Effective model selection is critical in symbolic regression (SR) to identify mathematical expressions that balance accuracy and complexity, and have low expected error on unseen data. Many modern implementations of genetic programming (GP) for SR generate a set of Pareto optimal candidate solutions, but reliable automatic selection of solutions that generalize well remains an open issue. Current literature offers various information-theoretic and Bayesian approaches, yet comprehensive comparisons of their performance across different data regimes are limited. This study presents a systematic empirical comparison of widely used selection criteria: the Akaike information criterion (AIC), the corrected AIC (AICc), the Bayesian information criterion (BIC), minimum description length (MDL), as well as Efron's bootstrap estimate for the in-sample prediction error on seven synthetic datasets with Gaussian noise. We rank candidate expressions generated by perturbing ground-truth functions to assess generalization error and selection probability of the ground-truth expression. Our findings reveal that MDL consistently identifies models with the lowest test error and the shortest length across most datasets. While no single criterion dominates all results, MDL and BIC produced the highest probability of selecting the ground-truth expressions.

Regression analysis is used for prediction and to understand the effect of independent variables on dependent variables. Symbolic regression (SR) automates the search for non-linear regression models, delivering a set of hypotheses that balances accuracy with the possibility to understand the phenomena. Many SR implementations return a Pareto front allowing the choice of the best trade-off. However, this hides alternatives that are close to non-domination, limiting these choices. Equality graphs (e-graphs) allow to represent large sets of expressions compactly by efficiently handling duplicated parts occurring in multiple expressions. The e-graphs allow to efficiently store and query all solution candidates visited in one or multiple runs of different algorithms and open the possibility to analyze much larger sets of SR solution candidates. We introduce rEGGression, a tool using e-graphs to enable the exploration of a large set of symbolic expressions which provides querying, filtering, and pattern matching features creating an interactive experience to gain insights about SR models. The main highlight is its focus in the exploration of the building blocks found during the search that can help the experts to find insights about the studied phenomena. This is possible by exploiting the pattern matching capability of the e-graph data structure.

The search for symbolic regression models with genetic programming (GP) has a tendency of revisiting expressions in their original or equivalent forms. Repeatedly evaluating equivalent expressions is inefficient, as it does not immediately lead to better solutions. However, evolutionary algorithms require diversity and should allow the accumulation of inactive building blocks that can play an important role at a later point. The equality graph is a data structure capable of compactly storing expressions and their equivalent forms allowing an efficient verification of whether an expression has been visited in any of their stored equivalent forms. We exploit the e-graph to adapt the subtree operators to reduce the chances of revisiting expressions. Our adaptation, called eggp, stores every visited expression in the e-graph, allowing us to filter out from the available selection of subtrees all the combinations that would create already visited expressions. Results show that, for small expressions, this approach improves the performance of a simple GP algorithm to compete with PySR and Operon without increasing computational cost. As a highlight, eggp was capable of reliably delivering short and at the same time accurate models for a selected set of benchmarks from SRBench and a set of real-world datasets.

Symbolic regression searches for analytic expressions that accurately describe studied phenomena. The main promise of this approach is that it may return an interpretable model that can be insightful to users, while maintaining high accuracy. The current standard for benchmarking these algorithms is SRBench, which evaluates methods on hundreds of datasets that are a mix of real-world and simulated processes spanning multiple domains. At present, the ability of SRBench to evaluate interpretability is limited to measuring the size of expressions on real-world data, and the exactness of model forms on synthetic data. In practice, model size is only one of many factors used by subject experts to determine how interpretable a model truly is. Furthermore, SRBench does not characterize algorithm performance on specific, challenging sub-tasks of regression such as feature selection and evasion of local minima. In this work, we propose and evaluate an approach to benchmarking SR algorithms that addresses these limitations of SRBench by 1) incorporating expert evaluations of interpretability on a domain-specific task, and 2) evaluating algorithms over distinct properties of data science tasks. We evaluate 12 modern symbolic regression algorithms on these benchmarks and present an in-depth analysis of the results, discuss current challenges of symbolic regression algorithms and highlight possible improvements for the benchmark itself.

Symbolic regression (SR) searches for parametric models that accurately fit a dataset, prioritizing simplicity and interpretability. Despite this secondary objective, studies point out that the models are often overly complex due to redundant operations, introns, and bloat that arise during the iterative process, and can hinder the search with repeated exploration of bloated segments. Applying a fast heuristic algebraic simplification may not fully simplify the expression and exact methods can be infeasible depending on size or complexity of the expressions. We propose a novel agnostic simplification and bloat control for SR employing an efficient memoization with locality-sensitive hashing (LHS). The idea is that expressions and their sub-expressions traversed during the iterative simplification process are stored in a dictionary using LHS, enabling efficient retrieval of similar structures. We iterate through the expression, replacing subtrees with others of same hash if they result in a smaller expression. Empirical results shows that applying this simplification during evolution performs equal or better than without simplification in minimization of error, significantly reducing the number of nonlinear functions. This technique can learn simplification rules that work in general or for a specific problem, and improves convergence while reducing model complexity.

Parent selection plays an important role in evolutionary algorithms, and many strategies exist to select the parent pool before breeding the next generation. Methods often rely on average error over the entire dataset as a criterion to select the parents, which can lead to an information loss due to aggregation of all test cases. Under ϵ-lexicase selection, the population goes to a selection pool that is iteratively reduced by using each test individually, discarding individuals with an error higher than the elite error plus the median absolute deviation (MAD) of errors for that particular test case. In an attempt to better capture differences in performance of individuals on cases, we propose a new criteria that splits errors into two partitions that minimize the total variance within partitions. Our method was embedded into the FEAT symbolic regression algorithm, and evaluated with the SRBench framework, containing 122 black-box synthetic and real-world regression problems. The empirical results show a better performance of our approach compared to traditional ϵ-lexicase selection in the real-world datasets while showing equivalent performance on the synthetic dataset.

Symbolic regression (SR) searches for analytical expressions representing the relationship between explanatory and response variables. Current SR methods assume a single dataset extracted from a single experiment. Nevertheless, frequently, the researcher is confronted with multiple sets of results obtained from experiments conducted with different set-ups. Traditional SR methods may fail to find the underlying expression since the parameters of each experiment can be different. In this work we present Multiview Symbolic Regression (MvSR), which takes into account multiple datasets simultaneously, mimicking experimental environments, and outputs a general parametric solution. This approach fits the evaluated expression to each independent dataset and returns a parametric family of functions f(x; θ) simultaneously capable of accurately fitting all datasets. We demonstrate the effectiveness of MvSR using data generated from known expressions, as well as real-world data from astronomy, chemistry and economy, for which an a priori analytical expression is not available. Results show that MvSR obtains the correct expression more frequently and is robust to hyperparameters change. In real-world data, it is able to grasp the group behaviour, recovering known expressions from the literature as well as promising alternatives, thus enabling the use MvSR to a large range of experimental scenarios.

Program synthesis is the process of generating a computer program following a set of specifications, which can be a high-level description of the problem and/or a set of input-output examples. The synthesis can be modeled as a search problem in which the search space is the set of all the programs valid under a grammar. As the search space is vast, brute force is usually not viable and search heuristics, such as genetic programming, also have difficulty navigating it without any guidance. In this paper we present HOTGP, a new genetic programming algorithm that synthesizes pure, typed, and functional programs. HOTGP leverages the knowledge provided by the rich data-types associated with the specification and the built-in grammar to constrain the search space and improve the performance of the synthesis. The grammar is based on Haskell's standard base library (the synthesized code can be directly compiled using any standard Haskell compiler) and includes support for higher-order functions, Λ-functions, and parametric polymorphism. Experimental results show that, when compared to 6 state-of-the-art algorithms using a standard set of benchmarks, HOTGP is competitive and capable of synthesizing the correct programs more frequently than any other of the evaluated algorithms.

Overparameterized models in regression analysis are often harder to interpret and can be harder to fit because of ill-conditioning. Genetic programming is prone to overparameterized models as it evolves the structure of the model without taking the location of parameters into account. One way to alleviate this is rewriting the expression and merging the redundant fitting parameters. In this paper we propose the use of equality saturation to alleviate overparameterization. We first notice that all the tested GP implementations suffer from overparameterization to different extents and then show that equality saturation together with a small set of rewriting rules is capable of reducing the number of fitting parameters to a minimum with a high probability. Compared to one of the few available alternatives, Sympy, it produces much better and consistent results. These results lead to different possible future investigations such as the simplification of expressions during the evolutionary process, and improvement of the interpretability of symbolic models.

Online social networks are at the limelight of the public debate, where antagonistic groups compete to impose conflicting narratives and polarize the discussions. This article proposes an approach for measuring network polarization and political sectarianism in Twitter based on user interaction networks. Centrality metrics identify a small group of influential users (polarizers and unpolarizers) who influence a larger group of users (polarizees and unpolarizees) according to their ideological stance (left, right, and undefined). This network polarization is computed by the Bayesian probability using typical actions such as following, tweeting, retweeting, and replying. The measurement of political sectarianism also uses Bayesian probability and words extracted from the tweets to quantify the intensity of othering, aversion, and moralization in the debate. We collected Twitter data from 33 conflicted political events in Brazil during 2020, strongly influenced by the COVID-19 pandemic. Based on our methodology and polarization score, our results reveal that the approach based on user interaction networks leads to an increasing understanding of polarized conflicts in Twitter. Also, a small number of polarizers is enough to represent the polarization and sectarianism of Twitter events.

Symbolic Regression searches for a parametric model with the optimal value of the parameters that best fits a set of samples to a measured target. The desired solution has a balance between accuracy and interpretability. Commonly, there is no constraint in the way the functions are composed in the expression or where the numerical parameters are placed, which can potentially lead to expressions that require a nonlinear optimization to find the optimal parameters. The representation called Interaction-Transformation alleviates this problem by describing expressions as a linear regression of the composition of functions applied to the interaction of the variables. One advantage is that any model that follows this representation is linear in its parameters, allowing an efficient computation. More recently, this representation was extended by applying a univariate function to the rational function of two Interaction-Transformation expressions, called Transformation-Interaction-Rational ( TIR ). The use of this representation was shown to be competitive with the current literature of Symbolic Regression. In this article, we make a detailed analysis of these results using the SRBench benchmark. For this purpose, we split the datasets into different categories to understand the algorithm behavior in different settings. We also test the use of nonlinear optimization to adjust the numerical parameters instead of Ordinary Least Squares. We find through the experiments that TIR has some difficulties handling high-dimensional and noisy datasets, especially when most of the variables are composed of random noise. These results point to new directions for improving the evolutionary search of TIR expressions.

Symbolic Regression searches for a function form that approximates a dataset often using Genetic Programming. Since there is usually no restriction to what form the function can have, Genetic Programming may return a hard to understand model due to non-linear function chaining or long expressions. A novel representation called Interaction-Transformation was recently proposed to alleviate this problem. In this representation, the function form is restricted to an affine combination of terms generated as the application of a single univariate function to the interaction of selected variables. This representation obtained competing solutions on standard benchmarks. Despite the initial success, a broader set of benchmarking functions revealed the limitations of the constrained representation. In this paper we propose an extension to this representation, called Transformation-Interaction-Rational representation that defines a new function form as the rational of two Interaction-Transformation functions. Additionally, the target variable can also be transformed with an univariate function. The main goal is to improve the approximation power while still constraining the overall complexity of the expression. We tested this representation with a standard Genetic Programming with crossover and mutation. The results show a great improvement when compared to its predecessor and a state-of-the-art performance for a large benchmark.

Shape-constrained Symbolic Regression integrates prior knowledge about the function shape into the symbolic regression model. This can be used to enforce that the model has desired properties such as monotonicity, or convexity, among others. Shape-constrained Symbolic Regression can also help to create models with better extrapolation behavior and reduced sensitivity to noise. The constraint evaluation can be challenging because exact evaluation of constraints may require a search for the extrema of non-convex functions. Approximations via interval arithmetic allow to efficiently find bounds for the extrema of functions. However, interval arithmetic can lead to overly wide bounds and therefore produces a pessimistic estimation. Another possibility is to use sampling which underestimates the true range. Sampling therefore produces an optimistic estimation. In this paper we evaluate both methods and compare them on different problem instances. In particular we evaluate the sensitivity to noise and the extrapolation capabilities in combination with noise data. The results indicate that the optimistic approach works better for predicting out-of-domain points (extrapolation) and the pessimistic approach works better for high noise levels.

We investigate the addition of constraints on the function image and its derivatives for the incorporation of prior knowledge in symbolic regression. The approach is called shape-constrained symbolic regression and allows us to enforce, for example, monotonicity of the function over selected inputs. The aim is to find models which conform to expected behavior and which have improved extrapolation capabilities. We demonstrate the feasibility of the idea and propose and compare two evolutionary algorithms for shape-constrained symbolic regression: (i) an extension of tree-based genetic programming which discards infeasible solutions in the selection step, and (ii) a two-population evolutionary algorithm that separates the feasible from the infeasible solutions. In both algorithms we use interval arithmetic to approximate bounds for models and their partial derivatives. The algorithms are tested on a set of 19 synthetic and four real-world regression problems. Both algorithms are able to identify models which conform to shape constraints which is not the case for the unmodified symbolic regression algorithms. However, the predictive accuracy of models with constraints is worse on the training set and the test set. Shape-constrained polynomial regression produces the best results for the test set but also significantly larger models.

In explainable AI, one aspect of a prediction's explanation is to measure each predictor's importance to the decision process. The importance can measure how much variation a predictor promotes locally or how much the predictor contributes to the deviation from a reference point (Shapley value). If we have the ground truth analytical model, we can calculate the former using the Partial Effect, calculated as the predictor's partial derivative. Also, we can estimate the latter by calculating the average partial effect multiplied by the difference between the predictor and the reference value. Symbolic Regression is a gray-box model for regression problems that returns an analytical model approximating the input data. Although it is often associated with interpretability, few works explore this property. This paper will investigate the use of Partial Effect with the analytical models generated by the Interaction-Transformation Evolutionary Algorithm symbolic regressor (ITEA). We show that the regression models returned by ITEA coupled with Partial Effect provide the closest explanations to the ground-truth and a close approximation to Shapley values. These results open up new opportunities to explain symbolic regression models compared to the approximations provided by model-agnostic approaches.

Symbolic regression aims to hypothesize a functional relationship involving explanatory variables and one or more dependent variables, based on examples of the desired input-output behavior. Genetic programming is a meta-heuristic commonly used in the literature to achieve this goal. Even though Symbolic Regression is sometimes associated with the potential of generating interpretable expressions, there is no guarantee that the returned function will not contain complicated constructs or even bloat. The Interaction-Transformation (IT) representation was recently proposed to alleviate this issue by constraining the search space to expressions following a simple and comprehensive pattern. In this paper, we resort to Simulated Annealing to search for a symbolic expression using the IT representation. Simulated Annealing exhibits an intrinsic ability to escape from poor local minima, which is demonstrated here to yield competitive results, particularly in terms of generalization, when compared with state-of-the-art Symbolic Regression techniques, that depend on population-based meta-heuristics, and committees of learning machines.

Interaction-Transformation (IT) is a new representation for Symbolic Regression that reduces the space of solutions to a set of expressions that follow a specific structure. The potential of this representation was illustrated in prior work with the algorithm called SymTree. This algorithm starts with a simple linear model and incrementally introduces new transformed features until a stop criterion is met. While the results obtained by this algorithm were competitive with the literature, it had the drawback of not scaling well with the problem dimension. This article introduces a mutation-only Evolutionary Algorithm, called ITEA, capable of evolving a population of IT expressions. One advantage of this algorithm is that it enables the user to specify the maximum number of terms in an expression. In order to verify the competitiveness of this approach, ITEA is compared to linear, nonlinear, and Symbolic Regression models from the literature. The results indicate that ITEA is capable of finding equal or better approximations than other Symbolic Regression models while being competitive to state-of-the-art nonlinear models. Additionally, since this representation follows a specific structure, it is possible to extract the importance of each original feature of a data set as an analytical function, enabling us to automate the explanation of any prediction. In conclusion, ITEA is competitive when comparing to regression models with the additional benefit of automating the extraction of additional information of the generated models.

The balance between approximation error and model complexity is an important trade-off for Symbolic Regression algorithms. This trade-off is achieved by means of specific operators for bloat control, modified operators, limits to the size of the generated expressions and multi-objective optimization. Recently, the representation Interaction-Transformation was introduced with the goal of limiting the search space to simpler expressions, thus avoiding bloating. This representation was used in the context of an Evolutionary Algorithm in order to find concise expressions resulting in small approximation errors competitive with the literature. Particular to this algorithm, two parameters control the complexity of the generated expression. This paper investigates the influence of those parameters w.r.t. the goodness-of-fit. Through some extensive experiments, we find that the maximum number of terms is more important to control goodness-of-fit but also that there is a limit to the extent that increasing its value renders any benefits. Second, the limit to the minimum and maximum value of the exponent has a smaller influence to the results and it can be set to a default value without impacting the final results.

Symbolic Regression techniques stand out from other regression analysis tools because of the possibility of generating powerful but yet simple expressions. These simple expressions may be useful in many practical situations in which the practitioner wants to interpret the obtained results, finetune the model, or understand the generating phenomena. Despite this possibility, the current state-of-the-art algorithms for Symbolic Regression usually require a high computational budget while having little guarantees regarding the simplicity of the returned expressions. Recently, a new Data Structure representation for mathematical expressions, called Interaction-Transformation (IT), was introduced together with a search-based algorithm named SymTree that surpassed a subset of the recent Symbolic Regression algorithms and even some state-of-the-art nonlinear regression algorithms, while returning simple expressions as a result. This paper introduces a lightweight tool based on this algorithm, named Lab Assistant. This tool runs on the client-side of any compatible Internet browser with JavaScript. Alongside this tool, two algorithms using the IT representation are introduced. Some experiments are performed in order to show the potential of the Lab Assistant to help practitioners, professors, researchers and students willing to experiment with Symbolic Regression. The results showed that this tool is competent to find the correct expression for many well known Physics and Engineering relations within a reasonable average time frame of a few seconds. This tool opens up lots of possibilities in Symbolic Regression research for low-cost devices to be used in applications where a high-end computer is not available.

Text Mining is a challenging task due to the lack of a naturally structured representation and the high dimensionality induced by the feature extraction techniques commonly used. Different feature extractions can lead to multiple views that can capture different aspects of the text documents being analyzed. The combination of these features can lead to a better accuracy in classification tasks but, also, an undesirable increase in the number of features. In this work, we investigate the use of a feature extraction technique called DCDistance used as a multiple feature extraction for text documents combined with a Genetic Algorithm based feature selection, hereby called MVDCD. The results show that the main advantage of MVDCD is that the dimensionality is reduced by more than 90% while significantly increasing the classification accuracy when compared to vanilla DCDistance and other feature selections techniques. A side effect of the use of DCDistance and MVDCD is the possibility of model interpretability, as the extracted features are explicit.

Computational Intelligence in Games comprises many challenges such as the procedural level generation, evolving adversary difficulty and the learning of autonomous playing agents. This last challenge has the objective of creating an autonomous playing agent capable of winning against an opponent on an specific game. Whereas a human being can learn a general winning strategy (i.e., avoid the obstacles and defeat the enemies), learning algorithms have a tendency to overspecialize for a given training scenario (i.e., perform an exact sequence of actions to win), not being able to face variations of the original scenario. To further study this problem, we have applied three variations of Neuroevolution algorithms to the EvoMan game playing learning framework with the main objective of developing an autonomous agent capable of playing in different scenarios than those observed during the training stages. This framework is based on the bosses fights of the well known game called Mega Man. The experiments show that the evolved agents are not capable of winning every challenge imposed to them but they are still capable of learning a generalized behavior.

Many practical problems are described by an objective-function with the intent to optimize a single goal. This leads to the important research topic of nonlinear optimization, that seeks to create algorithms and computational methods that are capable of finding a global optimum of such functions. But, many functions are multimodal, having many different global optima. Also, given the impossibility to create an exact model of a real-world problem, not every global (or local) optima is feaseable to be conceived. As such, it is interesting to find as many alternative optima in order to find one that is feaseable given unmodelled constraints. This paper proposes a methodology that, given a local optimum, it finds nearby local optima with similar objective-function values. This is performed by maximizing the approximation error of a Linear Interpolation of the function. The experiments show promising results regarding the number of detected peaks when compared to the state-of-the-art, though requiring a higher number of function evaluations on average.

Most datasets obtained in real-world applications are typically unlabeled, requiring a manual labor of classifying a sample of such data or the application of unsupervised learning. Clustering is typically used to devise how data are grouped together before sampling the data to be labeled. Most clustering algorithms often assumes that the number of clusters is known and that a given instance from the dataset belongs to only one cluster. The Fainet algorithm is a bioinspired fuzzy clustering algorithm that finds fuzzy partitions and dynamically estimates the number of clusters. The results from the literature showed that, given a correct parameters set, this algorithm can outperform most clustering methods from the literature. However, in order to obtain such optimal set, a typical user should first acquire a knowledge of the dataset being studied. This work proposes dynamic rules to finetune the parameters set on-the-fly. The advantages of the proposed method is that the parameters not only adapts to the dataset characteristics but also to how close the solutions are from the optima. The results show that the method greatly improves the prototypes representativeness while optimizing the estimated number of clusters.

The analysis of complex networks is an important research topic that helps us understand the underlying behavior of complex systems and the interactions of their components. One particularly relevant analysis is the detection of communities formed by such interactions. Most community detection algorithms work as optimization tools that minimize a given quality function, while assuming that each node belongs to a single community. However, most complex networks contain nodes that belong to two or more communities, which are called bridges. The identification of bridges is crucial to several problems, as they often play important roles in the system described by the network. By exploiting the multimodality of quality functions, it is possible to obtain distinct optimal communities where, in each solution, each bridge node belongs to a distinct community. This paper proposes a technique that tries to identify a set of (possibly) overlapping communities by combining diverse solutions contained in a pool, which correspond to disjoint community partitions of a given network. To obtain the pool of partitions, an adapted version of the immune-inspired algorithm named cob-aiNet[C] was adopted here. The proposed methodology was applied to four real-world social networks and the obtained results were compared to those reported in the literature. The comparisons have shown that the proposed approach is competitive and even capable of overcoming the best results reported for some of the problems.

In some data mining applications the analyzed data can be classified as simultaneously belonging to more than one class, this characterizes the multi-label classification problem. Numerous methods for dealing with this problem are based on decomposition, which essentially treats labels (or some subsets of labels) independently and ignores interactions between them. This fact might be a problem, as some labels may be correlated to local patterns in the data. In this paper, we propose to enhance multi-label classifiers with the aid of biclusters, which are capable of finding the correlation between subsets of objects, features and labels. We then construct binary features from these patterns that can be interpreted as local correlations (in terms of subset of features and instances) in the data. These features are used as input for multi-label classifiers. We experimentally show that using such constructed features can improve the classification performance of some decompositive multi-label learning techniques.

The Operation Planning of Hydroelectric Systems is a large, dynamic, stochastic, interconnected and nonlinear optimization problem. In this model, the minimization of penalized thermal complementation is considered as the objective function with the water discharge of hydroelectric plants at each period as the decision variables. An adaptation of two Evolutionary Metaheuristics, the Genetic Algorithm and the Differential Evolution, are proposed in this paper to solve this problem. These methods consider a set of solutions in order to perform exploration and exploitation of the search space allowing them to find several good quality solutions that can serve as alternatives to a given scenario. Tests performed with the Brazilian Subsystems and compared to one of the current used approaches show that the evolutionary methods can improve current solutions and can also bring the benefit of alternative solutions.

Text clustering is used on a variety of applications such as content-based recommendation, categorization, summarization, information retrieval and automatic topic extraction. Since most pair of documents usually shares just a small percentage of words, the dataset representation tends to become very sparse, thus the need of using a similarity metric capable of a partial matching of a set of features. The technique known as Co-Clustering is capable of finding several clusters inside a dataset with each cluster composed of just a subset of the object and feature sets. In word-document data this can be useful to identify the clusters of documents pertaining to the same topic, even though they share just a small fraction of words. In this paper a scalable co-clustering algorithm is proposed using the Locality-sensitive hashing technique in order to find co-clusters of documents. The proposed algorithm will be tested against other co-clustering and traditional algorithms in well known datasets. The results show that this algorithm is capable of finding clusters more accurately than other approaches while maintaining a linear complexity.

Artificial immune systems are composed of techniques inspired by immunology. The clonal selection principle ensures the organism adaptation to fight invading antigens by an immune response activated by the binding of antigens and antibodies. Since the immune response must correctly allocate the available resources in order to attack an antigen with its best available antibody while trying to learning an even better one, the reproduction rate of each immune cell must be carefully determined. This paper presents a novel fuzzy inference technique to calculate the suitable number of clones for immune inspired algorithms that uses the clonal selection process as the evolutionary process. More specifically, this technique is applied to the CLONALG algorithm for solving pattern recognition tasks and to the copt-aiNet algorithm for solving combinatorial optimization tasks, particularly the Traveling Salesman Problem. The obtained results show that the fuzzy approach makes it possible to automatically determine the number of clones in CLONALG and copt-aiNet, thus eliminating this key user-defined parameter.

Diversity maintenance is an important aspect in population-based metaheuristics for optimization, as it tends to allow a better exploration of the search space, thus reducing the susceptibility to local optima in multimodal optimization problems. In this context, metaheuristics based on the Artificial Immune System (AIS) framework, especially those inspired by the Immune Network theory, are known to be capable of stimulating the generation of diverse sets of solutions for a given problem, even though generally implementing very simple mechanisms to control the dynamics of the network. To increase such diversity maintenance capability even further, a new immune-inspired algorithm was recently proposed, which adopted a novel concentration-based model of immune network. This new algorithm, named cob-aiNet (Concentration-based Artificial Immune Network), was originally developed to solve real-parameter single-objective optimization problems, and it was later extended (with cob-aiNet[MO]) to deal with real-parameter multi-objective optimization. Given that both cob-aiNet and cob-aiNet[MO] obtained competitive results when compared to state-of-the-art algorithms for continuous optimization and also presented significantly improved diversity maintenance mechanisms, in this work the same concentration-based paradigm was further explored, in an extension of such algorithms to deal with single-objective combinatorial optimization problems. This new algorithm, named cob-aiNet[C], was evaluated here in a series of experiments based on four Traveling Salesman Problems (TSPs), in which it was verified not only the diversity maintenance capabilities of the algorithm, but also its overall optimization performance.

Biclustering is usually referred to as the process of finding subsets of rows and columns from a given dataset expressing a relationship. Each subset is a bicluster and corresponds to a sub-matrix whose elements tend to present a high degree of coherence with each other, that may lead to novel discoveries regarding the objects in the dataset. This coherence leads to the possibility of obtaining representative values for rows (subset of objects) and columns (subset of attributes) of each bicluster. In the literature, it is usually studied the additive coherence among elements, i.e. each element is represented by the sum of its respective representative values. But in a given dataset, it is also possible to find multiplicative relations, i.e. each element being represented by the multiplication of its respective representative values, and that may reveal distinct knowledge contained in the objects of the dataset. So, in this paper, a swarm based approach, named SwarmBcluster, is adapted to find both additive and multiplicative coherent biclusters from a dataset, in an attempt to enrich the amount of information provided by the biclusters. Experiments are performed considering two well known datasets and it is found that the multiplicative coherence biclusters improve the quality of the data analysis and may contribute to reduce the influence of noise.

Although the missing data problem has been studied for many years, it is still a relevant and challenging problem nowadays. Data can be missing for a variety of reasons, and there are several techniques capable of processing missing data. A parcel of them tries to estimate the missing values. This technique is called imputation. Recently, it was proposed a biclustering algorithm, based on Swarm Intelligence, named SwarmBCluster, to impute missing data. As it is a novel and promising algorithm, this paper intends to investigate the influence of its parameters on the performance. To achieve this objective, this paper will compare SwarmBCluster with other two imputation algorithms and, after that, it will perform a sensitivity analysis. The quality of the imputations is measured with the Root Mean Squared Error (RMSE). The experiments showed that SwarmBCluster presents good results concerning the RMSE metric and that the proper choice of parameters can considerably improve the performance of the algorithm.

Immune-inspired algorithms based on the Immune Network theory have been frequently claimed to be capable of maintaining diversity among the candidate solutions in their population. However, no specific study on this aspect to verify how the intrinsic diversity mechanisms of such immune algorithms behave, when compared to other approaches from the literature, has been made yet. Therefore, in this work we have addressed this issue, by taking the opt-aiNet algorithm (a popular immune-inspired algorithm developed for real-parameter optimization) and comparing its results with those of a modified version, in which the mechanisms associated with diversity maintenance were replaced by a traditional fitness sharing approach. Besides, two distance metrics were also considered for both algorithms: the traditional Euclidean distance, and the Line Distance, a metric proposed in the literature as capable of identifying whether two solutions belong to distinct local optima. The experiments were performed on six benchmark problems from the literature, each of them with distinct characteristics, and the results have shown that the original immune-inspired mechanisms of opt-aiNet are indeed more capable of stimulating the diversity of solutions, and also requiring a smaller amount of computational resources.

Biclustering is usually referred to as the process of finding subsets of rows and columns from a given dataset. Each subset is a bicluster and corresponds to a sub-matrix whose elements tend to present a high degree of coherence with each other. In order to find such structures, the δ-biclustering problem was formulated, being denoted as the problem of finding a set of biclusters limited by a maximum degree of coherence, measured by a mean-squared residue, while maximizing the bicluster total size. Additionally, it is expected a reduced overlap among the biclusters in the set, in other words, a minimization of the number of common elements shared by them. This also leads to a high coverage of the original dataset given the number of biclusters found. Most algorithms intended to find such biclusters focus only on the mean-squared residue and/or the bicluster size. This usually leads to a set of biclusters that do not fully cover the whole data and, as a consequence, shares a high overlap among them. This may generate redundant information on some portions of the dataset and lack of information on other portions. Also, some methods introduce noise into the dataset in order to promote a better coverage, but sometimes misleading the search. In this paper, a swarm-based approach, named SwarmBcluster, is created to effectively find biclusters without introducing noise and with the main objective of achieving maximum coverage. Experiments were performed considering two well-known datasets and a comparative analysis considering other approaches indicates that SwarmBcluster is capable of finding a set of biclusters with high coverage, while maintaining a high average volume and also obeying the coherence constraint imposed.

The biclustering technique was developed to avoid some of the drawbacks presented by standard clustering techniques. Given that biclustering requires the optimization of at least two conflicting objectives and that multiple independent solutions are desirable as the outcome, a few multi-objective evolutionary algorithms for biclustering were proposed in the literature. However, apart from the individual characteristics of the biclusters that should be optimized during their construction, several other global aspects should also be considered, such as the coverage of the dataset and the overlap among biclusters. These requirements will be addressed in this work with the MOM-aiNet+ algorithm, which is an improvement of the original multi-objective multipopulation artificial immune network denoted MOM-aiNet. Here, the MOM-aiNet+ algorithm will be described in detail, its main differences from the original MOM-aiNet will be highlighted, and both algorithms will be compared, together with three other proposals from the literature.

In many real-world scenarios, in contrast to standard benchmark optimization problems, we may face some uncertainties regarding the objective function. One source of these uncertainties is a constantly changing environment in which the optima change their location over time. New heuristics or adaptations to already available algorithms must be conceived in order to deal with such problems. Among the desirable features that a search strategy should exhibit to deal with dynamic optimization are diversity maintenance, a memory of past solutions, and a multipopulation structure of candidate solutions. In this paper, an immune-inspired algorithm that presents these features, called dopt-aiNet, is properly adapted to deal with six newly proposed benchmark instances, and the obtained results are outlined according to the available specifications for the competition at the Congress on Evolutionary Computation 2009.

This work introduces an ant-inspired algorithm for optimization in continuous search spaces that is based on the generation of random vectors with multivariate Gaussian pdf. The proposed approach is called MACACO -- Multivariate Ant Colony Algorithm for Continuous Optimization -- and is able to simultaneously adapt all the dimensions of the random distribution employed to generate the new individuals at each iteration. In order to analyze MACACO's search efficiency, the approach was compared to a pair of counterparts: the Continuous Ant Colony System (CACS) and the approach known as Ant Colony Optimization in en (ACOR). The comparative analysis, which involves well-known benchmark problems from the literature, has indicated that MACACO outperforms CACS and ACOR in most cases as the quality of the final solution is concerned, and it is just about two times more costly than the least expensive contender.

Collaborative filtering (CF) is a method to perform automated suggestions for a user based on the opinion of other users with similar interest. Most of the CF algorithms do not take into account the existent duality between users and items, considering only the similarities between users or only the similarities between items. The authors have proposed in a previous work a bio-inspired methodology for CF, namely BIC-aiNet, capable of clustering rows and columns of a data matrix simultaneously. The usefulness and performance of the methodology are reported in the literature. Now, the authors carry out more rigorous comparative experiments with BIC-aiNet and other techniques found in the literature, as well as evaluate the scalability of the algorithm in several datasets of different sizes. The results indicate that our proposal is able to provide useful recommendations for the users, outperforming other methodologies for CF.

Collaborative filtering (CF) is a method to perform automated suggestions for a user based on the opinion of other users with similar interest. Most of the CF algorithms do not take into account the existent duality between users and items, considering only the similarities between users or only the similarities between items. In this paper we propose a novel methodology for the CF capable of dealing with this situation. By proposing an immune-inspired bi clustering technique to carry out clustering of rows and columns at the same time, our algorithm is able to group similarities between users and items. In order to evaluate the proposed methodology, we have applied it to Movie Lens dataset which contains user's ratings to a large set of movies. The results indicate that our proposal is able to provide useful recommendations for the users, outperforming other methodologies for CF reported in the literature.

Multimodal optimization algorithms inspired by the immune system are generally characterized by a dynamic control of the population size and by diversity maintenance along the search. One of these proposals, denoted copt-aiNet (artificial immune network for combinatorial optimization), is used to deal with combinatorial problems like the Traveling Salesman Problem (TSP) and other permutation problems. In this paper, the copt-aiNet algorithm is extended and adapted to be applied to an important issue of modern data mining, the biclustering problem. The biclustering approach consists in simultaneously ordering the rows and columns of a given matrix, so that similar elements are grouped together. To illustrate the performance of the proposed method, two bitmap images are scrambled and used as input to the algorithm, and the biclustering procedure tries to restore the original image by grouping the pixels according to the similarity of colors in a neighborhood. Additionally, copt-aiNet is applied to gene expression data clustering, a classical problem of the bioinformatics literature, and its performance is compared with a hierarchical biclustering algorithm.

Multimodal optimization algorithms are being adapted to deal with dynamic optimization, mainly due to their ability to provide a faster reaction to unexpected changes in the optimization surface. The faster reaction may be associated with the existence of two important attributes in population-based algorithms devoted to multimodal optimization: simultaneous maintenance of multiple local optima in the population; and self-regulation of the population size along the search. The optimization surface may be subject to variations motivated by one of two main reasons: modification of the objectives to be fulfilled and change in parameters of the problem. An immune-inspired algorithm specially designed to deal with combinatorial optimization is applied here to solve time-varying TSP instances, with the cost of going from one city to the other being a function of time. The proposal presents favorable results when compared to the results produced by a high-performance ant colony optimization algorithm of the literature.

In this work, we propose an evolutionary-like approach to the problem of blind adaptive spatial filtering that is based on the decision-directed criterion and on the dopt-aiNet, an artificial immune network conceived to perform multimodal search in dynamic environments. The proposal was tested under static and time-varying undermodeled channel models, and, in all cases, its ability to find and track a solution close to the Wiener global optimum was attested. The obtained results reveal that the dopt-aiNet may decisively enhance the performance of adaptive arrays in scenarios built from elements that are representative of some aspects of real-world communication systems.

Multimodal optimization algorithms inspired by the immune system are generally characterized by a dynamic control of the population size and by diversity maintenance along the search. One of the most popular proposals is denoted opt-aiNet (artificial immune network for optimization) and is extended here to deal with time-varying fitness functions. Additional procedures are designed to improve the overall performance and the robustness of the immune-inspired approach, giving rise to a version for dynamic optimization, denoted dopt-aiNet. Firstly, challenging benchmark problems in static multimodal optimization are considered to validate the new proposal. No parameter adjustment is necessary to adapt the algorithm according to the peculiarities of each problem. In the sequence, dynamic environments are considered, and usual evaluation indices are adopted to assess the performance of dopt-aiNet and compare with alternative solution procedures available in the literature.