mot.library_functions.continuous_distributions package

Submodules

mot.library_functions.continuous_distributions.gamma module

class mot.library_functions.continuous_distributions.gamma.gamma_cdf

Bases: mot.library_functions.base.SimpleCLLibrary

Calculate the Cumulative Distribution Function of the Gamma function.

This computes: lower_incomplete_gamma(k, x/theta) / gamma(k)

With k the shape parameter, theta the scale parameter, lower_incomplete_gamma the lower incomplete gamma function and gamma the complete gamma function.

Function arguments:

• shape: the shape parameter of the gamma distribution (often denoted \(k\))
• scale: the scale parameter of the gamma distribution (often denoted \(\theta\))

class mot.library_functions.continuous_distributions.gamma.gamma_cdf_approx

Bases: mot.library_functions.base.SimpleCLLibrary

Approximate the Cumulative Distribution Function of the Gamma function.

This uses the approximation from Revfeim [1] to compute the cdf for x given the shape and scale parameters.

The approximation returns infinity for values near the tails of the distribution, i.e. where the cdf is near zero or near one.

Function arguments:

• x: the value at which to approximate the cdf
• shape: the shape parameter of the gamma distribution (often denoted \(k\))
• scale: the scale parameter of the gamma distribution (often denoted \(\theta\))

References

1. Revfeim, K. J. A. (1991). Approximation for the cumulative and inverse gamma distribution. Statistica Neerlandica, 45(3), 327–331.

class mot.library_functions.continuous_distributions.gamma.gamma_logpdf

Bases: mot.library_functions.base.SimpleCLLibrary

Computes the log of the Gamma probability density function using the shape and scale parameterization.

This computes the gamma PDF as:

\[\frac{-x}{\theta} + (k-1)\ln(x) - \ln(\Gamma(k)) - k * \ln(\theta)\]

With \(x\) the desired position, \(k\) the shape and \(\theta\) the scale.

class mot.library_functions.continuous_distributions.gamma.gamma_pdf

Bases: mot.library_functions.base.SimpleCLLibrary

Computes the Gamma probability density function using the shape and scale parameterization.

This computes the gamma PDF as:

\[{\frac{1}{\Gamma (k)\theta ^{k}}}x^{k-1}e^{-{\frac {x}{\theta }}}\]

With \(x\) the desired position, \(k\) the shape and \(\theta\) the scale.

class mot.library_functions.continuous_distributions.gamma.gamma_ppf

Bases: mot.library_functions.base.SimpleCLLibrary

Computes the inverse of the cumulative distribution function of the Gamma distribution.

This is the inverse of the Gamma CDF.

class mot.library_functions.continuous_distributions.gamma.gamma_ppf_approx

Bases: mot.library_functions.base.SimpleCLLibrary

Approximates the Gamma percentile point function.

This uses the approximation from Revfeim [1] to compute the ppf for y given the shape and scale parameters.

The approximation is not valid in the tails of the distribution, i.e. where the cdf is near zero or near one.

Function arguments:

• y: the value at which to approximate the ppf
• shape: the shape parameter of the gamma distribution (often denoted \(k\))
• scale: the scale parameter of the gamma distribution (often denoted \(\theta\))

References

1. Revfeim, K. J. A. (1991). Approximation for the cumulative and inverse gamma distribution. Statistica Neerlandica, 45(3), 327–331.

class mot.library_functions.continuous_distributions.gamma.igam

Bases: mot.library_functions.base.SimpleCLLibrary

Complemented incomplete Gamma integral

Also known as the regularized lower incomplete gamma function. Copied from Scipy (https://github.com/scipy/scipy/blob/master/scipy/special/cephes/igam.c), 05-05-2018:

/*                                                     igam.c
 *
 *     Incomplete Gamma integral
 *
 *
 *
 * SYNOPSIS:
 *
 * double a, x, y, igam();
 *
 * y = igam( a, x );
 *
 * DESCRIPTION:
 *
 * The function is defined by
 *
 *                           x
 *                            -
 *                   1       | |  -t  a-1
 *  igam(a,x)  =   -----     |   e   t   dt.
 *                  -      | |
 *                 | (a)    -
 *                           0
 *
 *
 * In this implementation both arguments must be positive.
 * The integral is evaluated by either a power series or
 * continued fraction expansion, depending on the relative
 * values of a and x.
 *
 * ACCURACY:
 *
 *                      Relative error:
 * arithmetic   domain     # trials      peak         rms
 *    IEEE      0,30       200000       3.6e-14     2.9e-15
 *    IEEE      0,100      300000       9.9e-14     1.5e-14
 */

class mot.library_functions.continuous_distributions.gamma.igam_fac

Bases: mot.library_functions.base.SimpleCLLibrary

Compute x^a * exp(-x) / gamma(a)

Copied from Scipy (https://github.com/scipy/scipy/blob/master/scipy/special/cephes/igam.c), 05-05-2018.

class mot.library_functions.continuous_distributions.gamma.igam_igamc_asymptotic_series

Bases: mot.library_functions.base.SimpleCLLibrary

Compute igam/igamc using DLMF 8.12.3/8.12.4.

Copied from Scipy (https://github.com/scipy/scipy/blob/master/scipy/special/cephes/igam.c), 05-05-2018.

The argument func should be 1 when computing for IGAM and 0 when computing for IGAMC.

class mot.library_functions.continuous_distributions.gamma.igam_series

Bases: mot.library_functions.base.SimpleCLLibrary

Compute igamc using DLMF 8.11.4

Copied from Scipy (https://github.com/scipy/scipy/blob/master/scipy/special/cephes/igam.c), 05-05-2018.

class mot.library_functions.continuous_distributions.gamma.igamc

Bases: mot.library_functions.base.SimpleCLLibrary

Complemented incomplete Gamma integral

Also known as the regularized upper incomplete gamma function. Copied from Scipy (https://github.com/scipy/scipy/blob/master/scipy/special/cephes/igam.c), 05-05-2018:

/*                                                  igamc()
 *
 *  Complemented incomplete Gamma integral
 *
 *
 *
 * SYNOPSIS:
 *
 * double a, x, y, igamc();
 *
 * y = igamc( a, x );
 *
 * DESCRIPTION:
 *
 * The function is defined by
 *
 *
 *  igamc(a,x)   =   1 - igam(a,x)
 *
 *                            inf.
 *                              -
 *                     1       | |  -t  a-1
 *               =   -----     |   e   t   dt.
 *                    -      | |
 *                   | (a)    -
 *                             x
 *
 *
 * In this implementation both arguments must be positive.
 * The integral is evaluated by either a power series or
 * continued fraction expansion, depending on the relative
 * values of a and x.
 *
 * ACCURACY:
 *
 * Tested at random a, x.
 *                a         x                      Relative error:
 * arithmetic   domain   domain     # trials      peak         rms
 *    IEEE     0.5,100   0,100      200000       1.9e-14     1.7e-15
 *    IEEE     0.01,0.5  0,100      200000       1.4e-13     1.6e-15
 */

class mot.library_functions.continuous_distributions.gamma.igamc_continued_fraction

Bases: mot.library_functions.base.SimpleCLLibrary

Compute igamc using DLMF 8.9.2.

Copied from Scipy (https://github.com/scipy/scipy/blob/master/scipy/special/cephes/igam.c), 05-05-2018.

class mot.library_functions.continuous_distributions.gamma.igamc_series

Bases: mot.library_functions.base.SimpleCLLibrary

Compute igamc using DLMF 8.7.3.

This is related to the series in igam_series but extra care is taken to avoid cancellation.

Copied from Scipy (https://github.com/scipy/scipy/blob/master/scipy/special/cephes/igam.c), 05-05-2018.

class mot.library_functions.continuous_distributions.gamma.igamci

Bases: mot.library_functions.base.SimpleCLLibrary

Copied from Scipy (https://github.com/scipy/scipy/blob/master/scipy/special/cephes/igami.c), 05-05-2018.

class mot.library_functions.continuous_distributions.gamma.igami

Bases: mot.library_functions.base.SimpleCLLibrary

Copied from Scipy (https://github.com/scipy/scipy/blob/master/scipy/special/cephes/igami.c), 05-05-2018.

mot.library_functions.continuous_distributions.normal module

class mot.library_functions.continuous_distributions.normal.normal_cdf

Bases: mot.library_functions.base.SimpleCLLibrary

Compute the Cumulative Distribution Function of the Gaussian distribution.

class mot.library_functions.continuous_distributions.normal.normal_logpdf

Bases: mot.library_functions.base.SimpleCLLibrary

Compute the log of the Probability Density Function of the Gaussian distribution.

class mot.library_functions.continuous_distributions.normal.normal_pdf

Bases: mot.library_functions.base.SimpleCLLibrary

Compute the Probability Density Function of the Gaussian distribution.

class mot.library_functions.continuous_distributions.normal.normal_ppf

Bases: mot.library_functions.base.SimpleCLLibrary

Computes the inverse of the cumulative distribution function of the Gaussian distribution.

This is the inverse of the Gaussian CDF.

Module contents