Covariance and Correlation

Consider the random variables Aj, X2. X with means mj, m2. and variances arf,a-f.cr. A question that arises frequently is, what is the average and the variance of the linear function 

The covariance, as defined by Eq. 2.38, suffers from the serious drawback that its value changes with the units used for the measurement of Xj,Xj. To eliminate this effect, the covariance is divided by the product of the standard deviations (t cry, and the resulting ratio is called the correlation coefficient 

Equations 2.40-2.44 will be applied in Sec. 2.15 for the calculation of the propagation of errors. 

We can see in Fig. 2.8(b) that high values of y tend to occur together with high values of x, and vice versa. In these cases, we say that the two random variables present a certain covariance, that is, a tendency to deviate conceitedly from their respective averages. We can obtain a numerical measure of the covariance from the products of the deviations (atj — Jc) and (jj — y) for each member of the sample. Since in this example the two deviations tend to have the same sign, be it positive or 

Note the analogy with the definition of the variance, Eq. (2.2). The denominator is again 72 — 1, because only ti — 1 of the n products of deviations are independent. Note also that Cov(x,x) is the same as the variance of x. 

Since the covariance depends on the units of measurement ofx and, it is difficult to use it as a standard to compare the degree of statistical association of different pairs of variables. If, for example, we were investigating oranges instead of beans, the numerical value of the covariance between mass and volume, measured in the same units used for the beans, would be much larger, but it would stUl mean the same thing a more or less constant density. 

Instead of trying to use the covariance itself as a standard for comparing the degree of statistical association of different pairs of variables, we apply a scale factor to it, dividing each individual deviation from the average by the standard deviation of the corresponding variable. This results in a sort of normalized covariance, which is called the correlation coefficient of the two variables (Eq. (2.9)). This definition forces the correlation coefficient of any pair of random variables to always be restricted to the [—1,+1] interval. The correlations of different pairs of variables are then measured on the same scale (which is dimensionless, as can be deduced from Eq. (2.9)) and can be compared directly. 

Statistically independent variables have a zero correlation coefficient. The converse, however, is not true, because the correlation coefficient measures only the linear association between two variables. A zero correlation coefficient only indicates that a hnear relation is not present. Other types of dependence may be present and not be reflected in the numerical value of the correlation coefficient. Exercise 2.11 illustrates one such possibihty. 

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A theorem, which we do not prove here, states that the nonzero eigenvalues of the product AB are identical to those of BA, where A is an nxp and where B is a pxn matrix [3]. This applies in particular to the eigenvalues of matrices of cross-products XX and X which are of special interest in data analysis as they are related to dispersion matrices such as variance-covariance and correlation matrices. If X is an nxp matrix of rank r, then the product X X has r positive eigenvalues in A and possesses r eigenvectors in V since we have shown above that ... 

Similar expressions to those in eqs. (29.72) and (29.73) can be derived for the variance-covariances and correlations between the rows of X ... 

Number of variables is larger than the number of objects 2.3.2 Estimating Covariance and Correlation... 

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... 

These operations have been employed extensively throughout the text see, for example, the calculation of covariance and correlation about the mean and the origin developed in Chapter 3. 

In the following we will thus present some basic statistical methods useful for determining turbulence quantities from experimental data, and show how these measurements of turbulence can be put into the statistical model framework. Usually, this involves separating the turbulent from the non-turhulent parts of the flow, followed by averaging to provide the statistical descriptor. We will survey some of the basic methods of statistics, including the mean, variance, standard deviation, covariance, and correlation (e.g., [66], chap 1 [154], chap 2 [156]). 

Theoretical studies on variability (e.g., covariance) and correlation of graph invariants were presented by Hollas [Hollas, 2002, 2003, 2005a, 2005b, 2005c, 2006 Hollas, Gutman et of, 2005]. 

A total of 10,000 patients were simulated. The simulated mean weight was 70 kg with a range of 45-90 kg. The simulated mean CrCL was 7.2 L/h with a range of 5.7-8.7 L/h. The correlation between weight and CrCL was 0.23. The means, covariance, and correlation for the simulated data were acceptable in simulating a patient with normal renal function. 

Notice that for a symmetric matrix, the corresponding lower off-diagonal elements are defined by defining any of the upper off-diagonal elements and vice versa. Covariance and correlation matrices are symmetric. 

TABLE III The variances, covariances, and correlation coefficients of the values of a selected group of constants based on the 2006 CODATA adjustment. The numbers in bold above the main diagonal are 10 times the numerical values of the relative ... 

Computations of the covariance and correlation matrix are prerequisites for the application of factorial methods. 

In earlier sections, we discussed the independence in a relationship between two r.v. s. If there is a relationship, it may be strong or weak. In this section we discuss two numerical measures of the strength of a relationship between two r. v. s, the covariance and correlation. 

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

For the present model, the means, covariances, and correlation functions can be found from the expanded jump moments (see [2]). In terms of the coefficients given in Eqs. (10) and (11) they are... 

The simplest use of statistical methods is to provide summary parameters characterising important statistical properties of input variables and of various measures of catalyst performance (such as yield or degree of conversion), or relationships between them. Such summary parameters are usually called descriptive statistics, their common representatives are mean, median, variance, standard deviation, covariance and correlation. 

Covariance and correlation matrix can be formed from the original input matrix X ... 

CJomputation of the covariance and correlation coefficients of common failure between set x and y, Cov 0xj0y) and p 0g y) respectively, indicates that the most diversity occurs between the rt and f set, which is a little surprising. The computed values are ... 

Definition Covariance and correlation of two random variables hetX and Fbe two random variables. The covariance of AT and Y is... 

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