Mean-variance model

Distribution and density Mean Variance Model Example... 

TABLE 7.3 NPD Functions, Means, Variances, and Moment of Some Model Batch and Flow Systems with Recirculation... 

For many mathematical operations, including addition, subtraction, multiplication, division, logarithms, exponentials and power relations, there are exact analytical expressions for explicitly propagating input variance and covariance to model predictions of output variance (Bevington, 1969). In analytical variance propagation methods, the mean, variance and covariance matrix of the input distributions are used to determine the mean and variance of the outcome. The following is an example of the exact analytical variance propagation approach. If w is the product of x times y times z, then the equation for the mean or expected value of w, E(w), is ... 

An uncertainty analysis involves the determination of the variation of imprecision in an output function based on the collective variance of model inputs. One of the five issues in uncertainty analysis that must be confronted is how to distinguish between the relative contribution of variability (i.e. heterogeneity) versus true certainty (measurement precision) to the characterization of predicted outcome. Variability refers to quantities that are distributed empirically - such factors as soil characteristics, weather patterns and human characteristics - which come about through processes that we expect to be stochastic because they reflect actual variations in nature. These processes are inherently random or variable, and cannot be represented by a single value, so that we can determine only their moments (mean, variance, skewness, etc.) with precision. In contrast, true uncertainty or model specification error (e.g. statistical estimation error) refers to an input that, in theory, has a single value, which cannot be known with precision due to measurement or estimation error. 

In the following we will thus present some basic statistical methods useful for determining turbulence quantities from experimental data, and show how these measurements of turbulence can be put into the statistical model framework. Usually, this involves separating the turbulent from the non-turhulent parts of the flow, followed by averaging to provide the statistical descriptor. We will survey some of the basic methods of statistics, including the mean, variance, standard deviation, covariance, and correlation (e.g., [66], chap 1 [154], chap 2 [156]). 

Many tools are available for analyzing experimentally designed data [Hoaglin Welsch 1978, Latorre 1984, Rao 1973, Searle et al. 1992, Weisberg 1985], Common to many of these approaches is that the estimated effects are treated as additive. This means that the effect of each factor is independent of the variation in other factors. In some situations, an additive model of main effects is not realistic because the factors do not affect the response independently. A well-working remedy to this is to allow interactions between the factors. Conceptually, traditional analysis of variance models start from main effects and seek to keep the number of interactions as low as possible and of the lowest possible order. 

Notice that nothing beyond the first two moments of Y is being assumed, i.e., only the mean and variance of the data are being defined and no distributional assumptions, such as normality, are being made. In residual variance model estimation, the goal is to understand the variance structure as a function of a set of predictors, which may not necessarily be the same as the set of predictors in the structural model (Davidian and Car-roll, 1987). Common, heteroscedastic error models are shown in Table 4.1. Under all these models, generic s is assumed to be independent, having zero mean and constant variance. 

Usually, variability increases as a systematic function of the mean response f(0 x) in which case a common choice of residual variance model is the power of the mean model... 

Figure 4.1 Variance of Y as a function of the mean response assuming a power of the mean residual variance model.

The choice of weights in WLS is based on either the observed data or on a residual variance model. If the variances are known and they are not a function of the mean, then the model can be redefined as... 

When no replicates are available, common weights that use the observed data include 1/Y or 1/Y2. These two weighting schemes in essence assume that the variance model is proportional to the mean or mean squared, respectively, and then crudely use the observation itself to estimate the mean. Although using observed data has a tremendous history behind it, using observed data as weights is problematic in that observed data are measured with error. A better estimate might be 1/Y or 1 /Y2 where the predicted values are used instead. In this manner, any measurement error or random variability in the data are controlled. 

ELS also produces inconsistent estimates of 0 when the wrong variance model is used, whereas GLS does not. The difference is because GLS estimates 0 by an estimating equation linear in Y, whereas ELS solves one that is quadratic in Y. For most pharmacokinetic data, however, where the variance is small relative to the range of the mean responses, ELS and GLS produce essentially equivalent results and the danger of the inconsistency vanishes. If, however, the variance is large relative to the range of responses, the two estimation methods may produce divergent results (Davidian, personal communication). 

True residual variance model Mixture distribution Mean value Between simulation CV(%) of parameter estimates ... 

Nonlinear mixed effects models consist of two components the structural model (which may or may not contain covariates) and the statistical or variance model. The structural model describes the mean response for the population. Similar to a linear mixed effects model, nonlinear mixed effects models can be developed using a hierarchical approach. Data consist of an independent sample of n-subjects with the ith subject having -observations measured at time points t i, t 2, . t n . Let Y be the vector of observations, Y = Y1 1, Yi,2,. ..Ynjl,Yn,2,. ..Yn,ni)T and let s... 

Analysis of variance for each dependent variable showed that in almost all cases, R2 coefficients higher than 0.83 were obtained (Table 2), which means that models were able to explain more than 83% of the observed responses. For the rate of gelation, thermal hysteresis and hardness, the lack of fit test was not significant. For Tge, and Tm, the lack of fit was significant, which means that the model may not have included all appropiate function of independent variables. According to Box and Draper,13 we considered the high coefficients R2 as evidence of the applicability of the model. 

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