SU387391A1 - DEVICE FOR EVALUATING THE DISTRIBUTION OF THE PROBABILITIES OF RANDOM PROCESSES - Google Patents

Description

one

The invention relates to specialized computing devices for measuring statistical characteristics of random processes.

The basis of a wide range of tasks of statistical research of technological processes, control and monitoring systems is the requirement of simultaneous evaluation of both conditional and unconditional functions and probability densities of the processes characterizing the objects of study. The determination of the characteristics of processes under conditions of normal operation of objects, when the observation time of processes is limited to the minimum possible, determined by the cost of collecting and processing statistical data, non-parametric methods for evaluating statistical characteristics are widely used. These methods make it possible to quickly evaluate the statistical characteristics of random processes over a limited observation interval. In particular, in combination with the methods of flow, its smoothing, they allow an approximate estimation of the statistical characteristics of relatively smooth nonstationary random processes using a single implementation.

A device is known for measuring the conditional probability density of one random process relative to the values of another random process. However, the known drying device only measures the conditional probability density and only stationary random processes under the condition of their fixed pairing. Measurements are carried out by the method of sequential statistical analysis, in each cycle of which only one ordinate of conditional probability density is measured, therefore the analysis time and, therefore, the duration of observation of processes in the object of study increases in proportion to the number of measured ordinates of conditional probability density .

The proposed device allows for the implementation of two relatively "smooth non-stationary random processes, whose duration is commensurate with the measurement time of one ordinate of conventional density

probabilities} of random processes in a known device, define conditional functions and conditional probability densities of the processes, so they are unconditional functions and probability densities. This is achieved by the fact that the proposed device additionally contains two smoothing blocks, an integrator matrix, a switch, an integrator, which are connected in such a way that in the process of entering the device a statistical

information is provided by its current, its smoothing, accumulation and subsequent calculation of the probability distribution functions and the probability densities of the studied processes. The drawing shows a block diagram of the proposed device. The device contains smoothing blocks I and 2, amplitude selectors 3 and 4, control unit 5, matrix of integrators 6, multiplication unit 7, switch 8, recorder 9, integrator 10 and multiplication unit 11. The signals under study are fed to smoothing blocks 1 to 2 each of which consists, for example, of an integrator of average values and an adder built on direct current amplifiers. The outputs of the smoothing units are connected to amplitude selectors 3 to 4 m with two inputs of the control unit 5. Amplitude selectors are ten channel devices with a common input. Each channel of the amplitude selector consists of an upper amplitude selector, a lower amplitude selector, two shaping devices and a subtractor. The outputs of the amplitude selectors are connected to two inputs of the integrator matrix 6. The third input of the integrator matrix is connected to the output of the control unit, and its output is connected to the switch 8, the second input of which is connected to the control unit. The integrator matrix, the control unit, and the switch are known devices. The switch output is connected to an integrator 10 built on a DC amplifier With deep negative feedback. The integrator outputs are connected to the control unit and to one of the inputs of the multiplication unit 11, the second input of which is connected to the control unit and the output to the recorder 9. The multiplication unit is a typical analog device that performs nonlinear signal conversion a and 3 at its inputs in accordance with the formula a. (A + G- (a-WG. As the recorder in the proposed device, any of the widely used digital printing devices can be used. The second input of the recorder through the device The multiplication 7 is connected to the output of the control unit of the device. The multiplication unit 7 is a typical DC amplifier with variable gain. The program of the proposed device is divided into two stages: 1) Primary processing and accumulation of the initial information. 2) Evaluation of the distribution probabilities of random processes. The operation of the device at the first stage consists in that the random processes X (t) and V under study (O arrive at smoothing blocks I and 2, where their current smoothing is carried out using the moving average method. Current values of mathematical expectations determined by integrators of average values , in accordance with the expressions m, (f), y (0-y) d where T is the current smoothing interval, the following is received: one of the inputs of the adders of the smoothing blocks, to the second input of which the studied processes X are fed (t and Y (0 smoothing blocks carried out simultaneously with the current process smoothing, the change of their dynamic range to a value of ± 50. At the same time, the gain factors / Ci and / C2 adders are fixed by the control unit 5. From the outputs of the adders, signals go to amplitude selectors 5 and 4, which convert them to voltage pulses proportional to the residence time of signals within the amplitude selector corridors. From the amplitude selector outputs, voltage pulses are fed to the inputs of the integrator matrix 6. The integrator matrix consists of 10 rows and 10 columns (The number of amplitude selector channels). The row inputs of the integrators located in the same row of the matrix are connected to the input bus of the row of the corresponding channel of the amplitude selector of the signal X (t) so that the input of the first row of the matrix is connected to the output of the first channel of the amplitude selector, the input bus of the second row is output of the second channel of the amplitude selector, and so on until the tenth row. Column inputs of integrators located in the same column of the integrator matrix are connected to the input bus of the column. The column input buses are connected to the corresponding 10 outputs of the amplitude Y selector signal (/). The implementation of this switching of the outputs of the amplitude selectors with the integrator matrix provides for the process of the input of the initial signals to integrate in the integrator matrix of counting pulses coming from the pulse generator of the control device to the third input of the integrator matrix. Thus, in the observation interval of the processes X (t) and Y (t), in the integrator matrix, the frequencies Vtj of the joint occurrence of events defined by the inequalities are fixed: | - gmin1 1L () | 1 gmax1, | / umin1 | U (01 Humax |, deAg nn, gmax, j max - MAXIM and minimum levels of restriction of the i-ro and / -H channels of the amplitude selectors, and in the pulse counter of the control unit the number of pulses N corresponding to the observation interval .

At the end of the primary processing and accumulation of information (in the form of Vjj frequencies) in the integrator matrix, the functions and densities of probability distributions of random processes are calculated.

Calculation of unconditional probability functions of random processes, or probabilities P of events defined by inequalities

/ (g,) (,

F (y) (,,

F (xi, yj) Plx (t) x „y (O yj,

The proposed device performs the following formulas:

10 i10 10

F (; c,) yyv., YV VV. ,,,

     2424

/ ten

( (one

F (xi, (/.). V 1 N Z4

a.l

a calculation of conditional functions of probability distribution of random processes

P (X, (() XC, g (and

P (({). X,. Ij (t}

; by formulas

(.

R (H.U}., Tj h

F (y.Xi) -. . N

Calculation of probability densities of random processes (conditional and unconditional), corresponding to the given probability distribution functions, is made according to the formulas:

ten

f (.}

Chg

ten

f (yj)

 {t i

f (XiVj)

ip

N &.

6

/ (3;)

ten

& X 2i; i l

/ (- "/. Jv)

 ./

/ (l- /) 10

& y vy / - - ,,.

where the wider corridors of amplitude selectors are 3 to 4 in accordance with the received number of channels (10) and the dynamic range of signals

(± 5b) & x 1

The operation program of the device is specified by the control unit. Consider the calculation of an estimate of the probability distribution in accordance with the given ratios. Calculation of the one-dimensional probability distribution function of a random process. Let us use the example of calculating the function F (Xi).

When this function is calculated by the control unit, switch 8 starts, one after the other, connecting the integrators of the first row of the matrix, starting with the first integrator, to the inputs of the integrator 10. At the end of the integrator round of the first row, the switch starts the second integrator round and ends Do not connect to the output of the matrix integrator located at the intersection of the f-th row and the tenth column. By this time, the output voltage of the intgoator W will be equal to, lr 10

Have v. .This is shashK

portioned

I / 1

but at the end of the switch's switch on the login of the multiplier; // block. To the second input of the unit, the riocTyriaeT voltage is raised, which is proportional to the level of the control unit. This voltage is generated by applying a voltage, which is linear, from the output of the counter to the input of the nonlinear element of the control unit, built on a DC amplifier and having a static characteristic as an inverse function. So at the entrances

block multiply voltages

(101

t /, U; E -, / and,

.fl; - N

which, after multiplying, are recorded by the recorder 9. At the same time, the recorder receives a voltage proportional to the magnitude of the argument of the distribution function F (x, i) generated by multiplication unit 7. The operation of the multiplication unit 7 is that the voltage proportional to the coefficient amplification / (, amplified by l,; times, where lg is a value proportional to the voltage in the middle of the t-ro corridor of the amplitude selector and determined in accordance with the following table:

In a similar way, the calculation of all the ordinates of the function F (xi),, 2, ..., 10 is carried out and the results of the calculations are recorded simultaneously with the registration of the real value of the argument, which is

CgHg.

Similarly, the function F (yi) is calculated. The only difference is in the VOLUME that the switchboard bypasses the matrix integrators in columns, starting with the integrator of the first row of the first column to the integrator of the tenth row of the / th column of the matrix.

The calculation of the two-dimensional Probability distribution function of a random process is performed as follows.

The switch, by command of the control unit, bypasses the matrix integrators line by line, starting with the integrator (the first row of the first column and ending with the integrator of the 1st row of the / th column). The switch switches to a new row of the matrix whenever it connects to the integrator the tenth output of the / of the matrix integrator. By the end of the roundabout of all the integrators located in the rectangle bounded by the rth row and the i -M column, the integrator 10 accumulates a voltage proportional to the value of 2 V which

 2j "p

after multiplying by the value - in the multiplication unit 11, as described above, is recorded by the recorder. At the same time, the arguments XfKi, yj-Kz, which enter the recorder from the control unit through multiplication unit 7, are registered.

Consider the calculation of conditional probability distribution functions using the example of the function F (Xilyj). In this case, the switchboard sequentially connects the integrator outputs of the matrix matrix to the input of the integrator 1- / to the input of the integrator 10. By the time of bypassing all the f-integrators of the first column, the recorder will record a value proportional to the product of 2 M output integrated with I

torus and - entering the multiplication block with

N

control unit. At the same time, the values of Xy-K and yj-Ki, which arrive at the recorder from the multiplication unit, are recorded. The calculation of the function P (nodes / Xr) is carried out in the same way as described by the switch bypass of integrators / and the string

The calculation of the probability distribution density, for example f (Xi), is carried out as follows. In the integrator 10, during the operation of the switch, the voltages accumulated in the integrators are summed.

ten

rth row of the matrix. The voltage HJ of the smart is supplied by the unit multiplying // by the voltage

proportional to the magnitude - incoming

from the control unit, and the result is recorded by the recorder. The second input of the recorder is fed from the multiplier 7, a voltage directly proportional to the value Xi-Ki, which is also fixed. The density calculation f (yi) is performed similarly. When calculating the two-dimensional probability density f (xi, Oz), the integrator 10 operates in a gain mode with a gain factor equal to one. In this case, the switchboard through the integrator 10 connects the output of the integrator i- / to the first input of the multiplication unit 11, and the second input of which from the control unit receives the voltage value--. The recorder records the result of the multiplication of the value of Vij and - simultaneously with the values of Xi-Ki and yj-Kz, coming from the multiplication unit 7.

Consider using the example of the function f (yj / Xi) the operation of the device when calculating the conditional probability densities of random processes. With the start of operation, the switch makes a sequential connection to the integrator of 10 integrators of the / th column of the matrix, after which

ten

proportional to 2 O. enters filament 1 of the control unit grader and is memorized.

After that, the switchboard through the integrator 10, which then acts as an amplifier with a gain factor "I, pushes the voltage v, -j from the output of the i- / matrix integrator to the first input of the multiplication unit, and to the second input of the multiplication unit

ten

voltage value ij obtained by the STS 1

ten

Lev of the passage of the voltage of the v / y from the output

control unit integrator via a non-linear element. The result of the calculation of / (i / jAt,) is recorded by the recorder simultaneously with the values Xi-Ki and yj-K2, arriving in

a logger from the output of multiplication unit 7. The calculation of the function f {Xi / yj) is performed in the same way.

Thus, the inclusion in the known device of additional smoothing blocks, the switch integrator matrix, the integrator, with their proper switching, allows the device to calculate both conditional and unconditional probability functions of stationary and relatively “smooth nonstationary random processes” processes. The simultaneous expansion of the class of random processes under investigation due to the accumulation of initial information in the integrator matrix is reduced compared with the known device, the duration of the observation time of random processes is necessary.

Subject invention

A device for estimating a probability distribution of random processes, comprising

amplitude selectors, control unit, the first output of which is connected to the integrator through the switch, the first output of which is connected to the control unit, the second output is connected to the first multiplication unit, the output of which is connected to the first input of the recorder, the second output of the control unit is connected through the second multiplication unit. the second input of the recorder, the third output of the unit up.

The control unit is connected to the first multiplication unit, characterized in that, in order to expand the class of solved tasks, it further comprises an integrator matrix and two smoothing blocks, whose inputs are connected to the device input terminals, the first inputs of the smoothing blocks are connected to the integrator matrix through appropriate amplitude selectors, the second inputs of the smoothing units are connected to the corresponding inputs of the control unit, the output of which is connected to the corresponding input of the integrator matrix, the output of which is connected switch.