Multivariate probability distribution

The sample of individuals is assumed to represent the patient population at large, sharing the same pathophysiological and pharmacokinetic-dynamic parameter distributions. The individual parameter 0 is assumed to arise from some multivariate probability distribution 0 / (T), where jk is the vector of so-called hyperparameters or population characteristics. In the mixed-effects formulation, the collection of jk is composed of population typical values (generally the mean vector) and of population variability values (generally the variance-covariance matrix). Mean and variance characterize the location and dispersion of the probability distribution of 0 in statistical terms. 

The Dirichlet distribution, often denoted Dir(a), is a family of continuous multivariate probability distributions parameterized by the vector a of positive real numbers. It is the multivariate generalization of the beta distribution and conjugate prior of the... 

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... 

Therefore, exact tests are considered that can be performed using two different approaches conditional and unconditional. In the first case, the total number of tumors r is regarded as fixed. As a result the null distribution of the test statistic is independent of the common probability p. The exact conditional null distribution is a multivariate hypergeometric distribution. 

The unconditional model treats the sum of all tumors as a random variable. Then the exact unconditional null distribution is a multivariate binomial distribution. The distribution depends on the unknown probability. 

A nonparametric approach can involve the use of synoptic data sets. In a synoptic data set, each unit is represented by a vector of measurements instead of a single measurement. For example, for synoptic data useful for pesticide fate, assessment could take the form of multiple physical-chemical measurements recorded for each of a sample of water bodies. The multivariate empirical distribution assigns equal probability (1/n) to each of n measurement vectors. Bootstrap evaluation of statistical error can involve sampling sets of n measurement vectors (with replacement). Dependencies are accounted for in such an approach because the variable combinations allowed are precisely those observed in the data, and correlations (or other dependency measures) are fixed equal to sample values. 

LDA is the first classification technique introduced into multivariate analysis by Fisher (1936). It is a probabilistic parametric technique, that is, it is based on the estimation of multivariate probability density fimc-tions, which are entirely described by a minimum number of parameters means, variances, and covariances, like in the case of the well-knovm univariate normal distribution. LDA is based on the hypotheses that the probability density distributions are multivariate normal and that the dispersion is the same for all the categories. This means that the variance-covariance matrix is the same for all of the categories, while the centroids are different (different location). In the case of two variables, the probability density fimction is bell-shaped and its elliptic section lines correspond to equal probability density values and to the same Mahala-nobis distance from the centroid (see Fig. 2.15A). 

It would now be most logical to let this probability between a and b be the RC, but in case of more than one independent variable with a multivariate error distribution it is a very complicated problem to calculate an almost always asymmetrical part of this distribution. To handle this problem the... 

The assumption of multivariate normal distribution underlying this method gives for the conditional (a posteriori) probability density of the category g... 

The distribution (6.6) is the multivariate generalization of the binomial distribution. Now consider an ensemble of similar systems in which the total N is not constant but distributed according to Poisson with average . Then the probability distribution in this grand ensemble is... 

So far we only considered transport of particles by diffusion. As mentioned in 1 the continuous description was not strictly necessary, because diffusion can be described as jumps between cells and therefore incorporated in the multivariate master equation. Now consider particles that move freely and should therefore be described by their velocity v as well as by their position r. The cells A are six-dimensional cells in the one-particle phase space. As long as no reaction occurs v is constant but r changes continuously. As a result the probability distribution varies in a way which cannot be described as a succession of jumps but only in terms of a differential operator. Hence the continuous description is indispensable, but the method of compounding moments can again be used. 

The hypergeometric distribution can be generalized to a multivariable form, the multivariate hypergeometric distribution, which can be used to extend Fisher s Exact Test to contingency tables larger than 2 by 2 and to multidimensional contingency tables. There is statistical software available to perform these calculations however, due to the complexity of the calculations and the large number of trial tables whose probability of occurrence must be calculated, this extension has received limited use. 

Probabilities of class membership and class boundaries or confidence intervals can be constructed, assuming a multivariate normal distribution. 

We start from the multivariate master equation for the probability distribution of number of particles in local elements. The local concentration of a species X in a volume element at the position r is denoted by x(f) and its statistical average is denoted by x which is assumed to be independent of the position due to the assumption of a homogeneous steady state. Then, the covariance of the fluctuations... 

Although P(x G(A)) can be estimated by analysing large numbers of samples, similarly for P(x g(B))> the procedure is still time consuming and requires large numbers of analyses. Fortunately, if the variables contributing to the vector pattern are assumed to possess a multivariate normal distribution, then these conditional probability values can be calculated from... 

For the confidence intervals, we merely compute the size of the ellipse containing a given probability of the multivariate t-distribution. That can be shown to be an F probability function [51. Given the number of estimated parameters, n, the confidence level, , and the number of data points,... 

The classical identification becomes difficult or inapplicable when we have systems with a high degree of nonlinearity, when only partial observations can be made, and when there are stochastic (random) elements, that is, the variables are given by probability distributions rather than by given numbers. Multivariate systems in which only relatively inaccurate and imprecise measurements can be made, usually with insufficient time and... 

In the case that objects of all classes obey a multivariate normal distribution, an optimal classification rule can be based on Bayes theorem. The assignment of a sample, x, characterized by p features to a class j of all classes g is based on maximizing the posterior probability ... 

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