Time sequence prediction method based on adaptive low-rank representation

Abstract

In order to overcome the defects of an existing low-rank model in non-stationary trend, pseudo-periodic mode and transient anomaly processing, the invention provides a time series prediction method based on self-adaptive low-rank representation, original time series data is decomposed into a low-rank component (L) and a sparse component (S) through robust principal component tracking (PCP) and principal component analysis (PCA) algorithms, and the low-rank component (L) and the sparse component (S) are subjected to low-rank prediction. And adaptive separation of a trend-periodic component and a residual (abnormal / transient) component is realized. According to the method, an orthogonal transformation matrix (A) is constructed, a data-driven orthogonal basis (B) and a Fourier orthogonal basis (UF, VF) are fused in the matrix, and time sequence data are mapped to a low-rank potential space with higher characterization capacity. The method adopts a convolution kernel norm minimization (CNNM) frame for prediction, a learnable dynamic search window is established under the frame, the size (wx, n) of the window is adaptively adjusted according to a sequence local feature (| un |), and through fusion of fast Fourier transform (FFT) decomposition and introduced motion vector information, the motion vector information of the motion vector is obtained. And accurate acquisition of multi-scale time sequence characteristics (such as periodic modes of different frequencies) is realized. Under the low-rank constraint, the method provided by the invention can effectively process the complex dynamic characteristics of high-dimensional sensing data, and meanwhile, the prediction precision of a non-stationary sequence is remarkably improved through a parameter adaptive mechanism.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of time series analysis and prediction, in particular to a time series prediction method based on observation adaptation and low-rank representation (LRP), which is particularly suitable for processing complex time series data with pseudo-periodicity, non-stationary trend and multi-scale dynamics. BACKGROUND

[0002] Traditional low-rank models construct the low-rank property of the cyclic matrix under the strict periodicity assumption, but the effect is significantly reduced when dealing with the conflict between trend and periodic components, phase shift or dynamic evolution. The decomposition method is difficult to handle high-dimensional pseudo-periodic data due to the linear assumption. Although the existing Laplacian convolution (LCR) representation improves the spatiotemporal data filling through graph regularization, its fixed kernel function and static rank assumption limit its adaptability in dynamic patterns. Studies have shown that when the time domain distance between non-key frames and key frames increases, it is difficult for traditional block matching methods to obtain accurate motion vectors, and the fixed search window strategy leads to computational redundancy and the decline of the quality of the hypothesis set. SUMMARY

[0003] The technical problem to be solved by the present application is how to adaptively construct a low-rank representation space while decoupling trend, periodic and dynamic components to achieve high-precision time series prediction. The technical solution is as follows:

[0004] Step 1, input sequence

[0005] Input the observed value P Ω (y) of the sequence to be predicted The motion vector of the forward adjacent frame When the sequence is initialized, the motion vector is initialized to 0.

[0006] Step 2, data matrix construction

[0007] Divide the training sequence into segments of length m to construct the generation matrix where n0=l-m+1, the optimal model size m∈[2h,10h] is determined by cross-validation, h is the prediction range, and the sample dimension ratio (SDR) is ≥ the hyperparameter τ.

[0008] Step 3, transformation matrix learning

[0009] Including the following sub-steps:

[0010] (1) Adopt principal component pursuit (PCP) to decompose Y into low-rank component L and sparse component S:

[0011]

[0012] wherein,

[0013] (2) Augmented matrix Solve the orthogonal transformation matrix B by ADMM algorithm:

[0014]

[0015] (3) Fourier orthogonal basis U F , Synthetic final transformation matrix

[0016] Step 4, Convolution low-rank prediction

[0017] (1) Project the observation value to the low-rank space:

[0018]

[0019] Where the kernel size k = βq (0 < β ≤ 1), the regularization parameter λ = 1000.

[0020] (2) Accelerate the calculation of the optimal matching block by FFT, and construct an adaptive rectangular search window:

[0021] w x,n = α (|u n | - min |u n |) + w x,min

[0022] Horizontally extended range [w x,min , w x,max ], where w x,min = max [5, min |u n |], coefficient

[0023] Step 5, multi-model collaborative prediction

[0024] Output the hypothesis set H = {h1,...,h L} of each non-overlapping block in the key frame, and fuse the external predictor (LSTM) results by weighted fusion:

[0025]

[0026] The weight ω i is determined by the inverse of the Euclidean distance, and γ is the adaptive coupling coefficient.

[0027] Preferably, the ADMM iteration in step 3.2 comprises:

[0028] a) Initialize B (0) = I, Z (0) = 0, W (0) = 0

[0029] b) update

[0030] c) threshold shrinkage

[0031] d) update Lagrange multiplier W (k+1) = W (k) + B (k+1) - Z (k+1)

[0032] e) convergence condition ||B (k+1) - Z (k+1) || F <∈.

[0033] Preferably, in step 4.1, when the kernel size k = q, the convolution operation is accelerated by fast Fourier transform, and the computational complexity is reduced from O(m 3 ) to O(mlogm).

[0034] The present application proposes a time series prediction method based on adaptive low-rank representation. Through robust principal component pursuit (PCP) and principal component analysis (PCA) algorithms, the original time series data is decomposed into low-rank component (L) and sparse component (S), realizing the adaptive separation of trend-periodic component and residual (abnormal / transient) component. The present application constructs an orthogonal transformation matrix (A); the matrix fuses data-driven orthogonal basis (B) and Fourier orthogonal basis (U F ,V F ), and maps the time series data to a low-rank latent space with better representation ability. The method uses the convolution kernel norm minimization (CNNM) framework for prediction, and establishes a learnable dynamic search window under this framework. The size (w x,n ) of the window is adaptively adjusted according to the local features (|u n |) of the sequence, and is fused with the introduced motion vector information through FFT decomposition, realizing accurate capture of multi-scale time series features (such as periodic patterns of different frequencies). Under the low-rank constraint, the method can effectively handle the complex dynamic characteristics of high-dimensional sensor data, and significantly improve the prediction accuracy of non-stationary sequences through parameter adaptive mechanism. BRIEF DESCRIPTION OF DRAWINGS

[0035] Figure 1 Four typical pseudo-periodic patterns (stationary periodic, trend interference, phase transition, multi-scale oscillation) are shown;

[0036] Figure 2 LRP algorithm flowchart, including PCP decomposition, transformation matrix learning and prediction fusion three stages. DETAILED DESCRIPTION

[0037] Example 1: Industrial sensor data prediction

[0038] Step 1: Input sequence: data preprocessing

[0039] Industrial sensor data contains four typical pseudo-periodic patterns as shown in Figure 1 Input 85 sets of sensor evaluation data (each sequence length l = 16), construct a training matrix Y of m = 16 x n = 85, and standardize.

[0040] Step 2: Data matrix construction

[0041] Construct the training matrix Standardize and verify SDR ≥ τ (τ = 4).

[0042] Step 3: Transform matrix learning

[0043] Perform PCP decomposition Y = L + S, initialize B (0) = I 2m , iterate 20 times by ADMM, the residual norm ||BY-E||1 decreases from the initial value 5.82 to 0.17, and the convergence threshold ∈ = 10 -3 . Finally, synthesize the transform matrix

[0044] Step 4: Convolution low-rank prediction

[0045] Calculate the current sequence spectral frequency τ = 0.65, detect μ2(Ay) = 0.83 > 0.7τ, activate the kernel shrinkage mechanism, set β = 0.5, and generate an initial convolution kernel of k = 8.

[0046] Step 5: Multi-model collaborative prediction

[0047] Fuse the LSTM external prediction results, weight γ = 0.3, regularization parameter initial value λ0= 1000, after 10 iterations the objective function value decreases to 1.2 x 10 -4 .

[0048] The result output: the average absolute percentage error (sMAPE) of the prediction performance and the true value is 0.00048, the normalized root mean square error (NRMSE) is 0.00048, the error is reduced by 89.7% compared with the CNNM method, and the phase delay is less than 0.2 sampling points.

[0049] Example 2: Energy load prediction

[0050] Step 1: Input sequence

[0051] Energy load data contains four typical pseudo-periodic patterns as shown in Figure 1Four typical pseudo-periodic patterns are shown, 19734 10-minute granularity sequences in the input energy dataset.

[0052] Step 2: Data matrix construction

[0053] Extract m = 1440 (24-hour period) to build 100 sub-matrices

[0054] Step 3: Transform matrix learning

[0055] PCP decomposition is performed Finally, the ADMM is solved to obtain the B matrix, generating

[0056] Step 4: Convolution low-rank prediction

[0057] Dynamic parameter adjustment is performed: μ2(Ay) = 0.82 > 0.7τ (τ = 1.1) is detected, automatically adjusting β from 0.5 to 0.3, generating a large-scale convolution kernel k = 432 to capture the daily periodic pattern. With 50% random missing, through the sparse constraint of the augmented matrix, the NRMSE stabilizes at 156.42 (fluctuation <± 2%), significantly better than LCR (NRMSE = 169.31) and CNNM (NRMSE = 163.99).

[0058] Step 5: Multi-model collaborative prediction

[0059] The external LSTM prediction results are integrated, and the fusion weights are estimated by weighted least squares (WLS). The final sMAPE decreases from 100.84 of the single model to 89.73, with a performance improvement of 10.3%.

Claims

1. The present invention proposes a time series prediction method based on adaptive low-rank representation. The technical solution is as follows: Step 1. Enter the sequence: Input the observation value P of the sequence to be predicted Ω (y), forward keyframe Motion vector of the forward adjacent frame The motion vectors are initialized to 0 when the sequence starts. Step 2: Data matrix construction: The training sequence Divide into segments of length m and construct the generator matrix Where n0 = l-m+1, and the optimal model size m∈[2h,10h] is determined by cross-validation, where h is the prediction range, ensuring that the sample dimension ratio SDR ≥ the hyperparameter τ. Step 3: Transformation matrix learning: The following sub-steps are included: (1) PCP is used to decompose Y into a low-rank component L and a sparse component S: in, (2) Constructing the augmented matrix Solve the orthogonal transformation matrix B using the alternating direction method of multipliers (ADMM) algorithm: (3) Calculate the Fourier orthogonal basis Composite final transformation matrix Step 4: Convolutional low-rank prediction: (1) Project the observations into a low-rank space: Among them, the kernel size k = βq (0 < β ≤ 1), and the regularization parameter λ = 1000. (2) Accelerate the calculation of the optimal matching block through FFT and construct an adaptive rectangular search window: w x,n =α(∣u n ∣-min∣u n ∣)+w x,min Horizontal expansion range [w x,min ,w x,max ], where w x,min =max[5,min|u n ∣], coefficient Step 5: Multi-model collaborative prediction: Output the hypothesis set H = {h1,...,h L }, by weighted fusion of the external predictor (long short-term memory network (LSTM)), we get the result: Among them, the weight ω i Determined by the inverse of the Euclidean distance, γ is the adaptive coupling coefficient.

2. The method according to claim 1, wherein The iterative process of the alternating direction multiplier method in step 3 includes: Initialize B (0) =I,Z (0) =0,W (0) =0 ·renew Threshold shrinkage operation Multiplier update W (k+1) =W (k) +B (k+1) -Z (k+1) Where: I represents the identity matrix, ρ>0 is the penalty factor, s λ / ρ (·) represents the contraction operator with a threshold of λ / ρ, ||·|| F represents the Frobenius norm of the matrix, and the superscript k represents the result of the k-th iteration.

3. The method according to claim 1, wherein The adaptive adjustment rule of the convolution kernel size in step 4 is: Where τ is the spectral frequency estimate of the current sequence, μ1 and μ2 are the first and second convolution coherence coefficients, respectively.

4. The method according to claim 1, wherein The online adjustment formula of the regularization parameter λ in step 5 is: Among them, α=0.9 is the smoothing coefficient, Ω t is the valid observation value at time t.

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