How to fit an elephant
Cited from John D. Cook
John von Neumann famously said
With four parameters I can fit an elephant, and with five I can make him wiggle his trunk.
By this he meant that one should not be impressed when a complex model fits a data set well. With enough parameters, you can fit any data set. It turns out you can literally fit an elephant with four parameters if you allow the parameters to be complex numbers. See this paper for details: “Drawing an elephant with four complex parameters”1.
Piotr also sent me the following Python code he’d written to implement the method in the paper. This code produced the image above.
Click to expand code.
""" Author: Piotr A. Zolnierczuk (zolnierczukp at ornl dot gov) Based on a paper by: Drawing an elephant with four complex parameters Jurgen Mayer, Khaled Khairy, and Jonathon Howard, Am. J. Phys. 78, 648 (2010), DOI:10.1119/1.3254017 """ import numpy as np import pylab # elephant parameters p1, p2, p3, p4 = (50 - 30j, 18 + 8j, 12 - 10j, -14 - 60j ) p5 = 40 + 20j # eyepiece def fourier(t, C): f = np.zeros(t.shape) A, B = C.real, C.imag for k in range(len(C)): f = f + A[k]*np.cos(k*t) + B[k]*np.sin(k*t) return f def elephant(t, p1, p2, p3, p4, p5): npar = 6 Cx = np.zeros((npar,), dtype='complex') Cy = np.zeros((npar,), dtype='complex') Cx[1] = p1.real*1j Cx[2] = p2.real*1j Cx[3] = p3.real Cx[5] = p4.real Cy[1] = p4.imag + p1.imag*1j Cy[2] = p2.imag*1j Cy[3] = p3.imag*1j x = np.append(fourier(t,Cx), [-p5.imag]) y = np.append(fourier(t,Cy), [p5.imag]) return x,y x, y = elephant(np.linspace(0,2*np.pi,1000), p1, p2, p3, p4, p5) pylab.plot(y,-x,'.') pylab.show()
Some comments:
- Actually the code above used five complex numbers, four for the contour and one for the eyepiece, thus ten parameters to fit the whole elephant.
- We need one additional parameter to make its trunk wiggle, along with some code modification. Animation, Download code
- Fourier strikes again!
