Recursive Order Research Articles

A construction for recursive linear orderings

AbstractWe re-express a previous general result in a way which seems easier to remember, using the terminology of infinite games. We show how this can be applied to construct recursive linear orderings, showing, for example, that if there is a linear ordering of type τ, then there is a recursive ordering of type ωβ · τ.

Efficient adaptive algorithms for ARX identification

Two efficient adaptive algorithmic families are developed for multichannel combiners characterized by unequal memory lengths for each input channel. The prewindowed, the covariance, and the sliding window case are addressed. The difference between the proposed methods lies in the way the Kalman gain vector is order-updated in each case. The first algorithm operates on both inputs simultaneously and thus utilizes block multichannel structured order recursions. The resulting scheme is called the diagonal-update algorithm. The second approach updates the Kalman gain in a two-step procedure by first reducing the size of one input and then the other input. The resulting method is called stairwise-update algorithm. Both algorithms are applicable to adaptive ARX (autoregressive exogenous) system identification, to adaptive control, and to the design of decision-feedback equalizers. Simulations are included.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Discrete variable stochastic approximation procedures and recursive autoregressive model identification

Two stochastic approximation procedures are proposed for finding a point attaining the maximum of a regression function defined and observable only at points on a set of discrete variables. The asymptotic convergence property of the procedures is discussed using the theorem of almost supermartingales. The procedures are applied to the recursive identification of autoregressive time series models. The identification procedure consists of a recursive order estimation stage and a recursive autoregressive parameter updating stage, and gives the true autoregressive model or the best autoregressive approximation model.

On a Higher Order Recursive Technique for Frequency Estimation

This paper considers the problem of frequency estimation using recursive schemes. The adaptive notch filter technique is reviewed and an algorithm based on the extended instrumental variable method is proposed. A new method for constructing the instrumental variables and a higher order recursive formulation are considered. Use of hypothesis testing analysis in this algorithm produces an adaptive procedure for constructing the instruments and estimating the frequencies.

Recursive order estimation of stochastic control systems

The predictive least squares criterion for order estimation is combined with an adaptive control strategy minimizing a quadratic cost and applied to multidimensional ARX systems. It is then shown that this combination enables us to estimate, recursively and in a strongly consistent way, both the order and the coefficients of the controlled system, while achieving asymptotically optimal cost.

Identifying Monetary Impacts on Agricultural Prices in VAR Models

Since the mid-1970s, a number of studies have used vector autoregressive (VAR) models to evaluate the magnitude and timing of macroeconomic impacts on agriculture. Use of the VAR method has been viewed as a way to obtain empirical evidence about these impacts that might not emerge from more traditional overidentified and less dynamic econometric models. The hope has been that by placing minimal restrictions on VAR models, the true structure of the economic relationships under investigation can be observed. While the aim is laudable, critics of VAR models have argued that it has had the unfortunate consequence of obscuring some important structural identification issues and seeming to promise that something can be obtained for nothing (Cooley and Leroy, Leamer). They point out that identifying restrictions are required to give economically interpretable meaning to the results of any simultaneous equation model (SEM). In standard practice, identification of VAR models is obtained implicitly by choice of a Choleski decomposition of the covariance matrix of one-step-ahead forecast errors, which is equivalent to imposing a recursive structure for the economy. Of concern is that the recursive structure may not be believable. Further, as Bernanke has observed, examination of ad hoc variants of the recursive order implies a strange set of prior beliefs in which the analyst holds strongly that the system is recursive but is not sure in what order the variables should be arranged. In response to these concerns, several recent papers have pointed out that identification of VAR models does not have to rely on the assumption that the contemporaneous structure is recursive (Bernanke, Sims). Rather, the distinguishing features of a VAR model are, first, that identification rests on the assumption that distinct, mutually orthogonal behavior shocks drive the economy, and, second, that lagged relationships among the endogenous and exogenous variables (if any) are left entirely unrestricted. Given these features, a VAR model must be identified solely by restrictions placed on the contemporaneous interactions among the variables, but the identifying restrictions do not have to preclude simultaneity. In this paper, we elaborate on the structural interpretation of VAR models and use this analysis to examine monetary impacts on agricultural prices. We discuss the identification problem and the implications of a recursive structure. We illustrate our points with respect to the three-variable model of money, industrial prices, and agricultural prices that has been used in several articles to evaluate the effects of monetary shocks on price dynamics (Bessler, Devadoss and Meyers). We then introduce a somewhat richer behavioral model of the macroeconomy and its effects. A simultaneous model is shown to offer a more believable identification of monetary policy and other macroeconomic shocks than a recursive model, and monetary impacts on agricultural prices are assessed.

ESTIMATION OF THE MOVING‐AVERAGE REPRESENTATION OF A STATIONARY PROCESS BY AUTOREGRESSIVE MODEL FITTING

Abstract. The Hannan‐Rissanen procedure for recursive order determination of an autoregressive moving‐average process provides ‘non‐parametric’ estimators of the coefficients b(u), say, of the moving‐average representation of a stationary process by auto‐regressive model fitting, and also that of the cross‐covariances, c(u), between the process and its linear innovations. An alternative ‘autoregressive’ estimator of the b(u) is obtained by inverting the autoregressive transfer function. Some uses of these estimators are discussed, and their asymptotic distributions are derived by requiring that the order k of the fitted autoregression approaches infinity simultaneously with the length T of the observed time series. The question of bias in estimating the parameters is also examined.

On Π1-automorphisms of recursive linear orders

If σ is the order type of a recursive linear order which has a nontrivial automorphism, we let denote the least complexity in the arithmetical hierarchy such that every recursive order of type σ has a nontrivial automorphism of complexity . In Chapter 16 of his book Linear orderings [R], Rosenstein discussed the problem of determining for certain order types σ. For example Rosenstein proved that , where ζ is the order type of the integers, by constructing a recursive linear order of type ζ which has no nontrivial Σ1-automorphism and showing that every recursive linear order of type ζ has a nontrivial Π1-automorphism. Rosenstein also considered linear orders of order type 2 · η, where 2 is the order type of a two-element chain and η is the order type of the rational numbers. It is easily seen that any recursive linear order of type 2 · η has a nontrivial ⊿2-automorphism; he showed that there is a recursive linear order of type 2 · η that has no nontrivial Σ1-automorphism. This left the question, posed in [R] and also by Lerman and Rosenstein in [LR], of whether or ⊿2. The main result of this article is that :

On Π1-automorphisms of recursive linear orders

If σ is the order type of a recursive linear order which has a nontrivial automorphism, we let denote the least complexity in the arithmetical hierarchy such that every recursive order of type σ has a nontrivial automorphism of complexity . In Chapter 16 of his book Linear orderings [R], Rosenstein discussed the problem of determining for certain order types σ. For example Rosenstein proved that , where ζ is the order type of the integers, by constructing a recursive linear order of type ζ which has no nontrivial Σ1-automorphism and showing that every recursive linear order of type ζ has a nontrivial Π1-automorphism. Rosenstein also considered linear orders of order type 2 · η, where 2 is the order type of a two-element chain and η is the order type of the rational numbers. It is easily seen that any recursive linear order of type 2 · η has a nontrivial ⊿2-automorphism; he showed that there is a recursive linear order of type 2 · η that has no nontrivial Σ1-automorphism. This left the question, posed in [R] and also by Lerman and Rosenstein in [LR], of whether or ⊿2. The main result of this article is that :

Efficient covariance ladder algorithms for finite arithmetic applications

This paper is concerned with the problem of constructing numerically robust covariance ladder estimation algorithms for finite arithmetic applications. Conventional least-squares (LS) ladder algorithms suffer from mixed time and order recursive update equations resulting in a poor numerical accuracy when implemented with finite arithmetic. In this paper, a more ‘direct’ and numerically robust computation approach of the ladder update recursions using the most recently introduced algebraic method of generalized residual energies (GRE's) is presented. The new algorithms separate time and order recursions in two independent subalgorithms with a highly modular structure. Besides the general framework, five algorithms of this type are presented. Based on these algorithms, a VLSI ladder chip-set is proposed. Fixed-point simulations of the new VLSI structures are performed for several types of input data. The experimental analysis shows that the new algorithms are superior over conventional techniques and can operate at a multiplier wordlength as low as 8 bits.

Stochastic Approximation Procedure on a Discrete Lattice and its Application to Recursive Time Series Identification

A stochastic approximation procedure is proposed for finding a point on a discrete lattice which gives the maximum of a regression function defined and observable only at points on the discrete lattice. Here, we call a discrete lattice as a set of points of integral multiples of some unit. Asymptotic convergence property of the proposed procedure is discussed. The procedure is applied to recursive identiifcation of autoregressive time series. The identification procedure consists of recursive order estimation stage and recursive updating stage of autoregressive parameters, and gives the true autoregressive model or the best autoregressive approximation model with probability one.

Fast design of multichannel FIR least-squares filters with optimum lag

This paper is mainly concerned with the efficient design of optimum lag FIR least-squares filters. These filters result when the minimization of the total mean-square error encompasses a time shift between the input signal and the desired response. The solution of this problem leads to a family of normal systems of equations whose associated matrix is the same, while the right-hand side members are parametrized by the lag ranging within a specified interval. The fast algorithm derived here for the determination of the optimum lag filter provides the solutions of the above family of systems by step-up, step-down, Levinson-like order recursions coupled with time updating Kalman-like relations and offers a recursive error capability as well. This is achieved by exploiting the near-to-Toeplitz structure of the matrix and a certain shift invariance property the right-hand side members possess. Finally motivated by the optimum lag problem, systems of equations with near-to-Toeplitz parameters are considered and a fast algorithm is developed. The proposed schemes find important applications in seismic signal processing, system identification, economic forecasting, etc.

On decoupling of linear recursions

We show how a well known algorithm to compute solutions of a second order recursion which are unstable both in forward and in backward direction, can be related to a number of other methods. They are: order reduction, invariant imbedding and decoupling based on triangularization. It is shown that these methods in this order form an increasingly general approach to solve the problem. In particular this means that the stability of the first three algorithms can be understood from the theory that has been established for the decoupling algorithm. In this way one does not need to investigate the stability of the large sparse system which is often related to the first method.

線形離散値システムにおけるプロセスと雑音特性の逐次形次数決定法

線形離散値システムにおけるプロセスと雑音特性の逐次形次数決定法

Dimensionally recursive order determination of linear discrete system

It is well known that the determinant ratio of the product moment matrix of observed input/output data gives a quick way of testing the order of a linear discrete system, and that the IPM (instrumental product moment) matrix is useful for eliminating the effect of the noise. This paper describes the dimensionally recursive algorithm for the determinant ratio associated with a new IPM matrix. This IPM matrix is constructed by using the lagged inputs themselves as the instrumental variables so that it would have a nesting structure. The algorithm is derived by taking advantage of its nesting structure. A particular quantity yielded naturally during the algorithm is shown to be a useful order test statistic, and a more robust order statistic is developed by using the parameter estimate given during the algorithm. By Monte Carlo simulations, the test procedures based on these statistics are confirmed to be valid in spite of the simple construction of the. IPM matrix even in the case where the noise is seriall...

Linear difference equations and generalized continued fractions

In one of his papers [5] Gautschi presents an algorithm for determining the minimal solution of a second-order homogeneous difference equation. The method is based on the connection between the existence of a minimal solution of such a difference equation and the convergence of a certain continued fraction. In the present paper, these results are generalized. For this purpose we use the concept of generalized continued fraction. The resulting algorithm is suitable for solving (n+1)-th order recursions (n≥1) for which there existn independent solutions that are dominated by each solution that does not belong to the space spanned by thesen solutions.

Ordinals connected with formal theories for transfinitely iterated inductive definitions

Let Th be a formal theory extending number theory. Call an ordinal ξ provable in Th if there is a primitive recursive ordering which can be proved in Th to be a wellordering and whose order type is &gt; ξ. One may define the ordinal ∣ Th ∣ of Th to be the least ordinal which is not provable in Th. By [3] and [12] we get , where IDN is the formal theory for N-times iterated inductive definitions. We will generalize this result not only to the case of transfinite iterations but also to a more general notion of ‘the ordinal of a theory’.For an X-positive arithmetic formula [X,x] there is a natural norm ∣x∣: = μξ (x ∈ Iξ), where Iξ is defined as {x: [∪μ&lt;ξIμ, x]} by recursion on ξ (cf. [7], [17]). By P we denote the least fixed point of [X,x]. Then P = ∪ξξ holds. If Th allows the formation of P, we get the canonical definitions ∥Th()∥ = sup{∣x∣ + 1: Th ⊢ x ∈ P} and ∥Th∥ = sup{∥Th()∥: P is definable in Th} (cf. [17]). If ≺ is any primitive recursive ordering define Q≺[X,x] as the formula ∀y(y ≺ x → y ∈ X) and ∣x∣≺:= ∣x∣O≺. Then ∣x∣≺ turns out to be the order type of the ≺ -predecessors of x and P is the accessible part of ≺. So Th ⊢ x ∈ P implies the provability of ∣x∣≺ in Th.

Second Order Polynomial Recursions

Divisibility properties of sequences of polynomials $\{ f_n (x)\} $ which satisfy a second order recursion of the form \[ f_{n + 1} (x) = a(x)f_n (x) + b(x)f_{n - 1} (x) \] are considered, with special emphasis on the sequence $\{ q_n (x)\} $ obtained when $a(x) = 1,b(x) = x$, and starting with $q_0 = 0,q_1 = 1$. It is shown that for $n > 2,q_n (x)$ is irreducible over the rationals if and only if n is prime. The further factorization of $q_n (x)$ over $GF(2)$ is characterized. In particular, $q_n (x)$ is irreducible over $GF(2)$ iff p is an odd prime such that $\alpha (p) = {{(p - 1)} / 2 }$, where $\alpha (k)$ is the least positive integer s such that $k| (4^s - 1) $.

Accurate estimates of solutions of second order recursions

Two important types of two dimensional matrix-vector and second order scalar recursions are studied. Both types possess two kinds of solutions (to be called forward and backward dominant solutions). For the directions of these solutions sharp estimates are derived, from which the solutions themselves can be estimated.

Properties preserved under recursion removal

In this paper we report research which is going on at present. Rather than presenting major results, we have selected a number of partial results and techniques in order to indicate a direction we think is promising. In the first section we review methods for converting a program with a calling structure involving recursion into one without such a structure. In the second section we discuss the separation of the aspects of control and storage in this recursion removal process. We show, by example, how the separation provides the possibility of preserving properties other than functional equivalence. These two sections provide a slight generalization of and complement work presented in the expository paper [S2]. In this third section we illustrate implications of invertibility of operations with respect to the recursion removal process. The central example of this section is the Ackerman functional, obtained by schematizing a version of the Ackerman function. With this example, we note that a recursion need not be convertible to a primitive recursion in order to be removable.