Shape code matrix

The shape types Tj are usually specified by various algebraic methods, for example, by a shape group or a shape matrix, or by some other algebraic or numerical means. The algebraic invariants or the elements of the matrices are numbers, and these numbers form a shape code. The (P,W)-shape similarity technique provides a nonvisual, algebraic, algorithmic shape description in terms of numerical shape codes, suitable for automatic, computer characterization and comparison of shapes and for the numerical evaluation of 3D shape similarity. 

Such a decreasing sequence is associated with each point of a grid on the (a,b) parameter map. The grid itself can be regarded as a matrix with the sequences of Betti numbers as elements. This matrix is a shape code for the molecule that can be used for storing shape information in molecular data banks and can be retrieved for shape similarity assessment by numerical methods. 

One of the most useful shape codes is based on shape matrices. As we have seen in Chapter 5, the N-neighbor relation N(D j, D j ) of various curvature domains and, given by Equation (5.8), leads to a shape matrix... 

The n diagonal elements ij(a,b) of the matrix s(a,b) are encoded by concatenating them into a single number. For example, a diagonal of elements (2, 2, 1, 2, 0, 2 ) is encoded as the decimal number 221202. (Since the only numerical values that occur along the diagonal are 0, 1, and 2, a ternary number system can also be used). The number so obtained is the second component C2(s(a,b)) of the shape code vector c(s(a,b)). 

The upper off-diagonal triangle of the shape matrix has elements 1 and 0 only, and these n(n-l)/2 numbers concatenated according to columns form a binary number that is the third component C3(s(a,b)) of the shape code vector c(s(a,b)). 

Note that if the element s(a,b)i,i of the shape matrix is different from zero, then the information on the dimension n of the matrix can be deduced from the second element C2(s(a,b)) of the shape code vector c(s(a,b)). The special case of s(a,b)i 1 = 0 seldom occurs, since this implies that the largest domain is a locally concave Dq domain relative to the curvature parameter b. Nevertheless, in order to avoid ambiguity in such cases, the dimension n is specified as the first component ci(s(a,b)) of the shape code vector c(s(a,b)). 

This shape code vector can be decoded easily by simply reversing the above process. In view of the relation n = C (s(0.0l,0)), the reconstruction of the diagonal elements of matrix s(0.0l,0) from C2(s(0.0l,0)) is a trivial task, whereas the... 

The global and local shape codes can be used for measuring global and local shape compexity, respectively. Let w(s(a,b,M)) and w(lli)(a,b,M)) denote the number of different entries of the n-dimensional global shape matrix s(a,b,M) and a k-dimensional local shape matrix lb(a,b,M), respectively. Simple global and local shape complexity measures of molecule M are defined as the following ratios ... 

The values of the Betti numbers at the grid points (a,b), or at the points [log(u), log 6 ] of the logarithmic map, form a matrix, M1" 1 1. In either of the direct or the logarithmic representations, this matrix is a numerical shape code for the fuzzy electronic density cloud of the molecule, representing the actual molecular shape. 

Consider, for example, two molecules, A and B, both in some fixed nuclear configuration, and calculate their shape codes in their matrix forms and... 

Michael Friendly s Corrgram (Friendly, 2002) uses a color- and shape-coded scatterplot matrix display (Tukey and Tukey, 1981) to show correlations between variables. Variables are permuted so that correlated variables are positioned adja-centiy. Guo (2003) and Guo et al. (2003) also evaluated all possible axis-parallel 2D projections according to the maximum conditional entropy to identify ones that are most useful to find clusters. They visualized the entropy values in a matrix display called the entropy matrix (MacEachren et al., 2003) that is also a color-coded scatterplot matrix. 

Time domain signal coding (TDSC) based on zero-crossings and wave shape. S-matrix is a one-dimensional matrix of the frequency of occurrence of waveshapes. A-matrix is a two-dimensional matrix of the frequency of occurrence of pairs of waveshapes (Chesmore, 2001 Chesmore and Ohya, 2004). Computationally simple to implement and can be implemented in real-time at well over 1 MHz. 

The minimum paths provide a molecular shape description, coded as a two- vay matrix table. The matrix may then be unfolded into a single array, which describes the molecular shape, as illustrated in Fig. 5.3. 

In practice, only a finite number of parameter pairs (a,b) are considered, for example, those at the grid points of the 41 x 21 grid of the parameter map (a,b) described above. The entire map of shape matrix codes can then be represented by three 861-dimensional vectors, C( ), C(2), and C(3), containing all first, all second, and all third components, respectively, of the individual c(s(a,b)) vectors. Alternatively, a single (3 x 861) - dimensional vector C can be assigned to the (a,b) parameter map, where C is obtained by concatenating the components of C(l), C(2), and C(3) into a single vector. 

The problems experienced with similarity queries with low chemical specificity identify a need for 2-D attribute sets which consider larger connectivity pattern units of the molecular graph. Indeed, they should describe the skeletal graph (i.e., a graph devoid of chemical embellishment) as a complete pattern unit. In 3-D, the molecular shape is completely defined by the full (A) distance matrix (e.g., Crippen ) involving all atom pairs in the structure, whether they are bonded or not. In 2-D, a similar description can be expressed (e.g., Bersohn ) by considering the minimum number of bonds that must be traversed to reach atom j from atom i. A number of authors (e.g., Randic and Wilkins Carhart, Smith and Ven-kataraghavan ) have used these inter-nodal bond separations (INBSs) to derive atom codes for structure-activity studies. They would also seem to form a valid basis for 2-D similarity analyses. 

The advantage of the finite difference method is the simple computer implementation of the procedures and thus, it is easy to write ovra codes and to implement or consider new features. The drawback can be the consideration of the boundary conditions for complex shaped geometries and the consideration of the symmetry of the stiffiiess matrix might be difficult. Thus, many applications of the finite difference method are restricted to simple geometries. To overcome these problems, the so called finite difference energy method was developed (Bushnell et al. 1971) where the displacement derivatives in the total potential energy of a system are approximated by finite differences and the minimum condition of the potential energy is used to calculate the unknown displacements. 

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