Time series anomaly detection method based on combined wavelet transform and topological persistence entropy

Abstract

The present invention discloses a time series anomaly detection method combining wavelet transform and topological persistence entropy, comprising: decomposing the time series to be detected into wavelet components of different series based on wavelet functions and decomposition series; using the wavelet components obtained after decomposition as variable sequences, embedding the variable sequences into three-dimensional space through Takens embedding theorem, and reconstructing point clouds of different wavelet components into three-dimensional space; inversely transforming the point cloud data of the reconstructed wavelet components to obtain a reconstructed time series; dividing the reconstructed time series into windows, and calculating the persistence entropy of each window; if the calculated persistence entropy of the window exceeds a preset persistence entropy threshold, the window is considered an anomaly window. The present invention extracts the characteristics and structural information of the time series based on wavelet analysis, and measures the complexity and regularity of the time series based on topological persistence entropy, with good detection accuracy and robustness.

Description

Technical Field

[0001] The present invention relates to the field of spatiotemporal data mining, and in particular to a time series anomaly detection method combining wavelet transform and granular topological persistence entropy. Background Art

[0002] Time series anomaly detection aims to identify data records or events that are significantly different from other normal data.

[0003] Currently, there is a time series anomaly detection solution based on wavelets. Specifically, wavelet entropy is used to detect abnormal windows in time series. First, the original time series is divided into several fixed-size time windows, and the PyWavelets library in Python is used to calculate the wavelet entropy of each time window. Finally, a wavelet entropy threshold is set, and windows larger than the threshold are considered abnormal windows.

[0004] Currently, deep learning techniques are being used to learn various patterns within time series data. Deep learning models use autoencoders based on convolutional neural networks (CNNs) or recurrent neural networks (RNNs) to detect anomalies in time series data. However, the occurrence of anomalous data is negligible compared to non-anomalous data, which poses a problem for machine learning classifiers used for anomaly detection.

[0005] The disclosure of the above background technology content is only used to assist in understanding the inventive concept and technical solution of the present invention. It does not necessarily belong to the prior art of this patent application, nor does it necessarily provide technical guidance. In the absence of clear evidence that the above content has been disclosed before the filing date of this patent application, the above background technology should not be used to evaluate the novelty and creativity of this application. Summary of the Invention

[0006] The purpose of the present invention is to propose a technical solution that combines the multi-scale characteristics of wavelet analysis and the long-term dependence analysis of persistence entropy to effectively capture abnormal situations in time series data.

[0007] In order to achieve the above object, the technical solution adopted by the present invention is as follows:

[0008] A time series anomaly detection method based on a combined wavelet transform and topological persistence entropy includes the following steps:

[0009] Based on the wavelet function and decomposition series, the time series to be detected is decomposed into wavelet components of different series;

[0010] The wavelet components obtained after decomposition are used as variable sequences, and the variable sequences are embedded into three-dimensional space by Takens embedding theorem, and point clouds of different wavelet components are reconstructed into three-dimensional space;

[0011] Perform inverse transformation on the point cloud data of the reconstructed wavelet component to obtain the reconstructed time series;

[0012] Divide the reconstructed time series into windows and calculate the persistence entropy of each window;

[0013] The calculated persistence entropy of each window is compared with a preset persistence entropy threshold. If the persistence entropy of a window exceeds the persistence entropy threshold, the window is an abnormal window.

[0014] Furthermore, based on any one of the above-mentioned technical solutions or a combination of multiple technical solutions, the wavedec function is selected to decompose the time series to be detected into wavelet components of different series. The wavedec function is used for the wavelet decomposition of the time series:

[0015] x(t)=∑ k c j，k ψ j，k (t)+∑ k d j，k φ j，k (t);

[0016] Among them, x(t) is the original signal c of the time series to be detected j，k and d j，k is the wavelet coefficient, ψ j，k (t) is the wavelet function, whose scale is j, translation is k, φ j，k (t) is the parent wavelet function with scale j and translation k.

[0017] Furthermore, according to any one of the above technical solutions or a combination of multiple technical solutions, Daubechies 4 wavelet function is selected, and the decomposition level is set to 5;

[0018] Each level of decomposition will obtain a set of low-frequency wavelet coefficients and a set of high-frequency wavelet coefficients;

[0019] Select high-frequency wavelet coefficients or low-frequency wavelet coefficients for delayed embedding of time series.

[0020] Furthermore, based on any one of the above technical solutions or a combination of multiple technical solutions, the variable sequence is embedded into the three-dimensional space by takingns embedding theorem as follows:

[0021] Determine the dimension of the dynamic system that generates the time series as d, determine a positive integer m and a delay parameter τ to construct the mapping function Map the time series into the delayed embedding space, where the mapping function is: y(t) = [x(t), x(t-τ), x(t-2τ), ..., x(t-(m-1)τ)], where y(t) is a point in the delayed embedding space, x(t-τ) is the value of the time series x(t) after a time delay of τ, x(t-2τ) is the value of the time series x(t) after a time delay of 2τ, and x(t-(m-1)τ) is the value of the time series x(t) after a time delay of (m-1)τ.

[0022] Furthermore, based on any one of the above technical solutions or a combination of multiple technical solutions, the Gaussian process regression model in the Scikit-Leam library in Python is used to inversely transform the reconstructed wavelet component point cloud data back into a time series, including:

[0023] Create a radial basis function as the kernel function;

[0024] Use the first dimension of the point cloud data as the target variable for training;

[0025] Make predictions at the new time point to obtain the reconstructed time series.

[0026] Furthermore, based on any one of the above technical solutions or a combination of multiple technical solutions, the Gaussian process regression model is completely defined by a mean function and a kernel function, wherein the following linear mean function is selected as the mean function:

[0027] (x) = β0 + β1x1 + β2x2 + ... + β k x k , where (x) represents the linear mean function, x=(x1, x2, ..., x k ) represents the input variable vector, β0, β1, β2, ..., β k Represents the coefficients of the linear mean function.

[0028] Furthermore, based on any one of the above technical solutions or a combination of multiple technical solutions, the following radial basis function is selected as the kernel function:

[0029] K(x, x') = exp(-||x-x'||^2 / (2*l^2)), where K(x, x') represents the kernel function value between the input variables x and x', ||x-x'|| represents the Euclidean distance, l is the length parameter of the kernel function, a represents the lower limit of the interval, and exp is an exponential function with the natural constant e as the base.

[0030] Furthermore, according to any one of the above-mentioned technical solutions or a combination of multiple technical solutions, the reconstructed time series is divided into sliding windows of fixed size, and the persistence entropy of each sliding window is calculated by the following steps:

[0031] Takens each sliding window into a point cloud;

[0032] Set the resolution threshold V1<V2…<V K , and construct VR filtering:

[0033] Based on filtering, generate persistence graphs corresponding to each sliding window;

[0034] Define a persistence pair set P in the persistence graph, where each persistence pair (p, q) represents the appearance time and disappearance time of a feature, and calculate its persistence length l=qp, which represents the duration of the feature in the persistence graph;

[0035] Calculating the frequency distribution of the persistence length l in the persistence graph to obtain a frequency distribution function f(l);

[0036] The probability of the persistence length l is calculated according to the frequency distribution function, and the persistence entropy is calculated by the following formula:

[0037] H = -∑p(l)*log(p(l)), where H represents the persistence entropy and p(l) represents the probability of persistence length l.

[0038] Furthermore, based on any one of the technical solutions or a combination of multiple technical solutions described above, the probability of the persistence length l is calculated by the following formula: p(l) = f(l) / ∑f(l); wherein p(l) represents the probability of the persistence length l, and f(l) represents the frequency distribution function.

[0039] Furthermore, based on any one of the aforementioned technical solutions or a combination of multiple technical solutions, the time series segment corresponding to the window determined to be the abnormal window is deleted to obtain a new reconstructed time series.

[0040] The beneficial effects of the technical solution provided by the present invention are as follows: the characteristics and structural information of the time series are extracted based on wavelet analysis, and then the complexity and regularity of the time series are measured using persistence entropy, effectively capturing anomalies in the time series data. BRIEF DESCRIPTION OF THE DRAWINGS

[0041] In order to more clearly illustrate the embodiments of the present application or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are only some embodiments recorded in this application. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0042] Figure 1An overall detection flow chart of time series anomaly detection provided by an exemplary embodiment of the present invention;

[0043] Figure 2 A basic flow chart of a time series anomaly detection method using a combined wavelet transform and topological persistence entropy provided for an exemplary embodiment of the present invention;

[0044] Figure 3 A schematic diagram of five-level wavelet components after decomposing a time series by wavelet is provided as an exemplary embodiment of the present invention;

[0045] Figure 4 A schematic diagram of an embedded point cloud and a reconstructed point cloud of different wavelet components provided for an exemplary embodiment of the present invention;

[0046] Figure 5 A schematic diagram of a reconstructed time series of a Gaussian process regression model provided as an exemplary embodiment of the present invention;

[0047] Figure 6 Sub-figure (a) is a schematic diagram of the third window point cloud of the reconstructed time series provided by an exemplary embodiment of the present invention, and sub-figure (b) is the persistence map PD corresponding to the window point cloud of sub-figure (a);

[0048] Figure 7 A persistence entropy sequence diagram corresponding to a time window of a reconstructed time series provided by an exemplary embodiment of the present invention;

[0049] Figure 8 Results of the time series anomaly detection model based on granular topological persistence entropy. DETAILED DESCRIPTION

[0050] In order to enable those skilled in the art to better understand the solutions of the present invention, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the embodiments described are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts should fall within the scope of protection of the present invention.

[0051] It should be noted that the terms "first", "second", etc. in the description and claims of the present invention and the above-mentioned drawings are used to distinguish similar objects and are not necessarily used to describe a specific order or sequence. It should be understood that the numbers used in this way can be interchanged where appropriate, so that the embodiments of the present invention described herein can be implemented in an order other than those illustrated or described herein. In addition, the terms "including" and "having" and any variations thereof are intended to cover non-exclusive inclusions. For example, a process, method, apparatus, product or device that includes a series of steps or units is not necessarily limited to those steps or units clearly listed, but may include other steps or units that are not clearly listed or inherent to these processes, methods, products or devices.

[0052] The present invention relates to a time series anomaly detection model that combines wavelet transform and topological persistence entropy, which fully utilizes the multi-scale analysis capability of wavelet transform and the stability measurement of topological data analysis to achieve effective detection of abnormal windows in time series. The overall detection flow chart is as follows: Figure 1 shown.

[0053] In one embodiment of the present invention, a time series anomaly detection method combining wavelet transform and topological persistence entropy is provided. Figure 2 As shown, the time series anomaly detection method includes the following steps:

[0054] The first step is to decompose the time series to be detected into wavelet components of different series based on wavelet function and decomposition series;

[0055] The time series to be tested in this embodiment is the monthly average hourly temperature data for Shushan District, Hefei City, Anhui Province from 2007 to 2022. This original time series contains a total of 4608 data points. Wavelet decomposition is performed using the PyWavelets library in Python. First, a suitable wavelet function and decomposition series are selected. The original time series is convolved with the selected wavelet function to obtain wavelet coefficients. Wavelet decomposition can be achieved through multi-level decomposition, and each level of decomposition produces a set of low-frequency and high-frequency wavelet coefficients. The low-frequency coefficients represent the smooth component of the signal, while the high-frequency coefficients represent the detail component of the signal. In the subsequent steps of this embodiment, the high-frequency components of the wavelet are selected for delayed embedding of the time series.

[0056] The PyWavelets library in Python is used to perform wavelet decomposition on the original time series. The specific algorithm steps are as follows:

[0057] 1.1) Select an appropriate wavelet function and decomposition level, and convolve the original time series with the selected wavelet function to obtain wavelet coefficients. In this embodiment, the Daubechies 4 wavelet function is selected as the wavelet basis function, and a five-level decomposition is set. Each level of decomposition produces a set of low-frequency and high-frequency wavelet coefficients. The low-frequency coefficients represent the smooth component of the signal, while the high-frequency coefficients represent the detail component of the signal. In this embodiment, the high-frequency component of the wavelet is selected for delay embedding of the time series.

[0058] 1.2) Select the wavedec function for wavelet decomposition of the original time series, decomposing the time series into wavelet coefficients of different scales, that is, decomposing it into wavelet components of different series. The specific wavedec function is used for the wavelet decomposition of the time series as follows:

[0059] x(t)=∑ k c j，k ψ j，k (t)+∑ k d j，k φ j，k (t);

[0060] Among them, x(t) is the original signal c of the time series to be detected j，k and d j，k is the wavelet coefficient, ψ j，k (t) is the wavelet function, whose scale is j, translation is k, φ j，k (t) is the parent wavelet function with scale j and translation k. The time series is decomposed into wavelet coefficients of different scales, such as Figure 3 shown.

[0061] The second step is to use the wavelet components obtained after decomposition as a variable sequence, and embed the variable sequence into the same three-dimensional space through Takens embedding theorem, as follows: Figure 4 As shown; and reconstruct the point clouds of different wavelet components into three-dimensional space;

[0062] This part mainly uses the high-frequency components of wavelets of different series as variable sequences, and embeds these variable sequences into 3D space through Takens embedding theorem to reconstruct point cloud data of different components. Takens embedding theorem can be described as: for a time series x(t) generated by a dynamic system, if the dimension of the dynamic system is d, then there exists a positive integer m and an appropriate delay parameter τ, so that a mapping function can be constructed Mapping time series into a delayed embedding space.

[0063] The mapping function That is, y(t) = [x(t), x(t-τ), x(t-2τ), …, x(t-(m-1)τ)], where y(t) is a point in the delayed embedding space, x(t-τ) is the value of the time series x(t) after a time delay of τ, x(t-2τ) is the value of the time series x(t) after a time delay of 2τ, and x(t-(m-1)τ) is the value of the time series x(t) after a time delay of (m-1)τ. The key to Takens' theorem is to choose an appropriate delay parameter τ and embedding dimension m. In this embodiment, τ = 10 and m = 3. Using Takens' theorem, different wavelet coefficient components are embedded into the same three-dimensional space, and the different wavelet component point clouds are reconstructed into a three-dimensional space.

[0064] The third step is to perform inverse transformation on the point cloud data of the reconstructed wavelet component to obtain the reconstructed time series;

[0065] The inverse transformation of the reconstructed point cloud data can be performed to convert it back to a time series using the inverse embedding method. This example uses the Gaussian process regression model in the Scikit-learn library in Python to perform the inverse transformation of the point cloud data. First, a radial basis function is created as the kernel function, and then the first dimension of the point cloud data is used as the target variable for training. Next, a prediction is made at the new time point to obtain the reconstructed time series, such as Figure 5 shown.

[0066] The reconstructed point cloud data is inversely transformed, and the Gaussian process regression model in the Scikit-leam library in Python is used to perform the inverse transformation of the point cloud data and convert it back into a time series. The Gaussian process regression model is a non-parametric regression method used to model the potential functional relationship of the data. It is based on the concept of Gaussian process and regards the data as an infinite-dimensional vector sampled from a Gaussian distribution. Gaussian process regression can be fully defined by a mean function and a covariance function (or kernel function). In this embodiment, the linear mean function is selected as the mean function and the radial basis function is selected as the kernel function, where the linear mean function can be expressed as:

[0067] (x) = β0 + β1x1 + β2x2 + ... + β k x k , where (x) represents the linear mean function, x=(x1, x2, ..., x k ) represents the input variable vector, β0, β1, β2, ..., β k Represents the coefficients of the linear mean function.

[0068] The radial basis function is usually expressed as a Gaussian kernel function, which has the form:

[0069] K(x, x') = exp(-||x - x'||^2 / (2*l^2)), where K(x, x') represents the kernel function value between the input variables x and x', ||x - x'|| represents the Euclidean distance, l represents the length parameter of the kernel function, a represents the lower limit of the interval, and exp represents an exponential function with the natural constant e as its base. In this embodiment, l = 2.3.

[0070] Step 4: Divide the reconstructed time series into windows and calculate the persistence entropy of each window;

[0071] The Giotto-tda library in Python is used to calculate the persistence entropy of each window. Persistence entropy is a metric in topological data analysis that measures the stability and persistence of topological structures. The specific algorithm steps are as follows:

[0072] 4.1) Divide the inverse transformed time series into sliding windows of fixed size. In the embodiment of the present invention, the window is divided into 205 fixed windows.

[0073] 4.2) Takens each window and embed it into a point cloud, see Figure 6 Subfigure (a) in , which schematically shows the third window point cloud data of the reconstructed time series.

[0074] 4.3) Based on piecewise linear filtering, the continuous homology of each sliding window is calculated. In this embodiment, VR filtering is selected. First, the resolution threshold V1<V2…<V K , and construct VR filtering:

[0075] Then, the persistence graph of the corresponding window is generated, and the persistence entropy of each persistence graph is calculated to characterize the abnormality of the time series. Specifically, x is calculated in the form of persistence graph PD(i), i=1,...,n i Local topological summary of ;

[0076] 4.4) Calculation of persistence entropy based on persistence graph PD(i). First, define the persistence pair set P in the persistence graph, where each persistence pair (p, q) represents the appearance time and disappearance time of a feature, such as Figure 6 As shown in the subgraph (b) in Figure 6 The persistence graph PD(3) of subgraph (a) in [1] is used to calculate its persistence length l = qp, which represents the duration of the feature in the persistence graph.

[0077] Secondly, the frequency distribution of the persistence length l in the persistence graph is calculated, that is, the number of times different persistence lengths l appear in the persistence graph is counted to obtain a frequency distribution function f(l);

[0078] Finally, the probability of persistence length l is calculated according to the frequency distribution function: p(l) = f(l) / ∑f(l); where p(l) represents the probability of persistence length l and f(l) represents the frequency distribution function; and the persistence entropy is calculated using the following formula:

[0079] H = -∑p(l)*log(p(l)), where H represents the persistence entropy and p(l) represents the probability of persistence length l.

[0080] The fifth step is to compare the calculated persistence entropy of each window with the preset persistence entropy threshold. If the persistence entropy of a window exceeds the persistence entropy threshold, the window is an abnormal window, such as Figure 7 shown.

[0081] If a window is determined to be an abnormal window, the time series segment corresponding to the abnormal window is deleted to obtain a new reconstructed time series.

[0082] Comparative Example 1

[0083] Time series anomaly detection model based on single wavelet: Wavelet entropy is used to detect abnormal windows in time series. First, the original time series is divided into 205 fixed-size time windows, and the PyWavelets library in Python is used to calculate the wavelet entropy of each time window. Finally, a wavelet entropy threshold is set, and windows larger than the threshold are considered abnormal windows.

[0084] Comparative Example 2

[0085] A time series anomaly detection model based on single topological data analysis: First, the original time series is divided into 205 fixed-size time windows. Based on this, each time window is embedded into a point cloud, and a persistence graph is generated for each window. Finally, the persistence entropy is calculated from the persistence graph, and a persistence entropy threshold is set. Windows larger than the threshold are considered anomaly windows.

[0086] The ROC curves of the embodiment of the present invention, comparative example 1 and comparative example 2 are as follows: Figure 8 As shown, the accuracy of the anomaly detection results of the embodiment of the present invention is about 92%, the accuracy of the anomaly detection results of Example 1 is about 76%, and the accuracy of the anomaly detection results of Example 2 is about 79%. The time series anomaly detection model of the joint wavelet transform and topological persistence entropy in this embodiment also shows obvious advantages in anomaly detection robustness.

[0087] It should be noted that, in this document, relational terms such as first and second, etc., are used only to distinguish one entity or operation from another entity or operation, and do not necessarily require or imply the existence of any such actual relationship or order between these entities or operations. Moreover, the terms "comprises," "comprising," or any other variants thereof are intended to cover non-exclusive inclusion, so that a process, method, article, or device comprising a series of elements includes not only those elements, but also other elements not explicitly listed, or elements inherent to such process, method, article, or device. In the absence of further limitations, an element defined by the phrase "comprising a ..." does not exclude the presence of other identical elements in the process, method, article, or device comprising the element.

[0088] The above is only a specific implementation method of the present application. It should be pointed out that for ordinary technicians in this technical field, several improvements and modifications can be made without departing from the principles of the present application. These improvements and modifications should also be regarded as the scope of protection of the present application.

Claims

1. A time series anomaly detection method combining wavelet transform and topological persistence entropy, characterized in that: The following steps are involved: Based on the wavelet function and decomposition series, the time series to be detected is decomposed into wavelet components of different series; the time series to be detected is temperature data, the wavedec function is selected to decompose the time series to be detected into wavelet components of different series, and the Daubechies wavelet function is selected for decomposition; The wavelet components obtained after decomposition are used as variable sequences, and the variable sequences are embedded into three-dimensional space by Takens embedding theorem, and point clouds of different wavelet components are reconstructed into three-dimensional space; Use the Gaussian process regression model in the Scikit-learn library in Python to inversely transform the point cloud data of the reconstructed wavelet component to obtain the reconstructed time series; The reconstructed time series is divided into sliding windows of fixed size, and the persistence entropy of each sliding window is calculated by the following steps: each sliding window is embedded into a point cloud by Takens; a resolution threshold V1 is set <V2…<V K , and construct VR filtering: Based on filtering, a persistence graph corresponding to each sliding window is generated; a persistence pair set P in the persistence graph is defined, wherein each persistence pair (p, q) represents the appearance time and disappearance time of a feature, and its persistence length l = qp is calculated, which represents the duration of the feature in the persistence graph; the frequency distribution of the persistence length l in the persistence graph is calculated to obtain a frequency distribution function f(l); the probability of the persistence length l is calculated based on the frequency distribution function: p(l) = f(l) / ∑f(l); wherein p(l) represents the probability of the persistence length l, and f(l) represents the frequency distribution function, and the persistence entropy is calculated using the following formula: H = -∑p(l)*log(p(l)), wherein H represents the persistence entropy; The calculated persistence entropy of each window is compared with a preset persistence entropy threshold. If the persistence entropy of a window exceeds the persistence entropy threshold, the window is an abnormal window.

2. The time series anomaly detection method based on joint wavelet transform and topological persistence entropy according to claim 1 is characterized in that: The wavedec function is used for the wavelet decomposition of time series: Among them, x(t) is the original signal c of the time series to be detected j,k and d j,k is the wavelet coefficient, ψ j,k (t) is the wavelet function, whose scale is j, translation is k, φ j,k (t) is the parent wavelet function with scale j and translation k.

3. The time series anomaly detection method based on joint wavelet transform and topological persistence entropy according to claim 2 is characterized in that: Select Daubechies 4 wavelet function and set the decomposition level to 5; Each level of decomposition will obtain a set of low-frequency wavelet coefficients and a set of high-frequency wavelet coefficients; Select high-frequency wavelet coefficients or low-frequency wavelet coefficients for delayed embedding of time series.

4. The time series anomaly detection method based on joint wavelet transform and topological persistence entropy according to claim 1 is characterized in that: The variable sequence is embedded into the three-dimensional space by Takens embedding theorem as follows: Determine the dimension of the dynamic system that generates the time series as d, determine a positive integer m and a delay parameter τ to construct a mapping function F: Map the time series into the delayed embedding space, where the mapping function F is: is: y(t) = [x(t), x(t-τ), x(t-2τ), …, x(t-(m-1)τ)], where y(t) is a point in the delayed embedding space, x(t-τ) is the value of the time series x(t) after a time delay of τ, x(t-2τ) is the value of the time series x(t) after a time delay of 2τ, and x(t-(m-1)τ) is the value of the time series x(t) after a time delay of (m-1)τ.

5. The time series anomaly detection method based on joint wavelet transform and topological persistence entropy according to claim 1 is characterized in that: The Gaussian process regression model in the Scikit-learn library in Python is used to inversely transform the reconstructed wavelet component point cloud data back into a time series, including: Create a radial basis function as the kernel function; Use the first dimension of the point cloud data as the target variable for training; Make predictions at the new time point to obtain the reconstructed time series.

6. The time series anomaly detection method based on joint wavelet transform and topological persistence entropy according to claim 5 is characterized in that: The Gaussian process regression model is fully defined by a mean function and a kernel function, wherein the following linear mean function is selected as the mean function: (x) = β0 + β1x1 + β2x2 + ... + β k x k , where (x) represents the linear mean function, x=(x1,x2,...,x k ) represents the input variable vector, β0,β1,β2,...,β k Represents the coefficients of the linear mean function.

7. The time series anomaly detection method based on joint wavelet transform and topological persistence entropy according to claim 6 is characterized in that: The following radial basis function is selected as the kernel function: K(x,x')=exp(-||x–x′||^2 / (2*l^2)), where K(x,x') represents the kernel function value between the input variables x and x', ||x–x′|| represents the Euclidean distance, l represents the length parameter of the kernel function, a represents the lower limit of the interval, and exp represents the exponential function with the natural constant e as the base.

8. The time series anomaly detection method based on joint wavelet transform and topological persistence entropy according to any one of claims 1 to 7, characterized in that: The time series segment corresponding to the window determined to be an abnormal window is deleted to obtain a new reconstructed time series.

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