Cosine correlation analysis

Cosine correlation analysis (CCA) is a multivariate technique that assesses similarity in spectral datasets and image datasets [52]. Other similarity analysis techniques include Euclidean distance [49] and spectral angle mapping [53] which have been used for spectral library searching and image contrast generation in remote sensing applications, respectively. 

Cosine correlation analysis assesses chemical heterogeneity without the need for training sets. Cosine correlation analysis identifies differences in spectral shape and efficiently provides chemical-based image contrast that is independent of absolute intensity, which makes it well suited to Raman and fluorescence chemical imaging. 

Figure 22 Cosine correlation analysis of Raman CPO-coated TPO images acquired in the C—H stretching region (2800-3000 cm" ) employing pure component LCTF Raman spectra as reference vectors. The Raman images were collected with a 20X objective (NA 0.46) (A) bright-field image (B) CPO correlation image (C) EPR correlation image (D) PP correlation image of coated TPO.

In electroencephalogram (EEG) waveforms, it may not be easy to visually estimate whether there is interdependence between the waveforms from two different leads. Correlation analysis is used to find common periodicities of two functions (waveforms) fi(t) and f2(t). We have seen that if we multiply two sine waves of the same frequency, the DC value of the product is proportional to the cosine of the phase difference cp between them (Eq. 8.24). We can therefore calculate the product as a function of delaying one of the waveforms with respect to the other, and look for maxima corresponding to cp = 0. Mathematically, for each time t, the correlation value c(t) can be found by summing up the products of one of the waveforms and a time displaced version of the other ... 

A correlation analysis is applied to the individual sine and cosine functions in the manner similar to the derivation for Equations (F28) and (F29), which in turn generates a number of synchronous and asynchronous spectra, that is, cospectra 4>s quad-spectra j/, for individual Fourier components with different frequencies s. 

The manner in which sample-to-sample resemblance is defined is a key difference between the various hierarchical clustering techniques. Sample analyses may be similar to one another in a variety of ways and reflect interest in drawing attention to different underlying processes or properties. The selection of an appropriate measure of similarity is dependent, therefore, on the objectives of the research as set forth in the problem definition. Examples of different similarity measures or coefficients that have been used in compositional studies are average Euclidean distance, correlation, and cosine. Many others that could be applied are discussed in the literature dealing with cluster analysis (15, 18, 19, 36, 37). 

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