Gaussian distribution covariance

Just as in everyday life, in statistics a relation is a pair-wise interaction. Suppose we have two random variables, ga and gb (e.g., one can think of an axial S = 1/2 system with gN and g ). The g-value is a random variable and a function of two other random variables g = f(ga, gb). Each random variable is distributed according to its own, say, gaussian distribution with a mean and a standard deviation, for ga, for example, (g,) and oa. The standard deviation is a measure of how much a random variable can deviate from its mean, either in a positive or negative direction. The standard deviation itself is a positive number as it is defined as the square root of the variance ol. The extent to which two random variables are related, that is, how much their individual variation is intertwined, is then expressed in their covariance Cab ... [Pg.157]

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... [Pg.154]

The vector nk describes the unknown additive measurement noise, which is assumed in accordance with Kalman filter theory to be a Gaussian random variable with zero mean and covariance matrix R. Instead of the additive noise term nj( in equation (20), the errors of the different measurement values are assumed to be statistically independent and identically Gaussian distributed, so... [Pg.307]

As was shown, the conventional method for data reconciliation is that of weighted least squares, in which the adjustments to the data are weighted by the inverse of the measurement noise covariance matrix so that the model constraints are satisfied. The main assumption of the conventional approach is that the errors follow a normal Gaussian distribution. When this assumption is satisfied, conventional approaches provide unbiased estimates of the plant states. The presence of gross errors violates the assumptions in the conventional approach and makes the results invalid. [Pg.218]

This shows that for a Gaussian distribution the covariance matrix defined in (3.8) is equal to A-1. It follows that a Gaussian distribution is fully determined by the averages of the variables and their covariance matrix. In particular, if the variables are uncorrelated, A-1 is diagonal and hence also A, so that the variables are also independent. Thus, provided that it is known that the joint distribution is Gaussian, uncorrelated implies independent (compare the Exercise in 3). This independence can always be achieved by a linear, and even by an orthogonal, transformation of the variables. [Pg.24]

Having found the moments and covariance we are now able to construct the corresponding Gaussian distribution... [Pg.212]

It should be emphasized that for the Markovian copolymers, the knowledge of these structure parameters will suffice for finding the probabilities of any sequences LZ, i.e., for a comprehensive description of the structure of the chains of such copolymers at their given average composition. As for the CD of the Markovian copolymers, for any fraction of Z-mers it is described at Z 1 by the normal Gaussian distribution with covariance matrix, which is controlled along with Z only by the values of structure parameters (Lowry, 1970). The calculation of their dependence on time and on the kinetic parameters of a reaction system enables a complete statistical description of the chemical structure of a Markovian copolymer. It is obvious therewith to which extent a mathematical modeling of the processes of the synthesis of linear copolymers becomes simpler when the sequence of units in their macromolecules is known to obey Markov statistics. [Pg.172]

Consider the several random variables Xj, j = 1,... n. Suppose that these Xj are distributed according to a multi-variable gaussian distribution with means [xj) and covariances [8xj8x = [xjX ) — [xj) (x ). Show that... [Pg.67]

In the literature (Chalons et al, 2010), only a bivariate EQMOM with four abscissas represented by weighted Gaussian distributions with a diagonal covariance matrix has been considered. However, it is likely that brute-force QMOM algorithms can be developed for other distribution functions. Using the multi-Gaussian representation as an example, the approximate NDF can be written as... [Pg.93]

Note that, in order for the Gaussian distributions to be well defined, all of the covariance matrices in Eq. (6.126) must be nonnegative. This leads to the condition that 4 > (1 + en)E2 > 0, or, equivalently, that 2 > yU2i > 0, which is always true. [Pg.249]

The factorial methods in this chapter are also called second-order transformations, because only two moments, mean and covariance, are needed to describe the Gaussian distribution of the variables. Other second-order transformations are FA, independent component analysis (ICA), and multivariate curve resolution (MCR). [Pg.144]

For large N, the posterior PDF can be approximated by Gaussian distribution and its covariance matrix is given by the inverse of the Hessian matrix. [Pg.66]

This optimization problem can be solved by the MATLAB function fminsearch [171]. It has been shown numerically for the globally identifiable case with a large number of data points that the updated PDF can be well approximated by a Gaussian distribution 0(9 9, H(9 ) ) with mean 9 and covariance matrix H(9 )- -, where U(9 ) denotes the Hessian oiJ(9) calculated ate = 9 ... [Pg.108]

In the case where a non-informative prior is used, the first term can be simply neglected. Furthermore, the updated PDF of the parameter vector 0 can be well approximated by a Gaussian distribution G(0 0, Ti(0 ) ) with mean 0 and covariance matrix H(0 ), where H 0 ) denotes the Hessian matrix of the objective function J calculated at = ... [Pg.115]

Then, the joint PDF p(yi, y2,.. , yNp 0, C) follows an NgNp-vaiisite Gaussian distribution with zero mean and covariance matrix ... [Pg.172]

Therefore, the conditional probability density p(yn yn-Np> yn-Np+i> > y -i. O follows an Ao-variate Gaussian distribution with mean y and covariance matrix... [Pg.173]

The posterior PDF in Equation (5.18) can be well approximated by a Gaussian distribution centered at the optimal (most probable) parameters (X, 0, ) and with covariance matrix T(X, 0, 9) equal to the inverse of the Hessian of the objective function 7(X, 0,9) = -Inp(X, 0,9 X, f,C) calculated at the optimal parameters [19], This covariance matrix is given by ... [Pg.201]

Gaussian distribution of random vector x with mean jx and covariance matrix E... [Pg.311]

Often, a xj-dimensJonaI Gaussian distribution is used which is defined by the mean vector of a class and the covariance matrix C380, 389, 391,... [Pg.81]

Thus the prior distribution of f is iV(0, Q) = Ai(0, fflRR ). However, each measurement contains noise, which we assume to be Gaussian with zero mean and variance cr. The vector of data points also has Gaussian distribution E(t) A (0, Q + ffv). We denote the covariance matrix of t by C =Q + ffyl-The distribution of the joint probability of observing tjv+i having previously observed t can be written as... [Pg.25]

We assume that the errors in the measurements are statistically independent, scaled by the weights, u , in such a way that they have equal variance ((T ) and come from a Gaussian distribution. In case of these reasonable assumptions weighted least squares coincides with the maximum likelihood estimate. The (weighted) experimental errors of the measurements are given by Y 0)y as in (6.2). This means that the covariance matrix of the experimental errors is given by ... [Pg.232]

We assume of course H > 0 (else v = 0 according to Remark (ii) to Section 9.2). On the other hand, we can assume H < I see (9.2.33). But then rankA = H < I hence A is not of full row rank and the random variable v does not have a probability density as introduced in Appendix E. The covariance matrix of random variable v is thus (positive semidefinite but) singular (not regular). If e is Gaussian, the adjustment vector v has a degenerate Gaussian distribution we shall not examine its properties. We have anyway the mean... [Pg.313]

A number of methods allow the estimation of probability densities, (a) A multivariate Gaussian distribution can be assumed the parameters are the class mean and the covariance matrix, (b) The p-dimensional probability density is estimated by the product of the probability densities of the p features, assuming they are independent, (c) The probability density at location x is estimated by a weighted sum of (Gaussian) kernel functions that have their centers at some prototype points of the class (neural network based on radial ba.sis functions, RBF ). (d) The probability density at location x is estimated from the neighboring objects (with known class memberships or known responses) by applying a voting scheme or by interpolation (KNN, Section 5.2). [Pg.357]

The above quantification cannot be appUed to the mode shape because it is vector valued and is subject to the norm constraint. One way to quantify the uncertainty of the mode shape is through the Expected modal assurance criterion (MAC), defined analogously as in the deterministic case. According to the posterior distribution, which is approximated by a Gaussian distribution, given the measured data, the mode shape is a Gaussian vector with mean vector equal to the most probable mode shape and a covariance matrix Co . The latter is given by... [Pg.221]

As the true coefficient of projection vectors are not available, decision making is based on corresponding estimates and and since these are random quantities, on their distributions. In this context, under mild assumptions, the estimators are shown to be asymptotically (for long data records, i.e., N oo) Gaussian distributed, with mean equal to the true coefficient of projection vector and covariance that... [Pg.1843]

First, consider the generic performance function G(X), and let fx(x) denote the joint probability density function of X. Recall X = X / = 1 to n], and let fix. and crx, denote the mean and standard deviation of respectively. Further, the covariance of X, and Xj is denoted by Cov(X Xj). The first-order second-moment (shortly, referred to as FOSM) method approximates G to be a Gaussian distribution, using only the mean and covariance of X. [Pg.3651]

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