Portability | Haskell 2011 + TypeFamilies |
---|---|

Stability | provisional |

Maintainer | Edward Kmett <ekmett@gmail.com> |

Safe Haskell | Safe-Infered |

- newtype L r = L {}
- (@@) :: (Num r, Ord r) => L r -> [r] -> r
- (@!) :: Num r => L r -> Vector r -> r
- (@#) :: Num r => L r -> Int -> [r]
- breakdown :: (Num r, Eq r) => L r -> Int
- trimean :: Fractional r => L r
- midhinge :: Fractional r => L r
- iqr :: Fractional r => L r
- iqm :: Fractional r => L r
- lscale :: Fractional r => L r
- trimmed :: Fractional r => Rational -> L r -> L r
- winsorized, winsorised :: Fractional r => Rational -> L r -> L r
- jackknifed :: Fractional r => L r -> L r
- mean :: Fractional r => L r
- total :: Num r => L r
- lmin :: Num r => L r
- lmax :: Num r => L r
- midrange :: Fractional r => L r
- nthSmallest :: Num r => Int -> L r
- nthLargest :: Num r => Int -> L r
- quantile :: Fractional r => Rational -> L r
- median :: Fractional r => L r
- tercile :: Fractional r => Rational -> L r
- t1, t2 :: Fractional r => L r
- quartile :: Fractional r => Rational -> L r
- q1, q3, q2 :: Fractional r => L r
- quintile :: Fractional r => Rational -> L r
- qu1, qu4, qu3, qu2 :: Fractional r => L r
- percentile :: Fractional r => Rational -> L r
- permille :: Fractional r => Rational -> L r
- hdquantile :: Fractional r => Rational -> L r
- quantileBy :: Num r => Estimator r -> Rational -> L r
- type Estimator r = Rational -> Int -> Estimate r
- data Estimate r = Estimate !Rational (IntMap r)
- r1 :: Num r => Estimator r
- r2 :: Fractional r => Estimator r
- r3 :: Num r => Estimator r
- r4 :: Fractional r => Estimator r
- r5 :: Fractional r => Estimator r
- r6 :: Fractional r => Estimator r
- r7 :: Fractional r => Estimator r
- r8 :: Fractional r => Estimator r
- r9 :: Fractional r => Estimator r
- r10 :: Fractional r => Estimator r

# L-Estimator

L-estimators are linear combinations of order statistics used by `robust`

statistics.

Num r => VectorSpace (L r) | |

Num r => AdditiveGroup (L r) |

## Applying an L-estimator

(@@) :: (Num r, Ord r) => L r -> [r] -> rSource

Calculate the result of applying an L-estimator after sorting list into order statistics

(@!) :: Num r => L r -> Vector r -> rSource

Calculate the result of applying an L-estimator to a *pre-sorted* vector of order statistics

## Analyzing an L-estimator

(@#) :: Num r => L r -> Int -> [r]Source

get a vector of the coefficients of an L estimator when applied to an input of a given length

## Robust L-Estimators

trimean :: Fractional r => L rSource

Tukey's trimean

breakdown trimean = 25

midhinge :: Fractional r => L rSource

midhinge = trimmed 0.25 midrange breakdown midhinge = 25%

iqr :: Fractional r => L rSource

interquartile range

breakdown iqr = 25% iqr = trimmed 0.25 midrange

iqm :: Fractional r => L rSource

interquartile mean

iqm = trimmed 0.25 mean

lscale :: Fractional r => L rSource

Direct estimator for the second L-moment given a sample

## L-Estimator Combinators

trimmed :: Fractional r => Rational -> L r -> L rSource

Calculate a trimmed L-estimator. If the sample size isn't evenly divided, linear interpolation is used as described in http://en.wikipedia.org/wiki/Trimmed_mean#Interpolation

winsorized, winsorised :: Fractional r => Rational -> L r -> L rSource

Calculates an interpolated winsorized L-estimator in a manner analogous to the trimmed estimator. Unlike trimming, winsorizing replaces the extreme values.

jackknifed :: Fractional r => L r -> L rSource

Jackknifes the statistic by removing each sample in turn and recalculating the L-estimator, requires at least 2 samples!

## Trivial L-Estimators

mean :: Fractional r => L rSource

The average of all of the order statistics. Not robust.

breakdown mean = 0%

midrange :: Fractional r => L rSource

midrange = lmax - lmin breakdown midrange = 0%

## Sample-size-dependent L-Estimators

nthSmallest :: Num r => Int -> L rSource

nthLargest :: Num r => Int -> L rSource

## Quantiles

### Common quantiles

quantile :: Fractional r => Rational -> L rSource

Compute a quantile with traditional direct averaging

median :: Fractional r => L rSource

The most robust L-estimator possible.

breakdown median = 50

tercile :: Fractional r => Rational -> L rSource

t1, t2 :: Fractional r => L rSource

terciles 1 and 2

breakdown t1 = breakdown t2 = 33%

quartile :: Fractional r => Rational -> L rSource

q1, q3, q2 :: Fractional r => L rSource

quantiles, with breakdown points 25%, 50%, and 25% respectively

quintile :: Fractional r => Rational -> L rSource

percentile :: Fractional r => Rational -> L rSource

breakdown (percentile n) = min n (100 - n)

permille :: Fractional r => Rational -> L rSource

### Harrell-Davis Quantile Estimator

hdquantile :: Fractional r => Rational -> L rSource

The Harrell-Davis quantile estimate. Uses multiple order statistics to approximate the quantile to reduce variance.

### Compute a quantile using a specified quantile estimation strategy

quantileBy :: Num r => Estimator r -> Rational -> L rSource

Compute a quantile using the given estimation strategy to interpolate when an exact quantile isn't available

# Sample Quantile Estimators

r2 :: Fractional r => Estimator rSource

.. with averaging at discontinuities

r3 :: Num r => Estimator rSource

The observation numbered closest to Np. NB: does not yield a proper median

r4 :: Fractional r => Estimator rSource

Linear interpolation of the empirical distribution function. NB: does not yield a proper median.

r5 :: Fractional r => Estimator rSource

.. with knots midway through the steps as used in hydrology. This is the simplest continuous estimator that yields a correct median

r6 :: Fractional r => Estimator rSource

Linear interpolation of the expectations of the order statistics for the uniform distribution on [0,1]

r7 :: Fractional r => Estimator rSource

Linear interpolation of the modes for the order statistics for the uniform distribution on [0,1]

r8 :: Fractional r => Estimator rSource

Linear interpolation of the approximate medans for order statistics.

r9 :: Fractional r => Estimator rSource

The resulting quantile estimates are approximately unbiased for the expected order statistics if x is normally distributed.

r10 :: Fractional r => Estimator rSource

When rounding h, this yields the order statistic with the least expected square deviation relative to p.