Online Mining Maximal Frequent Structures in Continuous Landmark Melody Streams
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(2) •••. User Query Streams. User Query Processor. Music ID. Melody Sequence. Buffer. H.-F. Li et al. / Pattern Recognition Letters 26 (2005) 1658–1674. •••. Melody (Sequence) Streams. Music Database. 1659. Melody Stream Processor Maximal Melody Structure Streams. Summary Data Structure in Main Memory. Fig. 1. Computation model for music melody streams.. stream model, knowledge discovery has two major characteristics (Babcock et al., 2002). First, the volume of a continuous stream over its lifetime could be huge and fast changing. Second, the continuous queries (not just one-shot queries) require timely answers, and the response time is short. Hence, it is not possible to store all the data in main memory or even in secondary storage. This motivates the design of in-memory summary data structure with small memory footprints that can support both one-time and continuous queries. In other words, data stream mining algorithms have to sacrifice the exactness of its analysis result by allowing some counting error. Although several techniques have been developed recently for discovering and analyzing the content of static music data (Bakhmutora et al., 1997; Hsu et al., 2001; Shan and Kuo, 2003; Yoshitaka and Ichikawa, 1999; Zhu et al., 2001), new techniques are needed to analyze and discover the content of streaming music data. Thus, this paper studies a new problem of how to discover the maximal melody structures in a continuous unbounded melody stream. The problem comes from the context of online music-downloading services (such as Kuro at www.music.com.tw), where the streams in question are streams of queries, i.e., music-downloading requests, sent to the server, and we are interested in finding the maximal melody structures requested by most customers during some period of time. With the computation model of music melody streams presented in Fig. 1, the melody stream processor and the summary data structure are two major components in the. music melody streaming environment. The user query processor receives user queries in the form of hTimestamp, Customer-ID, Music-IDi, and then transforms the queries into music data (i.e., melody sequences) in the form of hTimestamp, CustomerID, Music-ID, Melody-Sequencei by retrieving the music database. Note that a buffer can be optionally set for temporary storage of recent music melodies from the music melody streams. In this paper, we present a novel algorithm MMSLMS (Maximal Melody Structures of Landmark Melody Streams) for mining the set of all maximal melody structures in a landmark melody stream. Moreover, the music melody data and patterns are represented as sets of chord-sets (Type I Melody structures) or strings of chord-sets (Type II Melody structures). While providing a general framework of music stream mining, algorithm MMSLMS has two major features, namely one scan of music melody streams for online frequency collection, and prefix-tree-based compact pattern representation. With these two important features, MMSLMS is provided with the capability to work continuously in the unbounded streams for an arbitrary long time with bounded resources, and to quickly answer users queries at any time.. 2. Preliminaries 2.1. Music terminologies In this section, we describe several features of music data used in this paper. For the basic.
(3) 1660. H.-F. Li et al. / Pattern Recognition Letters 26 (2005) 1658–1674. terminologies on music, we refer to (Jones, 1974). A chord is a sounding combination of three or more notes at the same time. A note is a single symbol on a musical score, indicating the pitch and duration of what is to be sung and played. A chord-set is a set of chords (Shan and Kuo, 2003). Definition 1. The type I melody structure is represented as a set of chord-sets. The type II melody structure is represented as a string of chordsets. 2.2. Problem statement Let W = {i1,i2, . . . , in} be a set of chord-sets, called items for simplicity. A melody sequence S with m chord-sets is denoted by S = hx1x2 xmi, where xi 2 W, "i = 1,2, . . . , m. A block is a set of melody sequences. Definition 2. A landmark melody stream LMS = [B1,B2, . . . , BN), is an infinite sequence of blocks, where each block Bi is associated with a block identifier i, and N is the identifier of the ‘‘latest’’ block BN. The current length of LMS, written as jLMSj, is N. The blocks arrive in some order (implicitly by arrival time or explicitly by timestamp), and may be seen only once. Definition 3. A set Y W is called an item-set, i.e., a set of chord-sets. k-item-set is represented by (y1y2 yk). The support of an item-set Y, denoted as r(Y), is the number of melody sequences containing Y as a subset in the LMS seen so far. An item-set is frequent if its support is greater than or equal to minsup Æ jLMSj, where minsup is a user-specified minimum support threshold in the range of [0, 1], and jLMSj is the current length of the landmark melody stream LMS. Definition 4. A string Z is called an item-string, i.e., a string of chord-sets. A k-item-string is represented by hz1z2 zki, where zi 2 W, "i = 1,2, . . ., k. The support of an item-string Z, denoted as r(Z), is the number of melody sequences containing Z as a substring in the LMS seen so far. An item-string is frequent if its support is greater than or equal to minsup Æ jLMSj,. where minsup is a user-specified minimum support threshold in the range of [0, 1], and jLMSj is the current length of the landmark melody stream seen so far. Definition 5. A frequent item-set (or item-string) is called maximal if it is not a subset (or substring) of any other frequent item-set (or itemstring). In fact, the total number of maximal melody structures is smaller than that of frequent melody structure. Hence, the type of maximal melody structures is more suitable for the performance requirements of music stream mining. Definition 6. (Problem Definition of Online Mining Maximal Melody Structures in Continuous Landmark Melody Streams.) Given a landmark melody stream LMS = [B1,B2, . . . , BN) and the user specified minimum support, minsup, in the range of [0, 1], the problem of online mining maximal melody substructures is to discover the set of all maximal melody structures, i.e., maximal item-sets or maximal item-strings, in single one scan of the landmark music stream. 2.3. Main performance requirements of music melody stream mining The main performance challenges of mining melody streams are: (1) Online, one-pass algorithm: each sequence in the landmark melody stream is examined once. (2) Bounded-storage: limited memory for storing crucial, compressed information in summary data structure. (3) Real-time: per item processing time must be low. The proposed MMSLMS algorithm possesses all of these characteristics, while none of previously published methods (Bakhmutora et al., 1997; Hsu et al., 2001; Shan and Kuo, 2003; Yoshitaka and Ichikawa, 1999; Zhu et al., 2001) can claim the same..
(4) H.-F. Li et al. / Pattern Recognition Letters 26 (2005) 1658–1674. 3. Online mining maximal frequent structures in landmark melody streams 3.1. Chord-set memory border In this section, the upper bound on the number of candidate maximal melody structures is discussed, and an efficient algorithm for chord-set memory border construction is proposed. Theorem 1. Given a set of k frequent chord-sets from a landmark melody stream, an upper bound of the amount of maximal frequent melody structures is C kdk=2e . Proof. Assume that there are k frequent chordsets, i.e., k frequent items, in the current landmark melody stream. The solution space of mining all frequent item-sets in the worst case is C k1 þ C k2 þ þ C ki þ þ C kdk=2e þ þ C kk , where C k1 is the total number of frequent 1-item-sets, C ki is that of frequent i-item-sets, and C kk is that of frequent k-item-sets. We observe that the value of C kdk=2e is the maximum value among all the binominal coefficient C ki ; 8i ¼ 1; 2; . . . ; k, in mining all frequent i-item-sets. In other words, the number of frequent dk/2e-item-sets is a maximum. We will prove the number of maximal frequent itemsets can not be greater than the value C kdk=2e , i.e., C kdk=2e is the upper bound. We prove it by contradiction. Assume that the value of C kdk=2e is not the maximum number of maximal frequent item-sets, i.e., a larger upper bound U exists, where U > C kdk=2e . Consider that there are one or more frequent melody structures with length L, where L > dk/2e. If F is a frequent melody structure with length dk/2e + i and it is maximal, where i = 1,2, . . ., k dk/2e, then all of the substructures of F are frequent, which is based on the antimonotone Apriori heuristic (Agrawal and Srikant, 1994): if any i-item-set (or i-item-string) is not frequent, its (i + 1)-item-set (or (i + 1)-item-string) can never be frequent, but not maximal, which is based on the definition 5: a frequent item-set (or item-string) is called maximal if it is not a subset (or substring) of any other frequent item-set (or itemstring). In other words, it means that when one. 1661. maximal frequent structure with length L, where L > dk/2e, is added, at most L frequent melody structures with length L 1, are decremented from the current collection of maximal frequent melody structures found so far. Hence, the maximum number of maximal melody structures is changed from U to U 0 , where U 0 = U + 1 L, which is not greater than C kdk=2e . This conflicts with the assumption of U > C kdk=2e and results in a contradiction. Thus the statement is proven to be true. Therefore, we conclude that the maximum number of maximal melody structures is C kdk=2e in the problem of online mining maximal melody structures in a landmark melody stream. h. Example 1. Assume that there are five frequent items (i.e., frequent 1-item-sets) a, b, c, d, and e in the landmark melody stream as shown in Fig. 2. Let MF denote the total number of maximal frequent item-sets. At this point, a, b, c, d and e are maximal and MF ¼ C 51 . Based on the Apriori heuristic, C 52 frequent 2-item-sets are discovered in the worst case. In this case, these frequent 2-item-sets are also maximal and those frequent 1-item-sets are not maximal any more. The current MF is C 51 þ C 52 C 51 ¼ C 52 . Next, C 53 frequent 3-item-sets are found in the worst case. These frequent 3-item-sets are maximal but the sub-sets of the maximal 3-item-sets, i.e., frequent 2-item-sets, are not maximal any more. Now, the MF becomes C 52 þ C 53 C 52 ¼ C 53 . At this time, suppose the frequent 4-item-set abcd exists in this instance and it is also a maximal 4-itemset. The frequent subsets, with length three, of abcd, i.e., abc, abd, acd and bcd, are not maximal any more. Now, the MF becomes C 53 þ 1 4 ¼ 7, i.e., abcd, ace, ade, bcd, bce, bde, cde are maximal frequent item-sets. The new MF is smaller than the upper bound C 5d5=2e . Hence, we can find that if one or more frequent item-sets with length L, where L > d5/2e, are added into the collection of maximal frequent item-sets found so far, the value of MF would be changed and would be less than C 5d5=2e . Consequently, the C 5d5=2e is the upper bound of the number of maximal melody structures..
(5) 1662. H.-F. Li et al. / Pattern Recognition Letters 26 (2005) 1658–1674. a. b. c. d. C 15. e. ab. ac. ad. ae. bc. bd. be. cd. ce. de. C 52. abc. abd. abe. acd. ace. ade. bcd. bce. bde. cde. C 53. abcd. abce. abde. acde. bcde. C 54 C 55. abcde Fig. 2. Item-set enumeration lattice with respect to five items: a, b, c, d and e.. The key property of algorithm MMSLMS is derived from the recent work (Karp et al., 2003) for finding frequent elements in streaming data. The basic scheme of mining chord-sets from music data streams is generalized from the well-known algorithm (Fisher and Salzberg, 1982) for determining whether a value (majority element) occurs more than n/2 times, i.e., minsup = 0.5, in a data stream of length n. The method can be extended to an arbitrary value of minsup. The scheme is processed as follows. At any given time, a superset of k probably frequent chord-sets with at most 1/minsup times is maintained. Initially, the set is empty. As a chord-set is read from the melody sequence in the current block, two operations are performed as follows. First, if the current chord-set is not contained in the superset and some entries are free, it is inserted into the superset with a count set to one. Second, if the chord-set is already in the superset, its count is incremented by one. However, if the superset is full, the count of each entry in the superset is decremented by one, and the chord-sets whose frequencies are just one are dropped. The method thus identifies at most k candidates for having appeared more than n/(k + 1) times, and uses O(1/minsup) memory entries.. 3.2. The proposed algorithm: MMSLMS Algorithm MMSLMS has three modules: MMSLMS-buffer, MMSLMS-summary, and MMSLMS-mine. MMSLMS-buffer repeatedly reads in a block of melody sequences into available main memory. All compressed and essential information about the maximal melody structures will be maintained in the MMSLMS-summary. Finally, MMSLMS-mine finds the maximal melody structures by a depth-first manner in the current MMSLMS-summary. Therefore, the challenges of online mining landmark melody streams lie in the design of a space-efficient representation of the in-memory summary data structure and a fast discovery algorithm for finding maximal melody structures in real time. 3.2.1. MMSLMS-summary First of all, the in-memory data structure MMSLMS-summary is defined and the constructing process of MMSLMS-summary is discussed. Then we use a running example to illustrate. Definition 7. A MMSLMS-summary is an extended prefix-tree-based summary data structure defined below..
(6) H.-F. Li et al. / Pattern Recognition Letters 26 (2005) 1658–1674. 1. MMSLMS-summary consists of a CMB (Chordset Memory Border), and a set of MPI-trees (Maximal Prefix-Item trees of item-suffixes) denoted as MPI-trees(item-suffixes). 2. Each node in the MPI-tree(item-suffix) consists of four fields: item-id, support, block-id and node-link, where item-id is the item identifier of the inserting item, support registers the number of melody sequences represented by a portion of the path reaching the node with the item-id, the value of block-id assigned to a new node is the block identifier of the current block, and node-link links up a node with the next node with the same item-id in the same MPI-tree or null if there is none. 3. Each entry in the CMB consists of four fields: item-id, support, block-id, and head of node-link (a pointer links to the root node of the MPI-tree with the same item-id), abbreviated as head-link, where item-id registers which item identifier the entry represents, support records the number of transactions containing the item carrying the item-id, the value of block-id assigned to a new entry is the block identifier of current block, and head-link points to the root node of the MPI-tree(item-suffix). Notice that each entry with item-id in the CMB is an item-suffix and it is also the root node of the MPI-tree(item-id). 4. Each MPI-tree(item-suffix) has a specific CMBtable (Chord-set Memory Border table) with respect to the item-suffix (denoted as CMBtable(item-suffix)). The CMB-table(item-suffix) is composed of four fields, namely item-id, support, block-id, and head-link. The CMBtable(item-suffix) operates the same as the CMB except that the field head-link links to the first node carrying the item-id in the MPI-tree(itemsuffix). Notice that jCMB-table(item-suffix)j = jCMBj in the worst case, where jCMBj denotes the total number of entries in the CMB. The construction of MMSLMS-summary is described as follows. First of all, MMSLMS reads a melody sequence S from the current block. Then, MMSLMS projects the sequence S into many subsequences and inserts these subsequences into the CMB and MPI-trees. In details, each melody. 1663. sequence S, such as hx1,x2, . . . , xmi, in the current block should be projected by inserting m itemsuffix melody subsequences into the MMSLMSsummary. In other words, the melody sequence S = hx1,x2, . . . , xmi is converted into m melody subsequences; that is, hx1,x2, . . . , xmi, hx2,x3, . . . , xmi, . . . , hxm1,xmi, and hxmi. The m melody subsequences are called item-suffix sequences, since the first item of each melody subsequence is an itemsuffix of the original melody sequence S. This step is called sequence projection, and is denoted as Sequence-Projection (S) = {x1jS,x2jS, . . . , xijS, . . . , xmjS}, where xijS = hxi,xi+1, . . . , xmi, "i = 1,2, . . . , m. Furthermore, the cost of sequence projection of a melody sequence with length m is (m2 + m)/ 2, i.e., m + (m 1) + + 2 + 1. After Sequence-Projection (S), MMSLMS algorithm removes the original melody sequence S from the MMSLMS-buffer. Next, the set of items in these item-suffix sequences are inserted into the CMB and the MPI-trees(item-suffixes) as a branch, and the CMB-table(item-suffixes) are updated according to the item-suffixes. If an itemset (or item-string) share a prefix with an item-set (or item-string) already in the tree, the new itemset (or item-string) will share a prefix of the branch representing that item-set (or item-string). In addition, a support counter is associated with each node in the tree. The counter is updated when an item-suffix sequence causes the insertion of a new branch. In order to limit the memory size of the summary data structure MMSLMS-summary, a space pruning technique is performed. Let the minimum support threshold be minsup, in the range of [0, 1], and the current length of the landmark melody stream be N. The rule for space pruning is as follows. A melody structure E is deleted if E.support < minsup Æ N. E is called an infrequent melody structure. After pruning all infrequent melody structures from the CMB, CMB-table-(itemsuffix) and MPI-trees, the MMSLMS-summary contains all information about frequent melody structures of the landmark melody stream generated so far. Example 2 below illustrates the algorithm step by step. Note that the h i of sequences are omitted for clear presentation..
(7) 1664. H.-F. Li et al. / Pattern Recognition Letters 26 (2005) 1658–1674. Example 2. Let a block Bj of the landmark melody stream LMS be hacdefi, habei, hcefi, hacdfi, hcefi and hdfi, and the minimum support threshold be 0.5 (i.e., minsup = 0.5), where a, b, c, d, e and f are chord-sets (i.e., items) in a landmark melody stream seen so far. MMSLMS algorithm constructs the MMSLMS-summary with respect to the incoming block Bj and prunes all item-sets that are infrequent from the current MMSLMS-summary in the following steps. Note that each node or entry represented as (f1:f2:f3) is composed of three fields: item-id, support, and block-id. For example, (a:2:j) indicates that, from block Bj, item a appeared twice. Step 1: MMSLMS reads current block Bj into main memory for constructing the MMSLMSsummary. (a) First melody sequence acdef: First of all, MMSLMS algorithm reads the first melody sequence acdef and calls the Sequence-Projection (acdef). Then MMSLMS inserts the item-suffix sequences acdef, cdef, def, ef, and f into. the CMB, [MPI-tree(a), CMB-table(a)], [MPI-tree(c), CMB-table(c)], [MPItree(d), CMB-table(d)], [MPI-tree(e), CMB-table(e)], and [MPI-tree(f), CMB-table(f)], respectively. The result is shown in Fig. 3. In the following sub-steps, as demonstrated in Fig. 4 through Fig. 9, the head-links of each CMB-table (item-suffix) are omitted for concise presentation. (b) Second melody sequence abe: MMSLMS reads the second melody sequence abe and calls the Sequence-Projection (abe). Next, MMSLMS inserts the item-suffix sequences abe, be and e into the CMB, [MPI-tree(a), CMB-table(a)], [MPItree(b), CMB-table(b)] and [MPItree(e), CMB-table(e)], respectively. The result is shown in Fig. 4. (c) Third melody sequence cef: MMSLMS reads the third melody sequence cef and calls the Sequence-Projection (cef). Then, MMSLMS inserts the item-suffix sequences cef, ef and f into the CMB,. Fig. 3. MMSLMS-summary construction after inserting first melody sequence acdef in block Bj. In the following sub-steps, as demonstrated in Fig. 4 through Fig. 9, the head-links of each CMB-table (item-suffix) are omitted for concise presentation..
(8) H.-F. Li et al. / Pattern Recognition Letters 26 (2005) 1658–1674. 1665. Fig. 4. MMSLMS-summary construction after inserting second melody sequence abe.. [MPI-tree(c), CMB-table(c)], [MPItree(e), CMB-table(e)] and [MPItree(f), CMB-table(f)], respectively. The result is shown in Fig. 5. (d) Fourth melody sequence acdf: MMSLMS reads the fourth melody sequence acdf and calls the Sequence-Projection (acdf). Next, MMSLMS inserts the item-suffix sequences acdf, cdf, df and f into the CMB, [MPI-tree(a), CMB-table(a)], [MPI-tree(c), CMBtable(c)], [MPI-tree(d), CMB-table(d)] and [MPI-tree(f), CMB-table(f)], respectively. The result is shown in Fig. 6. (e) Fifth melody sequence cef: MMSLMS reads the fifth melody sequence cef and calls the Sequence-Projection (cef). Then, MMSLMS inserts the item-suffix sequences cef, ef and f into the CMB, [MPI-tree(c), CMB-table(c)], [MPItree(e), CMB-table(e)] and [MPItree(f), CMB-table(f)], respectively. The result is shown in Fig. 7.. (f) Sixth melody sequence df: MMSLMS reads the sixth melody sequence df and calls the Sequence-Projection (df). Next, MMSLMS inserts the item-suffix sequences df and f into the CMB, [MPI-tree(d), CMB-table(d)] and [MPItree(f), CMB-table(f)], respectively. The result is shown in Fig. 8. Step 2: After computing the current block Bj, all infrequent melody structures are pruned by MMSLMS from the current MMSLMSsummary. At this time, MMSLMS deletes the MPI-tree(b) and its corresponding CMB-table(b), and prunes the entry b from the CMB, since item b is an infrequent item; that is, r(b) < minsup Æ jLMSj, where r(b) = 1 and minsup Æ jLMSj = 0.5 Æ 6 = 3. Next, MMSLMS reconstructs the MPI-tree(a) by eliminating the information about the infrequent item b. The result is shown in Fig. 9. The description stated above is the constructing process of MMSLMS-summary with respect to the.
(9) 1666. H.-F. Li et al. / Pattern Recognition Letters 26 (2005) 1658–1674. Fig. 5. MMSLMS-summary construction after inserting third melody sequence cef.. Fig. 6. MMSLMS-summary construction after inserting third melody sequence acdf.. incoming block over a landmark melody stream. The MMSLMS-summary construction algorithm is depicted in Fig. 10.. 3.2.2. MMSLMS-mine In this section, the module, called MMSLMSmine, of mining maximal melody item-sets and.
(10) H.-F. Li et al. / Pattern Recognition Letters 26 (2005) 1658–1674. 1667. Fig. 7. MMSLMS-summary construction after inserting fifth melody sequence cef.. Fig. 8. MMSLMS-summary construction after inserting sixth melody sequence df.. maximal melody item-strings from the current MMSLMS-summary is discussed (Fig. 11).. First of all, given an entry id (from left to right, for example) in the current CMB, MMSLMS-mine.
(11) 1668. H.-F. Li et al. / Pattern Recognition Letters 26 (2005) 1658–1674. Fig. 9. Current MMSLMS-summary after pruning all infrequent melody structures with respect to infrequent item b.. generates candidate maximal melody structures by a top-down approach. The top-down method uses the frequent items (i.e., chord-sets) of CMB-table(id) and item id to generate the candidates. The generating order of these candidates is determined by the size of item-set, from item-set size 1 + jCMB-table(id)j down to size 2. Note that, the generating order ends in 2-item-sets because all frequent entries in the current CMB-table are frequent 1-item-sets. Then MMSLMS-mine checks these candidates whether they are frequent or not by traversing the MPI-tree(id). The MPI-tree traversing principle is described as follows. First, MMSLMS-mine generates a candidate maximal melody item-set, (j + 1)-item-set, containing the item id and all items of the CMB-table(id), where jCMB-table(id)j = j. Second, MMSLMS-mine traverses the MPI-tree via the node-links of the frequent candidate. After that if the candidate is not a frequent item-set, MMSLMS-mine generates substructure candidates with j-item-sets. Next, MMSLMS-mine executes the same MPI-tree traversing scheme for item-set counting. The process stops when MMSLMS-mine finds all maximal. frequent melody structures from the current MMSLMS-summary. Moreover, MMSLMS-mine stores these maximal melody structures into a temporal pattern list, called MMSLMS-list. Notice that MMSLMS-mine can find the set of frequent 2-itemsets by combining the item-suffix id with the frequent items of the CMB-table(id). Example 3. This example illustrates the mining of the maximal melody item-sets from the current MMSLMS-summary in Fig. 9. Let the minimum support threshold be 0.5, i.e., minsup = 0.5. (1) Now, we start the maximal melody item-set mining scheme from the frequent item a. At this moment, the frequent item-set is the only 1-item-set (a), since the support of items c, d, e and f in the CMB-table (a) are less than minsup Æ jLMSj, where jLMSj = jBjj = 6. (2) Next, MMSLMS-mine starts on the second entry c for maximal melody item-set mining. MMSLMS-mine generates a candidate maximal 3-item-set (cef), and traverses the MPItree(c) for counting its support. As a result,.
(12) H.-F. Li et al. / Pattern Recognition Letters 26 (2005) 1658–1674. 1669. Algorithm 1 (MMSLMS-summary Construction) Input: A landmark melody stream, LMS = [B1, B2, …, BN), and a user-specified minimum support threshold, minsup. Output: A current MMSLMS-summary. 1: CMB = ∅ /*initialize the CMB to empty.*/ 2: foreach block Bj do /* j = 1, 2, …, N */ 3: foreach melody sequence S = <x1, x2, …, xm> ∈ Bj (j = 1, 2, …, N) do 4: foreach item xi S do /*the CMB maintenance*/ 5: if xi ∉ CMB then 6: create a new entry (xi, 1, j, head-link) into the CMB; /* the entry form is (item-id, support, block-id, head-link)*/ 7: else /* the entry already in the CMB*/ 8: xi.support = xi.support + 1; /* increment the support of item-id xi by one*/ 9: end if 10: end for 11: call Sequence-Projection(S); /* project the sequence with every prefix-item xi for the construction of MPI-tree(xi)*/ 12: end for 13: call MMSLMS-summary-pruning(MMSLMS-summary, minsup, |LMS|); 14: end for Subroutine Sequence-Projection Input: A melody sequence S = <x1, x2, …, xm> and the current block-id j; Output: MPI-trees(xi), ∀i=1, 2, …, m; 1: foreach item xi (i =1, 2, …, m) do 2: MPI-tree-maintenance([xi|X], MPI-tree(xi), j); /* X = x1, x2, …, xm is the original melody sequence */ /* [xi|X] is an item-suffix melody sequence with item-suffix xi*/ 3: end for Subroutine MPI-tree-maintenance Input: An item-suffix melody sequence <xi, xi+1, …, xm> (i=1, 2, …, m), MPI-tree(xi) and the current block-id j; Output: The modified MPI-tree(xi), where i=1, 2, ..., m; 1: foreach item xi do /* i=1, 2, …, m */ 2: if MPI-tree has a child Y such that Y.item-id = xi.item-id then 3: Y.support = Y.support +1; /*increment Y’s support by one*/ 5: else 6: create a new node Y = (item-id, 1, j, node-link); /* initialize the Y’s support to 1, and link its parent link to MPI-tree, and its node-link linked to the nodes with same item-id via the node-link structure. */ 7: end if 8: end for Fig. 10. Algorithm of MMSLMS-summary construction.. the candidate (cef) is a maximal frequent item-set, since its support is 3, and it is not a sub-structure of any other maximal melody structures within the MMSLMS-list. Now, MMSLMS-mine stores the maximal item-set (cef) into the MMSLMS-list.. (3) MMSLMS-mine starts on the third entry d and generates a maximal frequent 2-itemset (df). We store this item-set (df) into the MMSLMS-list because it is not a sub-structure of any other maximal melody structures within the current MMSLMS-list..
(13) 1670. H.-F. Li et al. / Pattern Recognition Letters 26 (2005) 1658–1674. Subroutine MMSLMS-summary-pruning Input: An MMSLMS-summary, a user-specified minimum support threshold, minsup, and the current length of LMS, |LMS|; Output: An MMSLMS-summary which contains the set of all frequent melody structures. 1: foreach entry xi (i=1, 2, …, d) ∈ CMB, where d =|CMB| do. 2: 3: 4: 5: 6: 7: 8:. if xi .support < minsup |LMS| then /* xi is not a frequent item */ delete those nodes (item-id = xi) via node-link structure; merge the fragmented sub-trees; /* a simple way is to reinsert or to join the remainder sub-trees into the MPI-tree*/; delete MPI-tree(xi); delete the entry xi from the CMB; end if end for Fig. 10 (continued). Algorithm 2 (MMSLMS-mine) Input: A current MMSLMS-summary, the current length of landmark melody stream |LMS|, and a minimum support threshold minsup. Output: A temporal-pattern-list, MMSLMS-list, of maximal melody structures. 1: MMSLMS-list = ∅; 2: foreach entry e in the current CMB do 3: do generate a candidate maximal melody structure E with size |E| /* |E| = 1+|CMB-table(e) */ 4: counting E.support by traversing the MPI-tree(e); 5: if E.support minsup |LMS| then 6: if E ∉ MMSLMS-list and E is not a substructure of any other maximal frequent structures contained into the MMSLMS-list then 7: add E into the MMSLMS-list; 8: remove E’s substructures from the MMSLMS-list; 9: end if 10: else /* if E is not a frequent melody structure*/ 11: enumerate E into melody substructures with size |E| —1; 12: end if 13: until MMSLMS-mine find the set of all maximal frequent structures with respect to the item e; 14: end for Fig. 11. Algorithm of MMSLMS-mine.. (4) On the fourth entry e, since its maximal melody item-set (ef) is a sub-structure of previous maximal melody item-set (cef), MMSLMS-mine does not store it into the MMSLMS-list. (5) Finally, MMSLMS-mine computes the entry f, and generates a maximal frequent 1-itemset (f) directly, since the CMB-table(f) is. empty. MMSLMS-mine does not store it into the MMSLMS-list, because it is a substructure of a generated maximal item-set (cef). In conclusion, the Maximal Type I Melody Structures determined by algorithm MMSLMS are (a), (cef) and (df). Now, we describe the mining.
(14) H.-F. Li et al. / Pattern Recognition Letters 26 (2005) 1658–1674. principle of maximal melody item-strings, i.e., Maximal Type II Melody Structures, as below. MMSLMS-mine generates maximal melody itemstrings from the current MMSLMS-summary as shown in Fig. 9 by a depth-first-search (DFS) approach. Hence, the Maximal Type II Melody Structures determined by algorithm MMSLMS are hai, hci, hdi and hefi. Note that hfi is not maximal melody item-string since it is a sub-string of the existing maximal melody 2-item-string hefi. Based on the algorithm MMSLMS-mine in Fig. 10, we have the following lemma. Lemma 2. A melody structure is a maximal melody structure if and only if it is generated by algorithm MMSLMS-mine. Proof. Algorithm MMSLMS-mine is composed of two major steps: frequent melody structure selection (step 1) and maximal melody structure verification (step 2). These steps are performed in sequence. First of all, in the step of frequent melody structure selection, MMSLMS-mine finds frequent melody structure based on the Apriori property if any length i-item-set (or i-item-string) is not frequent, its length (i + 1)-item-set (or (i + 1)-itemstring) can never be frequent. That means MMSLMS-mine does not miss any frequent melody structures. Next, in step 2, MMSLMS-mine checks the frequent melody structures generated from step 1 against the maximal melody structures of the MMSLMS-list, a temporal pattern list of maximal melody structures. If this frequent melody structure is a sub-structure (i.e., sub-set or substring) of any other structures within the MMSLMS-list, then it is not a maximal melody structure according to the Definition 5; otherwise it is a candidate maximal melody structure before the next execution of step 2. Repeating step 1 and step 2, MMSLMS-mine can generate all the maximal melody structures contained in the MMSLMS-list. Hence, we have the lemma: a melody structure is a maximal melody structure if and only if it is generated by algorithm MMSLMS-mine. Space complexity analysis: The space requirement of MMSLMS-summary consists of two parts: the working space needed to create a CMB and the CMB-tables, and the storage space needed to. 1671. maintain the set of MPI-trees. Assume that CMB contains k frequent chord-sets such as e1,e2, . . . , ei, . . . , ek at any time. Based on the Theorem 1, we know that there are at most C kdk=2e maximal frequent chord-sets in the landmark melody stream seen so far. If we construct the MMSLMSsummary for all these maximal frequent melody structures, the maximum height of all the MPItrees is dk/2e. There are 1 þ C k1 þ C k1 þ þ 1 2 k1 C dk=2e1 nodes in the MPI-tree(e1), where the value 1 indicates the root node e1 of the MPI-tree(e1), þ C k1 þ þ C k1 and C k1 1 2 dk=2e1 are internal and leaf nodes of the MPI-tree(e1). Moreover, there are 1 þ C k2 þ C k2 þ þ C k2 nodes in the 1 2 dk=2e1 ki ki MPI-tree(e2), . . ., 1 þ C ki 1 þ C 2 þ þ C dk=2e1 kðk1Þ nodes in the MPI-tree(ei), 1 þ C 1 nodes in the MPI-tree(ek1), and 1 (root) node in the MPItree(ek). Thus, the total number of nodes of MPItrees in the MMSLMS-summary is. þ C k1 þ þ C k1 ð1 þ C k1 1 2 dk=2e1 Þ þ C k2 þ þ C k2 þ ð1 þ C k2 1 2 dk=2e1 Þ þ ki ki þ ð1 þ C ki 1 þ C 2 þ þ C dk=2e1 Þ þ kðk1Þ. þ ð1 þ C 1. Þþ1. ¼ ðC k1 þ C k1 þ C k1 þ þ C k1 0 1 2 dk=2e1 Þ þ C k2 þ C k2 þ þ C k2 þ ðC k2 0 1 2 dk=2e1 Þ þ ki ki ki þ ðC ki 0 þ C 1 þ C 2 þ þ C dk=2e1 Þ þ kðk1Þ. þ ðC 0. kðk1Þ. þ C1. Þ þ C kk 0 :. This number equals C k1 þ C k2 þ þ C kdk=2e based on Pascals Identity: let x and y be positive integers with x P y. Then C xþ1 ¼ C xy1 þ C xy . y Moreover, the worst case working space requires at most (k2 + k)/2 entries, which is based on the process of Sequence-Projection. Thus, the space requirement of MMSLMS-summary is ðk 2 þ kÞ=2 þ C k1 þ C k2 þ þ C kdk=2e . Finally, the upper bound of space requirement is O(2k). h The worst case space complexity of algorithm MMSLMS can be analyzed in terms of melody sequence size as described below. Assume that the average melody sequence size is m, the current.
(15) 1672. H.-F. Li et al. / Pattern Recognition Letters 26 (2005) 1658–1674. length of the landmark melody stream is N, and the minimum support threshold is minsup. The space requirement of algorithm MMSLMS is composed of two parts, working space and storage space. The working space is used to store the CMB and CMB-tables and the storage space is used to store the MPI-trees. The working space requirement is m + (m 1) + (m 2) + + 1 and the storage space requirement is also about m + (m 1) + (m 2) + + 1. Hence, the space requirement of MMSLMS for inserting a melody sequence with average size m into MMSLMS-summary is 2[m + (m 1) + (m 2) + + 1] = m2 + m. Hence, the space requirement of the stream generated so far in the worst case is N Æ (m2 + m). Note that in the analysis, we assume that the sminsup Æ N is just one and therefore every item of the incoming melody sequence is a frequent item, which is the worst case. However, we know that the value of N increases as time progresses. Hence, the pruning mechanism of MMSLMS-summary is deployed to limit the memory requirement not to exceed an upper bound. From the space complexity analysis, it is not surprising to find that the space complexity grows exponentially into the number of frequent items in the CMB, as all frequent item-sets are represented in the data structure. It is also the solution space of the problem. Time complexity analysis: From the construction process of MMSLMS-summary, we can see that exactly one scan of a landmark melody stream is required. The cost (denoted by Time-cost(S)) of inserting a melody sequence S into the MMSLMSsummary by sequence projection is jSj + (jSj 1) + + 1 = (jSj2 + jSj)/2; that is O(jfreq(S)j2), where freq(S) is the set of frequent items in the melody sequence S. Note that jfreq(S)j 6 jSj, where jSj denotes the size of the melody sequence S. Because the items within the CMB are frequent items, therefore, the cost of inserting a melody sequence S can be stated in terms of the size of CMB. Time-cost(S) = O(jS 0 j2), where jS 0 j is the number of chord-sets of melody sequence S within the CMB. In the worst case, if the melody sequence S contains all the frequent items within the CMB, Time-cost(S) = O(jCMBj2).. 4. Experimental results In this section, we first describe the data and experiment set-up used to evaluate the performance of the proposed algorithm, and then report our experimental results. 4.1. Synthetic data and experiment set-up To evaluate the performance of MMSLMS algorithm, two experiments are performed. The experiments were carried out on the IBM synthetic market-basket test data generator proposed by Agrawal and Srikant (1994). Two data streams, denoted by S10.I5.D1000K and S30.I15.D1000K, of size 1 million melody sequences each are studied. The first one, S10.I5.D1000K with 1 K unique items, has an average melody sequence size of 10 with average maximal potentially frequent structure size of 5. The second one, S30.I15.D1000K with 10 K unique items, has an average melody sequence size of 30 with average maximal potentially frequent structure size of 15. In all experiments, the melody sequences of each datasets are looked up in sequence to simulate the environment of a landmark melody stream. All the experiments are performed on a 1066-MHz Pentium III processor with 128 megabytes main memory, running on Microsoft Windows XP. In addition, all the programs are written in Microsoft/Visual C++ 6.0. 4.2. Experimental results In the first experiment, two primary factors, memory and execution time, are examined in the online mining of a landmark melody stream, since both should be bounded online as time advances. In Fig. 12(a), the execution time grows smoothly as the dataset size increases. This is because the average execution time of dataset S10.I5 and S30.I15 are about 12 and 25 s per block respectively, where a block is composed of 50,000 melody sequences. In other words, the computation time of dataset S10.I5 by algorithm MMSLMS is 12 s every 50,000 melody sequences, and for dataset S30.I15 is 25 s every 50,000 melody sequences. Hence, it grows smoothly as the dataset size increases. The memory usage in Fig. 12(b) for both.
(16) H.-F. Li et al. / Pattern Recognition Letters 26 (2005) 1658–1674. 1673. Fig. 12. Required resources for synthetic datasets: (a) execution time and (b) memory.. Fig. 13. (a) Linear scalability of the data stream size and (b) relative error of mining results.. synthetic datasets is stable as time progresses, indicating the feasibility of algorithm MMSLMS. Note that the synthetic landmark melody stream is partitioned into blocks with size 50 K. In the second experiment, we investigate the scalability and relative error of algorithm MMSLMS with respect to varying minimum supports. The relative error is defined as the difference between the measured support and the actual support estimation divided by the actual support. In Fig. 13(a), the execution time grows smoothly as the dataset increases (assume minsup = 0.01%) indicating linear scalability. Fig. 13(b) shows that the relative error decreases as minisup decreases, i.e., as the size of CMB decreases. Generally, the more frequent items are. maintained in the CMB, the more accurate the mining result is.. 5. Conclusions In this paper, we proposed a single-pass algorithm, MMSLMS, to discover and maintain all maximal melody structures in a landmark model that contains all the melody sequences in a data stream. In the MMSLMS algorithm, an efficient in-memory summary data structure, MMSLMS-summary, is developed to record all maximal frequent structures in the current landmark model. In addition, MMSLMS uses a space-efficient scheme, the Chord-set Memory Border (CMB), to guarantee.
(17) 1674. H.-F. Li et al. / Pattern Recognition Letters 26 (2005) 1658–1674. the upper-bound of space requirements of mining maximal melody sequences in a streaming environment. Theoretical analysis and experimental results with synthetic data show that MMSLMS algorithm can meet the performance requirements of data stream mining: one-scan, bounded-space and real time. Further work includes online mining maximal melody structures in count-based and timebased sliding window that contains the most recent melody sequences in a data stream. Acknowledgements The authors thank the reviewers precious comments for improving the quality of the paper. The research is supported by National Science Council of R.O.C. under grant no. NSC93-2213-E-009-043.. References Agrawal, R., Srikant, R., 1994. Fast algorithms for mining association rules. In: Proceedings of 20th International Conference on Very Large Data Bases, pp. 487–499.. Babcock, B., Babu, S., Datar, M., Motwani, R., Widom, J., 2002. Models and issues in data stream systems. In: Proceedings of 21th ACM Symposium on Principles of Database Systems, pp. 1–16. Bakhmutora, V., Gusev, V.U., Titkova, T.N., 1997. The search for adaptations in song melodies. Computer Music Journal 21 (1), 58–67. Fisher, M.J., Salzberg, S.L., 1982. Finding a majority among n votes: solution to problem 81-5. Journal of Algorithms 3 (4), 362–380. Hsu, J.L., Liu, C.C., Chen, A.L.P., 2001. Discovering nontrivial repeating patterns in music data. IEEE Transactions on Multimedia 3 (3), 311–325. Jones, G.T., 1974. Music Theory. Harper & Row, Publishers, New York. Karp, R.M., Papadimitrious, C.H., Shanker, S., 2003. A simple algorithm for finding frequent elements in streams and bags. ACM Transactions on Database Systems 28 (1), 51– 55. Shan, M.-K., Kuo, F.-F., 2003. Music style mining and classification by melody. IEICE Transactions on Information and Systems E86-D (4), 655–659. Yoshitaka, A., Ichikawa, T., 1999. A survey on content-based retrieval for multimedia databases. IEEE Transactions on Knowledge and Data Engineering 11 (1), 81–93. Zhu, Y., Kankanhalli, M.S., Xu, C., 2001. Pitch tracking and melody slope matching for song retrieval. In: Proceedings of the Second IEEE Pacific Rim Conference on Multimedia: Advances in Multimedia Information, pp. 530–537..
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