statistics-0.10.2.0: A library of statistical types, data, and functions

Portability | portable
Stability | experimental
Maintainer | bos@serpentine.com
Safe Haskell | None

Statistics.Quantile

Contents

- Quantile estimation functions
- Parameters for the continuous sample method
- References

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 ≤ k ≤ q.

Synopsis

- weightedAvg :: Vector v Double => Int -> Int -> v Double -> Double
- data ContParam = ContParam !Double !Double
- continuousBy :: Vector v Double => ContParam -> Int -> Int -> v Double -> Double
- midspread :: Vector v Double => ContParam -> Int -> v Double -> Double
- cadpw :: ContParam
- hazen :: ContParam
- s :: ContParam
- spss :: ContParam
- medianUnbiased :: ContParam
- normalUnbiased :: ContParam

# Quantile estimation functions

weightedAvg

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.

data ContParam

Parameters a and b to the `continuousBy` function.

Constructors

ContParam !Double !Double |

continuousBy

Arguments

:: Vector v Double |
=> ContParam | Parameters a and b.
-> 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. This is the method used by most statistical software, such as R, Mathematica, SPSS, and S.

midspread

Arguments

:: Vector v Double |
=> ContParam | Parameters a and b.
-> 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

# Parameters for the continuous sample method

cadpw :: ContParam

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.

hazen :: 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.

s :: 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.

spss :: 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.

medianUnbiased :: 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.

normalUnbiased :: 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.

# References

- Weisstein, E.W. Quantile. MathWorld. http://mathworld.wolfram.com/Quantile.html
- Hyndman, R.J.; Fan, Y. (1996) Sample quantiles in statistical packages. American Statistician 50(4):361–365. http://www.jstor.org/stable/2684934