DDHFm (version 1.1.4)

ddhft.np.2: Data-Driven Haar-Fisz transformation

Description

Forward Data-Driven Haar-Fisz transform

Usage

ddhft.np.2(data, Ccode=TRUE)

Value

A list containing the following components

hft

The data-driven Haar-Fisz transform of the input sequence

mu

The mu's obtained from the input sequence used for estimating the mean-variance relationship

sigma

The estimated standard deviation as a function of the mean, the result of the isotonic regression fit of sigma2 on sigma

sigma2

The local multiscale standard deviations associated with each mean

factors

The numbers that divide the detail coefficients to standardize variance (obtained from the mean-variance estimation)

Arguments

data

A vector of size $2^J$ containing the data to variance stabilize and Gaussianize.

Ccode

If TRUE then fast C code is used, otherwise R code is used

Author

Piotr Fryzlewicz <[e-mail]>

Details

Performs the data-driven Haar-Fisz transform on sequence data. This consists of (i) the Haar wavelet transform of sequence; (ii) estimation of mean-variance relationship between finest level smoothing and detail wavelet coefficients using isotonic regression (see isotone); (iii) divide wavelet detail coefficients by smooth ones subjected to the estimated mean-variance relationship; (iv) perform the inverse Haar wavelet transform of the modified coefficients.

The aim is to variance stabilize and Gaussianize the sequence data which is only assumed to be positive and possess an underlying increasing mean-variance relationship.

References

Fisz, M. (1955), The limiting distribution of a function of two independent random variables and its statistical application, Colloquium Mathematicum, 3, 138-146.

Delouille, V., Fryzlewicz, P. and Nason, G.P. (2005), A data-driven Haar-Fisz transformation for multiscale variance stabilization. Technical Report, 05:06, Statistics Group, Department of Mathematics, University of Bristol

See Also

ddhft.np.inv

Examples

#
# Generate example Poisson data set.
#
# Intensity function is steps from 1 to 32 in steps of 4 with each intensity
# lasting for 128 observations. Then sample Poisson with these intensities
#
v <- rpois(1024, lambda=rep(seq(from=1, to=32, by=4), rep(1024/8,8)))
#
# Let's take a look at this
#
if (FALSE) ts.plot(v)
#
# Ok. So mean of intensity clear increasing, but variance increasing too
#
# Now do data-driven Haar-Fisz
#
vhft <- ddhft.np.2(v)
#
# Now plot the variance stabilized series
#
if (FALSE) ts.plot(vhft$hft)
#
# The variance of the observations is much closer to 1. For example, let's
# look a the variance of the original series and the transformed one
#
# For the first intensity of 1
#
var(v[1:128])
#[1] 0.6628322
#
var(vhft$hft[1:128])
#[1] 1.025151
#
#
# And for second intensity of 5
#
#
var(v[129:256])
#[1] 4.389518
var(vhft$hft[129:256])
#[1] 1.312953
#
# So both transformed variances near to 1
#
# Now plot the estimated variance-mean relationship
#
if (FALSE) plot(vhft$mu, vhft$sigma)
if (FALSE) lines(vhft$mu, sqrt(vhft$mu))
#
# This is an approximately square root function (because you expect the
# sd of Poisson to be the square root of the mean).
#