Autocorrelation matrix

Here, the force F is a linear combination of the components of R it also has a Gaussian distribution and autocorrelation matrix that satisfies the same properties of R t) as shown in eq. (3), with I (the nxn unit matrix) replacing M [71] ... 

Exercise. Let Y be an r-variable Gaussian Markov process. Apply a linear transformation so that Pi(y, t) exp[— %y2. Then derive for the autocorrelation matrix... 

As a second conclusion from the moment equations we compute the autocorrelation matrix. Using (5.10a) and (5.12) one first finds... 

The multivariate autocorrelation matrix Rxx %) (for centered values) is defined as ... 

Then, for each topological distance k, a square symmetric general atom-type matrix, called atom-type autocorrelation matrix, denoted as ATAC, of size Nat x Nat, vvhere Nat is the total number of defined atom types, can be defined as [Authors, This Book]... 

Each uth row sum of the atom-type autocorrelation matrix is the number of vertices of any type located at distance k from vertices of type u if all the graph vertices are assigned an atom type, then the Wiener-type index of the matrix gives the graph distance count of order k. Note that since diagonal elements of atom-type autocorrelation matrices can be different from zero, the —> Wiener operator has to be applied as the following ... 

For a 3D-QSAR autocorrelation matrix, the distances are calculated from the 3D structures of the molecules. Both points on a CoMFA-like lattice and points on the molecular surface have been used for these distance calcula-tions.2 7 208 Similarly, the 3D autocorrelation properties are based on properties at these points (e.g., electrostatic or hydrophobic potential). Wagener et al. used a point density of 10 points/A on the van der Waals surface for the property calculation for the autocorrelation matrix, they considered distances from 1 to 13 A and a distance interval of 1 A to produce an autocorrelation vector of length 12.2° These 12 properties were then analyzed by principal components analysis, a Kohonen map, and a feed-forward multilayer neural... 

This expression is derived as the Fourier transform of a time-dependent one-particle autocorrelation function (26) (i.e. propagator), and cast in matrix form G(co) over a suitable molecular orbital (e.g. HF) basis, by means of the related set of one-electron creation (ai" ") and annihilation (aj) operators. In this equation, the sums over m and p run over all the states of the (N-1)- and (N+l)-electron system, l P > and I P " respectively. Eq and e[ represent the energy of the... 

The rate coefficient kernel of eqn. (369) and the initial condition term of eqn. (370) are those given by Northrup and Hynes [ 103]. These expressions are cast into the form of correlation functions (cf. the velocity autocorrelation function) and have a close similarity to the matrix elements in quantum mechanic applications. While they are quite easy to derive and... 

The stochastic quantity Y(t) may consist of several components Yj(t) (j = 1,2,..., r). The autocorrelation function is then replaced with the correlation matrix... 

Exercise. Determine the relation between the 2 x 2 correlation matrix of a complex process and its complex autocorrelation function. 

Exercise. For one-step processes W is a tridiagonal matrix. With the aid of (3.8) a similarity transformation can be constructed which makes the matrix symmetric, as in (V.6.15). Prove in this way that any finite one-step process has a complete set of eigenfunctions, and that its autocorrelation function consists of a sum of exponentials.510... 

However, if the initial state is a thermal state, such as the canonical den-sity matrix p - (1/Z)exp(- 3//), the autocorrelation is no longer given by a single quantum amplitude but becomes a sum of quantum amplitudes in which quantum phases are randomized. In the classical limit ft - 0, the leading expression becomes the purely classical autocorrelation function with the dynamics being ruled by the classical Liouvillian operator ci = Hci> ... 

The velocity autocorrelation function can be obtained from the relaxation equation [Eq. (76)], where Cv(z) = Cjt(q = 0z). Here the suffix s stands for single-particle property. For zero wavenumber, there is no contribution from the frequency matrix [that is, D v(q = 0) = 0] and the memory function matrix becomes diagonal. If we write (z) = Tfj (q = 0z), then the VACF in the frequency plane can be written as... 

With this consideration die relaxation equation will give rise to a set of coupled equations involving the time autocorrelation function of the density and the longitudinal current fluctuation, and also there will be cross terms that involve the correlation between the density fluctuation and the longitudinal current fluctuation. This set of coupled equations can be written in matrix notation, which becomes identical to that derived by Gotze from the Liouvillian resolvent matrix [3]. 

The multivariate autocorrelation function should contain the total variance of these autocorrelation matrices in dependence on the lag x. Principal components analysis (see Section 5.4) is one possibility of extracting the total variance from a correlation matrix. The total variance is equal to the sum of positive eigenvalues of the correlation matrices. This function of matrices is, therefore, reduced into a univariate function of multivariate relationships by the following instruction ... 

The multivariate autocorrelation function of the measured values compared with the highest randomly possible correlation value shows significant correlation up to Lag 7. So, the range of multivariate correlation is more extended than that of univariate correlation (see Section 9.1.3.3.1). This fact must be understood because the computation of the MACF includes the whole data matrix with all interactions between the measured parameters. For characterization of the multivariate heavy metal load of the test area only 14 samples in the screen are necessary. 

Each rotational state is coupled to all other states through the potential matrix V defined in (3.22). Initial conditions Xj(I 0) are obtained by expanding — in analogy to (3.26) — the ground-state wavefunction multiplied by the transition dipole function in terms of the Yjo- The total of all one-dimensional wavepackets Xj (R t) forms an R- and i-dependent vector x whose propagation in space and time follows as described before for the two-dimensional wavepacket, with the exception that multiplication by the potential is replaced by a matrix multiplication Vx-The close-coupling equations become computationally more convenient if one makes an additional transformation to the so-called discrete variable representation (Bacic and Light 1986). The autocorrelation function is simply calculated from... 

Let us then derive an expression for the matrix element of the flux operator in the coordinate representation, an expression we need in order to develop the time autocorrelation function of the flux operator in the coordinate representation. We use the axiom for the matrix element of the momentum operator in the coordinate representation, and obtain... 

In this work, we will most frequently use the symbols c = c(nr) when referring to both the time signals and the autocorrelation functions Cn = C(nr). The set exp (—ia>kt) is exceedingly nonorthogonal, and this property causes numerical difficulties in all methods for nonlinear fittings of experimental time signals c to the form (42) as discussed in Ref. [46]. If instead of the matrix element (4>0 US 0), we consider a more general scalar product... 

From Eqs. (5.2) and (S.5) we see that the diagonal matrix elements of the resolvent and the Laplace transform of an autocorrelation functions are related by... 

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