| Copyright | (c) 2009 2010 Bryan O'Sullivan |
|---|---|
| License | BSD3 |
| Maintainer | bos@serpentine.com |
| Stability | experimental |
| Portability | portable |
| Safe Haskell | None |
| Language | Haskell98 |
Statistics.Sample.Powers
Contents
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
Instances
| Eq Powers # | |
| Data Powers # | |
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 # gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c Powers # toConstr :: Powers -> Constr # dataTypeOf :: Powers -> DataType # dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c Powers) # dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c Powers) # gmapT :: (forall b. Data b => b -> b) -> Powers -> Powers # gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> Powers -> r # gmapQr :: (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> Powers -> r # gmapQ :: (forall d. Data d => d -> u) -> Powers -> [u] # gmapQi :: Int -> (forall d. Data d => d -> u) -> Powers -> u # gmapM :: Monad m => (forall d. Data d => d -> m d) -> Powers -> m Powers # gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> Powers -> m Powers # gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> Powers -> m Powers # | |
| Read Powers # | |
| Show Powers # | |
| Generic Powers # | |
| ToJSON Powers # | |
Defined in Statistics.Sample.Powers | |
| FromJSON Powers # | |
| Binary Powers # | |
| type Rep Powers # | |
Defined in Statistics.Sample.Powers | |
Constructor
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 kth
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
The number of elements in the original Sample. This is the
sample's zeroth simple power.
The sum of elements in the original Sample. This is the
sample's first simple power.
Statistics of location
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 #
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.
Standard deviation. This is simply the square root of the maximum likelihood estimate of the variance.
varianceUnbiased :: Powers -> Double #
Functions over central moments
centralMoment :: Int -> Powers -> Double #
Compute the kth central moment of a sample. The central moment is also known as the moment about the mean.
skewness :: 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.
kurtosis :: 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.
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 kth central moments in one pass. quantblog. http://quantblog.wordpress.com/2009/02/07/compute-kth-central-moments-in-one-pass/