Spatial representation of symbolic sequences through iterative function systems

Jeffrey proposed (1990) a graphic representation of DNA sequences using Barnsley's iterative function systems. In spite of further developments in this direction, the proposed graphic representation of DNA sequences has been lacking a rigorous connection between its spatial scaling characteristics and the statistical characteristics of the DNA sequences themselves. We 1) generalize Jeffrey's graphic representation to accommodate (possibly infinite) sequences over an arbitrary finite number of symbols; 2) establish a direct correspondence between the statistical characterization of symbolic sequences via Renyi entropy spectra (1959) and the multifractal characteristics (Renyi generalized dimensions) of the sequences' spatial representations; 3) show that for general symbolic dynamical systems, the multifractal f/sub H/-spectra in the sequence space coincide with the f/sub H/-spectra on spatial sequence representations.

[1]  Aleksandr Yakovlevich Khinchin,et al.  Mathematical foundations of information theory , 1959 .

[2]  Ramón Román-Roldán,et al.  Entropic feature for sequence pattern through iterated function systems , 1994, Pattern Recognit. Lett..

[3]  Rudolf H. Riedi,et al.  Conditional and Relative Multifractal Spectra , 1997 .

[4]  Ramón Román-Roldán,et al.  Application of information theory to DNA sequence analysis: A review , 1996, Pattern Recognit..

[5]  A. Rényi.  On the dimension and entropy of probability distributions , 1959 .

[6]  L. Barreira,et al.  On a general concept of multifractality: Multifractal spectra for dimensions, entropies, and Lyapunov exponents. Multifractal rigidity. , 1997, Chaos.

[7]  S. Basu,et al.  Chaos game representation of proteins. , 1997, Journal of molecular graphics & modelling.

[8]  H. J. Jeffrey.  Chaos game representation of gene structure. , 1990, Nucleic acids research.

[9]  Kenneth Falconer,et al.  Fractal Geometry: Mathematical Foundations and Applications , 1990 .

[10]  Meir Feder,et al.  A universal finite memory source , 1995, IEEE Trans. Inf. Theory.

[11]  P. A. P. Moran,et al.  Additive functions of intervals and Hausdorff measure , 1946, Mathematical Proceedings of the Cambridge Philosophical Society.

[12]  Y. Peres,et al.  Measures of full dimension on affine-invariant sets , 1996, Ergodic Theory and Dynamical Systems.

[13]  A. Fiser,et al.  Chaos game representation of protein structures. , 1994, Journal of molecular graphics.

[14]  Dana Ron,et al.  The Power of Amnesia , 1993, NIPS.

[15]  C. Beck,et al.  Thermodynamics of chaotic systems , 1993 .

[16]  Karel Culik,et al.  Affine automata and related techniques for generation of complex images , 1993, Theor. Comput. Sci..

[17]  Michael F. Barnsley,et al.  Fractals everywhere , 1988 .

[18]  J. Oliver,et al.  Entropic profiles of DNA sequences through chaos-game-derived images. , 1993, Journal of theoretical biology.

[19]  J. Rogers.  Chaos , 1876 .

[20]  Skolnick,et al.  Global fractal dimension of human DNA sequences treated as pseudorandom walks. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[21]  Wentian Li,et al.  The Study of Correlation Structures of DNA Sequences: A Critical Review , 1997, Comput. Chem..

[22]  H. Weiss,et al.  On the dimension of deterministic and random Cantor-like sets, symbolic dynamics, and the Eckmann-Ruelle Conjecture , 1996 .

[23]  Peter Tiño,et al.  Extracting finite-state representations from recurrent neural networks trained on chaotic symbolic sequences , 1999, IEEE Trans. Neural Networks.

[24]  K. Csiszȧr,et al.  An intron capture strategy used to identify and map a lysyl oxidase-like gene on chromosome 9 in the mouse. , 1997, Genomics.

[25]  Peter Grassberger,et al.  Information and Complexity Measures in Dynamical Systems , 1991 .

[26]  Karel Culik,et al.  Rational and Affine Expressions for Image Description , 1993, Discret. Appl. Math..

[27]  Ludwig Staiger SummaryWe.  Quadtrees and the Hausdorr Dimension of Pictures , 1989 .

[28]  H. J. Jeffrey.  Chaos game representation of gene structure. , 1990, Nucleic acids research.

[29]  R. Mantegna,et al.  Statistical mechanics in biology: how ubiquitous are long-range correlations? , 1994, Physica A.

[30]  P. Tiňo,et al.  Constructing finite-context sources from fractal representations of symbolic sequences , 1998 .

[31]  H. Weiss,et al.  A multifractal analysis of equilibrium measures for conformal expanding maps and Moran-like geometric constructions , 1997 .

[32]  James P. Crutchfield,et al.  Computation at the Onset of Chaos , 1991 .

[33]  K A Hill,et al.  The evolution of species-type specificity in the global DNA sequence organization of mitochondrial genomes. , 1997, Genome.

[34]  J. McCauley.  Chaos, dynamics, and fractals : an algorithmic approach to deterministic chaos , 1993 .