# Read e-book Essential Wavelets for Data Analysis Contents: Wavelets with Fourier Analysis. Statistical techniques for data analysis. Graphical Methods for Data Analysis. Statistical Techniques for Data Analysis. Essential behaviour analysis. Software for data analysis: Programming with R. Experimental Design and Data Analysis for Biologists.

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Actions Shares. Embeds 0 No embeds. No notes for slide. Wavelet transform and its applications in data analysis and signal and image processing 1. Introduction to Wavelet Transform: For the last two decades a newmathematical microscope has enabled scientists and engineers to viewinto the details of transient and time-variant phenomenon as was hithertonot possible using conventional tools.

We are familiar with Fourier series and Fourier transform which maps atime domain signal to frequency domain. Just as a prism breaks up whitelight into its different constituents, the Fourier transform breaks up a timedependent signal into its frequency components. Thus it can be called amathematical prism. The time information in the signal is distributedthroughout the frequency domain and is practically difficult to retrieve.

This is because the time information is stored in relative phases i. One faces difficulty usingFourier transform while analysing signals which have transients or arerapidly varying. One possible way to overcome the shortcomings of Fourier analysis is tohave a basis set whose elements are localised in time. It will be evenbetter if the basis function is similar to the function itself. This thought isemployed in wavelet transform. A wavelet is a small wave which oscillatesand decays in the time domain. Unlike Fourier Transform which uses onlysine and cosine waves, wavelet transform can use a variety of waveletseach fundamentally different from each other.

By scaling and translation of these two orthogonalfunctions we obtain the complete basis set. The energy of these functions is finite i. These are represented aslow pass or average coefficients and high pass or detail coefficients. Daughter wavelets are produced by scaling the mother wavelet and areorthogonal to the father wavelet, mother wavelet and all the successivedaughter wavelets.

The mother captures the variation in a broader scalewhereas the daughters zoom in on these variations at finer and finerscales. A given function can be expresses in terms of scaling and waveletfunctions as belowSome commonly used wavelet families are Harr, Daubechies, Symlets,Gaussian, Mexican hat, Morlet, Meyer etc. This was a very briefintroduction to wavelet transform.

Mathematical details have beenexcluded. In the next section we shall discuss some applications of wavelettransform. This is achieved bycomputing wavelet transform and then plotting a scalogram. The 2-Dscalogram of the above signal can be given as shown next. Scalogram -3 -3 Percentage of energy for each wavelet coefficient x 10 x 10 6 6 99 5 5 92 85 78 4 4 Scales a 71 64 57 3 3 50 43 2 2 36 29 22 1 1 15 8 1 Time or Space bscale freq The time is taken at an interval of 0.

De-noising data sets and signals:De-noising is a very common and very important aspect of anycommunication system. Fig 1c The general de-noising procedure comprises of the following steps: 1. Decompose: Choose a wavelet and a level N and compute the wavelet decomposition of the signal at level N. Threshold detail coefficients using soft thresholding 3.

Reconstruct the signal using the approximation coefficients and the modified detail coefficients. The above equations together describe j th level analysis filter bank. This calculation is repeated iterated up to scale J forming the multiresolution pyramidal algorithm; one stage is shown in Figure 5.

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As was mentioned, the lowpass filter is associated by scaling function and highpass filter by wavelet function. The filters for calculating the synthesis are the same by using orthogonal wavelet transform. Analysis and synthesis filters can have different length, as by using biorthogonal filter banks.

However, this algorithm can be used for orthogonal and nonorthogonal wavelets. For EN-signal originating from corrosion process, wavelet transform decompose it into approximation and detail signal components at different scales and locations. Wavelets have found many applications in different natural scientific disciplines, among them also in chemical engineering Radolphe et al.

The use of wavelets to study electrochemical noise transients was reported by Aballe Aballe et al. The wavelet analysis of electrochemical noise signals, where the signal was decomposed into wavelet-subbands was used for the characterization of pitting corrosion intensity Smulko et al. Wharton et al. Wavelet analysis based on the fractional energy contribution of smooth crystals and the lowest frequency detail crystal can provide information on the type and onset of corrosion general corrosion, metastable pitting, stable pitting in performed potentiostatic critical pitting temperature test for a superduplex stainless steel Kim, In study of the copper anode passivation by electrochemical noise analysis using wavelet transforms it has been found that during active dissolution the electrode surface is dominated by long time scale process and the change of the position of the maximum relative energy from D7 to D8 could be an indication of future passivation Lafront et al.

It was shown, that electrochemical potential noise analysis of Cu-BTA system using wavelet transformation can be used to achieve the inhibition efficiency Attarchi et al. Some other authors also reported about the fractal nature of corrosion processes and corresponding electrochemical noise signals. The electrochemical potential and current noise originating from the corrosion of carbon steel in distilled water was analyzed using multifractal analysis. The multifractal spectra are found to be qualitatively different for different temporal stages of the corrosion process Muniandy et al.

For little more detailed insight the short overview of this research is as follows. The discrete wavelet transform DWT decomposition of signal into on joint time position and frequency scale level depended amplitudes are presented with color lightness in time-frequency plane in Fig. Next, the DWT multiresolution decomposition of processes on 5 levels are shown in Fig.

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The crystals from D 1 to D 5 are the details of the signal, and A 5 is the approximation of the signal. Events with small time constants are taken into account by the fine scale coefficients, details D 1 , D 2. The information dealing with larger time constant events is included in details D 4 and D 5.

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Therefore, these kinds of plot allow the signal to be viewed over the full time range and considering different scales, which contains information about corrosion events occurring at a determined time — scale. Variances of details were calculated to detect the intensity of particular signal components on level j.

Understanding Wavelets, Part 2: Types of Wavelet Transforms

It is suggested that the events in the relatively higher frequency region may be associated with uniform corrosion. On the other hand, the events in the relatively lower frequency regions are responsible for pitting corrosion.

After short computation the value for Hurst parameter H , and fractal dimension D can be obtained:. It can be associated with the strength of persistence within a time series. The persistence defines the correlation between adjacent values within time series. The first process is stationary and the second is non-stationary. The obtained results indicating the presence of fractional Brownian motion in pitting corrosion with adjacent values in the time series being strongly correlated and fractional Gaussian noise in general corrosion, with adjacent values in the time series being weak correlated.

A summary of the wavelet — based fractal analysis is given in Table 1. We proposed also a new way for determination of persistence nature of the electrochemical noise on the basis of correlation coefficients between original signal and details R I i , D j , Table 2. The pitting corrosion is positively correlated with long memory effect.

Increasing correlation of signal I 2 to the D 2 detail and then decreasing to D 5 , indicates weak persistence and short memory effect of general corrosion processes.