Noise addition, correlation

For most potentiostats, the assumption Av >> 1 becomes invalid at frequencies above 1 to 10 kHz. In this case, the noise terms are still additive, but the interaction between the gain of the operational amplifier jmd the cell impedance results in additional correlation between the input and output chaimels. 

So the correlative processing gives additional (to the amplitude) variation of output signal and makes the signal processing a multiparamctric one. Such processing increases the sensitivity and reduces the noises. 

Evidence of localized corrosion can be obtained from polarization methods such as potentiodynamic polarization, EIS, and electrochemical noise measurements, which are particularly well suited to providing data on localized corrosion. When evidence of localized attack is obtained, the engineer needs to perform a careful analysis of the conditions that may lead to such attack. Correlation with process conditions can provide additional data about the susceptibility of the equipment to locaHzed attack and can potentially help prevent failures due to pitting or crevice corrosion. Since pitting may have a delayed initiation phase, careful consideration of the cause of the localized attack is critical. Laboratory testing and involvement of an... 

Now we are ready to consider what happens if the data are noisy. We will take the data we just used and add some noise to it. We will add normally distributed noise to each wavelength of each spectrum at a level of approximately 5%. It is important to understand that, within a given spectrum, the particular amount of noise added to each wavelength is independent of the noise added to the other wavelengths. And, of course, the noise we add to each spectrum is independent of the noise added to the other spectra. In other words, there is no correlation to the noise. Figure 39 contains a plot of the data before and after the addition of the noise. Figure 40 show two other views of the data after the additon of the noise. 

Tuziuti T, Yasui K, Sivakumar M, Iida Y, Miyoshi N (2005) Correlation between acoustic cavitation noise and yield enhancement of sonochemical reaction by particle addition. J Phys Chem A 109 4869 1872... 

This is for univariate data what happens in the case of multivariate (multiwavelength) spectroscopic analysis. The same thing, only worse. To calculate the effects rigorously and quantitatively is an extremely difficult exercise for the multivariate case, because not only are the errors themselves are involved, but in addition the correlation stmcture of the data exacerbates the effects. Qualitatively we can note that, just as in the univariate case, the presence of error in the absorbance data will bias the coefficient(s) toward zero , to use the formal statistical description. In the multivariate case, however, each coefficient will be biased by different amounts, reflecting the different amounts of noise (or error, more generally) affecting the data at different wavelengths. As mentioned above, these... 

Large data tables contain an amount of information which is partly hidden because the data complexity prevents ready interpretation. This is typical of NIR spectra collections. PCA is a projection method used to visualize all the information contained in the data table. It can be used to show in what respect one sample differs from another, which variables contribute most to this difference, and whether these variables contribute in the same way and are correlated or independent of each other. It also reveals sample patterns or groupings. In addition, it quantifies the amount of useful information, as opposed to noise or meaningless variation, contained in the data table. Principal components are defined only for the data set from which they were computed. They may also hold for other data of identical type, but this is not guaranteed, and it is certainly not true for different types of data. 

Table 12.3 compares the estimated analyte concentrations for DIED, PARAFAC, and PARAFAC x 3 noise (PARAFAC with the addition of a factor of three greater random errors) applied to the same calibration problem. Table 12.4 is analogous to Table 12.3, except that it also presents the squared correlation coefficients between the true and estimated X-way and Y-way profiles for all three species present in the six samples. It is first evident that PARAFAC slightly outperforms DTLD when applied to the same calibration problem. However, the improvement often lies in the third or fourth decimal place and is hardly significant when compared with the overall precision of the data. This near equivalence of DTLD and PARAFAC is rooted in the fact that DTLD performs admirably, and there is little room for... 

SA of SODEs describing chemically reacting systems was introduced early on, in the case of white noise added to an ODE (Dacol and Rabitz, 1984). In addition to expected values (time or ensemble average quantities), SA of variances or other correlation functions, or even the entire pdf, may also be of interest. In other words, in stochastic or multiscale systems one may also be interested in identifying model parameters that mostly affect the variance of different responses. In many experimental systems, the noise is due to multiple sources as a result, comparison with model-based SA for parameter estimation needs identification of the sources of experimental noise for meaningful conclusions. 

The analysis made in ref. 44 is based on the discussion of the related exact non-Markovian master equatioh and allows us to conclude that when the noise intensity 8 (0), Eq. (40), is constant the rates are exponentially enhanced with decreasing correlation time t and this is independent of the specific form of the nonlinear bistable flow /(x,a) and also of whether the random noise is additive or multiplicative. [Ilie only condition imposed is g(x) 0 in xi,x ,X2. ]... 

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