Weighting-Factor Analysis

For example, consider two weighting factors W for the cost basis and Wj for the weight basis, such that Wj+Wj =1. Let R be any rating factor (depth h, volume V, weight W, or cost C). 

For weight and cost to be rated together as both important. 

Material Comparison Table for Rectangles in Bending for PF Phenolics Rated 60% on Weight Basis and 40% on Cost Basis 

a material comparison table can be prepared using a combined strength and deflection basis, with weighting factors deflection basis, with weighting factors w and Wj, respectively, as follows  

For example, using Equation 6.11 with w, = 0.6 and Wj = 0.4, comparison of Table 6.3 is obtained for an equal strength basis (from Table 6.1). On the combined weight and cost basis, material 2 is nearly equal to material 1 as the best choice. 

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Although some of the models are capable of supporting the analysis of temporal and spatial aspects of the systems safety risk, the results are snapshots in time and do not lend themselves well to the profiling of risk Furthermore, apart from the Weighted Factors Analysis, none of the methodologies are eapable of taking into account the effects of both detrimental and benefieial facets of the systems behaviour towards safety. 

Hessami A. and Himter A. (2002) Formalization of weighted factors analysis, Knowledge-Based Systems, Issue 15, pp. 377-390. 

Hanning The Hanning correction provides the best capture of the individual frequency components of a signature. However, this weighting factor may distort the actual amplitude of the frequency components. Nevertheless, it is used for routine monitoring using FFT analysis. 

Hence is the fraction of the total sum of squares (or inertia) c of the data X that is accounted for by v,. The sum of squares (or inertia) of the projections upon a certain axis is also proportional to the variance of these projections, when the mean value (or sum) of these projections is zero. In data analysis we can assign different masses (or weights) to individual points. This is the case in correspondence factor analysis which is explained in Chapter 32, but for the moment we assume that all masses are identical and equal to one. 

In this respect, the weight coefficients are proportional to the column-sums. Distances of Chi-square form the basis of correspondence factor analysis (CFA) which is discussed in Chapter 32. 

Correspondence factor analysis can be described in three steps. First, one applies a transformation to the data which involves one of the three types of closure that have been described in the previous section. This step also defines two vectors of weight coefficients, one for each of the two dual spaces. The second step comprises a generalization of the usual singular value decomposition (SVD) or eigenvalue decomposition (EVD) to the case of weighted metrics. In the third and last step, one constructs a biplot for the geometrical representation of the rows and columns in a low-dimensional space of latent vectors. 

A factor analysis of the Raman spectra of a set of linear polyethylenes identified the existence of a third component in addition to the pure crystalline and pure amorphous components [78]. The characteristics of the Raman spectrum of the interphase were very similar to that of the crystalline spectrum indicating that the interphase retains a significant degree of order. Using the Raman method, the content of interphase in linear polyethylenes was found to increase with molecular weight [74—76,78]. For molecular weights below... 

Now, let us discuss the rate equations embodied in eq.(74). To do this, there is need of a statistical analysis. If the system is kept coupled to a thermostat at absolute temperature T, and assuming that w(i - >if) contains effects to all orders in perturbation theory, the rate of this unimolecular process per unit (state) reactant concentration k + is obtained after summation over the if-index is carried out with Boltzman weight factors p(if,T) ... 

Although satisfactory criteria for deciding whether data are better analyzed by distributions or multiexponential sums have yet to established, several methods for determining distributions have been developed. For pulse fluorometry, James and Ware(n) have introduced an exponential series method. Here, data are first analyzed as a sum of up to four exponential terms with variable lifetimes and preexponential weights. This analysis serves to establish estimates for the range of the preexponential and lifetime parameters used in the next step. Next, a probe function is developed with fixed lifetime values and equal preexponential factors. An iterative Marquardt(18) least-squares analysis is undertaken with the lifetimes remaining fixed and the preexponential constrained to remain positive. When the preexponential... 

Correction for nonconstant variance. To correct for nonconstant variance, it is necessary to weight standard measurements according to their local variance, S. For each standard concentration the variance is determined by repetitive analysis at that level, and a weighting factor, w = 1/s, is calculated. 

The Sturmian eigenfunctions in momentum space in spherical coordinates are, apart from a weight factor, a standard hyperspherical harmonic, as can be seen in the famous Fock treatment of the hydrogen atom in which the tridimensional space is projected onto the 3-sphere, i.e. a hypersphere embedded in a four dimensional space. The essentials of Fock analysis of relevance here are briefly sketched now. 

In some diseases a simple ordinal scale or a VAS scale cannot describe the full spectrum of the disease. There are many examples of this including depression and erectile dysfunction. Measurement in such circumstances involves the use of multiple ordinal rating scales, often termed items. A patient is scored on each item and the summation of the scores on the individual items represents an overall assessment of the severity of the patient s disease status at the time of measurement. Considerable amoimts of work have to be done to ensure the vahdity of these complex scales, including investigations of their reprodu-cibihty and sensitivity to measuring treatment effects. It may also be important in international trials to assess to what extent there is cross-cultural imiformity in the use and imderstand-ing of the scales. Complex statistical techniques such as principal components analysis and factor analysis are used as part of this process and one of the issues that need to be addressed is whether the individual items should be given equal weighting. 

Roscoe, B. A. Hopke, P. K. Comparison of Weighted and Unweighted Target Transformation Rotations in Factor Analysis, Computers and Chemistry, in press,... 

The effective dose equivalent (M ) is the formulation for the weighted dose equivalents in irradiated tissues or organs stipulated in 1977 by the International Commission on Radiological Protection [ICRP (1977a)]. He is based on an ICRP analysis of the risk information in the 1977 report of the United Nations Scientific Committee on the Effects of Atomic Radiation [UNSCEAR (1977)]. The formulation is given in Table 1.1, where Wi is the weighting factor for the relative radiosensitivity of the tissue and H-y is the dose equivalent in the irradiated tissue or organ. 

Recently, Tichy investigated 41) the dependencies of the steric constants, Es, v, L, Bj, B4, MV (molar volume), [P] (parachor), MR (molar refraction), MW (molecular weight), and % (molecular connectivity index) on lipophilicity, as it is measured by n 42) and f43) constants. The data were treated by factor analysis methods. 

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