Univariate experimentation

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... 

In the introduction to Part A we discussed the arch of knowledge [1] (see Fig. 28.1), which represents the cycle of acquiring new knowledge by experimentation and the processing of the data obtained from the experiments. Part A focused mainly on the first step of the arch a proper design of the experiment based on the hypothesis to be tested, evaluation and optimization of the experiments, with the accent on univariate techniques. In Part B we concentrate on the second and third steps of the arch, the transformation of data and results into information and the combination of information into knowledge, with the emphasis on multivariate techniques. 

A more subjective approach to the multiresponse optimization of conventional experimental designs was outlined by Derringer and Suich (22). This sequential generation technique weights the responses by means of desirability factors to reduce the multivariate problem to a univariate one which could then be solved by iterative optimization techniques. The use of desirability factors permits the formulator to input the range of property values considered acceptable for each response. The optimization procedure then attempts to determine an optimal point within the acceptable limits of all responses. 

To evaluate the quantitation capabilities of the experimental design, univariate calibration curves were constructed at 770 nm for four tested vapors as shown in Fig. 4.13. Upon exposure to the highest tested concentration of DCM and toluene... 

Exactly as in univariate analysis, once a model is created it can be applied to predict unknown samples. The difference with respect to the univariate case is that it is impossible to plot the model, because it is an equation in a multidimensional space. Hence, plots reporting predicted values vs. experimental values for standard samples of a training set are used to evaluate models reliability (validation). 

Figure 6,6 Experimentally observed univariant dTIdP equilibria for several silicates of petrogenetic interest. Melting curves are satisfactorily reproduced by Simon equation with parameters listed in table 6.5.

Classic univariate regression uses a single predictor, which is usually insufficient to model a property in complex samples. Multivariate regression takes into account several predictive variables simultaneously for increased accuracy. The purpose of a multivariate regression model is to extract relevant information from the available data. Observed data usually contains some noise and may also include irrelevant information. Noise can be considered as random data variation due to experimental error. It may also represent observed variation due to factors not initially included in the model. Further, the measured data may carry irrelevant information that has little or nothing to do with the attribute modeled. For instance, NIR absorbance... 

If the temp, is above the f.p., and ice is accordingly excluded, nine univariant systems, with three solid phases, are theoretically possible. W. Meyerhoffer and A. P. Saunders have studied this system in more detail, and the transition temp, was found to be 4 4°, not 3 7° they worked at 0°, 4" 4°, 16°, and 25° W. C. Blasdale followed up the work at 50°, 75°, and 100°. W. C. Blasdale s experimental data for 0°, 25°, and 50° expressed in eq. mols. of the various salts per 1000 mols. of water, are plotted in Figs. 54 to 56, with respect to four axes representing the four component salts—... 

A laboratory-made electrolytic cell was designed as an electrolytic generator of the molecular hydride of Cd. The influence of several parameters on the recorded signal was evaluated by the experimental design and subsequently optimised univariately... 

This chapter constitutes an attempt to demonstrate the utility of multivariate statistics in several stages of the scientific process. As a provocation, it is suggested that the multivariate approach (in experimental design, in data description and in data analysis) will always be more informative and make generalizations more valid than the univariate approach. Finally, the multivariate strategy can be really enjoyable, not the least for its capacity to reveal hidden treasures in data that in a univariate analysis look like a set of random numbers. 

This equation makes it possible to determine the number of moles of the component in each phase or to use n as an independent variable rather than V, if we so choose. The molar volumes are properties of the separate phases, and consequently can be considered as functions of the temperature and pressure. However, the system is univariant, and consequently the pressure is a function of the temperature when the temperature is taken as the independent variable. The relation between the temperature and pressure may be determined experimentally or may be determined by means of the Clapeyron equation. The differential of each molar volume may then be expressed by... 

The derivatives (dP/dT)S3t and (dxt/dT)sat may be determined experimentally or by solution of the set of Gibbs-Duhem equations applicable to each phase, provided we have sufficient knowledge of the system. If the system is multivariant, a sufficient number of intensive variables—the pressure or mole fractions of the components in one or more phases—must be held constant to make the system univariant. Thus, for a divariant system either the pressure or one mole fraction of one of the phases must be held constant. When the pressure is constant, Equation (9.9) becomes... 

B) We have pointed out that experimental studies are usually arranged so that the system is univariant. The experimental measurements then involve the determination of the values of the dependent intensive variables for chosen values of the one independent variable. Actually, the values of only one dependent variable need be determined, because of the condition that the Gibbs-Duhem equations, applicable to the system at equilibrium, must be... 

The experimental strategy used here is to perform a series of small experiments instead of a single comprehensive experiment. An univariate search was made in which only one variable was changed at a time. The information obtained in the earlier experiments performed during the univariant... 

On the other hand, random errors do not show any regular dependence on experimental conditions, since they are generated by many small and uncontrolled causes acting at the same time, and can be reduced but not completely eliminated. Thus, random errors are observed when the same measurement is repeatedly performed. In the simplest case, the universe of random errors is described by a continuous random variable e following a normal distribution with zero mean, i.e., for a univariate variable, the probability density function is given by... 

In equilibrium, the quantity N of a given sorbate, which is absorbed on a given sorbent, depends on its partial pressure (fugacity) P in the gas phase and the temperature T. A basic phenomenological description is specification of the functional dependence between N, P, and T. Both experimental observations and theoretical or thermodynamic descriptions are often the case in univariant functional descriptions the relation between N and P at constant T (an isotherm), between N and T at constant P (an isobar), or between P and T at constant N (an isostere). 

In an attempt to use milder acidic conditions the prereduction of Se(IV) to Se(VI) was carried out with a mixture of HC1 and HBr (10 percent v/v each) instead of HC1 alone (>50 percent v/v) [52]. Experimental parameters were selected by a univariate optimization method. The main advantage of the MW heating was that it allowed for a strict control over the heating power as well as over the time the heating was applied. Seven samples of orange juice were analyzed. Selenium was present in five of them as a mixture of Se(IV) and Se(VI), Se(IV) being the predominant species with concentrations ranging from 5.20 0.08 to 9.50 0.09 pg 1 1. 

This approach to calibration, although widely used throughout most branches of science, is nevertheless not always appropriate in all applications. We may want to answer the question can the absorbance in a spectrum be employed to determine the concentration of a compound . It is not the best approach to use an equation that predicts the absorbance from the concentration when our experimental aim is the reverse. In other areas of science the functional aim might be, for example, to predict an enzymic activity from its concentration. In the latter case univariate calibration as outlined in this section results in the correct functional model. Nevertheless, most chemists employ classical calibration and provided that the experimental errors are roughly normal and there are no significant outliers, all the different univariate methods should result in approximately similar conclusions. 

Dnring the infancy of IPC, retention prediction commonly faced trial-and-error procednres that attempted to make the problem univariate, holding all experimental conditions constant except one. This one-at-a-time changing of parameters, without regard to parameter interactions, is still practiced and may, in a time consuming way, improve performance. The description of the dependence of retention on the mobile phase composition parameters is the focus of interest of model makers becanse an a priori retention prediction is highly desirable. Optimization is finding the nnique combination of values of adjustable parameters that yields the best performance possible for a set of requirements. 

Once the experimental work has been completed you then need to consider how to interpret the results, i.e. how to maximize the chemical information inherent in the data. Initial attempts are often centred around plotting the data, to visualize trends and to allow conclusions to be drawn. The simplest form of data visualization is simply to tabulate the results (Chapter 38). As an example, if a class of students has determined the melting point of naphthalene, it is a relatively simple matter to tabulate the data (see Table 43.1). One possibility for the data is then to calculate the mean and standard deviation. Another approach would be to plot the data as a histogram, as in Fig. 43.1, so we are then able to make a visual interpretation of the quality of this univariate (one-variable) data. 

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