Covariances

Whereas standard deviation and variance are pnrely one dimensional, covariance is measnred between two dimensions. Instead of taking squares of the distances of the one-dimensional data x, we take the product of distances for two variables, x and y. For each data item, multiply the difference of the x value and the mean of x by the difference between the y value and the mean of y. 

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. 

---

The primary purpose for expressing experimental data through model equations is to obtain a representation that can be used confidently for systematic interpolations and extrapolations, especially to multicomponent systems. The confidence placed in the calculations depends on the confidence placed in the data and in the model. Therefore, the method of parameter estimation should also provide measures of reliability for the calculated results. This reliability depends on the uncertainties in the parameters, which, with the statistical method of data reduction used here, are estimated from the parameter variance-covariance matrix. This matrix is obtained as a last step in the iterative calculation of the parameters. 

The off-diagonal elements of the variance-covariance matrix represent the covariances between different parameters. From the covariances and variances, correlation coefficients between parameters can be calculated. When the parameters are completely independent, the correlation coefficient is zero. As the parameters become more correlated, the correlation coefficient approaches a value of +1 or -1. 

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) ... 

Let u be a vector valued stochastic variable with dimension D x 1 and with covariance matrix Ru of size D x D. The key idea is to linearly transform all observation vectors, u , to new variables, z = W Uy, and then solve the optimization problem (1) where we replace u, by z . We choose the transformation so that the covariance matrix of z is diagonal and (more importantly) none if its eigenvalues are too close to zero. (Loosely speaking, the eigenvalues close to zero are those that are responsible for the large variance of the OLS-solution). In order to liiid the desired transformation, a singular value decomposition of /f is performed yielding... 

This covariance ftmetion vanishes as t-5 approaches because the initial density profile has a finite integral, that creates a vanishing density when it spreads out over the infinite volume. 

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... 

The important underlying components of protein motion during a simulation can be extracted by a Principal Component Analysis (PGA). It stands for a diagonalization of the variance-covariance matrix R of the mass-weighted internal displacements during a molecular dynamics simulation. 

Step 2 This ensemble is subjected to a principal component analysis (PCA) [61] by diagonalizing the covariance matrix C G x 7Z, ... 

The free energy differences obtained from our constrained simulations refer to strictly specified states, defined by single points in the 14-dimensional dihedral space. Standard concepts of a molecular conformation include some region, or volume in that space, explored by thermal fluctuations around a transient equilibrium structure. To obtain the free energy differences between conformers of the unconstrained peptide, a correction for the thermodynamic state is needed. The volume of explored conformational space may be estimated from the covariance matrix of the coordinates of interest, = ((Ci [13, lOj. For each of the four selected conform-... 

APPENDIX - SETMMARY OF VECTOR AND TENSOR ANALYSTS 8.2.4 Covariant and contravariant vectors... 

Similar to vectors, based on the transfomiation properties of the second tensors the following three types of covariant, contravariant and mixed components are defined... 

Note that convected derivatives of the stress (and rate of strain) tensors appearing in the rheological relationships derived for non-Newtonian fluids will have different forms depending on whether covariant or contravariant components of these tensors are used. For example, the convected time derivatives of covariant and contravariant stress tensors are expressed as... 

The matrix gp, represents the components of a covariant second-order tensor called the metric tensor , because it defines distance measurement with respect to coordinates To illustrate the application of this definition in the... 

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. 

Rollins, D.K. and J.F. Davis, Gross Error Detection when Variance-Covariance Matrices are Unknown, AlChE Journal, 39(8), 1993, 13.35-1341. (Unknown statistics)... 

Define the variance-covariance matrix for this vector to be Q = B (BJB B... 

Principal component analysis (PCA) takes the m-coordinate vectors q associated with the conformation sample and calculates the square m X m matrix, reflecting the relationships between the coordinates. This matrix, also known as the covariance matrix C, is defined as... 

---