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Visual interactive fitting of bounded Johnson distributions David J. DeBrota, Robert S. Dittus, Stephen D. Roberts and James R. Wilson SIMULATION 1989; 52; 199 DOI: 10.1177/003754978905200505 The online version of this article can be found at: http://sim.sagepub.com/cgi/content/abstract/52/5/199

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Key words:

interactive distribution

fitting, bounded

Johnson distribution, microcomputer graphics,

subjective density

estimation

Visual interactive fitting of bounded Johnson distributions David J. DeBrota For Health Care Indiana University School of Medicine 1001 West 10th Street

Stephen D. Roberts Regenstrief Institute

Regenstrief Institute

Indianapolis,

Indiana University School of Medicine & Purdue University School of Industrial Engineering

Indianapolis,

IN 46202

Robert S. Dittus

James R. Wilson Purdue University School of Industrial Engineering

Regenstrief Institute Indiana University School of Medicine 1001 West 10th Street

Indianapolis,

ROBERT S. DITTUS is Assistant Professor of Medicine at the Indiana University School of Medicine. He obtained his B.S.I.E. in Industrial Engineering from Purdue University in 1973 and his M.D. at Indiana University in 1978, where he also completed residency and chief residency in internal medicine. He received an M.P.H. in Epidemiology from the University of North Carolina at Chapel Hill in 1984 during his fellowship as a Robert Wood Johnson Clinical Scholac His research has focused

the methodology for medical decision making and the analysis of clinical practices. He has been recently named Teaching and Research Scholar of the American College of Physicians. on

a Professor of Industrial Engineering at Purdue University and Professor of Internal Medicine at Indiana University School

STEPHEN D. ROBERTS is

of Medicine. His academic and

teaching responsibilities are in simulation modeling. His methodological research is simulation language design and includes development of the INSIGHT simulation language. He received his B.S.I.E., M.S.LE., and Ph.D. in industrial engineering from Purdue University and has held research and faculty positions at the University of Florida. He is active in several professional societies, the chair of SIGSIM, and TIMS representative to the board of the Winter Simulation Conference. He is currently on sabbatical leave to the Wolverine Software Corporation.

JAMES R. WILSON is an Associate Professor in the School of Industrial Engineering at Purdue University. He received a B.A. in mathematics from Rice University in 1970, and he received M.S. and Ph.D. degrees in industrial engineering from Purdue University in 1977 and 1979 respecHe has been involved in various simulation studies while work-

ing as a research analyst for the Houston Lighting & Power Company (1970-72) and while serving as a U.S. Army officer (1972-75). From 1979 to 1984, he was an Assistant Professor in the Mechanical Engineering Department of the University of Texas at Austin. His current research interests

West

Lafayette,

IN 47907

IN 46202

DAVID J. DE BROTA is presently a General Internal Medicine Fellow at Indiana University School of Medicine. He received a B.S. in physics and chemistry from Butler University in 1979 and an M.D. degree from Indiana University in 1983. He became a diplomate of the American Board of Internal Medicine in 1987, and continues to practice both outpatient and inpatient medicine. His research interests include the physiciancomputer interface, simulation modeling, Bayesian statistics, interactive computer graphics, and decision analysis.

tively.

IN 46202

include simulation output analysis, variance reduction techniques,

and medical decision analysis. He is member of ASA, IIE, ORSA, SCS, and TIMS. He is currently serving as the President of TIMS/College on Simulation and as the Simulation Department Editor of Management Science.

ranking-and-selection procedures,

a

ABSTRACT method for specifying a bounded interactive visual, Johnson (S ) probability distribution when little or no data are B available for formally identifying and fitting an input process. Using subjective information, the modeler provides values for familiar characteristics of an envisioned target distribution. These numerical characteristics are transformed into parameter values for the probability density function. The parameters can then be indirectly manipulated, either by revising the desired numerical values of the function’s specifiable characteristics or by directly altering the shape of the displayed curve. Interaction with a visual display of the fitted density permits the modeler to conveniently obtain a more realistic representation of an input process than was previously possible. The techniques involved have been packaged into a public-domain microcomputer-based software system called VISIFIT. We present a

INTRODUCTION Probability distributions serve as input models for most stochastic simulations. When a random sample of an input process is available, we can use a variety of approaches to identify and fit an appropriate statistical distribution (Hahn and Shapiro 1967, Law and Vincent 1987). One often-recommended approach is to create a histogram of the sample data set, select candidate distributions suggested by the histogram’s shape, fit the associated parameter values to the data, and then choose the parameterized distribution that best represents the data set (Law and Kelton 1982, Roberts 1983). However, when we cannot or do not wish to make direct observations of a system, we must make use of subjective information such as the modeler’s intuition and his experience in working with the system under study or with similar systems.

199


In the absence of adequate data, a target distribution must be specified according to envisioned or desired characteristics. Unfortunately, even though modelers should employ all available subjective information, they more typically limit themselves to convenient or simple descriptions of the target distribution. Users of GPSS find it easy to use the uniform distribution, conveniently specified by just the mean (midpoint) and spread (half-range). When a more flexible representation is needed, the triangular distribution is frequently employed because it requires only a mode and two end points. It is very common to employ triangular

distributions for &dquo;first-cut&dquo; simulation models (Roberts 1983, Wilson et al. 1982). Distributions that must be specified by their moments, or by more abstract shape parameters, are less attractive because these parameters are difficult to estimate without data, and the dependence of common distributional characteristics on these parameters is more complex. If a variety of inputs are to be modeled, then one must be familiar not only with a sizeable repertoire of statistical distributions, but also with the various methods by which each distribution’s parameter values are chosen so as to match selected properties of a target input process. As a result, there is a tendency to rely on just a few &dquo;standard&dquo; distributions - those with a small number of easily determined parameters. This practice may ignore important information and compromise a model’s overall validity, as when a process is known to follow a skewed distribution but is modeled by a normal distribution for convenience. In this paper

visual interactive approach to fitting distributions when little or no sample data are available. We use a single, highly flexible distributional form-the bounded Johnson (SB)-and make the estimation of its parameters a relatively easy task. The modeler may describe an initial target distribution with a combination of several convenient characteristics. He may then interactively modify the candidate distribution on a graphical display screen, experimenting with different curve shapes until a satisfactory fit is obtained. This approach does not typically yield a &dquo;standard&dquo; distribution; but fortunately practitioners are often indifferent to the exact functional form employed, so long as the desired characteristics are achieved and the fitted distribution has an intuitively reasonable shape. Our principal goals are to relieve the modeler of the burdens imposed by distribution choice and parameter value determination and to permit all available subjective information to be easily captured in a distribution we

present

a

specification. This paper is organized as follows. In Section 2 we discuss the distributional form on which our fitting procedure is based. Section 3 details the information required to describe a subjective distribution unambiguously. Section 4 presents VISIFIQ our software for _VISuaI Interactive FITting of distributions. Finally, Section 5 contains our conclusions on the merits and future appl ications of our methodology.

statistical moments of the envisioned distribution, information not readily obtained without data. We chose to

use

the Johnson translation system, and

more

specifically the Johnson S8 distribution (the S8 referring to the Bounded System). This choice was motivated by several considerations. Recently Swain, Venkatraman, and Wilson (1988) have developed FITTR1, a software package for fitting all types of Johnson distributions to data using a variety of fitting criteria. The methods proposed in this paper complement that work by providing a means of obtaining a parameterization of a Johnson SB distribution with minimal data. In contrast to the absolute lambda distribution (Schmeiser and Deutsch 1977) the Johnson S8 density is smooth (continuously differentiable) everywhere and it is capable

of assuming a wide variety of intuitively appealing shapes. Finally, the univariate Johnson system easily extends to multivariate systems (Johnson 1949b, Johnson 1987) and both univariate and multivariate Johnson systems are available in the INSIGHT simulation language (Roberts 1983). We confined ourselves to the SB subsystem of the Johnson translation system because it matched well our notions of the general characteristics of many potential envisioned target distributions. S8 distributions are bounded and they are capable of matching the skewness and kurtosis of most practical distributions. Realworld measurements are always bounded, even if only by the limits of technology. The SB is capable of assuming U-shaped bimodal forms, but because we seek to model fundamental input processes, we consider only unimodal Johnson S8 distributions to be appropriate. The SB is defined as a translation of the standard normal distribution. If Z is a standard normal variate, then X has an SB distribution if

,

where y and 6 are shape parameters of the associated 58 distribu0), the location parameters is the lower end point of the 58 distribution, and the scale parameter X is the range of the 58 distribution (with X > 0), such that < X< ~ + À. Once estimates of the parameters ~, X, my, and 6 are obtained, variate generation for the 58 is easily accomplished by translation of samples from the standard normal distribution with the above equation. Parameter estimation, therefore, is the main issue in the use of 58 distributions for simulation input modeling. Now if X has an 58 distribution with parameters y, 6, X, and ~, then the &dquo;standardized&dquo; variate tion (with 6 >

lies between zero and one with the same shape parameters as X but with location parameter zero and scale parameter one. The density of Y is then given by gohnson 1949a):

2. CHOICE OF DISTRIBUTION FAMILY

There are several general distribution families that could be considered as a basis for specification including the Burr distribution (Burr 1973), the Lambda distribution (Tukey 1960) and its extensions (Ramberg and Schmeiser 1972, Ramberg and Schmeiser 1974, Schmeiser and Deutsch 1977), the Pearson distributions Johnson and Kotz 1970), and the Johnson translation system (Johnson 1949a). Also, general techniques for representing flexible input models based on series expansions (Johnson and Kotz 1970) such as the Cornish-Fisher expansion, the Gram-Charlier series, and the Edgeworth series might be considered. However these series expansions were not pursued because they depend on knowing 1

The VISIFIT sohware is in the public domain and is available from the authors upon request.

can take a surprising variety of shapes, as illustrated Figure la. They may be sharply peaked or &dquo;broad-shouldered;’ symmetric or skewed. Although S8 densities do not closely fit all triangular distributions, they can serve as an appealing alternatives, given reasonable assumptions about spread, such as assuming the standard deviation to be one-sixth of the range (see Figure lb). They can approximate normal distributions over finite ranges (Figure 1c),

S8 densities in

and can match up to the first four moments of unimodal beta distributions (Figure 1c). They are also capable of certain bimodal forms (Figure 2), but these are generally less useful for the description of real-world processes.

200


Figure 1c. Figure 1d. Figure 1. Unimodal SB Distributions. Figure la shows various symmetric and skewed SB densities. Figure lb shows S8 densities (with standard deviations equal to one-sixth of their ranges) matching the ranges and modes of triangular distributions . Figure 1c shows S8 densities approximating normal distributions. Figure ld shows S8 densities matching the ranges, means, and standard deviations of unimodal beta distributions. the 58 distribution has been difficult to work with because of the mathematically complex relationship of its shape to the parameters my and 6. There are no convenient explicit equations relating the mode or any of the moments of an S, distribution to its parameter values. Therefore, for the distribution to be useful, the shape parameters must be recast into familiar terms that correspond to the envisioned characteristics of a target distribution.

Historically

Figure 2. Bimodel SB Distributions.

3. SUBJECTIVE SPECIFICATION OF PROBABILITY DISTRIBUTIONS Describing a distribution in sufficient detail to permit its approximation by a parameterized functional form is a nontrivial task,

when restricting consideration to smooth, thin-tailed, unimodal densities. Typically, numerical measures of central tendency, variability, and other complex nuances of a density’s &dquo;shape&dquo; are employed. Familiar examples include the mean, standard deviation, skewness, and kurtosis. While these statistical descriptors are easy to obtain from raw data, they are difficult to estimate for an envisioned distribution. The mean of an asymmetric, bounded distribution rarely coincides with other common measures of central tendency such as the mode, median, and midrange; and inexperienced estimators are frequently unable to make the proper distinctions among these measures (Spencer 1963). Subjective estimates of means are influenced by distributional variance and skewness, and may be biased (Beach and Swenson 1966). Intuitive variability estimates are inappropriately correlated with the magnitude of the mean (Lathrop 1967). Descriptors defined in terms of a distribution’s higher moments are for practical purposes unavailable except by calculation from data. even

We believe that a target distribution’s mode is more easily specified than any other measure of central tendency. It is a natural, easily understood &dquo;best guess&dquo; of what one is most likely to see on any single realization of the target random variable. Unlike the mean, the mode is not necessarily tied to the behavior of the distribution

201


in its tails; and unlike the median, it is not necessarily tied to the degree of asymmetry in the distribution. For skewed distributions, estimates of the mode and median are demonstrably better than

(Peterson and Miller 1964). In addition to the end points and mode, which suffice for the triangular distribution, at least one other descriptor is necessary to uniquely specify the more complex functional form of the Johnson S8 distribution. Fortunately, percentile points for envisioned distributions can be subjectively estimated with accuracy (Kahneman, Slovic, and Tversky 1982). An S8 distribution can be uniquely determined by its end points together with (a) two percentile points, or (b) the mode and one percentile point. Doubilet et al. (1985) have developed a method for the estimation of a logisticnormal distribution (which is a Johnson S8 distribution with 1.0) from the mean and either the 5th or 95th ~ 0.0 and X estimates of the

=

mean

=

percentile point. In an extreme case, if one provides multiple percentile points, an approximation to the distribution’s cumulative distribution function or probability density function can be constructed directly, but such information demands are in most cases unrealistic. Some combinations of desired distributional characteristics describe &dquo;impossible&dquo; distributions, which cannot be approximated by the Joh nsonSB, or any other smooth unimodal density. One example is a distribution bounded between 0.0 and 1.0, with a mean of 0.2 and a mode of 0.4. Even when an S8 distribution

be found with the desired characteristics, the corresponding density may have a shape quite unlike what the modeler imagines,

distribution. These are subject to later modification, if desired. Next the modeler is prompted for values of any twoof the fol low ing characteristics:

(a) Mode

(b)

Mean

(c) Median

(d) Arbitrary percentile point(s) (e) Width of the central 95% of the distribution (f) Standard deviation

Significantly, the user is free to provide two arbitrary, asymmetric percentile points, such as the 10th and 25th percentile points. Unlike other algorithms (Mage 1980), there is no requirement that the four input points (two end points and two percentile points) must correspond to equidistant normal deviates. As a default, motivated by the three-parameter specification of a PERTtype estimate (Wilson et al. 1982), the standard deviation can be optionally chosen to be one-sixth of the range. VISIFIT also allows one to specify the parameters of a beta distribution, to which it fits an S8 with the same range, mean, and standard deviation. By accepting a variety of different descriptions, we minimize the need for processing information prior to its input. The modeler is free to use whatever is convenient, familiar, or easily understood.

can

with a distribution bounded between 0.0 and 1.0 with a mean of 045 and a mode of 0.1 (see Figure 3). If a modeler fails to describe the target distribution accurately (that is, if he specifies characteristics inappropriate for the envisioned distribution), then the only way that this can be detected in the absence of data is by visual inspection of the resulting density’s shape. as

When the desired characteristics are entered, VISIFIT computes the parameters of the S8 distribution that most closely matches those characteristics. Several miscellaneous numerical techniques are employed in this calculation, and all are detailed in

Appendix

A.

Interactive Curve Modification Once the parameterization of the fitted S8 density is complete, the user is immediately presented with the distribution’s actual

shape on a graphical display screen (see Figure 4). Such visual feedback will sometimes suggest to the user different values for the characteristics of the target random variable X than were originally chosen. From these revised specifications a new set of parameter values is generated, and then a new fitted density

1

I

Figure 3. 4. SOFTWARE DESIGN VISIFIT combines flexible numerical description with interactive

visual curve modification to capture and refine available subjective information into a parameterized Johnson SB density. Primary design goals were ease of use, high speed on inexpensive microcomputers, and the requirements of minimal information and information processing from the user.

Specifying the Desired Characteristics At the outset of the interaction with VISIFIT, the user must specify the upper (maximum) and lower (minimum) end points of the

Figure

4. VISIFIT’S

Graphical Display.

is presented to the user. Cyclic interaction permits the user to experiment with different curve shapes until a satisfactory one is obtained. VISIFIT also

provides a still simpler scheme of interactive curve shape modification that frees the user from having to deal with numerical input by providing single-keystroke commands that directly manipulate the shape of the displayed curve. The modeler can adjust the shape of a displayed Johnson S8 curve by

202


trial-and-error until he is satisfied with the way it looks. Motivated

by our belief in the universal ease of specifying the mode, width, and percentile points of a distribution we implemented various single-keystroke commands producing the following immediate effects: (a) Move the mode towards the upper bound Move the mode towards the lower bound (c) Increase the width of the curve (d) Decrease the width of the curve (e) Move the 2.5th percentile point to the right (f) Move the 2.5th percentile point to the left (g) Move the 97.5th percentile point to the left (h) Move the 97.5th percentile point to the right

(b)

The magnitude of the change (in the direction indicated by the choice of control key pressed) is determined by an adaptive seeking strategy detailed in Appendix B. The modeler need only indicate the direction of desired change from each displayed curve to the next. The curve can be updated approximately twice each second on an IBM AT-class machine with an 80287 numeric coprocessor, and thus the overalprocess of changing a curve, even drastically from an initial shape, takes at mosta few seconds in the hands of an experienced user. Modification of the end points may be accomplished in two ways. The scale of the Xaxis may be changed, preserving the shape of the distribution while altering the absolute values of the end points. This rescaling also changes the absolute values of the mode and width. Alternatively, the absolute values of the mode and width may be preserved during a change in the end points, in which case a new curve with a visually different shape is obtained. Software

Implementation Details

programming for this project was done in Microsoft C (Microsoft Corp.), but some routines were written in Intel assembly

Most

language to speed up screen handling. VISIFIT runs on IBM or compatible microcomputers and requires either an IBM EGA (Extended Graphics Adapter), IBM VGA (Video Graphics Array), Hercu les Graphics Card?or compatible graphics display. It can make calls to the GraphiC subroutine library3 (Scientific Endeavors Corp.) to provide device-independent hard copy output, of which an example is Figure 3. Even on an IBM AT-compatible 80286-based microcomputer with a 10 MHz clock rate and an 80287 floating-point coprocessor, several seconds may be necessary for execution of the nonlinear

optimization algorithm that yields the y and 6 values for a specified pair of distribution characteristics (see Appendix A). Because the interactive curve-modification process requires greater speed, we utilize an approximation scheme for obtaining values ofy and 6 without solving the nonlinear minimization problem. This involves two-dimensional linear interpolation between the points in a lattice of regularly-spaced values of target characteristics whose corresponding values of ~y and 6 have been precomputed. The interpolated values of ~y and 6 are then used to update the displayed S8 cu rve.

been made for using other, simpler distributions (Law and Kelton 1982). There are distribution plotting facilities in INSIGHT (Roberts 1983), and XCELL + (Conway, et al. 1987) which can display the generalized Lambda (Ramberg and Schmeiser 1972), but these require the input of distribution parameters or moments. None of these alternatives provide the flexibility, ease of use, and visual interactivity of VISIFIT

By extending the bivariate fitting procedure of Johnson (1949b),

methodology can be applied to multivariate populations by to obtain an 5B distribution for each marginal variate separately and (b) estimating the pairwise correlations among the marginal variates, basing the correlation estimates on any intuition or other information available. Subjectivelyestimating correlations may be questionable, but it follows the spirit of our methodology by requesting minimal information. Objective evidence is needed to support the hypothesis that people can meaningfully describe distributions in the &dquo;absence&dquo; of data. It is unclear which descriptive characteristics of distributions are most meaningful and can be most reliably estimated. Presumably, modelers can provide only a limited amount of detail our

(a) using VISIFIT

about processes for which data are unavailable, but the true extent of their information is unknown. If modelers really possess substantial amounts of intuitive information, then perhaps a completely flexible curve, such as a cubic spline, should be used to construct distributions.

Admittedly, our choice of the Johnson 5B distribution is ad hoc. Further research should explore its appropriateness for the general representation of subjectively described distributions. Computational advances may render other flexible distributional forms more attractive, despite increased complexity. Nevertheless, for any competing distributional form to be useful, data-based fitting systems and multivariate extensions must be developed to equal those available for the Johnson S,. Our experience, both in the classroom

setting and with several ap-

plications, suggests that VISIFIT presents a natural and easy-to-use paradigm. In view of the widespread use of the triangular and uniform distributions, we can with very little additional effort create more realistic input models for simulation and other stochastic models. ACKNOWLEDGMENTS

The authors wish to acknowledge Kent Snyder of FACTROL for his early contributions and to Bruce Schmeiser of Purdue University School of Industrial Engineering for several helpful observations. APPENDICES

Appendix A: Computational Methods The principal computational task that the program must perform is the determination of the values of the shape parameters y and 6 of the Johnson 5B distribution from the pair of characteristics input by the user. A search is made for the values of y and 6 that minimize the sum of the squared differences between the computed values of two user-specified characteristics X, my, 6) and X2 (7, 6) and the user’s desired values X* and xz for those characteristics:

CONCLUSIONS

Assuming that modelers sometimes know about their input without possessing data about it, we have developed the software package, VISIFIT, to assist in the specification of a Johnson S8 distribution using primarily subjective information. We are unaware of a comparable methodology, although suggestions have 2 3

Hercules Computer Technology, Berkeley, CA separate commercial product and is

GraphiC is a

not

The minimization scheme we employ is a two-dimensional implementation of the Nelder-Mead algorithm (Nelder and Mead 1965), as modified by Olson (1974), and its search is restricted to regions where the Johnson S8 is unimodal (Johnson 1949a) by

distributed with VISIFIT. 203


of nonlinear penalty functions. Note that the optimization &dquo;compromises&dquo; when confronted with descriptions that are impossible for the Johnson 58, For example, requesting a distribution means

Width. The width of an interval containing the central 95% of the distribution is simply

bounded between 0.0 and 1.0, with a mean of 0.2 and a mode of yields an 58 density with mode and mean both equal to 0.3.

0.4

Computational methods for obtaining y and 6 from the input characteristics for both the &dquo;standardized&dquo; 58 variate Yand the target variate Xare detailed in the following.

the distance between the locations of the 2.5th 97.5th percentile points of the variate X.

Mode. Given the shape parameters my and 6, VISIFIT determines the mode myof the standardized S, variate Yas the solution of the

equation ....,r

The value of my is found by a hybrid bisection-Newton method that is both robust and rapidly converging (Dew and James 1983). The mode mx of the target random variable X is then given by

Mean. The mean x y of the standardized S8 variate Ycan be veniently evaluated as follows:

con-

integral is evaluated via a 16-point Hermitian quadrature (Stroud and Secrest 1966). The mean x x of the target variate X is

This

then given

by

percentile and

Appendix B: Adaptive seeking strategy There are many different one-dimensional search processes that converge to a target point using repeated indications of the direction to the target. Bisection, golden section, and Fibonacci searches are but a few (Wilde 1964). Allfundamentally seek to repeatedly narrow an interval containing the target until it is located to the desired accuracy. While these search procedures can efficiently find stationary targets, they may fail if given noisy or incorrect direction information at any time during the search process. This can occur both with operator error, of which we wish to be tolerant, and with a drifting target. In our experience, the target shape of the user’s envisioned distribution may drift considerably as he is visually confronted with one curve after another. The modeler is often unsure about the exact target at the outset and changes hiss mind as the search proceeds. Like the above mentioned search strategies, ours always converges eventually to a stationary target number, given persistently correct directional information. It has the advantage, however, of &dquo;recovering&dquo; if the target at any point during the search process lies outside an interval that should contain it, earlier queries.

on

the basis of responses

to

Median and Arbitrary Percentile Points. For 0 < a < 1, the 100a percentage point of the distribution of Y is given by

The rules for an iteration of our strategy are as follows (the initial positions and initial step directions and distances may be arbitrary or

random).

Defining at each where za is the location of the corresponding percentile of the standardized normal distribution. The median (50th percentile) is thus given by

the direction (positive or negative) from the current point to the target d ¡ -1 as the direction from the last (previous) point to the target (determined in the (i - 1)st iteration) d ¡ -2 as the direction from the second-to-last point to the target (determined in the ( i - 2) nd iteration) x as the magnitude (distance) of the move made from the last point to the present one,

di

since m_5 0.0. The quantile za is computed from the inverse normal cumulative distribution function as approximated from the algorithm of Beasley and Springer (1977). The corresponding 100a percentage point of the target variate X is finally given by =

we

Standard Deviation. The standard deviation ized S8 variate Y is given by

Qy of the standard-

and the standard deviation or, of the target variate X is given We

by

initially used Algorithm AS 99.3 (Hill, Hill, and Holder 1976)

to evaluate ayfor given

values of y and 6, but we now use Hermiquadrature for this calculation also, since it executes considerably faster while maintaining acceptable accuracy. tian

iteration i:

as

apply the applicable rule at each iteration: (a) If di has the sign opposite to di-1, move distance 0.5x in the &dquo;new&dquo; direction di. (b) If di has the same sign as di-1, but the opposite of di-2, move distance x in direction di. (c) If di has the same sign as both di-, and di-2, move distance 2.0x in direction di.

The algorithm is general for unidimensional search given only direction information. Stopping criteria, which in our case is provided by the user’s acceptance of the current point, may be based on the magnitude of x if desired. Our actual implementation also incorporates a minimum step size and an arbitrary choice not to ever take a step that exceeds half the distance to the absolute bound point in that direction.

204


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