Univariate models, random variables

We now consider the case in which, again, the independent variable jc, is considered to be accurately known, but now we suppose that the variances in the dependent variable y, are not constant, but may vary (either randomly or continuously) with JC . To show the basis of the method we use the simple linear univariate model, written as Eq. (2-76). 

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... 

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) ... 

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. 

Classic univariate regression uses a single predictor, which is usually insufficient to model a property in complex samples. Multivariate regression takes into account several predictive variables simultaneously for increased accuracy. The purpose of a multivariate regression model is to extract relevant information from the available data. Observed data usually contains some noise and may also include irrelevant information. Noise can be considered as random data variation due to experimental error. It may also represent observed variation due to factors not initially included in the model. Further, the measured data may carry irrelevant information that has little or nothing to do with the attribute modeled. For instance, NIR absorbance... 

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