Matrix correlation coefficient

The off-diagonal elements of the variance-covariance matrix represent the covariances between different parameters. From the covariances and variances, correlation coefficients between parameters can be calculated. When the parameters are completely independent, the correlation coefficient is zero. As the parameters become more correlated, the correlation coefficient approaches a value of +1 or -1. 

CALC COVARRIANCE MATRIX AND CORRELATION COEFFICIENT MATRIX. 

Data Analysis Because of the danger of false conclusions if only one or two parameters were evaluated, it was deemed better to correlate every parameter with all the others, and to assemble the results in a triangular matrix, so that trends would become more apparent. The program CORREL described in Section 5.2 retains the sign of the correlation coefficient (positive or negative slope) and combines this with a confidence level (probability p of obtaining such a correlation by chance alone). 

Calculate the correlation coefficient r for every combination of columns, and display the results in a triangular matrix (an absolute value just under 1.00 indicates a strong correlation between the measurements in columns i and j a minus sign indicates that the slope is negative). 

Scaling is a very important operation in multivariate data analysis and we will treat the issues of scaling and normalisation in much more detail in Chapter 31. It should be noted that scaling has no impact (except when the log transform is used) on the correlation coefficient and that the Mahalanobis distance is also scale-invariant because the C matrix contains covariance (related to correlation) and variances (related to standard deviation). 

For example, let us take a look at the data of Table 35.5a. This table shows two very simple data sets, X and Y, each containing only two variables. Is there a relationship between the two data sets Looking at the matrix of correlation coefficients (Table 35.5b) we find that the so-called intra-set (or within-set) correlations are strong ... 

Optimizing the GC instrument is crucial for the quantitation of sulfentrazone and its metabolites. Before actual analysis, the temperatures, gas flow rates, and the glass insert liner should be optimized. The injection standards must have a low relative standard deviation (<15%) and the calibration standards must have a correlation coefficient of at least 0.99. Before injection of the analysis set, the column should be conditioned with a sample matrix. This can be done by injecting a matrix sample extract several times before the standard, repeating this conditioning until the injection standard gives a reproducible response and provides adequate sensitivity. 

Matrix formed by a set of correlation coefficients related to m variables in multivariate data sets, R = (rXi,Xj). It is relevant in multicomponent analysis. 

In chromatography techniques, selectivity can be proved by the existence of good separation between the analyte and the other components (such as the matrix, impurities, degradation product(s), and metabolites). A consequence of this requirement is that the resolution of the analyte from the other components should be more than 1.5-2.0. In order to detect the possibility of coelution of other substance(s), the purity of the analyte peak should also be determined. For instance, the UV-Vis spectrum of the analyte peak/spot can be used to determine 4the purity of the analyte peak/spot, in this case the correlation coefficient V (this term is used by the software of DAD System Manager Hitachi, and CATS from Camag). With the same meaning and mathematical equation, other terms are used, such as Match... 

Several software packages contain simple command lines for performing matrix computations directly and thus are conveniently capable of computing the correlation coefficient, for example as in r2 (Equation 59-8). 

In principle, the relationships described by equations 66-9 (a-c) could be used directly to construct a function that relates test results to sample concentrations. In practice, there are some important considerations that must be taken into account. The major consideration is the possibility of correlation between the various powers of X. We find, for example, that the correlation coefficient of the integers from 1 to 10 with their squares is 0.974 - a rather high value. Arden describes this mathematically and shows how the determinant of the matrix formed by equations 66-9 (a-c) becomes smaller and smaller as the number of terms included in equation 66-4 increases, due to correlation between the various powers of X. Arden is concerned with computational issues, and his concern is that the determinant will become so small that operations such as matrix inversion will be come impossible to perform because of truncation error in the computer used. Our concerns are not so severe as we shall see, we are not likely to run into such drastic problems. 

The robust estimator still provides a correct estimation of the covariance matrix on the other hand, the estimate J>C> provided by the conventional approach, is incorrect and the signs of the correlated coefficients have been changed by the outliers. 

Four samples from a Polynesian island gave the lead isotope compositions given in Table 4.3. Calculate the mean and standard deviation vectors, the covariance matrix and the correlation coefficient between the two isotope ratios. 

The covariance matrix of / can be computed from standard-deviations and the unique correlation coefficient through equation (4.2.18)... 

The ijth term of this matrix represents the correlation coefficient (loading) between the (th variable and the jth principal component. 

The loading matrix FA1/2 given by equation (4.4.26) can be found in Table 4.14. Lead isotopes have strong correlation coefficients on the first component. They are decoupled from Sr and Nd isotopes which strongly correlate and anticorrelate, respectively, with the second component. On a global scale, Pb isotopic variations in oceanic islands seem to be decoupled from Sr and Nd isotopic variations. 

From the definition of variance and correlation coefficient and equation (4.2.18), we find that the covariance matrix is... 

In order to get the correlation coefficients, this matrix must be pre-multiplied by the inverse of the diagonal matrix having the standard deviations of x on the diagonal line, and post-multiplied by the inverse of the diagonal matrix having the standard deviations of y on the diagonal line. 

In our previous analysis, correlation coefficients were used between the fold distributions in different genomes to construct a distance matrix and a corresponding cluster dendogram (Wolf et al., 1999). This clustering showed significant differences in the fold composition between eukaryotes and prokaryotes (bacteria and archaea) as well as between free-living and parasitic bacteria. 

The correlation coefficients can be arranged in a matrix like the covariances. The resulting correlation matrix (R, with l s in the main diagonal) is for autoscaled x-data identical to C. 

The distance between object points is considered as an inverse similarity of the objects. This similarity depends on the variables used and on the distance measure applied. The distances between the objects can be collected in a distance matrk. Most used is the euclidean distance, which is the commonly used distance, extended to more than two or three dimensions. Other distance measures (city block distance, correlation coefficient) can be applied of special importance is the mahalanobis distance which considers the spatial distribution of the object points (the correlation between the variables). Based on the Mahalanobis distance, multivariate outliers can be identified. The Mahalanobis distance is based on the covariance matrix of X this matrix plays a central role in multivariate data analysis and should be estimated by appropriate methods—mostly robust methods are adequate. 

Matrix B consists of q loading vectors (of appropriate lengths), each defining a direction in the x-space for a linear latent variable which has maximum Pearson s correlation coefficient between y and jf for j = 1,..., q. Note that the regression coefficients for all y-variables can be computed at once by Equation 4.52, however,... 

The canonical correlation coefficients can also be used for hypothesis testing. The most important test is a test for uncorrelatedness of the x- and y-variables. This corresponds to testing the null hypothesis that the theoretical covariance matrix between the x- and y-variables is a zero matrix (of dimension mx x mY). Under the assumption of multivariate normal distribution, the test statistic... 

An alternative clustering method for variables is to use the correlation coefficient matrix in which each variable is considered as an object, characterized by the correlation coefficients to all other variables. PCA and other unsupervised methods can be applied to this matrix to obtain an insight into the similarities between the original variables. 

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