Behavior univariant

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. 

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. 

As introduced earlier, inputs can be transformed to reduce their dimensionality and extract more meaningful features by a variety of methods. These methods perform a numeric-numeric transformation of the measured input variables. Interpretation of the transformed inputs requires determination of their mapping to the symbolic outputs. The inputs can be transformed with or without taking the behavior of the outputs into account by univariate and multivariate methods. The transformed features or latent variables extracted by input or input-output analysis methods are given by Eq. (5) and can be used as input to the interpretation step. 

Han, C., McGue, M.K., and lacono, W.G. (1999) Lifetime tobacco, alcohol and other substance use in adolescent Minnesota twins univariate and multivariate behavioral genetic analyses. Addiction 94 981-993. 

Univariant equilibrium for which there is one degree of freedom, represents the equilibrium between two co-existing phases. Since there is only one degree of freedom, choosing a value for one external variable, e.g. temperature, determines the remaining variable in a dependent manner, and the locus of points represented on the phase diagram for univariant behavior must lie on a line or curve. Thus the curves on the unary phase diagram represent solid-liquid, solid-vapor, solid-solid, and liquid-vapor equilibrium. 

Invariant behavior occurs at the intersection of three univariant curves. This intersection defines a point at which three phases are in equilibrium. At these so called triple or invariant points, there are no degrees of freedom and both temperature and pressure assume fixed values. 

In rivers and streams heavy metals are distributed between the water, colloidal material, suspended matter, and the sedimented phases. The assessment of the mechanisms of deposition and remobilization of heavy metals into and from the sediment is one task for research on the behavior of metals in river systems [IRGOLIC and MARTELL, 1985]. It was hitherto, usual to calculate enrichment factors, for instance the geoaccumulation index for sediments [MULLER, 1979 1981], to compare the properties of elements. Distribution coefficients of the metal in water and in sediment fractions were calculated for some rivers to find general aspects of the enrichment behavior of metals [FOR-STNER and MULLER, 1974]. In-situ analyses or laboratory experiments with natural material in combination with speciation techniques are another means of investigation [LANDNER, 1987 CALMANO et al., 1992], Such experiments manifest univariate dependencies for the metals and other components, for instance between different metals and nitrilotriacetic acid [FORSTNER and SALOMONS, 1991], but the interactions in natural systems are often more complex. 

As concerns the former, statistical tests on the measured data are usually adopted to detect any abnormal behavior. In other words, an industrial process is considered as a stochastic system and the measured data are considered as different realizations of the stochastic process. The distribution of the observations in normal operating conditions is different from those related to the faulty process. Early statistical approaches are based on univariate statistical techniques, i.e., the distribution of a monitored variable is taken into account. For instance, if the monitored variable follows a normal distribution, the parameters of interest are the mean and standard deviation that, in faulty conditions, may deviate from their nominal values. Therefore, fault diagnosis can be reformulated as the problem of detecting changes in the parameters of a stochastic variable [3, 30],... 

Diekhoff, G. (1992) Statistics for the Social and Behavioral Sciences Univariated, Bivariate, Multivariate. Dubuque Wm C. Brown Publishers. 

If the process is out-of-control, the next step is to find the source cause of the deviation (fault diagnosis) and then to remedy the situation. Fault diagnosis can be conducted by associating process behavior patterns to specific faults or by relating the process variables that have significant deviations from their expected values to various equipment that can cause such deviations as discussed in Chapter 7. If the latter approach is used, univariate charts provide readily the information about process variables with significant deviation. Since multivariate monitoring charts summarize the information from many process variables, the variables that inflate... 

Since yMst is a random variable, SPM tools can be used to detect statistically significant changes. histXk) is highly autocorrelated. Use of traditional SPM charts for autocorrelated variables may yield erroneous results. An alternative SPM method for autocorrelated data is based on the development of a time series model, generation of the residuals between the values predicted by the model and the measured values, and monitoring of the residuals [1]. The residuals should be approximately normally and independently distributed with zero-mean and constant-variance if the time series model provides an accurate description of process behavior. Therefore, popular univariate SPM charts (such as x-chart, CUSUM, and EWMA charts) are applicable to the residuals. Residuals-based SPM is used to monitor lhist k). An AR model is used for representing st k) ... 

When the supernatant phase is multicomponent, the system is no longer univariant. Although the conditions of Eq. (8.68) must still be satisfied, this does not ensure that the composition of the amorphous phase will remain fixed with changes in A. At constant pressure the equilibrium force need no longer depend solely on the temperature. Consequently, total melting does not have to occur at constant force, in analogy to the behavior of a closed system. 

For each of these PLE, a Weibull interpolation was fitted due to lager intervals of the PLE step function in comparison to the univariate PLE. The reason for that is obvious because by splitting the data into 25 disjoint clusters, fewer failures are taken into account for each PLE, leading to lager intervals. Also the last fitted Weibull distribution per cluster was used to extrapolate the failure behavior for further failure predictions. As can be seen in Figure 2, up to three different shape parameters for Weibull distributions are used for the inter- and extrapolation of each PLE, so the changing failure behavior can be taken into account for prediction purposes. 

ARIMA is a sophisticated univariate modeling technique. ARIMA is the abbreviation of Autoregressive integrated moving average (also known as the Box-Jenkins model). It was developed in 1970 for forecasting purposes and relies solely on the past behavior of the variable being forecasted. The model creates the value of F, with input from previous values of the same dataset. This input includes a factor of previous values as well as the elasticity of the... 

Figure 21.9 provides a general comparison of univariate and multivariate SPC techniques (Alt et al., 1998). When two variables, xi and X2, are monitored individually, the two sets of control limits define a rectangular region, as shown in Fig. 21.9. In analogy with Example 21.5, the multivariate control limits define the dark, ellipsoidal region that represents in-control behavior. Figure 21.9 demonstrates that the application of univariate SPC techniques to correlated multivariate data can result in two types of misclassification false alarms and out-of-control conditions that are not detected. The latter type of misclassification occurred at sample 8 for the two Shewhart charts in Fig. 21.8.

A joint Gaussian behavior of attribute vectors A with ha > 2 cannot be examined graphically, due to the dimensionality. However, if the attribute vector Afc follows a Gaussian distribution with parameters jj, and S, the univariate variable... 

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