JP3679817B2 - Nonlinear time series data prediction method - Google Patents

Description

[0001]
[Industrial application fields]
The present invention relates to a nonlinear time-series data prediction method , and in particular, estimates the dynamics from time-series data observed at equal sampling intervals using a local fuzzy reconstruction method, and shows the near future until the estimated dynamics disappear. It relates to the prediction method .
[0002]
[Prior art]
Conventionally, ARX models mainly based on autoregressive analysis have been used for prediction of time series data, but in recent years, local reconstruction using deterministic chaos theory has been aimed at elucidating chaos phenomena. The prediction of nonlinear time series data by the method has been studied. Examples of the local reconstruction method include Gramschmitt's orthogonal method, tessellation method, and the like. Below is an overview of deterministic chaos.
[0003]
A seemingly irregular, unstable and complex behavior can often be generated from a differential equation or a difference equation that is dominated by determinism (when the initial value is determined, all subsequent states are determined in principle). This is the deterministic chaos of the dynamical system.
[0004]
The concept of deterministic chaos revealed that irregular phenomena in the world are not necessarily only those that are dominated by non-determinism. In other words, there are not a few phenomena in the world that are producing irregular behavior according to determinism.
[0005]
Common to dynamical systems that produce chaotic behavior is that their equations are non-linear.
[0006]
By the way, if the behavior of certain time-series data is chaotic, it can be considered that the behavior follows a deterministic law. If it is possible to estimate the nonlinear deterministic law, the near future until the deterministic causality is lost due to the “sensitive dependence on the initial value” of chaos from the measurement data at a certain point in time. It is possible to predict the data.
[0007]
Prediction from the standpoint of such a deterministic dynamical system theory is based on the embedded theory of time-series data that “the state space and attractor of the original dynamical system are reconstructed from a single observation time-series data”. ing. The outline of this theory is as follows.
[0008]
A vector (y (t), y (t−τ), y (t−2τ),..., Y (t− (n−1) τ)) is created from the observed time series data y (t) ( τ is a time delay).
[0009]
The vector will indicate a point in n-dimensional reconstructed state space R ^n. Therefore, when t is changed, a trajectory can be drawn in this n-dimensional reconstruction state space.
[0010]
It is assumed that the target system is a deterministic dynamical system, and the observation time series data is obtained through an observation system corresponding to the C ¹ continuous mapping from the state space of this dynamical system to the one-dimensional Euclidean space R. In this case, this reconstructed trajectory is embedded in the original deterministic system if n is sufficiently large.
[0011]
That is, if the observed time series data is derived from the original dynamical attractor, the attractor that preserves the attractor's phase structure is reproduced in the reconstructed state space. n is usually called an “embedding dimension”, but since the reconstruction operation is “embedding”, when the dimension of the state space of the original dynamical system is m, If established, it has proven to be sufficient.
[0012]
[Expression 1]
n ≧ 2m + 1 (1)
However, this is a sufficient condition, and depending on the data, even if it is less than 2m + 1, it may be embedded. Further, it is also shown that the reconstruction operation is a one-to-one mapping if n> 2d (where d is the box count dimension of the original dynamical attractor). The box count dimension is a kind of capacity dimension that is a fractal dimension.
[0013]
As described above, deterministic chaos makes it possible to make short-term predictions with high accuracy even for data that appears to be totally chaotic.
[0014]
[Problems to be solved by the invention]
However, when predicting time-series data, it is necessary to determine the order of the autoregressive equation, the correction coefficient due to various factors, etc. in the prediction by the ARX model of the conventional method, and it is difficult to linearly approximate the time-series data. In some cases, it may become unpredictable.
[0015]
Moreover, in the Gram Schmidt orthogonal system method of the local reconstruction method, when the selected neighborhood vector is not linearly independent, the prediction accuracy may be extremely deteriorated. Furthermore, even in the tessellation method, the calculation time may become extremely long as the number of embedding dimensions increases. Thus, these conventional methods still have problems in terms of prediction accuracy and calculation time.
[0016]
The present invention has been made under the above background, and an object of the present invention is to provide a nonlinear time-series data prediction method that always obtains a prediction accuracy of a certain level or more and has a short calculation time.
[0017]
[Means for Solving the Problems]
In order to solve the above problems, the present invention provides sampling means for taking in time-series data to be measured at equal sampling intervals at the request of the data vector generation means,
Time series data storage means for storing time series data sampled by the sampling means;
Data vector generating means for reading out data from the time series data storage means and generating a data vector for performing deterministic short-term prediction based on deterministic nonlinear dynamical system theory;
An attractor reconstructing means for receiving a data vector generated by the data vector generating means, embedding it in an n-dimensional state space with an embedding dimension n and a delay time τ, and reconstructing the attractor of the data vector;
Local fuzzy reconstruction means for generating local dynamics of the attractor reconstructed by the attractor reconstruction means;
In a non-linear time series data prediction method having data prediction value generation means for generating a prediction value of data from a processing result in the local fuzzy reconstruction means,
The local fuzzy reconstruction means includes:
A processing target data vector selection procedure for selecting a processing target data vector from the data vector generation means and outputting it to a neighborhood vector detection procedure and an axis component detection procedure;
A neighborhood vector detection procedure for receiving a processing target data vector selected by the processing target data vector selection procedure and detecting a plurality of neighborhood vectors located in the vicinity of the processing subject data vector from the attractors reconstructed by the attractor reconstruction means When,
A vector detection procedure after a predetermined step for detecting a vector after a predetermined step of the plurality of neighboring vectors from the attractor reconstructing means and outputting it to an axis component determination procedure;
Each axis component of each vector is received from the processing target data vector selection procedure and the neighborhood vector detection procedure, and an axis component is determined by detecting a difference in each axis component value between the processing target data vector and a plurality of the neighborhood vectors. Axis component detection procedure output to the procedure,
A neighborhood vector with the smallest difference between the axis component values obtained from the axis component detection procedure is extracted, and the difference between the axis component values after the predetermined step between the neighborhood vector and the processing target data vector is reduced. An axis component determination procedure for determining each axis component value after a predetermined step of the processing target data vector,
A non-linear time-series data prediction method is provided.
[0018]
The axis component determination procedure generates a membership function for each axis of the neighborhood vector, and determines each axis component value after a predetermined step of the processing target data vector by fuzzy inference using the membership function. There is also provided a non-linear time series data prediction method characterized by:
[0019]
[Action]
When the prediction target follows deterministic chaos, an appropriate time delay τ is set to the time series data y (t) and the data vector x (t) = (y (t), y (t−τ), y (t −2τ),... Y (t− (n−1) τ)) [where t = {(n−1) τ + 1} to N], and an attractor is placed on the n-dimensional reconstruction space by an embedding operation. When reconstructed, this attractor draws a periodic trajectory. In the above equation, N indicates the number of data vectors determined by the time series length, the embedding dimension, and the delay time.
[0020]
From this, a data vector ahead of a predetermined step can be predicted. Particularly in short-term prediction, it is possible to predict a data vector with high accuracy.
[0021]
Since the data vector of the predetermined step ahead includes the data of the predetermined step ahead as its component, the data of the predetermined step ahead can be predicted by extracting this component.
[0022]
In the present invention, a plurality of vectors (neighboring vectors) of data vectors to be processed are detected, and for the axis component to be processed, the difference between the axis component value of each neighboring vector and the axis component value of the processing target data vector. Is detected.
[0023]
Further, the axis component value after a predetermined step of each neighboring vector is detected, and the predetermined step of the axis component of the processing target data vector is reduced so that the difference from the original data vector with the small difference is small. Later component values are determined.
[0024]
Therefore, the axis component value after a predetermined step of the processing target vector is greatly affected by a vector having a close component value in the original vector, and is not greatly affected by a vector having a component value not very close.
[0025]
In this way, by performing data vector prediction for each axis, it is possible to perform prediction that reflects local vector motion, and to perform highly accurate prediction.
[0026]
In particular, a membership function having a peak at each axis component value of the neighborhood vector is generated, and a component value after a predetermined step of the axis component of the processing target data vector is determined by fuzzy inference using this membership function. By doing so, it is possible to perform a highly accurate prediction reflecting the nonlinear behavior of the attractor.
[0027]
By using local fuzzy inference in this way (local fuzzy reconstruction method), for example, social phenomena such as water supply demand, power demand, and gas demand, changes in temperature, sunspot number, etc. Can predict the behavior of time-series data until it loses its initial value dependency (that is, before the estimated dynamics is lost). .
[0028]
【Example】
Example 1
FIG. 1 shows a functional block diagram of a nonlinear time series data prediction method according to the present embodiment.
[0029]
In this figure, reference numeral 1 denotes a data vector generation unit (time series data storage means and data vector generation means), which takes in data from the time series data y (t) as shown in FIG. 2 at equal sampling intervals. Then, a data vector for performing deterministic short-term prediction based on the deterministic nonlinear dynamical system theory is generated.
[0030]
The attractor reconstructing unit 2 (attractor reconstructing means) creates an n-dimensional state space database by reconstructing data vector attractors in the n-dimensional state space using Takens embedding theory.
[0031]
A local fuzzy reconstruction unit 9 generates local dynamics of the attractor and creates a database thereof. The local fuzzy reconstruction unit 9 includes: a processing target data vector selection unit 3 (processing target data vector selection unit) ; a neighborhood vector detection unit 4 (neighboring vector detection unit) ; a predetermined (s) post-step vector detection unit 5 (after a predetermined step) The vector detection means is composed of an axis component detection section 6 (axis component detection means) and a fuzzy inference section 7 (axis component determination means) .
[0032]
Reference numeral 8 denotes a data prediction value generation unit (data prediction value generation means) , which generates a prediction value of data obtained from the processing result in the local reconstruction unit.
[0033]
The function of each part will be described in detail below.
[0034]
The data vector generation unit 1 creates a database of time series data y (t) based on the observed values, and further generates a vector x (t) represented by the following equation.
[0035]
[Expression 2]
x (t) = (y (t), y (t−τ), y (t−2τ),... y (t− (n−1) τ))
However, t = {(n-1) τ + 1} to N
The attractor reconstruction unit 2 embeds time-series data in an n-dimensional state space with an embedding dimension n and a delay time τ. This operation becomes the reconfiguration of the attractor. Τ is a time delay, and N is the number of data vectors determined by the time series length, the embedding dimension, and the delay time .
[0036]
As described above, this vector represents one point in the n-dimensional reconstruction state space R ⁿ , and when t is changed, a trajectory can be drawn in the n-dimensional reconstruction state space. An example is shown in FIG.
[0037]
In order to reproduce the attractor in the reconstructed state space, it is proved that the dimension n is sufficient if n ≧ 2m + 1 as described above, where m is the dimension of the original dynamical state space. N and τ are determined according to the conditions.
[0038]
When short-term prediction is performed, in practice, n = 3 to 4 is usually sufficient. In this way, the values of n and τ are set, and the state space and the attractor are reconfigured by the embedding operation.
[0039]
When the data to be predicted follows deterministic chaos, the attractor becomes a smooth manifold as shown in FIG. 4, and data can be predicted using this attractor.
[0040]
In the local fuzzy reconstruction unit 9, a data vector to be processed (usually a data vector obtained by the latest observation) is z (T), and a predicted data vector z of the data vector z (T + s) after the s steps. “(T + s) is obtained as follows.
[0041]
The processing target data vector selection unit 3 selects a data vector [usually the latest data vector z (T)] to be processed and outputs it to the neighborhood vector detection unit 4.
The neighborhood vector detection unit 4 detects a plurality of neighborhood vectors x (i) in the vicinity of z (T) from the data vectors in the attractor reconstruction unit 2 [iεN (z (T)), N (z (T)): set of indices i in the neighborhood x (i) of z (T)].
[0042]
The axial component detection unit 6 detects axial components of z (T) and x (i) and outputs them to the fuzzy inference unit 7.
[0043]
The vector detection unit 5 after s step detects the vector x (i + s) after the s step for each x (i) and outputs it to the fuzzy inference unit 7.
[0044]
The fuzzy inference unit 7 compares the component values of z (T) and x (i) for each axis, and calculates the prediction vector z ″ (T + s) from the comparison result and the component value of each x (i + s). Each component value is determined, and the prediction vector z ″ (T + s) obtained thereby is output to the data prediction value generation unit 8.
[0045]
The trajectory of the attractor is usually a smooth manifold, and it is preferable to determine the axis component nonlinearly rather than obtaining the axis component linearly as described above.
[0046]
The axis component determination process using fuzzy theory is shown below.
[0047]
If the data is generated according to deterministic laws and the value of s is sufficiently small, the process of x (i) reaching x (i + s), which is data s steps ahead, is deterministic. It is thought to be based on the dynamism that was followed.
[0048]
And this dynamism can be expressed by the following linguistic expressions.
[0049]
[Equation 3]
IF x (T) is x (i) THEN x (T + s) is x (i + s)
Where x (T): a set representing a data vector near z (T) in the n-dimensional reconstruction state space
x (T + s): a set representing a data vector after s steps of x (T)
i∈N (z (T)), N (z (T)): set of indices i of neighborhood x (i) of z (T) x (i) is a data vector of neighborhood of z (T) If the step s is before losing deterministic causality due to the “sensitive dependence on the initial value” of chaos, the dynamics from the state z (T) to the state z (T + s) are changed from the state x (i). It can be assumed that the dynamics of the state x (i + s) is approximately equivalent.
[0050]
When the attractor embedded in the n-dimensional reconstructed state space is a smooth manifold, the vector distance from z (T) to z (T + s) depends on the vector distance from z (T) to x (i). Affected. That is, it can be considered that the closer the trajectory x (i) is to z (T), the greater the impact on the trajectory from z (T) to z (T + s), and the smaller the distance.
[0051]
by the way,
[0052]
[Expression 4]
x (i) = (y (i), y (i−τ),..., y (i− (n−1) τ))
x (i + s) = (y (i + s), y (i + s−τ),..., y (i + s− (n−1τ)) (1)
Therefore, focusing on the j-axis in the n-dimensional reconstructed state space, equation (1) is
[0053]
[Equation 5]
IF aj (T) is yj ( i) THEN aj (T + s) is y j (i + s) (j = 1~n) ... (2)
here,
aj (T): j-axis component in the n-dimensional reconstruction state space of the neighborhood value x (i) of z (T) aj (T + s): j-axis component in the n-dimensional reconstruction state space of x (j + s) n: embedded It can be expressed as the number of dimensions.
[0054]
The trajectory from z (T) to z (T + s) is affected by the vector distance from z (T) to x (i), but the attractor that is the trajectory of this vector is a smooth manifold. This effect is expressed in a non-linear form. Therefore, in order to make the influence non-linear, if equation (2) is expressed by a fuzzy function,
[0055]
[Formula 6]
IF aj (T) is y'j (i) THEN aj (T + s) is y'j (i + s)
However, (j = 1 to n) (3)
Normally, when the function y (i) is fuzzified, the “˜” symbol is used, but here the “′” symbol is used.
[0056]
Also,
[0057]
[Expression 7]
z (T) = (y (T), y (T−τ),..., y (T− (n−1) τ))
Therefore, the j-axis component in the n-dimensional reconstruction state space of z (T) is yj (T).
Therefore, when the predicted value of the data vector z (T + s) after s steps of the data vector z (T) is z ″ (T + s), the j-axis component is expressed as yj (T) in aj (T) of Equation (3). ) Can be obtained as aj (T + s) by substituting and fuzzy inference, and this method will be referred to as a "local fuzzy reconstruction method".
[0058]
As a specific example, a case where the embedding dimension n = 3, the delay time τ = 4, and the number N of data vectors included in the vicinity will be described below.
[0059]
Each data vector is
[0060]
[Equation 8]
z (T) = (y1 (T), y2 (T-4), y3 (T-8))
x (a) = (y1 (a), y2 (a-4), y3 (a-8))
x (b) = (y1 (b), y2 (b-4), y3 (b-8))
x (c) = (y1 (c), y2 (c-4), y3 (c-8))
z "(T + s) = (y1 (T + s), y2 (T + s-4), y3 (T + s-8))
x (a + s) = (y1 (a + s), y2 (a + s-4), y3 (a + s-8))
x (b + s) = (y1 (b + s), y2 (b + s-4), y3 (b + s-8))
x (c + s) = (y1 (c + s), y2 (c + s-4), y3 (c + s-8))
Then, the fuzzy rule shown by Formula (3) is represented as Formula (4) (5) (6).
[0061]
For the first axis of the reconstructed state space:
[0062]
[Equation 9]
IF a1 (T) is y'1 (a) THEN a1 (T + s) is y'1 (a + s)
IF a1 (T) is y'1 (b) THEN a1 (T + s) is y'1 (b + s)
IF a1 (T) is y'1 (c) THEN a1 (T + s) is y'1 (c + s) (4)
For the second axis of the reconstructed state space,
[0063]
[Expression 10]
IF a2 (T) is y'2 (a) THEN a2 (T + s) is y'2 (a + s)
IF a2 (T) is y'2 (b) THEN a2 (T + s) is y'2 (b + s)
IF a2 (T) is y'2 (c) THEN a2 (T + s) is y'2 (c + s) (5)
For the third axis of the reconstructed state space,
[0064]
[Expression 11]
IF a3 (T) is y'3 (a) THEN a3 (T + s) is y'3 (a + s)
IF a3 (T) is y'3 (b) THEN a3 (T + s) is y'3 (b + s)
IF a3 (T) is y'3 (c) THEN a3 (T + s) is y'3 (c + s) (6)
Also, since the membership functions are x (a), x (b), and x (c) are neighboring data vectors centered on z (T), the antecedents of fuzzy rules (4), (5), and (6) The membership function of each axis in the reconstruction state space in the section is as shown in FIG.
[0065]
The membership function of the consequent part may be a fuzzy function, but in this example, all are crisp expressions to improve the calculation speed.
[0066]
For the dynamics expressed by the above fuzzy rules and membership functions, a1 (T) = y1 (T), a2 (T) = y2 (T), and a3 (T) = y3 (T) are used as input data. When inferring,
[0067]
[Expression 12]
y ″ 1 (T + s) = a1 (T + s)
y "2 (T + s-4) = a2 (T + s)
y "3 (T + s-8) = a3 (T + s) (7)
Thus, the predicted value y ″ 1 (T + s) of s step ahead of the original time series data y1 (T) is obtained as a1 (T + s).
[0068]
Next, as a specific application example, an example in which the present apparatus is applied to the behavior prediction of a logistic map well known as time-series data that performs chaotic behavior will be described.
[0069]
The logistic map is shown by Robert May and is an irregular phenomenon obtained from the simple difference equation shown in Equation (8) , that is, a chaotic phenomenon supported by determinism.
[0070]
[Expression 14]
_{X n + 1 = f (X} n) = aX n (1-X n) ... (8)
(1 ≦ a ≦ 4)
In Equation (8), examining the X _n in the case where the n → ∞ by changing a,
When 1 ≦ a ≦ 2 and n → ∞, X _n converges to (1-1 / a), and when 2 ≦ a ≦ 3, when n → ∞, X _n vibrates to (1-1 / a) convergent 3 ≦ a ≦ 1 + √6, X n When n → ∞ does not converge (asymptotic to 2 cycles)
When 1 + √6 ≦ a ≦ 3.57 and n → ∞, X _n does not converge (asymptotic to 2 ^m period)
When n → ∞ with 3.57 ≦ a ≦ 4, X _n does not converge (chaos state)
It becomes. FIG. 6 shows time series data of the chaotic state of the logistic map, and FIG. 7 shows the behavior prediction result using this data.
[0071]
As shown in FIG. 7 (a), the correlation between the calculated value and the predicted value is very high, and as shown in FIG. 7 (c), the time series of the calculated value and the predicted value almost coincide with each other. Yes.
[0072]
Further, as shown in FIG. 7B, the correlation coefficient is 0.9 or more until the prediction step is about 7 steps, and it can be seen that a high correlation is obtained between the calculated value and the predicted value.
[0073]
In addition to the above logistic maps, deterministic nonlinear time series data using mathematical models such as time series data obtained from Lorenz's atmospheric circulation equation, water supply demand forecast, power demand forecast, gas demand forecast, expressway It can be applied to the prediction of social phenomena such as traffic prediction, the prediction of natural phenomena such as changes in temperature and changes in weather.
[0074]
【The invention's effect】
As described above, according to the present invention, it is possible to perform data prediction of an observation target controlled by deterministic chaos with high accuracy, and to shorten the time required for data prediction. Further, unlike the conventional method, it is not necessary to determine the order of the autoregressive equation, the correction coefficient due to various factors, and the like.
[0075]
Furthermore, prediction is possible even when it is difficult to linearly approximate time-series data, and since the reconstruction is performed locally, it is possible to prevent the prediction accuracy from being extremely deteriorated.
[Brief description of the drawings]
FIG. 1 is a functional block diagram of a nonlinear time series data prediction method .
FIG. 2 is an explanatory diagram of time-series data.
FIG. 3 is an explanatory diagram of a locus of a data vector.
FIG. 4 is an explanatory diagram of an attractor.
FIG. 5 is a graph showing a membership function.
FIG. 6 is an explanatory diagram of time-series data of a logistic map.
FIG. 7 is a graph showing a prediction result of a logistic map.
[Explanation of symbols]
DESCRIPTION OF SYMBOLS 1 ... Data vector production | generation part 2 ... Attractor reconstruction part 3 ... Processing object data vector production | generation part 4 ... Neighboring vector detection part 5 ... S step vector detection part 6 ... Axis component detection part 7 ... Fuzzy inference part 8 ... Data prediction value Generation unit 9: Local fuzzy reconstruction unit

Claims (2)

Sampling means for capturing time-series data to be measured at equal sampling intervals at the request of the data vector generation means;
Time series data storage means for storing time series data sampled by the sampling means;
Data vector generating means for reading out data from the time series data storage means and generating a data vector for performing deterministic short-term prediction based on deterministic nonlinear dynamical system theory;
An attractor reconstructing means for receiving a data vector generated by the data vector generating means, embedding it in an n-dimensional state space with an embedding dimension n and a delay time τ, and reconstructing the attractor of the data vector;
Local fuzzy reconstruction means for generating local dynamics of the attractor reconstructed by the attractor reconstruction means;
In a non-linear time series data prediction method having data prediction value generation means for generating a prediction value of data from a processing result in the local fuzzy reconstruction means,
The local fuzzy reconstruction means includes:
A processing target data vector selection procedure for selecting a processing target data vector from the data vector generation means and outputting it to a neighborhood vector detection procedure and an axis component detection procedure;
A neighborhood vector detection procedure for receiving a processing target data vector selected by the processing target data vector selection procedure and detecting a plurality of neighborhood vectors located in the vicinity of the processing subject data vector from the attractors reconstructed by the attractor reconstruction means When,
A vector detection procedure after a predetermined step for detecting a vector after a predetermined step of the plurality of neighboring vectors from the attractor reconstructing means and outputting the detected vector to an axis component determination procedure;
Each axis component of each vector is received from the processing target data vector selection procedure and the neighborhood vector detection procedure, and an axis component is determined by detecting a difference in each axis component value between the processing target data vector and a plurality of the neighborhood vectors. Axis component detection procedure output to the procedure,
A neighborhood vector with the smallest difference between the axis component values obtained from the axis component detection procedure is extracted, and the difference between the axis component values after a predetermined step between the neighborhood vector and the processing target data vector is reduced. An axis component determination procedure for determining each axis component value after a predetermined step of the processing target data vector,
A nonlinear time series data prediction method characterized by the above. The nonlinear time series data prediction method according to claim 1,
The axis component determination procedure generates a membership function for each axis of the neighborhood vector, and determines each axis component value after a predetermined step of the processing target data vector by fuzzy inference using the membership function. A nonlinear time series data prediction method characterized by

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