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 // Copyright ©2016 The Gonum Authors. All rights reserved. // Use of this source code is governed by a BSD-style // license that can be found in the LICENSE file. package distuv import ( "math" "golang.org/x/exp/rand" "gonum.org/v1/gonum/mathext" ) const logPi = 1.1447298858494001741 // http://oeis.org/A053510 // StudentsT implements the three-parameter Student's T distribution, a distribution // over the real numbers. // // The Student's T distribution has density function // Γ((ν+1)/2) / (sqrt(νπ) Γ(ν/2) σ) (1 + 1/ν * ((x-μ)/σ)^2)^(-(ν+1)/2) // // The Student's T distribution approaches the normal distribution as ν → ∞. // // For more information, see https://en.wikipedia.org/wiki/Student%27s_t-distribution, // specifically https://en.wikipedia.org/wiki/Student%27s_t-distribution#Non-standardized_Student.27s_t-distribution . // // The standard Student's T distribution is with Mu = 0, and Sigma = 1. type StudentsT struct { // Mu is the location parameter of the distribution, and the mean of the // distribution Mu float64 // Sigma is the scale parameter of the distribution. It is related to the // standard deviation by std = Sigma * sqrt(Nu/(Nu-2)) Sigma float64 // Nu is the shape prameter of the distribution, representing the number of // degrees of the distribution, and one less than the number of observations // from a Normal distribution. Nu float64 Src rand.Source } // CDF computes the value of the cumulative distribution function at x. func (s StudentsT) CDF(x float64) float64 { // transform to standard normal y := (x - s.Mu) / s.Sigma if y == 0 { return 0.5 } // For t > 0 // F(y) = 1 - 0.5 * I_t(y)(nu/2, 1/2) // t(y) = nu/(y^2 + nu) // and 1 - F(y) for t < 0 t := s.Nu / (y*y + s.Nu) if y > 0 { return 1 - 0.5*mathext.RegIncBeta(0.5*s.Nu, 0.5, t) } return 0.5 * mathext.RegIncBeta(s.Nu/2, 0.5, t) } // LogProb computes the natural logarithm of the value of the probability // density function at x. func (s StudentsT) LogProb(x float64) float64 { g1, _ := math.Lgamma((s.Nu + 1) / 2) g2, _ := math.Lgamma(s.Nu / 2) z := (x - s.Mu) / s.Sigma return g1 - g2 - 0.5*math.Log(s.Nu) - 0.5*logPi - math.Log(s.Sigma) - ((s.Nu+1)/2)*math.Log(1+z*z/s.Nu) } // Mean returns the mean of the probability distribution. func (s StudentsT) Mean() float64 { return s.Mu } // Mode returns the mode of the distribution. func (s StudentsT) Mode() float64 { return s.Mu } // NumParameters returns the number of parameters in the distribution. func (StudentsT) NumParameters() int { return 3 } // Prob computes the value of the probability density function at x. func (s StudentsT) Prob(x float64) float64 { return math.Exp(s.LogProb(x)) } // Quantile returns the inverse of the cumulative distribution function. func (s StudentsT) Quantile(p float64) float64 { if p < 0 || p > 1 { panic(badPercentile) } // F(x) = 1 - 0.5 * I_t(x)(nu/2, 1/2) // t(x) = nu/(t^2 + nu) if p == 0.5 { return s.Mu } var y float64 if p > 0.5 { // Know t > 0 t := mathext.InvRegIncBeta(s.Nu/2, 0.5, 2*(1-p)) y = math.Sqrt(s.Nu * (1 - t) / t) } else { t := mathext.InvRegIncBeta(s.Nu/2, 0.5, 2*p) y = -math.Sqrt(s.Nu * (1 - t) / t) } // Convert out of standard normal return y*s.Sigma + s.Mu } // Rand returns a random sample drawn from the distribution. func (s StudentsT) Rand() float64 { // http://www.math.uah.edu/stat/special/Student.html n := Normal{0, 1, s.Src}.Rand() c := Gamma{s.Nu / 2, 0.5, s.Src}.Rand() z := n / math.Sqrt(c/s.Nu) return z*s.Sigma + s.Mu } // StdDev returns the standard deviation of the probability distribution. // // The standard deviation is undefined for ν <= 1, and this returns math.NaN(). func (s StudentsT) StdDev() float64 { return math.Sqrt(s.Variance()) } // Survival returns the survival function (complementary CDF) at x. func (s StudentsT) Survival(x float64) float64 { // transform to standard normal y := (x - s.Mu) / s.Sigma if y == 0 { return 0.5 } // For t > 0 // F(y) = 1 - 0.5 * I_t(y)(nu/2, 1/2) // t(y) = nu/(y^2 + nu) // and 1 - F(y) for t < 0 t := s.Nu / (y*y + s.Nu) if y > 0 { return 0.5 * mathext.RegIncBeta(s.Nu/2, 0.5, t) } return 1 - 0.5*mathext.RegIncBeta(s.Nu/2, 0.5, t) } // Variance returns the variance of the probability distribution. // // The variance is undefined for ν <= 1, and this returns math.NaN(). func (s StudentsT) Variance() float64 { if s.Nu <= 1 { return math.NaN() } if s.Nu <= 2 { return math.Inf(1) } return s.Sigma * s.Sigma * s.Nu / (s.Nu - 2) }