wavethresh (version 4.6.1)

denproj: 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.

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.

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.

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")