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//! A fast generator of discrete, bounded //! [Zipf-distributed](https://en.wikipedia.org/wiki/Zipf's_law) random numbers. //! //! For a random variable `X` whose values are distributed according to this distribution, the //! probability mass function is given by //! //! ```ignore //! P(X = k) = H(N,s) * 1 / k^s for k = 1,2,...,N //! ``` //! //! `H(N,s)` is the normalizing constant which corresponds to the generalized harmonic number //! of order `N` of `s`. //! //! This implementation is effectively a direct port of Apache Common's //! [RejectionInversionZipfSampler][ref], written in Java. It is based on the method described by //! Wolfgang Hörmann and Gerhard Derflinger in [*Rejection-inversion to generate variates from //! monotone discrete distributions*](https://dl.acm.org/citation.cfm?id=235029) from *ACM //! Transactions on Modeling and Computer Simulation (TOMACS) 6.3 (1996)*. //! //! [ref]: https://github.com/apache/commons-rng/blob/6a1b0c16090912e8fc5de2c1fb5bd8490ac14699/commons-rng-sampling/src/main/java/org/apache/commons/rng/sampling/distribution/RejectionInversionZipfSampler.java // XXX: uncomment and run nightly to benchmark // #![feature(test)] // extern crate test; extern crate rand; use rand::Rng; #[cfg(test)] extern crate randomkit; /// Random number generator that generates Zipf-distributed random numbers using rejection /// inversion. #[derive(Clone, Copy)] pub struct ZipfDistribution { /// Number of elements num_elements: f64, /// Exponent parameter of the distribution exponent: f64, /// `hIntegral(1.5) - 1}` h_integral_x1: f64, /// `hIntegral(num_elements + 0.5)}` h_integral_num_elements: f64, /// `2 - hIntegralInverse(hIntegral(2.5) - h(2)}` s: f64, } impl ZipfDistribution { /// Creates a new [Zipf-distributed](https://en.wikipedia.org/wiki/Zipf's_law) /// random number generator. /// /// Note that both the number of elements and the exponent must be greater than 0. pub fn new(num_elements: usize, exponent: f64) -> Result<Self, ()> { if num_elements == 0 { return Err(()); } if exponent <= 0f64 { return Err(()); } let z = ZipfDistribution { num_elements: num_elements as f64, exponent, h_integral_x1: ZipfDistribution::h_integral(1.5, exponent) - 1f64, h_integral_num_elements: ZipfDistribution::h_integral( num_elements as f64 + 0.5, exponent, ), s: 2f64 - ZipfDistribution::h_integral_inv( ZipfDistribution::h_integral(2.5, exponent) - ZipfDistribution::h(2f64, exponent), exponent, ), }; // populate cache Ok(z) } } impl ZipfDistribution { fn next<R: Rng + ?Sized>(&self, rng: &mut R) -> usize { // The paper describes an algorithm for exponents larger than 1 (Algorithm ZRI). // // The original method uses // H(x) = (v + x)^(1 - q) / (1 - q) // as the integral of the hat function. // // This function is undefined for q = 1, which is the reason for the limitation of the // exponent. // // If instead the integral function // H(x) = ((v + x)^(1 - q) - 1) / (1 - q) // is used, for which a meaningful limit exists for q = 1, the method works for all // positive exponents. // // The following implementation uses v = 0 and generates integral number in the range [1, // num_elements]. This is different to the original method where v is defined to // be positive and numbers are taken from [0, i_max]. This explains why the implementation // looks slightly different. let hnum = self.h_integral_num_elements; loop { use std::cmp; let u: f64 = hnum + rng.gen::<f64>() * (self.h_integral_x1 - hnum); // u is uniformly distributed in (h_integral_x1, h_integral_num_elements] let x: f64 = ZipfDistribution::h_integral_inv(u, self.exponent); // Limit k to the range [1, num_elements] if it would be outside // due to numerical inaccuracies. let k64 = x.max(1.0).min(self.num_elements); // float -> integer rounds towards zero let k = cmp::max(1, k64 as usize); // Here, the distribution of k is given by: // // P(k = 1) = C * (hIntegral(1.5) - h_integral_x1) = C // P(k = m) = C * (hIntegral(m + 1/2) - hIntegral(m - 1/2)) for m >= 2 // // where C = 1 / (h_integral_num_elements - h_integral_x1) if k64 - x <= self.s || u >= ZipfDistribution::h_integral(k64 + 0.5, self.exponent) - ZipfDistribution::h(k64, self.exponent) { // Case k = 1: // // The right inequality is always true, because replacing k by 1 gives // u >= hIntegral(1.5) - h(1) = h_integral_x1 and u is taken from // (h_integral_x1, h_integral_num_elements]. // // Therefore, the acceptance rate for k = 1 is P(accepted | k = 1) = 1 // and the probability that 1 is returned as random value is // P(k = 1 and accepted) = P(accepted | k = 1) * P(k = 1) = C = C / 1^exponent // // Case k >= 2: // // The left inequality (k - x <= s) is just a short cut // to avoid the more expensive evaluation of the right inequality // (u >= hIntegral(k + 0.5) - h(k)) in many cases. // // If the left inequality is true, the right inequality is also true: // Theorem 2 in the paper is valid for all positive exponents, because // the requirements h'(x) = -exponent/x^(exponent + 1) < 0 and // (-1/hInverse'(x))'' = (1+1/exponent) * x^(1/exponent-1) >= 0 // are both fulfilled. // Therefore, f(x) = x - hIntegralInverse(hIntegral(x + 0.5) - h(x)) // is a non-decreasing function. If k - x <= s holds, // k - x <= s + f(k) - f(2) is obviously also true which is equivalent to // -x <= -hIntegralInverse(hIntegral(k + 0.5) - h(k)), // -hIntegralInverse(u) <= -hIntegralInverse(hIntegral(k + 0.5) - h(k)), // and finally u >= hIntegral(k + 0.5) - h(k). // // Hence, the right inequality determines the acceptance rate: // P(accepted | k = m) = h(m) / (hIntegrated(m+1/2) - hIntegrated(m-1/2)) // The probability that m is returned is given by // P(k = m and accepted) = P(accepted | k = m) * P(k = m) // = C * h(m) = C / m^exponent. // // In both cases the probabilities are proportional to the probability mass // function of the Zipf distribution. return k; } } } } #[allow(deprecated)] impl rand::distributions::Sample<usize> for ZipfDistribution { fn sample<R: Rng>(&mut self, rng: &mut R) -> usize { self.next(rng) } } #[allow(deprecated)] impl rand::distributions::IndependentSample<usize> for ZipfDistribution { fn ind_sample<R: Rng>(&self, rng: &mut R) -> usize { self.next(rng) } } impl rand::distributions::Distribution<usize> for ZipfDistribution { fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> usize { self.next(rng) } } use std::fmt; impl fmt::Debug for ZipfDistribution { fn fmt(&self, f: &mut fmt::Formatter) -> Result<(), fmt::Error> { f.debug_struct("ZipfDistribution") .field("e", &self.exponent) .field("n", &self.num_elements) .finish() } } impl ZipfDistribution { /// Computes `H(x)`, defined as /// /// - `(x^(1 - exponent) - 1) / (1 - exponent)`, if `exponent != 1` /// - `log(x)`, if `exponent == 1` /// /// `H(x)` is an integral function of `h(x)`, the derivative of `H(x)` is `h(x)`. fn h_integral(x: f64, exponent: f64) -> f64 { let log_x = x.ln(); helper2((1f64 - exponent) * log_x) * log_x } /// Computes `h(x) = 1 / x^exponent` fn h(x: f64, exponent: f64) -> f64 { (-exponent * x.ln()).exp() } /// The inverse function of `H(x)`. /// Returns the `y` for which `H(y) = x`. fn h_integral_inv(x: f64, exponent: f64) -> f64 { let mut t: f64 = x * (1f64 - exponent); if t < -1f64 { // Limit value to the range [-1, +inf). // t could be smaller than -1 in some rare cases due to numerical errors. t = -1f64; } (helper1(t) * x).exp() } } /// Helper function that calculates `log(1 + x) / x`. /// A Taylor series expansion is used, if x is close to 0. fn helper1(x: f64) -> f64 { if x.abs() > 1e-8 { x.ln_1p() / x } else { 1f64 - x * (0.5 - x * (1.0 / 3.0 - 0.25 * x)) } } /// Helper function to calculate `(exp(x) - 1) / x`. /// A Taylor series expansion is used, if x is close to 0. fn helper2(x: f64) -> f64 { if x.abs() > 1e-8 { x.exp_m1() / x } else { 1f64 + x * 0.5 * (1f64 + x * 1.0 / 3.0 * (1f64 + 0.25 * x)) } } #[cfg(test)] mod tests { use randomkit; #[test] fn generate() { use super::ZipfDistribution; use rand; let n = 1000000; let cdf_steps = 1000usize; // sample our distribution let mut rng = rand::thread_rng(); let us = ZipfDistribution::new(n, 1.07).unwrap(); let mut f1: Vec<_> = (0..n) .map(|_| { use rand::distributions::Distribution; us.sample(&mut rng) as u64 }) .collect(); f1.sort(); // sample "real"/slow numpy zipf distribution let mut rng = randomkit::Rng::new().unwrap(); let oracle = randomkit::dist::Zipf::new(1.07).unwrap(); let mut f2: Vec<_> = (0..n) .map(|_| { use randomkit::Sample; 1 + (n as f64 * oracle.sample(&mut rng) as f64 / isize::max_value() as f64) as u64 }) .collect(); f2.sort(); // compute ECDF of both sample populations let cdf = (0..cdf_steps).map(|i| { // number of things less than n*i/cdf_steps let t = (n as f64 * i as f64 / cdf_steps as f64) as u64; let f1i = match f1.binary_search(&t) { Ok(i) => i, Err(i) => i, }; let f2i = match f2.binary_search(&t) { Ok(i) => i, Err(i) => i, }; println!("{} {} {}", t, f1i as f64 / n as f64, f2i as f64 / n as f64); (f1i as f64 / n as f64, f2i as f64 / n as f64) }); // Two-sample Kolmogorov–Smirnov test // https://en.wikipedia.org/wiki/Kolmogorov–Smirnov_test#Two-sample_Kolmogorov.E2.80.93Smirnov_test let diff = cdf.map(|(f1, f2)| (f1 - f2).abs()); let sup = diff.fold(0f64, |max, diff| if diff > max { diff } else { max }); let alpha = 0.005f64; let c = (-0.5 * (alpha / 2.0).ln()).sqrt(); let threshold = c * ((2 * n) as f64 / (n * n) as f64).sqrt(); if sup <= threshold { println!("max diff was {}", sup); println!("rejection criterion is {}", threshold); if false { // TODO: currently fails -- however, it is unclear if the alpha parameter for the // two generators actually *mean* the same thing, and thus whether they can be // meaningfully compared. assert!(sup <= threshold); } } } }