CN108282424A - The tetradic Joint diagonalization algorithm of blind source separating is closed for four data set associatives - Google Patents

The tetradic Joint diagonalization algorithm of blind source separating is closed for four data set associatives Download PDF

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CN108282424A
CN108282424A CN201810079041.2A CN201810079041A CN108282424A CN 108282424 A CN108282424 A CN 108282424A CN 201810079041 A CN201810079041 A CN 201810079041A CN 108282424 A CN108282424 A CN 108282424A
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龚晓峰
毛蕾
林秋华
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Dalian University of Technology
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Abstract

The invention discloses a kind of tetradic Joint diagonalization algorithms for being used for four data set associatives and closing blind source separating, include the following steps：S1, observation signal：Prewhitening is carried out respectively to four data set observation signals；S2, target tensor：After prewhitening, one group of Cross fourth order cumulant tensor is constructed；S3, initialization factor matrix；S4, cost function convergence calculate：If after single pass, algorithmic statement then calculates and terminates, if algorithm is still not converged, using the factor matrix of this update gained as initial value, scanned next time, traversal update Jacobi spin matrix, updating factor matrix, until convergence.The tetradic Joint diagonalization algorithm of the present invention for being used for four data set associatives and closing blind source separating, the algorithm is based on orthogonal rotation transformation, least squares sense optimal solution thus can be obtained, is a kind of J BSS methods for not more than four data set signals.

Description

The tetradic Joint diagonalization algorithm of blind source separating is closed for four data set associatives
Technical field
The present invention relates to field of signal processing such as biomedicine, communication, voice, arrays, one kind can be used for four data sets The tetradic Joint diagonalization algorithm of joint blind source separating.
Background technology
Joint blind source separating (Joint Blind Source Separation, J-BSS) as it is a kind of it is emerging, be based on More data set fusion treatment technologies of data-driven are obtained in field of signal processing such as biomedicine, communication, voice, arrays A large amount of concerns [1]-[14].
Different [14]-[24] with traditional blind source separating (BSS), J-BSS usually assumes that following more data set signal models：
x(m)(t)=A(m)s(m)(t), m=1,2 ..., M, (1A)
Wherein, x (t)(m)∈CNWith s (t)(m)∈CRThe observation signal and source signal of than the m-th data collection, A are indicated respectively(m) ∈CN×RIndicate the hybrid matrix of than the m-th data collection.
Parameter M, N, R indicate data set number, observation signal port number and source signal number respectively.
J-BSS is intended to only known more data set observation signal x(m)(t) under the premise of, to more data set hybrid matrix A(m) And source signal s(m)(t) joint identification is carried out.
In above-mentioned model, more data set signals are often same target or physical phenomenon, via different observed patterns The multi-group data obtained.
Herein, " different observed patterns " are different equipment or experimental program, such as brain electric (EEG) and functional MRI (fMRI)[2]；Or under same equipment and scheme, different tested individuals, such as more subject fMRI [3], [7]；Either phase With equipment, scheme and tested individual, different transform domains observation (such as frequency domain, time domain, spatial domain, statistics domain [3], [10])。
When handling above-mentioned more data set signals, correlation and diversity between different sets are utilized due to J-BSS, passes through Feature of the observation object in multi-level, more sections, multi-modal meaning can be described more fully hereinafter in information after fusion treatment, Therefore there is performance advantage in the BSS of identification capability, separation accuracy etc..
Research about J-BSS can trace back to the thirties in last century, the canonical correlation analysis side that document [1] is proposed earliest Method (Canonical Correlation Analysis:CCA).In later longer period of time, CCA is widely used in Among the fusion pretreatment of two datasets signal.
In recent years, CCA is expanded to the situation for being more than two datasets, i.e., more set CCA (Multiset CCA:M- CCA) [6], and applied in mostly subject fMRI data fusions.Then, the coupling matrix decomposition model of M-CCA is further Develop into matrix broad sense Joint diagonalization (Generalized joint diagonalization:GJD) model [8].
In addition, by classical independent component analysis (Independent Component Analysis:ICA it) develops Independent vector analyze (Independent Vector Analysis:IVA) it is also one of the main method of J-BSS [26]- [28]。
Nearest one or two years, the joint independence subspace analysis (joint towards more set, multidimensional source signal separation problem independent subspace analysis:J-ISA) gradually [29] of interest by people.
In the studies above, GJD is a kind of important method of J-BSS.Such methods, will be most by calculating cross cumulant Be converted into one group according to collection signal and intercouple, and with can joint diagonalization structure matrix slice, to by above-mentioned knot Structure carries out algebraically fitting and realizes J-BSS.
Currently, the research work in relation to GJD is mostly based on signal second order cross covariance.And it only is mutually accumulated based on quadravalence The method [8] of amount, also only consider Fourth-order cross cumulant matrix level low order can joint diagonalization structure, fail fully Excavating its high-order can joint diagonalization structure.In addition, this method is not subject to profit to the time domain specification of signal (such as non-stationary) With, and then cause the precision of algorithm insufficient.
Invention content
According to technical problem set forth above, and provide a kind of tetradic for being used for four data set associatives and closing blind source separating Joint diagonalization algorithm does not utilize the defect of combined signal higher order statistical characteristic to make up the prior art, and then leads to algorithm Precision deficiency disadvantage.The technological means that the present invention uses is as follows：
A kind of tetradic Joint diagonalization algorithm for being used for four data set associatives and closing blind source separating, includes the following steps：
S1, observation signal：
Prewhitening is carried out respectively to four data set observation signals：
The whitening matrix that note than the m-th data integrates is W(m)∈CR×N, m=1 ..., 4；
Then after pre -whitening processing, than the m-th data collection observation signal is：
y(m)(t)=W(m)x(m)(t)=U(m)s(m)(t)∈CR, (1)
Wherein, unitary matrice U(m)@W(m)A(m)∈CR×RIt is the hybrid matrix of than the m-th data collection signal after prewhitening；
S2, target tensor：
After prewhitening, one group of Cross fourth order cumulant tensor of construction is as follows：
Tk==cum (y(1)(tk),y(2)(tk),y(3)(tk),y(4)(tk)), k=1 ..., K, (2)
Wherein, " cum () " for calculating Cross fourth order cumulant, is defined as：
Wherein, " E () " indicates conjugation for calculating mathematic expectaion, subscript " * ", according to definition it is found that each tensor Tk∈CR×R×R×RIndicate four groups of observation signals in moment tkCross fourth order cumulant tensor, k=1 ..., K；
Its formula indicates as follows：
Tk=Dk×1U(1)×2U(2)*×3U(3)*×4U(4), (4)
Wherein ×nIndicate that the n modular multiplications of tensor sum matrix are accumulated, n=1 ..., 4；One tetradicWith square Battle arrayN modular multiplications product be defined as：
D in formula (4)k=cum (s(1)(tk),s(2)(tk),s(3)(tk),s(4)(tk))∈CR×R×R×RFor four group data sets Source signal in moment tkCross fourth order cumulant tensor；
S3, initialization factor matrix；
S4, cost function convergence calculate：
First, Optimality Criteria：
Objective function λ is data tensor TkWith fitting tensor Dk×1U(1)×2U(2)*×3U(3)*×4U(4)Between it is square Error：
Wherein | | | |FIt indicates not this black norm of Luo Beini, by minimizing λ, calculates hybrid matrix and anticipate in least square Optimal estimation value under justice；
Notice U(m)For unitary matrice, m=1,2,3,4, according to guarantor's plasticity of unitary transformation, formula (7) is rewritten as：
Wherein, off () is set to 0 for being inputted the super diagonal element of tensor, and diag () to it for inputting tensor Nondiagonal element is set to 0,For constant.Therefore, λ is minimized etc. Valence is in rightIt is maximized, therefore the present invention passes through following standard Then estimated mixing matrix U(m), m=1,2,3,4：
WhereinU is indicated respectively(1),U(2),U(3),U(4)Estimated value；
Formula (7)-(9) show by maximizing Tk×1U(1)H×2U(2)T×3U(3)T×4U(4)HSuper diagonal element norm it is flat Fang He, i.e., to T1,...,TKJoint diagonalization is carried out, hybrid matrix U is obtained(1),U(2),U(3),U(4)Under least square meaning Optimal estimation value；
Secondly, Jacobi iteration：
Unitary matrice to be solved is write as a series of product of Jacobi spin matrixs by Jacobi iteration, and then passes through difference Optimize each Jacobi spin matrix, the maximization of final function to achieve the objective；
Specifically, numeration is according to tensor TkAnd hybrid matrix U(m)It is respectively in the updated value of preceding an iterationWithIt is respectively in the updated value of current iterationWith
It is iterating through Jacobi spin matrix each timeTo TkAnd U(m)It is updated, i.e.,：
Wherein Jacobi spin matrixIt is defined as follows：
Wherein' i ' indicates imaginary part unit；
According to definition,In addition to the element of (i, i), (i, j), (j, i), four positions (j, j) are not 0, member on diagonal line Element is except 1, remaining position is all 0；
Enable the value of coordinate i be taken from 1 to R, the value of j is taken by i to R, for certain a pair of fixed coordinate value (i, j), by (9) and (10) it is found thatOptimal solution obtained by solving following optimization problems, m=1,2,3,4：
It obtainsLater, according to formula (10) to TkAnd U(m)It is updated；Included when coordinate (i, j) all values of traversal Whole iteration be known as single pass；
If after single pass, algorithmic statement then calculates and terminates, if algorithm is still not converged, with this update gained Factor matrix is scanned next time as initial value, traversal update Jacobi spin matrix, updating factor matrix, until receiving Until holding back.
As preferably before step S1, doing following hypothesis：
(A1) independence in organizing：As 1≤r ≠ u≤R,WithStatistical iteration；
(A2) inter-class correlation：As 1≤r=u≤R,WithStatistical correlation；
(A3) source signal is non-gaussian, non-stationary signal；
(A4) the more data set models considered are positive definite, i.e. N >=R；
(A5) the number M=4 of data set.
As preferably in practice, under assumed condition (A1) and (A4), whitening matrix W(m)∈CR×NIt can be believed by observing Number x(m)(t) singular value decomposition of second order covariance matrix and obtain.
As preferably by hypothesis (A1), (A2) and (A3) is it is found that each tensor DkIt is super diagonal tensor, i.e.,：
Define U '(m)@U(m)D(m)P, m=1,2,3,4, wherein D(1),D(2),D(3),D(4)∈CR×RFor diagonal matrix, and it is full Sufficient D(1)D(2)*D(3)*D(4)=I, I are unit matrix, P ∈ CR×RFor ordinal matrix；
It is not difficult to learn, if by the U in formula (4)(m)Replace with U '(m), equation still sets up；
Therefore, U '(m)When not considering that amplitude/phase is fuzzy and sequence is fuzzy and hybrid matrix U(m)It is of equal value；
It, can be to hybrid matrix U by being solved to equation (4)(m)Estimated.
It is updated in Jacobi spin matrix as preferred traversal, it is involved in every single-step iterationEnclosed optimal solution；
The enclosed optimal solution of Jacobi spin matrix
According to the property of Jacobi spin matrix, unitary transformation Only change tensorValue in following eight part：
Here we are indicated using MATLAB symbolsSub- tensor, such asIndicate fixed tensorFirst index value be i, the sub- tensor that remaining index value is obtained when being not fixed；
, it can be seen that maximizing tensorSuper diagonal element Norm squared and, be equivalent to maximize above-mentioned eight sub- tensors and intersect the super of sub- tensor that constituted sizes are 2 × 2 × 2 × 2 Diagonal element norm squared and；
Therefore, Optimality Criteria (12) can be rewritten as：
In order to be further simplified optimization process, we replace (13) using alternately newer mode with following four step, Wherein each step is for alternately updating
Wherein
Next, we withFor, explain the solution procedure of (14)；
First according to definition：
Wherein,
According to formula (15), have：
Wherein Mi,j=[m1,i,j,...,mK,i,j,q1,i,j,...,qK,i,j]∈C3×2K；
When above-mentioned derivation shows to maximize formula (16),Corresponding to matrixMain feature vector, NoteEstimated value beThen haveAnd then it can be obtained by (11)
Next, being updatedAnd according to same solutionSimilar step It is rapid to obtainSimilarly, it calculates and obtainsWith
Judgement as preferred iteration stopping condition is：
Calculate the relative changing value of the diagonal element norm sum of adjacent the obtained tensor of twice sweep：
When it is less than threshold epsilon, then it is assumed that algorithmic statement, iteration ends；
Or, judging Jacobi spin matrix that Current Scan is obtained whether close to unit matrix, specifically, definitionWork as ξ(1)、ξ(2)、ξ(3)And ξ(4)When simultaneously less than threshold epsilon, then illustrateWithIt has been similar to unit matrix, it cannot more fresh target tensor, it is believed that algorithmic statement, iteration are whole Only.
Compared with prior art, the tetradic joint of the present invention for being used for four data set associatives and closing blind source separating Diagonalization algorithm is a kind of J-BSS methods for not more than four data set signals.This method calculates more data set signals It realizes more data set letters in Cross fourth order cumulant tensor in different time periods, and by carrying out tensor Joint diagonalization to it Number J-BSS.
In addition, we have proposed a kind of tensor Joint diagonalization algorithm based on Givens continuous rotations.Compared with technology [30] the tensor diagonalization algorithm of forms data collection BSS of being only applicable to proposed in is compared, the algorithm be suitable for the load of tensor because The different situation of submatrix, to be suitable for J-BSS.Compared with the nonopiate tensor diagonalization algorithm proposed in technology [8] It compares, which is based on orthogonal rotation transformation, it is thus possible to obtain least squares sense optimal solution.
The tetradic Joint diagonalization algorithm of the present invention for being used for four data set associatives and closing blind source separating, applies The time domain specification of signal, the present invention calculate it in Cross fourth order cumulant tensor in different time periods to more data set signals, improve The precision of algorithm.
The tetradic Joint diagonalization algorithm of the present invention for being used for four data set associatives and closing blind source separating, utilizes four A factor matrix constructs tensor, during algorithm, while alternately updating four factor matrixs, and algorithm is accurate, compared with Algorithm in technology only alternately updates first three factor matrix and compares, and the 4th factor matrix is obtaining first three factor matrix Afterwards, it finds out automatically.
The Optimality Criteria of the present invention directly optimizes four factor matrixs, and the algorithm of the prior art only optimizes first three Factor matrix.Due to Optimality Criteria difference, and optimised tensor is also different, so subsequent optimization process is all different, In the prior art since first three factor matrix has been found out, so the 4th factor matrix is obtained by restoring target tensor , but the factor matrix precision being obtained by is low.The present invention directly optimizes four factor matrixs, improves the 4th factor square The precision of battle array.
The present invention is divided into k sections when calculating tensor, by signal, amounts to and calculates k tensor, compared with algorithm signal in technology It is not segmented, calculates n tensor, that is, compared with the dimension tensor of factor matrix, more signal messages are utilized, from And improve arithmetic accuracy.
Description of the drawings
The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.
Fig. 1 is inventive algorithm flow chart.
Fig. 2-1 is the A-PI values of JTD and GOJD algorithms under noise free conditions in the embodiment of the present invention 1 with the change of scanning times Change curve (K=3, N=3).
Fig. 2-2 is the A-PI values of JTD and GOJD algorithms under noise free conditions in the embodiment of the present invention 1 with the change of scanning times Change curve (K=3, N=50).
Fig. 2-3 is the A-PI values of JTD and GOJD algorithms under noise free conditions in the embodiment of the present invention 1 with the change of scanning times Change curve (K=50, N=3).
Fig. 2-4 is the A-PI values of JTD and GOJD algorithms under noise free conditions in the embodiment of the present invention 1 with the change of scanning times Change curve (K=50, N=50).
Fig. 3-1 be in the embodiment of the present invention 2 A-PI with SNR change curves schematic diagram (number of snapshots=100000).
Fig. 3-2 be in the embodiment of the present invention 2 A-PI with number of snapshots change curve schematic diagram (SNR=5dB).
Eight tunnel ECG data figure in Fig. 4 embodiment of the present invention 3.
Fig. 5 is practical FECG Signal separators result figure in the embodiment of the present invention 3.
Specific implementation mode
As shown in Figure 1, a kind of tetradic Joint diagonalization algorithm for being used for four data set associatives and closing blind source separating, first , do following hypothesis：
(A1) independence in organizing：As 1≤r ≠ u≤R,WithStatistical iteration；
(A2) inter-class correlation：As 1≤r=u≤R,WithStatistical correlation；
(A3) source signal is non-gaussian, non-stationary signal；
(A4) the more data set models considered are positive definite, i.e. N >=R；
(A5) the number M=4 of data set.
In above-mentioned hypothesis, (A1) and (A2) is the assumed condition generally used in existing joint blind source separation method； Condition (A3) and (A4) then show that the method for the invention is suitable in the case of positive definite, and the joint of non-gaussian and non-stationary signal is blind Source detaches；Condition (A5) then limits the data set number that the method for the present invention can be handled and is no more than four, and the present invention is with four numbers It is obtained by similar derivation according to the case where illustrating proposed joint blind source separation algorithm for collection, be less than four data sets.
Specifically include following steps：
S1, observation signal：
Prewhitening is carried out respectively to four data set observation signals：
The whitening matrix that note than the m-th data integrates is W(m)∈CR×N, m=1 ..., 4；
Then after pre -whitening processing, than the m-th data collection observation signal is：
y(m)(t)=W(m)x(m)(t)=U(m)s(m)(t)∈CR, (1)
Wherein, unitary matrice U(m)@W(m)A(m)∈CR×RIt is the hybrid matrix of than the m-th data collection signal after prewhitening； In practice, under assumed condition (A1) and (A4), whitening matrix W(m)∈CR×NObservation signal x can be passed through(m)(t) second order association The singular value decomposition of variance matrix and obtain [8].
S2, target tensor：
After prewhitening, one group of Cross fourth order cumulant tensor of construction is as follows：
Tk==cum (y(1)(tk),y(2)(tk),y(3)(tk),y(4)(tk)), k=1 ..., K, (2)
Wherein, " cum () " for calculating Cross fourth order cumulant, is defined as：
Wherein, " E () " indicates conjugation for calculating mathematic expectaion, subscript " * ", according to definition it is found that each tensor Tk∈CR×R×R×RIndicate four groups of observation signals in moment tkCross fourth order cumulant tensor, k=1 ..., K；
Its formula indicates as follows：
Tk=Dk×1U(1)×2U(2)*×3U(3)*×4U(4), (4)
Wherein ×nIndicate that the n modular multiplications of tensor sum matrix are accumulated, n=1 ..., 4；One tetradicWith square Battle arrayN modular multiplications product be defined as：
D in formula (4)k=cum (s(1)(tk),s(2)(tk),s(3)(tk),s(4)(tk))∈CR×R×R×RFor four group data sets Source signal in moment tkCross fourth order cumulant tensor；
By hypothesis (A1), (A2) and (A3) is it is found that each tensor DkIt is super diagonal tensor, i.e.,：
Define U '(m)@U(m)D(m)P, m=1,2,3,4, wherein D(1),D(2),D(3),D(4)∈CR×RFor diagonal matrix, and it is full Sufficient D(1)D(2)*D(3)*D(4)=I, I are unit matrix, P ∈ CR×RFor ordinal matrix；
It is not difficult to learn, if by the U in formula (4)(m)Replace with U '(m), equation still sets up；
Therefore, U '(m)Do not considering that amplitude/phase is fuzzy (by diagonal matrix D(m)Characterization) and it is sequentially fuzzy (by sorting Matrix P characterization) when and hybrid matrix U(m)It is of equal value.It, can be to hybrid matrix U by being solved to equation (4)(m)Estimated.
The purpose of joint blind source separating, exactly in the case where not considering the ordinary fuzzy precondition of above-mentioned two class, to mixed moment Battle array is recognized, and then is estimated source signal.
It is different with traditional BSS, when m takes different index value, different U '(m)Relative to true mixed moment Battle array U(m)Ordinal matrix P be identical, it means that via the hybrid matrix and source signal ingredient estimated by J-BSS, It is sequentially that nature is aligned, this is also advantages one of of the J-BSS compared to BSS.
Construct tensor using four factor matrixs in the present invention, during algorithm, while alternately update four because Submatrix only alternately updates first three factor matrix compared with the algorithm in technology and compares, and the 4th factor matrix is obtaining After first three factor matrix, find out automatically.
The present invention is divided into k sections when calculating tensor, by signal, amounts to and calculates k tensor, compared with algorithm signal in technology It is not segmented, calculates n tensor, that is, the dimension tensor of factor matrix is compared, and more signal messages are utilized, to Improve arithmetic accuracy.
S3, initialization factor matrix；
S4, cost function convergence calculate：
First, Optimality Criteria：
Objective function λ is data tensor TkWith fitting tensor Dk×1U(1)×2U(2)*×3U(3)*×4U(4)Between it is square Error：
Wherein | | | |FIndicate that not this black norm (Frobenius norm) of Luo Beini calculates mixing by minimizing λ Optimal estimation value of the matrix under least square meaning；
Notice U(m)For unitary matrice, m=1,2,3,4, according to guarantor's plasticity of unitary transformation, formula (7) is rewritten as：
Wherein, off () is set to 0 for being inputted the super diagonal element of tensor, and diag () to it for inputting tensor Nondiagonal element is set to 0,For constant.Therefore, λ is minimized etc. Valence is in rightIt is maximized, therefore the present invention passes through following standard Then estimated mixing matrix U(m), m=1,2,3,4：
WhereinU is indicated respectively(1),U(2),U(3),U(4)Estimated value；
Formula (7)-(9) show by maximizing Tk×1U(1)H×2U(2)T×3U(3)T×4U(4)HSuper diagonal element norm it is flat Fang He, i.e., to T1,...,TKJoint diagonalization is carried out, hybrid matrix U is obtained(1),U(2),U(3),U(4)Under least square meaning Optimal estimation value；Next, it is proposed that a kind of tensor Joint diagonalization algorithm based on Jacobi iteration.
The Optimality Criteria of the present invention directly optimizes four factor matrixs, and algorithm only optimizes first three in the prior art Factor matrix.Due to Optimality Criteria difference, and optimised tensor is also different, so subsequent optimization process is all different, In the prior art since first three factor matrix has been found out, so the 4th factor matrix is obtained by restoring target tensor , but the factor matrix precision being obtained by is low.The present invention directly optimizes four factor matrixs, improves the 4th factor square The precision of battle array.
Secondly, Jacobi iteration：
Unitary matrice to be solved is write as a series of product of Jacobi spin matrixs by Jacobi iteration, and then passes through difference Optimize each Jacobi spin matrix, the maximization of final function to achieve the objective；
Specifically, numeration is according to tensor TkAnd hybrid matrix U(m)It is respectively in the updated value of preceding an iterationWithIt is respectively in the updated value of current iterationWithIt is iterating through Jacobi spin matrix each time(again Referred to as Givens spin matrixs) to TkAnd U(m)It is updated, i.e.,：
Wherein Jacobi spin matrixIt is defined as follows：
Wherein' i ' indicates imaginary part unit.According to definition,In addition to (i, I), the element of (i, j), (j, i), four positions (j, j) are not 0, and element is except 1 on diagonal line, remaining position is all 0；
Enable the value of coordinate i be taken from 1 to R, the value of j is taken by i to R, for certain a pair of fixed coordinate value (i, j), by (9) and (10) it is found thatOptimal solution obtained by solving following optimization problems, m=1,2,3,4：
It obtainsLater, according to formula (10) to TkAnd U(m)It is updated；Included when coordinate (i, j) all values of traversal Whole iteration be known as single pass (Sweep)；
If after single pass, algorithmic statement then calculates and terminates, if algorithm is still not converged, with this update gained Factor matrix is scanned next time as initial value, traversal update Jacobi spin matrix, updating factor matrix, until receiving Until holding back.
The judgement of iteration stopping condition is：
Calculate the relative changing value of the diagonal element norm sum of adjacent the obtained tensor of twice sweep：
When it is less than threshold epsilon, then it is assumed that algorithmic statement, iteration ends；
Or, judging Jacobi spin matrix that Current Scan is obtained whether close to unit matrix, specifically, definitionWork as ξ(1)、ξ(2)、ξ(3)And ξ(4)When simultaneously less than threshold epsilon, then illustrateWithIt has been similar to unit matrix, it cannot more fresh target tensor, it is believed that algorithmic statement, iteration are whole Only.
As can be seen from the above description, Jacobi iteration dissolves optimization problem (9) for a series of sub- optimization problems (12).By In each Jacobi spin matrix only there are two unknown parameter, optimal solution is expected to have simple closed solutions form.It connects down Come, we are involved in deriving per single-step iterationEnclosed optimal solution.
It is involved in every single-step iteration in traversal update Jacobi spin matrixEnclosed optimal solution；
The enclosed optimal solution of Jacobi spin matrix
According to the property of Jacobi spin matrix, unitary transformation Only change tensorValue in following eight part:
Here we are indicated using MATLAB symbolsSub- tensor, such asIndicate fixed tensorFirst index value be i, the sub- tensor that remaining index value is obtained when being not fixed；
It is not difficult to find out, maximizes tensorSuper diagonal element Norm squared and, be equivalent to maximize above-mentioned eight sub- tensors and intersect the super of sub- tensor that constituted sizes are 2 × 2 × 2 × 2 Diagonal element norm squared and；
Therefore, Optimality Criteria (12) can be rewritten as：
In order to be further simplified optimization process, we replace (13) using alternately newer mode with following four step, Wherein each step is for alternately updating
Wherein
And
Next, we withFor, explain the solution procedure of (14)；
First according to definition：
Wherein
According to formula (15), have：
Wherein Mi,j=[m1,i,j,...,mK,i,j,q1,i,j,...,qK,i,j]∈C3×2K；
When above-mentioned derivation shows to maximize formula (16),Corresponding to matrixMain feature vector, NoteEstimated value beThen haveAnd then it can be obtained by (11)
Next, being updatedAnd according to same solutionSimilar step obtains Similarly, it calculates and obtainsWith
Above-mentioned derivation assumes that J-BSS problems are complex value, when J-BSS problems are real value, also through similar calculating by its It is converted into the Joint diagonalization problem of the real value tetradic, and it is solved by the rotation of real value Jacobi, calculates step It is similar with complex value situation.
It should be noted that real value Jacobi spin matrix is defined as at this time：
And in formula (15)mk,i,j,qk,i,jExpression formula be given by：
Embodiment 1, it is random to generate super diagonal tensor DkAnd unitary matrice U(1),U(2),U(3),U(4), and it is straight according to formula (4) Connect fabric tensor
Performance indicator (the Performance Index that algorithm performance is defined by such as following formula:PI it) is evaluated：
Wherein,SubscriptIndicate Moore-Penrose pseudoinverses.Separating resulting is distinguished It calculates its PI and then is averaged.With this PI that is averaged (Average PI:A-PI) as the evaluation index of algorithm performance.
PI values when by calculating the end of scan each time corresponding to algorithm, can reflect convergence.Fig. 2-1 to Fig. 2-4 is depicted when K and N take different value, JTD algorithms in 10 independent experiments, A-PI values with scanning times variation Curve.
As reference, we draw under the same conditions, the A-PI of generalized orthogonal matrix diagonalization algorithm (GOJD, [8]) (its data matrix is by T for curvekThree-dimensional index obtains after=fixation).See, the JTD algorithms proposed show monotonic decreasing, The convergent pathway of linear convergence.Compared with GOJD algorithms, 5 scannings that are averaged of JTD algorithms can reach convergence, have and more preferably receive Hold back performance.
Fig. 2-1 to Fig. 2-4 is the A-PI values of JTD and GOJD algorithms under noise free conditions with the change curve of scanning times. Here the experimental result under the conditions of four kinds is shown：(a) as shown in Fig. 2-1, the size N and quantity K of tensor take smaller value；(b) As shown in Fig. 2-2, the size N of tensor takes higher value, quantity K to take smaller value；(c) as Figure 2-3, the size N of tensor take compared with Small value, quantity K take higher value；(d) as in Figure 2-4, the size N and quantity K of tensor take higher value.
In this experiment, four groups of observation signal x are generated according to formula (1A) for embodiment 2(m)(t), m=1,2,3,4：
Wherein hybrid matrix A(m)∈CN×RIt is randomly generated by program, noise item n(m)(t)∈CN×TFor zero mean unit variance White Gaussian noise.
Source signal is constructed by following formula：
sr(t)=Qrs′r(t), (22)
Whereins′r(t)∈C4To be segmented the random phase letter of amplitude modulation Number, phase and amplitude modulation coefficient are randomly generated by program.Matrix Qr∈R4×4It is randomly generated by program, for introducing between set Correlation.It is not difficult to learn, the source signal constructed is non-stationary non-Gaussian signal (assuming that A3), and the source of different data collection is believed Number meet independent assumed condition (assuming that A1, A2) in inter-class correlation and group.
In formula (21), σsAnd σnRespectively represent signal strength and noise intensity.It further can define signal-to-noise ratio (SNR) It is as follows：SNR@20lgσs/σn。
We detach above-mentioned more data set signals with the J-BSS algorithms based on JTD proposed.And by it under BSS the and J-BSS algorithms for stating same type are compared：
(a) eigenmatrix approximately joint diagonalization algorithm (Joint Approximate Diagonalization of Eigenmatrices:JADE)[16]；
(b) three rank tensor Joint diagonalization algorithm (Simultaneous Third-Order Tensor Diaognalization:STOTD)[30]；
(c) mostly set canonical correlation analysis algorithm (Multiset Canonical Correlation Analysis: MCCA)[6]；
(d) generalized orthogonal matrix Joint diagonalization algorithm (Generalized Orthogonal Joint Diagonalization:GOJD)[8]。
Wherein JADE and STOTD is the BSS algorithms based on fourth order cumulant, and MCCA and GOJD are J-BSS algorithms. In the experiment of GOJD algorithms, the algorithm based on fourth order cumulant and second order covariance has been respectively adopted in we, is denoted as GOJD4 respectively And GOJD2.
Make us with the estimated value (2) that sampling Cross fourth order cumulant is Cross fourth order cumulant defined in (2), calculating is public Formula is as follows：
WhereinK-th segmentation of the observation signal on time dimension after expression prewhitening, m=1 ..., 4.Here section length is L, and Duplication α ∈ between adjacent segment [0,1), it is defined as the ratio of the coincidence points and L of adjacent segment.
In this experiment, we fix the observation port number N=10 of each data set, and source signal number is R=5, A-PI curves Each of o'clock average obtained by the result of 200 Monte Carlo independent experiments.First, set time sampling number T= 100000, L=T/10, α=0.3.
SNR is enabled to be changed to 10dB from 0dB.Next, fixed SNR=5dB, enables time sampling points be changed to by 10000 100000.In above-mentioned two situations, the A-PI curves of institute's comparison algorithm are as shown in Fig. 3-1 and Fig. 3-2.See, is proposed JTD algorithm performances are substantially better than other algorithms in addition to GOJD2.
Compared with GOJD2, found out by Fig. 3-1, when SNR is 2-8dB, the separation accuracy higher of JTD algorithms.Fig. 3-2 Show the JTD arithmetic accuracy highers that (80000-100000) are proposed when number of snapshots are higher, it is on the contrary then be GOJD2 more excellent Gesture.
This is primarily due to second order covariance used by GOJD2 compared with the fourth order cumulant used by JTD, short fast In the case of bat, there is smaller limited sampling error.
As shown in Fig. 3-1 and Fig. 3-2, JTD, GJD, MCCA, JADE, STOTD detach computer composite signal when performance Comparison.
Embodiment 3, experiment are intended to, from actual acquisition, be mixed with pregnant woman's electrocardiosignal (Mother ECG, MECG), tire In youngster's electrocardiosignal (Fetal ECG, FECG) and the observation data of other interference signals, FECG signals are extracted.
The eight tunnel ECG datas used are derived from the DAISY databases 0 of Univ Louvain, and signal waveform is as shown in Fig. 4.Sampling frequency Rate is 250Hz, and sampling number 2500, the sampling time is 10s, as shown in figure 4, being eight tunnel ECG datas of actual acquisition, data It is derived from the DAISY databases of Univ Louvain.
According to the following formula, eight road observation signals are divided into four data sets, contain five road observation signals in each data set：
It will be segmented after the observation signal prewhitening of four data sets, be L=500 per segment length, Duplication is α=0.5, no Difficulty learns K=10 at this time.Cross fourth order cumulant is calculated to build K target tensor to each segment signal.With JTD algorithms to its into Row Joint diagonalization can get each source signal ingredient after obtaining the mixed matrix of solution by formula (1), and the results are shown in Figure 4,Table Show the estimated value of the source signal of than the m-th data collection, as shown in Figure 5, it can be seen that MECG signals and FECG signals are successfully separated.
As seen from the figure, the 1st road signal of each data set and the 2nd road signal are MECG, and first three data concentrates the 3rd road letter Number be FECG.The result shows that the J-BSS algorithms based on JTD can be successfully separated the FECG signals of actual acquisition.
The foregoing is only a preferred embodiment of the present invention, but scope of protection of the present invention is not limited thereto, Any one skilled in the art in the technical scope disclosed by the present invention, according to the technique and scheme of the present invention and its Inventive concept is subject to equivalent substitution or change, should be covered by the protection scope of the present invention.
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Claims (6)

1. a kind of tetradic Joint diagonalization algorithm for being used for four data set associatives and closing blind source separating, it is characterised in that including with Lower step：
S1, observation signal：
Prewhitening is carried out respectively to four data set observation signals：
The whitening matrix that note than the m-th data integrates is W(m)∈CR×N, m=1 ..., 4；
Then after pre -whitening processing, than the m-th data collection observation signal is：
y(m)(t)=W(m)x(m)(t)=U(m)s(m)(t)∈CR, (1)
Wherein, unitary matrice U(m)@W(m)A(m)∈CR×RIt is the hybrid matrix of than the m-th data collection signal after prewhitening；
S2, target tensor：
After prewhitening, one group of Cross fourth order cumulant tensor of construction is as follows：
Tk==cum (y(1)(tk),y(2)(tk),y(3)(tk),y(4)(tk)), k=1 ..., K, (2)
Wherein, " cum () " for calculating Cross fourth order cumulant, is defined as：
Wherein, " E () " indicates conjugation for calculating mathematic expectaion, subscript " * ", according to definition it is found that each tensor Tk∈CR ×R×R×RIndicate four groups of observation signals in moment tkCross fourth order cumulant tensor, k=1 ..., K；
Its formula indicates as follows：
Tk=Dk×1U(1)×2U(2)*×3U(3)*×4U(4), (4)
Wherein ×nIndicate that the n modular multiplications of tensor sum matrix are accumulated, n=1 ..., 4；One tetradicWith matrixN modular multiplications product be defined as：
D in formula (4)k=cum (s(1)(tk),s(2)(tk),s(3)(tk),s(4)(tk))∈CR×R×R×RFor the source of four group data sets Signal is in moment tkCross fourth order cumulant tensor；
S3, initialization factor matrix；
S4, cost function convergence calculate：
First, Optimality Criteria：
Objective function λ is data tensor TkWith fitting tensor Dk×1U(1)×2U(2)*×3U(3)*×4U(4)Between mean square error：
Wherein | | | |FIt indicates not this black norm of Luo Beini, by minimizing λ, calculates hybrid matrix under least square meaning Optimal estimation value；
Notice U(m)For unitary matrice, m=1,2,3,4, according to guarantor's plasticity of unitary transformation, formula (7) is rewritten as：
Wherein, off () is set to 0 for being inputted the super diagonal element of tensor, and diag () is for inputting it the non-right of tensor Angle member is set to 0,For constant；
Therefore, λ minimize and be equivalent to pairIt carries out most Bigization, therefore, the present invention pass through following criterion estimated mixing matrix U(m), m=1,2,3,4：
WhereinU is indicated respectively(1),U(2),U(3),U(4)Estimated value；
Formula (7)-(9) show by maximizing Tk×1U(1)H×2U(2)T×3U(3)T×4U(4)HSuper diagonal element norm squared With that is, to T1,...,TKJoint diagonalization is carried out, hybrid matrix U is obtained(1),U(2),U(3),U(4)Under least square meaning most Excellent estimated value；
Secondly, Jacobi iteration：
Unitary matrice to be solved is write as a series of product of Jacobi spin matrixs by Jacobi iteration, and then by being separately optimized Each Jacobi spin matrix, the maximization of final function to achieve the objective；
Specifically, numeration is according to tensor TkAnd hybrid matrix U(m)It is respectively in the updated value of preceding an iterationWithIts It is respectively in the updated value of current iterationWith
It is iterating through Jacobi spin matrix each timeTo TkAnd U(m)It is updated, i.e.,：
Wherein Jacobi spin matrixIt is defined as follows：
Wherein' i ' indicates imaginary part unit；
According to definition,In addition to the element of (i, i), (i, j), (j, i), four positions (j, j) are not 0, element is on diagonal line Except 1, remaining position is all 0；
The value of coordinate i is enabled to be taken from 1 to R, the value of j is taken by i to R, for certain a pair of fixed coordinate value (i, j), by (9) and (10) It is found thatOptimal solution obtained by solving following optimization problems, m=1,2,3,4：
It obtainsLater, according to formula (10) to TkAnd U(m)It is updated；Included when coordinate (i, j) all values of traversal is complete Portion's iteration is known as single pass；
If after single pass, algorithmic statement then calculates and terminates, if algorithm is still not converged, with the factor of this update gained Matrix is scanned next time as initial value, traversal update Jacobi spin matrix, updating factor matrix, until converging to Only.
2. the tetradic Joint diagonalization algorithm according to claim 1 for being used for four data set associatives and closing blind source separating, It is characterized in that：
Before step S1, following hypothesis is done：
(A1) independence in organizing：As 1≤r ≠ u≤R,WithStatistical iteration；
(A2) inter-class correlation：As 1≤r=u≤R,WithStatistical correlation；
(A3) source signal is non-gaussian, non-stationary signal；
(A4) the more data set models considered are positive definite, i.e. N >=R；
(A5) the number M=4 of data set.
3. the tetradic Joint diagonalization algorithm according to claim 2 for being used for four data set associatives and closing blind source separating, It is characterized in that：
Under assumed condition (A1) and (A4), whitening matrix W(m)∈CR×NObservation signal x can be passed through(m)(t) second order covariance square Battle array singular value decomposition and obtain.
4. the tetradic Joint diagonalization algorithm according to claim 3 for being used for four data set associatives and closing blind source separating, It is characterized in that：
By hypothesis (A1), (A2) and (A3) is it is found that each tensor DkIt is super diagonal tensor, i.e.,：
Define U '(m)@U(m)D(m)P, m=1,2,3,4, wherein D(1),D(2),D(3),D(4)∈CR×RFor diagonal matrix, and meet D(1) D(2)*D(3)*D(4)=I, I are unit matrix, P ∈ CR×RFor ordinal matrix；
It is not difficult to learn, if by the U in formula (4)(m)Replace with U '(m), equation still sets up；
Therefore, U '(m)When not considering that amplitude/phase is fuzzy and sequence is fuzzy and hybrid matrix U(m)It is of equal value；
It, can be to hybrid matrix U by being solved to equation (4)(m)Estimated.
5. the tetradic Joint diagonalization algorithm according to claim 1 for being used for four data set associatives and closing blind source separating, It is characterized in that：
It is involved in every single-step iteration in traversal update Jacobi spin matrixEnclosed optimal solution；
The enclosed optimal solution of Jacobi spin matrix
According to the property of Jacobi spin matrix, unitary transformationOnly change Become tensorValue in following eight part：
Here we are indicated using Matlab symbolsSub- tensor, such asIndicate fixed tensor First index value be i, the sub- tensor that remaining index value is obtained when being not fixed；
, it can be seen that maximizing tensorThe norm of super diagonal element Quadratic sum is equivalent to maximize above-mentioned eight sub- tensors and intersects the super diagonal of sub- tensor that constituted sizes are 2 × 2 × 2 × 2 Element norm squared and；
Therefore, Optimality Criteria (12) can be rewritten as：
In order to be further simplified optimization process, we replace (13) using alternately newer mode with following four step, wherein Each step is for alternately updating
Wherein
And
Next, we withFor, explain the solution procedure of (14)；
First according to definition：
Wherein,
According to formula (15), have：
Wherein Mi,j=[m1,i,j,...,mK,i,j,q1,i,j,...,qK,i,j]∈C3×2K；
When above-mentioned derivation shows to maximize formula (16),Corresponding to matrixMain feature vector, note Estimated value beThen haveAnd then it can be obtained by (11)
Next, being updatedAnd it is asked according to same SolutionSimilar step obtainsSimilarly, it calculates and obtainsWith
6. being used for the tetradic Joint diagonalization calculation that four data set associatives close blind source separating according to claim 1 or 5 Method, it is characterised in that：
The judgement of iteration stopping condition is：
Calculate the relative changing value of the diagonal element norm sum of adjacent the obtained tensor of twice sweep：
When it is less than threshold epsilon, then it is assumed that algorithmic statement, iteration ends；
Or, judging Jacobi spin matrix that Current Scan is obtained whether close to unit matrix, specifically, definitionWork as ξ(1)、ξ(2)、ξ(3)And ξ(4)When simultaneously less than threshold epsilon, then illustrate WithIt has been similar to unit matrix, it cannot more fresh target tensor, it is believed that algorithmic statement, iteration ends.
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