The Multivariate Normal Distribution

As we saw in Section 3.1.1, the familiar bell-shaped curve describes the sampling distributions of many experiments. Many distributions encountered in chemistry are approximately normal [3], Regardless of the form of the parent population, the central limit theorem tells us that sums and means of samples of random measurements drawn from a population tend to possess approximately bell-shaped distributions in repeated sampling. The functional form of the curve is described by Equation 3.19. 

The term Vo Jin is a normalization constant that sets the total area under the curve to exactly 1.0. The approximate area under the curve within 1 standard deviation is 0.68, and the approximate area under the curve within 2 standard deviations is 0.95. 

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  

By properly considering the distribution of all three variables simultaneously, we get more information than is obtained by considering each variable individually. This is the so-called multivariate advantage. This extra information is in the form of correlation between the variables. 

---

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

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.

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. 

Having described squared distances and the variance-covariance matrix, we are now in a position to introduce the multivariate normal distribution, which is represented in Equation 3.27,... 

The multivariate normal distribution Defining the bounds for a data set... 

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... 

It is worth mentioning at this stage that the three-stage hierarchical model used in Bayesian analyses when undertaken within the framework provided by WinBUGS requires that normal distributions are parameterized as mean and precision. Precision is the inverse of variance. For example, when defining the prior for the population parameter vector 0, the multivariate normal distribution would be parameterized as the mean vector fi and the inverse of the variance-covariance matrix X such that,... 

The multivariate normal distribution is completely defined by the mean and covariance of X. 

In the two-dimensional nonsingular case, the multivariate normal distribution reduces to the bivariate normal distribution. This bivariate normal distribution is a generalization of the familiar univariate normal distribution for a single r.v. X. Let jXx and cr be the mean and standard deviation of X, /Ay and ay be the mean and... 

Chew V Confidence, prediction and tolerance regions for the multivariate normal distribution. JoumalofAnjerijg Statistics Association 1966 61 605-617. 

In some cases, the multivariate normal distribution is useful. For a vector of random variables, X, with a mean vector jl and a covariance matrix Z, then the multivariate normal distribution is... 

Thus the likelihood of the random sample from the multivariate normal distribution is proportional to the likelihood of the mean vector, which is like a single observation from a multivariate normal distribution with mean vector and covariance matrix E/n. 

We suggest that when the posterior has multiple nodes a similar strategy be used. At each mode, we find the multivariate normal distribution that matches the curvature of the target at that mode. The candidate density should be a mixture of the multivariate Student s t distributions with low degrees of fieedom where each component of the mixture corresponds to the multivariate normal found for that node. Using this mixture density as the independent candidate density will give fast convergence to the posterior. 

Method Based on the Equivalent Determinant The second approach for estimating the critical limits of the cumulative potential to be used for class modelling was proposed by Forina et al. and is based on the concept of equivalent determinant [59,60]. By the term equivalent determinant, the authors indicate the determinant of the variance/covariance matrix of the multivariate normal distribution having the same value of the mean probability density as the one estimated by the density-based method. Without entering in the details of the procedure, it is possible to demonstrate... 

In most mathematical models, the covariance between parameters, as measured by Eq. (7.137), is nonzero, that is, the parameters are correlated with each other. Careful experimental design may reduce, but never completely eliminate, this correlation. The individual confidence intervals calculated by Eq. (7.145) do not reflect the covariance. To do so, it is necessary to construct the joint confidence region of parameters. Using the multivariate normal distribution of b [Eq. (7.138)], we form the standardized normal variable ... 

---