RU2467383C2 - Method and apparatus for predicting nonstationary time series - Google Patents

Abstract

FIELD: physics, radio.

SUBSTANCE: invention is related to radio engineering and can be used to predict data in systems for different purposes. The method of predicting nonstationary time series, based on an autoregressive model of a moving average with minimum mean-square error additionally involves operations for recirculating input data by one reading, oversampling the initial process in a logarithmic time scale, finding the energy spectrum of the obtained signal, determining response corresponding to entropy of the sampling energy spectrum, calculating the maximum value of entropy responses, finding a prediction for realisation, which corresponds to the maximum entropy value, oversamping the prediction result in an exponential time scale.

EFFECT: high noise-immunity of the prediction process in conditions of additive noise and marked nonstationary nature of the predicted process.

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Description

The present invention relates to the field of special radio engineering and can be used to predict data in systems for various purposes, including the prediction of economic processes. Known forecasting methods based on the methods of Wiener and Jaglom. The general theory of forecasting random processes is described in detail in [1]. The effectiveness of decisions depends on the volume of a priori data, the processes under study, and the methods of their presentation. A disadvantage of the known technical solutions is the use, as a rule, of the hypothesis of stationarity of predictive procedures, including in the analysis of broadband signals.

Known methods for predicting processes with discrete time in the case of rational spectra, as well as one-dimensional stationary processes with a constant step in time. It is noted that the multidimensional case in the general theory of forecasting is much more complicated than the one-dimensional one. As a rule, the basis of forecasting is the analysis of the time series [1, 2]. The most common methods are: least squares method [3] and its modifications, exponential smoothing method, probabilistic modeling method and adaptive smoothing method. In any case, the selection of the most suitable model for describing the forecasting process is necessary.

When forecasting processes, discrete samples of two types are formed:

1. The model of the functional dependence of the process is unknown. In this case, problems of evaluating the functional dependence model from a whole class of available models are solved.

2. The model of the functional dependence of the process is known and it is only necessary to evaluate the parameters of the model of the initial process (regression coefficients b ₀ , b ₁ , b ₂ , ...).

, параболическая - y=b₀+b₁x+b₂x², логарифмическая - y=b₀+b₁lgx, экспоненциальная - y=b₀exp(b₁x) и другие.The following mathematical models are used: linear - y = b ₀ + b ₁ x, hyperbolic - y = b ₀ + b ₁ / x, exponential - y = b ₀ + b ₁ ^x , power -

, parabolic - y = b ₀ + b ₁ x + b ₂ x ² , logarithmic - y = b ₀ + b ₁ logx, exponential - y = b ₀ exp (b ₁ x) and others.

The solution of mathematical equations of communication involves the calculation of the initial parameters of the model (free term b ₀ and regression coefficients b ₁ , b ₂ ...).

With all the variety of models, there is still a form of analytical dependence, which has received the widest distribution. It is the regression equation in the form of polynomials (polynomial) located in ascending degrees of the studied factor and simultaneously linear to all coefficients:

y = f (x) = b ₀ + b ₁ x + b ₂ x ² + ... + b _m x ^m , where b ₀ , b ₁ , b ₂ , ..., b _m are the coefficients to be determined.

To determine the coefficients of the regression equation b _m, different methods are used (graphical, average method), however, the least-squares method (least squares) is most widely used.

The search for the coefficients b ₀ and b _{1 is} carried out according to the least-squares method, using the following equations:

,

.

,

.

An assessment based on the least squares method involves the mandatory satisfaction of a number of assumptions, the failure of which can lead to errors.

MNCs are widely used to obtain specific forecasts, which is explained by its simplicity and ease of implementation on a computer. The disadvantage of this method is that the trend model is rigidly fixed, therefore a reliable forecast can be obtained only for a short lead-time. MNCs are mainly used for short-term forecasting. The method makes it significantly difficult to choose the right type of model, as well as the rationale and choice of weights in a weighted least-squares method. Closest to the proposed invention is a method and device for predicting time series based on an autoregressive model of a moving average with a minimum mean square error [1].

The basis of forecasting is a random time series - this is an ordered sequence of random variables, which are the values of the samples of the received signal from the target. One of the important tasks when working with time series is to predict future values of the time series. In practical applications, the time series are highly correlated, which allows us to expect satisfactory estimates of the forecast.

The considered method and device allows predicting the time process against the background of additive Gaussian interference, while the task is to predict the values of z _{n + l} , l≥1 at the current time t _n . Such a forecast at time t _n is preempted l. Three explicit forms of model representation are considered, expressed by the following relationships:

1) Using the difference equation:

2) As an infinite weighted sum of current and previous impulses a _j :

.

.

3) As an infinite weighted sum of previous observations plus a random impulse:

,

,

where π _j are weights.

, наилучший прогноз представляется в виде

, где веса ψ задаются заранее, находят значение среднеквадратической ошибки прогноза

, находят минимальное значение СКО прогноза приравниванием

, определяют результирующий прогноз

или

, где e_n+l - ошибка прогноза

с упреждением l отсчетов. Полученный способ позволяет осуществлять прогнозирование процесса, однако на практике влияние шума или другой посторонней информации приводит к уменьшению отношения сигнал-помеха. Основным недостатком способа является то, что при наличии модуляции входного процесса формируется дополнительная ошибка прогноза за счет отклонения процесса от стационарного случая.The essence of the method is as follows. A sample of the input process z _{n is} selected for which it is necessary to make a forecast with lead l, which is a linear function

, the best forecast is presented as

, where the weights ψ are set in advance, find the value of the mean square error of the forecast

find the minimum value of the standard deviation of the forecast by equating

determine the resulting forecast

or

where e _{n + l} is the forecast error

with lead l counts. The obtained method allows predicting the process, however, in practice, the influence of noise or other extraneous information leads to a decrease in the signal-to-noise ratio. The main disadvantage of this method is that in the presence of modulation of the input process, an additional forecast error is formed due to the deviation of the process from the stationary case.

The aim of the invention is to increase the noise immunity of the forecasting process under the condition of the influence of additive interference and the expressed unsteady nature of the predicted process. The goal is achieved by the fact that:

1. A method for predicting non-stationary time series, based on the autoregressive model of a moving average with a minimum standard error, additionally contains operations of recirculating the input data by one count, oversampling the initial process in a logarithmic time scale, finding the energy spectrum of the received signal, determining the response corresponding to the entropy of the energy spectrum oversampling, calculating the maximum value of the entropy responses , finding a forecast for the implementation corresponding to the maximum value of entropy, oversampling the forecast result in an exponential time scale.

2. A time series prediction device comprising an analog-to-digital converter, a synchronization unit connected to a second input of an analog-to-digital converter, the first input of which is an input of the device, a prediction unit, a synchronization unit connected to a second input of an analog-to-digital converter, whose first input is the input of the device, the prediction unit, further comprises a delay unit, the first input of which is connected to the output of the synchronization unit, the second input is connected to the analog output go-to-digital converter, the output is connected to the input of the oversampling unit of the original process on a logarithmic time scale, the output of which is connected to the first input of the buffer memory unit, connected by its first output to the input of the signal spectrum finding unit, the output of the latter is the input of the square block of the signal module calculation, output which is connected to the input of the block for finding the entropy of the sample, the output of which is connected to the input of the block for calculating the maximum value of entropy, the output of which is the second the input of the buffer memory block, the second output of which is the input of the forecasting unit, connected by its output to the input of the oversampling unit of the prediction result according to the exponential law.

Time Series Prediction Method

The essence of the proposed method is as follows.

, где Ω - начальная частота, k - масштабный множитель, характеризующий крутизну модулирующей функции, видно, что мгновенная частота

, где ψ - фаза сигнала, стремится к бесконечности при t→0. Поскольку операция сжатия (расширения) не инвариантна относительно операции сдвига во времени, для формирования процедуры экстраполяции процесса требуется информация о начале обрабатываемой реализации. При работе с таким сигналом необходимо найти местоположение нуля, определяющего начало отсчетов.Existing forecasting methods work best when adopting the hypothesis of a linear stationary process. Real processes are often affected by factors that contribute to the non-stationary component. An example would be explosive processes in hydrophysics, sudden default in economics, etc. The proposed method considers the possibility of expanding the use of known forecasting methods for such processes. For stationary (additive) processes, sampling with a constant step is used. For non-stationary processes, it is proposed to apply sampling of signals with a variable step with the obligatory designation of zero process formation. An example of an unsteady signal is a signal with a hyperbolic FM (GFM signal), widely known in radio engineering [4]. When recording a signal

, where Ω is the initial frequency, k is the scale factor characterizing the steepness of the modulating function, it can be seen that the instantaneous frequency

, where ψ is the phase of the signal, tends to infinity as t → 0. Since the compression (expansion) operation is not invariant with respect to the time shift operation, the formation of the process extrapolation procedure requires information on the beginning of the processed implementation. When working with such a signal, it is necessary to find the location of zero, which determines the origin of the samples.

In the proposed method, the following operations are performed:

Temporal compression of the predicted process is performed, which is necessary to ensure signal processing in real time [5]. At the same time, at each step, the implementation is updated by one count. Thus, a class of implementations is formed that differ from each other by shifts by one reference. To form a class of discrete samples, each implementation is subjected to a logarithm and discretization operation.

The signal realizations are oversampling on a logarithmic time scale t → lnt. Since the initial process arrives with a constant step, the discrete process is first restored, which is solved by the interpolation procedure. In non-stationary oversampling, it is assumed that implementation zero is known. Let a process be given in a discrete form through a unit sampling interval on an interval of length M-1, which corresponds to M-samples. We also assume that m runs over the values m∈ (0 ... M-1). With the logarithmic scaling of the signal with step q, we have:

,

.

,

.

. Процедуру интерполяции следует начинать с последнего отсчета реализации сигнала, рассматривая его значение как последний отсчет интерполированного процесса. При указанном подходе к интерполяции полностью сохраняется информация, содержащаяся в сигнале. Видно, что для проведения логарифмической дискретизации интервала N-1 требуется бесконечное число отсчетов. Поэтому ограничимся дискретизацией только части интервала, формируемого конечной суммой геометрической прогрессии

. Начало для узлов интерполяции k∈(0…K-1) определяется значением:this implies

. The interpolation procedure should begin with the last count of the implementation of the signal, considering its value as the last count of the interpolated process. With this approach to interpolation, the information contained in the signal is completely stored. It can be seen that for the logarithmic sampling of the N-1 interval, an infinite number of samples is required. Therefore, we restrict ourselves to discretizing only part of the interval formed by the finite sum of the geometric progression

. The beginning for the interpolation nodes k∈ (0 ... K-1) is determined by the value:

.

.

To verify, we see that as K → ∞ t ₀ → 0. When using the standard interpolation procedure, you must specify the values of the interpolation nodes from left to right. If we carry out the inversion of reference numbers according to the law k∈ (0 ... K-1): k ₁ = K-1-k, then the interpolation nodes are set by the values:

After resampling the processes for each implementation s _n (t), the spectral function S _n (f, T) is formed:

.

.

.Then form the energy spectrum:

.

To determine the start of implementation, the following procedure is proposed. It is proposed to use entropy (Kullback information) as a criterion for choosing an implementation [6]:

,

,

- нормированный энергетический спектр выборки,

, r_ss(n) - корреляционная функция процесса.Where

- normalized energy spectrum of the sample,

, r _ss (n) is the correlation function of the process.

A heuristic explanation for choosing such a criterion is that it minimizes the randomness or “whiteness” of the process.

Next, the maximum value of entropy is found in accordance with relation (1.1). This operation is reduced to comparing responses with a threshold voltage. If the threshold is exceeded, a decision is made that the maximum is found.

To implement the corresponding maximum value of entropy, the operation of oversampling the prediction result in the exponential time scale t → expt is used. This transformation is the inverse of the previously considered logarithmic. Note that the previously performed scaling of the hyperbolic process (non-stationary) on a logarithmic scale actually made it close to the tonal signal. Such a signal is, in fact, stationary and is well predicted by known methods. The exponential transformation allowed us to return to the original hyperbolic process.

The application of the proposed method allows to obtain a more accurate forecast than when using known procedures.

To verify the proposed technical solutions, a statistical experiment was conducted in which the efficiency of using the procedures of predictive functions was evaluated:

1. The classical forecasting procedure implemented in the MathCad environment, based on the hypothesis of stationarity of input data (samples are distributed evenly over time) and evaluated as an ARSS forecast.

2. Procedures using hyperbolic symmetry to implement the forecasting procedure.

In the experiment, a methodology for assessing the effectiveness of forecasting procedures was highlighted. The signal and noise were artificially formed with the given values of the characteristics. With this approach, knowing the characteristics of the signal and noise, a complete description of the effectiveness of these procedures was obtained. Unfortunately, in general, the technique was of limited use, because of which the selected processes are formed from well-known models.

The forecast error estimate is determined in accordance with the ratio:

,

,

- исходный процесс,

- результат прогноза, AA, BB - средние арифметические значения процессов, определяемые по формулам:Where

- source process

- the result of the forecast, AA, BB - the arithmetic mean values of the processes determined by the formulas:

,

,

,

,

,

- вариации процессов

и

, определяемые по формулам:

,

- process variations

and

defined by the formulas:

,

,

.

.

.For its formation, the input process was partitioned into two parts, where the first part of the process was considered as an input sample, and the second part was an unknown but formed future part of the process

.

Due to sectioning with overlapping by half the size of the section and the operation of averaging the difference of the estimates, we obtain a general averaged estimate of the forecast error:

where M / 2 means that there is an overlap of half the length of the sections.

The estimate (1.2) was calculated for forecasting procedures according to the first and second methods with a change in the upper frequency.

From figure 2. follows that:

1. With an increase in the frequency band, the prediction error decreases.

2. At frequencies ranging from zero to 0.1, a sharp increase in prediction error appears.

3. Starting from a frequency in the range of 0.1 and higher, a smooth decrease in prediction error is observed.

The result of the estimation of the averaged prediction correlation depending on the frequency band is shown in FIG. 3.

From figure 3 it is seen that:

1. With an increase in the frequency band, the prediction correlation coefficient increases.

2. At frequencies up to 0.1, a drop in the value of the prediction correlation coefficient to frequencies within 0.1 is observed.

3. Starting from a frequency in the range of 0.1 and higher, a smooth increase in the value of the prediction correlation coefficient is observed.

According to the second method, a signal model was selected based on actual data. In this case, the experiment had a limited character of estimates in terms of estimating forecast errors.

The considered approach made it possible to increase the volume of a priori data due to the formation of a sample for data prediction, i.e. on the time plane (Δ, t), 0≤Δ≤∞, -∞ <t <∞. Calculations show that the proposed operations can reduce the forecast error by an average of 20%.

Signal time series prediction device

A device that implements the proposed method for predicting time series is shown in figure 1 and contains the following blocks:

- analog-to-digital converter (ADC) - 1;

- delay unit - 2;

- oversampling unit - 3;

- block buffer memory - 4;

- block fast Fourier transform (FFT) - 5;

- unit for calculating the square of the spectrum modulus - 6;

- entropy calculation unit - 7;

- block finding the maximum - 8;

- forecasting unit - 9;

- oversampling unit - 10;

- synchronization block - 11.

As delay unit 2, digital delay lines with recirculation (RLS) can be used [18.5]. The Fast Fourier Transform Unit (FFT) 5 is designed to find the spectral density of a signal and is widely used in digital signal processing [7].

The unit for finding the maximum of entropy 8 can be implemented in software or in the form of a threshold block.

In General, the device operates as follows. In the delay block 2, the analyzed implementation is constantly updated by one sample at each processing step. In oversampling unit 3, the sequence is converted to a logarithmic time scale t → lnt. The current implementation is stored in the buffer memory unit 4. FFT blocks 5, calculation of the square of module 6 ensure that the energy spectrum of the implementation is found for subsequent calculation of the entropy in block 7 in accordance with relation (1.1). When the maximum of entropy is found in block 8, a signal is output to the buffer memory block 4, which means that it stores the implementation of the signal with a correctly defined "zero" that determines the reference point during sampling. The selected implementation goes to the prediction block 9. The forecast result is returned to the original scale using the oversampling block 10, which operates according to the exponential law t → expt.

LITERATURE

1. Boxing Jenkins. Time Series Analysis, p. 144-150.

2. Hennon E. Multidimensional time series. - M .: "MIR", 1974. - p.145-185.

3. Baraz V.R. Correlation and regression analysis of the relationship between indicators of commercial activity using the Excel program: a training manual / V.R. BARAZ. - Yekaterinburg: GOU VPO "USTU-UPI", p.41-50.

4. Rikhachek. Signals that are acceptable in terms of the Doppler effect. TIIER, t.54, N 6, 1966. - p. 39-41.

5. E. Oppenheim. The use of digital signal processing. - M.: Mir, 1980 .-- p. 417-418.

6. S. Kulbak. Information Theory and Statistics. - M.: Publishing. Science, 1967 .-- 408 p.

7. Korneev V., Kisilev A. Modern microprocessors. - M.: Publishing. Nolidzh, 1998. - p.136-138.

Claims (2)

1. A method for predicting non-stationary time series based on an autoregressive model of a moving average with a minimum standard error, characterized in that it further comprises operations of recirculating input data by one count, resampling the initial process in a logarithmic time scale, finding the energy spectrum of the received signal, determining the response, the corresponding entropy of the energy spectrum of the sample, calculating the maximum value of the responses of entropy, finding the forecast for the implementation corresponding to the maximum entropy oversampling results in exponential forecasting time. 2. A non-stationary time series forecasting device comprising an analog-to-digital converter, a synchronization unit connected to a second input of an analog-to-digital converter, the first input of which is an input of the device, a prediction unit, characterized in that it further comprises a delay unit, the first input of which is connected to the output of the synchronization unit, the second input is connected to the output of the analog-to-digital converter, the output is connected to the input of the oversampling unit of the original process by the logarithmic the time scale, the output of which is connected to the first input of the buffer memory block connected by its first output to the input of the signal spectrum finding unit, the output of the latter is the input of the square unit of the signal module calculation, the output of which is connected to the input of the sample entropy finding unit, the output of which is connected to the input of the block calculating the maximum value of entropy, the output of which is the second input of the buffer memory block, the second output of which is the input of the prediction block connected by its output with the input of the oversampling unit of the prediction result according to the exponential law.

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