CN111832623A - Echo state network time sequence prediction algorithm based on phase space reconstruction - Google Patents

Echo state network time sequence prediction algorithm based on phase space reconstruction Download PDF

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CN111832623A
CN111832623A CN202010533889.5A CN202010533889A CN111832623A CN 111832623 A CN111832623 A CN 111832623A CN 202010533889 A CN202010533889 A CN 202010533889A CN 111832623 A CN111832623 A CN 111832623A
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夏亦犁
徐杰
裴文江
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Abstract

The invention discloses an echo state network time sequence prediction algorithm based on phase space reconstruction, which comprises the following steps: (1) determining two parameters of delay time and embedding dimension of phase space reconstruction, reconstructing a time sequence into a plurality of groups of data by utilizing the phase space reconstruction, and mapping the data from a low-dimensional space to a high-dimensional characteristic space; (2) determining parameters of an echo state network reserve pool, initializing an echo state network connection weight matrix, and constructing an echo state network model; (3) by utilizing the characteristic that the echo state network model can carry out multi-input, training and reconstructing a plurality of groups of data by using the model, and carrying out direct multi-step prediction and iterative multi-step prediction; (4) and directly fusing and iterating the multi-step prediction results by using weighted average to obtain a fused multi-step prediction value. The invention has better prediction precision in the multi-step prediction of the time sequence and can accurately predict the change trend of the time sequence.

Description

Echo state network time sequence prediction algorithm based on phase space reconstruction
Technical Field
The invention relates to the technical field of neural networks and time sequence analysis and prediction, in particular to an echo state network time sequence prediction algorithm based on phase space reconstruction.
Background
The traditional echo state network overcomes the problem of low training efficiency of the recurrent neural network, avoids the problems of local optimization and the like in the training process, and on the other hand, along with the deep research, some limitations of the traditional echo state network gradually appear, and in many cases, for some complex time sequence prediction problems, satisfactory prediction precision cannot be obtained by simply using the traditional echo state network. Therefore, it is necessary to combine the echo state network with other prediction or signal preprocessing methods, and to propose a combined echo state network prediction method to improve the prediction accuracy.
Time series multi-step prediction estimates the sequence value for a longer period of time in the future based on the current value of the sequence, which is clearly more challenging than single-step prediction, which only needs to estimate the sequence value of the next step based on the current value of the sequence.
Disclosure of Invention
The technical problem to be solved by the invention is to provide an echo state network time sequence prediction algorithm based on phase space reconstruction, which can be applied to time sequence analysis and prediction.
In order to solve the above technical problem, the present invention provides an echo state network time series prediction algorithm based on phase space reconstruction, which comprises the following steps:
(1) determining two parameters of delay time and embedding dimension of phase space reconstruction, reconstructing a time sequence into a plurality of groups of data by utilizing the phase space reconstruction, and mapping the data from a low-dimensional space to a high-dimensional characteristic space;
(2) determining parameters of an echo state network reserve pool, initializing an echo state network connection weight matrix, and constructing an echo state network model;
(3) by utilizing the characteristic that the echo state network model can carry out multi-input, training and reconstructing a plurality of groups of data by using the model, and carrying out direct multi-step prediction and iterative multi-step prediction;
(4) and directly fusing and iterating the multi-step prediction results by using weighted average to obtain a fused multi-step prediction value.
Preferably, in the step (1), reconstructing the time series into a plurality of sets of data by using phase space reconstruction includes the following steps:
(11) with two discrete time sequences s1,…,s1}，{q1,…,q1What is needed isThe information entropy of the system S and the system Q is as follows:
Figure BDA0002536355240000021
Figure BDA0002536355240000022
in the above formula PS(si)，PQ(qi) Respectively representing events S in S and QiAnd q isiThe probability of (d);
recording events siAnd q isiHas a joint distribution probability of PS,Q(si,qi) The joint information entropy of system S and system Q is as follows:
Figure BDA0002536355240000023
a mutual information calculation formula of the systems S and Q can be obtained:
I(S,Q)＝H(S)+H(Q)-H(S,Q)
(12) providing a set of time sequences (u (1), …, u (t), …, u (N)), wherein N is the sequence length;
defining (S, Q) ═ u (I), (I + τ)), (1 ≦ I ≦ N- τ), where S represents u (I) and Q represents u (I + τ), then I (S, Q) in step (11) is a function of the delay time τ, denoted as I (τ);
the delay time tau is calculated by the following steps: first, τ is taken as a value densely, and mutual information corresponding to different τ values is respectively calculated:
I(τ)＝H(x)+H(x+τ)-H(x,x+τ)
when I (tau) reaches a minimum value point, the corresponding tau is the delay time of phase space reconstruction;
(13) constructing a vector yi(d) (i + (d-1) τ)), (1 ≦ i ≦ N- (d-1) τ), wherein τ is the delay time in step (11);
yi(d) there is a closest point of approach yn(i,d)(d) (N (i, d) ≠ i, N (i, d) ≦ N- (d-1) τ) 1, and the Euclidean distance between two points is as follows:
Ri(d)＝||yi(d)-yn(i,d)(d)||2
when the dimension increases to d +1, the distance between two points in the new space changes, the new distance being Ri(d +1), and
Figure BDA0002536355240000024
when the dimension becomes larger, the Euclidean distance between two points also becomes larger, such that
Figure BDA0002536355240000031
If a is1(i,d)＞RτThen yn(i,d)(d) Is yi(d) A false closest point of approach, here RτIs a threshold value;
calculating the proportion of the false nearest points to the whole data volume from the minimum value 2 of the dimension, and then gradually increasing the dimension m until the proportion of the false nearest points is less than 5%, wherein the dimension m is the embedding dimension;
(14) the time sequence { u (N) ═ 1,2, …, N }, after phase space reconstruction, an m-dimensional data combination can be obtained:
u(n)＝[u(n),u(n+τ),…,u(n+(m-1)τ)]。
preferably, the step (2) specifically comprises the following steps:
(21) the predictive effect of the echo state network is influenced by pool parameters including: the method comprises the following steps of (1) obtaining a reserve pool sparsity degree SD, a reserve pool unit scale IS, a reserve pool scale N and a spectrum radius SR of a reserve pool internal connection weight;
(22) in the initialization phase, the input connection right between the input layer and the reserve pool is randomly generated
Figure BDA0002536355240000032
Connection right W inside reserve tankresThe weight matrixes are always fixed and do not need to be updated through training, and the feedback connection weights between the reserve pool and the output layer
Figure BDA0002536355240000033
Is set to 0.
Preferably, the step (3) specifically comprises the following steps:
(31) setting the network input and output of the echo state as u (n) and y (n), respectively, the internal state updating equation of the reserve pool is as follows:
Figure BDA0002536355240000034
wherein f is an activation function of a reserve pool node, and specifically a hyperbolic tangent function is used;
updating of output connection right requires internal state vector data support, and internal state vectors (x) in the reserve pool are collected at each moment1(i),x2(i),…,xN(i) Group to form a matrix B; collecting output values y (n) at each moment to form a matrix T;
the model output equation is as follows:
y(t+1)＝Wout(xT(n),u(n),y(n))
in the formula WoutIs the output connection right;
to make the model predict the value
Figure BDA0002536355240000041
Minimum mean square error between the sum of the true values y (n), WoutThe update is to solve the following formula:
Figure BDA0002536355240000042
solving the above formula by a least square method, the following can be obtained:
(Wout)T＝B-1T
(32) direct multi-step prediction is that a multi-step predictor is directly constructed to satisfy
Figure BDA0002536355240000043
In the formula, h represents a prediction time interval, u (n) represents reconstructed data, u (n) is used as the model input in the step (31), and u (n + h) is used as the output to obtain a predicted value;
(33) iterative multi-step prediction firstly determines a single-step predictor satisfying
Figure BDA0002536355240000044
Taking u (n) as the model input in the step (31) and u (n +1) as the output, the predicted value at the next moment is obtained
Figure BDA0002536355240000045
Combining n data from the time 2 to the time n +1 to obtain an input value u (n +1) at the time n + 1; the above formula is used for multiple times to carry out single step prediction, the predicted value of a future time point is obtained each time, and after iteration is carried out for h times, the h-step predicted value of a time sequence is obtained
Figure BDA0002536355240000046
Preferably, in the step (4), a plurality of predicted values obtained in the step (3) are fused, that is
Figure BDA0002536355240000047
In the formula, f (-) represents a fusion algorithm, and a fusion algorithm with lower complexity is used, namely a plurality of predicted values are weighted and averaged:
Figure BDA0002536355240000048
a1、a2representing the weight.
The invention has the beneficial effects that: (1) the echo state network time sequence prediction algorithm based on phase space reconstruction can accurately predict the time sequence change trend; the phase space reconstruction is combined with the echo state network, the phase space reconstruction has good nonlinear analysis capability in processing the time sequence, and the echo state network has the advantages of low algorithm complexity, difficult falling into local optimum and the like in processing the time sequence, so that the problem of poor prediction accuracy of the existing model to the time sequence is solved; (2) experimental data show that when the prediction step length is 5, the average absolute error of the method is improved by 20% compared with the traditional echo state network; when the predicted step length is 15%, the average absolute error of the method is improved by 40% compared with the traditional echo state network; the invention has better prediction precision in the multi-step prediction of the time sequence and can accurately predict the change trend of the time sequence.
Drawings
FIG. 1 is a schematic diagram of the algorithm flow of the present invention.
Fig. 2 is a schematic diagram of an echo state network model according to the present invention.
FIG. 3 is a schematic diagram of the multi-step prediction error of the present invention.
Detailed Description
As shown in fig. 1, an echo state network time series prediction algorithm based on phase space reconstruction includes the following steps:
(1) determining two parameters of delay time and embedding dimension of phase space reconstruction, reconstructing a time sequence into a plurality of groups of data by utilizing the phase space reconstruction, and mapping the data from a low-dimensional space to a high-dimensional characteristic space;
(2) determining parameters of an echo state network reserve pool, initializing an echo state network connection weight matrix, and constructing an echo state network model;
(3) by utilizing the characteristic that the echo state network can input multiple inputs, training and reconstructing multiple groups of data by using a model, and performing direct multi-step prediction and iterative multi-step prediction;
(4) and directly fusing and iterating the multi-step prediction results by using weighted average to obtain a fused multi-step prediction value.
In the embodiment, the echo state network time sequence prediction algorithm based on phase space reconstruction is applied to voltage time sequence prediction to show the effectiveness of the echo state network time sequence prediction algorithm, and the advantage of the echo state network time sequence prediction algorithm is shown by comparing the prediction accuracy with the prediction accuracy of the traditional echo state network algorithm.
FIG. 1 is a drawing ofAnd (3) predicting the time sequence of the echo state network based on phase space reconstruction. FIG. 2 is a block diagram of an echo state network in which the output connection W from the pool to the output layeroutIs the only weight that needs to be updated by training, and WoutThe updating of (1) is to solve a simple linear regression problem, and the calculation amount is extremely small. The echo state network has a simpler structure than the traditional recurrent neural network and has an extremely simple network training process. Fig. 3 is a voltage multi-step prediction error graph including a 5-step prediction error, a 10-step prediction error, and a 15-step prediction error, where the abscissa represents a predicted time point and the ordinate represents an error value. It can be seen that the time sequence prediction effect of the echo state network model based on the phase space reconstruction is good, and the predicted value is consistent with the real value.
Table 1 is a comparison diagram of the voltage multi-step prediction accuracy between the method of the present invention and the conventional echo state network in this embodiment. Therefore, the method is obviously superior to the traditional echo state network method, and the prediction precision is improved more obviously along with the increase of the prediction step length. For example, when the prediction is carried out in 15 steps, the prediction accuracy of the method is improved by 40% compared with that of the traditional echo state network.
TABLE 1 comparison of voltage multistep prediction accuracy between the method of the present invention and the conventional echo state network
Figure BDA0002536355240000061

Claims (5)

1. An echo state network time sequence prediction algorithm based on phase space reconstruction is characterized by comprising the following steps:
(1) determining two parameters of delay time and embedding dimension of phase space reconstruction, reconstructing a time sequence into a plurality of groups of data by utilizing the phase space reconstruction, and mapping the data from a low-dimensional space to a high-dimensional characteristic space;
(2) determining parameters of an echo state network reserve pool, initializing an echo state network connection weight matrix, and constructing an echo state network model;
(3) by utilizing the characteristic that the echo state network model can carry out multi-input, training and reconstructing a plurality of groups of data by using the model, and carrying out direct multi-step prediction and iterative multi-step prediction;
(4) and directly fusing and iterating the multi-step prediction results by using weighted average to obtain a fused multi-step prediction value.
2. The echo state network time series prediction algorithm based on phase space reconstruction as claimed in claim 1, wherein in the step (1), the phase space reconstruction is used to reconstruct the time series into a plurality of groups of data, comprising the following steps:
(11) with two discrete time sequences s1,…,s1}，{q1,…,q1And information entropy of the system S and the system Q is as follows:
Figure FDA0002536355230000011
Figure FDA0002536355230000012
in the above formula PS(si)，PQ(qi) Respectively representing events S in S and QiAnd q isiThe probability of (d);
recording events siAnd q isiHas a joint distribution probability of PS,Q(si,qi) The joint information entropy of system S and system Q is as follows:
Figure FDA0002536355230000013
a mutual information calculation formula of the systems S and Q can be obtained:
I(S,Q)＝H(S)+H(Q)-H(S,Q)
(12) providing a set of time sequences (u (1), …, u (t), …, u (N)), wherein N is the sequence length;
defining (S, Q) ═ u (I), (I + τ)), (1 ≦ I ≦ N- τ), where S represents u (I) and Q represents u (I + τ), then I (S, Q) in step (11) is a function of the delay time τ, denoted as I (τ);
the delay time tau is calculated by the following steps: first, τ is taken as a value densely, and mutual information corresponding to different τ values is respectively calculated:
I(τ)＝H(x)+H(x+τ)-H(x,x+τ)
when I (tau) reaches a minimum value point, the corresponding tau is the delay time of phase space reconstruction;
(13) constructing a vector yi(d) (i + (d-1) τ)), (1 ≦ i ≦ N- (d-1) τ), wherein τ is the delay time in step (11);
yi(d) there is a closest point of approach yn(i,d)(d) (N (i, d) ≠ i, N (i, d) ≦ N- (d-1) τ) 1, and the Euclidean distance between two points is as follows:
Ri(d)＝||yi(d)-yn(i,d)(d)||2
when the dimension increases to d +1, the distance between two points in the new space changes, the new distance being Ri(d +1), and
Figure FDA0002536355230000021
when the dimension becomes larger, the Euclidean distance between two points also becomes larger, such that
Figure FDA0002536355230000022
If a is1(i,d)＞RτThen yn(i,d)(d) Is yi(d) A false closest point of approach, here RτIs a threshold value;
calculating the proportion of the false nearest points to the whole data volume from the minimum value 2 of the dimension, and then gradually increasing the dimension m until the proportion of the false nearest points is less than 5%, wherein the dimension m is the embedding dimension;
(14) the time sequence { u (N) ═ 1,2, …, N }, after phase space reconstruction, an m-dimensional data combination can be obtained:
u(n)＝[u(n),u(n+τ),…,u(n+(m-1)τ)]。
3. the echo state network time series prediction algorithm based on phase space reconstruction as claimed in claim 1, wherein the step (2) specifically comprises the following steps:
(21) the predictive effect of the echo state network is influenced by pool parameters including: the method comprises the following steps of (1) obtaining a reserve pool sparsity degree SD, a reserve pool unit scale IS, a reserve pool scale N and a spectrum radius SR of a reserve pool internal connection weight;
(22) in the initialization phase, the input connection right between the input layer and the reserve pool is randomly generated
Figure FDA0002536355230000031
Connection right W inside reserve tankresThe weight matrixes are always fixed and do not need to be updated through training, and the feedback connection weights between the reserve pool and the output layer
Figure FDA0002536355230000032
Is set to 0.
4. The echo state network time series prediction algorithm based on phase space reconstruction as claimed in claim 1, wherein the step (3) comprises the following steps:
(31) setting the network input and output of the echo state as u (n) and y (n), respectively, the internal state updating equation of the reserve pool is as follows:
Figure FDA0002536355230000033
wherein f is an activation function of a reserve pool node, and specifically a hyperbolic tangent function is used;
updating of output connection right requires internal state vector data support, and internal state vectors (x) in the reserve pool are collected at each moment1(i),x2(i),…,xN(i) Group to form a matrix B; collecting output values y (n) at each moment to form a matrix T;
the model output equation is as follows:
y(t+1)＝Wout(xT(n),u(n),y(n))
in the formula WoutIs the output connection right;
to make the model predict the value
Figure FDA0002536355230000034
Minimum mean square error between the sum of the true values y (n), WoutThe update is to solve the following formula:
Figure FDA0002536355230000035
solving the above formula by a least square method, the following can be obtained:
(Wout)T＝B-1T
(32) direct multi-step prediction is that a multi-step predictor is directly constructed to satisfy
Figure FDA0002536355230000036
In the formula, h represents a prediction time interval, u (n) represents reconstructed data, u (n) is used as the model input in the step (31), and u (n + h) is used as the output to obtain a predicted value;
(33) iterative multi-step prediction firstly determines a single-step predictor satisfying
Figure FDA0002536355230000041
Taking u (n) as the model input in the step (31) and u (n +1) as the output, the predicted value at the next moment is obtained
Figure FDA0002536355230000042
Combining n data from the time 2 to the time n +1 to obtain an input value u (n +1) at the time n + 1; the above formula is used for multiple times to carry out single step prediction, the predicted value of a future time point is obtained each time, and after iteration is carried out for h times, the h-step predicted value of a time sequence is obtained
Figure FDA0002536355230000043
5. The echo state network time series prediction algorithm based on phase space reconstruction as claimed in claim 1, characterized in that in step (4), the multiple predicted values obtained in step (3) are fused, i.e. the multiple predicted values are fused
Figure FDA0002536355230000044
In the formula, f (-) represents a fusion algorithm, and a fusion algorithm with lower complexity is used, namely a plurality of predicted values are weighted and averaged:
Figure FDA0002536355230000045
a1、a2representing the weight.
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Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN114118373A (en) * 2021-11-25 2022-03-01 福州大学 Multi-dimensional time sequence missing completion method based on echo state network
CN114613372A (en) * 2022-02-21 2022-06-10 北京富通亚讯网络信息技术有限公司 Error concealment technical method for preventing packet loss in audio transmission

Cited By (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN114118373A (en) * 2021-11-25 2022-03-01 福州大学 Multi-dimensional time sequence missing completion method based on echo state network
CN114613372A (en) * 2022-02-21 2022-06-10 北京富通亚讯网络信息技术有限公司 Error concealment technical method for preventing packet loss in audio transmission
CN114613372B (en) * 2022-02-21 2022-10-18 北京富通亚讯网络信息技术有限公司 Error concealment technical method for preventing packet loss in audio transmission

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