Multivariate distribution generally

The generalized sample variance describes the scatter in the multivariate distribution. A large volume indicates a large generalized variance and a large amount of scatter in the multivariate distribution. A small volume indicates a small generalized... 

In this section, we introduce a multinomial distribution, an important family of discrete multivariate distributions. This family generalized the binomial family to the situation in which each trial has n (rather than two) distinct possible outcomes. Then multivariate normal and Dirichlet distributions wUl be discussed. 

Example 2.29 Multivariate Distribution from Tossing a Die. Consider tossing a six-sided die 10 times. Suppose the die is unbalanced so that the probability of observing a 1 is 1/21, the probability of observing a 2 is 2/21, and, in general, the probability of observing an i is 1/21. Now consider the random vector (Xi,..., Xs), where Xj counts the number of times i comes up in the 10 tosses. Then what is the probability of roUing four 6s, three 5s, two 4s, and one 3 ... 

The problem to obtain tolerance regions for random vectors with normal multivariate distributions is easy to establish. Generalizing Eq. (1), it will consists in finding the tolerance region R which verifies the equation... 

Fundamentals of continuous thermodynamics as applied to homopolymers characterized by univariate distribution functions have been reviewed extensively [28, 29]. Hence, this chapter will provide the fundamentals in their most general form by considering systems composed of any number of polydisperse ensembles described by multivariate distribution functions and any number of solvents and by referring to the papers [28, 29],... 

Recent development in liquid chromatography analysis of complex polymers shows a clear trend to combine more than one LC separation mechanism/tech-nique together with multiple detection techniques. It is a quite natural direction for the analysis of complex polymers with multivariate distributions in molecular characteristics. The coupled LC techniques have gained wide attention recently in the characterization of complex polymers and there are a number of monographs and reviews on this topic [4,8-13]. In this chapter, recent advances in LC separation of polymers are reviewed. References are generally restricted to the works published after 1995 since most of the works prior to the mid-1990s have been well summarized cdready [8,10,11,13]. 

Scale- Up of Electrochemical Reactors. The intermediate scale of the pilot plant is frequendy used in the scale-up of an electrochemical reactor or process to full scale. Dimensional analysis (qv) has been used in chemical engineering scale-up to simplify and generalize a multivariant system, and may be appHed to electrochemical systems, but has shown limitations. It is best used in conjunction with mathematical models. Scale-up often involves seeking a few critical parameters. Eor electrochemical cells, these parameters are generally current distribution and cell resistance. The characteristics of electrolytic process scale-up have been described (63—65). 

The Mahalanobis distance representation will help us to have a more general look at discriminant analysis. The multivariate normal distribution for w variables and class K can be described by... 

There are two points of view to take into account when setting up a trmning set for developing a predictive multivariate calibration model. One viewpoint is that the calibration set should be representative for the population for which future predictions are to be made. This will generally lead to a distribution of objects in experimental space that has a higher density towards the center, tailing out to the boundaries. Another consideration is that it is better to spread the samples more or... 

The approach of Fisher (1938) was originally proposed for discriminating two populations (binary classification), and later on extended to the case of more than two groups (Rao 1948). Here we will first describe the case of two groups, and then extend to the more general case. Although this method also leads to linear functions for classification, it does not explicitly require multivariate normal distributions of the groups with equal covariance matrices. However, if these assumptions are not... 

PLS is related to principal components analysis (PCA) (20), This is a method used to project the matrix of the X-block, with the aim of obtaining a general survey of the distribution of the objects in the molecular space. PCA is recommended as an initial step to other multivariate analyses techniques, to help identify outliers and delineate classes. The data are randomly divided into a training set and a test set. Once the principal components model has been calculated on the training set, the test set may be applied to check the validity of the model. PCA differs most obviously from PLS in that it is optimized with respect to the variance of the descriptors. 

Exercise. Generalize the formalism to more than one species of dots and show that the multivariate analog of the Poisson distribution is merely a product of singlevariable distributions (2.6). 

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... 

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. 

The multivariate normal distribution is a generalization of the univariate normal distribution with p> 2 dimensions. Consider a 1 xp vector x,T obtained by measuring several variables for the ith observation and the corresponding vector of means for each variable ... 

For convenience, we normalized the univariate normal distribution so that it had a mean of zero and a standard deviation of one (see Section 3.1.2, Equation 3.5 and Equation 3.6). In a similar fashion, we now define the generalized multivariate squared distance of an object s data vector, x , from the mean, ju, where 2 is the variance-covariance matrix (described later) ... 

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. 

In much the same way as the more common univariate statistics assume a normal distribution of the variable under study, so the most widely used multivariate models are based on the assumption of a multivariate normal distribution for each population sampled. The multivariate normal distribution is a generalization of its univariate counterpart and its equation in matrix notation is... 

The Wishart distribution is a multivariate gamma distribution, which itself is a general case of a chi-squared distribution. The Wishart also has the desirable property that random samples of any matrix from this distribution will always be positive definite. This is useful for simulating variance-covariance matrices, which have a positive determinant, and ensuring correlations that lie between -1 and 1. 

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