Method Fourier transform, data reduction

Due to the fact that the first phase of manipulation of such data is usually a fast scanning of the entire collection, a highly compressed representation of uniformly coded data is essential in order to accelerate the handling. After the search reduces the collection to a smaller group in which the target object is supposed to be, the full (extended) representation of objects can be invoked if necessary for further manipulation. In the next sections we shall discuss the use of two methods, Fast Fourier Transformation (FFT) and Fast Hadamard Transformation (FHT), for the reduction of object representations and show by some examples in 1- and 2-dimensional patterns (spectra, images) how the explained procedures can be used... 

Fourier Transform Method. Another method of data reduction is to take a fast Fourier transform (FFT) of the wave (10). As indicated in Figure 7, the Fourier transform of a damped sine wave with a single frequency is a single maximum in the frequency domain at the frequency of the oscillation. The amplitude (H) of the transformed data as a function of angular frequency ([Pg.344]

In virtually all types of experiments in which a response is analyzed as a function of frequency (e.g., a spectrum), transform techniques can significantly improve data acquisition and/or data reduction. Research-level nuclear magnetic resonance and infra-red spectra are already obtained almost exclusively by Fourier transform methods, because Fourier transform NMR and IR spectrometers have been commercially available since the late 1960 s. Similar transform techniques are equally valuable (but less well-known) for a wide range of other chemical applications for which commercial instruments are only now becoming available for example, the first commercial Fourier transform mass spectrometer was introduced this year (1981) by Nicolet Instrument Corporation. The purpose of this volume is to acquaint practicing chemists with the basis, advantages, and applications of Fourier, Hadamard, and Hilbert transforms in chemistry. For almost all chapters, the author is the investigator who was the first to apply such methods in that field. 

PCA is primarily a mathematical method for data reduction and it does not assume that the data have any particular distribution. We have seen how PCA can be used to reduce the dimensionality of a data set and how it may thus reveal clusters. It has been used, for example, on the results of Fourier transform spectroscopy in order to reveal differences between hair from different racial groups and for classifying different types of cotton fibre. In another example the concentrations of a number of chlorobiphenyls were measured in specimens from a variety of marine mammals. A PCA of the results revealed differences between species, differences between males and females, and differences between young and adult individuals. PCA also finds application in multiple regression (see Section 8.10). 

Van Veen and De Loos-Vollebregt reviewed various chemometric procedures which had been developed during the last decade for ICP-OES analysis. In these procedures, data reduction by techniques such as digital filtering, numerical derivatives, Fourier transforms, correlation methods, expert systems, fuzzy logic, neural networks, PCA, PLS, projection methods, Kalman filtering, MLR and generalised standard additions were discussed. 

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