Multiple-output Case Research Articles

An observer design for linear time-delay systems

An observer design is proposed for linear systems with time delays. The key to the design is to find a generalized coordinate change such that, in the new coordinates, all the time-delay terms are injected by the system's output. The existence of such a coordinate change is guaranteed by a rank condition on the observability matrix. The novelty of the proposed design is clearly reflected in the multiple-output case, where a dimensional expansion in the coordinate change could become necessary, and hence is allowed.

Combining two approaches to efficiency assessment

The advent of data envelopment analysis (DEA) enabled the measurement of efficiency to be extended to the case of multiple outputs. Prior to DEA we had the parametric approach based on multiple regression. We highlight some difficulties associated with these two approaches and present a hybrid which overcomes them whilst maintaining the respective advantages of each. This hybrid models the efficient frontier using an algebraic expression; the resulting smooth representation allows all units to be naturally enveloped and hence slacks to be avoided. (Slacks are potential improvements for inefficient units which are not accounted for in the DEA (radial) score, and so have been problematic for DEA.) The approach identifies the DEA-efficient units and fits a smooth model to them using maximum correlation modelling. This new technique extends the method of multiple regression to the case where there are multiple variables on each side of the model equation (eg outputs and inputs). The resulting expression for the frontier permits managers to estimate the effect on their efficiency score of adjustments in one or more input or output levels.

Nonparametric frontier estimation: a robust approach

Most nonparametric methods for estimating production frontiers (data envelopment analysis and free disposal hall (FDH)) are based on envelopment techniques. Statistical inference based on these estimators is available but, by construction, they are very sensitive to extreme values or outliers. We propose a nonparametric estimator, which is more robust to these extreme values. It is based on a concept of expected minimum input function (or expected maximal output function). We show how this function is related to the efficient frontier itself. The resulting estimator is related to the FDH estimator but it will not envelop all the data. The asymptotic theory is provided. Our approach includes the multiple input and multiple output cases.

Dynamic factor demands for US cigarette manufacturing under rational expectations

The rational expectations approach to adjustment cost models for factor demand is used to develop a dynamic model for US cigarette manufacturing. In the present study dynamic production modelling is extended to the case of multiple outputs. This analysis is the first to address cigarette manufacturing allowing for the possible influence of quasi-fixed factors, multiple outputs and rational expectations. Short-, intermediate-, and long-run factor demands are estimated and the presence of adjustment costs tested for in US cigarette manufacturing. The results indicate that there are significant adjustment costs associated with adjusting tobacco stock but not with adjusting the capital stock. Cigarettes produced for exports appear to differ in their marginal cost of production from cigarettes produced for sale in the US market.

Dynamic factor demands for US cigarette manufacturing under rational expectations

The rational expectations approach to adjustment cost models for factor demand is used to develop a dynamic model for US cigarette manufacturing. In the present study dynamic production modelling is extended to the case of multiple outputs. This analysis is the first to address cigarette manufacturing allowing for the possible influence of quasi-fixed factors, multiple outputs and rational expectations. Short-, intermediate-, and long-run factor demands are estimated and the presence of adjustment costs tested for in US cigarette manufacturing. The results indicate that there are significant adjustment costs associated with adjusting tobacco stock but not with adjusting the capital stock. Cigarettes produced for exports appear to differ in their marginal cost of production from cigarettes produced for sale in the US market.

Characterizing an equitable allocation of shared costs: A DEA approach

In many applications to which DEA could be applied, there is often a fixed or common cost which is imposed on all decision making units. This would be the case, for example, for branches of a bank which can be accessed via the numerous automatic teller machines scattered throughout the country. A problem arises as to how this cost can be assigned in an equitable way to the various DMUs. In this paper we propose a DEA approach to obtain this cost allocation which is based on two principles: invariance and pareto-minimality. It is shown that the proposed method is a natural extension of the simple one-dimensional problem to the general multiple-input multiple-output case.

A Cost Function Approach to Dynamic Duality: An Application to the US Cigarette Manufacturing Industry

The dynamic duality econometric approach with the case of multiple outputs is applied to the US cigarette manufacturing industry to test for the presence of adjustment costs and quasifixed inputs with regard to stocks of capital and tobacco. Capital and tobacco stocks are found to be quasi‐fixed inputs and the empirical results indicate that there are significant adjustment costs associated with adjusting these inputs. Short‐ and long‐run own‐ and cross‐price elasticities of factor demands are estimated for domestic and imported tobaccoes, materials, tobacco stocks, and capital. Output demand elasticities are also estimated. The two outputs, cigarettes produced for export and for the US market, are examined for equality of marginal costs. No evidence of differences in marginal costs was found. There is evidence that government restrictions on advertising have negative effects on output demand.

Subspace-based techniques for blind separation of convolutive mixtures with temporally correlated sources

This contribution addresses the blind identification of multiple input multiple output linear finite impulse response systems having a number of inputs less than the number of outputs. Recent publications have proposed an efficient second-order identification method in the single input multiple output case. Based on a subspace analysis, it allows a perfect recovery of the system parameters and excitation in a noise-free environment. Some extensions to the case of multiple inputs are also available under quite specific conditions. In this paper we indicate how to extend the original subspace-based approach to the general case.

On the calculation of the scale elasticity in DEA models

An issue in the multiple-output case within the literature on piecewise linear frontier production functions has been how to determine the nature of the scale properties. It is shown that knowledge of scale efficiency does not permit calculation of scale elasticity or vice versa. For inefficient units there is a relationship between input-saving and output-increasing efficiency measures through the average value of the scale elasticity between the two different reference points on the frontier. A direct approach to determine the scale properties is to calculate the scale elasticity from the dual problem to the calculation of Farrell efficiency measures.

Bifurcation Analysis of Cascade Controlled Nonlinear Systems

It is increasingly understood that limitations to control system performance are due to nonlinear effects in the open-loop system. A plethora of nonlinear behaviors such as large amplitude unforced oscillations, ignition/extinction behavior, and chaotic oscillations are found in chemical processes, particularly chemical reactors. Due to the increasingly integrated nature of unit operations, it is important to design robust control strategies to minimize these types of behavior. In this paper the bifurcation behavior of the cascade control structure is examined, and previous results are extended to the case of multiple outputs being fed-back. An equation relating the feedback gains at the onset of a limit point instability is derived.

Direct separability in multi-output technologies: An application to the Italian agricultural sector

The hypothesis of direct weak separability of the transformation function places fewer restrictions on production technology than those deriving from alternative multistage procedures based on indirect separability assumptions. We derive an explicit form of the necessary and sufficient conditions for direct weak separability in the multiple output case. These conditions are expressed in elasticity terms so they can be readily maintained or tested in commonly used flexible functional forms. Parametric restrictions are derived for the multi-output translog profit function. The procedure is applied to the Italian agricultural sector. Copyright 1996 by Oxford University Press.

Optimal Nonlinear Estimation of Linear Stochastic Systems

This paper presents a new nonlinear method of simultaneous parameter and state estimation called pseudo-linear identification (PLID), for stochastic linear time-invariant discrete-time systems. No assumptions are required about pole or zero locations; nor about relative degree, except that the system transfer function must be strictly proper. Under standard gaussian assumptions, for completely controllable and observable systems, it is proved that PLID is the minimum mean-square-error estimator of the states and model parameters, conditioned on the input and output measurements. It is also proved, given persistent excitation, that the parameter estimates converge a.e. to the true parameter values. All results have been extended to the multiple-input, multiple-output case, but the single-input, single-output case is presented here to simplify notation.

Multiple Outputs and "Homotheticity"

Since Ronald W. Shephard introduced homothetic, scalar-valued production functions [4, 41], it has become well-known that a homothetic production function is completely characterized by linear expansion paths and a cost function in which the output variable is separable from the price vector. In this paper we show that the equivalence between linear expansion paths and cost function separability does not hold for a multiple output technology. Therefore, the very notion of homotheticity needs to be refined and separate concepts of homotheticity must be used to characterize linear expansion paths on the one hand, and cost function separability on the other. One of these characterizations was introduced by Fare and Shephard: they defined the class of input mix independent correspondences and showed that a technology possesses linear [input] expansion paths if and only if the correspondence is mix independent ray-homothetic [2, 135]. We will show that in the multiple output case, these correspondences do not necessarily possess cost functions in which the output vector is separable from the price vector. Given the distinction between technologies possessing linear expansion paths and separable cost functions, we introduce the class of correspondences and prove that the cost function is separable if and only if the correspondence is input-homothetic. We then show that input-homothetic correspondences are ray-homothetic-and therefore have linear expansion paths. In addition, we investigate the properties of multiple output technologies and show that homogeneous technologies possess linear expansion paths. However, there is no connection between homogeneous and input-homothetic correspondences. Therefore, the relationships proved in this paper can be represented as follows:

A neural network adaptive controller for a class of nonlinear systems

Adaptive control techniques can achieve impressive control results for systems which can be accurately modeled or estimated. This paper addresses the case of a nonlinear system for which no model exists, and for which the only available data are a set of inputs and measured state outputs. A back-propagation neural network is used to take the place of the system model in a standard adaptive control strategy. Computer simulation is presented which shows the success of this strategy for a discrete-time single-input, single-output ( SISO) nonlinear system. Results can be extended to the multiple-input, multiple-output ( MIMO) case.

Stochastic system identification with noisy input using cumulant statistics

Addresses the problem of estimating the parameters of stochastic linear systems when the measurements of the system input as well as the system output are noise contaminated. It is assumed that the input is non-Gaussian and the noises are Gaussian. The square root of the magnitude of the fourth cumulant of a generalized error signal is proposed as a performance criterion for parameter estimation. An optimization algorithm is presented. Strong consistency of the proposed parameter estimators is proved under certain sufficient conditions. Both single-input single-output and multiple-input multiple-output cases are investigated. Finally, simulation results are presented to illustrate the proposed approach. >

A Sure Bet on Symmetry

AbstractThis paper argues that assertions appearing in the literature about alleged failure of duality to recover all the desired information about the case of multiple outputs with fixed allocatable inputs are incorrect. The paper shows that through the use of shadow prices of fixed inputs, it is possible to apply the familiar Euler‐Legendre transformation and recover all the desired information. The principal avenue for achieving these results is the process of decoupling the primal from the dual problem.

An algorithm for identifying and selecting the primed implicants of a multiple-output Boolean function

A multiple-output Boolean minimization procedure is presented that generates a minimum cover with computational efficiency by extending the directed search algorithm to the multiple-output case. This procedure is applicable to manual execution as well as to automated execution, and to both conventional two-level gating structures and programmable logic arrays (PLAs).< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

SIMULTANEOUS‐EQUATIONS BIAS AND AGRICULTURAL PRODUCTION FUNCTION ESTIMATION

During the twenty years or so since some early estimations of production functions in agriculture there has been considerable discussion of many of the technical problems associated with such estimation. Although the simultaneous nature of production decisions, leading to the determination of input levels in the production function, was pointed out early in the piece by Marschak and Andrews, little attention has been paid to this aspect of the problem until comparatively recently. The only systematic treatment of these problems is that of Hoch, although there appeared recently in the Economic Record an extended controversy between Konijn and Soper. The purpose of the present paper is to demonstrate that when the profit maximisation conditions are cast in a slightly different form, Hoch's conclusions concerning the existence of simultaneous equations bias cannot be sustained, although there is similarity of assumptions throughout. The treatment initially will be in terms of farms producing a single output. However, the examination of the simultaneous equation problem will be extended to the multiple output case where it will be shown that problems of an entirely different nature arise. The Cobb-Douglas form of production function will be adhered to throughout.