Covariance matrix, definition

The difference between the rigorous definitions of the covariance matrix of x for the new and old cases is... 

Assuming that j are normally distributed and uncorrelated, with zero mean and known positive definite covariance matrix the parameter estimation problem can be formulated as minimizing with respect to zj and 0 ... 

Assuming that e is Gaussian with zero mean and positive definite covariance matrix I then... 

The covariance matrix is positive semi-definite and symmetric. Thus, it can be written in terms of eigenvalues and eigenvectors as... 

Let us call x the vector of Na and Cl concentrations, x the vector of sample means and S the symmetric, positive-definite covariance matrix, i.e., the 2 x 2 matrix with variances on the diagonal and the covariance between Na and Cl concentrations as off-diagonal terms. The equation of the ellipse to be drawn can be written... 

The covariance matrix is factored using the diagonal matrix A and the eigenvector matrix U as U UT. Since 5 is symmetric and positive-definite, the eigenvalues are positive and the eigenvectors orthogonal. The inverse S 1 of S can be expanded as UA 1UT and the transformation... 

Multiplying the last two equations and using the definition of the covariance matrix, we obtain... 

Both S, by definition, and F through equation (4.4.12), are centered, i.e., their expectation is a null matrix. Therefore the sample covariance matrix between the reduced data and the components is... 

If y1 Y2, and Y3 are normally distributed, the constant probability surfaces are ellipsoids centered at y (Figure 5.12) and the statistical projection y of y will be defined as the point where the plane is tangent to the innermost probability ellipsoid. Points on the same ellipsoid are by definition at the same statistical distance from y. If Sy is the covariance matrix of the vector y, the statistical distance c between y and y is given by... 

From the definition of variance and correlation coefficient and equation (4.2.18), we find that the covariance matrix is... 

Based on the definition of the covariance cjk in Equation 2.9, the sample covariance matrix C can be calculated for mean-centered X by... 

Calculate the variance-covariance matrix associated with the straight line relationship y, = Po + PiA i, + r, for the following data (see Section 11.2 for a definition of D) ... 

Because of Eq. (4.31), A is the variance-covariance matrix of the set of unknowns X, which we will refer to as the eigenparameter. The eigenparameters X are, by the definition of the variance-covariance matrix, not correlated. 

Suppose we change the assumptions of the model in Section 5.3 to AS5 (x ) are an independent and identically distributed sequence of random vectors such that x, has a finite mean vector, finite positive definite covariance matrix Zxx and finite fourth moments E[xjxj xixm] = for all variables. How does the proof of consistency and asymptotic normality of b change Are these assumptions weaker or stronger than the ones made in Section 5.2 ... 

Consider GMM estimation of a regression model as shown at the beginning of Example 18.8. Let Wj be the optimal weighting matrix based on the moment equations. Let W2 be some other positive definite matrix. Compare the asymptotic covariance matrices of the two proposed estimators. Show conclusively that the asymptotic covariance matrix of the estimator based on Wj is not larger than that based on W2. 

From the definition of d the variance-covariance matrix % is evaluated, taking into account the variance-covariance matrices of the input data x, and of the instrument readings yt [1]. 

Suppose that the variables BJ are to be determined by a least-squares fit of the relations, Eq. 16, to the measured values T exp (vector Yexp). Assume that the measurements Yexp are unbiased ( (Yexp) = Ytrue where E() represents the mean or expectation value) and that the measurement errors and their correlations are described by the positive-definite nxn variance-covariance matrix 0Y which can be written as the dyadic P... 

From what has been shown in the preceding sections (cf. Eqs. 61 and 73, 83), it is possible to present the molecular structure resulting from both the r -fit method and any of the r()-derived methods in a convenient and easily comparable form, as a structural description in both Cartesian and internal coordinates, and with consistent errors and correlations (for small and larger molecules). A detailed comparison would require a sufficiently large SDS to determine a complete molecular structure, but the requirements are still the least restrictive of all methods presented. The input data must include the covariance matrix of the rotational constants or moments. This matrix may have to be adequately modeled to avoid grossly different weighting of isotopomers which is usually not warranted. The definition of the input data set... 

A real, symmetric matrix A is called positive definite if x Ax > 0 for every conforming nonzero real vector x. Extend the result of (a) to show that the covariance matrix E in Eq. (4.C-1) is positive definite if the scalar random variables i ,.... Emu are linearly independent, that is, if there is no nonzero m-vector x such that x Eu vanishes over the sample space of the random vector . 

In the previous section, we utilized local PCA to represent the moving normal modes, which depicted the locally harmonic but globally anharmonic dynamics of proteins. The major difficulty we met within the local PCA was the fact that two principal modes determined in adjacent time-windows, e,variance-covariance matrix C, or the quasidegeneracy in C. From a statistical viewpoint, this difficulty can be attributed to statistical fluctuation in the estimation of the principal modes due to the small sampling size in the determination of local PCA. 

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