Nowadays, usage of recognition systems has addressed many applications in almost all fields. However, Most of classification algorithms have obtained good performance for specific problems; they lack enough robustness for other problems. Therefore, recent researches are directed to the combinational methods which have more power, robustness, resistance, accuracy and generality. Combination of Multiple Classifiers (CMC) can be considered as a general solution method for pattern recognition problems. Inputs of CMC are result of separate classifiers and its output is combination of their predictions  and .
We may see CMC under numerous names like hybrid methods, decision combination, multiple experts, and mixture of experts, classifier ensembles, cooperative agents, opinion pool, decision forest, classifier fusion, and combinational systems and so on. Combinational methods usually result in the improvement of classification, because classifiers with different features and methodologies can complete each other -. Kuncheva in [7, 35, 36, 37, 38] using Condorcet Jury theorem , has shown that combination of classifiers can usually operate better than single classifier. It means if more diverse classifiers are used in the ensemble, then error of them can considerably be reduced because classifiers with different features and methodologies can complete each other -. Different categorizations of combinational classifier systems are represented in --[39-44]. Valentini and Masouli divide methods of combining classifiers into two categories: generative methods, nongenerative methods. In generative methods, a set of base classifiers is created by a set of base algorithms or by manipulating dataset. This is done in order to reinforce diversity of base classifiers , . For a good coverage on combinational methods the reader is referred to , , and -.
Theoretical and empirical works showed that a good ensemble is one where the individual classifiers have both accuracy and diversity. In other words, the individual classifiers make their errors on difference parts of the input space  and . Many approaches have been proposed to construct such ensembles. One group of these methods obtains diverse individuals by training accurate classifiers on different training set, such as bagging, boosting, cross validation and using artificial training examples --[45-50]. Another group of these methods adopts different topologies, initial weight setting, parameter setting and training algorithm to obtain individuals. For example, Rosen in  adjusted the training algorithm of the network by introducing a penalty term to encourage individual networks to be decorrelated. Liu and Yao in  used negative correlation learning to generate negatively correlated individual neural network. The third group is named selective approach group where the diverse components are selected from a number of trained accurate networks. For example, Opitz and Shavlik in  proposed a generic algorithm to search for a highly diverse set of accurate networks. Lazarevic and Obradoric in  proposed a pruning algorithm to eliminate redundant classifiers; Navone et al. in  proposed another selective algorithm based on bias/variance decomposition; GASEN proposed by Zhou et al. in  and PSO based approach proposed by Fu et al. in  also were introduced to select the ensemble components.
In general, an ensemble is built in two steps, that is, generating multiple base classifiers and then combining their predictions. According to the styles of training the base classifiers, current ensemble learning algorithms can be roughly categorized into two classes, that is, algorithms where component learners must be trained sequentially, and algorithms where component learners could be trained in parallel. The representative of the first category is AdaBoost , which sequentially generates a series of base classifiers where the training instances wrongly predicted by a base classifier will play more important role in the training of its subsequent classifier. The representative of the second category is Bagging , which generates many samples from the original training set via bootstrap sampling [29,50,87] and then trains a base classifier from each of these samples, whose predictions are combined via majority voting.
Research on classification systems is an open problem in the pattern recognition yet. There are many ways to improve the performance of classifiers. The new classification systems try to investigate errors and propose a solution to compensate them . One of these approaches is combination of classifiers. Dietterich in [31,88,99] has proved that a combination of classifiers is usually better than a single classifier, by three kinds of reasoning: Statistical, computational and pictorial reasoning. However, there are many ways to combine classifiers; there is no proof to determine the best one . One of the most important characteristics of combination of classifiers is diversity. We try to preserve the differences between classifiers. In this way, we can investigate more aspects of data.
In section 2 we will briefly overview combining classifier levels. We will try in section 3 to obtain diverse classifiers using manipulation of dataset labels. And finally section 4 is paper’s conclusion.
2. Combining Classifiers
In general, creation of combinational classifiers may be in four levels. It means combining of classifiers may happen in four levels. Figure 1 depicts these four levels. In level four, we try to create different subset of data in order to make independent classifiers. Bagging and boosting are examples of this method , . In these examples, we use different subset of data instead of all data for training. In level three, we use subset of features for obtaining diversity in ensemble. In this method, each classifier is trained on different subset of features , -. In level two, we can use different kind of classifiers for creating the ensemble [32,100,129]. Finally, in the level one, method of combining (fusion) is considered.
Figure 1. Different levels of creation of classifier ensemble.
In the combining of classifiers, we aim to increase the performance of classification. There are several ways for combining classifiers. The simplest way is to find best classifier and use it as main classifier. This method is offline CMC. Another method that is named online CMC uses all classifier in ensemble, for example, by voting. We will show that combining method can improve the result of classification.
3. Proposed Method
In this article, classification problem for a particular kind of dataset is argued. The goal is to break each class data into smaller subclasses such that error rate in each subclass is less than a threshold. It has been assumed that a class of data can include more than one cluster. For example in Farsi handwritten optical character recognition problem, digit 5 is written at least in two kinds of shape (2 clusters). This problem is shown in Figure 2.
In , it is shown that changing labels of classes can improve classification performance. So initial digit ‘5’ class is divided into two subclasses, digit ‘5’ type 1 and digit ‘5’ type 2, in order to ease classification goal of learning digit ‘5’ initial class complicated boundaries.
According to [7,13,152], if we have some really independent classifiers better than random classifiers, the simple ensemble (majority vote) of them can outperform their average performance in accuracy. Generally even if we increase the number of those independent classifiers, we can reach to any arbitrary accuracy, even 100%. But the problem restricting us for this goal is our incapability in obtaining those really independent classifiers.
Figure 2. Data of class ‘5’ and ‘0’; 5 is in left and 0 is in right.
It implies that making an ensemble of classifiers cannot surely always lead to generating diverse outputs by those classifiers; indeed their mistakes usually coincide with each other as well as their correct results. We are looking to find these really independent and approximately accurate (at least more accurate than random) classifiers with a method that will be examined in following section.
3.2. Proposed Algorithm
In proposed solution, according to error rate of each class, the class is divided into some subclasses in order to ease learning of decision boundaries by classifier. For a better understanding have a look at Figure 3.
This problem in dimension more than 2 will be probably more crucial. In this article the presumption is that a class is composed of more than one cluster which means that in a classification process with c classes, the number of real classes may be different from c.
Figure 3. A dataset with 3 class in wich class 1 contain 2 subclass.
In following, you can see the pseudo code of the proposed algorithm.
Algorithm1(original data set);
relabel training set using clusters;
for i=1 to max_iteration
As you can see at the Figure, this method get dataset as input, and put it into three partitions: training set, test set and validation set. Here, the training set, test set and validation set contain 60%, 15% and 25% of entire dataset respectively. Then the data of each class is extracted from the original training dataset. Firstly we initial the number of cluster in each class to one. After that we repeat the following process as many as the predetermined number. This predetermined number is considered 10 here:
1. At first a classifier is trained on training data.
2. Using validation data, error rate of each class is approximated.
3. We increase the number of clusters in each class with error rate greater than a threshold, by one, and also then data of that class is clustered using fuzzy K-means [153-171]. If this clustering causes to creation of a sparse cluster, we will rollback the entire process of this section for that class. We decrease the number of clusters in that class, and then recluster those data with decreased number.
4. After that according to clustering in the previous section, the data are relabeled.
5. Finally the current classifier is added to the ensemble and this iteration is concluded.
After above loop, the outputs of all classifiers in the ensemble on test set are fused using majority vote mechanism, and the algorithm returns accuracy of ensemble and ensemble itself. All classifiers existing in the ensemble are support vector machine [172-191].
It can be said about time order of this algorithm that the method just multiplies a constant multiplicand in the time order of simple algorithm (training a simple classifier). Suppose that the time order of training a simple classifier on a dataset with n datapoints and c classes is O (f(n,c)), also assume that the time order of clustering on that dataset is O(g(n,c)) and also m to be the number of max_iteration. Then the time order of this method is Ω (m*f(n,c)+c*g(n/c,q))) where q is a number that in average and experimentally is less than c; provided that clustering is performed in each iteration. For simplicity assume that time order of clustering and training a classifier on a dataset are approximately the same. It is obvious that g(n,c) is not a linear function and g(n/c,q)<<g(n,c) where q<c. We also assumed that g(n,c)~f(n,c), then g(n/c,q)<<f(n,c). So we come to the conclusion that factor c*g(n/c,q) is negligible in compare to factor f(n,c). Consequently the time order of the method will be O(m*f(n,c)) which is worse than initial classifier time order just as little as a constant multiplicand. Of course this waste of time is completely tolerable against important achieved accuracy.
It was shown that the necessary diversity of an ensemble can be achieved by clustering data points of each multipart class. The method was explained above in detail and the result over real and non-real dataset prove the correctness of our claim. As it was mentioned before, this method is sensitive to shape of dataset. It cannot work well on those of datasets with very singular dense classes.
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