Univariate characteristics

These weights depend on several characteristics of the data. To understand which ones, let us first consider the univariate case (Fig. 33.7). Two classes, K and L, have to be distinguished using a single variable, Jt,. It is clear that the discrimination will be better when the distance between and (i.e. the mean values, or centroids, of 3 , for classes K and L) is large and the width of the distributions is small or, in other words, when the ratio of the squared difference between means to the variance of the distributions is large. Analytical chemists would be tempted to say that the resolution should be as large as possible. 

We will not repeat Anscombe s presentation, but we will describe what he did, and strongly recommend that the original paper be obtained and perused (or alternatively, the paper by Fearn [15]). In his classic paper, Anscombe provides four sets of (synthetic, to be sure) univariate data, with obviously different characteristics. The data are arranged so as to permit univariate regression to be applied to each set. The defining characteristic of one of the sets is severe nonlinearity. But when you do the regression calculations, all four sets of data are found to have identical calibration statistics the slope, y-intercept, SEE, R2, F-test and residual sum of squares are the same for all four sets of data. Since the numeric values that are calculated are the same for all data sets, it is clearly impossible to use these numeric values to identify any of the characteristics that make each set unique. In the case that is of interest to us, those statistics provide no clue as to the presence or absence of nonlinearity. 

As for testing other characteristics of a univariate calibration, there are also ways to test for statistical significance of the slope, to see whether unity slope adequately describes the relationship between test results and analyte concentration. These are described in the book Principles and Practice of Spectroscopic Calibration [10]. The Statistics are described there, and are called the Data Significance t test and the Slope Significance t test (or DST and SST tests ). Unless the DST is statistically significant, the SST is meaningless, though. 

The characteristics of aluminophosphate molecular sieves include a univariant framework composition with Al/P = 1, a high degree of structural diversity and a wide range of pore sizes and volumes, exceeding the pore sizes known previously in zeolite molecular sieves with the VPI-5 18-membered ring material. They are neutral frameworks and therefore have nil ion-exchange capacity or acidic catalytic properties. Their surface selectivity is mildly hydrophilic. They exhibit excellent thermal and hydrothermal stability, up to 1000 °C (thermal) and 600 °C (steam). 

Multivariate calibration tools are used to construct models for predicting some characteristic of future samples. Chapter 5 begins with a discussion of the reasons for choosing multivariate over univariate calibration methods. The most widely used multivariate calibration tools are then presented in two categories classical and inverse methods. 

An index of toxicity is intended to be a simple tool that allows integrating and summarizing several variables into a single value. Realistically, this cannot be inferred without a judgement by environmental protection experts who consider all parameters available for their classification. PLS regression helped calculate an index fitted to expert judgement. The loss of information owing to the transformation of a multivariate situation to a univariate one was thus minimized since it is an inherent characteristic of multivariate analytical tools. 

In order to document the key characteristics of the tourist, a number of themes will be explored here. First, the existence and nature of tourist stereotypes will be considered. Then, the role of the tourist as a social position in society will be pursued. This treatment, together with some important conceptual schemes that help organise the stereotypes and role-related studies, will be followed by a consideration of the rich range of univariate approaches frequently used to classify tourists. 

Univariate optimization (table 5.7a) is not a good method for the optimization of chromatographic selectivity. This will be clear from the table, since despite a fairly high number of experiments only a local optimum will be located on the kinds of response surfaces typically encountered in chromatography (see figure 5.3). Moreover, the local optimum may be of little value, because no overall impression of the response surface is obtained during the process. Once this has been established, the other (favourable) characteristics of this method are no longer relevant. 

The characteristic feature of all these processes is the coexistence of two phases. According to the phase rule, a two-phase system consisting of a single species is univariant, and its intensive state is determined by the specification of just one intensive property. Thus the latent heat accompanying a phase change is a function of temperature only, and is related to other system properties by an exact thermodynamic equation ... 

Shi, W., Bugrim, A., Nikolsky, Y., Nikolskya, T., Breennan, R.J. (2008). Characteristics of genomic signatures derived using univariate methods and mechanistically anchored functional descriptors for predicting drug- and xenobiotic-induced nephrotoxicity. Toxicol. Mech. Meth. 18 267-76. 

One of the outstanding characteristic features of the modem analytical instrumentation is the ability of providing analytical signals in the form of different order tensors, i.e. vectors (the first order tensors), matrices (the second order tensors) and even higher order tensors. The multivariate data supplied by such instruments embody more information than the traditional univariate signal and are more suitable for the qualitative and quantitative analysis. 

As an example, consider the system formed by liquid water in equilibrium with its own vapor. The pressure—temperature diagram for this system has been constructed over the range of 1-99°C [10] and is shown in Fig. 1. The characteristics of a univariant system (one degree of freedom) are evident in that for each definite temperature value, water exhibits a fixed and definite pressure value. 

Conversely, if one begins with the anhydrous lithium iodide and exposes the solid to water vapor, as long as the vapor pressure is less than any of the dissociation pressures, no hydrate phase can form. At the lowest dissociation pressure a univariant system is obtained, since upon formation of the hydrate phase there must be three phases in equilibrium. Since the experiment is being conducted at constant laboratory temperature, the pressure must also be constant. Continued addition of water vapor can only result in an increase in the amount of hydrate phase and a decrease in the amount of anhydrate phase present. When the anhydrate is completely converted, the system again becomes bivariant, and the pressure increases again with the amount of water added. The higher hydrate forms are in turn produced at their characteristic conversion pressures in an equivalent manner. 

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