statistics-0.10.0.0: A library of statistical types, data, and functions

Portabilityportable
Stabilityexperimental
Maintainerbos@serpentine.com

Statistics.Sample.KernelDensity.Simple

Contents

Description

Kernel density estimation code, providing non-parametric ways to estimate the probability density function of a sample.

The techniques used by functions in this module are relatively fast, but they generally give inferior results to the KDE function in the main Statistics.KernelDensity module (due to the oversmoothing documented for bandwidth below).

Synopsis

Simple entry points

epanechnikovPDFSource

Arguments

:: Vector v Double 
=> Int

Number of points at which to estimate

-> v Double

Data sample

-> (Points, Vector Double) 

Simple Epanechnikov kernel density estimator. Returns the uniformly spaced points from the sample range at which the density function was estimated, and the estimates at those points.

gaussianPDFSource

Arguments

:: Vector v Double 
=> Int

Number of points at which to estimate

-> v Double

Data sample

-> (Points, Vector Double) 

Simple Gaussian kernel density estimator. Returns the uniformly spaced points from the sample range at which the density function was estimated, and the estimates at those points.

Building blocks

Choosing points from a sample

newtype Points Source

Points from the range of a Sample.

Constructors

Points 

Instances

choosePointsSource

Arguments

:: Vector v Double 
=> Int

Number of points to select, n

-> Double

Sample bandwidth, h

-> v Double

Input data

-> Points 

Choose a uniform range of points at which to estimate a sample's probability density function.

If you are using a Gaussian kernel, multiply the sample's bandwidth by 3 before passing it to this function.

If this function is passed an empty vector, it returns values of positive and negative infinity.

Bandwidth estimation

type Bandwidth = DoubleSource

The width of the convolution kernel used.

bandwidth :: Vector v Double => (Double -> Bandwidth) -> v Double -> BandwidthSource

Compute the optimal bandwidth from the observed data for the given kernel.

This function uses an estimate based on the standard deviation of a sample (due to Deheuvels), which performs reasonably well for unimodal distributions but leads to oversmoothing for more complex ones.

epanechnikovBW :: Double -> BandwidthSource

Bandwidth estimator for an Epanechnikov kernel.

gaussianBW :: Double -> BandwidthSource

Bandwidth estimator for a Gaussian kernel.

Kernels

type Kernel = Double -> Double -> Double -> Double -> DoubleSource

The convolution kernel. Its parameters are as follows:

  • Scaling factor, 1/nh
  • Bandwidth, h
  • A point at which to sample the input, p
  • One sample value, v

epanechnikovKernel :: KernelSource

Epanechnikov kernel for probability density function estimation.

gaussianKernel :: KernelSource

Gaussian kernel for probability density function estimation.

Low-level estimation

estimatePDFSource

Arguments

:: Vector v Double 
=> Kernel

Kernel function

-> Bandwidth

Bandwidth, h

-> v Double

Sample data

-> Points

Points at which to estimate

-> Vector Double 

Kernel density estimator, providing a non-parametric way of estimating the PDF of a random variable.

simplePDFSource

Arguments

:: Vector v Double 
=> (Double -> Double)

Bandwidth function

-> Kernel

Kernel function

-> Double

Bandwidth scaling factor (3 for a Gaussian kernel, 1 for all others)

-> Int

Number of points at which to estimate

-> v Double

sample data

-> (Points, Vector Double) 

A helper for creating a simple kernel density estimation function with automatically chosen bandwidth and estimation points.

References

  • Deheuvels, P. (1977) Estimation non paramtrique de la densit par histogrammes gnraliss. Mhttp:archive.numdam.orgarticleRSA_1977__25_3_5_0.pdf>