An extension of Fibonaccian search to several variables

Krolak, Patrick D.; Cooper, Leon N.

doi:10.1145/367651.367694

Cited by 23 publications

(10 citation statements)

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“…In each linear search, the convergence limit is computed as limit(M+ 1) = limit(A/)/2'0, where limit(0) =1-0 There are several ways in which Fibonacci search can be extended to multivariate functions. Krolak and Cooper (1963) and Box et al (1969) have proposed methods in which a series of Fibonacci searches are nested within each other. The method used here is based on a series of sequential linear searches in which the value of the function is optimised with respect to one parameter at a time.…”

Section: The Minimum Is Contained In the Interval [D^-d^]mentioning

confidence: 99%

The Calibration of Gravity, Entropy, and Related Models of Spatial Interaction

1972

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This paper presents a methodology for deriving best statistics for the calibration of spatial interaction models, and several procedures for finding best parameter values are described. The family of spatial interaction models due to Wilson is first outlined, and then some existing calibration methods are briefly reviewed. A procedure for deriving best statistics based on the principle of maximum-likelihood is then developed from the work of Hyman and Evans, and the methodology is illustrated using the example of a retail gravity model. Five methods for solving the maximum-likelihood equations are outlined: procedures based on a simple first-order iterative process, the Newton-Raphson method for several variables, multivariate Fibonacci search, search using the Simplex method, and search based on quadratic convergence, are all tested and compared. It appears that the Newton-Raphson method is the most efficient, and this is further tested in the calibration of disaggregated residential location models. IntroductionAn essential part of the process of developing operational models of urban systems involves the choice of appropriate methods of parameter estimation or calibration. In the case of models based upon systems of linear equations researchers have been able to draw upon the well-developed field of linear statistical method, but for many models such linear calibration methods are unsuitable. Several models, whose structure is described by non-linear equation systems, cannot be calibrated using the relatively direct approach of linear estimation, and yet few researchers have attempted to apply appropriate methods of non-linear estimation. A particularly important class or family of models, for which this calibration problem is severe, deals with spatial interaction in geographic systems such as cities and regions. In this paper an attempt will be made to develop a calibration methodology, relevant to models of spatial interaction, and to apply and test several techniques for efficiently solving the model's equations. The types of model dealt with here have been described by various terms. Traditionally such models are called gravity models, in analogy with Newton's concept of gravitation; more recently, similar models have been derived using powerful methods of entropy maximisation from statistical mechanics (Wilson, 1970a). Also included within this class of spatial interaction models are models such as those based on intervening opportunities which can be derived heuristically. Already there are indications that the foundations of a general methodology for calibrating such models are being laid. Two important papers, by Hyman (1969) and Evans (1971), tackle^ the calibration problem as a problem of statistical estimation, although the results which these authors present are specific to trip distribution models as used in transportation planning. This paper builds on the work of Hyman and Evans by extending their calibration methodology to other models in the family of spatial interaction models, and the metho...

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Section: The Minimum Is Contained In the Interval [D^-d^]mentioning

confidence: 99%

The Calibration of Gravity, Entropy, and Related Models of Spatial Interaction

1972

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“…Consider the problem of searching for the minimum of a unimodal function f(z) occurring in the interval (7), and say it is greater than f (8). The fifth evaluation is of f(9) and say it is less than f(S).…”

Section: Relation To Space Searchingmentioning

confidence: 99%

“…The fifth evaluation is of f(9) and say it is less than f(S). The minimum thus lies in (8,10). The sixth step is the evaluation of f(9, 10) giving the slope off at 2 = 9 and so allowing a decision between (8,9) and (9,10).…”

Section: Relation To Space Searchingmentioning

confidence: 99%

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A “Classical” Approach to Central Place Dynamics

Curry

1969

Geographical Analysis

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“…Proc. oftkc [?I Kulikow-ski [2] have independently proposed an extension of Kiefer's unidimensional Fibonacci optimumseeking technique [ 3 ] to multivariable functions. Sugie required that the objective functionf(x) to be maximized be a strictly convex function of the vector x, i.e., that for all 0 <h < 1 and all x1 and X?,…”

Section: Bibliogr~phementioning

confidence: 99%

A multivariable dichotomous optimum-seeking method

Wilde

1965

IEEE Trans. Automat. Contr.

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Before any identification can be performed, it is necessary to determine the plant type. Since methods for doing this are available, it is assumed that this is known at the start of identification. The identification problem then is to determine [FIT Fyfor an!. vector u, where A( u ) is the steady-state operator.It is possible to measure the Jacobian matrices by experimentall>making S changes in the input (letting the system come to the steady-state each time) and recording the resulting outputs. Let si denote a n : Y vector with a one in the i-th position (start the count from zero) and zeros everywhere else. Then the ith column is Since an infinitesimal change cannot be made, a small but finite -, will be used. Thus the identification procedure can be summarized as follows:Step 1: Establish a steady-state with respect to the vector u, and call the output vector observed over a full steady-state period y.Step 2: Establish a steady-state with respect to the vectors ( u + y s i ) with y small for i = O , 1, 2, . . . '%'-I. C a l l the outputs observed over a full period yi, respectively.Step 3: The columns of the matrix Since Fy is specified by the form of the known performance functional, the required identification has been performed. The question arises whether the required information can be obtained without determining the entire matrix aA( u ) / a u .Observe that ( 1 i ) YOW for identification of type-zero plants if A u is chosen to be the vector \\-here RF, is Fy written in reversed order. (The reversal operator R is a square matrix with ones down the secondary diagonal and zeros elsewhere.) Then for small -/ A ( u + YAU) -A(u) -aA(u) R F~ f n. (19) au \Yhen the set of equations (19) are written in reversed order they become R n = BF, (20) where the i, k-th element of the matrix E is B ,al(.\--k-l rkfor 2, k = 0, . . . , -1 ' -1 Thus if the plant has the property that for all i and k , then E = [aA/auIT and the required identification can be performed by making only one variation in the input. Modifications of this simplified procedure can be obtained by first establishing a steady-state with respect to a vector v = Ru. If the plant has the property then using A v = F y will make j the required vector [ a~( u ) / a u ]~~; or if then choosingAv=RFY makes ( 2 3 ) hold. X similar simplification may be possible for type one system., but since in that case A u is an 2 1 ; vector and Fy is a -1 8 vector i t is not apparent how this ma>-be done. R j the required vector. I t ran be readily shown that when A( u ) is linear (it is already assumed to be timeinvariant) that properties (21) and COSCLVSIOSST o recapitulate, it has been shown that the self-optimization problem for unknown and in general nonlinear plants can be solved by an iterative procedure which alternates identification with optimization stages. I t was further pointed out that when a plant is dynamic, care must be taken to insure that the plant is characterized by a unique fixed operator. For type-zero and type-one plant: . . it was shown that by ...

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An extension of Fibonaccian search to several variables

Cited by 23 publications

References 3 publications

The Calibration of Gravity, Entropy, and Related Models of Spatial Interaction

The Calibration of Gravity, Entropy, and Related Models of Spatial Interaction

A “Classical” Approach to Central Place Dynamics

A multivariable dichotomous optimum-seeking method

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