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| {{Refimprove|date=December 2011}}
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| The '''polar method''' (attributed to [[George Marsaglia]], 1964<ref>[http://www.jstor.org/stable/2027592 A convenient method for generating normal variables, G. Marsaglia and T. A. Bray, SIAM Rev. 6, 260–264, 1964]</ref>) is a [[pseudo-random number sampling]] method for generating a pair of independent [[standard normal random variable]]s. While it is superior to the [[Box–Muller transform]]{{Citation needed|date=September 2011}}, the [[Ziggurat algorithm]] is even more efficient.<ref>http://doi.acm.org/10.1145/1287620.1287622 Gaussian Random Number Generators, D. Thomas and W. Luk and P. Leong and J. Villasenor, ACM Computing Surveys, Vol. 39(4), Article 11, 2007, {{doi|10.1145/1287620.1287622}}</ref>
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| Standard normal random variables are frequently used in [[computer science]], [[computational statistics]], and in particular, in applications of the [[Monte Carlo method]].
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| The polar method works by choosing random points (''x'', ''y'') in the square −1 < ''x'' < 1, −1 < ''y'' < 1 until
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| :<math> s=x^2+y^2 < 1, \,</math>
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| and then returning the required pair of normal [[random variable]]s as
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| :<math> x\sqrt{\frac{-2\ln(s)}{s}}\,,\ \ y\sqrt{\frac{-2\ln(s)}{s}}.</math>
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| ==Theoretical basis==
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| The underlying theory may be summarized as follows:
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| If ''u'' is uniformly distributed in the interval
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| 0 ≤ ''u'' < 1, then the point
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| (cos(2π''u''), sin(2π''u''))
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| is uniformly distributed on the unit circumference
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| ''x''<sup>2</sup> + ''y''<sup>2</sup> = 1, and multiplying that point by an independent
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| random variable ρ whose distribution is
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| :<math>\Pr(\rho<a)=\int_0^a re^{-r^2/2}\,dr </math>
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| will produce a point
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| :<math> \left(\rho\cos(2\pi u),\rho\sin(2\pi u)\right) </math>
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| whose coordinates are jointly distributed as two independent standard
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| normal random variables.
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| ==History==
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| This idea dates back to [[Pierre-Simon Laplace|Laplace]], whom [[Carl Friedrich Gauss|Gauss]] credits with finding the above
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| :<math>I=\int_{-\infty}^\infty e^{-x^2/2}\,dx </math>
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| by taking the square root of
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| :<math>I^2 = \int_{-\infty}^\infty\int_{-\infty}^\infty e^{-(x^2+y^2)/2}\,dx\,dy
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| =\int_0^{2\pi}\int_0^\infty re^{-r^2/2} \, dr \, d\theta.</math>
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| The transformation to polar coordinates makes evident that θ is
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| uniformly distributed (constant density) from 0 to 2π, and that the
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| radial distance ''r'' has density
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| :<math>re^{-r^2/2}. \, </math>
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| (''r''<sup>2</sup> has the appropriate [[chi square]] distribution.)
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| This method of producing a pair of independent standard normal variates by radially projecting a random point on the unit circumference to a distance given by the square root of a chi-square-2 variate is called the polar method for generating a pair of normal random variables,
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| ==Practical considerations==
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| A direct application of this idea,
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| :<math>x=\sqrt{-2\ln(u_1)}\cos(2\pi u_2),\quad y=\sqrt{-2\ln(u_1)}\sin(2\pi u_2)</math>
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| is called the [[Box Muller transform]], in which the chi variate is usually
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| generated as
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| :<math>\sqrt{-2\ln(u_1)};</math>
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| but that transform requires logarithm, square root, sine and cosine functions. On some processors, the cosine and sine of the same argument can be calculated in parallel using a single instruction.<ref>{{cite web|last=Kanter|first=David|title=Intel’s Ivy Bridge Graphics Architecture|url=http://www.realworldtech.com/ivy-bridge-gpu/5/|work=Real World Tech|accessdate=8 April 2013}}</ref> Notably for Intel based machines, one can use fsincos assembler instruction or the expi instruction (available e.g. in D), to calculate complex
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| : <math>\text{expi}(z) = e^{i z} = \cos(z) + i \sin(z), \, </math>
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| and just separate the real and imaginary parts.
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| The polar method, in which
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| a random point (''x'', ''y'') inside the unit circle
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| is projected onto the unit circumference by setting ''s'' = ''x''<sup>2</sup> + ''y''<sup>2</sup> and forming the point
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| :<math>\left( \frac{x}{\sqrt{s}}, \frac{y}{\sqrt{s}} \right), \, </math>
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| is a faster procedure. Some researchers argue that the conditional if instruction (for rejecting a point outside of the unit circle), can make programs slower on modern processors equipped with pipelining and branch prediction.{{Citation needed|date=January 2011}} Also this procedure requires about 21% more evaluations of the underlying random number generator (only <math>\pi/4 \approx 79%</math> of generated points lie inside of unit circle).
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| That random point on the circumference is then radially projected the required random distance by means of
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| :<math>\sqrt{-2\ln(s)}, \, </math>
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| using the same ''s'' because that ''s'' is independent of the random point on the circumference and is itself uniformly distributed from 0 to 1.
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| == Implementation ==
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| Simple implementation in [[Java (programming language)|Java]]:
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| <source lang="java">
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| private static double spare;
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| private static boolean isSpareReady = false;
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| public static synchronized double getGaussian(double mean, double stdDev) {
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| if (isSpareReady) {
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| isSpareReady = false;
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| return spare * stdDev + mean;
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| } else {
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| double u, v, s;
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| do {
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| u = Math.random() * 2 - 1;
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| v = Math.random() * 2 - 1;
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| s = u * u + v * v;
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| } while (s >= 1 || s == 0);
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| double mul = Math.sqrt(-2.0 * Math.log(s) / s);
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| spare = v * mul;
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| isSpareReady = true;
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| return mean + stdDev * u * mul;
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| }
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| }
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| </source>
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| == References ==
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| {{Reflist}}
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| [[Category:Monte Carlo methods]]
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| [[Category:Pseudorandom number generators]]
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| [[Category:Non-uniform random numbers]]
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