SIMCA principal component modeling

SIMCA EMX, P.O.Box 336, S-95125 Lulea, Sweden 2200. Multivariate data analysis by SIMCA (principal component models of classes) and PLS (partial least square) (ref. 20). 

Whenever data belongs to different known categories a principal component model can be calculated for each category. This technique is used in the method SIMCA for classification and modelling quantitative correlations between the model parameters (axes) and external properties can be established (ref. 9,10). 

The variation in the data not explained by the principal component model is called the residual variance. Classification in SIMCA is made by comparing the residual variance of a sample with the average residual variance of those samples that make up the class. This comparison provides a direct measure of the similarity of a sample to a particular class and can be considered as a measure of the goodness of fit of a sample for a particular class model. To provide a quantitative basis for this comparison, an / -statistic is used to compare the residual variance of the sample with the mean residual variance of the class [72], The F-statistic can also be used to compute an upper limit for the residual variance of those samples that belong to the class, with the final result being a set of probabilities of class membership for each sample. 

Disjoint principal components modelling [266] and SIMCA (soft independent modelling of class analogy) [261,262,267] are examples of PCR wherein principal components models are developed for individual groups of responses within a data set. For these methods, classification is based on quality of fit of an unknown response pattern to the model developed for a given analyte [268-270]. This approach differs from standard PCR, where principal components are derived from the data matrix as a whole. 

In the SIMCA method for classification,[15,16] separate principal components models are determined for each class. The idea behind this is that classes are fairly homogeneous and that the objects in a class are similar to each other, and it is very likely that a principal components model with few components is sufficient to describe the variation within a class. When a new object is projected down to the... 

M. Sjostrom and S. Wold, SIMCA A pattern recognition method based on principal components models, in Pattern Recognition in Practice (E.S. Gelsema and L.N. Kanal Eds), North-Holland, Amsterdam (1980), pp. 351-359. 

Kvalheim, O.M. Karstang, T.V. (1992). SIMCA-Classification by means of disjoint cross validated principal component models. In Multivariate Pattern Recognition in Chemometrics, illustrated by case studies, R.G. Brereton (Ed.), 209-245, Elsevier, ISBN 0444897844, Amsterdam, Netherland... 

The SIMCA method develops principal component models for each training set category. The main goal is the reliable classification of new samples. When a prediction is made in SIMCA, new samples insufficiently close to the PC space of a class are considered non-members. Table 4 shows classification for compounds from the training set. Here sample 9 was classified incorrectly since its activity is 4.2 (more active) but it is classified by SIMCA as less active. 

More often, the SIMCA method is used. This finds separate principal component models for each class. By using SIMCA, the object variable number ratio is less critical and the model is constructed around the projected, rather than the original, data. The basic steps of principal component calculations as needed for SIMCA have been outlined in the chapter on projection methods with the NIPALS algorithm (Example 5.1). 

In SIMCA, an independent principal component model (cf PCA) is estabhshed for each individual class of the test data set. The evaluation of the assignment of objects to these classes of an estabhshed model is performed by statistically backed distance measures. 

This approach was originally developed by Wold (1976) under the name disjoint principal components models, later termed simple modelling of class analogy (SIMCA) (see also Wold and Sjostrom, 1977 Wold et al., 1983). While biological applications of SIMCA have been limited (e.g. Wold, 1976 Dahl et al., 1984), the technique exhibits some of the attributes of much more advauced neural-net architectures (see following discussion). Moreover, because of its basis in... 

One of the most popular pattern recognition methods in chemistry is SIMCA, an acronym for soft/simple independent modeling of class analogy. The central idea is to represent each class of objects by a separate principal component model. Because a probability can be estimated for belonging to a certain class and because outliers can be detected the method is called soft. Classification methods such as discriminant analysis are called hard if they give a categorical answer about the class membership. 

Figure 9 SIMCA approximation of a class of objects X by a principal component model, n, number of objects p, number of features p, number of model components (in this scheme p is 2) m, mean vector of all objects B, loadings of the principal components used U, scores corresponding to B E, residual matrix...

Historically, SIMCA [43,44], proposed by Wold et al. in 1976, was the first class-modelling method introduced in the literature. Its key assumption is that the main systematic variability characterizing the samples from a category can be captured by a principal component model (see Chapter 4) of opportune dimensionality, built on training samples from that class. In detail, defining... 

In contrast, SIMCA uses principal components analysis to model object classes in the reduced number of dimensions. It calculates multidimensional boxes of varying size and shape to represent the class categories. Unknown samples are classified according to their Euclidean space proximity to the nearest multidimensional box. Kansiz et al. used both KNN and SIMCA for classification of cyanobacteria based on Fourier transform infrared spectroscopy (FTIR).44... 

The main classification methods for drug development are discriminant analysis (DA), possibly based on principal components (PLS-DA) and soft independent models for class analogy (SIMCA). SIMCA is based only on PCA analysis one PCA model is created for each class, and distances between objects and the projection space of PCA models are evaluated. PLS-DA is for example applied for the prediction of adverse effects by nonsteroidal anti-... 

The multivariate techniques which reveal underlying factors such as principal component factor analysis (PCA), soft Independent modeling of class analogy (SIMCA), partial least squares (PLS), and cluster analysis work optimally If each measurement or parameter Is normally distributed In the measurement space. Frequency histograms should be calculated to check the normality of the data to be analyzed. Skewed distributions are often observed In atmospheric studies due to the process of mixing of plumes with ambient air. 

After determining the underlying factors which affect local precipitation composition at an Individual site, an analysis of the slmlllarlty of factors between different sites can provide valuable Information about the regional character of precipitation and Its sources of variability over that spatial scale. SIMCA ( ) Is a classification method that performs principal component factor analysis for Individual classes (sites) and then classifies samples by calculating the distance from each sample to the PGA model that describes the precipitation character at each site. A score of percent samples which are correctly classified by the PGA models provides an Indication of the separability of the data by sites and, therefore, the uniqueness of the precipitation at a site as modeled by PGA. 

Figure 4.93. PCA of TEA SI MCA library samples (O) with unknowns (labeled with numbers). The first two principal components are used to make the TEA SIMCA model.

While KliN only uses physical closeness of samples to construct models, SIMCA uses the position and shape of the object formed by the samples in row space fordass definition. Modeling the object fonned by an individual class is accomplisfed with principal components analysis (PCA) (see Section 4.2.2). A multidimensional box is constructed for each class and the classification of fit ture samples is performed by determining within which box, if any, the sample belong (using an F test). 

To construct the multidimensional boxes, a training set of samples with known class ideniit) is obtained. The training set is divided into separate sets, one for each class, and principal components are calculated separately for each of the classes. The number of relevant principal components (rank) is determined for each class and the SIMCA models are completed by defining boundary regions for each of the PCA models. 

Once the class boundaries are defined, it is important to determine whether any of the classes in the training set overlap. This indicates the discriminating power of the SIMCA models and will impact the confidence that can be placed on future predictions. TTiere are various algorithmic measures of class overlap and the reader is referred to their software package documentation for details. In this chapter, class overlap is indicated when training set samples are predicted to be members of multiple classes. This is demonstrated in a two-dimensional example shown in Figure 4.59- Two classes are shown where class A is described by one principal component and class B is described by two principal components. The overlap of the classes is indicated because unknown Z is classified as belonging to both classes. 

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