Covariance indicator

The covariances between the parameters are the off-diagonal elements of the covariance matrix. The covariance indicates how closely two parameters are correlated. A large value for the covariance between two parameter estimates indicates a very close correlation. Practically, this means that these two parameters may not be possible to be estimated separately. This is shown better through the correlation matrix. The correlation matrix, R, is obtained by transforming the co-variance matrix as follows... 

This result shows that, by its transformation properties, Aljkl is equivalent to a covariant vector of rank two. This process of summing over a pair of contravariant and covariant indices is called contraction. It always reduces the rank of a mixed tensor by two and thus, when applied to a mixed tensor of rank two, the result is a scalar ... 

Let us briefly recall a few of the basics of the algebra of tensors. An nth rank tensor in m-dimensional space is an object with n indices and rrf components. For a general tensor a distinction is made between contravari-ant (upper) indices and covariant (lower) indices. A tensor of rank mi + m2 may have mi contravariant indices and m2 covariant indices. The order of the indices is significant. Tensors can be classified according to whether they are... 

The use of literate programming techniques means that we can clearly indicate contravariant and covariant indices in the IAIj]]X fragment describing code for btas. 

The Cl and N03 covaried, indicating a common source, which was found to be two fertilizers applied in constant proportions a nitrogen-rich fertilizer and a potassium chloride fertilizer. 

Even though we do not invoke the full machinery of tensor analysis (Butkov, 1968), it is useful to keep the distinction between contravariant and covariant components clear. To avoid conflicting notation we do not use upper and lower indices to denote contravariant and covariant indices. Instead, we will use the suffix ao (lower case letters) on tensors whose indices are all contravariant, and AO (capital letters) on tensors whose indices are all covariant. No special suffix is used in the MO basis. For example, using the two- and four-index trace operators the energy is... 

We observe that the covariance indicator (cov(x,y)) has an expression which is similar to the dispersion of a statistical selection datum near the mean value (Eq. (5.11)). It is important to specify that the notion of variance (or dispersion) differs completely from the notion of covariance. 

In equation 17, N i2i)N i2j) is the number of the electron pairs classically expected from the basin population whereas N(Qi, f2 ) is the actual number of pairs obtained by integration of the pair-electron function over the basins and i2j. The variance is then a measure of the quantum mechanical uncertainty of the basin population which can be interpreted as a consequence of the electron delocalization, whereas the pair covariance indicates how much the population flucmations of two given basins are correlated. Within the AIM framework, the atomic locahzation and delocalization indices X A) and 5(A, B) have been introduced and defined by equations 18 and 19 ... 

A positive covariance indicates that both dimensions increase together, whereas if the covariance is negative, then as one dimension increases, the other decreases. If the covariance is zero, it indicates that the two dimensions are independent of each other. 

Indicator variable for missingness Case deletion Model/predict covariate Indicator variable for missingness... 

The process of reducing the rank of a tensor by summing over a pair of contravariant and covariant indices is known as contraction, e.g. 

Tensor components comprising both contravariant and covariant indices such as are denoted mixed tensors of the corresponding rank. Each index features its corresponding transformation property, e.g.. 

As we have seen above, the tensor has components with three indices, For such a tensor it is possible to perform a mathematical operation, called conttaction, that gives a co-vector as the result. In other words, it can be shown that suitable linear combinations of components of the tensor behave as the components of a vector. For the tensor it is possible to contract over the conttavariant index and over one of the covariant indices, and the result is a (co)-vector, indicated as or , whose generic (i-nth) component is (Einstein convention). Therefore, the explicit form of the three components of p in the reference system R are ... 

In study design, the problems of small numbers of subjects and large numbers of potential covariates indicate the need for realistic power estimates, in order to ensure that the numbers are sufficient to enable a deficit to be detected. 

The summation convention for double indices, for example, k in Eq. (113), is assumed, as before. However, we no longer make distinction between covariant and contravariant sets.) We set ourselves the task to find anti-Hermitean operators Xf, such that... 

Lambs received saline, oST at 40 )-lg/kg BW, or the indicated dose of hGRF per kg BW four times per day for 42 or 56 days. Half of the lambs were withdrawn from treatment after 42 days. Carcass data shown are for lambs treated 56 days. Carcass composition data were analy2ed by analysis of variance using carcass weight as the covariate. Data are summarized in Ref. 85. 

The two Ti curves do not covary in the quasigap. If the bonding was only between different atoms then the curves should covary. This indicates bonding between similar atoms. As noted above, the dd interatomic matrix elements for the Tii-Tii interaction are about 80 % larger than that in A3, supporting the idea of direct dd bonding. 

Note that if the kriging weights Xi(z) are data values-independent, the indicator estimates (11) are not hence, the final estimate of the conditional cdf Fx(z (N)) is data values-dependent. For each cut-off Z], the kriging estimator (11) requires a different indicator covariance model ... 

These various covariance models are Inferred directly from the corresponding indicator data i(3 z ), i-l,...,N. The indicator kriging approach is said to be "non-parametric, in the sense that it draws solely from the data, not from any multivariate distribution hypothesis, as was the case for the multi- -normal approach. 

These indicator covariance models allow differentiation of the spatial correlation of high-valued concentrations (cut-off high) and low to background-valued concentrations low). In the particular case study underlying the Figure 3, it was found that high value concentration data were more spatially correlated (due to the plume of pollution) than lower value data. 

Polynomial regression with indicator variables is another recommended statistical method for analysis of fish-mercury data. This procedure, described by Tremblay et al. (1998), allows rigorous statistical comparison of mercury-to-length relations among years and is considered superior to simple hnear regression and analysis of covariance for analysis of data on mercury-length relations in fish. 

If the errors in a, b, c are not too large, a linearized error approximation (see later discussion) can then be made by expanding the squares and cross-products of the infinitesimally small derivatives to variances and covariances. Using to indicate the coefficient of variation, Cx/x, we obtain... 

The first contaminated area, along the Cinca River and downstream of Monzon, reveals the covariant nature of PBDE and HBCD concentrations in the fish examined here. The amount of each contaminant is highly correlated within each fish. A likely explanation of this correlation is that the contaminants are being acquired from the same source (Cinca River). Moreover, PBDEs and HBCD are bioavailable as indicated by large fish to sediment ratios. Some correlations were found between the concentration levels of BFRs and the length and weight of specimen fishes,... 

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