f noise + deterministic oscillations

Description Usage Arguments Details Value Note See Also Examples

The fracdet class represents a wavelet transform that is assumed to be modelled by a simple fractal-deterministic model with long-memory properties. The model consists on a linear superposition of a fractional brownian motion (fBm) B and a deterministic band-limited signal x. That is, the observed signal is Y = x + B. The fracdet receives as parameters the wavelet transform of Y and an R fitted model object (nls or lm) containing the estimation of the parameters characterizing the fBm (The Hurst exponent H and the "dispersion" parameter sigma ^ 2). The main method of the fracdet class is estimateDetSignal, which permits the estimation of the deterministic signal x using bayesian modelling techniques in the wavelet-domain.

1	fracdet(x, fbmPars)

`x`	A `wd` object representing the wavelet transform of the observed signal Y.
`fbmPars`	Either a `nls` or a `lm` object (see `nls` and `nls`) providing estimates of the parameters characterizing the fBm part of the model (The Hurst exponent H and the "dispersion" parameter sigma ^ 2). Thus, the names of the fitted variables are assumed to be `H` and `sigma2`.

In addition to the specific methods for the fracdet class, all the wd methods can be used with a fracdet object.

A S3 fracdet object that represents the observed Y signal in the wavelet domain. The fracdet object will also contain the estimates of the fBm parameters.

When computing the wavelet transform of the Y signal through the wd method (see wd) we recommend using the "symmetric" boundary handling method (bc == "symmetric") if H > 0.5.

waveletVar, getWaveletVar, getWaveletVarModel, getFittedWaveletVar, as.wd, estimateDetSignal

## Not run: 
### deterministic signal estimation
# set parameters for the example
set.seed(3)
nlevels = 14
n = 2 ^ nlevels
xlim = c(200, 400)
H = 0.3
# simulate a simple fbm + sinusal signal
x = 1.25 * cospi(2 * 1:n / 10)
y = fbmSim(n = n, H = H) + x
plot(y, xlim = xlim,
     ylim = range(y[xlim[[1]]:xlim[[2]]]),
     type = "l")
lines(x, lty = 2, col = 2)
legend("topright", lty = 1:2, col = 1:2,
       legend = c("Y", "x"), bty = "n")
# compute the wavelet transform and the wavelet coefficients' variances
wy = wd(y, bc = "symmetric")
vpr = waveletVar(wy)
# do you note the increase in variance in level 11?
plot(vpr, xlim = c(4, nlevels - 1),
     ylim = range(vpr[5:nlevels]))
# estimate the fBm parameters avoiding levels 11 and 12 (with deterministic
# contributions). Level 12 is also avoid as a precaution
model = estimateFbmPars(vpr, use_resolution_levels = c(5:10, 13))

# Create a fracdet object...
fd = fracdet(wy, model)
# ... and check the fbm parameters estimates
coef(fd)
# A plot of the fitted variances may be useful
plot(getWaveletVar(fd))
points(getFittedWaveletVar(fd),
      col = 2,
      pch = 2)
# we could also extract the nls-fit and use the predict function
# The nls-fit is performed in semilog-space. Thus, a transformation
# of the predicted values is required
nls_model = getWaveletVarModel(fd)
points(resolutionLevels(vpr),
       2 ^ predict(nls_model, newdata = data.frame(x = resolutionLevels(vpr))),
       col = 3,
       pch = 3)
# Estimate the deterministic signal (this may take a while) taking
# into account the deviations in level 11, 12. The estimateDetSignal uses
# this information using a Bayesian modelling approach
wx = estimateDetSignal(fd, estimate_from = 11:12)
x_est = wr(wx)
# compare the original signal and the estimation
old_par = par(mfrow = c(2,1))
plot(y, xlim = xlim,
     ylim = range(y[xlim[[1]]:xlim[[2]]]),
     type = "l")
lines(x, lty = 2, col = 2)
legend("topright", lty = 1:2, col = 1:2,
       legend = c("Y", "x"), bty = "n")
plot(x, type = "l", xlim = xlim,
     ylim = range(x) * c(1, 2))
lines(x_est, col = 2, lty = 2)
legend("topright", lty = 1:2, col = 1:2,
       legend = c("x", "x estimate"), bty = "n")
par(old_par)

## End(Not run)