Runoff prediction method for a watershed based on time-frequency decomposition and ensemble learning model

Abstract

The application discloses a watershed runoff prediction method based on time-frequency decomposition and an integrated learning model, and relates to the technical field of watershed runoff prediction. The watershed runoff prediction method based on time-frequency decomposition and the integrated learning model comprises the following steps: S1, acquiring watershed runoff data, and analyzing variation characteristics of the watershed runoff data to obtain an original runoff sequence; S2, constructing a two-stage time-frequency decomposition and integrated learning combined model based on the original runoff sequence; and S3, predicting the runoff by using the two-stage time-frequency decomposition and integrated learning combined model to obtain a watershed runoff prediction result.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of runoff prediction methods, in particular to a runoff prediction method for a river basin based on time-frequency decomposition and an integrated learning model. BACKGROUND

[0002] Water resources are basic natural resources, strategic economic resources, and controlling elements of ecology and environment. China has abundant total water resources, but the per capita share is small, with more in the south and less in the north, more in summer and autumn and less in winter and spring, and uneven in time and space. Due to climate change in recent years, the temporal and spatial distribution of water resources has become more uneven; with the advancement of industrialization, water pollution needs to be further addressed; industrial, agricultural, and residential water use needs to be reasonably allocated. How to reasonably develop, manage, and utilize limited water resources is a problem that relevant departments and researchers are concerned about and urgently need to solve. Water resource problems constrain the further development of the national economy, and it is urgent to analyze the existing hydrological situation and master the future changes in water resources.

[0003] Hydrological forecasting can predict the hydrological state in a certain future time period based on known information, which can provide a reference for the scientific use of water resources. In particular, medium and long-term runoff forecasting can provide an important basis for the scientific and reasonable development, utilization, and management of water resources. Medium and long-term runoff forecasting usually refers to hydrological forecasting with a prediction period exceeding the maximum concentration time of the basin and more than 3 days but less than 1 year. Common time scales include years, seasons, months, and decades. Medium and long-term runoff forecasting is beneficial to making reasonable decisions for flood prevention and drought relief departments due to its long prediction period, and is a fundamental task in disaster prevention and mitigation. In reservoir operation, having high-precision monthly runoff prediction values helps reservoir managers make scientific decisions.

[0004] With climate change and frequent extreme weather, floods, droughts, and other disasters occur frequently, human activities have changed the original state of the underlying surface, and the natural runoff pattern has changed, making runoff prediction work face greater challenges. There is no universal model suitable for all basins for runoff prediction models, and appropriate models need to be constructed based on the actual situation of the basin.

[0005] In recent years, computer technology has been developing rapidly and is becoming increasingly popular in various fields. Prediction methods based on artificial intelligence have high computational efficiency and excellent computational performance, providing new ideas for medium and long-term runoff prediction. At the same time, signal decomposition technology is constantly improving, providing a new approach for predicting nonlinear and non-stationary runoff sequences. Machine learning algorithms can extract key information from large amounts of measured data and discover internal patterns in the data to establish models. As a branch of machine learning, integrated learning has high computational efficiency and superior prediction performance.

[0006] At present, domestic and foreign researchers have established a large number of models to carry out research on medium and long term runoff prediction, but there are still some problems to be solved:

[0007] (1) In karst area basin, many researchers carry out medium and long term runoff prediction research by establishing artificial intelligence algorithm model, however, as a branch of machine learning model, the application of ensemble learning model in karst area basin runoff prediction is less.

[0008] (2) Some researchers use single ensemble learning model to carry out medium and long term runoff prediction in some non-karst areas, and lack of research on combined model combining time-frequency decomposition and ensemble learning model. SUMMARY

[0009] In view of the problems in the related art, the present application proposes a basin runoff prediction method based on time-frequency decomposition and ensemble learning model to overcome the above technical problems existing in the prior art.

[0010] To this end, the specific technical solutions adopted by the present application are as follows:

[0011] The basin runoff prediction method based on time-frequency decomposition and ensemble learning model comprises

[0012] Further, the basin runoff prediction method based on time-frequency decomposition and ensemble learning model comprises the following steps:

[0013] S1, obtaining basin runoff data and analyzing its variation characteristics to obtain original runoff sequence;

[0014] S2, constructing a two-stage time-frequency decomposition and ensemble learning combined model based on the original runoff sequence;

[0015] S3, predicting runoff by using the two-stage time-frequency decomposition and ensemble learning combined model to obtain basin runoff prediction results.

[0016] Further, the obtaining of basin runoff data and the analysis of its variation characteristics to obtain the original runoff sequence comprises the following steps:

[0017] S11, obtaining basin runoff data, and performing annual characteristic analysis and interannual characteristic analysis by using mathematical statistics method;

[0018] S12, respectively using linear regression method, Pettitt method and Morlet wavelet method to analyze runoff trend, runoff mutation and runoff period of the basin runoff data.

[0019] Further, the analysis of the runoff period is calculated by using the wavelet transform system formula, wherein the wavelet transform system formula is:

[0020]

[0021] where ω f (a,b) are wavelet transform coefficients, R is a real field, t is time, is a conjugate complex number, a is a scale shrinkage factor, and b is a time translation factor;

[0022] is a continuous wavelet, and is a family of functions after scaling and translation, and the calculation formula is as follows:

[0023]

[0024] Further, the constructing a two-stage time-frequency decomposition and integrated learning combined model based on the original runoff sequence comprises the following steps:

[0025] S21, decomposing the original runoff sequence by using a complementary ensemble empirical mode decomposition method to obtain subsequence components;

[0026] S22, decomposing the subsequence components by using a variational mode decomposition method to obtain new subsequences;

[0027] S23, predicting the new subsequences by using a pre-constructed light gradient boosting machine model to obtain new subsequence prediction values based on two-stage decomposition.

[0028] Further, the decomposing the original runoff sequence by using a complementary ensemble empirical mode decomposition method to obtain subsequence components comprises the following steps:

[0029] S211, adding positive and negative white noise n(t) to the original sequence x(t) to form new sequences x + (t) and x - (t):

[0030] x + (t) = x(t) + n(t)

[0031] x - (t) = x(t) - n(t)

[0032] S212, decomposing the new sequences x + (t) and x - (t) by using an empirical mode decomposition method to obtain imf1 and imf2, respectively;

[0033] S213, repeating S211 and S212 to obtain two groups of integrated imf1 and imf2:

[0034]

[0035]

[0036] S214, calculating a set of positive and negative noise mean values imf:

[0037]

[0038] S215, reconstructing a signal, defining c i (t) is expressed as c i (t) and residual r n (t), the decomposition result of the complementary set empirical mode decomposition is:

[0039]

[0040] In the formula, n represents the number of the set of positive and negative noise mean values.

[0041] Further, the subsequence components include five intrinsic mode function components and one trend item component.

[0042] Further, the constraint condition when the subsequence components are decomposed by the variational mode decomposition method includes:

[0043] The sum of the bandwidths of the center frequencies of each mode component is minimum;

[0044] The sum of all mode components is equal to the original signal.

[0045] Further, the expression of the constraint condition is:

[0046]

[0047] In the formula, {u k}, {w k} respectively correspond to the kth mode component and the center frequency after decomposition; is a gradient operation; δ(t) is a Dirac function; j is an imaginary unit; t is time; * is a convolution symbol; u k (t) is the tth data under the kth decomposed mode; is an estimated center frequency; ω k ={ω1,ω2,ω3,...,ω k} is each center frequency set; s.t. is a constraint condition; K is the number of modes to be decomposed; u k ={u1,u2,...,u k} is each mode function set; f(t) is an original signal.

[0048] Further, the step of predicting the runoff by using the two-stage time-frequency decomposition and integrated learning combined model to obtain the basin runoff prediction result includes the following steps:

[0049] S31, respectively, obtain a new sub-sequence prediction value based on two-stage decomposition and a one-stage decomposition other sequence prediction value;

[0050] S32, linearly sum the new sub-sequence prediction value based on two-stage decomposition and the one-stage decomposition other sequence prediction value to obtain a basin runoff prediction result.

[0051] Further, the calculation of the one-stage decomposition other sequence prediction value comprises the following steps:

[0052] Obtain other sub-sequence components decomposed by the complementary ensemble empirical mode decomposition method on the original runoff sequence, and predict the other sub-sequence components by using a pre-constructed lightweight gradient boosting machine model to obtain the other sequence prediction value decomposed by the complementary ensemble empirical mode decomposition method.

[0053] The beneficial effects of the present application are:

[0054] (1) By analyzing the time-frequency characteristic data of the basin runoff, and using the analysis result to construct a combined model based on two-stage time-frequency decomposition and integrated learning, the runoff can be predicted by using the constructed combined model to obtain the basin runoff prediction result. Compared with the traditional runoff analysis method, the present application can effectively improve the prediction accuracy of the decomposed sub-sequence components by two-stage time-frequency decomposition, thereby improving the overall prediction accuracy of the model, and providing a new idea for medium and long term runoff prediction.

[0055] (2) By combining the complementary ensemble empirical mode decomposition method with the variational mode decomposition method, the monthly runoff sequence can be decomposed into two stages of time-frequency, thereby effectively avoiding the phenomenon of mode aliasing to a certain extent, and the runoff sequence can be decomposed into more stable runoff components, thereby effectively improving the prediction accuracy.

[0056] (3) The combined model constructed by decomposition and prediction can effectively improve the prediction effect compared with a single prediction model, and the complementary ensemble empirical mode decomposition method and the variational mode decomposition method can effectively improve the applicability of the runoff sequence prediction accuracy. BRIEF DESCRIPTION OF DRAWINGS

[0057] In order to more clearly illustrate the technical solutions in the embodiments of the present application or the prior art, the following will briefly introduce the drawings needed in the embodiments. Obviously, the drawings described below are only some embodiments of the present application, and other drawings can be obtained by those skilled in the art without creative labor.

[0058] Figure 1 is a flow chart of a basin runoff prediction method based on time-frequency decomposition and integrated learning model according to an embodiment of the present application;

[0059] Figure 2 is a schematic view of 6 new sub-sequences generated by the variational mode decomposition in the watershed runoff prediction method based on time-frequency decomposition and ensemble learning model according to the embodiment of the present application. DETAILED DESCRIPTION

[0060] To further explain the embodiments, the present application provides drawings which are part of the disclosure of the present application, mainly used to illustrate the embodiments, and can explain the operating principle of the embodiments in conjunction with the related description of the specification. Those of ordinary skill in the art can understand other possible implementations and advantages of the present application by referring to these contents. The components in the drawings are not drawn to scale, and similar component symbols are generally used to represent similar components.

[0061] According to the embodiment of the present application, a watershed runoff prediction method based on time-frequency decomposition and ensemble learning model is provided.

[0062] The present application will be further described in conjunction with the drawings and specific embodiments, as shown in the drawings, the watershed runoff prediction method based on time-frequency decomposition and ensemble learning model according to the embodiment of the present application includes the following steps: Figure 1

[0063] S1, obtaining watershed runoff data and analyzing its variation characteristics to obtain the original runoff sequence;

[0064] The obtaining of the watershed runoff data and the analysis of its variation characteristics to obtain the original runoff sequence includes the following steps:

[0065] S11, obtaining watershed runoff data, and performing intra-annual characteristic analysis and inter-annual characteristic analysis by using mathematical statistics method;

[0066] S12, respectively using linear regression method, Pettitt method and Morlet wavelet method to analyze the runoff trend, runoff mutation and runoff period of the watershed runoff data.

[0067] Specifically, in the present embodiment, linear trend method is used to analyze the runoff sequence of each month respectively. Linear trend method reflects the linear trend by constructing a linear regression equation, and the calculation formula is as follows:

[0068] y=ax+b

[0069] In the formula, a is the regression coefficient, x is the monthly runoff, and b is the intercept.

[0070] Specifically, in the present embodiment, Pettitt method is used to analyze the mutation of monthly runoff sequence. Pettit detection method is used to detect unknown mutation points in time series. The calculation steps of this method are simple, and the result is clear. The specific principle of the algorithm is as follows:​

[0071] For a known time series {Xi, i = 1, 2, ..., n}, define a statistical function U. t :

[0072]

[0073] In the formula, x t x k Let t and k be the time series samples, respectively, and sgn(.) be the sign function.

[0074]

[0075] Using the maximum value of the Ut sequence, we define a statistic Kt to represent the most likely mutation point:

[0076] P = 2exp{-6K} t 2 / (n 3 +n 2 )}

[0077] Generally, when P < 0.5, the change is considered significant, meaning that the sequence data has abruptly changed at that point.

[0078] Specifically, the runoff period is calculated using a wavelet transform system formula, which is:

[0079]

[0080] In the formula: ω f (a,b) are wavelet transform coefficients, R is the real number domain, and t is time. Let a be a conjugate complex number, a be the scale contraction factor, and b be the time shift factor.

[0081] It is a continuous wavelet, which is The family of functions after scaling and translation is calculated using the following formulas:

[0082]

[0083] S2. Construct a combined model based on two-stage time-frequency decomposition and ensemble learning using the original runoff sequence;

[0084] The step of constructing a combined model based on two-stage time-frequency decomposition and ensemble learning using the original runoff sequence includes the following steps:

[0085] S21. The original runoff sequence was decomposed using the complementary set empirical mode decomposition method, resulting in 5 imf components and 1 trend term (Res).

[0086] Complementary ensemble empirical mode decomposition(CEEMD) is an improved decomposition method based on empirical mode decomposition(EMD) and ensemble empirical mode decomposition(EEMD), which solves the mode mixing problem of EMD decomposition and the problems of large reconstruction error and poor decomposition completeness of EEMD. CEEMD decomposition can effectively improve the decomposition efficiency by adding positive and negative auxiliary white noise in the original signal. The steps of CEEMD decomposition are as follows:

[0087] S211, adding positive and negative white noise n(t) in the original sequence x(t) to form new sequences x + (t) and x - (t):

[0088] x + (t)=x(t)+n(t)

[0089] x - (t)=x(t)-n(t)

[0090] S212, decomposing new sequences x + (t) and x - (t) by using empirical mode decomposition method to obtain imf1 and imf2 respectively;

[0091] S213, repeating S211 and S212 to obtain two groups of integrated imf1 and imf2:

[0092]

[0093]

[0094] S214, calculating the ensemble imf with the mean value of positive and negative noise:

[0095]

[0096] S215, reconstructing the signal, defining c i (t) as imf, and the original sequence x(t) is expressed as the sum of c i (t) and residual r n (t), then the decomposition result of complementary ensemble empirical mode decomposition is:

[0097]

[0098] In the formula, n represents the number of ensembles with positive and negative noise mean values;

[0099] The algorithm process of empirical mode decomposition method includes:

[0100] The maximum and minimum points of the sequence are found in the original sequence x(t), and the maximum points are connected to obtain the upper envelope function x max (t) using a fitting function, and the lower envelope function x min (t) is obtained in the same way; the mean value functions are calculated from the upper and lower envelope functions, and the mean value function m1(t) is obtained; the mean value function m1(t) is subtracted from the original sequence x(t) to obtain a new sequence function h1(t):

[0101]

[0102] h1(t)=x(t)-m1(t)

[0103] If h1(t) satisfies the IMF condition, then h1(t) is the first intrinsic mode function IMF1; if h1(t) satisfies the IMF condition, then

[0104]

[0105] c1(t) is removed from x(t) to obtain r1; the process is repeated to obtain n IMF components until the sequence residual component r n is a monotonic function:

[0106] r1(t)=x(t)-c1(t)

[0107] The final EMD decomposition result:

[0108]

[0109] S22, decompose the subsequence component IMF1 by variational mode decomposition to obtain six new subsequences (the subsequence is shown in Figure 2 ).

[0110] Variational mode decomposition (VMD) is a signal processing method proposed by Konstantin Dragomiretskiy in 2014 based on EMD and its variants. VMD overcomes the problems of end effect and mode component aliasing in EMD, has a solid and complete mathematical theory as support, does not need to be solved iteratively, avoids error accumulation caused by iteration, can reduce the complexity and nonlinearity of time series non-stationarity, and can obtain sub-sequences containing multiple different frequency scales and relatively stable subsequences. It is suitable for non-stationary sequences.

[0111] VMD is an adaptive, fully nonrecursive mode variational and sequential processing method. The adaptivity of the decomposition method is reflected in that the number of modal decompositions can be determined autonomously according to the specific situation, and then the optimal center frequency and limited bandwidth of each mode are adaptively matched to achieve effective separation of imfs and obtain effective decomposition components of the given signal. Konstantin Dragomiretskiy proves through experimental results that the method is more robust in terms of sampling and noise;

[0112] The decomposition process of VMD is the solution process of the variational problem, and the core idea of the algorithm is to construct and solve the variational problem. The constraint condition of the solution process is that the sum of the bandwidths of the center frequencies of each modal component is minimized; the sum of all modal components is equal to the original signal, and the variational constraint expression is:

[0113]

[0114] In the formula, {u k} and {w k} correspond to the kth modal component and the center frequency after decomposition, respectively; is the gradient operation; δ(t) is the Dirac function; j is the imaginary unit; t is the time; * is the convolution symbol; u k (t) is the tth data under the kth decomposed mode; is the estimated center frequency; ω k ={ω1,ω2,ω3,...,ω k} is the set of center frequencies; s.t. is the constraint condition; K is the number of modes to be decomposed; u k ={u1,u2,...,u k} is the set of modal functions; f(t) is the original signal.

[0115] There are six parameters in VMD decomposition, the meanings and value ranges of the parameters are shown in Table 1. Among the six parameters, the number of decomposition modes K is the key parameter. If K is too large, the component information will be redundant, and too many decomposition modes will lead to a complex prediction process. If K is too small, the information in the original sequence cannot be reflected, so the value of K should be appropriate. According to the number of decomposition modes of EMD and CEEMD as an empirical reference, the value of K of VMD is 6. The value of Alpha is 1.5-2.0 times the length of the sequence. The remaining four parameters (tau, DC, init, and tol).

[0116] Table 1 Parameter setting of VMD decomposition method

[0117]

[0118] S23, predicting the new sub-sequence by using the pre-constructed light gradient boosting machine model (LightGBM model) to obtain the new sub-sequence prediction value based on two-stage decomposition.

[0119] S3, predicting the runoff by using the two-stage time-frequency decomposition and ensemble learning combined model to obtain the runoff prediction result of the basin;

[0120] The step of predicting the runoff by using the two-stage time-frequency decomposition and ensemble learning combined model to obtain the runoff prediction result of the basin comprises the following steps:

[0121] S31, respectively obtaining the new sub-sequence prediction value based on two-stage decomposition and the other sequence prediction value of one-stage decomposition;

[0122] Specifically, the calculation of the other sequence prediction value of one-stage decomposition comprises the following steps:

[0123] Obtain other sub-sequence components (i.e. other sub-sequence components excluding imf1) decomposed from the original runoff sequence by using the complementary ensemble empirical mode decomposition method, and predict the other sub-sequence components by using the pre-constructed light gradient boosting machine model to obtain the other sequence prediction value decomposed by using the complementary ensemble empirical mode decomposition method.

[0124] S32, linearly summing the new sub-sequence prediction value based on two-stage decomposition (i.e. the prediction value of imf1) and the other sequence prediction value of one-stage decomposition to obtain the runoff prediction result of the basin.

[0125] Through the analysis of the time-frequency characteristic data of the runoff of the basin, and by using the analysis result to construct the two-stage time-frequency decomposition and ensemble learning combined model, the runoff can be predicted by using the constructed combined model to obtain the runoff prediction result of the basin. Compared with the traditional runoff analysis method, the two-stage time-frequency decomposition can effectively improve the prediction accuracy of the decomposed sub-sequence components, thereby improving the overall prediction accuracy of the model, and providing a new idea for medium and long term runoff prediction. The two-stage time-frequency decomposition of the monthly runoff sequence can be performed by using the complementary ensemble empirical mode decomposition method combined with the variational mode decomposition method, thereby effectively avoiding the mode aliasing phenomenon to some extent, decomposing the runoff sequence into more stable runoff components, and thereby effectively improving the prediction accuracy. The combined model constructed by the decomposition and prediction mode can effectively improve the prediction effect compared with the single prediction model, and the complementary ensemble empirical mode decomposition method and the variational mode decomposition method can effectively improve the applicability of the runoff sequence prediction accuracy.

[0126] The above merely provides the preferred embodiment of the present application, and is not used to limit the present application. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present application should be included in the protection scope of the present application.

Claims

1. A watershed runoff prediction method based on time-frequency decomposition and ensemble learning model, characterized in that, The watershed runoff prediction method based on time-frequency decomposition and ensemble learning model includes the following steps: S1. Obtain watershed runoff data and analyze its variation characteristics to obtain the original runoff sequence; S2. Construct a combined model based on two-stage time-frequency decomposition and ensemble learning using the original runoff sequence; S3. Runoff is predicted using a combined model based on two-stage time-frequency decomposition and ensemble learning to obtain watershed runoff prediction results; The method of constructing a combined model based on two-stage time-frequency decomposition and ensemble learning using the original runoff sequence includes the following steps: S21. The original runoff sequence is decomposed using the complementary set empirical mode decomposition method to obtain subsequence components; S22. The subsequence components are decomposed by variational mode decomposition to obtain new subsequences; S23. Use a pre-built lightweight gradient booster model to predict the new subsequence and obtain the new subsequence prediction value based on two-stage decomposition. The process of decomposing the original runoff sequence using the complementary set empirical mode decomposition method to obtain subsequence components includes the following steps: S211. Add positive and negative white noise n(t) to the original sequence x(t) to form a new sequence x. + (t) and x - (t): x + (t)=x(t)+n(t) x - (t)=x(t)-n(t); S212. Decompose the new sequence x using the empirical mode decomposition method. + (t) and x - (t), respectively, to obtain imf1 and imf2; S213, Repeat S211 and S212 to obtain two sets of ensembles imf1 and imf2: S214. Calculate the set imf with both positive and negative noise means: S215, Reconstruct the signal, define c i x(t) is an imf sequence, and the original sequence x(t) is represented as c. i (t) and residual r n The sum of (t) gives the result of the empirical mode decomposition of complementary sets: In the formula, n represents the number of sets with both positive and negative noise means; The method of predicting runoff using a combined model based on two-stage time-frequency decomposition and ensemble learning to obtain watershed runoff prediction results includes the following steps: S31. Obtain the new subsequence prediction value and the other sequence prediction values ​​based on the two-stage decomposition, respectively. S32. The new subsequence prediction value based on the two-stage decomposition is linearly summed with the other sequence prediction values ​​of the first decomposition to obtain the watershed runoff prediction result. The calculation of other sequence predictions for the first decomposition includes the following steps: The other subsequence components after the original runoff sequence is decomposed by complementary set empirical mode decomposition are obtained, and the other subsequence components are predicted using a pre-built lightweight gradient booster model, thus obtaining the predicted values ​​of other sequences decomposed by complementary set empirical mode decomposition.

2. The watershed runoff prediction method based on time-frequency decomposition and ensemble learning model according to claim 1, characterized in that, The process of acquiring watershed runoff data and analyzing its variation characteristics to obtain the original runoff sequence includes the following steps: S11. Obtain watershed runoff data and use mathematical statistics to conduct intra-annual and inter-annual characteristic analysis; S12. The linear regression method, Pettitt method and Morlet wavelet method were used to analyze the runoff trend, runoff mutation and runoff cycle of the watershed runoff data.

3. The watershed runoff prediction method based on time-frequency decomposition and ensemble learning model according to claim 2, characterized in that, The runoff period analysis is performed using wavelet transform system formulas, whereby the wavelet transform system formulas are: In the formula: ω f (a,b) are wavelet transform coefficients; R is the field of real numbers; t represents time; It is a conjugate complex number; a is the scaling contraction factor; b is the time shift factor; It is a continuous wavelet, which is A family of functions after scaling and translation. The calculation formula is:

4. The watershed runoff prediction method based on time-frequency decomposition and ensemble learning model according to claim 3, characterized in that, The subsequence components include five intrinsic mode function components and one trend term component.

5. The watershed runoff prediction method based on time-frequency decomposition and ensemble learning model according to claim 4, characterized in that, The constraints for decomposing subsequence components using variational mode decomposition include: The sum of the bandwidths of the center frequencies of each modal component is minimized; The sum of all modal components equals the original signal.

6. The watershed runoff prediction method based on time-frequency decomposition and ensemble learning model according to claim 5, characterized in that, The expression for the constraint condition is: In the formula, {u k }、{w k } These correspond to the k-th modal component and the center frequency after decomposition, respectively; This is for gradient calculation; δ(t) is the Diclave function; j is the imaginary unit; t represents time; * indicates the convolution symbol; u k (t) represents the t-th data point in the k-th decomposition mode; To estimate the center frequency; ω k ={ω1,ω2,ω3,...,ω k } represents the frequency set of each center; st represents the constraint condition; K is the number of modes that need to be decomposed; u k ={u1,u2,...,u k } represents the set of modal functions; f(t) is the original signal.

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