RU2645273C1 - Method of selecting trend of non-stationary process with adaptation of approximation intervals - Google Patents

Abstract

FIELD: physics.

SUBSTANCE: in the method of selecting the trend of the non-stationary process, the intervals of approximation are adapted in such a way that a discrete implementation of the non-stationary process representing the sum of the useful signal and noise, is consistently divided into intervals, each of which approximation is carried out by the least square method, the width of the first interval is selected equal to the portion of the total duration of a single discrete implementation, the approximation in this interval is carried out and, if a discrepancy of approximation exceeds a predetermined value, the approximation is repeated, setting the width of the interval equal to the value in its previous value, for each subsequent interval the width is specified equal to the value in the previous interval, the approximation is carried out and, if a discrepancy of approximation is obtained less than a predetermined value of a set number of consecutive times, the width of the subsequent interval is increased, and if a discrepancy of approximation is obtained greater than a predetermined value, the width of the current interval is selected equal to the part of its existing value and the approximation is repeated, moreover, if the width of the last interval extends beyond a single discrete implementation of the non-stationary process, the width of this interval is limited by the width of the remaining reapproximation interval.

EFFECT: reduction of processing time.

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Description

The present invention relates to the field of computer technology and can be used in control systems and signal processing, in particular in digital signal processing systems, in the operation of control and monitoring systems, in the processing of signals in automatic control systems, in calculations when highlighting any development trends indicators and deciding on the removal of the effects of random factors.

Let there be a time series, which is a sequence of t ₁ , t ₂ , ..., t _n n values y ₁ , y ₂ , ..., y _n obtained at equidistant times (where y _i = y (t _i ), i = 1 , 2 ..., n) of some function y = y (t), which describes the only implementation of a non-stationary process on the time interval t ₁ ... t _n .

The process under consideration consists of an additive mixture of some systematic component S (t), which is a useful signal or trend, and a random component u (t), which is noise or interference: y (t) = S (t) + u (t) .

Regarding the random component u (t), we assume that the mathematical expectation M _{u (t)} = 0, the variance D _{u (t} ) = σ ^2, and the values of u (t) at different times are uncorrelated (cov (ut, us) = 0, t ≠ s). The trend S (t) in the general case is a quasi-determined signal, which is considered as some non-stationary process, the analytical description of which is unknown.

The main task is to highlight a useful signal (trend) in conditions of insufficient a priori information about the statistical properties of additive noise and the form of the useful signal function in the presence of a single discrete implementation of the investigated non-stationary process. It is a priori assumed that at some not too small time intervals the function S (t) can be approximated fairly accurately by a low degree polynomial.

. Здесь S'(t) - простая аппроксимирующая функция, которая на интервале времени [t₁; t_n] хорошо аппроксимирует полезный сигнал S(t) (Бендат Дж., Пирсол А. Прикладной анализ случайных данных: пер. с англ. – М.: Мир, 1989, стр. 114). Например, для линейной зависимости S'(t)=a+bt параметры a, b определяют из условия минимизации суммы квадратов отклонений элементов ряда y₁, y₂, …, y_n исследуемого процесса от значений полинома S'(t) в соответствующих точках t_i, т.е. из условияThere is a method of trend highlighting, which consists in approximating a discrete implementation of y ₁ , y ₂ , ..., y _{n of} the process under study by the function S '(t) by using the least squares method that minimizes the residual

. Here S '(t) is a simple approximating function, which on the time interval [t ₁ ; t _n ] well approximates the useful signal S (t) (Bendat J., Pirsol A. Applied analysis of random data: transl. from English. - M .: Mir, 1989, p. 114). For example, for the linear dependence S '(t) = a + bt, the parameters a, b are determined from the condition of minimizing the sum of the squares of the deviations of the elements of the series y ₁ , y ₂ , ..., y _{n of} the process under study from the values of the polynomial S' (t) at the corresponding points t _i , i.e. from the condition

Differentiating this expression with respect to a and b and equating them to zero, we obtain a system of linear equations:

Solving the resulting system of equations, determine the values of the approximation coefficients:

The disadvantage of the described method is the computational complexity of the selection of the useful signal. This is due to the fact that when applying the method to complex signals of long duration, described by a time series with a large number of members n, the use of simple approximating functions is impossible, and the use of complex functions with a large number of parameters, for example, high degree polynomials, sharply increases the computational complexity. For large sample sizes and the absence of a priori information on the statistical properties of random noise and the useful signal model, instead of high polynomials, piecewise approximation of the unsteady process by simple functions at short intervals is also used, but this also increases the computational complexity of the method.

Closest to the proposed invention in terms of technical nature and the achieved result (prototype) is a method for highlighting a trend by multiplying estimates of its only initial implementation (Razots) and a device for its implementation, which consists in the fact that the only discrete implementation of a non-stationary process, representing the sum of the useful signal and noise, sequentially divided into intervals, on each of which the least squares approximation is carried out. In this case, the initial implementation of the non-stationary random process is preliminarily divided into intervals using a random number generator distributed according to a uniform law, the subsequent mentioned approximation of the initial implementation of the non-stationary random process is performed on each of the mentioned intervals, the division into intervals is repeated a specified number of times, then sequentially in each interval partitions of the initial discrete implementation of a non-stationary random process receive estimates of the trend of the useful signal and then according to the method of the arithmetic mean averaged received trend estimation (patent RU №2207622, IPC G06F 17/18, publ. 2003).

The disadvantage of the described method is the increased computational complexity of the process of extracting a useful signal (trend), due to the fact that the lack of adaptation of the number and length of the partitioning intervals to the dynamics of the signal behavior leads to the need to split the implementation of the signal into a large number of intervals, and the partitioning process and the approximation procedure at each of intervals should be repeated multiple times.

The proposed method solves the technical problem of reducing the computational complexity of the trend highlighting process.

To achieve a technical result in a method for identifying a trend in a non-stationary process with the adaptation of approximation intervals, which consists in the fact that the only discrete implementation of the non-stationary process, representing the sum of the useful signal and noise, is successively divided into intervals, on each of which the least squares approximation is performed, the width of the first interval, choose an equal part of the total duration of a single discrete implementation, approximate this interval and if the approximation discrepancy exceeds a predetermined value, the approximation is repeated by setting the interval width to an equal part of its previous value, for each subsequent interval the width equal to the value in the previous interval is set, the approximation is performed, and if the approximation discrepancy is less than the specified value, the set number of times in a row is obtained, increase the width of the subsequent interval, and if the residual of approximation is greater than the specified value, then the width of the current interval is chosen equal to part of its existing value variations and repeat the approximation, and if the width of the last interval goes beyond the limits of the only discrete implementation of the non-stationary process, then the width of this interval is limited by the width of the remaining unapproximated interval.

Reducing the computational complexity of the process of extracting a useful signal (trend) due to the reduction or exclusion of repeated repetition of the approximation of the initial implementation by a given polynomial at all intervals and using intervals with a close to optimal length in areas with different spectral widths is achieved due to the fact that the width of the first interval is chosen equal to of the total duration of a single discrete implementation, approximate this interval and, if the residual approximation exceeds the specified value, the approximation is repeated, setting the interval width to an equal part of its previous value, for each subsequent interval, set the width equal to the value in the previous interval, perform the approximation, and if the approximation discrepancy is less than the specified value, the set number of times in a row is increased, the width of the subsequent interval is increased, and if an approximation residual greater than a given value is obtained, then the width of the current interval is chosen equal to a part of its existing value and the approximation is repeated w, wherein if the width of the last interval beyond a single discrete implementation nonstationary process, the width of the slot width to limit remaining neapproksimirovannogo interval.

The proposed method for identifying the trend of a non-stationary process with the adaptation of the approximation intervals is as follows.

Initially, there is a unique discrete implementation of the investigated non-stationary process y ₁ , y ₂ , ..., y _n , where y _i = y (t _i ), i = 1, 2, ..., n, representing the sum of the useful signal (trend) and noise, those. y (t) = S (t) + u (t). A priori information about the process under study is that at some intervals Δ _j ⊂ [t ₁ , t _n ] the useful signal (trend) is described quite accurately by the polynomial, where Δ _j = [t _j-1 , t _j ], j⊂ [ 1, n]. To describe the method, it is believed that the degree of the polynomial is equal to unity, i.e. S (t) = a + bt, although in the general case this may be the most arbitrary function.

The method under consideration contains the following steps:

Step 0:

The number of members of the series m = int (n * k ₀ ) included in the first approximated interval is determined, where the int function returns an integer value from the argument, and the coefficient k ₀ lying on the interval (0, 1) is determined from the dynamic properties of the original signal, for example from the width of the spectrum of the original signal f _max : the wider the spectrum, the smaller the value of k ₀ and, in accordance with the Nyquist theorem, k ₀ = (t _n -t ₁ ) / [2f _max ]. If there is no information about f _max , then we set k ₀ = 0.5.

If m = n, then take k ₀ = k ₀ * k ₀ and repeat step 0.

The value of the residual ε is determined, which determines the required accuracy of the correspondence of the selected trend to the available discrete implementation. Obviously, its value should not be less than the standard deviation of the noise component u (t).

Step 1:

Remember the first segment of the original series {y _i }, where i = 1, ... m;

The least squares method approximates the elements of the series y ₁ , y ₂ , ..., y _{m by a} polynomial of the first degree: S ₁ (t) = a ₁ + b ₁ t.

The coefficients a ₁ , b _{1 of the} approximating polynomial S ₁ (t) = a ₁ + b ₁ t are determined from the condition of minimizing the sum of squares of deviations of the elements of the series y ₁ , y ₂ , ..., y _m from the values of the polynomial S ₁ (t) at the corresponding points , i.e. from the condition

To find the values of the coefficients a ₁ , b ₁ corresponding to this condition, use the formulas

. Если невязка ε₁>ε, то задают m=int(m*k₁) и повторяют шаг 1 для нового значения m, где величину k₁ выбирают из соображений скорости адаптации ширины интервала аппроксимации к характеру изменения динамики полезного сигнала (оптимально, если k₁ лежит в пределах от 0,5 до 0,8). Если невязка ε₁≤ε, то фиксируют индекс последнего члена рассмотренного ряда r=m, задают число аппроксимаций без повторного пересчета p=0, номер следующего интервала j=2 и переходят к шагу 2.The residual ε _{1 of the} approximating function S ₁ (t _i ) is found on the first interval of the series {y _i }, where i = 1, ..., m, by the formula

. If the discrepancy ε ₁ > ε, then set m = int (m * k ₁ ) and repeat step 1 for the new value of m, where the value of k _{1 is} chosen from considerations of the rate of adaptation of the width of the approximation interval to the nature of the dynamics of the useful signal (optimally, if k ₁ lies in the range from 0.5 to 0.8). If the discrepancy ε ₁ ≤ ε, then we fix the index of the last member of the considered series r = m, set the number of approximations without recounting p = 0, the number of the next interval j = 2, and go to step 2.

Step 2:

If at the next step it turns out that m + r> n, then m = nr is set and the next m members of the series {y _i } are considered, where i = r + 1, r + 2, ..., r + m, r is the index, set in the previous step. The elements of the series y _{r + 1} , y _{r + 2} , ..., y _{r + m are} approximated by the least squares method by a polynomial of the first degree: S _j (t) = a _j + b _j t.

находят коэффициенты a_j, b_j аппроксимирующего полинома j-го интервала, дополнительно используя условие, необходимое для непрерывности ломаной аппроксимирующей функции S_j(t_r)=S_j-1(t_r), откуда следует a_j=a_j-1+(b_j-1-b_j)t_r. Находят невязку ε_j аппроксимирующей функции S_j(t_i) на отрезке ряда {y_i}, где i=r+1, …, r+m, по формуле

. Если невязка ε_j>ε, то полагают p=0, задают m=int(m*k₁) и повторяют шаг 2 с r+1 новым значением m. Если невязка ε_j≤ε, то сохраняют значения a_j, b_j, фиксируют индекс последнего члена рассмотренного ряда r=r+m, подсчитывают число "удачных", не требующих повторного пересчета из-за невыполнения условия ε_j≤ε аппроксимаций p=p+1, и, если p превышает заданное число, например 3, полагают p=0 и увеличивают ширину последующего аппроксимируемого интервала m по формуле m=int(m*k₂), где величину k₂ выбирают из тех же соображений, что и k₁ (оптимально, если k₂ лежит в пределах от 1.2 до 2.0, или, например, k₂=1/k₁). Задают r=r+m и, если r<n, инкрементируют номер интервала j=j+1 и переходят к началу шага 2.From the condition of minimum sum of squared deviations

find the coefficients a _j , b _{j of the} approximating polynomial of the jth interval, additionally using the condition necessary for the continuity of the broken approximating function S _j (t _r ) = S _j-1 (t _r ), which implies a _j = a _j-1 + (b _j-1 -b _j ) t _r . Find the residual ε _{j of the} approximating function S _j (t _i ) on the segment of the series {y _i }, where i = r + 1, ..., r + m, by the formula

. If the discrepancy ε _j > ε, then put p = 0, set m = int (m * k ₁ ) and repeat step 2 with r + 1 the new value m. If the discrepancy is ε _j ≤ ε, then the values of a _j , b _j are stored, the index of the last member of the considered series is fixed r = r + m, the number of “successful” ones that do not require recounting due to the non-fulfillment of the condition ε _j ≤ ε of approximations p = p + 1, and if p exceeds a given number, for example 3, put p = 0 and increase the width of the next approximated interval m according to the formula m = int (m * k ₂ ), where the value of k _{2 is} chosen from the same considerations as k ₁ (optimally, if k ₂ lies in the range from 1.2 to 2.0, or, for example, k ₂ = 1 / k ₁ ). Set r = r + m and, if r <n, increment the interval number j = j + 1 and go to the beginning of step 2.

If the function of the useful signal (trend) is rather accurately described by a polynomial of a given degree on a certain interval and the dynamic properties of the useful signal (trend) do not change throughout the entire discrete implementation of the investigated non-stationary process, then the width of all intervals will be equal to the width selected in the first step. Therefore, the computational complexity of the method implementation is reduced, since the number of repeated least-squares approximation procedures is significantly reduced. In addition, the method has the adaptability property, due to which the length and number of intervals optimize when the dynamics of the change in the useful signal (trend) at the next step differs from its dynamic properties at the previous step. This also helps to reduce the computational complexity of the implementation of the method, because the length of the intervals is chosen as high as possible, which means that the total number of approximations will be minimal. For example, suppose that the original discrete implementation contains two sections, of which one occupies 10% of its length and has a frequency spectrum, the upper boundary of which is 10 times that of the other boundary. Suppose that for a good recovery of the trend, it is enough to divide each of these sections into 5 intervals. Then, when applying the described method, it will take only 5 + 5 = 10 partition intervals of the original discrete implementation. If we assume that the adaptation of the length of the interval requires repeating the approximation, for example, 5 times in the first interval and 5 times when moving to another section (the number of repetitions is equal to the ratio of the logarithm of the ratio of the maximum frequencies in the section to the logarithm of the coefficients k ₁ or k ₂ ), then the total the number of approximations will be equal to 20. Since the location of the extended frequency spectrum section is not known in advance, in the prototype method the time interval is divided into 50 intervals, based on the length of the high-frequency section. If at the same time the number of repeated partitions into sections, and hence the repeated approximations, is set equal to 10 to obtain an acceptable averaging, then the total number of approximations will be 500, i.e., it will be 25 times more than in the proposed method.

In addition, trend information in the proposed method can be stored in the form of a set of coefficient values at each of the partition intervals, which is many times less than the number of members of the sequence y _i at these intervals, while the prototype method does not have such a possibility.

Thus, the use of the proposed method for highlighting the trend of a non-stationary process with the adaptation of the approximation intervals leads to a decrease in the computational complexity of the trend highlighting process.

Claims (1)

A method for identifying a trend in a non-stationary process with adaptation of the approximation intervals, namely, that the only discrete implementation of the non-stationary process, representing the sum of the useful signal and noise, is successively divided into intervals, on each of which the least squares approximation is carried out, characterized in that the width of the first interval choose an equal part of the total duration of a single discrete implementation, approximate this interval, and if the discrepancy e the approximation exceeds a predetermined value, the approximation is repeated, setting the width of the interval equal to part of its previous value, for each subsequent interval, set the width equal to the value in the previous interval, perform the approximation, and if approximation mismatch is less than the specified value a specified number of times in a row, then increase the width of the subsequent interval, and if an approximation mismatch is greater than a given value, the width of the current interval is chosen equal to a part of its existing value cheniya and repeated approximation, and if the width of the last interval beyond a single discrete implementation nonstationary process, the width of the slot width to limit remaining neapproksimirovannogo interval.