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{-# LANGUAGE BangPatterns, FlexibleContexts, UnboxedTuples #-}
-- |
-- Module : Statistics.Sample.KernelDensity
-- Copyright : (c) 2011 Bryan O'Sullivan
-- License : BSD3
--
-- Maintainer : bos@serpentine.com
-- Stability : experimental
-- Portability : portable
--
-- Kernel density estimation. This module provides a fast, robust,
-- non-parametric way to estimate the probability density function of
-- a sample.
--
-- This estimator does not use the commonly employed \"Gaussian rule
-- of thumb\". As a result, it outperforms many plug-in methods on
-- multimodal samples with widely separated modes.
module Statistics.Sample.KernelDensity
(
-- * Estimation functions
kde
, kde_
-- * References
-- $references
) where
import Prelude hiding (const,min,max)
import Numeric.MathFunctions.Constants (m_sqrt_2_pi)
import Statistics.Function (minMax, nextHighestPowerOfTwo)
import Statistics.Math.RootFinding (fromRoot, ridders)
import Statistics.Sample.Histogram (histogram_)
import Statistics.Transform (dct, idct)
import qualified Data.Vector.Generic as G
import qualified Data.Vector.Unboxed as U
-- | Gaussian kernel density estimator for one-dimensional data, using
-- the method of Botev et al.
--
-- The result is a pair of vectors, containing:
--
-- * The coordinates of each mesh point. The mesh interval is chosen
-- to be 20% larger than the range of the sample. (To specify the
-- mesh interval, use 'kde_'.)
--
-- * Density estimates at each mesh point.
kde :: Int
-- ^ The number of mesh points to use in the uniform discretization
-- of the interval @(min,max)@. If this value is not a power of
-- two, then it is rounded up to the next power of two.
-> U.Vector Double -> (U.Vector Double, U.Vector Double)
kde n0 xs = kde_ n0 (lo - range / 10) (hi + range / 10) xs
where
(lo,hi) = minMax xs
range = hi - lo
-- | Gaussian kernel density estimator for one-dimensional data, using
-- the method of Botev et al.
--
-- The result is a pair of vectors, containing:
--
-- * The coordinates of each mesh point.
--
-- * Density estimates at each mesh point.
kde_ :: Int
-- ^ The number of mesh points to use in the uniform discretization
-- of the interval @(min,max)@. If this value is not a power of
-- two, then it is rounded up to the next power of two.
-> Double
-- ^ Lower bound (@min@) of the mesh range.
-> Double
-- ^ Upper bound (@max@) of the mesh range.
-> U.Vector Double -> (U.Vector Double, U.Vector Double)
kde_ n0 min max xs
| n0 < 1 = error "Statistics.KernelDensity.kde: invalid number of points"
| otherwise = (mesh, density)
where
mesh = G.generate ni $ \z -> min + (d * fromIntegral z)
where d = r / (n-1)
density = G.map (/r) . idct $ G.zipWith f a (G.enumFromTo 0 (n-1))
where f b z = b * exp (sqr z * sqr pi * t_star * (-0.5))
!n = fromIntegral ni
!ni = nextHighestPowerOfTwo n0
!r = max - min
a = dct . G.map (/ G.sum h) $ h
where h = G.map (/ len) $ histogram_ ni min max xs
!len = fromIntegral (G.length xs)
!t_star = fromRoot (0.28 * len ** (-0.4)) . ridders 1e-14 (0,0.1) $ \x ->
x - (len * (2 * sqrt pi) * go 6 (f 7 x)) ** (-0.4)
where
f q t = 2 * pi ** (q*2) * G.sum (G.zipWith g iv a2v)
where g i a2 = i ** q * a2 * exp ((-i) * sqr pi * t)
a2v = G.map (sqr . (*0.5)) $ G.tail a
iv = G.map sqr $ G.enumFromTo 1 (n-1)
go s !h | s == 1 = h
| otherwise = go (s-1) (f s time)
where time = (2 * const * k0 / len / h) ** (2 / (3 + 2 * s))
const = (1 + 0.5 ** (s+0.5)) / 3
k0 = U.product (G.enumFromThenTo 1 3 (2*s-1)) / m_sqrt_2_pi
_bandwidth = sqrt t_star * r
sqr x = x * x
-- $references
--
-- Botev. Z.I., Grotowski J.F., Kroese D.P. (2010). Kernel density
-- estimation via diffusion. /Annals of Statistics/
-- 38(5):2916–2957. <http://arxiv.org/pdf/1011.2602>
|
