Global attractivity in a recursive sequence

In this paper, we Study the invariant intervals, the globally attractivity of the two equilibrium points, and the oscillatory behavior of tile solutions of the difference equation x(n =) ax(n-1) - bx(n-2)/c + x(n-2), n = 1,2,......, where a, b. c > 0. (C) 2003 Elsevier Inc. All rights reserved.

Maximum induced subgraph of a recursive circulant

The recursive circulant RC(2(n), 4) enjoys several attractive topological properties. Let max_epsilon(G) (m) denote the maximum number of edges in a subgraph of graph G induced by m nodes. In this paper, we show that max_epsilon(RC(2n,4))(m) = Sigma(i)(r)=(0)(p(i)/2 + i)2(Pi), where p(0) > p(1) > ... > p(r) are nonnegative integers defined by m = Sigma(i)(r)=(0)2(Pi). We then apply this formula to find the bisection width of RC(2(n), 4). The conclusion shows that, as n-dimensional cube, RC(2(n), 4) enjoys a linear bisection width. (c) 2005 Elsevier B.V. All rights reserved.

Global asymptotic stability in a rational recursive sequence

In this paper, we study the global stability of the difference equation x(n) = a + bx(n-1) + cx(n-1)(2)/d - x(n-2), n = 1,2,....., where a, b greater than or equal to 0 and c, d > 0. We show that one nonnegative equilibrium point of the equation is a global attractor with a basin that is determined by the parameters, and every positive Solution of the equation in the basin exponentially converges to the attractor. (C) 2003 Elsevier Inc. All rights reserved.

On the recursive sequence Xn = axn-1+bxn-2/c+dxn-1xn-2

In this Paper, we study the invariant intervals, the global attractivity of the equilibrium points, and the asymptotic behavior of the solutions of the difference equation x(n) = ax(n-1) + bx(n-2) / c + dx(n-1)x(n-2), n =1, 2, ..., where a greater than or equal to 0, b, c, d > 0. (C) 2004 Elsevier Inc. All rights reserved.

Simplified recursive identifier for ARMA processes

Symmetrical behaviour of the covariance matrix and the positive-definite criterion are used to simplify identification of single-input/single-output systems using recursive least squares. Simulation results are obtained and these are compared with ordinary recursive least squares. The adaptive nature of the identifier is verified by varying the system parameters on convergence.

Adaptive recursive control of a soil testing load frame

Recursive Learning Control (RLC) has the potential to significantly reduce the tracking error in many repetitive trajectory applications. This paper presents an application of RLC to a soil testing load frame where non-adaptive techniques struggle with the highly nonlinear nature of soil. The main purpose of the controller is to apply a sinusoidal force reference trajectory on a soil sample with a high degree of accuracy and repeatability. The controller uses a feedforward control structure, recursive least squares adaptation algorithm and RLC to compensate for periodic errors. Tracking error is reduced and stability is maintained across various soil sample responses.

Finite sample weighting of recursive forecast errors

This paper proposes and tests a new framework for weighting recursive out-of-sample prediction errors according to their corresponding levels of in-sample estimation uncertainty. In essence, we show how to use the maximum possible amount of information from the sample in the evaluation of the prediction accuracy, by commencing the forecasts at the earliest opportunity and weighting the prediction errors. Via a Monte Carlo study, we demonstrate that the proposed framework selects the correct model from a set of candidate models considerably more often than the existing standard approach when only a small sample is available. We also show that the proposed weighting approaches result in tests of equal predictive accuracy that have much better sizes than the standard approach. An application to an exchange rate dataset highlights relevant differences in the results of tests of predictive accuracy based on the standard approach versus the framework proposed in this paper.

Zero attracting recursive least squares algorithms

The l1-norm sparsity constraint is a widely used technique for constructing sparse models. In this contribution, two zero-attracting recursive least squares algorithms, referred to as ZA-RLS-I and ZA-RLS-II, are derived by employing the l1-norm of parameter vector constraint to facilitate the model sparsity. In order to achieve a closed-form solution, the l1-norm of the parameter vector is approximated by an adaptively weighted l2-norm, in which the weighting factors are set as the inversion of the associated l1-norm of parameter estimates that are readily available in the adaptive learning environment. ZA-RLS-II is computationally more efficient than ZA-RLS-I by exploiting the known results from linear algebra as well as the sparsity of the system. The proposed algorithms are proven to converge, and adaptive sparse channel estimation is used to demonstrate the effectiveness of the proposed approach.

A constrained recursive least squares algorithm for adaptive combination of multiple models

In this paper, we develop a novel constrained recursive least squares algorithm for adaptively combining a set of given multiple models. With data available in an online fashion, the linear combination coefficients of submodels are adapted via the proposed algorithm.We propose to minimize the mean square error with a forgetting factor, and apply the sum to one constraint to the combination parameters. Moreover an l1-norm constraint to the combination parameters is also applied with the aim to achieve sparsity of multiple models so that only a subset of models may be selected into the final model. Then a weighted l2-norm is applied as an approximation to the l1-norm term. As such at each time step, a closed solution of the model combination parameters is available. The contribution of this paper is to derive the proposed constrained recursive least squares algorithm that is computational efficient by exploiting matrix theory. The effectiveness of the approach has been demonstrated using both simulated and real time series examples.

An effective recursive partitioning approach for the packing of identical rectangles in a rectangle

In this work, we deal with the problem of packing (orthogonally and without overlapping) identical rectangles in a rectangle. This problem appears in different logistics settings, such as the loading of boxes on pallets, the arrangements of pallets in trucks and the stowing of cargo in ships. We present a recursive partitioning approach combining improved versions of a recursive five-block heuristic and an L-approach for packing rectangles into larger rectangles and L-shaped pieces. The combined approach is able to rapidly find the optimal solutions of all instances of the pallet loading problem sets Cover I and II (more than 50 000 instances). It is also effective for solving the instances of problem set Cover III (almost 100 000 instances) and practical examples of a woodpulp stowage problem, if compared to other methods from the literature. Some theoretical results are also discussed and, based on them, efficient computer implementations are introduced. The computer implementation and the data sets are available for benchmarking purposes. Journal of the Operational Research Society (2010) 61, 306-320. doi: 10.1057/jors.2008.141 Published online 4 February 2009