Principal component analysis basis factors

The interpretation of a multivariate image is sometimes problematic because the cause for pictorial structures may be complex and cannot be interpreted on the basis of images of single species even if they are processed by filtering etc. In such cases, principal component analysis (PCA) may advantageously be applied. The principle of the PCA is like that of factor analysis which has been mathematically described in Sect. 8.3.4. It is represented schematically in Fig. 8.33. 

Principal components analysis. There are innumerable excellent descriptions of the mathematical basis of PCA26-30 and this article will provide only a general overview. It is important, first, not to be confused between algorithms which are a means to an end, and the end in itself. There are several PCA algorithms of which NIPALS (described in Appendix A2.1) and SVD are two of the most common. If correctly applied, they will both lead to the same answer (within computer precision), the best approach depending on factors such as computing power and the number of components to be calculated. 

In PLS, the response matrix X is decomposed in a fashion similar to principal component analysis, generating a matrix of scores, T, and loadings or factors, P. (These vectors can also be referred to as basis vectors.) A similar analysis is performed for Y, producing a matrix of scores, U, and loadings, Q. 

The extraction of the eigenvectors from a symmetric data matrix forms the basis and starting point of many multivariate chemometric procedures. The way in which the data are preprocessed and scaled, and how the resulting vectors are treated, has produced a wide range of related and similar techniques. By far the most common is principal components analysis. As we have seen, PCA provides n eigenvectors derived from a. nx n dispersion matrix of variances and covariances, or correlations. If the data are standardized prior to eigenvector analysis, then the variance-covariance matrix becomes the correlation matrix [see Equation (25) in Chapter 1, with Ji = 52]. Another technique, strongly related to PCA, is factor analysis. ... 

On the other hand, successful identification of bacterial spores has been demonstrated by using Fourier transform infrared photoacoustic and transmission spectroscopy " in conjunction with principal component analysis (PCA) statistical methods. In general, PCA methods are used to reduce and decompose the spectral data into orthogonal components, or factors, which represent the most coimnon variations in all the data. As such, each spectrum in a reference library has an associated score for each factor. These scores can then be used to show clustering of spectra that have common variations, thus forming a basis for group member classification and identification. 

To compare the scent profiles of individuals we selected 11 compounds and calculated their relative peak areas. Principal Component Analysis (PCA) was used to compare the individual peak areas of the four individuals. PCA is a multivariate statistical method which reduces the dimensions of a single group of data by producing a smaller number of abstract variables (Jolliffe, 1986). For this analysis we used multiple samples of each individual and calculated all factors on the basis of a correlation matrix. The resulting first and second factor accounted for a total of 99.17 % of the variance in proportional peak area. 

In this paper the PLS method was introduced as a new tool in calculating statistical receptor models. It was compared with the two most popular methods currently applied to aerosol data Chemical Mass Balance Model and Target Transformation Factor Analysis. The characteristics of the PLS solution were discussed and its advantages over the other methods were pointed out. PLS is especially useful, when both the predictor and response variables are measured with noise and there is high correlation in both blocks. It has been proved in several other chemical applications, that its performance is equal to or better than multiple, stepwise, principal component and ridge regression. Our goal was to create a basis for its environmental chemical application. 

Fourier transform infrared (FTIR) spectroscopy of coal low-temperature ashes was applied to the determination of coal mineralogy and the prediction of ash properties during coal combustion. Analytical methods commonly applied to the mineralogy of coal are critically surveyed. Conventional least-squares analysis of spectra was used to determine coal mineralogy on the basis of forty-two reference mineral spectra. The method described showed several limitations. However, partial least-squares and principal component regression calibrations with the FTIR data permitted prediction of all eight ASTM ash fusion temperatures to within 50 to 78 F and four major elemental oxide concentrations to within 0.74 to 1.79 wt % of the ASTM ash (standard errors of prediction). Factor analysis based methods offer considerable potential in mineral-ogical and ash property applications. 

You will better understand the goals of factor analysis considering first the highly idealized situation with error-free observations and only r < n linearly independent columns in the matrix X. As discussed in Section 1.1, all columns of X are then in an r—dimensional subspace, and you can write them as linear combinations of r basis vectors. Since the matrix X X has now r nonzero eigenvalues, there are exactly r nonvanishing vectors in the matrix Z defined by (1.111), and these vectors form a basis for the subspace. The corresponding principal components z, z2,. .., zr are the coordinates in this basis. In the real life you have measurement errors, the columns of X... 

Principle components analysis (PCA), a form of factor analysis (FA), is one of the most common unsupervised methods used in the analysis of NMR data. Also known as Eigenanalysis or principal factor analysis (PEA), this method involves the transformation of data matrix D into an orthogonal basis set which describes the variance within the data set. The data matrix D can be described as the product of a scores matrix T, and a loading matrix P,... 

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