Ten lectures on wavelets
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We present a study where wavelet approximation techniques and some relatedcomputational algorithms are applied to non-stationary high frequencyfinancial timesseries. Wavelets represent a novel instrument as far as concernedapplications in the finance setting, but have a great relevance in manydomains, from physics to statistics. Thus, while one goal of the paper isto compare the numerical performance of global and local function optimizers,another goal is to try to show that ad hoc wavelet-based functiondictionaries are very useful for financial modeling through signaldecomposition and approximation. Detecting the latent dependence featureswhich are typically found in high frequency financial returns is particularlyimportant for the scope of proposing models which are able to achievereliable results in parameter estimation and pointwise function prediction.We show that by pre-processing data with wavelet dictionaries we effectivelyaccount for hidden periodic components, whose discovery allows to attain andimprove the feature extraction power. We refer to sparse approximationthrough the Matching Pursuit algorithm, thus handling the negative effects ofcovariance non-stationarity at very high frequencies.