J. Cent. South Univ. (2021) 28: 1747-1764

DOI: https://doi.org/10.1007/s11771-021-4730-x

Rotation forest based on multimodal genetic algorithm

XU Zhe(徐喆)^{1, 3}, NI Wei-chen(倪维晨)², JI Yue-hui(吉月辉)^{1, 3}

1. School of Electrical and Electronic Engineering, Tianjin University of Technology, Tianjin 300384, China;

2. Academic Affairs Office, Tianjin University of Technology, Tianjin 300384, China;

3. Tianjin Key Laboratory for Control Theory & Applications in Complicated Industry Systems,Tianjin 300384, China

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2021

Abstract: In machine learning, randomness is a crucial factor in the success of ensemble learning, and it can be injected into tree-based ensembles by rotating the feature space. However, it is a common practice to rotate the feature space randomly. Thus, a large number of trees are required to ensure the performance of the ensemble model. This random rotation method is theoretically feasible, but it requires massive computing resources, potentially restricting its applications. A multimodal genetic algorithm based rotation forest (MGARF) algorithm is proposed in this paper to solve this problem. It is a tree-based ensemble learning algorithm for classification, taking advantage of the characteristic of trees to inject randomness by feature rotation. However, this algorithm attempts to select a subset of more diverse and accurate base learners using the multimodal optimization method. The classification accuracy of the proposed MGARF algorithm was evaluated by comparing it with the original random forest and random rotation ensemble methods on 23 UCI classification datasets. Experimental results show that the MGARF method outperforms the other methods, and the number of base learners in MGARF models is much fewer.

Key words: ensemble learning; decision tree; multimodal optimization; genetic algorithm

Cite this article as: XU Zhe, NI Wei-chen, JI Yue-hui. Rotation forest based on multimodal genetic algorithm [J]. Journal of Central South University, 2021, 28(6): 1747-1764. DOI: https://doi.org/10.1007/s11771-021-4730-x.

1 Introduction

Ensemble learning is one of the main research areas of machine learning [1-3], and it is widely used in many fields such as fault detection and diagnosis [4-6], image recognition [7, 8], face recognition [9, 10], and data stream analysis [11, 12]. The main feature that the ensemble model is more accurate than the single model has been proved by TUMER et al [13] and BROWN et al [14]. DIETTERICH [2] explained the reason for the enhanced performance of the ensemble model from the perspective that the ensemble model can find the optimal hypothesis in the hypothesis space.

Multiple types of decision trees are commonly used as base models of ensembles for these algorithms are simple, efficient, and fast [15-18]. Besides, the structural complexity of decision trees can be easily and accurately controlled by tuning the depth parameter. Furthermore, the decision trees are of high variance due to their susceptibility to the variation in the training data. This striking feature facilitates the generation of diversity among base　 models. Moreover, decision trees can work directly with the dataset consisting of continuous and categorical variables, and even the variables with missing values.

There are many tree-based ensemble approaches. The Random Forest (RF) is one of the most famous tree-based ensemble methods [19, 20]. It utilizes the feature selection and random subspace method for injecting diversity into ensembles. The Rotation Forest is another tree-based ensemble method [21]. It selects feature subsets from the feature space, similar to RF, but the chosen feature subsets will be rotated to inject randomness. The Random Rotation Ensemble (RRE) method was proposed by BLASER et al [22]. It achieves the random injection using the characteristic of the decision tree to be very sensitive to the rotation of the training samples. The advantage of this method is that the randomness injection is achieved by random sample rotation, rather than random sample selection, such as Random Forest, Rotation Forest, and Bagging algorithms [23]. This advantage is apparent when the amount of data is small because all but not part of the data can be used for modelling.

Since all the base models of the above algorithms are randomly generated, each of these ensemble models would require an infinite number of the random base models, theoretically, to ensure the convergence. In practice, the number of random base models is a problem-dependent parameter, which is and should be of considerable value in most cases.

In addition to the number of base models, the diversity and accuracy of base learners are also crucial to the success of the ensemble model [24]. However, the two factors are contradictory [25]. When the diversity among base learners increases, the accuracy of the base learners will decrease, and vice versa. So, it is essential and a challenge to balance between the two critical factors in ensemble modelling.

There are various selective ensemble algorithms proposed to find the optimal balance between the accuracy and the diversity of base learners to improve the accuracy of ensembles. GASEN is a selective ensemble method, which can obtain a good generalization result [26]. In this method, instead of all models, a set of models are selected as base models in terms of weight coefficients optimized by genetic algorithm to form an ensemble model. Similar to the GASEN method, optimized weight coefficients are used to choose base learners in the WAD method proposed by ZENG et al [27]. The optimization goal in WAD is not just the accuracy of the ensemble model, but the target combined with accuracy and diversity.

Both of the above methods have one feature in common: the base models are selected from all candidate modes according to the optimal solution of genetic algorithm (GA). The composition of candidate models is also an important issue, which is missing from the above methods. If the candidate models are randomly generated, it is necessary to have a large number of candidate models as a guarantee of maintaining an appropriate balance between the diversity and accuracy of base models. In fact, if the number of candidate models is too large in GA, it will increase the number of decision variables, increase the search space, and reduce the search speed.

Multi-objective optimization is used in Ref. [28] to select base learners, which takes the model accuracy and diversity as two optimization goals. In this method, the diversity among base models selected by the multi-objective algorithm can not be guaranteed because the base models are selected according to a one-to-many comparison approach. That is, the outputs of each model are compared with the average outputs of all the candidate models. Moreover, the models with high diversity and low accuracy are also selected as base models. However, these selected base models may not contribute much to improving the accuracy of the ensemble model.

To avoid the shortcomings of the previous methods and improve the accuracy of decision tree based ensemble algorithm, a multimodal genetic algorithm based rotation forest is proposed for classification problems in this paper, called multimodal genetic algorithm based Rotation Forest (MGARF). The proposed method aims at building a selective tree-based ensemble with diverse and reasonably accurate base models.

The main improvement consists of using the multimodal optimization algorithm as the tree selection method. The optimization result of multimodal optimization is a solution set composed of multiple local and global optima individuals. Furthermore, the diversity among individuals in the solution set can be guaranteed by using the diversity preservation strategy that is an inherent function of the multimodal optimization. Also, in MGARF, base models are determined by the optimization solutions, and there is a one-to-one correspondence between them. Thus, the choice of the multimodal optimization algorithm appears to be reasonable.

In comparison with the existing selective ensemble methods, candidate models of MGARF are dynamically generated under the directions of evolutionary operations, which are superior to the randomly pre-generated models. Moreover, the number of these candidate models is not limited to a fixed number, and the number of candidate models does not affect the search space of the optimization algorithm.

Multimodal optimization is a combination method of a global metaheuristic optimization called the core algorithm, and a niching technique used as a preserve preservation method to allow the optimization results to converge to multiple local and global optima solutions.

Several metaheuristic optimization methods have been used as the core algorithm, such as GA, differential evolution (DE), and particle swarm optimization (PSO). Both the GA and DE metaheuristic optimization methods belong to the evolutionary algorithm inspired by the biological view of organismal evolution in nature.

The GA algorithm was originally used as the core algorithm of multimodal optimization, and many niching methods were proposed in combination with GA to provide diverse niches, such as clearing [29], crowding [30], fitness sharing [31], and species conservation methods [32].

The sharing method is the first attempt to solve multimodal optimization using the niching method. This method shares similarity values measured between individuals within the same subgroup. The crowding method simulates competition among similar individuals for limited resources. In this method, a parent can be replaced by its similar offspring with better fitness. The species conservation method is proposed based on the notion of species, into which the population is divided according to their similarity. Within each species, there is a species seed, a dominating individual, and will be passed on to the next generation to be conserved. This niching method only introduces one parameter, species distance, as the upper limit on the distance within a species. This feature facilitates parameter tuning and makes it easier to obtain optimal solutions. Thus, this method is used as the multimodal optimization in our proposed method. Similarity evaluation is an essential part of the above niching methods, and it can be measured by the Hamming or Euclidean distance metric for binary-coded variables or real variables, respectively.

Multimodal optimization methods with DE and PSO as core algorithms have been a growing area of recent research, such as the sharing DE [33], crowding DE [33], species-based DE [34], BMDE [35], ANDE [36], and PSO based EDHC [37]. Many research works have recently been focusing on the adaptive capability of multimodal optimization. DSDE [38] and SDLCSDE [39] methods adaptively select an algorithm parameter. An adaptive mutation strategy is used in AGDE [40]. CLPSO-LS [41] was proposed to determine the adaptive starting criterion for the local search method. Other new multimodal optimization methods have surfaced recently, including the chaos optimization-based NCOA [42] and RCSA Cuckoo Search [43] that uses the reinforced cuckoo search algorithm.

The main contribution of this paper is the multimodal optimization based selective ensemble scheme, which may prove to be useful in practical applications of ensemble learning. Furthermore, a specifically designed chromosome encoding and decoding method based on householder reflection is proposed and proven to solve GA based multimodal optimization, which is the basis of this proposed scheme.

The remainder of this paper is organized as follows. Section 2 shows an overview of our framework and implementation details of the MGARF algorithm. Then, experimental methods and results are presented in Section 3, and Section 4 concludes the work.

2 Modelling methods

The schematic diagram of the proposed MGARF algorithm is shown in Figure 1. For simplicity, only the process flow in the last generation of the GA-based multimodal optimization is illustrated in this figure, for the processes in other generations that are omitted in this figure are similar to those in the last generation. The genetic operations and the operations of multimodal optimization that are omitted will be presented later in Section 2.2.

In this algorithm, the phenotypes to be optimized are the rotation angles of the original data, and each decision tree is constructed with its corresponding rotated data. The optimal set can be obtained according to the accuracy and similarity measurements of their corresponding decision trees [44]. The rotation forest is composed of all base learners whose corresponding rotation angles are in the optimal set.

Figure 1 Schematic diagram of optimization process in last generation

2.1 Characteristic of decision tree

A graphic example is used to illustrate the unique character of the decision tree. The classification results showed in Figure 2 are obtained from two different decision tree models, which are built using the same decision tree modelling method. The samples in Figure 2(b) are obtained by rotating the samples 15° clockwise in Figure 2(a). In Figure 2(a), two types of samples are separated by one decision boundary. However, there are two decision boundaries in Figure 2(b), and the data are not correctly separated. That is, although the ensemble model became complicated, the classification accuracy is decreased.

This example intuitively demonstrates that the decision tree is sensitive to the rotation of samples because the decision boundaries of the tree can only be parallel to the coordinate axes. This example also shows that this character can be used to inject randomness into the decision tree based ensemble model, and the diversity between trees will increase as a result.

2.2 Multimodal optimization method

Combined with niching techniques, the genetic algorithm can be used to solve multimodal optimization problems. Our proposed method combines species conservation strategy, one of the niching techniques, with GA as a multimodal optimization method to select optimal rotation angles. Similar to the multi-objective optimization algorithm, this combined algorithm can also obtain multiple solutions. However, unlike the multi-objective optimization algorithm, there is only one single optimization objective in the GA-based multimodal optimization method.

Figure 2 (a) Decision boundary before data rotation;(b) Decision boundaries after data rotation

2.2.1 Seeds identification method

Seeds are some representative individuals in a population, and the identification of seeds is the basis for the optimization process. The seeds identification method is presented in Algorithm 1.

Algorithm 1 Procedure of seeds identification

The fitness value of the individual p_i is calculated using Eq. (1) that is the misclassification rate corresponding to p_i:

              (1)

where I(·) is the indicator function; T(x_n, p_i) is the classification result of the base learner T at x_n with the rotation angle corresponding to p_i; y_n is the actual value for observation x_n; and N is the number of data.

Individuals with better fitness are preferentially selected as seeds in the seeds set S{s_j | j=1, …, N_s}, where N_s is the number of seeds. As the individuals in the optimization represent data rotation angles, Δα(a, b)=|α_a-α_b| is used to evaluate the difference between two individuals, where α_a and α_b are rotation angles corresponding to individuals a and b. A smaller angle difference indicates higher individual similarity. The computed difference between any two seeds is not allowed to be less than the similarity threshold σ, which ensures the diversity among seeds.

The search for a better value of σ for each specific dataset is time-consuming. Instead, the following two rules are applied to adjust σ adaptively in every optimization generation.

Rule 1: if N_s<ηN_p, σ=σ-δ

Rule 2: if N_s>ηN_p, σ=σ+δ

where η∈(0,1] is a parameter that needs to be set in advance; and δ is an increment for adjustment. It can be seen that N_s is inversely proportional to σ.

2.2.2 Species identification

In evolutionary algorithms, a species is a sub-population in which individuals have high similarity with each other, but the similarity among individuals from different species is low. The species identification method is to split the entire population into disjoint subpopulations E_j (j=0, …, l). Each subpopulation is a distinct species. The species E_j (j=1, …, l) are within the area centered at their respective seeds s_j (j=0, …, l) with a radius of σ/2. The species E₀ contains all individuals not included in species E_j (j=1, …, l). The pseudo-code description of this procedure is presented in Algorithm 2.

Algorithm 2 Procedure of species identification method

The similarity values between every two individuals are evaluated in a population to identify each species. If the similarity value between two individuals is less than σ, they belong to the same species; otherwise, they belong to different species.

2.2.3 Species conservation method

Seeds are the best-fit individuals in their species, and there is a certain difference between each seed, so all the seeds are candidates for the optimal solution. The species conversation method is used to transfer seeds to the following generations to avoid the destruction of good genes in seeds due to selection, crossover, and mutation operations. The species conservation procedure is shown in Algorithm 3.

Algorithm 3 Procedure of conserving species

In this procedure, s_i,k-1 is the i-th seed in the k-1 generation, and G_k-1 is the seed set that contains all the seeds in the k-1 generation. E_i,k is the i-th species in the k-th generation, and the seed s_i,k-1∈E_i,k. g_w is the individual in the k-th generation to be replaced by the seed in the k-1 generation.

2.2.4 Overall flow of MGARF algorithm

The procedure of the proposed method is shown in Algorithm 4.

Algorithm 4 Procedure of MGARF algorithm

In addition to genetic operations such as crossover and mutation, seeds identification, species identification, species conversation, and optimal solution identification are also included in this algorithm. The optimal solution set is composed of seeds, but not all the seeds are included. Instead, only a part of seeds with better fitness values is selected to form the optimal solution set in optimal solution identification. According to the fitness values, the top K seeds are chosen as the solution set. K is the value rounded up to the nearest integer of θ·N_s,θ∈(0,1] is a parameter to be determined; N_s is the number of individuals in G_s.

2.2.5 Analysis of computational complexity of MGARF algorithm

The complexity of the proposed algorithm is mainly determined by the fitness evaluation and the species conserving technique based multimodal operations.

The fitness evaluation is an essential part of the genetic algorithm, whose complexity is O(N_p·D), where N_p is the population size; and D is the number of generation iterations. The worst case is N_p·D_max, where D_max is the maximum number of generation iterations allowed in the termination condition.

The multimodal operations include seeds identification, adjustment of the parameter σ, species conservation, and optimal solution identification, among which seeds identification and species conservation are more computationally intensive. The main computational burden of these two procedures comes from the evaluation of the distance between two individuals. Thus, the times of distance measurements to be carried out are chosen to present the computational complexity in multimodal operations [32].

The lower and upper limits for execution times of distance calculation in seeds identification (O_s) and seed conservation (O_c) are shown in Eqs. (2) and (3).

                  (2)

                  (3)

where N_s is the number of species to be identified for each generation, and it is determined by σ; and σ is adjusted automatically according to the parameter η. It means that N_η, the upper bound on N_s, can be given based on η.

The upper bound on O_c is also linear with N_η. The total complexity O_sc, O_sc=O_s+O_c, is presented in Eq. (4):

               (4)

When the value of N_η increases to N_p, the upper bound of the complexity of O_sc is which is the worst case. Such a high computational complexity will not be reached under normal circumstances, and the normal computational complexity should be O(N_p).

2.3 Encoding and decoding method

Householder reflection is used to rotate the samples in multi-dimensional space [45]. This method is a linear transformation using the transformation matrix H that is calculated as:

                (5)

where ||·|| is L2-norm; w is an n-dimensional unit vector; w^* is the conjugate transpose of the vector w; I∈R^n×n is an identity matrix; and H∈C^n×n is a unitary matrix. The vector v∈Cⁿ and its image vector Hv are mirror-symmetric about the hyperplane with normal vector w, as in Figure 3.

The angle between v and Hv can be regarded as the rotation angle from v to Hv, and the magnitude of the rotation angle can be considered as determined by w.

Thus, the normal vector w is encoded into real value chromosomes in the GA-based optimization for searching among all the possible rotation angles. Specifically, consider a given data pair (x, y). There are q numeric attributes in the p-dimensional predictor x, and y is a categorical variable. In the rotation process of variable x, only the q numeric features are rotated, as the rotation of categorical features is ineffective for decision tree modelling. According to the q numeric attributes in x, each individual is encoded as a q-1 dimensional vector c={c_i|i=1, …, q-1}∈R^q-1, and c_i∈[-1, 1], i=1, …, q-1. In the decoding process, the vector c is used to calculate the q-dimensional vector d:

                  (6)

where d_q∈[0, 1]. It is possible to obtain the normal vector w=d/||d|| from its corresponding chromosome using this encoding and decoding method.

Figure 3 Vector v and Hv are mirror-symmetric about hyperplane s

Theorem 1: This encoding and decoding method enables a one-to-one mapping between the chromosome and the normal vector.

Proof: Suppose there are two different encoding vectors b and a corresponding to the same q-dimensional normal vector w. Let r and s be the decoded vectors of b and a, respectively. Due to Eq. (6), we have b_i=r_i and a_i=s_i (i=1, …, q-1). Because w is the normalized vector of r and s, it implies r=ks, and b=ka, where k is a real number and

Since and we have:

                (7)

Combining b_j=ka_j, j∈[1, q-1], we have that

                (8)

we can get

                (9)

k=1 can be obtained from Eq. (9), that is r=s and b=a. It can be concluded that the uniqueness of this chromosome encoding and decoding method can be guaranteed.

Lemma 1: The normalized q-dimensional normal vectors obtained through this encoding and decoding method can represent all the q-dimensional unit vectors with the q-th dimension within the range [0, 1].

Proof: Given the encoding vector b with q-1 dimensions and its decoded vector r with q dimensions, all the elements in b and the first q-1 elements in r are in the range of [-1, 1], and the range of the q-th element r_q in r is [0, 1] as described above. For b_i, b_j∈b (i, j∈[1, q-1], and i≠j), the ratio of b_i to b_j is:

                             (10)

where is the average of all the elements in b. and the relative value of r_q andis:

                 (11)

Since , the range of r_q/is:

                            (12)

Combine Eq. (10) with Eq. (12), and we get r_q/b_i∈(-∞, ∞), i∈[1, q-1], which means that r_s/r_t∈(-∞, ∞) (s, t∈[1, q] and s≠t), as b_i=r_i, i∈[1, q-1]. It indicates that the ratios of any two elements in r range from -∞ to ∞. If the range of r_q is the same as those of other elements in r, the vector w obtained by normalizing r can represent any q-dimensional unit vectors. The actual range of r_q is [0, 1]. Therefore, only these unit vectors corresponding to w_q∈[0, 1] can be represented through this encoding and decoding method.

Theorem 2: With this chromosome encoding and decoding scheme, all possible rotation angles can be represented.

Proof: The proof of Theorem 2 relies on Lemma 1. Since ww^*=(-w)(-w)^* and the relation between w and H from Eq. (5), we can get that both w and -w correspond to the same matrix H that determines the rotation angle. It means the rotation angle corresponds to the decoded vector [r₁, r₂, …, r_q-1, -r_q] can also be obtained from [-r₁, -r₂, …, -r_q-1, r_q]. Thus, the value range of r_q only needs to be [0, 1] instead of [-1, 1]. And because of Lemma 1, this encoding and decoding scheme does not reduce the range of rotation angles but avoids the repetition of the rotation angles.

3 Experiments and results

Random Forest and Random Rotation Ensemble methods were used for comparison to evaluate the performance of the proposed MGARF method. Many applications suggest that RF is a state of art ensemble learning algorithm [46]. Each decision tree in the RRE model is built after a random rotation of the entire training data. Unlike the proposed MGARF method, RF and RRE combine all the trained decision trees as base learners without selection. These methods are all ensemble learning techniques, and the base learners within these methods were constructed using classification and regression trees (CART) [47, 48]. In other words, the focus of the following experiments is on comparing the performance of CART based ensemble algorithms.

3.1 Datasets and data preprocessing

The experiments are based on twenty-three datasets obtained from the UCI repository [49]. The details of the datasets are shown in Table 1, where N is number of observations; f is number of features; n is number of numeric features; c is number of categorical features; k is number of classes.

Each dataset was randomly split into a training set, a testing set, and a validation set. The validation set consisted of 20% of the original data, 70% of the remaining samples in the original dataset were used as the training set, and the rest of the data were used as the testing set.

For the MGARF method, base learners were built on the set of training data; fitness values in the optimization process were calculated using the testing set and the final ensemble models were evaluated using the validation set. For RF and RRE methods, base learners were also built on the training set, and the validation set was used to test the ensemble models.

Table 1 Datasets for experiments

For numeric features, a basic scaling method was used in the data preprocessing step. This scaling method scales the numeric value x_i,j, the i-th dimension of the j-th sample, according to the maximum and minimum values of the i-th dimension to range between 0 to 1. The scaled value of x_i,j can be calculated as:

    (13)

The maximum and minimum values are all in-sample values. If a sample exceeds the in-sample range, it will be set to 0 or 1, depending on which value is closer to it. Then the following procedure was performed to distinguish between numeric and categorical variables and to handle missing data.

1) Any column with non-numeric values or with the number of distinct numeric values less than 10 was treated as a categorical variable, and the other columns were treated as numerical variables.

2) For categorical missing data, a new category was created. The missing numerical data were set as the median of its column values.

3) Every categorical variable with C categories was transformed into dummy variables with C-1 0 or 1 values [50].

3.2 Stopping criteria and parameters

The following two stopping conditions were employed for the multimodal optimization procedure of the MGARF algorithm.

One is that |f_a(k)-f_a(k-1)|l holds for n consecutive generations and k>G_min; and the another is k >G_max.

The first condition evaluates whether the possibility of achieving a considerable improvement in the subsequent generations is low. It is a phenotypical termination rule, and the average fitness f_a(·) is used, where f_a(k) is the average fitness of the current generation (k-th generation). The parameter S_l is a threshold; G_min represents the minimum number of generation iterations allowed; and G_max is the upper limit on the number of generations.

For all datasets, the main parameters for MGARF were set to the values listed in Table 2.

The parameter η and θ are two important selection rates affecting the number of base learners in MGARF. In the following experiments, the η and θ were set to 0.7 in MGARF-0.7 and 0.8 in MGARF-0.8. The numbers of base models in MGARF-0.7 and MGARF-0.8 are about 49% and 64% of the population size, respectively. Thus, models built with these two values of η and θ can demonstrate the performance of MGARF models of moderate complexity.

Table 2 Main parameters for MGARF

In RF, the number of feature variables to select at random for each decision split is set to the square root of the total number of variables. A portion of samples was randomly selected with replacement for each bootstrap replica in RF. The size of each replica is 0.9N_t, where N_t is the total number of samples in the training set. There is no additional parameter in RRE.

The classification and regression trees (CART) base learners were grown without pruning. All experiments were repeated ten times.

3.3 Method and comparison

The exact average numbers of decision trees in models of MGARF-0.7 and MGARF-0.8 are listed in Table 3. The average numbers of trees in MGARF-0.7 and MGARF-0.8 are 49 and 64, respectively.

To compare the three algorithms with the same number of base learners, the numbers of trees in RF-64 and RRE-64 models were the same as that in the MGARF-0.8 model corresponding to the same dataset. The number 64 represents the overall average number of base learners in MGARF-0.8 models. In the hope of achieving better accuracy of RF and RRE models, the numbers of trees within MGARF-0.7 models were not chosen as the reference values, as their base learners are fewer than those within MGARF-0.8 models. RF-200, RF-500, RRE-200, and RRE-500 models were built to compare with ensemble models with multiple base learners. The numbers 200 and 500 indicate the numbers of trees.

The mean and standard deviation of the classification errors on benchmark datasets are listed in Table 4.

Table 5 presents the rankings of different methods, according to Table 4. The values ranging from 1 to 8 in each row are the rankings of the eight different models for a dataset. The smaller the ranking, the better the classification accuracy. If both the means and standard deviations of classification errors of different models in the same row were the same, then the rankings of these models were set to the average value of their original rankings.

Table 3 Average numbers of decision trees in MGARF models

Average rankings are listed in the last row of Table 5, representing the overall performance of different methods on all the datasets. Although the average rankings of MGARF methods were smaller than the others, the classification accuracy of MGARF models did not always outperform the others on every dataset.

Friedman’s test was used to make sure these methods are different significantly [51]. According to Table 5, The Chi-Squared value of Friedman’s test is 19.67, and the p-value is 0.0063 at the confidence of 95%. Thus, the null hypothesis can be rejected. Therefore, it means that the impact of these methods is different significantly.

Among three RF methods, RF-500 performed more accurately than the other two methods, RF-64 and RF-200, on 12 datasets, accounting for only 52% of all the 23 datasets. Similarly, among the three RRE methods, RRE-500 did not achieve the highest accuracy on all datasets. This result suggests that high uncertainty is introduced into RF and RRE models through the randomly generated base models. A large number of base models cannot guarantee the accuracy of the ensemble, for the RF will not converge until the number of base models approaches infinity. It shows that the selection of random base models in the ensembles is necessary to obtain more accurate ensemble models. Furthermore, the effectiveness of the base model selection scheme has been confirmed in the experiments. The results of both MGARF methods, MGARF-0.7 and MGARF-0.8, are better than those of the RF and RRE models in terms of the average ranking in Table 5.

Table 4 Mean and standard deviation of classification errors (%)

Tukey Kramer multiple comparison [52] was used to determine the significant differences between each method in Table 5 accurately, and the comparison results are shown in Figure 4.

In Figure 4, the x-axis and y-axis represent rankings and modelling methods, respectively. The average rankings of these methods are shown as circles. If the two methods are significantly different, the intervals, the line extending out from the circles, are not overlapped. Otherwise, the two methods are not significantly different. It can be seen that just MGARF-0.7 is significantly different from RF-64 and RRE-64. In general, the two MGARF methods with different parameters are not significantly different from all the other methods.

For RF and RRE models, the average rankings were improved with increasing the number of base learners, and the best rankings of the two ensemble methods were achieved by RF-500 and RRE-500 separately. It is consistent with the ensemble learning principle that the accuracy of the ensemble model tends to improve as the number of base learners increases. However, there are fewer base models in MGARF-0.7 than in MGARF-0.8, but the average ranking of MGARF-0.7 is better than that of MGARF-0.8. This observation is related to the selective integration feature of the MGARF method, which maintains diversity among base learners throughout the whole optimization process.

Table 5 Rankings of different methods correspond to Table 4

Figure 4 Multiple comparison of different modeling methods

The three algorithms, MGARF, RF, and RRE, were compared using the best case of each algorithm, and the rankings are listed in Table 6.

The p-value of the Friedman’s test of this comparison is 0.0125 at the confidence of 95%, and the multiple comparison with a 95% confidence interval is performed, and the results are shown in Figure 5 and Table 7. The intervals of MGARF do not overlap with the others in Figure 5. In the first two rows of Table 7, the value 0 does not fall within the upper and lower limits, and the p-values are small, 0.0199, and 0.0161, which indicates that the MGARF is significantly different from RF and RRE. Furthermore, the average ranking of MGARF was the smallest, and the numbers of base learners within MGARF models were no more than those within RF and RRE models. The MGARF algorithm has apparent advantages over RE and RRE algorithms in terms of classification accuracy and model complexity.

The computing speed tests were carried out using MATLAB 2018b on a Windows-based machine with 4 cores 3.2 GHz Intel I5 CPU, 32GB memory. Each method was run ten times on each dataset listed in Table 8. The average computing time is presented in this table.

Table 6 Rankings of best case of each algorithm

Figure 5 Multiple comparison of MGARF, RF and RRE algorithms

As different implementation codes were used, the execution time of RF and RRE methods are different for the same number of base models in the same dataset.

3.4 Effect of parameters

The parameter η is the ratio of the number of optimal individuals to the number of seeds, and θ is the selection rate for optimal individuals. They are two critical parameters in the MGARF algorithm affecting both model accuracy and the number of base learners. Both parameters have a value range of [0, 1]. The experiments of the effect of parameters θ and η in MGARF were carried out with the IRIS dataset. The parameters θ and η ranged from [0.1 1] in intervals of 0.1. We ran the experiments ten times for each combination of both parameters, and the mean classification errors of these experiments are listed in Table 9.

Table 7 Results of Tukey Kramer multiple comparison

Table 8 Average computing time for modeling (s)

Both parameters are proportional to the number of base learners. However, as the number of base learners increases, the accuracy of the ensemble model may not necessarily be improved. The best mean classification accuracy error was 4.00%, with only two base models when θ=0.1 and η=0.2. It is the most accurate classification result compared to all methods for the iris dataset in Table 4.

Since the selective property of the MGARF algorithm, this method did not produce large errors in this dataset, even with only one base model (θ=0.1 and η=0.1). The mean error of the entire table is 6.20%. Even if the search interval is increased to 0.3 or 0.5, the worst case (8.00%) can be avoided, and relatively high accurate ensembles can be obtained. Therefore, instead of using heuristic optimization algorithms to optimize the two parameters, the above simple grid search method can obtain a suitable result through multiply tests.

3.5 Diversity-error analysis

Kappa-error diagram [53] is commonly used to analyze the diversity-accuracy behavior of ensemble models, which is an effective way to explain why the ensemble models perform differently. In the Kappa-error diagram, the x-coordinate is the kappa value (pairwise diversity), and the y coordinate is the average individual error. In this paper, the average individual error is the average classification error calculated from the classification errors of two base learners.

The Cohen’s kappa (or kappa statistic) κ is a similarity coefficient [54, 55], which can be used to evaluate the similarities between base learners by using their classification results. There are several evaluation methods of model similarity, and the kappa coefficient is more accurate than others, especially when the sample imbalance occurs because it considers the agreement occurring by chance.

Two base learners determine the value of κ ranging from -1 to 1. A high value of κ indicates the low similarity of the two base learners. κ=1 means that the classification results of two base learners are in complete agreement, indicating that the two base learners are identical. When κ=0, the two classifiers are agreed by chance. κ<0 indicates that there is a specific disagreement between the two base learners.

There is a situation to be aware of when calculating κ. If all the outputs of the two models are the same, and the classification result of a model only contains one category. In other words, all the predicted values in the classification results of two base learners are the same. The divide-by-zero error will occur. In this case, κ can be set to 1 because of the similarity of the outputs of the two models.

Three representative datasets (Image-segmentation, Thyroid-disease, and E.coli) were selected to compare the ensemble models built by three modelling approaches (RRE, RF, and MGARF). The plots are shown in Figure 7.

The parameters θ and η of MGARF were both set to 0.8. For each dataset, three models were built with the same number of base learners, and the number was determined by the MGARF model. The statistical characteristics represented by a point in the figure are of two base learners. Thus, there will be n(n-1)/2 points representing all the pairwise characteristics in an ensemble model with n base learners. The numbers of base learners are listed in Table 10.

Table 9 Mean classification errors of 10 experiments on the iris dataset modeled by MGARF (%)

Figure 6 Kappa-error diagrams

Table 10 Number of base learners

The triangle marks the centroid of kappa-error points in each plot, and the classification error rates are listed at the top of plots. Unlike previous experiments, the classification error and number of base learners are of one ensemble model, not the average values of multiple models.

All plots displayed in a row correspond to the same dataset, and the scales of κ (x-axis) and error (y-axis) of these plots are the same. It is evident from the plots that the diversity and accuracy in ensemble models are indeed contradictory, and as one increases, the other decreases, and vice versa.

The MGARF method has the lowest model diversity for each dataset, while it has the highest pairwise model accuracy, giving it the lowest total classification error. This performance is due to the selective feature of the MGARF method, which removes some of the less accurate base models, and the diversity among the preserved base models is reduced. The RF method has the widest range of kappa values compared to RRE and MGARF, and its average kappa value is also the lowest, but it has a high level of error, resulting in a high overall classification error. The RRE method is neither as diverse as RF nor as accurate as MGARF, and its overall classification error is between RF and MGARF.

It can also be seen that the average accuracy of the RF models is affected by the number of base models. The RF did not perform well when the number of base models is low, such as the rankings, in Table 5, of RF-64, RF-200, RF-500 are 5.6, 4.7, and 4.4, respectively. The RRE method is identical to the RF method in this respect. However, the MGARF is different because the rankings of MGARF-0.7 and MGARF-0.8 are 3.1 and 3.4, respectively. This result indicates that the MGARF-0.7 with a smaller number of models achieved more accurate performance, and implies that the MGARF can maintain a good balance between diversity and accuracy of the base models. This performance is consistent with our expectation of selecting specific base models with the multimodal optimization method, and confirms that accuracy and diversity are keys to ensemble models.

Since the MGARF model can be simply and easily implemented, it can be applied to some general-purpose programmable controllers, such as PLC (Programmable Logic Controller) and MCU (Micro Controller Unit). They are designed to perform basic control and monitoring functions with limited computing resources. MGARF can be deployed directly on these devices to perform some online and offline tasks including data stream classification, fault detection, and diagnosis.

4 Conclusions

Based on the multimodal optimization, MGARF method is proposed in this paper. Multiple local and global optima can be obtained as optimal rotation angles by this algorithm. Instead of using random rotation angles that bring more uncertainty, more diverse and accurate base learners can be built with these optimal rotation angles.

In the GA-based multimodal optimization, the proposed encoding and decoding scheme plays a key role, for the full coverage of chromosome and the one-to-one correspondence between chromosomes and rotation angles can be guaranteed. Two parameters, θ and η, are introduced in the multimodal optimization to facilitate the adjustment of the model performance and the number of base learners that is an important factor in ensemble learning.

Performance improvements derived from the MGARF approach were observed through Tukey Kramer multiple comparisons. The MGARF method can achieve the highest classification accuracy over RF and RRE methods with a small number of base learners. Especially in the experiments investigating the effect of parameters, the MGARF model with just two trees achieved the best classification accuracy.

One limitation of the MGARF is that the training dataset must contain at least two numeric features. Otherwise, the ensembles will be invalid due to the loss of diversity.

The proposed method can also be applied to regression problems, for it only requires some simple adjustments to the integration method in the ensemble procedure.

Contributors

XU Zhe provided the concept, performed the experiments, and wrote the manuscript. NI Wei-chen helped with the data analysis and revision of the manuscript. JI Yue-hui contributed to manuscript preparation and the revision of the manuscript.

Conflict of interest

XU Zhe, NI Wei-chen and JI Yue-hui declare that they have no conflict of interest.

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(Edited by ZHENG Yu-tong)

中文导读

基于多峰遗传算法的旋转森林

摘要：在机器学习中，随机性是集成学习成功的一个关键因素，可通过旋转特征空间将随机性注入到基于树的集成学习模型中。随机地旋转特征空间是一种常见的做法，因此，需要大量的树来保证集成模型的性能。该方法在理论上是可行的，但它需要大量的计算资源，因此，本文提出了基于多峰遗传算法的旋转森林(MGARF)算法。该方法是一种基于树的分类集成学习算法，利用树的特性通过特征旋转注入随机性，利用多峰优化方法选择一个更多样化、更准确的基学习机子集。通过在23个UCI分类数据集上与原始随机森林和随机旋转集成方法进行比较，评估了所提出的MGARF算法的分类精度。实验结果表明，MGARF方法的性能优于其他方法，而且MGARF模型中的基学习机的数量更少。

关键词：集成学习；决策树；多峰优化；遗传算法

Foundation item: Project(61603274) supported by the National Natural Science Foundation of China; Project(2017KJ249) supported by the Research Project of Tianjin Municipal Education Commission, China

Received date: 2020-06-14; Accepted date: 2021-01-21

Corresponding author: XU Zhe, PhD, Lecturer; Tel: +86-18622570464; E-mail: xuzhe@tjut.edu.cn; ORCID: https://orcid.org/0000-0002-0973-0427