Robert Sabourin

Reza Davtalab, Rafael M. O. Cruz, Robert Sabourin

Most dynamic ensemble selection (DES) methods utilize the K-Nearest Neighbors (KNN) algorithm to estimate the competence of classifiers in a small region surrounding the query sample. However, KNN is very sensitive to the local distribution of the data. Moreover, it also has a high computational cost as it requires storing the whole data in memory and performing multiple distance calculations during inference. Hence, the dependency on the KNN algorithm ends up limiting the use of DES techniques for large-scale problems. This paper presents a new DES framework based on fuzzy hyperboxes called FH-DES. Each hyperbox can represent a group of samples using only two data points (Min and Max corners). Thus, the hyperbox-based system will have less computational complexity than other dynamic selection methods. In addition, despite the KNN-based approaches, the fuzzy hyperbox is not sensitive to the local data distribution. Therefore, the local distribution of the samples does not affect the system's performance. Furthermore, in this research, for the first time, misclassified samples are used to estimate the competence of the classifiers, which has not been observed in previous fusion approaches. Experimental results demonstrate that the proposed method has high classification accuracy while having a lower complexity when compared with the state-of-the-art dynamic selection methods. The implemented code is available at https://github.com/redavtalab/FH-DES_IJCNN.git.

Mariana A. Souza, Robert Sabourin, George D. C. Cavalcanti et al.

Class imbalance is a characteristic known for making learning more challenging for classification models as they may end up biased towards the majority class. A promising approach among the ensemble-based methods in the context of imbalance learning is Dynamic Selection (DS). DS techniques single out a subset of the classifiers in the ensemble to label each given unknown sample according to their estimated competence in the area surrounding the query. Because only a small region is taken into account in the selection scheme, the global class disproportion may have less impact over the system's performance. However, the presence of local class overlap may severely hinder the DS techniques' performance over imbalanced distributions as it not only exacerbates the effects of the under-representation but also introduces ambiguous and possibly unreliable samples to the competence estimation process. Thus, in this work, we propose a DS technique which attempts to minimize the effects of the local class overlap during the classifier selection procedure. The proposed method iteratively removes from the target region the instance perceived as the hardest to classify until a classifier is deemed competent to label the query sample. The known samples are characterized using instance hardness measures that quantify the local class overlap. Experimental results show that the proposed technique can significantly outperform the baseline as well as several other DS techniques, suggesting its suitability for dealing with class under-representation and overlap. Furthermore, the proposed technique still yielded competitive results when using an under-sampled, less overlapped version of the labelled sets, specially over the problems with a high proportion of minority class samples in overlap areas. Code available at https://github.com/marianaasouza/lords.

Eduardo V. L. Barboza, Robert Sabourin, Rafael M. O. Cruz

The k-Nearest Neighbors (kNN) algorithm has long been widely used in Machine Learning (ML) applications. However, the main concern when using it is the computational cost required for neighborhood search, which can make it unfeasible for large-scale applications. Optimization algorithms, such as the K-d tree, become an option in such scenarios. Under data streams, it can be challenging to maintain the properties of the K-d tree, as it requires inserting and deleting nodes on the fly. These operations can make maintaining the tree's balance and invariants difficult. Additionally, traditional K-d trees were initially designed for Minkowski-based distance functions. In this work, we describe an Online K-d tree and its adaptation to the Canberra distance that supports dynamic updates over data streams while preserving the structural invariants required for efficient traversal. Experimental analysis demonstrates that the Online K-d tree algorithm achieves faster processing time under data streams, and that adapting to the Canberra distance enabled effective subtree pruning, as evidenced by a minor loss in average accuracy and a substantial gain in instances processed per second. Our implementation can be found in our GitHub repository

Mathieu Beauchemin-Turcotte, Guy Gauthier, Robert Sabourin

This paper presents a new way to design a Fuzzy Terminal Iterative Learning Control (TILC) to control the heater temperature setpoints of a thermoforming machine. This fuzzy TILC is based on the inverse of a fuzzy model of this machine, and is built from experimental (or simulation) data with kriging interpolation. The Fuzzy Inference System usually used for a fuzzy model is the zero order Takagi Sugeno Kwan system (constant consequents). In this paper, the 1st order Takagi Sugeno Kwan system is used, with the fuzzy model rules expressed using matrices. This makes the inversion of the fuzzy model much easier than the inversion of the fuzzy model based on the TSK of order 0. Based on simulation results, the proposed fuzzy TILC seems able to give a very good initial guess as to the heater temperature setpoints, making it possible to have almost no wastage of plastic sheets. Simulation results show the effectiveness of the fuzzy TILC compared to a crisp TILC, even though the fuzzy controller is based on a fuzzy model built from noisy data.

Eduardo V. L. Barboza, Jean Paul Barddal, Robert Sabourin et al.

This work presents Causal Drift Generator (CaDrift), a time-dependent synthetic data generator framework based on Structural Causal Models (SCMs). The framework produces a virtually infinite combination of data streams with controlled shift events and time-dependent data, making it a tool to evaluate methods under evolving data. CaDrift synthesizes various distributional and covariate shifts by drifting mapping functions of the SCM, which change underlying cause-and-effect relationships between features and the target. In addition, CaDrift models occasional perturbations by leveraging interventions in causal modeling. Experimental results show that, after distributional shift events, the accuracy of classifiers tends to drop, followed by a gradual retrieval, confirming the generator's effectiveness in simulating shifts. The framework has been made available on GitHub.

Kecia G. de Moura, Rafael M. O. Cruz, Robert Sabourin

Handwritten Signature Verification (HSV) systems distinguish between genuine and forged signatures. Traditional HSV development involves a static batch configuration, constraining the system's ability to model signatures to the limited data available. Signatures exhibit high intra-class variability and are sensitive to various factors, including time and external influences, imparting them a dynamic nature. This paper investigates the signature learning process within a data stream context. We propose a novel HSV approach with an adaptive system that receives an infinite sequence of signatures and is updated over time. Experiments were carried out on GPDS Synthetic, CEDAR, and MCYT datasets. Results demonstrate the superior performance of the proposed method compared to standard approaches that use a Support Vector Machine as a classifier. Implementation of the method is available at https://github.com/kdMoura/stream_hsv.

Mariana A. Souza, Robert Sabourin, George D. C. Cavalcanti et al.

Dynamic selection techniques aim at selecting the local experts around each test sample in particular for performing its classification. While generating the classifier on a local scope may make it easier for singling out the locally competent ones, as in the online local pool (OLP) technique, using the same base-classifier model in uneven distributions may restrict the local level of competence, since each region may have a data distribution that favors one model over the others. Thus, we propose in this work a problem-independent dynamic base-classifier model recommendation for the OLP technique, which uses information regarding the behavior of a portfolio of models over the samples of different problems to recommend one (or several) of them on a per-instance manner. Our proposed framework builds a multi-label meta-classifier responsible for recommending a set of relevant model types based on the local data complexity of the region surrounding each test sample. The OLP technique then produces a local pool with the model that yields the highest probability score of the meta-classifier. Experimental results show that different data distributions favored different model types on a local scope. Moreover, based on the performance of an ideal model type selector, it was observed that there is a clear advantage in choosing a relevant model type for each test instance. Overall, the proposed model type recommender system yielded a statistically similar performance to the original OLP with fixed base-classifier model. Given the novelty of the approach and the gap in performance between the proposed framework and the ideal selector, we regard this as a promising research direction. Code available at github.com/marianaasouza/dynamic-model-recommender.

Rafael M. O. Cruz, Luiz G. Hafemann, Robert Sabourin et al.

DESlib is an open-source python library providing the implementation of several dynamic selection techniques. The library is divided into three modules: (i) \emph{dcs}, containing the implementation of dynamic classifier selection methods (DCS); (ii) \emph{des}, containing the implementation of dynamic ensemble selection methods (DES); (iii) \emph{static}, with the implementation of static ensemble techniques. The library is fully documented (documentation available online on Read the Docs), has a high test coverage (codecov.io) and is part of the scikit-learn-contrib supported projects. Documentation, code and examples can be found on its GitHub page: https://github.com/scikit-learn-contrib/DESlib.

Eduardo V. L. Barboza, Paulo R. Lisboa de Almeida, Alceu de Souza Britto et al.

Data streams pose challenges not usually encountered in batch-based ML. One of them is concept drift, which is characterized by the change in data distribution over time. Among many approaches explored in literature, the fusion of classifiers has been showing good results and is getting growing attention. DS methods, due to the ensemble being instance-based, seem to be an efficient choice under drifting scenarios. However, some attention must be paid to adapting such methods for concept drift. The training must be done in order to create local experts, and the commonly used neighborhood-search DS may become prohibitive with the continuous arrival of data. In this work, we propose IncA-DES, which employs a training strategy that promotes the generation of local experts with the assumption that different regions of the feature space become available with time. Additionally, the fusion of a concept drift detector supports the maintenance of information and adaptation to a new concept. An overlap-based classification filter is also employed in order to avoid using the DS method when there is a consensus in the neighborhood, a strategy that we argue every DS method should employ, as it was shown to make them more applicable and quicker. Moreover, aiming to reduce the processing time of the kNN, we propose an Online K-d tree algorithm, which can quickly remove instances without becoming inconsistent and deals with unbalancing concerns that may occur in data streams. Experimental results showed that the proposed framework got the best average accuracy compared to seven state-of-the-art methods considering different levels of label availability and presented the smaller processing time between the most accurate methods. Additionally, the fusion with the Online K-d tree has improved processing time with a negligible loss in accuracy. We have made our framework available in an online repository.

Marcos Monteiro, Alceu S. Britto, Jean P. Barddal et al.

This paper describes a classifier pool generation method guided by the diversity estimated on the data complexity and classifier decisions. First, the behavior of complexity measures is assessed by considering several subsamples of the dataset. The complexity measures with high variability across the subsamples are selected for posterior pool adaptation, where an evolutionary algorithm optimizes diversity in both complexity and decision spaces. A robust experimental protocol with 28 datasets and 20 replications is used to evaluate the proposed method. Results show significant accuracy improvements in 69.4% of the experiments when Dynamic Classifier Selection and Dynamic Ensemble Selection methods are applied.

Andre G. Hochuli, Alceu S. Britto, David A. Saji et al.

Over the last decades, most approaches proposed for handwritten digit string recognition (HDSR) have resorted to digit segmentation, which is dominated by heuristics, thereby imposing substantial constraints on the final performance. Few of them have been based on segmentation-free strategies where each pixel column has a potential cut location. Recently, segmentation-free strategies has added another perspective to the problem, leading to promising results. However, these strategies still show some limitations when dealing with a large number of touching digits. To bridge the resulting gap, in this paper, we hypothesize that a string of digits can be approached as a sequence of objects. We thus evaluate different end-to-end approaches to solve the HDSR problem, particularly in two verticals: those based on object-detection (e.g., Yolo and RetinaNet) and those based on sequence-to-sequence representation (CRNN). The main contribution of this work lies in its provision of a comprehensive comparison with a critical analysis of the above mentioned strategies on five benchmarks commonly used to assess HDSR, including the challenging Touching Pair dataset, NIST SD19, and two real-world datasets (CAR and CVL) proposed for the ICFHR 2014 competition on HDSR. Our results show that the Yolo model compares favorably against segmentation-free models with the advantage of having a shorter pipeline that minimizes the presence of heuristics-based models. It achieved a 97%, 96%, and 84% recognition rate on the NIST-SD19, CAR, and CVL datasets, respectively.

Victor L. F. Souza, Adriano L. I. Oliveira, Rafael M. O. Cruz et al.

SigNet is a state of the art model for feature representation used for handwritten signature verification (HSV). This representation is based on a Deep Convolutional Neural Network (DCNN) and contains 2048 dimensions. When transposed to a dissimilarity space generated by the dichotomy transformation (DT), related to the writer-independent (WI) approach, these features may include redundant information. This paper investigates the presence of overfitting when using Binary Particle Swarm Optimization (BPSO) to perform the feature selection in a wrapper mode. We proposed a method based on a global validation strategy with an external archive to control overfitting during the search for the most discriminant representation. Moreover, an investigation is also carried out to evaluate the use of the selected features in a transfer learning context. The analysis is carried out on a writer-independent approach on the CEDAR, MCYT and GPDS datasets. The experimental results showed the presence of overfitting when no validation is used during the optimization process and the improvement when the global validation strategy with an external archive is used. Also, the space generated after feature selection can be used in a transfer learning context.

Teruo M. Maruyama, Luiz S. Oliveira, Alceu S. Britto et al.

Usually, in a real-world scenario, few signature samples are available to train an automatic signature verification system (ASVS). However, such systems do indeed need a lot of signatures to achieve an acceptable performance. Neuromotor signature duplication methods and feature space augmentation methods may be used to meet the need for an increase in the number of samples. Such techniques manually or empirically define a set of parameters to introduce a degree of writer variability. Therefore, in the present study, a method to automatically model the most common writer variability traits is proposed. The method is used to generate offline signatures in the image and the feature space and train an ASVS. We also introduce an alternative approach to evaluate the quality of samples considering their feature vectors. We evaluated the performance of an ASVS with the generated samples using three well-known offline signature datasets: GPDS, MCYT-75, and CEDAR. In GPDS-300, when the SVM classifier was trained using one genuine signature per writer and the duplicates generated in the image space, the Equal Error Rate (EER) decreased from 5.71% to 1.08%. Under the same conditions, the EER decreased to 1.04% using the feature space augmentation technique. We also verified that the model that generates duplicates in the image space reproduces the most common writer variability traits in the three different datasets.

Simon Bernard, Hongliu Cao, Robert Sabourin et al.

Many classification problems are naturally multi-view in the sense their data are described through multiple heterogeneous descriptions. For such tasks, dissimilarity strategies are effective ways to make the different descriptions comparable and to easily merge them, by (i) building intermediate dissimilarity representations for each view and (ii) fusing these representations by averaging the dissimilarities over the views. In this work, we show that the Random Forest proximity measure can be used to build the dissimilarity representations, since this measure reflects similarities between features but also class membership. We then propose a Dynamic View Selection method to better combine the view-specific dissimilarity representations. This allows to take a decision, on each instance to predict, with only the most relevant views for that instance. Experiments are conducted on several real-world multi-view datasets, and show that the Dynamic View Selection offers a significant improvement in performance compared to the simple average combination and two state-of-the-art static view combinations.

Hongliu Cao, Simon Bernard, Robert Sabourin et al.

Multi-view learning is a learning task in which data is described by several concurrent representations. Its main challenge is most often to exploit the complementarities between these representations to help solve a classification/regression task. This is a challenge that can be met nowadays if there is a large amount of data available for learning. However, this is not necessarily true for all real-world problems, where data are sometimes scarce (e.g. problems related to the medical environment). In these situations, an effective strategy is to use intermediate representations based on the dissimilarities between instances. This work presents new ways of constructing these dissimilarity representations, learning them from data with Random Forest classifiers. More precisely, two methods are proposed, which modify the Random Forest proximity measure, to adapt it to the context of High Dimension Low Sample Size (HDLSS) multi-view classification problems. The second method, based on an Instance Hardness measurement, is significantly more accurate than other state-of-the-art measurements including the original RF Proximity measurement and the Large Margin Nearest Neighbor (LMNN) metric learning measurement.

Victor L. F. Souza, Adriano L. I. Oliveira, Rafael M. O. Cruz et al.

This paper investigates the presence of overfitting when using Binary Particle Swarm Optimization (BPSO) to perform the feature selection in a context of Handwritten Signature Verification (HSV). SigNet is a state of the art Deep CNN model for feature representation in the HSV context and contains 2048 dimensions. Some of these dimensions may include redundant information in the dissimilarity representation space generated by the dichotomy transformation (DT) used by the writer-independent (WI) approach. The analysis is carried out on the GPDS-960 dataset. Experiments demonstrate that the proposed method is able to control overfitting during the search for the most discriminant representation.

Victor L. F. Souza, Adriano L. I. Oliveira, Rafael M. O. Cruz et al.

High number of writers, small number of training samples per writer with high intra-class variability and heavily imbalanced class distributions are among the challenges and difficulties of the offline Handwritten Signature Verification (HSV) problem. A good alternative to tackle these issues is to use a writer-independent (WI) framework. In WI systems, a single model is trained to perform signature verification for all writers from a dissimilarity space generated by the dichotomy transformation. Among the advantages of this framework is its scalability to deal with some of these challenges and its ease in managing new writers, and hence of being used in a transfer learning context. In this work, we present a white-box analysis of this approach highlighting how it handles the challenges, the dynamic selection of references through fusion function, and its application for transfer learning. All the analyses are carried out at the instance level using the instance hardness (IH) measure. The experimental results show that, using the IH analysis, we were able to characterize "good" and "bad" quality skilled forgeries as well as the frontier region between positive and negative samples. This enables futures investigations on methods for improving discrimination between genuine signatures and skilled forgeries by considering these characterizations.

Andre G. Hochuli, Alceu S. Britto, Jean P. Barddal et al.

An end-to-end solution for handwritten numeral string recognition is proposed, in which the numeral string is considered as composed of objects automatically detected and recognized by a YoLo-based model. The main contribution of this paper is to avoid heuristic-based methods for string preprocessing and segmentation, the need for task-oriented classifiers, and also the use of specific constraints related to the string length. A robust experimental protocol based on several numeral string datasets, including one composed of historical documents, has shown that the proposed method is a feasible end-to-end solution for numeral string recognition. Besides, it reduces the complexity of the string recognition task considerably since it drops out classical steps, in special preprocessing, segmentation, and a set of classifiers devoted to strings with a specific length.

Luiz G. Hafemann, Robert Sabourin, Luiz S. Oliveira

Offline Handwritten Signature verification presents a challenging Pattern Recognition problem, where only knowledge of the positive class is available for training. While classifiers have access to a few genuine signatures for training, during generalization they also need to discriminate forgeries. This is particularly challenging for skilled forgeries, where a forger practices imitating the user's signature, and often is able to create forgeries visually close to the original signatures. Most work in the literature address this issue by training for a surrogate objective: discriminating genuine signatures of a user and random forgeries (signatures from other users). In this work, we propose a solution for this problem based on meta-learning, where there are two levels of learning: a task-level (where a task is to learn a classifier for a given user) and a meta-level (learning across tasks). In particular, the meta-learner guides the adaptation (learning) of a classifier for each user, which is a lightweight operation that only requires genuine signatures. The meta-learning procedure learns what is common for the classification across different users. In a scenario where skilled forgeries from a subset of users are available, the meta-learner can guide classifiers to be discriminative of skilled forgeries even if the classifiers themselves do not use skilled forgeries for learning. Experiments conducted on the GPDS-960 dataset show improved performance compared to Writer-Independent systems, and achieve results comparable to state-of-the-art Writer-Dependent systems in the regime of few samples per user (5 reference signatures).

Regis Antonio Saraiva Albuquerque, Albert Franca Josua Costa, Eulanda Miranda dos Santos et al.

We propose an online method for concept driftdetection based on dynamic classifier ensemble selection. Theproposed method generates a pool of ensembles by promotingdiversity among classifier members and chooses expert ensemblesaccording to global prequential accuracy values. Unlike currentdynamic ensemble selection approaches that use only local knowl-edge to select the most competent ensemble for each instance,our method focuses on selection taking into account the decisionspace. Consequently, it is well adapted to the context of driftdetection in data stream problems. The results of the experimentsshow that the proposed method attained the highest detection pre-cision and the lowest number of false alarms, besides competitiveclassification accuracy rates, in artificial datasets representingdifferent types of drifts. Moreover, it outperformed baselines indifferent real-problem datasets in terms of classification accuracy.

Luiz G. Hafemann, Robert Sabourin, Luiz S. Oliveira

The phenomenon of Adversarial Examples is attracting increasing interest from the Machine Learning community, due to its significant impact to the security of Machine Learning systems. Adversarial examples are similar (from a perceptual notion of similarity) to samples from the data distribution, that "fool" a machine learning classifier. For computer vision applications, these are images with carefully crafted but almost imperceptible changes, that are misclassified. In this work, we characterize this phenomenon under an existing taxonomy of threats to biometric systems, in particular identifying new attacks for Offline Handwritten Signature Verification systems. We conducted an extensive set of experiments on four widely used datasets: MCYT-75, CEDAR, GPDS-160 and the Brazilian PUC-PR, considering both a CNN-based system and a system using a handcrafted feature extractor (CLBP). We found that attacks that aim to get a genuine signature rejected are easy to generate, even in a limited knowledge scenario, where the attacker does not have access to the trained classifier nor the signatures used for training. Attacks that get a forgery to be accepted are harder to produce, and often require a higher level of noise - in most cases, no longer "imperceptible" as previous findings in object recognition. We also evaluated the impact of two countermeasures on the success rate of the attacks and the amount of noise required for generating successful attacks.

Jérôme Rony, Luiz G. Hafemann, Luiz S. Oliveira et al.

Research on adversarial examples in computer vision tasks has shown that small, often imperceptible changes to an image can induce misclassification, which has security implications for a wide range of image processing systems. Considering $L_2$ norm distortions, the Carlini and Wagner attack is presently the most effective white-box attack in the literature. However, this method is slow since it performs a line-search for one of the optimization terms, and often requires thousands of iterations. In this paper, an efficient approach is proposed to generate gradient-based attacks that induce misclassifications with low $L_2$ norm, by decoupling the direction and the norm of the adversarial perturbation that is added to the image. Experiments conducted on the MNIST, CIFAR-10 and ImageNet datasets indicate that our attack achieves comparable results to the state-of-the-art (in terms of $L_2$ norm) with considerably fewer iterations (as few as 100 iterations), which opens the possibility of using these attacks for adversarial training. Models trained with our attack achieve state-of-the-art robustness against white-box gradient-based $L_2$ attacks on the MNIST and CIFAR-10 datasets, outperforming the Madry defense when the attacks are limited to a maximum norm.

Rafael M. O. Cruz, Mariana A. Souza, Robert Sabourin et al.

Class-imbalance refers to classification problems in which many more instances are available for certain classes than for others. Such imbalanced datasets require special attention because traditional classifiers generally favor the majority class which has a large number of instances. Ensemble of classifiers have been reported to yield promising results. However, the majority of ensemble methods applied to imbalanced learning are static ones. Moreover, they only deal with binary imbalanced problems. Hence, this paper presents an empirical analysis of dynamic selection techniques and data preprocessing methods for dealing with multi-class imbalanced problems. We considered five variations of preprocessing methods and fourteen dynamic selection schemes. Our experiments conducted on 26 multi-class imbalanced problems show that the dynamic ensemble improves the AUC and the G-mean as compared to the static ensemble. Moreover, data preprocessing plays an important role in such cases.

Rafael M. O. Cruz, Robert Sabourin, George D. C. Cavalcanti

In dynamic selection (DS) techniques, only the most competent classifiers, for the classification of a specific test sample are selected to predict the sample's class labels. The more important step in DES techniques is estimating the competence of the base classifiers for the classification of each specific test sample. The classifiers' competence is usually estimated using the neighborhood of the test sample defined on the validation samples, called the region of competence. Thus, the performance of DS techniques is sensitive to the distribution of the validation set. In this paper, we evaluate six prototype selection techniques that work by editing the validation data in order to remove noise and redundant instances. Experiments conducted using several state-of-the-art DS techniques over 30 classification problems demonstrate that by using prototype selection techniques we can improve the classification accuracy of DS techniques and also significantly reduce the computational cost involved.

Rafael M. O. Cruz, Robert Sabourin, George D. C. Cavalcanti

In Dynamic Ensemble Selection (DES) techniques, only the most competent classifiers are selected to classify a given query sample. Hence, the key issue in DES is how to estimate the competence of each classifier in a pool to select the most competent ones. In order to deal with this issue, we proposed a novel dynamic ensemble selection framework using meta-learning, called META-DES. The framework is divided into three steps. In the first step, the pool of classifiers is generated from the training data. In the second phase the meta-features are computed using the training data and used to train a meta-classifier that is able to predict whether or not a base classifier from the pool is competent enough to classify an input instance. In this paper, we propose improvements to the training and generalization phase of the META-DES framework. In the training phase, we evaluate four different algorithms for the training of the meta-classifier. For the generalization phase, three combination approaches are evaluated: Dynamic selection, where only the classifiers that attain a certain competence level are selected; Dynamic weighting, where the meta-classifier estimates the competence of each classifier in the pool, and the outputs of all classifiers in the pool are weighted based on their level of competence; and a hybrid approach, in which first an ensemble with the most competent classifiers is selected, after which the weights of the selected classifiers are estimated in order to be used in a weighted majority voting scheme. Experiments are carried out on 30 classification datasets. Experimental results demonstrate that the changes proposed in this paper significantly improve the recognition accuracy of the system in several datasets.

Rafael M. O. Cruz, Robert Sabourin, George D. C. Cavalcanti

In this paper, we propose a novel dynamic ensemble selection framework using meta-learning. The framework is divided into three steps. In the first step, the pool of classifiers is generated from the training data. The second phase is responsible to extract the meta-features and train the meta-classifier. Five distinct sets of meta-features are proposed, each one corresponding to a different criterion to measure the level of competence of a classifier for the classification of a given query sample. The meta-features are computed using the training data and used to train a meta-classifier that is able to predict whether or not a base classifier from the pool is competent enough to classify an input instance. Three different training scenarios for the training of the meta-classifier are considered: problem-dependent, problem-independent and hybrid. Experimental results show that the problem-dependent scenario provides the best result. In addition, the performance of the problem-dependent scenario is strongly correlated with the recognition rate of the system. A comparison with state-of-the-art techniques shows that the proposed-dependent approach outperforms current dynamic ensemble selection techniques.

Rafael M. O Cruz, Robert Sabourin, George D. C. Cavalcanti

The key issue in Dynamic Ensemble Selection (DES) is defining a suitable criterion for calculating the classifiers' competence. There are several criteria available to measure the level of competence of base classifiers, such as local accuracy estimates and ranking. However, using only one criterion may lead to a poor estimation of the classifier's competence. In order to deal with this issue, we have proposed a novel dynamic ensemble selection framework using meta-learning, called META-DES. An important aspect of the META-DES framework is that multiple criteria can be embedded in the system encoded as different sets of meta-features. However, some DES criteria are not suitable for every classification problem. For instance, local accuracy estimates may produce poor results when there is a high degree of overlap between the classes. Moreover, a higher classification accuracy can be obtained if the performance of the meta-classifier is optimized for the corresponding data. In this paper, we propose a novel version of the META-DES framework based on the formal definition of the Oracle, called META-DES.Oracle. The Oracle is an abstract method that represents an ideal classifier selection scheme. A meta-feature selection scheme using an overfitting cautious Binary Particle Swarm Optimization (BPSO) is proposed for improving the performance of the meta-classifier. The difference between the outputs obtained by the meta-classifier and those presented by the Oracle is minimized. Thus, the meta-classifier is expected to obtain results that are similar to the Oracle. Experiments carried out using 30 classification problems demonstrate that the optimization procedure based on the Oracle definition leads to a significant improvement in classification accuracy when compared to previous versions of the META-DES framework and other state-of-the-art DES techniques.

Rafael M. O. Cruz, Dayvid V. R. Oliveira, George D. C. Cavalcanti et al.

Despite being very effective in several classification tasks, Dynamic Ensemble Selection (DES) techniques can select classifiers that classify all samples in the region of competence as being from the same class. The Frienemy Indecision REgion DES (FIRE-DES) tackles this problem by pre-selecting classifiers that correctly classify at least one pair of samples from different classes in the region of competence of the test sample. However, FIRE-DES applies the pre-selection for the classification of a test sample if and only if its region of competence is composed of samples from different classes (indecision region), even though this criterion is not reliable for determining if a test sample is located close to the borders of classes (true indecision region) when the region of competence is obtained using classical nearest neighbors approach. Because of that, FIRE-DES mistakes noisy regions for true indecision regions, leading to the pre-selection of incompetent classifiers, and mistakes true indecision regions for safe regions, leaving samples in such regions without any pre-selection. To tackle these issues, we propose the FIRE-DES++, an enhanced FIRE-DES that removes noise and reduces the overlap of classes in the validation set; and defines the region of competence using an equal number of samples of each class, avoiding selecting a region of competence with samples of a single class. Experiments are conducted using FIRE-DES++ with 8 different dynamic selection techniques on 64 classification datasets. Experimental results show that FIRE-DES++ increases the classification performance of all DES techniques considered in this work, outperforming FIRE-DES with 7 out of the 8 DES techniques, and outperforming state-of-the-art DES frameworks.

Rafael M. O. Cruz, Robert Sabourin, George D. C. Cavalcanti et al.

Dynamic ensemble selection systems work by estimating the level of competence of each classifier from a pool of classifiers. Only the most competent ones are selected to classify a given test sample. This is achieved by defining a criterion to measure the level of competence of a base classifier, such as, its accuracy in local regions of the feature space around the query instance. However, using only one criterion about the behavior of a base classifier is not sufficient to accurately estimate its level of competence. In this paper, we present a novel dynamic ensemble selection framework using meta-learning. We propose five distinct sets of meta-features, each one corresponding to a different criterion to measure the level of competence of a classifier for the classification of input samples. The meta-features are extracted from the training data and used to train a meta-classifier to predict whether or not a base classifier is competent enough to classify an input instance. During the generalization phase, the meta-features are extracted from the query instance and passed down as input to the meta-classifier. The meta-classifier estimates, whether a base classifier is competent enough to be added to the ensemble. Experiments are conducted over several small sample size classification problems, i.e., problems with a high degree of uncertainty due to the lack of training data. Experimental results show the proposed meta-learning framework greatly improves classification accuracy when compared against current state-of-the-art dynamic ensemble selection techniques.

Mariana A. Souza, George D. C. Cavalcanti, Rafael M. O. Cruz et al.

Dynamic Classifier Selection (DCS) techniques have difficulty in selecting the most competent classifier in a pool, even when its presence is assured. Since the DCS techniques rely only on local data to estimate a classifier's competence, the manner in which the pool is generated could affect the choice of the best classifier for a given sample. That is, the global perspective in which pools are generated may not help the DCS techniques in selecting a competent classifier for samples that are likely to be mislabelled. Thus, we propose in this work an online pool generation method that produces a locally accurate pool for test samples in difficult regions of the feature space. The difficulty of a given area is determined by the classification difficulty of the samples in it. That way, by using classifiers that were generated in a local scope, it could be easier for the DCS techniques to select the best one for the difficult samples. For the query samples in easy regions, a simple nearest neighbors rule is used. In the extended version of this work, a deep analysis on the correlation between instance hardness and the performance of DCS techniques is presented. An instance hardness measure that conveys the degree of local class overlap is then used to decide when the local pool is used in the proposed scheme. The proposed method yielded significantly greater recognition rates in comparison to a Bagging-generated pool and two other global pool generation schemes for all DCS techniques evaluated. The proposed scheme's performance was also significantly superior to three state-of-the-art classification models and statistically equivalent to five of them. Moreover, an extended analysis on the computational complexity of the proposed method and of several DS techniques is presented in this version. We also provide the implementation of the proposed technique using the DESLib library on GitHub.

Victor L. F. Souza, Adriano L. I. Oliveira, Robert Sabourin

The use of features extracted using a deep convolutional neural network (CNN) combined with a writer-dependent (WD) SVM classifier resulted in significant improvement in performance of handwritten signature verification (HSV) when compared to the previous state-of-the-art methods. In this work it is investigated whether the use of these CNN features provide good results in a writer-independent (WI) HSV context, based on the dichotomy transformation combined with the use of an SVM writer-independent classifier. The experiments performed in the Brazilian and GPDS datasets show that (i) the proposed approach outperformed other WI-HSV methods from the literature, (ii) in the global threshold scenario, the proposed approach was able to outperform the writer-dependent method with CNN features in the Brazilian dataset, (iii) in an user threshold scenario, the results are similar to those obtained by the writer-dependent method with CNN features.

Hongliu Cao, Simon Bernard, Laurent Heutte et al.

Cancer diagnosis and treatment often require a personalized analysis for each patient nowadays, due to the heterogeneity among the different types of tumor and among patients. Radiomics is a recent medical imaging field that has shown during the past few years to be promising for achieving this personalization. However, a recent study shows that most of the state-of-the-art works in Radiomics fail to identify this problem as a multi-view learning task and that multi-view learning techniques are generally more efficient. In this work, we propose to further investigate the potential of one family of multi-view learning methods based on Multiple Classifiers Systems where one classifier is learnt on each view and all classifiers are combined afterwards. In particular, we propose a random forest based dynamic weighted voting scheme, which personalizes the combination of views for each new patient for classification tasks. The proposed method is validated on several real-world Radiomics problems.

Andre G Hochuli, Luiz E S Oliveira, Alceu S Britto et al.

This paper presents segmentation-free strategies for the recognition of handwritten numeral strings of unknown length. A synthetic dataset of touching numeral strings of sizes 2-, 3- and 4-digits was created to train end-to-end solutions based on Convolutional Neural Networks. A robust experimental protocol is used to show that the proposed segmentation-free methods may reach the state-of-the-art performance without suffering the heavy burden of over-segmentation based methods. In addition, they confirmed the importance of introducing contextual information in the design of end-to-end solutions, such as the proposed length classifier when recognizing numeral strings.

Rafael M. O. Cruz, Hiba H. Zakane, Robert Sabourin et al.

Multiple classifier systems focus on the combination of classifiers to obtain better performance than a single robust one. These systems unfold three major phases: pool generation, selection and integration. One of the most promising MCS approaches is Dynamic Selection (DS), which relies on finding the most competent classifier or ensemble of classifiers to predict each test sample. The majority of the DS techniques are based on the K-Nearest Neighbors (K-NN) definition, and the quality of the neighborhood has a huge impact on the performance of DS methods. In this paper, we perform an analysis comparing the classification results of DS techniques and the K-NN classifier under different conditions. Experiments are performed on 18 state-of-the-art DS techniques over 30 classification datasets and results show that DS methods present a significant boost in classification accuracy even though they use the same neighborhood as the K-NN. The reasons behind the outperformance of DS techniques over the K-NN classifier reside in the fact that DS techniques can deal with samples with a high degree of instance hardness (samples that are located close to the decision border) as opposed to the K-NN. In this paper, not only we explain why DS techniques achieve higher classification performance than the K-NN but also when DS should be used.

Felipe N. Walmsley, George D. C. Cavalcanti, Dayvid V. R. Oliveira et al.

In Machine Learning, ensemble methods have been receiving a great deal of attention. Techniques such as Bagging and Boosting have been successfully applied to a variety of problems. Nevertheless, such techniques are still susceptible to the effects of noise and outliers in the training data. We propose a new method for the generation of pools of classifiers based on Bagging, in which the probability of an instance being selected during the resampling process is inversely proportional to its instance hardness, which can be understood as the likelihood of an instance being misclassified, regardless of the choice of classifier. The goal of the proposed method is to remove noisy data without sacrificing the hard instances which are likely to be found on class boundaries. We evaluate the performance of the method in nineteen public data sets, and compare it to the performance of the Bagging and Random Subspace algorithms. Our experiments show that in high noise scenarios the accuracy of our method is significantly better than that of Bagging.

Dayvid V. R. Oliveira, George D. C. Cavalcanti, Thyago N. Porpino et al.

Dynamic Ensemble Selection (DES) techniques aim to select locally competent classifiers for the classification of each new test sample. Most DES techniques estimate the competence of classifiers using a given criterion over the region of competence of the test sample (its the nearest neighbors in the validation set). The K-Nearest Oracles Eliminate (KNORA-E) DES selects all classifiers that correctly classify all samples in the region of competence of the test sample, if such classifier exists, otherwise, it removes from the region of competence the sample that is furthest from the test sample, and the process repeats. When the region of competence has samples of different classes, KNORA-E can reduce the region of competence in such a way that only samples of a single class remain in the region of competence, leading to the selection of locally incompetent classifiers that classify all samples in the region of competence as being from the same class. In this paper, we propose two DES techniques: K-Nearest Oracles Borderline (KNORA-B) and K-Nearest Oracles Borderline Imbalanced (KNORA-BI). KNORA-B is a DES technique based on KNORA-E that reduces the region of competence but maintains at least one sample from each class that is in the original region of competence. KNORA-BI is a variation of KNORA-B for imbalance datasets that reduces the region of competence but maintains at least one minority class sample if there is any in the original region of competence. Experiments are conducted comparing the proposed techniques with 19 DES techniques from the literature using 40 datasets. The results show that the proposed techniques achieved interesting results, with KNORA-BI outperforming state-of-art techniques.

Luiz G. Hafemann, Robert Sabourin, Luiz S. Oliveira

Methods for learning feature representations for Offline Handwritten Signature Verification have been successfully proposed in recent literature, using Deep Convolutional Neural Networks to learn representations from signature pixels. Such methods reported large performance improvements compared to handcrafted feature extractors. However, they also introduced an important constraint: the inputs to the neural networks must have a fixed size, while signatures vary significantly in size between different users. In this paper we propose addressing this issue by learning a fixed-sized representation from variable-sized signatures by modifying the network architecture, using Spatial Pyramid Pooling. We also investigate the impact of the resolution of the images used for training, and the impact of adapting (fine-tuning) the representations to new operating conditions (different acquisition protocols, such as writing instruments and scan resolution). On the GPDS dataset, we achieve results comparable with the state-of-the-art, while removing the constraint of having a maximum size for the signatures to be processed. We also show that using higher resolutions (300 or 600dpi) can improve performance when skilled forgeries from a subset of users are available for feature learning, but lower resolutions (around 100dpi) can be used if only genuine signatures are used. Lastly, we show that fine-tuning can improve performance when the operating conditions change.

Hongliu Cao, Simon Bernard, Laurent Heutte et al.

Breast cancer is one of the most common types of cancer and leading cancer-related death causes for women. In the context of ICIAR 2018 Grand Challenge on Breast Cancer Histology Images, we compare one handcrafted feature extractor and five transfer learning feature extractors based on deep learning. We find out that the deep learning networks pretrained on ImageNet have better performance than the popular handcrafted features used for breast cancer histology images. The best feature extractor achieves an average accuracy of 79.30%. To improve the classification performance, a random forest dissimilarity based integration method is used to combine different feature groups together. When the five deep learning feature groups are combined, the average accuracy is improved to 82.90% (best accuracy 85.00%). When handcrafted features are combined with the five deep learning feature groups, the average accuracy is improved to 87.10% (best accuracy 93.00%).

Hongliu Cao, Simon Bernard, Laurent Heutte et al.

Radiomics is a term which refers to the analysis of the large amount of quantitative tumor features extracted from medical images to find useful predictive, diagnostic or prognostic information. Many recent studies have proved that radiomics can offer a lot of useful information that physicians cannot extract from the medical images and can be associated with other information like gene or protein data. However, most of the classification studies in radiomics report the use of feature selection methods without identifying the machine learning challenges behind radiomics. In this paper, we first show that the radiomics problem should be viewed as an high dimensional, low sample size, multi view learning problem, then we compare different solutions proposed in multi view learning for classifying radiomics data. Our experiments, conducted on several real world multi view datasets, show that the intermediate integration methods work significantly better than filter and embedded feature selection methods commonly used in radiomics.

Rafael M. O. Cruz, Robert Sabourin, George D. C. Cavalcanti

Class-imbalance refers to classification problems in which many more instances are available for certain classes than for others. Such imbalanced datasets require special attention because traditional classifiers generally favor the majority class which has a large number of instances. Ensemble of classifiers have been reported to yield promising results. However, the majority of ensemble methods applied too imbalanced learning are static ones. Moreover, they only deal with binary imbalanced problems. Hence, this paper presents an empirical analysis of dynamic selection techniques and data preprocessing methods for dealing with multi-class imbalanced problems. We considered five variations of preprocessing methods and four dynamic selection methods. Our experiments conducted on 26 multi-class imbalanced problems show that the dynamic ensemble improves the F-measure and the G-mean as compared to the static ensemble. Moreover, data preprocessing plays an important role in such cases.

Saman Bashbaghi, Eric Granger, Robert Sabourin et al.

Face recognition (FR) systems for video surveillance (VS) applications attempt to accurately detect the presence of target individuals over a distributed network of cameras. In video-based FR systems, facial models of target individuals are designed a priori during enrollment using a limited number of reference still images or video data. These facial models are not typically representative of faces being observed during operations due to large variations in illumination, pose, scale, occlusion, blur, and to camera inter-operability. Specifically, in still-to-video FR application, a single high-quality reference still image captured with still camera under controlled conditions is employed to generate a facial model to be matched later against lower-quality faces captured with video cameras under uncontrolled conditions. Current video-based FR systems can perform well on controlled scenarios, while their performance is not satisfactory in uncontrolled scenarios mainly because of the differences between the source (enrollment) and the target (operational) domains. Most of the efforts in this area have been toward the design of robust video-based FR systems in unconstrained surveillance environments. This chapter presents an overview of recent advances in still-to-video FR scenario through deep convolutional neural networks (CNNs). In particular, deep learning architectures proposed in the literature based on triplet-loss function (e.g., cross-correlation matching CNN, trunk-branch ensemble CNN and HaarNet) and supervised autoencoders (e.g., canonical face representation CNN) are reviewed and compared in terms of accuracy and computational complexity.

Luiz G. Hafemann, Robert Sabourin, Luiz S. Oliveira

Verifying the identity of a person using handwritten signatures is challenging in the presence of skilled forgeries, where a forger has access to a person's signature and deliberately attempt to imitate it. In offline (static) signature verification, the dynamic information of the signature writing process is lost, and it is difficult to design good feature extractors that can distinguish genuine signatures and skilled forgeries. This reflects in a relatively poor performance, with verification errors around 7% in the best systems in the literature. To address both the difficulty of obtaining good features, as well as improve system performance, we propose learning the representations from signature images, in a Writer-Independent format, using Convolutional Neural Networks. In particular, we propose a novel formulation of the problem that includes knowledge of skilled forgeries from a subset of users in the feature learning process, that aims to capture visual cues that distinguish genuine signatures and forgeries regardless of the user. Extensive experiments were conducted on four datasets: GPDS, MCYT, CEDAR and Brazilian PUC-PR datasets. On GPDS-160, we obtained a large improvement in state-of-the-art performance, achieving 1.72% Equal Error Rate, compared to 6.97% in the literature. We also verified that the features generalize beyond the GPDS dataset, surpassing the state-of-the-art performance in the other datasets, without requiring the representation to be fine-tuned to each particular dataset.

Luiz G. Hafemann, Robert Sabourin, Luiz S. Oliveira

Research on Offline Handwritten Signature Verification explored a large variety of handcrafted feature extractors, ranging from graphology, texture descriptors to interest points. In spite of advancements in the last decades, performance of such systems is still far from optimal when we test the systems against skilled forgeries - signature forgeries that target a particular individual. In previous research, we proposed a formulation of the problem to learn features from data (signature images) in a Writer-Independent format, using Deep Convolutional Neural Networks (CNNs), seeking to improve performance on the task. In this research, we push further the performance of such method, exploring a range of architectures, and obtaining a large improvement in state-of-the-art performance on the GPDS dataset, the largest publicly available dataset on the task. In the GPDS-160 dataset, we obtained an Equal Error Rate of 2.74%, compared to 6.97% in the best result published in literature (that used a combination of multiple classifiers). We also present a visual analysis of the feature space learned by the model, and an analysis of the errors made by the classifier. Our analysis shows that the model is very effective in separating signatures that have a different global appearance, while being particularly vulnerable to forgeries that very closely resemble genuine signatures, even if their line quality is bad, which is the case of slowly-traced forgeries.

Julien-Charles Lévesque, Christian Gagné, Robert Sabourin

In this paper, we bridge the gap between hyperparameter optimization and ensemble learning by performing Bayesian optimization of an ensemble with regards to its hyperparameters. Our method consists in building a fixed-size ensemble, optimizing the configuration of one classifier of the ensemble at each iteration of the hyperparameter optimization algorithm, taking into consideration the interaction with the other models when evaluating potential performances. We also consider the case where the ensemble is to be reconstructed at the end of the hyperparameter optimization phase, through a greedy selection over the pool of models generated during the optimization. We study the performance of our proposed method on three different hyperparameter spaces, showing that our approach is better than both the best single model and a greedy ensemble construction over the models produced by a standard Bayesian optimization.

Luiz G. Hafemann, Robert Sabourin, Luiz S. Oliveira

Automatic Offline Handwritten Signature Verification has been researched over the last few decades from several perspectives, using insights from graphology, computer vision, signal processing, among others. In spite of the advancements on the field, building classifiers that can separate between genuine signatures and skilled forgeries (forgeries made targeting a particular signature) is still hard. We propose approaching the problem from a feature learning perspective. Our hypothesis is that, in the absence of a good model of the data generation process, it is better to learn the features from data, instead of using hand-crafted features that have no resemblance to the signature generation process. To this end, we use Deep Convolutional Neural Networks to learn features in a writer-independent format, and use this model to obtain a feature representation on another set of users, where we train writer-dependent classifiers. We tested our method in two datasets: GPDS-960 and Brazilian PUC-PR. Our experimental results show that the features learned in a subset of the users are discriminative for the other users, including across different datasets, reaching close to the state-of-the-art in the GPDS dataset, and improving the state-of-the-art in the Brazilian PUC-PR dataset.

Rafael M. O. Cruz, Robert Sabourin, George D. C. Cavalcanti

Dynamic ensemble selection (DES) techniques work by estimating the level of competence of each classifier from a pool of classifiers. Only the most competent ones are selected to classify a given test sample. Hence, the key issue in DES is the criterion used to estimate the level of competence of the classifiers in predicting the label of a given test sample. In order to perform a more robust ensemble selection, we proposed the META-DES framework using meta-learning, where multiple criteria are encoded as meta-features and are passed down to a meta-classifier that is trained to estimate the competence level of a given classifier. In this technical report, we present a step-by-step analysis of each phase of the framework during training and test. We show how each set of meta-features is extracted as well as their impact on the estimation of the competence level of the base classifier. Moreover, an analysis of the impact of several factors in the system performance, such as the number of classifiers in the pool, the use of different linear base classifiers, as well as the size of the validation data. We show that using the dynamic selection of linear classifiers through the META-DES framework, we can solve complex non-linear classification problems where other combination techniques such as AdaBoost cannot.

Luiz G. Hafemann, Robert Sabourin, Luiz S. Oliveira

The area of Handwritten Signature Verification has been broadly researched in the last decades, but remains an open research problem. The objective of signature verification systems is to discriminate if a given signature is genuine (produced by the claimed individual), or a forgery (produced by an impostor). This has demonstrated to be a challenging task, in particular in the offline (static) scenario, that uses images of scanned signatures, where the dynamic information about the signing process is not available. Many advancements have been proposed in the literature in the last 5-10 years, most notably the application of Deep Learning methods to learn feature representations from signature images. In this paper, we present how the problem has been handled in the past few decades, analyze the recent advancements in the field, and the potential directions for future research.

Guillaume Corriveau, Raynald Guilbault, Antoine Tahan et al.

Numerous genotypic diversity measures (GDMs) are available in the literature to assess the convergence status of an evolutionary algorithm (EA) or describe its search behavior. In a recent study, the authors of this paper drew attention to the need for a GDM validation framework. In response, this study proposes three requirements (monotonicity in individual varieties, twinning, and monotonicity in distance) that can clearly portray any GDMs. These diversity requirements are analysed by means of controlled population arrangements. In this paper four GDMs are evaluated with the proposed validation framework. The results confirm that properly evaluating population diversity is a rather difficult task, as none of the analysed GDMs complies with all the diversity requirements.

George S. Eskander, Robert Sabourin, Eric Granger

An offline signature-based fuzzy vault (OSFV) is a bio-cryptographic implementation that uses handwritten signature images as biometrics instead of traditional passwords to secure private cryptographic keys. Having a reliable OSFV implementation is the first step towards automating financial and legal authentication processes, as it provides greater security of confidential documents by means of the embedded handwritten signatures. The authors have recently proposed the first OSFV implementation which is reviewed in this paper. In this system, a machine learning approach based on the dissimilarity representation concept is employed to select a reliable feature representation adapted for the fuzzy vault scheme. Some variants of this system are proposed for enhanced accuracy and security. In particular, a new method that adapts user key size is presented. Performance of proposed methods are compared using the Brazilian PUCPR and GPDS signature databases and results indicate that the key-size adaptation method achieves a good compromise between security and accuracy. While average system entropy is increased from 45-bits to about 51-bits, the AER (average error rate) is decreased by about 21%.

Albert H. R. Ko, Robert Sabourin, Alceu S. Britto et al.

The Ensemble of Classifiers (EoC) has been shown to be effective in improving the performance of single classifiers by combining their outputs, and one of the most important properties involved in the selection of the best EoC from a pool of classifiers is considered to be classifier diversity. In general, classifier diversity does not occur randomly, but is generated systematically by various ensemble creation methods. By using diverse data subsets to train classifiers, these methods can create diverse classifiers for the EoC. In this work, we propose a scheme to measure data diversity directly from random subspaces, and explore the possibility of using it to select the best data subsets for the construction of the EoC. Our scheme is the first ensemble selection method to be presented in the literature based on the concept of data diversity. Its main advantage over the traditional framework (ensemble creation then selection) is that it obviates the need for classifier training prior to ensemble selection. A single Genetic Algorithm (GA) and a Multi-Objective Genetic Algorithm (MOGA) were evaluated to search for the best solutions for the classifier-free ensemble selection. In both cases, objective functions based on different clustering diversity measures were implemented and tested. All the results obtained with the proposed classifier-free ensemble selection method were compared with the traditional classifier-based ensemble selection using Mean Classifier Error (ME) and Majority Voting Error (MVE). The applicability of the method is tested on UCI machine learning problems and NIST SD19 handwritten numerals.