# Module Owl_stats

Statistics: random number generators, PDF and CDF functions, and hypothesis tests. The module also includes some basic statistical functions such as mean, variance, skew, and etc.

###### Randomisation functions
val shuffle : 'a array -> 'a array

shuffle x return a new array of the shuffled x.

val choose : 'a array -> int -> 'a array

choose x n draw n samples from x without replecement.

val sample : 'a array -> int -> 'a array

sample x n draw n samples from x with replacement.

###### Basic statistical functions
val sum : float array -> float

sum x returns the summation of the elements in x.

val mean : float array -> float

mean x returns the mean of the elements in x.

val var : ?mean:float -> float array -> float

var x returns the variance of elements in x.

val std : ?mean:float -> float array -> float

std x calculates the standard deviation of x.

val sem : ?mean:float -> float array -> float

sem x calculates the standard error of x, also referred to as standard error of the mean.

val absdev : ?mean:float -> float array -> float

absdev x calculates the average absolute deviation of x.

val skew : ?mean:float -> ?sd:float -> float array -> float

skew x calculates the skewness (the third standardized moment) of x.

val kurtosis : ?mean:float -> ?sd:float -> float array -> float

kurtosis x calculates the Pearson's kurtosis of x, i.e. the fourth standardized moment of x.

val central_moment : int -> float array -> float

central_moment n x calculates the n th central moment of x.

val cov : ?m0:float -> ?m1:float -> float array -> float array -> float

cov x0 x1 calculates the covariance of x0 and x1, the mean of x0 and x1 can be specified by m0 and m1 respectively.

val concordant : 'a array -> 'b array -> int

TODO

val discordant : 'a array -> 'b array -> int

TODO

val corrcoef : float array -> float array -> float

corrcoef x y calculates the Pearson correlation of x and y. Namely, corrcoef x y = cov(x, y) / (sigma_x * sigma_y).

val kendall_tau : float array -> float array -> float

kendall_tau x y calculates the Kendall Tau correlation between x and y.

val spearman_rho : float array -> float array -> float

spearman_rho x y calculates the Spearman Rho correlation between x and y.

val autocorrelation : ?lag:int -> float array -> float

autocorrelation ~lag x calculates the autocorrelation of x with the given lag.

val percentile : float array -> float -> float

percentile x p returns the p percentile of the data x. p is between 0. and 100. x does not need to be sorted beforehand.

val quantile : float array -> float -> float

quantile x p returns the p quantile of the data x. x does not need to be sorted beforehand. When computing several quantiles on the same data, it is more efficient to "pre-apply" the function, as in let f = quantile x in List.map f  0.1 ; 0.5 , in which case the data is sorted only once.

• raises Invalid_argument

if p is not between 0 and 1.

val first_quartile : float array -> float

first_quartile x returns the first quartile of x, i.e. 25 percentiles.

val third_quartile : float array -> float

third_quartile x returns the third quartile of x, i.e. 75 percentiles.

val interquartile : float array -> float

interquartile x returns the interquartile range of x which is defined as the first quartile subtracted from the third quartile.

val median : float array -> float

median x returns the median of x.

val min : float array -> float

min x returns the minimum element in x.

val max : float array -> float

max x returns the maximum element in x.

val minmax : float array -> float * float

minmax x returns both (minimum, maximum) elements in x.

val min_i : float array -> int

min_i x returns the index of the minimum in x.

val max_i : float array -> int

max_i x returns the index of the maximum in x.

val minmax_i : float array -> int * int

minmax_i x returns the indices of both minimum and maximum in x.

val sort : ?inc:bool -> float array -> float array

sort x sorts the elements in the x in increasing order if inc = true, otherwise in decreasing order if inc=false. By default, inc is true. Note a copy is returned, the original data is not modified.

val argsort : ?inc:bool -> float array -> int array

argsort x sorts the elements in x and returns the indices mapping of the elements in the current array to their original position in x.

The sorting is in increasing order if inc = true, otherwise in decreasing order if inc=false. By default, inc is true.

val rank : ?ties_strategy:[ Average | Min | Max ] -> float array -> float array

Computes sample's ranks.

The ranking order is from the smallest one to the largest. For example rank [|54.; 74.; 55.; 86.; 56.|] returns [|1.; 4.; 2.; 5.; 3.|]. Note that the ranking starts with one!

ties_strategy controls which ranks are assigned to equal values:

• Average the mean of ranks should be assigned to each value. Default.
• Min the minimum of ranks is assigned to each value.
• Max the maximum of ranks is assigned to each value.
type histogram = Owl_base_stats.histogram

Type for computed histograms, with optional weighted counts and normalized counts.

val histogram : [ Bins of float array | N of int ] -> ?weights:float array -> float array -> histogram

histogram bins x creates a histogram from values in x. If bins matches  N n it will construct n equally spaced bins from the minimum to the maximum in x. If bins matches  Bins b, b is taken as the sorted array of boundaries of adjacent bin intervals. Bin boundaries are taken as left-inclusive, right-exclusive, except for the last bin which is also right-inclusive. Values outside the bins are dropped silently.

histogram bins ~weights x creates a weighted histogram with the given weights which must match x in length. The bare counts are also provided.

Returns a histogram including the n+1 bin boundaries, n counts and weighted counts if applicable, but without normalisation.

val histogram_sorted : [ Bins of float array | N of int ] -> ?weights:float array -> float array -> histogram

histogram_sorted bins x is like histogram but assumes that x is sorted already. This increases efficiency if there are less bins than data. Undefined results if x is not in fact sorted.

val normalise : histogram -> histogram

normalize hist calculates a probability mass function using hist.weighted_counts if present, otherwise using hist.counts. The result is stored in the normalised_counts field and sums to one.

val normalise_density : histogram -> histogram

normalize_density hist calculates a probability density function using hist.weighted_counts if present, otherwise using hist.counts. The result is normalized as a density that is piecewise constant over the bin intervals. That is, the sum over density times corresponding bin width is one. If bins are infinitely wide, their density is 0 and the sum over width times density of all finite bins is the total weight in the finite bins. The result is stored in the density field.

val pp_hist : Stdlib.Format.formatter -> histogram -> unit

Pretty-print summary information on a histogram record

val ecdf : float array -> float array * float array

ecdf x returns (x',f) which are the empirical cumulative distribution function f of x at points x'. x' is just x sorted in increasing order with duplicates removed. The function does not support nan values in the array x.

val z_score : mu:float -> sigma:float -> float array -> float array

z_score x calculates the z score of a given array x.

val t_score : float array -> float array

t_score x calculates the t score of a given array x.

val normlise_pdf : float array -> float array

TODO

val tukey_fences : ?k:float -> float array -> float * float

tukey_fences ?k x returns a tuple of the lower and upper boundaries for values that are not outliers. k defaults to the standard coefficient of 1.5. For first and third quartiles Q1 and Q3, the range is computed as follows:

.. math:: (Q1 - k*(Q3-Q1), Q3 + k*(Q3-Q1))

val gaussian_kde : ?bandwidth:[ Silverman | Scott ] -> ?n_points:int -> float array -> float array * float array

gaussian_kde x is a Gaussian kernel density estimator. The estimation of the pdf runs in O(sample_size * n_points), and returns an array tuple (a, b) where a is a uniformly spaced points from the sample range at which the density function was estimated, and b is the estimates at these points.

Bandwidth selection rules is as follows: * Silverman: use rule-of-thumb for choosing the bandwidth. It defaults to 0.9 * min(SD, IQR / 1.34) * n^-0.2. * Scott: same as Silverman, but with a factor, equal to 1.06.

The default bandwidth value is Scott.

###### MCMC: Markov Chain Monte Carlo
val metropolis_hastings : (float array -> float) -> float array -> int -> float array array

TODO: metropolis_hastings f p n is Metropolis-Hastings MCMC algorithm. f is pdf of the p

val gibbs_sampling : (float array -> int -> float) -> float array -> int -> float array array

TODO: gibbs_sampling f p n is Gibbs sampler. f is a sampler based on the full conditional function of all variables

###### Hypothesis tests
type hypothesis = {
1. reject : bool;
2. p_value : float;
3. score : float;
}

Record type contains the result of a hypothesis test.

type tail = 
1. | BothSide
2. | RightSide
3. | LeftSide
(*

Types of alternative hypothesis tests: one-side, left-side, or right-side.

*)
val pp_hypothesis : Stdlib.Format.formatter -> hypothesis -> unit

Pretty printer of hypothesis type

val z_test : mu:float -> sigma:float -> ?alpha:float -> ?side:tail -> float array -> hypothesis

z_test ~mu ~sigma ~alpha ~side x returns a test decision for the null hypothesis that the data x comes from a normal distribution with mean mu and a standard deviation sigma, using the z-test of alpha significance level. The alternative hypothesis is that the mean is not mu.

The result (h,p,z) : h is true if the test rejects the null hypothesis at the alpha significance level, and false otherwise. p is the p-value and z is the z-score.

val t_test : mu:float -> ?alpha:float -> ?side:tail -> float array -> hypothesis

t_test ~mu ~alpha ~side x returns a test decision of one-sample t-test which is a parametric test of the location parameter when the population standard deviation is unknown. mu is population mean, alpha is the significance level.

val t_test_paired : ?alpha:float -> ?side:tail -> float array -> float array -> hypothesis

t_test_paired ~alpha ~side x y returns a test decision for the null hypothesis that the data in x – y comes from a normal distribution with mean equal to zero and unknown variance, using the paired-sample t-test.

val t_test_unpaired : ?alpha:float -> ?side:tail -> ?equal_var:bool -> float array -> float array -> hypothesis

t_test_unpaired ~alpha ~side ~equal_var x y returns a test decision for the null hypothesis that the data in vectors x and y comes from independent random samples from normal distributions with equal means and equal but unknown variances, using the two-sample t-test. The alternative hypothesis is that the data in x and y comes from populations with unequal means.

equal_var indicates whether two samples have the same variance. If the two variances are not the same, the test is referred to as Welche's t-test.

val ks_test : ?alpha:float -> float array -> (float -> float) -> hypothesis

ks_test ~alpha x f returns a test decision for the null hypothesis that the data in vector x comes from independent random samples of the distribution with CDF f. The alternative hypothesis is that the data in x comes from a different distribution.

The result (h,p,d) : h is true if the test rejects the null hypothesis at the alpha significance level, and false otherwise. p is the p-value and d is the Kolmogorov-Smirnov test statistic.

val ks2_test : ?alpha:float -> float array -> float array -> hypothesis

ks2_test ~alpha x y returns a test decision for the null hypothesis that the data in vectors x and y come from independent random samples of the same distribution. The alternative hypothesis is that the data in x and y are sampled from different distributions.

The result (h,p,d): h is true if the test rejects the null hypothesis at the alpha significance level, and false otherwise. p is the p-value and d is the Kolmogorov-Smirnov test statistic.

val var_test : ?alpha:float -> ?side:tail -> variance:float -> float array -> hypothesis

var_test ~alpha ~side ~variance x returns a test decision for the null hypothesis that the data in x comes from a normal distribution with input variance, using the chi-square variance test. The alternative hypothesis is that x comes from a normal distribution with a different variance.

val jb_test : ?alpha:float -> float array -> hypothesis

jb_test ~alpha x returns a test decision for the null hypothesis that the data x comes from a normal distribution with an unknown mean and variance, using the Jarque-Bera test.

val fisher_test : ?alpha:float -> ?side:tail -> int -> int -> int -> int -> hypothesis

fisher_test ~alpha ~side a b c d fisher's exact test for contingency table | a, b | | c, d |

The result (h,p,z) : h is true if the test rejects the null hypothesis at the alpha significance level, and false otherwise. p is the p-value and z is prior odds ratio.

val runs_test : ?alpha:float -> ?side:tail -> ?v:float -> float array -> hypothesis

runs_test ~alpha ~v x returns a test decision for the null hypothesis that the data x comes in random order, against the alternative that they do not, by running Wald–Wolfowitz runs test. The test is based on the number of runs of consecutive values above or below the mean of x. ~v is the reference value, the default value is the median of x.

val mannwhitneyu : ?alpha:float -> ?side:tail -> float array -> float array -> hypothesis

mannwhitneyu ~alpha ~side x y Computes the Mann-Whitney rank test on samples x and y. If length of each sample less than 10 and no ties, then using exact test (see paper Ying Kuen Cheung and Jerome H. Klotz (1997) The Mann Whitney Wilcoxon distribution using linked list Statistica Sinica 7 805-813), else usning asymptotic normal distribution.

val wilcoxon : ?alpha:float -> ?side:tail -> float array -> float array -> hypothesis

TODO

###### Discrete random variables

The _rvs functions generate random numbers according to the specified distribution. _pdf are "density" functions that return the probability of the element specified by the arguments, while _cdf functions are cumulative distribution functions that return the probability of all elements less than or equal to the chosen element, and _sf functions are survival functions returning one minus the corresponding CDF function. log versions of functions return the result for the natural logarithm of a chosen element.

val uniform_int_rvs : a:int -> b:int -> int

uniform_rvs ~a ~b returns a random uniformly distributed integer between a and b, inclusive.

val binomial_rvs : p:float -> n:int -> int

binomial_rvs p n returns a random integer representing the number of successes in n trials with probability of success p on each trial.

val binomial_pdf : int -> p:float -> n:int -> float

binomial_pdf k ~p ~n returns the binomially distributed probability of k successes in n trials with probability p of success on each trial.

val binomial_logpdf : int -> p:float -> n:int -> float

binomial_logpdf k ~p ~n returns the log-binomially distributed probability of k successes in n trials with probability p of success on each trial.

val binomial_cdf : int -> p:float -> n:int -> float

binomial_cdf k ~p ~n returns the binomially distributed cumulative probability of less than or equal to k successes in n trials, with probability p on each trial.

val binomial_logcdf : int -> p:float -> n:int -> float

binomial_logcdf k ~p ~n returns the log-binomially distributed cumulative probability of less than or equal to k successes in n trials, with probability p on each trial.

val binomial_sf : int -> p:float -> n:int -> float

binomial_sf k ~p ~n is the binomial survival function, i.e. 1 - (binomial_cdf k ~p ~n).

val binomial_logsf : int -> p:float -> n:int -> float

binomial_loggf k ~p ~n is the logbinomial survival function, i.e. 1 - (binomial_logcdf k ~p ~n).

val hypergeometric_rvs : good:int -> bad:int -> sample:int -> int

hypergeometric_rvs ~good ~bad ~sample returns a random hypergeometrically distributed integer representing the number of successes in a sample (without replacement) of size ~sample from a population with ~good successful elements and ~bad unsuccessful elements.

val hypergeometric_pdf : int -> good:int -> bad:int -> sample:int -> float

hypergeometric_pdf k ~good ~bad ~sample returns the hypergeometrically distributed probability of k successes in a sample (without replacement) of ~sample elements from a population containing ~good successful elements and ~bad unsuccessful ones.

val hypergeometric_logpdf : int -> good:int -> bad:int -> sample:int -> float

hypergeometric_logpdf k ~good ~bad ~sample returns a value equivalent to a log-transformed result from hypergeometric_pdf.

val multinomial_rvs : int -> p:float array -> int array

multinomial_rvs n ~p generates random numbers of multinomial distribution from n trials. The probability mass function is as follows.

.. math:: P(x) = \fracn!{x_1! \cdot\cdot\cdot x_k!

}

p_

^x_1 \cdot\cdot\cdot p_k^x_k

p is the probability mass of k categories. The elements in p should all be positive (result is undefined if there are negative values), but they don't need to sum to 1: the result is the same whether or not p is normalized. For implementation details, refer to :cite:davis1993computer.

val multinomial_pdf : int array -> p:float array -> float

multinomial_rvs x ~p return the probability of x given the probability mass of a multinomial distribution.

val multinomial_logpdf : int array -> p:float array -> float

multinomial_rvs x ~p returns the logarithm probability of x given the probability mass of a multinomial distribution.

val categorical_rvs : float array -> int

categorical_rvs p returns the value of a random variable which follows the categorical distribution. This is equavalent to only one trial from multinomial_rvs function, so it is just a simple wrapping.

###### Continuous random variables
val std_uniform_rvs : unit -> float

TODO

val uniform_rvs : a:float -> b:float -> float

TODO

val uniform_pdf : float -> a:float -> b:float -> float

TODO

val uniform_logpdf : float -> a:float -> b:float -> float

TODO

val uniform_cdf : float -> a:float -> b:float -> float

TODO

val uniform_logcdf : float -> a:float -> b:float -> float

TODO

val uniform_ppf : float -> a:float -> b:float -> float

TODO

val uniform_sf : float -> a:float -> b:float -> float

TODO

val uniform_logsf : float -> a:float -> b:float -> float

TODO

val uniform_isf : float -> a:float -> b:float -> float

TODO

val exponential_rvs : lambda:float -> float

TODO

val exponential_pdf : float -> lambda:float -> float

TODO

val exponential_logpdf : float -> lambda:float -> float

TODO

val exponential_cdf : float -> lambda:float -> float

TODO

val exponential_logcdf : float -> lambda:float -> float

TODO

val exponential_ppf : float -> lambda:float -> float

TODO

val exponential_sf : float -> lambda:float -> float

TODO

val exponential_logsf : float -> lambda:float -> float

TODO

val exponential_isf : float -> lambda:float -> float

TODO

val exponpow_rvs : a:float -> b:float -> float

.. math:: p(x) dx = (1/(2 a Gamma(1+1/b))) * exp(-|x/a|^b) dx

val exponpow_pdf : float -> a:float -> b:float -> float

TODO

val exponpow_logpdf : float -> a:float -> b:float -> float

TODO

val exponpow_cdf : float -> a:float -> b:float -> float

TODO

val exponpow_logcdf : float -> a:float -> b:float -> float

TODO

val exponpow_sf : float -> a:float -> b:float -> float

TODO

val exponpow_logsf : float -> a:float -> b:float -> float

TODO

val gaussian_rvs : mu:float -> sigma:float -> float

TODO

val gaussian_pdf : float -> mu:float -> sigma:float -> float

TODO

val gaussian_logpdf : float -> mu:float -> sigma:float -> float

TODO

val gaussian_cdf : float -> mu:float -> sigma:float -> float

TODO

val gaussian_logcdf : float -> mu:float -> sigma:float -> float

TODO

val gaussian_ppf : float -> mu:float -> sigma:float -> float

TODO

val gaussian_sf : float -> mu:float -> sigma:float -> float

TODO

val gaussian_logsf : float -> mu:float -> sigma:float -> float

TODO

val gaussian_isf : float -> mu:float -> sigma:float -> float

TODO

val gamma_rvs : shape:float -> scale:float -> float

TODO

val gamma_pdf : float -> shape:float -> scale:float -> float

TODO

val gamma_logpdf : float -> shape:float -> scale:float -> float

TODO

val gamma_cdf : float -> shape:float -> scale:float -> float

TODO

val gamma_logcdf : float -> shape:float -> scale:float -> float

TODO

val gamma_ppf : float -> shape:float -> scale:float -> float

TODO

val gamma_sf : float -> shape:float -> scale:float -> float

TODO

val gamma_logsf : float -> shape:float -> scale:float -> float

TODO

val gamma_isf : float -> shape:float -> scale:float -> float

TODO

val beta_rvs : a:float -> b:float -> float

TODO

val beta_pdf : float -> a:float -> b:float -> float

TODO

val beta_logpdf : float -> a:float -> b:float -> float

TODO

val beta_cdf : float -> a:float -> b:float -> float

TODO

val beta_logcdf : float -> a:float -> b:float -> float

TODO

val beta_ppf : float -> a:float -> b:float -> float

TODO

val beta_sf : float -> a:float -> b:float -> float

TODO

val beta_logsf : float -> a:float -> b:float -> float

TODO

val beta_isf : float -> a:float -> b:float -> float

TODO

val chi2_rvs : df:float -> float

TODO

val chi2_pdf : float -> df:float -> float

TODO

val chi2_logpdf : float -> df:float -> float

TODO

val chi2_cdf : float -> df:float -> float

TODO

val chi2_logcdf : float -> df:float -> float

TODO

val chi2_ppf : float -> df:float -> float

TODO

val chi2_sf : float -> df:float -> float

TODO

val chi2_logsf : float -> df:float -> float

TODO

val chi2_isf : float -> df:float -> float

TODO

val f_rvs : dfnum:float -> dfden:float -> float

TODO

val f_pdf : float -> dfnum:float -> dfden:float -> float

TODO

val f_logpdf : float -> dfnum:float -> dfden:float -> float

TODO

val f_cdf : float -> dfnum:float -> dfden:float -> float

TODO

val f_logcdf : float -> dfnum:float -> dfden:float -> float

TODO

val f_ppf : float -> dfnum:float -> dfden:float -> float

TODO

val f_sf : float -> dfnum:float -> dfden:float -> float

TODO

val f_logsf : float -> dfnum:float -> dfden:float -> float

TODO

val f_isf : float -> dfnum:float -> dfden:float -> float

TODO

val cauchy_rvs : loc:float -> scale:float -> float

TODO

val cauchy_pdf : float -> loc:float -> scale:float -> float

TODO

val cauchy_logpdf : float -> loc:float -> scale:float -> float

TODO

val cauchy_cdf : float -> loc:float -> scale:float -> float

TODO

val cauchy_logcdf : float -> loc:float -> scale:float -> float

TODO

val cauchy_ppf : float -> loc:float -> scale:float -> float

TODO

val cauchy_sf : float -> loc:float -> scale:float -> float

TODO

val cauchy_logsf : float -> loc:float -> scale:float -> float

TODO

val cauchy_isf : float -> loc:float -> scale:float -> float

TODO

val t_rvs : df:float -> loc:float -> scale:float -> float

TODO

val t_pdf : float -> df:float -> loc:float -> scale:float -> float

TODO

val t_logpdf : float -> df:float -> loc:float -> scale:float -> float

TODO

val t_cdf : float -> df:float -> loc:float -> scale:float -> float

TODO

val t_logcdf : float -> df:float -> loc:float -> scale:float -> float

TODO

val t_ppf : float -> df:float -> loc:float -> scale:float -> float

TODO

val t_sf : float -> df:float -> loc:float -> scale:float -> float

TODO

val t_logsf : float -> df:float -> loc:float -> scale:float -> float

TODO

val t_isf : float -> df:float -> loc:float -> scale:float -> float

TODO

val vonmises_rvs : mu:float -> kappa:float -> float

TODO

val vonmises_pdf : float -> mu:float -> kappa:float -> float

TODO

val vonmises_logpdf : float -> mu:float -> kappa:float -> float

TODO

val vonmises_cdf : float -> mu:float -> kappa:float -> float

TODO

val vonmises_logcdf : float -> mu:float -> kappa:float -> float

TODO

val vonmises_sf : float -> mu:float -> kappa:float -> float

TODO

val vonmises_logsf : float -> mu:float -> kappa:float -> float

TODO

val lomax_rvs : shape:float -> scale:float -> float

TODO

val lomax_pdf : float -> shape:float -> scale:float -> float

TODO

val lomax_logpdf : float -> shape:float -> scale:float -> float

TODO

val lomax_cdf : float -> shape:float -> scale:float -> float

TODO

val lomax_logcdf : float -> shape:float -> scale:float -> float

TODO

val lomax_ppf : float -> shape:float -> scale:float -> float

TODO

val lomax_sf : float -> shape:float -> scale:float -> float

TODO

val lomax_logsf : float -> shape:float -> scale:float -> float

TODO

val lomax_isf : float -> shape:float -> scale:float -> float

TODO

val weibull_rvs : shape:float -> scale:float -> float

TODO

val weibull_pdf : float -> shape:float -> scale:float -> float

TODO

val weibull_logpdf : float -> shape:float -> scale:float -> float

TODO

val weibull_cdf : float -> shape:float -> scale:float -> float

TODO

val weibull_logcdf : float -> shape:float -> scale:float -> float

TODO

val weibull_ppf : float -> shape:float -> scale:float -> float

TODO

val weibull_sf : float -> shape:float -> scale:float -> float

TODO

val weibull_logsf : float -> shape:float -> scale:float -> float

TODO

val weibull_isf : float -> shape:float -> scale:float -> float

TODO

val laplace_rvs : loc:float -> scale:float -> float

TODO

val laplace_pdf : float -> loc:float -> scale:float -> float

TODO

val laplace_logpdf : float -> loc:float -> scale:float -> float

TODO

val laplace_cdf : float -> loc:float -> scale:float -> float

TODO

val laplace_logcdf : float -> loc:float -> scale:float -> float

TODO

val laplace_ppf : float -> loc:float -> scale:float -> float

TODO

val laplace_sf : float -> loc:float -> scale:float -> float

TODO

val laplace_logsf : float -> loc:float -> scale:float -> float

TODO

val laplace_isf : float -> loc:float -> scale:float -> float

TODO

val gumbel1_rvs : a:float -> b:float -> float

TODO

val gumbel1_pdf : float -> a:float -> b:float -> float

TODO

val gumbel1_logpdf : float -> a:float -> b:float -> float

TODO

val gumbel1_cdf : float -> a:float -> b:float -> float

TODO

val gumbel1_logcdf : float -> a:float -> b:float -> float

TODO

val gumbel1_ppf : float -> a:float -> b:float -> float

TODO

val gumbel1_sf : float -> a:float -> b:float -> float

TODO

val gumbel1_logsf : float -> a:float -> b:float -> float

TODO

val gumbel1_isf : float -> a:float -> b:float -> float

TODO

val gumbel2_rvs : a:float -> b:float -> float

TODO

val gumbel2_pdf : float -> a:float -> b:float -> float

TODO

val gumbel2_logpdf : float -> a:float -> b:float -> float

TODO

val gumbel2_cdf : float -> a:float -> b:float -> float

TODO

val gumbel2_logcdf : float -> a:float -> b:float -> float

TODO

val gumbel2_ppf : float -> a:float -> b:float -> float

TODO

val gumbel2_sf : float -> a:float -> b:float -> float

TODO

val gumbel2_logsf : float -> a:float -> b:float -> float

TODO

val gumbel2_isf : float -> a:float -> b:float -> float

TODO

val logistic_rvs : loc:float -> scale:float -> float

TODO

val logistic_pdf : float -> loc:float -> scale:float -> float

TODO

val logistic_logpdf : float -> loc:float -> scale:float -> float

TODO

val logistic_cdf : float -> loc:float -> scale:float -> float

TODO

val logistic_logcdf : float -> loc:float -> scale:float -> float

TODO

val logistic_ppf : float -> loc:float -> scale:float -> float

TODO

val logistic_sf : float -> loc:float -> scale:float -> float

TODO

val logistic_logsf : float -> loc:float -> scale:float -> float

TODO

val logistic_isf : float -> loc:float -> scale:float -> float

TODO

val lognormal_rvs : mu:float -> sigma:float -> float

TODO

val lognormal_pdf : float -> mu:float -> sigma:float -> float

TODO

val lognormal_logpdf : float -> mu:float -> sigma:float -> float

TODO

val lognormal_cdf : float -> mu:float -> sigma:float -> float

TODO

val lognormal_logcdf : float -> mu:float -> sigma:float -> float

TODO

val lognormal_ppf : float -> mu:float -> sigma:float -> float

TODO

val lognormal_sf : float -> mu:float -> sigma:float -> float

TODO

val lognormal_logsf : float -> mu:float -> sigma:float -> float

TODO

val lognormal_isf : float -> mu:float -> sigma:float -> float

TODO

val rayleigh_rvs : sigma:float -> float

TODO

val rayleigh_pdf : float -> sigma:float -> float

TODO

val rayleigh_logpdf : float -> sigma:float -> float

TODO

val rayleigh_cdf : float -> sigma:float -> float

TODO

val rayleigh_logcdf : float -> sigma:float -> float

TODO

val rayleigh_ppf : float -> sigma:float -> float

TODO

val rayleigh_sf : float -> sigma:float -> float

TODO

val rayleigh_logsf : float -> sigma:float -> float

TODO

val rayleigh_isf : float -> sigma:float -> float

TODO

val dirichlet_rvs : alpha:float array -> float array

dirichlet_rvs ~alpha returns random variables of K-1 order Dirichlet distribution, follows the following probability dense function.

.. math:: f(x_1,...,x_K; \alpha_1,...,\alpha_K) = \frac

\mathbf{B(\alpha)

}

\prod_=1^K x_i^\alpha_i - 1

The normalising constant is the multivariate Beta function, which can be expressed in terms of the gamma function:

.. math:: \mathbfB(\alpha) = \frac\prod_{i=1^K \Gamma(\alpha_i)

}

\Gamma(\sum_{i=1^K \alpha_i)

}

Note that x is a standard K-1 simplex, i.e. :math:\sum_i^K x_i = 1 and :math:x_i \ge 0, \forall x_i \in 1,K.

val dirichlet_pdf : float array -> alpha:float array -> float

TODO

val dirichlet_logpdf : float array -> alpha:float array -> float`

TODO