Error variance structures

To illustrate how different error variance structures may influence observed values, concentrations (C) were simulated from a 1-compartment model with first-order absorption... 

Figure 4.3 Residual plots from fitting a 1-compartment model using OLS to the data shown in the bottom plot of Fig. 4.2 having a constant plus proportional error variance structure.

On the other hand, MCCC considers the influence of the variation of one parameter on model output in the context of simultaneous variations of all other parameters. In this situation, is smaller than 1 in absolute value and its size depends on the relative importance of the variation of model output due to the parameter of interest and the variation of model output given by the sum total of all sources (namely, the variability in all structural parameter values plus the error variance). 

The calculations discussed in the previous section fit the noise-free amplitudes exactly. When the structure factor amplitudes are noisy, it is necessary to deal with the random error in the observations we want the probability distribution of random scatterers that is the most probable a posteriori, in view of the available observations and of the associated experimental error variances. 

In this section we briefly discuss an approximate formalism that allows incorporation of the experimental error variances in the constrained maximisation of the Bayesian score. The problem addressed here is the derivation of a likelihood function that not only gives the distribution of a structure factor amplitude as computed from the current structural model, but also takes into account the variance due to the experimental error. 

Under general hypotheses, the optimisation of the Bayesian score under the constraints of MaxEnt will require numerical integration of (29), in that no analytical solution exists for the integral. A Taylor expansion of Ao(R) around the maximum of the P(R) function could be used to compute an analytical expression for A and its first and second order derivatives, provided the spread of the A distribution is significantly larger than the one of the P(R) function, as measured by a 2. Unfortunately, for accurate charge density studies this requirement is not always fulfilled for many reflexions the structure factor variance Z2 appearing in Ao is comparable to or even smaller than the experimental error variance o2, because the deformation effects and the associated uncertainty are at the level of the noise. 

An independent method to identify the stochastic errors of impedance data is described in Chapter 21. An alternative approach has been to use the method of maximum likelihood, in which the regression procedure is used to obtain a joint estimate for the parameter vector P and the error structure of the data. The maximum likelihood method is recommended under conditions where the error structure is unknown, but the error structure obtained by simultaneous regression is severely constrained by the assumed form of the error-variance model. In addition, the assumption that the error variance model can be obtained by minimizing the objective function ignores the differences eimong the contributions to the residual errors shown in Chapter 21. Finedly, the use of the regression procedure to estimate the standard deviation of the data precludes use of the statistic... 

A second approach has been to use the regression procedure to obtain an estimate for the error structure of the data. A sequential regression is employed in which the parameters for an assumed error structure model, e.g., equations (21.19) and (21.20), are obtained directly from regression to the data. In more recent work, the error variance model was replaced by... 

A critical component of comodeling multiple outputs is the appropriate weighting of individual observations. The weights must be appropriate for small and large responses within an output and the relative weights must be appropriate between outputs. Failure of the former standard can lead to regions of systematic error in the fitted function and failure in the latter standard can cause some of the outputs to inappropriately dominate the determination of fitted parameters. However, error variance model selection, as for structural model development, should be guided by parsimony stay as simple as possible. 

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. 

The smoothing parameter X may be readily determined in the ideal case when the error variance of the data is known. In the more common case of unknown data error structure, X will have to be determined in a somewhat subjective manner by visual inspection of the fits combined with some residual analysis. More appropriately, X may be determined using cross-validation principles enabling a more objective and automatic procedure. °... 

In this equation, the so-called common factors,/ which span the common factor space replace the components in equation (8), and the are again the loadings representing the correlations between common factors and the original variables. The q, are called unique factors their squared loadings, the uniquenesses comprise that part of the data variance which is attributable to unique variable properties not involved in the common correlation structure. FA is the method of choice in all cases where such unique properties of variables occur, and this is to be expected when a data matrix contains variables quite different in nature and meaning which are only loosely interrelated. In such matrices, error variances... 

Multivariate techniques are required to assess multiple sources of variance. Yet, there has generally been a paucity of multivariate studies in experimental psychology (Harris, 1992). Multiple indicators of constructs have become salient due to increasing interest in structural equation modeling in biobehavioral research where multiple indicators are necessary to estimate latent variables and their associated error variance. In addition, the occasions dimension holds particular special significance for psychophysiology because most studies involve repeated measurements to some degree (Vasey Thayer, 1987). 

The Bayesian spectral density approach approximates the spectral density matrix estimators as Wishart distributed random matrices. This is the consequence of the special structure of the covariance matrix of the real and imaginary parts of the discrete Fourier transforms in Equation (3.53) [295]. Another approximation is made on the independency of the spectral density matrix estimators at different frequencies. These two approximations were verified to be accurate at the frequencies around the peaks of the spectmm. The spectral density estimators in the frequency range with small spectral values will become dependent since aliasing and leakage effects have a greater impact on their values. Therefore, the likelihood function is constructed to include the spectral density estimators in a limited bandwidth only. In particular, the loss of information due to the exclusion of some of the frequencies affects the estimation of the prediction-error variance but not the parameters that govern the time-frequency structure of the response, e.g., the modal frequencies or stiffness of a structure. 

To apply the Bayesian model updating procedure, prior distributions with large standard deviations are selected for the model parameters. Lognormal (LN) distributions with mean 1 and standard deviation of the logarithm equal to 1 are adopted for the stiffness parameters LN distributions with mean 0.02 and standard deviation of the logarithm equal to 1 are adopted for the damping ratios LN distributions with mean —1.651 and standard deviation of the logarithm equal to 1 are adopted for the prediction-error variances for the structural accelerations. 

It is assumed that the structural eigenvectors explain successively less variance in the data. The error eigenvalues, however, when they account for random errors in the data, should be equal. In practice, one expects that the curve on the Scree-plot levels off at a point r when the structural information in the data is nearly exhausted. This point determines the number of structural eigenvectors. In Fig. 31.15 we present the Scree-plot for the 23x8 table of transformed chromatographic retention times. From the plot we observe that the residual variance levels off after the second eigenvector. Hence, we conclude from this evidence that the structural pattern in the data is two-dimensional and that the five residual dimensions contribute mostly noise. 

This is a statistical test designed by Malinowski [43] which compares the variance contributed by a structural eigenvector with that of the error eigenvectors. Let us suppose that is the variance contributed by the last structural eigen-... 

For tablet formulations a response is usually described as a function of the mixture composition. The mixture components are also the cause of a complication of the robustness problem the amount of instability caused by errors made in the composition results in a variance/covariance structure of the mixture variables which depends on the mixture composition itself. The relation between the variance/covariance structure of the mixture variables and the mixture composition itself can be derived using partial derivatives which is shown below. 

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