CN112784686A - Adaptive signal analysis method based on continuous multivariable variational modal decomposition - Google Patents
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Abstract
The invention discloses a self-adaptive signal analysis method based on continuous multivariate variational modal decomposition, which is suitable for self-adaptive decomposition, denoising and filtering of multi-channel stationary or non-stationary signals. According to the method, the decomposition scale and the fussy parameter optimization are not required to be set by priori knowledge, and the multichannel signals can be continuously and adaptively decomposed into the eigenmodes with different time scales according to the internal time-frequency domain characteristics of the signals. The method has practical value in engineering for denoising, adaptive filtering and pattern recognition of the multivariable signals.
Description
Technical Field
The invention relates to the analysis and processing of multivariate data, in particular to the analysis of non-stationary signals, and more particularly to a method for decomposing eigenmodes common to multi-channel signals to achieve a preprocessing of the signals.
Background
The statistics of non-stationary signals are time-varying functions in that the frequency of the signal varies with time. The non-stationary signals widely exist in the real world, and the statistical data in engineering signals, biological signals, medical signals and economics have non-stationarity, so that the method has important significance for processing and researching the non-stationary signals.
Because the frequency of non-stationary signals is time-varying, the characteristics of the signal must be analyzed simultaneously from both time and frequency domain perspectives. Common methods of signal decomposition include fourier transform, wavelet transform, Empirical Mode Decomposition (EMD), and the like. The filter based on Fourier transform can extract information of different frequency bands of the signal and analyze the frequency domain characteristics of the signal; the wavelet transform decomposes the signal into sequences with different scales (resolutions), and has the capability of characterizing local characteristics of the signal in time domain and frequency domain; empirical mode decomposition can decompose a non-stationary signal into a plurality of time sequences with narrow bandwidths, realize local stabilization and extract the intrinsic modes of the signal at different time scales. Wherein the fourier transform is only suitable for processing of stationary signals; the wavelet transformation needs to select a mother wavelet function according to prior knowledge, is not self-adaptive, is restricted by an inaccurate principle like Fourier transformation, and cannot obtain both the resolution of a time domain and a frequency domain; empirical mode decomposition can adaptively extract the intrinsic mode of a signal, but lacks of mathematical theory explanation, and the problem of mode aliasing occurs when an intermittent signal containing interruption disturbance is decomposed.
The Variable Mode Decomposition (VMD) methods proposed in recent years, which are similar to the EMD, are narrow-bandwidth eigenmodes for decomposing signals, but the VMD provides a sufficient mathematical theoretical explanation, decomposes signals into preset K time scale modes through repeated iterative optimization, and solves the problem of intermittent signal mode aliasing. However, the variational modal decomposition needs prior knowledge to judge the number K of eigenmodes, and the penalty factor α in the objective function needs to be optimized, which is not self-adaptive. Continuous variational modal decomposition (SVMD) is a latest algorithm proposed on the basis of variational modal decomposition, and the number of eigenmodes is not required to be preset, and only according to the intrinsic characteristics of signals, the eigenmodes on each time scale are extracted in a self-adaptive manner. The method is already applied to preprocessing of the electrocardiosignal, compared with the VMD, the SVMD denoising effect is better, the calculation complexity is low, and the method is more suitable for real-time online analysis. However, SVMD can only be used for signal decomposition of a single channel, and a multi-dimensional expansion method is not available. At present, the Multivariate Empirical Mode Decomposition (MEMD) based on EMD and the Multivariate Variational Mode Decomposition (MVMD) based on VMD inherit the advantages of the univariate method, but the defects of the univariate method cannot be avoided. The MEMD also has the problems of lack of theoretical explanation, long time consumption and modal aliasing, and the MVMD has the problems of more parameters and is not self-adaptive. If the SVMD can be subjected to multi-dimensional expansion and applied to multi-channel signals, the advantages of the SVMD algorithm can be inherited, and simultaneously, the extraction of multi-dimensional signal common modes, the selection of redundant channels, the multi-dimensional filtering of redundant information and the like can be realized.
Disclosure of Invention
The invention aims to solve the technical problems of adaptively decomposing a multi-channel signal, effectively extracting common modes of the multi-channel signal on different time scales, and realizing adaptive filtering and noise interference removal.
In order to solve the technical problems, the invention adopts the following technical means:
a self-adaptive signal analysis method based on continuous multivariate variational modal decomposition comprises the following steps:
(1) and constructing an enhanced Lagrange function as an objective function by using a secondary penalty term and a Lagrange multiplier, wherein in order to ensure that the original signal can be reconstructed after decomposition, the constraint condition is that the sum of the L-th scale, the mode of L-1 scales before decomposition and the unprocessed part is the original signal, and the objective function is as follows:
wherein, the original signal f (t) ═ f1(t)，f2(t)，...，fC(t)]，fk(t) is the original k channel signal, uLkIs the L-th time scale mode decomposed by the channel, fuk(t) is the fraction of the k-th channel that has not been processed by the first L-scale modal decomposition, ωLIs the center frequency, λ, of all channels on the lth time scalek(t) is the Lagrangian multiplier;
the same theoretical basis as the EMD, VMD, SVMD and MVMD algorithms, in order to realize the local stabilization of the non-stationary signal, one scale is decomposed, and the signal mode must be ensured to be narrow bandwidth; therefore, the eigenmode mean band in the objective function is needed to decompose the signals of all channels at various scalesWidth minimum, i.e. total bandwidth J on the Lth scale1The minimum, namely:
the same theoretical basis as SVMD algorithm, to avoid mode aliasing between different scales, when decomposing the mode of L-th scale, the part f of all channels where the signals are not decomposeduk(t) and the decomposed first L-1 scale modes do not contain information in the frequency band of the mode, so that the part f of the objective function that leaves all channel signals unresolveduk(t) and decomposed mode uik(t) center frequency ω at L-th scaleLAt minimum frequency response, i.e. ωLTotal frequency response J of all channels2The minimum, namely:
to avoid the occurrence of mode aliasing problems, it is also necessary to ensure that the mode of the L-th scale is at each center frequency ω of the already decomposed first L-1 scalesiThe frequency response at (i 1, 2.. gtoreq., L) is small, so the overall response J in the target function at the center frequency of the first L-1 scales of the L-th scale of the mode of all channel signals3The minimum, namely:
(3) formula (4) whereinL(t)，βi(t) are filters with frequency responses of:
filters are respectively at omegaLAnd ωiThe response is maximal at a frequency, thereby extracting a narrow bandwidth signal centered at that frequency;
the method is the same as the MVMD method in terms of the target, and all the channels are extracted from the common mode on the same time scale, so that similar eigenmodes with the same center frequency are decomposed on the same scale, and the eigenmode center frequency omega of all the channel signals of the L-th scale in the formulas (2) and (5) is requiredLThe same;
(2) optimizing the target function, converting a time domain signal in the target function into a frequency domain according to the energy conservation theorem of the time domain and the frequency domain, and solving the minimization problem by using an alternating direction multiplier method;
(3) all the decomposed eigenmodes are sequenced according to the sequence of the central frequencies from small to large, the larger the value of the scale L is, the higher the modal frequency of the scale is, so that the eigenmodes of different frequency bands can be selected through the value of the central frequency and the value of the L to carry out feature extraction and subsequent analysis, and the eigenmodes of different frequency bands obtained after decomposition by the method also realize the self-adaptive filtering of signals.
As a further improvement to the technical scheme:
the step (2) optimizes the target function, because the penalty factor alpha in the target function has a large influence on the decomposition of the eigen mode, and the optimal alpha values of different types of signals are greatly different during the decomposition, a heuristic method is introduced, a value range of alpha is set, the alpha is gradually increased from the minimum value to the maximum value along with the optimization iteration, the mode with the strongest energy on the current time scale is found out, and because the modes of all channels on the same time scale are similar, the same penalty factor alpha is used during the optimization iteration of each channel.
The optimization process of the objective function in the step (2) is to convert a time domain signal in the objective function into a frequency domain according to the energy conservation theorem of the time domain and the frequency domain, solve the minimization problem by using an alternating direction multiplier method, and sequentially extract the modes of all channels in each time scale until the extracted modes are noises, wherein the specific optimization process is as follows:
(1) setting a range [ alpha ] of penalty factorsmin，αmax]Convergence threshold ε1＝10-6Energy threshold ε2＝10-3Iteration parameter tau, initializing L ← 0;
(2) l ← L +1, initialization
n←0，m←1，α1←αmin；
(3) Iteration: n ← n +1, for all ω ≧ 0 and for each channel k (k ═ 1, 2.
Until convergence: for each channel k, the following are satisfied:
(4) initialization
n←0，
α1←αmin+emM ← m +1, repeat step (3) until αm≤αmax；
(5) Repeating the steps (2), (3) and (4) until convergence:
compared with the prior art, the invention adopts the technical scheme, which has the outstanding characteristics that: (1) the problem of mode aliasing in multi-channel non-stationary signal decomposition is solved; (2) the method has the advantages of few parameters, low calculation complexity and no need of an additional parameter optimization process; (3) is an adaptive decomposition method.
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Fig. 1 shows a three-channel signal containing gaussian noise and the result of decomposition of SMVMD and MEMD. Wherein, FIG. 1(a) shows a complex cosine signal f containing white Gaussian noise1And f2And a noise-free signal f3The components (a); FIG. 1(b) shows SMVMD converting f1、f2、f3Comparing the decomposed intrinsic mode BIMF with the original signal component; FIG. 1(c) shows that the MEMD will be1、f2、f3The decomposed eigenmode IMF is compared to the original signal content.
FIG. 2 shows the three-channel non-stationary discontinuous signal and the decomposition results of SMVMD and MEMD, wherein FIG. 2(a) shows the composite signal f of cosine signal and discontinuous signal4And a composite cosine signal f5And f6The components (a); FIG. 2(b) shows SMVMD converting f4、f5、f6Comparing the decomposed intrinsic mode BIMF with the original signal component; FIG. 2(c) shows that the MEMD will be f4、f5、f6And decomposing the intrinsic mode IMF of the first four scales.
FIG. 3 shows the decomposition result of the noisy non-stationary discontinuous signal and SMVMD, wherein FIG. 3(a) shows the non-stationary discontinuous signal f synthesized by the cosine signal and the discontinuous signal7And a composite cosine signal f8And f9The components (a); FIG. 3(b) shows SMVMD converting f7、f8、f9And comparing the decomposed intrinsic mode with the original signal component.
Fig. 4 is a histogram of the distribution of center frequencies of each scale after decomposing the electroencephalographic data of the subject D4 at pre-seizure and inter-seizure phases using SMVMD.
Detailed Description
The present invention will be further described with reference to the following examples.
Firstly, simulation data is used for verifying the effects of adaptive decomposition, denoising and anti-modal aliasing of the method, and then the actual electroencephalogram signal is applied to verify that the method is suitable for early prediction of epileptic seizure:
1. and testing the robustness of the algorithm to noise, and comparing the decomposition effects of SMVMD and MEMD. Constructing a complex cosine signal f containing white Gaussian noise1And f2And a noise-free signal f3The components of the material are as follows:
f1＝4cos(10πt)+2cos(20πt)+η1，
f2＝3cos(10πt)+3cos(50πt)+η2，
f3＝2cos(10πt)+3cos(20πt)+4cos(50πt)，
wherein eta1And η2Is white Gaussian noise, η1～N(0，0.12)，η2N (0, 0.12), the signal waveform is shown in FIG. 1 (a). SMVMD method sets convergence threshold epsilon in simulation experiment and real electroencephalogram data processing1＝10-6Energy threshold ε2＝10-3The iteration parameter τ is 0. The bandwidth of each mode of the simulation data is extremely narrow, so that the value ranges of the penalty factors alpha in the simulation test are all [500,60000 ]]. SMVMD will f1、f2、f3The decomposed pair of eigenmode BIMF and original signal components is shown in fig. 1 (b). In time domain, f after SMVMD decomposition1BIMF1 and BIMF2 components, f2The BIMF1 and BIMF3 components are substantially identical to their respective effective components, and f3The three BIMF components are completely identical to their effective components and are not affected by noise in the first two channel signals. The MEMD is affected by Gaussian noise, the signal of three channels is decomposed into 9 eigenmodes, the 3 rd to 5 th (IMF3, IMF4 and IMF5) are main modes, and the MEMD divides f1、f2、f3Decomposed pairs of intrinsic mode IMF and original signal componentsSuch as shown in fig. 1 (c). The mode of the MEMD decomposition in three dimensions is observed to be different from the effective component of the original signal in comparison with the SMVMD decomposition result. Furthermore, SMVMD adaptively resolves three modes of the signal, but MEMD needs to screen out the valid modes from the resolved 9 modes. Simulation results show that the SMVMD algorithm is superior to the MEMD algorithm in the aspect of noise interference resistance.
2. The test algorithm compares the modal separation effect of the SMVMD and MEMD methods on the non-stationary discontinuous signals with the effect of inhibiting modal aliasing. A three-channel signal was constructed with the following composition:
f5＝3cos(10πt)+4cos(20πt)，f6＝2cos(10πt)+3cos(20πt)+3cos(40πt)
the waveform is shown in FIG. 2(a), in which f4Is a composite signal of a cosine signal and a discontinuity signal, f5And f6Is a complex cosine signal. SMVMD and MEMD decomposition are respectively carried out on the three-channel signal, the pair of the BIMF of the eigenmode decomposed by the SMVMD and the original signal component is shown in fig. 2(b), and the IMF of the eigenmode of the first four scales decomposed by the MEMD is shown in fig. 2 (c). SMVMD adaptively decomposes 4 modes, which are basically consistent with the mode of the original signal, and breaks the signal f4Good separation without affecting the non-intermittent signal f5And f6The mode of (a). The mode decomposed by the MEMD algorithm has 7 scales, wherein the first 4 modes are main modes, but the IMF1 has serious mode aliasing phenomenon, and all the modes of the original signal are not completely separated. Therefore SMVMD has better modal aliasing rejection performance than MEMD.
3. The test algorithm constructs a three-channel signal for the modal separation effect of noise-containing non-stationary intermittent signals, and the components are as follows:
f8＝2cos(10πt)+3cos(20πt)+4cos(50πt)，
f9＝3cos(20πt)+2cos(50πt)+η2
the waveform is shown in FIG. 3(a), in which f7Non-stationary intermittent signals, f, synthesized for cosine signals and intermittent signals8And f9Is a complex cosine signal. SMVMD will f7、f8、f9The comparison of the decomposed eigenmodes with the original signal components is shown in FIG. 3(b), where SMVMD removes f7And f9White noise of the channel will be f7Different scale modal separation of channel discontinuity signals without affecting f8The mode of the channel.
In a simulation experiment, the decomposition effects of the MVMD method and the SMVMD method are similar, but the MVMD has to optimize two parameters: as shown in table 1, SMVMD parameters in the three simulation experiments are the same and do not need special optimization, but the optimal parameter α of MVMD in the three experiments is different, and the optimal decomposition effect can be obtained only by setting the correct number of decomposition scales and optimizing the parameter α.
TABLE 1 parameter α of MVMD and SMVMD in simulation experiments
4. The SMVMD method is applied to the analysis of real electroencephalogram data. The electroencephalogram data used by the invention come from an international electrophysiology Portal IEEG-Portal cloud platform (https:// www.ieeg.org /), and 4 intracranial electroencephalograms (iEEG) before and after the epileptic seizure of dogs (D1, D2, D3 and D4) in the database are selected. The electroencephalogram is recorded for a long time, wherein the recording time of D1 is 45 days, the time for continuously acquiring the electroencephalogram of the other three electroencephalograms exceeds one year, and the naturally occurring partial epilepsy has 107 times. The early prediction before the epileptic seizure is realized by pattern recognition of electroencephalogram signals in the early stage of the epileptic seizure (65 minutes before the epileptic seizure to 5 minutes before the epileptic seizure) and in the inter-seizure period (more than 7 days away from the epileptic seizure). The electroencephalogram signals are obtained by implanting a NeuroVista dynamic monitoring device into the brain for collection, have 16 channels in total, are symmetrically placed on the left hemisphere and the right hemisphere of the brain, and have the sampling rate of 400 Hz. The electroencephalogram data of each period of early-stage seizure and interval-of-seizure are continuously recorded for 1 hour, so that overfitting caused by the problem of class imbalance during model training is avoided, the number of the two types of electroencephalogram data is the same, and the electroencephalogram data of the interval-of-seizure are randomly selected. Firstly, segmenting electroencephalogram data, wherein each sample data is an electroencephalogram signal of 2 seconds, and the total number of sampling points is 800. Samples containing significant electromyographic interference were removed, and the number of samples is shown in table 2. As the features extracted at the later stage have significant difference in a high-frequency gamma wave band, 30Hz-80Hz band-pass filtering is carried out on the sample data, and SMVMD decomposition is applied to obtain eigenmode functions of different scales. Wherein the value range of the SMVMD parameter alpha is set as [200,500 ]. Taking the tested D4 as an example, the histogram of the central frequency distribution of the eigenmode of different scales resolved from the data of the pre-seizure and the inter-seizure periods is shown in fig. 4(a) and 4(b), respectively, and the scales with the central frequencies in the ranges of 30Hz to 40Hz, 50Hz to 60Hz and 60Hz to 70Hz are the main modes, wherein only one scale is in the range of 60Hz to 70Hz, and the features of the two types of samples in the eigenmode are significantly different, so the mode of the scale is selected to extract the features.
Features are then extracted, the 17 features extracted including: average cross-correlation coefficient between left and right half brains, average cross-sample entropy (maximum matching template length is 4, there are 4 features), average mutual recursion (including repetition rate, certainty index, average diagonal length, maximum diagonal length, diagonal length entropy, hierarchy, maximum vertical line length, type 1 repetition time, type 2 repetition time, repetition time entropy, transitivity, total 11 features) and average energy of all channels.
And finally, using a Support Vector Machine (SVM) for classification prediction, selecting a Gaussian kernel function with the kernel scale of 1, and using 25% for testing by adopting a leave-out method. And evaluating the prediction performance by continuously recording the electroencephalogram signals of a period of time, and when the number of samples predicted to be in the early stage of the epileptic seizure in a period of time exceeds 50%, judging that the period of time is in the early stage of the epileptic seizure. An optimal time window length is selected from 2s (one sample) to 5min (150 samples) to simultaneously ensure the sensitivity and high accuracy of prediction. Sensitivity TP/(TP + FN), specificity TN/(TN + FP), and error prediction rate FPR were calculated, respectively. Wherein TP is the true rate, namely the number of data segments correctly identified as the period of onset, TN is the true negative rate, namely the number of data segments correctly identified as the period of onset, FP is the false positive rate, namely the number of data segments incorrectly identified as the period of onset, FN is the false negative rate, namely the number of data segments incorrectly identified as the period of onset, and FPR is the number of times of incorrectly predicted epileptic seizure in the electroencephalography per hour. As shown in table 2, the average prediction accuracy (sensitivity) for the four pre-seizure stages tested by the pattern recognition method was 96.40%, the average prediction accuracy (specificity) for the inter-seizure stage was 98.69%, the average error prediction rate was 0.35/h, and the detection time windows were all within 150 s. The method has high sensitivity and low error rate for epilepsy prediction.
TABLE 2 results of prediction of four epilepsy tested
The above description is only for the purpose of illustrating the preferred embodiments of the present invention and is not to be construed as limiting the scope of the present invention, which is defined in the appended claims.
Claims (3)
1. A self-adaptive signal analysis method based on continuous multivariate variational modal decomposition is characterized by comprising the following steps: (1) and constructing an enhanced Lagrange function as an objective function by using a secondary penalty term and a Lagrange multiplier, wherein in order to ensure that the original signal can be reconstructed after decomposition, the constraint condition is that the sum of the L-th scale, the mode of L-1 scales before decomposition and the unprocessed part is the original signal, and the objective function is as follows:
wherein, the original signal f (t) ═ f1(t)，f2(t)，...，fC(t)]，fk(t) is the original k channel signal, uLkIs the L-th time scale mode decomposed by the channel, fuk(t) is the fraction of the k-th channel that has not been processed by the first L-scale modal decomposition, ωLIs the center frequency, λ, of all channels on the lth time scalek(t) is the Lagrangian multiplier;
the same theoretical basis as the EMD, VMD, SVMD and MVMD algorithms, in order to realize the local stabilization of the non-stationary signal, one scale is decomposed, and the signal mode must be ensured to be narrow bandwidth; therefore, the average bandwidth of eigenmodes of the signals of all channels decomposed on each scale needs to be minimized in the objective function, i.e. the total bandwidth J on the L-th scale1The minimum, namely:
the same theoretical basis as SVMD algorithm, to avoid mode aliasing between different scales, when decomposing the mode of L-th scale, the part f of all channels where the signals are not decomposeduk(t) and the decomposed first L-1 scale modes do not contain information in the frequency band of the mode, so that the part f of the objective function that leaves all channel signals unresolveduk(t) and decomposed mode uik(t) center frequency ω at L-th scaleLAt minimum frequency response, i.e. ωLTotal frequency response J of all channels2The minimum, namely:
to avoid the occurrence of mode aliasing problems, it is also necessary to ensure that the mode of the L-th scale is at each center frequency ω of the already decomposed first L-1 scalesiThe frequency response at (i 1, 2.. gtoreq., L) is small, so the mode of the L-th scale of all channel signals is centered in the first L-1 scales in the objective functionTotal response at frequency J3The minimum, namely:
(3) formula (4) whereinL(t)，βi(t) are filters with frequency responses of:
filters are respectively at omegaLAnd ωiThe response is maximal at a frequency, thereby extracting a narrow bandwidth signal centered at that frequency;
(2) optimizing the target function, converting a time domain signal in the target function into a frequency domain according to the energy conservation theorem of the time domain and the frequency domain, and solving the minimization problem by using an alternating direction multiplier method;
(3) all the decomposed eigenmodes are sequenced according to the sequence of the central frequencies from small to large, the larger the value of the scale L is, the higher the modal frequency of the scale is, so that the eigenmodes of different frequency bands can be selected through the value of the central frequency and the value of the L to carry out feature extraction and subsequent analysis, and the eigenmodes of different frequency bands obtained after decomposition by the method also realize the self-adaptive filtering of signals.
2. The adaptive signal analysis method according to claim 1, wherein: the step (2) optimizes the target function, because the penalty factor alpha in the target function has a large influence on the decomposition of the eigen mode, and the optimal alpha values of different types of signals are greatly different during the decomposition, a heuristic method is introduced, a value range of alpha is set, the alpha is gradually increased from the minimum value to the maximum value along with the optimization iteration, the mode with the strongest energy on the current time scale is found out, and because the modes of all channels on the same time scale are similar, the same penalty factor alpha is used during the optimization iteration of each channel.
3. The adaptive signal analysis method according to claim 1, wherein: the optimization process of the objective function in the step (2) is to convert a time domain signal in the objective function into a frequency domain according to the energy conservation theorem of the time domain and the frequency domain, solve the minimization problem by using an alternating direction multiplier method, and sequentially extract the modes of all channels in each time scale until the extracted modes are noises, wherein the specific optimization process is as follows:
(1) setting a range [ alpha ] of penalty factorsmin，αmax]Convergence threshold ε1＝10-6Energy threshold ε2＝10-3Iteration parameter tau, initializing L ← 0;
(2) l ← L +1, initialization
n←0，m←1，α1←αmin；
(3) Iteration: n ← n +1, update all ω ≧ 0 and each channel k (k ═ 1, 2
Until convergence: for each channel k, the following are satisfied:
(4) initialization
n←0，
α1←αmin+emM ← m +1, repeat step (3) until αm≤αmax；
(5) Repeating the steps (2), (3) and (4) until convergence:
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