Principal components analysis vector

Principal component analysis (PCA) takes the m-coordinate vectors q associated with the conformation sample and calculates the square m X m matrix, reflecting the relationships between the coordinates. This matrix, also known as the covariance matrix C, is defined as... 

In a general way, we can state that the projection of a pattern of points on an axis produces a point which is imaged in the dual space. The matrix-to-vector product can thus be seen as a device for passing from one space to another. This property of swapping between spaces provides a geometrical interpretation of many procedures in data analysis such as multiple linear regression and principal components analysis, among many others [12] (see Chapters 10 and 17). 

In the previous section we have developed principal components analysis (PCA) from the fundamental theorem of singular value decomposition (SVD). In particular we have shown by means of eq. (31.1) how an nxp rectangular data matrix X can be decomposed into an nxr orthonormal matrix of row-latent vectors U, a pxr orthonormal matrix of column-latent vectors V and an rxr diagonal matrix of latent values A. Now we focus on the geometrical interpretation of this algebraic decomposition. 

G. H. Dunteman, Principal Components Analysis. Sage Publications, Newbury Park, CA, 1989. L.L. Thurstone, Multiple-factor Analysis. A Development and Expansion of the Vectors of Mind. Univ. Chicago Press, Chicago, 1947. 

The application of principal components regression (PCR) to multivariate calibration introduces a new element, viz. data compression through the construction of a small set of new orthogonal components or factors. Henceforth, we will mainly use the term factor rather than component in order to avoid confusion with the chemical components of a mixture. The factors play an intermediary role as regressors in the calibration process. In PCR the factors are obtained as the principal components (PCs) from a principal component analysis (PC A) of the predictor data, i.e. the calibration spectra S (nxp). In Chapters 17 and 31 we saw that any data matrix can be decomposed ( factored ) into a product of (object) score vectors T(nxr) and (variable) loadings P(pxr). The number of columns in T and P is equal to the rank r of the matrix S, usually the smaller of n or p. It is customary and advisable to do this factoring on the data after columncentering. This allows one to write the mean-centered spectra Sq as ... 

Principal component analysis (PCA) is aimed at explaining the covariance structure of multivariate data through a reduction of the whole data set to a smaller number of independent variables. We assume that an m-point sample is represented by the nxm matrix X which collects i=l,...,m observations (measurements) xt of a column-vector x with j=, ...,n elements (e.g., the measurements of n=10 oxide weight percents in m = 50 rocks). Let x be the mean vector and Sx the nxn covariance matrix of this sample... 

M.-L. O Connell, T. Howley, A.G. Ryder, M.N. Leger and M.G. Madden, Classification of a target analyte in solid mixtures using principal component analysis, support vector machines, and Raman spectroscopy, Proc. SPIE-Int. Soc. Opt. Eng., 5826, 340-350 (2005). 

Technique 2 Elgenanalysls. It Is well known that the structure of a data set can be uncovered by performing an elgenanalysls of Its covariance matrix.(14) This Is often called principal component analysis. That Is, we arrange the M measurement made on each of N objects as a column vector and combine them to form an M x N matrix, A. A matrix B, resembling the covariance matrix of this data set, Is an M x M matrix AA whose elements are given by... 

Principal component analysis is a simple vector space transform, allowing the dimensionality of a data set to be reduced, while at the same time minimizing... 

Key Words 2D-QSAR traditional QSAR 3D-QSAR nD-QSAR 4D-QSAR receptor-independent QSAR receptor-dependent QSAR high throughput screening alignment conformation chemometrics principal components analysis partial least squares artificial neural networks support vector machines Binary-QSAR selecting QSAR descriptors. 

Multivariate models have been successful in identifying source contributions in urban areas. They are not independent of Information on source composition since the chemical component associations they reveal must be verified by source emissions data. Linear regressions can produce the typical ratio of chemical components in a source but only under fairly restrictive conditions. Factor and principal components analysis require source composition vectors, though it is possible to refine these source composition estimates from the results of the analysis (6.17). 

Color cluster rotation, which is described in Section 6.6, views pixels of the input image as a cloud of points. A principal component analysis is done to determine the main axis of this cloud of points. The main axis is rotated onto the gray vector. For the input data in Helson s experiments, there are only two different colors sensed by the sensor. The two colors line up along the axis ei, which is defined by the illuminant. 

Principal component analysis (PCA) is commonly used to identify those analytes that are most different from the control samples and provides for a visual characterization of the data set. Following data reduction, PCA is used to find linear combinations (eigenvectors) of the original resolved peaks most different from controls, and these vectors are used to create visually characterize data sets. The PCA Eigenvectors have several desirable properties, including (a) the combinations are not correlated and (b) they can be rank-ordered (from most to least). 

The wa in equation (6) are the PLS loading weights. They are explained in the theory in references 53 - 62. Equation (7) shows how X is decomposed bilinearly (as in principal component analysis) with its own residual Epls A. T is the matrix with the score vectors as columns, P is the matrix having the PLS loadings as columns. Also the vectors of P and wa can be used to construct scatter plots. These can reveal the data structure of the variable space and relations between variables or groups of variables. Since PLS mainly looks for sources of variance, it is a very good dirty data technique. Random noise will not be decomposed into scores and loadings, and will be stored in the residual matrices (E and F), which contain only non-explained variance . 

Principal component analysis is ideally suited for the analysis of bilinear data matrices produced by hyphenated chromatographic-spectroscopic techniques. The principle component models are easy to construct, even when large or complicated data sets are analyzed. The basis vectors so produced provide the fundamental starting point for subsequent computations. Additionally, PCA is well suited for determining the number of chromatographic and spectroscopically unique components in bilinear data matrices. For this task, it offers superior sensitivity because it makes use of all available data points in a data matrix. 

Recall from Chapter 4, Principal Component Analysis, that a mean-centered data matrix with n rows of mixture spectra recorded at m wavelengths, where each mixture contains up to k constituents, can be expressed as a product of k vectors representing concentrations and k vectors representing spectra for the pure constituents in the mixtures, as shown in Equation 5.20. 

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. 

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