More about independent variables

A closed system containing a single substance in a single phase has two independent variables, as we can see by the fact that the state is completely defined by values of T and p or of T and V. 

A closed single-phase system containing a mixture of several nonreacting substances, or a mixture of reactants and products in reaction equiUbrium, also has two independent variables. Examples are 

The systems in these two examples contain more than one substance, but only one component. The number of components of a system is the minimum number of substances or mixtures of fixed composition needed to form each phase. A system of a single pure substance is a special case of a system of one component. In an open system, the amount of each component can be used as an independent variable. 

Consider a system with more than one uniform phase. In principle, for each phase we could independently vary the temperature, the pressure, and the amount of each substance or component. There would then be 2 - - C independent variables for each phase, where C is the number of components in the phase. 

There usually are, however, various equiUbria and other conditions that reduce the number of independent variables. For instance, each phase may have the same temperature and the same pressure equilibrium may exist with respect to chemical reaction and transfer between phases (Sec. 2.4.4) and the system may be closed. (While these various conditions do not have to be present, the relations among T, p,V, and amounts described by an equation of state of a phase are always present.) On the other hand, additional independent variables are required if we consider properties such as the surface area of a liquid to be relevant.  

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Most of our attention in this chapter has been devoted to models using three independent variables, as opposed to the two variables used in more traditional factor analysis and in most global analysis. This has the disadvantages of a requirement to identify three or more appropriate independent variables and to perform a larger number of measurements. It has the advantage of providing a richer data set, the analysis of which can yield results that are more precise than those provided by two-variable factor analysis and that are more independent of specific physical models than global analysis. In those circumstances for which a PARAFAC model is appropriate, the components can be resolved with no other information about their properties. 

Equation (10a) is somewhat inconvenient first, because we prefer to use pressure rather than volume as our independent variable, and second, because little is known about third virial coefficients It is therefore more practical to substitute... 

The profits from using this approach are dear. Any neural network applied as a mapping device between independent variables and responses requires more computational time and resources than PCR or PLS. Therefore, an increase in the dimensionality of the input (characteristic) vector results in a significant increase in computation time. As our observations have shown, the same is not the case with PLS. Therefore, SVD as a data transformation technique enables one to apply as many molecular descriptors as are at one s disposal, but finally to use latent variables as an input vector of much lower dimensionality for training neural networks. Again, SVD concentrates most of the relevant information (very often about 95 %) in a few initial columns of die scores matrix. 

This level of simplicity is not the usual case in the systems that are of interest to chemical engineers. The complexity we will encounter will be much higher and will involve more detailed issues on the right-hand side of the equations we work with. Instead of a constant or some explicit function of time, the function will be an explicit function of one or more key characterizing variables of the system and implicit in time. The reason for this is that of cause. Time in and of itself is never a physical or chemical cause—it is simply the independent variable. When we need to deal with the analysis of more complex systems the mechanism that causes the change we are modeling becomes all important. Therefore we look for descriptions that will be dependent on the mechanism of change. In fact, we can learn about the mechanism of... 

In general, a model will express a relationship between an independent variable (input by the operator) and one or more dependent variables (output, produced by the model). A ubiquitous form of equation for such input/output functions are curves of the rectangular hyperbolic form. It is worth illustrating some general points about models with such an example. Assume that a model takes on the general form... 

For operation in fixed-point DSP chips, the independent variable h = pm/2 — pjj" is generally confined to the interval [-1, 1). Note that having the table go all the way to zero at the maximum negative pressure hA+ = -1 is not physically reasonable (0.8 would be more reasonable), but it has the practical benefit that when the lookup-table input signal is about to clip, the re flection coefficient goes to zero, thereby opening the feedback loop. 

When 60 patients (22 men, 38 women) who had taken lithium for 1 year or more (mean 6.9 years mean serum concentration 0.74 mmol/1) were interviewed about adverse effects, 60% complained of polyuria-polydipsia syndrome (serum creatinine concentrations were normal) and 27% had hypothyroidism requiring treatment (108). Weight gain was more common in women (47 versus 18%) as were hypothyroidism (37 versus 9%) and skin problems (16 versus 9%), while tremor was more common in men (54 versus 26%). Weight gain of over 5 kg in the first year of treatment was the only independent variable predictive of hypothyroidism. 

For simplicity, the usual notational distinction between independent variables Xiu and observations piu has been maintained in this chapter, with the experimental settings Xiu regarded as perfectly known. If the settings are imprecise, however, it is more natural to regard them as part of the observations y u, this leads to various error-in-variables estimation methods. Full estimation of E is then not possible (Solari 1969), and some assumptions about its elements are necessary to analyze the data. Conventional error-in-variables treatments use least squares, with a scalar variance... 

Data set variables can be distinguished by their role in the models as independent and dependent variables. Independent variables (or explanatory variables, predictor variables) are those variables assumed to be capable of taking part of a function suitable to model the response variable. Dependent variables (or response variables) are variables (often obtained from experimental measures) for which the interest is to find a statistical dependence on one or more independent variables. Independent variables constitute the data matrix X, whereas dependent variables are collected into a matrix Y with n rows and r columns (r= 1 when only one response variable is defined) (Figure D2). Moreover, additional information about the belonging of objects to different classes can be stored in a class vector c, which consists of integers from 1 to G each integer indicates a class and G is the total number of classes. 

The correlation coefficient gives the dependent and independent variables equal weight which is usually not true in scientific measurements. The r value tends to give more confidence in the goodness of fit than warranted. The fit must be quite poor before r becomes smaller than about 0.98 and is reaUy very poor when less than 0.9. 

Differential equations that contain partial derivatives of several independent variables are called partial differential equations. The differential equations that we have been discussing contain ordinary derivatives and are called ordinary differential equations. Ordinary differential equations occur that contain more than one dependent variable, but you must have one equation for each dependent variable and must solve them simultaneously. We will not discuss simultaneous differential equations, but you can read about such equations in some of the books listed at the end of the book, and Mathematica is capable of solving simultaneous differential equations. 

Whenever one is fitting a model to data, it is helpful to present information about the individual residuals, r = y - fi , in a way that allows the user to get a visual sense of the quality of the fit. When a single experimental variable is used, it is common to plot the individual residuals (divided by the estimated standard deviations if they are available) against the value of that variable. When there are two or more independent variables, as with multilinear models, a similar graph can be made for each... 

A disadvantage of this calibration method is the fact that the calibration coefficients (elements of the P matrix) have no physical meaning, since they do not reflect the spectra of the individual components. The usual assumptions about errorless independent variables (here, the absorbances) and error-prone dependent variables (here, concentrations) are not valid. Therefore, if this method of inverse calibration is used in coimection with OLS for estimating the P coefficients, there is only a slight advantage over the classical /C-matrix approach, due to the fact that a second matrix inversion is avoided. However, in coimection with more soft modeling methods, such as PCR or PLS, the inverse calibration approach is one of the most frequently used calibration tools. 

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