| Portability | portable |
|---|---|
| Stability | experimental |
| Maintainer | bos@serpentine.com |
Statistics.Math
Description
Mathematical functions for statistics.
- choose :: Int -> Int -> Double
- logBeta :: Double -> Double -> Double
- chebyshev :: Vector v Double => Double -> v Double -> Double
- chebyshevBroucke :: Vector v Double => Double -> v Double -> Double
- factorial :: Int -> Double
- logFactorial :: Int -> Double
- incompleteGamma :: Double -> Double -> Double
- logGamma :: Double -> Double
- logGammaL :: Double -> Double
- log1p :: Double -> Double
Functions
choose :: Int -> Int -> DoubleSource
Compute the binomial coefficient n ` k. For
values of k > 30, this uses an approximation for performance
reasons. The approximation is accurate to 12 decimal places in the
worst case
choose`
Example:
7 `choose` 3 == 35
Beta function
Chebyshev polynomials
A Chebyshev polynomial of the first kind is defined by the following recurrence:
t 0 _ = 1 t 1 x = x t n x = 2 * x * t (n-1) x - t (n-2) x
Arguments
| :: Vector v Double | |
| => Double | Parameter of each function. |
| -> v Double | Coefficients of each polynomial term, in increasing order. |
| -> Double |
Evaluate a Chebyshev polynomial of the first kind. Uses Clenshaw's algorithm.
Arguments
| :: Vector v Double | |
| => Double | Parameter of each function. |
| -> v Double | Coefficients of each polynomial term, in increasing order. |
| -> Double |
Evaluate a Chebyshev polynomial of the first kind. Uses Broucke's
ECHEB algorithm, and his convention for coefficient handling, and so
gives different results than chebyshev for the same inputs.
Factorial
factorial :: Int -> DoubleSource
Compute the factorial function n!. Returns ∞ if the
input is above 170 (above which the result cannot be represented by
a 64-bit Double).
logFactorial :: Int -> DoubleSource
Compute the natural logarithm of the factorial function. Gives 16 decimal digits of precision.
Gamma function
Compute the normalized lower incomplete gamma function γ(s,x). Normalization means that γ(s,∞)=1. Uses Algorithm AS 239 by Shea.
logGamma :: Double -> DoubleSource
Compute the logarithm of the gamma function Γ(x). Uses Algorithm AS 245 by Macleod.
Gives an accuracy of 10–12 significant decimal digits, except
for small regions around x = 1 and x = 2, where the function
goes to zero. For greater accuracy, use logGammaL.
Returns ∞ if the input is outside of the range (0 < x ≤ 1e305).
logGammaL :: Double -> DoubleSource
Compute the logarithm of the gamma function, Γ(x). Uses a Lanczos approximation.
This function is slower than logGamma, but gives 14 or more
significant decimal digits of accuracy, except around x = 1 and
x = 2, where the function goes to zero.
Returns ∞ if the input is outside of the range (0 < x ≤ 1e305).
Logarithm
log1p :: Double -> DoubleSource
Compute the natural logarithm of 1 + x. This is accurate even
for values of x near zero, where use of log(1+x) would lose
precision.
References
- Broucke, R. (1973) Algorithm 446: Ten subroutines for the manipulation of Chebyshev series. Communications of the ACM 16(4):254–256. http://doi.acm.org/10.1145/362003.362037
- Clenshaw, C.W. (1962) Chebyshev series for mathematical functions. National Physical Laboratory Mathematical Tables 5, Her Majesty's Stationery Office, London.
- Lanczos, C. (1964) A precision approximation of the gamma function. SIAM Journal on Numerical Analysis B 1:86–96. http://www.jstor.org/stable/2949767
- Macleod, A.J. (1989) Algorithm AS 245: A robust and reliable algorithm for the logarithm of the gamma function. Journal of the Royal Statistical Society, Series C (Applied Statistics) 38(2):397–402. http://www.jstor.org/stable/2348078
- Shea, B. (1988) Algorithm AS 239: Chi-squared and incomplete gamma integral. Applied Statistics 37(3):466–473. http://www.jstor.org/stable/2347328