Negative covariance

Some variables often have dependencies, such as reservoir porosity and permeability (a positive correlation) or the capital cost of a specific equipment item and its lifetime maintenance cost (a negative correlation). We can test the linear dependency of two variables (say x and y) by calculating the covariance between the two variables (o ) and the correlation coefficient (r) ... 

In case of correlated parameters, the corresponding covariances have to be considered. For example, correlated quantities occur in regression and calibration (for the difference between them see Chap. 6), where the coefficients of the linear model y = a + b x show a negative mutual dependence. 

Just as in everyday life, in statistics a relation is a pair-wise interaction. Suppose we have two random variables, ga and gb (e.g., one can think of an axial S = 1/2 system with gN and g ). The g-value is a random variable and a function of two other random variables g = f(ga, gb). Each random variable is distributed according to its own, say, gaussian distribution with a mean and a standard deviation, for ga, for example, (g,) and oa. The standard deviation is a measure of how much a random variable can deviate from its mean, either in a positive or negative direction. The standard deviation itself is a positive number as it is defined as the square root of the variance ol. The extent to which two random variables are related, that is, how much their individual variation is intertwined, is then expressed in their covariance Cab ... 

If two random variables are nncorrelated, then both their covariance Cab and their correlation coefficient rab are equal to zero. If two random variables are fully correlated, then the absolute value of their covariance is C,J = cacb, and the absolute value of their correlation coefficient is unity rab = 1. A key point to note for our EPR linewidth theory to be developed is that two fully correlated variables can be fully positively correlated rab = 1, or fully negatively correlated rab = -1. Of course, if two random variables are correlated to some extent, then 0 < Cab < oacb, and 0 < IrJ < 1. 

Equation 41-A3 can be checked by expanding the last term, collecting terms and verifying that all the terms of equation 41-A2 are regenerated. The third term in equation 41-A3 is a quantity called the covariance between A and B. The covariance is a quantity related to the correlation coefficient. Since the differences from the mean are randomly positive and negative, the product of the two differences from their respective means is also randomly positive and negative, and tend to cancel when summed. Therefore, for independent random variables the covariance is zero, since the correlation coefficient is zero for uncorrelated variables. In fact, the mathematical definition of uncorrelated is that this sum-of-cross-products term is zero. Therefore, since A and B are random, uncorrelated variables ... 

Negative covariances may occur in some intervals and result in nonsense values, such as negative pinterval The convention in such cases is to assume p., = 0 in intervals to the left of the hitmax, and p., = 1 in inter-vals to the right of the hitmax. Also, covariance in an interval can be so high as to result in a negative number in the square root term. The convention in this case is to assume that the square root term = 0. 

When a f>, sap may be negative. This generally occurs, for example, when the covariance ((p a(p p) is negative. 

The extension of simple relationships such as (5.60) to multiple-step chemistry has proven to be elusive. For example, (5.60) implies that the covariance between two scalars is always negative.33 However, this need not be the case. For example, if B were produced by another reaction between A and R 34... 

Consistent with these definitions, DNS data (Yeung 1994 Yeung 1996) show that Taa is always positive, while Taa is always negative. Analogous definitions and remarks hold for the scalar-covariance transfer function, i.e., for Tap and T p. 

Assume also that we have obtained responses of yn = 4.5 and y,2 = 6.0. The first-order relationship through these points is shown as a solid line in Figure 7.6. Again, for purposes of illustration, suppose the experiments are repeated twice, and values of >>13 = 4.35, y, = 6.15, yi, = 4.65, and y,6 = 5.85 are obtained the corresponding relationships are shown as dashed lines in Figure 7.6. For these sets of data, the slope and the y,-intercept do not tend to increase and decrease together. Instead, each parameter estimate varies oppositely as the other the covariance is negative. 

Figure 7.6 Illustration of negative covariance if the slope (6,) increases, the intercept (6 ) decreases if the slope decreases, the intercept increases.

Finally, the effect of the position of the third experiment on the covariance associated with hg nd h, is seen in Figure 8.6 to equal zero at x,3 = 0. If the third experiment is located at x, < 0, then the estimates of the slope and intercept vary together in the same way (the covariance is positive see Section 7.4). If the third experiment is located at x, > 0, the estimates of the slope and intercept vary together in opposite ways (the covariance is negative). 

So one can consider V as the eigenvector representation of E and QT as the eigenvector representation of C. Unfortunately, these matrices in their present form have little physical significance. The matrix QT whose rows represent eigenvectors of the covariance matrix must necessarily be orthogonal to each other and therefore must be negative at some points. The concept of a negative concentration is novel but unacceptable. 

The last term in Equation C-2 reflects the fact that uncertainties in m and b are not independent of each other. The quantity smh is called the covariance and it can be positive or negative. 

The off diagonal tern has expected value 0. Each tain in the sum in the lower right has expected value a2, so, after collecting terns, taking the negative, and inverting, we obtain the asymptotic covariance mahix,... 

Figure 8. Plot of average collagen SI3C and apatite Sl80. Assuming the alignment indicates environmentally-related covariation of the two coordinates, SI3C values that are less negative relative to the alignment would indicate higher consumption of animal protein.

In the trace-element data, the first principal component accounts for over 50% of the variance. Aluminum and most other elements correlate positively with the first principal component, a pattern consistent with simple dilution (22,23) - in this case, by quartz sand temper. In contrast, the second principal component (accounting for an additional 15% of the variance) represents the heavy mineral sand component (Ti, Hf, Zr), which negatively covaries with cobalt, manganese, antimony, and arsenic. The Qo and Qm clays from the lowlands are broadly similar in composition (Figure 5). The Qc deposit differs significantly, i.e., the low PC2 scores indicate high concentrations of the characteristic of heavy mineral sands (Ti, Hf, Zr). The Qk and Tp samples span range of composition, but are represented by only 2 samples each. 

---