yK      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKL M N O P Q R S T U V W X Y Z [ \ ] ^ _ ` a b c d e f g h i j k lmnopqrstuvwxyz{|}~       !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~                                          !!!!""############ # # # # ################### #!#"###$#%#&#'$($)%*%+%,%-%.%/%0&1&2&3'4'5'6'7'8(9(:(;)<)=)>)?)@*A+B+C+D+E,F,G,H,I-J-3.(c) 2009 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableNone=Weights for affecting the importance of elements of a sample.FSample with weights. First element of sample is data, second is weight Sample data.(c) 2011 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableNone:#Discrete cosine transform (DCT-II).jDiscrete cosine transform (DCT-II). Only real part of vector is transformed, imaginary part is ignored.>Inverse discrete cosine transform (DCT-III). It's inverse of  only up to scale parameter: (idct . dct) x = (* length x)sInverse discrete cosine transform (DCT-III). Only real part of vector is transformed, imaginary part is ignored. Inverse fast Fourier transform. 2Radix-2 decimation-in-time fast Fourier transform. KL MNOP  KL MNOP2014 Bryan O'SullivanBSD3None :Two-dimensional mutable matrix, stored in row-major order. 2Two-dimensional matrix, stored in row-major order.Rows of matrix.Columns of matrix.aIn order to avoid overflows during matrix multiplication, a large exponent is stored separately. Matrix data.       (c) 2014 Bryan O'SullivanBSD3NoneDAllocate new matrix. Matrix content is not initialized hence unsafe.SGiven row and column numbers, calculate the offset into the flat row-major vector. eGiven row and column numbers, calculate the offset into the flat row-major vector, without checking.  Number of rowNumber of columns !  ! !  !(c) 2011 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableNone05">The result of searching for a root of a mathematical function.#fThe function does not have opposite signs when evaluated at the lower and upper bounds of the search.$hThe search failed to converge to within the given error tolerance after the given number of iterations.%A root was successfully found.&]Returns either the result of a search for a root, or the default value if the search failed.':Use the method of Ridders to compute a root of a function.The function must have opposite signs when evaluated at the lower and upper bounds of the search (i.e. the root must be bracketed)."#$%&Default value.Result of search for a root.'Absolute error tolerance.&Lower and upper bounds for the search.Function to find the roots of.()*+,-./"%#$&'"#$%&' "#$%&'()*+,-.//(c) 2009 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableSafeQRSTUVWXYZ[\]^_`QRSTUVQRSTUVW0(c) 2011 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableNone%(c) 2009, 2010, 2011 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableNone:OT 5Sort a vector.6Sort a vector.7&Sort a vector using a custom ordering.8-Partially sort a vector, such that the least k elements will be at the front.9Return the indices of a vector.:Zip a vector with its indices.;8Compute the minimum and maximum of a vector in one pass.<Efficiently 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.=Multiply a number by itself.>Simple for loop. Counts from start to end-1.?&Simple reverse-for loop. Counts from start-1 to end (which must be less than start).ab5678 The number k of least elements.9:;<=>?@ 56789:;<=>?@ ;5678:9<=@>? ab56789:;<=>?@(c) 2009 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableNone: A Parameters a and b to the D function.CO(n log n). Estimate the kth q:-quantile of a sample, using the weighted average method.SThe following properties should hold: * the length of the input is greater than 0! * the input does not contain NaN * k "e 0 and k "d q"otherwise an error will be thrown.DO(n log n). Estimate the kth q-quantile of a sample x, using the continuous sample method with the given parameters. This is the method used by most statistical software, such as R, Mathematica, SPSS, and S.EO(n log 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]) ==> 1.333333F2California Department of Public Works definition, a=0, bl=1. Gives a linear interpolation of the empirical CDF. This corresponds to method 4 in R and Mathematica.GHazen's definition, a=0.5, bn=0.5. This is claimed to be popular among hydrologists. This corresponds to method 5 in R and Mathematica.H9Definition used by the SPSS statistics application, with a=0, b]=0 (also known as Weibull's definition). This corresponds to method 6 in R and Mathematica.I6Definition used by the S statistics application, with a=1, b;=1. The interpolation points divide the sample range into n-1@ intervals. This corresponds to method 7 in R and Mathematica.JMedian unbiased definition, a=1/3, bm=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.KNormal unbiased definition, a=3/8, b=3/8. An approximately unbiased estimate if the empirical distribution approximates the normal distribution. This corresponds to method 9 in R and Mathematica. ABCk, the desired quantile.q, the number of quantiles.x, the sample data.D Parameters a and b.k, the desired quantile.q, the number of quantiles.x, the sample data.E Parameters a and b.q, the number of quantiles.x, the sample data.FGHIJKc ABCDEFGHIJK CABDEFGIHJK ABCDEFGHIJKc (c) 2011 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableNone:LO(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 N; function. To specify these parameters directly, use the M function.MO(n)% Compute a histogram over a data set.PInterval (bin) sizes are uniform, based on the supplied upper and lower bounds.NO(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.L"Number of bins (must be positive).Sample data (cannot be empty).M]Number of bins. This value must be positive. A zero or negative value will cause an error.PLower bound on interval range. Sample data less than this will cause an error.Upper 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. Sample data.N"Number of bins (must be positive).Sample data (cannot be empty).LMNLMNLMN1(c) 2013 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableNone:dedede -2011 Aleksey Khudyakov, 2014 Bryan O'SullivanBSD3NoneMOConvert from a row-major list.P-create a matrix from a list of lists, as rowsQ Convert from a row-major vector.R/create a matrix from a list of vectors, as rowsS2create a matrix from a list of vectors, as columnsT#Convert to a row-major flat vector.U!Convert to a row-major flat list.V#Convert to a list of lists, as rowsW%Convert to a list of vectors, as rowsX(Convert to a list of vectors, as columnsYGenerate matrix using functionZ/Generate symmetric square matrix using function[8Create the square identity matrix with given dimensions.\FCreate a square matrix with given diagonal, other entries default to 0]=Return the dimensions of this matrix, as a (row,column) pair.fAvoid overflow in the matrix.^EMatrix-matrix multiplication. Matrices must be of compatible sizes (note: not checked)._Matrix-vector multiplication.`Raise matrix to n7th power. Power must be positive (/note: not checked).a=Element in the center of matrix (not corrected for exponent).b)Calculate the Euclidean norm of a vector.cReturn the given column.dReturn the given row.f)Apply function to every element of matrixg.Indicate whether any element of the matrix is NaN.hSGiven row and column numbers, calculate the offset into the flat row-major vector.ieGiven row and column numbers, calculate the offset into the flat row-major vector, without checking.ONumber of rows.Number of columns.(Flat list of values, in row-major order.PQNumber of rows.Number of columns.(Flat list of values, in row-major order.RSTUVWXYNumber of rowsNumber of columnsFunction which takes row and column as argument.ZNumber of rows and columnsFunction which takes row and column4 as argument. It must be symmetric in arguments: f i j == f j i[\]f^_`abcdeRow.Column.fghij$ >OPQRSTUVWXYZ[\]^_`abcdefghij$ QOPRSTUWXVYZ[\]a^_j`bcdf>eghiOPQRSTUVWXYZ[\]f^_`abcdefghij 2014 Bryan O'SullivanBSD3NonekO(r*c)Q Compute the QR decomposition of a matrix. The result returned is the matrices (q,r).kgkkkg+(c) 2008 Don Stewart, 2009 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableNone:lO(n)N Range. The difference between the largest and smallest elements of a sample.mO(n)Y Arithmetic mean. This uses Kahan-Babuaka-Neumaier summation, so is more accurate than n) unless the input values are very large.nO(n){ Arithmetic mean. This uses Welford's algorithm to provide numerical stability, using a single pass over the sample data. Compared to mP, this loses a surprising amount of precision unless the inputs are very large.oO(n)d Arithmetic mean for weighted sample. It uses a single-pass algorithm analogous to the one used by n.pO(n)H Harmonic mean. This algorithm performs a single pass over the sample.qO(n): Geometric mean of a sample containing no negative values.r 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.s 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.tZCompute 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 v 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.uCompute 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 v 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.vvMaximum likelihood estimate of a sample's variance. Also known as the population variance, where the denominator is n.whUnbiased estimate of a sample's variance. Also known as the sample variance, where the denominator is n-1.xCalculate 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.yCalculate 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.z^Standard deviation. This is simply the square root of the unbiased estimate of the variance.{jStandard error of the mean. This is the standard deviation divided by the square root of the sample size.|-Weighted variance. This is biased estimation.}3Maximum likelihood estimate of a sample's variance.~)Unbiased estimate of a sample's variance.hStandard deviation. This is simply the square root of the maximum likelihood estimate of the variance.CCovariance of sample of pairs. For empty sample it's set to zerowCorrelation coefficient for sample of pairs. Also known as Pearson's correlation. For empty sample it's set to zero.Pair two samples. It's like h3 but requires that both samples have equal size. ijklmnlmnopqrstuvwxyz{o|p}~qlmnopqrstuvwxyz{|}~lmnopqrstuvwxyz|{}~ijklmnlmnopqrstuvwxyz{o|p}~q (c) 2011 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableNone:]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.]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.PThe 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.PThe 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. Lower bound (min) of the mesh range. Upper bound (max) of the mesh range. (c) 2009 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableNone05: 7The 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)The width of the convolution kernel used.Points from the range of a Sample./Bandwidth estimator for an Epanechnikov kernel.*Bandwidth estimator for a Gaussian kernel.KCompute 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._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.@Epanechnikov kernel for probability density function estimation.<Gaussian kernel for probability density function estimation.eKernel density estimator, providing a non-parametric way of estimating the PDF of a random variable.}A helper for creating a simple kernel density estimation function with automatically chosen bandwidth and estimation points.Simple 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.Simple 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.Number of points to select, nSample bandwidth, h Input dataKernel function Bandwidth, h Sample dataPoints at which to estimateBandwidth functionKernel functionDBandwidth scaling factor (3 for a Gaussian kernel, 1 for all others)%Number of points at which to estimate sample data%Number of points at which to estimate Data sample%Number of points at which to estimate Data samplerr(c) 2009, 2010 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableNone05: O(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.5The order (number) of simple powers collected from a sample. Compute the k_th central moment of a sample. The central moment is also known as the moment about the mean.vMaximum 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 23o module, but the number is essentially free to compute if you have already collected a sample's simple powers. Requires  with  at least 2.hStandard deviation. This is simply the square root of the maximum likelihood estimate of the variance.hUnbiased estimate of a sample's variance. Also known as the sample variance, where the denominator is n-1. Requires  with  at least 2.ZCompute 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.Compute 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.'The number of elements in the original Sample-. This is the sample's zeroth simple power.$The sum of elements in the original Sample,. This is the sample's first simple power.0The arithmetic mean of elements in the original Sample.?This is less numerically robust than the mean function in the 23o module, but the number is essentially free to compute if you have already collected a sample's simple powers.sn, the number of powers, where n >= 2. s4None:tCalculate rank of every element of sample. In case of ties ranks are averaged. Sample should be already sorted in ascending order.Rank is index of element in the sample, numeration starts from 1. In case of ties average of ranks of equal elements is assigned to each$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]u_Compute rank of every element of vector. Unlike rank it doesn't require sample to be sorted.vSplit tagged vector wxyz{|tEquivalence relationVector to rankuvtuvwxyz{|tuv5(c) 2011 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableNone}MAn unchecked, non-integer-valued version of Loader's saddle point algorithm.~Returns [x, x^2, x^3, x^4, ...]bReturns an upper bound according to theorem 2 of "Sharp Bounds on the Entropy of the Poisson Law"KReturns the average of the upper and lower bounds accounding to theorem 2.KCompute 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.OCompute the entropy of a poisson distribution using the best available method.}~}}~(c) 2009 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableNone<=TpEstimate distribution from sample. First parameter in sample is distribution type and second is element type.Estimate 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.DGenerate discrete random variates which have given distribution. c is superclass because it's always possible to generate real-valued variates from integer valuesCGenerate discrete random variates which have given distribution..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./Returns the entropy of a distribution, in nats.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.  should return < if entropy is undefined for the chosen parameter values.CReturns the entropy of a distribution, in nats, if such is defined.Type class for distributions with variance. If distibution have finite variance for all valid parameter values it should be instance of this type class.Minimal complete definition is  or gType class for distributions with variance. If variance is undefined for some parameter values both  and  should return Nothing.Minimal complete definition is  or Type class for distributions with mean. If distribution have finite mean for all valid values of parameters it should be instance of this type class.(Type class for distributions with mean.  should return , if it's undefined for current value of data%Continuous probability distributuion.Minimal complete definition is  and either  or .@Probability density function. Probability that random variable X& lies in the infinitesimal interval [x,x+x ) equal to  density(x)"x<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 1-complement of quantile: "complQuantile x "a quantile (1 - x)Natural logarithm of density."Discrete probability distribution.Probability of n-th outcome.(Logarithm of probability of n-th outcomeuType class common to all distributions. Only c.d.f. could be defined for both discrete and continuous distributions.KCumulative 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 -" = 0+One's complement of cumulative distibution: (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.OGenerate variates from continuous distribution using inverse transform rule.+Backwards compatibility with genContinuous.Approximate 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.(Sum probabilities in inclusive interval.% Distribution Probability p Initial guessLower bound on intervalUpper bound on interval##(C) 2012 Edward Kmett, BSD-style (see the file LICENSE)Edward Kmett <ekmett@gmail.com> provisionalDeriveDataTypeableNone05The beta distributionAlpha shape parameterBeta shape parameterACreate beta distribution. Both shape parameters must be positive.ACreate beta distribution. Both shape parameters must be positive.Create beta distribution. Both shape parameters must be non-negative. So it allows to construct improper beta distribution which could be used as improper prior.Create beta distribution. Both shape parameters must be non-negative. So it allows to construct improper beta distribution which could be used as improper prior.Shape parameter alphaShape parameter betaShape parameter alphaShape parameter betaShape parameter alphaShape parameter betaShape parameter alphaShape parameter beta (c) 2009 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableNone05The binomial distribution.Number of trials. Probability.pConstruct binomial distribution. Number of trials must be non-negative and probability must be in [0,1] rangepConstruct binomial distribution. Number of trials must be non-negative and probability must be in [0,1] rangeNumber of trials. Probability.Number of trials. Probability.(c) 2011 Aleksey KhudyakovBSD3bos@serpentine.com experimentalportableNone05Cauchy-Lorentz distribution.Central value of Cauchy-Lorentz distribution which is its mode and median. Distribution doesn't have mean so function is named after median.Scale parameter of Cauchy-Lorentz distribution. It's different from variance and specify half width at half maximum (HWHM).Cauchy distributionCauchy distributionDStandard Cauchy distribution. It's centered at 0 and and have 1 FWHM Central pointScale parameter (FWHM) Central pointScale parameter (FWHM)          (c) 2010 Alexey KhudyakovBSD3bos@serpentine.com experimentalportableNone05Chi-squared distribution Get number of degrees of freedomUConstruct chi-squared distribution. Number of degrees of freedom must be positive.UConstruct chi-squared distribution. Number of degrees of freedom must be positive. !"#$ !"#$(c) 2016 Andr Szabolcs SzelpBSD3a.sz.szelp@gmail.com experimentalportableNone05("The discrete uniform distribution.)a,, the lower bound of the support {a, ..., b}*b,, the upper bound of the support {a, ..., b}+IConstruct discrete uniform distribution on support {1, ..., n}. Range n must be >0.,?Construct discrete uniform distribution on support {a, ..., b}.()*+Range,Lower boundary (inclusive)Upper boundary (inclusive)-./0123456789(*)+,()*+,)*()*+,-./0123456789(c) 2009 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableNone05<=?#Create an exponential distribution.@#Create an exponential distribution.A5Create 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.=>?Rate parameter.@Rate parameter.ABCDEFGHIJKLMNO=>?@=>?@>=>?@ABCDEFGHIJKLMNO(c) 2011 Aleksey KhudyakovBSD3bos@serpentine.com experimentalportableNone05SF distributionSTUVWXYZ[\]^_`abcdeSUTVWXY STUVXWYTUSTUVWXYZ[\]^_`abcde(c) 2009, 2011 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableNone05iThe gamma distribution.jShape parameter, k.kScale parameter, .lMCreate gamma distribution. Both shape and scale parameters must be positive.mMCreate gamma distribution. Both shape and scale parameters must be positive.nQCreate gamma distribution. Both shape and scale parameters must be non-negative.oQCreate gamma distribution. Both shape and scale parameters must be non-negative.ijklShape parameter. kScale parameter, .mShape parameter. kScale parameter, .nShape parameter. kScale parameter, .oShape parameter. kScale parameter, .pqrstuvwxyz{|ikjlmno ijklmnojkijklmnopqrstuvwxyz{|(c) 2009 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableNone05Distribution over [0..]Distribution over [1..]Create geometric distribution.Create geometric distribution.Create geometric distribution.Create geometric distribution.+ Success rate Success rate Success rate Success rate '(c) 2009 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableNone05mlkmlk (c) 2015 Mihai MaruseacBSD3mihai.maruseac@maruseac.com experimentalportableNone05<= Location.Scale.Create an Laplace distribution.Create an Laplace distribution.Create Laplace distribution from sample. No tests are made to check whether it truly is Laplace. Location of distribution estimated as median of sample.LocationScaleLocationScale(c) 2009 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableNone05<=The normal distribution.IStandard normal distribution with mean equal to 0 and variance equal to 1+Create normal distribution from parameters.WIMPORTANT: prior to 0.10 release second parameter was variance not standard deviation.+Create normal distribution from parameters.WIMPORTANT: prior to 0.10 release second parameter was variance not standard deviation.MVariance 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)Mean of distribution"Standard deviation of distributionMean of distribution"Standard deviation of distribution(c) 2009, 2011 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableNone05Create Poisson distribution.Create Poisson distribution.(c) 2013 John McDonnell;BSD3bos@serpentine.com experimentalportableNone059:;.Linear transformation applied to distribution. "LinearTransform   _ x' =  + xLocation parameter.Scale parameter.Distribution being transformed.,Apply linear transformation to distribution. (Get fixed point of linear transformation Fixed pointScale parameter Distribution            (c) 2011 Aleksey KhudyakovBSD3bos@serpentine.com experimentalportableNone05Student-T distributionECreate Student-T distribution. Number of parameters must be positive.ECreate Student-T distribution. Number of parameters must be positive. 0Create an unstandardized Student-t distribution. Number of degrees of freedom5Central value (0 for standard Student T distribution)Scale parameter!"#$%&'()*+,   !"#$%&'()*+,(c) 2011 Aleksey KhudyakovBSD3bos@serpentine.com experimentalportableNone050 Uniform distribution from A to B1Low boundary of distribution2Upper boundary of distribution3Create uniform distribution.4Create uniform distribution.0123456789:;<=>?@AB0213401234120123456789:;<=>?@AB(c) 2009 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableNone05:<=DRT(FLower limit. They are usually given for large quantities when it's not possible to measure them. For example: proton half-lifeH Lower limitI/Confidence level for which limit was calculatedJxUpper limit. They are usually given for small non-negative values when it's not possible detect difference from zero.L Upper limitM/Confidence level for which limit was calculatedN1Data types which could be multiplied by constant.PvConfidence interval. It assumes that confidence interval forms single interval and isn't set of disjoint intervals.RcLower error estimate, or distance between point estimate and lower bound of confidence interval.ScUpper error estimate, or distance between point estimate and upper bound of confidence interval.T<Confidence level corresponding to given confidence interval.UNormal 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.XTA 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: U& for normally distributed errors and PO 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 q5 allow to easily construct estimate from same inputs.ZPoint estimate.[!Confidence interval for estimate.\Newtype wrapper for p-value.]Confidence 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.^Create 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_Same as ^ but returns Nothing7 instead of error if parameter is out of [0,1] rangemkCLE 0.95 -- same as cl95 Just (mkCLFromSignificance 0.05)`Create 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 0.05aSame 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)bIGet confidence level. This function is subject to rounding errors. If 1 - CL is needed use c insteadcGet significance level.d90% confidence levele95% confidence levelf99% confidence levelgAConstruct PValue. Throws error if argument is out of [0,1] range.hConstruct PValue. Returns Nothing# if argument is out of [0,1] range.i Get p-valuejP-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).ksP-value expressed in sigma for one-tail hypothesis. This correspond to probability of obtaining value less than N.l"Express confidence level in sigmasm=Express confidence level in sigmas for one-tailed hypothesis.n"Create estimate with normal errorso Synonym for np&Create estimate with asymmetric error.q&Create estimate with asymmetric error.rGet confidence intervalsGet asymmetric errors cl95 > cl90TruekFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnPoint estimate1 erroroPoint estimate1 errorpCentral estimateHLower and upper errors. Both should be positive but it's not checked.Confidence level for intervalqCPoint estimate. Should lie within interval but it's not checked."Lower and upper bounds of intervalConfidence level for intervalrstuvwxyz{|}~1FGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrs1]bc^_`adefjklm\ighXYZ[UVWPQRSTJKLMFGHInoqprsNOKFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~ (c) 2009, 2010 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableNone02345:4An estimator of a property of a sample, such as its m.AThe use of an algebraic data type here allows functions such as  and  bootstrapBCA1 to use more efficient algorithms when possible.A 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.Run an  over a sample.6Single threaded and deterministic version of resample.O(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).Create vector using resamplesO(n) or O(n^2)c Compute a statistical estimate repeatedly over a sample, each time omitting a successive element.O(n)( Compute the jackknife mean of a sample.O(n)F Compute the jackknife variance of a sample with a correction factor c;, so we can get either the regular or "unbiased" variance.O(n)5 Compute the unbiased jackknife variance of a sample.O(n), Compute the jackknife variance of a sample.O(n)6 Compute the jackknife standard deviation of a sample. Drop the kth element of a vector.:Split a generator into several that can run independently."Estimation functions.Number of resamples to compute.Original sample.Estimation functions.Number of resamples to compute.Original sample.!2014 Bryan O'SullivanBSD3NonezPerform 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)./Compute the ordinary least-squares solution to A x = b.Solve the equation R x = b.Compute 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.{Bootstrap a regression function. Returns both the results of the regression and the requested confidence interval values.%Balance units of work across workers.tNon-empty list of predictor vectors. Must all have the same length. These will become the columns of the matrix A solved by .GResponder vector. Must have the same length as the predictor vectors.A& has at least as many rows as columns.b# has the same length as columns in A.R& is an upper-triangular square matrix.b* is of the same length as rows/columns in R.Predictors (regressors). Responders.Regression coefficients.Number of resamples to compute.Confidence level.Regression function.Predictor vectors.Responder vector."(c) 2009, 2011 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableNoneuBias-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"XBasic bootstrap. This method simply uses empirical quantiles for confidence interval.Confidence levelFull data sampleFEstimates obtained from resampled data and estimator used for this.Confidence vectorLEstimate from full sample and vector of estimates obtained from resamples2#None025 YTest type for test which compare positional (mean,median etc.) information of samples.XTest whether samples differ in position. Null hypothesis is samples are not differentqTest if first sample (A) is larger than second (B). Null hypothesis is first sample is not larger than second.+Test if second sample is larger than first.Result of statistical test.SProbability of getting value of test statistics at least as extreme as measured.Statistic used for test.>Distribution of test statistics if null hypothesis is correct.Result of hypothesis testing"Null hypothesis should be rejected"Data is compatible with hypothesis 4Check whether test is significant for given p-value. significant if parameter is , not significant otherwiser              $None:'Generic 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.(|Chi squared test for data with normal errors. Data is supplied in form of pair (observation with error, and expectation).'Number 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.Observation and expectation.(+Number of additional degrees of freedom.Observation and expectation.  '('('(%(c) 2011 Aleksey KhudyakovBSD3bos@serpentine.com experimentalportableNone:)ACheck 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.* Variant of )' which uses CFD in form of function.+Two 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.,!Calculate Kolmogorov's statistic Df for given cumulative distribution function (CDF) and data sample. If sample is empty returns 0.-!Calculate Kolmogorov's statistic Df for given cumulative distribution function (CDF) and data sample. If sample is empty returns 0..!Calculate Kolmogorov's statistic DB for two data samples. If either of samples is empty returns 0./PCalculate 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.) Distribution Data sample*CDF of distribution Data sample+Sample 1Sample 2, CDF functionSample- DistributionSample. First sample Second sample/Size of the sampleD value  )*+,-./)*+,-./)*+,-./&(c) 2014 Danny NavarroBSD3bos@serpentine.com experimentalportableNone0Kruskal-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).1The Kruskal-Wallis Test.8In textbooks the output value is usually represented by K or H*. This function already does the ranking.2oPerform Kruskal-Wallis Test for the given samples and required significance. For additional information check 1". This is just a helper function.It uses  Chi-Squaredc distribution for aproximation as long as the sizes are larger than 5. Otherwise the test returns .012  012201012'(c) 2010 Neil BrownBSD3bos@serpentine.com experimentalportableNone3The 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 4e and the related functions for testing significance, but this function is exposed for completeness.4The 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 3I). 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 6.5cCalculates 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"6: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 5Z) 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.7{Perform Mann-Whitney U Test for two samples and required significance. For additional information check documentation of 4 and 6". 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.345The sample size>The p-value (e.g. 0.05) for which you want the critical value.The critical value (of U).60Perform one-tailed test (see description above).=The samples' size from which the (U ,U ) values were derived.(The p-value at which to test (e.g. 0.05)The (U , U ) values from 4.Return 3 if the sample was too small to make a decision.70Perform one-tailed test (see description above).(The p-value at which to test (e.g. 0.05) First sample Second sampleReturn 3 if the sample was too small to make a decision.  345677456334567(None:OT8vTwo-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.9Two-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.:VPaired two-sample t-test. Two samples are paired in a within-subject design. Returns Nothing# if sample size is not sufficient.8one- or two-tailed testSample ASample B9one- or two-tailed testSample ASample B:one- or two-tailed testpaired samplesone- or two-tailed t statisticsdegree of freedomp-value  89:89:89:)(c) 2010 Neil BrownBSD3bos@serpentine.com experimentalportableNone$;3Calculate (n,T z,T {) values for both samples. Where n4 is reduced sample where equal pairs are removed. uThe 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.<oTests 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.=6Obtains 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.>Works 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. .Normal approximation for Wilcoxon T statistics?The Wilcoxon matched-pairs signed-rank test. The samples are zipped together: if one is longer than the other, both are truncated to the 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.;  <How to compare two samples#The p-value at which to test (e.g.  mkPValue 0.05)The (n,T z, T {) values from ;.Return 3 if the sample was too small to make a decision.=The sample sizeThe p-value (e.g.  mkPValue 0.05() for which you want the critical value.WThe critical value (of T), or Nothing if the sample is too small to make a decision.>The sample size3The value of T for which you want the significance.The significance (p-value). ?Perform one-tailed test.Sample of pairsReturn 3 if the sample was too small to make a decision.  ;<=>??;<>=;  <=> ?*None:@O(nlogn)_ Compute the Kendall's tau from a vector of paired data. Return NaN when number of pairs <= 1.@  @@@  +None:A:Pearson correlation for sample of pairs. Exactly same as B=Compute pairwise pearson correlation between rows of a matrixC0compute spearman correlation between two samplesD>compute pairwise spearman correlation between rows of a matrixABCDABCDABCDABCD,None$EXCalculate confidence intervals for Poisson-distributed value using normal approximationF|Calculate confidence intervals for Poisson-distributed value for single measurement. These are exact confidence intervalsGkCalculate 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.HPClopper-Pearson confidence interval also known as exact confidence intervals.EFGNumber of trialsNumber of successesHNumber of trialsNumber of successesEFGHFEHGEFGH-(c) 2009 Bryan O'SullivanBSD3bos@serpentine.com experimentalportableNone:IoCompute the autocovariance of a sample, i.e. the covariance of the sample against a shifted version of itself.J{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.IJIJIJIJ678.9.:.3;<=>?@ABBCCDEFGHIJKLMNOPQRSTUVWXYWZ[\]^_`abcdefghijklmnopqrsSttuvwxyz{|} ~  T U            !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&''()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghiijkllmnopqqrstuuvwwxyz{|}~                                          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Statistics.Distribution.Binomial%Statistics.Distribution.CauchyLorentz"Statistics.Distribution.ChiSquared'Statistics.Distribution.DiscreteUniform#Statistics.Distribution.Exponential%Statistics.Distribution.FDistributionStatistics.Distribution.Gamma!Statistics.Distribution.Geometric&Statistics.Distribution.HypergeometricStatistics.Distribution.LaplaceStatistics.Distribution.NormalStatistics.Distribution.Poisson!Statistics.Distribution.Transform Statistics.Distribution.StudentTStatistics.Distribution.UniformStatistics.ResamplingStatistics.RegressionStatistics.Resampling.BootstrapStatistics.Test.TypesStatistics.Test.ChiSquared!Statistics.Test.KolmogorovSmirnovStatistics.Test.KruskalWallisStatistics.Test.MannWhitneyUStatistics.Test.StudentTStatistics.Test.WilcoxonTStatistics.Correlation.KendallStatistics.CorrelationStatistics.ConfidenceIntStatistics.AutocorrelationStatistics.Types.InternalStatistics.InternalStatistics.Function.ComparisonStatistics.Sample.Internal 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uniformDistrE$fContGenUniformDistribution!$fMaybeEntropyUniformDistribution$fEntropyUniformDistribution"$fMaybeVarianceUniformDistribution$fMaybeMeanUniformDistribution$fVarianceUniformDistribution$fMeanUniformDistribution$fContDistrUniformDistribution!$fDistributionUniformDistribution$fBinaryUniformDistribution$fFromJSONUniformDistribution$fToJSONUniformDistribution$fReadUniformDistribution$fShowUniformDistribution$fEqUniformDistribution$fDataUniformDistribution$fGenericUniformDistribution LowerLimit lowerLimitllConfidenceLevel UpperLimit upperLimitulConfidenceLevelScalescaleConfInt confIntLDX confIntUDX confIntCL NormalErr normalErrorEstimateestPointestErrorPValueCLmkCLmkCLEmkCLFromSignificancemkCLFromSignificanceEconfidenceLevelsignificanceLevelcl90cl95cl99mkPValue mkPValueEpValuenSigmanSigma1 getNSigma getNSigma1estimateNormErr±estimateFromErrestimateFromIntervalconfidenceInterval asymErrors$fNFDataLowerLimit$fToJSONLowerLimit$fFromJSONLowerLimit$fBinaryLowerLimit$fNFDataUpperLimit$fToJSONUpperLimit$fFromJSONUpperLimit$fBinaryUpperLimit$fScaleEstimate$fScaleConfInt$fScaleNormalErr$fNFDataConfInt$fToJSONConfInt$fFromJSONConfInt$fBinaryConfInt$fNFDataNormalErr$fToJSONNormalErr$fFromJSONNormalErr$fBinaryNormalErr$fNFDataEstimate$fToJSONEstimate$fFromJSONEstimate$fBinaryEstimate$fNFDataPValue$fFromJSONPValue$fToJSONPValue$fBinaryPValue $fReadPValue $fShowPValue$fOrdCL $fNFDataCL $fFromJSONCL $fToJSONCL $fBinaryCL$fReadCL$fShowCL$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 EstimatorVarianceUnbiasedStdDevFunction Bootstrap fullSample resamplesResample fromResampleestimate resampleSTresampleresampleVector jackknife jackknifeMeanjackknifeVarianceUnbjackknifeVariancejackknifeStdDevsplitGen$fToJSONBootstrap$fFromJSONBootstrap$fBinaryBootstrap$fBinaryResample$fToJSONResample$fFromJSONResample $fEqResample$fReadResample$fShowResample$fDataResample$fGenericResample $fEqBootstrap$fReadBootstrap$fShowBootstrap$fGenericBootstrap$fFunctorBootstrap$fFoldableBootstrap$fTraversableBootstrap$fDataBootstrap olsRegressolsrSquarebootstrapRegress bootstrapBCAbasicBootstrap PositionTest SamplesDifferAGreaterBGreaterTesttestSignificancetestStatisticstestDistribution TestResult SignificantNotSignificant isSignificant significant$fNFDataPositionTest$fToJSONPositionTest$fFromJSONPositionTest$fBinaryPositionTest $fNFDataTest $fToJSONTest$fFromJSONTest $fBinaryTest$fNFDataTestResult$fToJSONTestResult$fFromJSONTestResult$fBinaryTestResult$fEqTestResult$fOrdTestResult$fShowTestResult$fDataTestResult$fGenericTestResult$fEqTest $fOrdTest $fShowTest $fDataTest $fGenericTest $fFunctorTest$fEqPositionTest$fOrdPositionTest$fShowPositionTest$fDataPositionTest$fGenericPositionTestchi2test chi2testContkolmogorovSmirnovTestkolmogorovSmirnovTestCdfkolmogorovSmirnovTest2kolmogorovSmirnovCdfDkolmogorovSmirnovDkolmogorovSmirnov2DkolmogorovSmirnovProbabilitykruskalWallisRank kruskalWalliskruskalWallisTestwilcoxonRankSums mannWhitneyUmannWhitneyUCriticalValuemannWhitneyUSignificantmannWhitneyUtest studentTTest welchTTest pairedTTestwilcoxonMatchedPairSignedRankwilcoxonMatchedPairSignificant wilcoxonMatchedPairCriticalValuewilcoxonMatchedPairSignificancewilcoxonMatchedPairTestkendallpearsonpearsonMatByRowspearmanspearmanMatByRowpoissonNormalCI poissonCInaiveBinomialCI binomialCIautocovarianceautocorrelation dctWorker idctWorkermfftfihalvevectorOK defaultShow1 defaultShow2 defaultShow3defaultReadPrecM1defaultReadPrecM2defaultReadPrecM3expectbaseGHC.ReadReadreadList readsPrecreadPrec readListPrecGHC.ShowShow showsPrecshowshowListMMmodErr robustSumVar avoidOverflow innerProduct&vector-0.12.0.1-JOtbAphig5lG2Gp1CUS1DjData.Vector.GenericzipT1TVrobustSumVarWeightedfastVar^ errorShortrank rankUnsorted splitByTagsRankrankCntrankValrankNumrankVec alyThm2UpperalyThm2 directEntropypoissonEntropyzipCoefficientsupperCoefficients4lowerCoefficients4upperCoefficients6lowerCoefficients6upperCoefficients8lowerCoefficients8upperCoefficients10lowerCoefficients10upperCoefficients12lowerCoefficients12GHC.BaseNothingGHC.ErrerrorPBDerrMsgerrMsgIUEDF _pdfFactorerrMsgRGDGD0errMsg0HDLDND ndPdfDenom ndCdfDenomPDlogDensityUnscalederrMkCL errMkPValueV_CLMV_CL D:R:VectorCL0V_PValue MV_PValueD:R:VectorPValue0 V_Estimate MV_EstimateD:R:VectorEstimate0 V_NormalErr MV_NormalErrD:R:VectorNormalErr0 V_ConfInt MV_ConfIntD:R:VectorConfInt0 V_UpperLimit MV_UpperLimitD:R:VectorUpperLimit0 V_LowerLimit MV_LowerLimitD:R:VectorLowerLimit0jackknifeVariance_dropAtpfxSumLpfxSumR singletonErrsolvebalance:<ghc-prim GHC.TypesTruesumWithalookup significance tStatisticstStatisticsPaired coefficients normalApproxsummedCoefficients numOfTiesBy mergeSortmerge