Variance-covariance method

If the underlying process is normal, the simulated distribution must converge to a normal distribution. In this situation, Monte Carlo analysis theoretically should yield exactly the same result as the multifactor variance-covariance method. The VaR estimated from the sample quantile must (not considering sampling variation) converge to the value of ao, where... 

The primary purpose for expressing experimental data through model equations is to obtain a representation that can be used confidently for systematic interpolations and extrapolations, especially to multicomponent systems. The confidence placed in the calculations depends on the confidence placed in the data and in the model. Therefore, the method of parameter estimation should also provide measures of reliability for the calculated results. This reliability depends on the uncertainties in the parameters, which, with the statistical method of data reduction used here, are estimated from the parameter variance-covariance matrix. This matrix is obtained as a last step in the iterative calculation of the parameters. 

The power algorithm [21] is the simplest iterative method for the calculation of latent vectors and latent values from a square symmetric matrix. In contrast to NIPALS, which produces an orthogonal decomposition of a rectangular data table X, the power algorithm decomposes a square symmetric matrix of cross-products X which we denote by C. Note that Cp is called the column-variance-covariance matrix when the data in X are column-centered. 

The expression x (J)P(j - l)x(j) in eq. (41.4) represents the variance of the predictions, y(j), at the value x(j) of the independent variable, given the uncertainty in the regression parameters P(/). This expression is equivalent to eq. (10.9) for ordinary least squares regression. The term r(j) is the variance of the experimental error in the response y(J). How to select the value of r(j) and its influence on the final result are discussed later. The expression between parentheses is a scalar. Therefore, the recursive least squares method does not require the inversion of a matrix. When inspecting eqs. (41.3) and (41.4), we can see that the variance-covariance matrix only depends on the design of the experiments given by x and on the variance of the experimental error given by r, which is in accordance with the ordinary least-squares procedure. 

However, there is a mathematical method for selecting those variables that best distinguish between formulations—those variables that change most drastically from one formulation to another and that should be the criteria on which one selects constraints. A multivariate statistical technique called principal component analysis (PCA) can effectively be used to answer these questions. PCA utilizes a variance-covariance matrix for the responses involved to determine their interrelationships. It has been applied successfully to this same tablet system by Bohidar et al. [18]. 

Important methods of data analysis base on evaluation of the covariance matrix (variance-covariance matrix)... 

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 ... 

The training set will be used to calculate the multivariate mean and variance-covariance matrix however, before calculating these parameters, we will graphically examine the training set to see if it contains measurements that are approximately normally distributed. This can be accomplished by several methods, the simplest being to plot histograms of the individual variables. Use the MATLAB hist command to... 

Global Two-Stage Method. An extensive description of the method is provided by Steimer et al. The global two-stage (GTS) approach has been shown, through simulation, to provide unbiased estimates of the population mean parameters and their variance-covariance, whereas the estimates of the variances were upwardly biased if the STS approach was used. These simulations were done under the ideal situation that the residual error was normally distributed with a known variance. However, it is a well-known fact that the asymptotic covariance matrix used in the calculations is approximate and under less ideal conditions, the approximation can be poor. ... 

When the principal components method was derived, matrix differentiation was used to determine the principal component vector p which minimized the sum of squared deviation. For this the symmetric mean centred variance-covariance matrix (X X) (X - X) was involved. 

Linear discriminant analysis (LDA) is a classification method that uses the distance between the incoming sample and the class centroid to classify the sample. For LDA using Mahalanobis distances, the classification metric uses the pooled variance-covariance matrix to weight the Mahalanobis distance ) between the incoming... 

Covariates are incorporated into the simulation as distributions that are either simulated stochastically or resampled from an existing database (18). Correlation between covariates is handled during stochastic simulations using multivariate distributions with appropriate variance-covariance structure. Alternatively, covariates resampled from a sufficiently large existing database carry all relevant covariates from an individual into a simulated individual and so capture inherent correlation. Regardless of the method, the simulated outputs for covariates need to be checked to ensure that they reflect the expected trial population and are consistent with trial inclusion and exclusion criteria. 

A proportional error, a constant additive error, and a combination of both error models were evaluated for the residual error model. Between-subject random effects were explored on the clearance of parent drug and metabolite, the volume of distribution of the parent drug, and the absorption rate constant. An exponential model was preferred. Interoccasion random effects were explored on the clearance of the parent drug and of the metabolite, the volume of distribution of the parent drug, and the absorption rate constant. An exponential model was preferred. The joint distribution of the between-subject random effect, the interoccasion random effects, and the residual error were assumed normal with mean 0 and variance-covariance matrices O for the between-subject and interoccasion random effects, and I, for the residual error to be estimated. The FO method was used for the estimation of the parameters. 

Standard errors and confidence intervals for functions of model parameters can be found using expectation theory, in the case of a linear function, or using the delta method (which is also sometimes called propagation of errors), in the case of a nonlinear function (Rice, 1988). Begin by assuming that 0 is the estimator for 0 and X is the variance-covariance matrix for 0. For a linear combination of observed model parameters... 

Newton-Raphson approach, as opposed to other methods, such as an expectation-maximization approach, is that the matrix of second derivatives of the objective function, evaluated at the optima, is immediately available. By denoting this matrix, H, 2H, is an asymptotic variance-covariance matrix of the estimated parameters G and R. Another method to estimate G and R is the noniterative MIVQUEO method. Using Monte Carlo simulation, Swallow and Monahan (1984) have shown that REML and ML are better estimators than MIVQUEO, although MIVQUEO is better when REML and ML methods fail to converge. 

In most cases, the fixed effect parameters are the parameters of interest. However, adequate modeling of the variance-covariance structure is critical for assessment of the fixed effects and is useful in explaining the variability of the data. Indeed, sometimes the fixed effects are of little interest and the variance components are of primary importance. Covariance structures that are overparameterized may lead to poor estimation of the standard errors for estimates of the fixed effects (Altham, 1984). However, covariance matrices that are too restrictive may lead to invalid inferences about the fixed effects because the assumed covariance structure does not exist and is not valid. For this reason, methods need to be available for testing the significance of the variance components in a model. 

Use statistical and graphical methods to compare various variance-covariance structures and select one of them. 

The method of merging separate band-by-band fits can simplify the fitting process with no sacrifice in accuracy of parameter determination. One of the reasons for considering a simultaneous multiband fit is that parameter correlations and standard errors can be significantly reduced. The merge process takes the band-by-band fitted parameters and variance-covariance matrices and combines them to obtain global constants that are identical to those that would have been obtained from a global fit. 

If overhtting occurs, then the prediction ability will be much worse than the classihcation ability. To avoid it, it is very important that the sample size is adequate to the problem and to the technique. A general rule is that the number of objects should be more than hve times (at least, no less than three times) the number of parameters to be estimated. LDA works on a pooled variance-covariance matrix this means that the total number of objects should be at least hve times the number of variables. QDA computes a variance-covariance matrix for each category, which makes it a more powerful method than LDA, but this also means that each category should have a number of objects at least hve times higher than the number of variables. This is a good example of how the more complex, and therefore better methods, sometimes cannot be used in a safe way because their requirements do not correspond to the characterishcs of the data set. 

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