Multivariate data matrices

Case study 1 chromatographic peak profiles 4.2.3 Multivariate Data Matrices 

What might we want to ask about the data How many compounds are in the chromatogram would be useful information. Partially overlapping peaks and minor 

Parameter Inertsil ODS Inertsil ODS-2 Inertsil ODS-3 Rromasil C18 Rromasil C8 Symmetry C18 Supelco ABZ+ Purospher 

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As an example, consider a data matrix consisting of 10 rows (labelled from 1 to 10) and eight columns (labelled from A to H), as in Table 4.10. This could represent a portion of a two-way HPLC-DAD data matrix, the elution profile of which in given in Figure 4.15, but similar principles apply to all multivariate data matrices. We choose a small example rather than case study 1 for this purpose, in order to be able to demonstrate all the steps numerically. The calculations are illustrated with reference to the first two PCs, but similar ideas are applicable when more components are computed. 

Another important aspect of metabolomic data analysis is visualization of the results. After application of effective bioinformatics tools for data visualization, the multivariate data matrices can be more easily compared. 

For example, the objects may be chemical compounds. The individual components of a data vector are called features and may, for example, be molecular descriptors (see Chapter 8) specifying the chemical structure of an object. For statistical data analysis, these objects and features are represented by a matrix X which has a row for each object and a column for each feature. In addition, each object win have one or more properties that are to be investigated, e.g., a biological activity of the structure or a class membership. This property or properties are merged into a matrix Y Thus, the data matrix X contains the independent variables whereas the matrix Ycontains the dependent ones. Figure 9-3 shows a typical multivariate data matrix. 

Data set (multivariate) Data matrix, random sample (of observations)... 

FIGURE 2.1 Simple multivariate data matrix X with n rows (objects) and m columns (variables, features). An example (right) with m — 3 variables shows each object as a point in a three-dimensional coordinate system. 

One aim of chemometrics is to obtain these predictions after first treating the chromatogram as a multivariate data matrix, and then performing PCA. Each compound in the mixture is a chemical factor with its associated spectra and elution profile, which can be related to principal components, or abstract factors, by a mathematical transformation. 

Scaling die rows to a constant total is useful if the absolute concentrations of samples cannot easily be controlled. An example might be biological extracts the precise amount of material might vary unpredictably, but the relative proportions of each chemical can be measured. This method of scaling introduces a constraint which is often called closure. The numbers in the multivariate data matrix are proportions and... 

Relatively few molecular structures in the CSD exhibit molecular symmetry. However, many substructural fragments of interest in structure correlation studies are small and symmetric. This fact is recognized in Chapter 2, and various aspects of fragment symmetry are discussed there. We now examine the consequences of fragment symmetry on the search process itself and, hence, on the relative ordering of the Np geometrical parameters recorded for each fragment in the multivariate data matrix G Nf,Np). 

Figure 9-3. Multivariate data matriK X, containing n objects each represented by m features. The matrix Y contains the properties of the objects that are to be investigated.

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. 

Scaling is a very important operation in multivariate data analysis and we will treat the issues of scaling and normalisation in much more detail in Chapter 31. It should be noted that scaling has no impact (except when the log transform is used) on the correlation coefficient and that the Mahalanobis distance is also scale-invariant because the C matrix contains covariance (related to correlation) and variances (related to standard deviation). 

Any data matrix can be considered in two spaces the column or variable space (here, wavelength space) in which a row (here, spectrum) is a vector in the multidimensional space defined by the column variables (here, wavelengths), and the row space (here, retention time space) in which a column (here, chromatogram) is a vector in the multidimensional space defined by the row variables (here, elution times). This duality of the multivariate spaces has been discussed in more detail in Chapter 29. Depending on the chosen space, the PCs of the data matrix... 

The aim of all the foregoing methods of factor analysis is to decompose a data-set into physically meaningful factors, for instance pure spectra from a HPLC-DAD data-set. After those factors have been obtained, quantitation should be possible by calculating the contribution of each factor in the rows of the data matrix. By ITTFA (see Section 34.2.6) for example, one estimates the elution profiles of each individual compound. However, for quantitation the peak areas have to be correlated to the concentration by a calibration step. This is particularly important when using a diode array detector because the response factors (absorptivity) may considerably vary with the compound considered. Some methods of factor analysis require the presence of a pure variable for each factor. In that case quantitation becomes straightforward and does not need a multivariate approach because full selectivity is available. 

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 ... 

In multivariate data analysis frequently the covariance matrix S is used... 

Usually multivariate analytical information is represented in form of a data matrix ... 

Matrix formed by a set of correlation coefficients related to m variables in multivariate data sets, R = (rXi,Xj). It is relevant in multicomponent analysis. 

A data matrix is the structure most commonly found in environmental monitoring studies. In these data tables or matrices, the different analyzed samples are placed in the rows of the data matrix, and the measured variables (chemical compound concentrations, physicochemical parameters, etc.) are placed in the columns of the data matrix. The statistical techniques necessary for the multivariate processing of these data are grouped in a table or matrix, or use tools, formulations, and notations of the lineal algebra. 

Statistical properties of a data set can be preserved only if the statistical distribution of the data is assumed. PCA assumes the multivariate data are described by a Gaussian distribution, and then PCA is calculated considering only the second moment of the probability distribution of the data (covariance matrix). Indeed, for normally distributed data the covariance matrix (XTX) completely describes the data, once they are zero-centered. From a geometric point of view, any covariance matrix, since it is a symmetric matrix, is associated with a hyper-ellipsoid in N dimensional space. PCA corresponds to a coordinate rotation from the natural sensor space axis to a novel axis basis formed by the principal... 

Principal component analysis (PCA) is aimed at explaining the covariance structure of multivariate data through a reduction of the whole data set to a smaller number of independent variables. We assume that an m-point sample is represented by the nxm matrix X which collects i=l,...,m observations (measurements) xt of a column-vector x with j=, ...,n elements (e.g., the measurements of n=10 oxide weight percents in m = 50 rocks). Let x be the mean vector and Sx the nxn covariance matrix of this sample... 

As mentioned before, this chapter has two goals, (a) to refresh some basic matrix mathematics and (b) to familiarise the reader with the essentials of both Matlab and Excel, particularly with respect to multivariate data... 

For the time being let us assume that we know all the individual concentrations of four mixtures of three chemical components forming matrix C. Let us also suppose that we know the molar absorptivities of all three components at six wavelengths, matrix A. From those two matrices one can construct a multivariate measurement, matrix Y. In this or a similar way, most "experimental" data matrices used in later chapters will be simulated. A simple Matlab example ... 

It is probably more realistic to assume that we know neither the rate constants nor the absorption spectra for the above example. All we have is the measurement Y and the task is to determine the best set of parameters which include the rate constants ki and /cj and the molar absorptivities, the whole matrix A. This looks like a formidable task as there are many parameters to be fitted, the two rate constants as well as all elements of A. In Multivariate Data, Separation of the Linear and Non-Linear Parameters (p.162), we start tackling this problem. 

As outlined in Multivariate Data, Separation of the Linear and Non-Linear Parameters, (p.162), it is crucial to eliminate the linear parameters by calculating the matrix A of molar absorptivities as a function of C and thus the rate constants. In fact, the function SsqCalc ABC is almost identical to Rcalc ABC (p.167). The only difference concerns the sum of squares, ssq, which is now returned instead of the residuals. 

We continue considering multivariate data sets, e.g. a series of spectra measured as a function of time, reagent addition etc. In short, a matrix of... 

In Chapter 2, we approach multivariate data analysis. This chapter will be helpful for getting familiar with the matrix notation used throughout the book. The art of statistical data analysis starts with an appropriate data preprocessing, and Section 2.2 mentions some basic transformation methods. The multivariate data information is contained in the covariance and distance matrix, respectively. Therefore, Sections... 

A simple form of multivariate data is a rectangular table (matrix, spreadsheet) consisting of n rows, m columns, and each cell containing a numerical value. Each row corresponds to an object, for instance a sample each column corresponds to a particular feature of the objects (variable, for instance a measurement on the objects). We call these data the matrix X, with element xy in row i and column j. A column vector x,- contains the values of variable j for all objects a row vector, xj, is a transposed vector and contains all features for object i (Table 2.1). 

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