CN109344995B - Multi-step prediction method of chaotic time sequence based on density peak clustering - Google Patents

Multi-step prediction method of chaotic time sequence based on density peak clustering Download PDF

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CN109344995B
CN109344995B CN201810992936.5A CN201810992936A CN109344995B CN 109344995 B CN109344995 B CN 109344995B CN 201810992936 A CN201810992936 A CN 201810992936A CN 109344995 B CN109344995 B CN 109344995B
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谢国波
姚灼琛
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Abstract

The invention discloses a multi-step prediction method of a chaotic time sequence based on density peak clustering, which comprises the steps of selecting adjacent tracks for multiple times, determining phase point boundaries through a clustering algorithm, reselecting nearest neighbor tracks in a phase point simulation mode when the tracks enter the phase point boundaries, changing the conventional method that the adjacent tracks are obtained for multiple times according to target phase points once, dynamically distributing the boundaries of the phase points without artificially determining a clustering center by combining the clustering algorithm, reducing the influence of human factors, and improving the problem of multi-step prediction accuracy reduction caused by the conventional model that the tracks are obtained for one time.

Description

Multi-step prediction method of chaotic time sequence based on density peak clustering
Technical Field
The invention relates to a time sequence prediction method, in particular to a multi-step prediction method of a chaotic time sequence based on density peak value clustering.
Background
The chaos is apparent stochastic motion generated by a determined nonlinear power system, and has the characteristics of sensitivity to initial conditions, long-term difficulty in prediction and the like; the chaotic time sequence prediction has wide application prospect. Based on the Takens phase space reconstruction theory, a plurality of scholars put forward various chaos time sequence prediction models, and the Volterra filter is widely concerned by scholars at home and abroad due to the advantages of high training speed, strong nonlinear approximation capability, high prediction precision and the like. Through a series of researches on multi-step prediction of a high-dimensional chaotic time sequence by using a Volterra filter, the Volterra filter solves a kernel function through system identification, and numerical approximation is carried out on a chaotic orbit of a phase space so as to realize prediction, and a Volterra filter model is verified to be capable of accurately predicting a low-dimensional chaotic system.
The phase space reconstruction theory is the basis of chaos time sequence prediction, Packard et al and Takens propose to carry out phase space reconstruction on chaos time sequence by a delay coordinate method, and the Takens theorem proves that if an embedding dimension m is more than or equal to 2d +1, d is a system dynamics dimension, a reconstructed power system is equivalent to a prime power system in a topological sense, and chaotic attractors in 2 phase spaces are differentiated and identical. The essence of the chaos time series prediction is an inverse problem of the power system, namely, a dynamic model of the system is reconstructed through the state of the power system.
At present, research work on a Volterra filter mostly stays in enhancing the adaptivity of the Volterra filter, but the local prediction model still has too high dependence on the initial prediction point, the precision of the local prediction model is often large in fluctuation range aiming at different prediction starting points, and the reliability still needs to be further improved.
Disclosure of Invention
The invention aims to provide a multi-step prediction method of a chaotic time sequence based on density peak clustering, which is used for solving the problems of high dependence on a prediction starting point and low reliability of the existing prediction model and is used for the research process of a weather convection model and the like.
A multi-step prediction method of a chaos time sequence based on density peak clustering comprises the following steps:
step 1, performing phase space reconstruction aiming at a chaotic time sequence to obtain a phase point training set consisting of phase points;
clustering the phase point training set;
step 2, taking the last phase point in the phase point training set as a target phase point, traversing the orbit of the target phase point and the orbits of other phase points in the cluster from the cluster where the target phase point is located, judging the orbit similarity xi of the orbit of the target phase point and the orbits of other phase points, selecting two orbits which are positioned at two sides of the orbit of the target phase point and have the highest orbit similarity as a nearest orbit LA1And LB1；
At said nearest track LA1And LB1Respectively selecting the phase points A closest to the target phase point1And phase point B1Drawing a straight line A through the target phase point1B1Perpendicular line of (1) with a foot of N1Is a reaction of N1As the simulated phase point and in place of the target phase point, record A1N1And A1B1The ratio of (A) to (B) P;
step 3, when the nearest track LA1And LB1When entering the next cluster of the cluster, the nearest tracks L are respectively usedA1And LB1The starting phase point in the cluster is taken as the closest phase point A2And phase point B2According to said ratio P, using the phase point A2And phase point B2Calculating a new simulation phase point N2Then, the simulation phase point N is screened by calculating the similarity of the orbit2Of the nearest track LA2And LB2；
And 4, repeating the step 3 until all the clusters are traversed, and obtaining a nearest track set L ═ L { L } composed of all the screened nearest tracksAi，L Bi1,2, …, n, n is the total number of the nearest tracks; then sequentially splicing all adjacent nearest tracks according to the traversal sequence to obtain two complete nearest tracks L after splicingA＝LA1+LA2+…+LAn、LB＝LB1+LB2+…+LBn；
Step 5, approaching the track LA、LBAnd forming a training set by all the phase points, and training by a Volterra filter to obtain a multi-step predicted value of the time sequence.
Further, clustering the phase point training set, including:
step 1.1, calculating Euclidean distance d between any two phase points i and j in the phase point training setij；
Step 1.2, selecting and calculating distance
All Euclidean distances d calculated for step 1.1ijSorting the sequence in ascending order, setting a parameter percentage, calculating the distance dcIs the number of sequences in percentage of the parameter.
Step 1.3, determining the phase point of the clustering center
Finding out the distance between the phase point training set and the phase point i to be less than dcThen finding the value with the minimum Euclidean distance from the phase point i in the phase points with the density larger than that of the phase point i, and determining the phase point of the clustering center;
step 1.4, distributing the remaining phase points except the clustering center phase point, wherein the cluster to which the remaining phase points belong is the cluster of the clustering center phase point which is the nearest and has the density larger than that of the clustering center phase point;
and step 1.5, after the phase points are distributed, if isolated clustering center phase points exist, distributing the clustering center phase points as residual phase points according to the method in the step 1.4, thereby completing the clustering.
Further, the method for calculating the track similarity ξ i in the step 2 is as follows:
let XMFor the target phase point, { XiI-1, 2, …, k is k neighboring phase points within the neighborhood of the target phase point and limits temporal separation, i.e., XiDifferent chaotic orbits are classified; xM-pAnd Xi-pAre each XMAnd XiBacktracking a pth step point, p ═ 1,2, …; in the formula [. C]Represents the inner product of the vector, | | represents the distance norm of the vector, | represents the absolute value, and the target phase point X is calculated by the formula 3MAnd neighbor phase point XiDistance similarity of (2):
Figure GDA0003058921840000031
calculating a target phase point XMAnd neighbor point XiThe following formula is applied to the direction similarity of (1):
Figure GDA0003058921840000032
calculating the track similarity speed xi i:
Figure GDA0003058921840000033
in the above formula, j is a step length parameter, and q is a backtracking step length.
Further, the percentage of the parameters is 5%, and 5 cluster center phase points are selected.
Further, the value of q is 2.
Compared with the prior art, the invention has the following technical characteristics:
1. the method firstly modifies the existing screening standard of the similarity of the adjacent tracks of the phase space in a small range, provides a method for effectively blocking the training set based on the density peak value clustering algorithm, and further reduces the possibility of mistakenly selecting the pseudo adjacent tracks by adopting a method for screening the adjacent tracks in a segmented manner compared with the existing mode of selecting the adjacent tracks at a single time.
2. The method adopts a simulation form to construct the possible position of the target phase point of each cluster according to the space geometric characteristics of the target phase point, and realizes the selection of the adjacent track of each cluster by using the simulation phase point to replace the target phase point.
3. Experiments show that compared with the conventional multi-step prediction filter model, the method has the advantages of more excellent generalization performance, less influence from a prediction starting point, less required training set, good practical use value in the research of weather convection models and the like, and improvement on the prediction accuracy of the weather convection model.
Drawings
FIG. 1 is a phase point decision diagram;
FIG. 2 is a schematic diagram of the determination of the simulated phase point in step 2;
FIG. 3 is a diagram of clustering effect in a simulation experiment;
FIG. 4 is a multi-step prediction graph of Lorenz chaotic time series, wherein (a) is the multi-step prediction graph of the method of the present invention, (b) is the multi-step prediction graph of a global prediction model, and (c) is the multi-step prediction graph of an RBF neural network model;
FIG. 5 is a schematic flow chart of the method of the present invention.
Detailed Description
According to the method, the chaos time sequence is reconstructed according to a phase space reconstruction theory to obtain a phase point training set, then effective phase points capable of describing the next change of the time sequence are screened out and used as a filter for training, and therefore the predicted value of the chaos time sequence is obtained. The method comprises the following specific steps:
step 1, aiming at a chaotic time sequence, performing phase space reconstruction by adopting a delay coordinate method to obtain a phase point training set consisting of phase points; and then clustering the phase point training set to obtain a clustered training set.
In the embodiment, too many clustering centers not only affect the clustering efficiency, but also bring obstruction in the process of screening the nearest track, and through experiments, 5 clustering centers are obtained to be more suitable; the clustering algorithm in this embodiment is as follows:
step 1.1, calculating Euclidean distance d between any two phase points i and j in the phase point training setij；
Step 1.2, selecting and calculating distance
All Euclidean distances d calculated for step 1.1ijSorting the sequence in ascending order, setting a parameter percentage, calculating the distance dcIs the number of sequences in percentage of the parameter.
For example, in this embodiment, the percentage of the component is 5%, if dijA total of 200, the distance d is calculatedcThe 200 th × 5% of the sequence is 10 Euclidean distances dij。
Step 1.3, determining the phase point of the clustering center
Figure GDA0003058921840000041
Wherein:
Figure GDA0003058921840000042
Figure GDA0003058921840000043
the purpose of equation 1 is to find the number of phase points in the phase point training set whose distance from the phase point i is less than dc, and the purpose of equation 2 is to find the density rho of the phase point iiOf the large phase points, the value at which the euclidean distance from the phase point i is smallest. And δ for the highest density phase pointiThe maximum distance from all phase points to phase point i.
According to formula 1 and formula2, is constructed by rhoiIs the abscissa, deltaiAs shown in fig. 1, several independent points have a larger ρiAnd deltaiTherefore, these phase points are selected as the cluster centers. In this embodiment, the 5 facies points farthest from the (0,0) point in euclidean distance are used as the cluster center facies points.
Step 1.4, distributing the remaining phase points except the phase point of the center of the cluster, wherein the cluster to which the phase point belongs is the nearest neighbor and has the density rhoiClusters of cluster center phase points larger than this.
And step 1.5, after the phase points are distributed, possibly having no other phase points except the clustering center phase point in the clustering, and distributing the clustering center phase point as the residual phase point according to the method in the step 1.4, thereby finishing the clustering.
Clustering the phase point training set;
step 2, in the step, firstly, a target phase point is determined, then traversal is carried out from the cluster where the target phase point is located, and the nearest tracks L of two clamping and approaching target phase points (namely the two tracks are respectively located at the two sides of the target phase point track) are determined through the calculation of the track similarity xiA1And LB1(ii) a The calculation formula of the track similarity ξ i is derived as follows:
let XMFor the target phase point, { XiI-1, 2, …, k is k neighboring phase points within the neighborhood of the target phase point and limits temporal separation, i.e., XiDifferent chaotic orbits are classified; xM-pAnd Xi-pAre each XMAnd XiBacktracking a pth step point, p ═ 1,2, …; in the formula [. C]Represents the inner product of the vector, | | represents the distance norm of the vector, | represents the absolute value, and the target phase point X is calculated by the formula 3MAnd neighbor phase point XiDistance similarity of (2):
Figure GDA0003058921840000051
from equation 3, the distance similarity
Figure GDA0003058921840000052
And is
Figure GDA0003058921840000053
Larger means that the neighboring phase point is closer to the target phase point.
When calculating the track similarity, the vector directions of the track similarity are required to be determined to be approximately the same; calculating a target phase point XMAnd neighbor point XiThe following formula is applied to the direction similarity of (1):
Figure GDA0003058921840000054
equation 4 represents the directional relationship between vectors in phase space, and
Figure GDA0003058921840000064
；
Figure GDA0003058921840000065
the minimum value 0 is taken when the vectors are orthogonal.
After the track distance similarity and the direction similarity of the target phase point and the adjacent phase point are obtained through the formula 3 and the formula 4, the intersection of the target phase point and the adjacent phase point is taken, and the intersection is the track similarity xi i:
Figure GDA0003058921840000061
in the above equation, j is a step parameter, and q is a backtracking step, so that the objective is to find nearest neighboring tracks with similar directions, so q is taken to be 2. To examine the similarity of the tracks, for XMAnd XiAnd the backtracking orbit similarity of the p steps is measured, the phase point with smaller backtracking step length has larger prediction effect on the target phase point, so the backtracking phase points in the Euclidean distance and the vector direction are respectively weighted, and the weighting coefficients are respectively
Figure GDA0003058921840000062
And
Figure GDA0003058921840000063
meanwhile, the existing method of weighting the Euclidean distance similarity and the vector direction similarity is used as the track similarity.
However, the above method does not take into account when XiAnd XMThe Euclidean distance direction included angles are greatly different, and at the moment, if the number of the training tracks is large enough, the possibility of selection is provided for 'pseudo adjacent tracks', so that the prediction result is influenced. Therefore, the invention comprises the following steps:
taking the last phase point in the phase point training set as a target phase point, traversing the track of the target phase point and the tracks of other phase points in the cluster from the cluster where the target phase point is located, judging the track similarity xi of the target phase point track and the tracks of other phase points (calculating the other phase points as neighbor phase points by formulas 3 to 5), and selecting two tracks which are positioned at two sides of the target phase point track and have the highest track similarity as a nearest track LA1And LB1(ii) a That is, the nearest neighbor orbit is the largest two orbits similarity selected from all orbits similarity, and the two orbits also need to approach the target phase point, that is, located on both sides of the target phase point.
After the nearest track of the target phase point is determined, the target phase point is replaced by the simulation phase point in the scheme, and the segmented nearest track is sequentially determined, wherein the method comprises the following steps of:
at said nearest track LA1And LB1Respectively selecting the phase points A closest to the target phase point1And phase point B1Wherein phase point A1Is the nearest track LA1Phase point of (3), phase point B1Is the nearest track LB1Phase point of (3);
line A is drawn by passing the target phase point1B1Perpendicular line of (1) with a foot of N1Is a reaction of N1As the simulated phase point and in place of the target phase point, record A1N1And A1B1Ratio P, P ═ A1N1：A1B1(ii) a As shown in fig. 2.
Step 3, when the nearest track LA1And LB1When entering the next cluster of the cluster, the nearest tracks L are respectively usedA1Starting point phase point and L in the clusterB1The starting phase point in the cluster (i.e., the first phase point of the closest track in the cluster) is taken as the closest a2And phase point B2Calculating a new simulated phase point N according to the ratio P2I.e. P ═ A2N2：A2B2From which the analog phase point N can be determined2The position of (a);
then according to (simulation phase point N)2With other phase points in the cluster) orbit similarity xi, screening the simulation phase point N2Of the nearest track LA2And LB2The specific method is the same as that in step 2 for determining the nearest track of the target phase point.
And 4, repeating the step 3 until all the clusters are traversed, and obtaining a nearest track set L ═ L { L } composed of all the screened nearest tracksAi，L Bi1,2, …, n, n is the total number of the nearest tracks;
then sequentially splicing all adjacent nearest tracks according to the traversal sequence to obtain two complete nearest tracks L after splicingA＝LA1+LA2+…+LAn、LB＝LB1+LB2+…+LBn；
Step 5, approaching the track LA、LBAnd forming a training set by all the phase points, and training by a Volterra filter to obtain a multi-step predicted value of the time sequence.
The method selects the adjacent track for multiple times, determines the phase point boundary through the clustering algorithm, reselects the nearest track through the mode of simulating the phase point when the track enters the phase point boundary, thereby obtaining the adjacent track for multiple times, combines the clustering algorithm, artificially determines the clustering center, dynamically allocates the boundary of the phase point, reduces the influence of artificial factors, and improves the problem of the reduction of the multi-step prediction precision of the existing method.
Simulation experiment:
the algorithm is realized in a simulation mode under Matlab 2016a software, wherein the clustering effect is shown in the following figure 3.
Since the main purpose of clustering is to divide the training set into blocks so as to select the nearest track in segments, the training set is only required to be divided into a plurality of cluster classes along with the track.
In order to compare with multi-step predictions of other methods such as an RBF neural network model, a global prediction model and the like, the method uniformly adopts the x component of the Lorenz chaotic system as initial iteration with 12000 under the noise-free level, performs phase space reconstruction with a subsequent 3000 chaotic sequence, generates 2980 phase points in total and predicts the final 2980 phase point as a target phase point. The Lorenz chaotic time series multi-step prediction graph is shown in FIG. 4.
As can be seen from (a), (b) and (c) of FIG. 4, the effective step size of the global prediction model is less than 50 steps, the effective step size of the RBF neural network model is about 80 to 90 steps, the accuracy of the invention is very ideal at cycle lengths of 1-67, and the deviation begins to appear at cycle lengths of 68-134, but the total value is in accordance with the true value.
Compared with the prior art, the method expands the effective precision of local area prediction from one time of quasi-cycle length to two times of quasi-cycle length, namely, from 67 steps to 134 steps, and simultaneously, the required training set is also reduced from the original 6500 iterations to 3000 times. Most importantly, the method improves the problem that the accuracy of the original model is greatly reduced at different prediction starting points, and effectively improves the stability of the local multi-step prediction accuracy.

Claims (5)

1. A multi-step prediction method of a chaos time sequence based on density peak clustering is characterized by comprising the following steps:
step 1, performing phase space reconstruction aiming at a chaotic time sequence to obtain a phase point training set consisting of phase points;
clustering the phase point training set;
step 2, taking the last phase point in the phase point training set as a target phase point, traversing the orbit of the target phase point and the orbits of other phase points in the cluster from the cluster where the target phase point is located, judging the orbit similarity xi of the orbit of the target phase point and the orbits of other phase points, and selecting two orbits which are positioned at two sides of the orbit of the target phase point and have the highest orbit similarity as the target phase pointIs the nearest track LA1And LB1；
At said nearest track LA1And LB1Respectively selecting the phase points A closest to the target phase point1And phase point B1Drawing a straight line A through the target phase point1B1Perpendicular line of (1) with a foot of N1Is a reaction of N1As the simulated phase point and in place of the target phase point, record A1N1And A1B1The ratio of (A) to (B) P;
step 3, when the nearest track LA1And LB1When entering the next cluster of the cluster, the nearest tracks L are respectively usedA1And LB1The starting phase point in the cluster is taken as the closest phase point A2And phase point B2According to said ratio P, using the phase point A2And phase point B2Calculating a new simulation phase point N2Then, the simulation phase point N is screened by calculating the similarity of the orbit2Of the nearest track LA2And LB2；
And 4, repeating the step 3 until all the clusters are traversed, and obtaining a nearest track set L ═ L { L } composed of all the screened nearest tracksAi，LBi1,2, …, n, n is the total number of the nearest tracks; then sequentially splicing all adjacent nearest tracks according to the traversal sequence to obtain two complete nearest tracks L after splicingA＝LA1+LA2+…+LAn、LB＝LB1+LB2+…+LBn；
Step 5, approaching the track LA、LBAnd forming a training set by all the phase points, and training by a Volterra filter to obtain a multi-step predicted value of the time sequence.
2. The multi-step prediction method of chaotic time series based on density peak clustering as claimed in claim 1, wherein clustering the training set of phase points comprises:
step 1.1, calculating Euclidean distance d between any two phase points i and j in the phase point training setij；
Step 1.2, selecting and calculating distance
All Euclidean distances d calculated for step 1.1ijSorting the sequence in ascending order, setting a parameter percentage, calculating the distance dcIs the number of sequences in the parameter percentage;
step 1.3, determining the phase point of the clustering center
Finding out the distance between the phase point training set and the phase point i to be less than dcThen finding the value with the minimum Euclidean distance from the phase point i in the phase points with the density larger than that of the phase point i, and determining the phase point of the clustering center;
step 1.4, distributing the remaining phase points except the clustering center phase point, wherein the cluster to which the remaining phase points belong is the cluster of the clustering center phase point which is the nearest and has the density larger than that of the clustering center phase point;
and step 1.5, after the phase points are distributed, if isolated clustering center phase points exist, distributing the clustering center phase points as residual phase points according to the method in the step 1.4, thereby completing the clustering.
3. The multi-step prediction method of the chaotic time series based on the density peak clustering as claimed in claim 1, wherein the method for calculating the orbit similarity ξ i in step 2 is as follows:
let XMFor the target phase point, { XiI-1, 2, …, k is k neighboring phase points within the neighborhood of the target phase point and limits temporal separation, i.e., XiDifferent chaotic orbits are classified; xM-pAnd Xi-pAre each XMAnd XiBacktracking a pth step point, p ═ 1,2, …; in the formula [. C]Represents the inner product of the vector, | | represents the distance norm of the vector, | represents the absolute value, and the target phase point X is calculated by the formula 3MAnd neighbor phase point XiDistance similarity of (2):
Figure FDA0002986436190000021
calculating a target phase point XMAnd neighbor point XiDirection similarity of (1), fortuneUsing the following equation:
Figure FDA0002986436190000022
calculating the track similarity speed xi i:
Figure FDA0002986436190000023
in the above formula, j is a step length parameter, and q is a backtracking step length.
4. The multi-step prediction method of chaotic time series based on density peak clustering as claimed in claim 2, wherein the percentage of the parameter is 5%, and 5 cluster center phase points are selected.
5. The multi-step prediction method of the chaotic time series based on the density peak clustering as claimed in claim 3, wherein the value of q is 2.
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