!ɕ       !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~          !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~  !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!""""####$$$$$%%%%%&&'''''''''''''''''''''''(((((((((((( ( ( ( ( (((((())))**.None>*u statisticsO(nlogn)_ Compute the Kendall's tau from a vector of paired data. Return NaN when number of pairs <= 1.+(c) 2011 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableNone6 statisticsMAn unchecked, non-integer-valued version of Loader's saddle point algorithm. statisticsReturns [x, x^2, x^3, x^4, ...] statisticsbReturns an upper bound according to theorem 2 of "Sharp Bounds on the Entropy of the Poisson Law" statisticsKReturns the average of the upper and lower bounds accounding to theorem 2. statisticsKCompute entropy directly from its definition. This is just as accurate as alyThm1 for lambda <= 1 and is faster, but is slow for large lambda, and produces some underestimation due to accumulation of floating point error. statisticsOCompute the entropy of a poisson distribution using the best available method.%(c) 2009, 2010, 2011 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableNone>SXD=  statisticsSort a vector. statisticsSort a vector. statistics&Sort a vector using a custom ordering. statistics-Partially sort a vector, such that the least k elements will be at the front. statisticsReturn the indices of a vector.  statisticsZip a vector with its indices.  statistics8Compute the minimum and maximum of a vector in one pass.  statisticsEfficiently compute the next highest power of two for a non-negative integer. If the given value is already a power of two, it is returned unchanged. If negative, zero is returned.  statisticsMultiply a number by itself.  statisticsSimple for loop. Counts from start to end-1. statistics&Simple reverse-for loop. Counts from start-1 to end (which must be less than start). statistics The number k of least elements.       ,(c) 2009 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableSafeE !"#$%&'()*+,-(c) 2009 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableNone%2467>_ statistics Parameters  and  to the r function. Exact meaning of parameters is described in [Hyndman1996] in section "Piecewise linear functions" statisticsO(nlog n). Estimate the kth q_-quantile of a sample, using the weighted average method. Up to rounding errors it's same as  quantile s.GThe following properties should hold otherwise an error will be thrown.(the length of the input is greater than 0the input does not contain NaNk "e 0 and k "d q statisticsO(nlog n). Estimate the kth q-quantile of a sample xB, using the continuous sample method with the given parameters.HThe following properties should hold, otherwise an error will be thrown.input sample must be nonemptythe input does not contain NaN 0 "d k "d q statisticsO(knlog n). Estimate set of the kth q-quantile of a sample x, using the continuous sample method with the given parameters. This is faster than calling quantile repeatedly since sample should be sorted only onceHThe following properties should hold, otherwise an error will be thrown.input sample must be nonemptythe input does not contain NaN)for every k in set of quantiles 0 "d k "d q statisticsO(knlog n). Same as quantiles but uses . container instead of / one. statistics2California Department of Public Works definition, =0, l=1. Gives a linear interpolation of the empirical CDF. This corresponds to method 4 in R and Mathematica. statisticsHazen's definition, =0.5, n=0.5. This is claimed to be popular among hydrologists. This corresponds to method 5 in R and Mathematica. statistics9Definition used by the SPSS statistics application, with =0, ]=0 (also known as Weibull's definition). This corresponds to method 6 in R and Mathematica. statistics6Definition used by the S statistics application, with =1, ;=1. The interpolation points divide the sample range into n-1U intervals. This corresponds to method 7 in R and Mathematica and is default in R. statisticsMedian unbiased definition, =1/3, m=1/3. The resulting quantile estimates are approximately median unbiased regardless of the distribution of x6. This corresponds to method 8 in R and Mathematica. statisticsNormal unbiased definition, =3/8, =3/8. An approximately unbiased estimate if the empirical distribution approximates the normal distribution. This corresponds to method 9 in R and Mathematica. statisticsO(nlog n) Estimate median of sample statisticsO(nlog n). Estimate the range between q-quantiles 1 and q-1 of a sample x@, using the continuous sample method with the given parameters.IFor instance, the interquartile range (IQR) can be estimated as follows: @midspread medianUnbiased 4 (U.fromList [1,1,2,2,3]) ==> 1.333333 statisticsO(nlog n?). Estimate the median absolute deviation (MAD) of a sample x using B. It's robust estimate of variability in sample and defined as:G MAD = \operatorname{median}(| X_i - \operatorname{median}(X) |) # statisticsWe use / as default value which is same as R's default. statisticsk, the desired quantile. statisticsq, the number of quantiles. statisticsx, the sample data. statistics Parameters  and . statisticsk, the desired quantile. statisticsq, the number of quantiles. statisticsx, the sample data.0 statisticsn number of elements statisticsk, the desired quantile. statisticsq, the number of quantiles. statistics Parameters  and . statisticsx, the sample data. statistics Parameters  and . statisticsq, the number of quantiles. statisticsx, the sample data. statistics Parameters  and . statisticsx, the sample data. statistics Parameters  and . statisticsk, the desired quantile. statisticsq, the number of quantiles. statisticsx, the sample data.(c) 2011 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableNone>+ statisticsO(n)% Compute a histogram over a data set.)The result consists of a pair of vectors:!The lower bound of each interval.*The number of samples within the interval.eInterval (bin) sizes are uniform, and the upper and lower bounds are chosen automatically using the -; function. To specify these parameters directly, use the , function., statisticsO(n)% Compute a histogram over a data set.PInterval (bin) sizes are uniform, based on the supplied upper and lower bounds.- statisticsO(n) Compute decent defaults for the lower and upper bounds of a histogram, based on the desired number of bins and the range of the sample data.$The upper and lower bounds used are  (lo-d, hi+d), where 8d = (maximum sample - minimum sample) / ((bins - 1) * 2)8If all elements in the sample are the same and equal to x range is set to (x - |x| 10, x + |x|10) . And if x is equal to 0 range is set to (-1,1)A. This is needed to avoid creating histogram with zero bin size.+ statistics"Number of bins (must be positive). statisticsSample data (cannot be empty)., statistics]Number of bins. This value must be positive. A zero or negative value will cause an error. statisticsPLower bound on interval range. Sample data less than this will cause an error. statisticsUpper bound on interval range. This value must not be less than the lower bound. Sample data that falls above the upper bound will cause an error. statistics Sample data.- statistics"Number of bins (must be positive). statisticsSample data (cannot be empty).+,-+,-(c) 2013 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableNone>././(c) 2009 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableNone@AX0 statisticspEstimate distribution from sample. First parameter in sample is distribution type and second is element type.1 statisticsEstimate distribution from sample. Returns nothing is there's not enough data to estimate or sample clearly doesn't come from distribution in question. For example if there's negative samples in exponential distribution.2 statisticsDGenerate discrete random variates which have given distribution. 4c is superclass because it's always possible to generate real-valued variates from integer values4 statisticsCGenerate discrete random variates which have given distribution.6 statistics.Type class for distributions with entropy, meaning Shannon entropy in the case of a discrete distribution, or differential entropy in the case of a continuous one. If the distribution has well-defined entropy for all valid parameter values then it should be an instance of this type class.7 statistics/Returns the entropy of a distribution, in nats.8 statisticsType class for distributions with entropy, meaning Shannon entropy in the case of a discrete distribution, or differential entropy in the case of a continuous one. 9 should return 1< if entropy is undefined for the chosen parameter values.9 statisticsCReturns the entropy of a distribution, in nats, if such is defined.: statisticsType class for distributions with variance. If distribution have finite variance for all valid parameter values it should be instance of this type class.Minimal complete definition is ; or <= statisticsgType class for distributions with variance. If variance is undefined for some parameter values both > and ? should return Nothing.Minimal complete definition is > or ?@ statisticsType class for distributions with mean. If a distribution has finite mean for all valid values of parameters it should be instance of this type class.B statistics(Type class for distributions with mean. C should return 1, if it's undefined for current value of dataD statistics$Continuous probability distribution.Minimal complete definition is F and either E or H.E statistics@Probability density function. Probability that random variable X& lies in the infinitesimal interval [x,x+x ) equal to  density(x)"xF statistics<Inverse of the cumulative distribution function. The value x for which P(X"dx) = pA. If probability is outside of [0,1] range function should call 2G statistics1-complement of quantile: "complQuantile x "a quantile (1 - x)H statisticsNatural logarithm of density.I statistics"Discrete probability distribution.J statisticsProbability of n-th outcome.K statistics(Logarithm of probability of n-th outcomeL statisticsuType class common to all distributions. Only c.d.f. could be defined for both discrete and continuous distributions.M statisticsKCumulative distribution function. The probability that a random variable X is less or equal than x , i.e. P(X"dx8). Cumulative should be defined for infinities as well: 'cumulative d +" = 1 cumulative d -" = 0N statistics,One's complement of cumulative distribution: (complCumulative d x = 1 - cumulative d x(It's useful when one is interested in P(X>x) and expression on the right side begin to lose precision. This function have default implementation but implementors are encouraged to provide more precise implementation.O statisticsOGenerate variates from continuous distribution using inverse transform rule.P statistics+Backwards compatibility with genContinuous.Q statisticsApproximate the value of X for which P(x>X)=p.This method uses a combination of Newton-Raphson iteration and bisection with the given guess as a starting point. The upper and lower bounds specify the interval in which the probability distribution reaches the value p.R statistics(Sum probabilities in inclusive interval.Q statistics Distribution statistics Probability p statistics Initial guess statisticsLower bound on interval statisticsUpper bound on interval#0123456789:<;=>?@ABCDFEHGIJKLMNOPQR#LMNIJKDFEHGBC@A=>?:<;8967014523OPQR(c) 2011 Aleksey KhudyakovBSD3bos@serpentine.com experimentalportableNone27iS statistics Uniform distribution from A to BT statisticsLow boundary of distributionU statisticsUpper boundary of distributionV statisticsCreate uniform distribution.W statisticsCreate uniform distribution.SUTVWSVWTU(c) 2013 John McDonnell;BSD3bos@serpentine.com experimentalportableNone27=>? i statistics.Linear transformation applied to distribution. "LinearTransform   _ x' =  + xk statisticsLocation parameter.l statisticsScale parameter.m statisticsDistribution being transformed.n statistics,Apply linear transformation to distribution.o statistics(Get fixed point of linear transformationn statistics Fixed point statisticsScale parameter statistics Distributionijklmnoijklmon (c) 2011 Aleksey KhudyakovBSD3bos@serpentine.com experimentalportableNone27 statisticsStudent-T distribution statisticsECreate Student-T distribution. Number of parameters must be positive. statisticsECreate Student-T distribution. Number of parameters must be positive. statistics0Create an unstandardized Student-t distribution. statisticsNumber of degrees of freedom statistics5Central value (0 for standard Student T distribution) statisticsScale parameter (c) 2009, 2011 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableNone27 statisticsCreate Poisson distribution. statisticsCreate Poisson distribution. (c) 2009 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableNone27 statisticsm statisticsl statisticsk statisticsm statisticsl statisticsk (c) 2009 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableNone27 statisticsDistribution over [0..] statisticsDistribution over [1..] statisticsCreate geometric distribution. statisticsCreate geometric distribution. statisticsCreate geometric distribution. statisticsCreate geometric distribution. statistics Success rate statistics Success rate statistics Success rate statistics Success rate (c) 2009, 2011 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableNone27 statisticsThe gamma distribution. statisticsShape parameter, k. statisticsScale parameter, . statisticsMCreate gamma distribution. Both shape and scale parameters must be positive. statisticsMCreate gamma distribution. Both shape and scale parameters must be positive. statisticsQCreate gamma distribution. Both shape and scale parameters must be non-negative. statisticsQCreate gamma distribution. Both shape and scale parameters must be non-negative. statisticsShape parameter. k statisticsScale parameter, . statisticsShape parameter. k statisticsScale parameter, . statisticsShape parameter. k statisticsScale parameter, . statisticsShape parameter. k statisticsScale parameter, .(c) 2011 Aleksey KhudyakovBSD3bos@serpentine.com experimentalportableNone27 statisticsF distribution  (c) 2016 Andr Szabolcs SzelpBSD3a.sz.szelp@gmail.com experimentalportableNone27W statistics"The discrete uniform distribution. statisticsa,, the lower bound of the support {a, ..., b} statisticsb,, the upper bound of the support {a, ..., b} statisticsIConstruct discrete uniform distribution on support {1, ..., n}. Range n must be >0. statistics?Construct discrete uniform distribution on support {a, ..., b}. statisticsRange statisticsLower boundary (inclusive) statisticsUpper boundary (inclusive)(c) 2010 Alexey KhudyakovBSD3bos@serpentine.com experimentalportableNone27 .. statisticsChi-squared distribution/ statistics Get number of degrees of freedom0 statisticsUConstruct chi-squared distribution. Number of degrees of freedom must be positive.1 statisticsUConstruct chi-squared distribution. Number of degrees of freedom must be positive../01./01(c) 2011 Aleksey KhudyakovBSD3bos@serpentine.com experimentalportableNone27++C statisticsCauchy-Lorentz distribution.D statisticsCentral value of Cauchy-Lorentz distribution which is its mode and median. Distribution doesn't have mean so function is named after median.E statisticsScale parameter of Cauchy-Lorentz distribution. It's different from variance and specify half width at half maximum (HWHM).F statisticsCauchy distributionG statisticsCauchy distributionH statistics@Standard Cauchy distribution. It's centered at 0 and have 1 FWHMF statistics Central point statisticsScale parameter (FWHM)G statistics Central point statisticsScale parameter (FWHM)CEDFGHCDEFGH(c) 2009 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableNone273V statisticsThe binomial distribution.W statisticsNumber of trials.X statistics Probability.Y statisticspConstruct binomial distribution. Number of trials must be non-negative and probability must be in [0,1] rangeZ statisticspConstruct binomial distribution. Number of trials must be non-negative and probability must be in [0,1] rangeY statisticsNumber of trials. statistics Probability.Z statisticsNumber of trials. statistics Probability.VXWYZVYZWX(C) 2012 Edward Kmett, BSD-style (see the file LICENSE)Edward Kmett <ekmett@gmail.com> provisionalDeriveDataTypeableNone27Ck statisticsThe beta distributionl statisticsAlpha shape parameterm statisticsBeta shape parametern statisticsACreate beta distribution. Both shape parameters must be positive.o statisticsACreate beta distribution. Both shape parameters must be positive.p statisticsCreate beta distribution. Both shape parameters must be non-negative. So it allows to construct improper beta distribution which could be used as improper prior.q statisticsCreate beta distribution. Both shape parameters must be non-negative. So it allows to construct improper beta distribution which could be used as improper prior.n statisticsShape parameter alpha statisticsShape parameter betao statisticsShape parameter alpha statisticsShape parameter betap statisticsShape parameter alpha statisticsShape parameter betaq statisticsShape parameter alpha statisticsShape parameter betakmlnopqknopqlm(c) 2009, 2010 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableNone27>o  statisticsO(n) Collect the n simple powers of a sample.XFunctions computed over a sample's simple powers require at least a certain number (or order) of powers to be collected.To compute the kth  , at least k$ simple powers must be collected.For the &, at least 2 simple powers are needed.For #, we need at least 3 simple powers.For (, at least 4 simple powers are required.*This function is subject to stream fusion. statistics5The order (number) of simple powers collected from a sample. statistics Compute the k_th central moment of a sample. The central moment is also known as the moment about the mean. statisticsvMaximum likelihood estimate of a sample's variance. Also known as the population variance, where the denominator is n4. This is the second central moment of the sample.CThis is less numerically robust than the variance function in the -.o module, but the number is essentially free to compute if you have already collected a sample's simple powers. Requires  with  at least 2. statisticshStandard deviation. This is simply the square root of the maximum likelihood estimate of the variance. statisticshUnbiased estimate of a sample's variance. Also known as the sample variance, where the denominator is n-1. Requires  with  at least 2. statisticsZCompute the skewness of a sample. This is a measure of the asymmetry of its distribution.*A sample with negative skew is said to be  left-skewedU. Most of its mass is on the right of the distribution, with the tail on the left. Eskewness . powers 3 $ U.to [1,100,101,102,103] ==> -1.497681449918257*A sample with positive skew is said to be  right-skewed. ?skewness . powers 3 $ U.to [1,2,3,4,100] ==> 1.4975367033335198*A sample's skewness is not defined if its  is zero. Requires  with  at least 3. statisticsCompute the excess kurtosis of a sample. This is a measure of the "peakedness" of its distribution. A high kurtosis indicates that the sample's variance is due more to infrequent severe deviations than to frequent modest deviations.1A sample's excess kurtosis is not defined if its  is zero. Requires  with  at least 4. statistics'The number of elements in the original Sample-. This is the sample's zeroth simple power. statistics$The sum of elements in the original Sample,. This is the sample's first simple power. statistics0The arithmetic mean of elements in the original Sample.?This is less numerically robust than the mean function in the -.o module, but the number is essentially free to compute if you have already collected a sample's simple powers. statisticsn, the number of powers, where n >= 2.  /None>y3 statisticsCalculate rank of every element of sample. In case of ties ranks are averaged. Sample should be already sorted in ascending order.Rank is index of element in the sample, numeration starts from 1. In case of ties average of ranks of equal elements is assigned to each$rank (==) (fromList [10,20,30::Int])> fromList [1.0,2.0,3.0]'rank (==) (fromList [10,10,10,30::Int])> fromList [2.0,2.0,2.0,4.0]4 statistics_Compute rank of every element of vector. Unlike rank it doesn't require sample to be sorted.5 statisticsSplit tagged vector3 statisticsEquivalence relation statisticsVector to rank345(c) 2011 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableNone> statistics#Discrete cosine transform (DCT-II). statisticsjDiscrete cosine transform (DCT-II). Only real part of vector is transformed, imaginary part is ignored. statistics>Inverse discrete cosine transform (DCT-III). It's inverse of  only up to scale parameter: (idct . dct) x = (* length x) statisticssInverse discrete cosine transform (DCT-III). Only real part of vector is transformed, imaginary part is ignored. statisticsInverse fast Fourier transform. statistics2Radix-2 decimation-in-time fast Fourier transform.(c) 2011 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableNone> statistics]Gaussian kernel density estimator for one-dimensional data, using the method of Botev et al.,The result is a pair of vectors, containing:The coordinates of each mesh point. The mesh interval is chosen to be 20% larger than the range of the sample. (To specify the mesh interval, use .)%Density estimates at each mesh point. statistics]Gaussian kernel density estimator for one-dimensional data, using the method of Botev et al.,The result is a pair of vectors, containing:#The coordinates of each mesh point.%Density estimates at each mesh point. statisticsPThe number of mesh points to use in the uniform discretization of the interval  (min,max)X. If this value is not a power of two, then it is rounded up to the next power of two. statisticsPThe number of mesh points to use in the uniform discretization of the interval  (min,max)X. If this value is not a power of two, then it is rounded up to the next power of two. statistics Lower bound (min) of the mesh range. statistics Upper bound (max) of the mesh range.0(c) 2009 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableNone statistics=Weights for affecting the importance of elements of a sample. statisticsFSample with weights. First element of sample is data, second is weight statistics Sample data.+(c) 2008 Don Stewart, 2009 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableNone>  statisticsO(n)N Range. The difference between the largest and smallest elements of a sample. statisticsO(n)Y Arithmetic mean. This uses Kahan-Babuaka-Neumaier summation, so is more accurate than ) unless the input values are very large. statisticsO(n){ Arithmetic mean. This uses Welford's algorithm to provide numerical stability, using a single pass over the sample data. Compared to P, this loses a surprising amount of precision unless the inputs are very large. statisticsO(n)d Arithmetic mean for weighted sample. It uses a single-pass algorithm analogous to the one used by . statisticsO(n)H Harmonic mean. This algorithm performs a single pass over the sample. statisticsO(n): Geometric mean of a sample containing no negative values. statistics Compute the k_th central moment of a sample. The central moment is also known as the moment about the mean.WThis function performs two passes over the sample, so is not subject to stream fusion.For samples containing many values very close to the mean, this function is subject to inaccuracy due to catastrophic cancellation. statistics Compute the kth and jth central moments of a sample.WThis function performs two passes over the sample, so is not subject to stream fusion.For samples containing many values very close to the mean, this function is subject to inaccuracy due to catastrophic cancellation. statisticsZCompute the skewness of a sample. This is a measure of the asymmetry of its distribution.*A sample with negative skew is said to be  left-skewedU. Most of its mass is on the right of the distribution, with the tail on the left. :skewness $ U.to [1,100,101,102,103] ==> -1.497681449918257*A sample with positive skew is said to be  right-skewed. 4skewness $ U.to [1,2,3,4,100] ==> 1.4975367033335198*A sample's skewness is not defined if its  is zero.WThis function performs two passes over the sample, so is not subject to stream fusion.For samples containing many values very close to the mean, this function is subject to inaccuracy due to catastrophic cancellation. statisticsCompute the excess kurtosis of a sample. This is a measure of the "peakedness" of its distribution. A high kurtosis indicates that more of the sample's variance is due to infrequent severe deviations, rather than more frequent modest deviations.1A sample's excess kurtosis is not defined if its  is zero.WThis function performs two passes over the sample, so is not subject to stream fusion.For samples containing many values very close to the mean, this function is subject to inaccuracy due to catastrophic cancellation. statisticsvMaximum likelihood estimate of a sample's variance. Also known as the population variance, where the denominator is n. statisticshUnbiased estimate of a sample's variance. Also known as the sample variance, where the denominator is n-1. statisticsCalculate mean and maximum likelihood estimate of variance. This function should be used if both mean and variance are required since it will calculate mean only once. statisticsCalculate mean and unbiased estimate of variance. This function should be used if both mean and variance are required since it will calculate mean only once. statistics^Standard deviation. This is simply the square root of the unbiased estimate of the variance. statisticsjStandard error of the mean. This is the standard deviation divided by the square root of the sample size. statistics-Weighted variance. This is biased estimation. statistics3Maximum likelihood estimate of a sample's variance. statistics)Unbiased estimate of a sample's variance. statisticshStandard deviation. This is simply the square root of the maximum likelihood estimate of the variance. statisticsCCovariance of sample of pairs. For empty sample it's set to zero statisticswCorrelation coefficient for sample of pairs. Also known as Pearson's correlation. For empty sample it's set to zero. statisticsPair two samples. It's like 63 but requires that both samples have equal size.(c) 2017 Gregory W. SchwartzBSD3gsch@mail.med.upenn.edu experimentalportableNone>] statisticsO(n)* Normalize a sample using standard scores: z = \frac{x - \mu}{\sigma} Where  is sample mean and  is standard deviation computed from unbiased variance estimation. If sample to small to compute  or it's equal to 0 Nothing is returned.(c) 2009 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableNone27>  statistics7The convolution kernel. Its parameters are as follows:Scaling factor, 1/nh Bandwidth, h&A point at which to sample the input, pOne sample value, v statistics)The width of the convolution kernel used. statisticsPoints from the range of a Sample. statistics/Bandwidth estimator for an Epanechnikov kernel. statistics*Bandwidth estimator for a Gaussian kernel. statisticsKCompute the optimal bandwidth from the observed data for the given kernel.This function uses an estimate based on the standard deviation of a sample (due to Deheuvels), which performs reasonably well for unimodal distributions but leads to oversmoothing for more complex ones. statistics_Choose a uniform range of points at which to estimate a sample's probability density function.mIf you are using a Gaussian kernel, multiply the sample's bandwidth by 3 before passing it to this function.aIf this function is passed an empty vector, it returns values of positive and negative infinity. statistics@Epanechnikov kernel for probability density function estimation. statistics<Gaussian kernel for probability density function estimation. statisticseKernel density estimator, providing a non-parametric way of estimating the PDF of a random variable. statistics}A helper for creating a simple kernel density estimation function with automatically chosen bandwidth and estimation points. statisticsSimple Epanechnikov kernel density estimator. Returns the uniformly spaced points from the sample range at which the density function was estimated, and the estimates at those points. statisticsSimple Gaussian kernel density estimator. Returns the uniformly spaced points from the sample range at which the density function was estimated, and the estimates at those points. statisticsNumber of points to select, n statisticsSample bandwidth, h statistics Input data statisticsKernel function statistics Bandwidth, h statistics Sample data statisticsPoints at which to estimate statisticsBandwidth function statisticsKernel function statisticsDBandwidth scaling factor (3 for a Gaussian kernel, 1 for all others) statistics%Number of points at which to estimate statistics sample data statistics%Number of points at which to estimate statistics Data sample statistics%Number of points at which to estimate statistics Data sample(c) 2009 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableNone27@A statisticsThe normal distribution. statisticsIStandard normal distribution with mean equal to 0 and variance equal to 1 statistics+Create normal distribution from parameters.WIMPORTANT: prior to 0.10 release second parameter was variance not standard deviation. statistics+Create normal distribution from parameters.WIMPORTANT: prior to 0.10 release second parameter was variance not standard deviation. statisticsMVariance is estimated using maximum likelihood method (biased estimation).Returns NothingY if sample contains less than one element or variance is zero (all elements are equal) statisticsMean of distribution statistics"Standard deviation of distribution statisticsMean of distribution statistics"Standard deviation of distribution(c) 2009 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableNone27>@AHVXo"( statisticsLower limit. They are usually given for large quantities when it's not possible to measure them. For example: proton half-life statistics Lower limit statistics/Confidence level for which limit was calculated statisticsxUpper limit. They are usually given for small non-negative values when it's not possible detect difference from zero. statistics Upper limit statistics/Confidence level for which limit was calculated statistics1Data types which could be multiplied by constant. statisticsvConfidence interval. It assumes that confidence interval forms single interval and isn't set of disjoint intervals. statisticscLower error estimate, or distance between point estimate and lower bound of confidence interval. statisticscUpper error estimate, or distance between point estimate and upper bound of confidence interval. statistics<Confidence level corresponding to given confidence interval. statisticsNormal errors. They are stored as 1 errors which corresponds to 68.8% CL. Since we can recalculate them to any confidence level if needed we don't store it. statisticsTA point estimate and its confidence interval. It's parametrized by both error type e and value type a,. This module provides two types of error: & for normally distributed errors and O for error with normal distribution. See their documentation for more details. For example 144 5+ (assuming normality) could be expressed as FEstimate { estPoint = 144 , estError = NormalErr 5 }Or if we want to express  144 + 6 - 4 at CL95 we could write: Estimate { estPoint = 144 , estError = ConfInt { confIntLDX = 4 , confIntUDX = 6 , confIntCL = cl95 }Prior to statistics 0.14 Estimate% data type used following definition: data Estimate = Estimate { estPoint :: {-# UNPACK #-} !Double , estLowerBound :: {-# UNPACK #-} !Double , estUpperBound :: {-# UNPACK #-} !Double , estConfidenceLevel :: {-# UNPACK #-} !Double } Now type Estimate ConfInt Double# should be used instead. Function 5 allow to easily construct estimate from same inputs. statisticsPoint estimate. statistics!Confidence interval for estimate. statisticsNewtype wrapper for p-value. statisticsConfidence level. In context of confidence intervals it's probability of said interval covering true value of measured value. In context of statistical tests it's 1-" where  is significance of test.BSince confidence level are usually close to 1 they are stored as 1-CL2 internally. There are two smart constructors for CL:  and ' (and corresponding variant returning Maybe). First creates CL( from confidence level and second from 1 - CL or significance level.cl95mkCLFromSignificance 0.05BPrior to 0.14 confidence levels were passed to function as plain Doubles. Use  to convert them to CL. statisticsCreate confidence level from probability  or probability confidence interval contain true value of estimate. Will throw exception if parameter is out of [0,1] rangemkCL 0.95 -- same as cl95mkCLFromSignificance 0.05 statisticsSame as  but returns Nothing7 instead of error if parameter is out of [0,1] rangemkCLE 0.95 -- same as cl95 Just (mkCLFromSignificance 0.05) statisticsCreate confidence level from probability  or probability that confidence interval does not contain true value of estimate. Will throw exception if parameter is out of [0,1] range,mkCLFromSignificance 0.05 -- same as cl95mkCLFromSignificance 0.05 statisticsSame as  but returns Nothing7 instead of error if parameter is out of [0,1] range-mkCLFromSignificanceE 0.05 -- same as cl95 Just (mkCLFromSignificance 0.05) statisticsIGet confidence level. This function is subject to rounding errors. If 1 - CL is needed use  instead statisticsGet significance level. statistics90% confidence level statistics95% confidence level statistics99% confidence level  statisticsAConstruct PValue. Throws error if argument is out of [0,1] range.  statisticsConstruct PValue. Returns Nothing# if argument is out of [0,1] range.  statistics Get p-value  statisticsP-value expressed in sigma. This is convention widely used in experimental physics. N sigma confidence level corresponds to probability within N sigma of normal distribution.Note that this correspondence is for normal distribution. Other distribution will have different dependency. Also experimental distribution usually only approximately normal (especially at extreme tails).  statisticssP-value expressed in sigma for one-tail hypothesis. This correspond to probability of obtaining value less than N. statistics"Express confidence level in sigmas statistics=Express confidence level in sigmas for one-tailed hypothesis. statistics"Create estimate with normal errors statistics Synonym for  statistics&Create estimate with asymmetric error. statistics&Create estimate with asymmetric error. statisticsGet confidence interval statisticsGet asymmetric errors statistics cl95 > cl90True statisticsPoint estimate statistics1 error statisticsPoint estimate statistics1 error statisticsCentral estimate statisticsHLower and upper errors. Both should be positive but it's not checked. statisticsConfidence level for interval statisticsCPoint estimate. Should lie within interval but it's not checked. statistics"Lower and upper bounds of interval statisticsConfidence level for interval1     1     None247~ o statisticsYTest type for test which compare positional (mean,median etc.) information of samples.p statisticsXTest whether samples differ in position. Null hypothesis is samples are not differentq statisticsqTest if first sample (A) is larger than second (B). Null hypothesis is first sample is not larger than second.r statistics+Test if second sample is larger than first.s statisticsResult of statistical test.u statisticsSProbability of getting value of test statistics at least as extreme as measured.v statisticsStatistic used for test.w statistics>Distribution of test statistics if null hypothesis is correct.x statisticsResult of hypothesis testingy statistics"Null hypothesis should be rejectedz statistics"Data is compatible with hypothesis{ statistics4Check whether test is significant for given p-value.| statisticssignificant if parameter is 7, not significant otherwiseopqrstuvwxyz{|stuvw{xyz|opqrNone>SX( statisticsvTwo-sample Student's t-test. It assumes that both samples are normally distributed and have same variance. Returns Nothing' if sample sizes are not sufficient. statisticsTwo-sample Welch's t-test. It assumes that both samples are normally distributed but doesn't assume that they have same variance. Returns Nothing$ if sample sizes are not sufficient. statisticsVPaired two-sample t-test. Two samples are paired in a within-subject design. Returns Nothing# if sample size is not sufficient. statisticsone- or two-tailed test statisticsSample A statisticsSample B statisticsone- or two-tailed test statisticsSample A statisticsSample B statisticsone- or two-tailed test statisticspaired samples8 statisticsone- or two-tailed statistics t statistics statisticsdegree of freedom statisticsp-valueopqrstuvwxyz{|(c) 2014 Danny NavarroBSD3bos@serpentine.com experimentalportableNoneD statisticsKruskal-Wallis ranking.EAll values are replaced by the absolute rank in the combined samples.The samples and values need not to be ordered but the values in the result are ordered. Assigned ranks (ties are given their average rank). statisticsThe Kruskal-Wallis Test.8In textbooks the output value is usually represented by K or H*. This function already does the ranking. statisticsoPerform Kruskal-Wallis Test for the given samples and required significance. For additional information check ". This is just a helper function.It uses  Chi-Squaredd distribution for approximation as long as the sizes are larger than 5. Otherwise the test returns 1.opqrstuvwxyz{|(c) 2011 Aleksey KhudyakovBSD3bos@serpentine.com experimentalportableNone>W statisticsACheck that sample could be described by distribution. Returns Nothing is sample is emptyRThis test uses Marsaglia-Tsang-Wang exact algorithm for calculation of p-value. statistics Variant of ' which uses CDF in form of function. statisticsTwo sample Kolmogorov-Smirnov test. It tests whether two data samples could be described by the same distribution without making any assumptions about it. If either of samples is empty returns Nothing.9This test uses approximate formula for computing p-value. statistics!Calculate Kolmogorov's statistic Df for given cumulative distribution function (CDF) and data sample. If sample is empty returns 0. statistics!Calculate Kolmogorov's statistic Df for given cumulative distribution function (CDF) and data sample. If sample is empty returns 0. statistics!Calculate Kolmogorov's statistic DB for two data samples. If either of samples is empty returns 0. statisticsPCalculate cumulative probability function for Kolmogorov's distribution with n< parameters or probability of getting value smaller than d with n-elements sample.PIt uses algorithm by Marsgalia et. al. and provide at least 7-digit accuracy. statistics Distribution statistics Data sample statisticsCDF of distribution statistics Data sample statisticsSample 1 statisticsSample 2 statistics CDF function statisticsSample statistics Distribution statisticsSample statistics First sample statistics Second sample statisticsSize of the sample statisticsD valueopqrstuvwxyz{| None>W statisticsGeneric form of Pearson chi squared tests for binned data. Data sample is supplied in form of tuples (observed quantity, expected number of events). Both must be positive.PThis test should be used only if all bins have expected values of at least 5. statistics|Chi squared test for data with normal errors. Data is supplied in form of pair (observation with error, and expectation). statisticsNumber of additional degrees of freedom. One degree of freedom is due to the fact that the are N observation in total and accounted for automatically. statisticsObservation and expectation. statistics+Number of additional degrees of freedom. statisticsObservation and expectation.opqrstuvwxyz{|!(c) 2009, 2010 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableNone 24567>HV߃ statistics4An estimator of a property of a sample, such as its .AThe use of an algebraic data type here allows functions such as  and  bootstrapBCA1 to use more efficient algorithms when possible. statisticsA resample drawn randomly, with replacement, from a set of data points. Distinct from a normal array to make it harder for your humble author's brain to go wrong. statisticsRun an  over a sample. statistics6Single threaded and deterministic version of resample. statisticsO(e*r*s)d Resample a data set repeatedly, with replacement, computing each estimate over the resampled data.?This function is expensive; it has to do work proportional to e*r*s, where e( is the number of estimation functions, r- is the number of resamples to compute, and s$ is the number of original samples.To improve performance, this function will make use of all available CPUs. At least with GHC 7.0, parallel performance seems best if the parallel garbage collector is disabled (RTS option -qg). statisticsCreate vector using resamples statisticsO(n) or O(n^2)c Compute a statistical estimate repeatedly over a sample, each time omitting a successive element. statisticsO(n)( Compute the jackknife mean of a sample.9 statisticsO(n)F Compute the jackknife variance of a sample with a correction factor c;, so we can get either the regular or "unbiased" variance. statisticsO(n)5 Compute the unbiased jackknife variance of a sample. statisticsO(n), Compute the jackknife variance of a sample. statisticsO(n)6 Compute the jackknife standard deviation of a sample.: statistics Drop the kth element of a vector. statistics:Split a generator into several that can run independently. statisticsEstimation functions. statisticsNumber of resamples to compute. statisticsOriginal sample. statisticsEstimation functions. statisticsNumber of resamples to compute. statisticsOriginal sample."2014 Bryan O'SullivanBSD3None statisticszPerform an ordinary least-squares regression on a set of predictors, and calculate the goodness-of-fit of the regression.The returned pair consists of:6A vector of regression coefficients. This vector has one moreD element than the list of predictors; the last element is the y-intercept value.R(, the coefficient of determination (see  for details). statistics/Compute the ordinary least-squares solution to A x = b.; statisticsSolve the equation R x = b. statisticsCompute RS, the coefficient of determination that indicates goodness-of-fit of a regression.gThis value will be 1 if the predictors fit perfectly, dropping to 0 if they have no explanatory power. statistics{Bootstrap a regression function. Returns both the results of the regression and the requested confidence interval values.< statistics%Balance units of work across workers. statisticstNon-empty list of predictor vectors. Must all have the same length. These will become the columns of the matrix A solved by . statisticsGResponder vector. Must have the same length as the predictor vectors. statisticsA& has at least as many rows as columns. statisticsb# has the same length as columns in A.; statisticsR& is an upper-triangular square matrix. statisticsb* is of the same length as rows/columns in R. statisticsPredictors (regressors). statistics Responders. statisticsRegression coefficients. statisticsNumber of resamples to compute. statisticsConfidence level. statisticsRegression function. statisticsPredictor vectors. statisticsResponder vector.#None%T statisticsXCalculate confidence intervals for Poisson-distributed value using normal approximation statistics|Calculate confidence intervals for Poisson-distributed value for single measurement. These are exact confidence intervals statisticskCalculate confidence interval using normal approximation. Note that this approximation breaks down when p3 is either close to 0 or to 1. In particular if np < 5 or  1 - np < 5) this approximation shouldn't be used. statisticsPClopper-Pearson confidence interval also known as exact confidence intervals. statisticsNumber of trials statisticsNumber of successes statisticsNumber of trials statisticsNumber of successes$(c) 2010 Neil BrownBSD3bos@serpentine.com experimentalportableNone%< statistics3Calculate (n,T z,T {) values for both samples. Where n4 is reduced sample where equal pairs are removed.= statisticsuThe coefficients for x^0, x^1, x^2, etc, in the expression prod_{r=1}^s (1 + x^r). See the Mitic paper for details.We can define: f(1) = 1 + x f(r) = (1 + x^r)*f(r-1) = f(r-1) + x^r * f(r-1) The effect of multiplying the equation by x^r is to shift all the coefficients by r down the list.1This list will be processed lazily from the head. statisticsoTests whether a given result from a Wilcoxon signed-rank matched-pairs test is significant at the given level.hThis function can perform a one-tailed or two-tailed test. If the first parameter to this function is  TwoTailedr, the test is performed two-tailed to check if the two samples differ significantly. If the first parameter is  OneTailedm, the check is performed one-tailed to decide whether the first sample (i.e. the first sample you passed to L) is greater than the second sample (i.e. the second sample you passed to ). If you wish to perform a one-tailed test in the opposite direction, you can either pass the parameters in a different order to X, or simply swap the values in the resulting pair before passing them to this function. statistics6Obtains the critical value of T to compare against, given a sample size and a p-value (significance level). Your T value must be less than or equal to the return of this function in order for the test to work out significant. If there is a Nothing return, the sample size is too small to make a decision.wilcoxonSignificant tests the return value of  for you, so you should use wilcoxonSignificant for determining test results. However, this function is useful, for example, for generating lookup tables for Wilcoxon signed rank critical values. The return values of this function are generated using the method detailed in the Mitic's paper. According to that paper, the results may differ from other published lookup tables, but (Mitic claims) the values obtained by this function will be the correct ones. statisticsWorks out the significance level (p-value) of a T value, given a sample size and a T value from the Wilcoxon signed-rank matched-pairs test.See the notes on wilcoxonCriticalValue for how this is calculated.> statistics.Normal approximation for Wilcoxon T statistics statisticsThe Wilcoxon matched-pairs signed-rank test. The samples are zipped together: if one is longer than the other, both are truncated to the length of the shorter sample.For one-tailed test it tests whether first sample is significantly greater than the second. For two-tailed it checks whether they significantly differCheck  and  for additional information. statisticsHow to compare two samples statistics#The p-value at which to test (e.g.  mkPValue 0.05) statisticsThe (n,T z, T {) values from . statisticsReturn 13 if the sample was too small to make a decision. statisticsThe sample size statisticsThe p-value (e.g.  mkPValue 0.05() for which you want the critical value. statisticsWThe critical value (of T), or Nothing if the sample is too small to make a decision. statisticsThe sample size statistics3The value of T for which you want the significance. statisticsThe significance (p-value). statisticsPerform one-tailed test. statisticsSample of pairs statisticsReturn 13 if the sample was too small to make a decision.opqrstuvwxyz{|%(c) 2010 Neil BrownBSD3bos@serpentine.com experimentalportableNonev/ statisticsThe Wilcoxon Rank Sums Test.This test calculates the sum of ranks for the given two samples. The samples are ordered, and assigned ranks (ties are given their average rank), then these ranks are summed for each sample.The return value is (W , W ) where W is the sum of ranks of the first sample and W is the sum of ranks of the second sample. This test is trivially transformed into the Mann-Whitney U test. You will probably want to use e and the related functions for testing significance, but this function is exposed for completeness. statisticsThe Mann-Whitney U Test.This is sometimes known as the Mann-Whitney-Wilcoxon U test, and confusingly many sources state that the Mann-Whitney U test is the same as the Wilcoxon's rank sum test (which is provided as I). The Mann-Whitney U is a simple transform of Wilcoxon's rank sum test.bAgain confusingly, different sources state reversed definitions for U and U , so it is worth being explicit about what this function returns. Given two samples, the first, xs , of size n and the second, xs , of size n , this function returns (U , U ) where U = W - (n (n +1))/2 and U = W - (n (n +1))/2, where (W , W ) is the return value of wilcoxonRankSums xs1 xs2.Some sources instead state that U and U should be the other way round, often expressing this using U ' = n n - U (since U + U = n n ).FAll of which you probably don't care about if you just feed this into . statisticscCalculates the critical value of Mann-Whitney U for the given sample sizes and significance level.(This function returns the exact calculated value of U for all sample sizes; it does not use the normal approximation at all. Above sample size 20 it is generally recommended to use the normal approximation instead, but this function will calculate the higher critical values if you need them.The algorithm to generate these values is a faster, memoised version of the simple unoptimised generating function given in section 2 of "The Mann Whitney Wilcoxon Distribution Using Linked Lists" statistics:Calculates whether the Mann Whitney U test is significant.aIf both sample sizes are less than or equal to 20, the exact U critical value (as calculated by Z) is used. If either sample is larger than 20, the normal approximation is used instead.If you use a one-tailed test, the test indicates whether the first sample is significantly larger than the second. If you want the opposite, simply reverse the order in both the sample size and the (U , U ) pairs. statistics{Perform Mann-Whitney U Test for two samples and required significance. For additional information check documentation of  and ". This is just a helper function.One-tailed test checks whether first sample is significantly larger than second. Two-tailed whether they are significantly different. statisticsThe sample size statistics>The p-value (e.g. 0.05) for which you want the critical value. statisticsThe critical value (of U). statistics0Perform one-tailed test (see description above). statistics=The samples' size from which the (U ,U ) values were derived. statistics(The p-value at which to test (e.g. 0.05) statisticsThe (U , U ) values from . statisticsReturn 13 if the sample was too small to make a decision. statistics0Perform one-tailed test (see description above). statistics(The p-value at which to test (e.g. 0.05) statistics First sample statistics Second sample statisticsReturn 13 if the sample was too small to make a decision. opqrxyz| opqrxyz|&(c) 2009, 2011 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableNone statisticsuBias-corrected accelerated (BCA) bootstrap. This adjusts for both bias and skewness in the resampled distribution.tBCA algorithm is described in ch. 5 of Davison, Hinkley "Confidence intervals" in section 5.3 "Percentile method" statisticsXBasic bootstrap. This method simply uses empirical quantiles for confidence interval. statisticsConfidence level statisticsFull data sample statisticsFEstimates obtained from resampled data and estimator used for this. statisticsConfidence vector statisticsLEstimate from full sample and vector of estimates obtained from resamples?2'(c) 2015 Mihai MaruseacBSD3mihai.maruseac@maruseac.com experimentalportableNone27@A, statistics Location. statisticsScale. statisticsCreate an Laplace distribution. statisticsCreate an Laplace distribution. statisticsCreate Laplace distribution from sample. No tests are made to check whether it truly is Laplace. Location of distribution estimated as median of sample. statisticsLocation statisticsScale statisticsLocation statisticsScale((c) 2009 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableNone27@A> statistics#Create an exponential distribution. statistics#Create an exponential distribution. statistics5Create exponential distribution from sample. Returns Nothingz if sample is empty or contains negative elements. No other tests are made to check whether it truly is exponential. statisticsRate parameter. statisticsRate parameter.)None> statistics:Pearson correlation for sample of pairs. Exactly same as  statistics=Compute pairwise pearson correlation between rows of a matrix statistics0compute spearman correlation between two samples statistics>compute pairwise spearman correlation between rows of a matrix*(c) 2009 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableNone> statisticsoCompute the autocovariance of a sample, i.e. the covariance of the sample against a shifted version of itself. statistics{Compute the autocorrelation function of a sample, and the upper and lower bounds of confidence intervals for each element.NoteX: The calculation of the 95% confidence interval assumes a stationary Gaussian process.@12312456789:;<=>?@ABCDEEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyGz{|}~                                       ! " # $ % & ' ( ) * + , - . / 0 1 2 3 456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~opcu000.auop       !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~  !!t!n!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!""""####$$$$$%%%%%&&'''''''''' ' ' ' ' '''''''''(((((((((( (!("(#($(%(&('((()(*(+(,)-).)/)0*1*2+}++3+4+5+678978:78;78<78=7>?7>@7>A7>B,C,D,E,F,G,HIJK7LMN7OP7QR/S/T/UIVWXYZ[!\!]"^"_$`$a&bc*statistics-0.15.2.0-FCVAYbUNN6k6ys2hOZ1wTyStatistics.QuantileStatistics.FunctionStatistics.Correlation.KendallStatistics.Sample.HistogramStatistics.Sample.InternalStatistics.DistributionStatistics.Distribution.Uniform!Statistics.Distribution.Transform Statistics.Distribution.StudentTStatistics.Distribution.Poisson&Statistics.Distribution.Hypergeometric!Statistics.Distribution.GeometricStatistics.Distribution.Gamma%Statistics.Distribution.FDistribution'Statistics.Distribution.DiscreteUniform"Statistics.Distribution.ChiSquared%Statistics.Distribution.CauchyLorentz Statistics.Distribution.BinomialStatistics.Distribution.BetaStatistics.Sample.PowersStatistics.TransformStatistics.Sample.KernelDensityStatistics.TypesStatistics.SampleStatistics.Sample.Normalize&Statistics.Sample.KernelDensity.SimpleStatistics.Distribution.NormalStatistics.Test.TypesStatistics.Test.StudentTStatistics.Test.KruskalWallis!Statistics.Test.KolmogorovSmirnovStatistics.Test.ChiSquaredStatistics.ResamplingStatistics.RegressionStatistics.ConfidenceIntStatistics.Test.WilcoxonTStatistics.Test.MannWhitneyUStatistics.Resampling.BootstrapStatistics.Distribution.Laplace#Statistics.Distribution.ExponentialStatistics.CorrelationStatistics.Autocorrelation(Statistics.Distribution.Poisson.InternalStatistics.Internal StatisticsSampleStatistics.Test.InternalStatistics.Types.Internal1data-default-class-0.1.2.0-FeIQ5tLoVZBHMSgrT9zptQData.Default.ClassdefDefault-math-functions-0.3.3.0-KLIC19lJwiXKvIGbZ7lNxd Numeric.MathFunctions.ComparisonwithinkendallsortgsortsortBy partialSortindicesindexedminMaxnextHighestPowerOfTwosquareforrfor unsafeModify ContParam weightedAvgquantile quantiles quantilesVeccadpwhazenspsssmedianUnbiasednormalUnbiasedmedian midspreadmad continuousBy$fFromJSONContParam$fToJSONContParam$fBinaryContParam$fDefaultContParam$fShowContParam $fEqContParam$fOrdContParam$fDataContParam$fGenericContParam $fFunctorPair$fFoldablePair histogram histogram_range robustSumVarsum FromSample fromSample DiscreteGengenDiscreteVarContGen genContVarEntropyentropy MaybeEntropy maybeEntropyVariancevariancestdDev MaybeVariance maybeVariance maybeStdDevMeanmean MaybeMean maybeMean ContDistrdensity complQuantile logDensity DiscreteDistr probabilitylogProbability Distribution cumulativecomplCumulative genContinuous genContinousfindRootsumProbabilitiesUniformDistributionuniformAuniformB uniformDistr uniformDistrE$fContGenUniformDistribution!$fMaybeEntropyUniformDistribution$fEntropyUniformDistribution"$fMaybeVarianceUniformDistribution$fMaybeMeanUniformDistribution$fVarianceUniformDistribution$fMeanUniformDistribution$fContDistrUniformDistribution!$fDistributionUniformDistribution$fBinaryUniformDistribution$fFromJSONUniformDistribution$fToJSONUniformDistribution$fReadUniformDistribution$fShowUniformDistribution$fEqUniformDistribution$fDataUniformDistribution$fGenericUniformDistributionLinearTransformlinTransLocation linTransScale linTransDistr scaleAroundlinTransFixedPoint$fContGenLinearTransform$fEntropyLinearTransform$fMaybeEntropyLinearTransform$fVarianceLinearTransform$fMaybeVarianceLinearTransform$fMeanLinearTransform$fMaybeMeanLinearTransform$fContDistrLinearTransform$fDistributionLinearTransform$fFunctorLinearTransform$fBinaryLinearTransform$fToJSONLinearTransform$fFromJSONLinearTransform$fEqLinearTransform$fShowLinearTransform$fReadLinearTransform$fDataLinearTransform$fGenericLinearTransformStudentT studentTndfstudentT studentTEstudentTUnstandardized$fContGenStudentT$fMaybeEntropyStudentT$fEntropyStudentT$fMaybeVarianceStudentT$fMaybeMeanStudentT$fContDistrStudentT$fDistributionStudentT$fBinaryStudentT$fFromJSONStudentT$fToJSONStudentT$fReadStudentT$fShowStudentT $fEqStudentT$fDataStudentT$fGenericStudentTPoissonDistribution poissonLambdapoissonpoissonE!$fMaybeEntropyPoissonDistribution$fEntropyPoissonDistribution"$fMaybeVariancePoissonDistribution$fMaybeMeanPoissonDistribution$fMeanPoissonDistribution$fVariancePoissonDistribution"$fDiscreteDistrPoissonDistribution!$fDistributionPoissonDistribution$fBinaryPoissonDistribution$fFromJSONPoissonDistribution$fToJSONPoissonDistribution$fReadPoissonDistribution$fShowPoissonDistribution$fEqPoissonDistribution$fDataPoissonDistribution$fGenericPoissonDistributionHypergeometricDistributionhdMhdLhdKhypergeometrichypergeometricE($fMaybeEntropyHypergeometricDistribution#$fEntropyHypergeometricDistribution)$fMaybeVarianceHypergeometricDistribution%$fMaybeMeanHypergeometricDistribution$$fVarianceHypergeometricDistribution $fMeanHypergeometricDistribution)$fDiscreteDistrHypergeometricDistribution($fDistributionHypergeometricDistribution"$fBinaryHypergeometricDistribution$$fFromJSONHypergeometricDistribution"$fToJSONHypergeometricDistribution $fReadHypergeometricDistribution $fShowHypergeometricDistribution$fEqHypergeometricDistribution $fDataHypergeometricDistribution#$fGenericHypergeometricDistributionGeometricDistribution0 gdSuccess0GeometricDistribution gdSuccess geometric geometricE geometric0 geometric0E$fContGenGeometricDistribution"$fDiscreteGenGeometricDistribution#$fMaybeEntropyGeometricDistribution$fEntropyGeometricDistribution$$fMaybeVarianceGeometricDistribution $fMaybeMeanGeometricDistribution$fVarianceGeometricDistribution$fMeanGeometricDistribution$$fDiscreteDistrGeometricDistribution#$fDistributionGeometricDistribution$fBinaryGeometricDistribution$fFromJSONGeometricDistribution$fToJSONGeometricDistribution$fReadGeometricDistribution$fShowGeometricDistribution$fContGenGeometricDistribution0#$fDiscreteGenGeometricDistribution0$$fMaybeEntropyGeometricDistribution0$fEntropyGeometricDistribution0%$fMaybeVarianceGeometricDistribution0!$fMaybeMeanGeometricDistribution0 $fVarianceGeometricDistribution0$fMeanGeometricDistribution0%$fDiscreteDistrGeometricDistribution0$$fDistributionGeometricDistribution0$fBinaryGeometricDistribution0 $fFromJSONGeometricDistribution0$fToJSONGeometricDistribution0$fReadGeometricDistribution0$fShowGeometricDistribution0$fEqGeometricDistribution$fDataGeometricDistribution$fGenericGeometricDistribution$fEqGeometricDistribution0$fDataGeometricDistribution0$fGenericGeometricDistribution0GammaDistributiongdShapegdScale gammaDistr gammaDistrEimproperGammaDistrimproperGammaDistrE$fContGenGammaDistribution$fMaybeEntropyGammaDistribution $fMaybeVarianceGammaDistribution$fMaybeMeanGammaDistribution$fMeanGammaDistribution$fVarianceGammaDistribution$fContDistrGammaDistribution$fDistributionGammaDistribution$fBinaryGammaDistribution$fFromJSONGammaDistribution$fToJSONGammaDistribution$fReadGammaDistribution$fShowGammaDistribution$fEqGammaDistribution$fDataGammaDistribution$fGenericGammaDistribution FDistributionfDistributionNDF1fDistributionNDF2 fDistributionfDistributionRealfDistributionEfDistributionRealE$fContGenFDistribution$fMaybeEntropyFDistribution$fEntropyFDistribution$fMaybeVarianceFDistribution$fMaybeMeanFDistribution$fContDistrFDistribution$fDistributionFDistribution$fBinaryFDistribution$fFromJSONFDistribution$fToJSONFDistribution$fReadFDistribution$fShowFDistribution$fEqFDistribution$fDataFDistribution$fGenericFDistributionDiscreteUniform rangeFromrangeTodiscreteUniformdiscreteUniformAB$fMaybeEntropyDiscreteUniform$fEntropyDiscreteUniform$fMaybeVarianceDiscreteUniform$fMaybeMeanDiscreteUniform$fVarianceDiscreteUniform$fMeanDiscreteUniform$fDiscreteDistrDiscreteUniform$fDistributionDiscreteUniform$fBinaryDiscreteUniform$fFromJSONDiscreteUniform$fToJSONDiscreteUniform$fReadDiscreteUniform$fShowDiscreteUniform$fEqDiscreteUniform$fDataDiscreteUniform$fGenericDiscreteUniform ChiSquared chiSquaredNDF chiSquared chiSquaredE$fContGenChiSquared$fMaybeEntropyChiSquared$fEntropyChiSquared$fMaybeVarianceChiSquared$fMaybeMeanChiSquared$fVarianceChiSquared$fMeanChiSquared$fContDistrChiSquared$fDistributionChiSquared$fBinaryChiSquared$fFromJSONChiSquared$fToJSONChiSquared$fReadChiSquared$fShowChiSquared$fEqChiSquared$fDataChiSquared$fGenericChiSquaredCauchyDistributioncauchyDistribMediancauchyDistribScalecauchyDistributioncauchyDistributionEstandardCauchy $fMaybeEntropyCauchyDistribution$fEntropyCauchyDistribution$fContGenCauchyDistribution$fContDistrCauchyDistribution $fDistributionCauchyDistribution$fBinaryCauchyDistribution$fFromJSONCauchyDistribution$fToJSONCauchyDistribution$fReadCauchyDistribution$fShowCauchyDistribution$fEqCauchyDistribution$fDataCauchyDistribution$fGenericCauchyDistributionBinomialDistributionbdTrials bdProbabilitybinomial binomialE"$fMaybeEntropyBinomialDistribution$fEntropyBinomialDistribution#$fMaybeVarianceBinomialDistribution$fMaybeMeanBinomialDistribution$fVarianceBinomialDistribution$fMeanBinomialDistribution#$fDiscreteDistrBinomialDistribution"$fDistributionBinomialDistribution$fBinaryBinomialDistribution$fFromJSONBinomialDistribution$fToJSONBinomialDistribution$fReadBinomialDistribution$fShowBinomialDistribution$fEqBinomialDistribution$fDataBinomialDistribution$fGenericBinomialDistributionBetaDistributionbdAlphabdBeta betaDistr betaDistrEimproperBetaDistrimproperBetaDistrE$fContGenBetaDistribution$fContDistrBetaDistribution$fMaybeEntropyBetaDistribution$fEntropyBetaDistribution$fMaybeVarianceBetaDistribution$fVarianceBetaDistribution$fMaybeMeanBetaDistribution$fMeanBetaDistribution$fDistributionBetaDistribution$fBinaryBetaDistribution$fFromJSONBetaDistribution$fToJSONBetaDistribution$fReadBetaDistribution$fShowBetaDistribution$fEqBetaDistribution$fDataBetaDistribution$fGenericBetaDistributionPowerspowersorder centralMomentvarianceUnbiasedskewnesskurtosiscount$fBinaryPowers$fToJSONPowers$fFromJSONPowers $fEqPowers $fReadPowers $fShowPowers $fDataPowers$fGenericPowersCDdctdct_idctidct_ifftfftkdekde_WeightsWeightedSample welfordMean meanWeighted harmonicMean geometricMeancentralMoments meanVariancemeanVarianceUnb stdErrMeanvarianceWeighted fastVariancefastVarianceUnbiased fastStdDev covariance correlationpair standardizeKernel BandwidthPoints fromPointsepanechnikovBW gaussianBW bandwidth choosePointsepanechnikovKernelgaussianKernel estimatePDF simplePDFepanechnikovPDF gaussianPDF$fBinaryPoints$fToJSONPoints$fFromJSONPoints $fEqPoints $fReadPoints $fShowPoints $fDataPoints$fGenericPointsNormalDistributionstandard normalDistr normalDistrE$$fFromSampleNormalDistributionDouble$fContGenNormalDistribution $fMaybeEntropyNormalDistribution$fEntropyNormalDistribution$fVarianceNormalDistribution!$fMaybeVarianceNormalDistribution$fMeanNormalDistribution$fMaybeMeanNormalDistribution$fContDistrNormalDistribution $fDistributionNormalDistribution$fBinaryNormalDistribution$fFromJSONNormalDistribution$fToJSONNormalDistribution$fReadNormalDistribution$fShowNormalDistribution$fEqNormalDistribution$fDataNormalDistribution$fGenericNormalDistribution LowerLimit lowerLimitllConfidenceLevel UpperLimit upperLimitulConfidenceLevelScalescaleConfInt confIntLDX confIntUDX confIntCL NormalErr normalErrorEstimateestPointestErrorPValueCLmkCLmkCLEmkCLFromSignificancemkCLFromSignificanceEconfidenceLevelsignificanceLevelcl90cl95cl99mkPValue mkPValueEpValuenSigmanSigma1 getNSigma getNSigma1estimateNormErr±estimateFromErrestimateFromIntervalconfidenceInterval asymErrors$fOrdCL $fNFDataCL $fFromJSONCL $fToJSONCL $fBinaryCL$fReadCL$fShowCL$fNFDataPValue$fFromJSONPValue$fToJSONPValue$fBinaryPValue $fReadPValue $fShowPValue$fNFDataEstimate$fToJSONEstimate$fFromJSONEstimate$fBinaryEstimate$fNFDataNormalErr$fToJSONNormalErr$fFromJSONNormalErr$fBinaryNormalErr$fNFDataConfInt$fToJSONConfInt$fFromJSONConfInt$fBinaryConfInt$fScaleEstimate$fScaleConfInt$fScaleNormalErr$fNFDataUpperLimit$fToJSONUpperLimit$fFromJSONUpperLimit$fBinaryUpperLimit$fNFDataLowerLimit$fToJSONLowerLimit$fFromJSONLowerLimit$fBinaryLowerLimit$fEqCL$fDataCL $fGenericCL $fEqPValue $fOrdPValue $fDataPValue$fGenericPValue $fEqEstimate$fReadEstimate$fShowEstimate$fGenericEstimate$fDataEstimate $fEqNormalErr$fReadNormalErr$fShowNormalErr$fDataNormalErr$fGenericNormalErr $fReadConfInt $fShowConfInt $fEqConfInt $fDataConfInt$fGenericConfInt$fEqUpperLimit$fReadUpperLimit$fShowUpperLimit$fDataUpperLimit$fGenericUpperLimit$fEqLowerLimit$fReadLowerLimit$fShowLowerLimit$fDataLowerLimit$fGenericLowerLimit$fVectorVectorCL$fMVectorMVectorCL $fUnboxCL$fVectorVectorPValue$fMVectorMVectorPValue $fUnboxPValue$fVectorVectorEstimate$fMVectorMVectorEstimate$fUnboxEstimate$fVectorVectorNormalErr$fMVectorMVectorNormalErr$fUnboxNormalErr$fVectorVectorConfInt$fMVectorMVectorConfInt$fUnboxConfInt$fVectorVectorUpperLimit$fMVectorMVectorUpperLimit$fUnboxUpperLimit$fVectorVectorLowerLimit$fMVectorMVectorLowerLimit$fUnboxLowerLimit PositionTest SamplesDifferAGreaterBGreaterTesttestSignificancetestStatisticstestDistribution TestResult SignificantNotSignificant isSignificant significant$fNFDataTestResult$fToJSONTestResult$fFromJSONTestResult$fBinaryTestResult $fNFDataTest $fToJSONTest$fFromJSONTest $fBinaryTest$fNFDataPositionTest$fToJSONPositionTest$fFromJSONPositionTest$fBinaryPositionTest$fEqTestResult$fOrdTestResult$fShowTestResult$fDataTestResult$fGenericTestResult$fEqTest $fOrdTest $fShowTest $fDataTest $fGenericTest $fFunctorTest$fEqPositionTest$fOrdPositionTest$fShowPositionTest$fDataPositionTest$fGenericPositionTest studentTTest welchTTest pairedTTestkruskalWallisRank kruskalWalliskruskalWallisTestkolmogorovSmirnovTestkolmogorovSmirnovTestCdfkolmogorovSmirnovTest2kolmogorovSmirnovCdfDkolmogorovSmirnovDkolmogorovSmirnov2DkolmogorovSmirnovProbabilitychi2test chi2testCont EstimatorVarianceUnbiasedStdDevFunction Bootstrap fullSample resamplesResample fromResampleestimate resampleSTresampleresampleVector jackknife jackknifeMeanjackknifeVarianceUnbjackknifeVariancejackknifeStdDevsplitGen$fBinaryResample$fToJSONResample$fFromJSONResample$fToJSONBootstrap$fFromJSONBootstrap$fBinaryBootstrap $fEqResample$fReadResample$fShowResample$fDataResample$fGenericResample $fEqBootstrap$fReadBootstrap$fShowBootstrap$fGenericBootstrap$fFunctorBootstrap$fFoldableBootstrap$fTraversableBootstrap$fDataBootstrap olsRegressolsrSquarebootstrapRegresspoissonNormalCI poissonCInaiveBinomialCI binomialCIwilcoxonMatchedPairSignedRankwilcoxonMatchedPairSignificant wilcoxonMatchedPairCriticalValuewilcoxonMatchedPairSignificancewilcoxonMatchedPairTestwilcoxonRankSums mannWhitneyUmannWhitneyUCriticalValuemannWhitneyUSignificantmannWhitneyUtest bootstrapBCAbasicBootstrapLaplaceDistribution ldLocationldScalelaplacelaplaceE%$fFromSampleLaplaceDistributionDouble$fContGenLaplaceDistribution!$fMaybeEntropyLaplaceDistribution$fEntropyLaplaceDistribution"$fMaybeVarianceLaplaceDistribution$fMaybeMeanLaplaceDistribution$fVarianceLaplaceDistribution$fMeanLaplaceDistribution$fContDistrLaplaceDistribution!$fDistributionLaplaceDistribution$fBinaryLaplaceDistribution$fFromJSONLaplaceDistribution$fToJSONLaplaceDistribution$fReadLaplaceDistribution$fShowLaplaceDistribution$fEqLaplaceDistribution$fDataLaplaceDistribution$fGenericLaplaceDistributionExponentialDistributionedLambda exponential exponentialE)$fFromSampleExponentialDistributionDouble $fContGenExponentialDistribution%$fMaybeEntropyExponentialDistribution $fEntropyExponentialDistribution&$fMaybeVarianceExponentialDistribution"$fMaybeMeanExponentialDistribution!$fVarianceExponentialDistribution$fMeanExponentialDistribution"$fContDistrExponentialDistribution%$fDistributionExponentialDistribution$fBinaryExponentialDistribution!$fFromJSONExponentialDistribution$fToJSONExponentialDistribution$fReadExponentialDistribution$fShowExponentialDistribution$fEqExponentialDistribution$fDataExponentialDistribution $fGenericExponentialDistributionpearsonpearsonMatByRowspearmanspearmanMatByRowautocovarianceautocorrelation alyThm2UpperalyThm2 directEntropypoissonEntropybaseGHC.ReadReadreadList readsPrecreadPrec readListPrecGHC.ShowShow showsPrecshowshowList defaultShow1 defaultShow2 defaultShow3defaultReadPrecM1defaultReadPrecM2defaultReadPrecM3&vector-0.12.0.3-ChzWbiXyvuNAQj0dcU08SgData.Vector.Generic.BaseVector Data.FoldableFoldabletoPk GHC.MaybeNothingGHC.Errerrorrank rankUnsorted splitByTagsData.Vector.Genericzipghc-prim GHC.TypesTrue significancejackknifeVariance_dropAtsolvebalance coefficients normalApprox:<