Multivariate probability

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). 

In order to declare a multivariate probability P = 0.977 for testing the multivariate correlation coefficient, the following univariate probability is required ... 

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... 

We have introduced the random vector variable X, with the hypotheses (E.1.2), (E.1.2a and b), and (E.l.lOa). Thus, in fact, we have introduced N scalar random variables, with a common (multivariate) probability density. 

So far it has been assumed that there is only a single variable that governs the behaviour of the probability space. However, in many cases, it is useful to deal with multivariate probability spaces where multiple variables determine the outcome. In general, all of the univariate results generalise straightforwardly to the multivariate case. In order to simplify the presentation, all results will be derived first for the bivariate n = 2) situation. The extension to an arbitrary n simply requires adding additional integrations. 

There is insufficient space here to give a full treatment of the theory of random functions. Thus the following specialises the treatment and proceeds formally. There are many texts on basic multivariate probability theory and statistics, and so a basic knowledge of these subjects is assmned on the part of the reader. Texts on the theory of turbulence, such as [65], present the theory in sufficient detail for applied geostatistics. Backgroimd in statistical field theory as reviewed by [18, 19, 73] is very relevant. 

J. MacQueen, Some methods for classification and analysis of multivariate observations. In L. Le Cam and J. Neyman (eds.), Proceedings 5th Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley, CA, 1967, pp. 281-297. 

Assuming the distribution models are accurate and that they model all the possible behaviors in the data set, Bayes s theorem says that pup2, and p3 are the probabilities that the unknown sample is a member of class 1, 2, or 3, respectively. The distributions are modeled using multivariate Gaussian functions in a method known as expectation maximization. ... 

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. 

It is probably more realistic to assume that we know neither the rate constants nor the absorption spectra for the above example. All we have is the measurement Y and the task is to determine the best set of parameters which include the rate constants ki and /cj and the molar absorptivities, the whole matrix A. This looks like a formidable task as there are many parameters to be fitted, the two rate constants as well as all elements of A. In Multivariate Data, Separation of the Linear and Non-Linear Parameters (p.162), we start tackling this problem. 

Chapter 3 starts with the first and probably most important multivariate statistical method, with principal component analysis (PC A). PC A is mainly used for mapping or summarizing the data information. Many ideas presented in this chapter, like the selection of the number of principal components (PCs), or the robustification of PCA, apply in a similar way to other methods. Section 3.8 discusses briefly related methods for summarizing and mapping multivariate data. The interested reader may consult extended literature for a more detailed description of these methods. 

Application of multivariate statistics to fatty acid data from the Tyrolean Iceman and other mummies is a mosaic stone in the investigation of this mid-European ancestor, which is still a matter of research (Marota and Rollo 2002 Murphy et al. 2003 Nerlich et al. 2003). The iceman is on public display in the South Tyrol Museum of Archaeology in Bolzano, Italy, stored at —6°C and 98% humidity, the conditions as they probably were during the last thousands of years. 

Maximizing the posterior probabilities in case of multivariate normal densities will result in quadratic or linear discriminant rules. However, the mles are linear if we use the additional assumption that the covariance matrices of all groups are equal, i.e., X = = Xk=X- In this case, the classification rule is based on linear discriminant scores dj for groups j... 

Notably, in logistic regression-like other multivariate analyses-the effects of causal variables can be shown to influence probabilities of outcomes in response variables independently of one another. The likelihood of using a lawn care company is significantly higher for women than men, for example, no matter the income or education of individuals. 

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