Wavelet coefficient descriptors

Wavelet coefficient descriptors (WCDs) ° exemplify how the WT enhances the quality of current descriptor technology and enables the development of improved models. WCDs are an adaptation of the transferable atom equivalent (TAE) descriptors developed by Breneman and Rhem. TAE descriptors are derived by quantifying the distributions of multiple electronic properties computed on electronic van der Waals surfaces, which are defined as the 0.002-e-au isosurfaces (Figure 15). TAE descriptors encode the... 

Lockwood, L., Jr., Breneman, C. M., Embrechts, M. J., Bennett, K. P., and Arciniegas, F. (2000) Use of 2D, 3D, TAE and wavelet coefficient descriptors (WCDs) for generating self-organizing Kohonen maps for QSAR, QSPR, and ADME analyses. Abstr. Pap. Am. Chem. Soc. 220th, COMP-134. 

Each additional resolution — up to the highest resolution level J—decomposes the coarse coefficients and leaves the detail coefficients unchanged. The remaining coarse coefficient cannot be decomposed further it consists of just four components. J is determined by the size n of the original vector with J = log2(n) - 2. Consequently, a wavelet-transformed descriptor can be represented by either single-level (j = 1) or multilevel (/< = /) decomposition. 

Wavelet analysis was also proposed for variable reduction problems and, in particular, the wavelet coefficients obtained from discrete wavelet transforms (DWT) were proposed as a molecular representation in PEST descriptor methodology and their sums as molecular descriptors [Breneman, Sundhng et al., 2003 Lavine, Davidson et al, 2003]. 

An important property of wavelet bases is their lack of translational invariance. In other words, when a pattern is translated, its descriptors are not only translated but also modified. This is a direct consequence of the down-sampling procedure and leads to distorted reconstruction of the underlying signal features. A possible solution is to omit down-sampling, resulting in a redundant family of coefficients. 

The resolution level can be chosen arbitrarily between 1 and J. Any of the valid combinations of coarse and detail coefficients at a certain resolution level that lead to a descriptor of the same size are possible. For example, an original RDF descriptor with 256 components (i.e., / = 6) can be decomposed up to the resolution level j = 3, and the Wavelet transform (WLT) can be represented as + D -1- + )< ( This... 

FIGURE 5.21 Combination coarse- and detail-filtered transformed RDF descriptor performed at the highest resolution level (J = 6). The transform is an alternative representation of an RDF in the wavelet domain. This signal consists of the coefficients G ) + D -1- 0(S)... 

The coarse- or detail-filtered wavelet transform or a combination of the coefficients for an RDF descriptor at a certain resolution level is done right before any postprocessing, like normalization. 

The additional information primarily affects analysis methods that rely on the appearance of individual peaks rather than the shape of the entire descriptor. Thus, statistical parameters like correlation coefficients and skewness are affected to a minor extent. However, coarse-filtered wavelet transformations lead to an increase in valuable information. 

Having the three-dimensional coordinates of atoms in the molecules, we can convert these into Cartesian RDF descriptors of 128 components (B = 100 A ). To simplify the descriptor we can exclude hydrogen atoms, which do not essentially contribute to the skeleton structure. Finally, a wavelet transform can be applied using a Daubechies wavelet with 20 filter coefficients (D20) to compress the descriptor. A low-pass filter on resolution level 1 results in vectors containing 64 components. These descriptors can be encoded in binary format to allow fast comparison during descriptor search. 

The descriptors are transformed by Daubechies wavelet decomposition with 20 filter coefficients (D20). 

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