Multidimensional data

Thus the hash code is not used as a direct way to access data rather it serves as an index or key to the filed data entry (Figure 2-66). Since hash coding receives unique codes by reducing multidimensional data to only one dimension, information gets lo.st. Thi.s los.s prevents a recon.struction of the complete data from the hash code. 

Data analysis is one aspect of multidimensional analyses that must be optimized in the future. The analysis of chromatographic data beyond one dimension is still exceedingly problematic, especially in the analyses of highly complex mixtures. Better software may need to be developed in order to analyze two- and three-dimensional peaks due to their complexity. Three-dimensional data is only useful today in terms of fingerprinting and often that even requires extensive data analysis. A great deal of research must still be carried out to make the interpretation and quantification of multidimensional data easier. 

The title implies that in this first chapter techniques are dealt with that are useful when the observer concentrates on a single aspect of a chemical system, and repeatedly measures the chosen characteristic. This is a natural approach, first because the treatment of one-dimensional data is definitely easier than that of multidimensional data, and second, because a useful solution to a problem can very often be arrived at in this manner. 

When confronted with multidimensional data it is easy to plug the figures into a statistical package and have nice tables printed that purportedly accurately analyze and represent the underlying factors. Have the following questions been asked ... 

Principal Component Analysis (PCA). Principal component analysis is an extremely important method within the area of chemometrics. By this type of mathematical treatment one finds the main variation in a multidimensional data set by creating new linear combinations of the raw data (e.g. spectral variables) [4]. The method is superior when dealing with highly collinear variables as is the case in most spectroscopic techniques two neighbor wavelengths show almost the same variation. 

P.J. Lewi, Multidimensional data representation in medicinal chemistry. In Chemometrics. Mathematics and Statistics in Chemistry (B.R. Kowalski, Ed.), Reidel, Dordrecht, 1984, pp. 351-376. 

Also neural networks were proposed to solve the problem of faithful representation of multidimensional data in representation spaces of lower dimensions [17]. 

On the other hand, factor analysis involves other manipulations of the eigen vectors and aims to gain insight into the structure of a multidimensional data set. The use of this technique was first proposed in biological structure-activity relationship (i. e., SAR) and illustrated with an analysis of the activities of 21 di-phenylaminopropanol derivatives in 11 biological tests [116-119, 289]. This method has been more commonly used to determine the intrinsic dimensionality of certain experimentally determined chemical properties which are the number of fundamental factors required to account for the variance. One of the best FA techniques is the Q-mode, which is based on grouping a multivariate data set based on the data structure defined by the similarity between samples [1, 313-316]. It is devoted exclusively to the interpretation of the inter-object relationships in a data set, rather than to the inter-variable (or covariance) relationships explored with R-mode factor analysis. The measure of similarity used is the cosine theta matrix, i. e., the matrix whose elements are the cosine of the angles between all sample pairs [1,313-316]. 

The basic detection concepts can be presented for the "zerodimensional case where detection decisions and detection limits are established simply from the characteristics of the chemical signal (instrument response), without giving detailed attention to other dimensions such as time, wavelength, analyte concentration, etc. Actually, higher dimensional situations (multiparameter separations or detector responses) reduce to this case either through sequential classification schemes or via algorithms which operate directly on the multidimensional data. 

Multidimensional Data Intercomparisons. Estimation of reliable uncertainty intervals becomes quite complex for non-linear operations and for some of the more sophisticated multidimensional models. For this reason, "chemometric" validation, using common, carefully-constructed test data sets, is of increasing importance. Data evaluation intercomparison exercises are thus analogous to Standard Reference Material (SRM) laboratory intercomparisons, except that the final, data evaluation step of the chemical measurement process is being tested. 

Chemometric quality assurance via laboratory and method intercomparisons of standardized test data sets, finally, is becoming recognized as essential for establishing the validity of detection decisions and estimated detection limits, especially when treating multidimensional data with sophisticated algorithms including several chemical components. 

The paradigm shift from critical activities from later drug development to earlier discovery phases some years ago has effectively led to a change in lead optimization and added a new dimension of complexity, while it is envisioned that from a multidimensional, data-driven process more suitable candidates in accord with the therapeutic target product profiles may emerge for the treatment of currently unmet medical needs. 

Principal components analysis is used to obtain a lower dimensional graphical representation which describes a majority of the variation in a data set. With PCA, a new set of axes arc defined in which to plot the samples. They are constructed so that a maximum amount of variation is described with a minimum number of axes. Because it reduces the dimensions required to visualize the data, PCA is a powerftil method for studying multidimensional data sets. 

Online analysis processing mainly comprises the interactive exploration of multidimensional data sets, or data cubes, which are manipulated by operations from matrix algebra, for example, slice-and-dice, roll-up, and drill-down. Computing performance is related to data warehouse size and also data quality, for example, missing data, unsharpness, and redundancy. The multidimensionality issue is critical for extracting pertinent information and selecting the results to be stored and visualized. 

Often, relationships between measured process parameters and desired product attributes are not directly measurable, but must rather be inferred from measurements that are made. This is the case with several spectroscopic measurements including that of octane number or polymer viscosity by NIR. When this is the case, these latent properties can be related to the spectroscopic measurement by using chemometric tools such as PLS and PCA. The property of interest can be inferred through a defined mathematical relation.39 Latent variables allow a multidimensional data set to be reduced to a data set of fewer variables which describe the majority of the variance related to the property of interest. This data compression using the most relevant data also removes the irrelevant or noisy data from the model used to measure properties. Latent variables are used to extract features from data, and can result in better accuracy of measurement and a reduced measurement time.4... 

In this manner, both PCA and FA provide a projection of the objects from the high-dimensional feature space on to a space defined by a few factors they can also be used as a method for graphical representation of multidimensional data. 

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