Method for separating trend using method of sliding trend estimates multiplication of its single source realization and device for realization of said method

FIELD: computer science.

SUBSTANCE: device has physical value measurements results storage block, delay block, averaging-out block, control block, clock pulse generation block, block for approximation using method of least squares, block for storing trend estimates.

EFFECT: lesser error, higher efficiency.

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The invention can be classified as information-measuring devices and can be used in various fields of science and technology.

Consider a time seriesthat represents a sequence of N values of y₁, y₂, ... , y_nsome functions ƒ (t_i), resulting in evenly spaced time points t₁, t₂, ... , t_n.

Consider a time series is an additive mixture of some systematic component (the deterministic component of the useful signal, trend) S(t) and a random component (noise, measurement error and other) u(t):

y(t)=S(t)+u(t).

Relative to the random component u(t) let us assume that Mi_t=0, Du_t=σ²and, moreover, its value at different points in time are uncorrelated (i.e. cov(u_tu_s)=0, t≠ S), although these conditions are not significant.

The main task - the selection of the useful signal (trend) in the conditions of insufficient a priori information about the statistical characteristics of the additive noise and the function of the useful signal (trend) if there is only one realization of the measured process. To be known a priori that the allocated function of the useful signal is smooth ([1], p.61), i.e. the allocated function for some is too small intervals of time can be quite accurately approximated by some polynomial P(t). This problem may occur: in the work receiving / transmitting devices or far space communications; economic calculations when highlighting the main trends in the development of any performance and making decisions about the removal of the effects of random factors; in meteorology, in the measurement of various characteristics of the atmosphere, etc.

In cases where the useful signal has a definite structure (i.e. the function S(t) belongs to a known class of functions) and is defined by a finite number of parameters are used parametric methods of assessment (this includes methods of regression analysis, that is based on the classical theory of least squares). In those cases, when there is no information about the function of the useful signal, for estimating the trend used non-parametric methods, such as smoothing. It is known that the best way of smoothing averaging over the ensemble (collection) of realization of y_j(t) (or y_jt) the original process ([2], p.35-36). However, in practice, we typically have only a single realization of the measured process. In this case, it is advisable to use methods of smoothing (filtering)that “work” with a single implementation, or ways in which you “clone” an existing implementation of the studied p is ocess.

Known this way of highlighting useful component as averaging over the ensemble of realizations ([3], s). For its application it is necessary to have N realizations of the initial process. Each implementation represents the time series resultsobservations (measurements) of the process y(t)obtained at n equally spaced points in time t₁, t₂, ... , t_n. These observations can be represented in a matrix implementations:

where y_j1, y_j2, ... , y_jn- j-I implementation of the original process, which is a sum of functions of the useful signal S(t) and noise component u(t).

Averaging over the ensemble of realizations involves:

- storing N input implementations of The_j1, y_j2, ... , y_jn, (j=1, 2, ... , N)obtained at n equally spaced points in time;

- calculating the arithmetic mean value of these implementations in each moment of time;

- replacement of the values of the initial realizations of the random process obtained average rating.

When applying this method calculates the arithmetic average matrix columns implementations (1):

... ,

The result is a smoothed time series .

The characteristics of the method-analogue, coinciding with the characteristics of the proposed technical solution, the following: sampling the signal at a time, storing the digital signal, finding the arithmetic mean, the replacement of the original time series smoothed.

The disadvantages of this method are the counterpart:

for the application of the method-analogue necessary to have multiple implementations.

Barriers to achieving the desired technical result are as follows: details of the method-analogue is not allowed to handle only the implementation of the original process, and not allow it to be applied to the already smoothed values (unlike ways, working with a single implementation); the result of processing multiple implementations significantly depends on the number of implementations.

The structural scheme of the device that implements the method-analogue holds for n realizations of the n buffer units, the inputs of which are the device inputs and outputs are connected through switches to the inputs of the blocks storing the measurement results, the outputs of which are connected to the inputs of the arithmetic unit through the switches, the outputs of which are connected to the input of the storage register of the trend, and the output register is an information output device.

In the home is e No. 2000127308 from 30.10.2000 (patent No. 2207622 published in B. I. No. 18 27.06.2003,) had proposed a method of breeding estimates with a limited amount of a priori data and only the implementation of the initial process.

Consider the method involves:

1) memorization of the input implement;

2) split time period [t₁,t_n] (the duration of implementation) by m intervals, random numbers, subaltern uniform distribution law;

3) verification that the intervals of the partition include not less than l values of the initial implementation;

4) finding for each interval partitioning coefficientsapproximating polynomial(k∈ [1,n], j∈ [1,N], i∈ [1,m]) by the method of least squares;

5) repeat the procedures in paragraphs (2) - 4) N times;

6) finding smoothing (approximation) function as the arithmetic mean value of “piecewise quadratic” approximating functions in each section t_kobtained for each split time period [t₁, t_n].

First, using a generator (sensor) random numbers, uniformly distributed in the interval (0;1), get m-1 numbersThen according to the formula y=t₁+(t_n-t₁x implementing the display interval (0;1) on the interval (t₁;t_n), ucaut corresponding splitting numbers period (t₁;t_n) on m parts, where

Intervals partitioning mean:

Check the condition: every intervalmust contain at least l values from the set {y₁, y₂, ... , y_n}. If this condition is not met, then discarded the number ofand generated the following.

The presence of this condition means that at least l m≤ n. Further, repeated N times, the procedure of splitting the segment [t₁,t_n] into m intervals of random length (verifying the above conditions). The result is a set of splits time period [t₁;t_n]:

For each intervalby the method of least squares to find evaluationsthe coefficients of the approximating polynomial a+bt+ct².

The result of the described algorithm (“multiplication” of estimatesthe coefficients of this polynomial) is the set defined on the interval [t₁;t_n] smoothing functions S^(j)(t) (j=1, 2, ... , N), each of which is a “piecewise quadratic”:

An approximate function is as average arithmetic functions(for all N partitions of the segment [t₁;t_n]):

The characteristics of the method-analogue, coinciding with the characteristics of the proposed technical solution, the following: sampling the signal at a time, storing the digital signal, the approximation by the method of least squares, find the arithmetic mean, the replacement of the original time series smoothed.

The disadvantages of this method are:

- to handle the implementation, you need to remember the whole sample;

it is impossible to make the processing of the original implementation in real-time;

growth allocation errors with unlimited increase in the reproduction of the original implementation.

Barriers to achieving the desired technical result are as follows:

- due to the fact that before processing the original implementation it is necessary to have the whole sample, the method of propagation estimates in real time is extremely limited;

the assumption that on each interval partitioning the original implementation of the useful signal can be described by a polynomial of the second degree, leads to an increased allocation error signal and increase noise component, with a decrease in the length of the interval s is ing.

The structural scheme of the device that implements the method is similar, contains a buffer unit, the inlet of which is an information input device and an output connected to information inputs of the storage units of measurement results to the control inputs of which through the switches connected to the outputs of the block partitioning of the original implementation, which contains the generator of random numbers distributed according to the uniform law, the output of which is connected to the input of the block address associated values, the output of which is connected to the input of block ranking, the output of which is connected to the register storing the sample of random numbers, whose output is an information output unit is split; the outputs of the storage units are connected to the inputs of blocks approximation the outputs are connected to the inputs of registers storing estimates of the baseline function, the outputs are connected to inputs of the arithmetic summing device, the output of which is connected to the input of the storage register of the trend, whose output is an information output device. Synchronous operation of the device is provided by a clock.

The known method exponential smoothing time series[4], s). Its peculiarity lies in the fact that the procedure of finding the smoothed values are used only predshestvuyushchei the original series, taken with a certain “weight”and “weight” measurement decreases with distance from the point in time for which you are defining a smoothed value of a number. To use this method, only one implementationthe original process.

The method of exponential smoothing involves memorizing the input realization of y₁, y₂, ... , y_nrandom process, the choice of the smoothing parameter α characterizing the “weight” of the current (most recent) observation (0<α <1), the choice of values of Q₀characterizing the initial conditions, the calculation of the smoothed values of the time series by recurrence formulas

(where k=1, 2, ... , n), replacing the initial values y₁, y₂, ... , y_nthe time series of the smoothed values of Q₁, Q₂, ... , Q_n.

First, when using exponential smoothing for time series is determined by the initial value of Q₀the smoothed series and the smoothing parameter α . Depending on the choice of parameter α (in particular, if α close to zero) the initial value of Q₀the smoothed series can have a significant impact on the result of processing the time series. In practical recommendations on the use of exponential smoothing ([5], s) offered b shall use them as the initial values of Q ₀either the first value range, or the arithmetic average of the first few members of the series, for example, Q₀=(y₁+y₂+y₃)/3. On the other hand, the influence of selection decreases with increasing length of the row and becomes insignificant when a large number of measurements (observations).

After choosing Q₀and α calculates the smoothed values of the time series, which replaced the original values:

Q₁=α y₁+(1-α )Q₀,

Q₂=α y₂+(1-α )Q₁=α y₂+α (1-α )y₁+(1-α )²Q₀,

The characteristics of the method-analogue, coinciding with the characteristics of the proposed technical solution, the following: sampling the signal at a time, storing the digital signal, the representation of the values of the smoothed series in the form of a polynomial from the values of the original series, replacing the original time series smoothed.

The disadvantages of this method are:

- the uncertainty of the choice of the smoothing parameter α ; in some cases (unreasonably) to determine the value of α based on the length of the smoothed number: α =2/(n+1) ([5], s); in practice, the smoothing parameter often find using the “grid”, i.e. the possible values of the parameter break “mesh” with a certain step; for example, consider the grid C is acini from α =0.1 to α =to 0.9 in increments of 0.1, and then select α for which the sum of squared residuals is a minimum;

- the uncertainty of the choice of initial values of Q₀that often leads to the need for repeated re-application of the method of exponential smoothing with another choice α and Q₀.

Barriers to achieving the desired technical result are as follows: method of exponential smoothing is not “plug and play” method, because the choice of parameters α and Q₀is subjective and depends on experience and practical skills of the researcher.

The structural scheme of the device that implements the method is similar, comprises a generator of clock pulses, a switch, a control unit, a storage register, the adder block multiplication, the output register storing the smoothed series of the original implementation.

There is a median smoothing method ([6], p.56). To use this method, only one realization of y₁, y₂, ... , y_nthe original process. The main advantage of median smoothing is resistant to outliers. The basis of the method is the calculation of the moving median.

The median smoothing method involves the following steps:

- memorization of the segment the implementation of time serieswhere m<n (the width of the “sliding window");

- ranking (ordering ascending) the selection of a number of;

- definition of median(Central member) ranked segment of the time series;

- the replacement of the Central (to rank) from the values of the selected segmentfound a median;

- shift “sliding window” on one value to the right (i.e. the choice instead of cutsome other segmentfrom).

The algorithm runs until until you reach the right end of the time series.

A priori for a time series is determined by the smoothing interval, i.e. the “sliding window” with a length of m samples, where m<n. The number of samples it is recommended to take an odd (m=2P+1), although the median smoothing can be performed even when the width of the “window”. For the first m values of a time series is computed their median; this will be smoothed value of the time series in the middle of the smoothing interval. Note that the median number in the time interval is defined as the Central member of Aracinovo series - the sequence of values number of included in this time interval, sorted in ascending order, namely the median filtering operation sequence of signal values is characterized by the ratio

where a fixed value of p=1, 2, ... specifies the width of the “window”. Then the smoothing interval is shifted by one value to the right repeats the calculation of the median and again the Central importance of the smoothing interval is replaced by the calculated median. Thus, in order to find the value of the moving median point with sequence number j, is calculated median values in the time interval [j-p, j+p].

The characteristics of the method-analogue, coinciding with the characteristics of the proposed technical solution, the following: sampling of the signal in time, remembering only the first m values of the digital signal, the allocation of time segments, replacing the original time series smoothed.

The disadvantages of this method are:

the first p and last p values of the series are lost (not smoothed);

- due to non-linearity cannot strictly distinguish between the effect of median filtering on the signal and the noise (if any property of linearity of this problem can be solved relatively simply);

- median smoothing can be considered as an effective method is the processing of the time series (signal) in terms of impulse noise, but in the absence of explicit emissions, this method leads to a more “tooth” of the curve (the moving average smoothing);

- median smoothing is not possible to use weights, i.e. the adaptive capabilities of the method are not available.

Barriers to achieving the desired technical result are as follows:

- if the width of the smoothing window is equal to 2p+1, the first p and last p values of the series is not treated;

when the median filter is not the property of additivity (and, hence, the median filter does not possess the property of linearity):

med{y_j ⁽¹⁾+y_j ⁽²⁾}≠ med{y_j ⁽¹⁾}+md{y_j ⁽²⁾};

- median filter preserves monotonically changing parts of the signal (and therefore with a small width “sliding window smoothing is not efficient enough).

The structural scheme of the device that implements the method is similar, comprises a generator of clock pulses, a switch, a control unit, a register storing unit rank, the block selecting the middle value, the output register, which stores a smoothed series of the original implementation.

Known methods for selecting trend the closest to the technical nature of the claimed method is the moving average ([5], s, 170). This is one of the most simple is erodov mechanical smoothing of time series. Consider this method as a prototype. To apply this method, only one realization of y₁, y₂, ... , y_nthe original process.

The moving average method involves the following steps:

- memorizing the implementationrandom process, where m<n (the width of the “sliding window”) rows, for which to calculate the average valuefor several;

- replacement of the Central values offound average

- shift “sliding window” on one value to the right (i.e. the choice instead of cutthe number of the next segmentfrom).

The algorithm runs until until it reaches the right end of the row.

A priori for a time series is determined by the smoothing interval m, i.e. a natural number of m<n. If you want to smooth out small random fluctuations, the smoothing interval take more; the smoothing interval is reduced, if you want to keep smaller fluctuations. For the first m values of a time series is computed their average value is tion; this will be smoothed value of the time series in the middle of the smoothing interval. Then the smoothing interval is shifted by one value to the right repeats the calculation of the arithmetic mean, etc. For this reason, this method is called by the MovingAverage method, since when performing the procedure, there is a sliding window of width 2R+1 for all series from beginning to end. The width of the window, usually odd, since theoretical value calculated for the Central value. Hence we get the formula for calculation of the smoothed values of a time series:

where p=(m-1)/2 (m is an odd number).

As a result of this procedure is obtained n-m+1 smoothed values. Standard deviation σ_mthe smoothed series is:

where through σ marked deviation of the original members of the series. Therefore, the larger the smoothing interval, the stronger the averaged data and less volatile allocated trend. Most often, the smoothing produced by the three, five and seven members of the original series. You should consider the following moving average: looking at the number of periodic oscillations of constant length, then the smoothing based on a moving average with an interval of shlian what I equal to or a multiple of the period of the oscillations completely eliminated. Often smoothing based on a moving average so strongly converts a number that highlighted the tendency is manifested only in the most General terms, and more small, but important detail analysis (waves, curves, etc) disappear. After smoothing small waves can sometimes change the direction opposite to the place of “peaks” appear “pits” and Vice versa. All this calls for caution in the use of a simple moving average and makes the search for more accurate methods to select the trend.

The characteristics of the prototype method, coinciding with the characteristics of the proposed technical solution, the following: sampling of the signal in time, remembering only the first m values of the digital signal, replacing the original time series smoothed.

The disadvantages of this method are:

the first p and last p values of the series are lost (not smoothed); this disadvantage is particularly noticeable effect when the length of the number is small, or if it is necessary to extrapolate beyond the considered time interval to solve the problem of forecasting;

- the method is not effective enough, because it does not include the fine details of the trend.

- used for time series with linear trend;

the use of this ways is and calls the residuals autocorrelation, even if it was absent in the original series, i.e. in the smoothed time series occurs interdependence of neighboring values of the series (the effect of the Slutsk-Yul).

Barriers to achieving the desired technical result are as follows:

- if the width of the smoothing window is equal to 2p+1, the first p and last p values of the series is not treated;

- since the Central value of the window (segment) smoothing is calculated as the arithmetic average of the neighboring, then the new values of the time series become dependent;

- replacement of the Central value of the window (segment) smoothing the average of neighboring values is significant (significant only in the case when the Central value deviates significantly from the average, and will have virtually no effect on the result, when the Central value fluctuates around the average value.

The structural scheme of the device that implements the method is similar, comprises a generator of clock pulses, the switch control unit, the first and second registers, the adder, the output of which is connected to the information input of the first register, the output of which is connected to the first information input of the switch, the second output of which is input devices (ed. St. No. 1 193 688).

There is also a variation of the considered method is a prototype of the a - the weighted moving average method. It differs from the previous method of smoothing the fact that the values of the time series included in the smoothing interval, are summed with different weights ([5], s). Here to calculate the smoothed values of the time seriesthe formula applies a weighted arithmetic mean:

and weight ρ_kdetermined using the method of least squares. These weights are calculated for different degrees of approximating polynomial and different intervals smooth. So, for polynomials of the second and third-order numerical sequence of weights when the smoothing interval m=5 has the form{-3;12;17;12;-3}, and if m=7 has the form{-2;3;6;7;6;3;-2}.

For the weighted moving average drawback still is the inability to smooth the values of the time series at the ends. In addition, the use of this method without negative weights induces autocorrelation in the residuals, i.e. the effect of the Slutsk-Yul.

The proposed method comes from having a single discrete implementation of the process under investigationwhere y_k=y(t_k), k=1, 2, ... ,n, represent the sum of the useful and noise components, i.e. y(t)=S(t)+u(t). A priori information about the process under study is that the selected interval m< ⁿ/₂useful signal quite accurately described by a polynomial of the second degree:

S(t)=a+bt+ct²

Consider the method involves the following steps:

- memorizing the implementation(j=1, 2, ... , n-m+1) random process, where m<n/2 (m - width “sliding window”), which will be approximated by a polynomial of second degree least-squares;

- the replacement of a number ofthe values of the approximation functionresulting in a number of

- shift “sliding window” on one value to the right (i.e. the choice instead of cutthe number of the next segmentfrom);

- repeat the procedure above until you reach the right end of the row, i.e. j=n-m+1.

First, define the width of the “sliding window” m<n/2, which will be approximated by a polynomial of second degree least-squares. Determining a number m, remember the first part of the original series. From the condition of minimum of the sum of squared deviations:

determine the coefficientsapproximating p is linoma:

Find the estimation of the approximation functionon the segment numberand remember her. Shift-on-one countdown “sliding window” and get a new piece of the original series. Re-manufactured approximation by a polynomial of second degree by the method of least squares, find the assessmentand remember her. Thus, we get r=n-m+1 segments of the original series and r their assessments. Segments of the original series can be represented in the form of a matrix of dimension m× r:

Then the matrix of the estimates can be written as follows:

Find the approximating functionaveraging over the side diagonals:

The essence of the proposed method and device is illustrated by a drawing.

The device for selection of the trend of results of measurements of physical quantities by a sliding breeding evaluations (“MOLE”) his only discrete source implementation of contains (Fig.9), the buffer unit storing the results of measurement (voltage, current, resistance, temperature, etc.) 1, the inlet of which is an information input device, the output of which is connected to the block C the support 2, the output of which is connected to the input of the block approximation by the method of least squares 3, the output of which is an information input unit storing estimates of trend 4, the output of which is connected to the input of block averaging 5, the output of which is an information output device to the additional information input units 1, 2, 3, 4, 5 are connected to the control unit 6. Synchronous operation of the device is provided by a generator of clock pulses 7.

Device to highlight trend works in the following way. In the buffer block is written to the first m values of the original discrete implementation of the results of measurements (m is the width of the “moving window”), the value of m is set by the control unit. Upon completion of the write buffer block is the approximation of the received m values by the method of ordinary least squares (OLS) by a polynomial of second degree.

The obtained evaluations of the rst m values of the original discrete implementation are recorded in the storage unit estimates a trend. After each subsequent received on the input values of the dimensions in the block approximation of the OLS estimation of the last m samples recorded in the buffer unit, and write them in the storage unit of the trend with a shift of one sample relative to the beginning of implementation. The estimates obtained by the method of the arithmetic mean are summarized in each m is the moment in time discretization of the original discrete implementation of the measurement results, and the final assessment of trend output device.

This method of allocation trend is as follows. In the control unit 6 sets the width of the moving window m, the information which is supplied to the additional information input register storing dimension 1, of the delay block 2, block approximation 3, the storage unit assessments 4 and block averaging 5. In the register storage dimensions 1 is written to the first m values of a discrete sequence of measurement results. Using the delay unit 2 is being delayed data in block 3 m-1 clock cycles. After receiving the m-th values of the block data storage register 1 received in block 3, where they are re-approximation of the least-squares polynomial of the second degree. Estimates obtained in block 3, is recorded in block 4. After each subsequent received on the input values of the dimensions in the block approximation of the OLS estimation of the last m samples and record them with a shift of one sample relative to the beginning of the implementation in block 4. The data recorded in block 4, enter in block 5, where the method of the arithmetic mean in each moment of time is determined by the average estimates of trend output unit 5 is an information output device.

The technical result - the reduction of error evaluation function for the serious signal (trend) with a limited amount of a priori information about the statistical characteristics of the additive noise and the function of the useful signal (trend), which is achieved by processing in real time only the original implementation of the measurement results “reproduction” estimates of approximating polynomials and subsequent averaging of the estimates of the function of the useful signal.

Technical features of the application of the proposed method at the stage of “reproduction” estimates of approximating polynomials are as follows. Let y_kvalues of the time series that fall in the interval(j=1, 2, ... , n-m+1) and corresponding to times t_k. Then a, b, C coefficients of approximating polynomial:

determined from the condition of minimization of the sum of squared deviations of the elements of a number of y_kcaught in the gap(j=1, 2, ... , n-m+1), from the values of the polynomial S(t_k) at appropriate points, i.e. from the condition

Σ (y_k-S(t_k))²→min,

where the summation extends to all values of y_kcaught in a specified amount of the split. To determine estimateswe obtain a system of linear equations

Introducing the notation:

let's rewrite the system (2) (relative unknowns a, b, C)

Solving the system (3), we obtain:

where K=AD²-AnC+B²n-2BCD+C³.

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

- The average quadratic error evaluating the function of the useful signal (trend) is much less of estimation errors when using other previously considered methods with limited a priori information about the process under study.

- Proposed method allows for the processing of the original data in real time.

- Evaluation of the function of the useful signal (trend) regardless of the form of the source function of the useful signal (trend) and the statistical characteristics of the additive noise and adequately displays the basic regularities of changes in the useful signal (trend).

- The absence of edge effects.

Figure 1 shows the original function of the measured process, representing the additive mixture of the harmonic signal (trend) and noise, distributed Gaussian. As can be seen from the analyzed model, the original function of the useful signal (trend) is significantly different from the function, opisyvayuschaya by a polynomial of second degree. Despite this obtained about the child, shown in figure 2 is almost exactly the original function. The treatment was carried out under the following initial data: a window width of m is equal to 16.

Figure 3 shows a complex signal consisting of a parabola, sine wave, constant and exponent; figure 4 shows a noise component, and figure 5 - signal with additive noise. In the signal processing (6) of the claimed method (“MOLE”) selected signal (trend), which comparison with the original signal is shown in Fig.7. Schedule residues shown in Fig.

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Device for selection trend by the method of reproduction of the moving estimates his only source implementation of the containing block storing the results of measurement of physical quantities, the entrance of which is an information input device and an output connected to the input of the delay block, the block averaging, the output of which is an information output device, the control unit, the output of which is connected to the control inputs of the blocks storing the results of measurement of physical quantities, delays, averaging block clock pulses, the output of which is connected to the clock inputs of the blocks storing the results of measurement of physical quantities, delays, averaging, characterized in that the device entered the block approximation by the method of least squares, whose input is connected to the output of the delay block and an output connected to the input of the storage unit that estimates of the trend, the output of which is connected to the input of block averaging, to the control inputs of the blocks approximation least-squares and store ratings trend connected the output of the control unit and to the clock inputs of blocks approximation least-squares and store ratings trend connected the output of the block clock.