Manifest variables

The eigenvectors extracted from the cross-product matrices or the singular vectors derived from the data matrix play an important role in multivariate data analysis. They account for a maximum of the variance in the data and they can be likened to the principal axes (of inertia) through the patterns of points that represent the rows and columns of the data matrix [10]. These have been called latent variables [9], i.e. variables that are hidden in the data and whose linear combinations account for the manifest variables that have been observed in order to construct the data matrix. The meaning of latent variables is explained in detail in Chapters 31 and 32 on the analysis of measurement tables and contingency tables. 

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. 

The advantages of a reduction in the number of variables are numerous, e.g., the interpretation of the data becomes easier and the influence of noise is reduced because the latent variables are weighted averages of the original manifest variables. 

PLS can handle short and fat K N), long and lean N K), and square N K) data structures. There is no upper limit (other than computational speed and memory) on the number of rows and colunms that may be treated. If a good characterization of the observations (compounds) is the primary goal, this is accomplished by increasing the number of relevant variables. Similarly, the number of observations may be increased to enhance the description of the variables. The underlying latent factors that are captured by the PLS model become more stable and distinct the greater the number of meaningful manifest variables included. 

PLS and similar projection methods, however, are based on other assumptions, namely that variables are correlated (collinear) and possibly also noisy and incomplete. These correlations are, in turn, modeled as arising from a small set of latent variables, where all measured (manifest) variables are modeled as linear combinations of the latent variables. In molecular science, the latent variables are often interpretable... 

Still another manifestation of mixed-film formation is the absorption of organic vapors by films. Stearic acid monolayers strongly absorb hexane up to a limiting ratio of 1 1 [272], and data reminiscent of adsorption isotherms for gases on solids are obtained, with the surface density of the monolayer constituting an added variable. 

There is some confusion in using Bayes rule on what are sometimes called explanatory variables. As an example, we can try to use Bayesian statistics to derive the probabilities of each secondary structure type for each amino acid type, that is p( x r), where J. is a, P, or Y (for coil) secondary strucmres and r is one of the 20 amino acids. It is tempting to writep( x r) = p(r x)p( x)lp(r) using Bayes rule. This expression is, of course, correct and can be used on PDB data to relate these probabilities. But this is not Bayesian statistics, which relate parameters that represent underlying properties with (limited) data that are manifestations of those parameters in some way. In this case, the parameters we are after are 0 i(r) = p( x r). The data from the PDB are in the form of counts for y i(r), the number of amino acids of type r in the PDB that have secondary structure J.. There are 60 such numbers (20 amino acid types X 3 secondary structure types). We then have for each amino acid type a Bayesian expression for the posterior distribution for the values of xiiry. 

The manifestation of turbulent eddies is gustiness and is displayed in the fluctuations seen on a continuous record of wind or temperature. Figure 19-3 displays wind direction traces during (a) mechanical and (b) thermal turbulence. Fluctuations due to mechanical turbulence tend to be quite regular that is, eddies of nearly constant size are generated. The eddies generated by thermal turbulence are both larger and more variable in size than those due to mechanical turbulence. 

How well do GCMs simulate the spatial variability of climatic change Today s GCMs utilize data grids that partition the atmosphere into cells, each covering an area about the size of Colorado. A mean state of the atmosphere (temperature, humidity, cloud cover, for example) is computed for each cell. Consequently, any ou ut statistics (the prediction) has a lower spatial resolution (more genei ized, less detailed) than the real atmosphere is likely to manifest. 

Climatic change may occur as fluctuations in means and/or extremes of climatic variables. Most studies of climatic change, however, focus almost exclusively on trends in mean values of climatic variables (usually temperature). In fact, a climatic shift also may be manifest as an increase or decrease in the frequency of occurrence of extremes. That is, episodes of drought or excessive rainfall or record high or low temperatures may become more or less frequent. Changes in frequency of extremes may occur with little or no concurrent change in means. 

Dans les cas d exposition de breve a moyenne duree, on a constate que les points d aboutissement importants de I action toxique etaient le systeme nerveux, le developpement, le systeme immunitaire et le systeme endocrinien, mais cette action se manifeste a des degres variables selon les differents composes. 

It is often said that the property of acidity is manifest only in the presence of a base, and NMR studies of probe molecules became common following studies of amines by Ellis [4] and Maciel [5, 6] and phosphines by Lunsford [7] in the early to mid 80s. More recently, the maturation of variable temperature MAS NMR has permitted the study of reactive probe molecules which are revealing not only in themselves but also in the intermediates and products that they form on the solid acid. We carried out detailed studies of aldol reactions in zeolites beginning with the early 1993 report of the synthesis of crotonaldehyde from acetaldehyde in HZSM-5 [8] and continuing through investigations of acetone, cyclopentanone [9] and propanal [10], The formation of mesityl oxide 1, from dimerization and dehydration of... 

Both objects are much less complicated than the total A -particle wavefunction itself, since they only depend on three spatial variables. The electron density is manifestly positive (or zero) everywhere in space while the spin-density can be positive or negative. If, by convention, there are more spin-up than spin-down electrons, the positive part of the spin-density will prevail and there will usually be only small regions of negative spin-density that arise from spin-polarization. This spin-polarization is physically important and is already included in the UHF method but not in the ROHF method that, by construction, can only describe the... 

Airway hyperresponsiveness is defined as the exaggerated ability of the airways to narrow in response to a variety of stimuli. Although AHR exists in patients without asthma, it is a characteristic feature of asthma and appears to be directly related to airway inflammation and the severity of asthma.1,3 Treatment of airway inflammation with inhaled corticosteroids attenuates AHR in asthma but does not eliminate it.1 Clinically, AHR manifests as increased variability of airway function. Although not commonly used to diagnose asthma, AHR can be evaluated clinically using a methacholine or histamine bronchoprovocation test. 

ECF. Note that phosphorus is the major anion within the cells. Given this distribution, serum phosphate concentration does not accurately reflect total body phosphorus stores. Phosphorus is expressed in milligrams (mg) or millimoles (mmol), not as milliequivalents (mEq). Because phosphorus is the source of phosphate for adenosine triphosphate (ATP) and phospholipid synthesis, manifestations of phosphorus imbalance are variable. 

The general experimental approach used in 2D correlation spectroscopy is based on the detection of dynamic variations of spectroscopic signals induced by an external perturbation (Figure 7.43). Various molecular-level excitations may be induced by electrical, thermal, magnetic, chemical, acoustic, or mechanical stimulations. The effect of perturbation-induced changes in the local molecular environment may be manifested by time-dependent fluctuations of various spectra representing the system. Such transient fluctuations of spectra are referred to as dynamic spectra of the system. Apart from time, other physical variables in a generalised 2D correlation analysis may be temperature, pressure, age, composition, or even concentration. 

To address this situation, a data interpretation system was constructed to monitor and detect changes in the second stage that will significantly affect the product quality. It is here that critical properties are imparted to the process material. Intuitively, if the second stage can be monitored to anticipate shifts in normal process operation or to detect equipment failure, then corrective action can be taken to minimize these effects on the final product. One of the limitations of this approach is that disturbances that may affect the final product may not manifest themselves in the variables used to develop the reference model. The converse is also true—that disturbances in the monitored variables may not affect the final product. However, faced with few choices, the use of a reference model using the process data is a rational approach to monitor and to detect unusual process behavior, to improve process understanding, and to maintain continuous operation. 

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