shuffle x return a new array of the shuffled x.

choose x n draw n samples from x without replecement.

sample x n draw n samples from x with replacement.

sum x returns the summation of the elements in x.

mean x returns the mean of the elements in x.

var x returns the variance of elements in x.

std x calculates the standard deviation of x.

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

absdev x calculates the average absolute deviation of x.

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

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

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

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

concordant x y calculates the number of concordant pairs in the given arrays x and y. A pair of observations \((x_i, y_i)\) and \((x_j, y_j)\) is concordant if both elements in one pair are either greater than or less than both elements in the other pair, i.e., \((x_i > x_j) \land (y_i > y_j)\) or \((x_i < x_j) \land (y_i < y_j)\).

The function returns the count of such pairs.

• parameter x

  The first array of observations.

• parameter y

  The second array of observations.

• returns

  The number of concordant pairs.

discordant x y calculates the number of discordant pairs in the given arrays x and y. A pair of observations \((x_i, y_i)\) and \((x_j, y_j)\) is discordant if one element in one pair is greater than the corresponding element in the other pair, and the other element is smaller, i.e., \((x_i > x_j) \land (y_i < y_j)\) or \((x_i < x_j) \land (y_i > y_j)\).

The function returns the count of such pairs.

• parameter x

  The first array of observations.

• parameter y

  The second array of observations.

• returns

  The number of discordant pairs.

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

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

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

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

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.

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.

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

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

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

median x returns the median of x.

min x returns the minimum element in x.

max x returns the maximum element in x.

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

min_i x returns the index of the minimum in x.

max_i x returns the index of the maximum in x.

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

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.

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.

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

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.

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.

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.

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.

Pretty-print summary information on a histogram record

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.

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

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

normalise_pdf arr takes an array of floats arr representing a probability density function (PDF) and returns a new array where the values are normalized so that the sum of the array equals 1.

• parameter arr

  The input array representing the unnormalized PDF.

• returns

  A new array of the same length as arr, with values normalized to ensure the sum is 1.

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:

  (Q1 - k*(Q3-Q1), Q3 + k*(Q3-Q1))

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.

metropolis_hastings target_density initial_state num_samples performs the Metropolis-Hastings algorithm to generate samples from a target distribution.

The Metropolis-Hastings algorithm is a Markov Chain Monte Carlo (MCMC) method that generates a sequence of samples from a probability distribution for which direct sampling is difficult. The algorithm uses a proposal distribution to explore the state space, accepting or rejecting proposed moves based on the ratio of the target densities.

• parameter target_density

  A function that computes the density (up to a normalizing constant) of the target distribution at a given point.

• parameter initial_state

  The starting point of the Markov chain, represented as an array of floats.

• parameter num_samples

  The number of samples to generate.

• returns

  A 2D array where each row represents a sample generated by the Metropolis-Hastings algorithm.

gibbs_sampling conditional_sampler initial_state num_samples performs Gibbs sampling to generate samples from a multivariate distribution.

Gibbs sampling is a Markov Chain Monte Carlo (MCMC) algorithm used to generate samples from a joint distribution when direct sampling is difficult. It works by iteratively sampling each variable from its conditional distribution, given the current values of all other variables.

• parameter conditional_sampler

  A function that takes the current state of the variables (as a float array) and the index of the variable to sample, and returns a new value for that variable sampled from its conditional distribution.

• parameter initial_state

  The starting point of the Markov chain, represented as an array of floats.

• parameter num_samples

  The number of samples to generate.

• returns

  A 2D array where each row represents a sample generated by the Gibbs sampling algorithm.

Record type contains the result of a hypothesis test.

Pretty printer of hypothesis type

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

wilcoxon ?alpha ?side x y performs the Wilcoxon signed-rank test on the paired samples x and y.

The Wilcoxon signed-rank test is a non-parametric statistical hypothesis test used to compare two related samples, matched samples, or repeated measurements on a single sample to assess whether their population mean ranks differ.

• parameter alpha

  The significance level for the test, typically set to 0.05 by default. This parameter is optional.

• parameter side

  Specifies the alternative hypothesis. It determines whether the test is one-sided or two-sided. The default is usually a two-sided test.

• parameter x

  The first array of sample data.

• parameter y

  The second array of sample data.

• returns

  A hypothesis type, indicating whether to reject the null hypothesis or not based on the test.

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

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

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

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.

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.

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.

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

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

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.

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.

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

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

  P(x) = \frac{n!}{{x_1}! \cdot\cdot\cdot {x_k}!} p_{1}^{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`.

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

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

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

std_uniform_rvs () generates a random variate from the standard uniform distribution over the interval [0, 1\).

• returns

  A float representing a random sample from the standard uniform distribution.

uniform_rvs ~a ~b generates a random variate from the uniform distribution over the interval [a, b\).

• parameter a

  The lower bound of the interval.

• parameter b

  The upper bound of the interval.

• returns

  A float representing a random sample from the uniform distribution over [a, b\).

uniform_pdf x ~a ~b computes the probability density function (PDF) of the uniform distribution at the point x over the interval [a, b\).

• parameter x

  The point at which to evaluate the PDF.

• parameter a

  The lower bound of the interval.

• parameter b

  The upper bound of the interval.

• returns

  The PDF value at x.

uniform_logpdf x ~a ~b computes the natural logarithm of the probability density function (log-PDF) of the uniform distribution at the point x over the interval [a, b\).

• parameter x

  The point at which to evaluate the log-PDF.

• parameter a

  The lower bound of the interval.

• parameter b

  The upper bound of the interval.

• returns

  The log-PDF value at x.

uniform_cdf x ~a ~b computes the cumulative distribution function (CDF) of the uniform distribution at the point x over the interval [a, b\).

• parameter x

  The point at which to evaluate the CDF.

• parameter a

  The lower bound of the interval.

• parameter b

  The upper bound of the interval.

• returns

  The CDF value at x.

uniform_logcdf x ~a ~b computes the natural logarithm of the cumulative distribution function (log-CDF) of the uniform distribution at the point x over the interval [a, b\).

• parameter x

  The point at which to evaluate the log-CDF.

• parameter a

  The lower bound of the interval.

• parameter b

  The upper bound of the interval.

• returns

  The log-CDF value at x.

uniform_ppf q ~a ~b computes the percent-point function (PPF), also known as the quantile function or inverse CDF, of the uniform distribution for a given probability q over the interval [a, b\).

• parameter q

  The probability for which to compute the corresponding quantile.

• parameter a

  The lower bound of the interval.

• parameter b

  The upper bound of the interval.

• returns

  The quantile corresponding to q.

uniform_sf x ~a ~b computes the survival function (SF), which is one minus the cumulative distribution function (1 - CDF), of the uniform distribution at the point x over the interval [a, b\).

• parameter x

  The point at which to evaluate the SF.

• parameter a

  The lower bound of the interval.

• parameter b

  The upper bound of the interval.

• returns

  The SF value at x.

uniform_logsf x ~a ~b computes the natural logarithm of the survival function (log-SF) of the uniform distribution at the point x over the interval [a, b\).

• parameter x

  The point at which to evaluate the log-SF.

• parameter a

  The lower bound of the interval.

• parameter b

  The upper bound of the interval.

• returns

  The log-SF value at x.

uniform_isf q ~a ~b computes the inverse survival function (ISF), which is the inverse of the survival function (SF), of the uniform distribution for a given probability q over the interval [a, b\).

• parameter q

  The probability for which to compute the corresponding value from the ISF.

• parameter a

  The lower bound of the interval.

• parameter b

  The upper bound of the interval.

• returns

  The value corresponding to q from the ISF.

exponential_rvs ~lambda generates a random variate from the exponential distribution with rate parameter lambda.

• parameter lambda

  The rate parameter of the exponential distribution (must be positive).

• returns

  A float representing a random sample from the exponential distribution.

exponential_pdf x ~lambda computes the probability density function (PDF) of the exponential distribution at the point x with rate parameter lambda.

• parameter x

  The point at which to evaluate the PDF.

• parameter lambda

  The rate parameter of the exponential distribution (must be positive).

• returns

  The PDF value at x.

exponential_logpdf x ~lambda computes the natural logarithm of the probability density function (log-PDF) of the exponential distribution at the point x with rate parameter lambda.

• parameter x

  The point at which to evaluate the log-PDF.

• parameter lambda

  The rate parameter of the exponential distribution (must be positive).

• returns

  The log-PDF value at x.

exponential_cdf x ~lambda computes the cumulative distribution function (CDF) of the exponential distribution at the point x with rate parameter lambda.

• parameter x

  The point at which to evaluate the CDF.

• parameter lambda

  The rate parameter of the exponential distribution (must be positive).

• returns

  The CDF value at x.

exponential_logcdf x ~lambda computes the natural logarithm of the cumulative distribution function (log-CDF) of the exponential distribution at the point x with rate parameter lambda.

• parameter x

  The point at which to evaluate the log-CDF.

• parameter lambda

  The rate parameter of the exponential distribution (must be positive).

• returns

  The log-CDF value at x.

exponential_ppf q ~lambda computes the percent-point function (PPF), also known as the quantile function or inverse CDF, of the exponential distribution for a given probability q with rate parameter lambda.

• parameter q

  The probability for which to compute the corresponding quantile.

• parameter lambda

  The rate parameter of the exponential distribution (must be positive).

• returns

  The quantile corresponding to q.

exponential_sf x ~lambda computes the survival function (SF), which is one minus the cumulative distribution function (1 - CDF), of the exponential distribution at the point x with rate parameter lambda.

• parameter x

  The point at which to evaluate the SF.

• parameter lambda

  The rate parameter of the exponential distribution (must be positive).

• returns

  The SF value at x.

exponential_logsf x ~lambda computes the natural logarithm of the survival function (log-SF) of the exponential distribution at the point x with rate parameter lambda.

• parameter x

  The point at which to evaluate the log-SF.

• parameter lambda

  The rate parameter of the exponential distribution (must be positive).

• returns

  The log-SF value at x.

exponential_isf q ~lambda computes the inverse survival function (ISF), which is the inverse of the survival function (SF), of the exponential distribution for a given probability q with rate parameter lambda.

• parameter q

  The probability for which to compute the corresponding value from the ISF.

• parameter lambda

  The rate parameter of the exponential distribution (must be positive).

• returns

  The value corresponding to q from the ISF.

exponpow_pdf x ~a ~b computes the probability density function (PDF) of the exponential power distribution at the point x with shape parameters a and b.

• parameter x

  The point at which to evaluate the PDF.

• parameter a

  The scale parameter of the distribution.

• parameter b

  The shape parameter of the distribution.

• returns

  The PDF value at x.

exponpow_logpdf x ~a ~b computes the natural logarithm of the probability density function (log-PDF) of the exponential power distribution at the point x with shape parameters a and b.

• parameter x

  The point at which to evaluate the log-PDF.

• parameter a

  The scale parameter of the distribution.

• parameter b

  The shape parameter of the distribution.

• returns

  The log-PDF value at x.

exponpow_cdf x ~a ~b computes the cumulative distribution function (CDF) of the exponential power distribution at the point x with shape parameters a and b.

• parameter x

  The point at which to evaluate the CDF.

• parameter a

  The scale parameter of the distribution.

• parameter b

  The shape parameter of the distribution.

• returns

  The CDF value at x.

exponpow_logcdf x ~a ~b computes the natural logarithm of the cumulative distribution function (log-CDF) of the exponential power distribution at the point x with shape parameters a and b.

• parameter x

  The point at which to evaluate the log-CDF.

• parameter a

  The scale parameter of the distribution.

• parameter b

  The shape parameter of the distribution.

• returns

  The log-CDF value at x.

exponpow_sf x ~a ~b computes the survival function (SF), which is one minus the cumulative distribution function (1 - CDF), of the exponential power distribution at the point x with shape parameters a and b.

• parameter x

  The point at which to evaluate the SF.

• parameter a

  The scale parameter of the distribution.

• parameter b

  The shape parameter of the distribution.

• returns

  The SF value at x.

exponpow_logsf x ~a ~b computes the natural logarithm of the survival function (log-SF) of the exponential power distribution at the point x with shape parameters a and b.

• parameter x

  The point at which to evaluate the log-SF.

• parameter a

  The scale parameter of the distribution.

• parameter b

  The shape parameter of the distribution.

• returns

  The log-SF value at x.

gaussian_rvs ~mu ~sigma generates a random variate from the Gaussian (normal) distribution with mean mu and standard deviation sigma.

• parameter mu

  The mean of the distribution.

• parameter sigma

  The standard deviation of the distribution.

• returns

  A float representing a random sample from the Gaussian distribution.

gaussian_pdf x ~mu ~sigma computes the probability density function (PDF) of the Gaussian (normal) distribution at the point x with mean mu and standard deviation sigma.

• parameter x

  The point at which to evaluate the PDF.

• parameter mu

  The mean of the distribution.

• parameter sigma

  The standard deviation of the distribution.

• returns

  The PDF value at x.

gaussian_logpdf x ~mu ~sigma computes the natural logarithm of the probability density function (log-PDF) of the Gaussian (normal) distribution at the point x with mean mu and standard deviation sigma.

• parameter x

  The point at which to evaluate the log-PDF.

• parameter mu

  The mean of the distribution.

• parameter sigma

  The standard deviation of the distribution.

• returns

  The log-PDF value at x.