Data Analysis Principal components

KEYWORDS grain-size, normalization, compositional data, trends, principal component analysis... 

The rapid classification of polymeric species is an important problem in the area of analytical chemistry in general and of particular relevance to recycling and waste management. To accomplish classification tasks, a combination of spectral data and principal component analysis (PCA) is often employed. 

Ochre is very common in the Terminal Archaic-Early Formative archaeological site of Jiskairumoko, (Rio Have, Lake Titicaca Basin, southern Peru). Within the site, ochre was found on tools, palettes, and in burials and soil deposits within structures in several contexts, suggesting both symbolic and functional uses of ochre. Variations in the color and contexts imply possibilities for different uses of ochre.. Instrumental neutron activation analysis was used to analyze the ochre samples found in Jiskairumoko. Multivariate analysis of the elemental data by principal components analysis suggests trends in the data related to the compositional variation of ochres on the site. Further analysis of the ochre will lead to conclusions about the variation in composition of the ochres from Jiskairumoko and possible archaeological conclusions about ancient technologies and uses of ochre on the site. 

A principal component analysis is reasonable only when the intrinsic dimensionality is much smaller than the dimensionality of the original data. This is the case for features related by high absolute values of the correlation coefficients. Whenever correlation between features is small, a significant direction of maximum variance cannot be found (Fig. 3.7) all principal components participate in the description of the data structure hence a reduction of data by principal component analysis is not possible. 

Pre-Processing of Structural Data using Principal Component Analysis (PCA)... 

Fig. 5.11 Multivariate analysis (principal components scores plots of the first three principal components) of the response of the two-size CdSe nanocrystals sensor film (a) PC 1 vs. PC 2 and (B) PC 1 vs. PC 3. Regions numbered 1 and 2 are data points of dynamic response from replicate (n = 3) film exposures to methanol and toluene, respectively. Unlabeled data points result from times when the film was exposed to a blank (dry air). Reprinted with permission from Leach and Potyrailo26 copyright 2006 Materials Research Society...

Near-infrared (NIR) spectroscopy is becoming an important technique for pharmaceutical analysis. This spectroscopy is simple and easy because no sample preparation is required and samples are not destroyed. In the pharmaceutical industry, NIR spectroscopy has been used to determine several pharmaceutical properties, and a growing literature exists in this area. A variety of chemoinfometric and statistical techniques have been used to extract pharmaceutical information from raw spectroscopic data. Calibration models generated by multiple linear regression (MLR) analysis, principal component analysis, and partial least squares regression analysis have been used to evaluate various parameters. 

Principal components analysis (see also p. 16) involves an examination of set of data as points in n-dimensional space (corresponding to n original tests) and determines (first) the direction that accounts for the biggest variability in the data (first principal component). The process is repeated until n principal components are evaluated, but not all of these are of practical importance because some may be attributable purely to experimental error. The number of significant principal components shows the number of independent properties being measured by the tests considered. 

Like many other statistical methods for the evaluation of biomonitoring data, the above-depicted example of a trend analysis considers only a single variable. Although the multivariate procedures consider several measured variables at the same time, their results are often only limited meaningful. Cluster analyses can reveal structures in a given data set principal component analyses concentrate the information contents of many variables in a set of a few latent variables, which are difficult to interpret correctly. 

Component models deal with data that can be meaningfully arranged in a single block. A common method to deal with this type of data is principal component analysis and this technique is explained in some detail. A less common method which will prove useful in the following, called two-mode component analysis, is also discussed briefly. 

To get at the question of overall influence, the matrix of structural model parameters and variance components was subjected to principal component analysis. Principal component analysis (PCA) was introduced in the chapter on Nonlinear Mixed Effects Model Theory and transforms a matrix of values to another matrix such that the columns of the transformed matrix are uncorrelated and the first column contains the largest amount of variability, the second column contains the second largest, etc. Hopefully, just the first few principal components contain the majority of the variance in the original matrix. The outcome of PC A is to take X, a matrix of p-variables, and reduce it to a matrix of q-variables (q < p) that contain most of the information within X. In this PC A of the standardized parameters (fixed effects and all variance components), the first three principal components contained 74% of the total variability in the original matrix, so PCA was largely successfully. PCA works best when a high correlation exists between the variables in the original data set. Usually more than 80% variability in the first few components is considered a success. 

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