The Model Of Random Quotient Fractal And Its Application Of Protein Analysis

Fractal theory is a very active mathematic branch of modern nonlinear science, which has been applied to physics, geology, material science and engineering technology. The idea and method of fractal has made great achievements in many kinds of science fields, such as pattern recognition, simulation of nature image, manage of information and facture of art. Protein research based on fractal is a new method. In recent years, because GeneBank and PDB have grown very rapidly, bioinformatics has been born to analyze these enormous data and developed very rapidly. To manage the data, people have ever tried to analyze protein based on fractal dimensions, which only shows the total complex tread of protein fold.As a method of problem solving, which based on substantial theory, considering the problem from different aspects and multi-hierarchy in the process of problem solving, quotient space granularity theory is a kind of powerful tool in which it can decrease the difficulty of the problem and reduce the computational cost. Following with modern probability theory, random fractal which syncretizes probability theory, classical analysis, geometry and fractal have developed very rapidly. Random quotient fractal theory is a method that connects quotient space theory to random fractal, and the granularity analysis idea of quotient space is used to research on fractal phenomenon. This paper studies deeply the property of random iterated function systems, connected with quotient space granularity theory, and con tructs a random quotient fractal model. Further discuss the property and application of the model, main works and results include:1,Show respectively the relations of random iterated function system(random IFS), recurrent iterated function system(RIFS), vector recurrent iterated function system(VRIFS) and quotient space, and construct the random fractal model.In the quotient fractal space model, for common iterated function system it is proved that its quotient spaces have their each unique attractors based on the definition of fractal mapping. Besides the orbit of its quotient space is periodic, which indicates when we study the dynamics property on the side of quotient spaces we may. have some facile results.For random IFS, imitating the method of common IFS, it corresponds with a quotient chain, which forms a hierarchical structure. Using the measure property of the original space it shows that the quotient space has its self-similar measure, and the support of the measure is the unique attractor of quotient space. At the same time the ergodicity of the quotient space is proved using the ergodicity of the original space.For RIFS and VRIFS, first the graduation is decided by the fractal mappings followed with the transferring probabilities, and then the quotient chain can be formed. Next the distance is induced on the quotient chain, and the distance space is proved to be complete and compact. As a result, a hierarchical structure is built.The idea of three models comprises randomicity and hierarchy, so they are generally called random quotient fractal model. The model both embodies the granularity ideas and has the character of fractal geometry. Because of inducing measure analysis, aiming at some complicated problems we can consider the approaching demand of the quotient space of multi-granularity, hierarchy and different precisions, and then find the approximate solving and some relative dynamics properties. Moreover, the complexity of problem solving can be reduced by a long way based on the granularity idea of the quotient space. This is the reason that the model builds.2,Apply random quotient space to analyze protein sequences. It shows that fractal dimension is the corporate character of the fractal object in the different granularity. The process of calculating fractal dimension is the process of dividing the object. Multifractal is the common character of all the quotient spaces which belong to different granularity or the same granularity but different aspects. It denotes that these fractal methods can be used in the feature expression of protein sequences. The following experiments are designed: No 1. Put forward rescaled range analysis and computed Hurst exponents of proteins. It shows the validity of ANN, Markov Chain, and Bayes Statistics. No 2. The self-similar measure of random IFS composed by twenty compressed mappings is designed to simulate a real measure of protein sequence. It finds the fourth probability P₄ of twenty compressed mappings of all of 120 proteins is maximal, which may be good at denoting protein sequences. No 3. Compare distance sequences of protein with different spacing. The experiment shows the distance sequence that the spacing distance is four is available to be complicated by multifractal. No 4. Construct an index S, and it denotes multifractal can reflect the complexity of protein fold, when the fractal dimensions are similar. No 5. When the feature sequences of some physics and chemistry properties are analyzed by multifractal, it is indicated that relative probability sets can better reflect the multifractal property. No 6. Give the range of the weighted gene q of multifractal.3,Predict structure class of protein based on multifractal parameters of six kinds of physics and chemistry properties, Hurst exponent and P₄. Respectively apply PNN, SVM, COVER and two test methods to predict the data sets. The experiments show that our method is consistent with other reported results for the high homologous data set, while it obviously excels to other reported results for the low homologous data set.