Structures, Fixing Parameters

In the present organisation of the programs, the above tasks require the cumbersome rewriting of parameter lists and small but important changes in the different routines. Such processes are error prone and are better avoided. 

These simplifications in the parameter handling are best organised by using structures and cell arrays, as provided by Matlab to supplement the matrix as a basic data type. We introduce both, beginning with structures. 

Structures are Matlab arrays with named data containers called fields. The fields of a structure can contain any kind of data. For example, one field might contain a text string representing a name, another might contain a scalar representing a billing amount, a third might hold a matrix of medical test results, and so on. (also see Matlab Help on structures). 

Until now, we have dealt with kinetic models and rate constants as the nonlinear parameters to be fitted to spectrophotometric absorbance data. However, measurements can be of a different kind and particularly titrations (e.g. pH-titrations) are often used for quantitative chemical analyses. In such instances concentrations can also be parameters. In fact, any variable used to calculate the residuals is a potential parameter to be fitted. 

In the examples below, we use the structure s into which we put all the information needed in different parts of the programs. 

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Fig. 6.45 Two-dimensional phase diagram as a function of chain composition for a blend of two diblocks (

Equation 6.36 for the adiabatic potential is exact within the framework of the mean field description. However, the structure of the electric part P1 is too complex to disclose its analytic properties. Here we examine the adiabatic potential numerically following the Carlson theory of elliptic integrals [15-21], To proceed with numerical computation, it is necessary to enter a set of parameters designed to describe an experimental situation. It will not surprise the reader who has made it this far that we use values of the chemically fixed parameters specified by the n-butylammonium vermiculite gels [22], namely m+ = 74 mp and m = 36 mp. The average density n0 of the small ions is given by... 

In rare cases, e.g. when all sites in the crystal structure are partially occupied, the scale factor may become strongly correlated with the population parameters. This requires special attention and detailed consideration of all possibilities exceeds the scope of this text. One of the options is to use the known scale factor as a fixed parameter while refining population parameters of all crystallographic sites. This option is, however, seldom available because relative and not absolute intensities are customarily measured in a powder diffraction experiment. 

When the number of samples per individual was increased to three, regardless of where the middle point was collected in time, the structural model parameters remained unbiased but the bias in the variance components was removed. When the number of subjects was increased to 100 and then 150, the bias and precision in the structural model parameters remained unchanged, but improved the estimation of the variance components. Hence, under these conditions, neither more data per subject nor more subjects improved the estimates of the fixed effects in the model. What were affected were the variance components. Both more data within a subject and more subjects resulted in better variance component estimation. 

To get at the question of overall influence, the matrix of structural model parameters and variance components was subjected to principal component analysis. Principal component analysis (PCA) was introduced in the chapter on Nonlinear Mixed Effects Model Theory and transforms a matrix of values to another matrix such that the columns of the transformed matrix are uncorrelated and the first column contains the largest amount of variability, the second column contains the second largest, etc. Hopefully, just the first few principal components contain the majority of the variance in the original matrix. The outcome of PC A is to take X, a matrix of p-variables, and reduce it to a matrix of q-variables (q < p) that contain most of the information within X. In this PC A of the standardized parameters (fixed effects and all variance components), the first three principal components contained 74% of the total variability in the original matrix, so PCA was largely successfully. PCA works best when a high correlation exists between the variables in the original data set. Usually more than 80% variability in the first few components is considered a success. 

The search or optimization method, the third component, describes the computational procedure to search over parameters and structures. Issues here include the computational methods used to optimize the scoring function and to search related parameters such as the maximum number of iterations or convergence specifications for iterative algorithms. Typical search methods are greedy search, gradient-depen-dent search methods, or breadth search methods [4], One distinguishes between searches that involve only the optimization of the parameters in fixed structures, and the optimization of structures and parameters for data mining methods that include searches over parameter and structure. 

In the introduction to last year s review we commented on the now almost ubiquitous use of computational methods in gas-phase structure determinations. In the very large majority of cases, the results of these calculations are used in some way in the structural analysis, perhaps as computed amplitudes of vibration (sometimes scaled to reproduce experimental frequencies). Sometimes calculated differences between related parameters are used as hxed parameters in the rehnements, but there are still relatively few uses of the computed data as flexible restraints. In this present review we describe two cases where there have been independent studies of the same molecules, and the results include some parameters that do not agree, within the expected levels of precision. We have tried to identify the sources of the discrepancies, and it seems that the imphcit assumption that fixed parameters are absolutely correct may be a problem. It is certainly an issue that requires care. 

As mentioned above, the numeric characteristic of sohd state process is conversion. The concept of conversion is completely different from that of concentration. If the pressure and temperature are constant, the concentration is the only characteristic for the system. Even in a state of fixed parameter value, the conversion is not the single-value characteristic of multi state system. In various systems with same components, it is possible to achieve the same conversion. The difference between these systems is the structure of the reaction region, and in this region, the products are generated first, followed by aggregation of products. 

In case of a generic structure, PARAMETRIC TYPE is either a parametric fixed structure or a variant structure. The parameter types have to be class types (see Classes on page 28) or, in case of a generic variant structure, enumeration types. 

Homogeneity of data. Homogeneous data will be uniform in structure and composition, usually possible to describe with a fixed number of parameters. Homogeneous data is encountered in simple NDT inspection, e.g. quality control in production. Inhomogeneous data will contain various combinations of indications from construction elements, defects and noise sources. An example of inhomogenous data are ultrasonic B-scan images as described in [Hopgood, 1993] or as encountered in the ultrasonic rail-inspection system described later in this paper. 

We designed the way of determination of parameters of the structure, based on fixed changing... 

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