Citation: Shahid Latif, Firuza Mustafa. A nonparametric copula distribution framework for bivariate joint distribution analysis of flood characteristics for the Kelantan River basin in Malaysia[J]. AIMS Geosciences, 2020, 6(2): 171-198. doi: 10.3934/geosci.2020012
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In data clustering or cluster analysis, the goal is to divide a set of objects into homogeneous groups called clusters [10,18,20,26,12,1]. For high-dimensional data, clusters are usually formed in subspaces of the original data space and different clusters may relate to different subspaces. To recover clusters embedded in subspaces, subspace clustering algorithms have been developed, see for example [2,15,19,17,9,21,16,22,3,25,7,11,13]. Subspace clustering algorithms can be classified into two categories: hard subspace clustering algorithms and soft subspace clustering algorithms.
In hard subspace clustering algorithms, the subspaces in which clusters embed are determined exactly. In other words, each attribute of the data is either associated with a cluster or not associated with the cluster. For example, the subspace clustering algorithms developed in [2] and [15] are hard subspace clustering algorithms. In soft subspace clustering algorithms, the subspaces of clusters are not determined exactly. Each attribute is associated to a cluster with some probability. If an attribute is important to the formation of a cluster, then the attribute is associated to the cluster with high probability. Examples of soft subspace clustering algorithms include [19], [9], [21], [16], and [13].
In soft subspace clustering algorithms, the attribute weights associated with clusters are automatically determined. In general, the weight of an attribute for a cluster is inversely proportional to the dispersion of the attribute in the cluster. If the values of an attribute in a cluster is relatively compact, then the attribute will be assigned a relatively high value. In the FSC algorithm [16], for example, the attribute weights are calculated as
wlj=1∑dh=1(Vlj+ϵVlh+ϵ)1α−1, l=1,2,…,k,j=1,2,…,d, | (1) |
where
Vlj=∑x∈Cl(xj−zlj)2. | (2) |
Here
wlj=exp(−Vljγ)∑ds=1exp(−Vlsγ), k=1,2,…,n,l=1,2,…,d, | (3) |
where
One drawback of the FSC algorithm is that a positive value of
w1=e−10e−10+e−30=11+e−20=1, w2=e−30e−10+e−30=11+e20=0. |
If we use
w1=e−1e−1+e−3=11+e−2=0.88, w2=e−3e−1+e−3=11+e2=0.12. |
From the above example we see that choosing an appropriate value for the parameter
In this paper, we address the issue from a different perspective. Unlike the group feature weighting approach, the approach we employ in this paper involves using the log transformation to transform the distances so that the attribute weights are not dominated by a single attribute with the smallest dispersion. In particular, we present a soft subspace clustering algorithm called the LEKM algorithm (log-transformed entropy weighting
The remaining part of this paper is structured as follows. In Section 2, we give a brief review of the LAC algorithm [9] and the EWKM algorithm [21]. In Section 3, we present the LEKM algorithm in detail. In Section 4, we present numerical experiments to demonstrate the performance of the LEKM algorithm. Section 5 concludes the paper with some remarks.
In this section, we introduce the EWKM algorithm [21] and the LAC algorithm [9], which are soft subspace clustering algorithms using the entropy weighting.
Let
F(U,W,Z)=k∑l=1[n∑i=1d∑j=1uilwlj(xij−zlj)2+γd∑j=1wljlnwlj], | (4) |
where
k∑l=1uil=1, i=1,2,…,n, | (5a) |
uil∈{0,1}, i=1,2,…,n,l=1,2,…,k, | (5b) |
d∑j=1wlj=1, l=1,2,…,k, | (5c) |
and
wlj>0, l=1,2,…,k,j=1,2,…,d. | (5d) |
Like the
uil={1, if ∑dj=1wlj(xij−zlj)2≤∑dj=1uiswsj(xij−zsj)2 for 1≤s≤k,0, if otherwise, |
for
wlj=exp(−Vljγ)∑ds=1exp(−Vlsγ) |
for
Vlj=n∑i=1uil(xij−zlj)2. |
Given
zlj=∑ni=1uilxij∑ni=1uil |
for
The parameter
The LAC algorithm (Locally Adaptive Clustering) [9] and the EWKM algorithm are similar soft subspace clustering algorithms in that both algorithms discover subspace clusters via exponential weighting of attributes. However, the LAC algorithm differs from the EWKM algorithm in the definition of objective function. Clusters found by the LAC algorithm are referred to as weighted clusters. The objective function of the LAC algorithm is defined as
E(C,Z,W)=k∑l=1d∑j=1(wlj1|Cl|∑x∈Cl(xj−zlj)2+hwljlogwlj), | (6) |
where
Like the
Sl={x:d∑j=1wlj(xj−zlj)2<d∑j=1wsj(xj−zsj)2,∀s≠l} | (7) |
for
wlj=exp(−Vlj)/h∑ds=1exp(−Vls/h) | (8) |
for
Vlj=1|Sl|∑x∈Sl(xj−zlj)2. |
Given the set of clusters
zlj=1|Sl|∑x∈Slxj | (9) |
for
Comparing Equation (6) with Equation (4), we see that the distances in the objective function of the LAC algorithm are normalized by the sizes of the corresponding clusters. As a result, the dispersions (i.e.,
In this section, we present the LEKM algorithm. The LEKM algorithm is similar to the EWKM algorithm [21] and the LAC algorithm [9] in that the entropy weighting is used to determine the attribute weights.
Let
P(U,W,Z)=k∑l=1n∑i=1uild∑j=1wljln[1+(xij−zlj)2]+λk∑l=1n∑i=1uild∑j=1wljlnwlj=k∑l=1n∑i=1uil[d∑j=1wljln[1+(xij−zlj)2]+λd∑j=1wljlnwlj], | (10) |
where
Similar to the EWKM algorithm, the LEKM algorithm tries to minimize the objective function given in Equation (10) iteratively by finding the optimal value of
Theorem 3.1. Let
uil={1, if D(xi,zl)≤D(xi,zs) for all s=1,2,…,k;0, if otherwise, | (11) |
for
D(xi,zs)=d∑j=1wljln[1+(xij−zsj)2]+λd∑j=1wljlnwlj. |
Proof. Since
f(ui1,ui2,…,uik)=k∑l=1uilD(xi,zl) | (12) |
is minimized. Note that
k∑l=1uil=1. |
The function defined in Equation (12) is minimized if Equation (11) holds. This completes the proof.
Theorem 3.2. Let
wlj=exp(−Vljλ)∑ds=1exp(−Vlsλ) | (13) |
for
Vlj=∑ni=1uilln[1+(xij−zlj)2]∑ni=1uil. |
Proof. The weight matrix
d∑j=1wlj=1, l=1,2,…,k, |
is the matrix
f(W)=P(U,W,Z)+k∑l=1βl(d∑j=1wlj−1) =k∑l=1n∑i=1uil[d∑j=1wljln[1+(xij−zlj)2]+λd∑j=1wljlnwlj] +k∑l=1βl(d∑j=1wlj−1). | (14) |
The weight matrix
∂f(W)∂wlj=n∑i=1uil(ln[1+(xij−zlj)2]+λlnwlj+λ)+βl=0 |
for
∂f(W)∂βl=d∑j=1wlj−1=0 |
for
From Equation (13) we see that the attribute weights of the
Theorem 3.3. Let
zlj=∑ni=1uil[1+(xij−zlj)2]−1xij∑ni=1uil[1+(xij−zlj)2]−1 | (15) |
for
Proof. If the set of cluster centers
∂P∂zlj=wljn∑i=1uil[1+(xij−zlj)2]−1[−2(xij−zlj)]=0. |
Since
n∑i=1uil[1+(xij−zlj)2]−1[−2(xij−zlj)]=0, |
from which Equation (15) follows.
In the standard
zlj=∑ni=1uil[1+(xij−z∗lj)2]−1xij∑ni=1uil[1+(xij−z∗lj)2]−1 | (16) |
for
To find the optimal values of
![]() |
The LEKM algorithm requires four parameters:
Parameter | Default Value |
1 | |
100 |
In this section, we present numerical experiments based on both synthetic data and real data to demonstrate the performance of the LEKM algorithm. We also compare the LEKM algorithm with the EWKM algorithm and the LAC algorithm in terms of accuracy and runtime. We implemented all three algorithms in Java and used the same convergence criterion as shown in Algorithm 1.
In our experiments, we use the corrected Rand index [8,13] to measure the accuracy of clustering results. The corrected Rand index is calculated from two partitions of the same dataset and its value ranges from -1 to 1, with 1 indicating perfect agreement between the two partitions and 0 indicating agreement by chance. In general, the higher the corrected Rand index, the better the clustering result.
Since the all the three algorithms are
To test the performance of the LEKM algorithm, we generated two synthetic datasets. The first synthetic dataset is a 2-dimensional dataset with two clusters and is shown in Figure 1. From the figure we see that the cluster in the top is compact but the cluster in the bottom contains several points that are far away from the cluster center. We can consider this dataset as a dataset containing noises.
Table 2 shows the average corrected Rand index of 100 runs of the three algorithms on the first synthetic dataset. From the table we see that the LEKM algorithm produced more accurate results than the LAC algorithm and the EWKM algorithm. The EWKM produced the least accurate results. Since the dispersion of an attribute in a cluster is normalized by the size of the cluster in the LAC and LEKM algorithms, the LAC and LEKM algorithms are less sensitive to the parameter.
Parameter | EWKM | LAC | LEKM |
1 | 0.0351 (0.0582) | 0.0024 (0.0158) | 0.9154 (0.2704) |
2 | 0.0378 (0.0556) | 0.9054 (0.2322) | 0.9063 (0.2827) |
4 | 0.012 (0.031) | 0.8019 (0.2422) | 0.9067 (0.2815) |
8 | -0.0135 (0.0125) | 0.7604 (0.2406) | 0.9072 (0.2799) |
16 | -0.013 (0.0134) | 0.7527 (0.2501) | 0.9072 (0.2799) |
Table 3 shows the confusion matrices produced by the best run of the three algorithms on the first synthetic dataset. We run the EWKM algorithm, the LAC algorithm, and the LEKM algorithm 100 times on the first synthetic dataset with parameter 2 (i.e.,
1 | 2 | 1 | 2 | 1 | 2 | |||||
C2 | 35 | 25 | C2 | 59 | 0 | C2 | 60 | 0 | ||
C1 | 25 | 15 | C1 | 1 | 40 | C1 | 0 | 40 | ||
(a) | (b) | (c) |
Table 4 shows the attribute weights of the two clusters produced by the best runs of the three algorithms. As we can see from the table that the attribute weights produced by the EWKM algorithm are dominated by one attribute. The attribute weights of one cluster produced by the LAC algorithm is also affected by the noises in the cluster. The attribute weights of the clusters produced by the LEKM algorithm seem reasonable as the two clusters are formed in the full space and approximate the same attribute weights are expected.
Weight | Weight | Weight | ||||||||
C1 | 1 | 3.01E-36 | C1 | 0.8931 | 0.1069 | C1 | 0.5448 | 0.4552 | ||
C2 | 1 | 2.85E-51 | C2 | 0.5057 | 0.4943 | C2 | 0.5055 | 0.4945 | ||
(a) | (b) | (c) |
Table 5 shows the average runtime of the 100 runs of the three algorithms on the first synthetic dataset. From the table we see that the EWKM algorithm converged the fastest. The LAC algorithm and the LEKM algorithm converged in about the same time.
Parameter | EWKM | LAC | LEKM |
1 | 0.0005 (0.0005) | 0.0021 (0.0032) | 0.0016 (0.0009) |
2 | 0.0002 (0.0004) | 0.0018 (0.0026) | 0.0013 (0.0006) |
4 | 0.0002 (0.0004) | 0.0017 (0.0025) | 0.0014 (0.0011) |
8 | 0.0003 (0.0004) | 0.0018 (0.0026) | 0.0016 (0.0017) |
16 | 0.0002 (0.0004) | 0.0018 (0.0025) | 0.0016 (0.002) |
The second synthetic dataset is a 100-dimensional dataset with four clusters. Table 6 shows the sizes and dimensions of the four clusters. This dataset was also used to test the SAP algorithm developed in [13]. Table 7 summarizes the clustering results of the three algorithms. From the table we see that the LEKM algorithm produced the most accurate results when the parameter is small. When the parameter is large, the attribute weights calculated by the LEKM algorithm become approximately the same. Since the clusters are embedded in subspaces, assigning approximately the same weight to attributes prevents the LEKM algorithm from recovering these clusters.
Cluster | Cluster Size | Subspace Dimensions |
A | 500 | 10, 15, 70 |
B | 300 | 20, 30, 80, 85 |
C | 500 | 30, 40, 70, 90, 95 |
D | 700 | 40, 45, 50, 55, 60, 80 |
Parameter | EWKM | LAC | LEKM |
1 | 0.557 (0.1851) | 0.5534 (0.1857) | 0.9123 (0.147) |
2 | 0.557 (0.1851) | 0.5572 (0.1883) | 0.928 (0.1361) |
4 | 0.557 (0.1851) | 0.5658 (0.1902) | 0.6128 (0.1626) |
8 | 0.557 (0.1851) | 0.574 (0.2028) | 0.3197 (0.1247) |
16 | 0.5573 (0.1854) | 0.6631 (0.2532) | 0.2293 (0.0914) |
Table 8 shows the confusion matrices produced by the runs of the three algorithms with the lowest objective function value. From the table we see that only three points were clustered incorrectly by the LEKM algorithm. Many points were clustered incorrectly by the EWKM algorithm and the LAC algorithm. Figures 2, 3, and Figure 4 plot the attribute weights of the four clusters corresponding to the confusion matrices given in Table 8. From Figures 2 and 3 we can see that the attribute weights were dominated by a single attribute. Figure 4 shows that the LEKM algorithm was able to recover all the subspace dimensions correctly.
Table 9 shows the average runtime of 100 runs of the three algorithms on the second synthetic dataset. From the table we see that the LEKM algorithm is slower than the other two algorithms. Since the center calculation of the LEKM algorithm is more complicate than that of the EWKM algorithm and the LAC algorithm, it is expected that the LEKM algorithm is slower than the other two algorithms.
Parameter | EWKM | LAC | LEKM |
1 | 0.7849 (0.4221) | 1.1788 (0.763) | 10.4702 (0.1906) |
2 | 0.7687 (0.4141) | 0.8862 (0.4952) | 10.3953 (0.1704) |
4 | 0.7619 (0.4101) | 0.8412 (0.4721) | 10.5236 (0.2023) |
8 | 0.7567 (0.4074) | 0.8767 (0.4816) | 10.5059 (0.2014) |
16 | 0.7578 (0.4112) | 0.8136 (0.5069) | 10.4122 (0.189) |
In summary, the test results on synthetic datasets have shown that the LEKM algorithm is able to recover clusters from noise data and recover clusters embedded in subspaces. The test results also show that the LEKM algorithm is less sensitive to noises and parameter values that the EWKM algorithm and the LEKM algorithm. However, the LEKM algorithm is in general slower than the other two algorithm due to its complex center calculation.
To test the algorithms on real data, we obtained two cancer gene expression datasets from [8]1. The first dataset contains gene expression data of human liver cancers and the second dataset contains gene expression data of breast tumors and colon tumors. Table 10 shows the information of the two real datasets. The two datasets have known labels, which tell the type of sample of each data point. The two datasets were also used to test the SAP algorithm in [13].
Dataset | Samples | Dimensions | Cluster sizes |
Chen-2002 | 179 | 85 | 104, 76 |
Chowdary-2006 | 104 | 182 | 62, 42 |
The datasets are available at http://bioinformatics.rutgers.edu/Static/Supplements/CompCancer/datasets.htm
Table 11 and Table 12 summarize the average accuracy and the average runtime of 100 runs of the three algorithms on the Chen-2002 dataset, respectively. From the average corrected Rand index shown in Table 11 we see that the LEKM algorithm produced more accurate results than the EWKM algorithm and the LAC algorithm did. However, the LEKM algorithm was slower than the other two algorithm.
Parameter | EWKM | LAC | LEKM |
1 | 0.025 (0.0395) | 0.0042 (0.0617) | 0.2599 (0.2973) |
2 | 0.0203 (0.0343) | 0.0888 (0.1903) | 0.2563 (0.2868) |
4 | 0.0135 (0.0279) | 0.041 (0.1454) | 0.2743 (0.2972) |
8 | 0.0141 (0.0449) | 0.0484 (0.1761) | 0.2856 (0.2993) |
16 | 0.0002 (0.0416) | 0.0445 (0.1726) | 0.2789 (0.2984) |
Parameter | EWKM | LAC | LEKM |
1 | 0.0111 (0.0031) | 0.0162 (0.0083) | 0.102 (0.0297) |
2 | 0.0123 (0.0033) | 0.0124 (0.006) | 0.1035 (0.0286) |
4 | 0.0143 (0.006) | 0.0151 (0.0105) | 0.1046 (0.0316) |
8 | 0.0122 (0.0043) | 0.0137 (0.0089) | 0.1068 (0.0337) |
16 | 0.0144 (0.007) | 0.014 (0.0091) | 0.105 (0.0323) |
The average accuracy and runtime of 100 runs of the three algorithms on the Chowdary-2006 dataset are shown in Table 13 and Table 14, respectively. From Table 13 we see than the LEKM algorithm again produced more accurate clustering results than the other two algorithm did. When the parameter was set to be 1, the LAC produced better results than the EWKM algorithm did. For other cases, however, the EWKM algorithm produced better results than the LAC algorithm did. The LAC algorithm and the EWKM algorithm are much faster than the LEKM algorithm as shown in Table 14.
Parameter | EWKM | LAC | LEKM |
1 | 0.3952 (0.3943) | 0.5197 (0.2883) | 0.5826 (0.3199) |
2 | 0.3819 (0.3825) | 0.19 (0.2568) | 0.5757 (0.3261) |
4 | 0.3839 (0.3677) | 0.0772 (0.1016) | 0.5823 (0.3221) |
8 | 0.4188 (0.3584) | 0.0595 (0.0224) | 0.5756 (0.3383) |
16 | 0.4994 (0.3927) | 0.0625 (0.0184) | 0.582 (0.3363) |
Parameter | EWKM | LAC | LEKM |
1 | 0.0115 (0.0048) | 0.0109 (0.0042) | 0.1369 (0.0756) |
2 | 0.011 (0.0046) | 0.0156 (0.0093) | 0.1446 (0.0723) |
4 | 0.0103 (0.0042) | 0.0147 (0.0076) | 0.1514 (0.0805) |
8 | 0.0107 (0.005) | 0.0141 (0.0063) | 0.1524 (0.0769) |
16 | 0.0113 (0.0047) | 0.0138 (0.0068) | 0.1542 (0.0854) |
In summary, the test results on real datasets show that the LEKM algorithm produced more accurate clustering results on average than the EWKM algorithm and the LAC algorithm did. However, the LEKM algorithm was slower than the other two algorithms.
The EWKM algorithm [21] and the LAC algorithm [9] are two soft subspace clustering algorithms that are similar to each other. In both algorithms, the attribute weights of a cluster are calculated as exponential normalizations of the negative attribute dispersions in the cluster scaled by a parameter. Setting the parameter is a challenge when the attribute dispersions in a cluster have a large range. In this paper, we proposed the LEKM (log-transformed entropy weighting
We tested the performance of the LEKM algorithm and compared it with the EWKM algorithm and the LAC algorithm. The test results on both synthetic datasets and real datasets have shown that the LEKM algorithm is able to outperform the EWKM algorithm and the LAC algorithm in terms of accuracy. However, one limitation of the LEKM algorithm is that it is slower than the other two algorithm because updating the cluster centers in each iteration in the LEKM algorithm is more complicate than that in the other two algorithms.
Another limitation of the LEKM algorithm is that it is sensitive to initial cluster centers. This limitation is common to most of the
The authors would like to thank referees for their insightful comments that greatly improve the quality of the paper.
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1. | Tongfeng Sun, 2018, Chapter 15, 978-3-030-00827-7, 140, 10.1007/978-3-030-00828-4_15 | |
2. | Qi He, Zhenxiang Chen, Ke Ji, Lin Wang, Kun Ma, Chuan Zhao, Yuliang Shi, 2020, Chapter 49, 978-3-030-16656-4, 530, 10.1007/978-3-030-16657-1_49 | |
3. | Guojun Gan, Yuping Zhang, Dipak K. Dey, Clustering by propagating probabilities between data points, 2016, 41, 15684946, 390, 10.1016/j.asoc.2016.01.034 |
Parameter | Default Value |
1 | |
100 |
Parameter | EWKM | LAC | LEKM |
1 | 0.0351 (0.0582) | 0.0024 (0.0158) | 0.9154 (0.2704) |
2 | 0.0378 (0.0556) | 0.9054 (0.2322) | 0.9063 (0.2827) |
4 | 0.012 (0.031) | 0.8019 (0.2422) | 0.9067 (0.2815) |
8 | -0.0135 (0.0125) | 0.7604 (0.2406) | 0.9072 (0.2799) |
16 | -0.013 (0.0134) | 0.7527 (0.2501) | 0.9072 (0.2799) |
1 | 2 | 1 | 2 | 1 | 2 | |||||
C2 | 35 | 25 | C2 | 59 | 0 | C2 | 60 | 0 | ||
C1 | 25 | 15 | C1 | 1 | 40 | C1 | 0 | 40 | ||
(a) | (b) | (c) |
Weight | Weight | Weight | ||||||||
C1 | 1 | 3.01E-36 | C1 | 0.8931 | 0.1069 | C1 | 0.5448 | 0.4552 | ||
C2 | 1 | 2.85E-51 | C2 | 0.5057 | 0.4943 | C2 | 0.5055 | 0.4945 | ||
(a) | (b) | (c) |
Parameter | EWKM | LAC | LEKM |
1 | 0.0005 (0.0005) | 0.0021 (0.0032) | 0.0016 (0.0009) |
2 | 0.0002 (0.0004) | 0.0018 (0.0026) | 0.0013 (0.0006) |
4 | 0.0002 (0.0004) | 0.0017 (0.0025) | 0.0014 (0.0011) |
8 | 0.0003 (0.0004) | 0.0018 (0.0026) | 0.0016 (0.0017) |
16 | 0.0002 (0.0004) | 0.0018 (0.0025) | 0.0016 (0.002) |
Cluster | Cluster Size | Subspace Dimensions |
A | 500 | 10, 15, 70 |
B | 300 | 20, 30, 80, 85 |
C | 500 | 30, 40, 70, 90, 95 |
D | 700 | 40, 45, 50, 55, 60, 80 |
Parameter | EWKM | LAC | LEKM |
1 | 0.557 (0.1851) | 0.5534 (0.1857) | 0.9123 (0.147) |
2 | 0.557 (0.1851) | 0.5572 (0.1883) | 0.928 (0.1361) |
4 | 0.557 (0.1851) | 0.5658 (0.1902) | 0.6128 (0.1626) |
8 | 0.557 (0.1851) | 0.574 (0.2028) | 0.3197 (0.1247) |
16 | 0.5573 (0.1854) | 0.6631 (0.2532) | 0.2293 (0.0914) |
Parameter | EWKM | LAC | LEKM |
1 | 0.7849 (0.4221) | 1.1788 (0.763) | 10.4702 (0.1906) |
2 | 0.7687 (0.4141) | 0.8862 (0.4952) | 10.3953 (0.1704) |
4 | 0.7619 (0.4101) | 0.8412 (0.4721) | 10.5236 (0.2023) |
8 | 0.7567 (0.4074) | 0.8767 (0.4816) | 10.5059 (0.2014) |
16 | 0.7578 (0.4112) | 0.8136 (0.5069) | 10.4122 (0.189) |
Dataset | Samples | Dimensions | Cluster sizes |
Chen-2002 | 179 | 85 | 104, 76 |
Chowdary-2006 | 104 | 182 | 62, 42 |
Parameter | EWKM | LAC | LEKM |
1 | 0.025 (0.0395) | 0.0042 (0.0617) | 0.2599 (0.2973) |
2 | 0.0203 (0.0343) | 0.0888 (0.1903) | 0.2563 (0.2868) |
4 | 0.0135 (0.0279) | 0.041 (0.1454) | 0.2743 (0.2972) |
8 | 0.0141 (0.0449) | 0.0484 (0.1761) | 0.2856 (0.2993) |
16 | 0.0002 (0.0416) | 0.0445 (0.1726) | 0.2789 (0.2984) |
Parameter | EWKM | LAC | LEKM |
1 | 0.0111 (0.0031) | 0.0162 (0.0083) | 0.102 (0.0297) |
2 | 0.0123 (0.0033) | 0.0124 (0.006) | 0.1035 (0.0286) |
4 | 0.0143 (0.006) | 0.0151 (0.0105) | 0.1046 (0.0316) |
8 | 0.0122 (0.0043) | 0.0137 (0.0089) | 0.1068 (0.0337) |
16 | 0.0144 (0.007) | 0.014 (0.0091) | 0.105 (0.0323) |
Parameter | EWKM | LAC | LEKM |
1 | 0.3952 (0.3943) | 0.5197 (0.2883) | 0.5826 (0.3199) |
2 | 0.3819 (0.3825) | 0.19 (0.2568) | 0.5757 (0.3261) |
4 | 0.3839 (0.3677) | 0.0772 (0.1016) | 0.5823 (0.3221) |
8 | 0.4188 (0.3584) | 0.0595 (0.0224) | 0.5756 (0.3383) |
16 | 0.4994 (0.3927) | 0.0625 (0.0184) | 0.582 (0.3363) |
Parameter | EWKM | LAC | LEKM |
1 | 0.0115 (0.0048) | 0.0109 (0.0042) | 0.1369 (0.0756) |
2 | 0.011 (0.0046) | 0.0156 (0.0093) | 0.1446 (0.0723) |
4 | 0.0103 (0.0042) | 0.0147 (0.0076) | 0.1514 (0.0805) |
8 | 0.0107 (0.005) | 0.0141 (0.0063) | 0.1524 (0.0769) |
16 | 0.0113 (0.0047) | 0.0138 (0.0068) | 0.1542 (0.0854) |
Parameter | Default Value |
1 | |
100 |
Parameter | EWKM | LAC | LEKM |
1 | 0.0351 (0.0582) | 0.0024 (0.0158) | 0.9154 (0.2704) |
2 | 0.0378 (0.0556) | 0.9054 (0.2322) | 0.9063 (0.2827) |
4 | 0.012 (0.031) | 0.8019 (0.2422) | 0.9067 (0.2815) |
8 | -0.0135 (0.0125) | 0.7604 (0.2406) | 0.9072 (0.2799) |
16 | -0.013 (0.0134) | 0.7527 (0.2501) | 0.9072 (0.2799) |
1 | 2 | 1 | 2 | 1 | 2 | |||||
C2 | 35 | 25 | C2 | 59 | 0 | C2 | 60 | 0 | ||
C1 | 25 | 15 | C1 | 1 | 40 | C1 | 0 | 40 | ||
(a) | (b) | (c) |
Weight | Weight | Weight | ||||||||
C1 | 1 | 3.01E-36 | C1 | 0.8931 | 0.1069 | C1 | 0.5448 | 0.4552 | ||
C2 | 1 | 2.85E-51 | C2 | 0.5057 | 0.4943 | C2 | 0.5055 | 0.4945 | ||
(a) | (b) | (c) |
Parameter | EWKM | LAC | LEKM |
1 | 0.0005 (0.0005) | 0.0021 (0.0032) | 0.0016 (0.0009) |
2 | 0.0002 (0.0004) | 0.0018 (0.0026) | 0.0013 (0.0006) |
4 | 0.0002 (0.0004) | 0.0017 (0.0025) | 0.0014 (0.0011) |
8 | 0.0003 (0.0004) | 0.0018 (0.0026) | 0.0016 (0.0017) |
16 | 0.0002 (0.0004) | 0.0018 (0.0025) | 0.0016 (0.002) |
Cluster | Cluster Size | Subspace Dimensions |
A | 500 | 10, 15, 70 |
B | 300 | 20, 30, 80, 85 |
C | 500 | 30, 40, 70, 90, 95 |
D | 700 | 40, 45, 50, 55, 60, 80 |
Parameter | EWKM | LAC | LEKM |
1 | 0.557 (0.1851) | 0.5534 (0.1857) | 0.9123 (0.147) |
2 | 0.557 (0.1851) | 0.5572 (0.1883) | 0.928 (0.1361) |
4 | 0.557 (0.1851) | 0.5658 (0.1902) | 0.6128 (0.1626) |
8 | 0.557 (0.1851) | 0.574 (0.2028) | 0.3197 (0.1247) |
16 | 0.5573 (0.1854) | 0.6631 (0.2532) | 0.2293 (0.0914) |
Parameter | EWKM | LAC | LEKM |
1 | 0.7849 (0.4221) | 1.1788 (0.763) | 10.4702 (0.1906) |
2 | 0.7687 (0.4141) | 0.8862 (0.4952) | 10.3953 (0.1704) |
4 | 0.7619 (0.4101) | 0.8412 (0.4721) | 10.5236 (0.2023) |
8 | 0.7567 (0.4074) | 0.8767 (0.4816) | 10.5059 (0.2014) |
16 | 0.7578 (0.4112) | 0.8136 (0.5069) | 10.4122 (0.189) |
Dataset | Samples | Dimensions | Cluster sizes |
Chen-2002 | 179 | 85 | 104, 76 |
Chowdary-2006 | 104 | 182 | 62, 42 |
Parameter | EWKM | LAC | LEKM |
1 | 0.025 (0.0395) | 0.0042 (0.0617) | 0.2599 (0.2973) |
2 | 0.0203 (0.0343) | 0.0888 (0.1903) | 0.2563 (0.2868) |
4 | 0.0135 (0.0279) | 0.041 (0.1454) | 0.2743 (0.2972) |
8 | 0.0141 (0.0449) | 0.0484 (0.1761) | 0.2856 (0.2993) |
16 | 0.0002 (0.0416) | 0.0445 (0.1726) | 0.2789 (0.2984) |
Parameter | EWKM | LAC | LEKM |
1 | 0.0111 (0.0031) | 0.0162 (0.0083) | 0.102 (0.0297) |
2 | 0.0123 (0.0033) | 0.0124 (0.006) | 0.1035 (0.0286) |
4 | 0.0143 (0.006) | 0.0151 (0.0105) | 0.1046 (0.0316) |
8 | 0.0122 (0.0043) | 0.0137 (0.0089) | 0.1068 (0.0337) |
16 | 0.0144 (0.007) | 0.014 (0.0091) | 0.105 (0.0323) |
Parameter | EWKM | LAC | LEKM |
1 | 0.3952 (0.3943) | 0.5197 (0.2883) | 0.5826 (0.3199) |
2 | 0.3819 (0.3825) | 0.19 (0.2568) | 0.5757 (0.3261) |
4 | 0.3839 (0.3677) | 0.0772 (0.1016) | 0.5823 (0.3221) |
8 | 0.4188 (0.3584) | 0.0595 (0.0224) | 0.5756 (0.3383) |
16 | 0.4994 (0.3927) | 0.0625 (0.0184) | 0.582 (0.3363) |
Parameter | EWKM | LAC | LEKM |
1 | 0.0115 (0.0048) | 0.0109 (0.0042) | 0.1369 (0.0756) |
2 | 0.011 (0.0046) | 0.0156 (0.0093) | 0.1446 (0.0723) |
4 | 0.0103 (0.0042) | 0.0147 (0.0076) | 0.1514 (0.0805) |
8 | 0.0107 (0.005) | 0.0141 (0.0063) | 0.1524 (0.0769) |
16 | 0.0113 (0.0047) | 0.0138 (0.0068) | 0.1542 (0.0854) |