statistics-0.16.2.1: A library of statistical types, data, and functions
Copyright(c) 2009 Bryan O'Sullivan
LicenseBSD3
Maintainerbos@serpentine.com
Stabilityexperimental
Portabilityportable
Safe HaskellSafe-Inferred
LanguageHaskell2010

Statistics.Quantile

Description

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 ≤ kq.

Synopsis

Quantile estimation functions

Below is family of functions which use same algorithm for estimation of sample quantiles. It approximates empirical CDF as continuous piecewise function which interpolates linearly between points \((X_k,p_k)\) where \(X_k\) is k-th order statistics (k-th smallest element) and \(p_k\) is probability corresponding to it. ContParam determines how \(p_k\) is chosen. For more detailed explanation see [Hyndman1996].

This is the method used by most statistical software, such as R, Mathematica, SPSS, and S.

data ContParam Source #

Parameters α and β to the continuousBy function. Exact meaning of parameters is described in [Hyndman1996] in section "Piecewise linear functions"

Constructors

ContParam !Double !Double 

Instances

Instances details
FromJSON ContParam Source # 
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Defined in Statistics.Quantile

ToJSON ContParam Source # 
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Data ContParam Source # 
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Methods

gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> ContParam -> c ContParam #

gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c ContParam #

toConstr :: ContParam -> Constr #

dataTypeOf :: ContParam -> DataType #

dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c ContParam) #

dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c ContParam) #

gmapT :: (forall b. Data b => b -> b) -> ContParam -> ContParam #

gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> ContParam -> r #

gmapQr :: forall r r'. (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> ContParam -> r #

gmapQ :: (forall d. Data d => d -> u) -> ContParam -> [u] #

gmapQi :: Int -> (forall d. Data d => d -> u) -> ContParam -> u #

gmapM :: Monad m => (forall d. Data d => d -> m d) -> ContParam -> m ContParam #

gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> ContParam -> m ContParam #

gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> ContParam -> m ContParam #

Generic ContParam Source # 
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Defined in Statistics.Quantile

Associated Types

type Rep ContParam :: Type -> Type #

Show ContParam Source # 
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Binary ContParam Source # 
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Default ContParam Source #

We use s as default value which is same as R's default.

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Methods

def :: ContParam #

Eq ContParam Source # 
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Ord ContParam Source # 
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type Rep ContParam Source # 
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type Rep ContParam = D1 ('MetaData "ContParam" "Statistics.Quantile" "statistics-0.16.2.1-34qObKIlIVGJpKYv9daW0Y" 'False) (C1 ('MetaCons "ContParam" 'PrefixI 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'SourceUnpack 'SourceStrict 'DecidedStrict) (Rec0 Double) :*: S1 ('MetaSel ('Nothing :: Maybe Symbol) 'SourceUnpack 'SourceStrict 'DecidedStrict) (Rec0 Double)))

class Default a where #

A class for types with a default value.

Minimal complete definition

Nothing

Methods

def :: a #

The default value for this type.

Instances

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Default All 
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def :: All #

Default Any 
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def :: Any #

Default CClock 
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def :: CClock #

Default CDouble 
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def :: CDouble #

Default CFloat 
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def :: CFloat #

Default CInt 
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def :: CInt #

Default CIntMax 
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def :: CIntMax #

Default CIntPtr 
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def :: CIntPtr #

Default CLLong 
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def :: CLLong #

Default CLong 
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def :: CLong #

Default CPtrdiff 
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def :: CPtrdiff #

Default CSUSeconds 
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def :: CSUSeconds #

Default CShort 
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def :: CShort #

Default CSigAtomic 
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def :: CSigAtomic #

Default CSize 
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def :: CSize #

Default CTime 
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def :: CTime #

Default CUInt 
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def :: CUInt #

Default CUIntMax 
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def :: CUIntMax #

Default CUIntPtr 
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def :: CUIntPtr #

Default CULLong 
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def :: CULLong #

Default CULong 
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def :: CULong #

Default CUSeconds 
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def :: CUSeconds #

Default CUShort 
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def :: CUShort #

Default Int16 
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def :: Int16 #

Default Int32 
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def :: Int32 #

Default Int64 
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def :: Int64 #

Default Int8 
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def :: Int8 #

Default Word16 
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def :: Word16 #

Default Word32 
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def :: Word32 #

Default Word64 
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def :: Word64 #

Default Word8 
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def :: Word8 #

Default Ordering 
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def :: Ordering #

Default NewtonParam 
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Defined in Numeric.RootFinding

Methods

def :: NewtonParam #

Default RiddersParam 
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Defined in Numeric.RootFinding

Methods

def :: RiddersParam #

Default ContParam Source #

We use s as default value which is same as R's default.

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Defined in Statistics.Quantile

Methods

def :: ContParam #

Default Integer 
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def :: Integer #

Default () 
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def :: () #

Default Double 
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def :: Double #

Default Float 
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def :: Float #

Default Int 
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def :: Int #

Default Word 
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def :: Word #

(Default a, RealFloat a) => Default (Complex a) 
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def :: Complex a #

Default (First a) 
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def :: First a #

Default (Last a) 
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def :: Last a #

Default a => Default (Dual a) 
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def :: Dual a #

Default (Endo a) 
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def :: Endo a #

Num a => Default (Product a) 
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def :: Product a #

Num a => Default (Sum a) 
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def :: Sum a #

Integral a => Default (Ratio a) 
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def :: Ratio a #

Default a => Default (IO a) 
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def :: IO a #

Default (Maybe a) 
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def :: Maybe a #

Default [a] 
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def :: [a] #

Default r => Default (e -> r) 
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def :: e -> r #

(Default a, Default b) => Default (a, b) 
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def :: (a, b) #

(Default a, Default b, Default c) => Default (a, b, c) 
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def :: (a, b, c) #

(Default a, Default b, Default c, Default d) => Default (a, b, c, d) 
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Methods

def :: (a, b, c, d) #

(Default a, Default b, Default c, Default d, Default e) => Default (a, b, c, d, e) 
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Defined in Data.Default.Class

Methods

def :: (a, b, c, d, e) #

(Default a, Default b, Default c, Default d, Default e, Default f) => Default (a, b, c, d, e, f) 
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Defined in Data.Default.Class

Methods

def :: (a, b, c, d, e, f) #

(Default a, Default b, Default c, Default d, Default e, Default f, Default g) => Default (a, b, c, d, e, f, g) 
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Defined in Data.Default.Class

Methods

def :: (a, b, c, d, e, f, g) #

quantile Source #

Arguments

:: Vector v Double 
=> ContParam

Parameters α and β.

-> Int

k, the desired quantile.

-> Int

q, the number of quantiles.

-> v Double

x, the sample data.

-> Double 

O(n·log n). Estimate the kth q-quantile of a sample x, using the continuous sample method with the given parameters.

The following properties should hold, otherwise an error will be thrown.

  • input sample must be nonempty
  • the input does not contain NaN
  • 0 ≤ k ≤ q

quantiles :: (Vector v Double, Foldable f, Functor f) => ContParam -> f Int -> Int -> v Double -> f Double Source #

O(k·n·log n). Estimate set of the kth q-quantile of a sample x, using the continuous sample method with the given parameters. This is faster than calling quantile repeatedly since sample should be sorted only once

The following properties should hold, otherwise an error will be thrown.

  • input sample must be nonempty
  • the input does not contain NaN
  • for every k in set of quantiles 0 ≤ k ≤ q

quantilesVec :: (Vector v Double, Vector v Int) => ContParam -> v Int -> Int -> v Double -> v Double Source #

O(k·n·log n). Same as quantiles but uses Vector container instead of Foldable one.

Parameters for the continuous sample method

cadpw :: ContParam Source #

California Department of Public Works definition, α=0, β=1. Gives a linear interpolation of the empirical CDF. This corresponds to method 4 in R and Mathematica.

hazen :: ContParam Source #

Hazen's definition, α=0.5, β=0.5. This is claimed to be popular among hydrologists. This corresponds to method 5 in R and Mathematica.

spss :: ContParam Source #

Definition used by the SPSS statistics application, with α=0, β=0 (also known as Weibull's definition). This corresponds to method 6 in R and Mathematica.

s :: ContParam Source #

Definition used by the S statistics application, with α=1, β=1. The interpolation points divide the sample range into n-1 intervals. This corresponds to method 7 in R and Mathematica and is default in R.

medianUnbiased :: ContParam Source #

Median unbiased definition, α=1/3, β=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.

normalUnbiased :: ContParam Source #

Normal unbiased definition, α=3/8, β=3/8. An approximately unbiased estimate if the empirical distribution approximates the normal distribution. This corresponds to method 9 in R and Mathematica.

Other algorithms

weightedAvg Source #

Arguments

:: Vector v Double 
=> Int

k, the desired quantile.

-> Int

q, the number of quantiles.

-> v Double

x, the sample data.

-> Double 

O(n·log n). Estimate the kth q-quantile of a sample, using the weighted average method. Up to rounding errors it's same as quantile s.

The following properties should hold otherwise an error will be thrown.

  • the length of the input is greater than 0
  • the input does not contain NaN
  • k ≥ 0 and k ≤ q

Median & other specializations

median Source #

Arguments

:: Vector v Double 
=> ContParam

Parameters α and β.

-> v Double

x, the sample data.

-> Double 

O(n·log n) Estimate median of sample

mad Source #

Arguments

:: Vector v Double 
=> ContParam

Parameters α and β.

-> v Double

x, the sample data.

-> Double 

O(n·log n). Estimate the median absolute deviation (MAD) of a sample x using continuousBy. It's robust estimate of variability in sample and defined as:

\[ MAD = \operatorname{median}(| X_i - \operatorname{median}(X) |) \]

midspread Source #

Arguments

:: Vector v Double 
=> ContParam

Parameters α and β.

-> Int

q, the number of quantiles.

-> v Double

x, the sample data.

-> 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

Deprecated

continuousBy Source #

Arguments

:: Vector v Double 
=> ContParam

Parameters α and β.

-> Int

k, the desired quantile.

-> Int

q, the number of quantiles.

-> v Double

x, the sample data.

-> Double 

Deprecated: Use quantile instead

References