A new approach for data visualization problem

Subject Areas : Evolutionary Computing

MohammadReza Keyvanpour ¹ , Mona Soleymanpour ²

1 - Department of Computer Engineering, Alzahra University
2 - Qazvin Islamic Azad Branch

Received: 2015-08-09 Accepted : 2015-10-18 Published : 2015-11-01

Keywords: Quadric Assignment Problem (QAP), Artificial Bee Colony (ABC), Data visualization,

Abstract :

Data visualization is the process of transforming data, information, and knowledge into visual form, making use of humans’ natural visual capabilities which reveals relationships in data sets that are not evident from the raw data, by using mathematical techniques to reduce the number of dimensions in the data set while preserving the relevant inherent properties. In this paper, we formulated data visualization as a Quadric Assignment Problem (QAP), and then presented an Artificial Bee Colony (ABC) to solve the resulted discrete optimization problem. The idea behind this approach is to provide mechanisms based on ABC to overcome trapped in local minima and improving the resulted solutions. To demonstrate the application of ABC on discrete optimization in data visualization, we used a database of electricity load and compared the results to other popular methods such as SOM, MDS and Sammon's map. The results show that QAP-ABC has high performance with compared others.

References:

1. L. Xu, Y. Xu, T. W. S. Chow, "PolSOM: A new method for multidimensional data visualization", Pattern Recognition, Volume 43, Issue 4, pp. 1668–1675, 2010.
2. Y. Xu, L. Xu, T. W. S. Chow, "PPoSOM: A new variant of PolSOM by using probabilistic assignment for multidimensional data visualization", Neurocomputing, Volume 74, Issue 11, pp. 2018–2027, 2011.
3. F. S. Tsai, "Dimensionality reduction techniques for blog visualization", Expert Systems with Applications, Volume 38, Issue, pp. 2766–2773, 2011.
4. R. Abbiw-Jackson, B. Golden, S. Raghavan, and E. Wasil, "A divide-and-conquer local search heuristic for data visualization", Computers and operations research, Volume 33, Issue 11, pp. 3070–3087, 2006.
5. M. H. Ghaseminezhad, A. Karami, "A novel self-organizing map (SOM) neural network for discrete groups of data clustering", Applied Soft Computing, Volume 11, Issue 4, pp. 3771–3778, 2011.
6. P. Klement, V. Snášel, "Using SOM in the performance monitoring of the emergency call-taking system", Simulation Modelling Practice and Theory, Volume 19, Issue 1, pp. 98–109, 2011.
7. J. W. Sammon, "A nonlinear mapping for data structure analysis", IEEE Transactions on computers, Volume 18, Issue 5, pp. 401–409, 1969.
8. J. Sun, C. Fyfe, M. Crowe, "Extending Sammon mapping with Bregman divergences", Information Sciences, Volume 187, pp. 72–92, 2012.
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Full-Text:



A new approach for data visualization problem

 

 

Abstract

Data visualization is the process of transforming data, information, and knowledge into visual form, making use of humans’ natural visual capabilities which reveals relationships in data sets that are not evident from the raw data, by using mathematical techniques to reduce the number of dimensions in the data set while preserving the relevant inherent properties. In this paper, we formulated data visualization as a Quadric Assignment Problem (QAP), and then presented an Artificial Bee Colony (ABC) to solve the resulted discrete optimization problem. The idea behind this approach is to provide mechanisms based on ABC to overcome trapped in local minima and improving the resulted solutions. To demonstrate the application of ABC on discrete optimization in data visualization, we used a database of electricity load and compared the results to other popular methods such as SOM, MDS and Sammon's map. The results show that QAP-ABC has high performance with compared others.

 

 

Keywords: Data visualization, Quadric Assignment Problem (QAP), Artificial Bee Colony (ABC)

 

1. Introduction

Data visualization is a graphical presentation of multidimensional data making use of humans’ natural visual capabilities. More specifically, data visualization reveals relationships in data sets that are not evident from the raw data, by using mathematical techniques to reduce the number of dimensions in the data set while preserving the relevant inherent properties. Compared with the data appear in their original high dimensional space, we may be more capable of finding the possible relationship among the data points in the low dimensional space, i.e., 2-D. therefore, it has been widely applied to solve many problems, e.g. signal compression, pattern recognition, image processing, etc [1].  

Dimensionality reduction or projection techniques can transform large data of multiple dimensions into a smaller, more manageable set with special methods. Principal component analysis (PCA) and MultiDimensional Scaling (MDS) are two popular methods for data reduction and visualization. PCA is one of the most widely used methods because it is effective and robust for performing linear projection. Linear projection means the projection of data is conducted by multiplying each component of the original vector with a scalar. Thus, PCA is not the most suitable approach when one is dealing with highly nonlinear data. MDS is another classical projection but its final visualization map is difficult to perceive, when one is handling high dimensional and highly unsymmetrical data set. Sammon’s mapping is the earliest approach in nonlinear projecting data into low dimensional space for visualizing multivariate data. Sammon’s mapping tries to minimize the distances between input data in the original high dimensional space and the output data in the projected space. It is capable of preserving the topological structure of the original data and their corresponding inter point distances, but Sammon’s mapping is computationally demanding especially when one is handling huge numbers of data points. In addition, it requires re-computation when new data points are added [2, 3].

Discrete optimization techniques provide a possible alternative for the data visualization problem. Discretizing the data visualization problem may result in a large-scale quadratic assignment problem that is very difficult to solve to optimality [4]. The complexities of these problems are high and it is a NP Complete. One approach to overcome high large-scale quadratic assignment and its complexity is solving it by heuristics or intelligent optimization. As more and more real-world optimization problems become increasingly complex, algorithms with more capable optimizations are also increasing in demand. We formulate the data visualization problem as a Quadric Assignment Problem (QAP) and then use the Artificial Bee Colony (ABC) to solve it. The idea behind this approach is to provide mechanisms based on ABC to overcome trapped in local minima and improving the resulted solutions. To evaluate our method we use a real data set and compare our proposed hybrid approach which is named QAP-ABC with SOM, MDS and Sammon's Map as popular methods for data visualization.

The rest of the paper is organized as follows. Section 2 reviews relevant works in data visualization. Section 3 introduces modeling the problem as a Quadratic Assignment Problem and discusses Artificial Bee Colony (ABC) to solve this problem in detail. Section 4 presents a case study to evaluate the technique with experimental results, and then compares it with Self-Organizing Maps (SOM), Sammon's Map (SM) and MultiDimensional Scaling (MDS) in section 5. Section 6 concludes the paper.

 

2. Background

The most common data visualization methods allocate a representation for each data point in a lower-dimensional space and try to optimize these representations so that the distances between them are as similar as possible to the original distances of the corresponding data items. The methods differ in that how the different distances are weighted and how the representations are optimized. Linear mapping, like principle component analysis, is effective but cannot truly reflect the data structure. Non-linear mapping, like Sammon’s Mapping (SM), MultiDimensional Scaling (MDS) and Self-Organizing Map (SOM) [5, 6], requires more computation but is better at preserving the data structure.

2.1. Sammon’s Mapping (SM)

Sammon Jr. [7] introduced a method for nonlinear mapping of multidimensional data into a two- or three-dimensional space. This nonlinear mapping is a MultiDimensional Scaling (MDS) projection which preserves approximately the inherent structure of the data and thus is widely used in pattern recognition. It tries to match the pairwise distances in the lower-dimensional representations with their original distances [3].

Denote the data vector  (N is the number of data points) in a d-dimensional input space and the corresponding vector  in a l-dimensional out- put space (l=2 or 3). Define the distance between  and  in the input space as  and the distance between  and in the output space as . Distances are measured by Euclidian metric. Error E is defined as follows:

Error E is minimized such that the inter-point distances in the output space approximate the corresponding distances in the input space. After mapping, the inherent structure of input data is preserved and visualized in the output space. But Sammon’s nonlinear projection is unable to provide an explicit mapping function, making the projection of new added data require re- computation. Besides, the distance calculations of  times result in that the computational complexity  is large. Even with today’s fast computers, nonlinear optimization techniques are usually slow and inefficient for large data sets. In addition, since the error is minimized by the steepest descent procedure, it is easily trapped in local minima [2, 3, 8].

 

2.2. MultiDimensional Scaling (MDS)

Multidimensional Scaling (MDS) refers to a class of algorithms that visualize proximity relations of objects by distances between points in a low-dimensional Euclidean space. MDS algorithms are commonly used to visualize proximity data (i.e., pairwise dissimilarity values instead of feature vectors) by a set of representation points in a suitable embedding space [9, 10].

Let  be a set of n objects in a d-dimensional feature space, and let  denote the Euclidean distance between  and . Let  be the number of dimensions of the output space,  be a -dimensional point representing  in the output space, and  denote the Euclidean distance between  and  in the output space. Let  be an n × d-dimensional vector denoting the coordinates  as follows:

(1)

Where  for i = 1, 2, …, n and j = 1, 2, …, d. Error  is defined as follows:

(2)

The only difference between Sammon’s Mapping and the nonlinear metric MDS is that (excluding the constant normalizing factor) the errors in distance preservation are normalized by the distance in the original space. Due to normalization, the preservation of small distances will be emphasized [9, 11].

The minimization of Error  is a optimization problem which is non-convex and sensitive to local minima and the main limitation of most MDS applications is that it requires  memory as well as  computation [12]. Thus, though it is possible to run them with small data size without any trouble, it is impossible to execute them with a large number of data due to memory limitation; therefore, this challenge could be considered as being a memory-bound problem. Moreover, the MDS final visualization map is difficult to perceive, when one is handling high-dimensional and highly unsymmetrical data set [1, 13].

 

2.3. Self-Organizing Maps (SOM)

SOM, proposed by Kohonen [14], is a class of neural networks trained in an unsupervised manner, using competitive learning. It is a well-known method for mapping a high-dimensional space onto a low-dimensional one. It is popular to consider mapping onto a two-dimensional grid of neurons. The method allows putting complex data into order, based on their similarity, and shows a map by which the features of the data can be identified and evaluated. A variety of realizations of SOM has been developed [5, 6].

The SOM architecture consists of two fully connected layers: an input layer and a Kohonen layer. Neurons in the Kohonen layer are arranged in a one- or two-dimensional lattice. Fig. 1 displays the layout of a one-dimensional map where the output neurons are arranged in a one-dimensional lattice. The number of neurons in the input layer matches the number of attributes of the objects. In the input layer each neuron has a feed-forward connection to each neuron in the Kohonen layer [3].

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig. 1. The structure of Self-Organization Map (SOM)

Kohonen’s SOM and Sammon’s nonlinear mapping are topology- and distance-preserving mapping techniques commonly used for multivariate data projections. However, the computations for both techniques are high.

 

3. Data Visualization Problem: Definition and Formulation 

Discrete optimization techniques provide a possible alternative for the data visualization problem. Discretizing the data visualization problem may result in a large-scale Quadratic Assignment Problem that is very difficult to solve. Roselyn Abbiw-Jackson [4] used the divide and conquer algorithm to solve the Quadratic Assignment Problem (QAP). We concentrate solve data visualization problem as discrete optimization.

 

 

 

 

 

 

 

 

Fig. 2. Structure of discretizing the data visualization problem

 

Let  be a set with  points in -dimensional space (). The data visualization problem is locating any -dimensional point in  to -dimensional space ( and ) such that a relevant measure of distance is preserved (Fig. 2). We use discrete optimization techniques to solve it. Therefore, we approximate the continuous -dimensional space by a lattice  while each cell has a center point. On the other hand, the data visualization problem is similar to assigning  point to  cell (center) points. The decision variables are given by:

(3)

Therefore, data visualization problem can be written as follows:

(4)

Where,  is a matrix measuring the distance between  given instances,  is a new distance matrix between assigned instances in the  dimensional space and  is a function of the deviation between the differences between the instances in the original space and the new -dimensional space. Choices for  include the functions for Sammon's Mapping and classical scaling, and all objective functions for non-metric scaling. By using Sammon's Mapping as objective function, data visualization problem can be formulated as:

(5)

Where,  is distance (usually Euclidean distance) between original points in space of  and  is distance between lattice points in space of .

The number of cells is determined with regards to requirement of data visualization method. A problem with points spread out will require a larger grid than one with points clustered together. The larger the grid, the more accurate the final result. To scale the cells (in -dimensional space) and the given data set, we find the greatest distance between the pairs of points in given data set. Let this distance be  and the greatest distance in the chosen lattice  be . Then, we multiply all original distances between points in  by , so that our lattice is scaled to the given problem. The problem of assigning  points to  lattice points cannot be treated as a linear assignment problem because a linear assignment problem assumes that the cost of assignment of one point to a lattice point does not depend on the assignment of the other points. However, this is not the case for data visualization problems. Therefore, this problem is a Quadric Assignment Problem (QAP) whose objective function is convex and it can have many local solutions. On the other hand, space of this problem is very large and decision variables are internally depended.

Any solution method can be used to solve this problem but it should be effective in circumstance of problem. Proposed solution method here is Artificial Bee Colony named QAP-ABC for solving data visualization problem as a discrete optimization problem. The idea behind this approach is to provide mechanisms for improving the data visualization precise. To introduce the proposed method, it is necessary to give a brief review of Artificial Bee Colony and its framework.

 

3. Proposed Method: QAP-ABC

The Artificial Bee Colony (ABC) method is inspired by the intelligent foraging behavior of a honey bee colony in which bees try to maximize the nectar amount loaded into the hive by bees assigned a specific task. In the foraging behavior, there are three types of bees, each of which has a different search characteristic: these are employed bees, onlooker bees and scout bees. The employed bees are responsible for exploiting the sources in their memory whereas the onlooker bees go to exploit potentially rich sources depending on the information taken by communicating with the employed bees. Scout bees are responsible for exploring undiscovered sources by a random motivation or external clue [15, 16].

The ABC algorithm mimics this foraging behavior of honey bees in the context of a global optimization algorithm. From the perspective of a meta-heuristic, a population of solutions corresponds to the food sources to be discovered and the optimization task is to find the most profitable source by using the different search characteristics of the employed bees, onlooker bees and scout bees [17-19]. Each type of bee is a phase of the algorithm as given in Fig. 3.

 

(6)

Before introducing the ABC algorithm, it is necessary to give used notation in this algorithm. Let, : A uniform random index from  interval and it has to different from . : The number of food sources. : The uniform random index that is belong to : The dimension of the problem. : The fitness value of the solution i. 1: Initialize the population of solution 2: Evaluate the population 3: Cycle = 1 4: Repeat 5: Produce new solution  for the employed bees using fallowing equation:                    (7) 6: Evaluate the population 7: Apply the greedy selection process between  and 8: Calculate the probability values  as follows:                    (8) 9: Produce new solution  for the onlookers depending on pi 10: Evaluate the population 11: Apply the greedy selection process between  and 12: Determine the abandoned solution, if exists, and replace it with a new randomly produced solution  for the scout. 13: Memorize the best food source position achieved so far 14: Cycle = Cycle + 1 15: Until Cycle = Maximum Cycle Number or time = Maximum CPU time Fig. 3. Pseudo code of the basic ABC algorithm   The perturbation strategy in ABC approach by using of calculation of  provides the solution to move towards promising regions of the search space. This is the positive feedback of the ABC algorithm. By a greedy selection in step 7, the new solution is kept in the memory and the current one is discarded if the new solution is better; otherwise, it is discarded, and the current one is retained in the population and a counter associated with the current solution is incremented by one in order to count the number of non progressive local searches in the neighborhood of the current solution. This counter will be used to determine the exhausted sources to be used in the phase for scout bees. In the phase for employed bees, the local searches are conducted for all solutions in turn, while in the phase for onlooker bees they are carried out for the solutions chosen probabilistically. This probabilistic selection provokes high quality solutions to be chosen more, but also allows poor solutions to be selected. This selection scheme is also another positive feedback of the ABC algorithm [17, 20]. It is clear from the above explanation that there are three control parameters in the basic ABC: The number of food sources which is equal to the number of employed or onlooker bees (SN), the value of dimension of the problem (D) and the maximum cycle number (MCN).  In our proposed approach (QAP-ABC), we transform the data visualization problem to a Quadratic Assignment Problem (QAP) and apply ABC algorithm to optimize it. Then use the best bee in final phase of ABC algorithm for projection of original data points to a map and providing the output lattice. The Block Diagram of QAP-ABC is presented in Fig. 4.     Fig. 4. Block Diagram of QAP-ABC   As it is seen in Fig. 4, our proposed approach has three phases which detailed explained in fallows: 3.1. Preprocessing Since it is possible that the used data set has been comprised missed, inconsistence or noisy data, thus it is required to improve the quality of the actual data for mining by data pre processing. This also increases the mining efficiency by reducing the time required for mining the preprocessed data. Data preprocessing involves data cleaning, data integration, data transformation, data reduction, data normalization and etc. this phase is done same approach in [21].   3.2. Projection of Data Points in r-Dimensional Space to q-Dimensional Space After preprocessing data phase, high dimensional data points in r-dimensions ,the number r of attributes is large, projected to low dimensional (q-dimensional) space whereas the data can only be visualized in two or three dimensions. This reduction of dimensions should do in a way that preserves the structure of the relationships (that is, distances) between original data points. We concentrate to this phase in our article. The flowchart of our proposed approach to solve data visualization problem using of ABC algorithm is presented in Fig. 5.     Fig. 5. Flow chart of QAP-ABC 3.2. Representation of Projected Data points to a 2-Dimensiond Map We use the best bee resulted from ABC to project the high dimensional data points in r-dimensions to 2-dimensionals (q=2) space. As it is presented in Fig. 6, the best bee has 366*2=732 cells that ith pair of cells corresponds to the coordinate of ith original data points in output map which i=1, 2, 3, …, 366. The best resulted solution from ABC has minimum error, thus it guaranty to preserving the structure of the relationships between original data points in output map.                    (9) (x366, y366) (x365, y365) (x2, y2) (x1, y1)            y366 x366 y365 x365 … y2 x2 y1 x1   Best Solution     Fig. 6. The best solution resulted from ABC representation   4. Experimental Results This section evaluates the performance of the QAP-ABC, SOM, MDS and Sammon's Map to data visualization. Experiments were performed using an Intel Core 2 Duo, 2.1 GHz processor with 2 GB of RAM.   4.1. Data Set In order to compare the proposed technique (QAP-ABC) and the well-known methods such as Self-Organizing Maps algorithm (SOM), Sammon' Map and MDS, a real-life case is considered here. Used data set in this research includes 24-hour electrical load of 366 days starting from 20 March 2008 in Esfahan. Therefore, we have 366 instances with 24 attributes, which any attribute is as an hour (a label number for all the daily load curves).   4.2. Parameters Setting The dimension of the output map in this research is assumed to be two (), because most visible media (for example: paper, monitor panel and etc.) are 2-dimensional. The output map is a 60×60 grid square. Dimension of the output map is calculated by a heuristic function in SOM toolbox in MATLAB [22]. ABC settings: Except common parameter (maximum cycle number), the basic ABC used in this study employs two control parameters D and SN. The maximum cycle number is supposed 2000 and other ABC parameters is resulted from [18] as: In original ABC algorithm, 50% of the colony consists of employed and the rest 50% consists of onlooker bees. In other words, the ratio of employed and onlooker bees are the same, 1:1 [23]. Therefore, we use ratio 1:1 in this study. To reach the optimal parameters we run ABC with different parameters D and SN. The ABC program was run 30 times and average of observations register for any parameters. Table 1 and Fig. 7 show the results of original (ratio 1:1) ABC algorithms with different D and SN. Table 1.  Fitness values for QAP-ABC with SN=25 and various number of dimensions number of dimensions (D) Fitness 2 0.0613 5 0.0543 10 0.0513 20 0.0489 30 0.0496 50 0.0506 100 0.0519     Fig. 7. The sensitivity of ABC to number of dimensions (D) in QAP-ABC   The result of QAP-ABC with SN=25 and various number of dimensions (D) is shown in Fig. 7 and Table. 1. As it is shown in Fig. 7 and Table 1 more number of dimensions improves the fitness. The decreasing of fitness by increasing the number of dimensions continues until D reach to 20. Therefore we propose D=20 for ABC. To determine the SN, we run ABC program with D=20 and different SN (see Table 2 and Fig. 8).   Table 2. Fitness values for QAP-ABC with D=20 and various number of onlooker bees Onlooker Bees (SN) Employed bees Fitness 10 10 0.0541 25 25 0.0489 50 50 0.0439 75 75 0.0395 100 100 0.0351 125 125 0.0332 150 150 0.0327 175 175 0.0327 200 200 0.0327   Fig. 8. The sensitivity of ABC to number of onlooker bees (SN) in QAP-ABC   As it is shown in Table 2 and Fig. 8, increasing of the number of onlooker bees (SN) improve the fitness of resulted solutions. The fitness is unchanged for number of onlooker bees greater than 125. Thus we propose SN=125 in our study. SOM settings: The SOMs were trained using the batch training algorithm in two phases: (1) rough training phase which lasted 1000 iterations with an initial neighborhood radius equal to 5, a final neighborhood radius equal to 2, While a learning rate starts at 0.5 and end at 0.1, and (2) fine training phase which lasted 500 iteration cycles, While a learning rate starts at 0.1 and end at 0.02, we do set the neighborhood partitions started at 2 and end with 0.   5. Quality of Data Visualization To illustrate the effectiveness of the proposed method (QAP-ABC), the precise results of used data visualization methods for several data sets that are randomly selected from original data set are shown from Error! Reference source not found.. Number size of data sets is 50-100 % (increment 10) of total records number. All of methods run for 1500 seconds. As it is seen in Table 3, Sammon's Map couldn’t reach to solution in limited iteration (2000 iterations) for large data bases (above 70%) and defect in convergence. Moreover, it is similar results for MDS. The MDS is defected for above 60% to reach solution in regarded iteration.   Table 3. The data visualization precise of various approaches Dataset size QAP-ABC SOM Sammon's Map MDS Fitness Running Time Fitness Running Time Fitness Running Time Fitness Running Time 50 0.0261 3215 0.0354 732 0.0262 12040 0.0264 11510 60 0.0263 4587 0.0373 762 0.0265 12350 - - 70 0.0275 6051 0.0395 775 - - - - 80 0.0285 7562 0.0435 782 - - - - 90 0.031 8541 0.489 793 - - - - 100 0.0327 10578 0.0563 1015 - - - -   Fig. 9. To compare the results of SOM and QAP-ABC   As shown in Error! Reference source not found., Sammon's Map and MDS although have a high precise but those couldn't be success for large data set and don’t convergent and give "Iteration limit exceeded. Minimization of criterion did not converge" error after 2000 iterations. Because, used nonlinear optimization techniques for MDS and Sammon’s Map are usually inefficient for large data sets. In addition, since nonlinear optimization techniques need high computational time, MDS and Sammon’s Map are slow. Fitness value, which calculates the coefficient error of preserving of the approximately inherent structure of the data by dimension reduction, is 0.0327 and 0.0563 for QAP-ABC and SOM, respectively (Fig. 9). It can be clearly observed that the QAP-ABC has been better than SOM and other used methods and 41% improvement in the fitness values over SOM. In ABC, The onlooker ants select scout bees based on elite selection. The elite selection preserves the best solutions without any change in next step. The transforming of the best solutions to next step and the ignorance of the worst solutions cause to search increase for bees in the high fitness region of the state space, increases thus improving exploitation. Moreover, number of soldier bees is determined based on the fitness of scout bees. Thus the best scout bees have more soldier bees for search. It result concentrate to better promising region of search space (exploitation) and decrease the computational time and increase the convergence speed. This is because the QAP-ABC increases the exploitation from information of meet points by using of positive feedback. It should also be noted that QAP-ABC has higher computational time compared with SOM.   6. Conclusion In this work we suggested a new approach based on Artificial Bee Colony and Quadric Assignment Problem named QAP-ABC for data visualization problem. 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