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-- |
-- 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 Statistics.Sample (Sample, mean)
import qualified Data.Vector.Unboxed as U
-- | Compute the autocovariance of a sample, i.e. the covariance of
-- the sample against a shifted version of itself.
autocovariance :: Sample -> U.Vector Double
autocovariance a = U.map f . U.enumFromTo 0 $ l-2
where
f k = U.sum (U.zipWith (*) (U.take (l-k) c) (U.slice k (l-k) c))
/ fromIntegral l
c = U.map (subtract (mean a)) a
l = U.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 :: Sample -> (U.Vector Double, U.Vector Double, U.Vector Double)
autocorrelation a = (r, ci (-), ci (+))
where
r = U.map (/ U.head c) c
where c = autocovariance a
dllse = U.map f . U.scanl1 (+) . U.map square $ r
where f v = 1.96 * sqrt ((v * 2 + 1) / l)
l = fromIntegral (U.length a)
ci f = U.cons 1 . U.tail . U.map (f (-1/l)) $ dllse
square x = x * x
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