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Class type
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.
``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``.
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.
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.
``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
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.
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:
``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 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 runnign 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 probabilty mass function is as follows.
``p`` is the probabilty 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 probabilty of ``x`` given the probabilty mass of a multinomial distribution.
val multinomial_logpdf : int array->p:float array-> float
``multinomial_rvs x ~p`` returns the logarithm probabilty of ``x`` given the probabilty 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