Copyright	(c) 2009 2010 Bryan O'Sullivan
License	BSD3
Maintainer	bos@serpentine.com
Stability	experimental
Portability	portable
Safe Haskell	None
Language	Haskell2010

Statistics.Sample.Powers

Contents

Types
Constructor
Descriptive functions
Statistics of location
Statistics of dispersion
Functions over central moments
References

Description

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.

Synopsis

data Powers
powers :: Vector v Double => Int -> v Double -> Powers
order :: Powers -> Int
count :: Powers -> Int
sum :: Powers -> Double
mean :: Powers -> Double
variance :: Powers -> Double
stdDev :: Powers -> Double
varianceUnbiased :: Powers -> Double
centralMoment :: Int -> Powers -> Double
skewness :: Powers -> Double
kurtosis :: Powers -> Double

Types

data Powers Source #

Instances

Instances details

Eq Powers Source #
Instance details Defined in Statistics.Sample.Powers Methods (==) :: Powers -> Powers -> Bool Source # (/=) :: Powers -> Powers -> Bool Source #
Data Powers Source #
Instance details Defined in Statistics.Sample.Powers Methods gfoldl :: ( forall d b. Data d => c (d -> b) -> d -> c b) -> ( forall g. g -> c g) -> Powers -> c Powers Source # gunfold :: ( forall b r. Data b => c (b -> r) -> c r) -> ( forall r. r -> c r) -> Constr -> c Powers Source # toConstr :: Powers -> Constr Source # dataTypeOf :: Powers -> DataType Source # dataCast1 :: Typeable t => ( forall d. Data d => c (t d)) -> Maybe (c Powers ) Source # dataCast2 :: Typeable t => ( forall d e. ( Data d, Data e) => c (t d e)) -> Maybe (c Powers ) Source # gmapT :: ( forall b. Data b => b -> b) -> Powers -> Powers Source # gmapQl :: (r -> r' -> r) -> r -> ( forall d. Data d => d -> r') -> Powers -> r Source # gmapQr :: forall r r'. (r' -> r -> r) -> r -> ( forall d. Data d => d -> r') -> Powers -> r Source # gmapQ :: ( forall d. Data d => d -> u) -> Powers -> [u] Source # gmapQi :: Int -> ( forall d. Data d => d -> u) -> Powers -> u Source # gmapM :: Monad m => ( forall d. Data d => d -> m d) -> Powers -> m Powers Source # gmapMp :: MonadPlus m => ( forall d. Data d => d -> m d) -> Powers -> m Powers Source # gmapMo :: MonadPlus m => ( forall d. Data d => d -> m d) -> Powers -> m Powers Source #
Read Powers Source #
Instance details Defined in Statistics.Sample.Powers Methods readsPrec :: Int -> ReadS Powers Source # readList :: ReadS [ Powers ] Source # readPrec :: ReadPrec Powers Source # readListPrec :: ReadPrec [ Powers ] Source #
Show Powers Source #
Instance details Defined in Statistics.Sample.Powers Methods showsPrec :: Int -> Powers -> ShowS Source # show :: Powers -> String Source # showList :: [ Powers ] -> ShowS Source #
Generic Powers Source #
Instance details Defined in Statistics.Sample.Powers Associated Types type Rep Powers :: Type -> Type Source # Methods from :: Powers -> Rep Powers x Source # to :: Rep Powers x -> Powers Source #
ToJSON Powers Source #
Instance details Defined in Statistics.Sample.Powers Methods toJSON :: Powers -> Value Source # toEncoding :: Powers -> Encoding Source # toJSONList :: [ Powers ] -> Value Source # toEncodingList :: [ Powers ] -> Encoding Source #
FromJSON Powers Source #
Instance details Defined in Statistics.Sample.Powers Methods parseJSON :: Value -> Parser Powers Source # parseJSONList :: Value -> Parser [ Powers ] Source #
Binary Powers Source #
Instance details Defined in Statistics.Sample.Powers Methods put :: Powers -> Put Source # get :: Get Powers Source # putList :: [ Powers ] -> Put Source #
type Rep Powers Source #
Instance details Defined in Statistics.Sample.Powers type Rep Powers = D1 (' MetaData "Powers" "Statistics.Sample.Powers" "statistics-0.16.1.2-IkOne9g3oJ1vhHVSRLPUO" ' True ) ( C1 (' MetaCons "Powers" ' PrefixI ' False ) ( S1 (' MetaSel (' Nothing :: Maybe Symbol ) ' NoSourceUnpackedness ' NoSourceStrictness ' DecidedLazy ) ( Rec0 ( Vector Double ))))

Constructor

powers Source #

Arguments

:: Vector v Double
=> Int	n , the number of powers, where n >= 2.
-> v Double
-> 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.

To compute the k th centralMoment , at least k simple powers must be collected.
For the variance , at least 2 simple powers are needed.
For skewness , we need at least 3 simple powers.
For kurtosis , at least 4 simple powers are required.

This function is subject to stream fusion.

Descriptive functions

order :: Powers -> Int Source #

The order (number) of simple powers collected from a sample .

count :: Powers -> Int Source #

The number of elements in the original Sample . This is the sample's zeroth simple power.

sum :: Powers -> Double Source #

The sum of elements in the original Sample . This is the sample's first simple power.

Statistics of location

mean :: Powers -> Double Source #

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.

Statistics of dispersion

variance :: Powers -> Double Source #

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.

stdDev :: Powers -> Double Source #

Standard deviation. This is simply the square root of the maximum likelihood estimate of the variance.

varianceUnbiased :: Powers -> Double Source #

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.

Functions over central moments

centralMoment :: Int -> Powers -> Double Source #

Compute the k th central moment of a sample. The central moment is also known as the moment about the mean.

skewness :: Powers -> Double Source #

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.

kurtosis :: Powers -> Double Source #

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.

References

Besset, D.H. (2000) Elements of statistics. Object-oriented implementation of numerical methods ch. 9, pp. 311–331. http://www.elsevier.com/wps/product/cws_home/677916
Anderson, G. (2009) Compute k th central moments in one pass. quantblog . http://quantblog.wordpress.com/2009/02/07/compute-kth-central-moments-in-one-pass/