dissimilarity metric learning in the belief function framework the evidential k nearest neighbor ek nn method provided a global treatment of imperfect knowledge regarding the class membership of training patterns it has outperformed traditional k nn rules in many applications but still shares some of their basic limitations e g 1 classification accuracy depends heavily on how to quantify the dissimilarity between different patterns and 2 no guarantee for satisfactory performance when training patterns contain unreliable imprecise and or uncertain input features in this paper we propose to address these issues by learning a suitable metric using a low dimensional transformation of the input space so as to maximize both the accuracy and efficiency of the ek nn classification to this end a novel loss function to learn the dissimilarity metric is constructed it consists of two terms the first one quantifies the imprecision regarding the class membership of each training pattern while by means of feature selection the second one controls the influence of unreliable input features on the output linear transformation the proposed method has been compared with some other metric learning methods on several synthetic and real datasets it consistently led to comparable performance with regard to testing accuracy and class structure visualization

implementing dempster s rule for hierarchial evidence this article gives an algorithm for the exact implementation of dempster s rule in the case of hierarchical evidence this algorithm is computationally efficient and it makes the approximation suggested by gordon and shortliffe unnecessary the algorithm itself is simple but its derivation depends on a detailed understanding of the interaction of hierarchical evidence

some novel developments in finite element procedures for gradient dependent plasticity summary improved algorithms are proposed for a gradient plasticity theory in which the laplacian of an invariant plastic strain measure enters the yield function particular attention is given to the type of finite elements that can be used within the format of gradient dependent plasticity assuming a weak satisfaction of the yield function mixed finite elements are developed in which the invariant plastic strain measure and the displacements are discretized two families of finite elements are developed one in which the invariant plastic strain measure is interpolated using c continuous polynomials and one in which penalty enhanced c continuous interpolants are used the performance of both families of finite elements is assessed numerically in one dimensional and two dimensional boundary value problems the regularizing effect of the used gradient enhancement in computations of elastoplastic solids is demonstrated both for mesh refinement and for the directional bias of the grid lines

over nonlocal gradient enhanced plastic damage model for concrete abstract classical continuum models exhibit strong mesh dependency during softening one method to regularize the problem is to introduce a length scale parameter via the nonlocal formulation however standard nonlocal enhancement either by integral or gradient formulation may serve only as a partial localization limiter for many material models the over nonlocal formulation where the weight for the nonlocal value is greater than unity and the excesses compensated by assigning a negative weight to the local value is able to fully regularize certain material models when standard nonlocal enhancement fails to do so a plastic damage model for concrete is formulated with this over nonlocal enhancement via the gradient approach and the full regularizing capabilities demonstrated

second order analysis for thin structures the eective behavior of second order strain energy densities is obtained using relaxation and i convergence techniques the cosserat theory see e g 13 14 and 2 is recovered within a dimension reduction analysis for 3d non linear elastic thin domains with varying profiles homogeneous and inhomogeneous plate models with periodic profiles are treated

gradient enhanced damage for quasi brittle materials summary conventional continuum damage descriptions of material degeneration suffer from loss of well posedness beyond a certain level of accumulated damage as a consequence numerical solutions are obtained which are unacceptable from a physical point of view the introduction of higher order deformation gradients in the constitutive model is demonstrated to be an adequate remedy to this deficiency of standard damage models a consistent numerical solution procedure of the governing partial differential equations is presented which is shown to be capable of properly simulating localization phenomena

on the justification of the nonlinear inextensional plate model we consider a cylindrical three dimensional body made of a saint venant kirchhoff material and we let its thickness go to zero for a specific order of magnitude for the applied loads and under appropriate restrictions on the set of admissible deformations we show that the almost minimizers of the total energy converge toward deformations that minimize the nonlinear bending energy obtained by fox raoult and simo using formal asymptotic expansions our result is obtained by convergence arguments

two component eigenfunction expansion for open systems described by the wave equation i completeness of expansion the concept of eigenfunction expansions for the wave equation is generalized to open systems in which waves escape to the outside these non conservative systems are non hermitian in the usual sense it is shown that the natural framework is an eigenfunction expansion within a two component formalism that treats the wavefunction and its conjugate momentum together provided the system approaches spatial infinity rapidly without tails and possesses spatial discontinuities the expansion in terms of the eigenfunctions which are now quasinormal modes is shown to be valid

asymptotic analysis of linearly elastic shells iii justification of koiter s shell equations we consider as in part i a family of linearly elastic shells of thickness 2 all having the same middle surfaces r3 where r2 is a bounded and connected open set with a lipschitz continuous boundary and l3 r3 the shells are clamped on a portion of their lateral face whose middle line is 0 where 0 is any portion of withlength 0 0 we make an essential geometrical assumption on the middle surfaces and on the set 0 which states that the space of inextensional displacements begin gathered v f omega eta eta i in h 1 omega times h 1 omega times h 2 omega hfill eta i partial v eta 3 0 on gamma 0 gamma alpha beta eta 0 in omega hfill end gathered where gamma alpha beta are the components of the linearized change is metric tensor ofs contains non zero functions this assumption is satisfied in particular ifs is a portion of cylinder and 0 is contained in a generatrix ofs

a hierarchy of plate models derived from nonlinear elasticity by gamma convergence we derive a hierarchy of plate models from three dimensional nonlinear elasticity by convergence what distinguishes the different limit models is the scaling of the elastic energy per unit volume h where h is the thickness of the plate this is in turn related to the strength of the applied force h membrane theory derived earlier by le dret and raoult corresponds to 0 nonlinear bending theory to 2 von karman theory to 3 4 and linearized vk theory to 3 intermediate values of lead to certain theories with constraints a key ingredient in the proof is a generalization to higher derivatives of our rigidity result 29 which states that for maps v 0 1 3 3 the l2 distance of v from a single rotation is bounded by a multiple of the l2 distance from the set so 3 of all rotations

the combination of evidence in the transferable belief model a description of the transferable belief model which is used to quantify degrees of belief based on belief functions is given the impact of open and closed world assumption on conditioning is discussed the nature of the frame of discernment on which a degree of belief will be established is discussed a set of axioms justifying dempster s rule for the combination of belief functions induced by two distinct evidences is presented

two component eigenfunction expansion for open systems described by the wave equation ii linear space structure for a broad class of open systems described by the wave equation the eigenfunctions which are quasinormal modes provide a complete basis for simultaneously expanding outgoing wavefunctions in this paper the linear space structure associated with this expansion is developed under a modified inner product the time evolution operator is self adjoint even though energy is not conserved for the system alone thus the eigenfunctions are mutually orthogonal consequently the usual tools of eigenfunction expansions can be transcribed to these open systems as an example the time independent perturbation theory is developed in straightforward analogy with quantum mechanics giving the shift in both the real part and the imaginary part of the eigenvalues

nonlinear regularization operators as derived from the micromorphic approach to gradient elasticity viscoplasticity and damage the construction of regularization operators presented in this work is based on the introduction of strain or damage micromorphic degrees of freedom in addition to the displacement vector and of their gradients into the helmholtz free energy function of the constitutive material model the combination of a new balance equation for generalized stresses and of the micromorphic constitutive equations generates the regularization operator within the small strain framework the choice of a quadratic potential w r t the gradient term provides the widely used helmholtz operator whose regularization properties are well known smoothing of discontinuities at interfaces and boundary layers in hardening materials and finite width localization bands in softening materials the objective is to review and propose nonlinear extensions of micromorphic and strain damage gradient models along two lines the first one introducing nonlinear relations between generalized stresses and strains the second one envisaging several classes of finite deformation model formulations the generic approach is applicable to a large class of elastoviscoplastic and damage models including anisothermal and multiphysics coupling two standard procedures of extension of classical constitutive laws to large strains are combined with the micromorphic approach additive split of some lagrangian strain measure or choice of a local objective rotating frame three distinct operators are finally derived using the multiplicative decomposition of the deformation gradient a new feature is that a free energy function depending solely on variables defined in the intermediate isoclinic configuration leads to the existence of additional kinematic hardening induced by the gradient of a scalar micromorphic variable

combining multiple classifiers by averaging or by multiplying abstract in classification tasks it may be wise to combine observations from different sources not only it decreases the training time but it can also increase the robustness and the performance of the classification combining is often done by just weighted averaging of the outputs of the different classifiers using equal weights for all classifiers then results in the mean combination rule this works very well in practice but the combination strategy lacks a fundamental basis as it cannot readily be derived from the joint probabilities this contrasts with the product combination rule which can be obtained from the joint probability under the assumption of independency in this paper we will show differences and similarities between this mean combination rule and the product combination rule in theory and in practice

a framework for nonlinear shells based on generalized stress and strain measures a very general class of shell theories is established based on a unified variational principle and on expansions with respect to biorthogonal function systems the shell models derived are able to accommodate the flow theory of large strain thermoelastoplasticity including the effects due to unloading the specific case of a compressible neo hookean material with a von mises yield condition is discussed in detail

exactly solvable path integral for open cavities in terms of quasinormal modes we evaluate the finite temperature euclidean phase space path integral for the generating functional of a scalar field inside a leaky cavity provided the source is confined to the cavity one can first of all integrate out the fields on the outside to obtain an effective action for the cavity alone subsequently one uses an expansion of the cavity field in terms of its quasinormal modes qnms the exact exponentially damped eigenstates of the classical evolution operator which previously have been shown to be complete for a large class of models dissipation causes the effective cavity action to be nondiagonal in the qnm basis the inversion of this action matrix inherent in the gaussian path integral to obtain the generating functional is therefore nontrivial but can be accomplished by invoking a novel qnm sum rule the results are consistent with those obtained previously using canonical quantization

maximal confidence intervals of the interval valued belief structure and applications an unsolved problem in the interval valued belief structure is the existence of maximal confidence intervals referring to the credible and maximal interval valued beliefs assigned to focal elements in which every point can be reached by a combination rule in this study the existence of maximal confidence intervals of the interval valued belief structure was investigated and validated in addition some applications of the maximal confidence interval in constructing valid and normalized interval valued belief structures are discussed a numerical experiment was conducted to illustrate the existence of maximal confidence intervals several propositions and a theorem were further proposed to demonstrate the validity of the maximal confidence interval by using the concept of the maximal confidence interval a procedure was proposed to construct valid and normalized interval valued belief structures and some meaningful guidance for assigning interval valued beliefs was also suggested to the experts finally a series of well designed examples was presented as applications of the maximal confidence interval the proposed procedure and the suggestions they indicated that valid and normalized interval valued belief structures can be achieved accurately and efficiently in practical applications with the proposed procedure and guidance on the basis of the concept of the maximal confidence interval

an evaluation of higher order plasticity theories for predicting size effects and localisation conventional plasticity theories are unable to capture the observed increase in strength of metallic structures with diminishing size they also give rise to ill posed boundary value problems at the onset of material softening in order to overcome both deficiencies a range of higher order plasticity theories have been formulated in the literature the purpose of this paper is to compare existing higher order theories for the prediction of a size effect and the handling of localisation effects to this end size effect predictions for foils in bending are compared with existing experimental data furthermore a study of one dimensional harmonic incremental solutions from a uniform reference state allows one to assess the nature of material localisation as predicted by these competing higher order theories these analyses show that only one of the theories considered the fleck hutchinson strain gradient plasticity theory based upon the toupin mindlin strain gradient framework fleck n a hutchinson j w 1997 strain gradient plasticity adv appl mech 33 295 361 allows one to describe both phenomena the other theories show either nonphysical size effects or a pathologically localised post peak response

jordan blocks and generalized bi orthogonal bases realizations in open wave systems dissipation can sometimes be described by a non hermitian hamiltonian h whose left and right eigenvectors f j fj form a bi orthogonal basis bb for waves in a class of open systems this is known to lead to exact complete bb expansions if f j fj 0 for all j if not normalization seems impossible and many familiar formulae fail examples are given the problem is related to the merging of eigenmodes so that h can only be diagonalized to jordan blocks the resolution involves a generalized bb containing extra vectors whose dynamics are modified by polynomials in the time t the splitting of merged modes under a perturbation is also treated one thus obtains a non trivial extension of the bb formalism for open systems

the intricate global structure of buckled states of compressible columns singularity theory and global multiparameter bifurcation theory are combined to show that nonbifurcating buckled states of compressible nonlinearly elastic columns lie on branches having detailed qualitative properties these properties distinguish the branches globally together with a delicate asymptotic analysis they are used to show that the global disposition of solution branches both bifurcating and not is strikingly different from that for incompressible columns

finite strain logarithmic hyperelasto plasticity with softening a strongly non local implicit gradient framework this paper addresses the extension of a eulerian logarithmic finite strain hyperelasto plasticity model in order to incorporate an isotropic plastic damage variable that leads to softening and failure of the plastic material it is shown that a logarithmic elasto plastic model with a strongly non local degrading yield stress exactly preserves the structure of its infinitesimal counterpart the strongly non local nature of the model makes it an attractive framework for the numerical solution of softening plasticity problems consistent constitutive tangent operators are derived for the particular case of hyperelasto j2 plasticity which are exactly equal to the corresponding infinitesimal tangent operators the finite element implementation along with the geometrically nonlinear contributions and the incremental solution strategy is outlined a benchmark example is solved illustrating the main differences between the purely elasto plastic case and the case with plastic damage finally the main model characteristics and its practical use are emphasized

mass function derivation and combination in multivariate data spaces in this paper we consider the problem of how to derive mass functions systematically from data samples we also consider the ensuing problem of how to combine different mass functions derived in this way we show that a mass function can be efficiently and systematically derived from multivariate data we also demonstrate that combining mass functions obtained in this manner can be done easily the way of deriving and combining mass functions is illustrated with a simple example

the hamiltonian structure of nonlinear elasticity the material and convective representations of solids rods and plates it is our belief that a thorough understanding of the mathematical underpinnings of elasticity is crucial to its analytical and numerical implementation for example in the analysis of rotating structures the coupling of the equations for geometrically inexact models obtained by linearization or other approximations with those for rotating rigid bodies can easily lead to misleading artificial softening effects that can significantly alter numerical results see simo and vuquoc 1986c especially equations 3 and 5 in this paper we consider fully nonlinear geometrically exact models for rods plates and shells which take into account shear and torsion as well as the usual bending effects in traditional rod and plate models these models can be obtained either from the three dimensional theory by a systematic use of projection methods see e g antman 1972 and naghdi 1972 or by a direct approach within the context of cosserat continuum remarkably the two approaches lead to essentially the same form of the governing field equations in the present context we have chosen as a model problem a particular rod model which may be regarded as an extension of the classical kirchhoff love model see love 1944 to include shear deformations as in

an invariant infinitesimal theory of motions superposed on a given motion abstract the development is applicable to any material but special attention is given to elastic solids included as a special case is an infinitesimal theory of elasticity with the following properties 1 it is properly invariant under arbitrary not necessarily infinitesimal superposed rigid body motions 2 it reduces by specialization to the theory of rigid bodies undergoing finite motion and 3 it can be brought into correspondence with the classical linear elasticity through a suitable reinterpretation of the symbols in the constitutive equation of the latter author

justification of the nonlinear kirchhoff love theory of plates as the application of a new singular inverse method in the framework of isotropic homogeneous nonlinear elasticity for a st venant kirchhoff material we consider a three dimensional plate of thickness and periodic in the two other directions using a new method that we call the singular inverse method we prove the existence of a solution rescaled uniformly in for small forces and at the same time we prove the rigorous convergence of this rescaled solution to the solution of the nonlinear kirchhoff love plate model we also state a 3d 2d error estimate

geometrically nonlinear flexural vibrations of plates in plane boundary conditions and some symmetry properties abstract this study is devoted to the derivation of some properties of the von karman equations for geometrically nonlinear models of plates with a boundary of arbitrary shape for applications to nonlinear vibration and buckling an intrinsic formulation of the local partial differential equations in terms of the transverse displacement and an airy stress function as unknowns is provided classical homogeneous boundary conditions with vanishing prescribed forces and displacements are derived in terms of the airy stress function in the case of a boundary of arbitrary geometry a special property of this operator crucial for some energy conserving numerical schemes and called triple self adjointness is derived in the case of an edge of arbitrary shape it is shown that this property takes a simple form for some classical boundary conditions so that the calculations in some practical cases are also simplified the applications of this work are either semi analytical methods of solution using an expansion of the solution onto an eigenmode basis of the associated linear problem or special energy conserving numerical methods

matrix games with payoffs of belief structures imprecise matrix games such as interval valued matrix games and fuzzy matrix games have attracted much interest for a long time most of the previous studies on imprecise matrix games mainly focus on the fuzzy uncertainty of payoffs however the uncertainties of nonspecificity and discord involved in payoffs are not well addressed so far the purpose of this paper is to study the matrix game with such types of uncertainties in order to achieve that purpose we present a matrix game model with payoffs of belief structures so as to integrate discord and nonspecificity the proposed model can be used to express more imprecise interactions between players in the reality besides as another main contribution of the study an effective method for solving matrix games with belief structures payoffs is developed to help us find the equilibrium points of the kind of games at last an example is given to illustrate the proposed model and method

the combination of multiple classifiers using an evidential reasoning approach in many domains when we have several competing classifiers available we want to synthesize them or some of them to get a more accurate classifier by a combination function in this paper we propose a class indifferent method for combining classifier decisions represented by evidential structures called triplet and quartet using dempster s rule of combination this method is unique in that it distinguishes important elements from the trivial ones in representing classifier decisions makes use of more information than others in calculating the support for class labels and provides a practical way to apply the theoretically appealing dempster shafer theory of evidence to the problem of ensemble learning we present a formalism for modelling classifier decisions as triplet mass functions and we establish a range of formulae for combining these mass functions in order to arrive at a consensus decision in addition we carry out a comparative study with the alternatives of simplet and dichotomous structure and also compare two combination methods dempster s rule and majority voting over the uci benchmark data to demonstrate the advantage our approach offers

waves in open systems via a biorthogonal basis dissipative quantum systems are sometimes phenomenologically described in terms of a non hermitian hamiltonian h with different left and right eigenvectors forming a biorthogonal basis it is shown that the dynamics of waves in open systems can be cast exactly into this form thus providing a well founded realization of the phenomenological description and at the same time placing these open systems into a well known framework the formalism leads to a generalization of norms and inner products for open systems which in contrast to earlier works is finite without the need for regularization the inner product allows transcription of much of the formalism for conservative systems including perturbation theory and second quantization

an implicit tensorial gradient plasticity model formulation and comparison with a scalar gradient model many rate independent models for metals utilize the gradient of effective plastic strain to capture size dependent behavior this enhancement sometimes termed as explicit gradient formulation requires higher order tractions to be imposed on the evolving elasto plastic boundary and the resulting numerical framework is complicated an implicit scalar gradient model was thus developed in peerlings peerlings r h j 2007 on the role of moving elastic plastic boundaries in strain gradient plasticity model simul mater sci eng 15 109 120 that has only c0 continuity requirements and its implementation is straightforward however both explicit and implicit scalar gradient models can be problematic when the effective plastic strains do not have smooth profiles to address this limitation an implicit tensorial gradient model is proposed in this paper based on the generalized micromorphic framework it is also demonstrated that the scalar and tensorial implicit gradient models give similar results when the effective plastic strains fluctuate smoothly

variational convergence for nonlinear shell models with directors and related semicontinuity and relaxation results we use a variational convergence method to study the consistency of various cosserat hypotheses in shell theory with the limit nonlinear membrane model derived from three dimensional elasticity in the course of the analysis we introduce a generalization of quasiconvexity that is suitable for problems of the calculus of variations with two vectorial unknowns one of which appears through its gradient the other one through its value in a weak w 1 p l p framework

numerical algorithm for the determination of the potential of a conservative system from its normal mode spectra an efficient numerical algorithm is developed for determining the potential of a conservative system described by a klein gordon or schrodinger equation from its normal mode spectra the algorithm is based on first solving the linear problem of perturbative inversion by means of tikhonov regularization and then using a series of perturbative steps to approach the full potential by newton s method the algorithm is stable and convergence is rapid

conjunctive and disjunctive combination of belief functions induced by nondistinct bodies of evidence dempster s rule plays a central role in the theory of belief functions however it assumes the combined bodies of evidence to be distinct an assumption which is not always verified in practice in this paper a new operator the cautious rule of combination is introduced this operator is commutative associative and idempotent this latter property makes it suitable to combine belief functions induced by reliable but possibly overlapping bodies of evidence a dual operator the bold disjunctive rule is also introduced this operator is also commutative associative and idempotent and can be used to combine belief functions issues from possibly overlapping and unreliable sources finally the cautious and bold rules are shown to be particular members of infinite families of conjunctive and disjunctive combination rules based on triangular norms and conorms

susy transformations for quasinormal modes of open systems supersymmetry susy in quantum mechanics is extended from square integrable states to those satisfying the outgoing wave boundary condition in a klein gordon formulation this boundary condition allows both the usual normal modes and quasinormal modes with complex eigenvalues the simple generalization leads to three features the counting of eigenstates under susy becomes more systematic the linear space structure of outgoing waves nontrivially different from the usual hilbert space of square integrable states is preserved by susy and multiple states at the same frequency not allowed for normal modes are also preserved the existence or otherwise of susy partners is furthermore relevant to the question of inversion are open systems uniquely determined by their complex outgoing wave spectra

elastic surface substrate interactions a theory for three dimensional finite deformations of elastic solids with conforming elastic films attached to their bounding surfaces is described the gurtin murdoch theory incorporating elastic resistance of the film to strain is generalized to account for the effects of intrinsic flexural resistance this modification yields a model that can be used to describe equilibrium deformations in the presence of compressive surface stress fields an associated variational theory is given and material symmetry considerations are discussed the theory is illustrated by examples

an improved method to construct basic probability assignment based on the confusion matrix for classification problem the determination of basic probability assignment bpa is a crucial issue in the application of dempster shafer evidence theory classification is a process of determining the class label that a sample belongs to in classification problem the construction of bpa based on the confusion matrix has been studied however the existing methods do not make full use of the available information provided by the confusion matrix in this paper an improved method to construct the bpa is proposed based on the confusion matrix the proposed method takes into account both the precision rate and the recall rate of each class an illustrative case regarding the prediction of transmembrane protein topology is given to demonstrate the effectiveness of the proposed method

a justification of nonlinear properly invariant plate theories a single asymptotic derivation of three classical nonlinear plate theories is presented in a setting which preserves the frame invariance properties of three dimensional finite elasticity by a successive scaling of the external loading on the three dimensional body the nonlinear membrane theory the nonlinear inextensional theory and the von karman equations are derived as the leading order terms in the asymptotic expansion of finite elasticity the governing equations of the nonlinear inextensional theory are of particular interest where 1 plane strain kinematics and plane stress constitutive equations are derived simultaneously from the asymptotic analysis 2 the theory can be phrased as a minimization problem over the space of isometric deformations of a surface and 3 the local equilibrium equations are identical to those arising in the one director cosserat shell model furthermore it can be concluded that with a regular single scale asymptotic expansion it is not possible to obtain a system of plate equations in which finite membrane strain and finite bending strain occur simultaneously in the leading order term of an asymptotic analysis

nonlinearly elastic shell models a formal asymptotic approach ii the flexural model this paper is the sequel of part i in which the limiting displacement field of a thin shell when its thickness approaches zero is identified as the solution of a two dimensional nonlinear membrane shell model when the geometry of the middle surface of the shell and the boundary conditions allow non zero inextensional displacements the previous membrane limit model is not relevant in this case we show how to update the assumptions on the applied forces acting on the shell so that a limiting model can be derived by an asymptotic analysis furthermore we identify this limit as the two dimensional nonlinear flexural shell model

the role of non linear theories in transient dynamic analysis of flexible structures abstract transient dynamic analysis of flexible structures undergoing large motions is considered for rotating structures it is explicitly shown that appropriate account of the influence of centrifugal force on the bending stiffness requires the use of a geometrically non linear at least second order beam theory use of a first order linearized linear beam theory results in a spurious loss of bending stiffness for a rotating plane beam a set of linear partial differential equations of motion that includes all inertia effects coriolis centrifugal acceleration of revolution and coupling between extensional and flexural deformations is derived from the fully non linear beam theory by consistent linearization the analysis is subsequently extended to the more general case of a plate accomodating shear deformation and undergoing a general three dimensional rotating motion the discretization process of the resulting linear equations of motion for the beam and the plate is also discussed

the random subspace method for constructing decision forests much of previous attention on decision trees focuses on the splitting criteria and optimization of tree sizes the dilemma between overfitting and achieving maximum accuracy is seldom resolved a method to construct a decision tree based classifier is proposed that maintains highest accuracy on training data and improves on generalization accuracy as it grows in complexity the classifier consists of multiple trees constructed systematically by pseudorandomly selecting subsets of components of the feature vector that is trees constructed in randomly chosen subspaces the subspace method is compared to single tree classifiers and other forest construction methods by experiments on publicly available datasets where the method s superiority is demonstrated we also discuss independence between trees in a forest and relate that to the combined classification accuracy