# Method for prediction of measurement results and device for its realisation

FIELD: information technologies.

SUBSTANCE: device comprises unit of input realization storage, clock oscillator, control unit, unit of useful signal extraction, unit of storage of five last values of useful component assessment, unit of approximation with polynom of the first degree, unit of approximation with polynom of the second degree, unit of output realization storage. In device end values of assessment are approximated with the help of method of least squares with polynom of the first or second degree, then produced equation of assessment is used to calculate values in forecast points.

EFFECT: forecasting measurement results on the basis of useful signal extraction without end effects, under conditions of limited a priori information about useful and accidental component.

1 dwg

The invention relates to the field of computer engineering and can be used in control systems and signal processing.

Input the implementation of the results of measurements represents only a discrete sequence of Y_{1}, Y_{2}, ..., Y_{n}where Y_{k}=Y(t_{k}),.

A mathematical model of the measurement results can be presented in the form:

where S_{k}- a useful component (trend); u_{k}additive random component, distributed according to a Gaussian law with mean m=0 and variance σ^{2}. The estimation of forecastis carried out over a time horizon T,.

The main task is the prediction of the measurement results based on the methods of extraction of the useful signal without end effects in conditions of limited a priori information about the useful and the random component. The problem of forecasting the measurement results may occur: 1) in the operation of management and control; 2) in radio engineering in signal processing in automatic control systems; 3) in the economy - to identify trends and make prognosis in the absence of a model of the original process; 4) Metrology for measuring characteristics condition the atmosphere etc.

To solve the problem of forecasting the results of measurements are now widely used statistical forecasting methods. The application of mathematical-statistical methods of forecasting based on the methods of least squares, adaptive methods, methods of autoregressive requires a priori information about the function of the trend to correctly choose the degree of approximating polynomial. In cases when such information is not available, apply heuristic techniques that are not based on probability and statistical theory: the method of moving averages, exponential smoothing method. For measurements, the formalization of which cannot be done by the time of forecasting, uses heuristic methods based on expert judgement of experts in the given field of knowledge. Main processing procedures prognostic expert assessments are the consistency check, the cluster analysis and the finding of group opinion. Check the consistency of the opinions of experts expressed rankings, is performed using the Spearman correlation Kendall and Spearman or Kendall and Babington Smith.

The known method of extrapolation [Ashmanov S.A. Mathematical models and methods in Economics. M.: Izd. Moscow state University, 1981. - 158 S.], which in mathematical terms is a distribution Hara is tera change the function of the useful component of the field observations in the area, lying outside this interval.

The problem of extrapolation is formed like this: suppose that in the interval 1, 2, ..., n known values of the measurement results Y_{k}need to determine the values of the useful component in the point n+1 lying outside this interval.

Technological and economic forecasting using the method of extrapolation can be performed by a limited number of functions, which are divided into four classes:

Class 1: linear growth functions in the greater part of the interval with the slowdown at the end.

Class 2A: on the whole interval has increased exponentially. The equation of the curve for the function of this class iswhere a is the process value at k=0; and - process parameter.

Class 2B: S-curves, characterized by an initial exponential growth. An example of a function class 2B is a curve logical growth curve (pearl).

Class 3: function with double-exponential growth with a subsequent transition to a more gentle curve.

Class 4: a function with a slow exponential growth in early development, which is replaced by a more sudden rapid growth and, finally, the slowdown at the end of development.

The characteristics of the method-analogue, coinciding with the characteristics of the proposed technical solution, the following: storing digital signal representation of the values of Enki useful component in the form of a polynomial from the discrete values of the initial implementation of the measurement results.

The disadvantages of this method and device are:

- priori information about the choice of extrapolation functions;

growth forecast errors in the presence of a random component in the original measurements.

Barriers to achieving the desired technical result are as follows: the assumption of the rapid nature of the changes in the predictable functions limits the application of the method of extrapolation to only those periods of time during which the change function is not offered abrupt changes.

The known method exponential smoothing [Aivazian S.A., Mkhitaryan B.C. Applied statistics and the basics of econometrics. - M.: UNITY, 1998. - 465 S.]. The method makes it possible to obtain an estimate of the parameters of the trend not the average level of the process, and the trend prevailing at the time of final observation. The greatest application method has been to implement medium-term forecasts. The feature of the method of exponential smoothing is that in the process of finding useful assessment component uses only the previous values of the input and results of measurements taken with a certain "weight", and the value of the weights decreases to the beginning of the implementation. To use this method, only one realization of Y_{1}, Y_{2},
..., Y_{n}the original process.

The method of exponential smoothing involves memorizing the original discrete implementation of the results of measurements Y_{1}, Y_{2}, ..., Y_{n}random process, the choice of the smoothing parameter α (0<α<1), the values of S_{0}the calculation estimates the useful component of the recurrent formula:

,.

To use exponential smoothing of the measurement results is determined by the initial value ofevaluation of the useful component and the smoothing parameter α. Wrong choice of initial conditions can have a significant impact on the result of processing the original discrete implementation of the measurement results.

Evaluation of prediction coefficients of the polynomial are determined using exponential moving averages on theorem of brown-Meier.

The characteristics of the method-analogue, coinciding with the characteristics of the proposed technical solution, the following: storing digital signal, obtaining estimates of the useful component, the representation of the values assessment is a useful component in the form of a polynomial from the discrete values of the initial implementation of the measurement results.

The disadvantages of the known method and device:

- the uncertainty of the choice of the smoothing parameter α, which determines the assessment coeff the patients trend and forecast results;

- uncertainty selectingthat leads to the unreasonableness of the repeated re-application of the method of exponential smoothing for other values of α and;

growth forecast errors decrease of the accuracy of the initial conditions.

Barriers to achieving the desired technical result are as follows: method of exponential smoothing is not "plug and play" fashion, as the choice of the parameters α and

is subjective and depends on experience and practical skills of the researcher, the values of α andthere are functions in the form of signal, noise, sample size.

The structural scheme of the device that implements the method, comprises a generator of such pulses, the switch control unit, a storage register, the adder block multiplication, the output storage register evaluation useful component.

There is a method of adaptive forecasting Holt [Anderson T. Statistical analysis of time series. - M.: Nauka, 1976. - 343 C.], the characteristic feature of which is the ability to continuously take into account information about the dynamic characteristics of the results izmerenii, adapting to these changes, giving the more weight the higher the information value of available observations, the closer they are to the current moment of prediction.

In the way Holt used the ideology of exponential smoothing, we introduce two smoothing parameter α_{1}and α_{2}(α_{1}>0, α_{2}<1). The forecast in this way is determined by a linear trend:

,.

Update prediction coefficients produced by the formulas:

,

.

Thus, the prediction by the method of Holt is a function of past and current data parameters α_{1}and α_{2}and initial values a_{0}(0, α_{1}α_{2}), and_{1}(0, α_{1}α_{2}).

The characteristics of the method-analogue, coinciding with the characteristics of the proposed technical solution, the following: storing digital signal, obtaining estimates of the useful component, the representation of the values assessment is a useful component in the form of a polynomial from the discrete values of the initial implementation of the measurement results.

The disadvantages of the known method and device:

- the uncertainty of the choice of the parameters α_{1}and α_{2}that determine the estimated coefficients of the trend and the forecast;

growth forecast errors decrease of the accuracy of the initial conditions.

Reasons preventing Costigan the Yu desired technical result
are as follows: method of adaptive forecasting Holt is not "plug and play" fashion, as the choice of the parameters α_{1}and α_{2}is subjective and depends on experience and practical skills of the researcher.

The known method the heuristic prediction [Henan, E. the Analysis of time series. - M.: Statistics, 1964. - 215 S.], which is based on the opinions of experts in this field of knowledge and, as a rule, is used to predict flows, formalization, which cannot be done by the time of prediction.

The data processing method of expert forecasting involves the following steps:

1) is Determined by the sum of ranks of each of the prediction values of the measurement results:

,

where a_{i,k}- the rank assigned to each k-th value of the predicted values of i an expert, m is the number of experts involved in the assessment.

2) is Determined by the average of the sum of ranks:

.

3) is Determined by the sum of the squared deviations of the sums of ranks:

.

4) is Determined by the multiple coefficient of rank correlation (coefficient of concordancia), to assess the degree of consistency of the experts ' opinion:

.

Coefficie the t concordancia can vary from 0 to 1. If it is significantly different from 0, we can assume that between the experts ' opinions there is consent.

5) an assessment is Made of non-randomness consent of expert opinion using Pearson criterion when the number of degrees of freedom r=(T-1), given the significance level α=0.05 and a confidence level of p=α-1:

.

After the transformation is determined by the weight (importance) for each option, the forecasted values:

,.

The characteristics of the method-analogue, coinciding with the characteristics of the proposed technical solution, the following: storing digital signal, using standard criteria.

The disadvantages of the known method and device:

- the inability to automate the prediction of the measurement results;

the dependence of the forecast from professional experience and intuition of experts.

Barriers to achieving the desired technical result are as follows: forecast expert assessments reflect the individual judgment of experts and based on the mobilization of professional experience and intuition.

Closest to the invention is a method of least squares and device for piecewise linear approximation [Bendat J., Persol A. Application analysis SL the tea data: TRANS.
from English. - M.: Mir, 1989. - 540 S., copyright certificate №1624479]. To use this method, only one realization of Y_{1}, Y_{2}, ..., Y_{N}the original process.

The method of least squares allows for measurement result Y_{1}, Y_{2}, ..., Y_{N}the original process to obtain an estimate,by minimizing the target function of the form:

In practice as a modeluse the following functions: linearquadratic, powerdemonstration, exponentiallogisticsand several others.

In the case whenis a polynomial of the first degreethe coefficients a and b can be found by minimizing the target function of the form:

Differentiating expression (2) for a and b and equating them to zero, we obtain a system of linear equations:

.

The solution of the system is:

.

When evaluatingthe sum of the squared deviations of values from values i.e. monitoring) reference and measurement is minimal (2).

The choice of valuation modelin each case carried out on a number of statistical criteria, such as variance, correlation ratio, etc. it Should be noted that the criterion of the method of least squares is the criterion of approximation, and is not forecast. However, taking into account the hypothesis of the stability of the process in the future, it can be assumed that under these conditions the model, the most successful for the approximation will be better for prediction.

The characteristics of the prototype method, coinciding with the characteristics of the proposed technical solution, the following: memorizing discrete signal, the approximation by the method of least squares.

The disadvantages of the prototype method is:

- when using this method requires a priori information about the function of the useful signal;

- useful error component is along the implement in the General case of nonlinear dependence and reaches its maximum values at the boundaries of the interval of approximation;

- using the method of least squares is impossible to obtain a reliable forecast for a large period of time;

- when polinomialnoi valuation models useful component of a rigorous solution to the problem of minimizing the objective function of the method of least squares is not always there because of the nonlinearity re emeu system of equations.

Barriers to achieving the desired technical result is the following:

efficiency assessment of the useful component and the prognosis depends on the volume of sales, the statistical characteristics of the additive random component and the availability of a priori information on the functional dependency models are useful component.

The structural scheme of the device for piecewise linear approximation contains a group of series-connected registers, the first and second myCitadel, the adder, the first and second accumulating adders, and delay elements, a generator of clock pulses, two multipliers and two divider by a constant factor.

The aim of the invention is the prediction of the measurement results based on the methods of extraction of the useful signal without end effects in conditions of limited a priori information about the useful and random components.

The proposed method of predicting the results of measurements comes from having a single discrete implementation of the process under investigation Y_{1}, Y_{2}, ..., Y_{n}where Y_{k}=Y(t_{k}),. A simplified mathematical model of the measurement results is described by the expression (1).

Consider a method for predicting the results of measurements involves:

1) memorizing the input re the implementation of Y_{
1}, Y_{2}, ..., Y_{n};

2) the useful component (trend),based on the methods of extraction of the useful signal without end effects;

3) choose the degree of approximating polynomial for prediction (p_{1}=1, p_{2}=2);

4) the approximation values assessment,using the method of least squares the selected polynomial degree p and obtain estimates;

5) obtained by the evaluation equationcalculating values of forecastpointswhere T=3.

The proposed method allows to obtain short-term forecast of the measurement results with high accuracy not more than three values. A method for predicting the results of the measurements lies in the allocation of usable components (trend),based on the methods of extraction of the useful signal without end effects. As such methods, it is proposed to use: method of breeding evaluations [patent No. 2207622], piecewise multiplication estimates [patent No. 2207622] or two-criterion method [Marchuk V., Rumyantsev CE, Shrivel I.S. two-criterion method of processing measurement results // Aviakosmicheskaya the e instrumentation, No. 12. - M.: Austenitized, - 2006, pp.33-35], which are effective ways of one-dimensional treatment in conditions of a priori uncertainty [1-4].

The essence of the method of reproduction of the estimates is the multiplication of the estimated useful component by repeatedly splitting the original implementation at intervals of random length and the evaluation of these intervals is a useful component using the approximation linear or quadratic functions by the method of least squares and then averaged at each time point.

The way piecewise multiplication estimates based on the principle of moving the reproduction of the estimated useful component is the result of the synthesis of the simple moving average method and method of propagation estimates. By moving piecewise linear or piecewise quadratic approximation is the multiplication of the estimated useful component with subsequent averaging of multiplied values in each moment of time.

In two-criterion method uses the objective function when combining the criteria of minimizing the mean square error and mean square finite difference of the first order values of the measured process, which is:

,

where α>0 - set constant factor, characterizing the degree of priority odnogolosnogo over the other.

The use of these cues useful component allows to obtain an estimate without end effects, that is, with a minimum allocation error signal at the boundaries of the interval of approximation, which further is used to perform prediction on the terminal points of the obtained estimates. This is the approximation of the limit values assessment,using the method of least squares polynomial of the firstor second degreep and obtain estimates. Therefore, determined by the coefficients of the approximating polynomial of the limit values assessment. Next, on the obtained evaluation equationcalculate the forecast values in pixelswhere T=3.

The device for implementing the method of prediction of the measurement results (figure 1) contains the block storage input 1, an information input device, the outputs of which are connected to the inputs of the block selection signal 4, the outputs of which are connected to the first n inputs of the storage unit of output realization of 8, and outputs n-5 n are connected to the inputs of the storage unit of the last five evaluation values useful component 5, the outputs of which are connected to the ENES to the inputs of the block approximation by a polynomial of the first degree 6 and the inputs of the block approximation by a polynomial of second degree 7, the outputs of which are combined with the outputs of the block approximation by a polynomial of the first degree 6 and is connected to the input n+1 n+3 block storage output realization 8, the output of which is an information output device; a first output control unit 3 is connected to the input of the block selection signal 4, the second output control unit 3 is connected to the inverse allows the input of the block approximation by a polynomial of the first degree 6 and allows the input of the block approximation by a polynomial of second degree 7, the third output connected to the control input of the storage unit of the last five evaluation values useful component; the timing device is set to a clock generator 2.

The device for implementing the method of prediction of the measurement results is as follows. The values of the input and results of the measurements are sent to the input device and stored in a storage unit of the input implement. In the block selection signal is reduced dispersion value of the noise component by using one of the methods of selection of usable components without end effects, the control parameters of the method are performed by the control unit. These estimates are useful component is recorded in the storage unit of output implementation, and evaluation of n-5 to n are recorded in the block x is anene the last five evaluation values useful component, the choice is with n-5 to n is performed by the control unit. The value of the block of storage of the last five evaluation values useful component is fed to the input of the block approximation by a polynomial of the first degree and to the input of the block approximation by a polynomial of second degree, which produce an approximation of the obtained values using the method of least squares with the conclusion of three values of the forecast obtained by equation approximation. From the output of the storage unit of output implementation, which contains the assessment of the useful component and the forecast values, the output of the device.

This method of prediction of the measurement results is as follows. The values of the input implement is recorded in the storage unit input 1 of size n. In the ECU 3, the parameters for the selection method useful component, on the basis of which the block selection useful component 4 is the reduction of the variance of the magnitude of the random component and obtaining estimates of the useful component. The value obtained by the block selection signal 4, is recorded in the storage unit of output implementation 8 from 1 to n cells, and will also be recorded values with n-5 n in the storage unit of the last five evaluation values useful component 5, the selection of appropriate values of the production is carried out by the control unit 3. The block approximation by a polynomial of the first degree 6 or block approximation by a polynomial of second degree 7 is the calculation of the predicted values stored in the storage unit of the last five evaluation values useful component 5 using the method of least squares, the result is recorded in the storage unit of output implementation 8. The choice of the block approximation by a polynomial of the first degree 6 or approximation by a polynomial of second degree 7 is performed by the control unit 3. The synchronisation device is a clock generator 2.

The technical result is the prediction of the measurement results based on the methods of extraction of the useful signal without end effects in conditions of limited a priori information about the useful and the random component.

Through simulation it was found that the proposed method has the following advantages:

- allows you to obtain accurate short-term forecast of results of measurements through the use of cues useful component (trend) without end effects;

- allows you to get the forecast in terms of limited a priori information about the useful and the random component.

The forecasting device containing the storage unit of the input, the input information which is input the disorder, the storage unit of output, the output of which is an information output device, characterized in that the outputs of the storage unit of the input implement is connected to the inputs of the block selection signal, the outputs of which are connected to the first n inputs of the storage unit of output realization, and outputs n-5 n are connected to the inputs of the storage unit of the last five evaluation values useful component, the outputs of which are connected to the inputs of the block approximation by a polynomial of the first degree and the inputs of the block approximation by a polynomial of second degree, the outputs of which are combined with the outputs of the block approximation by a polynomial of the first degree and is connected to the input n+1 n+3 unit storing the output of the implementation; the control unit to specify the selection method useful component, select the last five evaluation values useful component, and selecting a block approximation by a polynomial of the first degree or block approximation by a polynomial of second degree, the first output control unit connected to the input of parameters, the selection method of the useful signal of the block selection signal, the second output control unit connected to the inverse allows the input of the block approximation by a polynomial of the first degree and allows the input of the block approximation by a polynomial of second degree, third output is otklyuchen to the control input of the storage unit of the last five evaluation values useful component; the timing device is set to a clock generator.

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