SU962978A1 - Device for determining random process characteristics - Google Patents

Description

(54) DEVICE FOR DETERMINING THE CHARACTERISTICS OF A RANDOM.Process

The invention relates to specialized means of computer technology and can be used in determining the statistical characteristics of random variables. The device is designed to determine quantiles for an arbitrary unknown distribution of random variables. A device for calculating the statistical moments of random processes is known, comprising a centering filter, two multipliers, two quadratures, an integrator, and two averagers II. This device allows you to determine the third and fourth central points of a random process, but does not allow you to determine the distribution quantiles. The closest to the value and technical essence to the proposed method is the quantile determination device, which determines the selected median, containing two counters, a block of elements And, two triggers, a differentiating element ,. two pulse generators; three elements AND; element NO 2j. The disadvantage of the prototype is the limited nature of its tasks. It can define only one quantile, the selective median, which reduces the efficiency of the devices in which it is used. The purpose of the invention is to expand the functionality by defining a predetermined number of quantiles required. The goal is achieved by the fact that a device containing a pulse counter, inversion number counter, pulse generator, A group of elements AND, the first time whose inputs are combined. A buffer memory register, a group of switches, a register, a comparison circuit, two switches, adders groups are added. , scale amplifiers and subtractors, the input of the buffer memory register is the device input, the output of the register of the buffer memory is connected to the trigger input of the pulse generator, and the information output is the buffer memory is connected to the information input of the register, the clock input of which is connected to the first output of the pulse generator, the second output of which is connected to the input of the pulse counter, the control inputs of the first and second switches, the information output -l of each register is connected to the corresponding inputs of the first switch and keys with the first inputs of AND elements of the group, the output of the first digit of the register is connected to the first input of the first switch of the group, the information output of the i-ro digit (i 2, N - 1) is connected with the second input () ro switch and the first-input i-ro switch, group bodies, the output of the last N-ro register bit is connected to the second input (Nl) -ro of the group switch, the first and second outputs of the i-ro switch (i 1, N -1) connected to the input of the i-ro and fi + l) -ro register bits, respectively; both outputs of the first KOMiviyTaTopa are connected to the corresponding inputs of the comparison circuit, the output of which is connected to the information input of the second switch and the information input of the inversion number counter whose setup input is connected to the output of the pulse counter, and The inversion counter output is connected to the pulse generator stop input and to the second inputs of the elements, AND groups, the i-th output (i 1, N-1) of the second switch is connected to the control input of the i-ro switch, the outputs, elements And groups are the first group of device outputs, the first and second inputs of the i-ro subtraction unit of the group (i 1, N-1} are connected to the i-ro and Cil outputs) ro elements AND groups, respectively, the output of each subtraction unit is connected via a corresponding scaling amplifier to one of the inputs of the corresponding group adder, outputs which are the second group of outputs of the device, another input of the i-ro adder group 1, N-1) is connected to the output of the i-ro element AND group. The drawing shows a block diagram of a device for determining characteristics of a random process. The device contains a register 1 of buffer memory, a generator of 2 pulses, switches 3, register 4, first switch 5, counter 6 pulses, counter 7 of the number of inversions, circuit 8, comparison, second switch 1 switch 9, group of elements 10, blocks 11 subtraction, scale amplifiers 12, adders 13, outputs 14 of a device for determining quantiles. The device implements the following algorithm for determining quantiles. At its input comes a sequence of independent samples from some general set x, the integral distribution law P (x) and the density of the probability of which f (x) are unknown. From this sequence, the device for determining quantiles periodically makes a sample of N independent values (x, x ,, .. and rearranges it into a variational series, i.e., ranks according to; vi | with condition X (7.,. yx (7 "NJ-Index i means the number of a member of the sample. in the ranked row Ci 1,2, .. ,, N). It is known that the ranked sample has special properties. The number of the sample member in the ranked row i allows in advance, without knowing the specific value of this member of the sample X (-) AND of the form F (x), it is sufficient to accurately estimate the corresponding value of F ( x.) P by expression (i) F (X (-7) –P; j, T, -T. The specific value of the X /.- J sample term is obtained after ranking. Thus, in aggregate, everything is provided for the formation of quantile Xp information. From expression (1) it is obvious that when determining quantiles x-, corresponding to given values, the values of P; (J 1,2, ..., k), with a known sample size N, h. in advance, determine the numbers of the sample members in the ranked row (i, which correspond to the quantile of interest. Equated to obtain (i) / P. | CN-M (2) because the given values of P; may be any, the value of (i), j may be non-integral. Since there are no fractional numbers, no sample is taken, the nearest smaller member number of the sample in the ranked row is determined. For this, it is necessary to take the integer part of expression (2) (i). P -CN + I, (3 where E3 is the integer part of. (I). However, the specific value of the sample member with the number thus defined will not correspond exactly to what is required by the quantile. To determine a more accurate value, interpolation between the nearest smaller and larger sample terms. The interpolation percent is calculated in advance by the difference of expressions (2) and (.3). Af Ci, j - E (i) / l

Then the quantile value, which is corrected for interpolation, corresponds to a given probability level P; can be determined from the expression

U); (SR -%)

Expression (5) represents a generalized algorithm for the operation of quantile determinants that provide a given level of probability P; .

From expression (5) it follows that for the definition of quantilles x) corresponding to a predetermined level of probability P; at. known number of samples N, it is possible and necessary using expressions (2) - (4) to calculate in advance the numbers,, C (x) j J 1 of the register bits containing the ranked sample, and the interpolation fraction between them & J.

After ranking the sample from these bits, the specific values of the sampling members x 1 x i (i) j 1 and the lead interpolation in accordance with expression (5), we obtain the desired value quantile xy. , In the case when out of expression. (2) the integer value of the sample member number in the ranked row is immediately obtained, and it follows from (5) that the desired value of quantile X (i) j-can be obtained directly from. bit number (i), register with ranked sample.

The number of outputs in interpolation and without interpolation depends on the given probability levels P; required quantiles.

This algorithm is implemented in a device for determining quantiles as follows. The input of the register 1 of the buffer memory with a certain speed receives a sequence of independent samples from some general set x. The integral distribution law F (x) and the probability density f (x) of which are unknown.

Register 1 of the buffer memory, after filling, generates a signal to the generator of 2 pulses. According to this signal, the generator of 2 pulses sends out 4 pulses of N pulses per clock register input. Under the influence of clock pulses, the sequence values from the register of register 1 of the buffer memory are successively received through the information input into register 4 with the volume of N bits. Thus, in the main register 4, an II is formed ordered by a sample of values

With J i. , . . ,,

The clock frequency of the clock pulses is chosen in such a way as to match the rate at which the source sequence arrives at the input of register 1 of the buffer memory with

the write speed of the sample in register 4 and the speed of its processing. Sample values simultaneously with recording in register 4 via information outputs of each bit of register 4

are mounted on the respective inputs of the first switch 5, the switches 3 and the corresponding m inputs of the code from the AND 10 group.

The generator of 2 pulses, having completed the issuance of a pack of N clock pulses, begins to produce a continuous succession of sync pulses fed to the sync inputs of the first

switch 5, the second switch 9 and the input of the counter 6.

Under the influence of a sequence of clock pulses and in time with. The procedure of ordering (ranking) of the sample begins. The clock frequency is chosen such as to minimize the sampling ranking time.

The ranking procedure is based on

pairwise permutation algorithm.

When first viewed, each sample value is compared with a neighboring value. The pair of sample values, for inequality 7, is interchanged. Thus, the most significant member of the sample will take the place of X (j) already at the first viewing. Pairwise comparison and rearrangement of the sampling members continues until the sampling ranking process, i.e. X (x. S. .. X (N)...

The fact that the inequality 7 is true for the pairwise view of a winnings is an inversion, and the function of the sum of inversions is determined by the expression

  . .

U (x., X) O, X, - x. .

The value of T is calculated for each Cycle of pairwise comparison and permutation of the members of the sample. The sign of the end of the sampling ranking is the value of.

We show on a simple numerical prier that the pairwise permutation algorithm leads to the ranking of the sample. Let the members of a random sample of volume N 4 take the following values 1, 4, 3, 2.

The result of permutations of the sample members depending on the inversions for each viewing cycle is shown in the table.

Claims (1)

1. USSR author's certificate №330463, cl. G 06 G 7/52, 1972. 2, USSR Author's Certificate No. 744606, cl. G Q6 F 15/36, 1977 (prototype) about