# Two-dimensional least square method and devie to this end

FIELD: computer engineering.

SUBSTANCE: invention relates to computer engineering and can be used in systems for controlling and processing multi-dimensional signals. The device comprises a unit for storing input realisation 1, unit for calculating first coefficient 2, unit for calculating second coefficient 3, unit for calculating third coefficient 4, approximation unit 5, unit for evaluating useful component 6 and clock generator 7. Approximation of values of initial discrete realisation of the measured process is done through minimisation of the objective function, which is a sum of mean square deviations of two-dimensional evaluation of useful component from the initial two-dimensional realisation of measurement results.

EFFECT: separate two-dimensional evaluation of useful component given a single realisation of the measured process.

1 dwg

The present invention relates to information-measuring device and can be used in computing, control systems and signal processing.

In General, a simplified mathematical model of the measurement results is a two-dimensional discrete sequence

Y_{i,j},type:

where S_{i,j}- low frequency, slowly varying useful component;

η_{i,j}additive noise component, distributed according to a Gaussian law with zero mean and constant variance;

N is the number of rows, M is the number of columns of the two-dimensional array of results.

The main task - obtaining a two-dimensional evaluation of the useful component.

A similar problem can occur: 1) in the work receiving / transmitting devices or far space communications; 2) in radio engineering in signal processing; 3) digital image processing; 4) in meteorology and Economics in the processing of measurement results.

There is a method of approximation of functions on the minimax criterion [L.M. Goldenberg Digital signal processing: a Handbook / Lamholders, Bedminister, Menolak. - M.: Radio and communication, 1985. - 312 S., ill.]. This method pozvolyaet one-dimensional implementation of the results of measurements Y_{
1}, Y_{2}, ..., Y_{N}the original process to find an estimateminimizera a function of the form:

In practice, when solving minimax problems of approximation is more appropriate to use a class of algorithms Remez and vallée Poussin. In these algorithms use the set of points satisfying the conditions of Chebyshev alternance. To find points of Chebyshev alternance at problem solving polynomial approximations numerical procedure with iterative in nature.

The characteristics of the method-analogue, coinciding with the characteristics of the proposed technical solution, the following: memorizing discrete signal, the solution approximation, replacing the original discrete implementation of the measurement results of the smoothed values.

The disadvantages of this method are

- the complexity of solving the problem of Chebyshev approximation;

- the lack of accurate methods for the determination of points of Chebyshev alterans;

the rate of convergence of Chebyshev approximation algorithm greatly depends on the efficiency of the search algorithm approximation at each iteration;

- the impossibility of approximating a two-dimensional implementation of the measurement results.

Barriers to achieving the desired technical result of the ATA, are as follows:

- function minimax criterion is not differentiable.

The known method exponential smoothing [Bendat J., Persol A. Applied analysis of random data. - M.: Mir, 1989. - 540]. Its peculiarity lies in the fact 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 initial value estimate of S_{0}the calculation estimates the useful component of the recurrent expression:

replace the original values of the measurement results Y_{1}, Y_{2}, ..., Y_{N}smoothed values

To use the exponential smoothing method of the measurement results is determined by the initial value of S_{0}evaluation of the useful component and the smoothing parameter α.

Signs of a device similar will be deleted the e with the characteristics of the proposed technical solution, the following: memorizing discrete signal representation of the estimates of the useful component in the form of a polynomial from the discrete values of the initial implementation of the measurement results, the replacement values of the initial implementation of the smoothed values.

The disadvantages of the known devices are:

- the uncertainty of the choice of the smoothing parameter α, in some cases (unreasonably) to determine the value of α based on the volume of smooth implementation α=2/(N+1);

- the uncertainty of the choice of the parameter S_{0}that leads to repeated application of the method of exponential smoothing for other values of α and S_{0}to achieve the desired effectiveness of the smoothing in the sense of a given criterion;

- the impossibility of approximating a two-dimensional implementation of the measurement results.

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 S_{0}is subjective and depends on experience and practical skills of the researcher, the values of α and S_{0}are functions of the waveform, noise, sample size.

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

The known method variables differences [Mckendall, Stewart, Multivariate statistical analysis and time series. - M.: Nauka, 1976. - 736]. To use this method, only one realization of Y_{1}, Y_{2}, ..., Y_{N}the original process.

If you have a useful component described by a polynomial (or locally represented by a polynomial) with an overlay of a random component (noise), you can exclude the polynomial part of the calculation of the successive differences of the series. The difference of a polynomial of order m are polynomial of order m-1. Further, if the number contains a polynomial of order p, then the transition to the differences, repeated (p+1) times, eliminate it and leaves the elements associated with the random component of the original series.

In the General case, we have:

ΔY_{k}=Y_{k+1}-Y_{k},

Δ^{2}Y_{k}=ΔY_{t+1}-ΔY_{k}=Y_{k+2}-Y_{k+1}-Y_{k+1}-Y_{k}=Y_{k+2}-2Y_{k+1}-Y_{k},

where σ^{2}the variance of the random component.

From the last relation is obtained

Therefore, the method of successive differences of the variable is to compute the first,second, third, etc. differences, the determination of the sum of squares, divided byetc. and detecting the moment when this ratio becomes constant. Thus, the evaluation order of the polynomial contained in the source range, and variance of the random component.

The characteristics of the method-analogue, coinciding with the characteristics of the proposed technical solution, the following: memorizing discrete signal, the solution approximation, replacing the original discrete implementation of the measurement results of the smoothed values.

The disadvantages of this method are:

- the impossibility of approximating a two-dimensional implementation of the measurement results;

- this method does not eliminate such systematic components as periodic oscillations with a small period.

Barriers to achieving the desired technical result are as follows:

the way variables difference does not exclude the exponential or cyclical component.

Closest to the invention is a method of least squares and device for piecewise linear approximation [Bendat J., Persol A. Applied analysis of random data: TRANS. from English. - M.: Mir, 1989. - 540 S., copyright certificate №1624479].

The method of least squares allows for measurement result Y_{1}, Y_{2}, what is,
Y_{N}the original process to obtain an estimateminimizing the target function of the form:

Whenis a polynomial of the first degreethe coefficients a and b can be found by minimizing the objective function of the form

Differentiating expression (2) for a and b and equating 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 the values of the realization of the measurements is minimal (2).

The disadvantages of this method are the counterpart:

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

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

- 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 of the solved system of equations;

the inability of approximate the two-dimensional realization of the measurement results.

Barriers to achieving the desired technical result are as follows:

- efficiency assessment is a useful component depends on the volume of sales, the statistical characteristics of the additive noise 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 proposed two-dimensional method of least squares and device for its implementation allow to extract useful component in the presence of a single two-dimensional discrete realization of the measured process.

The original measurements are discrete sequence of values of the measured physical quantity Y_{i,j}in accordance with the expression (1).

For a two-dimensional array of measurements Y_{i,j}are approximations of the original values of the plane, opisyvayuschaya equation of the first degree type:

To find the values of the coefficients a, b and C minimizes the target function of the form:

Differentiating the last expression in a, b and C and equating to zero, we obtain a system of linear equations:

The solution of the system is

The device for realization of two-dimensional method of least squares (Fig 1) contains the block storage input 1, input by the information input device, the output of the storage unit of the input 1 is connected to the input of the computing unit of the first factor of 2, the input of the computing unit of the second factor of 3 and the input of the computing unit of the third factor 4, the output of the computing unit of the first factor of 2, the unit for computing the second factor is 3 and the evaluation unit of the third factor 4 is connected to the input of the block approximation 5, the output of which is connected to the input of the storage unit estimates the useful component 6, the output of which is an information output device, the timing device is not working set a clock generator 7.

The device for realization of two-dimensional method of least squares is as follows.

The original input two-dimensional implementation of the measurement results is recorded in the storage unit of the input implement . In blocks calculate the first factor of 2, the calculation of the second factor of 3 and calculate the third factor 4 coefficients are calculated in accordance with expression (4), (5), (6), respectively, using the values from the storage unit of the input implement. In block approximation 5 estimation of the useful component in accordance with the expression (3). The assessment of the useful component is transmitted from the output of the block approximation 5 to the input of the storage unit estimates the useful component 6.

Two-dimensional method of least squares is implemented as follows.

Based on the input values and results of measurements are calculated coefficients a, b and C in accordance with the expression (4), (5) and (6) respectively. Next to a two-dimensional array of measurement results is the approximation of the original values of the plane, opisyvayuschaya equation of the first degree of the form (3).

The technical result - the allocation of two-dimensional assessment of the useful component in the presence of only a discrete realization of the measured process.

A device for allocating a two-dimensional evaluation of the useful component in the presence of only a discrete realization of the measured process that contains the block storage of measurement results, presented in the form of

Yi,j=Si,j+ηi,j, where i=1,N, j=1, M,

Yi,j is the measurement results;

Si,j-bass m is Glenna changing a useful component;

ηi,j is the additive noise component according to Gaussian law with zero mean and constant variance;

N - number of rows;

M - number of columns of the two-dimensional array of measurement results,

input by the information input device,

the output of the specified block of storage of the measurement results is connected to the input of the computing unit of the first coefficient in the form

the input of the computing unit of the second coefficient in the form

the input of the computing unit of the third coefficient in the form

the output of the computing unit of the first coefficient, the unit for computing a second coefficient and computing unit of the third factor is connected to the input of the block approximation for the estimation of the useful component in the form of

Si,j=Ai+Bj+C, where i=1,N, j=1,M,

the output of which is connected to the input of the storage unit estimates the useful component, the output of which is an information output device, and a clock generator designed to ensure synchronization operation.

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