近年來，由於科技的進步，音樂的取得越來越容易，音樂也變得越來越普遍化。在我們周圍，各種各樣的音樂變得更加複雜及大量。音樂的爆炸性增加，迫使我們需要新的技術和工具，可以智慧地並自動地將音樂轉換成有用的資訊。許多研究將音樂物件視為一個依時間排序連續離散的符號。重複樣式是指在音樂序列中經常出現的子序列，它通常可以用來表示一個音樂物件的主題，此外，重複樣式也可以使用在音樂分類上。許\\\多方法被提出來來尋找音樂物件中的重複樣式，例如M2P ( Mining Maximum-length Patterns )演算法，它建構一個有方向性的graph，並用深度優先搜尋的方法去搜尋此graph。它用字串比對演算法來決定一段路徑是否符合成為重複樣式，並找出音樂物件中最長的重複樣式。雖然M2P演算法在找出樣式上是一個直觀的方法，但它會產生大量的候選樣式並且花費大量時間在執行字串比對演算法。因此，在這論文中，我們提出了PJ ( Position Join )演算法來有效率地找出最長重複樣式。在建構graph階段，我們發現可以透過修改graph的資訊來避免使用字串比對演算法來決定一段路徑是否為重複樣式。我們把長度為二的重複樣式所出現的位置記錄在矩陣上。當搜尋graph時，利用所記錄的位置來計算路徑的頻率。此外，我們透過位置來記錄重複路徑，產生終端邊並且記錄曾經搜尋過的路徑的資訊。動態地透過終端邊來修改原有graph，可以避免在搜尋graph時重複搜尋某些路徑。根據模擬的結果，我們證明了我們提出的PJ演算法比M2P演算法更有效率。

In recent years, the music has become popular due to the evolution of the technology. Various kinds of music around us become complexities and huge. The explosive
growth in the music has generated the urgent need for new techniques and tools
that can intelligently and automatically transform the music into useful information.
Many researches consider the music object as an continuously discrete note in time
order. Repeating patterns are some subsequences which appear frequently in the
music sequence. The repeating patterns usually can represent the theme of a music
object. Moreover, it also can be utilized in music classification. Many methods have
been proposed for finding the repeating patterns in music objects, for example, the
M2P (Mining Maximum-length Patterns) method. It constructs a directed graph and
uses the depth-first search to traverse the graph. It calculates the paths by the string
matching algorithm to decide whether they are repeating pattern, and finds out the
maximum-length repeating pattern in a music sequence. Although the M2P method
is a straightforward method to find out the patterns, it consumes time in creating too
many candidate patterns and performing the string matching algorithm. Therefore,
in this thesis, we propose the PJ (Position-Join) method to efficiently find out the
maximum-length repeating pattern. In the constructing graph step, we find out that
we can modify the information in the graph, and avoid to use the string matching
algorithm to decide whether a path is repeating pattern. We record the positions of
length two repeating patterns in the matrix. While traversing the graph, we calculate the frequency by the information of positions. Moreover, we record the repeated
path by the positions. We create terminal edges, and record the information of paths
which have been traversed. We dynamically modify the graph by terminal edges. It
can avoid to traverse some paths repeatedly in traversing the graph step. From our
performance study based on the synthetic data and real music data, we show that
our proposed PJ method is more efficient than the M2P method.

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Basic Music Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 The Pitch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Temporal Features . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Music Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 The Repeating Patterns . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.1 Non-Trivial Repeating Patterns . . . . . . . . . . . . . . . . . 7
1.3.2 Polyphonic Repeating Patterns . . . . . . . . . . . . . . . . . 8
1.3.3 Approximate Repeating Patterns . . . . . . . . . . . . . . . . 9
1.3.4 Maximum-Length Repeating Patterns . . . . . . . . . . . . . . 9
1.4 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . 12
2. A Survey of Mining Repeating Patterns . . . . . . . . . . . . . . . . 13
2.1 Mining Non-Trivial Repeating Patterns . . . . . . . . . . . . . . . . . 13
2.1.1 The Correlative Matrix Method . . . . . . . . . . . . . . . . . 13
2.1.2 The String-Join Method . . . . . . . . . . . . . . . . . . . . . 15
2.1.3 The TRP Method . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 The T-PRPD with Bit-String Method . . . . . . . . . . . . . . . . . . 16
2.3 Ning-Han Liu et al.’s Method . . . . . . . . . . . . . . . . . . . . . . 19
2.4 The M2P Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3. The Position-Join Method . . . . . . . . . . . . . . . . . . . . . . . . 25
3.1 Notations and Definition . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 The Proposed Position-Join Method . . . . . . . . . . . . . . . . . . . 26
3.2.1 Constructing the Graph . . . . . . . . . . . . . . . . . . . . . 27
3.2.2 The P-Join Method . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2.3 Traversing the Graph . . . . . . . . . . . . . . . . . . . . . . . 37
3.2.4 The Terminal Edge . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3 A Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4. Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.1 Generation of Experimental Data . . . . . . . . . . . . . . . . . . . . 50
4.2 Synthetic Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.3 Real Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

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