Data matrix rank

Hugus, Z. Z Jr., El-Awady, A. A. The determination of the number of spedes present in a system a new matrix rank treatment of spedrophotometric data. f Phys. Chem. 1971, 75, 2954-2957. 

It is important to realize that closure may reduce the rank of the data matrix by one. This is the case with row-closure when n>p, and with colunm-closure when n < p. It is always the case with double-closure. This reduction of the rank by one is the result of a linear dependence between the rows or columns of the table that results from closure of the data matrix. 

For the same reason as for double-closure, double-centring always reduces the rank of the data matrix by one, as a result of the introduction of a linear dependence among the rows and columns of the data table. 

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. 

In order to apply RBL or GRAFA successfully some attention has to be paid to the quality of the data. Like any other multivariate technique, the results obtained by RBL and GRAFA are affected by non-linearity of the data and heteroscedast-icity of the noise. By both phenomena the rank of the data matrix is higher than the number of species present in the sample. This has been demonstrated on the PCA results obtained for an anthracene standard solution eluted and detected by three different brands of diode array detectors [37]. In all three cases significant second eigenvalues were obtained and structure is seen in the second principal component. 

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 ... 

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... 

Real data are never noise-free and in purely mathematical terms, the rank of a noisy data matrix is always the smaller of the number of rows or columns. So, the question obviously is, where do we stop What is the correct number of independent species or the correct rank of the matrix Y How many singular values are statistically relevant Most importantly for the chemist what is the practical or the chemical rank how many components are there in the system ... 

PCA decomposes a (centered) data matrix X into scores T and loadings P, see Chapter 3. For a certain number a of PCs which is usually less than the rank of the data matrix, this decomposition is... 

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). 

I niike PC.. wiih there is no need to determine the rank of the data matrix, niis is espeeialh- useft.il for looking at a snapshot view of the data when the inherent dimensionalit - of the data set exceeds three. 

Loadings Plot (Model and Sample Diag io tiL) The iouding.s can he used to help determine the optimal number of factors to consider for the model. For spectroscopic and chromatographic data, the point at which the loading displays random behavior can indicate the maximum number to consider. Numerical evaluation of the randomness of the loadings has been proposed as a method for determination of the rank of a data matrix for spectroscopic data... 

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. 

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... 

Of course, random measurement error is unavoidable when real data are used. If we now suppose A is a 50 x 50 data matrix (50 spectra digitized at 50 points) with some random error, the exact solution for Equation 4.2 would require 50 pairs or dyads of basis vectors, one row basis vector and one column basis vector for each pair. The additional 48 pairs of row and column vectors would be required to account for the random variation in A. Usually, we are not interested in building a model that includes the random errors. Fortunately, by using the appropriate mathematical operations, we can use our original two basis vectors to reduce the rank or dimensionality of A from 50 to 2 without any significant loss of information. This allows us to ignore the basis vectors that explain random error. This data compression capability of the PCA model is exploited frequently and is one of its most important features. 

When the true intrinsic rank of a data matrix (the number of factors) is properly determined, the corresponding eigenvectors form an orthonormal set of basis vectors that span the space of the original data set. The coordinates of a vector a in an m-dimensional space (for example, a 1 x m mixture spectrum measured at m wavelengths) can be expressed in a new coordinate system defined by a set of orthonormal basis vectors (eigenvectors) in the lower-dimensional space. Figure 4.14 illustrates this concept. The projection of a onto the plane defined by the basis vectors x and y is given by a. To find the coordinates of any vector on a normalized basis vector, we simply form the inner product. The new vector a, therefore, has the coordinates a, = aTx and a2 = aTy in the two-dimensional plane defined by x and y. 

The rank of a matrix is a mathematical concept that relates to the number of significant compounds in a dataset, in chemical terms to the number of compounds in a mixture. For example, if there are six compounds in a chromatogram, the rank of the data matrix from the chromatogram should ideally equal 6. However, life is never so simple. What happens is that noise distorts this ideal picture, so even though there may be only six compounds, either it may appear that the rank is 10 or more, or else the apparent rank might even be reduced if the distinction between the profiles for certain compounds are indistinguishable from the noise. If a 15 x 300 X matrix (which may correspond to 15 UV/vis spectra recorded at 1 nm intervals between 201 and 500 nm) has a rank of 6, the scores matrix T has six columns and the loadings matrix P has six rows. 

Any analytical data obtained by hyphenated instruments or by two-way spectroscopic techniques such as excitation-emission fluorescence spectroscopy are bilinear ones. The bilinear data matrix has a very useful property, namely the rank of such matrix obtained with any chemical mixture is equal to the number of chemical components in the mixture. Thus, theoretically, the rank of a data matrix of any pure chemical component is unit. It can be expressed by the product of two vectors ... 

By far, singular value decomposition (SVD) is the most popular algorithm to estimate the rank of the data matrix D. As a drawback of SVD, the threshold that separates significant contributions from noise is difficult to settle. Other eigenvalue-based and error functions can be utilized in a similar way, but the arbitrariness in the selection of the significant factors still persists. For this reason, additional assays may be required, especially in the case of complex data sets. 

The classification of the 12 substances is given in Table 6 with respect to the cases C (Coincidence among all four methods), S (the rank of a substance depends on the method and therefore on the formalisms to include participation. HDT, as a method based purely on the data matrix seems to be on the safe side) and case W ( Worst case, discrepancies arise because of the different approaches (non-metric (HDT) vs. metric (the other three methods)). 

Although Cl (poset) levels can be determined directly from the indicator data matrix without conversion to ranks, we proceed immediately to... 

The mathematical concept of rank is not very convenient in chemical modeling. Take, for example, ultraviolet spectra (100 wavelengths) measured on ten different samples, each of which contains the same absorbing species at different concentrations. The resulting data matrix X has size (10 x 100) and, if the Lambert-Beer law holds, is essentially of rank one. 

In the example above on ultraviolet spectroscopy there is one single chemical source of variation in X the concentration of the analyte. Due to the linearity of the system this single source of variation generates an X matrix of rank one. Sometimes the term chemical rank is used to indicate the number of chemical sources of variation in a data matrix. 

The basis of RAFA is the direct eigenvalue solution of the mixture matrix of responses based on a singular-value decomposition. Let N represent a data matrix of the pure component response of analyte k from the two-dimensional sensor array and M represent the sample response of the same instrument to the analyte of interest and all interfering components the solution of the rank annihilation problem can be stated as follows ... 

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