Model-free analysis

but not all model-free methods are based on Factor Analysis and we start this chapter with a fairly detailed and comprehensive discussion of this topic. 

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Chapter 5, Model-Free Analyses. Model-based data fitting analyses rely crucially on the choice of the correct model model-free analyses allow insight into the data without prior chemical knowledge about the process. Model-free analysis is based on restrictions imposed on the results of the analysis. The restrictions that are demanded by the physics of the measurement rather then by the scientist. Typical restrictions of this kind are that concentrations and molar absorptivities have to be positive. 

Only multivariate (e.g. multi-wavelength) data are amenable to model-free analyses. While this is a restriction, it is not a serious one. The goal of the analysis is to decompose the matrix of data into a product of two physically meaningful matrices, usually into a matrix containing the concentration profiles of the components taking part in the chemical process, and a matrix that contains their absorption spectra (Beer-Lambert s law). If there are no model-based equations that quantitatively describe the data, model-free analyses are the only method of analysis. Otherwise, the results of model-... 

There is a rich collection of publications describing novel methods for Model-Free Analyses. The selection presented here does not cover the complete range it attempts to select the more useful and interesting methods. Such a selection is always influenced by personal preferences and thus can be biased. 

Excel does not provide functions for the factor analysis of matrices. Further, Excel does not support iterative processes. Consequently, there are no Excel examples in Chapter 5, Model-Free Analyses. There are vast numbers of free add-ins available on the internet, e.g. for the Singular Value Decomposition. Alternatively, it is possible to write Visual Basic programs for the task and link them to Excel. We strongly believe that such algorithms are much better written in Matlab and decided not to include such options in our Excel collection. 

In any book, there are relevant issues that are not covered. The most obvious in this book is probably a lack of in-depth statistical analysis of the results of model-based and model-free analyses. Data fitting does produce standard deviations for the fitted parameters, but translation into confidence limits is much more difficult for reasonably complex models. Also, the effects of the separation of linear and non-linear parameters are, to our knowledge, not well investigated. Very little is known about errors and confidence limits in the area of model-free analysis. 

In many applications, such as chromatography, equilibrium titrations or kinetics, where series of absorption spectra are recorded, the individual rows in Y, C and R correspond to a solution at a particular elution time, added volume or reaction time. Due to the evolutionary character of these experiments, the rows are ordered and this particular property will be exploited by important model-free analysis methods described in Chapter 5, Model-Free Analyses. 

Generally, the multivariate data analysis attempts to find the best matrices C and A for a given measured Y. We discuss a wide range of methods for this task, in depth, in the two Chapters 4 and 5, Model-Based Analyses and Model-Free Analyses. 

The number of linearly independent columns (or rows) in a matrix is called the rank of that matrix. The rank can be seen as the dimension of the space that is spanned by the columns (rows). In the example of Figure 4-15, there are three vectors but they only span a 2-dimensional plane and thus the rank is only 2. The rank of a matrix is a veiy important property and we will study rank analysis and its interpretation in chemical terms in great detail in Chapter 5, Model-Free Analyses. 

We discuss this decomposition again in great depth in Chapter 5, Model-Free Analyses. For the moment we need to identify a few essential properties of the Singular Value Decomposition and in particular of the matrices U, S, and V. 

On a very different note, in Chapter 5, Model-Free Analyses, we introduce methods that attempt a model-free analysis of the data. Typically, a matrix Y is automatically decomposed into the product of the matrices C and A. These analyses usually are not as robust as the fitting discussed in this chapter, however, the results can guide the researcher in finding the correct model. 

The Singular Value Decomposition of a matrix Y into the product USV is full of rich and powerful information. The model-free analyses we discussed so far are based on the examination of the matrices of eigenvectors U and V. Evolving Factor Analysis, EFA, is primarily based on the analysis of the matrix S of singular values. 

The initial guess for the concentration profiles is computed as the combination of the forward and backward EFA graphs the smaller of each forward/backward pair is used. We display them in Figure 5-44 as we use these initial guesses in most other upcoming model-free analyses. ... 

This process is repeated for all individual columns of C to result in the complete matrix T (containing l s in the first row). Finally, the product UT is the complete matrix C. As always with model-free analyses, only the shapes of the concentration profiles are determined. They have to be normalised in some way. 

The method of Alternating Least-Squares, ALS, is very simple and exactly for that reason it can be very powerful. ALS has found widespread applications and it is an important method in the collection of model-free analyses. In contrast to most other model-free analyses, ALS is not based on Factor Analysis. 

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