Data arrays , multivariate

Multivariate Image Analysis Strong and Weak Multiway Methods Strong and weak -way methods analyze 3D and 2D matrices, respectively. Hyperspectral data cube structure is described using chemometric vocabulary [17]. A two-way matrix, such as a classical NIR spectroscopy data set, has two modes object (matrix lines) and V variables (matrix columns). Hyperspectral data cubes possess two object modes and one variable mode and can be written as an OOV data array because of their two spatial directions. 

Esbensen, K.H., Wold, S., and Geladi, P., Relationships between higher-order data array configurations and problem formulations in multivariate data analysis, J. Chemom., 3, 33 18, 1988. 

Data arrays (one-way data) consist of spectral values taken at a given time point or electrophoretic responses at a given wavelength over time. One-way data can be used for sample characterization, classification, and quantification with multivariate calibration methods (e.g., principal component analysis and partial least square regression). 

The handling of large data arrays on a multitude of endpoints and parameters and the extraction of the information they contain (as in a hazard profile) requires the application of multivariate statistics. In environmental studies these are chiefly used in two fields ... 

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. 

Arrays were introduced in the mid-eighties as a method to counteract the cross-selectivity of gas sensors. Their use has since become a common practice in sensor applications [1], The great advantage of this technique is that once arrays are matched with proper multivariate data analysis, the use of non-selective sensors for practical applications becomes possible. Again in the eighties, Persaud and Dodds argued that such arrays has a very close connection with mammalian olfaction systems. This conjecture opened the way to the advent of electronic noses [2], a popular name for chemical sensor arrays used for qualitative analysis of complex samples. 

It is worth remarking that a gas sensor array is a mere mathematical construction where the sensor outputs are arranged as components of a vector. Arrays can also be utilized to investigate the properties of chemical sensors, or even better, the peculiar behaviour of a sensor as a component of an array. In this chapter, the more common sensor array methodologies are critically reviewed, including the most general steps of a multivariate data analysis. The application of such methods to the study of sensor properties is also illustrated through a practical example. 

Keywords electronic nose principal component analysis pattern recognition chemical sensors sensor arrays olfaction system multivariate data analysis. 

Consider now multivariate data, e.g. measurements at many wavelengths instead of only one, say kinetics followed by a diode-array spectrophotometer. Assume the instrument records the spectra at 1024 wavelengths. Compared with monovariate data (single wavelength), there is a dramatic increase in the number of parameters to be fitted. In addition to the rate constant, there are now 1024 molar absorptivities for each reacting component that need to be fitted. The algorithm devised so far cannot cope with that number of parameters. 

More complex than vectors or matrices (X, X andy, X and Y) are three-way data or multiway data (Smilde et al. 2004). Univariate data can be considered as one-way data (one measurement per sample, a vector of numbers) two-way data are obtained for instance by measuring a spectrum for each sample (matrix, two-dimensional array, classical multivariate data analysis) three-way data are obtained by measuring a spectrum under several conditions for each sample (a matrix for each sample, three-dimensional array). This concept can be generalized to multiway data. 

The basic theory of Kohonen maps—and only this will be treated here—is mathematically simple. A typical Kohonen map consists of a rectangular (often quadratic) array of fields (squares, cells, nodes, neurons) with a typical size of 5 x 5 (25 fields) to 100 x 100 (10,000 fields). Each field k is characterized by a vector wk, containing the weights wki, wia, , with in being the number of variables of a multivariate data set X (Figure 3.18) the lengths of the weight vectors are, for instance,... 

Chemometric evaluation methods can be applied to the signal from a single sensor by feeding the whole data set into an evaluation program [133,135]. Both principle component analysis (PCA) and partial least square (PLS) models were used to evaluate the data. These are chemometric methods that may be used for extracting information from a multivariate data set (e.g., from sensor arrays) [135]. The PCA analysis shows that the MISiC-FET sensor differentiates very well between different lambda values in both lean gas mixtures (excess air) and rich gas mixtures (excess fuel). The MISiC-FET sensor is seen to behave as a linear lambda sensor [133]. It... 

The data processing of the multivariate output data generated by the gas sensor array signals represents another essential part of the electronic nose concept. The statistical techniques used are based on commercial or specially designed software using pattern recognition routines like principal component analysis (PCA), cluster analysis (CA), partial least squares (PLSs) and linear discriminant analysis (LDA). 

Multivariate image reconstruction of FTIR array data... 

D. Lincoln, A. F. Fell, N. H. Anderson, and D. England, Assessment of chromatographic peak purity of drugs by multivariate analysis of diode-array and mass spectrometric data, J. Pharm. Biomed. Anal., 70 837-844(1992). 

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