PCA Principal Component Analysis

Mathematically speaking, if X is an (IxJ) matrix that contains J variables for I reactions, PCA divides this matrix into a systematic part TPt (the PCA model), and a residuals part E (Eq. (6.10)). T (lx R), and V (JxR) are two smaller matrices, the size of which depends on R, the number of significant PCs. T is the scores matrix. It represents the spread of the reactions within the model space. P is the loadings matrix. It describes the relationships between the variables. 

PCs are ranked according to the fraction of variance of the dataset that they explain. The first PC is the most important (it explains the largest fraction of variance), and so forth. Selecting the correct number of PCs is crucial. Too few PCs will leave important information out of the model, but too many PCs will include noise, and decrease the model s robustness (if R J, the PCA is pointless). Each time you make a new PCA model, you should examine the residuals matrix E. If the residuals are structured, it means that some information is left out. You can also decide on the correct number of PCs by performing a cross-validation (see below), or by examining the percentage of the variance explained by the model. 

PC A [1, 3] seeks to And the low-dimensional subspace within the data that maximally preserves the covariance up to rotation. This maximum covariance subspace encapsulates the directions along which the data varies the most. Therefore, projecting the data onto this subspace can be thought of as projecting the data onto the subspace that retains the most information. An example embedding found using PCA is shown in Fig. 2.2a. 

The intuition behind PCA is that the largest eigenvector of the matrix F corresponds to the dimension in the high-dimensional space along which X varies the most. Similarly, the second largest eigenvector corresponds to the dimension with the second most variation, and so on. So the top -eigenvectors describe the d-dimensional subspace which contains the most variance. 

It was mentioned earlier that empirical multivariate modeling often reqnires a very large amount of data. These data can contain a very large number of samples (IN), a very large number of variables (M) per sample, or both. In the case of PAT, where spectroscopic analytical methods are often used, the number of variables collected per process sample can range from the hundreds to the thonsands  

PCA is a data compression method that reduces a set of data collected on M variables over N samples to a simpler representation that uses a much fewer number (A M) of compressed variables , called principal components (or PCs). The mathematical model for the PCA method is provided below  

In practice, the choice of an optimal number of PCs to retain in the PCA model (A) is a rather snbjective process, which balances the need to explain as much of the original data as possible with the need to avoid incorporating too much noise into the PCA model (overfitting). The issne of overfitting is discnssed later in Section 12.4. 

Note This dataset contains four descript ors (x variables) for each of 150 different iris sampf es that can be in one of three known classes Setosa,  

As the above example illustrates, PCA can be an effective exploratory tool. However, it can also be used as a predictive tool in a PAT context. A good example of this nsage is the case where one wishes to determine whether newly collected analyzer responses are normal or abnormal with respect to previously collected responses. An efficient way to perform snch analyses wonld be to construct a PCA model using the previously collected responses, and apply this model to any analyzer response (Xp) generated by a subse-qnently-collected sample. Such PCA model application involves hrst a mnltiplication of the response vector with the PCA loadings (P) to generate a set of PCA scores for the newly collected response  

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Step 2 This ensemble is subjected to a principal component analysis (PCA) [61] by diagonalizing the covariance matrix C G x 7Z, ... 

We have to apply projection techniques which allow us to plot the hyperspaces onto two- or three-dimensional space. Principal Component Analysis (PCA) is a method that is fit for performing this task it is described in Section 9.4.4. PCA operates with latent variables, which are linear combinations of the original variables. 

Kohonen network Conceptual clustering Principal Component Analysis (PCA) Decision trees Partial Least Squares (PLS) Multiple Linear Regression (MLR) Counter-propagation networks Back-propagation networks Genetic algorithms (GA)... 

Sections 9A.2-9A.6 introduce different multivariate data analysis methods, including Multiple Linear Regression (MLR), Principal Component Analysis (PCA), Principal Component Regression (PCR) and Partial Least Squares regression (PLS). 

I Principal Component Analysis (PCA) transforms a number of correlated variables into a smaller number of uncorrelated variables, the so-called principal components. 

The previously mentioned data set with a total of 115 compounds has already been studied by other statistical methods such as Principal Component Analysis (PCA), Linear Discriminant Analysis, and the Partial Least Squares (PLS) method [39]. Thus, the choice and selection of descriptors has already been accomplished. 

Spectral features and their corresponding molecular descriptors are then applied to mathematical techniques of multivariate data analysis, such as principal component analysis (PCA) for exploratory data analysis or multivariate classification for the development of spectral classifiers [84-87]. Principal component analysis results in a scatter plot that exhibits spectra-structure relationships by clustering similarities in spectral and/or structural features [88, 89]. 

The dimensionality of a data set is the number of variables that are used to describe eac object. For example, a conformation of a cyclohexane ring might be described in terms c the six torsion angles in the ring. However, it is often found that there are significai correlations between these variables. Under such circumstances, a cluster analysis is ofte facilitated by reducing the dimensionality of a data set to eliminate these correlation Principal components analysis (PCA) is a commonly used method for reducing the dimensior ality of a data set. 

The field points must then be fitted to predict the activity. There are generally far more field points than known compound activities to be fitted. The least-squares algorithms used in QSAR studies do not function for such an underdetermined system. A partial least squares (PLS) algorithm is used for this type of fitting. This method starts with matrices of field data and activity data. These matrices are then used to derive two new matrices containing a description of the system and the residual noise in the data. Earlier studies used a similar technique, called principal component analysis (PCA). PLS is generally considered to be superior. 

In general, two related techniques may be used principal component analysis (PCA) and principal coordinate analysis (PCoorA). Both methods start from the n X m data matrix M, which holds the m coordinates defining n conformations in an m-dimensional space. That is, each matrix element Mg is equal to q, the jth coordinate of the /th conformation. From this starting point PCA and PCoorA follow different routes. 

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... 

We are about to enter what is, to many, a mysterious world—the world of factor spaces and the factor based techniques, Principal Component Analysis (PCA, sometimes known as Factor Analysis) and Partial Least-Squares (PLS) in latent variables. Our goal here is to thoroughly explore these topics using a data-centric approach to dispell the mysteries. When you complete this chapter, neither factor spaces nor the rhyme at the top of this page will be mysterious any longer. As we will see, it s all in your point of view. 

Recall that, in order to generate an ILS calibration, we must have at least as many samples as there are wavelengths used in the calibration. Since we only have 15 spectra in our training sets but each spectrum contains 100 wavelengths, we were forced to find a way to reduce the dimensionality of our spectra to 15 or less. We have seen that principal component analysis (PCA) provides us with a way of optimally reducing the dimensionality of our data without degrading it, and with the added benefit of removing some noise. 

Principal Component Analysis (PCA). Principal component analysis is an extremely important method within the area of chemometrics. By this type of mathematical treatment one finds the main variation in a multidimensional data set by creating new linear combinations of the raw data (e.g. spectral variables) [4]. The method is superior when dealing with highly collinear variables as is the case in most spectroscopic techniques two neighbor wavelengths show almost the same variation. 

Principal component analysis (PCA) is a statistical method having as its main purpose the representation in an economic way the location of the objects in a reduced coordinate system where only p axes instead of n axes corresponding to n variables (p[Pg.94]

The concept of property space, which was coined to quanhtahvely describe the phenomena in social sciences [11, 12], has found many appUcahons in computational chemistry to characterize chemical space, i.e. the range in structure and properhes covered by a large collechon of different compounds [13]. The usual methods to approach a quantitahve descriphon of chemical space is first to calculate a number of molecular descriptors for each compound and then to use multivariate analyses such as principal component analysis (PCA) to build a multidimensional hyperspace where each compound is characterized by a single set of coordinates. 

A first introduction to principal components analysis (PCA) has been given in Chapter 17. Here, we present the method from a more general point of view, which encompasses several variants of PCA. Basically, all these variants have in common that they produce linear combinations of the original columns in a measurement table. These linear combinations represent a kind of abstract measurements or factors that are better descriptors for structure or pattern in the data than the original measurements [1]. The former are also referred to as latent variables [2], while the latter are called manifest variables. Often one finds that a few of these abstract measurements account for a large proportion of the variation in the data. In that case one can study structure and pattern in a reduced space which is possibly two- or three-dimensional. 

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. 

Principal coordinates analysis (PCoA) is applied to distance tables rather than to original data tables, as is the case with principal components analysis (PCA). 

To reduce intensity effects, the data were normalized by reducing the area under each spectrum to a value of 1 [42]. Principal component analysis (PCA) was applied to the normalized data. This method is well suited to optimize the description of the fluorescence data sets by extracting the most useful data and rejecting the redundant ones [43]. From a data set, PCA assesses principal components and their corresponding spectral pattern. The principal components are used to draw maps that describe the physical and chemical variations observed between the samples. Software for PCA has been written by D. Bertrand (INRA Nantes) and is described elsewhere [44]. 

NMR alone is insufficient to enable the full assignment of the beer spectra to be made. Application of Principal Component Analysis (PCA) to the spectral profiles of beers of differing type (ales and lagers) showed some distinction on the basis of the aliphatic and sugar compositions, whereas the PCA of the aromatic profiles... 

Techniques for multivariate input analysis reduce the data dimensionality by projecting the variables on a linear or nonlinear hypersurface and then describe the input data with a smaller number of attributes of the hypersurface. Among the most popular methods based on linear projection is principal component analysis (PCA). Those based on nonlinear projection are nonlinear PCA (NLPCA) and clustering methods. 

However, there is a mathematical method for selecting those variables that best distinguish between formulations—those variables that change most drastically from one formulation to another and that should be the criteria on which one selects constraints. A multivariate statistical technique called principal component analysis (PCA) can effectively be used to answer these questions. PCA utilizes a variance-covariance matrix for the responses involved to determine their interrelationships. It has been applied successfully to this same tablet system by Bohidar et al. [18]. 

Two examples of unsupervised classical pattern recognition methods are hierarchical cluster analysis (HCA) and principal components analysis (PCA). Unsupervised methods attempt to discover natural clusters within data sets. Both HCA and PCA cluster data. 

Subsequently 36 strains of aerobic endospore-forming bacteria, consisting of six Bacillus species and one Brevibacillus species could be discriminated using cluster analysis of ESMS spectra acquired in the positive ion mode (m/z 200-2000).57 The analysis was carried out on harvested, washed bacterial cells suspended in aqueous acidic acetonitrile. The cell suspensions were infused directly into the ionization chamber of the mass spectrometer (LCT, Micromass) using a syringe pump. Replicates of the experiment were performed over a period of six months to randomize variations in the measurements due to possible confounding factors such as instrumental drift. Principal components analysis (PCA) was used to reduce the dimensionality of the data, fol-... 

FIGURE 23.5 Effect of feeding captive male ring-necked pheasant (Ph. colchicus) young a high- or low-protein feed for the first three weeks of life on the expression of wattle coloration (mean+SE) at 20 (open circles) and 40 (filled circles) weeks of age. Coloration was determined using a principal components analysis (PCA) of tristimulus scores (hue, saturation, and brightness) obtained with a Colortron II reflectance spectrophotometer. 

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