## Wavelet methods

Wavelet methods are very popular in density estimation, non-parametric regression and in signal analysis because of their ability to adapt to the local irregularities of the curve. When estimating spatially inhomogeneous curves, wavelet methods significantly outperform other traditional nonparametric methods. Typical wavelet estimators are non-linear threshold-type estimators. They attain (possibly within a factor that is only logarithmic in the sample size) asymptotically optimal rates simultaneously for large class of error measures and function spaces.

My research on wavelet methods has been focused on improving their flexibility in curve estimation. I have been looking for a finer balance between stochastic and approximation terms in the mean integrated squared error decomposition. I have also been dealing with data-based recommendations for choosing the tuning parameters in the estimation procedure, with implementing some modifications in the wavelet estimators as to make them satisfy some additional shape constraints like non-negativity, integral equal to one etc.

I have also implemented an online adaptive procedure of multiple sampling which allows for economic sampling of signals. The technique involves increasing the sampling rate when high-frequency terms are incorporated in the wavelet estimator, and decreasing it when signal complexity is judged to have decreased. The size of the wavelet coefficients at suitable resolution levels is used in deciding how and when to switch rates. The figure here (pdf) shows in its first panel the true signal. An additive Gaussian white noise with constant variance was added at a rate of 100 Hz to it and the goal was online recovery of the signal. In the second panel, our dual-rate sampling procedure has been applied. It samples at a low rate of only 16 Hz most of the time. However, when guided by the size of the wavelet coefficients, a highenough frequency is indicated, the procedure switches automatically to a high sampling rate of 100 Hz. This has happened near time points of 58 and of 75. When the high frequency fluctuations are over, the procedure switches back automatically to low rate sampling. The superimposed dashed line in the second panel indicates sampling rate as a function of time. Where the line is at level 0 or 0.5 the rate was low or high, respectively. The third panel shows the performance of the constant-rate sampler that uses the same number of sampling points like our dual-rate sampler. It can be seen that, for the same cost of sampling, the dual-rate sampler can resolve much better the high-frequency aberrations near the time points of 58 and of 75. Sometimes, such high-frequency aberrations are the most important feature in the signal.

My recent interest in wavelet methods is in applying them for image analysis. 