NATURE OF THE CORRELATION MATRIX

The conditioning of the data matrix in Equation (4.25) provides information on the potential difficulties to be encountered in calculations based on (The condition number for any matrix X is defined as where 

The elements of the correlation matrix in Equation (4.26) are given by where r and q are integers in the range + ]- 

The diagonal elements correspond to r = g and the off-diagonal elements correspond to r q. We show in the following theorem that the elements of the correlation matrix are weighted sums of the energy contributions from different frequencies present in the input signal. 

Theorem 4.2 Let U e ) denote the DFT of the input signal u k) with W radians. Let denote the frequency response of the rth 

Proof Prom Parseval s theorem, we can write the following relationship for the diagonal elements of the correlation matrix 

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This chapter consists of six sections. Section 4.2 introduces the FSF model structure. Section 4.3 examines the properties of the FSF model with a fast data sampling rate. Section 4.4 introduces the concept of a reduced order FSF model. Section 4.5 discusses the use of least squares for estimating the FSF model parameters from input-output data. Section 4.6 excunines the nature of the correlation matrix that arises when using a least squares estimator with an FSF model and the relationship between the elements of this matrix and the energy content of the input signal. 

The nature of the membrane matrix has significant influence on water mobility in the membrane. Water diffusion coefficients are high in perfluorinated membranes with weak hydrophiUc interaction [2]. The coefficients are lower for membranes with hydrocarbon matrix with stronger hydrophihc interaction. A correlation exists between the self-diffusion coefficients of water and co-ions the water... 

Experiments were performed with various LiAl-X LDHs, with X = Br, NO3 and ISO4. As with the intercalation process, the nature of the anion exerts a powerful influence on the reaction. In the case of sulfate, the deintercalation reaction does not go to completion - only 40% of the available lithium sulfate was released. The deintercalation reaction initially proceeds very quickly, but the process is then halted. The rate of deintercalation is NOs" > Cl > Br . This series does not correspond with data on the anion selectivity for intercalation into Al(OH)3, which is S04 > Cl" > Br" > NO3". Neither is there a correlation of the release data with the heats of hydration of the anions. The series observed arises because the intercalation and deintercalation processes are a balance of a number of factors, including interactions between the guest ions and the host matrix. 

Multivariate methods, on the other hand, resolve the major sources by analyzing the entire ambient data matrix. Factor analysis, for example, examines elemental and sample correlations in the ambient data matrix. This analysis yields the minimum number of factors required to reproduce the ambient data matrix, their relative chemical composition and their contribution to the mass variability. A major limitation in common and principal component factor analysis is the abstract nature of the factors and the difficulty these methods have in relating these factors to real world sources. Hopke, et al. (13.14) have improved the methods ability to associate these abstract factors with controllable sources by combining source data from the F matrix, with Malinowski s target transformation factor analysis program. (15) Hopke, et al. (13,14) as well as Klelnman, et al. (10) have used the results of factor analysis along with multiple regression to quantify the source contributions. Their approach is similar to the chemical mass balance approach except they use a least squares fit of the total mass on different filters Instead of a least squares fit of the chemicals on an individual filter. 

Fewer intercomparison studies have been carried out for peroxy radicals than for OH. Two chemical amplification methods were compared during a measurement campaign in Brittany, France (Cantrell et al., 1996). Although the measurements tended to track one another, there is more scatter than might be expected, given the similar nature of the instruments. For example, a plot of the data from one instrument against those from the second had a slope of 0.71 but a correlation coefficient of only r = 0.36. In another study (Zenker et al., 1998), comparison of three chemical amplifier techniques to matrix isolation-ESR gave... 

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