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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=F, nT=20)
REQUIRED 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.
OPTIONAL ARGUMENTS
- 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 T 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 in the
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)).
REFERENCES
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
`denwd'
# 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")
AUTHOR
David Herrick