Selecting the number of components

It is necessary to decide the appropriate number of components to use in a model. The appropriate dimensionality of a model may even change depending on what the specified purpose of the model is. Hence, appropriate dimensionality of, e.g., a PARAFAC model, is not necessarily identical to the three-way pseudo-rank of the data array. Appropriate model dimensionality is not only a function of the data but also a function of the context and aim of the analysis. Hence, a suitable PARAFAC model for exploring a data set may have a rank different from a PARAFAC model where the scores are used for a subsequent regression model. 

There are a number of tools for choosing the number of components. Some are based on statistical assumptions and significance testing, some based on empirical rules and some 

In the following, the so-called scree plot for determining the number of components in PCA will be described. Afterwards, it will be shown how this method can be modified for finding the appropriate number of components in PARAFAC and in Tucker3 models. 

Multi-way Analysis With Applications in the Chemical Sciences 

A certain cutoff in the scree plot is used to determine which components are too small to be used. For exploratory purposes, it may suffice to choose a cutoff value that leads to, e.g., 80 % of the variation explained for noisy data, but for more quantitative purposes it is useful to have a more elaborate determination of the appropriate cutoff value. Usually the number of components is chosen where the plot levels off to a linear decreasing pattern (see also Horn [1965]). Thus, no more than the number of factors to the left of this point should be retained. 

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When cross-validation is used for selecting the number of components, the number yielding a minimum PRESS value is often chosen. In practice, a minimum in the PRESS values may not always be present or sometimes the minimum is only marginally better than simpler models. In such cases, it becomes more difficult to decide on the number of components... 

All PARAFAC loadings may and do often change when the model rank is increased. At the point where the correct number of components is used, these changes are often not very dramatic. This provides an empirical tool for selecting the number of components to use [Geladi et al. 2000], This is shown in Figure 7.6 for the peat example of Example 7.3. Figure 7.6 shows a bar-plot of sum of squares of the ordered PARAFAC components as model rank increases. The models remain stable up to rank four and some of the components increase dramatically in variation in the rank five model. This is an indication... 

The criterion of Cliff in 1987 was adopted for selecting the number of components that should be used for the analysis, which states that the eigenvalues of acceptable components should explain 70% of the total variance (Lopez Hidalgo, 1994). 

Sometimes we do not know whether some compound really exists. This causes no problem in selecting the number of components. 

The number of components N is incremented by one and the entire procedure is repeated to arrive at a PRESS(A + 1). In other words, for A + 1 = 2, two principal components would be extracted and used to predict the deleted data values. As N approaches the tme number of significant components, the prediction should improve, and thus the PRESS should decrease. As the significant number is passed, noise begins to be included within the model, which has low predictive abiUty. At this point, the PRESS should start to increase again. Thus the basic type of criterion used in cross-vaUdation to select is that when... 

There are three rules of thumb to guide us in selecting the number of calibration samples we should include in a training set. They are all based on the number of components in the system with which we are working. Remember that components should be understood in the widest sense as "independent sources of significant variation in the data." For example, a... 

A basic assumption of OPA is that the purest spectra are mutually more dissimilar than the corresponding mixture spectra. Therefore, OPA uses a dissimilarity criterion to find the number of components and the corresponding purest spectra. Spectra are sequentially selected, taking into account their dissimilarity. The dissimilarity of spectrum i is defined as the determinant of a dispersion matrix Y,. In general, matrices Y, consist of one or more reference spectra, and the spectrum measured at the /th elution time. 

The selection of a mobile diase for the separation of simple aixtures may not be a particuleurly difficult problem and can be arrived at quite quickly by trial and error. Solvent systems can be screened in parallel using either several development chambers or a device like the Camag Vario KS chamber, which allows the simultaneous evaluation of a number of solvents by allowing each of these to migrate along parallel channels scored on a single TLC plate [8]. However, whenever the number of components in a mixture exceeds all but a small fraction of the spot capacity for the TLC system, a more systematic method of solvent optimization is required. 

Therefore, a 4a separation (R = 1), in which peak retention times differ by four times the width at half-height, corresponds to a 2% area overlap between peaks.1 The maximum number of peaks that could be separated in a given time period assuming a given value of R, is defined as the peak capacity.1 The peak capacity must be greater — usually much greater — than the number of components in the mixture for a separation to succeed. The resolution of two compounds can also be written in terms of the number of plates of a column, N, the selectivity, a, and the capacity factors, k, and k j, as12... 

In general, the number of components N is selected at the point where the addition of a new component does not give relevant additional information within the context of the studied problem or, in other words, when this component explains experimental noise only. Those components explaining proportions of small variance are not investigated, and they are assumed to be mainly related to small background contributions or to noise and experimental error. The selected number... 

There are several important issues for PCA, like the explained variances of each PC which determine the number of components to select. Moreover, it is of interest if outliers have influenced the PCA calculation, and how well the objects are presented in the PCA space. These and several other questions will be treated below. 

A number of performance criteria are not primarily dedicated to the users of a model but are applied in model generation and optimization. For instance, the mean squared error (MSE) or similar measures are considered for optimization of the number of components in PLS or PC A. For variable selection, the models to be compared have different numbers of variables in this case—and especially if a fit criterion is used—the performance measure must consider the number of variables appropriate measures are the adjusted squared correlation coefficient, adjR, or the Akaike S information criterion (AIC) see Section 4.2.3. 

One must consider the number of product terms that should be included in a model. For chromatography data obtained from similar samples, it can be expected that the data will contain a high degree of correlation. In our experiments, two- or three-component models usually accounted for >90% of the variance in the data for a class of similar samples. Results from crossvalidation should be considered as the primary criteria in selecting the number of principal components to be extracted from a given data set (34). 

When factor analysis is applied to the fiber-reinforced composite, the results are indeterminate making the problem of estimating the number of components quite difficult. However, if one selects the spectra carefully, excellent results can be obtained including a determination of the fraction of glass 7,8). 

The choice, for the number of components or for any other quantity, from a final set of possibilities, hence extending the selection hierarchy also to quantities. 

Nbf is the number of degrees of freedom, Nc is the number of components, and Np is the number of phases in the system. The number of degrees of freedom represents the number of independent variables that must be specified in order to fix the condition of the system. For example, the Gibbs phase rule specifies that a two-component, two-phase system has two degrees of freedom. If temperature and pressure are selected as the specified variables, then all other intensive variables—in particular, the composition of each of the two phases—are fixed, and solubility diagrams of the type shown for a hypothetical mixture of R and S in Fig. 1 can be constructed. 

Because of the specific application area of GSC in the world of small molecules, the number of components to be separated is usually small. For most practical problems, therefore, specific stationary phases are readily available. Hence, GSC is not the most fertile soil for selectivity optimization. 

When the number of components or reactions is too large, or the mechanism is too complex to deduce with statistical certainty, then response surface models can be used instead. Methods for the statistical design of experiments can be applied, reducing the amount of experimental data that must be collected to form a statistically meaningful correlation of selectivity and yield to the main process parameters. See Montgomery (2001) for a good introduction to the statistical design of experiments. 

The Camag AMD system consists of two main components the AMD developing unit (Fig. Ic) and the microprocessor-based controller. This system provides an AMD under reproducible conditions. For the AMD microprocessor-based controller, the following parameters may be chosen the eluent composition, by selecting the number of solvent reservoir the number of developing steps the developing time for each step the number of preconditions the option of emptying the mixer after a selected step. 

Separation problems become substantially more difficult as the number of components increases much above 10. Such complexity is often characteristic of environmental and biological samples. Different chromatographic modes offer potentially unlimited selectivity, but the conditions for optimal selectivity are correspondingly more difficult to find. A systematic basis for the combining of independent selectivity mechanism can provide a major boost to the overall selectiv-... 

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