Analysis of Measurement Tables

Measurement tables are the raw data that result from measurements on a set of objects. For the sake of simplicity we restrict our arguments to measurements obtained by means of instraments on inert objects, although they equally apply to sensory observations and to living subjects. By convention, a measurement table is organized such that its rows correspond to objects (e.g. chemical substances) and that its columns refer to measurements (e.g. physicochemical parameters). Here we adopt the point of view that objects are described in the table by means of the measurements performed upon them. Objects and measurements will also be referred to in a more general sense as row-variables and column-variables. 

A special type of homogeneous measurements is found in a compositiorml table which describes chemical samples by means of the relative concentrations of their components. By definition, relative concentrations in each row of a compositional table add up to unity or to 100%. Such a table is said to be closed with respect to the rows. In general, closure of a table results when their rows or columns add up to a constant value. This operation is only applicable to homogeneous tables. Yet another type of homogeneous table arises when the rows or columns can be ordered according to a physical parameter, such as in a table of spectroscopic absorptions by chemical samples obtained at different wavelengths. 

Multivariate analysis of these different types of measurements (heterogeneous, homogeneous, compositional, ordered) may require special approaches for each of them. For example, compositional tables that are closed with respect to the rows, require a different type of analysis than heterogeneous tables where the columns are defined with different units. The basic approach of principal components 

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The eigenvectors extracted from the cross-product matrices or the singular vectors derived from the data matrix play an important role in multivariate data analysis. They account for a maximum of the variance in the data and they can be likened to the principal axes (of inertia) through the patterns of points that represent the rows and columns of the data matrix [10]. These have been called latent variables [9], i.e. variables that are hidden in the data and whose linear combinations account for the manifest variables that have been observed in order to construct the data matrix. The meaning of latent variables is explained in detail in Chapters 31 and 32 on the analysis of measurement tables and contingency tables. 

According to Andersen [12] early applications of LLM are attributed to the Danish sociologist Rasch in 1963 and to Andersen himself. Later on, the approach has been described under many different names, such as spectral map analysis [13,14] in studies of drug specificity, as logarithmic analysis in the French statistical literature [15] and as the saturated RC association model [16]. The term log-bilinear model has been used by Escoufier and Junca [ 17]. In Chapter 31 on the analysis of measurement tables we have described the method under the name of log double-centred principal components analysis. 

A measurement table is different from a contingency table. The latter results from counting the number of objects that belong simultaneously to various categories of two measurements (e.g. molar refractivity and partition coefficient of chemical compounds). It is also called a two-way table or cross-tabulation, as the total number of objects is split up in two ways according to the two measurements that are crossed with one another. The analysis of contingency tables is dealt with specifically in Chapter 32. 

PCA [12, 16] is a multivariate statistics method frequently applied for the analysis of data tables obtained from environmental monitoring studies. It starts from the hypothesis that in the group of original data, there is a set of reduced factors or dominant components (sources of variation) which influence the observed data variance in an important way, and that these factors or components cannot be directly measured (they are hidden factors), since no specific sensors exist for them or, in other words, they cannot be experimentally observed. 

Table 17.3. Ba+ Dipole and quadrupole polarizabilities, ad and aq, determined from an analysis of measured 6s n intervals and calculated using coulomb wavefunctions.a...

A quantitative measure of the robustness of an analytical procedure is the different levels of the intermediate precision and their comparison (analysis of variances) (Table 4). Of course, this approach only addresses random effects, which depend on the extent... 

Let us start with an ANOVA in case of a single factor, termed a one-way analysis of variance. Table 2.12 demonstrates the general scheme of the measurements of this type of ANOVA. 

Based on the comparative analysis of measured results of 10 in-situ tests and indoor test data (measured value and calculated value collapse settlement under overburden pressure) given in Table 1, some phenomenon can be found ... 

With the use of desktop computers and work stations now common, it is possible for corrosion researchers to put in their data and come out with analysis of variance tables for factorial experiments and multiple linetir regression analysis tables for uncontrolled experiments. Most such tables include a statistical test of significance for observed effects. That test (such as a F test) is a measure of the probability that an observed effect either exists or is caused by random error. 

Due to their simplicity of construction and use and the relatively sharp cut-off characteristics, cascade impactors have been widely used for the size classification and size-classified chemical analysis of aerosols. Table 6.1 lists the most important integrating sampling methods with their main characteristics. Table 6.2 gives the most important differential, size-resolving methods used to sample and measure atmospherie aerosol particles. The section of the particle size distribution and the modes that dominate the sensitivity of the methods are indicated. The upper and lower size limits are nominal values for the most commonly used forms of the techniques. Cost, complexity of operational requirements, calibration problems, and the demands of the particular evaluation to be used also affect the choice of methods. For example, chemical analysis usually requires that a sample be collected, then taken to the evaluation device. 

However, most of the time, due to the complexity of the food samples and the need to minimize the sample treatment, that is, the chemical separation of the interferences from the compounds of interest, more powerful instrumental measurements, such as 2D spectroscopies, for example EEM fluorescence spectra, or hyphenated separation techniques, for example gas chromatogra-phy/MS (GC/MS), HPLC/DAD or HPLC/MS, are used. As mentioned in previous sections, obtaining a data table per sample is a much more natural scenario for the application of MCR techniques. In these instances, the typical strategy is to perform the simultaneous analysis of data tables related to standards of known concentration together with tables from unknown samples, in which the concentration(s) of the analyte(s) are determined. Either using multiset analysis or multiway analysis, these determinations benefit from the so-called second-order advantage, which means that analytes can be determined in the presence of interferences, even if those are absent from the calibration samples [48]. The reason why this second-order advantage exists is that MCR techniques describe the information of the compounds in separate concentration profiles and spectra, that is, they make a mathematicar separation of the information related to aU compounds in the analysed sample, analytes and interferences. Afterwards, only the information of the profiles related to the analytes is used for quantitation purposes. 

Chemical Composition. Chemical compositional data iaclude proximate and ultimate analyses, measures of aromaticity and reactivity, elemental composition of ash, and trace metal compositions of fuel and ash. All of these characteristics impact the combustion processes associated with wastes as fuels. Table 4 presents an analysis of a variety of wood-waste fuels these energy sources have modest energy contents. 

Suppose we have two methods of preparing some product and we wish to see which treatment is best. When there are only two treatments, then the sampling analysis discussed in the section Two-Population Test of Hypothesis for Means can be used to deduce if the means of the two treatments differ significantly. When there are more treatments, the analysis is more detailed. Suppose the experimental results are arranged as shown in the table several measurements for each treatment. The goal is to see if the treatments differ significantly from each other that is, whether their means are different when the samples have the same variance. The hypothesis is that the treatments are all the same, and the null hypothesis is that they are different. The statistical validity of the hypothesis is determined by an analysis of variance. 

Two dimensionless variables play key roles in the analysis of single transition systems (and some multiple transition systems). These are the throughput parameter [see Eq. (16-129)] and the number of transfer units (see Table 16-13). The former is time made dimensionless so that it is equal to unity at the stoichiometric center of a breakthrough cui ve. The latter is, as in packed tower calculations, a measure of mass-transfer resistance. 

For sources having a large component of emissions from low-level sources, the simple Gifford-Hanna model given previously as Eq. (20-19), X = Cqju, works well, especially for long-term concentrations, such as annual ones. Using the derived coefficients of 225 for particulate matter and 50 for SO2, an analysis of residuals (measured minus estimated) of the dependent data sets (those used to determine the values of the coefficient C) of 29 cities for particulate matter and 20 cities for SOj and an independent data set of 15 cities for particulate matter is summarized in Table 20-1. For the dependent data sets, overestimates result. The standard deviations of the residuals and the mean absolute errors are about equal for particulates and sulfur dioxide. For the independent data set the mean residual shows... 

To analyze a turbine, it is neeessary to measure pressures and temperatures aeross the turbine, shaft vibration, and the temperature and pressure of the lubrieation system. Table 19-7 shows the effeet various parameters have on important funetions of the turbines. Analysis of these parameters will aid in the predietion of ... 

Interdiffusion of bilayered thin films also can be measured with XRD. The diffraction pattern initially consists of two peaks from the pure layers and after annealing, the diffracted intensity between these peaks grows because of interdiffusion of the layers. An analysis of this intensity yields the concentration profile, which enables a calculation of diffusion coefficients, and diffusion coefficients cm /s are readily measured. With the use of multilayered specimens, extremely small diffusion coefficients (-10 cm /s) can be measured with XRD. Alternative methods of measuring concentration profiles and diffusion coefficients include depth profiling (which suffers from artifacts), RBS (which can not resolve adjacent elements in the periodic table), and radiotracer methods (which are difficult). For XRD (except for multilayered specimens), there must be a unique relationship between composition and the d-spacings in the initial films and any solid solutions or compounds that form this permits calculation of the compo-... 

In addition to qualitative analysis of nearly all the elements of the periodic table, EEL spectra also enable determination of the concentration of a single element which is part of the transmitted volume and hence gives rise to a corresponding ionization edge. As in all comparable spectroscopic techniques, for quantification the net edge signal, which is related to the number N of excited atoms, must be extracted from the raw data measured. The net intensity 4 of the feth ionization shell of an individual element is directly connected to this number, N, multiplied by the partial cross-section of ionization ) and the intensity Iq of the incident electron beam, i.e. ... 

Ion-selective electrodes are a relatively cheap approach to analysis of many ions in solution. The emf of the selective electrode is measured relative to a reference electrode. The electrode potential varies with the logarithm of the activity of the ion. The electrodes are calibrated using standards of the ion under investigation. Application is limited to those ions not subject to the same interference as ion chromatography (the preferred technique), e.g. fluoride, hydrogen chloride (see Table 10.3). 

Ail the parameters on which various consents (or permissions to dispose of, waste streams) are based must be reliably measured and recorded. This is easier to achieve with gaseous emissions (Chapter 10) and liquid effluents than with heterogeneous solid wastes. Systematic analysis of solid wastes will cover as a minimum the information in Table 17.15. 

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