An adaptive sparse mode decomposition method based on machine learning

Abstract

The present invention proposes an adaptive sparse modal decomposition method based on machine learning. First, a discrete time series to be analyzed is input in the form of a vector; a Fourier transform matrix is ​​constructed as an additional known input; then a set of adaptive filter groups with variable peak centers and bandwidths are defined; a four-layer neural network is used as a solver of a non-convex least squares objective function; finally, a loss function is defined as an objective function, network weights are trained, and the peak center value and bandwidth of the adaptive filter group and the filter weight coefficient are obtained by weight calculation to obtain the sparsest intrinsic mode function, thereby completing the decomposition of the signal mode; the present invention combines the neural network and the signal decomposition method in machine learning to realize the adaptive sparse decomposition of the time series to be analyzed into the intrinsic mode function, and in the process, automatically learns and optimizes the solution of the non-convex least squares problem, which is particularly important for the adaptive sparse decomposition of nonlinear and non-stationary time series.

Description

Technical Field

[0001] The present invention belongs to the technical field of signal processing, and in particular, relates to an adaptive sparse modal decomposition method based on machine learning. Background Art

[0002] Signal modal decomposition technology decomposes any signal mode into a set of single component signals, each of which is presented in the form of an amplitude-frequency modulated signal, called an intrinsic mode function (IMF). Signal modal decomposition technology can analyze the structure based on the mode and capture the internal characteristics of the data, and is widely used in the fields of medicine, engineering, media, communications, etc.

[0003] In the field of mechanical engineering, the intrinsic mode function is often used to diagnose faults based on the dynamic characteristics of the machine; in the field of structural engineering, the intrinsic mode function is used to reveal the modal parameters of the structure for structural health monitoring; in the field of music and voice recognition, the intrinsic mode function is used to separate different instrument components, etc.

[0004] Traditional non-adaptive signal modal decomposition technology usually uses orthogonal basis to decompose the signal, so it lacks flexibility for data. Since the characteristics of signals in nature are mostly nonlinear and non-stationary, the decomposition results using traditional fixed basis functions lack accuracy, so there is an urgent need for the emergence of adaptive signal modal decomposition technology. Unlike non-adaptive signal modal decomposition, which often has only one solution, adaptive signal modal decomposition technology constructs an infinitely large adaptive dictionary composed of adaptive bases, and the decomposition results are often not unique.

[0005] Existing adaptive signal modal decomposition methods mainly include empirical mode decomposition (EMD), ensemble empirical mode decomposition (EEMD), local mean decomposition (LMD), Hilbert vibration decomposition (Hilbert vibration decomposition), variational mode decomposition (VMD), etc. EMD and EEMD were both proposed by Huang E et al. The former is widely used in the field of adaptive signal modal decomposition due to its convenience and simplicity. However, due to the existence of problems such as modal aliasing and end effect, its accuracy needs to be improved. The subsequent emergence of EEMD improved the robustness of EMD to white noise. The proposal of HVD combines Hilbert transform and low-pass filtering technology, but still lacks universality for all signal forms. The proposal of VMD provides a solid theoretical foundation for adaptive signal modal decomposition. Although it has higher accuracy, it is sensitive to numerical values ​​such as the number of eigenmode functions and penalty factors, which affects its widespread use.

[0006] Adaptive signal modal decomposition technology aims to extract the features that reflect the structural characteristics of the signal. Therefore, the combination of cutting-edge data mining technology and traditional signal modal decomposition technology can realize a new type of adaptive signal modal decomposition technology. At present, data mining technology based on machine learning has emerged in various fields, and has gradually entered the public eye with its ability to well approximate arbitrary nonlinear functions and solve optimization problems. Based on this, it becomes a feasible measure to introduce machine learning into adaptive signal modal decomposition. Therefore, adaptive modal decomposition technology can be regarded as an optimization problem in machine learning, that is, the signal to be analyzed is expanded in the frequency domain, and the weights are learned through neural network technology, so as to automatically extract the various eigenmode functions of the signal in the frequency domain. Summary of the invention

[0007] In order to solve the problem that the adaptive sparse signal modal decomposition method in the prior art is still not perfect, the present invention proposes a new adaptive sparse modal decomposition method (Adaptive Sparse Mode Decomposition, ASMD), which has data-driven intelligence and flexibility that other adaptive modal decomposition technologies do not have. It proposes an adaptive sparse modal decomposition method based on machine learning, which combines neural networks and modal decomposition methods in machine learning. It can realize automatic learning of data by neural networks and optimize the solution of non-convex least squares problems to obtain the sparsest intrinsic mode function, so as to achieve the purpose of signal modal decomposition.

[0008] The present invention is achieved through the following technical solutions:

[0009] An adaptive sparse mode decomposition method based on machine learning:

[0010] The method specifically comprises the following steps:

[0011] Step 1: Take the discrete time series to be analyzed as input in the form of a vector, and construct the Fourier transform matrix as an additional known input;

[0012] Step 2, defining a set of adaptive filter banks with variable peak center and bandwidth;

[0013] Step 3: construct a four-layer neural network as a solver of the non-convex least squares objective function; reconstruct and output the signal input in step 1;

[0014] Step 4: By defining a loss function as the objective function, the network weights are trained, and the peak center value and bandwidth of the adaptive filter group in step 2 and the filter weight coefficient are obtained by weight calculation; the sparsest eigenmode function is obtained to complete the decomposition of the signal mode.

[0015] Furthermore, in step 1,

[0016] Assume a discrete time series signal f(t), which is represented by the sum of multiple IMFs, each of which has N discrete points:

[0017]

[0018] Among them, u i (t) = IMF i ,i=1,...,M,u i (t)∈R N , r(t) is the residual term;

[0019] Construct a Fourier basis matrix B of the same dimension as the original signal f(t) as another known input:

[0020]

[0021] Among them, b i (t) is the i-th Fourier basis vector.

[0022] Furthermore, in step 2,

[0023] Select an adaptive filter bank A(ω) consisting of K redundant adaptive filters:

[0024]

[0025] Among them, a i(ω) is the i-th adaptive filter.

[0026] Furthermore, in step 3,

[0027] The neural network is a four-layer fully connected network, including a Fourier transform layer, an adaptive filter layer, a sparse layer and an output layer;

[0028] In the first layer of the neural network, the input signal is first converted to the frequency domain through Fourier transform. Therefore, the matrix F(ω) of the input signal after Fourier transform is expressed as:

[0029]

[0030] In the second layer of the neural network, the adaptive filter group extracts the components in the frequency domain of the input signal, and the matrix F consisting of K redundant sub-signals is obtained. red (ω) is:

[0031]

[0032] Among them, f i (ω) is the sub-signal component extracted by the i-th adaptive filter in F(ω);

[0033] In the third layer of the neural network, the extracted sub-components are sparsely extracted and the eigenmode function is used to reconstruct the original signal:

[0034]

[0035] Among them, F IMF (ω) is the reconstructed signal spectrum after sparsification, and W is a sparse matrix;

[0036] Finally, for the last layer of the neural network, the output reconstructed signal f rec (t) is expressed as:

[0037]

[0038] In the formula, B -1 is the inverse Fourier transform matrix.

[0039] Furthermore, in step 4,

[0040] The non-convex objective function to be optimized is defined as:

[0041] min||f(t)-f rec (t)||2+||W||1

[0042] Substituting the formula in step 3, the objective function further becomes:

[0043] min||f(t)-F IMF (ω)B -1 ||2+||W||1

[0044] =min||f(t)-F red (ω)WB -1 ||2+||W||1

[0045] =min||f(t)-A(ω)F(ω)WB -1 ||2+||W||1

[0046] That is, the loss function of the neural network is:

[0047]

[0048] Where i represents a discrete time point, N is the length of the discrete signal, are the original signal and reconstructed signal values ​​of the i-th point respectively; ω j is the weight coefficient of the jth adaptive filter;

[0049] The neural network back propagation optimization algorithm is stochastic gradient descent. The update of weight parameters is based on a single sample. The parameter update method is:

[0050]

[0051] Among them, α is the learning rate.

[0052] Beneficial effects of the present invention

[0053] The present invention is based on the deep learning technology in the field of machine learning, combined with the existing spectrum-based signal modal decomposition method, to construct a neural network least squares solver capable of realizing adaptive sparse modal decomposition, so as to achieve the purpose of obtaining the sparsest intrinsic mode function by inputting any signal to be analyzed. The invention has the advantages of high efficiency, accuracy and easy operation, and can be widely used in modal decomposition in many fields.

[0054] The present invention combines the neural network and modal decomposition method in machine learning, which can realize the automatic learning of data by the neural network and optimize the solution of non-convex least squares problems to obtain the sparsest intrinsic mode function to achieve the purpose of signal modal decomposition. BRIEF DESCRIPTION OF THE DRAWINGS

[0055] Figure 1 This is a framework diagram of the adaptive sparse modal decomposition method based on machine learning;

[0056] Figure 2 It is a schematic diagram of the time domain curve of the synthetic signal f1;

[0057] Figure 3It is a schematic diagram of the frequency domain curve of the synthetic signal f1;

[0058] Figure 4 It is a schematic diagram of the time domain curve of the intrinsic mode function of the synthetic signal f1 obtained by the ASMD method;

[0059] Figure 5 It is the signal component extraction diagram in the frequency domain of the adaptive filter of the synthetic signal f1 under the ASMD method;

[0060] Figure 6 is the sparse weight coefficient of the adaptive filter of the synthetic signal f1 under the ASMD method;

[0061] Figure 7 It is a schematic diagram of the time domain curve of Length-of-Day data;

[0062] Figure 8 It is a schematic diagram of the frequency domain curve of Length-of-Day data;

[0063] Fig. 9 It is a schematic diagram of the time domain curve of the eigenmode function of the Length-of-Day data obtained by the ASMD method;

[0064] Fig.10 This is the signal component extraction diagram in the frequency domain of the adaptive filter of Length-of-Day data under the ASMD method;

[0065] Fig.11 is the sparse weight coefficient of the adaptive filter for Length-of-Day data under the ASMD method. DETAILED DESCRIPTION

[0066] The technical solutions in the embodiments of the present invention will be described clearly and completely below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments 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 creative work are within the scope of protection of the present invention.

[0067] Combination Figures 1 to 11 .

[0068] An adaptive sparse mode decomposition method based on machine learning:

[0069] The method specifically comprises the following steps:

[0070] Step 1: Take the discrete time series to be analyzed as input in the form of a vector, and construct the Fourier transform matrix as an additional known input;

[0071] Step 2, defining a set of adaptive filter banks with variable peak center and bandwidth;

[0072] Step 3: construct a four-layer neural network as a solver of the non-convex least squares objective function; reconstruct and output the signal input in step 1;

[0073] Step 4: By defining a loss function as the objective function, the network weights are trained, and the peak center value and bandwidth of the adaptive filter group in step 2 and the filter weight coefficient are obtained by weight calculation; the sparsest eigenmode function is obtained to complete the decomposition of the signal mode.

[0074] In step 1,

[0075] Assume a discrete time series signal f(t), which is represented by the sum of multiple IMFs, each of which has N discrete points:

[0076]

[0077] Among them, u i (t) = IMF i ,i=1,...,M,u i (t)∈R N , r(t) is the residual term;

[0078] Construct a Fourier basis matrix B of the same dimension as the original signal f(t) as another known input:

[0079]

[0080] Among them, b i (t) is the i-th Fourier basis vector.

[0081] In step 2,

[0082] Select an adaptive filter bank A(ω) consisting of K redundant adaptive filters:

[0083]

[0084] Among them, a i (ω) is the i-th adaptive filter.

[0085] In step 3,

[0086] The neural network is a four-layer fully connected network (excluding the input layer), including a Fourier transform layer, an adaptive filtering layer, a sparse layer, and an output layer;

[0087] In the first layer of the neural network, the input signal is first converted to the frequency domain through Fourier transform. Therefore, the matrix F(ω) of the input signal after Fourier transform is expressed as:

[0088]

[0089] In the second layer of the neural network, the adaptive filter group extracts the components in the frequency domain of the input signal, and the matrix F consisting of K redundant sub-signals is obtained. red (ω) is:

[0090]

[0091] Among them, f i (ω) is the sub-signal component extracted by the i-th adaptive filter in F(ω);

[0092] In the third layer of the neural network, due to the redundancy of the number of adaptive filters, the redundancy of the extracted sub-components is caused, and the sum of the extracted sub-components does not obtain the reconstructed signal. At this time, it is necessary to perform sparse extraction on the extracted sub-components and use the eigenmode function that can best reflect the characteristics of the signal data to reconstruct the original signal:

[0093]

[0094] Among them, F IMF (ω) is the reconstructed signal spectrum after sparsification, and W is a sparse matrix;

[0095] Finally, for the last layer of the neural network, the output reconstructed signal f rec (t) is expressed as:

[0096]

[0097] In the formula, B -1 is the inverse Fourier transform matrix.

[0098] In step 4,

[0099] The non-convex objective function to be optimized is defined as:

[0100] min||f(t)-f rec (t)||2+||W||1

[0101] Substituting the formula in step 3, the objective function further becomes:

[0102] min||f(t)-F IMF (ω)B -1 ||2+||W||1

[0103] =min||f(t)-F red (ω)WB -1 ||2+||W||1

[0104] =min||f(t)-A(ω)F(ω)WB -1 ||2+||W||1

[0105] That is, the loss function of the neural network is:

[0106]

[0107] Where i represents a discrete time point, N is the length of the discrete signal, are the original signal and reconstructed signal values ​​of the i-th point respectively; ω j is the weight coefficient of the jth adaptive filter;

[0108] The neural network back propagation optimization algorithm is stochastic gradient descent. The update of weight parameters is based on a single sample, so it can converge to the global optimal value faster than the gradient descent algorithm. The parameter update method is:

[0109]

[0110] Among them, α is the learning rate.

[0111] Effect verification: Example 1 is a synthetic signal f1 in the form of:

[0112] f a =cos(30πt+10sin(2πt))

[0113] f b =sin(150πt+5sin(6πt))

[0114]

[0115] f d =(2+cos(8πt))cos(240πt)

[0116] f e =sin(500πt)

[0117] f1=f a +f b +f c +f d +f e

[0118] The synthetic signal f1 is a time-course signal formed by superposition of five IMFs in the time domain. The signal length is N=1024, the time normalization is 1, and the initial number of adaptive filters is 20. Figure 2 , Figure 3The time domain curve and frequency domain curve of f1 are shown respectively, where the original signal and the reconstructed signal obtained by the ASMD method are represented by light green and dark blue curves respectively, and the red curve oscillating at amplitude 0 is the error between the two. Figure 4 The time domain curves of the five eigenmode functions of f1 obtained by the ASMD algorithm are shown. Figure 5 The graphs of the 5 non-zero adaptive filters in the frequency domain under the ASMD algorithm analysis of the 20 initial filters are shown. The corresponding peak center frequency and filter number are marked in the figure. The blue curve is the Fourier transform result of the synthetic signal f1. It can be seen that the 5 filters can extract five different components with high accuracy (the filter and the synthetic signal spectrum are displayed in the form of double-sided spectrum). Figure 6 The changes in the corresponding sparse weight coefficient values ​​of the 20 initial adaptive filters during the iterative operation of the ASMD algorithm are shown. In the initial stage, the coefficients of all 20 adaptive filters are 1. When the number of iterations is 100, the operation is completed. At this time, only 5 filters have coefficients of about 1, and the other coefficients are close to 0, which reflects that the ASMD method has sparseness in extracting the intrinsic mode function and can accurately extract the sparsest effective components.

[0119] Example 2 is the identification of the eigenmode function of the standard Length-of-Day data. This data was proposed by Gross of the Jet Propulsion Laboratory in Pasadena, California, USA. It records the length of a day starting on January 20, 1962. One day is one data. Here, the first 700 days, i.e. 700 discrete points, are taken for analysis. As in the previous example, the number of initial redundant adaptive filters is 20.

[0120] Figure 7 , Figure 8 The time domain curve and frequency domain curve of the Length-of-Day data are shown respectively. The original signal and the reconstructed signal obtained by the ASMD method are represented by light green and dark blue curves respectively, and the red curve oscillating at an amplitude of 0 is the error between the two. Fig. 9 The time domain curves of the four intrinsic mode functions obtained by the ASMD algorithm are shown. Among them, the amplitude of the first intrinsic mode function IMF1 is close to 0 and can be ignored. Fig.10 The figure shows the non-zero frequency domain graphs of the 4 adaptive filters under the ASMD algorithm analysis of the 20 initial filters. The corresponding peak center frequency and filter number are marked in the figure. Similarly, since the amplitude of the highest frequency component is too small, the corresponding filter is ignored in the final result, that is, the number of eigenmode functions is considered to be 3. Fig.11The changes in the corresponding sparse weight coefficient values ​​of the 20 initial adaptive filters of the Length-of-Day data during the iterative operation of the ASMD algorithm are shown. In the initial stage, the coefficients of all 20 adaptive filters are 1. When the number of iterations is 300, the operation is completed. At this time, only 3 filters have coefficients of about 1, and the other coefficients are close to 0, which reflects that the ASMD method has sparseness in extracting intrinsic mode functions and can accurately extract the sparsest effective components.

[0121] The above is a detailed introduction to the adaptive sparse modal decomposition method based on machine learning proposed in the present invention, and the principles and implementation methods of the present invention are explained. The description of the above embodiments is only used to help understand the method of the present invention and its core idea; at the same time, for those skilled in the art, according to the idea of ​​the present invention, there will be changes in the specific implementation method and application scope. In summary, the content of this specification should not be understood as a limitation on the present invention.

Claims

1. An adaptive sparse modal decomposition method based on machine learning, characterized in that: The method specifically comprises the following steps: Step 1: Take the discrete time series to be analyzed as input in the form of a vector, and construct the Fourier transform matrix as an additional known input; Step 2, defining a set of adaptive filter banks with variable peak center and bandwidth; Step 3: construct a four-layer neural network as a solver of the non-convex least squares objective function; reconstruct and output the signal input in step 1; The neural network is a four-layer fully connected network, including a Fourier transform layer, an adaptive filter layer, a sparse layer and an output layer; In the first layer of the neural network, the input signal is first converted to the frequency domain through Fourier transform. Therefore, the matrix F(ω) of the input signal after Fourier transform is expressed as: B is the Fourier basis matrix; In the second layer of the neural network, the adaptive filter group extracts the components in the frequency domain of the input signal, and the matrix F consisting of K redundant sub-signals is obtained. red (ω) is: F red (ω)=A(ω)·F(ω)=[f1(ω),f2(ω),...,f K (oh)], Among them, f i (ω) is the sub-signal component extracted by the i-th adaptive filter in F(ω); A(ω) is the adaptive filter bank; In the third layer of the neural network, the extracted sub-components are sparsely extracted and the eigenmode function is used to reconstruct the original signal: Among them, F IMF (ω) is the reconstructed signal spectrum after sparsification, and W is a sparse matrix; Finally, for the last layer of the neural network, the output reconstructed signal f rec (t) is expressed as: In the formula, B -1 is the inverse Fourier transform matrix; Step 4: By defining a loss function as the objective function, the network weights are trained, and the peak center value and bandwidth of the adaptive filter group in step 2 and the filter weight coefficient are obtained by weight calculation; the sparsest eigenmode function is obtained to complete the decomposition of the signal mode; In step 4, the non-convex objective function to be optimized is defined as: min||f(t)-f rec (t)||2+||W||1 Where f(t) is a discrete time series signal; Substituting the formula in step 3, the objective function further becomes: min||f(t)-F IMF (ω)B -1 ||2+||W||1 =min||f(t)-F red (ω)WB -1 ||2+||W||1 =min||f(t)-A(ω)F(ω)WB -1 ||2+||W||1 That is, the loss function of the neural network is: Where i represents a discrete time point, N is the length of the discrete signal, and f i (t) is the original signal value at the i-th point is the reconstructed signal value; ω j is the weight coefficient of the jth adaptive filter; The neural network back propagation optimization algorithm is stochastic gradient descent. The update of weight parameters is based on a single sample. The parameter update method is: Among them, α is the learning rate.

2. The method according to claim 1, characterized in that: In step 1, Assume a discrete time series signal f(t), which is represented by the sum of multiple IMFs, each of which has N discrete points: in, r(t) is the residual term; Construct a Fourier basis matrix B of the same dimension as the original signal f(t) as another known input: Among them, b i (t) is the i-th Fourier basis vector.

3. The method according to claim 2, characterized in that: In step 2, Select an adaptive filter bank A(ω) consisting of K redundant adaptive filters: Among them, a i (ω) is the i-th adaptive filter.

Images