paper error propagation properties of recursive least squares adaptation algorithms the numerical properties of implementations of the recursive least squares identification algorithm are of great importance for their continuous use in various adaptive schemes here we investigate how an error that is introduced at an arbitrary point in the algorithm propagates it is shown that conventional ls algorithms including bierman s ud factorization algorithm are exponentially stable with respect to such errors i e the effect of the error decays exponentially the base of the decay is equal to the forgetting factor the same is true for fast lattice algorithms the fast least squares algorithm sometimes known as the fast kalman algorithm is however shown to be unstable with respect to such errors

the relative efficiency of liu type estimator in a partially linear model in this paper we study the partially linear model y x b f e we introduce a new liu type estimator in a partially linear model then we compared the new estimator with the two step estimator in the mean squared error sense finally we give a simulation study to explain the validity and feasibility of the approach

eigenvalue estimates of an indefinite block triangular preconditioner for saddle point problems in this paper an indefinite block triangular preconditioner for symmetric saddle point problems is discussed the bounds for all the real eigenvalues of the preconditioned matrix are provided and the bounds for the real and imaginary parts of all the complex eigenvalues of the preconditioned matrix are also provided the corresponding theoretical results of paper by bergamaschi 2012 are improved numerical experiments are reported to confirm our theoretical results

structure of indicator function classes with finite vapnik chervonenkis dimensions the vapnik chervonenkis vc dimension is used to measure the complexity of a function class and plays an important role in a variety of fields including artificial neural networks and machine learning one major concern is the relationship between the vc dimension and inherent characteristics of the corresponding function class according to sauer s lemma if the vc dimension of an indicator function class f is equal to d the cardinality of the set fs1n will not be larger than d 0dcnd therefore there naturally arises a question about the vc dimension of an indicator function class what kinds of elements will be contained in the function class f if f has a finite vc dimension in this brief we answer the above question first we investigate the structure of the function class f when the cardinality of the set fs1n reaches the maximum value d 0dcnd based on the derived result we then figure out what kinds of elements will be contained in f if f has a finite vc dimension

the polygon exploration problem we present an on line strategy that enables a mobile robot with vision to explore an unknown simple polygon we prove that the resulting tour is less than 26 5 times as long as the shortest watchman tour that could be computed off line our analysis is doubly founded on a novel geometric structure called angle hull let d be a connected region inside a simple polygon p we define the angle hull of d cal ah d to be the set of all points in p that can see two points of d at a right angle we show that the perimeter of cal ah d cannot exceed in length the perimeter of d by more than a factor of 2 this upper bound is tight

fixed point error analysis of stochastic gradient adaptive lattice filters a theoretical analysis is presented of the stochastic gradient adaptive lattice filter used as a linear one step predictor when the effects of finite precision arithmetic are taken into account only the fixed point implementation is considered both the unnormalized and normalized adaptation algorithms are analyzed expressions of the steady state mean squared values of the accumulated numerical errors in the computation of the reflection coefficients and the prediction errors of different orders have been developed the results show that the dominant term in the expressions for the mean squared values of the numerical errors is inversely proportional to the convergence parameter furthermore they indicate that the quantization errors associated with the reflection coefficients are more critical than those associated with representing the prediction error sequences signals with high correlation among samples produce larger numerical errors in the adaptive lattice filter the authors present several simulation examples that show close agreement with the theoretical results they also show that the gradient adaptive lattice filters have better numerical properties than their transversal counterparts

discretized vapnik chervonenkis dimension for analyzing complexity of real function classes in this paper we introduce the discretized vapnik chervonenkis vc dimension for studying the complexity of a real function class and then analyze properties of real function classes and neural networks we first prove that a countable traversal set is enough to achieve the vc dimension for a real function class whereas its classical definition states that the traversal set is the output range of the function class based on this result we propose the discretized vc dimension defined by using a countable traversal set consisting of rational numbers in the range of a real function class by using the discretized vc dimension we show that if a real function class has a finite vc dimension only a finite traversal set is needed to achieve the vc dimension we then point out that the real function classes which have the infinite vc dimension can be grouped into two categories type a and type b subsequently based on the obtained results we discuss the relationship between the vc dimension of an indicator output network and that of the real output network when both networks have the same structure except for the output activation functions finally we present the risk bound based on the discretized vc dimension for a real function class that has infinite vc dimension and is of type a we prove that with such a function class the empirical risk minimization erm principle for the function class is still consistent with overwhelming probability this is a development of the existing knowledge that the erm learning is consistent if and only if the function class has a finite vc dimension

vc dimension of exterior visibility in this paper we study the vapnik chervonenkis vc dimension of set systems arising in 2d polygonal and 3d polyhedral configurations where a subset consists of all points visible from one camera in the past it has been shown that the vc dimension of planar visibility systems is bounded by 23 if the cameras are allowed to be anywhere inside a polygon without holes here we consider the case of exterior visibility where the cameras lie on a constrained area outside the polygon and have to observe the entire boundary we present results for the cases of cameras lying on a circle containing a polygon vc dimension 2 or lying outside the convex hull of a polygon vc dimension 5 the main result of this paper concerns the 3d case we prove that the vc dimension is unbounded if the cameras lie on a sphere containing the polyhedron hence the term exterior visibility

visibility occlusion and the aspect graph this chapter studies the ways in which the topology of the image of a polyhedron changes with changing viewpoint we catalog the ways that the topological appearance or aspect can change this enables us to find maximal regions of viewpoints of the same aspect we use these techniques to construct the viewpoint space partition vsp a partition of viewpoint space into maximal regions of constant aspect and its dual the aspect graph here we present tight bounds on the maximum size of the vsp and the aspect graph and give algorithms for their construction first in the convex case and then in the general case in particular we give bounds on the maximum size of n 2 and n 6 under an orthographic projection viewing model and of n 3 and n 9 under a perspective viewing model the algorithms make use of a new representation of the appearance of polyhedra from all viewpoints called the aspect representation or asp we believe that this representation is one of the significant contributions of this paper

a floating point arithmetic error analysis of direct and indirect coefficient updating techniques for adaptive lattice filters the ways in which finite precision arithmetic effects can deleteriously manifest themselves in both the stochastic gradient and the recursive least squares adaptive lattice filters are discussed closed form expressions are derived for the steady state variance of the accumulated arithmetic error in a single adaptive lattice coefficient using a floating point stochastic arithmetic error analysis the analytical results show that the performance of adaptive lattice filters using a direct updating computational form is less sensitive to finite precision effects than that of adaptive lattice filters using an indirect updating computational form in addition a method for reducing the self generated noise is presented experimental results obtained on a 32 b floating point hardware implementation of the adaptive lattice filters and with computer simulations are included to verify the analytical results describing the effects of finite precision on adaptive lattice filters

numerically robust least squares lattice ladder algorithms with direct updating of the reflection coefficients new time recursive equations are derived for the reflection coefficients and the ladder gains in the a priori and a posteriori forms of the exact least squares ls lattice ladder filtering algorithms the numerical accuracy of the ls lattice ladder algorithms obtained by use of these new direct time update equations is analyzed and compared to the accuracy resulting from the conventional ls lattice ladder algorithms the analysis and a number of simulation results which are presented lead us to conclude that the new a priori and a posteriori forms of the ls lattice ladder algorithms yield superior performance

error accumulation effects for the a posteriori rlsl prediction filter this paper presents a numerical analysis of the a posteriori recursive least squares lattice filter with indirect updating the important problem of finite precision effects in adaptive rls lattice filters is addressed for this particular filter algorithm but the technique is applicable to many other versions new recursions for the filter residuals and the filter powers forward and backward are derived which describe the effects of arithmetic errors that originate at one stage m and are passed to the next stage m 1 these recursions permit an arithmetic error analysis at the zeroth filter stage to have meaning at all future filter stages effects from machine precision spl eta are linked to the forgetting factor spl lambda and time n to derive bounds for arithmetic error growth at the zeroth filter stage these results provide an explicit upper bound on the forgetting factor that ensures acceptable error propagation additionally we offer a computationally inexpensive way to monitor arithmetic error effects during the normal execution of the filter algorithm through a running error analysis

fixed point error analysis of the normalized ladder algorithm an attempt is made to analyze the fixed point error performance of the normalized ladder algorithm for autoregressive system identification assuming rounding arithmetic a preliminary simulation study of this algorithm has shown that the bias in the estimated reflection coefficients is much more predominant than the variance of the error in the estimate the study therefore is directed to find a model for predicting the bias in the estimated reflection coefficients the analysis shows that the roundoff errors associated with the square root operations in one of the algorithm equations are mainly responsible for the bias in the estimated reflection coefficients these errors arise because of the normalization procedure that makes the quantities under the square root operations very close to one two main results are presented in the paper 1 a simplified theoretical expression for predicting the average bias in the estimated reflection coefficients at any stage is derived 2 a recursive relation for the average error arising from the finite precision arithmetic in the squared residuals is derived this relation illustrates how the errors made in one stage affect the errors in the succeeding stages simulations are performed to check the theoretical models the experimental results agree very closely with the theoretical predictions

pure order recursive least squares ladder algorithms the new class of pure order recursive ladder algorithms porla is presented in this paper the new method obtains the true not approximate least squares ls ladder solution by performing two steps first the covariance matrix of the estimated signal is calculated time recursively and second the reflection coefficients of the ladder form are determined by a pure order recursive procedure initialized from the covariance matrix since time updates in the ladder recursions have been eliminated by the new approach error propagation does not occur and substantial improvements in numerical accuracy compared to conventional mixed time and order recursive ls ladder algorithms are efficiently achieved by the presented algorithms in contrast to conventional ls ladder algorithms fast initial convergence is not corrupted by roundoff error in the new method the true ls pure order recursive ladder algorithm is derived and extended to joint process estimation additionally four computationally efficient approximate ladder algorithms derived from the new approach are given one of them identically represents the well known makhoul covariance ladder algorithm 1977 which can now be computed without levinson recursions therefore this new formulation leads to an improved numerical performance a much simpler implementation scheme and drastically reduced computational costs compared to its widely used traditional counterpart finally dynamic range increasing power normalized versions of the algorithms are also given in the paper

restricted difference based liu estimator in partially linear model partially linear model is useful in statistical model as a multivariate nonparametric fitting method this paper deals with statistical inference for the partially linear model in the presence of multicollinearity when some additional linear restrictions are assumed to hold the corresponding restricted difference based liu estimator for the parametric component is constructed the asymptotically properties of the proposed estimators are discussed finally a simulation study is presented to explain the performance of the estimators

recursive least squares lattice algorithms a geometrical approach several time recursive least squares algorithms have been developed in recent years in this paper a geometrical formalism is defined which utilizes a nested family of metric spaces indexed by the data time interval this approach leads to a simplified derivation of the so called recursive least squares lattice algorithms recursive in time and order in particular it is found that the resulting structure provides a single framework which encompasses an entire family of fairly complex algorithms as well as providing geometrical insight into their behavior

how to learn an unknown environment i the rectilinear case we consider the problem faced by a robot that must explore and learn an unknown room with obstacles in it we seek algorithms that achieve a bounded ratio of the worst case distance traversed in order to see all visible points of the environment thus creating a map divided by the optimum distance needed to verify the map if we had it in the beginning the situation is complicated by the fact that the latter off line problem the problem of optimally verifying a map is np hard although we show that there is no such competitive algorithm for general obstacle courses we give a competitive algorithm for the case of a polygonal room with a bounded number of obstacles in it we restrict ourselves to the rectilinear case where each side of the obstacles and the room is parallel to one of the coordinates and the robot must also move either parallel or perpendicular to the sides in a subsequent paper we will discuss the extension to polygons of general shapes we also discuss the off line problem for simple rectilinear polygons and find an optimal solution in the l 1 metric in polynomial time in the case where the entry and the exit are different points

finite precision error analysis of qrd rls and star rls adaptive filters the qr decomposition based recursive least squares rls adaptive filtering qrd rls algorithm is suitable for vlsi implementation since it has good numerical properties and can be mapped onto a systolic array a new fine grain pipelinable star rls algorithm was developed the pipelined star rls algorithm pstar rls is useful for high speed applications the stability of qrd rls star rls and pstar rls has been proved but the performance of these algorithms in finite precision arithmetic has not yet been analyzed the authors determine expressions for the degradation in the performance of these algorithms due to finite precision by exploiting the steady state properties of these algorithms simple expressions are obtained that depend only on known parameters this analysis can be used to compare the algorithms and to decide the wordlength to be used in an implementation since floating or fixed point arithmetic representations may be used in practice both representations are considered the results show that the three algorithms have about the same finite precision performance with pstar rls performing better than star rls which does better than qrd rls these algorithms can be implemented with as few as 8 bits for the fractional part depending on the filter size and the forgetting factor used the theoretical expressions are found to be in good agreement with the simulation results

a nearly optimal sensor placement algorithm for boundary coverage locating visual sensors in 2d can be often modeled as an art gallery problem tasks such as surveillance require observing or covering the interior of a polygon with a minimum number of sensors or guards for other tasks such as inspection and image based rendering observing the boundaries of the environment is sufficient as interior covering ic also edge covering ec is np hard and no finite algorithm is known for its exact solution approximate ec solutions are provided by many heuristic algorithms but their performances with respect to optimality minimum number of sensors is unknown in this paper we propose a new ec sensors location technique the algorithm is incremental and converges toward the optimal solution it refines an initial approximation provided by an integer covering algorithm iec where each edge is observed entirely by at least one sensor a lower bound for the number of sensors specific of the polygon considered is used at each step for evaluating the quality of the current solution and a set of rules are provided for performing a local refinement to reduce the computational burden the algorithm has been implemented and tests over hundreds of random polygons show that it supplies solutions very close to and often coincident with the lower bound and then suboptimal or optimal in addition the approximate starting solutions provided by the iec algorithms are on the average close to optimum the tight lower bound can also be used for testing other ec sensor location algorithms running times allow dealing with polygons with up to a few hundreds of edges which appears adequate for many practical cases an enhanced version of the algorithm also taking into account range and incidence constraints has also been implemented and tested

efficiency of a liu type estimator in semiparametric regression models in this paper we consider the semiparametric regression model y x b f e recently hu 11 proposed ridge regression estimator in a semiparametric regression model we introduce a liu type combined ridge stein estimator lte in a semiparametric regression model firstly liu type estimators of both b and f are attained without a restrained design matrix secondly the lte estimator of b is compared with the two step estimator in terms of the mean square error we describe the almost unbiased liu type estimator in semiparametric regression models the almost unbiased liu type estimator is compared with the liu type estimator in terms of the mean squared error matrix a numerical example is provided to show the performance of the estimators

least squares order recursive lattice smoothers conventional least squares order recursive lattice lsorl filters use present and past data values to estimate the present value of a signal this paper introduces lsorl smoothers which use past present and future data for that purpose except for an overall delay needed for physical realization lsorl smoothers can substantially outperform lsorl filters while retaining all the advantages of an order recursive structure