Univariate, term

Multivariate statistical methods should be preferred for evaluating such multidimensional data sets since interactions and resulting correlations between the water compounds have to be considered. Fig. 8-1, which shows the univariate fluctuations in the concentrations of the analyzed compounds, illustrates the large temporal and local variability. Therefore in univariate terms objective assessment of the state of pollutant loading is hardly possible. 

Depending on how the previous measurements are combined in Eq. (7), univariate filtering methods can be categorized as linear or nonlinear. In terms of Eq. (7), linear filtering methods use a fixed scale parameter or are single-scale, whereas nonlinear filtering methods are multiscale. Figure 7 summarizes decompositions in terms of time and frequency. 

By means of these variances the limits of detection XkLD of the k analytes tmder investigation can be estimated in analogy to the univariate case (see Sect. 7.5), namely on the basis of the critical values ynetkyC of the net signals their vector is denoted yk,c in the last term of Eq. (6.116b) ... 

Example calculations for a bivariate system can be found in Marchisio and Fox (2006) and Zucca et al. (2006). We should note that for multivariate systems the choice of the moments used to compute the source terms is more problematic than in the univariate case. For example, in the bivariate case a total of 3 M moments must be chosen to determine am, bm and cm. In most applications, acceptable accuracy can be obtained with 3[Pg.283]

It is sometimes convenient to fix the pressure and decrease the degrees of freedom by one in dealing with condensed phases such as substances with low vapour pressure. The Gibbs phase rule for a ternary system at isobaric conditions is Ph + F = C + 1=4, and there are four phases present in an invariant equilibrium, three in univariant equilibria and two in divariant phase fields. Finally, three dimensions are needed to describe the stability field for the single phases e.g. temperature and two compositional terms. It is most convenient to measure composition in terms of mole fractions also for ternary systems. The sum of the mole fractions is unity thus, in a ternary system A-B-C ... 

If the heat capacity functions of the various terms in the reaction are known and their molar enthalpy, molar entropy, and molar volume at the 2) and i). of reference (and their isobaric thermal expansion and isothermal compressibility) are also all known, it is possible to calculate AG%x at the various T and P conditions of interest, applying to each term in the reaction the procedures outlined in section 2.10, and thus defining the equilibrium constant (and hence the activity product of terms in reactions cf eq. 5.272 and 5.273) or the locus of the P-T points of univariant equilibrium (eq. 5.274). If the thermodynamic data are fragmentary or incomplete—as, for instance, when thermal expansion and compressibility data are missing (which is often the case)—we may assume, as a first approximation, that the molar volume of the reaction is independent of the P and T intensive variables. Adopting as standard state for all terms the state of pure component at the P and T of interest and applying... 

Figure 5.71A shows univariant equilibrium curves for various molar amounts of ferrous component in the orthopyroxene mixture. The P-T field is split into two domains, corresponding to the structural state of the coexisting quartz (a and j3 polymorphs, respectively). If the temperature is known, the composition of phases furnishes a precise estimate of the P of equilibrium for this paragenesis. Equation 5.277 is calibrated only for the most ferriferous terms, and the geobarometer is applicable only to Fe-rich rocks such as charnockites and fayalite-bearing granitoids. 

The Clausius-Clapeyron equation describes the univariant equilibrium between crystal and melt in the P-Tfield. Because molar volumes and molar entropies of molten phases are generally greater than their crystalline counterparts, the two terms and AFfusion both positive and we almost invariably observe an... 

The precision of thermobarometric equations 9.130 and 9.131 (once T is known, P is also fixed by the water-vapor univariant curve) depends on the accuracy of the last term on the right, which becomes more precise as the fractional amount of gas in vapor Xg falls. Rearranging equations 9.130 and 9.131 with the introduction of mass distribution constants of the type defined in equation 9.102, Giggenbach (1980) transformed equations 9.130 and 9.131 into thermobarometric functions based on the chemistry of the fluid. 

The next step is to construct the regression model from the calibration data using one ariable plus an intercept. Tlie results of this (now univariate) model (see Table 5.10) indicate that the intercept is not a significant term in the regression. This is expected because the data are known to obey Beer s Law. Traditionally, in statistics the intercept term is left in the model even when found to be nonsignificant. 

In traditional method validation, assessment of the calibration has been discussed in terms of linear calibration models for univariate systems, with an emphasis on the range of concentrations that conform to a linear model (linearity and the linear range). With modern methods of analysis that may use nonlinear models or may be multivariate, it is better to look at the wider picture of calibration and decide what needs to be validated. Of course, if the analysis uses a method that does conform to a linear calibration model and is univariate, then describing the linearity and linear range is entirely appropriate. Below I describe the linear case, as this is still the most prevalent mode of calibration, but where different approaches are required this is indicated. 

Remark 1 Necessary and sufficient optimality conditions can be also expressed in terms of higher order derivatives assuming that the function /(jc) has such higher order derivatives. For instance, a necessary and sufficient condition for x G being a local minimum of a univariate function /(jc) which has (k + l)th derivative can be stated as... 

Aside from univariate linear regression models, inverse MLR models are probably the simplest types of models to construct for a process analytical application. Simplicity is of very high value in PAC, where ease of automation and long-term reliability are critical. Another advantage of MLR models is that they are rather easy to communicate to the customers of the process analytical technology, since each individual X-variable used in the equation refers to a single wavelength (in the case of NIR) that can often be related to a specific chemical or physical property of the process sample. 

This equation gives the change of pressure with a change of temperature at the maximum or minimum. The system becomes univariant. However, because of the cancellation of the terms containing the chemical potentials, the determination of the change of the composition of the phases at the maximum or minimum with change of temperature or pressure cannot be determined from the Gibbs-Duhem equations alone. In order to do so we introduce the equality of the differentials of the chemical potentials in the two phases for one of the components, so... 

Note that the matrix is symmetrical about the diagonal variances appear on the diagonal and covariances appear on the off-diagonal. If we were to neglect the covariance terms from the variance-covariance matrix, any resulting statistical analysis that employed it would be equivalent to a univariate analysis in which we consider each variable one at a time. At the beginning of the chapter we noted that considering all variables simultaneously yields more information, and here we see that it is precisely the covariance terms of the variance-covariance matrix that encodes this extra information. 

Now that some potentially reasonable wavelengths have been identified, several models will be built and compared. These include assorted combinations of univariate and multivariate models with and without intercept terms. For this section, derivative spectra are not considered. It should be noted that results equal to the inclusion of an intercept term could be obtained by mean centering. 

The body of this chapter discusses population-based univariate reference values and quantities derived from them. If, for example, we produce, treat, and use separate reference values for cholesterol and triglycerides in serum, we have two sets of univariate reference values. The term multivariate... 

The variable x in the preceding formulas denotes a quantity that varies. In our context, it signifies a reference value. If the variable by chance may take any one of a specified set of values, we use the term variate (i.e, a random variable). In this section, we consider distributions of single variates (i.e., univariate distributions). In a later section, we also discuss the joint distribution of two or more variates bivariate or multivariate distributions). 

The representation of this equation for anything greater than two variates is difficult to visualize, but the bivariate form (m = 2) serves to illustrate the general case. The exponential term in Equation (26) is of the form x Ax and is known as a quadratic form of a matrix product (Appendix A). Although the mathematical details associated with the quadratic form are not important for us here, one important property is that they have a well known geometric interpretation. All quadratic forms that occur in chemometrics and statistical data analysis expand to produce a quadratic smface that is a closed ellipse. Just as the univariate normal distribution appears bell-shaped, so the bivariate normal distribution is elliptical. 

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