-- Hoogle documentation, generated by Haddock -- See Hoogle, http://www.haskell.org/hoogle/ -- | A library of statistical types, data, and functions -- -- This library provides a number of common functions and types useful in -- statistics. We focus on high performance, numerical robustness, and -- use of good algorithms. Where possible, we provide references to the -- statistical literature. -- -- The library's facilities can be divided into four broad categories: -- -- @package statistics @version 0.13.3.0 module Statistics.Test.Types -- | Test type. Exact meaning depends on a specific test. But generally -- it's tested whether some statistics is too big (small) for -- OneTailed or whether it too big or too small for -- TwoTailed data TestType OneTailed :: TestType TwoTailed :: TestType -- | Result of hypothesis testing data TestResult -- | Null hypothesis should be rejected Significant :: TestResult -- | Data is compatible with hypothesis NotSignificant :: TestResult -- | Significant if parameter is True, not significant otherwiser significant :: Bool -> TestResult instance GHC.Generics.Generic Statistics.Test.Types.TestResult instance Data.Data.Data Statistics.Test.Types.TestResult instance GHC.Show.Show Statistics.Test.Types.TestResult instance GHC.Classes.Ord Statistics.Test.Types.TestResult instance GHC.Classes.Eq Statistics.Test.Types.TestResult instance GHC.Generics.Generic Statistics.Test.Types.TestType instance Data.Data.Data Statistics.Test.Types.TestType instance GHC.Show.Show Statistics.Test.Types.TestType instance GHC.Classes.Ord Statistics.Test.Types.TestType instance GHC.Classes.Eq Statistics.Test.Types.TestType instance Data.Aeson.Types.Class.FromJSON Statistics.Test.Types.TestType instance Data.Aeson.Types.Class.ToJSON Statistics.Test.Types.TestType instance Data.Aeson.Types.Class.FromJSON Statistics.Test.Types.TestResult instance Data.Aeson.Types.Class.ToJSON Statistics.Test.Types.TestResult -- | Fourier-related transformations of mathematical functions. -- -- These functions are written for simplicity and correctness, not speed. -- If you need a fast FFT implementation for your application, you should -- strongly consider using a library of FFTW bindings instead. module Statistics.Transform type CD = Complex Double -- | Discrete cosine transform (DCT-II). dct :: (Vector v CD, Vector v Double, Vector v Int) => v Double -> v Double -- | Discrete cosine transform (DCT-II). Only real part of vector is -- transformed, imaginary part is ignored. dct_ :: (Vector v CD, Vector v Double, Vector v Int) => v CD -> v Double -- | Inverse discrete cosine transform (DCT-III). It's inverse of -- dct only up to scale parameter: -- -- 
--   (idct . dct) x = (* length x)
--
idct :: (Vector v CD, Vector v Double) => v Double -> v Double -- | Inverse discrete cosine transform (DCT-III). Only real part of vector -- is transformed, imaginary part is ignored. idct_ :: (Vector v CD, Vector v Double) => v CD -> v Double -- | Radix-2 decimation-in-time fast Fourier transform. fft :: Vector v CD => v CD -> v CD -- | Inverse fast Fourier transform. ifft :: Vector v CD => v CD -> v CD -- | Basic matrix operations. -- -- There isn't a widely used matrix package for Haskell yet, so we -- implement the necessary minimum here. module Statistics.Matrix.Types type Vector = Vector Double type MVector s = MVector s Double -- | Two-dimensional matrix, stored in row-major order. data Matrix Matrix :: {-# UNPACK #-} !Int -> {-# UNPACK #-} !Int -> {-# UNPACK #-} !Int -> !Vector -> Matrix -- | Rows of matrix. [rows] :: Matrix -> {-# UNPACK #-} !Int -- | Columns of matrix. [cols] :: Matrix -> {-# UNPACK #-} !Int -- | In order to avoid overflows during matrix multiplication, a large -- exponent is stored separately. [exponent] :: Matrix -> {-# UNPACK #-} !Int -- | Matrix data. [_vector] :: Matrix -> !Vector -- | Two-dimensional mutable matrix, stored in row-major order. data MMatrix s MMatrix :: {-# UNPACK #-} !Int -> {-# UNPACK #-} !Int -> {-# UNPACK #-} !Int -> !(MVector s) -> MMatrix s debug :: Matrix -> String instance GHC.Classes.Eq Statistics.Matrix.Types.Matrix instance GHC.Show.Show Statistics.Matrix.Types.Matrix -- | Basic mutable matrix operations. module Statistics.Matrix.Mutable -- | Two-dimensional mutable matrix, stored in row-major order. data MMatrix s MMatrix :: {-# UNPACK #-} !Int -> {-# UNPACK #-} !Int -> {-# UNPACK #-} !Int -> !(MVector s) -> MMatrix s type MVector s = MVector s Double replicate :: Int -> Int -> Double -> ST s (MMatrix s) thaw :: Matrix -> ST s (MMatrix s) -- | Given row and column numbers, calculate the offset into the flat -- row-major vector. bounds :: MMatrix s -> Int -> Int -> (MVector s -> Int -> r) -> r -- | Allocate new matrix. Matrix content is not initialized hence unsafe. unsafeNew :: Int -> Int -> ST s (MMatrix s) unsafeFreeze :: MMatrix s -> ST s Matrix unsafeRead :: MMatrix s -> Int -> Int -> ST s Double unsafeWrite :: MMatrix s -> Int -> Int -> Double -> ST s () unsafeModify :: MMatrix s -> Int -> Int -> (Double -> Double) -> ST s () immutably :: NFData a => MMatrix s -> (Matrix -> a) -> ST s a -- | Given row and column numbers, calculate the offset into the flat -- row-major vector, without checking. unsafeBounds :: MMatrix s -> Int -> Int -> (MVector s -> Int -> r) -> r -- | Haskell functions for finding the roots of mathematical functions. module Statistics.Math.RootFinding -- | The result of searching for a root of a mathematical function. data Root a -- | The function does not have opposite signs when evaluated at the lower -- and upper bounds of the search. NotBracketed :: Root a -- | The search failed to converge to within the given error tolerance -- after the given number of iterations. SearchFailed :: Root a -- | A root was successfully found. Root :: a -> Root a -- | Returns either the result of a search for a root, or the default value -- if the search failed. fromRoot :: a -> Root a -> a -- | 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). ridders :: Double -> (Double, Double) -> (Double -> Double) -> Root Double instance GHC.Generics.Generic (Statistics.Math.RootFinding.Root a) instance Data.Data.Data a => Data.Data.Data (Statistics.Math.RootFinding.Root a) instance GHC.Show.Show a => GHC.Show.Show (Statistics.Math.RootFinding.Root a) instance GHC.Read.Read a => GHC.Read.Read (Statistics.Math.RootFinding.Root a) instance GHC.Classes.Eq a => GHC.Classes.Eq (Statistics.Math.RootFinding.Root a) instance Data.Aeson.Types.Class.FromJSON a => Data.Aeson.Types.Class.FromJSON (Statistics.Math.RootFinding.Root a) instance Data.Aeson.Types.Class.ToJSON a => Data.Aeson.Types.Class.ToJSON (Statistics.Math.RootFinding.Root a) instance Data.Binary.Class.Binary a => Data.Binary.Class.Binary (Statistics.Math.RootFinding.Root a) instance GHC.Base.Functor Statistics.Math.RootFinding.Root instance GHC.Base.Monad Statistics.Math.RootFinding.Root instance GHC.Base.MonadPlus Statistics.Math.RootFinding.Root instance GHC.Base.Applicative Statistics.Math.RootFinding.Root instance GHC.Base.Alternative Statistics.Math.RootFinding.Root -- | Types for working with statistics. module Statistics.Types -- | An estimator of a property of a sample, such as its mean. -- -- The use of an algebraic data type here allows functions such as -- jackknife and bootstrapBCA to use more efficient -- algorithms when possible. data Estimator Mean :: Estimator Variance :: Estimator VarianceUnbiased :: Estimator StdDev :: Estimator Function :: (Sample -> Double) -> Estimator -- | Sample data. type Sample = Vector Double -- | Sample with weights. First element of sample is data, second is weight type WeightedSample = Vector (Double, Double) -- | Weights for affecting the importance of elements of a sample. type Weights = Vector Double -- | Fast O(NlogN) implementation of Kendall's tau. -- -- This module implementes Kendall's tau form b which allows ties in the -- data. This is the same formula used by other statistical packages, -- e.g., R, matlab. -- -- 
--   \tau = \frac{n_c - n_d}{\sqrt{(n_0 - n_1)(n_0 - n_2)}}
--
-- -- where n_0 = n(n-1)/2, n_1 = number of pairs tied for the first -- quantify, n_2 = number of pairs tied for the second quantify, n_c = -- number of concordant pairs$, n_d = number of discordant pairs. module Statistics.Correlation.Kendall -- | O(nlogn) Compute the Kendall's tau from a vector of paired -- data. Return NaN when number of pairs <= 1. kendall :: (Ord a, Ord b, Vector v (a, b)) => v (a, b) -> Double -- | Constant values common to much statistics code. -- -- DEPRECATED: use module Constants from math-functions. -- | Deprecated: use module Numeric.MathFunctions.Constants from -- math-functions module Statistics.Constants -- | Useful functions. module Statistics.Function -- | Compute the minimum and maximum of a vector in one pass. minMax :: (Vector v Double) => v Double -> (Double, Double) -- | Sort a vector. sort :: Vector Double -> Vector Double -- | Sort a vector. gsort :: (Ord e, Vector v e) => v e -> v e -- | Sort a vector using a custom ordering. sortBy :: (Vector v e) => Comparison e -> v e -> v e -- | Partially sort a vector, such that the least k elements will be -- at the front. partialSort :: (Vector v e, Ord e) => Int -> v e -> v e -- | Zip a vector with its indices. indexed :: (Vector v e, Vector v Int, Vector v (Int, e)) => v e -> v (Int, e) -- | Return the indices of a vector. indices :: (Vector v a, Vector v Int) => v a -> v Int -- | 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. nextHighestPowerOfTwo :: Int -> Int -- | Compare two Double values for approximate equality, using -- Dawson's method. -- -- The required accuracy is specified in ULPs (units of least precision). -- If the two numbers differ by the given number of ULPs or less, this -- function returns True. within :: Int -> Double -> Double -> Bool -- | Multiply a number by itself. square :: Double -> Double unsafeModify :: MVector s Double -> Int -> (Double -> Double) -> ST s () -- | Simple for loop. Counts from start to end-1. for :: Monad m => Int -> Int -> (Int -> m ()) -> m () -- | Simple reverse-for loop. Counts from start-1 to end -- (which must be less than start). rfor :: Monad m => Int -> Int -> (Int -> m ()) -> m () -- | Commonly used sample statistics, also known as descriptive statistics. module Statistics.Sample -- | Sample data. type Sample = Vector Double -- | Sample with weights. First element of sample is data, second is weight type WeightedSample = Vector (Double, Double) -- | O(n) Range. The difference between the largest and smallest -- elements of a sample. range :: (Vector v Double) => v Double -> Double -- | O(n) Arithmetic mean. This uses Kahan-Babuška-Neumaier -- summation, so is more accurate than welfordMean unless the -- input values are very large. mean :: (Vector v Double) => v Double -> Double -- | O(n) Arithmetic mean. This uses Welford's algorithm to provide -- numerical stability, using a single pass over the sample data. -- -- Compared to mean, this loses a surprising amount of precision -- unless the inputs are very large. welfordMean :: (Vector v Double) => v Double -> Double -- | O(n) Arithmetic mean for weighted sample. It uses a single-pass -- algorithm analogous to the one used by welfordMean. meanWeighted :: (Vector v (Double, Double)) => v (Double, Double) -> Double -- | O(n) Harmonic mean. This algorithm performs a single pass over -- the sample. harmonicMean :: (Vector v Double) => v Double -> Double -- | O(n) Geometric mean of a sample containing no negative values. geometricMean :: (Vector v Double) => v Double -> Double -- | Compute the kth central moment of a sample. The central moment -- is also known as the moment about the mean. -- -- This 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. centralMoment :: (Vector v Double) => Int -> v Double -> Double -- | Compute the kth and jth central moments of a sample. -- -- This 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. centralMoments :: (Vector v Double) => Int -> Int -> v Double -> (Double, Double) -- | Compute 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-skewed. 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. -- -- 
--   skewness $ U.to [1,2,3,4,100]
--   ==> 1.4975367033335198
--
-- -- A sample's skewness is not defined if its variance is zero. -- -- This 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. skewness :: (Vector v Double) => v Double -> Double -- | Compute 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. -- -- A sample's excess kurtosis is not defined if its variance is -- zero. -- -- This 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. kurtosis :: (Vector v Double) => v Double -> Double -- | Maximum likelihood estimate of a sample's variance. Also known as the -- population variance, where the denominator is n. variance :: (Vector v Double) => v Double -> Double -- | Unbiased estimate of a sample's variance. Also known as the sample -- variance, where the denominator is n-1. varianceUnbiased :: (Vector v Double) => v Double -> Double -- | Calculate 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. meanVariance :: (Vector v Double) => v Double -> (Double, Double) -- | Calculate 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. meanVarianceUnb :: (Vector v Double) => v Double -> (Double, Double) -- | Standard deviation. This is simply the square root of the unbiased -- estimate of the variance. stdDev :: (Vector v Double) => v Double -> Double -- | Weighted variance. This is biased estimation. varianceWeighted :: (Vector v (Double, Double)) => v (Double, Double) -> Double -- | Maximum likelihood estimate of a sample's variance. fastVariance :: (Vector v Double) => v Double -> Double -- | Unbiased estimate of a sample's variance. fastVarianceUnbiased :: (Vector v Double) => v Double -> Double -- | Standard deviation. This is simply the square root of the maximum -- likelihood estimate of the variance. fastStdDev :: (Vector v Double) => v Double -> Double -- | Covariance of sample of pairs. For empty sample it's set to zero covariance :: (Vector v (Double, Double), Vector v Double) => v (Double, Double) -> Double -- | Correlation coefficient for sample of pairs. Also known as Pearson's -- correlation. For empty sample it's set to zero. correlation :: (Vector v (Double, Double), Vector v Double) => v (Double, Double) -> Double -- | Pair two samples. It's like zip but requires that both samples -- have equal size. pair :: (Vector v a, Vector v b, Vector v (a, b)) => v a -> v b -> v (a, b) -- | Basic matrix operations. -- -- There isn't a widely used matrix package for Haskell yet, so we -- implement the necessary minimum here. module Statistics.Matrix -- | Two-dimensional matrix, stored in row-major order. data Matrix Matrix :: {-# UNPACK #-} !Int -> {-# UNPACK #-} !Int -> {-# UNPACK #-} !Int -> !Vector -> Matrix -- | Rows of matrix. [rows] :: Matrix -> {-# UNPACK #-} !Int -- | Columns of matrix. [cols] :: Matrix -> {-# UNPACK #-} !Int -- | In order to avoid overflows during matrix multiplication, a large -- exponent is stored separately. [exponent] :: Matrix -> {-# UNPACK #-} !Int -- | Matrix data. [_vector] :: Matrix -> !Vector type Vector = Vector Double -- | Convert from a row-major vector. fromVector :: Int -> Int -> Vector Double -> Matrix -- | Convert from a row-major list. fromList :: Int -> Int -> [Double] -> Matrix -- | create a matrix from a list of lists, as rows fromRowLists :: [[Double]] -> Matrix -- | create a matrix from a list of vectors, as rows fromRows :: [Vector] -> Matrix -- | create a matrix from a list of vectors, as columns fromColumns :: [Vector] -> Matrix -- | Convert to a row-major flat vector. toVector :: Matrix -> Vector Double -- | Convert to a row-major flat list. toList :: Matrix -> [Double] -- | Convert to a list of vectors, as rows toRows :: Matrix -> [Vector] -- | Convert to a list of vectors, as columns toColumns :: Matrix -> [Vector] -- | Convert to a list of lists, as rows toRowLists :: Matrix -> [[Double]] -- | Generate matrix using function generate :: Int -> Int -> (Int -> Int -> Double) -> Matrix -- | Generate symmetric square matrix using function generateSym :: Int -> (Int -> Int -> Double) -> Matrix -- | Create the square identity matrix with given dimensions. ident :: Int -> Matrix -- | Create a square matrix with given diagonal, other entries default to 0 diag :: Vector -> Matrix -- | Return the dimensions of this matrix, as a (row,column) pair. dimension :: Matrix -> (Int, Int) -- | Element in the center of matrix (not corrected for exponent). center :: Matrix -> Double -- | Matrix-matrix multiplication. Matrices must be of compatible sizes -- (note: not checked). multiply :: Matrix -> Matrix -> Matrix -- | Matrix-vector multiplication. multiplyV :: Matrix -> Vector -> Vector transpose :: Matrix -> Matrix -- | Raise matrix to nth power. Power must be positive (/note: not -- checked). power :: Matrix -> Int -> Matrix -- | Calculate the Euclidean norm of a vector. norm :: Vector -> Double -- | Return the given column. column :: Matrix -> Int -> Vector -- | Return the given row. row :: Matrix -> Int -> Vector -- | Apply function to every element of matrix map :: (Double -> Double) -> Matrix -> Matrix -- | Simple for loop. Counts from start to end-1. for :: Monad m => Int -> Int -> (Int -> m ()) -> m () unsafeIndex :: Matrix -> Int -> Int -> Double -- | Indicate whether any element of the matrix is NaN. hasNaN :: Matrix -> Bool -- | Given row and column numbers, calculate the offset into the flat -- row-major vector. bounds :: (Vector -> Int -> r) -> Matrix -> Int -> Int -> r -- | Given row and column numbers, calculate the offset into the flat -- row-major vector, without checking. unsafeBounds :: (Vector -> Int -> r) -> Matrix -> Int -> Int -> r -- | Useful matrix functions. module Statistics.Matrix.Algorithms -- | O(r*c) Compute the QR decomposition of a matrix. The result -- returned is the matrices (q,r). qr :: Matrix -> (Matrix, Matrix) module Statistics.Correlation -- | Pearson correlation for sample of pairs. pearson :: (Vector v (Double, Double), Vector v Double) => v (Double, Double) -> Double -- | Compute pairwise pearson correlation between rows of a matrix pearsonMatByRow :: Matrix -> Matrix -- | compute spearman correlation between two samples spearman :: (Ord a, Ord b, Vector v a, Vector v b, Vector v (a, b), Vector v Int, Vector v Double, Vector v (Double, Double), Vector v (Int, a), Vector v (Int, b)) => v (a, b) -> Double -- | compute pairwise spearman correlation between rows of a matrix spearmanMatByRow :: Matrix -> Matrix -- | Type classes for probability distributions module Statistics.Distribution -- | Type class common to all distributions. Only c.d.f. could be defined -- for both discrete and continous distributions. class Distribution d where complCumulative d x = 1 - cumulative d x -- | Cumulative distribution function. The probability that a random -- variable X is less or equal than x, i.e. -- P(X≤x). Cumulative should be defined for infinities as -- well: -- -- 
--   cumulative d +∞ = 1
--   cumulative d -∞ = 0
--
cumulative :: Distribution d => d -> Double -> Double -- | 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. complCumulative :: Distribution d => d -> Double -> Double -- | Discrete probability distribution. class Distribution d => DiscreteDistr d where probability d = exp . logProbability d logProbability d = log . probability d -- | Probability of n-th outcome. probability :: DiscreteDistr d => d -> Int -> Double -- | Logarithm of probability of n-th outcome logProbability :: DiscreteDistr d => d -> Int -> Double -- | Continuous probability distributuion. -- -- Minimal complete definition is quantile and either -- density or logDensity. class Distribution d => ContDistr d where density d = exp . logDensity d logDensity d = log . density d -- | Probability density function. Probability that random variable -- X lies in the infinitesimal interval -- [x,x+δx) equal to density(x)⋅δx density :: ContDistr d => d -> Double -> Double -- | Inverse of the cumulative distribution function. The value x -- for which P(X≤x) = p. If probability is outside -- of [0,1] range function should call error quantile :: ContDistr d => d -> Double -> Double -- | Natural logarithm of density. logDensity :: ContDistr d => d -> Double -> Double -- | Type class for distributions with mean. maybeMean should return -- Nothing if it's undefined for current value of data class Distribution d => MaybeMean d maybeMean :: MaybeMean d => d -> Maybe Double -- | 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. class MaybeMean d => Mean d mean :: Mean d => d -> Double -- | Type class for distributions with variance. If variance is undefined -- for some parameter values both maybeVariance and -- maybeStdDev should return Nothing. -- -- Minimal complete definition is maybeVariance or -- maybeStdDev class MaybeMean d => MaybeVariance d where maybeVariance d = (*) <$> x <*> x where x = maybeStdDev d maybeStdDev = fmap sqrt . maybeVariance maybeVariance :: MaybeVariance d => d -> Maybe Double maybeStdDev :: MaybeVariance d => d -> Maybe Double -- | 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 variance or stdDev class (Mean d, MaybeVariance d) => Variance d where variance d = square (stdDev d) stdDev = sqrt . variance variance :: Variance d => d -> Double stdDev :: Variance d => d -> Double -- | 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. maybeEntropy should return -- Nothing if entropy is undefined for the chosen parameter -- values. class (Distribution d) => MaybeEntropy d -- | Returns the entropy of a distribution, in nats, if such is defined. maybeEntropy :: MaybeEntropy d => d -> Maybe Double -- | 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. class (MaybeEntropy d) => Entropy d -- | Returns the entropy of a distribution, in nats. entropy :: Entropy d => d -> Double -- | Generate discrete random variates which have given distribution. class Distribution d => ContGen d genContVar :: (ContGen d, PrimMonad m) => d -> Gen (PrimState m) -> m Double -- | Generate discrete random variates which have given distribution. -- ContGen is superclass because it's always possible to generate -- real-valued variates from integer values class (DiscreteDistr d, ContGen d) => DiscreteGen d genDiscreteVar :: (DiscreteGen d, PrimMonad m) => d -> Gen (PrimState m) -> m Int -- | Generate variates from continous distribution using inverse transform -- rule. genContinous :: (ContDistr d, PrimMonad m) => d -> Gen (PrimState m) -> m Double -- | 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. findRoot :: ContDistr d => d -> Double -> Double -> Double -> Double -> Double -- | Sum probabilities in inclusive interval. sumProbabilities :: DiscreteDistr d => d -> Int -> Int -> Double module Statistics.Distribution.Beta -- | The beta distribution data BetaDistribution -- | Create beta distribution. Both shape parameters must be positive. betaDistr :: Double -> Double -> BetaDistribution -- | Create beta distribution. This construtor doesn't check parameters. improperBetaDistr :: Double -> Double -> BetaDistribution -- | Alpha shape parameter bdAlpha :: BetaDistribution -> Double -- | Beta shape parameter bdBeta :: BetaDistribution -> Double instance GHC.Generics.Generic Statistics.Distribution.Beta.BetaDistribution instance Data.Data.Data Statistics.Distribution.Beta.BetaDistribution instance GHC.Show.Show Statistics.Distribution.Beta.BetaDistribution instance GHC.Read.Read Statistics.Distribution.Beta.BetaDistribution instance GHC.Classes.Eq Statistics.Distribution.Beta.BetaDistribution instance Data.Aeson.Types.Class.FromJSON Statistics.Distribution.Beta.BetaDistribution instance Data.Aeson.Types.Class.ToJSON Statistics.Distribution.Beta.BetaDistribution instance Data.Binary.Class.Binary Statistics.Distribution.Beta.BetaDistribution instance Statistics.Distribution.Distribution Statistics.Distribution.Beta.BetaDistribution instance Statistics.Distribution.Mean Statistics.Distribution.Beta.BetaDistribution instance Statistics.Distribution.MaybeMean Statistics.Distribution.Beta.BetaDistribution instance Statistics.Distribution.Variance Statistics.Distribution.Beta.BetaDistribution instance Statistics.Distribution.MaybeVariance Statistics.Distribution.Beta.BetaDistribution instance Statistics.Distribution.Entropy Statistics.Distribution.Beta.BetaDistribution instance Statistics.Distribution.MaybeEntropy Statistics.Distribution.Beta.BetaDistribution instance Statistics.Distribution.ContDistr Statistics.Distribution.Beta.BetaDistribution instance Statistics.Distribution.ContGen Statistics.Distribution.Beta.BetaDistribution -- | The binomial distribution. This is the discrete probability -- distribution of the number of successes in a sequence of n -- independent yes/no experiments, each of which yields success with -- probability p. module Statistics.Distribution.Binomial -- | The binomial distribution. data BinomialDistribution -- | Construct binomial distribution. Number of trials must be non-negative -- and probability must be in [0,1] range binomial :: Int -> Double -> BinomialDistribution -- | Number of trials. bdTrials :: BinomialDistribution -> Int -- | Probability. bdProbability :: BinomialDistribution -> Double instance GHC.Generics.Generic Statistics.Distribution.Binomial.BinomialDistribution instance Data.Data.Data Statistics.Distribution.Binomial.BinomialDistribution instance GHC.Show.Show Statistics.Distribution.Binomial.BinomialDistribution instance GHC.Read.Read Statistics.Distribution.Binomial.BinomialDistribution instance GHC.Classes.Eq Statistics.Distribution.Binomial.BinomialDistribution instance Data.Aeson.Types.Class.FromJSON Statistics.Distribution.Binomial.BinomialDistribution instance Data.Aeson.Types.Class.ToJSON Statistics.Distribution.Binomial.BinomialDistribution instance Data.Binary.Class.Binary Statistics.Distribution.Binomial.BinomialDistribution instance Statistics.Distribution.Distribution Statistics.Distribution.Binomial.BinomialDistribution instance Statistics.Distribution.DiscreteDistr Statistics.Distribution.Binomial.BinomialDistribution instance Statistics.Distribution.Mean Statistics.Distribution.Binomial.BinomialDistribution instance Statistics.Distribution.Variance Statistics.Distribution.Binomial.BinomialDistribution instance Statistics.Distribution.MaybeMean Statistics.Distribution.Binomial.BinomialDistribution instance Statistics.Distribution.MaybeVariance Statistics.Distribution.Binomial.BinomialDistribution instance Statistics.Distribution.Entropy Statistics.Distribution.Binomial.BinomialDistribution instance Statistics.Distribution.MaybeEntropy Statistics.Distribution.Binomial.BinomialDistribution -- | The Poisson distribution. This is the discrete probability -- distribution of a number of events occurring in a fixed interval if -- these events occur with a known average rate, and occur independently -- from each other within that interval. module Statistics.Distribution.Poisson data PoissonDistribution -- | Create Poisson distribution. poisson :: Double -> PoissonDistribution poissonLambda :: PoissonDistribution -> Double instance GHC.Generics.Generic Statistics.Distribution.Poisson.PoissonDistribution instance Data.Data.Data Statistics.Distribution.Poisson.PoissonDistribution instance GHC.Show.Show Statistics.Distribution.Poisson.PoissonDistribution instance GHC.Read.Read Statistics.Distribution.Poisson.PoissonDistribution instance GHC.Classes.Eq Statistics.Distribution.Poisson.PoissonDistribution instance Data.Aeson.Types.Class.FromJSON Statistics.Distribution.Poisson.PoissonDistribution instance Data.Aeson.Types.Class.ToJSON Statistics.Distribution.Poisson.PoissonDistribution instance Data.Binary.Class.Binary Statistics.Distribution.Poisson.PoissonDistribution instance Statistics.Distribution.Distribution Statistics.Distribution.Poisson.PoissonDistribution instance Statistics.Distribution.DiscreteDistr Statistics.Distribution.Poisson.PoissonDistribution instance Statistics.Distribution.Variance Statistics.Distribution.Poisson.PoissonDistribution instance Statistics.Distribution.Mean Statistics.Distribution.Poisson.PoissonDistribution instance Statistics.Distribution.MaybeMean Statistics.Distribution.Poisson.PoissonDistribution instance Statistics.Distribution.MaybeVariance Statistics.Distribution.Poisson.PoissonDistribution instance Statistics.Distribution.Entropy Statistics.Distribution.Poisson.PoissonDistribution instance Statistics.Distribution.MaybeEntropy Statistics.Distribution.Poisson.PoissonDistribution -- | The Cauchy-Lorentz distribution. It's also known as Lorentz -- distribution or Breit–Wigner distribution. -- -- It doesn't have mean and variance. module Statistics.Distribution.CauchyLorentz -- | Cauchy-Lorentz distribution. data CauchyDistribution -- | Central value of Cauchy-Lorentz distribution which is its mode and -- median. Distribution doesn't have mean so function is named after -- median. cauchyDistribMedian :: CauchyDistribution -> Double -- | Scale parameter of Cauchy-Lorentz distribution. It's different from -- variance and specify half width at half maximum (HWHM). cauchyDistribScale :: CauchyDistribution -> Double -- | Cauchy distribution cauchyDistribution :: Double -> Double -> CauchyDistribution standardCauchy :: CauchyDistribution instance GHC.Generics.Generic Statistics.Distribution.CauchyLorentz.CauchyDistribution instance Data.Data.Data Statistics.Distribution.CauchyLorentz.CauchyDistribution instance GHC.Read.Read Statistics.Distribution.CauchyLorentz.CauchyDistribution instance GHC.Show.Show Statistics.Distribution.CauchyLorentz.CauchyDistribution instance GHC.Classes.Eq Statistics.Distribution.CauchyLorentz.CauchyDistribution instance Data.Aeson.Types.Class.FromJSON Statistics.Distribution.CauchyLorentz.CauchyDistribution instance Data.Aeson.Types.Class.ToJSON Statistics.Distribution.CauchyLorentz.CauchyDistribution instance Data.Binary.Class.Binary Statistics.Distribution.CauchyLorentz.CauchyDistribution instance Statistics.Distribution.Distribution Statistics.Distribution.CauchyLorentz.CauchyDistribution instance Statistics.Distribution.ContDistr Statistics.Distribution.CauchyLorentz.CauchyDistribution instance Statistics.Distribution.ContGen Statistics.Distribution.CauchyLorentz.CauchyDistribution instance Statistics.Distribution.Entropy Statistics.Distribution.CauchyLorentz.CauchyDistribution instance Statistics.Distribution.MaybeEntropy Statistics.Distribution.CauchyLorentz.CauchyDistribution -- | The chi-squared distribution. This is a continuous probability -- distribution of sum of squares of k independent standard normal -- distributions. It's commonly used in statistical tests module Statistics.Distribution.ChiSquared -- | Chi-squared distribution data ChiSquared -- | Construct chi-squared distribution. Number of degrees of freedom must -- be positive. chiSquared :: Int -> ChiSquared -- | Get number of degrees of freedom chiSquaredNDF :: ChiSquared -> Int instance GHC.Generics.Generic Statistics.Distribution.ChiSquared.ChiSquared instance Data.Data.Data Statistics.Distribution.ChiSquared.ChiSquared instance GHC.Show.Show Statistics.Distribution.ChiSquared.ChiSquared instance GHC.Read.Read Statistics.Distribution.ChiSquared.ChiSquared instance GHC.Classes.Eq Statistics.Distribution.ChiSquared.ChiSquared instance Data.Aeson.Types.Class.FromJSON Statistics.Distribution.ChiSquared.ChiSquared instance Data.Aeson.Types.Class.ToJSON Statistics.Distribution.ChiSquared.ChiSquared instance Data.Binary.Class.Binary Statistics.Distribution.ChiSquared.ChiSquared instance Statistics.Distribution.Distribution Statistics.Distribution.ChiSquared.ChiSquared instance Statistics.Distribution.ContDistr Statistics.Distribution.ChiSquared.ChiSquared instance Statistics.Distribution.Mean Statistics.Distribution.ChiSquared.ChiSquared instance Statistics.Distribution.Variance Statistics.Distribution.ChiSquared.ChiSquared instance Statistics.Distribution.MaybeMean Statistics.Distribution.ChiSquared.ChiSquared instance Statistics.Distribution.MaybeVariance Statistics.Distribution.ChiSquared.ChiSquared instance Statistics.Distribution.Entropy Statistics.Distribution.ChiSquared.ChiSquared instance Statistics.Distribution.MaybeEntropy Statistics.Distribution.ChiSquared.ChiSquared instance Statistics.Distribution.ContGen Statistics.Distribution.ChiSquared.ChiSquared -- | The exponential distribution. This is the continunous probability -- distribution of the times between events in a poisson process, in -- which events occur continuously and independently at a constant -- average rate. module Statistics.Distribution.Exponential data ExponentialDistribution -- | Create an exponential distribution. exponential :: Double -> ExponentialDistribution -- | Create exponential distribution from sample. No tests are made to -- check whether it truly is exponential. exponentialFromSample :: Sample -> ExponentialDistribution edLambda :: ExponentialDistribution -> Double instance GHC.Generics.Generic Statistics.Distribution.Exponential.ExponentialDistribution instance Data.Data.Data Statistics.Distribution.Exponential.ExponentialDistribution instance GHC.Show.Show Statistics.Distribution.Exponential.ExponentialDistribution instance GHC.Read.Read Statistics.Distribution.Exponential.ExponentialDistribution instance GHC.Classes.Eq Statistics.Distribution.Exponential.ExponentialDistribution instance Data.Aeson.Types.Class.FromJSON Statistics.Distribution.Exponential.ExponentialDistribution instance Data.Aeson.Types.Class.ToJSON Statistics.Distribution.Exponential.ExponentialDistribution instance Data.Binary.Class.Binary Statistics.Distribution.Exponential.ExponentialDistribution instance Statistics.Distribution.Distribution Statistics.Distribution.Exponential.ExponentialDistribution instance Statistics.Distribution.ContDistr Statistics.Distribution.Exponential.ExponentialDistribution instance Statistics.Distribution.Mean Statistics.Distribution.Exponential.ExponentialDistribution instance Statistics.Distribution.Variance Statistics.Distribution.Exponential.ExponentialDistribution instance Statistics.Distribution.MaybeMean Statistics.Distribution.Exponential.ExponentialDistribution instance Statistics.Distribution.MaybeVariance Statistics.Distribution.Exponential.ExponentialDistribution instance Statistics.Distribution.Entropy Statistics.Distribution.Exponential.ExponentialDistribution instance Statistics.Distribution.MaybeEntropy Statistics.Distribution.Exponential.ExponentialDistribution instance Statistics.Distribution.ContGen Statistics.Distribution.Exponential.ExponentialDistribution -- | The gamma distribution. This is a continuous probability distribution -- with two parameters, k and ϑ. If k is integral, the -- distribution represents the sum of k independent exponentially -- distributed random variables, each of which has a mean of ϑ. module Statistics.Distribution.Gamma -- | The gamma distribution. data GammaDistribution -- | Create gamma distribution. Both shape and scale parameters must be -- positive. gammaDistr :: Double -> Double -> GammaDistribution -- | Create gamma distribution. This constructor do not check whether -- parameters are valid improperGammaDistr :: Double -> Double -> GammaDistribution -- | Shape parameter, k. gdShape :: GammaDistribution -> Double -- | Scale parameter, ϑ. gdScale :: GammaDistribution -> Double instance GHC.Generics.Generic Statistics.Distribution.Gamma.GammaDistribution instance Data.Data.Data Statistics.Distribution.Gamma.GammaDistribution instance GHC.Show.Show Statistics.Distribution.Gamma.GammaDistribution instance GHC.Read.Read Statistics.Distribution.Gamma.GammaDistribution instance GHC.Classes.Eq Statistics.Distribution.Gamma.GammaDistribution instance Data.Aeson.Types.Class.FromJSON Statistics.Distribution.Gamma.GammaDistribution instance Data.Aeson.Types.Class.ToJSON Statistics.Distribution.Gamma.GammaDistribution instance Data.Binary.Class.Binary Statistics.Distribution.Gamma.GammaDistribution instance Statistics.Distribution.Distribution Statistics.Distribution.Gamma.GammaDistribution instance Statistics.Distribution.ContDistr Statistics.Distribution.Gamma.GammaDistribution instance Statistics.Distribution.Variance Statistics.Distribution.Gamma.GammaDistribution instance Statistics.Distribution.Mean Statistics.Distribution.Gamma.GammaDistribution instance Statistics.Distribution.MaybeMean Statistics.Distribution.Gamma.GammaDistribution instance Statistics.Distribution.MaybeVariance Statistics.Distribution.Gamma.GammaDistribution instance Statistics.Distribution.MaybeEntropy Statistics.Distribution.Gamma.GammaDistribution instance Statistics.Distribution.ContGen Statistics.Distribution.Gamma.GammaDistribution -- | The Geometric distribution. There are two variants of distribution. -- First is the probability distribution of the number of Bernoulli -- trials needed to get one success, supported on the set 1,2... -- Sometimes it's referred to as the shifted geometric -- distribution to distinguish from another one. -- -- Second variant is probability distribution of the number of failures -- before first success, defined over the set 0,1... module Statistics.Distribution.Geometric data GeometricDistribution data GeometricDistribution0 -- | Create geometric distribution. geometric :: Double -> GeometricDistribution -- | Create geometric distribution. geometric0 :: Double -> GeometricDistribution0 gdSuccess :: GeometricDistribution -> Double gdSuccess0 :: GeometricDistribution0 -> Double instance GHC.Generics.Generic Statistics.Distribution.Geometric.GeometricDistribution0 instance Data.Data.Data Statistics.Distribution.Geometric.GeometricDistribution0 instance GHC.Show.Show Statistics.Distribution.Geometric.GeometricDistribution0 instance GHC.Read.Read Statistics.Distribution.Geometric.GeometricDistribution0 instance GHC.Classes.Eq Statistics.Distribution.Geometric.GeometricDistribution0 instance GHC.Generics.Generic Statistics.Distribution.Geometric.GeometricDistribution instance Data.Data.Data Statistics.Distribution.Geometric.GeometricDistribution instance GHC.Show.Show Statistics.Distribution.Geometric.GeometricDistribution instance GHC.Read.Read Statistics.Distribution.Geometric.GeometricDistribution instance GHC.Classes.Eq Statistics.Distribution.Geometric.GeometricDistribution instance Data.Aeson.Types.Class.FromJSON Statistics.Distribution.Geometric.GeometricDistribution instance Data.Aeson.Types.Class.ToJSON Statistics.Distribution.Geometric.GeometricDistribution instance Data.Binary.Class.Binary Statistics.Distribution.Geometric.GeometricDistribution instance Statistics.Distribution.Distribution Statistics.Distribution.Geometric.GeometricDistribution instance Statistics.Distribution.DiscreteDistr Statistics.Distribution.Geometric.GeometricDistribution instance Statistics.Distribution.Mean Statistics.Distribution.Geometric.GeometricDistribution instance Statistics.Distribution.Variance Statistics.Distribution.Geometric.GeometricDistribution instance Statistics.Distribution.MaybeMean Statistics.Distribution.Geometric.GeometricDistribution instance Statistics.Distribution.MaybeVariance Statistics.Distribution.Geometric.GeometricDistribution instance Statistics.Distribution.Entropy Statistics.Distribution.Geometric.GeometricDistribution instance Statistics.Distribution.MaybeEntropy Statistics.Distribution.Geometric.GeometricDistribution instance Statistics.Distribution.DiscreteGen Statistics.Distribution.Geometric.GeometricDistribution instance Statistics.Distribution.ContGen Statistics.Distribution.Geometric.GeometricDistribution instance Data.Aeson.Types.Class.FromJSON Statistics.Distribution.Geometric.GeometricDistribution0 instance Data.Aeson.Types.Class.ToJSON Statistics.Distribution.Geometric.GeometricDistribution0 instance Data.Binary.Class.Binary Statistics.Distribution.Geometric.GeometricDistribution0 instance Statistics.Distribution.Distribution Statistics.Distribution.Geometric.GeometricDistribution0 instance Statistics.Distribution.DiscreteDistr Statistics.Distribution.Geometric.GeometricDistribution0 instance Statistics.Distribution.Mean Statistics.Distribution.Geometric.GeometricDistribution0 instance Statistics.Distribution.Variance Statistics.Distribution.Geometric.GeometricDistribution0 instance Statistics.Distribution.MaybeMean Statistics.Distribution.Geometric.GeometricDistribution0 instance Statistics.Distribution.MaybeVariance Statistics.Distribution.Geometric.GeometricDistribution0 instance Statistics.Distribution.Entropy Statistics.Distribution.Geometric.GeometricDistribution0 instance Statistics.Distribution.MaybeEntropy Statistics.Distribution.Geometric.GeometricDistribution0 instance Statistics.Distribution.DiscreteGen Statistics.Distribution.Geometric.GeometricDistribution0 instance Statistics.Distribution.ContGen Statistics.Distribution.Geometric.GeometricDistribution0 -- | The Hypergeometric distribution. This is the discrete probability -- distribution that measures the probability of k successes in -- l trials, without replacement, from a finite population. -- -- The parameters of the distribution describe k elements chosen -- from a population of l, with m elements of one type, and -- l-m of the other (all are positive integers). module Statistics.Distribution.Hypergeometric data HypergeometricDistribution hypergeometric :: Int -> Int -> Int -> HypergeometricDistribution hdM :: HypergeometricDistribution -> Int hdL :: HypergeometricDistribution -> Int hdK :: HypergeometricDistribution -> Int instance GHC.Generics.Generic Statistics.Distribution.Hypergeometric.HypergeometricDistribution instance Data.Data.Data Statistics.Distribution.Hypergeometric.HypergeometricDistribution instance GHC.Show.Show Statistics.Distribution.Hypergeometric.HypergeometricDistribution instance GHC.Read.Read Statistics.Distribution.Hypergeometric.HypergeometricDistribution instance GHC.Classes.Eq Statistics.Distribution.Hypergeometric.HypergeometricDistribution instance Data.Aeson.Types.Class.FromJSON Statistics.Distribution.Hypergeometric.HypergeometricDistribution instance Data.Aeson.Types.Class.ToJSON Statistics.Distribution.Hypergeometric.HypergeometricDistribution instance Data.Binary.Class.Binary Statistics.Distribution.Hypergeometric.HypergeometricDistribution instance Statistics.Distribution.Distribution Statistics.Distribution.Hypergeometric.HypergeometricDistribution instance Statistics.Distribution.DiscreteDistr Statistics.Distribution.Hypergeometric.HypergeometricDistribution instance Statistics.Distribution.Mean Statistics.Distribution.Hypergeometric.HypergeometricDistribution instance Statistics.Distribution.Variance Statistics.Distribution.Hypergeometric.HypergeometricDistribution instance Statistics.Distribution.MaybeMean Statistics.Distribution.Hypergeometric.HypergeometricDistribution instance Statistics.Distribution.MaybeVariance Statistics.Distribution.Hypergeometric.HypergeometricDistribution instance Statistics.Distribution.Entropy Statistics.Distribution.Hypergeometric.HypergeometricDistribution instance Statistics.Distribution.MaybeEntropy Statistics.Distribution.Hypergeometric.HypergeometricDistribution -- | The normal distribution. This is a continuous probability distribution -- that describes data that cluster around a mean. module Statistics.Distribution.Normal -- | The normal distribution. data NormalDistribution -- | Create normal distribution from parameters. -- -- IMPORTANT: prior to 0.10 release second parameter was variance not -- standard deviation. normalDistr :: Double -> Double -> NormalDistribution -- | Create distribution using parameters estimated from sample. Variance -- is estimated using maximum likelihood method (biased estimation). normalFromSample :: Sample -> NormalDistribution -- | Standard normal distribution with mean equal to 0 and variance equal -- to 1 standard :: NormalDistribution instance GHC.Generics.Generic Statistics.Distribution.Normal.NormalDistribution instance Data.Data.Data Statistics.Distribution.Normal.NormalDistribution instance GHC.Show.Show Statistics.Distribution.Normal.NormalDistribution instance GHC.Read.Read Statistics.Distribution.Normal.NormalDistribution instance GHC.Classes.Eq Statistics.Distribution.Normal.NormalDistribution instance Data.Aeson.Types.Class.FromJSON Statistics.Distribution.Normal.NormalDistribution instance Data.Aeson.Types.Class.ToJSON Statistics.Distribution.Normal.NormalDistribution instance Data.Binary.Class.Binary Statistics.Distribution.Normal.NormalDistribution instance Statistics.Distribution.Distribution Statistics.Distribution.Normal.NormalDistribution instance Statistics.Distribution.ContDistr Statistics.Distribution.Normal.NormalDistribution instance Statistics.Distribution.MaybeMean Statistics.Distribution.Normal.NormalDistribution instance Statistics.Distribution.Mean Statistics.Distribution.Normal.NormalDistribution instance Statistics.Distribution.MaybeVariance Statistics.Distribution.Normal.NormalDistribution instance Statistics.Distribution.Variance Statistics.Distribution.Normal.NormalDistribution instance Statistics.Distribution.Entropy Statistics.Distribution.Normal.NormalDistribution instance Statistics.Distribution.MaybeEntropy Statistics.Distribution.Normal.NormalDistribution instance Statistics.Distribution.ContGen Statistics.Distribution.Normal.NormalDistribution -- | Transformations over distributions module Statistics.Distribution.Transform -- | Linear transformation applied to distribution. -- -- 
--   LinearTransform μ σ _
--   x' = μ + σ·x
--
data LinearTransform d LinearTransform :: {-# UNPACK #-} !Double -> {-# UNPACK #-} !Double -> d -> LinearTransform d -- | Location parameter. [linTransLocation] :: LinearTransform d -> {-# UNPACK #-} !Double -- | Scale parameter. [linTransScale] :: LinearTransform d -> {-# UNPACK #-} !Double -- | Distribution being transformed. [linTransDistr] :: LinearTransform d -> d -- | Get fixed point of linear transformation linTransFixedPoint :: LinearTransform d -> Double -- | Apply linear transformation to distribution. scaleAround :: Double -> Double -> d -> LinearTransform d instance GHC.Generics.Generic (Statistics.Distribution.Transform.LinearTransform d) instance Data.Data.Data d => Data.Data.Data (Statistics.Distribution.Transform.LinearTransform d) instance GHC.Read.Read d => GHC.Read.Read (Statistics.Distribution.Transform.LinearTransform d) instance GHC.Show.Show d => GHC.Show.Show (Statistics.Distribution.Transform.LinearTransform d) instance GHC.Classes.Eq d => GHC.Classes.Eq (Statistics.Distribution.Transform.LinearTransform d) instance Data.Aeson.Types.Class.FromJSON d => Data.Aeson.Types.Class.FromJSON (Statistics.Distribution.Transform.LinearTransform d) instance Data.Aeson.Types.Class.ToJSON d => Data.Aeson.Types.Class.ToJSON (Statistics.Distribution.Transform.LinearTransform d) instance Data.Binary.Class.Binary d => Data.Binary.Class.Binary (Statistics.Distribution.Transform.LinearTransform d) instance GHC.Base.Functor Statistics.Distribution.Transform.LinearTransform instance Statistics.Distribution.Distribution d => Statistics.Distribution.Distribution (Statistics.Distribution.Transform.LinearTransform d) instance Statistics.Distribution.ContDistr d => Statistics.Distribution.ContDistr (Statistics.Distribution.Transform.LinearTransform d) instance Statistics.Distribution.MaybeMean d => Statistics.Distribution.MaybeMean (Statistics.Distribution.Transform.LinearTransform d) instance Statistics.Distribution.Mean d => Statistics.Distribution.Mean (Statistics.Distribution.Transform.LinearTransform d) instance Statistics.Distribution.MaybeVariance d => Statistics.Distribution.MaybeVariance (Statistics.Distribution.Transform.LinearTransform d) instance Statistics.Distribution.Variance d => Statistics.Distribution.Variance (Statistics.Distribution.Transform.LinearTransform d) instance (Statistics.Distribution.MaybeEntropy d, Statistics.Distribution.DiscreteDistr d) => Statistics.Distribution.MaybeEntropy (Statistics.Distribution.Transform.LinearTransform d) instance (Statistics.Distribution.Entropy d, Statistics.Distribution.DiscreteDistr d) => Statistics.Distribution.Entropy (Statistics.Distribution.Transform.LinearTransform d) instance Statistics.Distribution.ContGen d => Statistics.Distribution.ContGen (Statistics.Distribution.Transform.LinearTransform d) -- | Student-T distribution module Statistics.Distribution.StudentT -- | Student-T distribution data StudentT -- | Create Student-T distribution. Number of parameters must be positive. studentT :: Double -> StudentT studentTndf :: StudentT -> Double -- | Create an unstandardized Student-t distribution. studentTUnstandardized :: Double -> Double -> Double -> LinearTransform StudentT instance GHC.Generics.Generic Statistics.Distribution.StudentT.StudentT instance Data.Data.Data Statistics.Distribution.StudentT.StudentT instance GHC.Read.Read Statistics.Distribution.StudentT.StudentT instance GHC.Show.Show Statistics.Distribution.StudentT.StudentT instance GHC.Classes.Eq Statistics.Distribution.StudentT.StudentT instance Data.Aeson.Types.Class.FromJSON Statistics.Distribution.StudentT.StudentT instance Data.Aeson.Types.Class.ToJSON Statistics.Distribution.StudentT.StudentT instance Data.Binary.Class.Binary Statistics.Distribution.StudentT.StudentT instance Statistics.Distribution.Distribution Statistics.Distribution.StudentT.StudentT instance Statistics.Distribution.ContDistr Statistics.Distribution.StudentT.StudentT instance Statistics.Distribution.MaybeMean Statistics.Distribution.StudentT.StudentT instance Statistics.Distribution.MaybeVariance Statistics.Distribution.StudentT.StudentT instance Statistics.Distribution.Entropy Statistics.Distribution.StudentT.StudentT instance Statistics.Distribution.MaybeEntropy Statistics.Distribution.StudentT.StudentT instance Statistics.Distribution.ContGen Statistics.Distribution.StudentT.StudentT -- | Variate distributed uniformly in the interval. module Statistics.Distribution.Uniform -- | Uniform distribution from A to B data UniformDistribution -- | Create uniform distribution. uniformDistr :: Double -> Double -> UniformDistribution -- | Low boundary of distribution uniformA :: UniformDistribution -> Double -- | Upper boundary of distribution uniformB :: UniformDistribution -> Double instance GHC.Generics.Generic Statistics.Distribution.Uniform.UniformDistribution instance Data.Data.Data Statistics.Distribution.Uniform.UniformDistribution instance GHC.Show.Show Statistics.Distribution.Uniform.UniformDistribution instance GHC.Read.Read Statistics.Distribution.Uniform.UniformDistribution instance GHC.Classes.Eq Statistics.Distribution.Uniform.UniformDistribution instance Data.Aeson.Types.Class.FromJSON Statistics.Distribution.Uniform.UniformDistribution instance Data.Aeson.Types.Class.ToJSON Statistics.Distribution.Uniform.UniformDistribution instance Data.Binary.Class.Binary Statistics.Distribution.Uniform.UniformDistribution instance Statistics.Distribution.Distribution Statistics.Distribution.Uniform.UniformDistribution instance Statistics.Distribution.ContDistr Statistics.Distribution.Uniform.UniformDistribution instance Statistics.Distribution.Mean Statistics.Distribution.Uniform.UniformDistribution instance Statistics.Distribution.Variance Statistics.Distribution.Uniform.UniformDistribution instance Statistics.Distribution.MaybeMean Statistics.Distribution.Uniform.UniformDistribution instance Statistics.Distribution.MaybeVariance Statistics.Distribution.Uniform.UniformDistribution instance Statistics.Distribution.Entropy Statistics.Distribution.Uniform.UniformDistribution instance Statistics.Distribution.MaybeEntropy Statistics.Distribution.Uniform.UniformDistribution instance Statistics.Distribution.ContGen Statistics.Distribution.Uniform.UniformDistribution -- | Fisher F distribution module Statistics.Distribution.FDistribution -- | F distribution data FDistribution fDistribution :: Int -> Int -> FDistribution fDistributionNDF1 :: FDistribution -> Double fDistributionNDF2 :: FDistribution -> Double instance GHC.Generics.Generic Statistics.Distribution.FDistribution.FDistribution instance Data.Data.Data Statistics.Distribution.FDistribution.FDistribution instance GHC.Read.Read Statistics.Distribution.FDistribution.FDistribution instance GHC.Show.Show Statistics.Distribution.FDistribution.FDistribution instance GHC.Classes.Eq Statistics.Distribution.FDistribution.FDistribution instance Data.Aeson.Types.Class.FromJSON Statistics.Distribution.FDistribution.FDistribution instance Data.Aeson.Types.Class.ToJSON Statistics.Distribution.FDistribution.FDistribution instance Data.Binary.Class.Binary Statistics.Distribution.FDistribution.FDistribution instance Statistics.Distribution.Distribution Statistics.Distribution.FDistribution.FDistribution instance Statistics.Distribution.ContDistr Statistics.Distribution.FDistribution.FDistribution instance Statistics.Distribution.MaybeMean Statistics.Distribution.FDistribution.FDistribution instance Statistics.Distribution.MaybeVariance Statistics.Distribution.FDistribution.FDistribution instance Statistics.Distribution.Entropy Statistics.Distribution.FDistribution.FDistribution instance Statistics.Distribution.MaybeEntropy Statistics.Distribution.FDistribution.FDistribution instance Statistics.Distribution.ContGen Statistics.Distribution.FDistribution.FDistribution -- | Functions for approximating quantiles, i.e. points taken at regular -- intervals from the cumulative distribution function of a random -- variable. -- -- The number of quantiles is described below by the variable q, -- so with q=4, a 4-quantile (also known as a quartile) has -- 4 intervals, and contains 5 points. The parameter k describes -- the desired point, where 0 ≤ k ≤ q. module Statistics.Quantile -- | O(n log n). Estimate the kth q-quantile of -- a sample, using the weighted average method. weightedAvg :: Vector v Double => Int -> Int -> v Double -> Double -- | Parameters a and b to the continuousBy function. data ContParam ContParam :: {-# UNPACK #-} !Double -> {-# UNPACK #-} !Double -> ContParam -- | O(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. continuousBy :: Vector v Double => ContParam -> Int -> Int -> v Double -> Double -- | O(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. -- -- For instance, the interquartile range (IQR) can be estimated as -- follows: -- -- 
--   midspread medianUnbiased 4 (U.fromList [1,1,2,2,3])
--   ==> 1.333333
--
midspread :: Vector v Double => ContParam -> Int -> v Double -> Double -- | California Department of Public Works definition, a=0, -- b=1. Gives a linear interpolation of the empirical CDF. This -- corresponds to method 4 in R and Mathematica. cadpw :: ContParam -- | Hazen's definition, a=0.5, b=0.5. This is claimed to be -- popular among hydrologists. This corresponds to method 5 in R and -- Mathematica. hazen :: ContParam -- | Definition 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. s :: ContParam -- | Definition 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. spss :: ContParam -- | Median unbiased definition, a=1/3, b=1/3. The resulting -- quantile estimates are approximately median unbiased regardless of the -- distribution of x. This corresponds to method 8 in R and -- Mathematica. medianUnbiased :: ContParam -- | Normal 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. normalUnbiased :: ContParam -- | The Laplace distribution. This is the continuous probability defined -- as the difference of two iid exponential random variables or a -- Brownian motion evaluated as exponentially distributed times. It is -- used in differential privacy (Laplace Method), speech recognition and -- least absolute deviations method (Laplace's first law of errors, -- giving a robust regression method) module Statistics.Distribution.Laplace data LaplaceDistribution -- | Create an Laplace distribution. laplace :: Double -> Double -> LaplaceDistribution -- | Create Laplace distribution from sample. No tests are made to check -- whether it truly is Laplace. Location of distribution estimated as -- median of sample. laplaceFromSample :: Sample -> LaplaceDistribution -- | Location. ldLocation :: LaplaceDistribution -> Double -- | Scale. ldScale :: LaplaceDistribution -> Double instance GHC.Generics.Generic Statistics.Distribution.Laplace.LaplaceDistribution instance Data.Data.Data Statistics.Distribution.Laplace.LaplaceDistribution instance GHC.Show.Show Statistics.Distribution.Laplace.LaplaceDistribution instance GHC.Read.Read Statistics.Distribution.Laplace.LaplaceDistribution instance GHC.Classes.Eq Statistics.Distribution.Laplace.LaplaceDistribution instance Data.Aeson.Types.Class.FromJSON Statistics.Distribution.Laplace.LaplaceDistribution instance Data.Aeson.Types.Class.ToJSON Statistics.Distribution.Laplace.LaplaceDistribution instance Data.Binary.Class.Binary Statistics.Distribution.Laplace.LaplaceDistribution instance Statistics.Distribution.Distribution Statistics.Distribution.Laplace.LaplaceDistribution instance Statistics.Distribution.ContDistr Statistics.Distribution.Laplace.LaplaceDistribution instance Statistics.Distribution.Mean Statistics.Distribution.Laplace.LaplaceDistribution instance Statistics.Distribution.Variance Statistics.Distribution.Laplace.LaplaceDistribution instance Statistics.Distribution.MaybeMean Statistics.Distribution.Laplace.LaplaceDistribution instance Statistics.Distribution.MaybeVariance Statistics.Distribution.Laplace.LaplaceDistribution instance Statistics.Distribution.Entropy Statistics.Distribution.Laplace.LaplaceDistribution instance Statistics.Distribution.MaybeEntropy Statistics.Distribution.Laplace.LaplaceDistribution instance Statistics.Distribution.ContGen Statistics.Distribution.Laplace.LaplaceDistribution -- | Resampling statistics. module Statistics.Resampling -- | 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. newtype Resample Resample :: Vector Double -> Resample [fromResample] :: Resample -> Vector Double -- | O(n) or O(n^2) Compute a statistical estimate repeatedly over a -- sample, each time omitting a successive element. jackknife :: Estimator -> Sample -> Vector Double -- | O(n) Compute the jackknife mean of a sample. jackknifeMean :: Sample -> Vector Double -- | O(n) Compute the jackknife variance of a sample. jackknifeVariance :: Sample -> Vector Double -- | O(n) Compute the unbiased jackknife variance of a sample. jackknifeVarianceUnb :: Sample -> Vector Double -- | O(n) Compute the jackknife standard deviation of a sample. jackknifeStdDev :: Sample -> Vector Double -- | O(e*r*s) 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). resample :: GenIO -> [Estimator] -> Int -> Sample -> IO [Resample] -- | Run an Estimator over a sample. estimate :: Estimator -> Sample -> Double -- | Split a generator into several that can run independently. splitGen :: Int -> GenIO -> IO [GenIO] instance GHC.Generics.Generic Statistics.Resampling.Resample instance Data.Data.Data Statistics.Resampling.Resample instance GHC.Show.Show Statistics.Resampling.Resample instance GHC.Read.Read Statistics.Resampling.Resample instance GHC.Classes.Eq Statistics.Resampling.Resample instance Data.Aeson.Types.Class.FromJSON Statistics.Resampling.Resample instance Data.Aeson.Types.Class.ToJSON Statistics.Resampling.Resample instance Data.Binary.Class.Binary Statistics.Resampling.Resample -- | The bootstrap method for statistical inference. module Statistics.Resampling.Bootstrap -- | A point and interval estimate computed via an Estimator. data Estimate Estimate :: {-# UNPACK #-} !Double -> {-# UNPACK #-} !Double -> {-# UNPACK #-} !Double -> {-# UNPACK #-} !Double -> Estimate -- | Point estimate. [estPoint] :: Estimate -> {-# UNPACK #-} !Double -- | Lower bound of the estimate interval (i.e. the lower bound of the -- confidence interval). [estLowerBound] :: Estimate -> {-# UNPACK #-} !Double -- | Upper bound of the estimate interval (i.e. the upper bound of the -- confidence interval). [estUpperBound] :: Estimate -> {-# UNPACK #-} !Double -- | Confidence level of the confidence intervals. [estConfidenceLevel] :: Estimate -> {-# UNPACK #-} !Double -- | Bias-corrected accelerated (BCA) bootstrap. This adjusts for both bias -- and skewness in the resampled distribution. bootstrapBCA :: Double -> Sample -> [Estimator] -> [Resample] -> [Estimate] -- | Multiply the point, lower bound, and upper bound in an Estimate -- by the given value. scale :: Double -> Estimate -> Estimate instance GHC.Generics.Generic Statistics.Resampling.Bootstrap.Estimate instance Data.Data.Data Statistics.Resampling.Bootstrap.Estimate instance GHC.Show.Show Statistics.Resampling.Bootstrap.Estimate instance GHC.Read.Read Statistics.Resampling.Bootstrap.Estimate instance GHC.Classes.Eq Statistics.Resampling.Bootstrap.Estimate instance Data.Aeson.Types.Class.FromJSON Statistics.Resampling.Bootstrap.Estimate instance Data.Aeson.Types.Class.ToJSON Statistics.Resampling.Bootstrap.Estimate instance Data.Binary.Class.Binary Statistics.Resampling.Bootstrap.Estimate instance Control.DeepSeq.NFData Statistics.Resampling.Bootstrap.Estimate -- | Functions for regression analysis. module Statistics.Regression -- | Perform an ordinary least-squares regression on a set of predictors, -- and calculate the goodness-of-fit of the regression. -- -- The returned pair consists of: -- -- olsRegress :: [Vector] -> Vector -> (Vector, Double) -- | Compute the ordinary least-squares solution to A x = b. ols :: Matrix -> Vector -> Vector -- | Compute R², the coefficient of determination that indicates -- goodness-of-fit of a regression. -- -- This value will be 1 if the predictors fit perfectly, dropping to 0 if -- they have no explanatory power. rSquare :: Matrix -> Vector -> Vector -> Double -- | Bootstrap a regression function. Returns both the results of the -- regression and the requested confidence interval values. bootstrapRegress :: GenIO -> Int -> Double -> ([Vector] -> Vector -> (Vector, Double)) -> [Vector] -> Vector -> IO (Vector Estimate, Estimate) -- | Functions for computing histograms of sample data. module Statistics.Sample.Histogram -- | O(n) Compute a histogram over a data set. -- -- The result consists of a pair of vectors: -- -- -- -- Interval (bin) sizes are uniform, and the upper and lower bounds are -- chosen automatically using the range function. To specify these -- parameters directly, use the histogram_ function. histogram :: (Vector v0 Double, Vector v1 Double, Num b, Vector v1 b) => Int -> v0 Double -> (v1 Double, v1 b) -- | O(n) Compute a histogram over a data set. -- -- Interval (bin) sizes are uniform, based on the supplied upper and -- lower bounds. histogram_ :: (Num b, RealFrac a, Vector v0 a, Vector v1 b) => Int -> a -> a -> v0 a -> v1 b -- | O(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 -- -- 
--   d = (maximum sample - minimum sample) / ((bins - 1) * 2)
--
-- -- If 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). This is -- needed to avoid creating histogram with zero bin size. range :: (Vector v Double) => Int -> v Double -> (Double, Double) -- | Kernel density estimation. This module provides a fast, robust, -- non-parametric way to estimate the probability density function of a -- sample. -- -- This estimator does not use the commonly employed "Gaussian rule of -- thumb". As a result, it outperforms many plug-in methods on multimodal -- samples with widely separated modes. module Statistics.Sample.KernelDensity -- | Gaussian kernel density estimator for one-dimensional data, using the -- method of Botev et al. -- -- The result is a pair of vectors, containing: -- -- kde :: (Vector v CD, Vector v Double, Vector v Int) => Int -> v Double -> (v Double, v Double) -- | Gaussian kernel density estimator for one-dimensional data, using the -- method of Botev et al. -- -- The result is a pair of vectors, containing: -- -- kde_ :: (Vector v CD, Vector v Double, Vector v Int) => Int -> Double -> Double -> v Double -> (v Double, v Double) -- | Kernel density estimation code, providing non-parametric ways to -- estimate the probability density function of a sample. -- -- The techniques used by functions in this module are relatively fast, -- but they generally give inferior results to the KDE function in the -- main KernelDensity module (due to the oversmoothing documented -- for bandwidth below). -- | Deprecated: Use Statistics.Sample.KernelDensity instead. module Statistics.Sample.KernelDensity.Simple -- | 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. epanechnikovPDF :: Vector v Double => Int -> v Double -> (Points, Vector Double) -- | 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. gaussianPDF :: Vector v Double => Int -> v Double -> (Points, Vector Double) -- | Points from the range of a Sample. newtype Points Points :: Vector Double -> Points [fromPoints] :: Points -> Vector Double -- | Choose a uniform range of points at which to estimate a sample's -- probability density function. -- -- If you are using a Gaussian kernel, multiply the sample's bandwidth by -- 3 before passing it to this function. -- -- If this function is passed an empty vector, it returns values of -- positive and negative infinity. choosePoints :: Vector v Double => Int -> Double -> v Double -> Points -- | The width of the convolution kernel used. type Bandwidth = Double -- | Compute 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. bandwidth :: Vector v Double => (Double -> Bandwidth) -> v Double -> Bandwidth -- | Bandwidth estimator for an Epanechnikov kernel. epanechnikovBW :: Double -> Bandwidth -- | Bandwidth estimator for a Gaussian kernel. gaussianBW :: Double -> Bandwidth -- | The convolution kernel. Its parameters are as follows: -- -- type Kernel = Double -> Double -> Double -> Double -> Double -- | Epanechnikov kernel for probability density function estimation. epanechnikovKernel :: Kernel -- | Gaussian kernel for probability density function estimation. gaussianKernel :: Kernel -- | Kernel density estimator, providing a non-parametric way of estimating -- the PDF of a random variable. estimatePDF :: Vector v Double => Kernel -> Bandwidth -> v Double -> Points -> Vector Double -- | A helper for creating a simple kernel density estimation function with -- automatically chosen bandwidth and estimation points. simplePDF :: Vector v Double => (Double -> Double) -> Kernel -> Double -> Int -> v Double -> (Points, Vector Double) instance GHC.Generics.Generic Statistics.Sample.KernelDensity.Simple.Points instance Data.Data.Data Statistics.Sample.KernelDensity.Simple.Points instance GHC.Show.Show Statistics.Sample.KernelDensity.Simple.Points instance GHC.Read.Read Statistics.Sample.KernelDensity.Simple.Points instance GHC.Classes.Eq Statistics.Sample.KernelDensity.Simple.Points instance Data.Aeson.Types.Class.FromJSON Statistics.Sample.KernelDensity.Simple.Points instance Data.Aeson.Types.Class.ToJSON Statistics.Sample.KernelDensity.Simple.Points instance Data.Binary.Class.Binary Statistics.Sample.KernelDensity.Simple.Points -- | Very fast statistics over simple powers of a sample. These can all be -- computed efficiently in just a single pass over a sample, with that -- pass subject to stream fusion. -- -- The tradeoff is that some of these functions are less numerically -- robust than their counterparts in the Sample module. Where this -- is the case, the alternatives are noted. module Statistics.Sample.Powers data Powers -- | O(n) Collect the n simple powers of a sample. -- -- Functions computed over a sample's simple powers require at least a -- certain number (or order) of powers to be collected. -- -- -- -- This function is subject to stream fusion. powers :: Vector v Double => Int -> v Double -> Powers -- | The order (number) of simple powers collected from a sample. order :: Powers -> Int -- | The number of elements in the original Sample. This is the -- sample's zeroth simple power. count :: Powers -> Int -- | The sum of elements in the original Sample. This is the -- sample's first simple power. sum :: Powers -> Double -- | The arithmetic mean of elements in the original Sample. -- -- This is less numerically robust than the mean function in the -- Sample module, but the number is essentially free to compute if -- you have already collected a sample's simple powers. mean :: Powers -> Double -- | Maximum likelihood estimate of a sample's variance. Also known as the -- population variance, where the denominator is n. This is the -- second central moment of the sample. -- -- This is less numerically robust than the variance function in the -- Sample module, but the number is essentially free to compute if -- you have already collected a sample's simple powers. -- -- Requires Powers with order at least 2. variance :: Powers -> Double -- | Standard deviation. This is simply the square root of the maximum -- likelihood estimate of the variance. stdDev :: Powers -> Double -- | Unbiased estimate of a sample's variance. Also known as the sample -- variance, where the denominator is n-1. -- -- Requires Powers with order at least 2. varianceUnbiased :: Powers -> Double -- | Compute the kth central moment of a sample. The central moment -- is also known as the moment about the mean. centralMoment :: Int -> Powers -> Double -- | Compute 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-skewed. Most of -- its mass is on the right of the distribution, with the tail on the -- left. -- -- 
--   skewness . 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 variance is zero. -- -- Requires Powers with order at least 3. skewness :: Powers -> Double -- | 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. -- -- A sample's excess kurtosis is not defined if its variance is -- zero. -- -- Requires Powers with order at least 4. kurtosis :: Powers -> Double instance GHC.Generics.Generic Statistics.Sample.Powers.Powers instance Data.Data.Data Statistics.Sample.Powers.Powers instance GHC.Show.Show Statistics.Sample.Powers.Powers instance GHC.Read.Read Statistics.Sample.Powers.Powers instance GHC.Classes.Eq Statistics.Sample.Powers.Powers instance Data.Aeson.Types.Class.FromJSON Statistics.Sample.Powers.Powers instance Data.Aeson.Types.Class.ToJSON Statistics.Sample.Powers.Powers instance Data.Binary.Class.Binary Statistics.Sample.Powers.Powers -- | Pearson's chi squared test. module Statistics.Test.ChiSquared -- | 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. chi2test :: (Vector v (Int, Double), Vector v Double) => Double -> Int -> v (Int, Double) -> TestResult -- | Test type. Exact meaning depends on a specific test. But generally -- it's tested whether some statistics is too big (small) for -- OneTailed or whether it too big or too small for -- TwoTailed data TestType OneTailed :: TestType TwoTailed :: TestType -- | Result of hypothesis testing data TestResult -- | Null hypothesis should be rejected Significant :: TestResult -- | Data is compatible with hypothesis NotSignificant :: TestResult -- | Kolmogov-Smirnov tests are non-parametric tests for assesing whether -- given sample could be described by distribution or whether two samples -- have the same distribution. It's only applicable to continous -- distributions. module Statistics.Test.KolmogorovSmirnov -- | Check that sample could be described by distribution. -- Significant means distribution is not compatible with data for -- given p-value. -- -- This test uses Marsaglia-Tsang-Wang exact alogorithm for calculation -- of p-value. kolmogorovSmirnovTest :: Distribution d => d -> Double -> Sample -> TestResult -- | Variant of kolmogorovSmirnovTest which uses CFD in form of -- function. kolmogorovSmirnovTestCdf :: (Double -> Double) -> Double -> Sample -> TestResult -- | Two sample Kolmogorov-Smirnov test. It tests whether two data samples -- could be described by the same distribution without making any -- assumptions about it. -- -- This test uses approxmate formula for computing p-value. kolmogorovSmirnovTest2 :: Double -> Sample -> Sample -> TestResult -- | Calculate Kolmogorov's statistic D for given cumulative -- distribution function (CDF) and data sample. If sample is empty -- returns 0. kolmogorovSmirnovCdfD :: (Double -> Double) -> Sample -> Double -- | Calculate Kolmogorov's statistic D for given cumulative -- distribution function (CDF) and data sample. If sample is empty -- returns 0. kolmogorovSmirnovD :: (Distribution d) => d -> Sample -> Double -- | Calculate Kolmogorov's statistic D for two data samples. If -- either of samples is empty returns 0. kolmogorovSmirnov2D :: Sample -> Sample -> Double -- | Calculate cumulative probability function for Kolmogorov's -- distribution with n parameters or probability of getting value -- smaller than d with n-elements sample. -- -- It uses algorithm by Marsgalia et. al. and provide at least 7-digit -- accuracy. kolmogorovSmirnovProbability :: Int -> Double -> Double -- | Test type. Exact meaning depends on a specific test. But generally -- it's tested whether some statistics is too big (small) for -- OneTailed or whether it too big or too small for -- TwoTailed data TestType OneTailed :: TestType TwoTailed :: TestType -- | Result of hypothesis testing data TestResult -- | Null hypothesis should be rejected Significant :: TestResult -- | Data is compatible with hypothesis NotSignificant :: TestResult module Statistics.Test.KruskalWallis -- | Kruskal-Wallis ranking. -- -- All 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). kruskalWallisRank :: [Sample] -> [Sample] -- | The Kruskal-Wallis Test. -- -- In textbooks the output value is usually represented by K or -- H. This function already does the ranking. kruskalWallis :: [Sample] -> Double -- | Calculates whether the Kruskal-Wallis test is significant. -- -- It uses Chi-Squared distribution for aproximation as long as -- the sizes are larger than 5. Otherwise the test returns -- Nothing. kruskalWallisSignificant :: [Int] -> Double -> Double -> Maybe TestResult -- | Perform Kruskal-Wallis Test for the given samples and required -- significance. For additional information check kruskalWallis. -- This is just a helper function. kruskalWallisTest :: Double -> [Sample] -> Maybe TestResult -- | Mann-Whitney U test (also know as Mann-Whitney-Wilcoxon and Wilcoxon -- rank sum test) is a non-parametric test for assesing whether two -- samples of independent observations have different mean. module Statistics.Test.MannWhitneyU -- | Perform Mann-Whitney U Test for two samples and required significance. -- For additional information check documentation of mannWhitneyU -- and mannWhitneyUSignificant. 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. mannWhitneyUtest :: TestType -> Double -> Sample -> Sample -> Maybe TestResult -- | The 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 -- wilcoxonRankSums). The Mann-Whitney U is a simple transform of -- Wilcoxon's rank sum test. -- -- Again confusingly, different sources state reversed definitions for U₁ -- and U₂, so it is worth being explicit about what this function -- returns. Given two samples, the first, xs₁, of size n₁ and the second, -- xs₂, of size n₂, this function returns (U₁, U₂) where U₁ = W₁ - -- (n₁(n₁+1))/2 and U₂ = W₂ - (n₂(n₂+1))/2, where (W₁, W₂) is the return -- value of wilcoxonRankSums xs1 xs2. -- -- Some sources instead state that U₁ and U₂ should be the other way -- round, often expressing this using U₁' = n₁n₂ - U₁ (since U₁ + U₂ = -- n₁n₂). -- -- All of which you probably don't care about if you just feed this into -- mannWhitneyUSignificant. mannWhitneyU :: Sample -> Sample -> (Double, Double) -- | Calculates 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" mannWhitneyUCriticalValue :: (Int, Int) -> Double -> Maybe Int -- | Calculates whether the Mann Whitney U test is significant. -- -- If both sample sizes are less than or equal to 20, the exact U -- critical value (as calculated by mannWhitneyUCriticalValue) 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 -- (U₁, U₂) pairs. mannWhitneyUSignificant :: TestType -> (Int, Int) -> Double -> (Double, Double) -> Maybe TestResult -- | The 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 (W₁, W₂) where W₁ is the sum of ranks of the first -- sample and W₂ 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 mannWhitneyU and the related functions for testing -- significance, but this function is exposed for completeness. wilcoxonRankSums :: Sample -> Sample -> (Double, Double) -- | Test type. Exact meaning depends on a specific test. But generally -- it's tested whether some statistics is too big (small) for -- OneTailed or whether it too big or too small for -- TwoTailed data TestType OneTailed :: TestType TwoTailed :: TestType -- | Result of hypothesis testing data TestResult -- | Null hypothesis should be rejected Significant :: TestResult -- | Data is compatible with hypothesis NotSignificant :: TestResult -- | The Wilcoxon matched-pairs signed-rank test is non-parametric test -- which could be used to whether two related samples have different -- means. -- -- WARNING: current implementation contain serious bug and couldn't be -- used with samples larger than 1023. -- https://github.com/bos/statistics/issues/18 module Statistics.Test.WilcoxonT -- | 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 differ -- -- Check wilcoxonMatchedPairSignedRank and -- wilcoxonMatchedPairSignificant for additional information. wilcoxonMatchedPairTest :: TestType -> Double -> Sample -> Sample -> Maybe TestResult wilcoxonMatchedPairSignedRank :: Sample -> Sample -> (Double, Double) -- | Tests whether a given result from a Wilcoxon signed-rank matched-pairs -- test is significant at the given level. -- -- This function can perform a one-tailed or two-tailed test. If the -- first parameter to this function is TwoTailed, the test is -- performed two-tailed to check if the two samples differ significantly. -- If the first parameter is OneTailed, the check is performed -- one-tailed to decide whether the first sample (i.e. the first sample -- you passed to wilcoxonMatchedPairSignedRank) is greater than -- the second sample (i.e. the second sample you passed to -- wilcoxonMatchedPairSignedRank). If you wish to perform a -- one-tailed test in the opposite direction, you can either pass the -- parameters in a different order to -- wilcoxonMatchedPairSignedRank, or simply swap the values in the -- resulting pair before passing them to this function. wilcoxonMatchedPairSignificant :: TestType -> Int -> Double -> (Double, Double) -> Maybe TestResult -- | 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. wilcoxonMatchedPairSignificance :: Int -> Double -> Double -- | Obtains 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 -- wilcoxonMatchedPairSignedRank 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 paper "Critical Values for the Wilcoxon Signed Rank -- Statistic", Peter Mitic, The Mathematica Journal, volume 6, issue 3, -- 1996, which can be found here: -- http://www.mathematica-journal.com/issue/v6i3/article/mitic/contents/63mitic.pdf. -- 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. wilcoxonMatchedPairCriticalValue :: Int -> Double -> Maybe Int -- | Test type. Exact meaning depends on a specific test. But generally -- it's tested whether some statistics is too big (small) for -- OneTailed or whether it too big or too small for -- TwoTailed data TestType OneTailed :: TestType TwoTailed :: TestType -- | Result of hypothesis testing data TestResult -- | Null hypothesis should be rejected Significant :: TestResult -- | Data is compatible with hypothesis NotSignificant :: TestResult -- | Functions for computing autocovariance and autocorrelation of a -- sample. module Statistics.Autocorrelation -- | Compute the autocovariance of a sample, i.e. the covariance of the -- sample against a shifted version of itself. autocovariance :: (Vector v Double, Vector v Int) => v Double -> v Double -- | Compute the autocorrelation function of a sample, and the upper and -- lower bounds of confidence intervals for each element. -- -- Note: The calculation of the 95% confidence interval assumes a -- stationary Gaussian process. autocorrelation :: (Vector v Double, Vector v Int) => v Double -> (v Double, v Double, v Double)