Compressive sensing sparse sampling method based on principal component analysis

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he 9th Iteratoal Symposum o ND Aerospace Compressve sesg sparse samplg method based o prcpal compoet aalyss More fo about ths artcle: http://www.dt.et/?

Equpmet echology, Najg Uversty of Iformato Scece ad echology, Najg, Cha 3 School of Computer ad Software, Najg Uversty of Iformato Scece ad echology, Najg, Cha 4 School of Iformato ad Cotrol, Najg

Structural health motorg techology, cludg motorg, detecto ad detfcato of structures, has become a key techology to detect the structural safety performace of aerospace structures.

1 he 9th Iteratoal Symposum o ND Aerospace Compressve sesg sparse samplg method based o prcpal compoet aalyss More fo about ths artcle: Yaje SUN,,3, Fehog GU 3, Sa JI,,3, Lhua WANG 4 Jagsu egeerg Cetre of Network Motorg, Najg Uversty of Iformato Scece ad echology, Najg, Cha Jagsu Collaboratve Iovato Ceter o Atmospherc Evromet ad Equpmet echology, Najg Uversty of Iformato Scece ad echology, Najg, Cha 3 School of Computer ad Software, Najg Uversty of Iformato Scece ad echology, Najg, Cha 4 School of Iformato ad Cotrol, Najg Uversty of Iformato Scece ad echology, Najg, Cha Correspodg author: Yaje Su, e-mal: syj@ust.edu.c Abstract he rapd developmet of aerospace has creased the techcal requremets for aerospace structures. Structural health motorg techology, cludg motorg, detecto ad detfcato of structures, has become a key techology to detect the structural safety performace of aerospace structures. he early detecto techques were lmted by the structure ad performace requremets of the examed materals. herefore, the detecto of complex compoets s very dffcult. Ultrasoc phased array techology uses mult-elemet trasducer wth dfferet shapes, ca mprove the shortage of the early detecto techology ad realze the detecto of the spected structure wth hgh-speed, omdrectoal ad mult agle. However, due to the omdrectoal ad mult agle characterstcs of ultrasoc phased array techology, the amout of formato obtaed by scag wll be eormous, ad wll crease the dffculty of formato processg. As a result, effectve processg method of large amouts of formato s crtcal. Compressve sesg theory poted out that oly whe the sgal s sparse the tme doma or a trasform doma, the sgal ca be sampled wth the samplg rate much less tha the samplg rate of tradtoal Nyqust samplg theorem, ad recostructed wth hgh probablty. hat s, the sparsty of sgal s the precodto of compressve sesg. Most of sgal s ot sparse atural socety, but ca be expressed as sparse form by some kd of sparse trasformato. Commoly used sparse trasform, such as Dscrete Fourer rasform, Dscrete Cose rasform, ad so o, wll lose some formato after sparse trasformato, because these trasform bases used the trasformato are geerally fxed. However, usg prcpal compoet aalyss method for data reducto, the ew varable wth low dmeso whch s lear correlato wth the orgal varable s selected to stead of the orgal varable wth hgh dmeso for the data reducto. So that the useful data of the orgal sgal ca be cluded the sparse sgal after dmesoalty reducto wth maxmze portablty. herefore, the sgal after sparse represetato ca reduce the loss of data as much as possble, ad s coducve to mprove the effcecy of sgal recostructo. Fally, the composte materal plate was used for the expermetal verfcato. he expermetal result shows that the sparse represetato of sgal based o prcpal compoet aalyss ca reduce sgal dstorto ad mprove sgal recostructo effcecy. Keywords: Prcpal compoet aalyss, Compressve sesg, Sparse represetato, Sgal recostructo. Itroducto Prcpal compoet aalyss (PCA) s oe of the most commoly used multvarate statstcal techques []. I 9, Pearso [] proposed frst the study of space a lear ad plae of the best smulato. he, mproved by Hotellg [3], the dmesoalty reducto ad feature extracto of multvarate statstcal methods was formed. Prcpal compoet aalyss uses a orthogoal mathematcal trasformato to covert the observed values of a set of possble depedet varables to the values of the varables that are ot learly related whch s called the prcpal compoet. he umber of prcpal compoets s less tha or equal to the umber of orgal varables. Oly whe the data s combed wth ormal dstrbuto, the prcpal compoet s depedet of each other. PCA s sestve to the correlato degree betwee the orgal varables, ad s also called Hotellg trasform, dscrete KL trasform or proper orthogoal decomposto dfferet felds.

2 he 9th Iteratoal Symposum o ND Aerospace By projectg the data to the low dmesoal space ad the most possble features of the orgal data, PCA ca be used to deal wth the data of hgh dmeso, ose ad hgh correlato, ad become the most wdely used techology the feld of process motorg. So far the developmet has become a kd of exploratory data aalyss ad predcto model of the effectve tool by feature extracto usg covarace or correlato matrx decomposto method, or the use of a set of data matrx sgal value. I recet decades, scholars have made a deep research o the characterstcs of PCA extracto ad dmeso reducto. It s the developmet of PCA theory ad t s wdely used dfferet dscples [4, 5, 6]. Wold et.al [7] used the method of cross examato to determe the umber of PCA prcpal compoet, ad the method based o the PCA for model predcto. Ku [8] troduces the method of "tme lag trasfer" to the feld of statstcal motorg, ad exteds the motorg method of the prevous statc PCA to the dyamc PCA method, whch s appled to the detecto of the dsturbace of the dyamc multvarable system. Compressve sesg (CS) poted out that as log as the sgal s sparse, the through the samplg rate s far lower tha that of the tradtoal Nyqust samplg theorem to collect sgals, ad the complete the recostructo of sgals by recostructo algorthm [9,,]. he theory must be premsed o the sparsty of the sgal, the prcpal compoet aalyss ca be used for data dmesoalty reducto, therefore, the compressed sesg method based o prcpal compoet aalyss s proposed ths paper, the ultrasoc phased array structural health motorg process, for sparse data represetato problem gves a better soluto.. Prcpal compoet aalyss Prcpal compoet aalyss method s based o the orgal data space, to reduce the dmeso of the orgal data space by costructg a ew set of latet varables, ad the from the mappg space sample ew major chages formato extracto, statstcal features, whch costtute the orgal data of the spatal characterstcs of uderstadg. he varable of the ew mappg space s composed of the lear combato of the orgal data varables, whch greatly reduces the dmeso of the projecto space. Because the statstcal characterstc vectors of the projecto space are orthogoal to each other, the correlato betwee varables s elmated, ad the complexty of the orgal process characterstc aalyss s smplfed. he basc dea s to fd a ew set of varables stead of the orgal oe, ad the ew varable s a lear combato of the orgal varables. From a optmzato pot of vew, the umber of ew varables s less tha the orgal varables, ad to maxmze the amout of useful formato to carry the orgal varable, ad the ew varables are ot related to each other. Its cotets clude the defto ad acqusto of ma elemets, as well as through the prcpal compoet of the data recostructo. hs method ca effectvely detfy the most mportat elemets ad structures the data, remove the ose ad redudacy, reduce the orgal complex data, ad reveal the smple structure behd the complex data. Assumg that X s a N M-dmeso matrx, each colum correspods to a varable, ad each row correspods to a sample. he the matrx X ca be decomposed to the sum of the outer product of M vectors, ad the outer product s the product of the two equal legth vectors, where must make the colum be multpled by the row, as show below. X t p t p tm pm () where t s the score vector, p s the load vector, ad the score vector of X s called the prcpal compoet of X. Each of the score vector s orthogoal, that s, for ay ad j, whe j, t ca meet t t. he each of the load vector s also orthogoal, ad the legth of each load vector s, as show below.

3 he 9th Iteratoal Symposum o ND Aerospace Gve the orgal data x x j N M dmesoal effects, ad the expresso s show as follow. where x j x j p p j, j () p p j, j, makg the stadardzato of x to elmate the x x, ad S j xj x j j x S j j j (3), j,,, m. Makg calculato of the correlato coeffcet matrx betwee the data varables after stadardzed operato, that s, the covarace matrx R, s show below. r r r m r r rm R (4) rm rm rmm where the elemet rjk represets the correlato coeffcet of the orgal varable x j ad x k, ad r r, as show below. jk kj r xk x x kj xj k jk,, xk x xkj xj k k, j,, m he elemets o the dagoal are correspodg to the varace of the observato varables, ad the o dagoal elemets are correspodg to the covarace betwee the observato varables. Assumg that the large varace of the dagoal s a sgal, ad the small varace s a ose, f the elemet o the dagoal s greater, the sgal s stroger, ad the mportace of the varable s hgher. Otherwse, f the elemet o the dagoal s smaller, t dcates that there may be ose or secodary varables. he elemet o the o dagoal correspods to the redudacy degree betwee the correlated observato varables. Jacob method s used to solve the characterstc equato I R, the egevalues of the covarace matrx ad the correspodg egevectors are obtaed. he make t accordg to the sze of the order, the characterstc value s recorded as,,, m, ad the correspodg feature vector s recorded as p, p,, pm. p p, p,, pm (6) he calculate the ma elemets t Xp, where the prcpal compoet t o behalf of the projecto of the data matrx x o the drecto of the load vector correspodg to the ma elemet. If the legth of projecto s greater, the degree of coverage or the scope of the chage s larger the drecto of p. Ad the, f t t tm, p represets the maxmum drecto of the chage of x, pm represets the smallest drecto of the chage of x. he calculatg the cotrbuto rate of each prcpal compoets s,,,, m, as well as m k k (5) 3

4 he 9th Iteratoal Symposum o ND Aerospace the cumulatve cotrbuto rate s k m k k k,,,, m. I geeral, the -th, -th, k-th prcpal compoet correspodg to the egevalues of,,, k wll be selected, where the cumulatve cotrbuto rate of egevalues s betwee 85% ad 95%. I addto, accordg to the eeds, the correspodg dmeso (that s, the umber of prcpal compoets) s selected to composto of the trasformato matrx, as show bellow. r r r k r r rk A (7) rm rm rmk Fally, the ew data after dmeso reducto s calculated as bellow. s A x (8) 3. Compressve sesg method based o prcpal compoet aalyss Compressve sesg s a ovel theory of samplg ad restorato for sparse sgal [9,,]. As log as the orgal sgal s sparsty the tme doma or uder some kd of orthogoal trasform, the sgal ca be sampled low samplg rate, ad the orgal sgal ca be recostructed wth hgh probablty. 3. Sparse represetato of sgal Compressve sesg theory s based o the premse that the sgal must be sparse. Whe makg sparse represetato of the sgal, the approprate sparse trasform base accordg to the sgal characterstcs s ecessary to be selected. he prcpal compoet aalyss method makes a kd of hgh dmesoal data reduce for low dmesoal data. he, lookg for a set of ew varables low dmesoal to replace the orgal varables hgh dmeso, where the ew varables must satsfy the codtos assocated wth the orgal varables. herefor, the ew varables ca carry the maxmum formato of orgal varable. Accordg to the prcpal compoet aalyss, ths method ca be used to make sparse represetato of the sgal. Compared wth the commoly used sparse represetato method, the sparse sgal obtaed by the proposed method s more closely related to the orgal sgal. Suppose a orgal x sgal wth the legth of N, the umber of sgal s M, a N M dmeso matrx ca be costructed wth the orgal sgal, because there are mutual relatoshp betwee the ampltude of each sgal each tme pot. Accordg to the prcpal compoet aalyss, the covarace matrx obtaed of N N dmeso ca be used as the sparse trasform based for sparse represetato of sgal. he, the orgal sgal x ca be expressed as below. x N or x (9) where [,,, N ] s the trasformato matrx of N N dmeso, s the sparse coeffcet vector obtaed by x accordg to the prcpal compoet aalyss, ad must meet the followg formula, or x, x () x 4

5 he 9th Iteratoal Symposum o ND Aerospace 3. Projecto observato of the sgal he core of the compressve sesg theory s the desg of the measuremet matrx, ad ca drectly determe whether the compressve sesg ca be mplemeted successfully. If the sgal x has a sparse represetato uder a orthogoal trasform, a measuremet matrx, R MN, whch s ot related to the trasform base ca be gve, ad a lear measuremet of M dmeso ca be obtaed, as show below. y, () Suppose the producto measuremet vector s y y, y,..., y ), the 5 ( M y x () Suffcet formato about the sgal x s cluded these lmted observatos, ad f makg, the formula () ca be coverted as follow. y x x (3) where s called the sesg matrx. I order to restore the orgal sgal wth hgh probablty, the producto measuremet matrx, whch s ot related to the sparse trasform based ad satsfed wth the restrcted sometry property, s eeded to be costructed to make producto trasformato of the sgal. Gauss radom measuremet matrx s ot related to the majorty of fxed orthogoal base ad satsfes the restrcted sometry property, so the Gauss matrx ca be used as the projecto observato matrx [,3,4]. For the ultrasoc phased array sgal, the Gauss radom measuremet matrx s multpled wth the sparse coeffcet of the phased array sgal, ad the observato vector of the sgal ca be obtaed. Suppose the measuremet matrx s dsplayed as below. M N dmeso, ad R MN, the the geeral term (, j) h j (4) Each elemet the matrx s depedet to the Gauss dstrbuto wth the mea value of, ad the varace of M. hs matrx s ot related to the vast majorty of sparse sgal, ad requres less measuremet values the recostructo. Gauss radom measuremet matrx s a matrx wth very strog radomcty, but the dsadvatage s hgh ucertaty. For a sgal wth a legth of N ad a sparse degree of K, oly M ck log( N K) measured values are eeded to recover the orgal sgal wth hgh probablty, where the c s a very small costat. 3.3 Sparse recostructo of the sgal Durg the process of compressve sesg, recostructg the sgal x from the observatos y s the verse problem relatve to compresso samplg, ad s called the sgal recostructo. By solvg the equato (3), the recostructed sgal ca be obtaed. hs problem s uderdetermed wth fte solutos. Cades et al. proved that the uderdetermed problem ca be solved by solvg the mmum l-orm, that s show below. m s.t. y x (5) Formula (5) s a lear programmg problem, ad s also a covex optmzato problem. Makg the recostructo error take to accout, t s coverted to a mmum l-orm problem, as dsplayed follow. m s.t. y (6) Durg the process of sgal recostructo, covex optmzato algorthm ad greedy teratve algorthm are commoly used [5,6,7]. Oe kd of algorthm s based o covex optmzato, maly by creasg the costrat to obta the sparsest. Ad commoly used algorthms are bass pursut algorthm ad gradet projecto sparse recostructo algorthm. he other kd of algorthm s based o greedy teratve algorthm, maly by the combato M

6 he 9th Iteratoal Symposum o ND Aerospace of local optmzato method to fd the o-zero coeffcets, order to approach the orgal sgal. Commoly used algorthms are matchg pursut algorthm ad orthogoal matchg pursut algorthm. 4. Expermet ad results I ths sesso, the composte plate works as the expermetal object. here are e pezoelectrc elemets the lear array arraged o the plate wth mm of adjacet equal terval. I sgal acqusto, data collecto pots are 4, ad samplg frequecy s fs=hz. Makg oe array elemet as a drve to trasmt sgal, ad the other eght elemets as the sesor to receve the reflecto sgal. Each array elemet stmulates the sgal turs, the each degree correspods to 98 sgals, ad 988 sets of data ca be obtaed. he 9 degree drecto of the data emtted by No. array elemet ad receved by No. array elemet s selected to work as the expermetal data, ad the processg method of other agles s cosstet wth ths. he tme doma waveform of the data set s show fgure. 5 ampltude/v tme/ms Fg. he waveform of orgal sgal tme doma At frst, prcpal compoet aalyss s used to deal wth the waveform obtaed by the 9 degree drecto of the phased array sgal emtted by No. array elemet ad receved by No. array elemet. he sparse represetato of the orgal sgal s obtaed, as show fgure. It ca be see that the sparse coeffcet of the phased array sgal after PCA trasform s mostly zero or close to zero, whch s cosstet wth the characterstc of sparse sgal. sparse coeffcet/v tme/ms Fg. he sparse coeffcet after prcpal compoet aalyss 6

7 he 9th Iteratoal Symposum o ND Aerospace he, the legth of utrasoc phased array data s N=, ad the umber of observatos M=4 s selected to complete the operato of sgal projecto observato, ad the waveform show fgure coeffcet/v observatos/ms Fg. 3 he sgal obtaed by projecto observato Fally, the bass pursut algorthm s used to deal wth the ultrasoc phased array sgal, ad the recostructed sgal obtaed are show fgure 4. 5 orgal recostructo ampltude/v -5 Elarged Fg.b tme/ms (a) he recostructo of the orgal sgal 3 orgal recostructo ampltude/v tme/ms (b) he local elargemet of recostructed sgal Fg. 4 he sgal recostructo based o bass pursut algorthm 7

8 he 9th Iteratoal Symposum o ND Aerospace 5. Expermetal error aalyss he recostructed sgal based o orthogoal matchg pursut algorthm has some dffereces the sgal waveform, compared wth the orgal phased array sgal. I order to aalyze the effect of the recostructo algorthm more accurately, the recostructed error s dsplayed umercally, as show table. he absolute error V ad the relatve error are calculated as below, V (7) V V V V % (8) V where V s the ampltude of recostructed sgal at the pot of maxmum ampltude devato, ad V s the ampltude of orgal phased array sgal at the same pot. able. Recostructo error algorthm BP he absolute error/v.87 he relatve error/%.39 where V. 87, ad.39%. he table shows the error comparso of some commo trasform base ad the prcpal compoet aalyss method. able. he error comparso orthogoal trasformato PCA DC DF he absolute error/v he relatve error/% where DC s Dscrete Cose rasform, ad DF s Dscrete Fourer rasform. From the aalyss of expermetal error, t dcates that the relatve error s relatvely lower tha commoly used method. hat s, the proposed method ca be appled to sgal sparse represetato of compressve sesg. 6. Cocluso hs paper research o the compressve sesg sparse samplg method based o prcpal compoet aalyss. hs method ot oly solves the dffculty storage ad processg due to the large amout of data obtaed by ultrasoc phased array structural health motorg, but also effectvely mproves the relatoshp betwee the orgal sgal ad the sgal after sparse represetato. Ad the expermetal result shows that prcpal compoet aalyss ca be used to recostruct the sgal obtaed from the phased array structure health motorg after sparse represetato of the sgal wth small recostructo error. I future research, we ca choose more optmzed projecto observato matrx, ad more effcet recostructo algorthm to recostruct the ultrasoc phased array sgal more effcet. Ackowledgemet hs work s supported by Natoal Natural Scece Foudato of Cha (Grat No. 5454, No. 6679), Jagsu Govermet Scholarshp for Overseas Studes ad the PAPD fud. 8

9 he 9th Iteratoal Symposum o ND Aerospace Refereces. S Lu, M Gu, Q Zhag, B L. Prcpal compoet aalyss algorthm vdeo compressed sesg. Iteratoal Joural for Lght ad Electro Optcs, 4, 5(3): Karl Pearso F.R.S. O les ad plaes of closest ft to systems of pots space. Phlosophcal Magaze, 9, (6): H Hotellg. Aalyss of a complex of statstcal varables to prcpal compoets. Joural of Educatoal Psychology, 933, 4(6): 47: R. Masero, G. Quer, D. Muaretto, M. Ross, J. Wdmer, M. Zorz. Data acqusto through jot compressve sesg ad prcpal compoet aalyss. IEEE Coferece o Global elecommucatos, 9 : Y Zhag, C Xu, C L, H Yu, J Cao. Wood defect detecto method wth PCA feature fuso ad compressed sesg. Joural of Forestry Research, 5, 6(3): A. Sgh, L.N. Sharma, S. Dadapat. Mult-chael ECG data compresso usg compressed sesg egespace. Computers Bology & Medce, 6, 73: S. Wold. Cross-valdatory estmato of the umber of compoets factor ad prcpal compoets models. echometrcs, 978, (4): W. Ku, R.H. Storer, C. Georgaks. Dsturbace detecto ad solato by dyamc prcpal compoet aalyss. Chemometrcs ad Itellget Laboratory Systems, 995, 3(): DONOHO D.L. Compressed sesg. IEEE rasactos o Iformato heory, 6, 5(4): H Y, Z Lu. Survey of compressed sesg. Cotrol ad Decso, 3, 8(): Y Su, F Gu. Compressve sesg of pezoelectrc sesor respose sgal for phased array structural health motorg. Iteratoal Joural of Sesor Networks, 7, 3(4): CANDES E. he restrcted sometry property ad ts mplcatos for compressed sesg. C.R. Math.Acad. Sc. Pars, 8, 346(9-): J L, Q Wag, Y She, B L. Collaboratve Costructo of Measuremet Matrx ad Recostructo Algorthm Compressve Sesg. Acta Electroca Sca, 3, 4(): Che, G Zhu, Y Lu. Optmzato research of Gauss matrces compressve sesg. Applcato Research of Computers, 4, 3(): J Feg, G Zhag, Y Lu. Improved Sparse Sgal Recostructo Algorthm Based o SL Norm. Joural of Data Acqusto ad Processg, 6, 3(): Y L, S He. Bld Sgal Separato Algorthm for No-egatve Matrx Factorzato Based o Projected Gradet. Computer Egeerg, 6, 4(): Y Zhu, Y Zhu, Y Peg. Adaptve Compressve Sesg ad rackg of Dyamc Sparse Spectrum. Joural of Sgal Processg, 6, 3(3):

Functions of Random Variables

Functions of Random Variables Fuctos of Radom Varables Chapter Fve Fuctos of Radom Varables 5. Itroducto A geeral egeerg aalyss model s show Fg. 5.. The model output (respose) cotas the performaces of a system or product, such as weght,