音樂及DNA序列之多重碎形分析 ; Multifractal Analyses of Music and Nucleotide Sequences in DNA
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自然界存在著釵h不規則和無條理的複雜景觀形物及現象,例如:雲和海岸線、瀑布的落水和飛沫、城市的噪音、股票價格的變動等,詳細研究會發現釵h現象中都具有自相似性,即在系統的整體與部分之間,這一部分與那一部分之間都具有相似性(Self Similarity),或稱之為碎形性(Fractal)的性質;音樂可以表達人們對自然界的感受,DNA序列為大自然一切生物生命現象與物種演化的統馭者,亦當不例外。儘管自然界的釵h體系表面上看來是極不規則且雜亂無章,但在不同的尺度給予觀察和分析時,往往可發現它們在尺度上存有不變性,為能尋找不受尺度影響且能適切表達自然現象的量化指標,Mandelbrot在七十年代提出的碎形理論(Fractal Theory)對解釋世上的這些繁雜、但又具有自相似性的現象提供了一個重要工具,此也構成了本文使用碎形理論來分析至解決問題的動機。在本文中吾人就嘗試以此工具針對音樂和DNA序列這二個主題來進行碎形方面的相關研究。 在音樂序列方面,吾人將五線譜上音符位置之起落視為一單變數之隨機行走(Random Walk),以一類似山岳高低起伏輪廓的曲線呈現音樂序列。藉由分數布朗運動(Fractional Brownian Motion)及傅立葉能量頻譜(Fourier Power Spectrum)之分析,得以探討音樂之碎形特性。本文之研究結果顯示音樂具備自然界中隨處可見之長程關聯性(Long-Range Correlation)及自我相似性(Self-Similarity),這或野i解釋音樂如何得以模擬大自然之諧和。此外,關於音樂之間的關聯性比較,除了採用統計上常用的計算線性相關係數的方式外,吾人還根據信息理論,計算兩音樂序列的互信息量,並以此區分出兩音樂之間非線性關聯性的強弱。檢視分析的結果可以發現,儘管兩序列的線性相關係數接近零,它們仍然可能有相當高程度的非線性關聯,因此當要判斷兩序列相關程度的時候,必須同時考慮到兩者的線性、非線性相關性。最後,本文採用多重碎形理論,以變換不同尺度觀察音樂序列局部性質的方式研究各種不同風格的樂曲旋律及節奏的自相似結構,藉此得到不同音樂風格在數學上的具體表示。由不同樂曲的多重碎形頻譜中 曲線確實呈現出一類似開口朝下拋物線輪廓的趨勢可以看出,音樂中存在著多重碎形結構。除此之外,由這些曲線的外型、開口大小也可以區分不同的音樂風格。 在DNA序列方面,吾人從碎形理論出發,針對Myosin Heavy Chain基因家族裡屬於各個不同等級生物體的一系列DNA序列,以多重碎形頻譜分析法(Multifractal Spectral Analysis)來探討其鹼基排列分布所顯現之幾何特性。研究的結果除了顯示出DNA序列與音樂序列相同皆呈現出多重碎形(Multifractal)之特性外,亦證實了鹼基的排列分布型態與物種演化層次有相當程度的關聯。更重要的是,本文中發現了DNA長鏈中帶遺傳訊息片段(Exon)與不帶遺傳訊息區域(Intron)兩者之間在局部碎形尺度比例指數(Hölder Exponent)分布上的差異,並用這樣的差異來辨認DNA序列中會譯製成蛋白質的片段。在本文中除了會說明造成此現象生物上的解釋,還測試了幾個非Myosin Heavy Chain基因的DNA序列並比較其計算結果與已知的Exon片段,證實用局部碎形尺度比例指數來辨認DNA序列中會譯製成蛋白質的片段的確具有簡單、所需計算時間短以及具備尺度不變性等多項優點,可提供生物資訊(Bio-Informatics)上另一個分析DNA序列的有效方法。 音樂是由音符或者可以說是符號所組成,由幾個簡單的音符按照作曲的意願可以組成優美動聽的樂曲,同樣是這幾個音符如果隨機排列,一定是令人難以忍受的噪音,那麼音樂的奧秘究意在哪裡?這是自中世紀,從Aristotle學派開始,一直是人們努力探求的方向。同樣地,生命體中DNA分子的鹼基序列記載著生長發育的全部指令,然而描繪出它的語言卻是異常地簡單,僅由四種字母組成,生命信息也是通過這些字母的線性排列來表達的,過去人們已做了大量的研究試圖揭示其中的有序結構和它產生的模式,但還有很多現象並不是很清楚,比如是生物在演化過程當中,其DNA序列之碎形維度分布頻譜(Spectrum of Fractal Dimension)等。近年來隨著碎形理論的發展,為處理不規則符號序列提供了一種新的有效的方法,相信本文的結果應能提供上述問題答案的一些線索及一個全新的思考方向。 ; 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 ...
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音樂及DNA序列之多重碎形分析 ; Multifractal Analyses of Music and Nucleotide Sequences in DNA
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Autor/in / Beteiligte Person: | 蘇致遠 ; Su, Zhi-Yuan ; 伍次寅 ; 臺灣大學:機械工程學研究所 |
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Veröffentlichung: | 2004 |
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