Matrix of correlations

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 ... 

Even if a correlation is below the conventional level of significance, consideration should be given to whether it might alter the risk estimate, and it may be prudent to include it. When measnred or estimated correlations are nsed to specify dependencies in Monte Carlo models, it is important to check that the matrix of correlations satisfies mathematical constraints (Table 2.3). 

Basic Concepts. The goal of factor and components analysis is to simplify the quantitative description of a system by determining the minimum number of new variables necessary to reproduce various attributes of the data. Principal components analysis attempts to maximally reproduce the variance in the system while factor analysis tries to maximally reproduce the matrix of correlations. These procedures reduce the original data matrix from one having m variables necessary to describe the n samples to a matrix with p components or factors (p[Pg.26]

Both component and factor analysis as defined by equations 17 and 18 aim at the identification of the causes of variation in the system. The analyses are performed somewhat differently. For the principal components analysis, the matrix of correlations defined by equation 10 is used. For the factor analysis, the diagonal elements of the correlation matrix that normally would have a value of one are replaced by estimates of the amount of variance that is within the common factor space. This problem of separation of variance and estimation of the matrix elements is discussed by Hopke et al. (4). 

The variance-covariance matrix can be normalized to give the matrix of correlation coefficients between variables. Recall that the correlation coefficient is the cosine of the angle, (j>, between two vectors. Because the correlation of any variable with itself is always perfect (py = 1), the diagonal elements of the correlation matrix, R, are always 1.00. 

The first step of FA is factor extraction the methods described below are the most commonly used. The extraction methods calculate a set of orthogonal factors (or components) that in combination reproduce the matrix of correlation. The criteria used to generate the factors are not homogeneous for all methods but the differences between their solutions may be quite small. 

Factor analysis comprises several closely related algorithms for transforming a matrix of correlations among a number of observed variables into a matrix of... 

Table 4 The matrix of correlations between the analytes determined from heart-tissue data...

Table 2 The matrix of correlations between objects from Table 1, (a). Samples A and D, B and C form new objects and a new correlation matrix can be calculated, (b). Sample E then joins AD and F joins BC to provide the final step and apparent correlation matrix, (c)...

Some of these spectra are illustrated in Figure 8 and the variation in absorbance at each wavelength as a function of tryptophan concentration is shown in Figure 9. No single wavelength measure exhibits an obvious linear trend with analyte concentration and a univariate calibration is unlikely to prove successful. The matrix of correlation coefficients between the variables, dependent and independent, is given in Table 12. The independent variable most highly correlated with tryptophan concentration is the measured absorbance at Xi2, Ai2, i.e. 

Table 14 Matrix of correlations between the residuals from Table 13 ...

The quantity and precision of the measurements reported here are sufficient to determine the matrix of correlation functions and, from this, a reaction pathway that is qualitatively consistent with the reaction mechanism established previously. The existence of unmeasured species did not compromise the analysis. The quantity and precision of the data were not excessive, and thus we expect the method to be generally applicable. 

Readers with experience in chemometrics will have noticed that, like principal components analysis (PCA), MDS is a dimensionality reduction method. For each molecule, a large number of attributes (similarity to each other molecule) is reduced to a much smaller number of coordinates in an abstract property space, which reproduce the original data within an established error. The pertinent difference is that PCA uses the matrix of correlations between a set of (redundant) properties, which are usually obtained from a table of those properties for an initial set of molecules. In contrast, MDS uses a matrix of similarities between each pair of molecules (or substituents). 

It should be obvious that the production of MV portfolios is not extraordinarily input intensive. Efficient frontiers for five asset portfolios require only five predicted returns, five standard deviations, and a five-by-five symmetrical matrix of correlation coefficients. Yet the process yields indispensable information that allows investors to select suitable portfolios. 

To make the propagation of model, the model will be replicated a number of times M equal to 10000. Due to the correlation of the output quantities it is necessary to determine the matrix of correlation coefficients of these magnitudes. 

Goal Given a matrix of correlations, correlation creates the two-dimensional correlation plot. 

FIGURE 4.1 Scree plot eigenvalues 2 of the matrix of correlation coefficients of 23 parameters for 28 solvents, in descending order. Four eigenvalues are greater than unity, with a distinct break before the fifth, suggesting that four independent properties of the solvents are significant. 

Figure 4.12 shows how the 9 parameters, for 13 non-HBD solvents, are disposed in the frame of the first three PCs. The three largest eigenvalues of the matrix of correlation coefficients were 6.53,1.58, and 0.87, the rest much smaller. Three supposedly hard parameters p, SB, and-AH(BFp appear together, with A fCHClj) and at a distance. All are far from C, as would be expected. B. again is off by itself (see Section 4.3). It is nearer than any other, suggesting that it measures mainly hard interactions, but not in the same manner as the other hard measures. This analysis reinforces the idea that basicity is not a simple property. 

The application of normal mode analysis to macromolecules such as proteins and nucleic acids has only recently become more common. Normal modes can be calculated either using harmonic analysis, where the second derivative matrix of the potential energy is calculated for a minimized structure, or using quasi-harmonic analysis, where the matrix of correlations of atomic displacements is calculated from a molecular dynamics (MD) trajectory. At temperatures below about 200 K, protein dynamics are primarily harmonic. Above this temperature there is appreciable non-harmonic motion which can be studied using quasi-elastic scattering techniques. There is evidence that such anharmonic motions are also important for protein function and quasi-harmonic analysis allows them to be incorporated implicitly to some extent within a harmonic model. 

A significance test (t test) is performed, as described in Sec. 7.2.3 [(Eq. (7.102)], on the parameters to test the null hypothesis that any one of the parameters might be qual to zero. The 95% confidence intervals of each measured variable are calculated. The variance-covariance matrix and the matrix of correlation coefficients of the parameters are calculated according to Eqs. (7.135) and (7.154), respectively. The analysis of variance of the regression results is performed as shown in Table 7.2. Finally, the randomness tests are applied to the residuals to test for the randomness of the distribution of these residuals. 

The equilibrium correlation functions for the conserved variables are defined by 5Ap r,t)5Ay r, t )), where (5A) = A - (A), and the brackets denote an average over the equilibrium distribution. In a stationary, translationally invariant system, the correlation functions depend only on the differences r - r and t — t, and the Fourier transform of the matrix of correlation functions is... 

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