One-variable-at-a-time

Here we change one variable at a time, and cycle over all the variables until a minimum is reached. We proceed as follows ... 

The simple-minded approach for minimizing a function is to step one variable at a time until the function has reached a minimum, and then switch to another variable. This requires only the ability to calculate the function value for a given set of variables. However, as tlie variables are not independent, several cycles through tlie whole set are necessary for finding a minimum. This is impractical for more than 5-10 variables, and may not work anyway. Essentially all optimization metliods used in computational chemistry tlius assume that at least the first derivative of the function with respect to all variables, the gradient g, can be calculated analytically (i.e. directly, and not as a numerical differentiation by stepping the variables). Some metliods also assume that tlie second derivative matrix, the Hessian H, can be calculated. 

Analytical methods often contain many different variables which need to be optimized to attain best performance. The different variables are not always independent. For example, pH and polarity in a solution may be interdependent. Optimization by changing one variable at a time, while... 

These are solved by matrix manipulations. Programs are in POLYMATH, CONSTANTINIDES AND CHAPRA CANALE. When the number of equations is not large, a manual procedure can be used to eliminate one variable at a time by reduction of the leading coefficients to unity and appropriate additions and subtractions of the equations. 

A set of linear equations can be solved by a variety of procedures. In principle the method of determinants is applicable to any number of equations but for large systems other methods require much less numerical effort. The method of Gauss illustrated here eliminates one variable at a time, ends up with a single variable and finds all the roots by a reverse procedure. 

To determine the order of reaction. It is always good research strategy to change only one variable at a time. That way, the measured response (if any) can be attributed unambiguously to the change in that variable. And the variable of choice in this example will be one or other of the two initial concentrations. 

The confidence region obtained using a simulated one variable at a time design was first examined, since this was the design used by the original experimenters. 

Fig. 32. Approximate 95% confidence region for Eq. (149) and one variable at a time design.

The known information regarding the influence of various factors is given below. Most of the investigators have tried to study the effect of one variable at a time, but often minor variations of other variables also occur alongside. For example, variation of viscosity by the use of glycerol solution varies the surface tension by a few dynes per centimeter. 

A stepwise variable selection method adds or drops one variable at a time. Basically, there are three possible procedures (Miller 2002) ... 

Environmental studies are often characterized by large numbers of variables measured on many samples. When poor understanding of the system exists one tends to rely upon the "measure everything and hope that the obvious will appear" approach. The problem is that in complex chemical systems significant patterns in the data are not always obvious when one examines the data one variable at a time. Interactions among the measured chemical variables tend to dominate the data and this useful information is not extracted by univariate approaches. 

Infrared data in the 1575-400 cm region (1218 points/spec-trum) from LTAs from 50 coals (large data set) were used as input data to both PLS and PCR routines. This is the same spe- tral region used in the classical least-squares analysis of the small data set. Calibrations were developed for the eight ASTM ash fusion temperatures and the four major ash elements as oxides (determined by ICP-AES). The program uses PLSl models, in which only one variable at a time is modeled. Cross-validation was used to select the optimum number of factors in the model. In this technique, a subset of the data (in this case five spectra) is omitted from the calibration, but predictions are made for it. The sum-of-squares residuals are computed from those samples left out. A new subset is then omitted, the first set is included in the new calibration, and additional residual errors are tallied. This process is repeated until predictions have been made and the errors summed for all 50 samples (in this case, 10 calibrations are made). This entire set of... 

In this chapter a number of preprocessing tools are discussed. They are divided into two ba.sic types depending on whether they operate on samples or variables. Sample preproces.sing tools operate on one sample at a time over all variables. Variable preprocessing tools operate on one variable at a time over all samples. Therefore, if a sample is deleted from a data. set, variable preprocessing calculations must be repeated, while the sample preprocessing calculations will not be affected. 

To examine the ruggedness of the factors that were selected one could test these factors one variable at a time, i.e. change the level of one factor and keep all other factors at nominal level. The result of this experiment is then compared to the result of experiments with all factors at nominal level. The difference between the two types of experiments gives an idea of the effect of the factor in the interval between the two levels. The disadvantage of this method is that a large number of experiments is required when the number of factors is large. 

In the optimization of tablet formulations, different approaches can be used. The one variable at a time method requires many experiments and there is no guarantee that an optimal formulation is achieved. Moreover the interaction between different factors, which may influence the tablet properties, will not be detected [10]. The use of an experimental design can be helpful in the optimization of tablet formulations. Mixture designs can be used to describe the response (tablet properties) as a function of the... 

There is a tendency among control and statistics theorists to refer to trial and error as one-variable-at-a-time (OVAT). The results are often treated as if only one variable were controlled at a time. The usual trial, however, involves variation in more than one controlled variable and almost always includes uncontrolled variations. The trial-and-error method is fortunately seldom a random process. The starting cycle is usually based on manufacturers specifications or experience with a similar process and/or material. Trial variations on the starting cycle are then made, sequentially or in parallel, until an acceptable cycle is found or until funds and/or time run out. The best cycle found, in terms of one or a combination of product qualities, is then selected. Because no process can be repeated exactly in all cases, good cure cycles include some flexibility, called a process window, based on equipment limitations and/or experience. 

Figure 4 Schematic of the model-based optimization process, where performance depends on two variables (Vi and V2). Model-based methods would explore the entire oval domain, seeking the global best. Common OVAT practices only explore a few points along orthogonal trajectories. Abbreviation OVAT, one variable at a time.

Current practices in industrial pharmacy can now be put in perspective. Typically, the method of choice is univariate one variable at a time (OVAT). One variable is examined for a few conditions, which, in practice, are selected within a safe subset of the permissible design space. A value of this parameter is selected and kept subsequently constant. Another variable is then examined, a value is chosen, and the process continues sequentially. Intuitively, unless the target function is essentially a plane, if the end result is anywhere near the global optimum, it is only by chance. A historical reason for this dated practice is that the regulatory framework greatly discouraged implementation of the virtuous cycle mentioned above, which... 

Interactions are very important in establishing robust analytical methodology. Many analytical chemists were taught at school, and alas in some instances at university, that THE way to conduct experiments was to vary one variable at a time and hold all others constant. This is probably the cardinal sin of experimentation. Analytical chemistry abounds with examples of where the level of one reagent non-linearly affects the effect of another. An easy way to look at this is to plot the CRF values observed for one factor at each of its levels for two levels of another. 

In principle, it should be possible to obtain the electronic energy levels of the molecules as a solution of the Schrodinger equation, if inter-electronic and internuclear cross-coulombic terms are included in the potential energy for the Hamiltonian. But the equation can be solved only if it can be broken up into equations which are functions of one variable at a time. A simplifying feature is that because of the much larger mass of the nucleus the motion of the electrons can be treated as independent of that of the nucleus. This is known as the Bom-Oppen-heimer approximation. Even with this simplification, the exact solution has been possible for the simplest of molecules, that is, the hydrogen molecule ion, H + only, and with some approximations for the H2 molecule. 

The transition must be shown to be reversible. If the protein concentration is low (less than 0.5 mg/ml), this can be shown to be true for RNase over a wide range of temperature but only at pH values below about 3. The transitions are frequently so sharp that accurate estimates of equilibrium constants can be made only over a narrow range of the variable. Thus comparisons of parameters usually involve alteration of more than one variable at a time (i.e., pH and temperature) in order to keep the measurements in an accessible range. Paying as much attention... 

Figure 1. Comparison of univariate (one-variable-at-a-time) and simplex optimization of a simple response surface with two interdependent parameters (density and temperature). Ellipses represent contours of the response surface. See text for additional discussion.

If neither binary gradient nor three-solvent isocratics are successful, some systems will next try to perform a three-solvent gradient optimization. This development is very difficult to visualize. Assuming simultaneous optimization of %B, %C, and flow rate hinge points, it takes a long, computation-intensive time to carry out. It would be nearly impossible to carry out manually. The key is continually to use the rule of one change only one variable at a time and to carefully select limits for evaluation. 

This study evaluated changes in the quantity of volatiles formed from rhamnose and proline as functions of three reaction conditions temperature, pH and relative concentration. The objective was to identify and quantify the presence of interactions between dependant variable reaction conditions. That is, for example, the effect of temperature on trends brought about by pH. Such interactions cannot be estimated when variables are studied one-variable-at-a-time. A further benefit would be attained if the models could provide insight into the chemical processes involved. 

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