1. Introduction
Techniques from statistical mechanics can be used for the investigation of the macroscopic properties of a physical system consisting of many elements. Recently, research activities utilizing statistical mechanical models or techniques for information processing have become increasingly popular.
Rose et al. [1] [2] proposed deterministic annealing (DA) as a deterministic variant of simulated annealing (SA) [3] . In DA, the minimization problem for an objective function is treated as the minimization of the free energy of a system. The DA approach tracks the function’s minimum with decreasing the system temperature, thus allowing the deterministic optimization of the objective function at each temperature. Hence, DA is more efficient than SA, but does not guarantee that the solution is the global optimal solution. From the viewpoint of statistical mechanics, the membership functions of the fuzzy c-means (FCM) clustering [4] with maximum entropy or entropy regularization methods [5] [6] can be seen as distribution functions from statistical mechanics. For example, FCM maximized with the Shannon entropy gives a membership function similar to the Boltzmann distribution function [1] .
Tsallis [7] , inspired by multi-fractal, non-extensively extended the Boltzmann? Gibbs statistics by postulating a generalized form of the entropy (the Tsallis entropy) with a generalization parameter q. The Tsallis entropy is proved to be applicable to the numerous systems [8] [9] . In the field of fuzzy clustering, a membership function was derived by maximizing the Tsallis entropy within the framework of FCM [10] [11] [12] . This membership function has a similar form to the statistical mechanical distribution function, and is suitable for use with annealing methods because it contains a parameter corresponding to the system temperature. Accordingly, the Tsallis entropy maximized FCM was successfully combined with the DA method as Tsallis-DAFCM in [13] .
One of the major challenges with using Tsallis-DAFCM is the determination of an appropriate value for
and the highest (or initial) annealing temperature,
, for a given data set. Especially, the determination of a suitable
value is a fundamental problem for systems where the Tsallis entropy is applied. Even in physics, quite a few systems are known in which
is calculable. In the previous study [13] , the values were experimentally determined, and only roughly optimized.
Accordingly, we presented a method that can determine both
and
simultaneously from a given data set without introducing additional parameters [14] . The membership function of Tsallis-DAFCM was approximated by a series expansion to simplify the function. Based on this simplified formula, both
and
could be estimated along with the membership function for a given data set. However, it was also found that the results from this method depend on the estimation of the radius of the distribution of the data or the location of clusters.
To overcome this difficulty, in this study, we propose a method that utilizes K-means [15] as a preprocessing step of the approximation method. That is, a data set is clustered by K-means roughly. We then estimate the radius of the distribution of the data set, and apply the approximation method to determine
and
.
Experiments are performed on numerical data and the Iris Data Set [16] , and the results show that the proposed method can be used to determine
and
automatically and algebraically from a data set. It is also confirmed that the data can be partitioned into clusters appropriately using these parameters.
2. FCM with Tsallis Entropy Maximization
Let
be a data set in p-dimensional real space, and let
be the
distinct clusters. Let
be the membership function, and let
(1)
be the objective function of FCM, where
.
On the other hand, the Tsallis entropy is defined as
(2)
where
is the probability of the
th event and
is a real number [7] . The Tsallis entropy reaches the Shannon entropy as
.
Next, we apply the Tsallis entropy maximization method to FCM [12] [13] . First, Equation (2) is rewritten as
(3)
Then, the objective function in Equation (1) is rewritten as
(4)
Under the normalization constraint of
(5)
the Tsallis entropy functional becomes
(6)
where
and
are the Lagrange multipliers. By applying the variational method, the stationary condition for the Tsallis entropy functional yields the following membership function for Tsallis-FCM [12] :
(7)
where
(8)
From Equation (7), the expression for
becomes
(9)
3. Approximation of Membership Function
The performance of Tsallis-DAFCM is superior to those of other entropy-based- FCM methods [12] . However, it is still unknown how to determine an appropriate
value and a highest annealing temperature
for a given data set. To tackle this problem, we first simplify the membership function using a series expansion.
3.1. Series Expansion of
in Equation (7) can be expanded to a power of
as follows:
(10)
When the temperature is high enough, if the series expansion up to the third order terms is used, Equation (10) becomes
(11)
where
(12)
3.2. Determination of
and
Based on the results in Section 3.1, we propose a method for determining both
and
simultaneously.
First, to ensure the convergence of Equation (10), we use the following expression for
:
(13)
where
and
denote the maximum number of iterations, and the number of iterations to be used in the calculation of
, respectively.
can be calculated as
.
Then, setting
and replacing
with the continuous variable
, Equation (11) becomes
(14)
where
(15)
From this equation,
can be determined as follows. By designating the range of the dataset as
, the maximum range of the distribution
is defined as
(16)
Furthermore, by assuming that the radius
of each cluster is between
, and
tends to
at
, Equation (14) can be solved for
. Consequently, we have the following formula for
.
(17)
It should be noted that in this equation, for simplicity,
is set to
(18)
because Equation (7) tends to
as
goes to
.
4. Proposed Algorithm
By combining the method presented in the previous section with Tsallis- DAFCM, we proposed the following fuzzy c-means clustering algorithm [14] . In this algorithm, the number of clusters in the data is assumed to be known in advance.
In the first algorithm shown in Figure 1, the parameters
and
for a given data set are determined (
is the maximum number of iteration. In Equation (17),
and
are approximated by
and
, respectively.).
The second algorithm is the conventional Tsallis-DAFCM algorithm [12] .
1) Set the temperature reduction rate
, and the thresholds for convergence
and
.
2) Generate c initial clusters at random locations. Set the current temperature
to
.
3) Calculate
using Equation (7).
4) Calculate the cluster centers using Equation (9).
5) Compare the difference between the current centers and the centers of the previous iteration obtained using the same temperature
. If the convergence condition
is satisfied, then go to Step 2.6. Otherwise re-
Figure 1. Processing flow of the conventional method.
turn to Step 2.3.
6) Compare the difference between the current centers and the centers of the previous iteration obtained using a lower temperature
. If the convergence condition
is satisfied, then stop. Otherwise decrease the temperature;
, and return to Step 2.3.
The experimental results in [14] confirmed that the first algorithm can determine
desirably. However, they also revealed that q from this algorithm strongly depends on the estimation of the radius r in Equation (17). Accordingly, as shown in Figure 2, the first algorithm is divided in two parts. The first one determines
. In the second part, the K-means algorithm is utilized to calculate r by assuming that each data point belongs to its nearest cluster.
5. Experiments
To examine the effectiveness of the proposed algorithm, we conducted two experiments.
Figure 2. Processing flow of the proposed method.
5.1. Experiment 1
The first experiment examined whether appropriate
and
values can be determined for a given data set, and the relation between the number of iterations
and the parameters
and
.
In this experiment, data sets containing (a) three clusters and (b) five clusters were used, as shown in Figure 3. Each cluster follows a normal distribution, and contains 2, 250 data points.
Dependencies of the maximum, minimum, mean and standard deviation of
, a mean radius of the data distribution and q for Figure 3(a) on the number of iterations N are summarized in Table 1, Table 2 and Table 7. Figure 4 shows the plots of the maximum, minimum, and mean of q. In these tables,
and
denote
and the mean of
and
, respectively.
and
, on the other hand, denote the maximum and mean radius of the distribution obtained by K-means, respectively.
In Table 7, the value of q for rmax for example is calculated using Equation (17) as
. Based on the results in Table 1, the value of q was calculated by
(a)(b)
Figure 3. Numerical data (
denotes the cluster number). (a)
; (b)
.
Figure 4. Maximum, minimum, and mean of
(
,
).
Table 1. Maximum, minimum, mean, and standard deviation of
(
).
Table 2. Maximum, minimum, mean, and standard deviation of
and
(
).
fixing
to its mean value 5.351e−06.
,
, and
for Figure 3(a) are 860.0, 430.0 and 286.7, respectively.
From Table 1, it can be seen that the maximum of
tends to increase and the minimum of
tends to decrease with increasing N. However, when N become 100 or more, the mean of
does not depend on N.
From Table 2, it can be seen that the mean of
and
hardly depends on N, though the standard deviation becomes larger when N become 1, 000 or more. This is caused by a very seldom misclassification of K-means.
Comparing the results in Table 7, it can be found that, when r is set to
or
, q has smaller standard deviations, and the magnitude of the change in the mean values of q is comparatively small. This shows that q can be calculated stably by performing K-means first. It is also can be found that the maximum of q increases with increasing N, because of the random locations of clusters. Even though
overestimates the mean radius of the clusters, clustering can be performed properly in this case.
Accordingly,
has little impact on clustering in this experiment.
Dependencies of the maximum, minimum, mean and standard deviation of
, a mean radius of the data distribution and
for Figure 3(b) on the number of iterations
are summarized in Table 3, Table 4 and Table 8. Figure 5 shows the plots of the maximum, minimum, and mean of
.
Figure 5. Maximum, minimum, and mean of
(
,
).
Table 3. Maximum, minimum, mean, and standard deviation of
(
).
Table 4. Maximum, minimum, mean, and standard deviation of
and
(
).
,
, and
for Figure 3(b) are 860.0, 430.0 and 258.0, respectively. Based on the results in Table 3, the value of
was calculated by fixing
to 3.608e−06.
Comparing these results with those in Table 1, Table 2 and Table 7, it can be found that
for
has larger standard deviations than those for
. This is caused by an increase in the number of combinations of data points and clusters.
In Table 8, it can be seen that
for
has the largest standard deviations. This is considered to be caused by the significant standard deviations of
shown in Table 2, suggesting a variation of the estimation of the radius of the distribution. On the other hand,
for
has the smallest standard deviations.
Substituting the values of
and
in Table 3 and Table 8 directly, Figure 6 and Figure 7 compare the membership function for the cluster
,
(19)
with
(20)
for
,
and
, and
and 10,000. In the equations,
is set to each of the cluster coordinates in Figure 3(b). The data projections on the xz and yz planes are also plotted.
The figures show no significant difference between
and
and between
and
.
Figure 6. Comparisons of the membership functions calculated by Equations (19) and (20) (
,
,
).
Figure 7. Comparisons of the membership functions calculated by Equations (19) and (20) (
,
,
).
Compared with the clusters in Figure 3(a), those in Figure 3(b) are not aligned in a straight line. However, the results for
are as accurate as those for
. As a result, the maximum error factor is considered to be
. Since the clusters in Figure 3(a) are aligned in a straight line,
cannot be determined optimally by locating clusters randomly as does in the algorithm in Figure 1.
From these results, it can be confirmed that
is sufficient to determine both
and
for the data sets in Figure 1.
5.2. Experiment 2
In this experiment, the Iris Data Set [16] , which comprises data from 150 iris flowers with four-dimensional vectors, is used. The three clusters to be detected are Versicolor, Virginia and Setosa, and the parameters in the algorithm in Figure 2 are set as follows:
, and
, and
.
,
, and
are 5.90, 2.95 and 1.97, respectively.
5.2.1. Determination of Parameters
The maximum, minimum, mean, and standard deviation of
,
,
and
are summarized in Table 5, Table 6 and Table 9. Figure 8 shows the plots of the maximum, minimum, and mean of
. Based on the results in Table 5, the value of
was calculated by fixing
to 1.076e-01.
From Table 5, it can be seen that a dependency of
on
is same as those in Table 1 and Table 3. Table 6 shows that the mean of
and
Figure 8. Maximum, minimum, and mean of
for the Iris data set (
).
Table 5. Maximum, minimum, mean, and standard deviation of
for the Iris data set.
Table 6. Maximum, minimum, mean, and standard deviation of
and
for the Iris data set.
can be calculated regardless of the value of
.
Table 9 shows that the standard deviations of q for
and
are smaller than those of
and
showing the effectiveness of the proposed method.
It can be found that these tables show that the proposed method gives similar results to those in the Section 5.1, and
to 10 is sufficient to determine
,
,
, and
. In the algorithm shown in Figure 1, it is unnecessary to repeat the calculations of the means of
and
the same number of times
.
It is also found that not only the estimations of the radius are important to improve the accuracy because
gives superior result compared with those of
. For this reason, a preprocessing method that can estimate the location of clusters quickly, such as the Canopy method [17] might be suitable for the proposed method to be more effective.
5.2.2. Clustering Accuracy
The maximum and mean number of data points misclassified by the previous method [14] , the proposed method, and Tsallis-DAFCM in 1, 000 trials are summarized in Table 10 and Figure 9.
is fixed to 1/1.076e−01 = 9.294. In Tsallis-DAFCM, as a typical value,
is changed from 1.2 to 2.8.
Even though the experiment was repeated 1000 times, the results obtained with the proposed method were almost identical.
By comparing the mean number of misclassified data points of the proposed method with those of the previous method, it can be confirmed the results from both methods are not significantly different when
and
or when
and
.
By comparing the mean number of misclassified data points of the proposed method with those of Tsallis-DAFCM, it can be confirmed the proposed method can get slightly better results. By examining the maximum number of misclassified, we see that Tsallis-DAFCM misclassifies data more often than does the proposed method.
These results confirm that appropriate values of
and
for the Iris Data Set can be estimated by the proposed method. Setting
is most suitable for this data set.
5.2.3. Computational Time
Figure 10 compares the mean of computational times of
and
, and clus-
Figure 9. Maximum, minimum, and mean numbers of misclassified data points for the Iris Data Set of the previous method, the proposed method and Tsallis-DAFCM (
,
).
(a)(b)(c)
Figure 10. Mean of computational times of
, and clustering for the Iris Data Set. (a)
; (b)
; (c) Clustering.
tering in 1000 trials (Executions were conducted on an Intel(R) Core(TM)2 Duo CPU E6550 @ 2.33 GHz).
Figure 10(a) shows that the computational time of
does not depend on
and increases proportionally to
because, as can be seen from Equations (12) and (13), the value of
is determined independently of
and
is calculated
times.
Figure 8(b), on the other hand, shows that the calculation of
for
sometimes takes time suggesting that, in this case,
becomes too large to give an appropriate
value.
Figure 10(c) shows that that when
is set to
or
, clustering can be conducted quickly and stably.
5.3. Evaluation of the Proposed Algorithm
From the experimental results in 5.1 and 5.2, the effectiveness of the proposed algorithm using K-means can be evaluated as follows:
1)
and
can be obtained with very small variances without consuming much computational time;
2)
can be determined with a very small variance using
without consuming;
3) Much computational time;
4) The numerical data sets and the Iris Data Set can be clustered desirably using
.
6. Conclusions
The Tsallis entropy is a q-parameter extension of the Shannon entropy. FCM with the Tsallis entropy maximization has a proper characteristic for clustering, especially when it is combined with DA as Tsallis-DAFCM. The extent of its membership function strongly depends on the parameter
and the initial annealing temperature
.
In this study, we proposed a method for approximating the membership function of Tsallis-DAFCM which, by using the K-means method as a preprocessing step, determines
and
automatically and algebraically from a given data set.
Experiments were performed on the numerical data sets and the Iris Data Set, and showed that the proposed method can more accurately and stably determine
and
algebraically than the previous method without consuming much computational time. It was also confirmed that the data can be partitioned into clusters appropriately using these parameters.
In the future, as described in 5.1, we first intend to explore ways to improve the accuracy of the estimates for
and
by using other rough clustering methods. We then intend to examine the effectiveness of the method using very complicated real world data set [18] .
Appendix
Table 7. Maximum, minimum, mean, and standard deviation of
(
,
).
Table 8. Maximum, minimum, mean, and standard deviation of
(
,
).
Table 9. Maximum, minimum, mean, and standard deviation of
for the Iris Data Set (
).
Table 10. Maximum, minimum, and mean numbers of misclassified data points for the Iris Data Set of the previous method, the proposed method and Tsallis-DAFCM (
,
).
Table 11. Computational times of
,
, and clustering for the Iris data set.