Cross-validation principal components

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... 

A crucial decision in PLS is the choice of the number of principal components used for the regression. A good approach to solve this problem is the application of cross-validation (see Section 4.4). 

M. Stone and R.J. Brooks, Continuum regression cross-validated sequentially constructed prediction embracing ordinary least sqaures, partial least squares, and principal component regression. J. Roy. Stat. Soc. B52 (1990) 237-269. 

To construct the reference model, the interpretation system required routine process data collected over a period of several months. Cross-validation was applied to detect and remove outliers. Only data corresponding to normal process operations (that is, when top-grade product is made) were used in the model development. As stated earlier, the system ultimately involved two analysis approaches, both reduced-order models that capture dominant directions of variability in the data. A PLS analysis using two loadings explained about 60% of the variance in the measurements. A subsequent PCA analysis on the residuals showed that five principal components explain 90% of the residual variability. 

Cross Validation for Determination of the Number of Principal Components... 

The likeness of samples within the class can be assessed by the proximity of samples to each other in plots derived from principal components models. The statistical technique of cross-validation (17) was used to... 

In this example, two principal components are arbitrarily selected. More or fewer may be necessary, and this is a function of a predetermined stopping rule for extraction of principal components from X. In SIMCA method, a cross validation technique (2) is used. 

For partial least-squares (PLS) or principal component regression (PCR), the infrared spectra were transferred to a DEC VAX 11/750 computer via the NIC-COM software package from Nicolet. This package also provided utility routines used to put the spectra into files compatible with the PLS and PCR software. The PLS and PCR program with cross-validation was provided by David Haaland of Sandia National Laboratory. A detailed description of the program and the procedures used in it has been given (5). 

Root Mean Square Ei ror of Cross-Validation for PCA Plot (Model Diagnostic) Figure 4.63 displays the RMSECV PCA vs. number of principal components for the class B data from a leave-one-out cross-validation calculation. The RMSECy PCA quickly drops and levels off at two principal components, consistent with the choice of a rank tv- o model. 

For a well-behaved sensor array, only a small subset k of n available PCs is sufficient to characterize the matrix. Once again, Principal Component Regression (PCR) is a data reduction tool. The robustness of the selection of k can be tested by cross-validation in which case data subsets are randomly selected and the error matrix H xn is calculated. 

Initially an optimised model was constructed using the data collected as outlined above by constructing a principal component (PC)-fed linear discriminant analysis (LDA) model (described elsewhere) [7, 89], The linear discriminant function was calculated for maximal group separation and each individual spectral measurement was projected onto the model (using leave-one-out cross-validation) to obtain a score. The scores for each individual spectrum projected onto the model and colour coded for consensus pathology are shown in Fig. 13.3. The simulation experiments used this optimised model as a baseline to compare performance of models with spectral perturbations applied to them. The optimised model training performance achieved 93% accuracy overall for the three groups. 

To assess whether a principal component is significant (that the shape of the point swarm differs sufficiently from that of a spheroid), cross-validation is employed [33-36]. In cross-validation, a part (say 1 /4th or 1 /5th) of the data is held out in a patterned way (deletions in diagonal bands in X as in... 

The optimal model is determined by finding the minimum error between the extracted concentrations and the reference concentrations. Cross-validation is also used to determine the optimal number of model parameters, for example, the number of factors in PLS or principal components in PCR, and to prevent over- or underfitting. Technically, because the data sets used for calibration and validation are independent for each iteration, the validation is performed without bias. When a statistically sufficient number of spectra are used for calibration and validation, the chosen model and its outcome, the b vector, should be representative of the data. 

An important issue in PCR is the selection of the optimal number of principal components kopt, for which several methods have been proposed. A popular approach consists of minimizing the root mean squared error of cross-validation criterion RMSECV,. For one response variable (q = 1), it equals... 

Principal component analysis is central to many of the more popular multivariate data analysis methods in chemistry. For example, a classification method based on principal component analysis called SIMCA [69, 70] is by the far the most popular method for describing the class structure of a data set. In SIMCA (soft independent modeling by class analogy), a separate principal component analysis is performed on each class in the data set, and a sufficient number of principal components are retained to account for most of the variation within each class. The number of principal components retained for each class is usually determined directly from the data by a method called cross validation [71] and is often different for each class model. 

Principal component analysis of the response matrix afforded one significant component (cross-validation) which described 82% of total variance in Y. As all responses are of the same kind (percentage yield) the data were not autoscaled prior to analysis. The response y1]L was deleted as it did not vary. The scores and loadings are also given in Table 14. The score values were used to fit a second-order... 

There is an approach in QSRR in which principal components extracted from analysis of large tables of structural descriptors of analytes are regressed against the retention data in a multiple regression, i.e., principal component regression (PCR). Also, the partial least square (PLS) approach with cross-validation 29 finds application in QSRR. Recommendations for reporting the results of PC A have been published 130). 

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