{-# LANGUAGE FlexibleContexts #-} -- | -- Module : Statistics.Autocorrelation -- Copyright : (c) 2009 Bryan O'Sullivan -- License : BSD3 -- -- Maintainer : bos@serpentine.com -- Stability : experimental -- Portability : portable -- -- Functions for computing autocovariance and autocorrelation of a -- sample. module Statistics.Autocorrelation ( autocovariance , autocorrelation ) where import Prelude hiding (sum) import Statistics.Function (square) import Statistics.Sample (mean) import Statistics.Sample.Internal (sum) import qualified Data.Vector.Generic as G -- | Compute the autocovariance of a sample, i.e. the covariance of -- the sample against a shifted version of itself. autocovariance :: (G.Vector v Double, G.Vector v Int) => v Double -> v Double autocovariance a = G.map f . G.enumFromTo 0 \$ l-2 where f k = sum (G.zipWith (*) (G.take (l-k) c) (G.slice k (l-k) c)) / fromIntegral l c = G.map (subtract (mean a)) a l = G.length a -- | Compute the autocorrelation function of a sample, and the upper -- and lower bounds of confidence intervals for each element. -- -- /Note/: The calculation of the 95% confidence interval assumes a -- stationary Gaussian process. autocorrelation :: (G.Vector v Double, G.Vector v Int) => v Double -> (v Double, v Double, v Double) autocorrelation a = (r, ci (-), ci (+)) where r = G.map (/ G.head c) c where c = autocovariance a dllse = G.map f . G.scanl1 (+) . G.map square \$ r where f v = 1.96 * sqrt ((v * 2 + 1) / l) l = fromIntegral (G.length a) ci f = G.cons 1 . G.tail . G.map (f (-1/l)) \$ dllse