Matrix rank analysis

According to Sections 2.1.1, 2.1.4, and 2.5.6 the relationship between the concentrations and the linear independent degrees of advancement can be given by (see Section 2.2.1.1) 

Thus the vector of the differences of concentrations is a product of a matrix for the general stoichiometric coefficients and the vector of the linear independent degrees of advancements. This relationship is valid for each time t,. For various times given as f (/ = 0,1,2./n) these measured differences in concentration can be arranged in a matrix 

In this case it is essential that all the concentrations are measured at the same reaction time, that means simultaneously. If this prerequisite cannot be experimentally achieved the measured values have to be synchronised with respect to equivalent wavelength -independent reaction times (see Fig. 4.6 in Section 4.2.2.3). Usually the concentration at the time tg is taken as a reference  

However, this procedure is not necessary in the case of the matrix rank 

Under these conditions the rank of the matrix in eq. (5.2) equals according to definition. That means 5 equals the number of linear independent concentrations and the number of linear independent partial steps of reaction according to eq. (5.1), respectively. The numerical rank analysis is explained by an example [15]. 

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An important simplifying consequence of the use of inverted concentration ratios is that the reaction is independent of O2 concentration, which means that unintended 02 contamination should not distort the data. Because of the complexity of the reaction, the relatively new technique of Matrix Rank Analysis was used to sort out the speciation. This analysis led to the identification of two sulfur-containing intermediates [Fe2(0H)S03]3+ and [Fe(S03]+. Other reactant species known to be present under these conditions include S02, HS03, Fe3+, Fe(OH)2+, and... 

Example 5.1 Example of a matrix rank analysis The following mechanism is assumed ... 

The number of linearly independent columns (or rows) in a matrix is called the rank of that matrix. The rank can be seen as the dimension of the space that is spanned by the columns (rows). In the example of Figure 4-15, there are three vectors but they only span a 2-dimensional plane and thus the rank is only 2. The rank of a matrix is a veiy important property and we will study rank analysis and its interpretation in chemical terms in great detail in Chapter 5, Model-Free Analyses. 

Evaluating of the Distribution of Components in the Peak by Local Rank Analysis. Complementary information about the evolution of the components inside the CE peak system can be obtained from local rank analysis. In this case, instead of estimating the rank of the whole D matrix, a succession of smaller submatrices derived from D is analyzed to get the evolution of the mathematical factors throughout the system. The most widely used evolutionary methods are as follows ... 

Definition of hazard, risk discussions on likelihood, consequence risk — register, matrix, ranking. Consequence ranking, preliminary hazard analysis tolerance point—ALARP refreshing on mathematics, fault tolerance, plant ageing, and basic functional safety fail safe operations in plants. 

In the case of flexible robots, several identification schemes have been studied. Some on-line identification schemes are based on input-output ARMA representations [15, 16, 17]. Another approach consists in elaborating a minimal identification model based on a knowledge model of the robot and in applying a least-squares method. There are two kinds of minimal identification model the first consists in applying the theorem of energy for the robot, the second comes from the dynamic model. More details on these two models applied in the case of one or two link-planar robots can be found in [18] and [19]. A set of standard parameters has been proposed. Its minimality has been demonstrated using a numerical rank analysis of the observation matrix which is constructed with a random sequence of points. 

Construction of non-random initial estimates of matrix C or S. Local rank analysis methods, such as evolving factor analysis (EFA) [1,2], or methods based on the selection of pure variables, such as simple-to-use-interactive self-modelling mixture analysis (SIMPLISMA) [18,19], can be used for this purpose. 

A theorem, which we do not prove here, states that the nonzero eigenvalues of the product AB are identical to those of BA, where A is an nxp and where B is a pxn matrix [3]. This applies in particular to the eigenvalues of matrices of cross-products XX and X which are of special interest in data analysis as they are related to dispersion matrices such as variance-covariance and correlation matrices. If X is an nxp matrix of rank r, then the product X X has r positive eigenvalues in A and possesses r eigenvectors in V since we have shown above that ... 

Basically, we make a distinction between methods which are carried out in the space defined by the original variables (Section 34.4) or in the space defined by the principal components. A second distinction we can make is between full-rank methods (Section 34.2), which consider the whole matrix X, and evolutionary methods (Section 34.3) which analyse successive sub-matrices of X, taking into account the fact that the rows of X follow a certain order. A third distinction we make is between general methods of factor analysis which are applicable to any data matrix X, and specific methods which make use of specific properties of the pure factors. 

In 1978, Ho et al. [33] published an algorithm for rank annihilation factor analysis. The procedure requires two bilinear data sets, a calibration standard set Xj and a sample set X . The calibration set is obtained by measuring a standard mixture which contains known amounts of the analytes of interest. The sample set contains the measurements of the sample in which the analytes have to be quantified. Let us assume that we are only interested in one analyte. By a PCA we obtain the rank R of the data matrix X which is theoretically equal to 1 + n, where rt is the number of interfering compounds. Because the calibration set contains only one compound, its rank R is equal to one. 

The application of principal components regression (PCR) to multivariate calibration introduces a new element, viz. data compression through the construction of a small set of new orthogonal components or factors. Henceforth, we will mainly use the term factor rather than component in order to avoid confusion with the chemical components of a mixture. The factors play an intermediary role as regressors in the calibration process. In PCR the factors are obtained as the principal components (PCs) from a principal component analysis (PC A) of the predictor data, i.e. the calibration spectra S (nxp). In Chapters 17 and 31 we saw that any data matrix can be decomposed ( factored ) into a product of (object) score vectors T(nxr) and (variable) loadings P(pxr). The number of columns in T and P is equal to the rank r of the matrix S, usually the smaller of n or p. It is customary and advisable to do this factoring on the data after columncentering. This allows one to write the mean-centered spectra Sq as ... 

Example Consider the following example from Noble and Daniel [Applied Linear Algebra, Prentice-Hall (1987)] with the MATLAB commands to do the analysis. Define the following real matrix with m = 3 and n = 2 (whose rank k = 1). 

Sections on matrix algebra, analytic geometry, experimental design, instrument and system calibration, noise, derivatives and their use in data analysis, linearity and nonlinearity are described. Collaborative laboratory studies, using ANOVA, testing for systematic error, ranking tests for collaborative studies, and efficient comparison of two analytical methods are included. Discussion on topics such as the limitations in analytical accuracy and brief introductions to the statistics of spectral searches and the chemometrics of imaging spectroscopy are included. 

A first question in model-free analysis is how many components are there in a system Or, in other words, what is the rank of the matrix Y In particular, what is the influence of noise Providing an answer to these questions is a first, extremely powerful result of SVD. 

As regards a contaminated soil, this type of analysis may not be possible because the various hydrocarbons cannot be extracted from the sample with equal efficiency. Volatile organic compounds require special procedures to achieve satisfactory recovery from the soil matrix. It thus becomes important to distinguish between those compounds that are considered to be volatile and those that rank as semi- or nonvolatile compounds. 

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). 

Eigenvectors reduce the dimensionality of the data matrix when the rank of the covariance matrix is E < V, so that V — E eigenvalues vanish, or when some eigenvectors are not significant, the use of some classification methods with the scores on the first eigenvectors, instead of the original variables, can avoid singular matrices or/and noticeably speed up data analysis. 

So, MCBA builds a covariance matrix of the residuals around the inner model and from this matrix it obtains a probability density function as bayesian analysis does, taking into account that the dimensionality of the inner space correspondingly reduces the rank of the covariance matrix from which a minor must be extracted. 

What factor analysis allows initially is a determination of the number of components required to reproduce the adsorbance or data matrix A. Factor analysis allows us to find the rank of the matrix A and the rank of A can be interpreted as being equal to the number of absorbing components. To find the rank of A, the matrix ATA is... 

Gas chromatography/olfactometry (GCO) methods have been developed as screening procedures to detect potent odorants in food extracts. The FD-factors or CHARM values determined in food extracts are not consequently an exact measure for the contribution of a single odorant to the overall food flavor for the following reasons. During GCO the complete amount of every odorant present in the extract is volatilized. However, the amount of an odorant present in the headspace above the food depends on its volatility from the food matrix. Furthermore, by AEDA or CHARM analysis the odorants are ranked according to their odor thresholds in air, whereas in a food the relative contribution of an odorant is strongly affected by its odor threshold in the food matrix. The importance of odor thresholds in aroma research has been recently emphazised by Teranishi et al. [58],... 

Seven simulated LC-UV/Vis DAS data matrices were constructed in MATLAB 5.0 (MathWorks Inc., Natick, MA). Each sample forms a 25 x 50 matrix. The simulated LC and spectral profiles are shown in Figure 12.3a and Figure 12.3b, respectively. Spectral and chromatographic profiles are constructed to have a complete overlap of the analyte profile by the interferents. Three of the samples represent pure standards of unit, twice-unit, and thrice-unit concentration. These standards are designated SI, S2, and S3, respectively. Three three-component mixtures of relative concentrations of interferent 1 analyte interferent 2 are 1 1.5 0.5, 2 0.5 2, and 2 2.5 1, and these are employed for all examples. These mixtures are designated Ml, M2, and M3, respectively. An additional two-component mixture, 2 2.5 0, is employed as an example for rank annihilation factor analysis. This sample is designated M4. In most applications, normally distributed random errors are added to each digitized channel of every matrix. These errors are chosen to have a mean of zero and a standard deviation of 0.14, which corresponds to 10% of the mean response of the middle standard. In the rank annihilation factor analysis (RAFA) examples, errors were chosen to simulate noise levels of 2.5 and 5% of the mean response of S2. In some PARAFAC examples, the noise level was chosen to be 30% of the mean response of the second standard. 

Dimensional analysis of this example is associated by a reduction of the rank of the matrix, because the base dimension of mass is only contained in the density, p. From this it does not follow that the density wouldn t be relevant here, but that it is already fully considered in the kinematic viscosity v, which is defined by v = p/p. Therefore... 

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