Covariant basis

For each coordinate 2 in the full space, we may define a covariant basis vector 0R /02 and a contravariant basis vector 02 /0R, which obey orthogonality and completeness relations... 

A generalized set of reciprocal vectors for a constrained system is defined here to be any set off contravariant basis vectors b, ..., b- and K covariant basis... 

To prove Eq. (2.182), we start with the Cartesian divergence, and expand in covariant basis vectors, and the Cartesian gradient in covariant basis vectors, to obtain... 

Obviously, the contravariant AO-basis is not equivalent with the covariant basis because of the non-orthogonality of the AO basis functions. If we try to transform the contravariant quantity again to the MO basis with the AO-MO coefficients, we will get the following equation... 

Embedded coordinates are used (recalling Note 2.5) with the covariant basis ... 

This shows that a covariant tensor is introduced on a contravariant basis e of the dual space, while a contravariant tensor a = ad e, (gi e j is given on a covariant basis ct which is defined by a(, tj) for tj V. Mixed tensors corresponding to (, u) and ( , are similarly introduced. 

The expression (18) features in the calculation of surface gradients. (An alternative derivation for the normal is also available through n = ti x t2.) It is useful to introduce the covariant basis vectors and in terms of the contravariant ones as follows ... 

Here C = aa ) is the covariance of the basis functions used to model the turbulence. Covariance matrices are positive semi-definite by dehnition which implies a C a > 0, and thus a dehned maximum of Pr a exists. 

The relationship between the noise and atmospheric covariances is also evident in Eq. 17. If the noise on the measurements is large the N term dominates the inverse which means only the large eigenvalues of C contribute to the inverse. Consequently only the low order modes are compensated and a smooth reconstruction results. When the data is very noisy then 1 and hence a tend to zero. If the data is very noisy, then no estimate of the basis coefficients is made. 

If we decide to only estimate a finite number of basis modes we implicitly assume the coefficients of all the other modes are zero and that the covariance of the modes estimated is very large. Thus QN Q becomes large relative to C and in this case Eq. 16 simplifies to a weighted least squares formula... 

The next step was to augment and expand the model to be able to predict the dose response for the comparator. Comparator data, from SBA (summary basis for approval data submitted to the FDA), yielded one model that predicted both candidate and comparator performance. The model accounted for age, disease baseline, and trial differences. Differences based on sex, weight, and other covariates were estimated to be negligible. The addition of the comparator data improved the predictive ability of the model for both drugs (Fig. 22.3). 

A later analysis (Emhart et al. 1987) related PbB levels obtained at delivery (maternal and cord blood) and at 6 months, 2 years, and 3 years of age to developmental tests (MDI, PDI, Kent Infant Development Scale [KID], and Stanford-Binet IQ) administered at 6 months, 1 year, 2 years, and 3 years of age, as appropriate. After controlling for covariates and confounding risk factors, the only significant associations of blood lead with concurrent or later development were an inverse association between maternal (but not cord) blood lead and MDI, PDI, and KID at 6 months, and a positive association between 6-month PbB and 6-month KID. The investigators concluded that, taken as a whole, the results of the 21 analyses of correlation between blood lead and developmental test scores were "reasonably consistent with what might be expected on the basis of sampling variability," that any association of blood lead level with measures of development was likely to be due to the dependence of both PbB and... 

Statistical properties of a data set can be preserved only if the statistical distribution of the data is assumed. PCA assumes the multivariate data are described by a Gaussian distribution, and then PCA is calculated considering only the second moment of the probability distribution of the data (covariance matrix). Indeed, for normally distributed data the covariance matrix (XTX) completely describes the data, once they are zero-centered. From a geometric point of view, any covariance matrix, since it is a symmetric matrix, is associated with a hyper-ellipsoid in N dimensional space. PCA corresponds to a coordinate rotation from the natural sensor space axis to a novel axis basis formed by the principal... 

In the previous development it was assumed that only random, normally distributed measurement errors, with zero mean and known covariance, are present in the data. In practice, process data may also contain other types of errors, which are caused by nonrandom events. For instance, instruments may not be adequately compensated, measuring devices may malfunction, or process leaks may be present. These biases are usually referred as gross errors. The presence of gross errors invalidates the statistical basis of data reconciliation procedures. It is also impossible, for example, to prepare an adequate process model on the basis of erroneous measurements or to assess production accounting correctly. In order to avoid these shortcomings we need to check for the presence of gross systematic errors in the measurement data. 

As discussed before, in a strict sense, there is always some degree of dependence between the sample data. An alternative approach is to make use of the covariance matrix of the constraint residuals to eliminate the dependence between sample data (or the influence of unsteady-state behavior of the process during sampling periods). This is the basis of the so-called indirect approach. 

The basis for calculating the correlation between two variables xj and xk is the covariance covariance matrix (dimension m x m), which is a quadratic, symmetric matrix. The cases j k (main diagonal) are covariances between one and the same variable, which are in fact the variances o-jj of the variables Xj for j = 1,..., m (note that in Chapter 1 variances were denoted as variance—covariance matrix (Figure 2.7). Matrix X refers to a data population of infinite size, and should not be confused with estimations of it as described in Section 2.3.2, for instance the sample covariance matrix C. 

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... 

The generalized Fisher theorems derived in this section are statements about the space variation of the vectors of the relative and absolute space-specific rates of growth. These vectors have a simple natural (biological, chemical, physical) interpretation They express the capacity of a species of type u to fill out space in genetic language, they are space-specific fitness functions. In addition, the covariance matrix of the vector of the relative space-specific rates of growth, gap, [Eq. (25)] is a Riemannian metric tensor that enters the expression of a Fisher information metric [Eqs. (24) and (26)]. These results may serve as a basis for solving inverse problems for reaction transport systems. 

The terms variant and covariant refer to the transformation properties of the quantities. A transformation may be defined by the transformation matrix T operating on the direct space basis a, such that... 

Any 37/-dimensional Cartesian vector that is associated with a point on the constraint surface may be divided into a soft component, which is locally tangent to the constraint surface and a hard component, which is perpendicular to this surface. The soft subspace is the /-dimensional vector space that contains aU 3N dimensional Cartesian vectors that are locally tangent to the constraint surface. It is spanned by / covariant tangent basis vectors... 

At both the trial level and the development plan level, statisticians should take time to review the case report forms (CRFs) to make sure, in particular, that the data being collected will be appropriate for the precise, unambiguous and unbiased measurement of primary and secondary endpoints. Other aspects of the data being collected should also be reviewed in light of the way they will be used in the analysis. For example, baseline data will form the basis of covariates to be used in any adjusted analyses, intermediate visit data may be needed for the use of... 

Detail regarding the methods of analysis for the primary endpoint(s) including specification of covariates to be the basis of any adjusted analyses... 

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