System-theoretical algorithmic solution to waiting times in semi-Markov queues Markov renewal processes with matrix-exponential semi-Markov kernels provide a generic tool for modeling auto-correlated interarrival and service times in queueing systems. In this paper, we study the steady-state actual waiting time distribution in an infinite capacity single-server semi-Markov queue with the auto-correlation in interarrival and service times modeled by Markov renewal processes with matrix-exponential kernels. Our approach is based on the equivalence between the waiting time distribution of this semi-Markov queue and the output of a linear feedback interconnection system. The unknown parameters of the latter system need to be determined through the solution of a SDC (Spectral-Divide-and-Conquer) problem for which we propose to use the ordered Schur decomposition. This approach leads us to a completely matrix-analytical algorithm to calculate the steady-state waiting time which has a matrix-exponential distribution. Besides its unifying structure, the proposed algorithm is easy to implement and is computationally efficient and stable. We validate the effectiveness and the generality of the proposed approach through numerical examples. In this paper, we study the steady-state waiting time in a single-server queue in which there is auto-correlation both in interarrival and service times. We model such auto-correlations by using a Markov renewal process or sequence (denoted by MRP) with finite state-space for both interarrivals and services with the associated semi-Markov kernels in matrixexponential form. In particular, we study the Lindley equation in continuous-time:. where A k (designating interarrival times) and B k (designating service times) are Markov renewal processes and the distribution of W = lim k→∞ W k needs to be obtained algorithmically (when it exists), which is the scope of the current paper. Above, for k ≥ 0, A k and B k are independent of each other and also of W k . The case of cross-correlation between A k and B k leads to dependent queues which are outside the scope of the current paper; see [3] for a review of the early literature and [4, 5] for more recent studies on this topic. The random variable W designates the steady-state waiting time for the queueing system described in (1) which is known to be the semi-Markov queue or the SM/SM/1 queue in short [6] . We note that the Lindley equation (1) perfectly describes the queue waiting time in a queueing system with renewal-type services using the first come first serve (FCFS) service discipline. However, the situation is different when the service times are of the more general Markov renewal type in which case the Lindley recurrence may come short of accommodating different models based on how the service process behaves when the queue is empty [7] . In the current paper, we focus on queues that are described solely by the Lindley equation (1) as in [6] . For the case of non-renewal service times, more general models, for instance the ones discussed in [7] , with more general descriptions than that of the recurrence (1) , are left for future research.. Queueing systems with arrivals governed by a semi-Markov process have been studied extensively since an SM/M/1 queue was first analyzed in [8] . On the other hand, [9] uses a matrix factorization method for the more general SM/G/1 queue. The Reference [10] studies the SM/PH/1 queue and introduces a non-linear matrix equation for calculating the waiting time distribution. However, the proposed iterative techniques for solving the matrix equation are relatively slow due to their linear convergence rates. The more general SM/SM/1 queue is studied in [6] by solving Wiener-Hopf equations, and waiting times are found without having to find queue occupancy probabilities first. However, the method is based on transform domain calculations and polynomial root finding; therefore it may not scale to large-scale problems. The work in [4] is similar to ours in the sense of solving the semi-Markov queue described by the Lindley equation, and [4] also allows dependence between interarrival and service times, but their models are relatively limited compared with ours and their methods again rely on transform domain calculations. Recently, a spectral decomposition approach is proposed in [11] for the SM/SM/1 queue using the calculation of eigenvalues and eigenvectors of a so-called coupling matrix and potential use of complex arithmetic. However, calculating eigenvectors is known to be error-prone, especially for closely located eigenvalues or eigenvalues with multiplicity.. The alternative approach is the matrix-analytical approach pioneered by Neuts which does not rely on calculating polynomial roots or eigenvectors. In the matrix-analytical approach, the queue occupancy is observed at certain embedded epochs and a structured Markov chain (of M/G/1 or G/M/1 type) is constructed for the queue length, for instance for the (B)MAP/G/1 queue [12, 13] . The key to the matrix-analytical approach is the solution to a nonlinear matrix equation which can be solved by quadratical convergence rate algorithms like the logarithmic reduction algorithm for [14] and the iterative scheme of [15] for QBD (Quasi Birth and Death) type Markov chains, and the cyclic reduction algorithm of [16] , the invariant subspace approach of [17] , and the technique proposed in [18] for M/G/1 type Markov chains. Once a solution for this matrix equation is obtained, one can then find the queue length probabilities recursively [19] . Given the steady-state queue length probabilities, the waiting time distribution and its moments can be obtained, although not in a very compact form [2] . We refer the reader to [20] and [13] for an extensive treatment of the matrix-analytical approach. Studies related to auto-correlated service times are less common and we now list a few from the existing literature. A MAP/MSP/1 queue is studied in [7] using matrix-analytical techniques in terms of the queue length, but waiting time results are not given. The Reference [21] studies a queue whose service speed changes according to an external environment governed by a Markov process.. The goal of the current paper is to provide a computationally efficient and stable algorithmic method to compute the actual waiting time distribution for the semi-Markov queue described by the Lindley recurrence (1) . This problem is the same as the one stated in [6] . The starting point of the current paper is the assumption of the semi-Markov kernels of interarrival times and service times having matrix-exponential representations. A semi-Markov kernel (or kernel matrix) denoted by F (t) is said to have a matrix-exponential (ME) representation if the kernel F (t) satisfies:. where F (t) is square of size n and T is of size m. In this representation, n is the number of states and m is the number of modes of the underlying MRP. Throughout this paper, we call such processes MRP-ME. An MRP-ME is a generalization of some wellknown processes like phase-type (PH-type) [22] and ME-type renewal processes [23] , Markovian arrival process (MAP) [24] , rational arrival process (RAP) [25] , and some batch arrival models, for instance the batch Poisson model. Assuming that n A (n B ) and m A (m B ) denote the number of states and the number of modes of the underlying MRP-ME for interarrival times (service times), our main result is that steady-state waiting time for the corresponding semi-Markov queue has a matrixexponential distribution with n A m B modes, and finding the coefficients of this distribution amounts to solving a particular spectral-divide-and-conquer (SDC) problem applied to a matrix of size n A m B +n B m A . Similar results have also been obtained in the discrete-time setting, but for renewal arrivals and services only in [26] . Given a matrix A, the SDC problem of interest in this paper is finding an orthogonal matrix Q such that. where the eigenvalues of A ++ are exactly the same as the eigenvalues of A with positive or zero real parts. The advantages of the proposed approach are:. • The approach benefits from being purely matrix-analytical, as explained in [22, 13] , and the generality of the MRP-ME model allows one to use a single unifying algorithm for different well-known queueing models. While doing so, we do not need to construct the embedded Markov chain as in the matrix-analytical approach of Neuts and we directly obtain the waiting time distribution and its moments using expressions that appear to be much simpler than the ones available in the literature. However, we note that we do not give expressions for the queue length in our approach.. • As the numerical engine, we propose to use the ordered Schur decomposition which is known to be the standard serial algorithm for solving the SDC problem in the numerical linear algebra literature due to its well-established numerical stability and computational efficiency [27] . The stability of the Schur decomposition approach stems from its ability to avoid the calculation of all eigenvectors. We calculate only one single eigenvector for which we have a closed form expression as opposed to numerically calculating all eigenvectors, as in [4] . All other steps of the proposed algorithm are based on standard vector and matrix operations and are very easy to implement using a linear algebra package like MATLAB.. • In the definition of the MRP-ME, we allow point masses at the origin, which allows us to model batch point processes as well, with the same structure. This flexibility has the potential to reduce the need for separate algorithms for queueing systems with batch arrivals or services.. • In case the interarrival and service MRP-MEs are renewal processes, i.e., n A = n B = 1, then the SDC problem is to apply to a matrix of additive size m A + m B as opposed to multiplicative size, which is a significant advantage for this special case.. • In the special case of n B = m A = 1, we propose a Householder transformation for the SDC problem which is computationally more efficient than ordered Schur decomposition.. The remainder of the paper is organized as follows. Section 2 provides preliminaries and notation used in the paper. Section 3 describes the ME distribution and Markov renewal processes with matrix-exponential kernels and how they relate to well-known stochastic models used in queueing literature. We introduce a state-space algebra for such distributions in Section 4. We provide our results on the SM/SM/1 queue in Section 5. In Section 6, we present our matrix-analytical algorithm for the SM/SM/1 queue. Section 7 addresses a modified Lindley recurrence within the same framework. Section 8 provides numerical examples to validate the effectiveness of the proposed approach. We conclude in the final section. In this paper, we introduce a stochastic model, namely the MRP-ME, which is a Markov renewal process with a matrixexponential kernel. This model is a superset of the well-known phase-type and matrix-exponential-type renewal processes, as well as the more general Markovian and rational arrival processes. An MRP-ME is also general enough to model batch arrivals (or services) as demonstrated in the numerical example involving the M X /ME/1 queue. We study the steady-state waiting time in a semi-Markov queue with infinite-capacity in which the interarrivals and services are both modeled with MRP-MEs. Without having to solve for the steady-state queue lengths by matrix geometric techniques, we introduce an algorithm that directly finds the waiting time distribution which is in matrix-exponential form. The algorithm to obtain the parameters of the matrix-exponential form is relatively easy to implement and the numerical engine relies on the ordered Schur decomposition whose various stable and efficient implementations exist in the literature. The numerical examples we present lead us to believe that the proposed algorithm is a promising candidate for a wide range of Markov renewal queueing problems. As future work, we list the possibility of allowing inter-dependence between arrivals and services, finite capacity queues, more general service models of Markov renewal type, and the study of relationship between the proposed model and the existing batch arrival and service models.

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The price of selfish behavior in bilateral network formation Given a collection of selfish agents who wish to establish links to route traffic among themselves, the set of equilibrium network topologies may appear quite different from the centrally enforced optimum. We study the quality (price of anarchy) of equilibrium networks in a game where links require the consent of both participants and are negotiated bilaterally and compare these networks to those generated by an earlier model due to Fabrikant et al. [6] in which links are formed unilaterally. We provide a characterization of stable and efficient networks in the bilateral network formation game, show that the set of stable networks is richer than those in the unilateral game, and that all stable networks of the unilateral game are also stable in the bilateral game. We also provide an upper and lower bound on the price of anarchy (tight in the size of the network n but not the link cost α) of the bilateral game and show that the worst-case price of anarchy of the bilateral model is worse than for the unilateral model. A careful empirical analysis demonstrates that the average price of anarchy is better in the bilateral connection game than in the unilateral game for small link costs but worse as links become more expensive. In the process, a powerful equivalence between link-based graph stability and two game-theoretic equilibrium notions is also discussed. The equivalence establishes necessary and sufficient conditions for an equilibrium in the bilateral game that helps provide a partial geometric characterization of equilibrium graphs. Many network settings involve strategic peer selection by independent agents which can be modeled game-theoretically as network formation. These networks are formed endogenously by the actions of agents selfishly maximizing an individual objective function. For this reason, the overall efficiency of such networks, which are the stable outcomes of decentralized strategic interactions, can be worse than the network(s) formed by a central authority maximizing aggregate utility. Understanding this tension between stability and efficiency lies at the heart of our investigation into network formation.. Koutsoupias and Papadimitriou [1] [2] coin the term price of anarchy to refer to the increase in cost caused by independent selfish behavior with respect to the centralized, social welfare-maximizing solution. The price of anarchy has received much attention in games dealing with various networking issues, such as load balancing [3] [4], routing [2] , and flow control [5] . This paper is part of a growing body of literature [6] [7] [8] [9] that studies the price of anarchy in network design, where individuals decide whether or not to form links in a shared network.. In many networking applications agents' interactions are regulated by an intermediary. For this reason, we are interested in how the rules governing the network formation process affect the game's equilibrium outcomes. We study how the quality of worst-case and average-case equilibrium networks differ between a network formation game where link creation requires bilateral consent as compared to a setting where links can be established unilaterally. We view the distinction between bilateral and unilateral formation as something that can be controlled through rules enforced by an intermediary.. The network design game that we study is based on a communication network model by Fabrikant et al. [6] in which agents are nodes and their strategy choices create an undirected graph. We augment the model by having links formed bilaterally (instead of unilaterally) and having the cost of links shared equally between participating nodes. The incorporation of consent represents a natural, realistic extension for a model where traffic routing costs are incurred at both ends of a link.. The bilateral connection game is described as follows. Each node simultaneously chooses a (possibly empty) subset of the other nodes in the game, and a link between any two nodes is established if it has the consent of both parties directly involved in the link. The intersection of these sets of edges is the resulting graph. The edges are undirected, meaning that they can be used to route traffic in both directions. Payoffs are construed as costs, meaning that agents are trying to minimize their respective costs of participating in the network, and the cost incurred by an agent is expressed quite generally as a trade-off between the cost of establishing links to other agents and its proximity to the rest of the network. We refer to the model by Fabrikant et al. [6] as the unilateral connection game, which differs from ours in that agents can unilaterally establish links and in the requisite solution concepts for the different models' evaluation. We presented a bilateral consent-driven model of network formation. The worst-case price of anarchy in the bilateral connection game is at least as bad as for the unilateral game where consent is not required. It is strictly worse contingent on a conjecture on the price of anarchy in the unilateral connection game put forth by Fabrikant et al. [6] that claims that the price of anarchy in the unilateral game is constant, whereas we prove a lower bound of Ω(log 2 α) for the worstcase price of anarchy in the bilateral game. Empirical analysis of equilibrium networks in the two games also suggests that the worst-case is strictly worse in the bilateral game. Finally, we show empirically that while the average efficiency of networks in the bilateral setting is better than in the unilateral setting when links are inexpensive, the situation is reversed when links become more expensive. Altogether, the comparative results suggest that worst-case equilibrium networks have a higher price of anarchy in a consent-driven network formation setting where players are limited in their ability to coordinate simultaneously than when consent is not required and players are unconstrained in their ability to act selfishly.. Our investigations into the role of consent in network formation suggest that the rules of engagement strongly affect the set of possible equilibria. We are currently investigating how bilateral and multilateral transfers between players may help mediate the price of anarchy in the connection game. We are also interested in extending the game to a setting where the network formation is dynamic and on-going. The dynamics of network formation can be controlled by an intermediary, subject to equilibrium constraints suggested by the dynamic network formation process. The best-response dynamic in the UCG is much more powerful than in the BCG. The problem of mutual blocking in the formation of new links in the BCG and the limited capacity to coordinate afforded to players under pairwise stability means that players are much less able to react to the positive externalities of other players' links, which is the root cause of the tension between stability and efficiency in the connection game for α > 1. This has two major consequences.. On the one hand, the set of pairwise stable network geometries in the BCG is much richer than the set of Nash equilibrium network shapes admitted in the UCG. For example, extremal and cage graphs, which feature prominently in many communication network designs, are pairwise stable in the BCG but not generally Nash supportable in the UCG. Fabrikant et al. [6] study a network creation game in the context of communication networks where links are generated by the unilateral actions of players and link costs are one-sided. Our bilateral network creation model may be 1 The incorporation of consent forces us to use a refinement of the Nash equilibrium.. better suited for modeling communication network design, given that interconnect costs are typically two-sided.. There is a growing interest in understanding how simple rules of engagement established by a system designer will affect the functioning of a networked system maintained endogenously by the strategic actions of participants [8] [10] . In order to understand the role of bilateral consent in strategic network formation, we compare equilibria and bounds on the price of anarchy in our bilateral network creation game with the unilateral model proposed by Fabrikant et al. [6] .. Anshelevich et al. [7] [8] study a different cost-sharing network connection game where, given an undirected graph structure G, players have a set of specified terminal nodes that they would like to see connected in the purchased network (which is necessarily a subgraph of G). In [8] , they study how fair cost allocation schemes affect the quality of the best Nash equilibrium network, and so understand the protocol constraining agents' interactions as suggesting an equilibrium outcome. Our model includes only local costsharing and our interaction protocol is restricted to a rule for how individual links are formed and severed, and does not propose a specific equilibrium. Of mention is that the socially efficient solution is stable for both connection games we consider.. Lopez-Pintado [11] and Melendex-Jimenez [12] both consider bilateral network formation models with local cost sharing, but their models differ from our own in the payoff structure and the equilibrium concepts applied. Payoffs in both these models are link-separable and derived from underlying (anti-)coordination games, while the benefits of connections in our bilateral game are formulated in terms of a QoS measure based on the entire network topology. Also, both papers use the Nash equilibrium solution concept whereas we rely on the pairwise Nash equilibrium and pairwise stability.. Sarangi et al. [13] investigate the effects on network architecture of different distance-based utility functions, but investigate the problem from the standpoint of Nash networks (as developed by Bala and Goyal [14] ) and only consider unilateral link costs. Finally Haller and Sarangi [15] provide characterizations of stable networks under one-sided and two-sided link formation costs for a class of link-based payoffs. We are concerned with a different payoff structure, provide a stronger characterization of stable networks for our specific domain, and focus on the price of anarchy.

O B O O B O O B O O O O O O O O O O O B O O B O O O O O B B O O O O O O O O O O O O O B B I I O B O O O B O B O O O O O O O O O O O O O O O O O O O O B O O O O O O O O B O O O O O O B O O O O O O O O B O B O O O O O O O O O O O O O O B O O O O O O O O B O O O O O O B O O O O O O B I O O B I I O O O B O O B O O O O O B O O O O B O O O O O B I I O O O B O O O O O O B O O B I O O O O B I I O O O O O B B O O O O B O O O O O O O B O O O O O B O O O O O B B O O O B O O O O O O O O O O O O O B O O O B O O O O O O B O B O O B O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O B O O O O O O O O B O O O O O O O O O B I I O O O O O O B O O O O B O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O B O O O O O O O O O B O O O B O O O O O O O O O O O O O O O O O O O O O O O B O B O O O B O O O O O B O O O O B O O O O B O B O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O B O B O O O O O O O O B B O O O O O O O O O O O O O O O O O B O O O B O O B O O O O O O O O B O B O O O O O O O O O O O O O O O O B O B O O O O O O O O O O O O B B O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B O O B O O O B O O O O O O O O O B O O B O O O O O O O O O O O B O O O O B O O O O O O O O B B O O O O O O O O O O B O O O O O O O O O O O O O O O O B O B O O O B I I O O O B B O O O O O O O O O B O O O O O O O O O O O O O O O B I I O O O B B O O O O O O O O O O O B I I O O O B O O O O O O B I O O O O O O O B I I O O O B B I O B O O O O O O O O O O O O O O O O B O O O O O O O O B O O O O O O O O O O O O O O B O O O O O O O B O O O O O O O O O O B O O O O B I I O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O B B O O O O O O O B O O O O O B O O O O O O O O B O O O O O O O O O B O O O O O B O B O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B O O O O O O O O O B O B O O B B O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O B B O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O B O B O O O O O O O B O O O O O O O O O O O O O O O B O B O O O O O O O O O O O O O O O O B O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O B I I O O O B O B O O O B O O O O O O O O O O O O O O O O B B B O O O O B O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O B O O O O B O O O O B B O O O O O O O O O O O B O O B O O O O O O O O O O O O O O O O B O O O O O O O O O O B O O O O O O O O O O O O B O O O O O O O O O O O B O O O O B O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O B O O O O O O O B O O O O O O O O O B O O B O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O B I I


Learning Bayesian network classifiers by risk minimization Bayesian networks (BNs) provide a powerful graphical model for encoding the probabilistic relationships among a set of variables, and hence can naturally be used for classification. However, Bayesian network classifiers (BNCs) learned in the common way using likelihood scores usually tend to achieve only mediocre classification accuracy because these scores are less specific to classification, but rather suit a general inference problem. We propose risk minimization by cross validation (RMCV) using the 0/1 loss function, which is a classificationoriented score for unrestricted BNCs. RMCV is an extension of classification-oriented scores commonly used in learning restricted BNCs and non-BN classifiers. Using small real and synthetic problems, allowing for learning all possible graphs, we empirically demonstrate RMCV superiority to marginal and class-conditional likelihood-based scores with respect to classification accuracy. Experiments using twenty-two real-world datasets show that BNCs learned using an RMCV-based algorithm significantly outperform the naive Bayesian classifier (NBC), tree augmented NBC (TAN), and other BNCs learned using marginal or conditional likelihood scores and are on par with non-BN state of the art classifiers, such as support vector machine, neural network, and classification tree. These experiments also show that an optimized version of RMCV is faster than all unrestricted BNCs and comparable with the neural network with respect to run-time. The main conclusion from our experiments is that unrestricted BNCs, when learned properly, can be a good alternative to restricted BNCs and traditional machine-learning classifiers with respect to both accuracy and efficiency. 249 Common approaches for learning a model are roughly partitioned into generative, discriminative, or a combination of both approaches [13] [14] [15] . Generative models (e.g., BN, density estimation) summarize data probabilistically and are more flexible, since the user can bring in conditional independence assumptions, priors, and hidden variables. Generative classifiers learn a model of the joint probability of the variables and the related class label, and use Bayes' theorem to compute the posterior probability of the class variable and make predictions. Discriminative models (e.g., NN and SVM) only learn from data to make accurate predictions by directly estimating the class posterior probability or via discriminant functions, and thus offer the user less flexibility in data representation and inference. The dilemma in the machine learning community regarding which approach of learning -generative or discriminative -is more appropriate for learning a BNC structure, has gained considerable attention in recent years. Most empirical studies demonstrate superiority of the discriminative approach with respect to the accuracy of the learned BNC [5] [6] [7] 16] . For some models, however, it is shown [13] that the choice of either of the approaches depends on the sample size; for small sample sizes, the generative approach, which relatively quickly approaches its asymptotic error, is favored, whereas the discriminative method is preferred for larger sample sizes. Classifiers combining generative and discriminative modeling use generative models, yet estimate the model structure and/or parameters to reduce the classification error.Several studies [4] [5] [6] [7] 10, 17] have demonstrated that BNC structures learned using generative scores do not usually contribute to high classification accuracy since there is lack of agreement between the score used for learning and the score used for evaluation, i.e., the classification accuracy. That is, classifiers based on structures having high values of the generative scores are not necessarily highly accurate. To address this issue, we propose risk minimization by cross validation (RMCV) for a classification-oriented score and S&S algorithm for learning unrestricted BNCs. Note that other uses of classificationoriented scores in learning unrestricted BNCs [7, 18] are in a somewhat different context. Moreover, RMCV is an extension to common use of classification-oriented scores in learning restricted-BNCs and non-BN classifiers. While commonly used S&S algorithms use likelihood-based scores suitable for general inference, RMCV minimizes an empirical estimation of the classification error rate, and thereby learns highly accurate BNCs. That is, RMCV performs discriminative learning of a generative (BN) model. This model does not need to estimate the true distribution, generate data from this distribution, or infer about any non-class variable. It needs to perform a discriminative classification task. RMCV learns generative models that are complicated, only to discriminate accurately among classes.In the beginning, we suggest and compare several variants of the RMCV score and algorithm. We further show that the RMCV score is better suited for classification than any other score commonly used for learning a BNC, i.e., compared with other scores and BNCs, the accuracy of an RMCV-based classifier increases monotonically with the improvement in the value of the RMCV score. Then, we compare the classifier learned using RMCV with likelihood-based, conditional-likelihood-based, and other BNCs, as well as with non-BN classifiers, such as NN, SVM, and CT. This involves nine leading BNCs and five state of the art non-BN classifiers in a most extensive comparison of BNCs and non-BN classifiers using twenty-two real-world datasets. The comparison demonstrates that an RMCV-based classifier is faster and significantly more accurate than all BNCs and comparable (usually favorably), with respect to accuracy, with the non-BN classifiers. These are encouraging news for researchers and practitioners who appreciate the benefits of the BN model, but once they encounter a classification task, replace the BN with a traditional classifier and thereby lose the BN benefits.We begin by reviewing BN in Section 2. In Section 3, we focus on learning a BNC using common scores. The RMCV score and algorithm are presented in Section 4, and classifiers learned using RMCV are experimentally compared to other BNCs and non-BN classifiers in Section 5. Section 6 describes recent studies in learning a BNC. Finally, we draw conclusions and summarize the study in Sections 7 and 8, respectively.A BN model B for a set of random variables X = {X 1 , X 2 , . . . , X n }, each having a finite set of mutually exclusive states, consists of two main components, B = (G, ). The structure G = (V, E) is a directed acyclic graph (DAG). V is a finite set of nodes of G corresponding to X, and E is a finite set of directed edges of G connecting V. is a set of parameters that quantify the structure. The parameters are local conditional probability distributions (or densities), P(X i = x i |Pa i , G), for each X i ∈ X conditioned on its parents in the graph, Pa i ⊂ X. In this study, we are interested only in discrete variable BNs and complete data.The joint probability distribution over X given G -assumed to encode this distribution -is the product of these local probability distributions [2, 3] ,where x is the assignment of states to the variables in X and x i is X i 's state.During inference, the conditional probability distribution of a subset of nodes in the graph (the 'hidden' nodes) given another subset of nodes (the 'observed' nodes) and the BN model is calculated. A common method for exact inference is the junction tree algorithm [19] , but when there is only one hidden node (e.g., the class node in classification), direct inference based on (1) and Bayes' rule is more feasible. Note that the computation of conditional probability distributions for inference depends on the graph. Thus, a structure, either based on expert knowledge or learned from the data, must first be obtained. One fundamental task of machine learning is classification, where instances (patterns) are assigned to their corresponding classes. This is a supervised learning task, where a training dataset of instances with labels representing instance classes is used to train a classifier. Since a Bayesian (belief) network (BN) [1] [2] [3] provides a graphical model for encoding relationships, such as dependencies and conditional independencies between variables, and for inferring probabilistically about variables, it is natural to use the BN for classification.. Indeed, along with the traditional classifiers based on the neural network (NN), the support vector machine (SVM), and the (decision) classification tree (CT), classifiers based on BN have recently been introduced and studied [4] [5] [6] [7] [8] [9] [10] [11] . Learning a Bayesian network classifier (BNC) requires learning the structure (graph) of the graphical model and its parameters so that the learned BN will excel in inference of a specific variable, that is the class variable, and not necessarily of all variables. When focusing on structure learning, exhaustively searching the space of possible graphs is infeasible [2] , and thus search and score (S&S) structure learning algorithms sub-optimally search the space and select the structure achieving the highest value of a score [2, 3, 12] . However, until very recently, all S&S structure learning algorithms used a generative score, and thereby led to learning a generative model that is not specific to classification, but to general inference.. The S&S approach to learning a structure from data [2, 12] comprises a search for the structure achieving the highest score, e.g., hill-climbing (HC) [3, 12] , and a score, generally the Bayesian score [2] , P(G|D) = P(D|G)P(G) P(D) = P(D, G) P(D) (2) for a structure G given a dataset D = {v 1 , v 2 , . . . , v N }, which is a random sample of N independent instances from the joint probability distribution of X. Likelihood-based (unsupervised) scores, such as MDL and K2, and structure learning algorithms that model the relations between variables without giving precedence to the class variable, often produce BNCs that are less accurate than optimal. Other algorithms that do consider the class variable attempt to estimate the class-conditional likelihood (supervised) score, for which exact computation is infeasible. Moreover, this attempt does not necessarily comply with expert predictions that are available in supervised learning.. In this work, a score called RMCV and an algorithm based on this score were suggested for learning accurate unrestricted BNCs. The score is based on risk minimization of the 0/1 loss function using cross validation, and thus is classificationoriented. Discriminative scores for learning a BNC, including the classification accuracy, increase the computational cost of learning, and thus the implementation of RMCV structure and parameter learning has been optimized. Our empirical evaluation using twenty-two classification problems usually demonstrated superiority of the proposed score and algorithm, with respect to accuracy and run-time, over other scores and algorithms. Thus, we believe that RMCV is a viable tool for learning accurate unrestricted BNCs and worthy of a place among other state of the art machine-learning classification algorithms. In recent years, learning a BNC has been studied intensively (for up-to-date reviews see [7] [8] [9] [10] ). Some studies focus on learning restricted DAGs, usually NBC-based (e.g., TAN), that implement classification primarily through the positioning of the class variable as a parent of all DAG nodes, regardless of the learned structure and learning algorithm [4, 39, 40] . Keogh and Pazzani [39] extend NBC to TAN by learning edges between non-class nodes that increase the classification accuracy, instead of the conditional mutual information as in the original TAN [4] . Cheng and Greiner [40] compare NBC and TAN with the unrestricted general BN, for which the structure is learned using node ordering and based on a spanning tree. Sierra and Larranaga [41] search structures within a space created using a genetic algorithm allowing, in the first case, a crossover between BNs and a mutation of a BN or that, in the second case, is restricted to structures based only on Markov blankets (MBs) of the class variable. The selected structure, in the first case, is the one maximizing the K2 metric [2] , and, in the second case, the one that maximizes the classification error (empirical risk under the 0/1 loss). Classifiers are learned for predicting the survival time of individuals diagnosed with malignant skin melanoma. Ling and Zhang [42] suggest learning a TAN that maximizes AUC rather than the classification accuracy. They consecutively add to NBC edges between a SuperParent [39] and its favorite child -each (separately) increases AUC the most -and keep adding such edges, as long as the AUC value of the derived BNC is improved. Ling and Zhang have found that BNCs learned according to this scheme provide both high AUC values and high classification accuracies. Recently, Pernkopf and Bilmes [10] considered the classification error as the ideal criterion for discriminative structure learning, assuming sufficient training data. Kontkanen et al. [5] , Grossman and Domingos [6] , and Guo and Greiner [7] drew a similar conclusion. However, for computational reasons, Pernkopf and Bilmes limit the use of this measure to learning only k-tree structures. These tree-restricted structures are also required to have edges pointing from the class node to each of the structure nodes and not vice versa (similar, e.g., to TAN). To further reduce computational complexity, Pernkopf and Bilmes suggest several heuristic methods based on conditional mutual information and/or the classification error to order the variables (parents first) and select node parents according to this order. Results show an advantage of discriminative structure learning over generative learning.. While TAN-based classifiers relax some of the strong conditional independency assumptions of NBC, they are still restricted BNCs. Other restrictions to NBC that facilitate possible curse-of-dimensionality and improve the classifier accuracy are imposed through mechanisms of feature selection that let only variables that increase accuracy to be connected to the class variable (e.g., [11, [43] [44] [45] [46] ). For example, in [44] a subset of the features is selected using topological orders based on conditional independence (CI) tests of increasing orders to maximize the amount of information provided by the structure.. Some studies that concern unrestricted DAGs concentrate on developing discriminative methods for parameter learning [9, 47] , comparing methods of discriminative and generative parameter learning [7, 16] , or comparing unrestricted, generatively-learned BNCs with restricted BNCs [48] . For example, Madden [48] finds that unrestricted BNCs learned using likelihood-based (MDL and BDeu) scores and careful parameter learning are comparable to TAN, and thus wonders why use unrestricted BNCs at all. He provides three reasons for that. First is the advantage of using an unrestricted BNC that models the whole distribution, demonstrates all node connections, and isolates variables that do not contribute to classification (as they are outside the class node MB). Second is the greater representational power of unrestricted BNCs over restricted classifiers (such as TAN). Third is the complexity of unrestricted BNCs that frequently is less than that of BNCs that are restricted to a fixed number of edges (e.g., between the class node and any non-class node of TAN), where some of these edges are quite often unnecessary.. Studies that are more relevant to our research are in discriminative (supervised) structure learning of the unrestricted BNC, where the class variable is in the focus of the criterion for learning the structure. Kontkanen et al. [5] -one of the pioneer studies in the field -demonstrates experimentally the advantage of supervised learning methods over unsupervised learning methods in learning a BNC. The authors make a distinction between the logarithmic loss function, commonly used in structure learning, and the 0/1 loss function that is more appropriate for optimizing a BNC. They also suggest using CV to estimate the criterion employed during learning and point to the connection between CV and an approximation of the factorization of the supervised (marginal) likelihood. Bilmes [49] introduces the explaining away residual discriminative measure for learning a dynamic BN for speech recognition. Grossman and Domingos [6] start from the empty structure, use an HC search, and learn a BNC using a CLL-based criterion that is either penalized using the MDL principle or balanced by a restriction of no more than two parents for a node (Table 4 ). Pernkopf and Bilmes [16] empirically compare both discriminative and generative parameter learning on both discriminatively and generatively structured BNCs. They found that discriminative structure learning of NBC, TAN, and the Bayesian multinet by optimizing the classification error produces the most accurate classifiers on almost half of the 25 datasets they examined.. Guo and Greiner [7] suggest different criteria for structure learning of unresticted BNCs and experimentally compare them for different complexities of the class variable MB. Using mainly synthetic data, Guo and Greiner compare (see Table 4 for a reference) likelihood-based criteria, such as MDL and a variant of the K2 metric called BDeu [12] , class-conditional likelihood-based criteria, such as CLL and CMDL, a bias-variance (BV) criterion that is the expected mean-square-error of the classifier, and the classification error (CE). Although CE was found to be one of the best discriminative learning criteria, the authors recommend using BV for discriminative structure learning due to its superior performance. We note that no search algorithm is proposed to accompany CE and thereby to turn it into a structure learning concept/algorithm. Second, the CE score is evaluated in some synthetic scenarios (e.g., after modifying the true graph) that may represent a naive search procedure but not a real scenario or a real structure-learning task. Third, no classifier based on the CE score is either evaluated in real classification problems (but only on synthetic networks or modifications of the ALARM network) or compared to other BN and non-BN classifiers. Finally, the correspondence between CE and the true classification accuracy (error) -and therefore the contribution of CE to finding accurate BNCs -is not checked.. Pernkopf [18] compares methods of learning restricted or unrestricted BNCs to the k-nearest neighbor classifier following a preliminary feature selection. He found that both paradigms are comparable. One of Pernkopf's suggested methods -the selective unrestricted BN using classical floating search (CFS-SUN) -is the algorithm most similar to our (independentlydeveloped) RMCV algorithm. CFS-SUN starts from the empty initial graph, and iteratively includes and excludes candidate variables to be connected to the structure based on a feature selection stage, as long as the BNC accuracy increases. Instead of focusing on a classification-based criterion for searching for a structure in a space of augmented DAGs (usually augmented NBC) or in the space of unrestricted DAGs, Acid et al. [8] focus on searching a structure in the space of partially directed acyclic graphs, even using the standard generative criteria (e.g., BDeu). The results in [8] demonstrate that attempts to improve the BNC accuracy should not necessarily concentrate only on finding discriminative scores, but should also explore more focused search spaces for the classifier using dedicated search procedures.. Other BNCs that have been investigated recently include multi-dimensional BNCs (MBCs) and Bayesian multinets. An MBC structure is partitioned to a class sub-graph (having a node representing each class), variable sub-graph, and a bridge (between class nodes and variables) sub-graph [50] . Borchani et al. [51] suggest an algorithm for learning each of the sub-graphs based on a wrapper greedy forward selection approach. A Bayesian multinet classifier is composed of a set of local BNs. Each local network represents a different class using a different set of independences [52] . By computing the joint probabilities of the local networks and the a priori probability of the class variable, and by using Bayes' theorem, classification of a test instance is made to the class that maximizes the class posterior probability. Learning a local network is based on maximization of the network log-likelihood [4, 53] , minimization of the corresponding Kullback-Leibler (KL) divergence [54] , maximization of the network (class) posterior probability [49] , or maximization of the classification error [55] .

O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O B O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O 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The bridge-connectivity augmentation problem with a partition constraint In this paper, we consider the augmentation problem of an undirected graph with k partitions of its vertices. The main issue is how to add a set of edges with the smallest possible cardinality so that the resulting graph is 2-edge-connected, i.e., bridge-connected, while maintaining the original partition constraint. To solve the problem, we propose a simple linear-time algorithm. To the best of our knowledge, the most efficient sequential algorithm runs in O(n(m+n log n) log n) time. However, we show that it can also run in O(log n) parallel time on an EREW PRAM using a linear number of processors, where n is the number of vertices in the input graph. If a simple graph exists, our main algorithm ensures that it is as simple as possible. A graph is said to be k-edge-connected if it remains connected after the removal of any set of edges whose cardinality is less than k. Finding the smallest set of edges to make an undirected graph k-edge-connected is a fundamental problem in many important applications; readers may refer to [1] [2] [3] for a comprehensive survey.. Augmentation problems in bipartite graphs are studied in [4] [5] [6] . The related 2-edge-connectivity augmentation problem arises naturally from research on the security of statistical data [7] [8] [9] [10] [11] . The information is stored in a cross-tabulated table and it is common practice to suppress some of the cells in the table to protect sensitive information. A basic concerning issue of this practice is its effectiveness, that is how to suppress a small number of cells to protect the information, so that the resulting table does not leak important information and the sensitive information will not be revealed to an adversary. This protection problem can be reduced to an augmentation problem in bipartite graphs [12, 6, 13, 14] . In addition, many algorithms have been developed to resolve the problem of making general graphs k-edge connected or k-vertex connected for various values of k [15, 16, 4, [17] [18] [19] . For example, a linear-time algorithm for the smallest bridge-connectivity augmentation problem in a general graph that does not have a partition constraint is proposed in [15] ; while a linear-time algorithm for the bridge-connectivity augmentation with a bipartite constraint is described in [20] . Jensen et al. [5] presented an algorithm that solves the k-edge-connectivity augmentation problem in a graph containing partition constraints in O(n(m + n log n) log n) time, where n is the number of vertices, and m is the number of distinct edges in the input graph.. In this paper, we focus on augmenting graphs with a partition constraint. Specifically, the constraint requires that the vertex set of an input graph must be partitioned into k disjoint vertex subsets, and each edge in the augmentation must be added between two different vertex subsets. We propose a linear-time algorithm that addresses the problem of adding the smallest number of edges to a graph with a given partition constraint to make it 2-edge-connected, or bridge-connected, while maintaining the constraint. 2-edge-connectivity augmentation of the graph in Fig. 1(a) is shown in Fig. 1(b) . Moreover, a smallest 2-edge-connectivity augmentation of (a) without a partition constraint is shown in Fig. 1 The main algorithm proposed in this paper produces a simple graph if such an augmentation solution exists. If this is not possible, it generates a multigraph. First, we prove that we can easily add edges to the input graph such that the resulting graph G has the property whereby each degree ≤ 1 block of BB(G ) contains a legal edge. We select a legal edge of each degree ≤ 1 block and apply the circular-swap function to all the selected legal edges to form a 2-edge-connected graph. The main algorithm can be parallelized to run in O(log n) parallel time using a linear number of EREW processors. We consider the following two cases. Case 1. e ∈ G i . Assuming that G is a 2-edge-connected component, then, according to Theorem 1, there is no bridge e in G i or G i , where 0 ≤ i ≤ α − 2. Without loss of generality, we assume that e ∈ G α−1 . Since G is a 2-edge-connected component. Hence, after removing e, a path still exists from u to v through u. ; v which is shown in Fig. 7(a) . This means G is connected and e is not a bridge in G i . ; u α−1 which is shown in Fig. 7(b) . This means G is connected, and e 1 and e 2 are not bridges. Hence, the theorem holds.

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Parametric free-form shape design with PDE models and reduced basis method We present a coupling of the reduced basis methods and free-form deformations for shape optimization and design of systems modelled by elliptic PDEs. The free-form deformations give a parameterization of the shape that is independent of the mesh, the initial geometry, and the underlying PDE model. The resulting parametric PDEs are solved by reduced basis methods. An important role in our implementation is played by the recently proposed empirical interpolation method, which allows approximating the nonaffinely parameterized deformations with affinely parameterized ones. These ingredients together give rise to an efficient online computational procedure for a repeated evaluation design environment like the one for shape optimization. The proposed approach is demonstrated on an airfoil inverse design problem. We have presented a method for efficient shape design of PDE-modelled systems. We combined freeform deformations to generate the parametric geometry and reduced basis methods to solve the resulting parameterized PDEs in an inverse problem framework of shape optimization. To demonstrate the applicability of the proposed approach we considered an inverse airfoil design problem in 2-d potential flow. It was confirmed that using free-form deformations to generate the parametric domain deformations a small number of design variables is sufficient to solve already quite challenging inverse airfoil design problems. The reduced basis methods can then be used to obtain an efficient online procedure for shape optimization, where several iterative evaluations of the state equations are needed and computational efficiency is appreciated.. The free-form deformations are independent of the mesh, geometry, and even the PDE model. Thus we believe that in the future it will be possible to extend the proposed methods to more realistic and interesting problems using different viscous flow models.

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View-invariant motion trajectory-based activity classification and recognition Abstract Motion trajectories provide rich spatio-temporal information about an object's activity. The trajectory information can be obtained using a tracking algorithm on data streams available from a range of devices including motion sensors, video cameras, haptic devices, etc. Developing view-invariant activity recognition algorithms based on this high dimensional cue is an extremely challenging task. This paper presents efficient activity recognition algorithms using novel view-invariant representation of trajectories. Towards this end, we derive two Affine-invariant representations for motion trajectories based on curvature scale space (CSS) and centroid distance function (CDF). The properties of these schemes facilitate the design of efficient recognition algorithms based on hidden Markov models (HMMs). In the CSS-based representation, maxima of curvature zero crossings at increasing levels of smoothness are extracted to mark the location and extent of concavities in the curvature. The sequences of these CSS maxima are then modeled by continuous density (HMMs). For the case of CDF, we first segment the trajectory into subtrajectories using CDF-based representation. These subtrajectories are then represented by their Principal Component Analysis (PCA) coefficients. The sequences of these PCA coefficients from subtrajectories are then modeled by continuous density hidden Markov models (HMMs). Different classes of object motions are modeled by one Continuous HMM per class where state PDFs are represented by GMMs. Experiments using a database of around 1750 complex trajectories (obtained from UCI-KDD data archives) subdivided into five different classes are reported. Object motion trajectory-based analysis and recognition has gained significant interest in scientific circles lately. Examples of the motion trajectory include tracking results from video trackers, sign language data measurements gathered from wired glove interfaces fitted with sensors, Global Positioning System (GPS) coordinates of satellite phones, cars using Car Navigation Systems (CNS), animal mobility experiments, etc. This spatio-temporal data embodies semantically rich information about the behavior of the object of interest, the action performed and the interaction among groups of objects [10] . Novel applications employing analysis of motion trajectory are emerging due to enhanced interest in homeland security as well as due to prevalence of multimedia gadgets in commercial and scientific endeavors. For example, in sign and gesture recognition, the signer moves his hands in specific pattern for a particular word. In sports video trajectory analysis and understanding can assist the players, coaches and sports analysts with strategies used on the field based on the motion patterns of players and their mutual interaction. Another important area is automatic video surveillance which is used, for example, in real-time observation of people and vehicles, in a busy environment, leading to a description of actions and mutual interactions. Psychological studies have shown that human beings can routinely discriminate and recognize this kind of object motion using motion pattern, even in large viewing distances or poor visibility conditions; whereas, other cues such as clothes, appearance, or hair style tend to vanish at large distances or poor visibility conditions [23] .. We observe that developing high-accuracy activity classification and recognition algorithms in multiple view situations is still an extremely challenging task. The object trajectory is typically modeled as a sequence of consecutive locations of the object on a coordinate system resulting in a vector in 2-D or 3-D Euclidean space. An object trajectory captured from different view-points leads to entirely different representations. Multiple representations of an object motion captured from different view points are related by perspective transformation exactly, and by generic affine transformation approximately. The set of affine transformations includes scaling, rotation, translation and shear. To satisfy the view independent requirement, the trajectory data has to be represented in an affine invariant feature space.. This paper presents efficient object activity recognition algorithms based on novel view-invariant representation of trajectories. Since we are addressing the problem of classification of activities based on trajectory data only, we cannot ensure the semantic interpretation of all actions resulting from those trajectories. This problem can be addressed by the addition of other cues along with motion trajectories in the design of activity recognition systems. We derive two Affine-invariant representations for motion trajectories based on curvature scale space (CSS) and centroid distance function (CDF). These feature spaces allow affine invariant representation of trajectories as temporal sequences of subtrajectories analogous to the characterization of words as a sequence of phonemes. We subsequently capture the interactions among subtrajectories using ergodic Hidden Markov Models (HMM). All HMM parameters including the topology are learnt from training datasets. The performance of classification of motion trajectories using CSS-and CDFbased view-invariant representations is analyzed and computer simulation experiments are reported. The comparisons are performed in terms of receiver operating characteristics (ROC) as well as accuracy values. Experiments are conducted on the Australian Sign Language (ASL) data set obtained from University of California at Irvine's Knowledge Discovery in Databases archive [18] . The ASL dataset used in our experiments has trajectories for five word classes as signed by five signers of varying skill levels. Each word class has 69 trajectories for a total of 345 original samples. The training is performed on half of these samples, while for testing an affined database containing rotated version of these trajectories at five different angles is generated. Thus from the original set of 345 trajectories, our training set contains 170 trajectories while the test set contains 1725 trajectories. Our results show that overall, CDF-based representations perform better in terms of ROC and accuracy values.. The remaining sections of this paper are organized as follows: Section 2 surveys related work on trajectory modeling and view-invariant feature spaces. Section 3 briefly describes the two view-invariant feature spaces used in this presentation. Section 4 presents the HMM-based viewinvariant action recognition using CSS and CDF-based representations. Section 5 provides a comparison of the above methods in terms of their retrieval performance. The details of the data sets used and the experiments conducted are also provided in Sect. 5. Section 6 provides a discussion of the results of the computer experiments. Finally, in Sect. 7, we present a brief summary and conclusion and outline future research in this area. In this paper, we have presented a detailed discussion on the topic of motion trajectory-based statistical modeling and classification of data in a view-invariant setting. Our aim in this presentation has been to motivate the need for, and challenges involved in, the classification and recognition of temporal data resulting from object tracking captured through multiple views. We cast the view-invariant activity recognition problem as affine-invariant image shape retrieval. This approach detects the CSS contour maxima and represents the trajectories based on the locations of these peaks. The limitation of this approach is that it does not model the data between segmentation points. This shortcoming is alleviated in the CDF+PCA based representation scheme. Here, the centroid distance function from the trajectory data is computed. The CDF-based representation is used to segment the trajectories and the PCA coefficients of the segmented data are then stored to train the HMMs on. We have based our experiments on ROC curves as well as accuracy values. The classification systems were tested on a standard ASL data set. The results are generated for various sizes from 690 to 1725 trajectories in this dataset.. Future research must focus on motion trajectory-based modeling and classification of video sequences that are robust to camera orientation and movement. Generalization of the proposed PCA representation to nonlinear transformations (e.g., Kernel PCA or Kernel Discriminant Analysis), are needed to deal with nonlinear classification. An important extension of our approach would be required to perform multiple motion trajectory-based classification for "semantic" retrieval from video sequences. It is also possible that the basis of our approach could be used for video sequence mining by detection and identification of motion trajectories in the video query. This section provides a survey of the related work from recent literature in the areas of trajectory modeling, viewinvariant representation and applications of trajectory-based representation and learning. Studies into human psychology have shown the extra-ordinary ability of human beings to recognize object motion even from minimal information system such as Moving Light Displays (MLDs). Such displays are obtained by making a video of moving subjects wearing reflective pads/light bulbs on their body joints in almost dark conditions. In spite of the paucity of information, human observers easily perceive not only motion but also the kind of motion; e.g., walking, running, dancing, cycling, etc. [23] . Based on this understanding, object motion has been an important feature for the representation and discrimination of one object from another in video applications. Earlier approaches in motion-based methods focused on object tracking from raw and compressed domain videos [16, 21, 37, 38] . Indexing and searching based on object motion as the dominant cue has attracted a lot of research activity in the past few years [2, 8, 11, 13, 14, 39] .. View-invariant representation has also been addressed in [35] for modeling and recognizing actions performed by individuals in video sequences. The representation is based on dynamic instants (segmentation points) of the trajectories. For each dynamic instant in the trajectory, frame number, location of the hand and "sign" of the instant (−ve for counter clockwise turn and +ve for clockwise turn) is stored. The matching is performed on trajectories with the same number of dynamic instants and same sign permutations. This approach though compact in representation is suboptimal and too much dependent on the segmentation process. Also, it has no room for partial trajectory processing. The system is tested on 47 instants of 16 different actions, so it is hard to judge the scalability issues of this system for large datasets. In our previous work on view-invariant representation, we addressed the problem of indexing and retrieval. We used the Fourier descriptor (FD) computed from centroid distance function [3] , as well as curvature scale space [4] . One important contribution of our work is that we have formulated the view-invariant trajectory representation as open-ended shape representation problem. Specifically, we argue that the only difference between an image shape and the trajectory of an object is that the former is a closed contour while the later is not. This allows us to take advantage of a wealth of recent work involving affine-invariant image shape description. A number of shape representation schemes for image analysis have been proposed under the affine transform setting [31, 42, 27] .. Semantics-based processing of trajectory data to extract high-level information, such as activity recognition, has gained interest quite recently [7, 29, 36] . Dagtas et al. [15] propose several motion description schemes that serve as intermediate spatiotemporal models for event-based retrieval of video. They present trajectory-based and trail-based models for motion-based indexing of videos. They provide the implementation details of PICTURESQUE, a video database retrieval system using a query-by-example framework. In [20] , the issue of recognizing a set of plays from American football videos is considered. Using a set of classes each representing a particular game plan and computation of perceptual features from trajectories, the propagation of uncertainty paradigm is implemented using automatically generated Bayesian network. The problem with above approaches is that they are highly domain-dependant, with domain knowledge and sensor dependence on video data being intimately woven into the systems. On similar lines, Nevatia et al. [19] have addressed the issue of activity recognition in single or multiple actor situations which exhibit some specific patterns of whole body motion. Nuria Oliver et al [30] present a system based on Coupled Hidden Markov Models (CHMM) for classifying the types of interactions between humans in a video surveillance setting. Yuri Ivanov et al. [22] address a similar problem using a stochastic context-free grammar parsing mechanism. Stauffer et al. [25] have developed an indexing and retrieval system for video footage in surveillance systems. Their prototype system, Spot, can answer natural language questions about moving objects that appear in a surveillance video footage, like "Did any cars leave the garage towards North." In our previous work on high level trajectory modeling for activity recognition, we have proposed a semisupervised learning approach based on Gaussian Mixture Model (GMM) [5] . In [6] we showed that although the GMM-based approach robustly captures the underlying complex statistical distribution of trajectory data, it fails to model the dynamic time warping aspect of it. For this purpose, we proposed a semisupervised HMM-based trajectory classifier for activity recognition. In the forthcoming sections, we present our view-invariant representation of the trajectories using CSS and CDF-based representation along with HMM-based classifier for activity recognition.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O B I I O O B I I O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O B I I O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O B I


Sections, categories and keywords as interest specification tools for personalised news services Abstract. Through a evaluation of system performance and user satisfaction for the Mercurio system -a system that sends personalised news selections via email -, the general applicability and usefulness of different methods of specifying user interest (sections, categories and keywords) are considered for the general case of digital news services. The specific characteristics distinguishing such systems from more general information systems are outlined and their effect is discussed. An evaluation blueprint for them is proposed starting from information retrieval procedures, existing work on search engine evaluation, and a close study of the working principles and the required evaluation according to the particular properties and conditions of the services under consideration. Actual evaluation results for system tests based both on real users and custom tailored test cases are presented and discussed. Conclusions cover the nature of the information handling tasks that digital news services are faced with, the relative merits of sections, categories, and key words with respect to this particular set of tasks, and the risks of careless application of recall and precision measures in systems such as these.The recent boom in the popularity of the Internet has resulted in a rapid expansion of the range of paradigms of information services available to the common user. One such paradigm is that of systems offering to send users a selection of the daily news by electronic mail. These systems are currently classed as information filtering applications, yet differ from more general information filtering or information retrieval systems in that their contents are completely renewed at regular intervals -daily in the case of most digital newspapers -and remain stable during intervening periods. This simple difference has considerable implications on the general characteristics of the databases that house these contents: they are perishable, throughout their lifetime they remain static, they are usually small, and they hold no claim to universal coverage of any information domain. Although existing systems of this sort are applying wellknown techniques from the fields of information retrieval and information filtering, these particular restrictions governing their operation may affect the general applicability and usefulness of different methods of information access. Furthermore, it is at present unclear to what extent generally accepted evaluation measures for information retrieval, such as recall and precision, can be meaningfully applied in these circumstances. The recent boom in the popularity of the Internet has resulted in a rapid expansion of the range of paradigms of information services available to the common user. One such paradigm is that of systems offering to send users a selection of the daily news by electronic mail. These systems are currently classed as information filtering applications, yet differ from more general information filtering or information retrieval systems in that their contents are completely renewed at regular intervals -daily in the case of most digital newspapers -and remain stable during intervening periods. This simple difference has considerable implications on the general characteristics of the databases that house these contents: they are perishable, throughout their lifetime they remain static, they are usually small, and they hold no claim to universal coverage of any information domain. Although existing systems of this sort are applying wellknown techniques from the fields of information retrieval and information filtering, these particular restrictions governing their operation may affect the general applicability and usefulness of different methods of information access. Furthermore, it is at present unclear to what extent generally accepted evaluation measures for information retrieval, such as recall and precision, can be meaningfully applied in these circumstances.. In order to explore these issues we have carried out thorough evaluations of a system that incorporates these basic methods (newspaper sections, categories and keywords) into the process of selecting the particular information items that are relevant to a given I The research work presented in this paper was partially funded by grant REF TS203/1999 of the ATYCA program, Ministerio de Industria y Energía, Spain user. The Mercurio system allows readers of the newspaper to receive a periodic e-mail message containing the news items that the system finds particularly relevant to the interests of the user, previously defined during registration.. The system constitutes an integrated and customisable option of combining document relevance information from sources of three different kinds: -A prior categorisation by the system owner (documents, i.e. news are sorted into sections by the editor) -Keyword search information over the content of the documents -Automatic categorisation results against an alternative (less domain specific) set of categories An additional control layer is provided that allows users to specify how important each of the different methods of selection is to his particular interests. As such, it presents a flexible, multidimensional and browsable user model, and a set of well founded techniques for user models and information items matching. These features make it an ideal test bed for the questions described above.. Evaluation is carried out in two distinct stages. During the initial stage, evaluation of the system is carried out according to standard practice in the field of information retrieval.. The results fall specially short with respect to determining relative performance of the different methods of information access involved. The second stage of the evaluation includes additional sets of experiments specially designed to shed light on this issue.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O