Static and dynamic consistent perturbation analysis for nonlinear inextensible planar frames HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt età la diffusion de documents scientifiques de niveau recherche, publiés ou non, emanant desétablissements d'enseignement et de recherche français ouétrangers, des laboratoires publics ou privés.the adjective 'numerical', used here to partially connote the algo rithm, should not be understood in the usual meaning of 'approx imation', but rather as an 'automatic procedure' to numerically evaluate the coefficients of some series expansions, that would be impossible to perform manually. For these reasons, the proce dure should be considered as 'asymptotically exact'.This approach, however, does not seem to be thoroughly inves tigated, yet. An attempt along this line was performed in [15] , where, by dealing with continuous beams on sliding supports, an algorithm transforming a nonlinear continuous problem into a nonlinear discrete map was illustrated. Of course the consistent perturbation method can only be applied to systems made of ele ments whose field equations can be asymptotically integrated for general non homogeneous boundary conditions, typically one dimensional elements. However, it is expected that it can be ap plied also for bi or three dimensional elements, in the framework of a semi variational (Kantorovich) approach, as those used, e.g. in [16, 17] .In this paper, the algorithm is detailed for a geometrically non linear planar frame, made of inextensible and shear indeformable beams, whose masses are lumped at their ends. The inextensible model appears physically reasonable and analytically suitable, since it reduces the number of field equations; in contrast, how ever, it calls for a proper treatment of the axial reactive stress, in the framework of a mixed displacement stress approach [18, 19] . Inextensible frames and pantographic type structures have also been used in the literature to develop higher gradient theories of materials endowed with microscopic structures [20, 21] . The gov erning equations are derived via a direct (equilibrium and compat ibility) approach, differently from [18, 19] , where a constrained variational problem was tackled; moreover, more general loading conditions are allowed here.The paper is organized as follows. In Section 2 the nonlinear field equations for a massless beam are asymptotically solved for prescribed displacements at the ends and longitudinal reactive stress, all taken as independent state variables for the element. This solution is asymptotically exact up to third order in the per turbation parameter, which is a measure of the amplitude of the response. In Section 3 a state relation for the element is drawn, and the relations are successively assembled for the frame. Here lumped masses are introduced and discrete nonlinear equations are derived, expressing equilibrium at joints and kinematic com patibility for the elements. In Section 4 the global equations are solved asymptotically, namely: (a) a straightforward expansion is used for static problems, in order to express the structural re sponse as a function of a load parameter; (b) the Multiple Scale Method is applied, to evaluate both the free and forced response of the frame, the latter relevant to harmonic loads in primary res onant conditions. Internal resonances have been so far excluded, but they could be easily accounted for. As a consequence, the re sponse is mono modal, but the contribution of the passive modes (in the Center Manifold perspective) is accounted for. In Section 5 some examples have been worked out and relevant results illus trated. Results provided by the proposed method are validated by a comparison with a FEM solution of both static and dynamic prob lems. Finally, in Section 6, some conclusions are drawn. Two Appendixes report computational details.A straight beam is considered, as an element of a planar frame. The beam is assumed internally constrained and massless. The continuous problem for the single element is formulated, and asymptotically solved.The (static) nonlinear elastic problem for a rectilinear beam, is formulated here. The beam is considered to be axially inextensible and shear indeformable, and modeled as an elastic, polar, one dimensional, internally constrained continuum. The field equilib rium equations, in vector form, turn out to be ( Fig. 1 By introducing the components with respect to the (a x , a y , a z ) element basis, with (a x , a y ) spanning the plane in which the beam bends, and a x aligned with the beam axis in the reference configu ration, it follows that ( Fig. 1 So that the equilibrium conditions (1) lead to three scalar equations: By summarizing, the problem is governed by a mixed differen tial algebraic problem in the unknowns u(s), R B , S B , constituted by:(i) a second order differential equation, obtained combining the first three Eqs. (6); (ii) the relevant boundary conditions Eqs. (7); (iii) the side algebraic conditions (8). By introducing the nondi mensional quantities: MðsÞ u 0 ðsÞð 11ÞSince the solution to Eqs. (7) (10) cannot be pursued in closed form, a perturbation approach is followed. It is based on the hypothesis that all the quantities involved, loads, displacements, active and reactive stresses, are first order quantities, whose order of magnitude is O(e), where 0 < e ( 1, is a small perturbation parameter, acting as bookkeeping, to be eliminated at the end of the analysis. Accordingly, displacements and reactions are ex panded in truncated n term series as:while loads are rescaled as:In the following, n = 3 will be taken and hats omitted. By substitut ing Eqs. (12) and (13) and, similarly, R B2 , R B3 , ... are undetermined at e 2 , e 3 , ... orders. However, as it will be clearer ahead, they will be determined by the joint equilibrium conditions when elements will be assembled.To limit algebra, the loads b x , b y will be assumed ahead to be constant along the beam axis. By performing integrations and accounting for boundary conditions, the e order Eqs. (14) lead to:Eq. (17 3 ) can be solved with respect to S B1 and this, in turn, substi tuted in Eq. (17 1 ), to obtain u 1 (s) expressed as a function of the no dal displacements and load; it results:or, in symbolic form: where F 3 ; S 3 ; U 3 are homogeneous cubic functions and F 1;1 ; S 1;1 ; U 1;1 are bilinear functions of their arguments (with the second order part of the loads equal to zero); moreover F 1;1 ðw 1 ; w 2 Þw T 2 ½@F 2 ðwÞ=@w w w 1 and analogous. The explicit expressions for F k ; S k ; U k are given in the Appendix A. Eqs. (18) (20), together with the series expansions (12), asymptotically solve the problem (10), to within an error smaller than O(e 3 ).The relations relevant to the continuous problem are now recast in discrete form. Then, they are assembled for the structure, and inertia forces acting on masses lumped at joints are taken into ac count at this stage. The static and dynamic problems for the frame are thus formulated.To formulate the problem in a discrete form, it needs to link the external forces acting at the ends of the beam, namely f :T , to the loads b and to the displacements at the same points, u :T , where all the compo nents are expressed in the element basis (Fig. 2) . However, due to the inextensibility condition, forces depend on the transversal disT and on the reactive stress R B , not on the longitudinal displacements u A , u B ; on the other hand, the kine matic constraint (10 4 ) must be appended to this relation as a scalar condition involving all the displacements and R B . Consequently, the relations sought for are of the following type: f fðv; R B ; bÞ 0 u B u A gðv; R B ; bÞ & ð22ÞThey will be named the state relations for the element. Their form suggests to introduce the following vectors:said of the dependent and the independent state variables, respec tively, which permit to write Eqs. (22) are the tangent system matrix and the load vector, respectively; moreover: L : where N 2 (q 1 , b), N 1,1 ((q 1 , b), (q 2 , 0)), N 3 (q 1 , b) are reported in the Appendix B.When a planar frame is considered, made of M elements and N joints, the independent state variables are a set of 3 N displace ments and M reactive stresses, i.e. q :...; R MB Þ T ; here and in the following, an underbar de notes a vector (or a tensor) whose components are evaluated in an extrinsic (global) basis (a x , a y , a z ), with (a x , a y ) spanning the plane of the frame. The relevant elastic problem is governed by a set of (3N + M) equations, made of 3 N joint equilibrium equations, plus M constraint equations. In order to write these equations, the ele ments state vectors q (24) (the apex e being introduced here for con venience), must be expressed in the global basis. However, while displacements u e and forces f e obey the usual rule for basis rota tions, the reactive force R e B must be leaved unaltered in such a change. Therefore, the state vectors for the element transform as:where:is a 7 Â 7 transformation matrix, including the well known 6 Â 6 rotation matrix R Finally, Eqs. (34) are assembled, to express the nodal equilibrium equations P e X eT Q e þ P 0 and the compatibility equations q e = X e q, where X e are collocation matrices and P nodal active forces. The state relation for the frame is therefore:in which a load parameter l, affecting all the forces, has been intro duced for later convenience. Finally, by using Eqs. (35) Eqs. (38) asymptotically govern the static problem for the frame. To tackle the dynamic problem, one has to model the inertia forces. Now, the mass density of the beam, m e , entails inertia forces that should be included in the field equations (1). These call for solving partial differential equations in space and time. A much simpler approach, however, consists in modelling the beams as massless and in lumping their total mass at the ends, although this requires a finer discretization. Lumped mass at the joints intro duce inertia forces in the load vector P, which therefore splits in an active and an inertial part, P: P a + P i . To use quantities consis tent with those in Eqs. (9) is the 7 Â 7 element mass matrix; accordingly, the global inertia force vector is P i ðtÞ P e MqðtÞ, with M : Eqs. (38) and (42), respectively governing the static and the dy namic problems for the frame, are solved.When Eqs. (38) The series q P k e k q k , accordingly, reads: The (undamped) harmonically forced response of the frame is now analyzed.All loads, acting at the nodes, P a (t), and distributed on the beams, b(t), are assumed to be synchronous, obeying to the law: P a ðtÞ P 0 cos Xt; bðtÞ b 0 cos Xt Table 1 Fundamental nonlinear frequency of the frames in Fig. 6 , when a/l = 0.15, vs the total number of elements. where underbars have been omitted. It should be noticed that, since the distributed loads enter quadratic and cubic nonlinearities, para metric excitation terms of frequency X or 2X and external excitation terms of frequency 3X are generated. If, in contrast, only nodal forces are applied to the frame (i.e. b 0 = 0), then, just the external excita tion (46 1 ) exists, while nonlinearities (46 2 4 ) are of autonomous type. The excitation frequency is here assumed to be nearly resonant with the k th natural frequency x k , i.e.:X : x k þ er; r Oð1Þð 47Þwhere r is a detuning parameter. Moreover, any internal reso nances occurring among the natural frequencies, as well as any combination resonances occurring among the external and natural frequencies, are excluded. Hence, the response of the system is mono modal (at e order, but all modes contributes at higher orders, as forced by the leading one). Since the external excitation is reso nant, it is assumed of soft type and therefore shifted at the e 3 order in the perturbation scheme while the nonlinear effects of the distributed loads shift at orders higher than e 3 . Eq. (48 1 ) admits the generating solution:where u k is the (real) kth natural mode, x k the associated linear natural frequency, A k (t 1 , t 2 ) a slow time depending complex ampli tude, and c.c. denotes complex conjugate terms. With Eq. (49), Eq. (48 2 ) reads:where an overbar indicates the complex conjugate. By removing secular terms, d 1 A k = 0 follows, and the Eq. (50) furnishes:where z 11 and z 11 are solution of:Substitution in Eq. (48 3 ) and imposition of solvability condition lead to: whereu k ÞÞ and c 3 = p 0 /2. The steady solution _ a _ c 0 reads:Eq. (55) supplies, in implicit form, the detuning vs the amplitude, r = r(a; l), with the load as a parameter, from which the forcing fre quency amplitude response, X = x k + r(a;l) is derived from Eq.(47). This, also furnishes the natural frequency amplitude response (also known as backbone curve), when the excitation amplitude l = 0 is zero, namely:Finally, the response to the harmonic excitation is evaluated by Eqs. (49) and (51), namely q = q 1 + q 2 , and making use of Eq. (47) and definition of c. In real form it reads:q au k cosðXt cÞþ 1 2 a 2 ðz 11 þ z 11 cos 2ðXt cÞÞ The algorithm developed above has been applied to few sample structures to check efficiency and for illustration purposes. Simple frames have been considered for both nonlinear static and dynamic analyses. Then, a larger structure has been considered and its dynamics studied.A simple planar frame, made of two mutually orthogonal beams, depicted in Fig. 3 , has been considered for static analysis. The beams have the same length l = 18 m, axial stiffness EA = 9.03 Â 10 8 N and bending stiffness EI = 3.17 Â 10 6 Nm 2 ; the inertia radius is q = 0.059 m, therefore the slenderness is k: l/ q = 300. A two beams system is considered, with two different boundary conditions and loads (CS and HH systems in Fig 3) . In the CS system (Fig. 3 a ) , the frame is clamped at one end (A, where the three boundary conditions u A =0, v A = 0 and u A = 0 are en forced) and simply supported to the other (C, where v C = 0 holds).The AB column is loaded by uniformly distributed forces per unit length p 1 = 1.50 Â 10 3 N/m, acting normally and tangentially to the beam axis and a force F 1 = 4.5 Â 10 3 N is applied at the connect ing joint B. In the HH system (Fig. 3 b ) , the frame is hinged at both ends (A and C, where u A =0,v A = 0 and u C =0,v C = 0 are respectively enforced). The BC beam is loaded by a uniformly distributed force per unit length p 2 = 4.00 Â 10 3 N/m, acting orthogonally to its axis, while a force F 2 = 36.00 Â 10 3 N is applied at the connecting joint B. Since the finite element formulated here is asymptotically exact, only two elements are necessary in the analysis, one for each beam, and therefore no convergence analysis needs to be carried out.By taking all the forces proportional to a load multiplier l, the load paths shown in Fig. 4 a,b are obtained, by using Eqs. (44). The plots display the horizontal displacement u B (in meters) of the joint B of the CS system, and the modulus of the vertical displacement v D (in meters) of the midspan joint D of the HH system, both vs the load multiplier l. Here, I denotes the linear path, II the solution truncated at the second order, and III the solution truncated at the third order. It is seen that the resulting effect of nonlinearities is hardening in the CS system and softening in the HH system.In order to validate the method illustrated here, results have been compared to those obtained with FEM models, implemented with the commercial code SAP2000. The FEM solutions have been obtained for few selected values of the load intensity, and the rel evant displacements reported in Fig. 4 by dots. Both an extensible and an inextensible model were considered, the first by accounting for the real axial stiffness of the beams, and the second by artifi cially magnifying the cross section area. However, it was found that relevant results were practically indistinguishable. Overall, the approximation offered by the third order asymptotic solutions is found to be excellent, although the magnitude of the displace ments is very large. It should be stressed that while the perturba tion approach requires solving three linear problems to build up the whole paths, the FEM solution requires solving a nonlinear prob lem for each value of the load multiplier.The deformed shapes of the frame, relevant to l = 1, are illus trated in Fig. 5 ; again, linear (I), quadratic (II), and cubic (III) solu tions are shown (Fig. 5 a,c ) , while the separated (incremental) contributions are illustrated in Fig. 5 b,d . In the first frame, consis tently with the hardening effect, nonlinearities reduce the the hor izontal displacement of the joint B; in the second one, which instead exhibits a softening behavior, nonlinearities increase the vertical displacement of the joint D.The dynamic behavior of the simple frame of Fig. 6 has been analyzed. The two beams have equal length l = 30 m, axial stiffness EA = 1.53 Â 10 9 N, bending stiffness EI = 34.17 Â 10 6 Nm 2 , and inertia radius q = 0,149 m, so that slenderness is k = 200. Moreover, the mass per unit length is m = 5.84 kg/m. The two structures differ in the boundary conditions, namely hinged hinged (HH) and clamped supported (CS). Due to the inertia forces acting along the beam, it is necessary to divide each beam in several elements, in order that the assumed lumped mass model can adequately approximate the continuous model. A preliminary error analysis was therefore carried out, by evaluating the nonlinear fundamental frequency X of both structures, as given by Eq. (56), for a given va lue of the nondimensional amplitude to length ratio, taken as a/ l = 0.15 (see Fig. 6 for the meaning of the amplitude, and for the shape of the 1st nonlinear normal mode). Each beams has been di vided in M/2 elements (hence N = M + 1 joints) and results reported in Table 1 . It is seen that the frequency converges when M = 16, so that the analysis has been carried out with this discretization.For the two frames, the relative correction (X x 1 )/x 1 of the fundamental natural frequency X with respect to the linear one x 1 , is shown in Fig. 7 vs the nondimensional amplitude a/l. While the HH frame possesses a strong softening behavior, the CS frame exhibits an almost linear, very weak hardening behaviour, simi larly to the fixed free beam, as well known in literature (see, e.g.[22]). The response of the HH and CS frames to a harmonic load is now addressed. A horizontal force P(t)=P 0 cos (Xt) is applied at the joint between the two beams, and the frequency response curve given by Eqs. (55) is plotted in Fig. 8 for different values of the nondi mensional excitation amplitude (2P o /c 1 x 1 l = 0, 0.001, 0.005, 0.01) and true or magnified cross section area.The previous semi analytical results were validated via compar ison with a FEM analysis carried out by the code SAP 2000. An har monically varying force of given X frequency was assigned and the nonlinear response recorded for approximately 500 s. After that, a FFT was performed to extract from the response the amplitude of the X frequency content. Both a script in Mathematica and the pro gram QtiPlot were used to check errors in FFT. Especially when a magnified value of the cross section area was considered, this pro cedure required extremely long computations on a PC with a mul ticore CPU (six cores, 12 threads, clock 3.2 GHz, 32 Gb RAM), of the order of 20 h to obtain a single point of the diagram! The relevant results, obtained for few points, both adopting the true or magni fied axial stiffness are reported in Fig. 8 . Again, an excellent agree ment with the consistent perturbation method is found. This latter, however, requires few minutes to evaluate the whole curve.During the forced motion, and according to Eq. (57), the struc ture experiences a linear X harmonic response and a second order response, consisting in a drift and a 2X harmonic motion, so that the resulting response is no more harmonic. Therefore, if some shots are taken during a period of oscillation, the shape of structure modifies itself. These are displayed in Fig. 9 when 2P o /c 1 x 1 l = 0.01, (X x 1 )/x 1 = 0.025, 0.0002 and t = (0, 0.3, 0.6, 0.9)p/X.The dynamics of a more complex frame (see Fig. 10 ), also known as Vierendeel beam, has finally been analyzed. Such a problem, that would be intractable by a traditional manual approach, is indeed useful to show the capability of the proposed numerical perturba tion method. The frame is made of 5 equal square meshes of side l/ 5, with l = 30 m the total length. All the 14 beams are equal, of axial stiffness EA = 1.53 Â 10 9 N, bending stiffness EI = 34.17 Â 10 6 Nm 2 , slenderness k = 40 and mass per unit length m = 5.84 kg/m. Both upper and bottom longerons are clamped on the left and supported on the right. Each beam has been divided in 4 elements, so that the whole structure is made of 56 elements and 54 joints. If symmetry of the structure with respect the horizontal axis is taken into ac count, then the number of joints reduces to 29.The analysis has been focused on several modes; here, only the results relevant to the 1st and 2nd mode are reported. For each considered mode the corrections of the natural frequency versus the amplitude of free oscillation have been evaluated by Eq. (56) and plotted in Fig. 11 . In the figure, the amplitude a represents the maximum vertical displacement of the linear mode, and n = 1, 2 is the number of the mode. It appears that the frame posses a softening behavior, stronger for higher modes.When a harmonic vertical force is applied in the middle of the upper beam, having nondimensional amplitude 2P o /c 1 x 1 l =0, 0.001, 0.005, 0.01, the frequency response curves of Fig. 12 follow from Eq. (56). Fig. 13 reports some shots of the evolution of the deformed con figuration at t = (0, 0.3, 0.6, 0.9)p/X, when the beam is forced clo sely to the 1 st and 2 nd mode, respectively (2P o /c 1 x 1 l = 0.01, (X x)/x = 0.0423, 0.1139)In this paper, a numerical perturbation method for analyzing statics and dynamics of general geometrically nonlinear planar frames, made of inextensible and shear indeformable beams, has been illustrated. The method is based on a 'consistent perturbation analysis', in which the displacement fields are not interpolated be tween the nodes, as in the FEM approach, but rather obtained by asymptotic integration of the field equations, under the simplifying assumption that masses are lumped at the ends. The numerical as pect of the algorithm does not consists in any approximation, but only in the fact that calculations, that would be manually impossi ble for their complexity, are instead rearranged in a way they can be made by a computer. Goal of the analysis is to evaluate the numerical values of the coefficients of the series expansions, which describe the response in terms of a control parameter (e.g. the load multiplier in a static problem, or the forcing frequency, in a dy namic problem). To achieve this goal, few linear problems (typically three) must be solved to obtain the whole curve, instead of a large number of nonlinear problems, as instead required by a continuation incremental iterative method.The paper addressed all the aspect of the problem, namely: (a) the formulation of the model, (b) the solution algorithm, (c) the discussion of the results.Formulations was aimed to find a third order relation between end forces and end displacements. Due to the internal constraints, the longitudinal component of the reactive internal contact forces also appeared in the set of the end forces, counterbalanced by an additional kinematic condition. Modeling was therefore carried out in the context of the mixed formulation of elasticity. Once the relations have been obtained for the element, they were assem bled, to furnish global equation for the frame, leading to perturba tion equations to be solved in cascade.The straightforward perturbation expansion method was used for solving static problems, and the multiple scale method for dy namic problems, both furnishing solutions which are analytical in the control parameter. Attention was focused on evaluating the frequency amplitude relationships, for nonlinear free vibrations as well as harmonic excitation.A number of sample systems were studied, consisting in simple planar frames and a larger Vierendel beam. Several aspects were addressed as: convergence of results, comparison between differ ent order asymptotic solutions, mechanical interpretation of the structural behaviour. Remarkably, the semi analytical results were found to be in excellent agreement with computationally expan sive FEM analyses. In this paper, a numerical perturbation method for analyzing statics and dynamics of general geometrically nonlinear planar frames, made of inextensible and shear indeformable beams, has been illustrated. The method is based on a 'consistent perturbation analysis', in which the displacement fields are not interpolated be tween the nodes, as in the FEM approach, but rather obtained by asymptotic integration of the field equations, under the simplifying assumption that masses are lumped at the ends. The numerical as pect of the algorithm does not consists in any approximation, but only in the fact that calculations, that would be manually impossi ble for their complexity, are instead rearranged in a way they can be made by a computer. Goal of the analysis is to evaluate the numerical values of the coefficients of the series expansions, which describe the response in terms of a control parameter (e.g. the load multiplier in a static problem, or the forcing frequency, in a dy namic problem). To achieve this goal, few linear problems (typically three) must be solved to obtain the whole curve, instead of a large number of nonlinear problems, as instead required by a continuation incremental iterative method.. The paper addressed all the aspect of the problem, namely: (a) the formulation of the model, (b) the solution algorithm, (c) the discussion of the results.. Formulations was aimed to find a third order relation between end forces and end displacements. Due to the internal constraints, the longitudinal component of the reactive internal contact forces also appeared in the set of the end forces, counterbalanced by an additional kinematic condition. Modeling was therefore carried out in the context of the mixed formulation of elasticity. Once the relations have been obtained for the element, they were assem bled, to furnish global equation for the frame, leading to perturba tion equations to be solved in cascade.. The straightforward perturbation expansion method was used for solving static problems, and the multiple scale method for dy namic problems, both furnishing solutions which are analytical in the control parameter. Attention was focused on evaluating the frequency amplitude relationships, for nonlinear free vibrations as well as harmonic excitation.. A number of sample systems were studied, consisting in simple planar frames and a larger Vierendel beam. Several aspects were addressed as: convergence of results, comparison between differ ent order asymptotic solutions, mechanical interpretation of the structural behaviour. Remarkably, the semi analytical results were found to be in excellent agreement with computationally expan sive FEM analyses. Finally, the cubic components are:

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Content-based video genre classification using multiple cues This paper presents an automatic video genre classification system, which utilizes several low-level audio-visual cues as well as cognitive and structural information to classify the types of TV programs and YouTube videos. Classification is performed using support vector machines. The system is integrated to our content-based video processing system and shares the same features that we have been using for high-level feature detection task in TRECVID evaluations. The proposed system is extensively evaluated using complete TV programs from Italian RAI TV channel, from French TV channels, and videos from YouTube on which 99.6%, 99%, and 92.4% correct classification rates are attained, respectively. These results show that the developed system can reliably determine TV programs' genre. It also provides a good basis for classifying genres of YouTube videos, which can be improved by using additional information, such as tags and titles, to obtain more robust results. Further experiments indicate that the quality of video does not influence the results significantly. It is found that the performance drop in classifying genres of YouTube videos is mainly due to the large variety of content contained in these videos. Automatic video genre classification is an important task in multimedia indexing. Several studies have been conducted on this topic [9] . Initial studies have focused on classifying genres of TV programs. Recently classifying genres of web videos has also attracted significant interest [14, 3, 10, 13] . A challenge "Robust, As-Accurate-As-Human Genre Classification for Video" has also been included within Multimedia Grand Challenge [1] .. The studies on TV genre classification mainly utilize audiovisual features. For example, in [7] color statistics, cut detection, camera motion, object motion, and audio are extracted from videos. From these properties further style attributes like camera panning and zooming, scene transitions, object motion, speech, and music are extracted. Finally the video is classified as news, car race, tennis, commercials, or cartoons, using its style profile derived from the style attributes. In [12] , they use camera motion and a dominant color feature as visual features and mel-frequency cepstral coefficients as audio features. The classification is performed using pseudo 2D hidden Markov models.. Besides audio-visual information, the studies on web video genre classification also benefit from additional information such as tags, titles, and surrounding text. In [14] , semantic features based on concept histograms and text features that are derived from the title, tags, and video description are used. Similarly, in the Multimedia Grand Challenge on genre classification [1] , semantic features, titles, tags, relevance from related videos, and user interest deduced from the user's videos are exploited [3, 10, 13] .. A comprehensive overview of the studies on TV genre classification can be found in [8, 9] . In these papers, the authors also present a robust TV genre classification system. In their first system [8] , they utilize visual features based on color -hue, saturation, value, luminance-, texture -Tamura features contrast, directionality-, and temporal activity. The audio features consist of average speech rate and normalized duration values of several audio classes, e.g. silence, noise, and speech. They also utilize structural features, that contain average shot length and distribution of the shot lengths along the video, and cognitive features that contain information about average number of faces per video and their position. They use a multi-layer perceptron for classification. In this study, the authors built a very large dataset containing 262 complete TV programs of around 110 hours. They achieved 92% accuracy over seven genres. The work in [9] extends their previous work with modifications in structural and cognitive features. With these modifications, their performance increased from 92% to 95% on the same dataset.. In this study, we developed an automatic video genre classification system which uses the features that are already extracted for detecting high-level features (HLF) in videos [5, 6] . This brings two main advantages. The first advantage is since the features are already extracted for HLF detection, there is only a classification step as an overhead to our content analysis system. The other advantage is that knowing the genre of the processed video could give cues about the available high-level features in the video, which in turn could lead to a higher performance in the high-level feature detection task. In our study, the used low level visual features are: HSV color histogram, color moments, autocorrelogram, co-occurence texture, wavelet texture grid, and edge histogram. Mel-frequency cepstral coefficients, fundamental frequency, signal energy, and zero crossing rate are the features extracted from audio. In addition to these audio-visual features, cognitive and structural features as suggested in [8] are also exploited. Extensive experiments have been conducted on three different datasets. The experimental results on Italian and French TV programs have shown that the system can reliable classify genres of TV programs. It has been observed that classifying genres of YouTube videos poses a more difficult problem due to the diversity in the content of these videos. However, using audio-visual features still provides a robust basis, whereas further improvement in the performance could be realized by using a training dataset that covers more variety and exploiting additional information, such as tags.. The organization of the paper is as follows. In Sections 2 and 3, low level visual features and audio features are briefly explained, respectively. Cognitive and structural features are shortly described in Section 4. In Section 5, information about the classification process is conveyed. Experimental results are presented and discussed in Section 6. Finally, in Section 7, conclusions are given. In this paper we presented a content-based automatic video genre classification system. The proposed system exploits low level audio and visual features and combines them with cognitive and structural information. In the system, as low level visual features, color and texture descriptors are used. Cognitive information is based on the number of faces per frame and their distribution over video and different locations. Structural information contains average shot duration and distribution of shot lengths. Mel-frequency cepstral coefficients, fundamental frequency, signal energy, and zero crossing rate are computed to represent the audio information of each video.. The proposed approach has been extensively tested on two different TV program datasets, and on YouTube videos. Very high correct classification rates are achieved on TV programs: 99.6% on the RAI dataset and 99% on the Quaero dataset. It has been found that due to high variation in content, it is more difficult to classify genres of YouTube videos. The achieved correct classification rate on these videos is 92.4%. For example, some YouTube football videos show a best-of some football player with music in the background, which makes it more difficult to distinguish it from a commercial or music video. However, with the use of training data that can cover more variety and utilizing text information, the performance is expected to match the ones obtained on the TV program datasets.. Examining each feature individually, color moments, wavelet texture grid, and edge histogram are found to be most useful visual cues; whereas mel-frequency cepstral coefficients have proven to be the most promising audio feature.. In summary, the developed system performs genre classification very efficiently without causing a significant computational load to our high-level feature detection system. Moreover, provided genre information is expected to provide additional cues which can be used to improve the HLF detection system's performance. We used three different datasets to evaluate the proposed automatic video genre classification system. Experiments have been conducted both on complete TV programs and videos from YouTube. Number of videos and represented genres can be viewed in Table 1 .

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


A decision aid for intensity-modulated radiation-therapy plan selection in prostate cancer based on a prognostic Bayesian network and a Markov model Objective-The prognosis of cancer patients treated with intensity modulated radiation therapy (IMRT) is inherently uncertain, depends on many decision variables, and requires that a physician balance competing objectives: maximum tumor control with minimal treatment complications.Methods-In order to better deal with the complex and multiple objective nature of the problem we have combined a prognostic probabilistic model with multi-attribute decision theory which incorporates patient preferences for outcomes.Results-The response to IMRT for prostate cancer was modeled. A Bayesian network was used for prognosis for each treatment plan. Prognoses included predicting local tumor control, regional spread, distant metastases, and normal tissue complications resulting from treatment. A Markov model was constructed and used to calculate a quality-adjusted life-expectancy which aids in the multi-attribute decision process.Conclusions-Our method makes explicit the tradeoffs patients face between quality and quantity of life. This approach has advantages over current approaches because with our approach risks of health outcomes and patient preferences determine treatment decisions. Radiation therapy (RT) for the treatment of cancer has undergone a major change in the past decade. Intensity modulated radiation therapy (IMRT), predicated on technical and physical advances, provides the capability of significantly improved dose localization. This capability allows the physician to prescribe a higher tumor dose with a reduced risk of complications. The competing goals of maximizing tumor control while minimizing normal tissue complications is a complicated decision which is typically made using radiation beam intensity optimization algorithms as de facto decision making algorithms, guided by physician experience, but without explicit representation of physician knowledge.. Optimization algorithms select values of the large number of beam intensities that comprise an IMRT treatment. These algorithms minimize the value of an objective function that attempts to embody the goals of the treatment. Multiple objectives are used for a particular tissue, such as limits on the maximum dose, the minimum dose and the degree of dose uniformity. The total number of objectives used in a particular case depends on the type and location of the tumor but typically ranges from 5 to 20. An important limitation with the current approach is that the multiple objective problem of IMRT is transformed to a single objective function by adding many functions together, each weighted by a scalar importance factor. The values of the importance factors are not known a priori, so a trial-and-error process is required. This process takes extra time, and reduces the solution space that can be explored.. Our approach to improving this process is to couple the results of a multiobjective optimization to a decision aid consisting of a Bayesian network used for prognosis and a Markov model. The multiobjective optimization yields a set of treatment plans (specifically, a Pareto optimal set [1] ) that span the range of possible dose distributions. This range includes plans that provide good coverage of the tumor with significantly high doses to normal tissues to plans that better spare the normal tissues at the expense of lower doses to the tumor.. The Bayesian network -also known as a belief network -provides probabilities for outcomes related to treatment of the tumor and for each of the outcomes related to normal tissue. The network is designed to model the clinical decision making process involved in choosing among different treatment plans, and it predicts the probability that a given plan will result in control of the disease, and treatment complications. Conditional probability tables are derived from prospective clinical trials, results from the research literature, and the beliefs of practicing clinicians. The Bayesian network produces probabilities over some fixed period of time corresponding to the length of the study from which they are derived. We transform these prognostic probabilities to constant annual probabilities for the state transitions in the Markov model. We use utilities from the literature and evaluate the Markov model with a cohort simulation to compute a quality adjusted life expectancy (QALE) for each IMRT treatment plan. This work builds on our original introduction of the Bayesian network to predict disease control using IMRT for prostate cancer [2, 3] . In this paper, we describe a more mature version of the network, and extend it with a Markov model. We have developed a prognostic Bayesian network for prostate cancer IMRT plan selection and extended it to calculate a quality adjusted life expectancy using an established, well understood method, a Markov model evaluated with a cohort simulation. By developing a Bayesian network and a Markov model which work in concert we are able to combine clinical reports with physician experience to make detailed predictions as the efficacy of individual plans, and we are able to rank each plan with a QALE calculation. In the context of multiobjective optimization this model provides a more sophisticated method of ranking treatment plans than is available through current standard approaches. The method is the first of its kind in IMRT to simultaneously consider empirical probabilities, physician degree of belief, and patient preferences. Previous IMRT plan ranking systems have relied on a subset of the parameters we have included, such as cold spots within the PTV, NTCP, EUD or modified versions of these metrics [31, 32] . Only one ranking system seeks to include the beliefs of practicing physicians [33] . None of these works include both clinical results and physician experience, and patient preferences are not considered. These previous publications rank IMRT plans by means of a newly introduced metric, the interpretation of which tends to be opaque to the clinician. The method reported here is the first decision aid in IMRT to combine the major dosimetric parameters of a treatment plan with clinical results, physician experience, and patient preferences.. Previous decision models in prostate cancer have utilized a Markov model and have focused on comparisons of RT vs. radical prostatectomy [34, 35, 29] or on the cost effectiveness of Xray radiation therapy [36] or proton radiation therapy [37] . The transition probabilities for these models were taken from the literature and represented average outcomes based only on a patient's pre-treatment characteristics. Prognosis in these models was meant to assist with the decision between treatment modalities. We consider patients who have already elected to receive RT, and our decision model intends to determine which of several RT treatment plans is preferable for a particular patient.. A Bayesian network is well suited to assist with prognosis in IMRT plan selection because physicians draw upon many sources of information to predict an outcome. Clinical trials report on the predictive power of a set of variables chosen before the study begins. Retrospective studies mine past results for predictive variables, many of which are different than those covered in clinical trials. Physicians must combine these sources and they supplement that information with their subjective degree of belief in outcomes.. The Bayesian network described is a model of the radiation therapy process from the perspective of a clinician prescribing radiation therapy. It does not attempt to model the basic causal biological and radiobiological processes of cancer, although there are some such elements. For example, the PSA value and Gleason score are measures of abnormal antigen production and numbers of mitotically abnormal cells which do provide a connection between therapy and the basic processes. Most of the nodes in the model, however, reflect empirical observations that may or may not have direct biological corollaries. It is anticipated that as understanding of the biological basis of prostate cancer grows, our model will develop in parallel.. Prognosis is of central importance in health care, although there have been few previous efforts at developing prognostic Bayesian networks in cancer [38] [39] [40] . The uncertainty present in predicting the future, however, makes the Bayesian network formalism well suited to this task, and the usefulness of Bayesian networks for medical prognostication is clearly recognized [41, 42] . Part of the problem may lie in the fact that there are often competing outcomes to a cancer treatment, including side effects and complications that can be severe and cannot be neglected when designing a therapy regime for a patient. There are no simple, established methods to use a Bayesian network to rank results with multiple outcomes. Influence diagrams are one attempt to combine utilities and probabilities of outcomes, but as others have noted influence diagrams do not easily lend themselves to the time-dependent task of cost accounting with disease states and utilities [43] .. The limitations of the Bayesian network and Markov model are complemented by the strengths of the other in our decision aid. The Bayesian network calculates the probabilities of the set of possible outcomes. The Markov model uses transition probabilities calculated from those probabilities to predict life expectancies. The use of QALE allows us to combine the disparate outcomes in a simple, clinically useful metric.. Our model shows that for prostate cancer treatment decisions the benefits reaped from dose escalation could be worth the additional cost of incurring a higher probability of treatment complications, at least up to the highest doses for which our model has conditional probabilities. This reflects the way in which physicians act in the field of prostate cancer plan selection, and serves as a valuable proof of principle for our decision aid. Future work will include applying our Bayesian network and Markov model-based QALE ranking to cases of IMRT for head-and-neck cancer, which generally have lower survival probabilities, more severe possible complications, and a considerably more complex decision structure.

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


Survey on liver CT image segmentation methods The segmentation of liver using computed tomography (CT) data has gained a lot of importance in the medical image processing field. In this paper, we present a survey on liver segmentation methods and techniques using CT images, recent methods presented in the literature to obtain liver segmentation are viewed. Generally, liver segmentation methods are divided into two main classes, semi-automatic and fully automatic methods, under each of these two categories, several methods, approaches, related issues and problems will be defined and explained. The evaluation measurements and scoring for the liver segmentation are shown, followed by the comparative study for liver segmentation methods, pros and cons of methods will be accentuated carefully. In this paper, we concluded that automatic liver segmentation using CT images is still an open problem since various weaknesses and drawbacks of the proposed methods can still be addressed. In this paper, semi and automatic liver segmentation in CT images relevant methods and approaches are reviewed. Although, several segmentation techniques have been experimented (e.g. gray level based techniques, learning techniques, model fitting techniques, probabilistic atlases, and level set etc), the problem is still open. The gray level based techniques so far obtain the most promising results in the reviewed methods, but they are not robust to database variations. Gray level estimation does not consider the high variability of CT intensity values. Therefore, the performance could decrease significantly when larger and complex data sets are used. In addition to that, most methods require manual intervention and require some critical parameters to be experimentally estimated, all these facts affects on the method robustness in a negative manner. Learning techniques are strongly dependent on the training set, which must be chosen carefully. Model based techniques and probabilistic atlases suffer from the same difficulty, since they need a lot of data to be properly collected, and manually segmented to produce the model; this is because the obtained model is strongly affected by the training set and by user errors and biases. The initial placement of the model affects the results of the segmentation. Besides, processing non standard liver shapes increases the chances for the algorithms to fail. Regarding to level set approaches, their main drawback is the difficulty in defining a proper speed function and its parameters. Furthermore, a com-parison among different methods is not meaningful because of the lack of a common dataset and a unique measurement. In addition to that, the employed datasets in most researches are very small.

B O B O O B O O B O B 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 B B O O O O O O O O O O O O O O B B O O O B 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 B 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 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 B B 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 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Stochastic adaptive control model for traffic signal systems An adaptive control model of a network of signalized intersections is proposed based on a discrete-time, stationary, Markov decision process. The model incorporates probabilistic forecasts of individual vehicle actuations at downstream inductance loop detectors that are derived from a macroscopic link transfer function. The model is tested both on a typical isolated traffic intersection and a simple network comprised of five four-legged signalized intersections, and compared to full-actuated control. Analyses of simulation results using this approach show significant improvement over traditional full-actuated control, especially for the case of high volume, but not saturated, traffic demand.Far fewer well-tested examples exist of real-time adaptive traffic control systems that react to actual traffic conditions on-line, the most notable among these being the well-known OPAC algorithm (Gartner, 1983) , and RHODES TM , a real-time traffic-adaptive signal control system that uses a traffic flow arrivals algorithm (Head, 1995) based on detector information to predict future traffic volume.In general, two issues must be addressed to achieve real-time adaptive traffic control: (1) development of a mathematical model for the control of the stochastic, highly nonlinear traffic system, and (2) design of an appropriate control law such that the behavior of the system meets certain performance indices (e.g., mini mum queue length, minimum delay time, etc.). Mathematical models used for the representation of traffic phe nomena on signalized surface street networks can be classified into the following three generalized categories:(1) store-and-forward models (Hakimi, 1969; Singh and Tamura, 1974; D'Ans and Gazis, 1976) , (2) disper sion-and-store models (Cremer and Schoof, 1989; Chang et al., 1994) , and (3) kinematic wave models (Ste phanedes and Chang, 1993; Lo, 2001) .There are two fundamental approaches for on-line optimization: binary choice logic and the sequential approach. In the binary choice logic approach, time is divided into successive small intervals, and a binary decision is made either to extend the current signal phase by one interval, or to terminate it. Examples of this approach include Miller's algorithm, traffic optimization logic (TOL), modernized optimized vehicle actua tion strategy (OVA), stepwise adjustment of signal timing (SAST), etc. (Lin, 1989; Lin and Vijayakumar, 1989) . The drawback of this approach is that it only considers a very short future time interval (usually 3-6 s) for the decision, and thus cannot guarantee the overall optimization of the signal operation. In the sequential approach, the length of a decision-making stage is relatively longer (from 50 to 100 s) to more clo sely approach the long-term optimal control. In OPAC, developed explicitly for real-time traffic control, the alternative disadvantages of the binary and sequential approaches are mitigated by incorporating a rollinghorizon approach; however, its application formally is limited to isolated intersections. Artificial neural net works (ANN) also have been applied to finding the solution for traffic control problems (Nakatsuji and Kaku, 1991) through an assumed mapping between the control variables (e.g., the split) and the objective function (e.g., the queue length); the neural network is trained off-line, using the nonlinear mapping ability of ANN, to realize this relationship. Then the signal optimization is performed on-line, using the self-orga nization property of an ANN. The training algorithm is a stepwise method (combination of a Cauchy machine and the ''back-propagation'' algorithm). However, this approach is valid only when the traffic sys tem is in steady state.Although most existing adaptive signal control strategies incorporate an implicit recognition that traffic conditions are time variant due to random processes, they generally adopt explicitly deterministic control models. Additionally, most employ heuristic control strategies without an embedded traffic flow model. Alter natively, the random nature of the traffic system lends itself more directly to a stochastic control approach. In the work reported here, a stochastic traffic signal control scheme, based on Markovian decision control, is introduced. The objective is to develop a real-time adaptive control strategy that explicitly incorporates the random nature of the traffic system in the control. A Markov control model is first developed; then the signal control problem is formulated as a decision-making problem for the Markov model. This approach is tested both on a typical isolated traffic intersection and a simple network comprised of five four-legged signalized intersections, and compared to full-actuated control. Analyses of simulation results using this approach show significant improvement over traditional full-actuated control, especially for the case of high volume traffic demand.A stochastic process x(t) is called Markov (Papoulis, 1984) if its future probabilities are determined by its most recent values; i.e., if for every n and t 1 < t 2 � � � < t n P ðxðt n Þ 6 x n jxðtÞ8t 6 t n�1 Þ ¼ P ðxðt n Þ 6 x n jxðt n�1 ÞÞ:The adaptive control algorithm proposed is based on a discrete-time, stationary, Markov control model (also known as a Markov decision process or Markov dynamic programming) defined on (X, A, P, R), where At a signalized intersection, traffic signals typically operate in one of three different control modes: pretimed control, semi-actuated control and full-actuated control (Wilshire et al., 1985) . In pre-timed control, all of the control parameters are fixed and preset off-line. Off-line techniques (e.g., the various versions of the TRANSYT (Robertson, 1969) family of software packages) are useful in generating the parameters for fixed timing plans for conventional pre-timed urban traffic control systems based on the deterministic traffic conditions during different time periods of the day (e.g., peak hours, off-peak hours). In actuated (both semiand full-)control, the control signal is adjusted in accordance with a ''closed-loop, on-line'' control strategy based on real-time traffic demand measures obtained from detectors; while the controllers themselves respond to the fluctuations of the traffic flows in the network, the base parameters do not. Alternatively, a class of con trol algorithms that includes SCOOT (Split, Cycle And Offset Optimization Technique) (Hunt et al., 1982; Robertson and Bretherton, 1991) and SCATS (Sydney Coordinated Adaptive Traffic System) (Lowrie, 1982) are generally considered to be ''on-line'' algorithms, in which the control strategy is to ''match'' the cur rent traffic conditions obtained from detectors to the ''best'' pre-calculated off-line timing plan.. 1. X, a Borel 2 space, is the state space and every element in the space x 2 X is called a state; 2. A, also a Borel space, is defined as the set of all possible controls (or alternatives). Each state x 2 X is asso ciated with a non-empty measurable subset A(x) of A whose elements are the admissible alternatives when the system is in state x; 3. P, a probability measure space in which an element p ij k denotes the transition probability from state i to state j under alternative action k; and 4. R, a measurable function, also called a one-step reward.. Selection of a particular alternative results in an immediate reward and a transition probability to the next state. The total expected discounted reward over an infinite period of time is defined as. where r( AE ) is the one-step transition reward, b t (0 6 b t 6 1) is the discount factor, and a is the policy. The opti mal reward v * , or the supremum (least upper bound) of V, is defined as v � ðx; a � Þ ¼ sup½V ðx; aÞ�:. It can be obtained by solving a functional equation (also called the dynamic programming equation, or DPE):. where T is a contraction operator 3 and " # N X TvðxÞ ¼ max qðx; aÞ þ b vðxÞp a : ð4Þ ij a2A j¼1. The expected one-step transition reward q(x, a), is defined as N X qðx; aÞ ¼ r a a ð5Þ ij p ij : j¼1. The unique solution of the above DPE can be calculated iteratively by the successive approximation method (Hernandez-Lerma, 1989) 4 :. ij a2A j¼1. Therefore, for a specific control problem, once the transition matrix and the reward matrix are defined, then by maximizing the total expected reward, a policy for choosing an alternative for each state can be obtained. This represents the optimal strategy that should be followed. Traffic signal control is a major ATMS component and its enhancement arguably is the most efficient way to reduce surface street congestion. The objective of the research presented here has been to present a more effective systematic approach to achieve real-time adaptive signal control for traffic networks. In this research, the problem of finding optimal traffic signal timing plans has been solved as a decisionmaking problem for a controlled Markov process. Controlled Markov processes have been used extensively to analyze and control complicated stochastic dynamical systems; its probabilistic, decision-making features match almost perfectly with the design features of a traffic signal control system. The Markovian model devel oped herein as the system model for signal control incorporates Robertson's platoon dispersion traffic model between intersections and employs the value iteration algorithm to find the optimal decision for the controlled Markov process. Analysis of computer simulation results indicates that this systematic approach is more effi cient than the traditional full-actuated control, especially under the conditions of high traffic demand.. There are, of course, significant limitations to the present approach. Most notable is that as the size of the traffic network increases, i.e., the number of nodes/intersections and/or links increases, the dimension of the Markovian control model increases dramatically, requiring more memory space and computation time. This dimensionality issue is very important to real-time implementation, where processing speed is crucial. In the current formulation, one potential solution to this problem is alluded to by decomposing the network into sets of inter-linked network kernels of five intersections that could be handled by distributed/parallel processing protocols; however, no attempt has been made to thoroughly investigate the issues of such decomposition algorithms. Further, before any attempt to implement the results, a comprehensive sensitivity analysis needs to be conducted to study the effect of the various parameters employed in the simulation testing on both the performance of the model as well as on the objective function. Finally, for field testing, the original C language code must first be rewritten into assembly language; then the firmware can be loaded, or ''burned in'' to the PROM (Programmable Read-Only Memory) chip of the controller.

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


Uncovering functional dependencies in MDD-compiled product catalogues A functional dependency is a logical relationship amongst the attributes that define a table of data. Specifically, a functional dependency holds when the values of a subset of the attributes in a dataset determine the values of one or more other attributes. Uncovering such dependencies is utilized in many domains, such as database design. We demonstrate that it can also be utilized in a recommendation context when datasets represent product catalogues. State-of-the-art approaches to discovering functional dependencies require a tabular representation of the data. However, product catalogues can sometimes be defined implicitly, for example, as a set of solutions to a combinatorial problem. Such combinatorial catalogues can have a very large number of products, thus making standard approaches to uncovering functional dependencies inapplicable. In this paper we present the first approach to computing functional dependencies over compiled knowledge representations which can often be small even for huge catalogues. In particular, we develop efficient algorithms that operate over decision diagrams, which allow us to handle catalogues that are out of reach for current approaches. We apply our algorithms to tabular and combinatorial benchmarks and detect a number of properties that could be considered as anomalies in product catalogues. Uncovering functional dependencies is an important problem in many artificial intelligence (AI) domains. Many AI datasets are represented in tabular form, defined in terms of a set of attributes. A dataset might be represented as a database or as a spreadsheet. Determining that one or more attributes is functionally determined by values of other attributes can be of critical importance, e.g. in Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. analyzing the effect of chemical compounds on cancerogenity or studying the shopping habits of the customers.. We suggest that uncovering functional dependencies can also be useful in the context of recommender systems. In particular, various forms of anomalies can be introduced if care is not taken when the catalogue is updated, or if the raw catalogue data is "cleaned" either by vendors before deploying it in an online e-commerce application, or by researchers before subjecting it to various forms of experimental evaluation. We show how one class of such anomalies can be detected using functional dependencies. For instance, ignoring seemingly "redundant" attributes can lead to catalogues where identical product specifications have different prices. Detecting this anomalous case reduces to verifying that the price attribute is functionally determined by the remaining attributes.. However, some product catalogues cannot be represented in a tabular form, but can be defined implicitly, as solutions to combinatorial problems. We refer to such cases as combinatorial catalogues. For example, in the holiday travel domain a solution is any itinerary that fits transportation schedules and accommodation availability. In the product configuration domain a solution is any product that is possible to manufacture in a factory. There could be a huge number of such solutions, making it infeasible to represent the catalogue in a table. Techniques depending on a tabular representation of all solutions would, therefore, be inapplicable.. In this paper we, therefore, present an approach to uncovering functional dependencies when a catalogue is compiled into a multivalued decision diagram (MDD). MDDs are a multi-valued extension of binary decision diagrams [1] which have been proved to be useful for compactly representing solution sets in many combinatorial domains, such as planning, diagnosis, probabilistic reasoning, and product configuration. While the size of an MDD is, in the worst-case, linear in the number of solutions, they can often be exponentially smaller. The algorithms we develop in this paper have quadratic time complexity in the size of the MDD, thus offering potentially dramatic improvements in the complexity of reasoning over large solution spaces. Therefore, we have an approach that outperforms the state-of-the-art whenever a catalogue with a large number of solutions can be represented using a small MDD. In the experimental section we show that such catalogues frequently occur in the product configuration domain. To the best of our knowledge, this is the first approach to functional dependency detection and exploitation over compiled knowledge representations.. The remainder of this paper is organised as follows. Section 2 presents the necessary technical background required throughout the paper. We develop MDD-based algorithms for uncovering various forms of functional dependencies in Section 3. We report on a number of experiments in Section 4 and conclude in Section 5. We presented an approach to computing functional dependencies over an MDD representation of a product catalogue. Unlike state-of-the-art approaches, we do not require that we operate over a tabular representation of data, and hence can handle huge catalogues that have small MDD representations. Using functional dependencies as an analytical tool we discovered that a set of publicly available product catalogues exhibits specific characteristics some of which can be regarded as anomalies in the catalogue definition or in the preprocessing step of the catalogues. Solution Sets. We consider contexts where we are given a set of variables (product attributes) X = {x1, . . . , xn} and an associated set of finite domains D1, . . . , Dn such that each variable xi takes a value from the corresponding set of possible values Di. We are also given a set of solutions (available products) Sol ⊆ D1 × . . . × Dn, which can be defined either explicitly as a set of items in a product catalogue, or implicitly as the set of solutions to a constraint satisfaction problem X, D, C = def {c1, . . . , cm} , in which each ci ∈ C restricts the set of possible choices one can make over a subset of the variables in X. Consider the following example which demonstrates each representation form for a T-shirt catalogue.

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


Explicit error bounds for lazy reversible Markov Chain Monte Carlo Abstract. We prove explicit, i.e., non-asymptotic, error bounds for Markov Chain Monte Carlo methods, such as the Metropolis algorithm. The problem is to compute the expectation (or integral) of f with respect to a measure π which can be given by a density ̺ with respect to another measure. A straight simulation of the desired distribution by a random number generator is in general not possible. Thus it is reasonable to use Markov chain sampling with a burn-in. We study such an algorithm and extend the analysis of Lovasz and Simonovits (1993) to obtain an explicit error bound.

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