Factor Analysis common factors

If sulfuric acid, H2SO4, is added to an aqueous solution of formic acid, carbon monoxide bubbles out rapidly. This also occurs if phosphoric add, HjPO, is added instead. The common factor is that both of these acids release hydrogen ions, H+. Yet, careful analysis shows that the concentration of hydrogen ion is constant during the rapid decomposition of formic acid. Evidently, hydrogen ion acts as a catalyst in the decomposition of formic acid. 

The resulting data of the Box-Behnken design were used to formulate a statistically significant empirical model capable of relating the extent of sugar 3deld to the four factors. A commonly used empirical model for response surface analysis is a quadratic polynomial of the type... 

The goal of factor analysis (FA) and their essential variant principal component analysis (PCA) is to describe the structure of a data set by means of new uncorrelated variables, so-called common factors or principal components. These factors characterize frequently underlying real effects which can be interpreted in a meaningful way. 

The main difference between factor analysis and principal component analysis is the way in which the variances of Eq. (8.20) are handled. Whereas the interest of FA is directed on the common variance var Xij)comm and both the other terms are summarized as unique variance... 

The factor extraction according to Eq. (8.20) in the course of which the number of common factors are estimated by rank analysis and coefficients of the factors (factor loadings) are calculated. 

An important application field of factor and principal component analysis is environmental analysis. Einax and Danzer [1989] used FA to characterize the emission sources of airborne particulates which have been sampled in urban screening networks in two cities and one single place. The result of factor analysis basing on the contents of 16 elements (Al, B, Ba, Cr, Cu, Fe, Mg, Mn, Mo, Ni, Pb, Si, Sn, Ti, V, Zn) determined by Optical Atomic Emission Spectrography can be seen in Fig. 8.17. In Table 8.3 the common factors, their essential loadings, and the sources derived from them are given. 

Let us dispose of the most common answer first. This answer is the one given in most of the discussions about the relative merits of the two formulations, e.g. [2], and is essentially a practical one we use the Inverse Beer s Law formulation because by doing so, we need to only determine the concentration(s) of the analyte(s) of interest. In the Beer s law formulation, you must determine the concentrations of all components in a mixture, whether they are of interest or not. Of course, there is benefit to that also as Malinowski points out, you can determine the number of components in a mixture and their spectra, as well as their concentrations, by proper application of the techniques of factor analysis in such a case [3],... 

McPhie, P. (2004). CD studies on films of amyloid proteins and polypeptides Quantitative g-factor analysis indicates a common folding motif. Biopolymers 75, 140-147. 

There is room for further analysis in many traditional areas, as pointed out above during the discussion of enolization. Also, it is noted that the employment of transition state pKf values is very close to the use of the proton activating factors and deprotonating factors, introduced by Stewart (Stewart and Srinivasan, 1978 Stewart, 1985). It is to be hoped that the two approaches can be consolidated in a common view of acid-base catalysis. 

On the other hand, factor analysis involves other manipulations of the eigen vectors and aims to gain insight into the structure of a multidimensional data set. The use of this technique was first proposed in biological structure-activity relationship (i. e., SAR) and illustrated with an analysis of the activities of 21 di-phenylaminopropanol derivatives in 11 biological tests [116-119, 289]. This method has been more commonly used to determine the intrinsic dimensionality of certain experimentally determined chemical properties which are the number of fundamental factors required to account for the variance. One of the best FA techniques is the Q-mode, which is based on grouping a multivariate data set based on the data structure defined by the similarity between samples [1, 313-316]. It is devoted exclusively to the interpretation of the inter-object relationships in a data set, rather than to the inter-variable (or covariance) relationships explored with R-mode factor analysis. The measure of similarity used is the cosine theta matrix, i. e., the matrix whose elements are the cosine of the angles between all sample pairs [1,313-316]. 

In complex systems that involve multiple Fe-bearing species and phases, such as those that are typical of biologic systems (Tables 1 and 2), it is often difficult or impossible to identify and separate all components for isotopic analysis. Commonly only the initial starting materials and one or more products may be analyzed for practical reasons, and this approach may not provide isotope fractionation factors between intermediate components but only assess a net overall isotopic effect. In the discussions that follow on biologic reduction and oxidation, we will conclude that significant isotopic fractionations are likely to occur among intermediate components. 

More commonly, we are faced with the need for mathematical resolution of components, using their different patterns (or spectra) in the various dimensions. That is, literally, mathematical analysis must supplement the chemical or physical analysis. In this case, we very often initially lack sufficient model information for a rigorous analysis, and a number of methods have evolved to "explore the data", such as principal components and "self-modeling analysis (21), cross correlation (22). Fourier and discrete (Hadamard,. . . ) transforms (23) digital filtering (24), rank annihilation (25), factor analysis (26), and data matrix ratioing (27). 

It is also beyond the graphical representation capabilities commonly used. Factor analysis is one of the pattern recognition techniques that uses all of the measured variables (features) to examine the interrelationships in the data. It accomplishes dimension reduction by minimizing minor variations so that major variations may be summarized. Thus, the maximum information from the original variables is included in a few derived variables or factors. Once the dimen-... 

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. 

The advantage of utilizing the standardized form of the variable is that quantities of different types can be included in the analysis including elemental concentrations, wind speed and direction, or particle size information. With the standardized variables, the analysis is examining the linear additivity of the variance rather than the additivity of the variable itself. The disadvantage is that the resolution is of the deviation from the mean value rather than the resolution of the variables themselves. There is, however, a method to be described later for performing the analysis so that equation 16 applies. Then, only variables that are linearly additive properties of the system can be included and other variables such as those noted above must be excluded. Equation 17 is the model for principal components analysis. The major difference between factor analysis and components analysis is the requirement that common factors have the significant values of a for more than one variable and an extra factor unique to the particular variable is added. The factor model can be rewritten as... 

Both component and factor analysis as defined by equations 17 and 18 aim at the identification of the causes of variation in the system. The analyses are performed somewhat differently. For the principal components analysis, the matrix of correlations defined by equation 10 is used. For the factor analysis, the diagonal elements of the correlation matrix that normally would have a value of one are replaced by estimates of the amount of variance that is within the common factor space. This problem of separation of variance and estimation of the matrix elements is discussed by Hopke et al. (4). 

Prior Applications. The first application of this traditional factor analysis method was an attempt by Blifford and Meeker (6) to interpret the elemental composition data obtained by the National Air Sampling Network(NASN) during 1957-61 in 30 U.S. cities. They employed a principal components analysis and Varimax rotation as well as a non-orthogonal rotation. In both cases, they were not able to extract much interpretable information from the data. Since there is a very wide variety of sources of particles in 30 cities and only 13 elements measured, it is not surprising that they were unable to provide much specificity to their factors. One interesting factor that they did identify was a copper factor. They were unable to provide a convincing interpretation. It is likely that this factor represents the copper contamination from the brushes of the high volume air samples that was subsequently found to be a common problem ( 2). 

Hopke, et al. (4) and Gaarenstroom, Perone, and Moyers (7) used the common factor analysis approach in their analyses of the Boston and Tucson area aerosol composition, respectively. In the Boston data, for 90 samples at a variety of sites, six common factors were identified that were interpreted as soil, sea salt, oil-fired power plants, motor vehicles, refuse incineration and an unknown manganese-selenium source. The six factors accounted for about 78 of the system variance. There was also a high unique factor for bromine that was interpreted to be fresh automobile exhaust. Large unique factors for antimony and selenium were found. These factors may possibly represent emission of volatile species whose concentrations do not oovary with other elements emitted by the same source. 

Multivariate methods, on the other hand, resolve the major sources by analyzing the entire ambient data matrix. Factor analysis, for example, examines elemental and sample correlations in the ambient data matrix. This analysis yields the minimum number of factors required to reproduce the ambient data matrix, their relative chemical composition and their contribution to the mass variability. A major limitation in common and principal component factor analysis is the abstract nature of the factors and the difficulty these methods have in relating these factors to real world sources. Hopke, et al. (13.14) have improved the methods ability to associate these abstract factors with controllable sources by combining source data from the F matrix, with Malinowski s target transformation factor analysis program. (15) Hopke, et al. (13,14) as well as Klelnman, et al. (10) have used the results of factor analysis along with multiple regression to quantify the source contributions. Their approach is similar to the chemical mass balance approach except they use a least squares fit of the total mass on different filters Instead of a least squares fit of the chemicals on an individual filter. 

A substantial amount of confusion (9,10.13,14) has recently developed as to an approach s dependence on conservation of mass. As Cooper and Watson ( ) have noted, the F j factors refer to the source chemistry as it arrives at the receptor. It is assumed with the conservation of mass that the Fj j as might be measured at a receptor, is the same as have been measured at the source. As noted above, this may not be valid depending on the source and the method used for source sampling. The chemical mass balance method incorporates the F j directly in its calculations and as a result is often perceived as having a greater dependence on this assumption than methods such as factor analysis which do not use Fy values in their calculations. Factor analysis methods, however, identify abstract factors, which explain variability. It is impossible to attribute a common... 

Since the sensory data collected involved degree of sample difference from a reference, it was felt that the analytical data should be analyzed in a similar manner. In cases where some peaks making up a multicomponent mixture are known to be specific to that mixture, this is a relatively simple matter. In such cases, the peak areas of the known components can be compared to a reference and average percent difference calculated. However, if it is not possible to pick out peaks that are clearly specific to a single multicomponent mixture, a more sophisticated technique such as factor analysis is required. There are circumstances where all peaks are common to each multicomponent mixture, i.e. qualitatively similar but quantitatively different. Also there are cases where peaks are found only in one of the multicomponent mixtures, but it is not clear to which mixture they belong. In these cases factor analysis is required to extract patterns that are characteristic of the specific multicomponent mixtures. Analytical concentrations of each of the multicomponent mixtures are then calculated as a set of factor scores where each score is directly proportional to the actual concentration of each multicomponent mixture. 

Using principal components analysis, VOCs can be grouped into common factors that display similar source emission profiles. From the main database of eight buildings, we have grouped different VOCs into components (sources). Table 10.6 shows the extracted components using the PCA analysis of the VOC database (Zuraimi et al., 2006). 

When analyzing real data sets one has to find common factor structures which explain the main part of the variance of the data. Therefore in factor analysis the total variance of the data is divided by the reduced factor solution into the three parts ... 

The communality is introduced as a mathematical measure of this common feature variance. The communality is the part of the variance of one feature which is described by the common factor solution in the factor analysis. High communalities, hj, mean that this feature variance is highly explained by the factor solution. Low communalities for one feature detect either a specific feature variance or high random error. 

The aim of the application of factor analysis (FA) to environmental problems is to characterize the complex changes which occur to all the features observed in partial systems of the natural environment. These common factors explain the complex state of the environment more comprehensively and causally and so enable extraction of the essential part of the information contained in a set of data. 

The exclusive consideration of common factors seems to be promising, especially for such environmental analytical problems, as is shown by the variance splitting of the investigated data material (Tab. 7-2). Errors in the analytical process and feature-specific variances can be separated from the common reduced solution by means of estimation of the communalities. This shows the advantage of the application of FA, rather than principal components analysis, for such data structures. Because the total variance of the data sets has been investigated by principal components analysis, it is difficult to separate specific factors from common factors. Interpretation with regard to environmental analytical problems is, therefore at the very least rendered more difficult, if not even falsified for those analytical results which are relatively strongly affected by errors. 

The purpose of application of factor analysis (FA) is the characterization of complex changes of all observed features in partial systems of the environment by determination of summarized factors which are more comprehensive and causally explicable. The method extracts the essential information from a data set. The exclusive consideration of common factors in the reduced factor analytical solution seems to be particularly promising for the analytical process. The specific variances of the observed features will be separated from the reduced factor analytical results by means of the estimation of the communalities. They do not falsify the influence of the main pollution sources (see also Tab. 7-2). The mathematical fundamentals of FA are explained in detail in Section 5.4.3 (see also [MALINOWSKI, 1991 WEBER, 1986]). 

In many chemical studies, the measured properties of the system can be regarded as the linear sum of the fundamental effects or factors in that system. The most common example is multivariate calibration. In environmental studies, this approach, frequently called receptor modeling, was first applied in air quality studies. The aim of PCA with multiple linear regression analysis (PCA-MLRA), as of all bilinear models, is to solve the factor analysis problem stated below ... 

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