History of the Science of Modeling an Elephant

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This is a compilation of the amusing but enlightening scientific history on the question of how many parameters does it take to fit (or model) and elephant. We start with Freeman Dyson attributing the famous quote (regarding fitting an elephant) to John von Neumann by way of Enrico Fermi. There is then a vague attribution of this quote to several famous physicists in Brown and Sethna (2003) in Physical Review E. I then added Wei (1975) as the first somewhat serious approach to modeling an elephant. This is discussed in Burnham and Anderson (2002). Mayer et al. (2010) finally showed in their paper that indeed an elephant can be modeled with 4 parameters and with a 5th parameter its trunk can be made to wiggle. This paper was mentioned in Nature Physics in 2010, which also has a superior color presentation of Mayer et al.'s elephantine graph. Lastly, there was an amusing fictional intellectual history of modeling an elephant given by J. Dobaczewski in a presentation given at an actual scientific conference on nuclear structure physics in Finland.

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<p>essay turning points</p> <p>A meeting with Enrico FermiHow one intuitive physicist rescued a team from fruitless research.Freeman Dyson</p> <p>O</p> <p>ne of the big turning points in my life was a meeting with Enrico Fermi in the spring of 1953. In a few minutes, Fermi politely but ruthlessly demolished a programme of research that my students and I had been pursuing for several years. He probably saved us from several more years of fruitless wandering along a road that was leading nowhere. I am eternally grateful to him for destroying our illusions and telling us the bitter truth. Fermi was one of the great physicists of our time, outstanding both as a theorist and as an experimenter. He led the team that built the first nuclear reactor in Chicago in 1942. By 1953 he was head of the team that built the Chicago cyclotron, and was using it to explore the strong forces that hold nuclei together. He made the first accurate measurements of the scattering of mesons by protons, an experiment that gave the most direct evidence then available of the nature of the strong forces . At that time I was a young professor of theoretical physics at Cornell University, responsible for directing the Crossed paths: A discussion with Enrico research of a small army of graduate Fermi (above) made Freeman Dyson students and postdocs. I had put them (right) change his career direction. to work calculating mesonproton scattering, so that their theoretical calculations a package of our theoretical could be compared with Fermis measure- graphs to show to Fermi. ments. In 1948 and 1949 we had made When I arrived in Fermis similar calculations of atomic processes,using office, I handed the graphs to the theory of quantum electrodynamics,and Fermi, but he hardly glanced found spectacular agreement between experi- at them. He invited me to sit ment and theory.Quantum electrodynamics down, and asked me in a is the theory of electrons and photons friendly way about the health interacting through electromagnetic forces. of my wife and our newBecause the electromagnetic forces are weak, born baby son, now fifty we could calculate the atomic processes years old. Then he delivered precisely. By 1951, we had triumphantly his verdict in a quiet, even voice. There are finished the atomic calculations and were two ways of doing calculations in theoretical looking for fresh fields to conquer. We physics, he said. One way, and this is the decided to use the same techniques of calcu- way I prefer, is to have a clear physical picture lation to explore the strong nuclear forces. of the process that you are calculating. The We began by calculating mesonproton other way is to have a precise and selfscattering, using a theory of the strong forces consistent mathematical formalism. You known as pseudoscalar meson theory. By the have neither. I was slightly stunned, but spring of 1953, after heroic efforts, we had ventured to ask him why he did not consider plotted theoretical graphs of mesonproton the pseudoscalar meson theory to be a selfscattering.We joyfully observed that our consistent mathematical formalism. He calculated numbers agreed pretty well with replied, Quantum electrodynamics is a Fermis measured numbers. So I made good theory because the forces are weak, an appointment to meet with Fermi and and when the formalism is ambiguous we show him our results. Proudly, I rode the have a clear physical picture to guide us.With Greyhound bus from Ithaca to Chicago with the pseudoscalar meson theory there is no2004 Nature Publishing Group</p> <p>physical picture, and the forces are so strong that nothing converges. To reach your calculated results, you had to introduce arbitrary cut-off procedures that are not based either on solid physics or on solid mathematics. In desperation I asked Fermi whether he was not impressed by the agreement between our calculated numbers and his measured numbers. He replied, How many arbitrary parameters did you use for your calculations? I thought for a moment about our cut-off procedures and said, Four. He said, I remember my friend Johnny von Neumann used to say, with four parameters I can fit an elephant, and with five I can make him wiggle his trunk. With that, the conversation was over. I thanked Fermi for his time and trouble,and sadly took the next bus back to Ithaca to tell the bad news to the students.Because it was important for the students to have their names on a published paper, we did not abandon our calculations immediately. We finished them and wrote a long paper that was duly published in the Physical Review with all our names on it. Then we dispersed to find other lines of work. I escaped to Berkeley, California, to start a new career in condensed-matter physics. Looking back after fifty years, we can clearly see that Fermi was right. The crucial discovery that made sense of the strong forces was the quark. Mesons and protons are little bags of quarks. Before Murray GellMann discovered quarks, no theory of the strong forces could possibly have been adequate. Fermi knew nothing about quarks, and died before they were discovered. But somehow he knew that something essential was missing in the meson theories of the 1950s. His physical intuition told him that the pseudoscalar meson theory could not be right. And so it was Fermis intuition, and not any discrepancy between theory and experiment, that saved me and my students from getting stuck in a blind alley. Freeman Dyson is at the Institute for Advanced Study, Einstein Drive, Princeton, New Jersey 08540, USA.297</p> <p>NATURE | VOL 427 | 22 JANUARY 2004 | www.nature.com/nature</p> <p>HEKA DAVIS PHOTO/AIP EMILIO SEGR VISUAL ARCHIVES</p> <p>UNIV. CHICAGO/AIP EMILIO SEGR VISUAL ARCHIVES</p> <p>There is a famous aphorism in physics: Give me four parameters and I can fit an elephant. Give me five and I can wag its tail [5]. [5] We have tried to find the appropriate attribution for this quote but have been unsuccessful. Variants of the statement (differing in the number of parameters) have been attributed to C.F. Gauss, Niels Bohr, Lord Kelvin, Enrico Fermi, and Richard Feynman. Excerpted from: Kevin S. Brown and James P. Sethna. 2003. Statistical mechanical approaches to models with many poorly known parameters. Physical Review E 68: 021904.</p> <p>Excerpted from: Kenneth P. Burnham and David R. Anderson. 2002. Model Selection and Multimodel Inference: A Practical Information-theoretic Approach. 2nd ed. New York: Springer.1.4 Inference and the Principle of Parsimony 29</p> <p>ously. The advent of Markov chain Monte Carlo methods (Gilks et al. 1996, Gamerman 1997) may soon give rise to a general but practical framework for spatiotemporal modeling; model selection will be an important component of such a framework. A step towards this general framework was made by Buckland and Elston (1993), who modeled changes in the spatial distribution of wildlife. There are many other examples where modeling of data plays a fundamental role in the biological sciences. Henceforth, we will exclude only modeling that cannot be put into a likelihood or quasi-likelihood (Wedderburn 1974) framework and models that do not explicitly relate to empirical data. All least squares formulations are merely special cases that have an equivalent likelihood formulation in usual practice. There are general information-theoretic approaches for models well outside the likelihood framework (Qin and Lawless 1994, Ishiguo et al. 1997, Hurvich and Simonoff 1998, and Pan 2001a and b). There are now model selection methods for nonparametric regression, splines, kernel methods, martingales, and generalized estimation equations. Thus, methods exist for nearly all classes of models we might expect to see in the theoretical or applied biological sciences.</p> <p>1.41.4.1</p> <p>Inference and the Principle of ParsimonyAvoid Overtting to Achieve a Good Model Fit</p> <p>Consider two analysts studying a small set of biological data using a multiple linear regression model. The rst exclaims that a particular model provides an excellent t to the data. The second notices that 22 parameters were used in the regression and states, Yes, but you have used enough parameters to t an elephant! This seeming conict between increasing model t and increasing numbers of parameters to be estimated from the data led Wel (1975) to answer * the question, How many parameters does it take to t an elephant? Wel nds that about 30 parameters would do reasonably well (Figure 1.2); of course, had he t 36 parameters to his data, he could have achieved a perfect t. Wels nding is both insightful and humorous, but it deserves further interpretation for our purposes here. His standard is itself only a crude drawingit even lacks ears, a prominent elephantine feature; hardly truth. A better target would have been a large, digitized, high-resolution photograph; however, this, too, would have been only a model (and not truth). Perhaps a real elephant should have been used as truth, but this begs the question, Which elephant should we use? This simple example will encourage thinking about full reality, true models, and approximating models and motivate the principle of parsimony in the following section. William of Occam suggested in the fourteenth century that one shave away all that is unnecessarya dictum often referred to as Occams razor. Occams razor has had a long history</p> <p>*Wei (1975); see previous.</p> <p>30</p> <p>1. Introduction</p> <p>FIGURE 1.2. How many parameters does does it take to t an elephant? was answered by Wel (1975). He started with an idealized drawing (A) dened by 36 points and used least squares Fourier sine series ts of the form x(t) 0 + i sin(it/36) and y(t) 0 + i sin(it/36) for i 1, . . . , N. He examined ts for K 5, 10, 20, and 30 (shown in BE) and stopped with the t of a 30 term model. He concluded that the 30-term model may not satisfy the third-grade art teacher, but would carry most chemical engineers into preliminary design.</p> <p>in both science and technology, and it is embodied in the principle of parsimony. Albert Einstein is supposed to have said, Everything should be made as simple as possible, but no simpler. Success in the analysis of real data and the resulting inference often depends importantly on the choice of a best approximating model. Data analysis in the biological sciences should be based on a parsimonious model that provides an accurate approximation to the structural information in the data at hand; this should not be viewed as searching for the true model. Modeling and model selection are essentially concerned with the art of approximation (Akaike 1974).</p> <p>Drawing an elephant with four complex parametersJurgen MayerMax Planck Instilflte of Molecular Cell Biology and Genetics, Pjotenilauerstr. 108, 01307 Dresden, Germany</p> <p>Khaled KhairyEuropean Molecular Biology La.boratory, MeyerhofstrafJe. 1, 69117 Heidelberg, Gennany</p> <p>Jonathon HowardMax Planck Institute of Molecular Cell Biology and Genetics, Pfotellhauerstr. 108, 01307 Dresden, Germany</p> <p>(Received 20 August 2008; accepted 5 October 2009)</p> <p>We define four complex numbers representing the parameters needed to specify an elephantine shape. The real and imaginary parts of these complex numbers are the coefficients of a Fourier coordinate expansion, a powerful tool for reducing the data required to define shapes. 2010American Association of PhYSics Teacllel'.r.</p> <p>[DOl: 10.1119/1.3254017]</p> <p>A turning point in Freeman Dyson's life occurred during a meeting in the Spring of 1953 when Enrico Fermi criticized the complexity of Dyson's model by quoting Johnny von Neumann:' ''With four parameters I can fit an elephant, and with five [ can make him wiggle his trunk." Since then it has become a well-known saying among physicists, but nobody has successfully implemented it. To parametrize an elephant, we note that its perimeter can be described as a set of points (x(t),y(t)), where t is a parameter that can be interpreted as the elapsed time while going along the path of the contour. If the speed is uniform, t becomes the arc length. We expand x and y separately2 as a</p> <p>trace out elJiptical corrections analogous to Ptolemy's epicycles.' Visualization of the corresponding elJipses can be found at Ref. 6. We now use this tool to fit an elephant with four parameters. Wei' tried this task in 1975 using a least-squares Fourier sine series but required about 30 terms. By analyzing the picture in Fig. 1(a) and eliminating components with amplitudes less than 10% of the maximum amplitude, we obtained an approximate spectrum. The remaining amplitudes were</p> <p>(a)</p> <p>Fourier series x(t) = 2: (A% cos(kt) + B:l sin(kt)) ,k=O</p> <p>2Or-----------------.. -----..j40t</p> <p>Pattern</p> <p>(1)</p> <p>I ,</p> <p>i!!</p> <p>:.. GOt</p> <p>yet) = 2: (A~ cos(kt) +B~ sin(kt)) ,k=O</p> <p>i 8O~!</p> <p>1</p> <p>i:,</p> <p>i</p> <p>(2)</p> <p>100150 100 150</p> <p>where Ai, 11k, Ak. and EX are the expansion coefficients. The lower indices k apply to the kth term in the expansion, and the upper indices denote the x or y expansion, respectively. Using this expansion of the x and y coordinates, We can analyze shapes by tracing the boundary and calculating the coefficients in the expansions (using standard methods from Fourier analysis). By truncating the expansion, the shape is smoothed. Truncation leads to a huge reduction in the information necessary to express a certain shape compared to a pixelated image, for example. Szekely et al.' used this approach to segment magnetic resonance imaging data. A similar approach was used to analyze the shapes of red blood cells: with a spherical harmonics expansion serving as a 3D generalization of the Fourier coordinate expansion. The coefficients represent the best fit to the given shape in the following sense. The k=O component corresponds to the center of mass of the perimeter. The k= I component Corresponds to the best fit ellipse. The higher order components648Am. I. Phys. 78 (6). Iune 20tO</p> <p>120L. __ .. _.. _"'"_._ .... ______ .__ .... ,. __ .___ .......... __ ._.... _____ .1</p> <p>200</p> <p>x</p> <p>(b)100</p> <p>50</p> <p>Y-50</p> <p>-1B00 -</p> <p>-50</p> <p>x</p> <p>0</p> <p>50</p> <p>100</p> <p>Fig. 1. (a) Outline of an eJephnnt. (b) Three snapshots of the wiggling trunk.Q)</p> <p>http://oupl.org/.ljp</p> <p>20 I0 American Association of Physics Teachers</p> <p>648</p> <p>Table I. The five complex parameters PI,'" ,Ps that encode the elephant including its wiggling trunk. ParameterPI::::50-30i Pl=18+8iP3=12-IOi</p> <p>Real part81=50 ill=18</p> <p>Imaginary partB{=-30B~=88~=-1O</p> <p>the special case of fitting an elephant, it is even possible to lower it to four complex parameters and therein implement a well-known saying.</p> <p>ACKNOWLEDGMENTS Many thanks to Jean-Yvcs Tinevez and Marija Zanie, as well as the anonymous reviewers, for revising and improving this article.IF. Dyson, "A meeting with Enrico Fermi," Nature (London) 427(6972), 297 (2004). 2F. P. Kuhl and C. R. Giardina, "Elliptic Fourier features of a closed contour," Comput. Graph. Image Process. 18, 236-258 (1982). J G. Szekely, A. Kelemen, C. Brec...</p>