Multifractal Analyses of Music and Nucleotide Sequences in DNA
Date Issued
2004
Date
2004
Author(s)
Su, Zhi-Yuan
DOI
zh-TW
Abstract
Many important spatial patterns and physical phenomena of nature are either irregular or fragmented to such an extreme degree that shapes such as coastlines, mountains and clouds are not easily described by traditional Euclidean geometry. Nevertheless, they often possess a remarkable simplifying invariance under changes of magnification. The statistical self-similarity is the essential quality of fractals in nature. They all have fractal – or, more generally, scaling – properties. If music is imitating the characteristic way our world changes in time as Greek philosophers generally agreed and DNA is considered one of the most sophisticated masterpieces ever created by Great Nature, then without exception it would also show the ubiquity of fractal character in its structure. Mandelbrot’s fractal geometry has blossomed tremendously in the past 20 years and has helped provide both a description and a mathematical model for many of seemingly complex forms found in nature. With such a focus, by utilizing fractal theory, this study attempts to analyze music and DNA sequences.
About music, by converting the sequence of musical notes on the staffs into a one-variable random walk (music walk), such a graph may resemble a mountain landscape, with jagged ridges of all length scales from very bumps to enormous peaks. Quantitative measures of the correlation and fractal properties of music are performed using the fractional Brownian motion (FBM) and Fourier power spectrum analyses. The results show that music exhibits the ubiquity of a long-range correlation (fractal) over decades of notes similar to those found in many naturally occurring fluctuating phenomena. This underlying structure might explain why music sounds pleasing and how music imitates the harmony of nature. Furthermore, the quantities that can measure the correlation between two music scores are the correlation coefficient commonly used in the statistics and mutual information firstly applied in communication system. The difference is that correlation coefficient evaluates linear correlation, but mutual information evaluates non-linear correaltion. The results of analyses show me that two music scores could have close relations, even though the correlation coefficient is nearly equal to zero. For this reason, when we need to determine correlation degree of two sequences, we should think about linear and non-linear correlations at the same time. Finally, we use a multifractal methodology to analyze the melody and rhythm sequences of music with different styles. The multifractal spectrums ( – curve) of music scores are calculated. By seeing the familiar, inverted, downward-opening parabola shape, it implies that the melody and rhythm sequences of music are multifractals. Moreover, we can also distinguish the styles of different music by observing the shape and opening of the parabola.
About DNA, the same as music, we have demonstrated the multifractal aspects of the base sequences in DNA chains by using the multifractal spectrum method. Specific analysis of the Myosin Heavy Chain type II gene family belonging to several different species has indicated correlation between fractal properties and evolutionary order. In particular, the study has also shown a difference in the local scaling exponent (Hölder exponent) for coding (exon) and non-coding (intron) segments of DNA. In the dissertation, we not only explore possible biological interpretations for such difference, but also propose that it might be served as an effective tool for identifying coding regions in a gene of unknown function by analyzing other DNA sequences excluding Myosin Heavy Chain gene and comparing the calculation results with known exon locations recorded in the GenBank. It is corroborated that a simple method for locating protein-coding regions in a long DNA strand can be devised. The present study can provide a different way, other than the usual molecular biology approach, of analyzing DNA sequences.
Notes (Maybe you could think that they are just symbols) compose music. Composer can compose pleasing music by using several notes, but the same notes played by a monkey are merely annoyed noises. What is the secret of music? This is the issue that humans want to know since the Middle Ages. Similarly, DNA is a long double helical chain composed of a large number of nucleotides, each carrying one of four bases conventionally symbolized by the four letters: A, T, G and C. The sequential order of these four bases along the DNA chain encodes important genetic information concerning instructions of critical life activities and inheritable features of a living organism. Although, in the past, many people have attempted to realize the sequential structure and how it formulates by doing a lot of researches, still we don’t know the answers, such as the relations between biological evolution and DNA sequence’s spectrum of fractal dimension, etc. Recently, fractal theory has become central tools in analyzing irregular symbolic sequences. It is hoped that the results of this dissertation not only give a part of answers, they also provide different ideas.
Subjects
DNA序列
蛋白質編碼區域預測
外顯子
多重碎形頻譜
相關性分析
頻譜分析
多重碎形
lder指數
碎形
Hö
局部碎形分布尺度比例指數
長程關聯性
互信息
音樂
Hurst指數
碎形布朗運動
Hurst Exponent
Mutual Information
DNA
Nucleotide Sequence
Local Scaling Exponent
FBM
Exon
Correlation Analysis
Protein Coding Region Prediction
Music
Spectral Analysis
Multifractal
Long-Range Correlation
lder Exponent
Multifractal Spectrum
Fractal
Fractional Brownian Motion
SDGs
Type
thesis
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