Model univariate distribution

The expressions for InpR associated with the equation-of-state models and the univariate distribution function are given in the papers [56-59]. 

The normal model can take a variety of forms depending on the choice of noninformative or infonnative prior distributions and on whether the variance is assumed to be a constant or is given its own prior distribution. And of course, the data could represent a single variable or could be multidimensional. Rather than describing each of the possible combinations, I give only the univariate normal case with informative priors on both the mean and variance. In this case, the likelihood for data y given the values of the parameters that comprise 6, J. (the mean), and G (the variance) is given by the familiar exponential... 

The location-and-scale model states that the n univariate observations x are independent and identically distributed (i.i.d.) with distribution function F[Qr - 6)/o], where F is known. Typically F is the standard Gaussian distribution function O. 

In univariate statistics a key question discussed previously was to evaluate how close the values of p and m are for a certain population and an experimental m. The answer to this question is used as a model for significance tests. One main tool used to evaluate statistical data is the distribution function, which describes the distribution of measurements about their mean. In other words, the distribution function gives the... 

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... 

Considerable work has been focused on determining the asymptotic null distribution of -2 log-likelihood -ILL) when the alternative hypothesis is the presence of two subpopulations. In the case of two univariate densities mixed in an unknown proportion, the distribution of -ILL has been shown to be the same as the distribution of [max(0, Y)f, where Y is a standard normal random variable (28). Work with stochastic simulations resulted in the proposal that -2LL-c is distributed with d degrees of freedom, where d is equal to two times the difference in the number of parameters between the nonmixture and mixture model (not including parameters used for the probability models) and c=(n-l-p- gl2)ln (31). In the expression for c, n is the number of observations, p is the dimensionality of the observation, and g is the number of subpopulations. So for the case of univariate observations (p = 1), two subpopulations (g = 2), and one parameter distinguishing the mixture submodels (not including the mixing parameter), -2LL-(n - 3)/n with two... 

The selection of the structural PK model and residual error models was based on the goodness-of-fit plots and on the difference in NONMEM objective function (approximately -2 x log likelihood) between hierarchical models (i.e., the likelihood ratio test). This difference is asymptomatically distributed with a degree of freedom equal to the number of additional parameters of the full compared to the reduced model. A p-value of 0.05 was chosen for one additional parameter, corresponding to a difference in the objective function of 3.84. Potential covariates were selected by univariate analysis, testing the addition of each covariate on each of the relevant PK parameters. When a set of covariates, identified by the... 

Since yMst is a random variable, SPM tools can be used to detect statistically significant changes. histXk) is highly autocorrelated. Use of traditional SPM charts for autocorrelated variables may yield erroneous results. An alternative SPM method for autocorrelated data is based on the development of a time series model, generation of the residuals between the values predicted by the model and the measured values, and monitoring of the residuals [1]. The residuals should be approximately normally and independently distributed with zero-mean and constant-variance if the time series model provides an accurate description of process behavior. Therefore, popular univariate SPM charts (such as x-chart, CUSUM, and EWMA charts) are applicable to the residuals. Residuals-based SPM is used to monitor lhist k). An AR model is used for representing st k) ... 

Statistical indices are fundamental numerical quantities measuring some statistical property of one or more variables. They are applied in any statistical analysis of data and hence in most of Q S AR methods as well as in some algorithms for the calculation of molecular descriptors. The most important univariate statistical indices are indices of central tendency and indices of dispersion, the former measuring the center of a distribution, the latter the dispersion of data in a distribution. Among the bivariate statistical indices, the correlation measures play a fundamental role in all the sciences. Other important statistical indices are the diversity indices, which are related to the injbrmationcontentofavariahle,the —> regressiowparameters, used for regression model analysis, and the —> classification parameters, used for classification model analysis. 

In the present work, two methods are chosen to conduct variable selection. The first is t-test, which is a simple univariate method that determines whether two samples from normal distributions could have the same mean when standard deviations are unknown but assumed to be equal. The second is subwindow permutation analysis (SPA) which was a model population analysis-based approach proposed in our previous work [14]. The main characteristic of SPA is that it can output a conditional P value by implicitly taking into account synergistic effects among multiple variables. With this conditional P value, important variables or conditionally important variables can be identified. The source codes in Matlab and R are freely available at [46]. We apply these two methods on a type 2 diabetes mellitus dataset that contains 90 samples (45 healthy and 45 cases) each of which is characterized by 21 metabolites measured using a GC/MS instrument. Details of this dataset can be found in reference [32]. 

In this section we describe the six discrete probability distributions and five continuous probability distributions that occur most frequently in bioinformatics and computational biology. These are called univariate models. In the last three sections, we discuss probability models that involve more than one random variable called multivariate models. 

As with univariate regression, an analysis of the residuals is important in evaluating the model. The residuals should be randomly and normally distributed. Figure 8.11 shows a plot of the residuals against the fitted values for Cl , the residuals do not show any particular pattern. Figure 8.12 plots the predicted values against the measured values. The points are reasonably close to a straight line with no obvious outliers. 

Since the components in the same system share a common stress, it is reasonable to assume that the deterioration of components are dependent. In this paper, we use Levy copula to model the dependence between components for a wide range of dependence. To introduce Levy copula, we firstly recall that according to the Sklar s Theorem, for univariate continuous cumulative distribution function and H a multivariate joint cumulative distribution function there exists a unique function C such that... 

This is that/f(xj,X2,x ) has the same dependence structure with C u, U2,uf) regardless its marginal functions. Hence, copulas allow to separate the univariate margins and the multivariate dependence structure in the continuous multivariate distribution functions. To model the dependence structure for stochastic processes, Cont Tankov (2004) define the dependence structure of Levy measure by Levy copula. Thus Levy copula retains the dependence information of a Levy measure. Let X = (X, ..., X ) be a Levy process. Then there exists a Levy copula Q such that the tail integral of X satisfies ... 

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