Analysis factor

Principal component analysis (PCA), as described in Chapter 4, is an unsupervised learning method which aims to identify principal components, combinations of variables, which best characterize a data set. Best here means in terms of the information content of the components (variance) and that they are orthogonal to one another. Each principal component (PC) is a linear combination of the original variables as shown in eqn (4.1) and repeated below 

The principal components do not (necessarily) have any physical meaning, they are simply mathematical constructs calculated so as to comply with the conditions of PCA of the explanation of variance and orthogonality. Thus, PCA is not based on any statistical model. Factor analysis (FA), on the other hand, is based on a statistical model which holds that any given data set is based on a number of factors. The factors themselves are of two types common factors and unique factors. Each variable in a data set is composed of a mixture of the common factors and a single unique factor associated with that variable. Thus, for any variable Xu in an A-dimen-sional data set we can write 

Takagi and co-workers (1989) applied FA to gas chromatography retention data for 190 solutes measured using 21 different stationary phases. Three factors were found to be sufficient to explain about 98 per cent of the variance of the retention data physicochemical meanings to these factors were ascribed as shown below  

The attempted physicochemical interpretation of the factors highlights a common problem with PCA and FA, along with the question of how 

Plot of factor scores for meat and fish samples. C, B, P- hand-deboned chicken, beef, and pork M, m, mechnically debonded pork and chick i F, cod fish f, cod fish in presence of cryoprotectants (from Li-Chan et at. 1987, with permission of the Institute of 

Causal factor analysis is used when there are multiple problems with a long causal factor chain of events. A causal factor chain is a sequence of events that shows, step by step, the events that took place in order for the accident to occur. Causal factor analysis puts all the necessary and sufficient events and causal factors for an accident in a logical, chronological sequence. It analyzes the accident and evaluates evidence during an investigation. It is also used to help prevent similar accidents in the future and to validate the accuracy of pre-accidental system analysis. It is used to help 

FIGURE 9.1 An event and causal factor tree (chart). (Source Courtesy of the National Aeronautical and Space Administration.) 

On the downside, causal factor analysis is time consuming and requires the investigator to be familiar with the process for it to be effective. As can be seen later in this chapter, the accident scene may need to be revisited a number of times and areas that are not directly related to the accident may need to viewed, in order to have a complete event and causal factor chain. It requires a broad perspective of the accident in order to identify any hidden problems that would have caused the accident. 

The collection of facts should begin immediately after an accident occurs. Start with an accident site walk-through, interviews, and actual physical material collection. This will increase the accuracy of the information that is collected and help eliminate any uncertainty or vagueness. It wiU also help when it comes time to put together the event s causal factor chain. At this stage, the details will prove necessary. It is important to have as much information as possible in order to have an ideal accident investigation, including the events and the conditions at the time. 

Direct causes are the basic contributing factors that directly cause the accident to occur. For example, it might be determined that the immediate events or conditions leading up to an accident might be traced to the explosion of a pressurized vessel. The direct cause should be stated in one sentence. Then briefly explain what happened. A direct cause would be the release of energy or hazardous material. (Examples of the release of energy and examples of hazardous material can be found in Chapter 7.) Another example of a direct cause statement would be The electrician made contact between the metal rod and the exposed 220-volt wire. State the cause as simply as humanly possible. 

The first step of FA is factor extraction the methods described below are the most commonly used. The extraction methods calculate a set of orthogonal factors (or components) that in combination reproduce the matrix of correlation. The criteria used to generate the factors are not homogeneous for all methods but the differences between their solutions may be quite small. 

Graphical representation is also important to visualize the results attained by the methods that extracted the factors. The objective is not to improve the fit between the observed and reproduced correlation matrices but as an aid to the interpretation of scientific results, making them more understandable. Eigenvector rotation is the most commonly used, and the four types of rotation are as follows  

On the other hand, if the correlation matrix has variables that are 100% redundant, then the inverse of the matrix cannot be computed it is the so-called ill conditioning matrix. This happens when there are high intercorrelated variables (e.g. a variable that is the sum of two other variables). The statistical 

Statisticians do not always distinguish between factor analysis and principal components analysis, but for chemists factors often have a physical significance, whereas PCs are simply abstract entities. However, it is possible to relate PCs to chemical information, such as elution profiles and spectra in HPLC-DAD by 

Relationship between PCA and factor analysis in coupled chromatography 

Factor analysis is often called by a number of alternative names such as rotation or transformation , but is a procedure used to relate the abstract PCs to meaningful chemical factors, and die influence of Malinowski in the 1980s introduced diis terminology into chemometrics. 

Principal component analysis is used to reduce the information in many variables into a set of weighted linear combinations of those variables it does not differentiate between common and unique variance. If latent variables have to be determined, which contribute to the common variance in a set of measured variables, factor analysis (FA) is a valuable statistical method, since it attempts to exclude unique variance from the analysis. 

FA is a statistical approach that can be used to analyze interrelationships among a large number of variables and to explain these variables in terms of their common underlying dimensions, or factors. This statistical approach involves finding a way of condensing the information contained in a number of original variables into a smaller set of dimensions, or factors, with a minimum loss of information [52]. 

There are three stages in factor analysis (1) generation of a correlation matrix for all the variables consisting of an array of correlation coefficients of the variables (2) extraction of factors from the correlation matrix based on the correlation coefficients of the variables and (3) rotation of factors to maximize the relationship between the variables and some of the factors. FA requires a set of data points in matrix form, and the data must be bilinear — that is, the rows and columns must be independent of each other. 

The extraction of 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 

Factor analysis is the name given to eigen-analysis of a data matrix with the intended aim of reducing the data set of n variables to a specified number, p, of fewer linear combination variables, or factors, with which to describe the data. Thus, p is selected to be less than n and, hopefully, the new data matrix will be more amenable to interpretation. The final interpretation of the meaning and significance of these new factors lies with the user and the context of the problem. 

A full description and derivation of the many factor analysis methods reported in the analytical literature is beyond the scope of this book. We will limit ourselves here to the general and underlying features associated with the technique. A more detailed account is provided by, for example, Hopke and others. 

Steps (a) to (c) or (d) are as for principal components analysis. However, as the final aim is usually to interpret the results of the analysis in terms of chemical or spectroscopic properties, the method adopted at each step should be selected with care and forethought. A simple example will serve to illustrate 

Each of these transformations can be expressed in matrix form as a transform of the data matrix X into a new matrix Y followed by calculating the 

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.  

The Essentials of Factor Analysis , 2nd Edn, Cassel Educational, London, UK, 1990. P.K. Hopke, in Methods of Environmental Data Analysis , ed. C.N. Hewitt, Elsevier, Essex, UK, 1992. 

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Malinowski E R and Howery D G 1980 Factor Analysis In Chemistry (New York Wiley)... 

An alternative to principal components analysis is factor analysis. This is a technique which can identify multicollinearities in the set - these are descriptors which are correlated with a linear combination of two or more other descriptors. Factor analysis is related to (and... 

Multiple linear regression is strictly a parametric supervised learning technique. A parametric technique is one which assumes that the variables conform to some distribution (often the Gaussian distribution) the properties of the distribution are assumed in the underlying statistical method. A non-parametric technique does not rely upon the assumption of any particular distribution. A supervised learning method is one which uses information about the dependent variable to derive the model. An unsupervised learning method does not. Thus cluster analysis, principal components analysis and factor analysis are all examples of unsupervised learning techniques. 

R. J. Rummel, Applied Factor Analysis, Northwestern University Press, Evanston, lU., 1970. 

State-of-the-art for data evaluation of complex depth profile is the use of factor analysis. The acquired data can be compiled in a two-dimensional data matrix in a manner that the n intensity values N(E) or, in the derivative mode dN( )/d , respectively, of a spectrum recorded in the ith of a total of m sputter cycles are written in the ith column of the data matrix D. For the purpose of factor analysis, it now becomes necessary that the (n X m)-dimensional data matrix D can be expressed as a product of two matrices, i. e. the (n x k)-dimensional spectrum matrix R and the (k x m)-dimensional concentration matrix C, in which R in k columns contains the spectra of k components, and C in k rows contains the concentrations of the respective m sputter cycles, i. e. ... 

Since the introduction of factor analysis for evaluation of depth profile data by Gaarenstroom [2.12], many papers have been published [e.g. 2.13-2.20]. With the help of factor analysis, three results can directly be obtained ... 

Performance-influencing factors analysis is an important part of the human reliability aspects of risk assessment. It can be applied in two areas. The first of these is the qualitative prediction of possible errors that could have a major impact on plant or personnel safety. The second is the evaluation of the operational conditions under which tasks are performed. These conditions will have a major impact in determining the probability that a particular error will be committed, and hence need to be systematically assessed as part of the quantification process. This application of PIFs will be described in Chapters 4 and 5. 

We now consider a type of analysis in which the data (which may consist of solvent properties or of solvent effects on rates, equilibria, and spectra) again are expressed as a linear combination of products as in Eq. (8-81), but now the statistical treatment yields estimates of both a, and jc,. This method is called principal component analysis or factor analysis. A key difference between multiple linear regression analysis and principal component analysis (in the chemical setting) is that regression analysis adopts chemical models a priori, whereas in factor analysis the chemical significance of the factors emerges (if desired) as a result of the analysis. We will not explore the statistical procedure, but will cite some results. We have already encountered examples in Section 8.2 on the classification of solvents and in the present section in the form of the Swain et al. treatment leading to Eq. (8-74). 

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. 

Malinowski, E.R., et. al. Factor Analysis in Chemistry, 2nd edition, John Wiley and Sons, New York, 1991. 

Antoon, M.K., et. al. "Factor Analysis Applied to Fourier Transform Infrared Spectra", Appl. Spec. 1979, (33) 351-357. 

Bulmer, J.T., et. al. "Factor Analysis as a Complement to Band Resolution Techniques. I. The Method and its Application to Self-Association of Acetic Acid",./. Phys. Chem. 1973, (77) 256-262. 

Culler, S.R., et. at. "Factor Analysis Applied to a Silane Coupling Agent on E-Glass Fiber System", Appl. Spec. 1984 (38) 495-500. 

Malinowski, E.R., "Statistical F-Tests for Abstract Factor Analysis and Target Testing 1,/. Chemo. 1987 (1) 49-60... 

Malinowski, Eit. "Theory of Error in Factor Analysis", Anal. Chem. 1977, (49) 606-612. 

Factor Analysis, 78 Factor spaces, 78 Factorial design, 29 Factors, 94... 

Ihteiference factor Specific rotation factor analysis... 

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