Transformed variables

When we draw a scatter plot of all X versus Y data, we see that some sort of shape can be described by the data points. From the scatter plot we can take a basic guess as to which type of curve will best describe the X—Y relationship. To aid in the decision process, it is helpful to obtain scatter plots of transformed variables. For example, if a scatter plot of log Y versus X shows a linear relationship, the equation has the form of number 6 above, while if log Y versus log X shows a linear relationship, the equation has the form of number 7. To facilitate this we frequently employ special graph paper for which one or both scales are calibrated logarithmically. These are referred to as semilog or log-log graph paper, respectively. 

The least-squares procedure can be appHed to the transformed variables of any of the equations in Table 2, where a simple transformation of one or both of the variables results in a linearized expression. The sums for equations 83 and 84 must be formed from the transformed variables rather than from the original data. 

Using block diagram algebra and Laplace transform variables, the controlled variable C(.s) is given by... 

The transformed variables describe the system composition with or without reaction and sum to unity as do Xi and yi. The condition for azeotropy becomes X, = Y,. Barbosa and Doherty have shown that phase and distillation diagrams constructed using the transformed composition coordinates have the same properties as phase and distillation region diagrams for nonreactive systems and similarly can be used to assist in design feasibility and operability studies [Chem Eng Sci, 43, 529, 1523, and 2377 (1988a,b,c)]. A residue curve map in transformed coordinates for the reactive system methanol-acetic acid-methyl acetate-water is shown in Fig. 13-76. Note that the nonreactive azeotrope between water and methyl acetate has disappeared, while the methyl acetate-methanol azeotrope remains intact. Only... 

It frequently happens that we plot or analyze data in terms of quantities that are transformed from the raw experimental variables. The discussion of the propagation of error leads us to ask about the distribution of error in the transformed variables. Consider the first-order rate equation as an important example ... 

The Laplace transformation converts a function of t, F(t), into a function of s, f s), where s is the transform variable. The quantity/(s) is called the Laplace transform of F(t). Equation (3-66) shows several equivalent symbolic representations of the Laplace transform of the function y = F(t). 

Some authors use the letter p instead of s for the transform variable. 

What are the units of the Laplace transform variable s when applied to a first-order reaction ... 

Table 6-1 lists the experimental quantities, k, T, ct, the transformed variables x, y, and the weights w. (It is necessary, in least-squares calculations, to carry many more digits than are justified by the significant figures in the data at the conclusion, rounding may be carried out as appropriate.) The sums required for the solution of the normal equations are... 

Using (7.30), one can easily perform integration, required by (7.27). Substituting the result into (7.26), taking account of initial condition (7.28) we have the following matrix equation relative to the Fourier-transformed variables dq ((o)... 

Figure 2. Typical photochemical smog cycle in which hydrocarbons HC are consumed, NO is photooxidized to N02, and O3 accumulates. A. Typical variables. B. Showing transformed variable [03-NO ]. Adapted from Moshiri (75).

Plot the transformed variable versus time. A straight line is a visually appealing demonstration that the correct value of n has been found. Figure 7.3 shows these plots for the data of Example 7.4. The central line in Figure 7.3 is for n = 1.53. The upper line shows the curvature in the data that results from assuming an incorrect order of n = 2, and the lower line is for n=l. 

Thus, Tis a linear function of the new independent variables, X, X2,. Linear regression analysis is used to ht linear models to experimental data. The case of three independent variables will be used for illustrative purposes, although there can be any number of independent variables provided the model remains linear. The dependent variable Y can be directly measured or it can be a mathematical transformation of a directly measured variable. If transformed variables are used, the htting procedure minimizes the sum-of-squares for the differences... 

However, a correlation with road test ratings can often be improved if an independent term of log V is added to the transformation variable log a-jv. The friction or side force coefficient can then be written as... 

Linearization. In preliminary screening of reaction mechanisms, it is very useful to construct plots of experimental data transformed in such a way that the plot of the dependent (transformed) variable versus the independent (transformed) variable is a straight line if the rate equation being the basis of transformation has been chosen properly. This is illustrated with the rate expression for a-th order kinetics ... 

A straight line on a plot of transformed variables should not be considered as proof that the kinetic model corresponds to the true reaction mechanism. It only shows that the particular kinetic model can be successfully used to describe the relationship between the reaction rate and... 

There are several good reasons to focus on linear models. Theory may indicate that a linear relation is to be expected, e.g. Lambert-Beer s law of the linear relationship between concentration and absorbance. Even when a linear relation does not hold strictly it can be a sufficiently good local approximation. Finally, one may try and find a transformation of the individual variables (e.g. a logarithmic transformation), in order to obtain an acceptable linear model for the transformed variables. Thus, we simplify eq. (36.1) to... 

The above equations suggest that the unknown parameters in polynomials A( ) and B() can be estimated with RLS with the transformed variables yn and un k. Having polynomials A( ) and B(-) we can go back to Equation 13.1 and obtain an estimate of the error term, e , as... 

C(z l) which are used for the computation of the transformed variables yn and un )c at the next sampling interval. 

More typically, we have an indication that a transformed variable f y) has constant error variance and will wish to use this information to weight y appropriately. For example, we may suspect logy has constant error variance and wish to fit y. More typically, we might feel that y has constant error-variance and wish to fit 1/y. 

Now, if the transformed variable is normalized to Zw by the Jacobian of the transformation, we obtain... 

The main idea is to replace the x-variables with new variables which are transformations of the x-variables. The transformed variables can be used instead of the original ones or can be added (augmented variable set). The relation between the y-variable and the derived x-variables is established by linear models of the form... 

The use of Laplace transfonnations yields some very useful simplifications in notation and computation. Laplace-transforming the linear ordinary differential equations describing our processes in terms of the independent variable t converts them into algebraic equations in the Laplace transform variable s. This provides a very convenient representation of system dynamics. 

The equivalence between the stochastic variables and the At-transformed variables is given by the relation... 

Included one with log-transformed variables and the other with both log-transformations and with each variable standardized by size (Table IV). In general, the standard errors of the estimates were large relative to the mean which Indicates a relatively poor fit for the models tested here. 

A final measure of location is the harmonic mean. This is rarely used explicitly although again it may be implicitly used. For example, when considering the analysis of heart rate data, many statisticians would recommend that the reciprocal of the heart rate be analysed rather than the heart rate itself. Again, this has to do with an attempt to make the distribution of the transformed variable be more s)unmetric. The... 

The criterion of mean-unbiasedness seems to be occasionally overemphasized. For example, the bias of an MLE may be mentioned in such a way as to suggest that it is an important drawback, without mention of other statistical performance criteria. Particularly for small samples, precision may be a more important consideration than bias, for purposes of an estimate that is likely to be close to the true value. It can happen that an attempt to correct bias results in lowered precision. An insistence that all estimators be UB would conflict with another valuable criterion, namely parameter invariance (Casella and Berger 1990). Consider the estimation of variance. As remarked in Sokal and Rohlf (1995), the familiar sample variance (usually denoted i ) is UB for the population variance (a ). However, the sample standard deviation (s = l is not UB for the corresponding parameter o. That unbiasedness cannot be eliminated for all transformations of a parameter simply results from the fact that the mean of a nonlinearly transformed variable does not generally equal the result of applying the transformation to the mean of the original variable. It seems that it would rarely be reasonable to argue that bias is important in one scale, and unimportant in any other scale. 

Some statistical tests are specific for evaluation of normality (log-normality, etc., normality of a transformed variable, etc.), while other tests are more broadly applicable. The most popular test of normality appears to be the Shapiro-Wilk test. Specialized tests of normality include outlier tests and tests for nonnormal skewness and nonnormal kurtosis. A chi-square test was formerly the conventional approach, but that approach may now be out of date. 

Since the step response (F curve) is the time integral of the pulse response (C curve), the Laplace transform of the step response for the general boundary conditions as used by van der Laan will be given by dividing equation (27) by the transform variable, p,... 

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