In this paper, we derive models of the selection pressure in XCS for proportionate (roulette wheel) selection and tournament selection. We show that these models can explain the empirical results that have been previously presented in the literature. We validate the models on simple problems showing that, (i) when the model assumptions hold, the theory perfectly matches the empirical evidence; (ii) when the model assumptions do not hold, the theory can still provide qualitative explanations of the experimental results.
Modeling XCS in class imbalances: Population size and parameter settingskknsastry
This paper analyzes the scalability of the population size required in XCS to maintain niches that are infrequently activated. Facetwise models have been developed to predict the effect of the imbalance ratio—ratio between the number of instances of the majority class and the minority class that are sampled to XCS—on population initialization, and on the creation and deletion of classifiers of the minority class. While theoretical models show that, ideally, XCS scales linearly with the imbalance ratio, XCS with standard configuration scales exponentially. The causes that are potentially responsible for this deviation from the ideal scalability are also investigated. Specifically, the inheritance procedure of classifiers’ parameters, mutation, and subsumption are analyzed, and improvements in XCS’s mechanisms are proposed to effectively and efficiently handle imbalanced problems. Once the recommendations are incorporated to XCS, empirical results show that the population size in XCS indeed scales linearly with the imbalance ratio.
Let's get ready to rumble redux: Crossover versus mutation head to head on ex...kknsastry
This paper analyzes the relative advantages between crossover and mutation on a class of deterministic and stochastic additively separable problems with substructures of non-uniform salience. This study assumes that the recombination and mutation operators have the knowledge of the building blocks (BBs) and effectively exchange or search among competing BBs. Facetwise models of convergence time and population sizing have been used to determine the scalability of each algorithm. The analysis shows that for deterministic exponentially-scaled additively separable, problems, the BB-wise mutation is more efficient than crossover yielding a speedup of Θ(<em>l</em> log<em>l</em>), where <em>l</em> is the problem size. For the noisy exponentially-scaled problems, the outcome depends on whether scaling on noise is dominant. When scaling dominates, mutation is more efficient than crossover yielding a speedup of Θ(<em>l</em> log<em>l</em>). On the other hand, when noise dominates, crossover is more efficient than mutation yielding a speedup of Θ(<em>l</em>).
Analyzing probabilistic models in hierarchical BOA on traps and spin glasseskknsastry
The hierarchical Bayesian optimization algorithm (hBOA) can solve nearly decomposable and hierarchical problems of bounded difficulty in a robust and scalable manner by building and sampling probabilistic models of promising solutions. This paper analyzes probabilistic models in hBOA on two common test problems: concatenated traps and 2D Ising spin glasses with periodic boundary conditions. We argue that although Bayesian networks with local structures can encode complex probability distributions, analyzing these models in hBOA is relatively straightforward and the results of such analyses may provide practitioners with useful information about their problems. The results show that the probabilistic models in hBOA closely correspond to the structure of the underlying optimization problem, the models do not change significantly in subsequent iterations of BOA, and creating adequate probabilistic models by hand is not straightforward even with complete knowledge of the optimization problem.
Automated alphabet reduction with evolutionary algorithms for protein structu...kknsastry
This paper focuses on automated procedures to reduce the dimensionality of protein structure prediction datasets by simplifying the way in which the primary sequence of a protein is represented. The potential benefits of this procedure are faster and easier learning process as well as the generation of more compact and human-readable classifiers. The dimensionality reduction procedure we propose consists on the reduction of the 20-letter amino acid (AA) alphabet, which is normally used to specify a protein sequence, into a lower cardinality alphabet. This reduction comes about by a clustering of AA types accordingly to their physical and chemical similarity. Our automated reduction procedure is guided by a fitness function based on the Mutual Information between the AA-based input attributes of the dataset and the protein structure feature that being predicted. <br />
To search for the optimal reduction, the Extended Compact Genetic Algorithm (ECGA) was used, and afterwards the results of this process were fed into (and validated by) BioHEL, a genetics-based machine learning technique. BioHEL used the reduced alphabet to induce rules for protein structure prediction features. BioHEL results are compared to two standard machine learning systems. Our results show that it is possible to reduce the size of the alphabet used for prediction from twenty to just three letters resulting in more compact, i.e. interpretable, rules. Also, a protein-wise accuracy performance measure suggests that the loss of accuracy accrued by this substantial alphabet reduction is not statistically significant when compared to the full alphabet.
Using STELLA to Explore Dynamic Single Species Models: The Magic of Making Hu...Lisa Jensen
The use of formal, mathematical models allows stakeholders, decision makers and scientists to better visualize interactions and relationships within ecological systems. This study uses STELLA, a modeling tool, to simulate simple population dynamics for the humpback whale (Megaptera novaengliae) to better understand the impacts of reproductive and mortality rates as well as alternative solution algorithms used to drive the model. A wide range of population dynamics occurred as a result of varying time increments for calculating populations and use of available solution algorithms. Populations are most likely to achieve equilibrium when reproduction and mortality result in approximately the same number of individuals.
ORDINARY LEAST SQUARES REGRESSION OF ORDERED CATEGORICAL DATA- INBeth Larrabee
- The document describes a simulation study that evaluated the performance of ordinary least squares regression (OLSLR) for analyzing ordered categorical response (OCR) variables.
- Across different frequency distributions and numbers of categories for the OCR, the empirical type I error rate for OLSLR was close to the nominal 0.05 level.
- Empirical power for OLSLR increased as the number of categories in the OCR increased, but this trend slowed for OCRs with 5 or more categories. For most scenarios, OLSLR power was similar to probit regression power.
A Hybrid Immunological Search for theWeighted Feedback Vertex Set ProblemMario Pavone
In this paper we present a hybrid immunological inspired algorithm (HYBRID-IA) for solving the Minimum Weighted Feedback Vertex Set (MWFVS) problem. MWFV S is one of the most interesting and challenging combinatorial optimization problem, which finds application in many fields and in many real life tasks. The proposed algorithm is inspired by the clonal selection principle, and therefore it takes advantage of the main strength characteristics of the operators of (i) cloning; (ii) hypermutation; and (iii) aging. Along with these operators, the algorithm uses a local search procedure, based on a deterministic approach, whose purpose is to refine the solutions found so far. In order to evaluate the efficiency and robustness of HYBRID-IA several experiments were performed on different instances, and for each instance it was compared to three different algorithms: (1) a memetic algorithm based on a genetic algorithm (MA); (2) a tabu search metaheuristic (XTS); and (3) an iterative tabu search (ITS). The obtained results prove the efficiency and reliability of HYBRID-IA on all instances in term of the best solutions found and also similar performances with all compared algorithms, which represent nowadays the state-of-the-art on for MWFV S problem.
The Likelihood Ratio Test in Structural Equation Models: A Multi-Group Approa...Dr. Amarjeet Singh
In this study a multi-group approach is used
within the structural equations modelling framework,
with the purpose of validating the temporal stability of
the material deprivation measurement model, as well
the impact of the existence of children in the household.
For the purposes of model comparison, the use of the
likelihood ratio test is discussed, when the normal
distribution of the data cannot be assumed and the
robust Satorra-Bentler scaled-corrected chi-square
estimator is used. The statistical package LISREL 8.8 is
used for estimating models.
Modeling XCS in class imbalances: Population size and parameter settingskknsastry
This paper analyzes the scalability of the population size required in XCS to maintain niches that are infrequently activated. Facetwise models have been developed to predict the effect of the imbalance ratio—ratio between the number of instances of the majority class and the minority class that are sampled to XCS—on population initialization, and on the creation and deletion of classifiers of the minority class. While theoretical models show that, ideally, XCS scales linearly with the imbalance ratio, XCS with standard configuration scales exponentially. The causes that are potentially responsible for this deviation from the ideal scalability are also investigated. Specifically, the inheritance procedure of classifiers’ parameters, mutation, and subsumption are analyzed, and improvements in XCS’s mechanisms are proposed to effectively and efficiently handle imbalanced problems. Once the recommendations are incorporated to XCS, empirical results show that the population size in XCS indeed scales linearly with the imbalance ratio.
Let's get ready to rumble redux: Crossover versus mutation head to head on ex...kknsastry
This paper analyzes the relative advantages between crossover and mutation on a class of deterministic and stochastic additively separable problems with substructures of non-uniform salience. This study assumes that the recombination and mutation operators have the knowledge of the building blocks (BBs) and effectively exchange or search among competing BBs. Facetwise models of convergence time and population sizing have been used to determine the scalability of each algorithm. The analysis shows that for deterministic exponentially-scaled additively separable, problems, the BB-wise mutation is more efficient than crossover yielding a speedup of Θ(<em>l</em> log<em>l</em>), where <em>l</em> is the problem size. For the noisy exponentially-scaled problems, the outcome depends on whether scaling on noise is dominant. When scaling dominates, mutation is more efficient than crossover yielding a speedup of Θ(<em>l</em> log<em>l</em>). On the other hand, when noise dominates, crossover is more efficient than mutation yielding a speedup of Θ(<em>l</em>).
Analyzing probabilistic models in hierarchical BOA on traps and spin glasseskknsastry
The hierarchical Bayesian optimization algorithm (hBOA) can solve nearly decomposable and hierarchical problems of bounded difficulty in a robust and scalable manner by building and sampling probabilistic models of promising solutions. This paper analyzes probabilistic models in hBOA on two common test problems: concatenated traps and 2D Ising spin glasses with periodic boundary conditions. We argue that although Bayesian networks with local structures can encode complex probability distributions, analyzing these models in hBOA is relatively straightforward and the results of such analyses may provide practitioners with useful information about their problems. The results show that the probabilistic models in hBOA closely correspond to the structure of the underlying optimization problem, the models do not change significantly in subsequent iterations of BOA, and creating adequate probabilistic models by hand is not straightforward even with complete knowledge of the optimization problem.
Automated alphabet reduction with evolutionary algorithms for protein structu...kknsastry
This paper focuses on automated procedures to reduce the dimensionality of protein structure prediction datasets by simplifying the way in which the primary sequence of a protein is represented. The potential benefits of this procedure are faster and easier learning process as well as the generation of more compact and human-readable classifiers. The dimensionality reduction procedure we propose consists on the reduction of the 20-letter amino acid (AA) alphabet, which is normally used to specify a protein sequence, into a lower cardinality alphabet. This reduction comes about by a clustering of AA types accordingly to their physical and chemical similarity. Our automated reduction procedure is guided by a fitness function based on the Mutual Information between the AA-based input attributes of the dataset and the protein structure feature that being predicted. <br />
To search for the optimal reduction, the Extended Compact Genetic Algorithm (ECGA) was used, and afterwards the results of this process were fed into (and validated by) BioHEL, a genetics-based machine learning technique. BioHEL used the reduced alphabet to induce rules for protein structure prediction features. BioHEL results are compared to two standard machine learning systems. Our results show that it is possible to reduce the size of the alphabet used for prediction from twenty to just three letters resulting in more compact, i.e. interpretable, rules. Also, a protein-wise accuracy performance measure suggests that the loss of accuracy accrued by this substantial alphabet reduction is not statistically significant when compared to the full alphabet.
Using STELLA to Explore Dynamic Single Species Models: The Magic of Making Hu...Lisa Jensen
The use of formal, mathematical models allows stakeholders, decision makers and scientists to better visualize interactions and relationships within ecological systems. This study uses STELLA, a modeling tool, to simulate simple population dynamics for the humpback whale (Megaptera novaengliae) to better understand the impacts of reproductive and mortality rates as well as alternative solution algorithms used to drive the model. A wide range of population dynamics occurred as a result of varying time increments for calculating populations and use of available solution algorithms. Populations are most likely to achieve equilibrium when reproduction and mortality result in approximately the same number of individuals.
ORDINARY LEAST SQUARES REGRESSION OF ORDERED CATEGORICAL DATA- INBeth Larrabee
- The document describes a simulation study that evaluated the performance of ordinary least squares regression (OLSLR) for analyzing ordered categorical response (OCR) variables.
- Across different frequency distributions and numbers of categories for the OCR, the empirical type I error rate for OLSLR was close to the nominal 0.05 level.
- Empirical power for OLSLR increased as the number of categories in the OCR increased, but this trend slowed for OCRs with 5 or more categories. For most scenarios, OLSLR power was similar to probit regression power.
A Hybrid Immunological Search for theWeighted Feedback Vertex Set ProblemMario Pavone
In this paper we present a hybrid immunological inspired algorithm (HYBRID-IA) for solving the Minimum Weighted Feedback Vertex Set (MWFVS) problem. MWFV S is one of the most interesting and challenging combinatorial optimization problem, which finds application in many fields and in many real life tasks. The proposed algorithm is inspired by the clonal selection principle, and therefore it takes advantage of the main strength characteristics of the operators of (i) cloning; (ii) hypermutation; and (iii) aging. Along with these operators, the algorithm uses a local search procedure, based on a deterministic approach, whose purpose is to refine the solutions found so far. In order to evaluate the efficiency and robustness of HYBRID-IA several experiments were performed on different instances, and for each instance it was compared to three different algorithms: (1) a memetic algorithm based on a genetic algorithm (MA); (2) a tabu search metaheuristic (XTS); and (3) an iterative tabu search (ITS). The obtained results prove the efficiency and reliability of HYBRID-IA on all instances in term of the best solutions found and also similar performances with all compared algorithms, which represent nowadays the state-of-the-art on for MWFV S problem.
The Likelihood Ratio Test in Structural Equation Models: A Multi-Group Approa...Dr. Amarjeet Singh
In this study a multi-group approach is used
within the structural equations modelling framework,
with the purpose of validating the temporal stability of
the material deprivation measurement model, as well
the impact of the existence of children in the household.
For the purposes of model comparison, the use of the
likelihood ratio test is discussed, when the normal
distribution of the data cannot be assumed and the
robust Satorra-Bentler scaled-corrected chi-square
estimator is used. The statistical package LISREL 8.8 is
used for estimating models.
Stability of Individuals in a Fingerprint System across Force LevelsITIIIndustries
This research studied the question: “Are all
individual’s performance stable in a fingerprint recognition
system?” The fingerprints of 154 individuals, provided at
different force levels, were examined using the biometric
menagerie tool, first coined by Doddington et al. in 1998. The
Biometric Menagerie illustrates how each person in a given
dataset performs in a biometric system, by using their genuine
and impostor scores, and providing them a classification based
upon those scores. This research examined the biometric
menagerie classifications across different force levels in a
fingerprint recognition study to uncover if individuals performed
the same over five force levels. The study concluded that they did
not, and a new metric has been created to quantify this
phenomenon. As a result of this discovery, the new metric,
Stability Score Index is described to showcase the movement of
individuals in the menagerie.
Simulation Study of Hurdle Model Performance on Zero Inflated Count DataIan Camacho
The document summarizes a simulation study that evaluates the performance of hurdle models on zero-inflated count data under different scenarios. It finds that hurdle models can omit significant predictors but their performance decreases substantially with multicollinearity, with about 50% larger errors and biased parameter estimates. The study generates data with different sample sizes from 100 to 1 million cases and introduces multicollinearity and omission of predictors to evaluate hurdle model adequacy.
Investigations of certain estimators for modeling panel data under violations...Alexander Decker
This document investigates the efficiency of four methods for estimating panel data models (pooling, first differencing, between, and feasible generalized least squares) when the assumptions of homoscedasticity, no autocorrelation, and no collinearity are jointly violated. Monte Carlo simulations were conducted under varying conditions of heteroscedasticity, autocorrelation, collinearity, sample size, and time periods. The results showed that in small samples, the feasible generalized least squares estimator is most efficient when heteroscedasticity is severe, regardless of autocorrelation and collinearity levels. However, when heteroscedasticity is low to moderate with moderate autocorrelation, first differencing and feasible generalized least squares
The Influence of Age Assignments on the Performance of Immune AlgorithmsMario Pavone
How long a B cell remains, evolves and matures inside a population plays a crucial role on the capability for an immune algorithm to jump out from local optima, and find the global optimum. Assigning the right age to each clone (or offspring, in general) means to find the proper balancing between the exploration and exploitation. In this research work we present an experimental study conducted on an immune algorithm, based on the clonal selection principle, and performed on eleven different age assignments, with the main aim to verify if at least one, or two, of the top 4 in the previous efficiency ranking produced on the one-max problem, still appear among the top 4 in the new efficiency ranking obtained on a different complex problem. Thus, the NK landscape model has been considered as the test problem, which is a mathematical model formulated for the study of tunably rugged fitness landscape. From the many experiments performed is possible to assert that in the elitism variant of the immune algorithm, two of the best age assignments previously discovered, still continue to appear among the top 3 of the new rankings produced; whilst they become three in the no elitism version. Further, in the first variant none of the 4 top previous ones ranks ever in the first position, unlike on the no elitism variant, where the previous best one continues to appear in 1st position more than the others. Finally, this study confirms that the idea to assign the same age of the parent to the cloned B cell is not a good strategy since it continues to be as the worst also in the new
efficiency ranking.
The document discusses mixed membership models for document clustering. It begins by introducing mixed membership models, which aim to discover multiple cluster memberships for each document rather than assigning documents to a single cluster like traditional clustering models. It then provides an example of applying mixed membership models to a sample document, showing how the document could have membership in both a science and technology topic cluster. The document continues building towards introducing Latent Dirichlet Allocation as a technique for mixed membership modeling of documents.
Gamma and inverse Gaussian frailty models: A comparative studyinventionjournals
Frailty models have become very popular during the last three decades and their applications are numerous. The main goal of this manuscript is to compare two frailty models (gamma frailty model and inverse Gaussian frailty model) each of which has a log-logistic distribution to be its baseline hazard function. A real data set is applied for the two considered frailty models in order to deal with models comparison. It has been concluded that the gamma frailty model is the best model fits this data set. Then the inverse Gaussian frailty model, which provides a better fit of the considered data set than the Cox’s model.
The document describes using a genetic algorithm to find the maximum values of single-variable functions. It discusses:
1) How genetic algorithms work by simulating biological evolution to optimize solutions.
2) Testing the genetic algorithm on continuous and non-continuous functions that are difficult to optimize with traditional methods, such as multimodal, non-differentiable functions.
3) The genetic algorithm was able to find maximum values close to the real maximum for complex test functions, demonstrating its effectiveness at optimizing these difficult single-variable functions.
A Mathematical Model for the Genetic Variation of Prolactin and Prolactin Rec...IJERA Editor
The Weibull distribution is a widely used model for studying fatigue and endurance life in engineering devices and materials. Recent advances in Weibull theory have also created numerous specialized Weibull applications. Modern computing technology has made many of these techniques accessible across the engineering spectrum. Despite its popularity, and wide applicability the traditional 2 – parameter and 3- parameter Weibull distribution is unable to capture all the lifetime phenomenon for instance the data set which has a non – monotonic failure rate function. Recently several generalization of Weibull distribution has been studied. An approach to the construction of flexible parametric model is to embed appropriate competing models into a larger model by adding shape parameter. Some recent generalizations of Weibull distribution including the Exponentiated Weibull, Extended Weibull, Modified Weibull are discussed and references [5] therein, along with their reliability functions. In this paper a new generalization of Weibull distribution called the transmuted Weibull distribution is utilized for our medical application. For example, Prolactin and prolactin receptors are present in normal breast tissue, benign breast disease, breast cancer cell lines, and breast tumour tissue, leading to speculation that the proliferative and antiapoptotic effects of prolactin in breast epithelial cells could be a factor in breast carcinogenesis. In this paper, a test for the distribution of prolactin concentrations in controls, by menopausal status and relationships with Serum Prolactin Levels and Breast Cancer Risk was investigated in the application part, by using Transmuted Weibull Distribution. As a result the mathematical curves for the Probability Density Function, Reliability Function and Hazard Rate Function are obtained for the corresponding medical curve given in the application part.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
A controversial genetic restoration mechanism has been proposed for the model organism Arabidopsis thaliana. This theory proposes that genetic material from non-parental ancestors is used to restore genetic information that was inadvertently corrupted during reproduction. We evaluate the effectiveness of this strategy by adapting it to an evolutionary algorithm solving two distinct benchmark optimization problems. We compare the performance of the proposed strategy with a number of alternate strategies – including the Mendelian alternative. Included in this comparison are a number of biologically implausible templates that help elucidate likely reasons for the relative performance of the different templates. Results show that the proposed non- Mendelian restoration strategy is highly effective across the range of conditions investigated – significantly outperforming the Mendelian alternative in almost every situation.
Review of Limayem, M., S. G. Hirt, and C. M. K. Cheung (2007), “How Habits Limits the Predictive Power of Intention: The Case of Information Systems Continuance,” MIS Quarterly, Vol. 31, No. 4, 705-737.
O poema descreve a cruz como o caminho para o céu, inspiração e salvação. Através da cruz, Jesus levou a luz para os mortais e aponta o caminho para a paz eterna no lar celestial. A cruz será o farol que guiará a alma do poeta em sua vida obscura rumo à mansão celestial.
Earthquakes occur when continental plates collide or move against each other, causing destruction of infrastructure through collapsed buildings and landslides, as well as loss of life from tsunamis generated by undersea quakes. Developing nations affected by earthquakes require outside assistance to recover from damage, which global cooperation makes possible as seen by international aid to Haiti following its 2010 earthquake.
A hunter with dogs, gun and cart pursues a red-legged partridge across hills and plains. As the partridge tries to evade the hunter and his barking dogs, the hunter shoots it down with three gunshots, killing the partridge. The short poem describes the futile attempts of the partridge to escape from the hunter and his dogs and guns.
Erindi flutt 21. febrúar 2009 á fundi hjá Hugmyndaráðuneytinu. Erindið fjallar um gögn og hagnýtingu þeirra, aðgengi að gögnum hins opinbera, gegnsæi í viðskiptum og hvernig opnara aðgengi að gögnum hefði getað varað okkur mun fyrr við efnahagshruninu.
Stability of Individuals in a Fingerprint System across Force LevelsITIIIndustries
This research studied the question: “Are all
individual’s performance stable in a fingerprint recognition
system?” The fingerprints of 154 individuals, provided at
different force levels, were examined using the biometric
menagerie tool, first coined by Doddington et al. in 1998. The
Biometric Menagerie illustrates how each person in a given
dataset performs in a biometric system, by using their genuine
and impostor scores, and providing them a classification based
upon those scores. This research examined the biometric
menagerie classifications across different force levels in a
fingerprint recognition study to uncover if individuals performed
the same over five force levels. The study concluded that they did
not, and a new metric has been created to quantify this
phenomenon. As a result of this discovery, the new metric,
Stability Score Index is described to showcase the movement of
individuals in the menagerie.
Simulation Study of Hurdle Model Performance on Zero Inflated Count DataIan Camacho
The document summarizes a simulation study that evaluates the performance of hurdle models on zero-inflated count data under different scenarios. It finds that hurdle models can omit significant predictors but their performance decreases substantially with multicollinearity, with about 50% larger errors and biased parameter estimates. The study generates data with different sample sizes from 100 to 1 million cases and introduces multicollinearity and omission of predictors to evaluate hurdle model adequacy.
Investigations of certain estimators for modeling panel data under violations...Alexander Decker
This document investigates the efficiency of four methods for estimating panel data models (pooling, first differencing, between, and feasible generalized least squares) when the assumptions of homoscedasticity, no autocorrelation, and no collinearity are jointly violated. Monte Carlo simulations were conducted under varying conditions of heteroscedasticity, autocorrelation, collinearity, sample size, and time periods. The results showed that in small samples, the feasible generalized least squares estimator is most efficient when heteroscedasticity is severe, regardless of autocorrelation and collinearity levels. However, when heteroscedasticity is low to moderate with moderate autocorrelation, first differencing and feasible generalized least squares
The Influence of Age Assignments on the Performance of Immune AlgorithmsMario Pavone
How long a B cell remains, evolves and matures inside a population plays a crucial role on the capability for an immune algorithm to jump out from local optima, and find the global optimum. Assigning the right age to each clone (or offspring, in general) means to find the proper balancing between the exploration and exploitation. In this research work we present an experimental study conducted on an immune algorithm, based on the clonal selection principle, and performed on eleven different age assignments, with the main aim to verify if at least one, or two, of the top 4 in the previous efficiency ranking produced on the one-max problem, still appear among the top 4 in the new efficiency ranking obtained on a different complex problem. Thus, the NK landscape model has been considered as the test problem, which is a mathematical model formulated for the study of tunably rugged fitness landscape. From the many experiments performed is possible to assert that in the elitism variant of the immune algorithm, two of the best age assignments previously discovered, still continue to appear among the top 3 of the new rankings produced; whilst they become three in the no elitism version. Further, in the first variant none of the 4 top previous ones ranks ever in the first position, unlike on the no elitism variant, where the previous best one continues to appear in 1st position more than the others. Finally, this study confirms that the idea to assign the same age of the parent to the cloned B cell is not a good strategy since it continues to be as the worst also in the new
efficiency ranking.
The document discusses mixed membership models for document clustering. It begins by introducing mixed membership models, which aim to discover multiple cluster memberships for each document rather than assigning documents to a single cluster like traditional clustering models. It then provides an example of applying mixed membership models to a sample document, showing how the document could have membership in both a science and technology topic cluster. The document continues building towards introducing Latent Dirichlet Allocation as a technique for mixed membership modeling of documents.
Gamma and inverse Gaussian frailty models: A comparative studyinventionjournals
Frailty models have become very popular during the last three decades and their applications are numerous. The main goal of this manuscript is to compare two frailty models (gamma frailty model and inverse Gaussian frailty model) each of which has a log-logistic distribution to be its baseline hazard function. A real data set is applied for the two considered frailty models in order to deal with models comparison. It has been concluded that the gamma frailty model is the best model fits this data set. Then the inverse Gaussian frailty model, which provides a better fit of the considered data set than the Cox’s model.
The document describes using a genetic algorithm to find the maximum values of single-variable functions. It discusses:
1) How genetic algorithms work by simulating biological evolution to optimize solutions.
2) Testing the genetic algorithm on continuous and non-continuous functions that are difficult to optimize with traditional methods, such as multimodal, non-differentiable functions.
3) The genetic algorithm was able to find maximum values close to the real maximum for complex test functions, demonstrating its effectiveness at optimizing these difficult single-variable functions.
A Mathematical Model for the Genetic Variation of Prolactin and Prolactin Rec...IJERA Editor
The Weibull distribution is a widely used model for studying fatigue and endurance life in engineering devices and materials. Recent advances in Weibull theory have also created numerous specialized Weibull applications. Modern computing technology has made many of these techniques accessible across the engineering spectrum. Despite its popularity, and wide applicability the traditional 2 – parameter and 3- parameter Weibull distribution is unable to capture all the lifetime phenomenon for instance the data set which has a non – monotonic failure rate function. Recently several generalization of Weibull distribution has been studied. An approach to the construction of flexible parametric model is to embed appropriate competing models into a larger model by adding shape parameter. Some recent generalizations of Weibull distribution including the Exponentiated Weibull, Extended Weibull, Modified Weibull are discussed and references [5] therein, along with their reliability functions. In this paper a new generalization of Weibull distribution called the transmuted Weibull distribution is utilized for our medical application. For example, Prolactin and prolactin receptors are present in normal breast tissue, benign breast disease, breast cancer cell lines, and breast tumour tissue, leading to speculation that the proliferative and antiapoptotic effects of prolactin in breast epithelial cells could be a factor in breast carcinogenesis. In this paper, a test for the distribution of prolactin concentrations in controls, by menopausal status and relationships with Serum Prolactin Levels and Breast Cancer Risk was investigated in the application part, by using Transmuted Weibull Distribution. As a result the mathematical curves for the Probability Density Function, Reliability Function and Hazard Rate Function are obtained for the corresponding medical curve given in the application part.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
A controversial genetic restoration mechanism has been proposed for the model organism Arabidopsis thaliana. This theory proposes that genetic material from non-parental ancestors is used to restore genetic information that was inadvertently corrupted during reproduction. We evaluate the effectiveness of this strategy by adapting it to an evolutionary algorithm solving two distinct benchmark optimization problems. We compare the performance of the proposed strategy with a number of alternate strategies – including the Mendelian alternative. Included in this comparison are a number of biologically implausible templates that help elucidate likely reasons for the relative performance of the different templates. Results show that the proposed non- Mendelian restoration strategy is highly effective across the range of conditions investigated – significantly outperforming the Mendelian alternative in almost every situation.
Review of Limayem, M., S. G. Hirt, and C. M. K. Cheung (2007), “How Habits Limits the Predictive Power of Intention: The Case of Information Systems Continuance,” MIS Quarterly, Vol. 31, No. 4, 705-737.
O poema descreve a cruz como o caminho para o céu, inspiração e salvação. Através da cruz, Jesus levou a luz para os mortais e aponta o caminho para a paz eterna no lar celestial. A cruz será o farol que guiará a alma do poeta em sua vida obscura rumo à mansão celestial.
Earthquakes occur when continental plates collide or move against each other, causing destruction of infrastructure through collapsed buildings and landslides, as well as loss of life from tsunamis generated by undersea quakes. Developing nations affected by earthquakes require outside assistance to recover from damage, which global cooperation makes possible as seen by international aid to Haiti following its 2010 earthquake.
A hunter with dogs, gun and cart pursues a red-legged partridge across hills and plains. As the partridge tries to evade the hunter and his barking dogs, the hunter shoots it down with three gunshots, killing the partridge. The short poem describes the futile attempts of the partridge to escape from the hunter and his dogs and guns.
Erindi flutt 21. febrúar 2009 á fundi hjá Hugmyndaráðuneytinu. Erindið fjallar um gögn og hagnýtingu þeirra, aðgengi að gögnum hins opinbera, gegnsæi í viðskiptum og hvernig opnara aðgengi að gögnum hefði getað varað okkur mun fyrr við efnahagshruninu.
JNAutomotive_2009 AAA Aggressive Driving Research UpdateJN Automotive
The AAA Foundation was established in 1947 as a 501(c)(3) nonprofit research affiliate of AAA/CAA with a North American focus. Its mission is to identify traffic safety problems, foster research for solutions, and disseminate information and education resources, funded through donations. Surveys found Americans and over three-quarters of those surveyed see aggressive driving as a serious traffic safety issue, though nearly half admitted speeding over 15 mph, reflecting a "do as I say, not as I do" attitude. The Foundation's analysis of NHTSA fatal crash data from 2003-2007 found that as many as 56% of fatal crashes involved unsafe driving behaviors typically tied to aggression like speeding, reckless driving, and failure to
The document discusses how data centers and networks need to evolve to address the convergence of large video packets and billions of small IoT packets. A stratified structure with three levels - centralized hubs, regional edge data centers, and micro data centers located close to end users - is proposed to better support this new architecture. This hierarchical structure would improve processing capabilities, distribute infrastructure throughout locations based on customer demand, and maximize uptime at each mission critical level. Planning also needs to shift from tactical to more strategic, long-term thinking to accommodate evolving technical, network, application and user requirements over the next 5-10 years.
Farmers Insurance offers career opportunities for insurance agents. Agents can choose to start as Reserve Business Builders, receiving training and leads. Within 4 months of training, agents can convert to Career Agents and receive subsidies for commissions, office space, and licensing for up to 2 years. Career Agents establish their own agency and receive ongoing support through Farmers University. Top performing agents are recognized through programs like Blue Vase Society, Leaders Roundtable, Championship, and Presidents Council. Farmers Insurance provides agents flexibility and support to build a successful insurance business.
This document discusses machine intelligence and machine learning. It covers topics such as behavior-based AI vs knowledge-based AI, supervised vs unsupervised learning, classification vs prediction, and decision tree induction for classification. Decision trees are built using an algorithm that selects the attribute that best splits the data at each step to create partitions. Pruning techniques are used to avoid overfitting.
This document discusses the use of machine learning in official statistics. It begins by contrasting the traditional "data modeling" approach used in statistics with the newer "algorithmic modeling" approach used in machine learning. It then discusses how machine learning can be applied both in traditional statistical production processes based on primary survey data, as well as in new multi-source production processes that incorporate alternative data sources like big data. Specifically, machine learning can be used for tasks like imputation, outlier detection, and estimation. The document concludes that machine learning represents a paradigm shift for official statistics and is particularly well-suited for new data sources, as it prioritizes prediction over interpretability and generalizability.
Cost Optimized Design Technique for Pseudo-Random Numbers in Cellular Automataijait
In this research work, we have put an emphasis on the cost effective design approach for high quality pseudo-random numbers using one dimensional Cellular Automata (CA) over Maximum Length CA. This work focuses on different complexities e.g., space complexity, time complexity, design complexity and searching complexity for the generation of pseudo-random numbers in CA. The optimization procedure for
these associated complexities is commonly referred as the cost effective generation approach for pseudorandom numbers. The mathematical approach for proposed methodology over the existing maximum length CA emphasizes on better flexibility to fault coverage. The randomness quality of the generated patterns for the proposed methodology has been verified using Diehard Tests which reflects that the randomness quality
achieved for proposed methodology is equal to the quality of randomness of the patterns generated by the maximum length cellular automata. The cost effectiveness results a cheap hardware implementation for the concerned pseudo-random pattern generator. Short version of this paper has been published in [1].
The document discusses sequential (1,4) and (2,4) systems based on sequential order statistics for estimating parameters. It defines sequential (k,n) systems where component failures alter the failure rates of remaining components. It derives the probability density functions of sequential order statistics for (2,4) and (1,4) systems assuming absolute continuous lifetime distributions. It also evaluates the mean time before failure and derives minimum risk equivariant estimators for location and scale parameters of these systems.
Our Journal actively promotes a level playing field, providing resources and support to scholars from diverse communities. We believe that by removing barriers to entry, we contribute to a more vibrant and robust statistical community.
This study introduces and compares different methods for estimating the two parameters of generalized logarithmic series distribution. These methods are the cuckoo search optimization, maximum likelihood estimation, and method of moments algorithms. All the required derivations and basic steps of each algorithm are explained. The applications for these algorithms are implemented through simulations using different sample sizes (n = 15, 25, 50, 100). Results are compared using the statistical measure mean square error.
USING CUCKOO ALGORITHM FOR ESTIMATING TWO GLSD PARAMETERS AND COMPARING IT WI...ijcsit
This study introduces and compares different methods for estimating the two parameters of generalized logarithmic series distribution. These methods are the cuckoo search optimization, maximum likelihood estimation, and method of moments algorithms. All the required derivations and basic steps of each algorithm are explained. The applications for these algorithms are implemented through simulations using different sample sizes (n = 15, 25, 50, 100). Results are compared using the statistical measure mean square error.
This document compares different estimation methods for parameters of the generalized logarithmic series distribution (GLSD), including cuckoo search optimization (CSO), maximum likelihood estimation (MLE), and method of moments (MOM). The CSO algorithm is introduced and applied to estimate the two GLSD parameters. Simulation results using different sample sizes show that CSO performs best for small sample sizes while MLE is best for large sample sizes, based on mean square error. The document concludes that CSO is the best estimator for small sample sizes.
A Comparative Analysis of Feature Selection Methods for Clustering DNA SequencesCSCJournals
Large-scale analysis of genome sequences is in progress around the world, the major application of which is to establish the evolutionary relationship among the species using phylogenetic trees. Hierarchical agglomerative algorithms can be used to generate such phylogenetic trees given the distance matrix representing the dissimilarity among the species. ClustalW and Muscle are two general purpose programs that generates distance matrix from the input DNA or protein sequences. The limitation of these programs is that they are based on Smith-Waterman algorithm which uses dynamic programming for doing the pair-wise alignment. This is an extremely time consuming process and the existing systems may even fail to work for larger input data set. To overcome this limitation, we have used the frequency of codons usage as an approximation to find dissimilarity among species. The proposed technique further reduces the complexity by extracting only the significant features of the species from the mtDNA sequences using the techniques like frequent codons, codons with maximum range value or PCA technique. We have observed that the proposed system produces nearly accurate results in a significantly reduced running time.
Analysis On Classification Techniques In Mammographic Mass Data SetIJERA Editor
Data mining, the extraction of hidden information from large databases, is to predict future trends and behaviors, allowing businesses to make proactive, knowledge-driven decisions. Data-Mining classification techniques deals with determining to which group each data instances are associated with. It can deal with a wide variety of data so that large amount of data can be involved in processing. This paper deals with analysis on various data mining classification techniques such as Decision Tree Induction, Naïve Bayes , k-Nearest Neighbour (KNN) classifiers in mammographic mass dataset.
An Automatic Clustering Technique for Optimal ClustersIJCSEA Journal
This document presents a new automatic clustering algorithm called Automatic Merging for Optimal Clusters (AMOC). AMOC is a two-phase iterative extension of k-means clustering that aims to automatically determine the optimal number of clusters for a given dataset. In the first phase, AMOC initializes a large number of clusters k using k-means. In the second phase, it iteratively merges the lowest probability cluster with its closest neighbor, recomputing metrics each time to evaluate if the merge improved clustering quality. The algorithm stops merging once no improvements are found. Experimental results on synthetic and real datasets show AMOC finds nearly optimal cluster structures in terms of number, compactness and separation of clusters.
An Improved Iterative Method for Solving General System of Equations via Gene...Zac Darcy
Various algorithms are known for solving linear system of equations. Iteration methods for solving the
large sparse linear systems are recommended. But in the case of general n× m matrices the classic
iterative algorithms are not applicable except for a few cases. The algorithm presented here is based on the
minimization of residual of solution and has some genetic characteristics which require using Genetic
Algorithms. Therefore, this algorithm is best applicable for construction of parallel algorithms. In this
paper, we describe a sequential version of proposed algorithm and present its theoretical analysis.
Moreover we show some numerical results of the sequential algorithm and supply an improved algorithm
and compare the two algorithms.
An Improved Iterative Method for Solving General System of Equations via Gene...Zac Darcy
Various algorithms are known for solving linear system of equations. Iteration methods for solving the
large sparse linear systems are recommended. But in the case of general n× m matrices the classic
iterative algorithms are not applicable except for a few cases. The algorithm presented here is based on the
minimization of residual of solution and has some genetic characteristics which require using Genetic
Algorithms. Therefore, this algorithm is best applicable for construction of parallel algorithms. In this
paper, we describe a sequential version of proposed algorithm and present its theoretical analysis.
Moreover we show some numerical results of the sequential algorithm and supply an improved algorithm
and compare the two algorithms.
Volume of data available in the digital world is increasing every day at a greater speed. Due to enhancement of various technologies and new algorithms, extraction of essential data from huge volume of data is not a tough task nowadays but our goal is the extraction of patterns and knowledge from large amounts of data. Different sources are available for collecting the reviews about a product. To enhance the quality of the products and services these reviews provides different features of the products. Models can use one or more classifiers in trying to determine the probability of a set of data belonging to another set, say spam or 'ham'. Depending on definitional boundaries, modeling is synonymous with, the field of machine learning, as it is more commonly referred to in academic or research and development contexts. In this paper we identified and discussed about three algorithms which are efficient in identifying essential patterns in the available huge volume of data.
This document discusses predicting the secondary structure of proteins using machine learning algorithms. The researchers used 57 features of 700 amino acids to train logistic regression, naive Bayes, decision tree, and random forest models. Random forest achieved the best accuracy of 78.76% for a dataset of 1000 samples. The results show that modern machine learning algorithms can efficiently and accurately predict protein secondary structures. Room for improvement remains in adding new informative features to further boost prediction accuracy.
This research paper demonstrates the invention of the kinetic bands, based on Romanian mathematician and statistician Octav Onicescu’s kinetic energy, also known as “informational energy”, where we use historical data of foreign exchange currencies or indexes to predict the trend displayed by a stock or an index and whether it will go up or down in the future. Here, we explore the imperfections of the Bollinger Bands to determine a more sophisticated triplet of indicators that predict the future movement of prices in the Stock Market. An Extreme Gradient Boosting Modelling was conducted in Python using historical data set from Kaggle, the historical data set spanning all current 500 companies listed. An invariable importance feature was plotted. The results displayed that Kinetic Bands, derived from (KE) are very influential as features or technical indicators of stock market trends. Furthermore, experiments done through this invention provide tangible evidence of the empirical aspects of it. The machine learning code has low chances of error if all the proper procedures and coding are in play. The experiment samples are attached to this study for future references or scrutiny.
This document describes a study that compared the repeatability of gait data from two different lower limb models - one based on anatomical landmarks (AL model) and the other using a functional method to define hip joint centers and mean helical knee axes (FUN model). Ten subjects participated in gait analyses using both models. The FUN model produced slightly more repeatable hip and knee joint kinematic and kinetic data compared to the AL model. A foot calibration rig was also developed to define the foot segment independently of anatomical landmarks, improving repeatability.
A Mathematical Programming Approach for Selection of Variables in Cluster Ana...IJRES Journal
The document presents a mathematical programming approach for selecting important variables in cluster analysis. It formulates a nonlinear binary model to minimize the distance between observations within clusters, using indicator variables to select important variables. The model is applied to a sample dataset of 30 observations across 5 variables, correctly identifying variables 3, 4 and 5 as most important for clustering the observations into two groups. The results are compared to an existing variable selection heuristic, with the mathematical programming approach achieving a 100% correct classification versus 97% for the other method.
MediaEval 2015 - UNED-UV @ Retrieving Diverse Social Images Task - Postermultimediaeval
This paper details the participation of the UNED-UV group at the 2015 Retrieving Diverse Social Images Task. This year, our proposal is based on a multi-modal approach that firstly applies a textual algorithm based on Formal Concept Analysis (FCA) and Hierarchical Agglomerative Clustering (HAC) to detect the latent topics addressed by the images to diversify them according to these topics. Secondly, a Local Logistic Regression model, which uses the low level features and some relevant and non-relevant samples, is adjusted and estimates the relevance probability for all the images in the database.
http://ceur-ws.org/Vol-1436/
http://www.multimediaeval.org
Similar to Modeling selection pressure in XCS for proportionate and tournament selection (20)
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Effective and efficient multiscale modeling is essential to advance both the science and synthesis in a wide array of fields such as physics, chemistry, materials science, biology, biotechnology and pharmacology. This study investigates the efficacy and potential of using genetic algorithms for multiscale materials modeling and
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The document describes research into using efficient estimation of distribution algorithms (EDAs) like the compact genetic algorithm (cGA) to solve optimization problems involving billions of variables. Key aspects discussed include the cGA's memory and computational efficiency through techniques like parallelization and vectorization. The researchers were able to solve a noisy OneMax problem involving over 33 million variables to optimality and a problem with 1.1 billion variables with relaxed convergence. The document argues this research is important because many real-world problems involving nanotechnology, biology, and information systems require solving optimization problems at massive scales.
Empirical Analysis of ideal recombination on random decomposable problemskknsastry
This paper analyzes the behavior of a selectorecombinative genetic algorithm (GA) with an ideal crossover on a class of random additively decomposable problems (rADPs). Specifically, additively decomposable problems of order k whose subsolution fitnesses are sampled from the standard uniform distribution U[0,1] are analyzed. The scalability of the selectorecombinative GA is investigated for 10,000 rADP instances. The validity of facetwise models in bounding the population size, run duration, and the number of function evaluations required to successfully solve the problems is also verified. Finally, rADP instances that are easiest and most difficult are also investigated.
Population sizing for entropy-based model buliding In genetic algorithms kknsastry
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Let's get ready to rumble redux: Crossover versus mutation head to head on ex...kknsastry
This paper analyzes the relative advantages between crossover and mutation on a class of deterministic and stochastic additively separable problems with substructures of non-uniform salience. This study assumes that the recombination and mutation operators have the knowledge of the building blocks (BBs) and effectively exchange or search among competing BBs. Facetwise models of convergence time and population sizing have been used to determine the scalability of each algorithm. The analysis shows that for deterministic exponentially-scaled additively separable, problems, the BB-wise mutation is more efficient than crossover yielding a speedup of Θ(l logl), where l is the problem size. For the noisy exponentially-scaled problems, the outcome depends on whether scaling on noise is dominant. When scaling dominates, mutation is more efficient than crossover yielding a speedup of Θ(l logl). On the other hand, when noise dominates, crossover is more efficient than mutation yielding a speedup of Θ(l).
Modeling selection pressure in XCS for proportionate and tournament selectionkknsastry
In this paper, we derive models of the selection pressure in XCS for proportionate (roulette wheel) selection and tournament selection. We show that these models can explain the empirical results that have been previously presented in the literature. We validate the models on simple problems showing that, (i) when the model assumptions hold, the theory perfectly matches the empirical evidence; (ii) when the model assumptions do not hold, the theory can still provide qualitative explanations of the experimental results.
Modeling XCS in class imbalances: Population sizing and parameter settingskknsastry
This paper analyzes the scalability of the population size required in XCS to maintain niches that are infrequently activated. Facetwise models have been developed to predict the effect of the imbalance ratio—ratio between the number of instances of the majority class and the minority class that are sampled to XCS—on population initialization, and on the creation and deletion of classifiers of the minority class. While theoretical models show that, ideally, XCS scales linearly with the imbalance ratio, XCS with standard configuration scales exponentially.
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Silicon Cluster Optimization Using Extended Compact Genetic Algorithmkknsastry
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A Practical Schema Theorem for Genetic Algorithm Design and Tuningkknsastry
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This paper studies fitness inheritance as an efficiency enhancement technique for genetic and evolutionary algorithms. Convergence and population-sizing models are derived and compared with experimental results. These models are optimized for greatest speed-up and the optimal inheritance proportion to obtain such a speed-up is derived. Results on OneMax problems show that when the inheritance effects are considered in the population-sizing model, the number of function evaluations are reduced by 20% with the use of fitness inheritance. Results indicate that for a fixed population size, the number of function evaluations can be reduced by 70% using a simple fitness inheritance technique.
Efficient Cluster Optimization Using A Hybrid Extended Compact Genetic Algori...kknsastry
The document discusses the benefits of meditation for reducing stress and anxiety. Regular meditation practice can help calm the mind and body by lowering heart rate and blood pressure. Studies have shown that meditating for just 10-20 minutes per day can have significant positive impacts on both mental and physical health over time.
Modeling Tournament Selection with Replacement Using Apparent Added Noisekknsastry
This paper analyzes the effects of tournament selection with replacement on the convergence time and population sizing for selectorecombinative genetic algorithms. This paper empirically demonstrates that the run duration remains the same and is not affected whether the tournament selection is performed with or without replacement. However, the population size required is more if tournament selection is performed with replacement rather than without replacement to attain the same level of accuracy. An approximate population-sizing model is derived based on apparent added noise for the case of tournament selection with replacement. The proposed model is verified with experimental results.
Evaluation Relaxation as an Efficiency-Enhancement Technique: Handling Varian...kknsastry
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Analysis of Mixing in Genetic Algorithms: A Surveykknsastry
Ensuring building-block (BB) mixing is critical to the success of genetic and evolutionary algorithms. There has been a growing interest in analyzing and understanding BB mixing and it is necessary to organize and categorize representative literature. This paper presents an exhaustive survey of studies on one or more aspects of mixing. In doing so, a classification of the literature based on the role of recombination operators assumed by those studies is developed. Such a classification not only highlights the significant results and unifies existing work, but also provides a foundation for future research in understanding mixing in genetic algorithms.
How Well Does A Single-Point Crossover Mix Building Blocks with Tight Linkage?kknsastry
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Scalability of Selectorecombinative Genetic Algorithms for Problems with Tigh...kknsastry
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- These are slides of the talk given at IEEE International Conference on Software Testing Verification and Validation Workshop, ICSTW 2022.
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See how organizational priorities and strategic approaches to data security and privacy are evolving around the globe.
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A tale of scale & speed: How the US Navy is enabling software delivery from l...sonjaschweigert1
Rapid and secure feature delivery is a goal across every application team and every branch of the DoD. The Navy’s DevSecOps platform, Party Barge, has achieved:
- Reduction in onboarding time from 5 weeks to 1 day
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- Maintenance of superior security standards and inherent policy enforcement with Authorization to Operate (ATO)
Development teams can ship efficiently and ensure applications are cyber ready for Navy Authorizing Officials (AOs). In this webinar, Sigma Defense and Anchore will give attendees a look behind the scenes and demo secure pipeline automation and security artifacts that speed up application ATO and time to production.
We will cover:
- How to remove silos in DevSecOps
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Threats to mobile devices are more prevalent and increasing in scope and complexity. Users of mobile devices desire to take full advantage of the features
available on those devices, but many of the features provide convenience and capability but sacrifice security. This best practices guide outlines steps the users can take to better protect personal devices and information.
National Security Agency - NSA mobile device best practices
Modeling selection pressure in XCS for proportionate and tournament selection
1. Modeling Selection Pressure in XCS for
Proportionate and Tournament Selection
Albert Orriols-Puig, Kumara Sastry,
Pier Luca Lanzi, David E. Goldberg,
and Ester Bernad´-Mansilla
o
IlliGAL Report No. 2007004
February 2007
Illinois Genetic Algorithms Laboratory
University of Illinois at Urbana-Champaign
117 Transportation Building
104 S. Mathews Avenue Urbana, IL 61801
Office: (217) 333-2346
Fax: (217) 244-5705
2. Modeling Selection Pressure in XCS for
Proportionate and Tournament Selection
Albert Orriols-Puig1,2 , Kumara Sastry1 , Pier Luca Lanzi1,3 ,
David E. Goldberg1 , and Ester Bernad´-Mansilla2
o
1
Illinois Genetic Algorithm Laboratory (IlliGAL)
Department of Industrial and Enterprise Systems Engineering
University of Illinois at Urbana-Champaign
Urbana, IL 61801
2
Group of Research in Intelligent Systems
Computer Engineering Department
Enginyeria i Arquitectura La Salle — Ramon Llull University
Quatre Camins, 2. 08022, Barcelona, Catalonia, Spain.
3
Artificial Intelligence and Robotics Laboratory (AIRLab)
Dip. di Elettronica e Informazione
Politecnico di Milano
P.za L. da Vinci 32, I-20133, Milano, Italy
aorriols@salle.url.edu, ksastry@uiuc.edu, lanzi@elet.polimi.it,
deg@uiuc.edu, esterb@salle.url.edu
February 11, 2007
Abstract
In this paper, we derive models of the selection pressure in XCS for proportionate (roulette
wheel) selection and tournament selection. We show that these models can explain the empirical
results that have been previously presented in the literature. We validate the models on simple
problems showing that, (i) when the model assumptions hold, the theory perfectly matches the
empirical evidence; (ii) when the model assumptions do not hold, the theory can still provide
qualitative explanations of the experimental results.
1 Introduction
Learning Classifier Systems are complex, fascinating machines introduced more than 30 years ago
by John H. Holland, the father of genetic algorithms (Holland, J.H., 1975). They combine different
paradigms (genetic algorithms for search and reinforcement learning for estimation) which are
applied to a general-purpose rule-based representation to solve problems online. Because they
combine different paradigms, they are also daunting to study. In fact, the analysis of how a learning
classifier system works requires knowledge about how the genetic and the reinforcement components
1
3. work and, most important, about how such components interact on the underlying representation.
As a consequence, few theoretical models have been developed (e.g., (Horn, Goldberg, & Deb, 1994;
Bull, 2001)) and learning classifier systems are more often studied empirically.
Recently, facetwise modeling (Goldberg, 2002) has been successfully applied to develop bits
and pieces of a theory of XCS (Wilson, 1995), the most influential learning classifier system of the
last decade. Butz et al. (Butz, Kovacs, Lanzi, & Wilson, 2004) modeled different generalization
pressures in XCS so as to provided the theoretical foundation of Wilson’s generalization hypothe-
sis (Wilson, 1998); they also derived population bounds that ensure effective genetic search in XCS.
Later, Butz et al. (Butz, Goldberg, & Lanzi, 2004) applied the same approach to derive a bound
for the learning time of XCS until maximally accurate classifiers are found. More recently, Butz et
al. (Butz, Goldberg, Lanzi, & Sastry, 2004) presented a Markov chain analysis of niche support in
XCS which resulted in another population size bound to ensure effective problem substainance.
In this paper, we propose a step futher toward the understanding of XCS (Wilson, 1995) and
present a model of selection pressure under proportionate (roulette wheel) selection (Wilson, 1995)
and tournament selection (Butz, Sastry, & Goldberg, 2005). These selection schemes have been
empirically compared by Butz et al. (Butz, Goldberg, & Tharakunnel, 2003; Butz, Sastry, & Gold-
berg, 2005) and later by Kharbat et al. (Kharbat, Bull, & Odeh, 2005) leading to different claims
regarding the superiority of one selection scheme versus the other. In genetic algorithms, these se-
lection schemes have been exhaustively studied through the analysis of takeover time (see (Goldberg
& Deb, 1990; Goldberg, 2002) and references therein). In this paper, we follow the same approach
as (Goldberg & Deb, 1990; Goldberg, 2002) and develop theoretical models of selection pressure in
XCS for proportionate and tournament selection. Specifically, we perform a takeover time analysis
to estimate the time from an initial proportion of best individuals until the population is converged
or substantially converged to the best. We start from the typical assumption made in takeover
time analysis (Goldberg & Deb, 1990): XCS has converged to an optimal solution, which in XCS
typically consists of a set of non-overlapping niches. We write differential equations that describe
the change in proportion of the best classifier in one niche for roulette wheel and tournament selec-
tion. Initially, we focus on classifier accuracy, later we extend the model taking into account also
classifier generality. We solve the equations and derive a closed form solution of takeover time in
XCS for the two selection schemes (Section 3). In Section 4, we use these equations to determine
the conditions under which proportionate and tournament selection (i) produce the same initial
growth of the best classifiers in the niche, or (ii) result in the same takeover time. In Section 5,
we introduce two artificial test problems which we use in Section 6 to validate the models showing
that, when the assumptions of non-overlapping niches hold, the models perfectly match the empir-
ical evidence while they accurately approximate the empirical results, when such an assumption is
violated. Finally, in Section 7, we extend the models to include classifier generality and show that,
again, the models fit empirical evidence when the model assumptions hold.
2 The XCS Classifier System
We briefly describe XCS giving all the details that are relevant to this work. We refer the reader
to (Wilson, 1995; Butz & Wilson, 2002) for more detailed descriptions.
Knowledge Representation. In XCS, classifiers consist of a condition, an action, and four main
parameters: (i) the prediction p, which estimates the payoff that the system expects when the
classifier is used; (ii) the prediction error , which estimates the error affecting the prediction p;
(iii) the fitness F , which estimates the accuracy of the payoff prediction given by p; and (iv) the
numerosity num, which indicates how many copies of classifiers with the same condition and the
2
4. same action are present in the population.
Performance Component. At time t, XCS builds a match set [M] containing the classifiers
in the population [P] whose condition matches the current sensory input st ; if [M] contains less
than θnma actions, covering takes place and creates a new classifier that matches s t and has a
random action. For each possible action a in [M], XCS computes the system prediction P (s t , a)
which estimates the payoff that XCS expects if action a is performed in st . The system prediction
P (st , a) is computed as the fitness weighted average of the predictions of classifiers in [M] which
advocate action a. Then, XCS selects an action to perform; the classifiers in [M] which advocate
the selected action form the current action set [A]. The selected action a t is performed, and a scalar
reward R is returned to XCS.1
Parameter Updates. The incoming reward R is used to update the parameters of the classifiers
in [A]. First, the classifier prediction pk is updated as, pk ← pk + β(R − pk ). Next, the error k is
updated as, k ← k + β(|R − p| − k ). To update the classifier fitness F , the classifier accuracy κ
is first computed as,
1 if ε < ε0
κ= (1)
α(ε/ε0 )−ν otherwise
the accuracy κ is used to compute the relative accuracy κ as, κ = κ/ [A] κi . Finally, the classifier
fitness F is updated towards the classifier’s relative accuracy κ as, F ← F + β(κ − F ).
Genetic Algorithm. On regular basis (dependent on the parameter θga ), the genetic algorithm is
applied to classifiers in [A]. It selects two classifiers, copies them, and with probability χ performs
crossover on the copies; then, with probability µ it mutates each allele. The resulting offspring
classifiers are inserted into the population and two classifiers are deleted to keep the population
size constant.
Two selection mechanisms have been introduced so far for XCS: proportionate, roulette wheel,
selection (Wilson, 1995) and tournament selection (Butz, Sastry, & Goldberg, 2005). Roulette wheel
selects a classifier with a probability proportional to its fitness. Tournament randomly chooses the
τ percent of the classifiers in the action set and among these it selects the classifier with higher
“microclassifier fitness” fk computed as fk = Fk /nk (Butz, Sastry, & Goldberg, 2005; Butz, 2003).
3 Modeling Takeover Time
The analysis of takeover time in XCS poses two main challenges. First, while in genetic algorithms
the fitness of an individual is usually constant, in XCS classifier fitness changes over time based on
the other classifiers that appear in the same evolutionary niche. Second, while in genetic algorithms
the selection and the replacement of individuals are usually performed over the whole population,
in XCS selection is niche based, while deletion is population based.
To model takeover time (Goldberg & Deb, 1990), we have to assume that XCS has already
converged to an optimal solution, which in XCS consists of non overlapping niches (Wilson, 1995).
Accordingly, we can focus on one niche without taking into account possible interactions among
overlapping niches. We consider a simplified scenario in which a niche contains two classifiers, cl 1
and cl 2 ; classifier cl k has fitness Fk , prediction error k , numerosity nk , and microclassifier fitness
fk = Fk /nk . In this initial phase, we focus on classifier accuracy and hypothesize that cl 1 be the
1
In this paper we focus on XCS viewed as a pure classifier, i.e., applied to single-step problems. A complete
description is given elsewhere (Wilson, 1995; Butz & Wilson, 2002).
3
5. “best” classifier in the niche, because is the most accurate, i.e., κ1 ≥κ2 , while we assume that cl 1 and
cl 2 are equally general and thus they have the same reproductive opportunities (this assumption
will be relaxed later in Section 7). Finally, we assume that deletion selects classifiers randomly from
the same niche. Although this assumption may appear rather strong, if the niches are activated
uniformly this assumption will have little effect as the empirical validation of the model will show
(Section 6).
3.1 Roulette Wheel Selection
Under Roulette Wheel Selection (RWS), the selection probability of a classifier depends on the
ratio of its fitness Fi over the fitness of all classifiers in the action set. Without loss of generality,
we assume that classifier fitness is a simple average of classifier’s relative accuracies and compute
the fitness of classifiers cl 1 and cl 2 as,
κ 1 n1 1
F1 = =
κ 1 n1 + κ 2 n2 1 + ρnr
κ 2 n2 ρnr
F2 = =
κ 1 n1 + κ 2 n2 1 + ρnr
where nr = n2 /n1 and ρ is the ratio between the accuracy of cl 2 and the accuracy of cl 1 (ρ = κ1 /κ2 ).
The probability Ps of selecting the best classifier cl 1 in the niche is computed as,
F1 1
Ps = =
F1 + F 2 1 + ρnr
Once selected, a new classifier is created and inserted in the population while one classifier is
randomly deleted from the niche with probability Pdel (cl j ) = nj /n with n = n1 + n2 .
We can now model the evolution of the numerosity of the best classifier cl 1 at time t, n1,t ,
which will (i) increase in the next generation if cl 1 is selected by the genetic algorithm and another
classifier is selected for deletion; (ii) decrease if cl 1 is not selected by the genetic algorithm but cl 1
is selected for deletion; (iii) remain the same, in all the other cases. More formally,
1 n1
n1,t + 1 1+ρnr 1 − n
1
1 − 1+ρnr n1
n1,t − 1
n1,t+1 = n
n1,t otherwise
where n is the niche size. This model is relaxed by assuming that only one classifier can be deleted
when sampling the niche it belongs. Grouping the equations above, we obtain,
n1,t
1
n1,t+1 = n1,t + −
1 + ρnr n
which can be rewritten in terms of the proportion Pt of classifiers cl 1 in the niche (i.e., Pt = n1,t /n).
Using the equality nr = (1 − Pt )/Pt we derive,
1 1 1
Pt+1 = Pt + · − Pt
1−Pt
n 1+ρ P n
t
assuming Pt+1 − Pt ≈ dp/dt we have,
1 Pt (1 − ρ) − Pt2 (1 − ρ)
dp
≈ Pt+1 − Pt = (2)
dt n Pt (1 − ρ) + ρ
4
6. that is,
Pt (1 − ρ) + ρ 1−ρ
dp = dt (3)
Pt (1 − Pt ) n
which can be solved by integrating each side of the equation, between P0 , the initial proportion of
cl 1 , and the final proportion P of cl 1 up to which cl 1 has taken over,
P
Pt (1 − ρ) + ρ 1−ρ t(1 − ρ)
dp = dt = (4)
Pt (1 − Pt ) N N
P0
The left integral can be solved as follows:
P P
Pt (1 − ρ) + ρ ρ(1 − Pt ) + Pt t(1 − ρ)
dp = dp = (5)
Pt (1 − Pt ) Pt (1 − Pt ) n
P0 P0
P P
ρ 1 t(1 − ρ)
dp + dp = (6)
P0 P t P0 (1 − Pt ) n
P 1−P t(1 − ρ)
ρ ln − ln = (7)
P0 1 − P0 n
from which we derive the takeover time of cl 1 in roulette wheel selection,
N P 1 − P0
t∗ = ρ ln + ln (8)
rws
1−ρ P0 1−P
The takeover time t∗ under the assumption that the two classifiers are equally general depends
rws
(i) on the ratio ρ between the accuracy of cl 2 and the accuracy of cl 1 , as well as, (ii) on the initial
proportion of cl 1 in the population. A higher ρ implies an increase in the takeover time. When
ρ = 1 (i.e., cl 1 and cl 2 are equally accurate), t∗ tends to infinity, that is, both cl 1 and cl 2 will
rws
remain in the population. When ρ is smaller, the takeover time decreases. For a ρ close to zero,
1−P0
t∗ can be approximated by t∗ ρ→0 ≈ n ln 1−P . The takeover time also depends on the initial
rws
proportion P0 of cl 1 in the population. A lower P0 increases the term inside the brackets, and so
it increases the takeover time. In contrast, an higher P0 results in a lower takeover time.
3.2 Tournament Selection
To model takeover time for tournament selection in XCS we assume a fixed tournament size of
s, instead of the typical variable tournament size (Butz, 2003). The underline assumption of the
takeover time analysis is that the system already converged to an optimal solution made of non
overlapping niches. Thus, there is basically no difference between a fixed tournament size and a
variable one. Tournament selection randomly chooses s classifiers in the action set and selects the
one with higher fitness. As before, we assume that cl 1 is the best classifier in the niche, which
in terms of tournament selection translates into requiring that f1 > f2 , where fi is the fitness of
the microclassifiers associated to cl i . In this case, the numerosity n1 of cl 1 will (i) increase if cl 1
participates in the tournament and another classifier is selected to be deleted; (ii) decrease if cl 1
does not participate in the tournament but is selected by the deletion operator; (iii) remain the
same, otherwise. More formally,
s
1 − 1 − n1 1 − n1
n1,t + 1 n n
s
1 − n1 n1
n1,t − 1
n1,t+1 = n n
n1,t otherwise
5
7. By grouping the above equations we derive the expected numerosity of cl 1 ,
n1 s n1 n1 s n1
nt+1 = nt + 1 − 1 − 1− − 1−
n n n n
which we rewrite in terms of the proportion P of cl 1 in the niche (i.e., Pt = n1,t /n) as,
1
(1 − Pt ) 1 − (1 − Pt )s−1
Pt+1 = Pt + (9)
n
dp
Assuming ≈ Pt+1 − Pt , we derive,
dt
dp 1
(1 − Pt )[1 − (1 − Pt )s−1 ] ,
≈ Pt+1 − Pt = (10)
dt n
that is,
(1 − Pt )s−2
dt 1
= dp + dp
1 − (1 − Pt )s−1
n 1 − Pt
Integrating each side of the equation we obtain,
P P
(1 − Pt )s−2
dt 1
= dp + dp
1 − (1 − Pt )s−1
n 1 − Pt
P0 P0
i.e.,
1 − (1 − P )s−1
t 1 − P0 1
= ln + ln
1 − (1 − P0 )s−1
n 1−P s−1
so that the takeover time of cl 1 for tournament selection is,
1 − (1 − P )s−1
1 − P0 1
t∗ S = n ln + ln (11)
T
1 − (1 − P0 )s−1
1−P s−1
Given our assumptions, takeover time for tournament selection depends on the initial proportion
of the best classifier P0 and the tournament size s. Both logarithms on the right hand side take
positive values if P > P0 , indicating that the best classifier will always take over the population
regardless of its initial proportion in the population. When P < P0 , both logarithms result in
negative values, and so does the takeover time.
4 Comparison
We now compare the takeover time for roulette wheel selection and tournament selection and we
study the salient differences between the models. For this purpose, we compute the values of s
and ρ for which, (i) the two selection schemes have the same initial increase of the proportion of
the best classifier, and for which (ii) the two selection schemes have the same takeover time. This
analysis results in two expressions that permit to compare the behavior of roulette wheel selection
and tournament selection in different scenarios.
First, we analyze the relation between the ratio of classifier accuracies ρ and the selection
pressure s in tournament selection to obtain the same initial increase in the proportion of the best
classifier with both selection schemes. Taking the equations 2 and 10, and replacing P t by P0 we
get that the initial increase of the best classifier in each selection scheme is:
1 − ρ P0 (1 − P0 )
∆RW S =
n (1 − ρ)P0 + ρ
1
(1 − P0 )[1 − (1 − P0 )s−1 ]
∆T S =
n
6
8. By requiring ∆RW S = ∆T S , we obtain,
1 − ρ P0 (1 − P0 ) 1 − P0
[1 − (1 − P0 )s−1 ],
=
n (1 − ρ)P0 + ρ n
that is,
Pt (1 − ρ)
= 1 − (1 − Pt )s−1 ,
(1 − ρ)Pt + ρ
ln ρ − ln [(1 − ρ)P0 + ρ]
s=1+ (12)
ln(1 − P0 )
This equation indicates that, in tournament selection, the tournament size s which regulates the
selection pressure has to increase as the ratio of accuracies ρ decreases to have the same initial
increase of the best classifier in the population as roulette wheel selection. For example, for ρ = 0.01
and P0 = 0.01, tournament selection requires s = 70 to produce the same initial increase of the best
classifier as roulette wheel selection. On the other hand, low values of s result in the same effect
as roulette wheel selection with high values of ρ. Equation 12 indicates that, even with a small
tournament size, tournament selection produces a stronger pressure towards the best classifier in
scenarios in which slightly inaccurate but initially highly numerous classifiers are competing against
highly accurate classifiers. On the other hand, when the competition involves highly inaccurate
classifiers, a larger tournament size s is required to obtain the same selection pressure provided by
roulette wheel selection.
We now analyze the conditions under which both selection schemes result in the same takeover
time. For this purpose, we equate t∗ S in Equation 8 with t∗ S in Equation 11,
RW T
t∗ S = t ∗ S (13)
RW T
P ∗ )s−1
P∗
n 1 − P0 1 − P0 1 1 − (1 −
ρ ln + ln = n ln + ln (14)
1 − (1 − P0 )s−1
1−ρ P0 1 − P∗ 1 − P∗ s−1
Which can be rewriten as follows:
1 − (1 − P ∗ )s−1
P ∗ (1 − P0 )
ρ 1
ln = ln .
1 − (1 − P0 )s−1
1−ρ P0 (1 − P ∗ ) s−1
By approximating (1 − x)s−1 by its first order Taylor series at the point 0, that is, (1 − x)s−1 =
1 + (s − 1)x, and by futher simplifications, we derive,
1 − (1 − P ∗ )s−1 P∗
1 1
ln = ln (15)
1 − (1 − P0 )s−1
s−1 s−1 P0
Reorganizing, we obtain the following relation:
ln(P ∗ ) − ln(P0 )
1−ρ
s=1+ (16)
ρ ln[P ∗ (1 − P0 )] − ln[P0 (1 − P ∗ )]
Given an initial proportion of the best classifier, the equation is guided by the term 1−ρ . As the
ρ
accuracy-ratio ρ decreases, s needs to increase polynomially. On the other hand, higher values of ρ
require a low tournament size s. Thus, Equation 16 reaches the same conclusions of Equation 12:
tournament selection is better than roulette wheel in scenarios where highly accurate classifiers
compete with slightly inaccurate ones.
7
9. 5 Test Problems
To validate the models of takeover time for roulette wheel and proportionate selection, we designed
two test problems. The single-niche problem fully agrees with the model assumption. It consists
of a niche with a highly accurate classifier cl 1 and a less accurate classifier cl 2 . The covering is off
and the population is initialized with N · P0 copies of cl 1 and N · (1 − P0 ) copies of cl 2 , where N
is the population size. The prediction error 1 of the best classifier cl 1 is always set to zero while
the prediction error 2 of cl 2 is set as,
ρν
2= 0 (17)
α
where ρ is the ratio between the accuracy of cl 2 and the accuracy of cl 1 (i.e., ρ = κ2 /κ1 ; Section 3).
The multiple-niche problem consists of a set of m niches each one containing one maximally
accurate classifier and one classifier that belongs to all niches. The population contains P 0 · N
copies of maximally accurate classifiers (equally distributed in the m niches) and (1 − m · P 0 ) · N
copies of the classifier that appears in all the niches. The parameters of classifiers are updated as
in the case of the single niche in the previous section, permitting to vary the ratio of accuracies
ρ, and so, validating the model under different circumstances. This test problem violates two
model assumptions. First, the size of the different niches differ from the population size since the
problem consists of more than one niche. While in the model we assumed that the deletion would
always select a classifier in the same niche, in this case deletion can select any classifier from the
population. Second, the niches are overlapping since there is a classifier that belongs to all the
niches. This also means that the sum of all niche sizes will be greater than the population size, i.e.,
m
1 (ni,1 + ni,2 ) > N , where ni,1 is the numerosity of the maximally accurate classifier in niche i
and ni,1 is the numerosity of the less accurate classifier in the niche i.
6 Experimental Validation
We validated the takeover time models using the single-niche problem and the multiple-niche
problem. Our results show a full agreement between the model and the experiments in the single-
niche problem, when all the model assumptions hold. They also show that the theory correctly
predicts the practical results in problems where the initial assumptions do not hold, if either the
accuracy-ratio is low enough in roulette wheel selection or the tournament size is high enough in
tournament selection.
Single-Niche. At first, we applied XCS on the single-niche problem and analyzed the proportion
of the best classifier cl 1 under roulette wheel selection and tournament selection. Figure 1 compares
the proportion of the best classifier in the niche as predicted by our model (reported as lines) and
the empirical data (reported as dots) for roulette wheel selection and tournament selection with
s = {2, 3, 9} when ρ = 0.01. The empirical data are averages over 50 runs. Figure 1 shows a
perfect match between the theory and the empirical results. It also shows that, as predicted by
the models and discussed in Section 4, roulette wheel produces a faster increase in the proportion
of the best classifier for ρ = 0.01. To obtain a similar increase with tournament selection we need
a higher tournament size: in fact, the second best increase in the proportion of the best classifier
is provided by tournament selection with s = 9. Finally, as indicated by the theory, the time to
achieve a certain proportion of the best classifier in tournament selection increases as the selection
pressure decreases.
We performed another set of experiments and compared the proportion of the best classifier
in the niche for an accuracy ratio ρ of 0.5 and 0.9. Figure 2 compares the proportion of cl 1 as
8
10. Takeover time for ρ = 0.01
1
0.9
0.8
0.7
Proportion of the Best Classifier
Empirical RWS
0.6 Model RWS
Empirical TS s=2
Model TS s=2
0.5
Empirical TS s=3
Model TS s=3
Empirical TS s=9
0.4
Model TS s=9
0.3
0.2
0.1
0
0 2000 4000 6000 8000 10000 12000 14000 16000
Activation Time of the Niche
Figure 1: Empirical and theoretical takeover time when ρ = 0.01 for roulette wheel selection and
tournament selection with s=2, s=3 and s=9.
predicted by our model (reported as lines) and the empirical data (reported as dots) for (a) ρ = 0.5
and (b) ρ = 0.9; the empirical data are averages over 50 runs. The results show a good match
between the theory and the empirical results. As predicted by the model, tournament selection is
not influenced by the increase of ρ: the increase in the proportion of cl 1 for ρ = 0.5 (Figure 2a)
is basically the same as the increase obtained when ρ = 0.9 (Figure 2b). Thus, coherently to
what empirically shown in (Butz, Sastry, & Goldberg, 2005), tournament selection demonstrates
its robustness in maintaining and increasing the proportion of the best classifier even when there is
a small difference between the fitness of the most accurate classifier (cl 1 ) and the fitness of the less
accurate one (cl 2 ). In contrast, roulette wheel selection is highly influenced by the accuracy ratio ρ:
as the ratio approaches 1, i.e., the accuracy of the two classifiers become very similar, the increase
in the proportion of the best classifier becomes smaller and smaller. In fact, when ρ = 0.90, after
16000 activations of the niche the best classifier cl 1 has taken over only the 5% of the niche.
Multiple-niche. In the next set of experiments, we validated our model of takeover time on the
multiple-niche problem, where the assumptions about non-overlapping niches and about deletion
being performed in the niche are violated. For this purpose, we ran XCS on the multiple-niche
problem with two niches (m = 2); each niche contains one maximum accurate classifier and there
is one, less accurate, overlapping classifier that participates in both niches.
Figure 3a compares the proportion of the best classifier in the niche for roulette wheel selection
for an accuracy ratio ρ of 0.01 and 0.20. The plots indicate that, for small values of the accuracy
ratio ρ, our model (reported as lines) slightly underestimates the empirical takeover time (reported
as dots). As the accuracy ratio ρ increases, the model tends to overestimate the empirical data
(Figure 3b). This behavior can be easily explained. The lower the accuracy ratio (Figure 3a), the
higher the pressure toward the highly accurate classifiers, and consequently, the faster the takeover
time. When ρ is 0.20 and 0.30, the difference between the model and the empirical results is
visible only at the beginning, while it basically disappears as the number of activations increases.
For higher values of ρ, the overgeneral, less accurate, cl 2 has more reproductive opportunities in
both niches it participates. These results indicate that (i) the model for roulette wheel selection is
accurate in general scenarios if the ratio of accuracies is small (i.e., when there is a large proportion
of accurate classifiers in the niche); (ii) in situations where there is a small proportion of the best
classifier in one niche competing with other slightly inaccurate and overgeneral classifiers (above
9
11. Takeover time for ρ = 0.50 Takeover time for ρ = 0.90
1 1
0.9 0.9
0.8 0.8
Empirical RWS
Model RWS
0.7 0.7
Proportion of the Best Classifier
Proportion of the Best Classifier
Empirical TS s=2
Model TS s=2
Empirical RWS
Empirical TS s=3
0.6 0.6
Model RWS
Model TS s=3
Empirical TS s=2
Empirical TS s=9
Model TS s=2
0.5 0.5 Model TS s=9
Empirical TS s=3
Model TS s=3
Empirical TS s=9
0.4 0.4
Model TS s=9
0.3 0.3
0.2 0.2
0.1 0.1
0 0
0 2000 4000 6000 8000 10000 12000 14000 16000 0 2000 4000 6000 8000 10000 12000 14000 16000
Activation Time of the Niche Activation Time of the Niche
(a) (b)
Figure 2: Empirical and theoretical takeover time for roulette wheel selection and tournament
selection with s=2, s=3 and s=9 when (a) ρ = 0.50 and (b) ρ = 0.90.
a certain threshold of ρ), the overgeneral classifier may take over the population removing all the
copies of the best classifier. It is interesting to note that, as the number of niches increases from
m = 2 to m = 16, the agreement between the theory and the experiments gently degrades (see
appendix A for more results).
Figure 4a compares the proportion of the best classifier as predicted by our model and as
empirically determined in tournament selection with s = 9. As in roulette wheel selection for
a small ρ, the results for tournament selection show that the empirical takeover time is slightly
faster than the one predicted by the theory. Again, this behavior is due to the presence of the
overgeneral classifier in both niches, causing a higher pressure toward its deletion. Increasing the
tournament size s produces little variations in either the empirical results or the theoretical values,
and so the conclusions extracted for s = 9 can be extended for a higher s. On the other hand,
decreasing s causes the empirical values to go closer to the theoretical values, since the pressure
toward the deletion of the overgeneral classifier decreases. Figure 4b reports the proportion of one
of the maximum accurate classifiers for s ∈ {2, 3}. The results show that the theory accurately
predicts the empirical values for s = 3, but for s = 2 the difference between the model and the
data is large. In this case, a small tournament size combined with the presence of the overgeneral
classifier in both niches produces a strong selection pressure toward the overgeneral classifier that
delays the takeover time.
The results in the multi-niche problem confirm what empirically shown in (Butz, Sastry, &
Goldberg, 2005): tournament selection is more robust than roulette wheel selection. Under perfect
conditions both schemes perform similarly, which is coherent to what shown in (Kharbat, Bull, &
Odeh, 2005). However, the takeover time of the best classifier is delayed in roulette wheel selection
for an higher accuracy ratio ρ. The empirical results indicate that, with roulette wheel selection,
the best classifier was eventually removed from the population when ρ ≥ 0.5. On the other hand,
tournament selection demonstrated theoretically and practically to be more robust, since it does
not depend on the individual fitness of each classifier. In all the experiments with tournament
selection, the best classifier could take over the niche, and only for the extreme case (i.e., for s = 2)
10
12. Takeover time in RWS for Overlapping Niches and δ={0.01,0.20} Takeover time in RWS for Overlapping Niches and ρ={0.30,0.40,0.50}
1
1
0.9
0.9
0.8
0.8
Proportion of the Best Classifier
Proportion of the Best Classifier
Empirical RWS δ=0.01 Empirical RWS ρ=0.30
0.7
0.7
Model RWS δ=0.01 Model RWS ρ=0.30
Empirical RWS δ=0.20 Empirical RWS ρ=0.40
Model RWS δ=0.20 Model RWS ρ=0.40
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2 0.2
0.1 0.1
0 2000 4000 6000 8000 10000 12000 14000 16000 0 2000 4000 6000 8000 10000 12000 14000 16000
Activation Time of the Niche Activation Time of the Niche
(a) (b)
Figure 3: Takeover time for roulette wheel selection when (a) ρ = {0.01, 0.20} and (b) ρ ∈
{0.30, 0.40}.
the experiments considerably disagreed with the theory.
7 Modeling Generality
Finally, we extend our model of takeover time with classifier generality. As before, we model the
proportion of the best classifier cl 1 in one niche, however, in this case cl 1 is not only the maximally
accurate classifier in the niche, but also the maximally general with respect to niche; thus cl 1 is now
the best classifier in the most typical XCS sense (Wilson, 1998). We focus on one niche and assume
that cl 1 , being the maximally general and maximally accurate classifier for the niche, appears in the
niche with probability 1 so that cl 2 appears in the niche with a relative probability ρm . Similarly
to what we previously did in Section 3, we can model the numerosity of the classifier cl 1 at time
t, n1,t as follows. The numerosity n1,t of cl 1 in the next generation will (i) increase when both
cl 1 and cl 2 appear in the niche and cl 1 is selected by the genetic algorithm while cl 2 is selected
for deletion; (ii) increase when only cl 1 appears in the niche and another classifier is deleted; (iii)
decrease only if both cl 1 and cl 2 are in the niche, cl 1 is not selected by the genetic algorithm but
it is selected for deletion. Otherwise, the numerosity of cl 1 will remain the same. More formally,
1
1 − n1 + (1 − ρm ) 1 − n1
n1,t + 1 ρm 1+ρn n n
r
1 n1
n1,t+1 =
n1,t − 1 ρm 1 − 1+ρnr n
n1,t otherwise
As done before, we can group these equations to obtain,
n1,t n1,t
1
n1,t+1 = n1,t + ρm − + (1 − ρm ) 1 −
1 + ρnr n n
11
13. Takeover time in TS s=9 for Overlapping Niches Takeover time in TS s={2,3} for Overlapping Niches
1 1
0.9 0.9
0.8 0.8
Proportion of the Best Classifier
Proportion of the Best Classifier
Empirical TS s=9
Model TS s=9
0.7 0.7
Empirical TS s=2
0.6 0.6
Model TS s=2
Empirical TS s=3
0.5 0.5 Model TS s=3
0.4 0.4
0.3 0.3
0.2 0.2
0.1 0.1
0 2000 4000 6000 8000 10000 12000 14000 16000 0 2000 4000 6000 8000 10000 12000 14000 16000
Activation Time of the Niche Activation Time of the Niche
(a) (b)
Figure 4: Takeover time for tournament selection when (a) s = 9, (b) s ∈ {2, 3}.
we rewrite n1,t in terms of the proportion Pt of cl 1 in the niche (Pt = n1,t /n with nr = (1 − Pt )/Pt )
and obtain,
1 1 − Pt
Pt+1 = Pt + · [(1 − ρ)Pt + (1 − ρm ))ρ]
n ρ + (1 − ρ)Pt
Assuming Pt+1 − Pt ≈ dp/dt,
dp 1 1 − Pt
≈ Pt+1 − Pt = · [(1 − ρ)Pt + (1 − ρm ))ρ]
dt n rho + (1 − ρ)Pt
which can be simplified as follows:
ρ + (1 − ρ)P 1
dP = dt
(1 − P ) [(1 − ρ)P + ρ(1 − ρm )] n
Integrating the above equation with initial conditions t = 0, Pt = P0 , we get the takeover time as:
n 1 − P0
t∗
rws = ln +
1 − ρρm 1−P
(1 − ρ)P + ρ(1 − ρm )
+ ρρm ln (18)
(1 − ρ)P0 + ρ(1 − ρm )
which depends on the ratio of accuracies ρ, the initial proportion of the best classifier P 0 and the
generality of the inaccurate classifier, represented by ρm . In the previous model, cl 1 would not
take over the niche when it was as accurate as cl 2 (Section 3). In this case, cl 1 will take over the
population if it is either more accurate or more general than cl 2 . Otherwise, if cl1 and cl2 are
equally accurate and general, both will persist in the population (t∗ S → ∞), coherently with
RW
the previous model. Low values of ρm suppose quicker takeover times than high values of ρm . For
ρm = 1, i.e., the classifiers cl1 and cl2 are equally general, Equation 18 is equivalent to Equation 8.
12
14. Similarly, we extend the model of takeover time for tournament selection taking classifier gen-
erality into account. In this case, the numerosity n1,t of cl 1 at time t is,
n1,t + 1 ρm 1 − 1 − n1 s ·
n
· 1 − n1 + (1 − ρm ) 1 − n1
n n
n1,t+1 =
n1,t − 1 ρm 1 − n1 s n1
n n
n otherwise.
1,t
Grouping these equations, we obtain:
n1 s n1 n1 n1 s n1
nt+1 = nt + ρm 1 − 1 − 1− + (1 − ρm ) 1 − − ρm 1 − , (19)
n n n n n
As we did before, we can group these equations and we can rewrite n 1,t+1 in terms of Pt as,
1
(1 − Pt ) 1 − ρm (1 − Pt )s−1
Pt+1 = Pt +
n
dp
which, by assuming ≈ Pt+1 − Pt :
dt
dp 1
(1 − Pt )[1 − ρm (1 − Pt )s−1 ]
≈ Pt+1 − Pt = (20)
dt n
dt 1
= dp (21)
(1 − Pt )[1 − ρm (1 − Pt )s−1 ]
n
ρm (1 − Pt )s−2
dt 1
= dp + dp (22)
1 − ρm (1 − Pt )s−1
n 1 − Pt
Integrating each side of the expression we obtain:
P P
ρm (1 − Pt )s−2
dt 1
= dp + dp (23)
1 − ρm (1 − Pt )s−1
n 1 − Pt
P0 P0
1 − ρm (1 − P )s−1
t 1 − P0 1
= ln + ln (24)
1 − ρm (1 − P0 )s−1
n 1−P s−1
Thus, the takeover time of the classifier cl1 in tournament selection is:
1 − ρm (1 − P )s−1
1 − P0 1
t∗ S = n ln + ln (25)
T
1 − ρm (1 − P0 )s−1
1−P s−1
Takeover time for tournament selection now depends on the tournament size s, the initial proportion
of the best classifier P0 , and also on the generality of the inaccurate classifier ρm . For low ρm or
high s, the value of the second logarithm in the right term of the equation diminishes so that
the takeover time mainly depends logarithmically on P0 . High values of ρm imply slower takeover
times; for ρm = 1, this model equates to Equation 11.
We validated the new models of takeover time for roulette wheel selection and tournament
selection on the single-niche problem for a matching ratio ρm of 0.1 and 0.9. The experimental
design is essentially the same one used in the previous experiments (Section 6), but in this case
the less accurate and less general classifier cl 2 appears in the niche with probability ρm . Figure 5
compares the proportion of the best classifier cl 1 in the niche as predicted by the theory and
empirically determined. Both the plots for roulette wheel selection (Figure 5) and tournament
selection (Figure 6) show a perfect match between the theory (reported as lines) and the empirical
data (reported as dots).
13
15. Takeover time for ρ = 0.10 Takeover time for ρ = 0.90
m m
1 1
0.9 0.9
0.8 0.8
Proportion of the Best Classifier
Proportion of the Best Classifier
0.7 0.7
Empirical RWS ρ=0.10
0.6 0.6
Model RWS ρ=0.10
Empirical RWS ρ=0.30
0.5 0.5
Model RWS ρ=0.30
Empirical RWS ρ=0.10
Empirical RWS ρ=0.50
Model RWS ρ=0.10
0.4 0.4
Model RWS ρ=0.50
Empirical RWS ρ=0.30
Empirical RWS ρ=0.70
Model RWS ρ=0.30
Model RWS ρ=0.70
0.3 0.3
Empirical RWS ρ=0.50
Empirical RWS ρ=0.90
Model RWS ρ=0.50
Model RWS ρ=0.90
0.2 0.2
Empirical RWS ρ=0.70
Model RWS ρ=0.70
Empirical RWS ρ=0.90
0.1 0.1
Model RWS ρ=0.90
0 0
0 1000 2000 3000 4000 5000 6000 7000 8000 0 0.5 1 1.5 2 2.5 3
Activation Time of the Niche Activation Time of the Niche 4
x 10
(a) (b)
Figure 5: Takeover time for roulette wheel selection when (a) ρm is 0.1 and (b) ρm is 0.9.
8 Conclusions
We have derived theoretical models for the selection pressure in XCS under roulette wheel and tour-
nament selection. We have shown that our models are accurate in very simplified scenarios, when
the models’ assumptions hold, and they can qualitatively explain the behavior of the two selection
mechanisms in more complex scenarios, when the models’ assumptions do not hold. Overall, our
models confirm what empirically shown in (Butz, Sastry, & Goldberg, 2005): tournament selection
is more robust than roulette wheel selection. Under perfect conditions both schemes perform sim-
ilarly, which is coherent to what shown in (Kharbat, Bull, & Odeh, 2005). However, the selection
pressure is weaker in roulette wheel selection when the classifiers in the niche have similar accura-
cies. On the other hand, tournament selection turns out to be more robust both theoretically and
practically, since it does not depend on the individual fitness of each classifier.
Acknowledgements
This work was sponsored by Enginyeria i Arquitectura La Salle, Ramon Llull University, as well
as the support of Ministerio de Ciencia y Tecnolog´ under project TIN2005-08386-C05-04, and
ıa
Generalitat de Catalunya under Grants 2005FI-00252 and 2005SGR-00302.
This work was also sponsored by the Air Force Office of Scientific Research, Air Force Materiel
Command, USAF, under grant FA9550-06-1-0096, the National Science Foundation under grant
DMR-03-25939 at the Materials Computation Center. The U.S. Government is authorized to
reproduce and distribute reprints for government purposes notwithstanding any copyright notation
thereon. The views and conclusions contained herein are those of the authors and should not
be interpreted as necessarily representing the official policies or endorsements, either expressed or
implied, of the Air Force Office of Scientific Research, the National Science Foundation, or the U.S.
Government.
14
16. Takeover time for ρ = 0.10 Takeover time for ρ = 0.90
m m
1 1
0.9 0.9
0.8 0.8
Proportion of the Best Classifier
Proportion of the Best Classifier
0.7 0.7
0.6 0.6
0.5 0.5
0.4 0.4
Empirical TS s=2 Empirical TS s=2
Model TS s=2 Model TS s=2
Empirical TS s=3 Empirical TS s=3
0.3 0.3
Model TS s=3 Model TS s=3
Empirical TS s=10 Empirical TS s=10
0.2 0.2
Model TS s=10 Model TS s=10
Empirical TS s=70 Empirical TS s=70
Model TS s=70 Model TS s=70
0.1 0.1
0 0
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Activation Time of the Niche Activation Time of the Niche
(b)
Figure 6: Takeover time for tournament selection when (a) ρm is 0.1 and (b) ρm is 0.9.
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A Experimental Validation on the 16-Niche Problem
Finally, we aim at analyzing if the agreement between the theory and the empirical observations
degrade if the number of overlapping niches increases. For this purpose, we ran XCS on the
multiple-niche problem fixing the number of niches to m=16. That is, the problem consisted of 16
niches, one maximum accurate classifier per niche, and one less accurate, overlapping classifier that
participates in all the 16 niches. Figure 7(a) and 7(b) compare the proportion of the best classifier
in the niche for roulette wheel selection for ρ={0.01,0.20,0.30,0.40}. The plots show that, for small
values of the accuracy ratio ρ, the empirical takeover time (reported as dots) is slightly slower than
the one predicted by our model (reported as lines). For higher values of ρ, the difference between the
model and the empirical results is more pronounced. So, we find a similar behavior to the multiple-
niche problem with two niches. But now, increasing the number of niches also delays the takeover
time, and the agreement between the model and the empirical results degrades faster as ρ increases.
This difference can be easily explained. As m increases, the ovegeneral classifier participates in a
higher proportion of action sets than any of the m best classifiers. Thus, the overgeneral classifier
will receive an increasing proportion of reproductive events with respect to the best classifiers as
m increases. Lower values of ρ produce a strong pressure toward accurate classifiers, and so, even
increasing the number of niches, the model nicely approximates the empirical results.
Figure 8(a) and 8(b) compare the proportion of the best classifier in the niche for tournament
selection for s={2,3,9}. For s=9, the theory accurately predics the empirical results. Increasing
the tournament size produces very little variations. For s=3, theory and practical results slightly
differ at the beginning of the run, but this difference disappears as the number of activations
increases. For s=2, the difference is large; that small tournament size combined with the presence
of the overgeneral classifier in all niches produces a strong selection pressure toward the overgeneral
classifier. As happened in roulette wheel selection, augmenting the number of overlapping niches
increases the selection pressure toward the overgeneral classifier, since the proportion of the action
sets that the overgeneral participates in with respect to the best classifier increases with the number
of niches. For higher values of s, i.e., s=9, the strong selection pressure toward accurate classifiers
can counterbalance this effect.
16
18. Takeover time in RWS for 16 Niches with Overlapping Classifiers and ρ={0.01,0.20} Takeover time in RWS for 16 Niches with Overlapping Classifiers and ρ={0.30,0.40,0.50}
1 1
0.9 0.9
0.8 0.8
Proportion of the Best Classifier
Proportion of the Best Classifier
Empirical RWS ρ=0.01 Empirical RWS ρ=0.30
0.7 0.7
Model RWS ρ=0.01 Model RWS ρ=0.30
Empirical RWS ρ=0.20 Empirical RWS ρ=0.40
0.6 0.6
Model RWS ρ=0.20 Model RWS ρ=0.40
0.5 0.5
0.4 0.4
0.3 0.3
0.2 0.2
0.1 0.1
0 0
0 2000 4000 6000 8000 10000 12000 14000 16000 0 0.5 1 1.5 2 2.5 3 3.5 4
Activation Time of the Niche Activation Time of the Niche 4
x 10
(a) (b)
Figure 7: Takeover time for roulette wheel selection when (a) ρ = {0.01, 0.20} and (b) ρ ∈
{0.30, 0.40} in the 16-niche problem.
Takeover time in TS s=9 for 16 Niches with Overlapping Classifiers Takeover time in TS s={2,3} for 16 Niches with Overlapping Classifiers
1 1
0.9 0.9
0.8 0.8
Empirical TS s=2
Proportion of the Best Classifier
Proportion of the Best Classifier
Model TS s=2
Empirical TS s=9
0.7 0.7
Empirical TS s=3
Model TS s=9
Model TS s=3
0.6 0.6
0.5 0.5
0.4 0.4
0.3 0.3
0.2 0.2
0.1 0.1
0 0
0 2000 4000 6000 8000 10000 12000 14000 16000 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Activation Time of the Niche Activation Time of the Niche 4
x 10
(a) (b)
Figure 8: Takeover time for tournament selection when (a) s=9 and (b) s={2,3}
17