Principal components modeling utility

The advantage of utilizing the standardized form of the variable is that quantities of different types can be included in the analysis including elemental concentrations, wind speed and direction, or particle size information. With the standardized variables, the analysis is examining the linear additivity of the variance rather than the additivity of the variable itself. The disadvantage is that the resolution is of the deviation from the mean value rather than the resolution of the variables themselves. There is, however, a method to be described later for performing the analysis so that equation 16 applies. Then, only variables that are linearly additive properties of the system can be included and other variables such as those noted above must be excluded. Equation 17 is the model for principal components analysis. The major difference between factor analysis and components analysis is the requirement that common factors have the significant values of a for more than one variable and an extra factor unique to the particular variable is added. The factor model can be rewritten as... 

The 0-NLPCA network has 8-6-10-12 neurons in each layer, yielding a prototype model with 6 principal components (PCs). For comparison, the linear PCA was also applied to the same data. As a performance criterion, the root mean square of error (RMSE) was evaluated to compare the prediction ability of the developed PCA and O-NLPCA models on the training and validation data. While the linear PCA gave 0.3021 and 0.3227 RMSE on training and validation data sets, respectively, the O-NLPCA provided 0.2526 and 0.2244 RMSE. This suggests that to capture the same amount of information, the linear PCA entails utilization of more principal components than its nonlinear counterpart. As a result, the information embedded in the nonlinear principal components addresses the underlying events more efficiently than the linear ones. 

In chemometrics, PCR and PLS seem to be the most widely used method for building a calibration model. Recently, we developed a method, called elastic component regression (ECR), which utilizes a tuning parameter a [0,l] to supervise the decomposition of X-matrix [36], which falls into the category of continuum regression [37-40]. It is demonstrated theoretically that the elastic component resulting from ECR coincides with principal components of PC A when a = 0 and also coincides with PLS components when a = 1. In this context, PCR and PLS occupy the two ends of ECR and a (0,l) will lead to an infinite number of transitional models which collectively uncover the model path from PCR to PLS. The source codes implementing ECR in MATLAB are freely available at [41]. In this section, we would like to compare the predictive performance of PCR, PLS and an ECR model with a = 0.5. 

Due to the absence of hydrogen donor capabilities [31], cyanopropyl silica phases are less retentive in normal-phase liquid chromatography than under-ivatized silica or other NP packing materials. Therefore, very few applications have been reported that utilize cyanopropyl-bonded silica in the HILIC mode [32,33]. The limited number of applications may also be attributed to the mechanical instabiUty of cyanopropyl-bonded silica when operated with solvents of intermediate polarity. This instabihty is caused by a decrease in the adhesion of particles to each other that maintain the integrity of the column bed in either nonpolar or highly polar solvents [25]. Dinh et al. [34] performed a multivariate modeling of column selectivity by principal component analysis of chromatographic data from polar compounds of various structures on 20 commercially available HILIC columns and verified the low potential of cyanopropyl-bonded silica columns due to insufficient hydrophilicity. 

With respect to the appHed regression methodologies, RR is similar to PCR in that the independent variables are transformed to their principal components (PCs). However, while PCR utilizes only a subset of the PCs, RR retains them all but downweights them based on their eigenvalues. With PLS, a subset of the PCs is also used, but the PCs are selected by considering both the independent and dependent variables. For each model developed, the cross-validated R was obtained using the leave-one-out (LOO) approach and can be calculated as follows ... 

The literature of multivariate classification shows that several types of methods have found utility in application to chemical problems. Excellent discussions of the major methods can be found in Strouf ° and Tou and Gon-zalez. The most frequently used methods include parametric approaches involving linear and quadratic discriminant analysis based on the Bayesian approach,nonparametric linear discriminant development methods,and those methods based on principal components analysis such as SIMCA (Soft Independent Modeling by Class Analogy). 

XRF and scattering (EDXRFS) spectroscopy method for direct rapid analysis of trace bioavailable macronutrients (i.e. C, N, Na, Mg, P) in soils. Chemo-metric techniques, namely principal component analysis (PCA), partial least squares (PLS) and artificial neural networks (ANNs), were utilized for pattern recognition based on fluorescence and regions of Compton scatter peaks, and to develop multivariate quantitative calibration models based on Compton scatter peaks, respectively. 

The previous section alludes to the most common problems in quantitative Raman spectroscopic calibrations Most models require that all components in a system to be known and modeled in the calibration data to accurately predict any one component. Inverse calibration techniques such as inverse multiple linear regression (inverse MLR), principal component regression (PCR) and partial least squares (PLS also known as principal latent structures) avoid this problem by forcing the calibration steps to utilize only the spectral features which are either changing (PCR) or directly correlated to the property of interest (PLS). More so, not all components in a sample need to be known to perform an inverse calibration. The basic form of an inverse calibration centers around an equation of the form... 

Then, the drift Brownian motion is utilized to establish the reliability model for the first principal component data. The parameter estimation results... 

An important hybrid approach has also developed in recent years that makes use of block-structured or modular models. These models are composed of parametric and/or nonparametric components properly connected to represent reliably the input-output relation. The model specification task for this class of models is more demanding and may utilize previous parametric and/or nonparametric modeling results. A promising variant of this approach, which derives from the general Volterra-Wiener formulation, employs principal dynamic modes as a canonical set of filters to represent a broad class of nonlinear dynamic systems. Another variant of the modular approach that has recently acquired considerable popularity but will not be covered in this review is the use of artificial neural networks to represent input-output nonlinear mappings in the form of connectionist models. These connectionist models are often fully parametrized, making this approach affine to parametric modeling, as well. 

As we have learned earlier, in the case of the 2D deformation model, the square is transformed into a parallelogram and can also undergo rotation. The strain is referred to as uniform if it is the same at all points of the body. Let us consider several particular examples of uniform strain utilizing an analogy with stressed-state tensors. The concept of a principal coordinate system in which there are no shear components, as well as the rules for transforming tensors to the principal coordinate system are the same for both 8 and Oij. 

---