multicriteria sorting using a valued indifference relation under a preference disaggregation paradigm abstract most multicriteria decision methods need the definition of a significant amount of preferential information from a decision agent the preference disaggregation analysis paradigm infers the model s parameter values from holistic judgments provided by a decision agent here a new method for inferring the parameters of a fuzzy outranking model for multicriteria sorting is proposed this approach allows us to use most of the preferential information contained in a reference set the central idea is to characterize the quality of the model by measuring discrepancies and concordances amongst i the preference relations derived from the outranking model and ii the preferential information contained in the reference set the model s parameters are inferred from a multiobjective optimization problem according to some additional preferential information from a decision agent once the model has been fitted sorting decisions about new objects are performed by using a fuzzy indifference relation this proposal performs very well in some examples

nonparametric modeling of neural point processes via stochastic gradient boosting regression statistical nonparametric modeling tools that enable the discovery and approximation of functional forms e g tuning functions relating neural spiking activity to relevant covariates are desirable tools in neuroscience in this article we show how stochastic gradient boosting regression can be successfully extended to the modeling of spiking activity data while preserving their point process nature thus providing a robust nonparametric modeling tool we formulate stochastic gradient boosting in terms of approximating the conditional intensity function of a point process in discrete time and use the standard likelihood of the process to derive the loss function for the approximation problem to illustrate the approach we apply the algorithm to the modeling of primary motor and parietal spiking activity as a function of spiking history and kinematics during a two dimensional reaching task model selection goodness of fit via the time rescaling theorem model interpretation via partial dependence plots ranking of covariates according to their relative importance and prediction of peri event time histograms are illustrated and discussed additionally we use the tenfold cross validated log likelihood of the modeled neural processes 67 cells to compare the performance of gradient boosting regression to two alternative approaches standard generalized linear models glms and bayesian p splines with markov chain monte carlo mcmc sampling in our data set gradient boosting outperformed both bayesian p splines in approximately 90 of the cells and glms 100 because of its good performance and computational efficiency we propose stochastic gradient boosting regression as an off the shelf nonparametric tool for initial analyses of large neural data sets e g more than 50 cells more than 105 samples per cell with corresponding multidimensional covariate spaces e g more than four covariates in the cases where a functional form might be amenable to a more compact representation gradient boosting might also lead to the discovery of simpler parametric models

a geometric approach to leveraging weak learners adaboost is a popular and effective leveraging procedure for improving the hypotheses generated by weak learning algorithms adaboost and many other leveraging algorithms can be viewed as performing a constrained gradient descent over a potential function at each iteration the distribution over the sample given to the weak learner is proportional to the direction of steepest descent we introduce a new leveraging algorithm based on a natural potential function for this potential function the direction of steepest descent can have negative components therefore we provide two techniques for obtaining suitable distributions from these directions of steepest descent the resulting algorithms have bounds that are incomparable to adaboost s the analysis suggests that our algorithm is likely to perform better than adaboost on noisy data and with weak learners returning low confidence hypotheses modest experiments confirm that our algorithm can perform better than adaboost in these situations

theories of causal ordering abstract this paper is a response to iwasaki and simon 14 which criticizes de kleer and brown 8 we argue that many of their criticisms particularly concerning causality modeling and stability originate from the difference of concerns between engineering and economics our notion of causality arises from considering the interconnections of components not equations when no feedback is present the ordering produced by our qualitative physics is similar to theirs however when feedback is present our qualitative physics determines a causal ordering around feedback loops as well causal ordering is a general technique not only applicable to qualitative reasoning therefore we also explore the relationship between causal ordering and propagation of constraints upon which the methods of qualitative physics are based

efficient margin maximizing with boosting adaboost produces a linear combination of base hypotheses and predicts with the sign of this linear combination the linear combination may be viewed as a hyperplane in feature space where the base hypotheses form the features it has been observed that the generalization error of the algorithm continues to improve even after all examples are on the correct side of the current hyperplane the improvement is attributed to the experimental observation that the distances margins of the examples to the separating hyperplane are increasing even after all examples are on the correct side we introduce a new version of adaboost called adaboost that explicitly maximizes the minimum margin of the examples up to a given precision the algorithm incorporates a current estimate of the achievable margin into its calculation of the linear coefficients of the base hypotheses the bound on the number of iterations needed by the new algorithms is the same as the number needed by a known version of adaboost that must have an explicit estimate of the achievable margin as a parameter we also illustrate experimentally that our algorithm requires considerably fewer iterations than other algorithms that aim to maximize the margin

the composition and validation of heterogeneous control laws we present a method for creating and validating a nonlinear controller by the composition of heterogeneous local control laws appropriate to different operating regions like fuzzy logic control these methods apply even in the presence of incomplete knowledge of the structure of the system the boundaries of the operating regions or even the control action to take unlike fuzzy logic control these methods can be analyzed by a combination of classical and qualitative methods each operating region of the system has a classical control law which provides high resolution control and can be analyzed by classical methods operating regions are defined by fuzzy set membership functions the global control law is the weighted average of the local control laws where the weights are provided by the operating region membership functions a heterogeneous control law can be analyzed even in the presence of incomplete knowledge by representing it as a qualitative differential equation and using qualitative simulation to predict the set of possible behaviors of the system by expressing the desired guarantee as a statement in a modal temporal logic the validity of the guarantee can be automatically checked against the set of possible behaviors we demonstrate heterogeneous controllers and our qualitative methods for proving their properties first for a simple level controller for a water tank and second for a highly nonlinear chemical reactor

discontinuity induced bifurcations of equilibria in piecewise smooth and impacting dynamical systems a rich variety of dynamical scenarios has been shown to occur when a fixed point of a non smooth map undergoes a border collision this paper concerns a closely related class of discontinuity induced bifurcations those involving equilibria of n dimensional piecewise smooth flows specifically transitions are studied which occur when a boundary equilibrium that is one lying within a discontinuity manifold is perturbed it is shown that such equilibria can either persist under parameter variations or can disappear giving rise to different bifurcation scenarios conditions to classify among the possible simplest scenarios are given for piecewise smooth continuous filippov and impacting systems also we investigate the possible birth of other attractors e g limit cycles at a boundary equilibrium bifurcation

a multiple criteria sorting method where each category is characterized by several reference actions the electre tri nc method this paper presents electre tri nc a new sorting method which takes into account several reference actions for characterizing each category this new method gives a particular freedom to the decision maker in the co construction decision aiding process with the analyst to characterize the set of categories while there is no constraint for introducing only one reference action as typical of each category like in electre tri c almeida dias et al 2010 as in such a sorting method this new sorting method is composed of two joint rules electre tri nc also fulfills a certain number of natural requirements additional results on the behavior of the new method are also provided in this paper namely the ones with respect to the addition or removal of the reference actions used for characterizing a certain category a numerical example illustrates the manner in which electre tri nc can be used by a decision maker a comparison with some related sorting procedures is presented and it allows to conclude that the new method is appropriate to deal with sorting problems

a boosting approach to flexible semiparametric mixed models in linear mixed models the influence of covariates is restricted to a strictly parametric form with the rise of semi and non parametric regression also the mixed model has been expanded to allow for additive predictors the common approach uses the representation of additive models as mixed models an alternative approach that is proposed in the present paper is likelihood based boosting boosting originates in the machine learning community where it has been proposed as a technique to improve classification procedures by combining estimates with reweighted observations likelihood based boosting is a general method which may be seen as an extension of l2 boost in additive mixed models the advantage of boosting techniques in the form of componentwise boosting is that it is suitable for high dimensional settings where many explanatory variables are present it allows to fit additive models for many covariates with implicit selection of relevant variables and automatic selection of smoothing parameters moreover boosting techniques may be used to incorporate the subject specific variation of smooth influence functions by specifying random slopes on smooth effects this results in flexible semiparametric mixed models which are appropriate in cases where a simple random intercept is unable to capture the variation of effects across subjects

probabilistic classification vector machines in this paper a sparse learning algorithm probabilistic classification vector machines pcvms is proposed we analyze relevance vector machines rvms for classification problems and observe that adopting the same prior for different classes may lead to unstable solutions in order to tackle this problem a signed and truncated gaussian prior is adopted over every weight in pcvms where the sign of prior is determined by the class label i e 1 or 1 the truncated gaussian prior not only restricts the sign of weights but also leads to a sparse estimation of weight vectors and thus controls the complexity of the model in pcvms the kernel parameters can be optimized simultaneously within the training algorithm the performance of pcvms is extensively evaluated on four synthetic data sets and 13 benchmark data sets using three performance metrics error rate err area under the curve of receiver operating characteristic auc and root mean squared error rmse we compare pcvms with soft margin support vector machines svmsoft hard margin support vector machines svmhard svm with the kernel parameters optimized by pcvms svmpcvm relevance vector machines rvms and some other baseline classifiers through five replications of twofold cross validation f test i e 5 times 2 cross validation f test over single data sets and friedman test with the corresponding post hoc test to compare these algorithms over multiple data sets we notice that pcvms outperform other algorithms including svmsoft svmhard rvm and svmpcvm on most of the data sets under the three metrics especially under auc our results also reveal that the performance of svmpcvm is slightly better than svmsoft implying that the parameter optimization algorithm in pcvms is better than cross validation in terms of performance and computational complexity in this paper we also discuss the superiority of pcvms formulation using maximum a posteriori map analysis and margin analysis which explain the empirical success of pcvms

multicriteria assignment method proaftn methodology and medical application abstract this paper presents a new fuzzy multicriteria classification method called proaftn for assigning alternatives to predefined categories this method belongs to the class of supervised learning algorithms and enables to determine the fuzzy indifference relations by generalising the indices concordance and discordance used in the electre iii method then it assigns the fuzzy belonging degree of the alternatives to the categories we also present a clinical application of the proposed method in the cytopathological diagnosis of acute leukaemia

boosting with the l2 loss this article investigates a computationally simple variant of boosting l2boost which is constructed from a functional gradient descent algorithm with the l2 loss function like other boosting algorithms l2boost uses many times in an iterative fashion a prechosen fitting method called the learner based on the explicit expression of refitting of residuals of l2boost the case with symmetric linear learners is studied in detail in both regression and classification in particular with the boosting iteration m working as the smoothing or regularization parameter a new exponential bias variance trade off is found with the variance complexity term increasing very slowly as m tends to infinity when the learner is a smoothing spline an optimal rate of convergence result holds for both regression and classification and the boosted smoothing spline even adapts to higher order unknown smoothness moreover a simple expansion of a smoothed 0 1 loss function is derived to reveal the importance of the d

analysis of boosting algorithms using the smooth margin function we introduce a useful tool for analyzing boosting algorithms called the smooth margin function a differentiable approximation of the usual margin for boosting algorithms we present two boosting algorithms based on this smooth margin coordinate ascent boosting and approximate coordinate ascent boosting which are similar to freund and schapire s adaboost algorithm and breiman s arc gv algorithm we give convergence rates to the maximum margin solution for both of our algorithms and for arc gv we then study adaboost s convergence properties using the smooth margin function we precisely bound the margin attained by adaboost when the edges of the weak classifiers fall within a specified range this shows that a previous bound proved by ratsch and warmuth is exactly tight furthermore we use the smooth margin to capture explicit properties of adaboost in cases where cyclic behavior occurs

rough set approach to multi attribute decision analysis abstract we review the methodology of the rough set analysis of multi attribute decision problems rough set theory proved to be a useful tool for analysis of a vague description of decision situations it answers two basic questions related to multi attribute decision problems one about explanation of a decision situation and another about prescription of some decisions basing on analysis of a decision situation we define four classes of multi attribute decision problems depending on the structure of their representation its interpretation and the kind of questions related then we characterize the rough set methodology for each particular class of decision problems we use simple practical examples to illustrate this presentation a review of related literature is made throughout the paper

global qualitative description of a class of nonlinear dynamical systems in this paper we propose a methodology to derive a qualitative description of the behavior of a system from an incompletely known nonlinear dynamical model the model is written as an algebraic structure with unknown parameters and or functions under some hypotheses we obtain a graph describing the possible transitions between regions defined by the trends of the state variables and their relative positions a qualitative simulation of the model can be compared with on line data for fault detection purpose we give the example of a nonlinear biological model in dimension three for the growth of cells in a bioreactor

a new approach to multi criteria sorting based on fuzzy outranking relations the theseus method abstract in this paper we propose the theseus method a new approach based on fuzzy outranking relations to multi criteria sorting problems compared with other outranking based methods theseus is inspired by another view of multi criteria classification problems it utilizes a new way of evaluating the assignment of an object to an element of the set of ordered categories that were previously defined this way is based on comparing every possible assignment with the information from various preference relations that are derived from a fuzzy outranking relation defined on the universe of objects the appropriate assignment is determined by solving a simple selection problem the capacity of a reference set for making appropriate assignments is related to a good characterization of the categories a single reference action characterizing a category may be insufficient to achieve well determined assignments in this paper the reference set capacity to perform appropriate assignments is characterized by some new concepts this capacity may be increased when more objects are added to the reference set theseus is a method for handling the preference information contained in such larger reference sets

on the doubt about margin explanation of boosting margin theory provides one of the most popular explanations to the success of adaboost where the central point lies in the recognition that margin is the key for characterizing the performance of adaboost this theory has been very influential e g it has been used to argue that adaboost usually does not overfit since it tends to enlarge the margin even after the training error reaches zero previously the minimum margin bound was established for adaboost however breiman 1999 9 pointed out that maximizing the minimum margin does not necessarily lead to a better generalization later reyzin and schapire 2006 37 emphasized that the margin distribution rather than minimum margin is crucial to the performance of adaboost in this paper we first present the kth margin bound and further study on its relationship to previous work such as the minimum margin bound and emargin bound then we improve the previous empirical bernstein bounds audibert et al 2009 maurer and pontil 2009 2 30 and based on such findings we defend the margin based explanation against breiman s doubts by proving a new generalization error bound that considers exactly the same factors as schapire et al 1998 39 but is sharper than breiman s 1999 9 minimum margin bound by incorporating factors such as average margin and variance we present a generalization error bound that is heavily related to the whole margin distribution we also provide margin distribution bounds for generalization error of voting classifiers in finite vc dimension space

a primal dual convergence analysis of boosting boosting combines weak learners into a predictor with low empirical risk its dual constructs a high entropy distribution upon which weak learners and training labels are uncorrelated this manuscript studies this primal dual relationship under a broad family of losses including the exponential loss of adaboost and the logistic loss revealing weak learnability aids the whole loss family for any e 0 o ln 1 e iterations suffice to produce a predictor with empirical risk e close to the infimum the circumstances granting the existence of an empirical risk minimizer may be characterized in terms of the primal and dual problems yielding a new proof of the known rate o ln 1 e arbitrary instances may be decomposed into the above two granting rate o 1 e with a matching lower bound provided for the logistic loss

on the bayes risk consistency of regularized boosting methods the probability of error of classification methods based on convex combinations of simple base classifiers by boosting algorithms is investigated the main result of the paper is that certain regularized boosting algorithms provide bayes risk consistent classifiers under the sole assumption that the bayes classifier may be approximated by a convex combination of the base classifiers nonasymptotic distribution free bounds are also developed which offer interesting new insight into how boosting works and help explain its success in practical classification problems

statistical behavior and consistency of classification methods based on convex risk minimization we study how closely the optimal bayes error rate can be approximately reached using a classification algorithm that computes a classifier by minimizing a convex upper bound of the classification error function the measurement of closeness is characterized by the loss function used in the estimation we show that such a classification scheme can be generally regarded as a nonmaximum likelihood conditional in class probability estimate and we use this analysis to compare various convex loss functions that have appeared in the literature furthermore the theoretical insight allows us to design good loss functions with desirable properties another aspect of our analysis is to demonstrate the consistency of certain classification methods using convex risk minimization this study sheds light on the good performance of some recently proposed linear classification methods including boosting and support vector machines it also shows their limitations and suggests possible improvements

convexity classification and risk bounds many of the classification algorithms developed in the machine learning literature including the support vector machine and boosting can be viewed as minimum contrast methods that minimize a convex surrogate of the 0 1 loss function the convexity makes these algorithms computationally efficient the use of a surrogate however has statistical consequences that must be balanced against the computational virtues of convexity to study these issues we provide a general quantitative relationship between the risk as assessed using the 0 1 loss and the risk as assessed using any nonnegative surrogate loss function we show that this relationship gives nontrivial upper bounds on excess risk under the weakest possible condition on the loss function that it satisfies a pointwise form of fisher consistency for classification the relationship is based on a simple variational transformation of the loss function that is easy to compute in many applications we also present a refined version of this result in the

the rate of convergence of adaboost the adaboost algorithm was designed to combine many weak hypotheses that perform slightly better than random guessing into a strong hypothesis that has very low error we study the rate at which adaboost iteratively converges to the minimum of the exponential loss unlike previous work our proofs do not require a weak learning assumption nor do they require that minimizers of the exponential loss are finite our first result shows that the exponential loss of adaboost s computed parameter vector will be at most e more than that of any parameter vector of l1 norm bounded by b in a number of rounds that is at most a polynomial in b and 1 e we also provide lower bounds showing that a polynomial dependence is necessary our second result is that within c e iterations adaboost achieves a value of the exponential loss that is at most e more than the best possible value where c depends on the data set we show that this dependence of the rate on e is optimal up to constant factors that is at least 1 e rounds are necessary to achieve within e of the optimal exponential loss

combinations of weak classifiers to obtain classification systems with both good generalization performance and efficiency in space and time we propose a learning method based on combinations of weak classifiers where weak classifiers are linear classifiers perceptrons which can do a little better than making random guesses a randomized algorithm is proposed to find the weak classifiers they are then combined through a majority vote as demonstrated through systematic experiments the method developed is able to obtain combinations of weak classifiers with good generalization performance and a fast training time on a variety of test problems and real applications theoretical analysis on one of the test problems investigated in our experiments provides insights on when and why the proposed method works in particular when the strength of weak classifiers is properly chosen combinations of weak classifiers can achieve a good generalization performance with polynomial space and time complexity

the adaboost flow we introduce a dynamical system that we call the adaboost flow the flow is defined by a system of odes with control we show that three algorithms of the adaboost family i the adaboost algorithm of schapire and freund ii the arc gv algorithm of breiman and iii the confidence rated prediction of schapire and singer can be embedded in the adaboost flow the nontrivial part of the adaboost flow equations coincides with the equations of dynamics of the nonperiodic toda system written in terms of spectral variables we provide a novel invariant geometrical description of the adaboost algorithm as a gradient flow on a foliation defined by level sets of the potential function we propose a new approach for constructing boosting algorithms as a continuous time gradient flow on measures defined by various metrics and potential functions finally we explain the similarity of the adaboost algorithm with the perelman s construction for the ricci flow 2015 wiley periodicals inc

solving stiff differential equations with the method of patches we introduce a new method for solving very stiff sets of ordinary differential equations the basic idea is to replace the original nonlinear equations with a set of equally stiff equations that are piecewise linear and therefore can be solved exactly we demonstrate the value of the method on small systems of equations for which some other methods are inefficient or produce spurious solutions estimate error bounds and discuss extensions of the method to larger systems of equations and to partial differential equations

birth and death of the double scroll abstract the double scroll is a chaotic attractor observed from an extremely simple electronic circuit made of 4 linear circuit elements 2 capacitors 1 inductor and 1 resistor and a 2 terminal nonlinear resistor characterized by a 3 segment piecewise linear v i characteristic this nonlinear resistor can be built in the laboratory using 2 transistors and diodes the qualitative behavior of the double scroll attractor has been described elsewhere where each cross section of the double scroll attractor is shown to consist of 2 tightly wound spirals our objective in this paper is to present a bifurcation analysis of this circuit in terms of 2 basic parameters various bifurcation phenomena such as hopf bifurcation period doubling cascades rossler s spiral and screw type attractors periodic windows shilnikov phenomenon double scroll and boundary crisis are described in particular special emphasis is given to the mechanisms leading to the birth and the death of the double scroll attractor all bifurcation phenomena described in this paper are observed not only experimentally but also confirmed by digital computer simulations moreover an analytical foundation of the double scroll family of equations has been developed and used to explain in a rigorous way all observed bifurcation phenomena in addition rossler s spiral type and screw type attractors have been observed from the same circuit where the nonlinear resistor has only one break point i e it is described by a 2 segment piecewise linear v i characteristic this means that extremely complicated non periodic chaotic waveforms can arise in the simplest third order uncoupled electrical circuit in which all elements except one a resistor are linear and passive and in which the constitutive relation of the nonlinear resistor is made of the simplest conceivable nonlinearity namely 2 straight line segments

complexities of convex combinations and bounding the generalization error in classification we introduce and study several measures of complexity of functions from the convex hull of a given base class these complexity measures take into account the sparsity of the weights of a convex combination as well as certain clustering properties of the base functions involved in it we prove new upper confidence bounds on the generalization error of ensemble voting classification algorithms that utilize the new complexity measures along with the empirical distributions of classification margins providing a better explanation of generalization performance of large margin classification methods

estimation of a regression function by maxima of minima of linear functions in this paper estimation of a regression function from independent and identically distributed random variables is considered estimates are defined by minimization of the empirical l2 risk over a class of functions which are defined as maxima of minima of linear functions results concerning the rate of convergence of the estimates are derived in particular it is shown that for smooth regression functions satisfying the assumption of single index models the estimate is able to achieve up to some logarithmic factor the corresponding optimal one dimensional rate of convergence hence under these assumptions the estimate is able to circumvent the so called curse of dimensionality the small sample behavior of the estimates is illustrated by applying them to simulated data

generalized additive modeling with implicit variable selection by likelihood based boosting the use of generalized additive models in statistical data analysis suffers from the restriction to few explanatory variables and the problems of selection of smoothing parameters generalized additive model boosting circumvents these problems by means of stagewise fitting of weak learners a fitting procedure is derived which works for all simple exponential family distributions including binomial poisson and normal response variables the procedure combines the selection of variables and the determination of the appropriate amount of smoothing penalized regression splines and the newly introduced penalized stumps are considered as weak learners estimates of standard deviations and stopping criteria which are notorious problems in iterative procedures are based on an approximate hat matrix the method is shown to be a strong competitor to common procedures for the fitting of generalized additive models in particular in high dimensional settings with many nuisance predictor variables it performs very well

homoclinic hyperchaos an explicit example abstract a four variable 2 domain piecewise linear c 1 ordinary differential equation is indicated explicitly two parameter sets and initial conditions implying a homoclinic return to a generalized saddle focus are given with finite precision the numerically larger eigenvalue is negative the two positive eigenvalues one for the complex conjugate pair the other real are equal only under this singular condition does the extended neighborhood in parameter space in which topological hyperchaos survives include the homoclinic case itself this does not diminish the power of the homoclinicity criterion the second set of parameters and initial conditions is valid for arbitrarily high dimension numbers it implies existence of maximum hyperchaos in an extended neighborhood analytical and numerical studies on both the c 1 and the c version of the equation are possible

supervised projection approach for boosting classifiers in this paper we present a new approach for boosting methods for the construction of ensembles of classifiers the approach is based on using the distribution given by the weighting scheme of boosting to construct a non linear supervised projection of the original variables instead of using the weights of the instances to train the next classifier with this method we construct ensembles that are able to achieve a better generalization error and are more robust to noise presence it has been proved that adaboost method is able to improve the margin of the instances achieved by the ensemble moreover its practical success has been partially explained by this margin maximization property however in noisy problems likely to occur in real world applications the maximization of the margin of wrong instances or outliers can lead to poor generalization we propose an alternative approach where the distribution of the weights given by the boosting algorithm is used to get a supervised projection then the supervised projection is used to train the next classifier using a uniform distribution of the training instances the proposed approach is compared with three boosting techniques namely adaboost gentleboost and madaboost showing an improved performance on a large set of 55 problems from the uci machine learning repository and less sensitiveness to noise in the class labels the behavior of the proposed algorithm in terms of margin distribution and bias variance decomposition is also studied

causality in device behavior abstract this paper shows how formal characterizations of causality and of the method of comparative statics long used in econometrics thermodynamics and other domains can be applied to clarify and make rigorous the qualitative causal calculus recently proposed by de kleer and brown 2 the formalization shows exactly what assumptions are required to carry out causal analysis of a system of interdependent variables in equilibrium and to propagate disturbances through such a system the intuitive concepts of causality captured by de kleer and brown provide a rough approximation to the standard analytic techniques that are used in the treatment of simultaneous algebraic and differential equations

a dynamic systems perspective on qualitative simulation abstract this paper presents a simple geometric interpretation of qualitative simulation qs based on the phase space representation of dynamic systems theory qs consists of two steps transition analysis determines the sequence of qualitative states that a system traverses and global interpretation derives its long term behavior i recast transition analysis as a search problem in phase space and replace the assorted transition rules with two algebraic conditions the first condition determines transitions between arbitrarily shaped regions in phase space as opposed to qs which only handles n dimensional rectangles it also provides more accurate results by considering only the boundaries between regions the second condition determines whether nearby trajectories approach a fixed point asymptotically it obtains better results than qs by exploiting local stability properties i recast global interpretation of dissipative systems as a search for attractors in phase space and present a global interpretation strategy for the subset of these systems whose local behavior determines global behavior uniquely although limited in scope this strategy handles many systems that appear in the qs literature

boosting as a regularized path to a maximum margin classifier in this paper we study boosting methods from a new perspective we build on recent work by efron et al to show that boosting approximately and in some cases exactly minimizes its loss criterion with an l1 constraint on the coefficient vector this helps understand the success of boosting with early stopping as regularized fitting of the loss criterion for the two most commonly used criteria exponential and binomial log likelihood we further show that as the constraint is relaxed or equivalently as the boosting iterations proceed the solution converges in the separable case to an l1 optimal separating hyper plane we prove that this l1 optimal separating hyper plane has the property of maximizing the minimal l1 margin of the training data as defined in the boosting literature an interesting fundamental similarity between boosting and kernel support vector machines emerges as both can be described as methods for regularized optimization in high dimensional predictor space using a computational trick to make the calculation practical and converging to margin maximizing solutions while this statement describes svms exactly it applies to boosting only approximately

a refined margin analysis for boosting algorithms via equilibrium margin much attention has been paid to the theoretical explanation of the empirical success of adaboost the most influential work is the margin theory which is essentially an upper bound for the generalization error of any voting classifier in terms of the margin distribution over the training data however important questions were raised about the margin explanation breiman 1999 proved a bound in terms of the minimum margin which is sharper than the margin distribution bound he argued that the minimum margin would be better in predicting the generalization error grove and schuurmans 1998 developed an algorithm called lp adaboost which maximizes the minimum margin while keeping all other factors the same as adaboost in experiments however lp adaboost usually performs worse than adaboost putting the margin explanation into serious doubt in this paper we make a refined analysis of the margin theory we prove a bound in terms of a new margin measure called the equilibrium margin emargin the emargin bound is uniformly sharper than breiman s minimum margin bound thus our result suggests that the minimum margin may be not crucial for the generalization error we also show that a large emargin and a small empirical error at emargin imply a smaller bound of the generalization error experimental results on benchmark data sets demonstrate that adaboost usually has a larger emargin and a smaller test error than lp adaboost which agrees well with our theory

boosting for high dimensional linear models we prove that boosting with the squared error loss l 2 boosting is consistent for very high dimensional linear models where the number of predictor variables is allowed to grow essentially as fast as o exp sample size assuming that the true underlying regression function is sparse in terms of the 11 norm of the regression coefficients in the language of signal processing this means consistency for de noising using a strongly overcomplete dictionary if the underlying signal is sparse in terms of the l 1 norm we also propose here an aic based method for tuning namely for choosing the number of boosting iterations this makes l 2 boosting computationally attractive since it is not required to run the algorithm multiple times for cross validation as commonly used so far we demonstrate l 2 boosting for simulated data in particular where the predictor dimension is large in comparison to sample size and for a difficult tumor classification problem with gene expression microarray data

electre tri nb a new multiple criteria ordinal classification method abstract this paper presents a new method for multiple criteria ordinal classification sorting problems in an ordinal classification problem the categories also called classes in related literature are pre defined and ordered from the best to the worst or from the worst to the best the actions not necessarily known a priori are assigned to the different ordered categories several electre type methods were designed to deal with this kind of problem however none of them proposes a characterization of the categories through a set of limiting profiles this is the novelty of the current method which may be considered as an extension of electre tri b it fulfills a set of structural requirements uniqueness of the assignments independence monotonicity homogeneity conformity and stability with respect to merging and splitting operations all these features will be presented in the current paper and illustrated through three examples

automatic analysis of one parameter planar ordinary differential equations by intelligent numeric simulation abstract this paper describes research on automating the analysis of physical systems modeled by ordinary differential equations it describes a program called poincare that analyzes one parameter autonomous planar systems at the level of experts through a combination of theoretical dynamics numerical simulation and geometric reasoning the input is the system a bounding box for the system state a bounding interval for the parameter and error tolerances poincare partitions the parameter interval into open subintervals of equivalent behavior bounded by bifurcation points classifies the bifurcation points and constructs representative phase diagrams for the subintervals it detects local and global generic one parameter bifurcations it constructs the phase diagrams by identifying fixed points saddle manifolds and limit cycles and partitioning the remaining trajectories into open regions of uniform asymptotic behavior it obtains the region boundaries by numeric simulation guided by theoretical knowledge it determines the behavior near fixed points from the jacobian of the system scans outward to find basins of attractors and limit cycles and stops at the bounding box

electre tri c a multiple criteria sorting method based on characteristic reference actions in this paper a new sorting method following a decision aiding constructive approach is proposed this method is called electre tri c as a sorting method a set of categories must be defined to represent the way in which the actions that are going to be assigned to each of them should further be processed this method is appropriate to deal with decision aiding contexts where the categories are completely ordered and each of them is defined through a single characteristic reference action the set of characteristic actions should be co constructed through an interactive process between the analyst and the decision maker electre tri c has been conceived to verify a set of natural structural requirements conformity homogeneity monotonicity and stability which can be viewed as its fundamental properties this method is composed of two joint rules called descending rule and ascending rule which must be used conjointly and not separately each one of these rules selects only one category or a range of possible categories for a possible assignment of an action this assignment depends on the comparison of such an action to the characteristic actions according to a chosen credibility level numerical examples are also presented in order to illustrate the main theoretical results provided by the method

local and global behavior near homoclinic orbits we study the local behavior of systems near homoclinic orbits to stationary points of saddle focus type we explicitly describe how a periodic orbit approaches homoclinicity and with the help of numerical examples discuss how these results relate to global patterns of bifurcations

multicriteria sorting using a valued preference closeness relation abstract to define a metric or a similarity relation with good properties is a crucial issue in most analogy and rule induction oriented methods for multicriteria sorting here we propose a valued indifference closeness relation which is inspired on concordance and discordance measures the criterion weights are obtained from the preferential information embedded in a reference set this proposal performs very well in several practical problems

boosted regression trees with errors in variables in this article we consider nonparametric regression when covariates are measured with error estimation is performed using boosted regression trees with the sum of the trees forming the estimate of the conditional expectation of the response both binary and continuous response regression are investigated an approach to fitting regression trees when covariates are measured with error is described and the boosting algorithms consist of its repeated application the main feature of the approach is that it handles situations where multiple covariates are measured with error some simulation results are given as well as its application to data from the framingham heart study

automatic qualitative analysis of dynamic systems using piecewise linear approximations abstract this paper explores automating the qualitative analysis of physical systems it describes a program called plr for piecewise linear reasoner that formalizes an analysis strategy employed by experts plr takes parameterized ordinary differential equations as input and produces a qualitative description of the solutions for all initial values it approximates intractable nonlinear systems with piecewise linear ones analyzes the approximations and draws conclusions about the original systems it chooses approximations that are accurate enough to reproduce the essential properties of their nonlinear prototypes yet simple enough to be analyzed completely and efficiently it derives additional properties such as boundedness or periodicity by theoretical methods plr s analysis depends on abstract properties of systems rather than on specific numeric values this makes its conclusions more robust and enables it to handle parameterized equations transparently i demonstrate plr on several common nonlinear systems and on published examples from mechanical engineering

on the dual formulation of boosting algorithms we study boosting algorithms from a new perspective we show that the lagrange dual problems of l1 norm regularized adaboost logitboost and soft margin lpboost with generalized hinge loss are all entropy maximization problems by looking at the dual problems of these boosting algorithms we show that the success of boosting algorithms can be understood in terms of maintaining a better margin distribution by maximizing margins and at the same time controlling the margin variance we also theoretically prove that approximately l1 norm regularized adaboost maximizes the average margin instead of the minimum margin the duality formulation also enables us to develop column generation based optimization algorithms which are totally corrective we show that they exhibit almost identical classification results to that of standard stagewise additive boosting algorithms but with much faster convergence rates therefore fewer weak classifiers are needed to build the ensemble using our proposed optimization technique

chaos in a three dimensional single loop feedback system with a piecewice linear feedbackk function abstract this paper describes chaotic behaviour in a single loop feedback system of low order the system is particularly simple and because it is piecewise linear and hence can be undersdtood globally a description of the flow much more complete than usual can be given an orbit structure is suggested for the system after a minor approximation is considered the approximation gives two equivalent discrete one dimensional maps of the interval to itself and these maps are discussed in the light of recent work the validity of the approximation is confirmed by computer simulation

multicriteria fuzzy classification procedure procftn methodology and medical application in this paper we introduce a new classification procedure for assigning objects to predefined classes named procftn this procedure is based on a fuzzy scoring function for choosing a subset of prototypes which represent the closest resemblance with an object to be assigned it then applies the majority voting rule to assign an object to a class we also present a medical application of this procedure as an aid to assist the diagnosis of central nervous system tumours the results are compared with those obtained by other classification methods reported on the same data set including decision tree production rules neural network k nearest neighbor multilayer perceptron and logistic regression our results are very encouraging and show that the multicriteria decision analysis approach can be successfully used to help medical diagnosis

the dynamics of adaboost cyclic behavior and convergence of margins in order to study the convergence properties of the adaboost algorithm we reduce adaboost to a nonlinear iterated map and study the evolution of its weight vectors this dynamical systems approach allows us to understand adaboost s convergence properties completely in certain cases for these cases we find stable cycles allowing us to explicitly solve for adaboost s output using this unusual technique we are able to show that adaboost does not always converge to a maximum margin combined classifier answering an open question in addition we show that non optimal adaboost where the weak learning algorithm does not necessarily choose the best weak classifier at each iteration may fail to converge to a maximum margin classifier even if optimal adaboost produces a maximum margin also we show that if adaboost cycles it cycles among support vectors i e examples that achieve the same smallest margin

rough sets methodology for sorting problems in presence of multiple attributes and criteria abstract we consider a sorting classification problem in the presence of multiple attributes and criteria called the mac however the domains of attributes are not preference ordered while the domains of criteria scales are totally ordered by preference relations among the attributes we distinguish between qualitative attributes and quantitative attributes in order to construct a comprehensive preference model that could be used to support the sorting task we consider preferential information of the decision maker dm in the form of assignment examples i e exemplary assignments of some reference actions to the decision classes the preference model inferred from these examples is a set of if then decision rules the rules are derived from rough approximations of decision classes made up of reference actions they satisfy conditions of completeness and dominance and manage with possible ambiguity inconsistencies in the set of examples our idea of rough approximations involves three relations together indiscernibility similarity and dominance defined on qualitative and quantitative attributes and on criteria respectively the usefulness of this approach is illustrated by an example

periodical solutions in a pulse modulated model of endocrine regulation with time delay a hybrid mathematical model of endocrine regulation obtained by augmenting the classical continuous smith model with a pulse modulated feedback to describe episodic pulsatile secretion is considered conditions for existence and local orbital stability of periodical solutions with m impulses in the least period m cycles are derived an important implication of the performed analysis is that the nonlinear dynamics of the pulse modulated system and not the delay itself cause the sustained closed loop oscillations furthermore simulation and bifurcation analysis indicate that increasing the time delay in the system in hand typically but not always leads to less complex dynamic pattern in the closed loop system by giving rise to stable cycles of lower periodicity

boosting with early stopping convergence and consistency boosting is one of the most significant advances in machine learning for classification and regression in its original and computationally flexible version boosting seeks to minimize empirically a loss function in a greedy fashion the resulting estimator takes an additive function form and is built iteratively by applying a base estimator or learner to updated samples depending on the previous iterations an unusual regularization technique early stopping is employed based on cv or a test set this paper studies numerical convergence consistency and statistical rates of convergence of boosting with early stopping when it is carried out over the linear span of a family of basis functions for general loss functions we prove the convergence of boosting s greedy optimization to the infinimum of the loss function over the linear span using the numerical convergence result we find early stopping strategies under which boosting is shown to be consistent based on i i d samples and we obtain bounds on the rates of convergence for boosting estimators simulation studies are also presented to illustrate the relevance of our theoretical results for providing insights to practical aspects of boosting as a side product these results also reveal the importance of restricting the greedy search step sizes as known in practice through the work of friedman and others moreover our results lead to a rigorous proof that for a linearly separable problem adaboost with e 0 step size becomes an l 1 margin maximizer when left to run to convergence

a note on the correctness of the causal ordering algorithm in this paper we examine in detail the algorithm of simon h a simon causal ordering and identifiability in w c hood t c koopmans eds studies in econometric method cowles commission for research in economics monograph no 14 john wiley sons inc new york 1953 pp 49 74 chapter iii called the causal ordering algorithm coa used for constructing the causal ordering of a system given a complete specification of the system in terms of a set of structural equations that govern the variables in the system this algorithm constructs a graphical characterization of the model in a form that we call a partial causal graph simon argued in h a simon causal ordering and identifiability in w c hood t c koopmans eds studies in econometric method cowles commission for research in economics monograph no 14 john wiley sons inc new york 1953 pp 49 74 chapter iii and subsequent papers that a graph so generated explicates causal structure among variables in the model we formalize this claim further by proving that any causal model based on a one to one correspondence between equations and variables must be consistent with the coa

semiring induced valuation algebras exact and approximate local computation algorithms local computation in join trees or acyclic hypertrees has been shown to be linked to a particular algebraic structure called valuation algebra there are many models of this algebraic structure ranging from probability theory to numerical analysis relational databases and various classical and non classical logics it turns out that many interesting models of valuation algebras may be derived from semiring valued mappings in this paper we study how valuation algebras are induced by semirings and how the structure of the valuation algebra is related to the algebraic structure of the semiring in particular c semirings with idempotent multiplication induce idempotent valuation algebras and therefore permit particularly efficient architectures for local computation also important are semirings whose multiplicative semigroup is embedded in a union of groups they induce valuation algebras with a partially defined division for these valuation algebras the well known architectures for bayesian networks apply we also extend the general computational framework to allow derivation of bounds and approximations for when exact computation is not feasible

dimensionality reduction via sparse support vector machines we describe a methodology for performing variable ranking and selection using support vector machines svms the method constructs a series of sparse linear svms to generate linear models that can generalize well and uses a subset of nonzero weighted variables found by the linear models to produce a final nonlinear model the method exploits the fact that a linear svm no kernels with l1 norm regularization inherently performs variable selection as a side effect of minimizing capacity of the svm model the distribution of the linear model weights provides a mechanism for ranking and interpreting the effects of variables starplots are used to visualize the magnitude and variance of the weights for each variable we illustrate the effectiveness of the methodology on synthetic data benchmark problems and challenging regression problems in drug design this method can dramatically reduce the number of variables and outperforms svms trained using all attributes and using the attributes selected according to correlation coefficients the visualization of the resulting models is useful for understanding the role of underlying variables

boosting iterative stochastic ensemble method for nonlinear calibration of subsurface flow models abstract a novel parameter estimation algorithm is proposed the inverse problem is formulated as a sequential data integration problem in which gaussian process regression gpr is used to integrate the prior knowledge static data the search space is further parameterized using karhunen loeve expansion to build a set of basis functions that spans the search space optimal weights of the reduced basis functions are estimated by an iterative stochastic ensemble method isem isem employs directional derivatives within a gauss newton iteration for efficient gradient estimation the resulting update equation relies on the inverse of the output covariance matrix which is rank deficient in the proposed algorithm we use an iterative regularization based on the l2 boosting algorithm l2 boosting iteratively fits the residual and the amount of regularization is controlled by the number of iterations a termination criteria based on akaike information criterion aic is utilized this regularization method is very attractive in terms of performance and simplicity of implementation the proposed algorithm combining isem and l2 boosting is evaluated on several nonlinear subsurface flow parameter estimation problems the efficiency of the proposed algorithm is demonstrated by the small size of utilized ensembles and in terms of error convergence rates

towards a practical theory of reformulation for reasoning about physical systems reformulation is ubiquitous in problem solving and is especially common in modeling physical systems in this paper we examine reformulation techniques in the context of reasoning about physical systems this paper does not present a general theory of reformulation but it studies a number of known reformulation techniques to achieve a broad understanding of the space of available reformulations in doing so we present a practical framework for specifying classifying and evaluating various reformulation techniques applicable to this class of problems our framework provides the terminology to specify the conditions under which a particular reformulation technique is applicable the cost associated with performing the reformulation and the effects of the reformulation with respect to the problem encoding

rough sets theory for multicriteria decision analysis the original rough set approach proved to be very useful in dealing with inconsistency problems following from information granulation it operates on a data table composed of a set u of objects actions described by a set q of attributes its basic notions are indiscernibility relation on u lower and upper approximation of either a subset or a partition of u dependence and reduction of attributes from q and decision rules derived from lower approximations and boundaries of subsets identified with decision classes the original rough set idea is failing however when preference orders of attribute domains criteria are to be taken into account precisely it cannot handle inconsistencies following from violation of the dominance principle this inconsistency is characteristic for preferential information used in multicriteria decision analysis mcda problems like sorting choice or ranking in order to deal with this kind of inconsistency a number of methodological changes to the original rough sets theory is necessary the main change is the substitution of the indiscernibility relation by a dominance relation which permits approximation of ordered sets in multicriteria sorting to approximate preference relations in multicriteria choice and ranking problems another change is necessary substitution of the data table by a pairwise comparison table where each row corresponds to a pair of objects described by binary relations on particular criteria in all those mcda problems the new rough set approach ends with a set of decision rules playing the role of a comprehensive preference model it is more general than the classical functional or relational model and it is more understandable for the users because of its natural syntax in order to workout a recommendation in one of the mcda problems we propose exploitation procedures of the set of decision rules finally some other recently obtained results are given rough approximations by means of similarity relations rough set handling of missing data comparison of the rough set model with sugeno and choquet integrals and results on equivalence of a decision rule preference model and a conjoint measurement model which is neither additive nor transitive

population theory for boosting ensembles tree ensembles are looked at in distribution space that is the limit case of infinite sample size it is shown that the simplest kind of trees is complete in d dimensional l 2 p space if the number of terminal nodes t is greater than d for such trees we show that the adaboost algorithm gives an ensemble converging to the bayes risk

greedy function approximation a gradient boosting machine function estimation approximation is viewed from the perspective of numerical optimization in function space rather than parameter space a connection is made between stagewise additive expansions and steepest descent minimization a general gradient descent boosting paradigm is developed for additive expansions based on any fitting criterion specific algorithms are presented for least squares least absolute deviation and huber m loss functions for regression and multiclass logistic likelihood for classification special enhancements are derived for the particular case where the individual additive components are regression trees and tools for interpreting such treeboost models are presented gradient boosting of regression trees produces competitive highly robust interpretable procedures for both regression and classification especially appropriate for mining less than clean data connections between this approach and the boosting methods of freund and shapire and friedman hastie and tibshirani are discussed

generalization bounds and complexities based on sparsity and clustering for convex combinations of functions from random classes a unified approach is taken for deriving new generalization data dependent bounds for several classes of algorithms explored in the existing literature by different approaches this unified approach is based on an extension of vapnik s inequality for vc classes of sets to random classes of sets that is classes depending on the random data invariant under permutation of the data and possessing the increasing property generalization bounds are derived for convex combinations of functions from random classes with certain properties algorithms such as svms support vector machines boosting with decision stumps radial basis function networks some hierarchies of kernel machines or convex combinations of indicator functions over sets with finite vc dimension generate classifier functions that fall into the above category we also explore the individual complexities of the classifiers such as sparsity of weights and weighted variance over clusters from the convex combination introduced by koltchinskii and panchenko 2004 and show sparsity type and cluster variance type generalization bounds for random classes

different paradigms for choosing sequential reweighting algorithms analyses of the success of ensemble methods in classification have pointed out the important role played by the margin distribution function on the training and test sets while it is acknowledged that one should generally try to achieve high margins on the training set the more precise shape of the empirical margin distribution function one should favor in practice is subject to different approaches we first present two concurrent philosophies for choosing the empirical margin profile the minimax margin paradigm and the mean and variance paradigm the best known representative of the first paradigm is the adaboost algorithm and this philosophy has been shown by several other authors to be closely related to the principle of the support vector machine we show that the second paradigm is very close in spirit to fisher s linear discriminant in a feature space we construct two boosting type algorithms very similar in their form dedicated to one or the other philosophy we consequently derive by interpolation a very simple family of iterative reweighting algorithms that can be understood as different trade offs between the two paradigms and argue from experiments that this can allow for a suitable adaptivity to different classification problems particularly in the presence of noise or excessive complexity of the base classifiers

an introduction to kernel based learning algorithms this paper provides an introduction to support vector machines kernel fisher discriminant analysis and kernel principal component analysis as examples for successful kernel based learning methods we first give a short background about vapnik chervonenkis theory and kernel feature spaces and then proceed to kernel based learning in supervised and unsupervised scenarios including practical and algorithmic considerations we illustrate the usefulness of kernel algorithms by discussing applications such as optical character recognition and dna analysis

an empirical study of using rotation forest to improve regressors this paper investigates the performance of rotation forest ensemble method in improving the generalization ability of a base predictor for solving regression problems through conducting experiments on several benchmark data sets which is also compared with that of bagging random forest adaboost r2 and a single regression tree the sensitivity of rotation forest to the choice of parameters included in it is also studied on the considered regression data sets adaboost r2 is seen to generally outperform rotation forest and both of them are better than random forest and a single tree with respect to bagging and rotation forest it seems that there is not a clear winner between them furthermore pruning the tree seems to have some bad effect on the performance of all the considered methods

causality tests and observationally equivalent representations of econometric models abstract this article deals with the problem of observational equivalence of alternative econometric equation systems on the same data as a practical obstacle to econometricians success in applying definitions of causality by simon 1953 strotz and wold 1960 and granger 1969 to economic time series

support vector machinery for infinite ensemble learning ensemble learning algorithms such as boosting can achieve better performance by averaging over the predictions of some base hypotheses nevertheless most existing algorithms are limited to combining only a finite number of hypotheses and the generated ensemble is usually sparse thus it is not clear whether we should construct an ensemble classifier with a larger or even an infinite number of hypotheses in addition constructing an infinite ensemble itself is a challenging task in this paper we formulate an infinite ensemble learning framework based on the support vector machine svm the framework can output an infinite and nonsparse ensemble through embedding infinitely many hypotheses into an svm kernel we use the framework to derive two novel kernels the stump kernel and the perceptron kernel the stump kernel embodies infinitely many decision stumps and the perceptron kernel embodies infinitely many perceptrons we also show that the laplacian radial basis function kernel embodies infinitely many decision trees and can thus be explained through infinite ensemble learning experimental results show that svm with these kernels is superior to boosting with the same base hypothesis set in addition svm with the stump kernel or the perceptron kernel performs similarly to svm with the gaussian radial basis function kernel but enjoys the benefit of faster parameter selection these properties make the novel kernels favorable choices in practice

prediction games and arcing algorithms the theory behind the success of adaptive reweighting and combining algorithms arcing such as adaboost freund schapire 1996a 1997 and others in reducing generalization error has not been well understood by formulating prediction as a game where one player makes a selection from instances in the training set and the other a convex linear combination of predictors from a finite set existing arcing algorithms are shown to be algorithms for finding good game strategies the minimax theorem is an essential ingredient of the convergence proofs an arcing algorithm is described that converges to the optimal strategy a bound on the generalization error for the combined predictors in terms of their maximum error is proven that is sharper than bounds to date schapire freund bartlett and lee 1997 offered an explanation of why adaboost works in terms of its ability to produce generally high margins the empirical comparison of adaboost to the optimal arcing algorithm shows that their explanation is n

handling multicriteria preferences in cluster analysis in the framework of multicriteria decision aid a lot of interest has been devoted to sorting problems in which the set of categories is pre defined besides preference oriented multicriteria clustering has received little attention usual geometric and related metrics are not well suited for this problem here we propose a clustering method based on a valued indifference relation inspired by outranking methods we suggest a method based on comparing cluster centers and an average net flow score of clusters to build a complete ranking of the set of clusters that is a way of defining a set of ordered categories for sorting purposes the new approach performs very well in some examples

some theory for generalized boosting algorithms we give a review of various aspects of boosting clarifying the issues through a few simple results and relate our work and that of others to the minimax paradigm of statistics we consider the population version of the boosting algorithm and prove its convergence to the bayes classifier as a corollary of a general result about gauss southwell optimization in hilbert space we then investigate the algorithmic convergence of the sample version and give bounds to the time until perfect separation of the sample we conclude by some results on the statistical optimality of the l2 boosting

aspects of bifurcation theory for piecewise smooth continuous systems abstract systems that are not smooth can undergo bifurcations that are forbidden in smooth systems we review some of the phenomena that can occur for piecewise smooth continuous maps and flows when a fixed point or an equilibrium collides with a surface on which the system is not smooth much of our understanding of these cases relies on a reduction to piecewise linearity near the border collision we also review a number of codimension two bifurcations in which nonlinearity is important

lorenz like systems and classical dynamical equations with memory forcing an alternate point of view for singling out the origin of chaos an alternate view for the emergence of chaos in lorenz like systems is presented in this paper for such purpose the lorenz problem is reformulated in a classical mechanical form and it turns out to be equivalent to the problem of a damped and forced one dimensional motion of a particle in a two well potential with a forcing term depending on the memory of the particle past motion the dynamics of the original lorenz system in the proposed particle phase space can then be rewritten in terms of a one dimensional first exit time problem the emergence of chaos turns out to be due to the discontinuous solutions of the transcendental equation ruling the time for the particle to cross the intermediate potential wall the whole problem is tackled analytically deriving a piecewise linearized lorenz like system that preserves all the essential properties of the original model