RESEARCH Open Access

Clustering algorithm for audio signalsbased on the sequential Psim matrix andTabu SearchWenfa Li1, Gongming Wang2* and Ke Li1

Abstract

Audio signals are a type of high-dimensional data, and their clustering is critical. However, distance calculationfailures, inefficient index trees, and cluster overlaps, derived from the equidistance, redundant attribute, and sparsity,respectively, seriously affect the clustering performance. To solve these problems, an audio-signal clusteringalgorithm based on the sequential Psim matrix and Tabu Search is proposed. First, the audio signal similarity iscalculated with the Psim function, which avoids the equidistance. The data is then organized using a sequentialPsim matrix, which improves the indexing performance. The initial clusters are then generated with differentialtruncation and refined using the Tabu Search, which eliminates cluster overlap. Finally, the K-Medoids algorithm isused to refine the cluster. This algorithm is compared to the K-Medoids and spectral clustering algorithms usingUCI waveform datasets. The experimental results indicate that the proposed algorithm can obtain better Macro-F1and Micro-F1 values with fewer iterations.

Keywords: Audio signal clustering, Sequential Psim matrix, Tabu Search, Heuristic search, K-Medoids, Spectral clustering

1 IntroductionAudio signal clustering forms the basis for speech recog-nition, audio synthesis, audio retrieval, etc. Audio signalsare considered as high-dimensional data, with dimen-sionalities of more than 20 [1]. Their clustering is under-taken based on this consideration and solving theproblems in high-dimensional data clustering, in this re-gard, is highly beneficial.There are three types of clustering algorithms for high-

dimensional data: attribute reduction [2], subspace clus-tering [3], and co-clustering [4]. The first method reducesthe data dimensionality with attribute conversion or re-duction and then, performs clustering. The effect of thismethod is heavily dependent on the degree of dimensionreduction; if it is considerable, useful information may belost, and if it is less, clustering cannot be done effectively.The second method divides the original space into severaldifferent subspaces and searches for cluster in the sub-space. When the dimensionality is high and the accuracy

requirement rigorous, the number of subspaces rapidly in-creases. Thus, searching for a cluster in a subspace be-comes a bottleneck and may lead to failure [5]. The thirdmethod implements clustering iteratively with respect tothe content and feature alternately. The clustering resultis adjusted as per the semantic relationship between thetheme and characteristic, realizing a balance between dataand attribute clustering. This method has two stages,resulting in a high time complexity. In addition to theabove three methods, clustering algorithms for high-dimensional data includes hierarchical clustering [6], par-allel clustering [7], knowledge-driven clustering [8], etc.However, these methods also have similar problems.Equidistance, the redundant attribute, and sparsity are

the fundamental factors affecting the clustering perform-ance of high-dimensional data [9]. Equidistance renders thedistance between any two points in a high-dimensionalspace approximately equal, leading to a failure in the clus-tering algorithm, based on the distance. The redundant at-tribute increases the dimensionality of the high-dimensional data and the complexity of the index structure,decreasing the efficiency of building and retrieving theindex structure. Sparsity enables uniform data distribution,

* Correspondence: gongmingwang@126.com2Institute of Biophysics, Chinese Academy of Sciences, No. 15 Datun Road,Beijing, ChinaFull list of author information is available at the end of the article

© The Author(s). 2017 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, andreproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link tothe Creative Commons license, and indicate if changes were made.

Li et al. EURASIP Journal on Audio, Speech, and Music Processing (2017) 2017:26 DOI 10.1186/s13636-017-0123-3

and some clusters may overlap with each other, affectingthe clustering precision.It is reported that some dimensional components of

the high-dimensional data are non-related noise thathide the real distance, resulting in equidistance. ThePsim function can find and eliminate noise in all the di-mensions [10]. The Tabu Search [11] is a heuristic globalsearch algorithm. All the possible existent clusters arecombinatorically optimized using the Tabu Search suchthat a non-overlap cluster is selected.To solve the clustering problems owing to equidis-

tance, the redundant attribute, and sparsity, an efficientaudio signal clustering algorithm is proposed, by integra-tion with the Psim matrix and Tabu Search. First, for allthe points in the high-dimensional space, the Psimvalues between them and the corresponding locationnumbers are stored in a Psim matrix. Next, a sequentialPsim matrix is generated by sorting the elements in eachrow of the Psim matrix. Further, the initial clusters aregenerated with differential truncation and refined withthe Tabu Search. Finally, the initial clusters are iterativelyrefined with the K-Medoids algorithm, until all the clus-ter medoids are stable.

2 Related works2.1 Psim matrixTraditional similarity measurement methods (e.g., the Eu-clidean distance, Jaccard coefficient [12], and Pearson co-efficient [12]) fail in high-dimensional space because inthese methods, equidistance is a common phenomenon inhigh-dimensional space; hence, the calculated distance isnot the real distance. To solve this problem, the Hsimfunction [13] was proposed; however, the relative differ-ence and noise distribution were not considered. TheGsim function [14] was also proposed and the relative dif-ferences of the properties in different dimensions were an-alyzed, but the weight discrepancy was ignored. Theproposed Close function [15] can reduce the influence ofcomponents in certain dimensions, whose variances arelarger; however, the relative difference was not consideredand it would be affected by noise. The Esim [16] functionwas proposed by improving the Hsim and Close functionsand considering the influence of the property on the simi-larity. In every dimension, the Esim component has apositive correlation. All the dimensions are divided intonormal and noisy. In a noisy dimension, noise is the mainingredient. When it is similar and larger than the signal, ina normal dimension, Esim is invalid. The secondary meas-urement method [17] is used to calculate the similarity byconsidering the property distribution, space distance, etc.However, the noise distribution and weight are not takeninto account. In addition, its formula is complicated andthe calculation is slow. In high-dimensional space, a largedifference exists in certain dimensionalities [10], even

though the data is similar. This difference occupies a largeportion of the similarity calculation; hence, all the calcula-tion results are similar. Therefore, the Psim function [10]was proposed to diminish the influence of noise on thesimilarity data; experimental results indicate that thismethod is suitable.When using the Psim function to measure the similar-

ity, the data component in every dimension must besorted and the value range divided into several intervals.The similarity between X and Y in the j-th dimension isadded to the Psim function, if and only if, their datacomponents are in the same interval.In an n-dimensional space, the Psim value between X

and Y is as follow:

Psim X;Yð Þ ¼X

j∈Ds X;Yð Þ

1−jXj−Y jjlj−uj

� �Ds X;Yð Þj j

n

where Xj and Yj are the data components of X and Y inthe j-th dimension. Ds(X,Y) is a subscript set of Xj andYj, which are in the same interval [uj, lj], and |Ds(X,Y)|is the number of elements in Ds(X,Y). The above is theoutline of the Psim function; a detailed introduction canbe found in [10].Data organization is critical in a clustering algorithm.

In the traditional method, the data space is separatedusing an index tree and mapped onto the index-treenodes. The commonly used index trees are the R tree[18], cR tree [19], VP tree [20], M tree [21], SA tree [22],etc. The partitioning of the data space is the foundationfor building an index tree, but its complexity increaseswith the increase in dimensionality. Thus, it is difficultto build index trees for high-dimensional data. Inaddition, the retrieval efficiency of the index tree fallssharply with the increase in dimensionality. The retrievalfunction works effectively, when the dimensionality isless than 16; however, it weakens rapidly, for dimension-alities greater than 16, even down to the level of a linearsearch [23]. A sequential Psim matrix is used to solvethis problem. First, all the Psim values between points,S1, S2, ⋯, SM, are calculated to build a Psim matrix,PsimMat, with a size, M ×M. PsimMat(i, t) is composedof three properties: i, t, and Psim(Si, St). Next, the se-quential Psim matrix, SortPsimMat, is generated by sort-ing the elements in every row of PsimMat in thedescending order of the Psim value. The elements in thei-th row represent the similarities between Si and theother points. From left to right, the Psim values grad-ually diminish, indicating a decrease in the similarity. Itcan be seen that the sequential Psim matrix is not af-fected by the dimensionality and can represent the simi-larity distribution of all the points. Therefore, it issuitable for high-dimensional data clustering.

(1)

Li et al. EURASIP Journal on Audio, Speech, and Music Processing (2017) 2017:26 Page 2 of 9

2.2 Differential truncationThe elements in every row of SortPsimMat are regardedas a sequence, A, whose length is M. The sequential Psimdifferential matrix, DeltaPsimMat, is generated with a dif-ferential operation on the sequence, A. The size of DeltaP-simMat is M × (M − 1). The elements in the i-th row ofSortPsimMat represent the similarities between Si and theother points. Several points corresponding to the frontierelements in this row, from the left, would form a clustercentered at Si because the similarity between the elementsinside the cluster is higher than that of those outside.Thus, the similarity differences between the elements in-side the cluster are lesser than that of the others. Assum-ing that the cluster centered at Si has pi elements, the leftpi − 1 elements in the i-th row of DeltaPsimMat are lesserthan the differential threshold, ΔAmax. Thus, a reasonableΔAmax is set up and all the elements that are less thanΔAmax in the i-th row of DeltaPsimMat are found to forma cluster centered at Si.

2.3 Tabu SearchAfter differential truncation, the intersection of some ofthe clusters may not be null. Thus, the overlapping ele-ments should be eliminated by refinement. The clustersthat are to be refined are called the imminent-refiningcluster sets, and the initial values are the clusters afterdifferential truncation. The clusters that have been re-fined are called the refined cluster sets and their initialvalues are null. The refinement of the cluster is an itera-tive process. Considering the average Psim of the re-mainder elements in the i-th row of SortPsimMat, afterdifferential truncation, as the similarity of a cluster cen-tered at Si, the operation in every iteration is as follows:First, the similarity of every cluster is calculated. Next,the cluster with the highest similarity is added into therefined cluster set and the element in the other cluster isdeleted, if it is in the selected cluster. However, there is aproblem. After deleting the overlapping elements, thesimilarity of the cluster in the imminent-refining clusterset may be greater than that of the selected cluster. Tosolve this problem, Tabu Search is used for refinement.Tabu Search is an expansion of the neighborhood

search, a global optimum algorithm [11], and is mainlyused for combinatorial optimization. A roundaboutsearch can be avoided using the Tabu rule and aspirationcriterion, for improving the global search efficiency. Thismethod can accept an inferior solution and has a strong“climbing” ability; it has a higher probability of obtaininga global optimal solution.The main process of Tabu Search is as follows: Initially,

a random initial solution is regarded as the current solu-tion, and several neighboring solutions are considered asthe candidate solutions. Further, if the objective functionvalue of a certain candidate solution meets the aspiration

criterion, the current solution is replaced by this candidatesolution and added to the Tabu list. Else, the best choiceof a non-Tabu object is considered as the new current so-lution. In addition, the corresponding solution must beadded into the Tabu list [24]. The above steps are re-peated, until the terminate criterion is satisfied.In order to use Tabu Search for refining the cluster, an

appropriate Tabu object, Tabu list, aspiration criterion, andterminate criterion are required. The Tabu objects are theelements in the refined cluster set and are saved into theTabu list to prevent the Tabu Search from falling into thelocal optimum. The Tabu length is set as the number ofclusters after differential truncation. In every iterationprocess, the selected cluster is considered as the Tabu ob-ject. After eliminating the overlapping elements, the cluster,whose similarity is higher than that of the previously se-lected cluster, is considered as the better cluster and it re-places the previously selected cluster. The previouslyselected cluster is removed from the Tabu list and addedinto the imminent-refining cluster set. The above “eliminat-ing the overlapping elements—searching for a better clus-ter” process is repeated, until a better cluster can no longerbe found. Then, the previously selected cluster is consid-ered as the optimal cluster of this iteration. The search forthe better cluster of the next iteration then commences,until the imminent-refining cluster set is null.

3 Clustering algorithm3.1 Problem descriptionThe dataset of M audio signals with a length, n, is con-sidered as the point set, S = {S1, S2,⋯, SM}, of n-dimen-sional space, where Si = {Si1,⋯, Sij,⋯, Sin}, i = 1, 2, ⋯,M, j = 1, 2, ⋯, n, and Sij are the j-th property of Si.The goal is to search for sets, C1, C2, ⋯, CK, that meet

the following two requirements:

1. C1∪ C2∪ ∪ CK = S2. Cv∩ Cw = φ, for any 1 ≤ v ≠w ≤ K.

3.2 Framework of the clustering algorithmThe proposed clustering algorithm has four steps, asshown in Fig. 1. First, the sequential Psim matrix is builtto represent the similarity between the points in the set,S. Next, the initial cluster is generated by integrationwith the differential truncation and heuristic search. Fur-ther, the initial cluster is refined with Tabu Search. Fi-nally, the expected cluster is generated by clustering,based on the K-Medoids.

3.3 Clustering algorithm procedure3.3.1 Construction of a sequential Psim matrixThe Psim values between all the points in the set, S, arecalculated using Eq. 1 and saved into the Psim matrix,PsimMat. Then, the elements in every row of PsimMat

Li et al. EURASIP Journal on Audio, Speech, and Music Processing (2017) 2017:26 Page 3 of 9

are sorted with quicksort to obtain the sequential Psimmatrix, SortPsimMat. The above is a brief introduction;the detailed procedure can be found in [25].

3.3.2 Initial cluster generationFirst, the Laplacian matrix, L, is generated by PsimMat,and its eigenvalue distribution is used to determine thenumber of expected clusters [26]. Next, the differentialthreshold, ΔAmax, is initialized. Let Cmax be the maximaltime for searching the cluster set. The upper bound ofCmax is the combinatorial number, CK

M , where K is thenumber of clusters. Searching CK

M times is time-consuming because the magnitude of CK

M is M!. Inaddition, the K expected clusters maybe not found bysearching CK

M times. Thus, Cmax is set to Cmax =M and aheuristic search is implemented. Finally, the collision listof the clusters, TBLC, is set to null and i = 1. Then, theinitial cluster can be generated with the following steps.

Step 1: The elements in the i-th row of DeltaPsimMatare visited from left to right, until the pi-th element isgreater than the differential threshold, ΔAmax, for thefirst time.Step 2: The points corresponding to the left pi ‐ 1elements in the i-th row of SortPsimMat are used toconstruct a cluster, Ci

T , centered at Si.Step 3: If i <M, then i = i + 1; go to Step 1; else, c = 1.Step 4: K clusters, Ci1

T ;Ci1T ;⋯;CiK

T , are selected from Mclusters, C1

T ;C2T ;⋯;CM

T , to ensure that the setcomposed of K centers of the selected cluster are not inthe Tabu list,TBLC.Step 5: If the union of K clusters is equal to the set,S, then the set, Ci ¼ C0

i ;C1i ;⋯;CK

i

� �, is considered

as the initial cluster set, where Cvi ¼ Civ

T . Otherwise,the set CI is added into the TBLC; go to Step 6.

Step 6: If c ≥ Cmax, then i = 1, increase ΔAmax, clearTBLC, and go to Step 1. Otherwise, c = c + 1; go toStep 4.

3.3.3 Refinement of the initial clusterThe initial cluster set, CI, is considered as the imminent-refining cluster set, CRefining. The refined cluster set, CRe-

fined and the Tabu list are both null. The maximal searchtime is Fmax and a heuristic algorithm is used for thesearch. Fmax is set to Fmax = K because Fmax is propor-tional to the size of CI. The refinement procedure for CI

is as follows.

Step 1: The number of iterations, r = 0, and the numberof searches in the current iteration, f = 0.Step 2: The similarity of every cluster in CRefining iscalculated, and the cluster with the highest similarity isconsidered as the better cluster, COptimal, and movedinto the refined cluster set, CRefined. In addition, theselected cluster and similarity are added toTBLF.Step 3: The element in the reminder cluster, CRefining, isdeleted, if it is in the cluster, COptimal. Then, thesimilarity of every cluster in CRefining is calculated, andthe cluster with the highest similarity is expressed asCMaxPsim.Step 4: If the average similarity of every cluster inCMaxPsim is not more than those in COptimal or f ≥ Fmax,then go to Step 5. Otherwise, f = f + 1; go to Step 6.Step 5: If r ≥ K, then the refinement of the initial clusteris terminated. Otherwise, r = r + 1, f = 0; go to Step 2.Step 6: Cluster COptimal is moved back to CRefining fromCRefined and the corresponding information in the TBLFis deleted.Step 7: Cluster CMaxPsim is considered as the bettercluster, COptimal, and moved into CRefined. In addition,the corresponding information is added into theTBLF.Step 8: Go to Step 2.

3.3.4 Clustering based on iterative partitioningThe cluster, after Tabu Search, has the basic clustercharacteristics. For further improvement, clusteringbased on K-Medoids [27] is implemented.

3.4 Convergence analysisThe proposed clustering algorithm has four steps, andthe corresponding convergence analysis is as follows:

1. Construction of a sequential Psim matrixThe Psim matrix, PsimMat, is generated by runningEq. 1 M ×M times; the sequential Psim matrix,SortPsimMat, is generated by sorting the elementsin every row of PsimMat. The above operation canbe completed within a limited time; hence, this stepconverges.

Fig. 1 Framework of the proposed clustering algorithm

Li et al. EURASIP Journal on Audio, Speech, and Music Processing (2017) 2017:26 Page 4 of 9

2. Generating the initial clusterFirst, the number of expected clusters can bedetermined by spectral clustering in a limited time.Next, with the increase in the differential threshold,ΔAmax, the number of elements in every clusterincreases. Thus, the union, Ci1

T∪Ci1T∪L∪C

iKT , gets

closer to the set, S, gradually. Thus, this stepconverges.

3. Refinement of the initial clusterThis step iterates K times. In every iterationprocedure, the calculation of the K averagesimilarities of the cluster is carried out K times, atmost. The above operation can be completed inlimited time; thus, this step converges.

4. Clustering based on iterative partitioningThis step is based on the K-Medoids clustering algo-rithm, which converges naturally. Thus, this stepalso converges. The above statements show that thefour steps can be completed in limited time. Thus,the proposed clustering algorithm converges.

3.5 Time complexity analysisConsidering multiplication and comparison as the twobasic operations, the corresponding complexity analysisis as follows:

1. Construction of a sequential Psim matrixThe complexity of this step is reported to beO(M2 · n) [25].

2. Generating the initial clusterThe size of the Laplacian matrix, L, in Section 3.3.2is M ×M. Its top K eigenvalues are calculated withthe power iteration method [28], and the timecomplexity is O(K ·M2). The optimal number rangeof the clusters, Kopt, is 1≤Kopt≤

ffiffiffiffiffiM

p. Hence, the time

complexity for the calculation of the eigenvalues inthe Laplacian matrix, L, is O(M2.5). Further, theinitial cluster is generated by iterating several times.The analysis in every iteration process is as follows:First, the differential threshold, ΔAmax, is increasedand the elements in every row of DeltaPsimMat arevisited. The time complexity is O(M2), accordingly.Then, K clusters are selected and tested whethertheir union is equal to the set, S. The maximalcomparison time for calculating the union of twoclusters is M2n because the maximal number ofelements in a cluster is M and the dimension of anelement is n. Thus, the maximal comparison timefor the union of K clusters is (K ‐ 1)M2n. In addition,the selection operation of the K clusters areperformed Cmax =M times, at most. Therefore, thetime complexity of the selection process is O((K −1)M2n · Cmax) =O((K − 1)M2n ·M) =O(KM3n). Theoptimal number of clusters, Kopt, is less than

ffiffiffiffiffiM

p

[29]; i.e., O(KM3n) =O(M3.5n). Let the maximaliterations be H. Hence, the time complexity forgenerating the initial cluster is O(H ·M3.5n).

3. Refinement of the initial clusterIn this step, the basic operation is the calculation ofthe K similarities of the cluster. The maximalnumber of elements in every cluster is M. Thus, thenumber of addition operations is K ·M. This basicoperation is carried out K2 times, at most. Therefore,the total number of addition operations is K3 ·M;i.e., the time complexity in this step is O(K3M). Theupper bound of the optimal number of clusters isKopt ¼

ffiffiffiffiffiM

p[29]. Thus, the time complexity can be

expressed as O(M2.5).4. Clustering based on iterative partitioning

This step should be iteratively carried out Q times.In every iteration process, there are three basicoperations: the construction of K clusters, thecomputation of the K medoids of the clusters, andthe calculation of the objective functions Eq and E�

q .During these three basic operations, the Psim valueis calculated as K ·M, M and M times, respectively.Thus, the total number of Psim calculations in thisstep is Q · (KM +M +M) =Q(K + 2)M. From Eq. 1, itcan be seen that the time complexity of the Psimcalculation is O(n). Therefore, the time complexityof this step is O(Q(K + 2)M · n) =O(QKMn), whichis briefly expressed as O(QM1.5n) by virtue of theproperty [29] of the optimal number of clusters, 1≤Kopt≤

ffiffiffiffiffiM

p.

To sum up the above statements, the time complexityof the proposed clustering algorithm is O(M2 ⋅ n) +O(M2 ⋅ n) +O(M ⋅ n log n) +O(M2.5) +O(H ⋅M3.5n) +O(M2.5) +O(QM1.5n) =O(M2 ⋅ n) +O(M ⋅ n log n) +O(M2.5) +O(H ⋅M3.5n) +O(QM1.5n). Generally, the dif-ference in the magnitudes of M and n is negligible, i.e.,M > > log n. Thus, O(M ⋅ n log n) and O(M2.5) can be ig-nored, relative to O(M2 ⋅ n). The magnitudes of H and Qare the same because they are both iterations. Thus,O(QM1.5n) can be ignored relative to O(H ⋅M3.5n).Therefore, the time complexity can be briefly expressedas O(M2 ⋅ n) +O(H ⋅M3.5n) =O(H ⋅M3.5n).As can be seen from the above analysis, this algorithm

is a polynomial time algorithm, which can be carried outin a normal machine and condition.

4 Experiment4.1 OverviewIn the following experiment, the hardware includes anAMD Athlon(tm) II X2-250 processor and a Kingston 4Gmemory; the software used includes the Win7 operatingsystem and Microsoft Visual Studio 2012. The audio sig-nal data [30] is downloaded from the UCI database. This

Li et al. EURASIP Journal on Audio, Speech, and Music Processing (2017) 2017:26 Page 5 of 9

dataset is composed of 5000 audio vectors with lengths of41, and each one is produced by mixing a normal wave-form with noise; there are three categories.The number of clusters is determined using the spec-

tral clustering algorithm. Then, the test data is clusteredten times with the proposed clustering algorithm, basedon the Psim matrix and Tabu Search (PM-TS clusteringalgorithm), the K-Medoids clustering algorithm [29], andthe spectral clustering algorithm [26]. In the process ofeach clustering, the iterations, Macro-F1 and Micro-F1[31], are calculated. In addition, their average in tenclustering processes is required. Finally, these algorithmsare compared based on the above results.

4.2 Selection criteria for the compared algorithmIn our experiment, there are three criteria for selecting thecompared clustering algorithm: the selected algorithm mustbe widely recognized by academia and industry, it shouldbe suitable for high-dimensional data clustering and con-verge stably, and must be relevant to our algorithm.Based on the above criteria, the K-Medoids and the

spectral clustering algorithms were selected. Both arewidely used by academia and industry, can converge sta-bly, and are strongly related to our algorithm. In addition,they are also related to each other. The detailed analysis isas follows:

1. Correlation analysis of the K-Medoids clusteringalgorithm

The K-Medoids clustering algorithm is one of thefew clustering algorithms that is theoretically con-vergent. At the beginning of this algorithm, the ini-tial cluster is randomly selected, and subsequently,iterative clustering is done by the center of thenearest point close to the center of the cluster. The-oretically, iterative clustering is the key to conver-gence and not the initial cluster. The focus of ourstudy is to propose a clustering algorithm suitablefor high-dimensional data; convergence is a pre-requisite to be satisfied. Therefore, the iterativeclustering strategy of the K-Medoids clustering al-gorithm is adopted for convergence. In addition,the randomly selected initial cluster of the K-Medoids clustering is replaced by a refined non-overlapping cluster. Thus, our algorithm is derivedfrom the K-Medoids clustering algorithm.

2. Correlation analysis of the spectral clusteringalgorithmFirst, the spectral clustering algorithm procedure issimilar to that of the proposed and K-Medoids clus-tering algorithms. The clustering function can becompleted only by the adjacency matrix that storesthe similarity of the points and not by the vectorthat records the point coordinates, as in the K-Means clustering algorithm. Next, the spectral clus-tering algorithm is based on graph theory. The datapoints and their similarities are represented as the

Fig. 2 Running time for producing sequential Psim matrix ten times

Fig. 3 Searching times for producing initial clusters ten times

Fig. 4 Overlap of the initial cluster and that of the refined clusterafter running Tabu Search ten times

Fig. 5 Recall of the initial cluster and that of the refined cluster afterrunning Tabu Search ten times

Li et al. EURASIP Journal on Audio, Speech, and Music Processing (2017) 2017:26 Page 6 of 9

vertex and weight of the edge, respectively. Theeigenvector of the adjacency matrix is extractedfrom its Laplace matrix and is subsequently used forclustering. Because the number of eigenvectors isconsiderably lesser than the dimension of the points,it can be regarded as a dimensionality reductionclustering algorithm. Our algorithm is also of thesame type because the sparse and noisy dimensioncomponents do not participate in the computation.Hence, both the algorithms are similar with respectto the reduction in dimensionality. Finally, the num-ber of clusters used in the proposed and K-Medoidsclustering algorithms is calculated with the eigen-value decomposition of the Laplace matrix in thespectral clustering algorithm, i.e., some of the resultsof the spectral clustering algorithm are useful for theproposed and K-Medoids clustering algorithms; thus,these algorithms are strongly related to the spectralclustering algorithm.

4.3 Tabu Search analysisTabu Search, the core of our algorithm, can solve theoverlap problem of the initial cluster to improve its qual-ity. Ten Sequential Psim Matrix are generated, and therunning time is shown in Fig. 2. After that, the corre-sponding initial clusters are generated in accordance withthe method described in Section 3.3.2 and refined withthe method in Section 3.3.3 to produce ten sets of refined

clusters. The searching times for producing the initialclusters are shown in Fig. 3. The overlap, recall, precision,and running time of the initial and refined clusters are cal-culated, respectively, as shown in Figs. 4, 5, 6 and 7.It can be seen that 35–45% of the elements, which

were overlapping in the initial cluster, are eliminated byTabu Search. The recall and precision of the initial clus-ter were approximately 39 and 34%, respectively. AfterTabu Search, the recall reduced to approximately 33%,but the precision increased to approximately 39%. Thisis owing to the elimination of certain correct classifiedelements, while deleting the overlapping elements in thecluster, leading to a reduction in the recall. However, thenumber of error classified elements deleted by TabuSearch is more. Therefore, the proportion of correctclassified elements in the cluster increases. The search-ing times for producing the initial clusters is the randomnumber from 1 to 5000, because the maximal searchingtimes Cmax is M = 5000. But in most cases, the expectedinitial clusters can be found within 1500 times, and thecorresponding running time is less than 9 s. The upperbound of running time for refining cluster is total timeto construct all the permutations of cluster. In our ex-periment, the operation time is less than 0.5 s due to thelimited number of clusters (only three clusters).The experimental results are averaged and presented

in Table 1; they illustrate the role of Tabu Search in therefinement of the initial cluster. After Tabu search, theprecision increased from 33.9 to 38.4%, although the re-call reduced to 33.4%, satisfying the equilibrium distribu-tion of the precision and recall. In addition, severaloverlapping elements (approximately 40%) in the initialcluster are completely deleted with Tabu Search. The

Fig. 6 Precision of the initial cluster and that of the refined clusterafter running Tabu Search ten times

Fig. 7 Running time of the initial cluster and that of the refinedcluster after running Tabu Search ten times

Table 1 Average performances of the initial and refined clusterswith Tabu Search

Initial cluster Refined cluster

Overlap (%) 40.7 0

Recall (%) 39 33.4

Precision (%) 33.9 38.4

Running time (s) 7.36 0.49

Fig. 8 Iterations of the three algorithms, run ten times

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average running time for producing the initial clusters is7.36 s, and the one in refinement is 0.49 s. It can be seenthat the time load from cluster optimization is accept-able. Therefore, it can improve the quality of the clusterto a certain extent.

4.4 Stability analysisFirst, the method for determining the number of clus-ters, in Section 3.3.2, is applied to the test data. The re-sult is three, which corresponds to the existing data.Then, this dataset is clustered ten times with the PM-TSclustering algorithm, K-Medoids clustering algorithm[29], and the spectral clustering algorithm [26]. The cor-responding results are depicted in Figs. 8, 9, 10 and 11.It can be seen that the iterations for the PM-TS clus-

tering algorithm are lesser than those of the K-Medoidsand spectral clustering algorithms, indicating that ourproposed method can obtain a more precise initial clus-ter and converges faster. In addition, the whole runningspeed of PM-TS clustering algorithm is faster than theone of K-Medoids and spectral clustering algorithms. Inmost cases, the clustering accuracy (Macro-F1 andMicro-F1) and stability (variations in Macro-F1 andMicro-F1) are both of the order, PM-TS clustering algo-rithm > spectral clustering algorithm > K-Medoids clus-tering algorithm. The above results demonstrate theadvantages of the PM-TS clustering algorithm in termsof the speed, accuracy, and stability. In some cases, theclustering accuracies of the K-Medoids and spectralclustering algorithms are less than 50%, indicating a

clustering failure. However, the PM-TS clustering algo-rithm did not have similar issues, exhibiting its validity.

4.5 Whole-performance analysisThe experimental results are averaged and presented inTable 2; they illustrate the better performance of thePM-TS clustering algorithm compared to the K-Medoidsand spectral clustering algorithms. On the one hand, itsiterations are lesser and the convergence is fast as wellas the whole running speed. On the other hand, it hasno failure cases and Macro-F1/Micro-F1 increase morethan 13 and 10%, respectively. To sum up the above ana-lysis, the failure in the distance calculation, the ineffi-cient index tree, and the cluster overlap, derived fromthe characteristics of the high-dimensional data, can becorrected using the PM-TS clustering algorithm.

5 ConclusionsAudio signal clustering is critical in media computing.The key to improving its performance is the solving ofthe problems that exist in high-dimensional data cluster-ing, such as failures in the distance calculation, ineffi-cient index trees, cluster overlaps, etc. To address theseproblems, a clustering algorithm integrated with the se-quential Psim matrix, differential truncation, and TabuSearch is proposed. Compared to the other clustering al-gorithms, its characteristics are as follows: In high-dimensional space, the sequential Psim matrix is used tocalculate the distance and organize data. Differentialtruncation and Tabu Search are used to obtain the initialcluster with a high accuracy. Experimental results indi-cate that the performance of this algorithm is better than

Fig. 9 Whole running time of the three algorithms, run ten times Fig. 11 Micro-F1 of the three algorithms, run ten times

Fig. 10 Macro-F1 of the three algorithms, run ten times

Table 2 Average performances of the three algorithms

K-Medoidsclustering

Spectralclustering

PM-TSclustering

Iterations 14.2 21.8 12.8

Whole runningtime (s)

27.16 54.58 19.48

Macro-F1 (%) 50.71 46.94 59.81

Micro-F1 (%) 53.54 51.58 61.65

Li et al. EURASIP Journal on Audio, Speech, and Music Processing (2017) 2017:26 Page 8 of 9

that of the K-Medoids and spectral clustering algo-rithms. Several heuristic methods used in this algorithmhave a potential for improvement. Thus, our future workincludes the determination of more effective initial pa-rameters, evaluation functions, and convergence criteria,for improving the accuracy of the results.

AcknowledgementsWe would like to thank Editage (www.editage.com) for English languageediting and publication support.

FundingThis work is partly supported by the National Nature Science Foundation ofChina (No. 61502475) and the Importation and Development of High-CaliberTalents Project of the Beijing Municipal Institutions (No. CIT &TCD201504039).

Availability of data and materialsThe dataset supporting the conclusions of this article is available in the UCIdatabase, http://archive.ics.uci.edu/ml/datasets/Waveform+Database+Generator+%28Version+2%29.

Authors’ contributionsWL has conducted the research, analyzed the data, and authored the paper.GW has performed the overall design, providing new methods or models,and has written/revised the paper. All authors read and approved the finalmanuscript.

Ethics approval and consent to participateNot applicable.

Consent for publicationNot applicable.

Competing interestsThe authors declare that they have no competing interests.

Publisher’s NoteSpringer Nature remains neutral with regard to jurisdictional claims inpublished maps and institutional affiliations.

Author details1College of Information Technology, Beijing Union University, No. 97Beisihuan East Road, Beijing, China. 2Institute of Biophysics, Chinese Academyof Sciences, No. 15 Datun Road, Beijing, China.

Received: 6 April 2017 Accepted: 23 November 2017

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