MEASURES AND THEIR DIMENSION SPECTRUMS FOR COOKIE-CUTTER SETS IN R~d

0. IntroductionMany theoretical physicists and mathematicians have studied the Hausdorff dimensions,measures and multifractal decompositions of fractals and obtained a lot of satisfactory results. Particularlyl there hajs given thorough and detailed study for self-similar fractals(of. e.g. [1--7]). mom the dynamical system point of view self-similar sets are regarded asthe attractors of iterated function systems consisting of self-similar cofltraction mappings.However, the researches for measu…     

Multifractal analysis of complex networks

Complex networks have recently attracted much attention in diverse areas of science and technology.Many networks such as the WWW and biological networks are known to display spatial heterogeneity which can be characterized by their fractal dimensions.Multifractal analysis is a useful way to systematically describe the spatial heterogeneity of both theoretical and experimental fractal patterns.In this paper,we introduce a new box-covering algorithm for multifractal analysis of complex networks.This algorithm is used to calculate the generalized fractal dimensions D q of some theoretical networks,namely scale-free networks,small world networks,and random networks,and one kind of real network,namely protein-protein interaction networks of different species.Our numerical results indicate the existence of multifractality in scale-free networks and protein-protein interaction networks,while the multifractal behavior is not clear-cut for small world networks and random networks.The possible variation of D q due to changes in the parametersof the theoretical network models is also discussed.     

Fractals in DNA sequence analysis

Fractal methods have been successfully used to study many problems in physics,mathematics,engineering,finance,and even in biology,There has been an increasing interest in unravelling the mysteries of DNA;for example,how can we distinguish coding and noncoding sequences,and the problems of classification and evolution relationship of organisms are key problems in bioinformatics,Although much research has been carried out by taking into consideration the long-range correlations in DNA sequences,and the global fractal dimension has been used in these works by other people,the models and methods are somewhat rough and the results are not satisfactory.In recent years,our group has introduced a time series model(statistical point of view)and a visual representation (geometrical point of view) to DNA sequence analysis.We have also used fractal dimension,correlation dimension,the Hurst exponent and the dimension spectrum (multifractal analysis)to discuss problems in this field.In this paper,we introduce these fractal models and methods and the results of DNA sequence analysis.     

Relationships of exponents in multifractal detrended fluctuation analysis and conventional multifractal analysis

Multifractal detrended fluctuation analysis (MF-DFA) is a relatively new method of multifractal analysis. It is extended from detrended fluctuation analysis (DFA), which was developed for detecting the long-range correlation and the fractal properties in stationary and non-stationary time series. Although MF-DFA has become a widely used method, some relationships among the exponents established in the original paper seem to be incorrect under the general situation. In this paper, we theoretically and experimentally demonstrate the invalidity of the expression τ(q)=qh(q)-1 stipulating the relationship between the multifractal exponent τ(q) and the generalized Hurst exponent h(q). As a replacement, a general relationship is established on the basis of the universal multifractal formalism for the stationary series as τ(q)=qh(q)-qH'-1, where H' is the nonconservation parameter in the universal multifractal formalism. The singular spectra, α and f(α), are also derived according to this new relationship.     

Protein structural classification and family identification by multifractal analysis and wavelet spectrum

Family identification is helpful for predicting protein functions. It has been known from the literature that longer sequences of base pairs or amino acids are required to study patterns in biological sequences. Since most protein sequences are relatively short, we randomly concatenate or link the protein sequences from the same family or superfamily together to form longer protein sequences. The 6-letter model, 12-letter model, 20-letter model, the revised Schneider and Wrede scale hydrophobicity, solvent accessibility and stochastic standard state accessibility are used to convert linked protein sequences into numerical sequences. Then multifractal analyses and wavelet analysis are performed on these numerical sequences. The parameters from these analyses can be used to construct parameter spaces where each linked protein is represented by a point. The four classes of proteins, namely the α,β, α+βand α /β classes, are then distinguished in these parameter spaces. The Fisher linear discriminant algorithm is used to assess the discriminant accuracy. Numerical results indicate that the discriminant accuracies are satisfactory in separating these classes. We find that the linked proteins from the same family or superfamily tend to group together and can be separated from other linked proteins. The methods are helpful for identifying the family of an unknown protein.     

Chaos game representation of functional protein sequences, and simulation and multifractal analysis of induced measures

Investigating the biological function of proteins is a key aspect of protein studies. Bioinformatic methods become important for studying the biological function of proteins. In this paper, we first give the chaos game representation (CGR) of randomly-linked functional protein sequences, then propose the use of the recurrent iterated function systems (RIFS) in fractal theory to simulate the measure based on their chaos game representations. This method helps to extract some features of functional protein sequences, and furthermore the biological functions of these proteins. Then multifractal analysis of the measures based on the CGRs of randomly-linked functional protein sequences are performed. We find that the CGRs have clear fractal patterns. The numerical results show that the RIFS can simulate the measure based on the CGR very well. The relative standard error and the estimated probability matrix in the RIFS do not depend on the order to link the functional protein sequences. The estimated probability matrices in the RIFS with different biological functions are evidently different. Hence the estimated probability matrices in the RIFS can be used to characterise the difference among linked functional protein sequences with different biological functions. From the values of the D_q curves, one sees that these functional protein sequences are not completely random. The D_q of all linked functional proteins studied are multifractal-like and sufficiently smooth for the C_q (analogous to specific heat) curves to be meaningful. Furthermore, the D_q curves of the measure \mu based on their CGRs for different orders to link the functional protein sequences are almost identical if q\geq 0. Finally, the C_q curves of all linked functional proteins resemble a classical phase transition at a critical point.     

Predicting the subcellular location of apoptosis proteins based on recurrence quantification analysis and the Hilbertben Huang transform

Apoptosis proteins play an important role in the development and homeostasis of an organism. The elucidation of the subcellular locations and functions of these proteins is helpful for understanding the mechanism of programmed cell death. In this paper, the recurrent quantification analysis, Hilbert-Huang transform methods, the maximum relevance and minimum redundancy method and support vector machine are used to predict the subcellular location of apoptosis proteins. The validation of the jackknife test suggests that the proposed method can improve the prediction accuracy of the subcellular location of apoptosis proteins and its application may be promising in other fields.