Trait statrs::distribution::Continuous

pub trait Continuous<T, K> {
    fn pdf(&self, x: T) -> K;
    fn ln_pdf(&self, x: T) -> K;
}

The Continuous trait provides an interface for interacting with continuous statistical distributions

Remarks

All methods provided by the Continuous trait are unchecked, meaning they can panic if in an invalid state or encountering invalid input depending on the implementing distribution.

Required methods

fn pdf(&self, x: T) -> K

Returns the probability density function calculated at x for a given distribution. May panic depending on the implementor.

Examples

use statrs::distribution::{Continuous, Uniform};

let n = Uniform::new(0.0, 1.0).unwrap();
assert_eq!(1.0, n.pdf(0.5));

fn ln_pdf(&self, x: T) -> K

Returns the log of the probability density function calculated at x for a given distribution. May panic depending on the implementor.

Examples

use statrs::distribution::{Continuous, Uniform};

let n = Uniform::new(0.0, 1.0).unwrap();
assert_eq!(0.0, n.ln_pdf(0.5));

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Implementors

impl Continuous<f64, f64> for Beta

fn pdf(&self, x: f64) -> f64

Calculates the probability density function for the beta distribution at x.

Formula

let B(α, β) = Γ(α)Γ(β)/Γ(α + β)

x^(α - 1) * (1 - x)^(β - 1) / B(α, β)

where α is shapeA, β is shapeB, and Γ is the gamma function

fn ln_pdf(&self, x: f64) -> f64

Calculates the log probability density function for the beta distribution at x.

Formula

let B(α, β) = Γ(α)Γ(β)/Γ(α + β)

ln(x^(α - 1) * (1 - x)^(β - 1) / B(α, β))

where α is shapeA, β is shapeB, and Γ is the gamma function

impl Continuous<f64, f64> for Cauchy

fn pdf(&self, x: f64) -> f64

Calculates the probability density function for the cauchy distribution at x

Formula

1 / (πγ * (1 + ((x - x_0) / γ)^2))

where x_0 is the location and γ is the scale

fn ln_pdf(&self, x: f64) -> f64

Calculates the log probability density function for the cauchy distribution at x

Formula

ln(1 / (πγ * (1 + ((x - x_0) / γ)^2)))

where x_0 is the location and γ is the scale

impl Continuous<f64, f64> for Chi

fn pdf(&self, x: f64) -> f64

Calculates the probability density function for the chi distribution at x

Formula

(2^(1 - (k / 2)) * x^(k - 1) * e^(-x^2 / 2)) / Γ(k / 2)

where k is the degrees of freedom and Γ is the gamma function

fn ln_pdf(&self, x: f64) -> f64

Calculates the log probability density function for the chi distribution at x

Formula

ln((2^(1 - (k / 2)) * x^(k - 1) * e^(-x^2 / 2)) / Γ(k / 2))

impl Continuous<f64, f64> for ChiSquared

fn pdf(&self, x: f64) -> f64

Calculates the probability density function for the chi-squared distribution at x

Formula

1 / (2^(k / 2) * Γ(k / 2)) * x^((k / 2) - 1) * e^(-x / 2)

where k is the degrees of freedom and Γ is the gamma function

fn ln_pdf(&self, x: f64) -> f64

Calculates the log probability density function for the chi-squared distribution at x

Formula

ln(1 / (2^(k / 2) * Γ(k / 2)) * x^((k / 2) - 1) * e^(-x / 2))

impl Continuous<f64, f64> for Erlang

fn pdf(&self, x: f64) -> f64

Calculates the probability density function for the erlang distribution at x

Remarks

Returns NAN if any of shape or rate are INF or if x is INF

Formula

(λ^k / Γ(k)) * x^(k - 1) * e^(-λ * x)

where k is the shape, λ is the rate, and Γ is the gamma function

fn ln_pdf(&self, x: f64) -> f64

Calculates the log probability density function for the erlang distribution at x

Remarks

Returns NAN if any of shape or rate are INF or if x is INF

Formula

ln((λ^k / Γ(k)) * x^(k - 1) * e ^(-λ * x))

where k is the shape, λ is the rate, and Γ is the gamma function

impl Continuous<f64, f64> for Exponential

fn pdf(&self, x: f64) -> f64

Calculates the probability density function for the exponential distribution at x

Formula

λ * e^(-λ * x)

where λ is the rate

fn ln_pdf(&self, x: f64) -> f64

Calculates the log probability density function for the exponential distribution at x

Formula

ln(λ * e^(-λ * x))

where λ is the rate

impl Continuous<f64, f64> for FisherSnedecor

fn pdf(&self, x: f64) -> f64

Calculates the probability density function for the fisher-snedecor distribution at x

Remarks

Returns NaN if freedom_1, freedom_2 is INF, or x is +INF or -INF

Formula

sqrt(((d1 * x) ^ d1 * d2 ^ d2) / (d1 * x + d2) ^ (d1 + d2)) / (x * β(d1
/ 2, d2 / 2))

where d1 is the first degree of freedom, d2 is the second degree of freedom, and β is the beta function

fn ln_pdf(&self, x: f64) -> f64

Calculates the log probability density function for the fisher-snedecor distribution at x

Remarks

Returns NaN if freedom_1, freedom_2 is INF, or x is +INF or -INF

Formula

ln(sqrt(((d1 * x) ^ d1 * d2 ^ d2) / (d1 * x + d2) ^ (d1 + d2)) / (x *
β(d1 / 2, d2 / 2)))

where d1 is the first degree of freedom, d2 is the second degree of freedom, and β is the beta function

impl Continuous<f64, f64> for Gamma

fn pdf(&self, x: f64) -> f64

Calculates the probability density function for the gamma distribution at x

Remarks

Returns NAN if any of shape or rate are INF or if x is INF

Formula

(β^α / Γ(α)) * x^(α - 1) * e^(-β * x)

where α is the shape, β is the rate, and Γ is the gamma function

fn ln_pdf(&self, x: f64) -> f64

Calculates the log probability density function for the gamma distribution at x

Remarks

Returns NAN if any of shape or rate are INF or if x is INF

Formula

ln((β^α / Γ(α)) * x^(α - 1) * e ^(-β * x))

where α is the shape, β is the rate, and Γ is the gamma function

impl Continuous<f64, f64> for InverseGamma

fn pdf(&self, x: f64) -> f64

Calculates the probability density function for the inverse gamma distribution at x

Formula

(β^α / Γ(α)) * x^(-α - 1) * e^(-β / x)

where α is the shape, β is the rate, and Γ is the gamma function

fn ln_pdf(&self, x: f64) -> f64

Calculates the probability density function for the inverse gamma distribution at x

Formula

ln((β^α / Γ(α)) * x^(-α - 1) * e^(-β / x))

where α is the shape, β is the rate, and Γ is the gamma function

impl Continuous<f64, f64> for LogNormal

fn pdf(&self, x: f64) -> f64

Calculates the probability density function for the log-normal distribution at x

Formula

(1 / xσ * sqrt(2π)) * e^(-((ln(x) - μ)^2) / 2σ^2)

where μ is the location and σ is the scale

fn ln_pdf(&self, x: f64) -> f64

Calculates the log probability density function for the log-normal distribution at x

Formula

ln((1 / xσ * sqrt(2π)) * e^(-((ln(x) - μ)^2) / 2σ^2))

where μ is the location and σ is the scale

impl Continuous<f64, f64> for Normal

fn pdf(&self, x: f64) -> f64

Calculates the probability density function for the normal distribution at x

Formula

(1 / sqrt(2σ^2 * π)) * e^(-(x - μ)^2 / 2σ^2)

where μ is the mean and σ is the standard deviation

fn ln_pdf(&self, x: f64) -> f64

Calculates the log probability density function for the normal distribution at x

Formula

ln((1 / sqrt(2σ^2 * π)) * e^(-(x - μ)^2 / 2σ^2))

where μ is the mean and σ is the standard deviation

impl Continuous<f64, f64> for Pareto

fn pdf(&self, x: f64) -> f64

Calculates the probability density function for the Pareto distribution at x

Formula

if x < x_m {
    0
} else {
    (α * x_m^α)/(x^(α + 1))
}

where x_m is the scale and α is the shape

fn ln_pdf(&self, x: f64) -> f64

Calculates the log probability density function for the Pareto distribution at x

Formula

if x < x_m {
    -INF
} else {
    ln(α) + α*ln(x_m) - (α + 1)*ln(x)
}

where x_m is the scale and α is the shape

impl Continuous<f64, f64> for StudentsT

fn pdf(&self, x: f64) -> f64

Calculates the probability density function for the student's t-distribution at x

Formula

Γ((v + 1) / 2) / (sqrt(vπ) * Γ(v / 2) * σ) * (1 + k^2 / v)^(-1 / 2 * (v
+ 1))

where k = (x - μ) / σ, μ is the location, σ is the scale, v is the freedom, and Γ is the gamma function

fn ln_pdf(&self, x: f64) -> f64

Calculates the log probability density function for the student's t-distribution at x

Formula

ln(Γ((v + 1) / 2) / (sqrt(vπ) * Γ(v / 2) * σ) * (1 + k^2 / v)^(-1 / 2 *
(v + 1)))

where k = (x - μ) / σ, μ is the location, σ is the scale, v is the freedom, and Γ is the gamma function

impl Continuous<f64, f64> for Triangular

fn pdf(&self, x: f64) -> f64

Calculates the probability density function for the triangular distribution at x

Formula

if x < min {
    0
} else if min <= x <= mode {
    2 * (x - min) / ((max - min) * (mode - min))
} else if mode < x <= max {
    2 * (max - x) / ((max - min) * (max - mode))
} else {
    0
}

fn ln_pdf(&self, x: f64) -> f64

Calculates the log probability density function for the triangular distribution at x

Formula

ln( if x < min {
    0
} else if min <= x <= mode {
    2 * (x - min) / ((max - min) * (mode - min))
} else if mode < x <= max {
    2 * (max - x) / ((max - min) * (max - mode))
} else {
    0
} )

impl Continuous<f64, f64> for Uniform

fn pdf(&self, x: f64) -> f64

Calculates the probability density function for the continuous uniform distribution at x

Remarks

Returns 0.0 if x is not in [min, max]

Formula

1 / (max - min)

fn ln_pdf(&self, x: f64) -> f64

Calculates the log probability density function for the continuous uniform distribution at x

Remarks

Returns f64::NEG_INFINITY if x is not in [min, max]

Formula

ln(1 / (max - min))

impl Continuous<f64, f64> for Weibull

fn pdf(&self, x: f64) -> f64

Calculates the probability density function for the weibull distribution at x

Formula

(k / λ) * (x / λ)^(k - 1) * e^(-(x / λ)^k)

where k is the shape and λ is the scale

fn ln_pdf(&self, x: f64) -> f64

Calculates the log probability density function for the weibull distribution at x

Formula

ln((k / λ) * (x / λ)^(k - 1) * e^(-(x / λ)^k))

where k is the shape and λ is the scale

impl<'a> Continuous<&'a [f64], f64> for Dirichlet

fn pdf(&self, x: &[f64]) -> f64

Calculates the probabiliy density function for the dirichlet distribution with given x's corresponding to the concentration parameters for this distribution

Panics

If any element in x is not in (0, 1), the elements in x do not sum to 1 with a tolerance of 1e-4, or if x is not the same length as the vector of concentration parameters for this distribution

Formula

(1 / B(α)) * Π(x_i^(α_i - 1))

where

B(α) = Π(Γ(α_i)) / Γ(Σ(α_i))

α is the vector of concentration parameters, α_i is the ith concentration parameter, x_i is the ith argument corresponding to the ith concentration parameter, Γ is the gamma function, Π is the product from 1 to K, Σ is the sum from 1 to K, and K is the number of concentration parameters

fn ln_pdf(&self, x: &[f64]) -> f64

Calculates the log probabiliy density function for the dirichlet distribution with given x's corresponding to the concentration parameters for this distribution

Panics

If any element in x is not in (0, 1), the elements in x do not sum to 1 with a tolerance of 1e-4, or if x is not the same length as the vector of concentration parameters for this distribution

Formula

ln((1 / B(α)) * Π(x_i^(α_i - 1)))

where

B(α) = Π(Γ(α_i)) / Γ(Σ(α_i))

α is the vector of concentration parameters, α_i is the ith concentration parameter, x_i is the ith argument corresponding to the ith concentration parameter, Γ is the gamma function, Π is the product from 1 to K, Σ is the sum from 1 to K, and K is the number of concentration parameters

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