R: Calculate empirical scaling function coefficients of a p.d.f.

denproj {wavethresh} | R Documentation

## Calculate empirical scaling function coefficients of a p.d.f.

### Description

Calculates empirical scaling function coefficients of the probability density function from sample of data from that density, usually at some "high" resoloution.

### Usage

    denproj(x, tau=1, J, filter.number=10, family="DaubLeAsymm", covar=FALSE, nT=20)

### Arguments

`x` | Vector containing the data. This can be of any length.
`J` | The resolution level at which the empirical scaling function coefficients are to be calculated.
`tau` | This parameter allows non-dyadic resolutions to be used, since the resolution is specified as `tau * 2J`.
`filter.number` | The filter number of the wavelet basis to be used.
`family` | The family of wavelets to use, can be "DaubExPhase" or "DaubLeAsymm".
`covar` | Logical variable. If TRUE then covariances of the empirical scaling function coefficients are also calculated.
`nT` | The number of iterations to be performed in the Daubechies-Lagarias algorithm, which is used to evaluate the scaling functions of the specified wavelet basis at the data points.

### Details

This projection of data onto a high resolution wavelet space is described in detail in Chapter 3 of Herrick (2000). The maximum and minimum values of `k` for which the empirical scaling function coefficient is non-zero are determined and the coefficients calculated for all k between these limits as `sum(phiJk(xi))/n`. The scaling functions are evaluated at the data points efficiently, using the Daubechies-Lagarias algorithm (Daubechies & Lagarias (1992)). Coded kindly by Brani Vidakovic.

Herrick, D.R.M. (2000) Wavelet Methods for Curve and Surface Estimation. PhD Thesis, University of Bristol.

Daubechies, I. & Lagarias, J.C. (1992). Two-Scale Difference Equations II. Local Regularity, Infinite Products of Matrices and Fractals. SIAM Journal on Mathematical Analysis, 24(4), 1031–1079.

### Value

A list with components:

`coef` | A vector containing the empirical scaling function coefficients. This starts with the first non-zero coefficient, ends with the last non-zero coefficient and contains all coefficients, including zeros, in between.
`covar` | Matrix containing the covariances, if requested.
`klim` | The maximum and minimum values of k for which the empirical scaling function coefficients cJk are non-zero.
`p` | The primary resolution `tau * 2J`.
`filter` | A list containing the filter.number and family specified inthe function call.
`n` | The length of the data vector x.
`res` | A list containing the values of `p`, `tau` and `J`.

### Author(s)

David Herrick

### See Also

`Chires5`, `Chires6`, `denwd`, `denwr`

### Examples

    # Simulate data from the claw density and find the
    # empirical scaling function coefficients
    data <- rclaw(100)
    datahr <- denproj(data, J=8, filter.number=4,family="DaubLeAsymm")

[Package wavethresh version 4.6.8 Index]