Matrix second moment

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... 

The superscript used in the coefficient matrices in (6.192) is a reminder that the statistics must be evaluated at the notional-particle location. For example, e = e(X r), and the scalar standard-deviation matrix and scalar correlation matrix p are computed from the location-conditioned scalar second moments X )(X, t). 

The second moments may be combined into an r x r matrix . More important is the covariance matrix... 

The equations for the second moments are conveniently written in matrix form,... 

The stochastic differential equation (5.2) also generates equations for the second moments (ip q) p" (qr)), see (XVI.2.18). These moments constitute the density matrix. An easier way to find the equation for the density matrix is by writing (5.5) by means of the density matrix,... 

Experience has shown the covariance matrix 0rm to be conspicuously different when the same problem was treated by either the rs-method (without enforcing the first and second moment conditions) or any of the r0-derived methods. For the former method the errors of the coordinates were much less correlated. This, as well as the better condition number of the normal equation system, is no doubt a... 

Arai)2 = (ft rl (pi) - (coordinate-system-dependent. If we want to choose coordinate-system-independent quantities, we can use the second moment matrix... 

Normalized distance distribution moments were used to define —> molecular profiles and the second moment Dx takes part in defining the —> hyper-Wiener index) moreover, the index D2 was demonstrated to be equal to half the trace of the distance matrix D raised to the second power [Diudea, 1996a Diudea and Gutman, 1998]. 

One effect of the use of a diagonal weight matrix is to make the solution to Eq. (18) depend on the particular linear combination of moments of inertia used. Thus, use of principal moments of inertia, or principal planar second moments, or differences in either of these all lead to different structural parameters even if the same set of rotational constants is used to derive the moments. By comparing structural parameters derived from each set of moments, some insight into the effect of the correlations between the moments and the magnitudes of the model errors may be obtained. 

With H>, = 1, M is a coordinate second-moment matrix with eigenvectors e, ei, 3. The corresponding eigenvalues M, M2, M3 (in decreasing order) are the sums of squares of deviations of the atoms from the planes normal to these vectors. Thus is normal to the best plane, ei is normal to the worst plane, and 2 is perpendicular to e, and ej. 

The left-hand side of the last equation involves the variance-covariance matrix of the observed distribution of structural parameters, and the right-hand side involves the compliance matrix C = for the system in question as well as model provides a conceptual basis for relating observed distributions P x -xo) to the energy surfaces associated with small x, but its actual application is beset with obstacles. 

Once r has been calculated, the moments of the distribution function can be found. For chains in free solution, Flory has developed an elegant generator matrix technique that by-passes r and calculates the second moment directly. Unfortunately, for attached polymer chains, such as terminally... 

From second moment of radical from pentadeutero pyridine in DCl—DjO matrix. 

However, since the power spectra of the excitation forces are available and the structure Is assumed to be linear, the power spectral method can be applied to determine first and second moment of the response. Hence, carring out a Fourier transform of these matrix equations, the mechanical transfer functions of the structure con be established, in order to reduce the computational effort only the mechanical transfer functions of the top floor deflections and the first floor load effects are determined explicitly. Considering these transfer functions the matrix of the power spectra of the response quantities can be determined as follows. 

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