JPH06290164A - Predicting method for time series data - Google Patents

Abstract

PURPOSE:To improve the precision of the non-linear prediction of chaos time series data using local approximation by using a distance evaluation formula weighted in accordance with the uncertainty of a coordinate component. CONSTITUTION:In the topological space of buried dimension (d) using delay time coordinate, a reference point X0 is expressed as X0=(x0, X-1, x-2,..., x-d+1), and it contains (d-1) pieces of past data besides a present value x0, and when the measurement error of x0 is made epsilon0, an error epsilon contained in x-j becomes an equation I. It is a Lyaqunov index corresponding to the component remarked presently, and in the case of xi<0, a situation is clear from the definition of the Lyaqunov index, and in the case of xi>0, the uncertainty expands in the positive direction of proper time, an there is possibility that the present value is being observed somewhere in a range expanding in the way of an exponential function. Then, as considering its upper and lower limits, the uncertainty contained in x-j is expressed by th equation I. Accordingly, weight corresponding to this uncertainty is expressed by an equation II, and in the case of Euclid distance, it is corrected like an equation III.

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a method of predicting time series data which behaves chaotically.

[0002]

2. Description of the Related Art Even if seemingly random time series data has chaotic fluctuations, there are rules in the randomness, and it is possible to make predictions by using them. Physical about the basics of forecasting
Review Letters, 59, 845-848 (1987) (Phys. Re
v. Lett., 59, 845-848 (1987)), Physikady, 3
5, 335-345 (1989) (Physica D, 35, 335-345 (1989)
), Nature, 344,734-742 (1990) (Nature, 344,
734-742 (1990).

Here, time series data x _-N , x _{-N + 1} , x _{-N + 2} , ..., x _-2 , x _-1 , x ₀ which behaves chaotically are when given, consider the case of predicting the n steps ahead of x _n. A known method, measurement of the correlation dimension, calculation of the Lyapunov exponent, a method by nonlinear prediction, etc. are used to determine whether or not the time series data is chaotic when given.

[0004] In the conventional method constitutes a d-dimensional phase space by implantation from the time-series data, i.e. x _-i to a point X _-i in phase space

[0005]

[Equation 1]

Correspond to Current time series data value x ₀
The point X ₀ corresponding to is used as a reference point, a point in the vicinity of this reference point is searched from a point composed of past data, and local fitting or the like is performed to obtain a predicted value.

[0007]

The accuracy of the prediction by the conventional method is said to depend on the number of points composed of past data globally (Physical Review Letters above). On the other hand, it is natural that the local prediction accuracy depends on the quality of the points near the reference point, but this point is not paid much attention to. There was a problem that I could not get it.

[0008]

In the conventional prediction method, a normal Euclidean distance is used as a distance evaluation formula when searching for a point near a reference point. It is expressed by the following equation.

[0009]

[Equation 2]

Where d ₀ is the optimum embedding dimension.
In the usual case, the optimum value of the embedding dimension cannot be known in advance. The optimum value is the embedding dimension that gives the best prediction accuracy when the past data is bisected and the second half is predicted by the conventional method based on the first half.

In the conventional method using Equation 1, since each component of coordinates is treated equally, it is impossible to find a neighboring point suitable for prediction. This is because the coordinates constitute past data at different times, and the distance may be evaluated by the following formula, for example, in consideration of the uncertainty when the current value is used as a reference.

[0012]

[Equation 3]

Where K is the Lyapunov exponent of the chaotic orbit
with λ _i

[0014]

[Equation 4]

It can be expressed as The Lyapunov index is obtained by a known method. If there is not enough past data, K can be used as an adjustment parameter to determine the best prediction accuracy. The determination of the best prediction accuracy is performed similarly to the optimization of the embedding dimension. The use of Equation 3 makes it possible to calculate the j-th component of coordinates when calculating the distance.

[0016]

[Equation 5]

It corresponds to weighting with and conversion to new coordinates.

By using Equation 3, it becomes possible to find a neighboring point according to the purpose of prediction. That is, search for X _-m that minimizes the distance from X _0, and find its first component x _-m
The use of the predicted value for x _n <x _n> is

[0019]

[Equation 6]

It is expressed as follows. This is a nonparametric zeroth-order local approximation. It is also possible to use the centroids X _G of some neighboring points instead of the nearest point X _-m . Nonparametric means that no parameter is used to express the predicted value <x _n >.

In the first-order approximation of the Jacobian method, which is one of the parametric local approximations,

[0022]

[Equation 7]

It becomes Where A _-m is the Jacobian matrix determined from the information near the point X _-m . As a result, the prediction accuracy can be greatly improved. Also, in the usual parametric first-order approximation

[0024]

[Equation 8]

[0025] a ₀ and a _k are parameters that should be determined by the fitting method. In the parametric case, it is generally possible to expand by a d-dimensional h-th order polynomial.

In this case, each parameter is determined from the information of the set of points {X _-k } near X ₀ selected by the _equation 3 and the set of points {X _{-k + n} } after n steps of them. be able to. This corresponds to the direct method discussed in Physicadi above. When predicting n steps ahead, the prediction can be performed in one step as described above, or it can be performed by an iterative method in which the prediction one step ahead is repeated n times as discussed in this document.

Further, it is possible to use not only Equation 3 when selecting a point near X ₀ , but also Equation 3 when determining the coefficients appearing in Equation 7 and Equation 8. It is number 5
It corresponds to performing the prediction in the coordinate space weighted by.

From the viewpoint of coordinate transformation for weighting according to equation 5, the norm is not limited to the Euclidean norm, and other norms such as the maximum norm can be used.

[0029]

[Operation] How the distance evaluation formula given above improves the prediction result will be described. In the phase space of the embedding dimension d using the delay time coordinate, the reference point X ₀ is

[0030]

[Equation 9]

It is expressed as It contains d-1 past data in addition to the current value x ₀ . If the measurement error of x ₀ is ε ₀ , the error ε contained in x _-j is

[0032]

[Equation 10]

It becomes This is the Lyapunov exponent corresponding to the component currently being focused on. When λ _i <0, it is clear from the definition of Lyapunov exponent, which is shown in Fig. 1 (a). When λ _i > 0, the uncertainty essentially expands in the positive direction of time (broken line in Fig. 1 (b)). The current value may be observed anywhere in the exponentially spread range.
Considering the upper and lower limits, the uncertainty included in x _-j is expressed as in Eq. Therefore, the weight corresponding to this uncertainty is expressed by Expression 5, and in the case of Euclidean distance, it is corrected as Expression 3.

[0034]

【Example】

(Example 1) Time series data having a chaotic behavior is obtained by a Henon map. That is, x _{i + 1} = 1.0 + y _i -1.4 x _i ² y _{i + 1} = 0.3 x _i , with (x ₀ , y ₀ ) = (0.3, 0.3) as the initial value i =
Using {x _i } from 101 to i = 900 for past data i
= 901 to i = 1100 are predicted. Prediction accuracy is calculated using the correlation coefficient between the actual calculated value and the predicted value.
For the prediction, direct methods such as nonparametric zero-order approximation and Jacobian first-order approximation were used.

FIGS. 2A and 2B show the results obtained by the conventional method and the method of the present invention, respectively. For the embedding dimensions 1 to 7, prediction was performed up to 10 steps. In this system, the original dynamical system is two-dimensional, so we get the best prediction result at d = 2. On the other hand, in the case of the present invention, such dependency on the embedding dimension is weakened. Weighting was performed with K = 1.98. With such weighting,
For example, in the prediction one step ahead at d = 2, the correlation coefficient is
It has been improved from 0.999891 to 0.999965.

The primary prediction was also examined.
The results of the conventional method and the method of the present invention are shown in FIGS. 3 (a) and 3 (b). As a whole, the result is worse than the 0th-order prediction,
Compared with the conventional method, the method of the present invention greatly improves the result.

(Embodiment 2) An example in which the present invention is applied to power demand forecasting will be described. Of the data of the actual power consumption of 1400 electric power taken in one hour of the sampling cycle, the first half 1150 points were used as the past data, and the latter half 250 points were predicted.
The embedding dimension was d = 3 and the delay was τ = 4. Predictions were made using a parametric local first-order approximation. n
= 1 The result is shown in Fig. 4. FIG. 4 (a) shows the result when the conventional method is used, and FIG. 4 (b) shows the result when the equation 3 is used as K = 0.25. The actual usage amount is shown by a solid line and the prediction result is shown by a broken line. The results are better when using Equation 3.
The average error was 4.5% when the conventional method was used and 2.9% when the present invention was used.

(Example 3) An example in which the present invention is applied to a control system for alloy plating will be described. The plating apparatus considered here is a current regulation type. In alloy plating, component ratio distribution and film thickness are given as target values. Regarding the film thickness control, the conventional control is sufficiently effective, and here, the control of the component ratio distribution will be mainly taken up. The metal ions in the plating solution deposit faster as the potential of the electrode is lower than the redox potential of itself. Therefore, when the potential changes periodically, the component ratio changes periodically in the thickness direction of the plating layer. If the potential is constant, the component ratio will also be constant. Therefore, the component ratio distribution can be controlled by controlling the potential.

The structure is shown in FIG. The controller 501 controls the current-regulating alloy plating apparatus including the galvanostat 502 and the electrolytic cell 503. The electrolytic cell 503 includes therein a reference electrode 504, a working electrode 505, and a counter electrode 506 made of an alloy component metal. Further, the inside of the tank is filled with an electrolyte solution containing metal ions as an alloy component. Counter electrode 50
Plating proceeds in a form in which the metal No. 6 is eluted and deposited on the working electrode 505. At this time, the working electrode 505 and the counter electrode 506
Between the current value set by the controller 501
i flows. Potential of working electrode 505 with respect to the reference electrode E = E _W
-E _R fluctuates with changes in the electrode surface condition.

This potential E is sent to the controller 501 via the galvanostat 502. In this example, when the potential of the working electrode oscillates chaotically, the current value i
The desired periodic potential fluctuation is obtained by slightly changing. There are innumerable unstable periodic states near the chaotic state. When the one periodic state is realized as the potential of the working electrode, it forms one component ratio distribution. It is possible to know in advance what kind of component ratio distribution these unstable periodic points correspond to by analyzing the Poincare section. The control algorithm is as shown in FIG. 6 in order to speed up the transition from the chaotic state to the target periodic state.

The potential E which is the output from the controlled object 601
Is sampled and discretized in the Poincare section by the discretization means 602. Below, the discretized potential E _i is regarded as the output. The component ratio distribution is designated as the designation of the periodic points to be stabilized. Therefore, the range of potential that can be stabilized
I _t = [E _min , E _max ] is determined. Control command device 603
Sends typical potentials E ⁰ and I _t at the target periodic point. At the current value i = i ₀ before the start of control, the potential E _i changes chaotically. The current value i is adjusted within the range of [i ₀ -Δi, i ₀ + Δi] and controlled as follows.

At the decision step 604, the potential E _i becomes I
When it is included in _t , the control for speeding up is not necessary, and the current value i is corrected by the stabilization algorithm 605 so that it can be stabilized at the target periodic point in the next step. Stabilization can be achieved by adding a correction δi proportional to the deviation of the periodic point from the typical potential E ⁰ .

At the decision step 604, the potential E _i becomes I
Predict if not included in _t . Current value i
The predicting means 606 estimates the section I _k of the potential after k steps when changing in the range of [i ₀ −Δi, i ₀ + Δi]. The correction value δi according to the speedup algorithm 607
Ask for. When I _n has the first overlap with I _t , δ
i adjustable transition to I _t after n steps of. as a result
The stabilization algorithm 605 becomes effective in the nth step, and the target periodic point can be stabilized in the next step.
OGY, respectively for stabilization and acceleration algorithms
Use the SOGY algorithm.

The film thickness is controlled so that the amount of electricity flowing is monitored and the plating is completed when the target value is reached.

The results of applying the present invention to Cu-Ag alloy plating will be described. Current value 1.25 in this system
At mA, the potential E showed chaotic fluctuations. Current value i
Was controlled in the range of [1.2, 1.3] (mA).
For the prediction, a parametric first-order local approximation was used, and 2000 points were used for i = 1.2, 1.25, and 1.3 mA as past data.

As a result, an alloy plating layer as shown in FIGS. 7 (1) and 7 (2) was produced. (1) shows the case of the present invention, and (2) shows the case of the present invention. These are prepared by forming a plating layer having a component ratio distribution a by controlling the transition from the chaotic state to the cycle 1 state. The shaded area corresponds to the transitional state from the chaotic state to the period 1 state, but it is preferable that the number is as small as possible. Figure 8 (1) shows the corresponding potential fluctuation
Shown in (2). In the drawing, where the start and the end are indicated by arrows, new control commands are issued and the control targets are reached.

From these results, it became clear that it is possible to reduce the portion corresponding to the transient state by making a prediction for distance evaluation in consideration of the uncertainty included in the coordinate component.

[0048]

According to the present invention, it is possible to remove the influence of uncertainty due to the nature of chaotic trajectories, and improve the accuracy of nonlinear prediction of chaotic time series data using local approximation.

[Brief description of drawings]

FIG. 1 is a diagram for explaining the uncertainty included in a coordinate component due to the nature of chaos.

FIG. 2 is a diagram showing a result of comparing the result of 0th-order prediction in Example 1 with a conventional method.

FIG. 3 is a diagram showing a result of comparison of a result of primary prediction in Example 1 with a conventional method.

FIG. 4 is a diagram showing a result of applying the present invention to power demand prediction.

FIG. 5 is a configuration diagram of a current-regulating alloy plating system when a working electrode potential is monitored.

6 is a block diagram showing the contents of control in the system of FIG.

FIG. 7 is a schematic sectional view of the produced alloy plating layer.

FIG. 8 is a diagram showing corresponding potential fluctuations when the alloy plated layer of FIG. 7 is produced.

[Simple explanation of symbols]

501 ... Controller, 502 ... Galvanostat, 5
03 ... Electrolyzer, 504 ... Reference electrode R, 505 ... Working electrode W,
506 ... Counter electrode C, 507 ... Current value i, 508 ... Reference electrode potential E _R , 509 ... Working electrode potential E _W , 510 ... Working electrode potential E (= E _W -E _R ), 601 based on the reference electrode potential ... Control object, 602 ... Discretizing means using Poincare section, 6
03 ... Control command device, 604 ... Judgment step, 605 ...
Periodic point stabilization algorithm, 606 ... Prediction means, 60
7 ... Algorithm for speeding up.

 ─────────────────────────────────────────────────── ─── Continuation of the front page (72) Inventor Hiroaki Okuhira 292 Yoshida-cho, Totsuka-ku, Yokohama-shi, Kanagawa Prefectural Institute of Technology Hitachi, Ltd.

Claims (5)

[Claims] 1. A non-linear prediction method characterized by using a distance evaluation formula weighted according to the uncertainty of a coordinate component in the prediction of a time series having a chaotic behavior. 2. When predicting a time series having a chaotic behavior, a distance evaluation formula weighted according to the uncertainty of a coordinate component is used when searching for a neighboring point of a reference point from past data. Characteristic nonlinear prediction method. 3. The non-linear prediction method according to claim 1, wherein non-parametric local approximation is used. 4. The non-linear prediction method according to claim 1 or 2, wherein parametric local polynomial approximation is used. 5. The non-linear prediction method according to claim 4, wherein the polynomial order is first-order.