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Kleinberg’s Hyper-Richness Based Fuzzy Partition Clustering for Efficient Bi-Temporal Data

Clustering is the process used for partitioning the total dataset into different classes of similar objects. The group contains knowledge about their members and also helps to understand the structure of the dataset very easily. Clustering the bitemporal data is one of the major tasks in data mining since the bitemporal datasets are very large with various attribute counts. Hence the accurate clustering is still challenging tasks. In order to improve the clustering accuracy with less complexity, Kleinberg’s Hyper-richness Bitemporal property based fuzzy c means partition Clustering (KHBP-FCMPC) technique is introduced. The KHBP-FCMPC technique partition the bitemporal dataset into number of possible groups with an improved performance rate based on a distance metric. At first, the ‘c’ numbers of clusters are initialized. The KHBP-FCMPC technique uses the core data point module and authority sector module to minimize the execution time of clustering the data points. Core data point module served as the centroid of the cluster. Each cluster contains one core data point. After that, the distance is computed with the membership function. The authority sector module assigns the data points into the cluster with minimum distance. After that, the centroid is updated and the process iterated until the convergence is met. Finally, the Kleinberg’s Hyper-richness Bitemporal property is applied to verify the total dataset equals the partition of all the data points. This property used to group the entire data points into the cluster with higher accuracy. Experimental evaluation is carried out using a temporal dataset with different factors such as clustering accuracy, false positive rate, time complexity and space complexity with a number of data points. The experimental results show that the proposed KHBP-FCMPC technique increases the bitemporal data clustering accuracy with less false positive rate, time complexity as well as space complexity. Based on the results observations, KHBP-FCMPC technique is more efficient than the state-of-the-art methods.


INTRODUCTION
Clustering is the data mining method which partitions a whole dataset into different groups based on their similarity. The data within the groups are more similar other than the data in different clusters. In general, clustering of temporal data is more complex since the dimensionality of the dataset is considerably larger. Temporal data is the data which is stored related to the time instances. The temporal data comprises two attributes such as transaction time and valid time. Transaction time records the time period during which the information stored in the temporal database. Valid time is the time for which an event is true in the real world. These two attributes are united to form bitemporal data. In this research, the clustering techniques are used for bitemporal relational data analysis.
In general, the various clustering technique is presented such as the density-based technique, partitioning techniques, Hierarchical techniques, Grid-based methods and so on. Among the several techniques, partitioning based clustering technique is used for bitemporal relational data analysis. The advantage of the partition-based clustering algorithm is it uses an iterative method for generating the clusters where the entire data are grouped. The Bi-weighted ensemble approach was introduced in [1] for partitioning the temporal data through Hidden Markov Model-based approaches. The ensemble clustering approach increased the performances of temporal data clustering but failed to attain high accuracy with less complexity. A Parametric Link among Multinomial Mixtures (PLMM) model was introduced in [2] for partitioning the temporal categorical data. Though the method increased the clustering accuracy, the time complexity was not minimized.
An algorithm termed as variational Expectation-Maximization (VEM-DyMix) was developed in [3] for grouping the temporal data. The algorithm increased the clustering and estimation accuracy with minimal computation time but the space complexity was not minimized. A generalized k means clustering technique was introduced in [4] for grouping the similar temporal data through weighted and kernel time. Though the technique minimizes the time complexity, the clustering performance was not increased. Clustering a temporal network was presented in [5] through the topological similarity measure. The approach increased the accurate clustering results but the false positive rate was not minimized in an efficient manner. A finite mixture model was introduced in [6] for dynamic clustering of spatiotemporal data with minimal error.
The model discovered the level-based clusters in spatiotemporal data but the time and space complexity were not minimized. A Hidden Markov Model-based hybrid meta-clustering ensemble method was introduced in [7] for increasing the temporal data clustering analysis. The technique failed to solve the more time-consuming problems during clustering analysis. The temporal human activity patterns were recognized using a clustering technique based on temporal similarity [8]. The performance of clustering accuracy was not improved for efficient activity recognition.
The Creating Discriminative models were developed in [9] for categorization and partitioning the time series data. Though the model increases the clustering accuracy, the time complexity was not minimized. A novel hybrid clustering algorithm was introduced in [10] based on the similarity of time series data. The algorithm attained accurate clustering results but the false positive rate was not minimized. The major issues are identified from the above-said literature such as minimum clustering accuracy, high time and space complexity, high false positive rate, failure to attain accurate clustering results and so on.
In order to overcome the major issues, an efficient technique called, Kleinberg's Hyper-richness Bitemporal property based fuzzy c means partition Clustering (KHBP-FCMPC) is introduced. The major contributions of the proposed KHBP-FCMPC technique compared to the existing works are summarized as follows,  To acquire the high clustering accuracy with minimal complexity, KHBP-FCMPC technique is introduced. The KHBP-FCMPC technique uses hyper-richness property based fuzzy 'C' means partition clustering. The 'C' number of clusters and centroids are initialized. The Kleinberg's Hyper-richness uses the core data point module as the centroid of the clusters. The authority sector module computes the distance between the centroid and data points. Then the gradient descent function used to find the minimum distance between the centroid and data points. After that, the authority sector module groups similar data points into the cluster. This process minimizes the time and space complexity.  To increase the clustering accuracy and minimize the false positive rate, the Kleinberg's Hyper-richness property is applied. The property is employed to correctly group all the similar data points into the cluster.
The paper is ordered as follows. Section 2 reviews the related works. In section 3, the description of the KHBP-FCMPC technique is presented with a neat diagram. Experimental evaluation of proposed KHBP-FCMPC technique and state-of-art methods are discussed with the temporal dataset in section 4. Section 5 provides the experimental results and discussion of certain parameters with the table and graphical representation. Finally, section 6 concludes the proposed work.

II. RELATED WORKS
A novel spatial and temporal similarity measurements based clustering technique was introduced in [11]. The accurate data clustering was not performed with minimal time complexity. An enhanced spatiotemporal clustering technique was introduced in [12] for grouping the spatiotemporal data with less runtime. But the technique failed to minimize the false positive rate during the clustering process.
A weighted clustering ensemble algorithm was developed in [13] for grouping the temporal data with high quality. Though the ensemble algorithm minimizes the computational complexity, the algorithm was complex on processing the bitemporal dataset correctly without any prior knowledge. A hierarchical aligned cluster analysis (HACA) was performed in [14] for grouping and visualizing the time series data. This clustering technique failed to minimize the computational complexity in terms of both space and time. An adaptive method was introduced in [15] for grouping the spatiotemporal events by finding the nearest neighbor. The adaptive method failed to redefine the distance between the events for grouping the entire data. Though the method minimizes the time complexity, the space complexity was not minimized. A hierarchical trajectory clustering technique was introduced in [16] for spatiotemporal periodic pattern mining. But, the technique was not suitable for multi-level time periods and complex datasets. A temporal-constrained sub-trajectory cluster analysis was performed in [17] using partition based clustering technique. The cluster analysis was not directly applicable to large datasets. A mixed fuzzy clustering (MFC) algorithm was developed in [18] for grouping the data with dynamic time warping (DTW) distance. Though the algorithm increases the clustering accuracy, the performance of the false positive rate remained unsolved. The Temporal Pattern Miner (TPMiner) and Probabilistic Temporal Pattern Miner (P-TPMiner) were developed in [19] to effectively determine the temporal pattern and probabilistic temporal patterns respectively. The methods outperform well with less runtime and memory usage. But the accurate clustering was not improved. A dynamic stochastic block model was introduced in [20] for grouping the temporal data across the various time steps. But the model was not efficient for grouping the temporal data with less complexity.
HMM-based hybrid meta-clustering ensemble was designed in [21] to solve the problems of initialization and model selection for temporal data clustering. But the model did not reduce computational cost. Sequential Subspace Clustering via Temporal Smoothness [22] for minimizing the exponential variance of short term historical data. Efficient Data Stream Clustering with Sliding Windows Based on Locality-Sensitive Hashing was introduced in [23] for improving data stream clustering over sliding windows. But a fixed number of clusters should be specified before clustering. The issues are identified from the above-said methods are addressed by introducing a novel technique called KHBP-FCMPC technique. The processes of KHBP-FCMPC technique are explained in the next section.

III. KLEINBERG'S HYPER-RICHNESS BASED FUZZY C MEANS PARTITION CLUSTERING WITH BITEMPORAL DATA
Bitemporal data clustering is the process of partitioning the huge dataset into dissimilar groups over the time instances. The data points within the group are similar to each other. The processing of large dataset is very difficult using conventional data processing tools. In this case, partitions the large dataset into different groups is essential for identifying the complete knowledge about the data elements and also help the users to understand the structure of the dataset. The conventional clustering technique increases the complexity and not much efficient for processing the large dataset. Therefore, an effective clustering technique is required to group the similar type of data with less complexity. Based on this motivation, Kleinberg's Hyper-Richness Bitemporal property based fuzzy c means Partition Clustering (KHBP-FCMPC) technique is introduced. In KHBP-FCMPC technique, the author Jon Kleinberg's defines the richness property while clustering the data points. The richness property defines that any partition of the bitemporal data is obtained by modifying the distance function. Based on this property, the clustering is performed to attain higher accuracy and minimal complexity. The architecture of the proposed KHBP-FCMPC technique is illustrated in figure 1.   Figure 1 illustrates an architecture diagram of KHBP-FCMPC technique to group the bitemporal data with high accuracy. Initially, the bitemporal data points are collected from the large dataset. After collecting the data points, the numbers of clusters are initialized randomly for partitioning the data points into different groups. While partitioning the data points, the richness property is applied to find more similar data. The KHBP-FCMPC technique uses the core data point module (i.e. cluster centroid) and authority sector module to minimize the data clustering time. Core data point module served as the center of the cluster. Authority sector module authorizes (i.e. allows) the data points into the cluster based on the distance measure. The clustering process is described as follows.
Let us consider the number of bitemporal data collected from the bitemporal dataset . (1) From (1), represents the number of bitemporal data, represents the bitemporal dataset. Initially, 'c' numbers of clusters are initialized. For each cluster, the centroid is computed for grouping data points into the cluster. In KHBP-FCMPC technique, the membership grades are allocated to each data points which specify the degree to which data points belongs to the cluster. The centroid of the cluster defines the mean of entire data points in that cluster. Therefore, the centroid is computed using the following mathematical equations, From (2), denotes a cluster centroid, any data point ' ' has a set of coefficients giving the degree of being in the cluster , represents the fuzzifier determines the level of cluster fuzziness. The centroid of the cluster is a core data point module. The objective function of the KHBP-FCMPC technique is expressed as follows, From (3), denotes a objective function, represents the membership value, represents the fuzzifier, represents the bitemporal data, denotes a cluster centroid, represents the Euclidean distance between the data ( and the cluster centroid . The above equation (3) shows that the clustering algorithm computes the objective function by the multiplication of the membership values . The membership is computed based on the distance function, From (4), denotes a membership values, represents the bitemporal data, , denotes a centroid of the cluster. The KHBP-FCMPC technique uses the gradient descent function to minimize the objective function which is defined as follows, From (5), arguments of the minimum are abbreviated as , represents the objective function. After that, the authority sector module in the proposed technique assigns the data which is closer to the centroid of the cluster. Therefore, the data which is closest to the centroid is said to be a member of that specific cluster. Finally, the cluster centroid is updated to group all the data points into the cluster. After that, the proposed technique uses the Kleinberg's Hyper-Richness Bitemporal property to groups the entire data points into the cluster by changing the distances between the data point and cluster centroid.

Hyper-Richness Bitemporal property:
Let us consider the total bitemporal dataset is and it also considers the pairs ( where represents the bitemporal data and denotes a distance function. The richness property defines the total dataset is equal to the set of all possible partitions which is expressed as follows, From (6), represents the total dataset there exist distance functions , denotes a several possible partitions.
is hyper-rich if it is capable of producing all possible partitions i.e. total bitemporal dataset is equal to set of all possible partitions . Hence it is called as hyper-Richness Bitemporal property. By applying this property, the entire similar bipemporal data are correctly grouped into any one of the clusters. As a result, the clustering accuracy is improved and minimized the false positive rate. The flow diagram of the proposed KHBP-FCMPC technique is illustrated in the following figure.   Figure 2 shows the flow process of the proposed KHBP-FCMPC technique to partitions the bitemporal data with higher accuracy based on the distance. The minimum distance among the data point and specific cluster centroid have a high probability to group similar data into the particular cluster. The initial cluster centroid is updated and groups each data points until the convergence is met.
The algorithmic process of KHBP-FCMPC technique is described as follows, Input : Number of temporal data Output: Improves the clustering accuracy Begin 1. Initialize 'c' number of clusters 2. Compute the cluster centroid 3.
Compute the objective function based on fuzzy membership function 4.
Find minimum distance 5.
Group data points into the clusters 6.
Update the cluster centroid 7.
If convergence is not met then 8.
check hyper-richness of the data points 11. end if 12.
Group all the data points into the cluster End Algorithm 1: Kleinberg's Hyper-richness Bitemporal property based fuzzy c means partition Clustering Algorithm 1 describes the clustering of bitemporal data with high accuracy. Initially, 'c' numbers of clusters are randomly initialized. After that, cluster centroid is computed based on mean of all data points. Then the objective function is computed based on distance function and membership value. The minimum distance between the bitemporal data points and centroid is grouped into the specific cluster. This process is iterative until the convergence is met. Finally, the richness property is verified with the total dataset and the partitions. As a result, the data point within the cluster has high similarity and between the clusters has less similarity. The verification result shows the entire data are correctly grouped into the clusters resulting in increases the accuracy.

IV. EXPERIMENTAL EVALUATION
An experimental evaluation of KHBP-FCMPC technique and Bi-weighted ensemble approach [1] and Parametric Link among Multinomial Mixtures (PLMM) [2] are implemented using Java language. The Activity Recognition from Single Chest-Mounted Accelerometer Data Set is used for the experimental evaluation. This dataset is employed for purpose of activity recognition with Bitemporal data. The dataset is collected from the UCI machine learning repository (https://archive.ics.uci.edu/ml/ datasets/Activity+Recognition+from+Single+ChestMounted +Accelerometer#). This dataset gathers the temporal data points from a wearable accelerometer fixed on the chest. Bitemporal data denotes the valid time and transaction time. The dataset characteristics are univariate, sequential and time-Series. The attributes characteristics are real and the association tasks performed by the dataset are clustering and classification. The uncalibrated accelerometer data are gathered from 15 participants presenting 7 dissimilar activities. This dataset provides the challenges for detection and validation of people through the various motion patterns. The data files are separated for each participant. Each file comprises the five columns such as sequential number, x acceleration, y acceleration, z acceleration, and labels. The labels are represented by the numbers. Totally seven different labels are presented for recognizing the activities. In this dataset, the labels are considered as an output result. The experiments are carried out with different parameters such as clustering accuracy, false positive rate, time complexity, and space complexity. Totally ten various runs are performed with a number of temporal data. For the experimental consideration, the numbers of temporal data are collected from 1000 to 10000. The experimental results are discussed in the next section.

V. PERFORMANCE RESULTS AND DISCUSSION
The results attained from the experimental evaluation of three clustering methods KHBP-FCMPC technique and Bi-weighted ensemble approach [1] and PLMM [2] are discussed in this section with a table and graphical results. The various metrics such as clustering accuracy, false positive rate, time complexity and space complexity are considered for evaluating performance results of three different clustering methods. For each subsection, sample mathematical calculation is presented.

5.1
Performance results of clustering accuracy Clustering accuracy is defined as the ratio of a number of similar data points are correctly grouped into the cluster to the total number of data points. The clustering accuracy is mathematically formulated as follows, From (7), represents the clustering accuracy. The accuracy is measured in percentage (%). Higher the temporal data point clustering accuracy, more efficient the method is said to be. The sample mathematical computation of clustering accuracy is presented as follows, Sample mathematical calculation for clustering accuracy:  Proposed KHBP-FCMPC technique: Number of data points correctly grouped is 860 and the total number of data points is 1000. Then the clustering accuracy is calculated as follows,  Existing Bi-weighted ensemble approach: Number of data points correctly grouped is 780 and the total number of data points is 1000. Then the clustering accuracy is calculated as follows,  Existing PLMM: Number of data points correctly grouped is 730 and the total number of data points is 1000. Then the clustering accuracy is calculated as follows,  Table 2 illustrates the temporal data clustering accuracy using three different clustering techniques namely KHBP-FCMPC, Bi-weighted ensemble approach [1] and PLMM [2]. The clustering accuracy is computed based on the number of temporal data points. For the experimental consideration, ten different temporal data points are considered from 1000 to 10000. The above-reported results show that the temporal data clustering accuracy of KHBP-FCMPC technique is considerably increased when compared to existing clustering techniques. The results are plotted in the graph as shown in figure 3. The graphical results clearly report that the clustering accuracy of KHBP-FCMPC technique is considerably increased when compared to the existing clustering techniques. This improvement is achieved by a partition based clustering technique. The time series data are collected from the activity recognition dataset. Then the seven clusters are initialized and the centroid of each cluster is computed. Followed by, the distance between the data points and the centroid is calculated to group the similar bitemporal data points into the clusters. The Kleinberg's hyper-richness bitemporal property is applied while clustering the data points to verify all the similar data points are correctly grouped into the clusters. By this way, the seven human activities are correctly recognized with higher accuracy. The ten different results are obtained with various input bitemporal data points. The clustering accuracy results of KHBP-FCMPC technique is compared to the clustering accuracy of conventional techniques. The comparison results clearly report that the clustering accuracy is significantly improved by 7% and 13% when compared to existing Bi-weighted ensemble approach [1] and PLMM [2] respectively.

Performance results of false positive rate
The false positive rate is defined as the ratio of a number of data points are incorrectly grouped into the cluster to the total number of data points. The mathematical formula for computing the false positive rate is expressed as follows,   1000  14  22  27  2000  16  24  28  3000  15  23  26  4000  12  17  22  5000  11  14  18  6000  13  18  23  7000  10  13  18  8000  11  14  19  9000  9  16  21  10000 10 13 16 The performance results of false positive rate versus a number of data points are illustrated in table 3. The false positive rate is calculated to find numbers of data are incorrectly grouped into clusters. The above table shows that the ten different results of false positive rate versus a number of data points. Therefore, the incorrect clustering results of the proposed KHBP-FCMPC technique are significantly minimized when compared to existing clustering techniques. Let us consider the number of data points is 1000, the number of data points are incorrectly grouped is 140 by applying KHBP-FCMPC technique resulting false positive rate is 14%. Similarly, the number of data points are incorrectly grouped are 220 and 270 by applying Bi-weighted ensemble approach [1] and PLMM [2]. The resultant false positive rates are 22% and 27% respectively. Similarly, the nine remaining results are computed using the three different techniques. The performance results are shown in the two-dimensional graph.  Figure 4 illustrates the experimental results of the false positive rate based on the number of data points using Activity Recognition dataset. By using the Activity Recognition dataset, the different activities performed by the human are identified through the clustering process. In figure 3, three different color lines such as violet, red and green indicate the false positive rate of three different clustering methods namely KHBP-FCMPC technique, Biweighted ensemble approach [1] and PLMM [2] respectively. The graphical results show that the KHBP-FCMPC technique minimizes the false positive rate than the existing methods. This is because the KHBP-FCMPC technique uses the gradient descent function to find the minimum distance between the centroid and data points. This function minimizes the incorrect data point clustering. In addition, the hyper-richness property is applied for verifying the entire bitemporal dataset is correctly partitioned into the seven different classes. In this case, the entire uncalibrated accelerometer data are correctly grouped into the seven types of clusters resulting in minimizes the clustering accuracy.
Totally ten various experimental results are obtained with different bitemporal data. The performance results of three methods illustrate that the false positive rate of KHBP-FCMPC technique is minimized by 29% and 44% when compared to existing Bi-weighted ensemble approach [1] and PLMM [2] respectively.

5.3
Performance results of time complexity Time complexity is defined as the amount of time required to group the similar data point into the cluster. The time complexity is mathematically computed using the following equations, (9) From (9), represents the time complexity, denotes a number of data points, is the time taken for grouping the one data point. The time complexity is measured in the unit of milliseconds (ms). The calculation of time complexities using three different methods are presented as follows.

Sample Mathematical calculation for time complexity:
 Proposed KHBP-FCMPC: Number of data is 1000 and the time taken for grouping the one data is 0.025ms , then the time complexity is mathematically calculated as follows,  Existing Bi-weighted ensemble approach: Number of data is 1000 and the time taken for grouping the one data is 0.028ms, then the time complexity is mathematically calculated as follows,     Figure 5 shows the performance results of time complexity using human activity recognition dataset. The above graphical results clearly illustrate that the proposed KHBP-FCMPC technique minimizes the time complexity while clustering the data points when compared to existing methods. During the clustering process, the KHBP-FCMPC technique uses the two module namely core data point module and authority sector module. The core data point module act as a centroid of the cluster. Then the authority sector module uses the gradient descent function to find the distance between the data points and centroid of the cluster.  The authority sector module assigns the data points into the cluster based on the distance measure. These two modules help to minimize the time complexity. The average of results minimizes the time complexity by 9% and 20% when compared to existing Bi-weighted ensemble approach [1] and PLMM [2] respectively.

Performance results of space complexity
Space complexity is computed as an amount of storage space consumed for storing similar temporal data points. The formula for computing the space complexity is expressed as follows, From (10), represents the space complexity, denotes a number of data points. Space complexity is measured in the unit of Mega bytes (MB). The method consumes less storage space for storing the similar data. The calculations for space complexity with three different methods are presented as follows. Sample mathematical calculation for space complexity  Proposed KHBP-FCMPC: Number of data points are 1000 and space for storing the one data point is 0.011MB. Then the space complexity is calculated as follows,  Existing Bi-weighted ensemble approach: Number of data points are 1000 and space for storing the one data point is 0.013. Then the space complexity is calculated as follows,  Existing PLMM: Number of data points are 1000 and space for storing the one data point is 0.015MB. Then the space complexity is calculated as follows,  Table 5 describes the performance results of space complexity with respect to a number of data points. The bitemporal data points are taken from 1000 to 10000. The above table value clearly illustrates that the space complexity of the KHBP-FCMPC technique is significantly minimized than the existing methods. The results are illustrated in figure 6. Figure 6 depicts the experimental results of the space complexity with respect to a number of data points. The above figure confirms that the amount of storage space of the proposed KHBP-FCMPC technique is considerably reduced. While processing the large volume of data points, high storage space is required resulting in increasing the dimensionality. Therefore the proposed algorithm partitions the large dataset into different clusters for minimizing the dimensionality. As a result, the less amount of memory space is exploited for storing bitemporal data points. This is because of the KHBP-FCMPC technique only groups the similar data points into the clusters. The similar data point is identified through the distance measure. Then the gradient descent function only finds the closest data points from the cluster centroid. Thus, the similar temporal (i.e., time series) data are grouped into dissimilar clusters. This process of KHBP-FCMPC technique minimizes the space complexity.

Figure 6: Performance results of space complexity versus number of data points
Let us consider, a number of data points is 1000, an amount of space consumed for storing the bitemporal data points are 11MB using KHBP-FCMPC technique. The space complexity of the Bi-weighted ensemble approach [1] and PLMM [2] are 13MB and15MB respectively. Similarly, the nine remaining runs are performed and take the average of ten results. The average results show that the space complexity of KHBP-FCMPC technique is considerably minimized by 13% and 23% when compared to state-of-theart methods. The above experimental results show that the KHBP-FCMPC technique improves the computation of bitemporal data in terms of high clustering accuracy and less false positive rate, time complexity as well as space complexity.