```-- |
-- Module    : Statistics.Autocorrelation
-- Copyright : (c) 2009 Bryan O'Sullivan
--
-- 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
```