Cosine coefficients

In a subsequent study, we examined the influence of seven similarity indices on the enrichment of actives using the topological CATS descriptor and the 12 COBRA datasets [31]. In particular, we evaluated to what extent different similarity measures complement each other in terms of the retrieved active compounds. Retrospective screening experiments were carried out with seven similarity measures Manhattan distance, Euclidian distance, Tanimoto coefficient, Soergel distance, Dice coefficient, cosine coefficient, and spherical distance. Apart from the GPCR dataset, considerable enrichments were achieved. Enrichment factors for the same datasets but different similarity measures differed only slightly. For most of the datasets the Manhattan and the Soergel distance... 

Tanimoto coefficient Dice coefficient Cosine coefficient... 

There are other coefficients like the Dice coefficient. Cosine coefficient, simple matching coefficient, and Tversky similarity coefficient. 

Viscoelastic parameters Ki, K2 and q of the model are determined by fitting the experimental curve V(t) to the sum of cosines with constant coefficients ... 

For the Fourier coefficients of the modulus and the phase we note that, because of the time-inversion invariance of the amplitude, the former is even in f and the latter is odd. Therefore the former is representable as a cosine series and the latter as a sine series. Formally,... 

The resulting similarity measures are overlap-like Sa b = J Pxi ) / B(r) Coulomblike, etc. The Carbo similarity coefficient is obtained after geometric-mean normalization Sa,b/ /Sa,a Sb,b (cosine), while the Hodgkin-Richards similarity coefficient uses arithmetic-mean normalization Sa,b/0-5 (Saa+ b b) (Dice). The Cioslowski [18] similarity measure NOEL - Number of Overlapping Electrons (Eq. (10)) - uses reduced first-order density matrices (one-matrices) rather than density functions to characterize A and B. No normalization is necessary, since NOEL has a direct interpretation, at the Hartree-Fodt level of theory. 

Empirically, the Dice coefficient has worked better than cosine similarity in retrieving actives and is the standard choice for use with the ap and tt descriptors. 

As the scalar product of two vectors is related to the cosine of the angle included by these vectors by Eq. (4), a frequently used similarity measure is the cosine coefficient (Eq. (5)). 

Cosine similarity coefficient Also known as the Carbo index c. EllWiB s... 

The frequency-dependent coefficients in this equation are given separate names and symbols to facilitate discussion. Remember it is these coefficients that determine the behavior of the system the trigonometric functions merely describe the oscillations. The following can be said of the coefficient of the cosine term ... 

The first Fourier transformation of the FID yields a complex function of frequency with real (cosine) and imaginary (sine) coefficients. Each FID therefore has a real half and an imaginary half, and when subjected to the first Fourier transformation the resulting spectrum will also have real and imaginary data points. When these real and imaginary data points are arranged behind one another, vertical columns result. This transposed data... 

The summation for the coefficient of the P2 term likewise includes phase insensitive contributions where I m = Im, but also now one has terms for which I = I 2, which introduce a partial dependence on relative phase shifts— specifically on the cosine of the relative phase shift. Again, this conclusion has long been recognized for example, by an explicit factor cos(ri j — ri i) in one term of the Cooper-Zare formula for the photoelectron (3 parameter in a central potential model [43]. 

Instead of this methodology, we have chosen to use Fourier analysis of the entire peak shape. By this procedure all of the above problems are eliminated. In particular, we focus on the cosine coefficients of the Fourier series representing a peak. The instrumental effects are readily removed, and the remaining coefficient of harmonic number, (n), A, can be written as a product ... 

Information concerning size distribution and strain profile can be obtained from the cosine Fourier coefficients, which describe the symmetric peak broadening. 

Fourier coefficients, represent the frequencies of the cosine functions. In the remainder of this chapter we use the shorthand notation F(n) for [A - jB ]. 

A shift or translation of f(r) by results in a modulation of the Fourier coefficients by exp(-ycofo). Without shift f(r) is transformed into F(fo). After a shift by tQ, fit - tg) is transformed into exp(-ja)tQ) F(sine wave) of the Fourier coefficients. The frequency cotg of the... 

The phase spectrum 0(n) is defined as 0(n) = arctan(A(n)/B(n)). One can prove that for a symmetrical peak the ratio of the real and imaginary coefficients is constant, which means that all cosine and sine functions are in phase. It is important to note that the Fourier coefficients A(n) and B(n) can be regenerated from the power spectrum P(n) using the phase information. Phase information can be applied to distinguish frequencies corresponding to the signal and noise, because the phases of the noise frequencies randomly oscillate. 

Because x, as well as w are normalized, represents the cosine or correlation coefficient between the two vectors. In a variant of ART, Fuzzy ART, a fuzzy similarity measure is used instead of the cosine similarity measure [14]. 

The average incident tangential momentum is muh while the average scattered tangential momentum is muf. If the gas molecule equilibrates with the surface and the scattered momentum is zero, we have Knudsen cosine scattering and complete accommodation of the incident gas molecule with the surface. On the other extreme, if specular reflection occurs, the incident momentum is retained upon scattering and mut = muf. The momentum accommodation coefficient, / is introduced to describe the type of scattering that does occur, and it is defined by... 

The term Fourier coefficient originates from the theory of Fourier series, in which periodic functions are expanded based on a set of sine- and cosine-functions. The expansion coefficients are called Fourier coefficients. 

It should be recognized that the discrete Fourier coefficients G(x, y, co) are represented by complex numbers. The real part Re(G(x, y, to)) of the complex number represents the amplitude of the cosine part of the sinusoidal function and the imaginary part Im(G(x, y, co)) represents the amplitude of the sine wave. 

Better procedures for determining particle sizes from X-ray diffraction are based on line profile analysis with Fourier transform methods. The average size is obtained from the first derivative of the cosine coefficients and the distribution of particle sizes from the second derivative. When used in this way, XRD offers a fundamental advantage over electron microscopy, because it samples a much larger portion of the catalyst. The reader is referred to publications by Cohen and coworkers for more details and examples [4,10,11],... 

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