Distribution, normal multivariate

The conceptually simplest model, which for reasons explained later is called UNEQ, is based on the multivariate normal distribution. Suppose we have carried... 

We also make a distinction between parametric and non-parametric techniques. In the parametric techniques such as linear discriminant analysis, UNEQ and SIMCA, statistical parameters of the distribution of the objects are used in the derivation of the decision function (almost always a multivariate normal distribution... 

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... 

We have assumed that the prior information can be described by the multivariate normal distribution, i.e., k is normally distributed with mean kB and co-variance matrix VB. 

We will begin with the concept of the multivariate normal distribution. 

Figure 1-1 Development of the concept of the Multivariate Normal Distribution (this one shown having three dimensions) - see text for details. The density of points along a cross-section of the distribution in any direction is also an MND, of lower dimension.

Vector e has a multivariate normal distribution. Mah and Tamhane (1982) proposed the use of the test on the estimates,... 

As was indicated in Section 7.2, the vector of measurement adjustments, e, has a multivariate normal distribution with zero mean and covariance matrix V. Thus, the objective function value of the least square estimation problem (7.21), ofv = eT l> 1 e, has a central chi-square distribution with a number of degrees of freedom equal to the rank of A. 

A multivariate normal distribution data set with the variance and mean given by this i and x was generated by the Monte Carlo method to simulate the process sampling data. The data size was 1000 and it was used to investigate the performance of the indirect method. 

The confidence intervals defined for a single random variable become confidence regions for jointly distributed random variables. In the case of a multivariate normal distribution, the equation of the surface limiting the confidence region of the mean vector will now be shown to be an n-dimensional ellipsoid. Let us assume that X is a vector of n normally distributed variables with mean n-column vector p and covariance matrix Ex. A sample of m observations has a mean vector x and an n x n covariance matrix S. 

We assume that a random variable vector Y of (here upper-case is used to indicate not a matrix but an ordered set of m random variables) distributed as a multivariate normal distribution has been measured through an adequate analytical protocol (e.g., CaO concentration, the 87Sr/86Sr ratio,...). The outcome of this measurement is the data vector jm. Here ym is the mean of a large number of measurements with expected... 

We now proceed to m observations. The ith observation provides the estimates xi of the independent variables Xj and the estimate y, of the dependent variable Y. The n estimates xtj of the variables Xj provided by this ith observation are lumped together into the vector xt. We assume that the set of the (n+1) data (i/,y,) associated with the ith observation represent unbiased estimates of the mean ( yf) of a random (n + 1)-vector distributed as a multivariate normal distribution. The unbiased character of the estimates is equivalent to... 

In Sections 1.6.3 and 1.6.4, different possibilities were mentioned for estimating the central value and the spread, respectively, of the underlying data distribution. Also in the context of covariance and correlation, we assume an underlying distribution, but now this distribution is no longer univariate but multivariate, for instance a multivariate normal distribution. The covariance matrix X mentioned above expresses the covariance structure of the underlying—unknown—distribution. Now, we can measure n observations (objects) on all m variables, and we assume that these are random samples from the underlying population. The observations are represented as rows in the data matrix X(n x m) with n objects and m variables. The task is then to estimate the covariance matrix from the observed data X. Naturally, there exist several possibilities for estimating X (Table 2.2). The choice should depend on the distribution and quality of the data at hand. If the data follow a multivariate normal distribution, the classical covariance measure (which is the basis for the Pearson correlation) is the best choice. If the data distribution is skewed, one could either transform them to more symmetry and apply the classical methods, or alternatively... 

If it can be assumed that the multivariate data follow a multivariate normal distribution with a certain mean and covariance matrix, then it can be shown that the squared Mahalanobis distance approximately follows a chi-square distribution... 

If the data majority is multivariate normally distributed, the squared score distances can be approximated by a chi-square distribution, y2, with a degrees of freedom. 

Note that the approximation by the chi-square distribution is only possible for multivariate normally distributed data which somehow is in conflict if outliers are present that should be identified with this measure. We recommend that robust PCA is used whenever diagnostics is done because robust methods tolerate deviations from multivariate normal distribution. 

Outliers may heavily influence the result of PCA. Diagnostic plots help to find outliers (leverage points and orthogonal outliers) falling outside the hyper-ellipsoid which defines the PCA model. Essential is the use of robust methods that are tolerant against deviations from multivariate normal distributions. 

The canonical correlation coefficients can also be used for hypothesis testing. The most important test is a test for uncorrelatedness of the x- and y-variables. This corresponds to testing the null hypothesis that the theoretical covariance matrix between the x- and y-variables is a zero matrix (of dimension mx x mY). Under the assumption of multivariate normal distribution, the test statistic... 

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... 

If the assumptions (multivariate normal distributions with equal group covariance matrices) are fulfilled, the Fisher rule gives the same result as the Bayesian rule. However, there is an interesting aspect for the Fisher rule in the context of visualization, because this formulation allows for dimension reduction. By projecting the data... 

Although model-based clustering seems to be restrictive to elliptical cluster forms resulting from models of multivariate normal distributions, this method has several advantages. Model-based clustering does not require the choice of a distance measure, nor the choice of a cluster validity measure because the BIC measure can be... 

Model-based clustering assumes that each cluster can be modeled by a multivariate normal distribution (with varying parameters). If the clusters can be well modeled in this way, the method is powerful, and can estimate an optimum number of clusters. Especially for higher-dimensional data it is computer time demanding. 

Such a measure of the separation between classes will work best when It can be assumed that the classes approximate multivariate normal distributions. That Is a reasonable assumption for the classes modeled by the output of the FCV algorithms. 

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

The other components were independent of sampling design, and they collect the random errors with a multivariate normal distribution, where some significant correlations among variables are still present. 

Consider, sampling from a multivariate normal distribution with mean vector /i = /n,..., Hm) and... 

In most models developed for pharmacokinetic and pharmacodynamic data it is not possible to obtain a closed form solution of E(yi) and var(y ). The simplest algorithm available in NONMEM, the first-order estimation method (FO), overcomes this by providing an approximate solution through a first-order Taylor series expansion with respect to the random variables r i,Kiq, and Sij, where it is assumed that these random effect parameters are independently multivariately normally distributed with mean zero. During an iterative process the best estimates for the fixed and random effects are estimated. The individual parameters (conditional estimates) are calculated a posteriori based on the fixed effects, the random effects, and the individual observations using the maximum a posteriori Bayesian estimation method implemented as the post hoc option in NONMEM [10]. 

The remaining chapters of the book introduce some of the advanced topics of chemometrics. The coverage is fairly comprehensive, in that these chapters cover some of the most important advanced topics. Chapter 6 presents the concept of robust multivariate methods. Robust methods are insensitive to the presence of outliers. Most of the methods described in Chapter 6 can tolerate data sets contaminated with up to 50% outliers without detrimental effects. Descriptions of algorithms and examples are provided for robust estimators of the multivariate normal distribution, robust PC A, and robust multivariate calibration, including robust PLS. As such, Chapter 6 provides an excellent follow-up to Chapters 3, 4, and 5. 

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