79 lines
2.8 KiB
JavaScript
79 lines
2.8 KiB
JavaScript
/**
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* @license Apache-2.0
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*
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* Copyright (c) 2018 The Stdlib Authors.
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*
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* Licensed under the Apache License, Version 2.0 (the "License");
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* you may not use this file except in compliance with the License.
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* You may obtain a copy of the License at
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*
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* http://www.apache.org/licenses/LICENSE-2.0
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*
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* Unless required by applicable law or agreed to in writing, software
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* distributed under the License is distributed on an "AS IS" BASIS,
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* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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* See the License for the specific language governing permissions and
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* limitations under the License.
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*/
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'use strict';
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// MODULES //
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var factory = require( './factory.js' );
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// MAIN //
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/**
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* Generates a standard normally distributed random number.
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*
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* ## Method
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*
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* The basic Ziggurat method works as follows:
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*
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*
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* ```tex
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* x_{C-1}(r) \left[ f(0) - f\left( x_{C-1}(r) \right) \right] - V(r) = 0
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* ```
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*
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* where
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*
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* ```tex
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* V(r) = r \; f(r) + \int_r^\infty \; f(x) \; dx
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* ```
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*
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* and \\( r \\) denotes the right-most \\( x_1 \\).
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*
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* - We then use the following rejection algorithm:
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*
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* - Draw a box \\( B_i \\) at random with probability \\( \tfrac{1}{C} \\).
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* - Draw a random number from the box as \\( z = U_0 x_i \\) for \\( i > 0 \\) and \\( z = U_0 V / f(x_1) \\).
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* - If \\( z < x_{i+1} \\), accept \\( z \\).
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* - If \\( i = 0 \\), accept a \\( v \\) by transforming the tail of the normal distribution to the unit interval and then use rejection technique by Marsaglia, G. (1964) to generate a standard normal variable. Otherwise, if \\( i > 0 \\) and \\( U_1 \left[ f(x_i) - f(x_{i+1})\right] < f(z) - f(x_{i+1}) \\) accept \\( z \\).
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* - Go back to the first step.
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*
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* - The improved version by Doornik (2005) changes step four in order to correct a deficiency of the original Ziggurat algorithm. The updated version requires the generation of two random numbers, a uniform variable drawn from \\( U(-1,1) \\) and the last seven bits of a random integer.
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*
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* ## References
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*
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* - Doornik, Jurgen A. 2005. "An Improved Ziggurat Method to Generate Normal Random Samples." <https://www.doornik.com/research/ziggurat.pdf>.
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* - Marsaglia, George, and Wai Wan Tsang. 2000. "The Ziggurat Method for Generating Random Variables." _Journal of Statistical Software_ 5 (1): 1–7. doi:[10.18637/jss.v005.i08](http://dx.doi.org/10.18637/jss.v005.i08).
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* - Marsaglia, George. 1964. "Generating a Variable from the Tail of the Normal Distribution." _Technometrics_ 6 (1): 101–2. doi:[10.1080/00401706.1964.10490150](http://dx.doi.org/10.1080/00401706.1964.10490150).
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*
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*
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* @name randn
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* @type {PRNG}
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* @returns {number} pseudorandom number
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*
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* @example
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* var r = randn();
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* // returns <number>
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*/
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var randn = factory();
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// EXPORTS //
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module.exports = randn;
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