3. [30 points] We have discussed wavelets as a means to extract information from

Question

3. [30 points] We have discussed wavelets as a means to extract information from a dataset (e.g., considering a dataset as a signal and attempting to understand the information in the signal).

3.1. What is a wavelet and how does a wavelet analysis differ from a classical Fourier analysis?

3.2. How does one dilate and translate with a wavelet? Why does one consider these transformations?

3.3. Why is wavelet analysis of use, particularly with data that has both a periodic and (quasi)periodic structure? (Hint: consider the temperature analysis presented in class.)

Explanation / Answer

3.1. Wavelet: It is a wave like oscillation with amplitude that begins at zero, increases, and then decreases back to zero. It can typically be imagined as a "brief oscillation" like one might see recorded by a seismograph. Generally, wavelets are firmly crafted to have specific properties that make them useful for signal processing. These are combined using a reverse, shift, multiplication and integration technique called convolution, with parts of a known signal to extract information from the unknown signal.

3.2. Wavelet theory generated by a set of filters by dilation and translation of generating wavelet. A set of wavelengths are generated from mother wavelet. The translation parameter ranges over the length of the input signal. The values of the input series that affect the wavelet coefficients at rising scales increases is shown in the function conofinf().

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