Multivariate distributions

In the previous sections our discussions has concentrated on just two r.v. s. We refer to their joint distribution as a bivariate distribution. When more than two r.v. s, say Xi, X2. Xk, are jointly distributed, we can similarly define their joint p.m.f. (or p.d.f.), denoted by/(xi, X2. x ), referred to as a multivariate distribution. The properties of a multivariate distribution are namral extensions of the bivariate distribution properties. Examples of a multivariate distribution are multinomial, multivariate, normal, and Dirichlet distributions. 

The marginal p.m.f. or p.d.f. can also be computed by summing or integrating the joint p.m.f. or p.d.f. over all possible values of the other coordinates. For example, the marginal distribution of Xj is given by 

are said to be mutually independent if and only if their joint distribution factors into the product of their marginal distributions, that is, 

These A, s are called eigenvalues, which represent the dispersion conesponding to each eigenvector direction. 

Analogously, we can define the correlation matrix of the X, s. This matrix, defined by R, has all diagonal entries equal to 1 and the (r, y)th off-diagonal entry 

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Polymer Particle Balances (PEEK In the case of multiconponent emulsion polymerization, a multivariate distribution of pjarticle propierties in terms of multiple internal coordinates is required in this work, the polymer volume in the piarticle, v (continuous coordinate), and the number of active chains of any type, ni,n2,. .,r n (discrete coordinates), are considered. Therefore... 

The first approach consists of assuming some multivariate distribution model for the random function P(x), xeA A convenient... 

These various covariance models are Inferred directly from the corresponding indicator data i(3 z ), i-l,...,N. The indicator kriging approach is said to be "non-parametric, in the sense that it draws solely from the data, not from any multivariate distribution hypothesis, as was the case for the multi- -normal approach. 

Electron density statistics. At high resolution we know the shape of the electron density of an atom, in which case we only need to know its exact location to reconstruct the electron density in its immediate vicinity. At lower resolution we can impose an expected shape on the uni- or multivariate distributions of electron density within the protein region in a procedure that is known as histogram matching. 

It is desirable to characterize dependencies by fitting a multivariate distribution. However, some measurements are missing for some units. 

In a A -component mixture -1 independent variables are present. The multivariate distribution will then be formed using the variances of k- ... 

In SIMCA the distribution of the object in the inner model space is not considered, so the probability density in the inner space is constant and the overall PD appears as shown in Figs. 29, 30 for the enlarged and reduced SIMCA models. In CLASSY, Kernel estimation is used to compute the PD in the inner model space, whereas the errors in the outer space are considered, as in SIMCA, uncorrelated and with normal multivariate distribution, so that the overall distribution, in the inner and outer space of a one-dimensional model, looks like that reported in Fig. 31. Figures 32, 33 show the PD of the bivariate normal distribution and Kernel distribution (ALLOC) for the same data matrix as used for Fig. 31. Although in the data set of French wines no really important differences have been detected between SIMCA (enlarged model), ALLOC and CLASSY, it seems that CLASSY should be chosen when the number of objects is large and the distribution on the components of the inner model space is very different from a rectangular distribution. 

H., Peck, C. C., Mould, D. R. Simulation of correlated continuous and categorical variables using a single multivariate distribution. J Pharmacokinet Pharmacodyn 2006 [Epub ahead of print]. 

The X matrix is the p x p variance-covariance matrix, which is a measure of the degree of scatter in the multivariate distribution. 

The population parameters 2 and [x completely specify the properties of a multivariate distribution. Usually it is impossible determine the population parameters therefore, one usually tries to estimate them from a small finite sample of size n, where n is the number of observations, The population mean vector, p, is approximated by the sample mean vector, x, which is simply the mean of each column in the data matrix X shown in Figure 3.4. As n becomes large, the approximation in Equation 3.31 becomes better. 

In the multivariate distribution, it is assumed that measurements of objects or aliquots of material (e.g., a single trial) produces vectors, x having a multivariate normal distribution. The measurements of the p variables in a single object, such as xT = x,xi>1,. .., x J will usually be correlated. In fact, this is expected to be the case. The measurements from different objects, however, are assumed to be independent. The independence of measurements from object to object or from trial to trial may not hold when an instrument drifts over time, as with sets of p wavelengths in a spectrum. Violation of the tentative assumption of independence can have a serious impact on the quality of statistical inferences. As a consequence of these assumptions, we can make the following statements about data sets that meet the above criteria ... 

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... 

Oja, H., Descriptive statistics for multivariate distributions, Stat. Probab. Lett., 1, 327-332, 1983. 

This supervised classification method, which is the most used, accepts a normal multivariate distribution for the variables in each population ((Ai,..., A ) Xi) ), and calculates the classification functions minimising the possibility of incorrect classification of the observations of the training group (Bayesian type rule). If multivariate normality is accepted and equality of the k covariance matrices ((Ai,..., Xp) NCfti, X)), Linear Discriminant Analysis (LDA) calculates... 

Assuming a normal multivariate distribution, with the same covariance matrices, in each of the populations, (X, X2,..., Xp) V(7t , 5), the multivariate analysis of variance MANOVA) for a single factor with k levels (extension of the single factor ANOVA to the case of p variables), permits the equality of the k mean vectors in p variables to be tested Hq = jl = 7 2 = = where ft. = fl, fif,..., fVp) is the mean vector of p variables in population Wi. The statistic used in the comparison is the A of Wilks, the value of which can be estimated by another statistic with F-distribution. If the calculated value is greater than the tabulated value, the null hypothesis for equality of the k mean vectors must be rejected. To establish whether the variables can distinguish each pair of groups a statistic is used with the F-distribution with p and n — p — k + i df, based on the square of Mahalanobis distance between the centroids, that permits the equality of the pairs of mean vectors to be compared Hq = jti = ft j) (Aflfl and Azen 1979 Marti n-Alvarez 2000). 

The variable x in the preceding formulas denotes a quantity that varies. In our context, it signifies a reference value. If the variable by chance may take any one of a specified set of values, we use the term variate (i.e, a random variable). In this section, we consider distributions of single variates (i.e., univariate distributions). In a later section, we also discuss the joint distribution of two or more variates bivariate or multivariate distributions). 

With the STS approach estimates of individual parameters are combined as if the set of estimates were a true Wsample from a multivariate distribution. It has been recommended as a very simple and valuable approach for pooling individual estimates of PK parameters derived from experimental PK studies (29). The advantage of the STS approach is its simplicity, but the validity of its results should not be overemphasized. However, it has been shown from simulation studies that the STS approach tends to overestimate parameter dispersion (the variance-covariance matrix) (20, 30). 

Covariates are incorporated into the simulation as distributions that are either simulated stochastically or resampled from an existing database (18). Correlation between covariates is handled during stochastic simulations using multivariate distributions with appropriate variance-covariance structure. Alternatively, covariates resampled from a sufficiently large existing database carry all relevant covariates from an individual into a simulated individual and so capture inherent correlation. Regardless of the method, the simulated outputs for covariates need to be checked to ensure that they reflect the expected trial population and are consistent with trial inclusion and exclusion criteria. 

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