Pearson correlation matrix

Although for binary data other distance, metrics are in general more appropriate (e.g., Tanimoto metrics), for simplicity we can compute the standardized (to the mean and standard deviation of the distribution) Pearson correlation matrix, which contains the correlation coefficients between each of the five assays. These data can then be used to duster the chemicals based on their correlation as a metric of similarity The groupings depicted in Fig. 6-14(b)... 

In Sections 1.6.3 and 1.6.4, different possibilities were mentioned for estimating the central value and the spread, respectively, of the underlying data distribution. Also in the context of covariance and correlation, we assume an underlying distribution, but now this distribution is no longer univariate but multivariate, for instance a multivariate normal distribution. The covariance matrix X mentioned above expresses the covariance structure of the underlying—unknown—distribution. Now, we can measure n observations (objects) on all m variables, and we assume that these are random samples from the underlying population. The observations are represented as rows in the data matrix X(n x m) with n objects and m variables. The task is then to estimate the covariance matrix from the observed data X. Naturally, there exist several possibilities for estimating X (Table 2.2). The choice should depend on the distribution and quality of the data at hand. If the data follow a multivariate normal distribution, the classical covariance measure (which is the basis for the Pearson correlation) is the best choice. If the data distribution is skewed, one could either transform them to more symmetry and apply the classical methods, or alternatively... 

In order to eliminate parameters that are correlated to each other, we calculate their Pearson correlation coefficients (25). Linearly uncorrelated parameters have Pearson correlation coefficients close to zero and likely describe different aspects of the phenotype under study (exception for non-linearly correlated parameters which cannot be scored using Pearson s coefficient). We have developed an R template in KNIME to calculate Pearson correlation coefficients between parameters. Redundant parameters that yield Pearson correlation coefficients above 0.4 are eliminated. It is important to visually inspect the structure of the data using scatter matrices. A Scatter Plot and a Scatter Matrix node from KNIME exist that allow color-coding the controls for ease of viewing. 

Bivariate statistics. The objective here is to look for possible relationships between pairs of variables. Pearson s correlation has traditionally been the most used, although the analysis of the correlation matrix should be studied before the use of most multivariate statistical procedures. 

TABLE 3 Correlation matrix and factor analysis of four variables — Pearson (r) with reliabilities partialled out... 

Figure 16.11. Portion of a scatter plot matrix illustrating pairwise plots of log ratios across a series of microarray experiments. Each scatter plot is used to calculate a Pearson correlation coefficient between the two sets of measurements (Khan et al., 1998).

Cox and Clifford (1982) have proposed a way of presenting correlation coefficient data for a suite of rocks in a diagrammatic form. Their method, which is purely descriptive, uses the Pearson product-moment coefficient of correlation and is an attempt to utilize and display graphically the large amount of information contained in a correlation matrix, without resorting to plotting the enormous number of... 

Furthermore, given the large quantity of multivariate data available, it was necessary to reduce the number of variables. Thus, if two any descriptors had a high Pearson correlation coefficient (r > 0.8), one of the two was randomly excluded from the matrix, since theoretically they describe the same property to be modeled (biological response). Therefore it is sufficient to use only one of them as an independent variable in a predictive model (Ferreira, 2002). Moreover those descriptors that showed the same values for most of the samples were eliminated too. 

The PCA results show the score plot (Fig. 2) relative to the first and second principal components. In PCI, there is a distinct separation of the compounds into two classes. More active compounds are on the left side, while less active are on the right side. They were chosen among all data set (1700 descriptors) and they are assumed to be very important to investigate anticancer mechanism involving artemisinins. Table 1 displays the values computed for these four descriptors. This step was crucial since a matrix with 1700 columns was reduced to only 4 columns. No doubt it is more appropriate to deal with a smaller matrix. The first three principal components, PCI, PC2 and PC3 explained 43.6%, 28.7% and 20.9% of the total variance, respectively. The Pearson correlation coefficient between the variables is in general low (less than 0.25, in absolute values) exception occurs between Mor29m and ICS, which is -0.65 ... 

In addition, since we have more than one variable, it is possible to calculate a product-moment (Pearson) correlation coefficient for each pair of variables. These are summarized in the correlation matrix in Table 8.2, obtained using Minitab. 

The T matrix contains row (object) projections into t (score) vectors. The p vectors are obtained by projecting the X matrix columns (variables) in the loadings matrix P. In PCA, the score of a compound for a variable is a linear combination, t = pi 1 + + Pm Xm, where p are the correlation (direction) coefficients of the principal component plane contained in the p (loading) vector. Two linear combinations are uncorrelated if the Pearson correlation for the corresponding scores is zero. The variance of a linear combination is the sample variance of the corresponding a scores. The first PC is, then, the linear combination oi maximum variance, when the condition pi + + = 1 is... 

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,... 

Principal Components Analysis (PCA) is a multivariable statistical technique that can extract the strong correlations of a data set through a set of empirical orthogonal functions. Its historic origins may be traced back to the works of Beltrami in Italy (1873) and Jordan in Prance (1874) who independently formulated the singular value decomposition (SVD) of a square matrix. However, the first practical application of PCA may be attributed to Pearson s work in biology [226] following which it became a standard multivariate statistical technique [3, 121, 126, 128]. 

Pearson s linear correlation coefficient, 239 perturbation analysis, pairwise decomposition scheme Frobenius norm, 235 RTB Hessian matrix, 234 symmetric positive semidefinite (SPSD) matrix, 237... 

Independency The correlations among L, a, b, h, C, Sl, Sa, Sb, Sh, and Sc are analyzed using matrix plots, a two-dimensional matrix of individual plots and Pearson coefficient generated by MINITAB 17. 

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