The factor analysis method

The basic model of the factor analysis method as applied here assumes that the x-ray emission Intensity of any specified element Is a linear sum of the quantity of that element found In the minerals present at that sample location ... 

In the Factor Analysis method [D] is subdivided into two matrices, applying Principal Component Analysis (PCA). This is a pure mathematical solution. There is no relation between the obtained matrices and the physical parameters. However, it is possible to relate the abstract solution resulting from the PCA to a physical relevant solution, using some pre-knowledge. 

By using the factor analysis method, the data can be displayed graphically as shown in Figure 3. In this analysis, a trend is observed... 

By using the factor analysis method, the mutagenicity data were applied only on water after disinfection. Then, comparisons were made between treatment lines (Figure 4). By using the same interpretation (i.e., assigning weights of 3, 2, or 1) as that used for the ozone and GAC treatment, the conclusions based on ordered classifications of treatment lines for each month and for the year are the following Treatment line 1 is the best line of the pilot plant and yields a 92 relative ideal complete treatment line, treatment line 2 is 66 , and treatment line 4 is 41 (Table VII). 

If the nature of the major sources influencing a particular receptor is unknown, statistical factor analysis methods can be combined with ambient measurements to estimate the source composition. Assuming that for a particular location several ambient particulate samples are collected and analyzed for several elements, the resulting data will probably include information about the fingerprints of the sources affecting the location. Principal-component analysis (PCA) is one of the factor analysis methods used to unravel the hidden source information from a rich ambient measurement data set. Factor analysis models are mathematically complex, and their results are often difficult to interpret. 

The factor analysis methods described above can be used to explore mixing, and to identify components. However, they do not yield estimates of the end-member compositions. Miesch (1981) proposed an alternative method, starting with Q-mode factor analysis, which aims to identify and quantify sensible end-member compositions based on the structure of the data. The concept is developed as follows ... 

The improvement in computer technology associated with spectroscopy has led to the expansion of quantitative infrared spectroscopy. The application of statistical methods to the analysis of experimental data is known as chemometrics [5-9]. A detailed description of this subject is beyond the scope of this present text, although several multivariate data analytical methods which are used for the analysis of FTIR spectroscopic data will be outlined here, without detailing the mathematics associated with these methods. The most conunonly used analytical methods in infrared spectroscopy are classical least-squares (CLS), inverse least-squares (ILS), partial least-squares (PLS), and principal component regression (PCR). CLS (also known as K-matrix methods) and PLS (also known as P-matrix methods) are least-squares methods involving matrix operations. These methods can be limited when very complex mixtures are investigated and factor analysis methods, such as PLS and PCR, can be more useful. The factor analysis methods use functions to model the variance in a data set. 

Chemistry is full of calculations. Our basic goal is to help you develop the knowledge and strategies you need to solve these problems. In this chapter, you will review the Metric system and basic problem solving techniques, such as the Unit Conversion Method. Your textbook or instructor may call this problem solving method by a different name, such as the Factor-Label Method and Dimensional Analysis. Check with your instructor or textbook as to for which SI (Metric) prefixes and SI-English relationships will you be responsible. Finally, be familiar with the operation of your calculator. (A scientific calculator will be the best for chemistry purposes.) Be sure that you can correctly enter a number in scientific notation. It would also help if you set your calculator to display in scientific notation. Refer to your calculator s manual for information about your specific brand and model. Chemistry is not a spectator sport, so you will need to Practice, Practice, Practice. 

In this section, we will introduce one of the two common methods for solving problems. (You will see the other method in Chapter 5.) This is the Unit Conversion Method. It will be very important for you to take time to make sure you fully understand this method. You may need to review this section from time to time. The Unit Conversion Method, sometimes called the Factor-Label Method or Dimensional Analysis, is a method for simplifying chemistry problems. This method uses units to help you solve the problem. While slow initially, with practice it will become much faster and second nature to you. If you use this method correctly, it is nearly impossible to get the wrong answer. For practice, you should apply this method as often as possible, even though there may be alternatives. 

Dalton s law Dalton s law states that in a mixture of gases (A + B + C. . . ) the total pressure is simply the sum of the partial pressures (the pressures associated with each individual gas), decomposition reactions Decomposition reactions are reactions in which a compound breaks down into two or more simpler substances, diamagnetism Diamagnetism is the repulsion of a molecule from a magnetic field due to the presence of all electrons in pairs, dilute Dilute is a qualitative term that refers to a solution that has a relatively small amount of solute in comparison to the amount of solvent, dimensional analysis Dimensional analysis, sometimes called the factor label method, is a method for generating a correct setup for a mathematical problem. 

The unit analysis method involves analyzing the units and setting up conversion factors. You match and arrange the units so that they divide out to give the desired unit in the answer. Then you multiply and divide the numbers that correspond to the units. 

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). 

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... 

The factor analysis technique used was unable to distinguish separate soil and road sources. Ca appeared with Al, Si, K, Ti, and Fe on a factor that can be characterized only as "crustal," including both soil and road materials. It appears that a chemical element balance should always be used as a check on factor analysis results, at least until a more sophisticated factor analysis method, such as target transformation factor analysis (14), can be shown not to require it. 

Recently, Tichy investigated 41) the dependencies of the steric constants, Es, v, L, Bj, B4, MV (molar volume), [P] (parachor), MR (molar refraction), MW (molecular weight), and % (molecular connectivity index) on lipophilicity, as it is measured by n 42) and f43) constants. The data were treated by factor analysis methods. 

In order to compare the results between two laboratories for the same sample or, for example, two instruments for the same analysis method, it is essential to know whether the standard deviation, i of the first set of results is significantly different from that of the second set, s2. This is accomplished by using the variance equality test. In this test, an F factor is calculated, which is the ratio of the two variances such that F > 1 ... 

In the third sample (a-WC/a-W2C), when the number of spectra selected for the factor analysis of W was lowered, a new factor appeared. This new factor, was assigned to a-WC. This compound is present only close to the surface of the sample, and is probably represented by only one vector of the matrix used for factor analysis. As the other component (a-W2C, is present in all the other spectra, its weight in the matrix is considerably greater than that of a-WC. Moreoever the shapes of the two spectra are quite similar. It seems that the limit of detection of the Indicator Function is reached, since if the weight of a-WC is increased, the method will detect it as a component. This limitation of the Indicator Function was already theoretically shown by Palacio17 and is confirmed here now experimentally. 

Dimensional analysis is also known as unit analysis or the factor label method. 

The simplest way to carry out calculations that involve different units is to use the dimensional-analysis method. In this method, a quantity described in one unit is converted into an equivalent quantity with a different unit by using a conversion factor to express the relationship between units ... 

The dimensional-analysis method gives the right answer only if the equation is set up so that the unwanted units cancel. If the equation is set up in any other way, the units won t cancel properly, and you won t get the right answer. Thus, if you were to multiply your height in inches by the incorrect conversion factor inches per meter, you would end up with an incorrect answer expressed in meaningless units ... 

The known information is the speed in kilometers per hour the unknown is the speed in miles per hour. Find the appropriate conversion factor, and use the dimensional-analysis method to set up an equation so the km units cancel. 

When all of the individual component spectra are not known, implicit calibration methods are often adopted. Among these, factor analysis methods such as principal component regression (PCR)24 and partial least squares (PLS)25 are frequently used because they can function under conditions in which the number of spectra used for calibration is less than the number of wavelengths sampled. For example, a calibration set may include 30 spectra with each spectrum having 500 data points (wavelengths). 

Principal component regression and partial least squares are two widely used methods in the factor analysis category. PCR decomposes the matrix of calibration spectra into orthogonal principal components that best capture the variance in the data. These new variables eliminate redundant information and, by using a subset of these principal components, filter noise from the original data. With this compacted and simplified form of the data, equation (12.7) may be inverted to arrive at b. 

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