CN103439070B - A kind of separation method of bridge Long-term Deflection effect - Google Patents

Abstract

The open a kind of bridge Long-term Deflection separation method of the present invention, belongs to Bridge Health Monitoring Technology field.This separation method, according to the characteristic of each composition of deflection of bridge span, utilizes M LS SVM model of fit to set up temperature and the non-linear relation of temperature amount of deflection effect.The training step of this M LS SVM includes: by the temperature data empirical mode decomposition of bridge, the temperature data after decomposing is as the input signal of M LS SVM；With subtraction clustering algorithm, sample data is classified；By sorted sample data, each sub-LS SVM model is trained, thus sets up each sub-LS SVM model；Use principal components regression method to carry out the predicted value of comprehensive multiple models, eliminate the data dependence between local system, improve robustness and the generalization ability of system, ultimately form M LS SVM model.The precision utilizing the match value of the Long-term Deflection that this separation method obtains is high.

Description

A kind of separation method of bridge Long-term Deflection effect

Technical field

The present invention relates to the separation method of a kind of bridge Long-term Deflection effect, especially relate to a kind of based on many least squares Support vector machine (multiple least square support vector machine, M-LS-SVM) bridge Long-term Deflection The separation method of effect, belongs to Bridge Health Monitoring Technology field.

Background technology

The amount of deflection of bridge is one of important indicator of reflection bridge structure safe, utilizes various deflection metrology method, obtains The ANOMALOUS VARIATIONS signal of structural behaviour, finds the potential safety hazard of large scale structure early, it has also become the research direction of civil structure, but Although modern computing means constantly make progress, Accurate Prediction greatly across the Long-term Deflection of prestressed concrete be not the most an appearance Easy work.

The signal that Long Period Health Monitoring obtains usually contains the common effect of many factors, such as environmental effect, load effect With the shrinkage and creep effect etc. of concrete, the research to signals such as large scale structure measured displacements, strains shows, temperature is impact letter The main factor of number change, the damage signal of structure often " is flooded " by its institute, so that cannot be directly according to measured signal pair The safe condition of structure makes correct evaluation.Liu Xia equality is in " in bridge deflection monitoring based on LS-SVM, temperature effects separates " Disclosed in utilize the method for LS-SVM model separation temperature effects, its analogue signal obtained and measured signal Calculation results Showing, temperature amount of deflection based on LS-SVM separates, and match value and actual value maximum relative error are less than 5%.

Summary of the invention

It is an object of the invention to provide a kind of separation method based on M-LS-SVM bridge Long-term Deflection effect, utilizing should The match value that separation method obtains is less with actual value maximum relative error, i.e. precision is higher.

The separation method of bridge Long-term Deflection effect of the present invention includes following two step:

1) train M-LS-SVM disjunctive model by sample data, set up M-LS-SVM disjunctive model；Sample data is divided into M- The input sample data of LS-SVM model and output sample data, input sample data is the temperature difference of the denoising after filtering of actual measurement Data, output sample data is removal load effect and the deflection of bridge span effect of noise after actual measurement;

2) Long-term Deflection is isolated by M-LS-SVM disjunctive model.

Above-mentioned M-LS-SVM disjunctive model is set up by following steps:

The sample data needed for M-LS-SVM is set up in a, acquisition；

B, the temperature data of bridge empirical mode decomposition (Empirical Mode Decomposition, EMD) is entered Row decomposes, and the temperature data after decomposing is as the input signal of M-LS-SVM；

C, with subtraction clustering algorithm, sample data is classified；

D, by sorted sample data to each sub-least square method supporting vector machine (least square support Vector machine, LS-SVM) model is trained, thus sets up each sub-LS-SVM model；

E, employing principal components regression (principal component regression, PCR) method carry out comprehensive all sub-LS- The predicted value of SVM model, eliminates the data dependence between local system, improves robustness and the generalization ability of system, finally Form M-LS-SVM model.

The step obtaining the sample data needed for setting up M-LS-SVM includes:

A) at time point t₁, temperature difference T₁, in the case of vehicular load, measure amount of deflection s of bridge_{Without car load 1}(t₁,T₁)；

B) at time point t₂, temperature difference T₁, in the case of vehicular load, measure amount of deflection s of bridge_{Without car load 2}(t₂,T₁)；Same Measuring under the temperature difference of sample is the change in order to avoid altitude temperature difference effect；

C) time point t it is positioned at₁With time point t₂Between the Long-term Deflection of time point t obtained by mathematical model one:

L (t, T)=s(t, T)-(T-T₁)×δ(t)

Wherein, s (t, T) is time point t, temperature difference T, there is filtered amount of deflection during car load；L(t, T) it is time point t, temperature Long-term Deflection during difference T, δ (t) is a coefficient；Be can get by mathematical model one and multiple be positioned at time point t₁With time point t₂Between The Long-term Deflection value of time point.

The Long-term Deflection utilizing mathematical model one to calculate time point t can produce error, and the error occurred uses linear correction Method is modified, whereinGo to revise L(t, T with Δ), the Long-term Deflection value of the most each time point Can accurately determine.

For p data point X in q dimension space₁,X₂,...,X_p, with subtraction clustering algorithm, sample data is classified Step include:

A) data point X is obtained by mathematical model two_i(i=1,2 ..., p) the density index D at place_i

$D_i=\underset{j=1}{\overset{p}{\mathrm{\Σ}}}\exp [-\frac{{\left|\right|X_i-X_j\left|\right|}^2}{{\left(0.5\mathrm{\η}\right)}^2}]=\underset{j=1}{\overset{p}{\mathrm{\Σ}}}\exp [-\underset{k=1}{\overset{q}{\mathrm{\Σ}}}\frac{{(x_{i.k}-x_{j.k})}^2}{{\left(0.5\mathrm{\η}\right)}^2}]$

$\mathrm{\η}=\frac{1}{2}\underset{k}{\min }\left\{\underset{i}{\max }\right||X_i-X_k|\left|\right\}$

Wherein, η is an X_iCluster radius, the density index of this point is contributed little by the data point beyond radius；By number Strong point X_i(i=1,2 ..., the 1st cluster centre X p) it is respectively defined as from high to low by density index_c1, the 2nd cluster centre X_c2..., pth cluster centre X_cp；And define D_ck(k=1,2 ..., p) it is data point X_ckDensity index；

B) density index for each data point is modified by mathematical model three

$D_i=D_{i-1}-D_{\mathrm{ck}}\exp [-\frac{{\left|\right|X_i-X_{\mathrm{ck}}\left|\right|}^2}{{\left(0.5\mathrm{\β}\right)}^2}]$

β=(1 ~ 2) η

β defines the neighborhood that density index function substantially reduces, β=(1 ~ 2) η can avoid selected center each other away from Close to too；Select data point X that density index after correction is the highest_ck+1As new cluster centre；Work as D_iBy this data point during ＜ 0 Density index be set to zero；

C) judgeWhether setting up, if being false, then forwarding step b) to；If setting up, exiting, and thus terminating to gather Class process；ε is for a setup parameter, and this parameter determines the initialization cluster centre number of final generation, by ensureing to be chosen The heart can preferably represent input data and number of clusters is suitable, and the value of ε should exist with cluster centre number here Between, whereinFor rounding symbol, p is the number of samples that need to carry out classifying.

The step isolating Long-term Deflection by M-LS-SVM disjunctive model includes:

A, to measure relevant bridge signal (including dynamic deflection, the temperature difference etc.) be filtered, temperature signal EMD is divided Solve；

B, using temperature difference data as M-LS-SVM mode input, the output result obtained be exactly corresponding bridge temperature effect Should；

C, by actual measurement dynamic deflection signal and bridge temperature effects signal subtraction, obtain bridge Long-term Deflection signal.

Relative to prior art, present invention separation method based on M-LS-SVM bridge Long-term Deflection effect, the length obtained The match value of phase amount of deflection is less with the actual value maximum relative error of Long-term Deflection, and precision is high.

Accompanying drawing explanation

Below in conjunction with accompanying drawing and embodiment, the present invention is described further.

Fig. 1 is M-LS-SVM model of the present invention.

Fig. 2 is the temperature signal that background bridge is actual.

Fig. 3 a is the temperature amount of deflection effect data that background bridge is actual.

Fig. 3 b is the temperature amount of deflection effect data of LS-SVM matching.

Fig. 4 a is the dynamic deflection value that background bridge is actual.

Fig. 4 b is background bridge Long-term Deflection LS-SVM matched curve and Midas value of calculation.

Fig. 4 c is the mistake by the Long-term Deflection value of temperature amount of deflection effect after M-LS-SVM separation with the long-term deflection data of reality Difference.

Detailed description of the invention

The signal obtained in deflection of bridge structure long term monitoring, contains vehicular load, wind load, concrete shrinkage Xu The effect of the effect such as change, loss of prestress, variations in temperature, structural damage and environment noise, therefore, the structural response that monitoring obtains Increment signal and temperature signal are the superpositions of multiple effect.Consider to affect the principal element of amount of deflection, according to PCA, Deflection of bridge structure response signal Y and temperature signal T may be considered following several effect sum:

$\left\{\begin{array}{c}Y=Y_T+Y_P+Y_L+Y_R\\ T=T_d+T_f+T_y\end{array}\right.$

Wherein: Y_TIt is temperature amount of deflection effect, Y_PIt is load amount of deflection effect, Y_LIt is that structure Long-term Deflection (includes concrete shrinkage The amount of deflection effect of creep deflection, loss of prestress and structural damage etc.), Y_RIt it is the test error amount of deflection effect of system；T_dIt it is day temperature Difference, T_fIt is abrupt temperature drop, T_yBeing annual range of temperature, in formula, various effects and the temperature difference are all the functions of time t.In deflection of bridge structure group In Fen, the frequency ratio of load effect is higher, if regarding hour as the second, load amount of deflection frequency is generally hundreds of hertz；Temperature difference per day effect Referring to that the temperature of a day changes and cause the knots modification of structural deflection, its frequency ratio load amount of deflection frequency is much smaller, if regard hour as Second, then the amplitude of Long-term Deflection and annual range of temperature deflection signals is mainly distributed on the low frequency part of less than 0.01 hertz, temperature effects In annual range of temperature to affect yardstick close with the time scale of structure Long-term Deflection, and there is lap.

Such as Fig. 1, the separation method of bridge Long-term Deflection effect of the present invention includes two big steps:

1) train M-LS-SVM disjunctive model by sample data, thus set up M-LS-SVM disjunctive model；

2) Long-term Deflection is isolated by M-LS-SVM disjunctive model.

Above-mentioned M-LS-SVM disjunctive model is set up by following steps:

A, in order to set up M-LS-SVM disjunctive model, the sample data setting up M-LS-SVM model, sample data need to be obtained Being divided into input sample data and the output sample data of M-LS-SVM model, input sample data is the denoising after filtering of actual measurement Temperature difference data, output sample data is to remove load effect and the deflection of bridge span effect of noise after actual measurement;M-LS-is set up in acquisition The step of the sample data needed for SVM includes:

A) at time point t₁, temperature difference T₁, in the case of vehicular load, measure amount of deflection s of bridge_{Without car load 1}(t₁,T₁)；

B) at time point t₂, temperature difference T₁, in the case of vehicular load, measure amount of deflection s of bridge_{Without car load 2}(t₂,T₁)；Same Measuring under the temperature difference of sample is the change in order to avoid altitude temperature difference effect；

C) time point t it is positioned at₁With time point t₂Between the Long-term Deflection of time point t obtained by mathematical model one:

L (t, T)=s(t, T)-(T-T₁)×δ(t)

Wherein, s (t, T) is time point t, temperature difference T, there is filtered amount of deflection during car load；L(t, T) it is time point t, temperature Long-term Deflection during difference T, δ (t) is a coefficient；δ (t) can calculate in the following manner: in setting one day, Long-term Deflection is constant, permissible At t₁′、t₁' ' time point (t₁' and t₁' time point is on the same day) measure dynamic deflection and the temperature difference, by t₁' and t₁The data of ' ' time point all generations Enter mathematical model one, coefficient δ (t) can be solved.

Be can get by mathematical model one and multiple be positioned at time point t₁With time point t₂Between the Long-term Deflection value of time point；Utilize Mathematical model one calculates the Long-term Deflection of time point t can produce error, uses linear correction method to be modified the error occurred, WhereinGo to revise L(t, T with Δ), the Long-term Deflection value of the most each time point can accurately determine.

B, by temperature difference data T (t) of bridge with EMD decompose, catabolic process is as follows:

Step1: initialize, make r₁(t)=T (t), i=1, k=0；

Step2: the IMF of acquisition n-th order:

1. initialize, make h₁(t)=r₁(t)；

2. h is found out_kAll maximum of (t) and minimum point；

3. respectively maximum point and minimum point are fitted by cubic spline functions, seek upper and lower envelope e₊ (t) and e_-(t)；

4. upper lower envelope average m is calculated_k(t)；

⑤h_k+1(t)=h_k(t)-m_k(t)；

6. judgeWhether it is not more than given thresholding μ, takes 0.1≤μ≤0.8, if being not more than, then c_i(t)=h_k(t)；Not so, k=k+1 is made to forward to 2.；

Step3:r_k+1(t)=r_k(t)-c_k+1(t), it is judged that whether surplus is monotonic function or constant, if the most whole EMD catabolic process terminates.

Data after decomposition are as the input value of M-LS-SVM model.

C, for p data point X in q dimension space₁,X₂,...,X_p, sample data carried out point with subtraction clustering algorithm The step of class includes:

A) data point X is obtained by mathematical model two_i(i=1,2 ..., p) the density index D at place_i

$D_i=\underset{j=1}{\overset{p}{\mathrm{\Σ}}}\exp [-\frac{{\left|\right|X_i-X_j\left|\right|}^2}{{\left(0.5\mathrm{\η}\right)}^2}]=\underset{j=1}{\overset{p}{\mathrm{\Σ}}}\exp [-\underset{k=1}{\overset{q}{\mathrm{\Σ}}}\frac{{(x_{i.k}-x_{j.k})}^2}{{\left(0.5\mathrm{\η}\right)}^2}]$

$\mathrm{\η}=\frac{1}{2}\underset{k}{\min }\left\{\underset{i}{\max }\right||X_i-X_k|\left|\right\}$

Wherein, η is an X_iCluster radius, the density index of this point is contributed little by the data point beyond radius；By number Strong point X_i(i=1,2 ..., the 1st cluster centre X p) it is respectively defined as from high to low by density index_c1, the 2nd cluster centre X_c2..., pth cluster centre X_cp；And define D_ck(k=1,2 ..., p) it is data point X_ckDensity index；

B) density index for each data point is modified by mathematical model three

$D_i=D_{i-1}-D_{\mathrm{ck}}\exp [-\frac{{\left|\right|X_i-X_{\mathrm{ck}}\left|\right|}^2}{{\left(0.5\mathrm{\β}\right)}^2}]$

β=(1 ~ 2) η

β defines the neighborhood that density index function substantially reduces, β=(1 ~ 2) η can avoid selected center each other away from Close to too；Select data point X that density index after correction is the highest_ck+1As new cluster centre；Work as D_iBy this data point during ＜ 0 Density index be set to zero；

C) judgeIf whether setting up. it is false, then forwards to b)；If setting up, exiting, and thus terminating to cluster Journey；ε is for a setup parameter, and this parameter determines the initialization cluster centre number of final generation, for ensureing selected center energy Enough preferably representative input data and number of clusters are suitable, and the value of ε should exist with cluster centre number hereBetween, WhereinFor rounding symbol, p is the number of samples that need to carry out classifying.

By above-mentioned subtractive clustering process, determine cluster centre and submodel number n, and sample data is divided into n portion Point.

D, by sorted sample data, each sub-LS-SVM model is trained, thus sets up each sub-LS-SVM model.

E, all sub-LS-SVM models are connected into M-LS-SVM model by power, be represented by

Y=W^TY=w₁Y₁+w₂Y₃+L+w_nY_n

The traditional method that its weight W solves is to use method of least square, i.e. W=H^*Y_MLSSVM, wherein H^*=(H^TH)^-1H^T, permissible Obtain by H is carried out singular value decomposition, but due to trained multiple LS-SVM models reflection be same nonlinear dependence System, it is serious relevant each other, uses method of least square to be difficult to preferably be connected weights, the most whole model pre- Survey precision will decline.Use PCR method to carry out the predicted value of comprehensive multiple models, eliminate the data dependence between local system, The robustness of raising system and generalization ability.Use PCR method that H is written as form:Its Middle t_i=u_iλ_iWith i-th pivot component and the load component that v is respectively matrix H, for unit orthogonal vectors, and t_i=Hv_i, λ_iIt is The singular value of matrix H, u_iAnd v_iIt is respectively and singular value λ_iCorresponding left eigenvector and right characteristic vector.Owing to matrix H exists Dependency, typically negates mapping and calculates as the front k item of Main change part, thus structural matrix H_k: H ≈ H_K=TV^T, wherein T =[t₁,t₂,...,t_k], V=[v₁,v₂,...,v_k].By W=H^*Y_MLSSVM, have Y_MLSSVM=HW=TV^TW.Note W_K=V^TW, permissible Calculate W_kLeast square solution be

W_k=(T^TT)^-1T^TY_MLSSVM

Owing to V is orthogonal vectors, there is V^T=V^-1, then obtain connection weight value matrix

W=VW_k=V (T^TT)^-1T^TY_MLSVM

The step isolating Long-term Deflection by M-LS-SVM disjunctive model includes:

1, the relevant bridge signal (including dynamic deflection, the temperature difference etc.) measured is filtered, temperature signal EMD is divided Solve；

2, using temperature difference data as M-LS-SVM mode input, the output result obtained is exactly corresponding bridge temperature effect Should；

3, dynamic deflection signal and the bridge temperature effects signal subtraction of actual measurement is bridge Long-term Deflection signal.

It is applied to concrete bridge monitoring below in conjunction with the present invention illustrate.

Background bridge is respectively positioned on China's highway, is built up the end of the year 1996.Background bridge superstructure is that variable cross-section in advance should Power Concrete Continuous Box Beam, is combined as 75m+125m+75m across footpath, uses symmetrical cantilever pouring method construction.

In order to set up M-LS-SVM model, it is contemplated that the Long-term Deflection of 10 years will be calculated, so adopting in units of 12 hours The temperature of sample 365 days (1 year) and temperature effects data, totally 730 groups of data, will wherein 365 groups of temperature and temperature effects data make For training set, other 365 groups of data are as test set.

The data gathered first pass around Filtering Processing, it is therefore an objective to filter Load Effects and the noise jamming of high frequency.Fig. 2 is to survey The temperature data measured, in order to obtain multidimensional input temp data, improves the robustness of M-LS-SVM model, and the present invention first uses EDM algorithm separation temperature data.Signal decomposition can be some according to the feature of signal itself by empirical mode decomposition adaptively Individual intrinsic mode function sum, separates the temperature data EMD method measured, and is to set up multi input, the LS-of single output SVM model, increases the robustness of model.By subtractive clustering computing, 365 groups of data of training are divided into 4 input spaces, Namely establishing 4 sub-LS-SVM models, use PCR method to carry out the output valve of comprehensive multiple submodels, Fig. 3 a is actual Temperature amount of deflection effect data, Fig. 3 b is the temperature amount of deflection effect data of LS-SVM matching.Fig. 4 a is actual dynamic deflection value, figure 4b is background bridge Long-term Deflection LS-SVM matched curve and Midas value of calculation；Fig. 4 c is by temperature amount of deflection after M-LS-SVM separation The Long-term Deflection value of effect and the error amount of the long-term deflection data of reality, match value and actual value maximum relative error are 2.90%, Less than 3%.

Embodiment described above only have expressed one embodiment of the present invention, and it describes more concrete and detailed, but also Therefore the restriction to the scope of the claims of the present invention can not be interpreted as.It should be pointed out that, for those of ordinary skill in the art For, without departing from the inventive concept of the premise, it is also possible to make some deformation and improvement, these broadly fall into the guarantor of the present invention Protect scope.Therefore, the protection domain of patent of the present invention should be as the criterion with claims.

Claims (4)

1. the separation method of a bridge Long-term Deflection effect, it is characterised in that include following two step:

1) train M-LS-SVM disjunctive model by sample data, set up M-LS-SVM disjunctive model；

2) Long-term Deflection is isolated by M-LS-SVM disjunctive model；

Described M-LS-SVM disjunctive model is set up by following steps:

The sample data needed for M-LS-SVM is set up in a, acquisition；

B, being decomposed by the temperature difference data empirical mode decomposition of bridge, the temperature difference data after decomposing are as M-LS-SVM's Input signal；

C, with subtraction clustering algorithm, sample data is classified；

D, by sorted sample data, each sub-LS-SVM model is trained, thus sets up each sub-LS-SVM model；

E, employing principal components regression method carry out the predicted value of comprehensive all sub-LS-SVM models, form M-LS-SVM model；

The step obtaining the sample data needed for setting up M-LS-SVM includes:

A) at time point t₁, temperature difference T₁, in the case of vehicular load, measure amount of deflection s of bridge_{Without car load 1}(t₁,T₁)；

B) at time point t₂, temperature difference T₁, in the case of vehicular load, measure amount of deflection s of bridge_{Without car load 2}(t₂,T₁)；Same Measuring under the temperature difference is the change in order to avoid altitude temperature difference effect；

C) time point t it is positioned at₁With time point t₂Between the Long-term Deflection of time point t obtained by mathematical model one:

L (t, T)=s (t, T)-(T-T₁)×δ(t)

Wherein, s (t, T) is time point t, temperature difference T, there is filtered amount of deflection during car load；L (t, T) is time point t, during temperature difference T Long-term Deflection, δ (t) is a coefficient；Be can get by mathematical model one and multiple be positioned at time point t₁With time point t₂Between time point Long-term Deflection value；

The step isolating Long-term Deflection by M-LS-SVM disjunctive model includes:

A, to measure relevant bridge signal be filtered, temperature signal empirical mode decomposition is decomposed；

B, using temperature difference data as M-LS-SVM mode input, the output result obtained is exactly corresponding bridge temperature effects；

C, by actual measurement dynamic deflection signal and bridge temperature effects signal subtraction, obtain bridge Long-term Deflection signal.

Separation method the most according to claim 1, it is characterised in that: utilize mathematical model one to calculate scratching for a long time of time point t Degree can produce error, uses linear correction method to be modified, wherein the error occurredGo with Δ Revise L (t, T).

Separation method the most according to claim 1, it is characterised in that: for p data point X in q dimension space₁,X₂,…, X_p, with subtraction clustering algorithm, the step that sample data carries out classifying is included:

A) data point X is obtained by mathematical model two_i(i=1,2 ..., p) the density index D at place_i

$D_i=\underset{j=1}{\overset{p}{\mathrm{\Σ}}}\exp \[-\frac{\left|\right|X_i-X_j|{|}^2}{{\left(0.5\mathrm{\η}\right)}^2}\]=\underset{j=1}{\overset{p}{\mathrm{\Σ}}}\exp \[-\underset{k=1}{\overset{q}{\mathrm{\Σ}}}\frac{{(x_{i.k}-x_{j.k})}^2}{{\left(0.5\mathrm{\η}\right)}^2}\]$

$\mathrm{\η}=\frac{1}{2}\underset{k}{min}\left\{\underset{i}{min}\right||X_i-X_k|\left|\right\}$

Wherein, η is an X_iCluster radius, the density index of this point is contributed little by the data point beyond radius；By data point X_i (i=1,2 ..., the 1st cluster centre X p) it is respectively defined as from high to low by density index_c1, the 2nd cluster centre X_c2..., Pth cluster centre X_cp；And define D_ck(k=1,2 ..., p) it is data point X_ckDensity index；

B) density index for each data point is modified by mathematical model three

$D_i=D_{i-1}-D_{ck}\exp \[-\frac{\left|\right|X_i-X_{ck}|{|}^2}{{\left(0.5\mathrm{\β}\right)}^2}\]$

β=(1～2) η

β defines the neighborhood that density index function substantially reduces, and β=(1～2) η can avoid the distance each other at selected center The nearest；Select data point X that density index after correction is the highest_ck+1As new cluster centre；Work as D_iBy this data point during ＜ 0 Density index is set to zero；

C) judgeWhether setting up, if being false, then forwarding step b) to；If setting up, exiting, and thus terminating to cluster Journey；ε is for a setup parameter, and this parameter determines the initialization cluster centre number of final generation, for ensureing selected center energy Enough preferably representative input data and number of clusters are suitable, and the value of ε should exist with cluster centre number hereBetween, WhereinFor rounding symbol, p is the number of samples that need to carry out classifying.

Separation method the most according to claim 2, it is characterised in that use principal components regression method to carry out comprehensive all sub-LS- The predicted value of SVM model, the process forming M-LS-SVM model includes:

A) all sub-LS-SVM models are connected into M-LS-SVM model by power, be represented by

Y=W^TY=w₁Y₁+w₂Y₃+L+w_nY_n

W=H^*Y_MLSSVM

H^*=(H^TH)^-1H^T

$H=t_1{v}_1^T+t_2{v}_2^T+...+t_n{v}_n^T$

Wherein t_i=u_iλ_iWith i-th pivot component and the load component that v is respectively matrix H, for unit orthogonal vectors；And t_i= Hv_i, λ_iIt is the singular value of matrix H, u_iAnd v_iIt is respectively and singular value λ_iCorresponding left eigenvector and right characteristic vector；

B) front k item structural matrix H is taken_k: H ≈ H_K=TV^T, wherein T=[t₁,t₂,…,t_k], V=[v₁,v₂,…,v_k], make W_K= V^TW, calculates W_kLeast square solution be

W_k=(T^TT)^-1T^TY_MLSSVM

Owing to V is orthogonal vectors, there is V^T=V^-1, then obtain connection weight value matrix

W=VW_k=V (T^TT)^-1T^TY_MLSVM。