Sequence hypercube

Uncertainty - analyzes the uncertainty of a system, sequence, or end state using either the Monte Carlo or Latin Hypercube simulation technique. 

Latin hypercube Hammersley sequence sampling (LHSS), 26 1013-1015 Latin hypercube sampling (LHS), 26 1005, 1007-1011, 1012, 1013, 1014, 1015 future trends in, 26 1047 in process synthesis and design,... 

Sequences. See also Hammersley sequence sampling (HSS) Latin hypercube entries Sobol sequence quasi-Monte Carlo, 26 1011, 1013 quasirandom, 26 1016, 1036, 1048 Sequence tagged sites (STS), 12 513, 515 Sequencing batch reactor (SBR)... 

We can also consider the number of boundaries for objects in different dimensions. A line segment has two boundary points. A square is bounded by four line segments. A cube is bounded by six squares. Following this trend, we would expect a hypercube to be bounded by eight cubes. This sequence follows an arithmetic progression (2, 4, 6, 8. ..). 

Figure 9. Sequence spaces of biopolymers and hypercubes. Genotypes may be ordered according to their Hamming distances. The sequence spaces of binary...

Figure 1. Lexicographic ordering of sequences through successive duplications of sequence space. As shown, for binary sequences the sequence space of dimension n can be constructed by duplication of sequence space of dimension n — 1, This iterative procedure is used in Appendix 2 to construct mutation matrix in such a way that eigenvalues and eigenvectors can be computed easily [8]. Each of 2 points specifies binary (R, Y) sequence. If, in addition, two alternative base classes (R = G or A, F = C or U) are specified, then to each of points in binary sequence space another subspace of binary specification is added, yielding total of 4 points or dimension of hypercube of 2v.

Figure 21. Mutant space high-value contour near local optimum. Diagram is multiply branched tree with different macromolecular sequences at vertices. Each line joins neighboring sequences whose values are within 0.5 of locally optimum sequence at lower center for linearized fitness function of type 2 [Eqn. (IV.7)] and reference fold that is cruciform, like tRNA, for sequence of length 72. Over 1300 branches shown extending up to 10 mutant shells away from central optimum. Better sequence (labeled optimum) was found in tenth mutant shell. Non-random sampling of mutant sequences demonstrated typical of population sampling in quasispecies model. Note small number of ridges that penetrate deeply into surrounding mutant space. (Additional connected paths due to hypercube topology of mutant space not shown.)...

Stable foci occur when two or more variables switch in some sequence but approach zero as the number of switchings approaches infinity, while all other variables (if any) converge to some (possibly nonzero) value. Thus, stable foci typically have an associated cyclic sequence of orthants through which an approaching trajectory passes. A necessary condition for a stable focus is a cycle on the A-cube digraph. We say that a cycle has dimension A if A is the dimension of the smallest hypercube on which the given cycle can be drawn. 

We define uniformity in higher dimensions as follows. Suppose we define n-tuples JJ = (ui+j,. .. ui+B) and divide the n-dimensional unit hypercube into many equal subvolumes. A sequence is uniform if in the limit of an infinite sequence all the subvolumes have an equal number of occurrences of random -tuples. For a random sequence this will be true for all values of and all partitions into subvolumes, although in practice we test only for small values of n. 

Fig. 5.10 A comparison of samples produced by different sampling methods for a 2-parameter model (a) 1,024 random sampling points, (b) 1,024 Latin hypercube sampling points, (c) 1,024 points of the Halton sequence and (d) 1,024 points of the Sobol sequence. All sampling procedures are based on a uniform distribution in the domain [0, 1] x [0, 1]. Adapted from (Ziehn 2008)...

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