JP5273613B2 - Random number generation apparatus, random number generation method, random number generation program, and simulation apparatus - Google Patents

Description

  The present invention relates to a random number generation device, a random number generation method, a random number generation program, and a simulation device, and in particular, a random number generation device suitable for generating random numbers that preferably have symmetry in distribution, such as uniform random numbers and normal random numbers, The present invention relates to a random number generation method, a random number generation program, and a simulation apparatus using the generated random numbers.

  The Monte Carlo method is a simulation technique using random numbers, and has been studied in various fields. Many researches have been made on the generation of uniform random numbers and normal random numbers, and the Mersenne Twister (hereinafter abbreviated as “MT”) method is particularly well known as a method for generating uniform pseudorandom numbers. On the other hand, as a method for generating normal random numbers, for example, Non-Patent Document 1 introduces a method by Shibuya.

Takao Tsuda, “Monte Carlo Method and Simulation”, revised edition, Baifukan Co., Ltd., November 15, 1977, pp. 60-61

  However, the conventional random number generation method generally has a large amount of calculation, and in order to perform a complicated simulation on a computer, reducing the amount of calculation as much as possible leads to an increase in speed. Further, in the case of uniform random numbers and normal random numbers, the improvement of the uniform distribution of random numbers and the method applied to the distribution leads to high speed and high accuracy of the Monte Carlo simulation.

  The present invention has been made in view of the above points, and in particular, a random number generation device, a random number generation method, and a random number suitable for generating random numbers that preferably have symmetry in distribution, such as uniform random numbers and normal random numbers. An object of the present invention is to provide a generation program and a simulation apparatus using the generated random number.

In order to solve the above problem, a random number generation device according to the present invention includes a source random number acquisition unit that acquires N (N is a natural number of 2 or more) source random numbers, and a code from the N source random numbers. And a random number output unit that outputs at least a part of the random numbers generated by the random number generation unit as random numbers, the random number generation unit including the N primitives When the random numbers are represented as (x ₁ , x ₂ ,... X _N ), N numbers of random numbers are generated by performing an inversion process of at least one sign of the N number of primitive random numbers. Among the random numbers, the number of primitive random numbers for which the sign inversion process is performed is changed, and random numbers are generated until the sign inversion process for all of the N primitive random numbers is finally performed.

  In the random number generation device according to the present invention, the random number generation unit sequentially performs “sign inversion processing” on the obtained N primitive random numbers to generate a new random number. With this processing, it is possible to maintain and improve the accuracy of simulation while reducing the amount of calculation required for generating random numbers and the amount of calculation required for simulation.

The “sign inversion process” will be described in detail below. For example, when N = 2, two random numbers x ₁ and y ₁ are acquired as the original random numbers, and the origin If the coordinates are (0, 0), the _{x 1} to -x _1, it includes processing for converting _{y 1} to -y _1. If the origin coordinate is (0.5, 0.5) and two random numbers x ₁ and y ₁ are generated within the range of 0 to 1, respectively, x ₁ is set. The process of converting y ₁ to 1-y ₁ into 1-x ₁ is also an example of “sign inversion process”. In the case of N = 3, three random numbers x ₁ , y ₁ , and z ₁ are acquired as primitive random numbers. By sequentially performing “code inversion processing” for each of the three random numbers. Seven new points (21 random numbers) on the three-dimensional coordinates are generated. That is, the random number generator may generate (N * 2 ^N −N) random numbers from the N primitive random numbers.

The random number generation unit acquires N primitive random numbers generated by the primitive random number acquisition unit within a range from 0 to a (a is a positive real number), and generates a random number within a range from 0 to a. In the case of generating, (a−x _n ) can be calculated as a sign inversion process for the _nth primitive random number x _n . The random number output unit may be configured to output at least a part of the original random number acquired as the random number as a random number.

  For the generation of random numbers by such “sign inversion processing”, for example, in the case of three-dimensional multiple integration or the simulation of a three-variable model, it is preferable to generate random numbers with N = 3. This is because it is possible to improve the uniform distribution and the fit to the multi-dimensional probability distribution for all three coordinate axes. In the case of performing the calculation of four-dimensional multiple integration, it is preferable to set N = 4.

  The application range of the random number generation method according to the present invention is not limited to multiple integration, and can be used for various Monte Carlo simulations. In particular, it is suitable for calculations in which random numbers having symmetry in distribution, such as uniform random numbers and normal random numbers, are preferably used. A random number generation program according to the present invention is a computer program that causes a computer to execute random number generation according to the present invention. The recording medium according to the present invention is a computer-readable recording medium in which the computer program according to the present invention is recorded. Examples of the recording medium include various recording media such as ROM, CD-ROM, and DVD. The random number generation according to the present invention can also be realized by hardware using an electronic circuit.

  According to the random number generation device and the like according to the present invention, the random number distribution can be optimized while reducing the amount of calculation.

It is a block diagram for demonstrating the whole structure of the random number generator in embodiment of this invention. It is a figure for demonstrating the processing content of the random number generation part in the 1st Embodiment of this invention. It is a figure for demonstrating the case where an origin is set to (0.5, 0.5) in 1st Embodiment. It is a figure for demonstrating the processing content of the random number generation part in 2nd Embodiment. It is a figure for demonstrating the specific way of thinking of the random number generation in 2nd Embodiment. When N = 4, it is a figure for demonstrating a mode that 60 new random numbers are produced | generated from four primitive random numbers. It is a figure which shows the result of the multiple integral calculation using the produced | generated random number.

  Hereinafter, embodiments of the present invention will be described with reference to the drawings. FIG. 1 is a block diagram for explaining a schematic configuration of the random number generation device according to the present embodiment. The random number generation device 100 according to the present embodiment includes a primitive random number acquisition unit 101, a random number generation unit 102, and a random number output unit 103.

  The primitive random number obtaining unit 101 obtains a primitive random number that is a basis for random number generation in the random number generation device according to the present embodiment. In the case of a uniform random number, the original random number may be a random number generated by the MT method, for example, or may be a pseudo random number or a natural random number generated by another method. In the case of normal random numbers, normal random numbers by Shibuya may be used.

  The random number generation unit 102 generates more than N random numbers from the N random numbers acquired by the original random number acquisition unit 101 by sign inversion processing. The processing of the random number generation unit 102 will be described in detail later.

  The random number output unit 103 outputs the random number generated by the random number generation unit 102. The random number output from the random number output unit 103 can include the original random number acquired by the original random number acquisition unit 101. The random numbers output from the random number output unit 103 are sequentially used for, for example, Monte Carlo simulation.

(Embodiment 1)
As a first embodiment of the present invention, a case where the original random number acquisition unit 101 sequentially acquires two random numbers (N = 2) will be described. FIG. 2 is a diagram for explaining the processing contents of the random number generation unit 102 of the present embodiment.

In the example of the figure, it is assumed that the primitive random number acquisition unit 101 has acquired two primitive random numbers (x ₁ , y ₁ ). The random number generation unit 102 generates six new random numbers from the two random numbers by code inversion processing. That is, when the origin of the xy coordinates is (0, 0), six new random numbers of (x ₁ , -y ₁ ) (-x ₁ , y ₁ ) (-x ₁ , -y ₁ ) Is generated.

  The random number output unit 103 adds two primitive random numbers to the six random numbers newly generated by the random number generation unit 102 and outputs a total of eight random numbers. Thereafter, the primitive random number acquisition unit 101 further acquires two primitive random numbers, generates six new random numbers in the same manner as described above, and the random number output unit 103 outputs eight random numbers. By repeating this process, a large amount of random numbers can be generated sequentially. Since the sign inversion process such as the process of the random number generation unit 102 according to the present embodiment requires a smaller amount of calculation compared to the generation process of the original random number, the calculation amount decreases as the number of necessary random numbers increases. The effect is increased.

In the case where the primitive random number generator 101 acquires a primitive random number generated between 0 and 1, and the random number output unit 103 also outputs a random number between 0 and 1, FIG. The origin at may be set to (0.5, 0.5) (0.5 = 1/2). FIG. 3 is a diagram for explaining a case where the origin is set to (0.5, 0.5). In this case, where the primitive random number acquired by the primitive random number acquisition unit 101 is located in the first to fourth quadrants is indefinite, but in any quadrant, six new random numbers are entered. It is clear that can be easily generated. Even if the origin is other than (0, 0), the original random number is in the range from 0 to a (a is a positive real number), and the random number generation unit 102 selects a random number in the range from 0 to a. When outputting, the value of the origin is set to a / 2, and the value _xn of the _nth primitive random number may be converted into the value of (a− _xn ). The conversion of _xn into (a− _xn ) is also included in the “sign inversion process”. The definition of “sign inversion processing” can be considered in the same manner even when the number N of original random numbers increases. The uniform random number η in the section (a, b) can be converted as follows: η = (b−a) × ξ + a using the uniform random number ξ generated in the range from 0 to 1. . Further, “sign inversion processing” may be performed with (a + b) / 2 as the origin. That is, “sign inversion processing” is possible even with a uniform random number in the section (a, b). Even in the case of a regular random number, if the origin is converted so as to be a symmetric point, the “sign inversion process” can be easily realized. In general, in the case of normal random numbers, the position of the peak of the distribution of normal random numbers actually required is often not 0, but the peak value as a target value obtained empirically (or measured). It is also possible to perform “sign inversion processing” by regarding as the origin.

  The advantage of using such a random number generation method is to improve the uniform distribution of uniform random numbers, and to improve the method of applying to random numbers (how to apply to multidimensional probability distributions) for normal random numbers. It is a point that can be. That is, it is possible to significantly reduce the amount of calculation while maintaining the quality as a random number used for simulation, which can be applied to both the area integration type Monte Carlo method such as multiple integration and the event reproduction type Monte Carlo method.

(Embodiment 2)
In the first embodiment, the case where the source random number acquisition unit 101 acquires two source random numbers each has been described. In the second embodiment, a case will be described in which the source random number acquisition unit 101 acquires three source random numbers (N = 3).

FIG. 4 is a diagram for explaining the processing contents of the random number generation unit 102 in the present embodiment. In the present embodiment, the primitive random number acquisition unit 101 acquires three primitive random numbers (x ₁ , y ₁ , z ₁ ).

The random number generation unit 102 generates 21 random numbers from the three primitive random numbers. 21 corresponds to 3 * 2 ³ -3. This number is a value obtained by multiplying the number N of the obtained primitive random numbers by the Nth power of 2, and subtracting the number N of the primitive random numbers. FIG. 5 is a diagram for explaining a specific concept of random number generation in the present embodiment.

Since there are three primitive random numbers, first, the sign inversion process is performed for any one (for example, z ₁ ). Next, for example, the sign inversion process is performed for y ₁ , and the process is performed as z ₁ and −z ₁ . Furthermore, it performs inversion processing of the code also x _1, increases the primitive random sequential inversion processing of the code is executed.

Finally, the sign inversion process is performed for all three primitive random numbers to obtain random numbers (-x ₁ , -y ₁ , -z ₁ ). As a result of the above processing, a total of 21 (= 7 * 3) random numbers are generated except the original random number. As the number N of the source random numbers acquired by the source random number acquisition unit 101 is increased, the amount of calculation required to generate a large number of random numbers is reduced as compared with normal random number generation. It is made. Random number generation method (code) when the original random number is generated between 0 and 1 and the random number is generated between 0 and 1, that is, when the origin is set to (0.5, 0.5, 0.5) The concept of the inversion process is the same as in the first embodiment.

In the first and second embodiments, the source random number acquisition unit 101 acquires two or three source random numbers, and the random number generation unit 102 has 2 * 2 ² −2 (= 6), or The case where 3 * 2 ³ -3 (21) random numbers are generated has been described. However, the number of primitive random numbers acquired by the primitive random number acquisition unit 101 is not limited to two or three, and it is of course possible to increase the number of random number generation dimensions in the random number generation unit 102 to four or more. Theoretically, as the value of N is increased, the amount of calculation is greatly reduced to (1/2 ^N ). However, the processing time varies depending on the specific programming method.

  For example, when N = 4, as shown in FIG. 6, 60 new random numbers are generated from 4 primitive random numbers. However, the larger the numerical value of N, the higher the ratio of outputting the same absolute value as a random number. This may not be a big problem when used in a so-called quasi-Monte Carlo method, and is rather preferable in the simulation of a four-dimensional multiple integration or a four-variable model as described below. If used for other purposes, an embodiment in which the same absolute value is thinned out and output from the random numbers generated by the random number generation unit 102 is also possible.

  FIG. 7 is a diagram illustrating a numerical calculation example of multiple integration to which the random number generation according to the present embodiment is applied. In the figure, “normal random number” means a program using a random number generation program attached to Fujitsu Fortran & C Academic Package V4.0, and MT random number is the Mersenne Twister random number. Calculation example 1 shows the result when the above embodiment (N = 3) is applied to three-dimensional multiple integration, and calculation example 2 shows the above embodiment (N = 4) for four-dimensional multiple integration. The result when applying is shown. In Calculation Example 1, a large accuracy improvement effect is seen. In addition, it can be seen that the calculation example 2 has a large effect of reducing the calculation time in addition to the accuracy improvement effect. It was shown that the calculation amount reduction effect of the present invention increases as the number of dimensions represented by N increases.

  The present invention can be applied to, for example, random number generation used in the Monte Carlo method.

101 Primitive random number acquisition unit 102 Random number generation unit 103 Random number output unit

Claims (20)

A source random number acquisition unit for acquiring N (N is a natural number of 2 or more) source random numbers;
A random number generator for generating a random number from the N primitive random numbers by means of sign inversion;
A random number output unit that outputs at least a part of the random number generated by the random number generation unit as a random number;
The random number generator
When the N primitive random numbers are represented as (x ₁ , x ₂ ,... X _N ), N number of random numbers is generated by performing an inversion process of at least one code of the N primitive random numbers. , Change the number of primitive random numbers for which the sign inversion process is performed among N primitive random numbers, and finally generate random numbers until the sign inversion process for all N primitive random numbers is performed. Random number generator characterized. The random number generator
The random number generation device according to claim 1, wherein (N * 2 ^N −N) random numbers are generated from the N primitive random numbers. The random number generator
When the primitive random number acquisition unit acquires N primitive random numbers generated within a range from 0 to a (a is a positive real number), and generates a random number within a range from 0 to a, n The random number generation device according to claim 1, wherein (a−x _n ) is calculated as a sign inversion process with respect to the first primitive random number x _n . When the primitive random number acquisition unit acquires N primitive random numbers generated within a range from a to b, and the random number generation unit generates a random number within a range from a to b, (a + b) The random number generation device according to claim 1 or 2, wherein a sign inversion process with / 2 as an origin is performed. The said random number generation part performs the inversion process of the code | symbol which used the preset target value as the origin, when the said primitive random number acquisition part acquires a regular random number. Random number generator. The random number output unit
6. The random number generation device according to claim 1, wherein at least a part of the acquired original random number is output as a random number. The primitive random number obtaining unit obtains two primitive random numbers,
The random number generation device according to claim 1, wherein the random number generation unit generates six random numbers from the two acquired primitive random numbers. The primitive random number obtaining unit obtains three primitive random numbers,
The random number generation device according to any one of claims 1 to 6, wherein the random number generation unit generates 21 random numbers from the obtained three primitive random numbers. The random number generation device includes:
The random number generation device according to any one of claims 1 to 6, further comprising: a dimension number designation input receiving unit that receives a designation input of the value of N. The random number output unit
The random number generation device according to claim 1, wherein the same value is thinned out from the random number generated by the random number generation unit and output as a random number. A primitive random number obtaining step of obtaining N (N is a natural number of 2 or more) primitive random numbers;
A random number generation step of generating a random number from the N primitive random numbers by a sign inversion process;
In the random number generation step,
When the N primitive random numbers are represented as (x ₁ , x ₂ ,... X _N ), N number of random numbers is generated by performing an inversion process of at least one code of the N primitive random numbers. , Change the number of primitive random numbers for which the sign inversion process is performed among N primitive random numbers, and finally generate random numbers until the sign inversion process for all N primitive random numbers is performed. Characteristic random number generation method. In the random number generation step,
The random number generation method according to claim 10, wherein (N * 2 ^N −N) random numbers are generated from the N primitive random numbers. In the random number generation step,
When N primitive random numbers generated in the range from 0 to a (a is a positive real number) are acquired in the primitive random number acquisition unit step, and random numbers are generated in the range from 0 to a, n 13. The random number generation method according to claim 11, wherein (a−x _n ) is calculated as a sign inversion process for the first primitive random number x _n . Primitive random number acquisition processing for acquiring N (N is a natural number of 2 or more) primordial random numbers;
Causing the computer to execute a random number generation process for generating a random number from the N primitive random numbers by a sign inversion process;
In the random number generation process,
When the N primitive random numbers are represented as (x ₁ , x ₂ ,... X _N ), N number of random numbers is generated by performing an inversion process of at least one code of the N primitive random numbers. , Among the N primitive random numbers, changing the number of primitive random numbers for which the sign inversion process is performed, and finally generating a random number until the sign inversion process for all N primitive random numbers is performed. A random number generation program that is executed by a computer. In the random number generation process,
The random number generation program according to claim 14, wherein (N * 2 ^N −N) random numbers are generated from the N primitive random numbers. In the random number generation process,
In the primitive random number acquisition process, when N primitive random numbers generated within a range from 0 to a (a is a positive real number) are acquired and a random number is generated within a range from 0 to a, the nth The random number generation program according to claim 14, wherein (a−x _n ) is calculated as a sign inversion process for the original random number _xn .   The computer-readable recording medium which recorded the random number generation program in any one of Claim 14 to 16. An electronic circuit component including a random number generation unit that generates a random number from N (N is a natural number of 2 or more) primitive random numbers by a sign inversion process,
The random number generator
When the N primitive random numbers are represented as (x ₁ , x ₂ ,... X _N ), N number of random numbers is generated by performing an inversion process of at least one code of the N primitive random numbers. , Change the number of primitive random numbers for which the sign inversion process is performed among N primitive random numbers, and finally generate random numbers until the sign inversion process for all N primitive random numbers is performed. Features electronic circuit components. In the simulation apparatus which performs simulation using the random number generated by the random number generation apparatus according to claim 8,
A simulation apparatus characterized in that a simulation calculation object is a three-dimensional multiple integral calculation. In the simulation apparatus which performs a simulation using the random number generated by the random number generation apparatus according to claim 1,
An N-dimensional multiple integration or a simulation apparatus characterized by using N random numbers generated by the random number generation unit by acquiring N primitive random numbers when performing simulation calculation of a mathematical model modeled on N variables.

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