Projected Variance method

The Projected Variance (PV) method describes robustness as the variance of the response induced by the variance in the independent variable(s) propagated through the response surface. This method was first described by Box [17]. Vuchkov et al. [18] has used this method in case of second... 

In the method of linear discriminant analysis, one therefore seeks a linear function of the variables, D, which maximizes the ratio between both variances. Geometrically, this means that we look for a line through the cloud of points, such that the projections of the points of the two groups are separated as much as possible. The approach is comparable to principal components, where one seeks a line that explains best the variation in the data (see Chapter 17). The principal component line and the discriminant function often more or less coincide (as is the case in Fig. 33.8a) but this is not necessarily so, as shown in Fig. 33.8b. 

Accountability of the variances and covariances of the responses makes this optimization procedure particularly noteworthy. From the formulator s viewpoint, the distance criterion could lead to an unacceptable optimum if the formulation levels at the optimum produce response values in an undesirable property range. Khuri and Conlon mentioned the possible use of this procedure for multiresponse mixture optimization although no elaboration or examples were given. The reliance of this method on the gradient projection technique could present difficulties with component level compensation if applied to formulations. 

Methods of robust PCA are less sensitive to outliers and visualize the main data structure one approach for robust PCA uses a robust estimation of the covariance matrix, another approach searches for a direction which has the maximum of a robust variance measure (projection pursuit). 

Four pairs of structures with identical descriptors merge at a distance of zero. From the chemist s point of view clustering appears more satisfying than the linear projection method PCA (with only 47.6% of the total variance preserved by the first two PCA scores). A number of different clustering algorithms have been applied to the 20 standard amino acids by Willet (1987). 

PLS is related to principal components analysis (PCA) (20), This is a method used to project the matrix of the X-block, with the aim of obtaining a general survey of the distribution of the objects in the molecular space. PCA is recommended as an initial step to other multivariate analyses techniques, to help identify outliers and delineate classes. The data are randomly divided into a training set and a test set. Once the principal components model has been calculated on the training set, the test set may be applied to check the validity of the model. PCA differs most obviously from PLS in that it is optimized with respect to the variance of the descriptors. 

While thousands of analyses of archaeological bronzes have been reported in the literature, the basis for comparing them, especially those from different laboratories, is shaky. A round-robin project of chemical analyses was attempted to improve the situation. Two ancient bronze objects were milled to a fine powder, sieved, and mixed to a homogeneous mass. Samples of 500 mg each drawn randomly from this mass were circulated, and results were returned from 21 laboratories. Forty-eight elements were analyzed some laboratories did only one element, some did as many as 42. The coefficient of variance (or relative standard deviation) ranges from 4% for Cu up to over 200% for some trace elements. The results are tabulated, and methods are suggested to narrow the spread of results in the next run of this program. 

Laboratories review the SOW to establish whether their capacity, capability, and accreditations meet the project requirements. If any of the requirements are not met, the laboratory will specify the exceptions in a letter accompanying the bid proposal. Typically, exceptions include the absence of necessary accreditations and variances from the desired RLs and acceptability criteria. If the planning team has selected the RLs and the acceptability criteria based on current analytical methods and standard laboratory practices, the variances will not be significant and will not affect data quality or usability. 

The principal component plot of the objects allows a visual cluster analysis. The distances between data points in the projection, however, may differ considerably from the actual distance values. This will be the case when variances of the third and following principal components cannot be left out of consideration. A serious interpretation should include the application of at least another cluster analysis method (ref. 11,12). 

In the analysis of the effect on the calculated quantity of random errors in measured quantities it is unfortunate that the only model susceptible to an exact statistical treatment is the linear one (II). Here we have attempted to characterize the frequency distribution of the error in the calculated vapor composition by the standard methods and have not included a co-variance term for each pair of dependent variables (12). Our approach has given a satisfactory result for the methanol-water-sodium chloride system but it has not been tested on other systems and perhaps of more importance, it has not been possible, so far, to confirm the essential correctness of the method by an independent procedure. Work is currently being undertaken on this project. 

Once the project is under construction, a project cost control system should be implemented. This system provides a status report listing construction status, current budget, committed cost, expenditures to date, cost to be committed, final cost, and variance from budget. This cost system provides management with a method to plan and track the various phases of a packaging project. 

However, by definition, these univariate methods of hypothesis testing are inappropriate for multispecies toxicity tests. As such, these methods are an attempt to understand a multivariate system by looking at one univariate projection after another, attempting to find statistically significant differences. Often the power of the statistical tests is quite low due to the few replicates and the high inherent variance of many of the biotic variables. 

An eel extract and a sewage sludge were prepared and half the extracts cleaned-up by a single laboratory. The second half of the extracts were cleaned-up by each laboratory using the different methods later employed in this certification project in an optimised form. Both sets of samples were analysed for the seven CB congeners to determine the level of variance amongst the clean-up procedures. 

Theory. The variance criterion (i.e., maximizing the variance in the data) of classical PCA is very sensitive to outlying samples. As a consequence, the real structure of the data cannot always be revealed. To overcome this problem, rPCA (9-13) was introduced, which aims to obtain PCs that are less influenced by outliers. Additionally, robust methods should be able to detect the outlying observations. These goals are achieved by applying a more robust parameter (than variance) as projection index. 

Theory. PP is also a variable reduction method, very similar to PCA. In fact, PP can be considered a generalization of classical PCA (6,14-18). While in PCA the PCs are determined by maximizing variance, in PP, the latent variables, called the projection pursuit features (PPFs), are obtained by optimizing a given projection index that describes the inhomogeneity of the data, instead of its variance (6,18). In the literature, many PP indices have been described. 

The practical implication is that this information has to be retrieved with sufficient accuracy for small values of p, before the signal disappears in the statistical noise. The projection time p can be kept small by using optimized basis states constructed to reduce the overlap of the linear space spanned by the basis states u(> with the space spanned by the eigenstates beyond the first n of interest. We shall describe, mostly qualitatively, how this can be done by a generalization of a method used for optimization of individual basis states [3,21-23], namely, minimization of variance of the configurational eigenvalue, the local energy in quantum Hamiltonian problems. 

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