Univariant

The mathematical requirements for unique determination of the two slopes mi and ni2 are satisfied by these two measurements, provided that the second equation is not a linear combination of the first. In practice, however, because of experimental error, this is a minimum requirement and may be expected to yield the least reliable solution set for the system, just as establishing the slope of a straight line through the origin by one experimental point may be expected to yield the least reliable slope, inferior in this respect to the slope obtained from 2, 3, or p experimental points. In univariate problems, accepted practice dictates that we... 

Dichromate-permanganate determination is an artificial problem because the matrix of coefficients can be obtained as the slopes of A vs. x from four univariate least squares regression treatments, one on solutions containing only at... 

Subtracting the slope matrix obtained by the multivariate least squares tieatment from that obtained by univariate least squares slope matiix yields the error mahix... 

At this point the system has throe phases (CUSO4 CuS04,Hj0 HjO vapour) and the number of components is two (anhydrous salt water). Hence by the phase rule, F + F = C + 2, t.e., 3+F = 2 + 2, or F=l. The system is consequently univariant, in other words, only one variable, e.g., temperature, need be fixed to define the system completely the pressure of water vapour in equilibrium with CUSO4 and CuS04,Hj0 should be constant at constant temperature. 

In its simplest form, a direct, univariate cahbration method proceeds by assuming there is a mathematical model that relates analytical instmment response to concentration. Traditionally in analytical chemistry, the model is assumed to be a linear relation between response r and concentration c ... 

The normal model can take a variety of forms depending on the choice of noninformative or infonnative prior distributions and on whether the variance is assumed to be a constant or is given its own prior distribution. And of course, the data could represent a single variable or could be multidimensional. Rather than describing each of the possible combinations, I give only the univariate normal case with informative priors on both the mean and variance. In this case, the likelihood for data y given the values of the parameters that comprise 6, J. (the mean), and G (the variance) is given by the familiar exponential... 

Because there is only one independent variable, the subscript has been omitted. We now note that Zx/n = x and Zy/n = y, so we find Eqs. (2-75) as the normal equations for unweighted univariate least-squares regression. 

We now consider the case in which, again, the independent variable jc, is considered to be accurately known, but now we suppose that the variances in the dependent variable y, are not constant, but may vary (either randomly or continuously) with JC . To show the basis of the method we use the simple linear univariate model, written as Eq. (2-76). 

Carrying through the treatment as before yields Eqs. (2-78) as the normal equations for weighted linear univariate least-squares regression. 

If the reaction series cannot be correlated with one of these univariate LFER, it may be possible to fit the data to Eq. (7-30). a multivariate LFER. Examples of this approach are given by Ehrenson et al. ... 

Univariate LSERs may possess the conventional LEER form, as exemplified by Eq. (8-67), the Grunwald-Winstein equation, or they may simply be plots of log k against a solvent parameter such as Z, (30), or ir. Brownstein developed an LEER form for the latter type of correlation, writing... 

VALLEY is a steady-state, univariate Gaussian plmne dispersion algoridun designed for estimating either 24-hour or aimual concentrations resulting from emissions from up to 50 (total) point and area sources. 

A more sophisticated version of the sequential univariate search, the Fletcher-Powell, is actually a derivative method where elements of the gradient vector g and the Hessian matrix H are estimated numerically. 

Moments 92. Common Probability Distributions for Continuous Random Variables 94. Probability Distributions for Discrete Random Variables. Univariate Analysis 102. Confidence Intervals 103. Correlation 105. Regression 106. 

The consequences of the stability condition are clearly demonstrated by considering the univariant equilibrium... 

Barrer s discussion4 of his analog of Eq. 28 merits some comment. Equation 28 expresses the equilibrium condition between ice and hydrate. As such it is valid for all equilibria in which the two phases coexist and not only for univariant equilibria corresponding with a P—7" line in the phase diagram. (It holds, for instance, in the entire ice-hydratell-gas region of the ternary system water-methane-propane considered in Section III.C.(2).) In addition to Eq. 28 one has Clapeyron s equation... 

Carson and Katz5 studied another part of the methane-propane-water system. These authors investigated its behavior when an aqueous liquid, a hydrocarbon liquid, a gas, and some solid were present. It was found that the system was univariant so that the solid consisted of a single phase only. This phase is a hydrate which proved to contain methane and propane in various ratios. They then concluded that these hydrates behaved as solid solutions. It is clear that Carson and Katz measured a part of the four-phase line HllL1L2G. 

Fig. 8. The system CHCla-HaS-HaO. The univariant equilibria of the single-component systems have been indicated by dotted lines, those of the binary systems by thin lines, and those of the ternary system by heavy lines. The latter are approximate only, except for the lower half of the four-phase line HnLxL%G measured by von Stackelberg and Friihbuss.4 ...

The difference in behavior reported by Von Stackelberg is not thought to be an essential one. Whether decomposition of a hydrate containing two solutes by removal of its vapor through pumping will be a univariant process depends on the number and compositions of the phases formed. In the system of Fig. 10, for instance, a mixed HtS-propane hydrate will exhibit a constant decomposition pressure on pumping at — 3°C if it contains HaS and propane in the azeotropic" ratio of approximately 3 1. 

Uniform positive background, 255 United atom model, 276 Univariant, equilibrium, 19, 23 system, 91... 

However, it is not obvious that when we work with multivariate data, our training set must span the concentration ranges of interest in a multivariate (as opposed to univariate) way. It is not sufficient to create a series of samples where each component is varied individually while all other components are held constant. Our training set must contain data on samples where all of the various components (remember to understand "components" in the broadest sense) vary simultaneously and independently. More about this shortly. 

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