unwrap(); |
| <span class="macro">assert_eq!</span>((lnorm.norm.mean, lnorm.norm.std_dev), (-core::f64::INFINITY, <span class="number">0.0</span>)); |
| |
| <span class="kw">let </span>lnorm = LogNormal::from_mean_cv(<span class="number">1.0</span>, <span class="number">0.0</span>).unwrap(); |
| <span class="macro">assert_eq!</span>((lnorm.norm.mean, lnorm.norm.std_dev), (<span class="number">0.0</span>, <span class="number">0.0</span>)); |
| |
| <span class="kw">let </span>e = core::f64::consts::E; |
| <span class="kw">let </span>lnorm = LogNormal::from_mean_cv(e.sqrt(), (e - <span class="number">1.0</span>).sqrt()).unwrap(); |
| <span class="macro">assert_almost_eq!</span>(lnorm.norm.mean, <span class="number">0.0</span>, <span class="number">2e-16</span>); |
| <span class="macro">assert_almost_eq!</span>(lnorm.norm.std_dev, <span class="number">1.0</span>, <span class="number">2e-16</span>); |
| |
| <span class="kw">let </span>lnorm = LogNormal::from_mean_cv(e.powf(<span class="number">1.5</span>), (e - <span class="number">1.0</span>).sqrt()).unwrap(); |
| <span class="macro">assert_almost_eq!</span>(lnorm.norm.mean, <span class="number">1.0</span>, <span class="number">1e-15</span>); |
| <span class="macro">assert_eq!</span>(lnorm.norm.std_dev, <span class="number">1.0</span>); |
| } |
| <span class="attribute">#[test] |
| </span><span class="kw">fn </span>test_log_normal_invalid_sd() { |
| <span class="macro">assert!</span>(LogNormal::from_mean_cv(-<span class="number">1.0</span>, <span class="number">1.0</span>).is_err()); |
| <span class="macro">assert!</span>(LogNormal::from_mean_cv(<span class="number">0.0</span>, <span class="number">1.0</span>).is_err()); |
| <span class="macro">assert!</span>(LogNormal::from_mean_cv(<span class="number">1.0</span>, -<span class="number">1.0</span>).is_err()); |
| } |
| } |
| </code></pre></div> |
| </section></div></main><div id="rustdoc-vars" data-root-path="../../" data-current-crate="rand_distr" data-themes="ayu,dark,light" data-resource-suffix="" data-rustdoc-version="1.66.0-nightly (5c8bff74b 2022-10-21)" ></div></body></html> |