Cross-covariance

The cross-covariance matrix I., is then estimated by, and the PLS weight vectors ra are computed as in the SIMPLS algorithm, but now starting with instead of S. In analogy with Equation 6.32, the x-loadings py are defined as p = / (vf X/ ) Then the deflation of the scatter matrix is performed as... 

I Quantify fluoresence at PM and calculate Y cross-covariance at PM for each image... 

Unlike the variance, the cross-covariance may have positive or negative values. If Xi and X2 are not correlated, ctx x2 = 0. The techniques discussed in Section 3.2 can be used to demonstrate the properties of the variance given in Table 3.2. 

If the errors are not correlated, i.e., if the cross-covariance terms are equal to zero, the variance of / can be expressed as... 

ACC transforms = Auto-Cross-Covariance transforms —> autocorrelation descriptors... 

These are autocovariances and cross-covariances calculated from sequential data with the aim of transforming them into uniform-length descriptors suitable for QSAR modeling. ACC transforms were originally proposed to describe peptide sequences [Wold, Jonsson et al, 1993 Sjbstrbm, Rannar et al., 1995 Andersson, Sjostrom et al., 1998 Nystrom, Andersson et al., 2000]. To calculate ACC transforms, each amino acid position in the peptide sequence is defined in terms of three orthogonal z-scores, derived from a Principal Component Analysis (PC A) of 29 physico-chemical properties of the 20 coded amino acids. 

Then, for each peptide sequence, auto- and cross-covariances with lags k= 1, 2,. .., K, are calculated as... 

McCabe techniques of variable reduction are based on the calculation of the conditional covariance (or correlation) matrix of the excluded variables McCabe, 1984 1410 /id. This matrix represents the residual information left in the variable that are not selected, after the effect of the most relevant variables has been removed. It is a square symmetric matrix of order q = p — k, vhere p is the total number of variables and k is the number of retained variables, and is derived from the covariance (correlation) matrix of the retained variables Sj (of size k x k), the covariance (correlation) matrix of the deleted variables Sjj (of size q X q), and the cross-covariance (correlation) matrix bet veen the t vo sets of variables Srd (of size k X q) ... 

Sjdstrdm, M., Rarmar, S. and Wieslander, A. (1995) Polypeptide sequence property relationships in Escherichia coli based on auto cross covariances. Chemom. Intell. Lab. Syst., 29, 295-305. 

The cross-correlation between two measured biosignals x(n) and y(n) is defined statistically as Eyx(IO = Ely(n)x(n + h)], where the operator E represents the statistical expectation or mean and k is the amount of time signal jc(n) is delayed with respect to y(n). Given two time sequences of x( ) and y(n) each of N points, the commonly used estimate of cross-covariance c,Jik) is as follows ... 

For two stationary time series, X and Y, the cross-covariance, Yxy x), is defined as... 

The cross-covariance frmction for the two time series does not depend on t. Joint stationarity implies that the cross-correlation function can be written as... 

Estimating the Autocovariance and Cross-Covariance and Correlation Functions... 

The autocovariance and cross-covariance can be estimated using the following formulae ... 

The cross-correlation plot shown in Eig. 5.4 between the mean summer and spring temperatures has a similar format to the previously considered autocorrelation plot. The confidence interval is, as was previously noted, the same as for the autocorrelation plot. The salient feature is the 4 largest lags at —20, —16, 3, and 4. At this point, it would be useful to comment briefly about the meaning of these values. Since the formula for computing the cross-covariance can be written as or equivalently yt = Xf we can see that positive values correspond to a relationship between past values of x (or in our case, the mean spring temperature)... 

The presented formula assumes that there are no repeated roots in the decomposition of the function. If there are repeated roots, then the value can be obtained by eitho- taking the limit of the above equation as two of the roots approach each other or looking at Appendix A3 of (Shardt 2012a), which presents a detailed method for the symbolic computation of the cross-covariance for two arbitrary time series. 

The degree of interaction between two signals can also be determined from cross-power spectrum, coherence (van den Schaaf et al., 1999b), or cross-covariance function (Greon et al., 1997). In case of wide solid... 

Cross-Correlation Coefficient n The cross-covariance normalized hy the product of the standard deviations of the two sections from the two random variable sequences used to calculate the cross-covariance. The cross-correlation coefficient is a widely used measure of the correlation between two sections of two different... 

The cross-covariance of a random variable with itself is referred to as the autocovariance, which is often used interchangeably and confused with the autocorrelation which is actually the cross-correlation of a random variable with itself. The cross-covariance is related to and often confused with the cross-correlation. 

The cross-covariance normalized by the product of the standard deviations of the two sections is the cross-... 

Cross covariance is defined as the product of the process output signal with the process input signal, while lagging it by / = 1,2,... sampling intervals. 

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