projectCurveWavelets: Function to project a curve on a wavelet basis.
In mattiaciollaro/MeanShift: Clustering via the Mean Shift Algorithm

Description Usage Arguments Details Value Author(s) References See Also Examples

View source: R/ext_functional.R

This function performs the Discrete Wavelet Transform (DWT) on a numeric vector representing a curve (i.e. a "functional" datum) observed on a grid and thresholds the wavelet coefficients, thus yielding a denoised and compressed representation of the same curve.

1 2	projectCurveWavelets( x, y, irreg.grid=FALSE, grid.length=NULL, filter.number=10, family="DaubLeAsymm", bc="periodic", verbose=FALSE, ... )

`x`	a numeric vector of x coordinates at which the curve is observed.
`y`	a numeric vector of y coordinates representing the curve. `x` and `y` must have the same length.
`irreg.grid`	logical. TRUE if `x` is not an equispaced grid.
`grid.length`	a positive power of 2 or NULL (default). In order to apply the DWT, `length(x)` must be a positive power of 2. By default, if `grid.length=NULL` and `length(x)` is not a power of 2, `x` is extended to an equispaced grid whose length is positive power of 2 and `y` is extended interpolated on the extended grid. If `projectCurveWavelets` is used on multiple curves, `grid.length` should be set manually to ensure that all the discretized curves have the same length before the DWT is applied on each of them.
`filter.number`	an integer specifying the smoothness of the wavelet used in the wavelet decomposition of `y`. See the functions `wd` and `irregwd` of wavethresh for details.
`family`	a character string specifying the family of wavelets used in the wavelet decomposition of `y`. See the functions `wd` and `irregwd` of wavethresh for details.
`bc`	a character string specifying how to handle the boundary condition. See the functions `wd` and `irregwd` of wavethresh for details.
`verbose`	logical. Controls the printing of "informative" messages whilst the computation progresses. Such messages are generally annoying so it is turned off by default.
`...`	further arguments to control the thresholding of the wavelet coefficients. See `threshold.wd` and `threshold.irregwd` of the wavethresh package for details. By default, `projectCurvesWavelets` uses the default values of `threshold.wd` and `threshold.irregwd` to perform the thresholding of the wavelet coefficients.

The function normalizes the input grid to the standard unit interval, i.e. the minimum and the maximum values of x.grid are respectively 0 and 1.

projectCurveWavelet is designed to be used as a preliminary step towards functional clustering using the mean shift algorithm. Given a sample of curves, projectCurveWavelet can be used to represent each curve as a sparse vector of coefficients. These coefficients can be fed as a matrix to msClustering or bmsClustering and clustered via the mean shift algorithm or the blurring mean shift algorithm.

The function outputs a list with names

`coefficients`	a numeric vector of thresholded wavelet coefficients.
`y.wdT`	an object of class `wd` or `irregwd`. See `threshold.wd` and `threshold.irregwd` of the wavethresh package for details.
`y.wavelet`	a numeric vector with the reconstruction of `y` after the application of the DWT and the thresholding of the wavelet coefficients.
`x.grid`	the extended and equispaced grid of x values associated to `y.wavelet`.

Mattia Ciollaro and Daren Wang

Nason, G. (2010) Wavelet methods in statistics with R.

wavethresh wd irregwd threshold.wd threshold.irregwd wr msClustering bmsClustering

## generate a noisy curve observed on a regular grid
set.seed( 1 )
n.grid <- 1000
x <- seq( 2, 8, length=n.grid )
sigma.epsilon1 <- 2
sigma.epsilon2 <- 2.5
sigma.epsilon3 <- 3
sigma.epsilon4 <- 1
epsilon <- rnorm( 1000, sd=rep( c( sigma.epsilon1,
sigma.epsilon2, sigma.epsilon3, sigma.epsilon4 ),
rep( 250, 4 ) ) )
y <- x*sin( 3*x ) + 0.3*x^2 + epsilon

## project on wavelet basis with soft universal thresholding
## of the wavelet coefficients
wave <- projectCurveWavelets( x, y, type="soft", policy="universal" )

## plot wavelet reconstruction of the curve
## Not run: 
x.norm <- ( x - min( x ) ) / ( max( x ) - min( x ) )
plot( x.norm, y )
lines( wave$x.grid, wave$y.wavelet, col=2, lwd=3 )
## End(Not run)

## inspect wavelet coefficients
wave.coeffs <- wave$coefficients
print( length( wave.coeffs ) )  ## 1023 coefficients
print( sum( wave.coeffs != 0 ) )  ## only 12 are non-zero