SU734769A1 - Generator of markov's train of random numbers with logarithmically normal distribution - Google Patents

Description

a single parameter characterizing a Markov sequence of such random numbers. An independent change in the parameters of the generated sequence of random numbers with a log-normal distribution is very important for the practical use of the generator in simulating various physical and technical processes.

The purpose of the invention is to expand the functionality of the generator by ensuring the independence of the adjustment of the statistical parameters of the Markov generator of random numbers with a log-normal distribution.

To achieve this goal, a Markov random number generator with log-normal distribution, containing a clock generator, a generator of normally distributed random numbers, whose input is connected to the output of a clock generator, an autoregression unit, the first input of which is connected to the output of a generator of normally distributed random numbers, the first multiplier, the first input of which is connected to the output of the autoregression unit, the first adder, the first input of which is connected It is connected to the output of the first multiplier, an exponential converter, the input of which is connected to the output of the first adder, and the output is the output of the generator, a functional converter, the output of which is connected to the second input, and the input is to the third input of the autoregressive unit, first, second and third sensors numbers, the second and third adders are entered, the subtraction unit, the second and third multipliers, the first and second division blocks, the quad, the square root extraction block, the first, second and third logarithms, and the output of the first The numbers are connected to the inputs of the quadrant and the first logarithm, the output of which is connected to the first input of the subtraction unit, the output of which is connected to the second input of the first adder, the output of the second number sensor is connected to the first input of the first division unit, the second input is connected to the output of the quad, and the output of the first division unit is connected to the input of the second adder and to the first input of the third multiplier, the output of the second adder is connected via the second logarithm unit to the first input of the second division unit, and with the input of the square root extraction unit, the output of which is connected to the second input of the first multiplier and the input of the second multiplier, the output of which is connected to the second input of the subtraction unit, the output of the third number sensor is connected to the second input of the third multiplier, which output is connected through a series-connected third adder and third The logarithm unit is connected to the second input of the second division unit, the output of which is connected to the third input of the autoregressive unit.

The block diagram of the generator is shown in Fig. 1, where 1 is the clock pulse generator, 2 is the generator of normally distributed random numbers, 3 is the autoregression unit, 4 is the first multiplier, 5 is the first adder, 6 is the exponential transform. binder, 7 - output terminal, 8 - functional converter, 9 - first

0 sensor of numbers, 10 - second sensor of numbers, AND - third sensor of numbers, 12 - quad, 13 - first logarithm block, 14 - subtraction block, 15 - first division block, 16 - second adder, 17 - second logarithm block, 18 - extraction unit.

square root, 19 is the second multiplier, 20 is the third multiplier, 21 is the third adder, 22 is the third logarithm block, 23 is the second division block.

0 The output of the generator 1 clock pulses is connected to the input of the generator 2 normal independent random numbers, the output of which is connected to the first input of the autoregression unit 3. The output of the autoregression unit 3 is connected to the first input of the first multiplier 4, the output of which is connected to the first input of the first adder 5. The output of the first adder 5 is connected to the input of the exponential converter 6, you-. the course of which is connected to the output terminal

0 7. The output of the first sensor 9 numbers is connected to the inputs of the Quad 12 and the first logarithmic unit 13. The output of the last is connected to the first input of the subtraction unit 14, the output of which is connected to the second input of the first adder 5. The output of the second

 The sensor 10 numbers connected to the first input of the first block 15 division, the second input of which is connected to the output of the quad 12, and the output to the input of the second adder 16 and the first input of the third multiplier 20.

Q The output of the second adder 16 is connected via the second block 17 of the logarithm to the first input of the second division unit 23 and to the input of the square root extraction unit 18, the output of which is connected to the second input of the first multiplier 4 and to the input

5 of the second multiplier 19. The output of the latter is connected to the second input of the subtraction unit 14. The output of the third number sensor 11 is connected to the second input of the third multiplier 20, the output of which is connected through the serially connected third adder 21 and the third logarithmic unit 22 to the first input of the second division unit 23. The output of the division unit 23 is connected to the second input of the autoregression unit 3 via the functional converter 8, and to the third

5 input block autoregression 3 directly.

The proposed device operates as follows.

The principle of operation of the device consists in simultaneously changing the parameters of a sequence of normal Markov numbers so that in this case only one parameter of the Markov sequence of log-normal random numbers is changed. The structure of the proposed device is determined by the relationship between the average value, the variance and the correlation coefficient of the neighboring numbers in the normal sequence and the average value, the variance and the correlation coefficient of the adjacent numbers of the logarithmically normal sequence.

The clock generator 1 synchronizes the operation of the entire device and, in particular, starts generator 2, which, in time with generator 1, generates a sequence of normal independent random numbers with an average value O and with a dispersion of one unit. In block 3, the autoregression of these independent normal random numbers forms a Markov sequence of random numbers according to the autoregression scheme, with the first autoregressive coefficient determined by the number at the second input of block 3, and the second autoregressive coefficient by the number at the third input of block 3. To obtain a stationary sequence of random numbers At the output terminal 7, the following condition is met: the sum of squares of numbers on the second and third inputs of the autoregression unit 3 is equal to one. The Markov random numbers from the output of block 3 are multiplied in the first multiplier 4 by some positive number, which results in a change in their variance and in the first adder 5 they add a certain real number, which results in a change in their average value. In converter 6, an exponential transformation of the number X occurs at its input to the number exp X at its output. The number C at the second input of the autoregression unit 3 is determined from the number C g at its third input with the help of the functional converter 8, which performs the conversion C1 (1-C i) -j. The sequence parameters at the output terminal 7 are determined by the numbers on the second inputs of the first multiplier 4 and the first adder 5 and the number on the third input of the autoregression unit 3.

In the first sensor 9, the average value of the numbers in the sequence taken from the output terminal 7 and having a lognormal distribution is set, in the second sensor 10 the value of the dispersion of these numbers is set and in the third sensor 11 - the value of the correlation coefficient between neighboring numbers in the sequence removed from the output terminals 7. In blocks 12 through 23, simultaneous non-linear conversion of these numbers takes place for input into blocks 3, 4 and 5 so that on favorable

terminal 7 was obtained a sequence of numbers with parameters that are harnessed in sensors 9, 10, 11.

The desired value of the standard deviation from the numbers in the normal sequence depends on the mean and variance of the numbers in a log-normal sequence. Therefore, the variance value stored in sensor 10 is divided in the first block 15 of dividing by the square of the average value formed in quadrant 12. The resulting quotient is added one in the second adder 16, after which the second logarithm of the total 17 is formed in the second block and 18, the square root of the logarithm is extracted. The resulting number is multiplied by the normal random number in the first multiplier 4.

The desired mean value of the numbers in the normal sequence also depends on the mean and variance of the numbers in a log-normal sequence. Therefore, in block 13 of the logarithm, the natural logarithm of the average value stored in sensor 9 is formed, and in block 14 the dispersion of the normal sequence, previously multiplied in the second multiplier 19 by 0.5, is subtracted. The normal number in the first adder 5 is shifted to the obtained value. The desired value at the third input of the autoregressive unit 3 depends on all three parameters of the logarithmically normal sequence. The correlation coefficient between adjacent numbers in a logarithmically normal sequence, stored in sensor 11, is multiplied in the third

S multiplier 20 by the quotient of the variance per square of the average from the output of the first division block 15. In the third adder 21, the unit is added to the resulting product, in the third block 22 the natural logarithm of the sum is formed, which in the second division block 23 is divided by the variance of the normal sequence. The number at the output of dividing unit 23 is the required second autoregression coefficient for the numbers stored in the sensors 9, 10, 11.

The positive effect from the application of the proposed device is to increase the degree of automation of modeling technical systems and physical phenomena, where the proposed devices are used, for example, in input generators

0 impacts or generators of unwanted disturbances on the system.

Claims (2)

Input and perturbing effects have their own physical parameters arising from the properties of the object being modeled. These parameters are different from the parameters of the original signal sequence converted in the generator to obtain a signal sequence for modeling input and disturbance effects. The proposed device automatically generates the desired values of the parameters of the original signal sequence based on the specified values of the parameters of the Markov signal sequence with a log-normal distribution. This is an important circumstance because, in essence, the modeling of systems and phenomena implies the use of multiple sets of initial data and the transition from one set of initial data to another set must be fast. The economic effect of the application of the proposed device can be assessed as follows. Determining the values of the parameters of the original sequence takes a manual way on the order of 4-8 minutes. In the case of application of the proposed device, the time for determining the parameters of the initial sequence is practically determined only by the time of setting the generator knobs to the desired values of the parameters of the signal sequence with a log-normal distribution and is several seconds. Thus, simulations with a single data set will save about 4–8 min. Assuming that the simulation with one set of initial data lasts about half an hour, we get that in an 8-hour working day, 64-128 minutes or 13-27 / 0 working hours of an engineer engaged in modeling will be saved. The load on equipment, usually expensive, is increased by the same amount. The above estimate gives an idea of the order of magnitude of the economic effect. Thus, the proposed generator of Markov sequences, random numbers with a log-normal distribution, has new, broader capabilities than existing generators and makes it possible to more effectively solve problems in the field of digital modeling and statistical tests. Claim Generator Markov random number sequence with log-normal distribution, containing clock generator, generator of normally distributed random numbers, the INPUT of which is connected to the output of the clock generator, autoregression unit, the first input of which is connected to the output of the generator of normally distributed random numbers, the first multiplier, the first input of which is connected to the output of the autoregression unit, the first adder, the first input of which is connected to the output of the first exponential transducer whose input is connected to the output of the first adder, and the output is the output of the generator, a functional transducer whose output is connected to the second input, and the input to the third input of the autoregressive unit, the first, second and third number sensors, different that, in order to expand the functionality of the generator by ensuring the independence of adjustments of the statistical parameters of the generator, the second and third adders, the subtraction unit, the second and third multipliers, first and second division blocks, quad, square root extraction block, first, second and third logarithms, the output of the first number sensor connected to the inputs of the quad and the first logarithm, the output of which is connected to the first input of the subtractor, the output of which is connected to ; about the second input of the first adder; the output of the second number sensor is connected to the first input of the first dividing unit, the second input of which is connected to the quad output, and the output of the first dividing unit is connected to the second input the adder and the first input of the third multiplier, the output of the second adder via the second logarithmic unit is connected to the first input of the second division unit and to the input of the square root extraction unit, the output of which is connected to the second input of the first multiplier and to the input of the second multiplier, whose output is connected to the second the input of the subtraction unit, the output of the third number sensor is connected to the second input of the third multiplier, the output of which is connected through a serially connected third adder and a third logarithmic unit to a second input of the second dividing unit, whose output is connected to the third input autoregressive block. Sources of information taken into account in the examination 1. USSR author's certificate in application number 2172038 / 18-24, 1975. 2. Bykov V.V. Digital modeling in statistical radio engineering. “Soviet Radio, M. 1971, p. 124 (prototype).