ENSEMBLE learning has been widely used in data and

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1 IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 9, NO. 5, SEPTEMBER Sparse Kernel-Based Hyperspectral Anoaly Detecton Prudhv Gurra, Meber, IEEE, Heesung Kwon, Senor Meber, IEEE, andtothyhan Abstract In ths letter, a novel enseble-learnng approach for anoaly detecton s presented. The proposed technque as to optze an enseble of kernel-based one-class classfers, such as support vector data descrpton (SVDD) classfers, by estatng optal sparse weghts of the subclassfers. In ths ethod, the features of a gven ultvarate data set representng noralcy are frst randoly subsapled nto a large nuber of feature subspaces. An enclosng hypersphere that defnes the support of the noralcy data n the reproducng kernel Hlbert space (RKHS) of each respectve feature subspace s estated usng standard SVDD. The jont hypersphere n the RKHS of the cobned kernel s learned by optally cobnng the weghted ndvdual kernels whle posng the l1 constrant on the cobnng weghts. The jont hypersphere representng the optal copact support of the ultvarate data n the jont RKHS s then used to test a new data pont to deterne f t belongs to the noralcy data or not. A perforance coparson between the proposed algorth and regular SVDD s reported usng hyperspectral age data as well as general ultvarate data. Index Ters Hyperspectral anoaly detecton, sparse kernelbased enseble learnng (SKEL), support vector data descrpton (SVDD). I. INTRODUCTION ENSEMBLE learnng has been wdely used n data and pattern classfcaton because an enseble decson based on explotng a large degree of freedo avalable n hghdensonal ultvarate data corrupted by nose and outlers can generally provde ore robust generalzaton perforance than the regular decson by a sngle classfer [1]. Ths s partcularly true n the case of hyperspectral classfcaton and detecton probles [2], [3]. A novel enseble-learnng technque called sparse kernelbased enseble learnng (SKEL) [2], [3] has been recently developed by two of the current authors for hyperspectral classfcaton probles that use support vector achnes (SVMs) as subclassfers. In SKEL, subdecson functons were frst learned wthn the respectve randoly selected feature subspaces, and an optal sparse cobnaton of the subdecson functons fro a large nuber of SVMs was subsequently estated through the l1-nor-constraned optzaton of the kernel weghts. Ths s based on ultple kernel learnng Manuscrpt receved May 24, 2011; revsed October 5, 2011 and Deceber 9, 2011; accepted January 27, Date of publcaton March 15, 2012; date of current verson May 29, P. Gurra and H. Kwon are wth the U.S. Ary Research Laboratory, Adelph, MD USA (e-al: pkgurra@gal.co; heesung.kwon@us.ary.l). T. Han s wth Aglex Technologes, Inc., Chantlly, VA USA (e-al: Tothy.Han@aglex.co). Color versons of one or ore of the fgures n ths paper are avalable onlne at Dgtal Object Identfer /LGRS (MKL) algorth developed n [5]. A sgnfcant proveent n detecton perforance was reported by usng SKEL for hyperspectral as well as ultvarate data. In ths letter, the prncple of SKEL s extended to hyperspectral anoaly detecton where the exaples of the reference sgnatures of the objects of nterest are no longer avalable n advance. The proposed technque s called sparse kernelbased anoaly detecton (SKAD) where ntally, a large nuber of one-class classfers based on the support vector data descrpton (SVDD) ethod [6] are used as subclassfers. SVDD s one of the state-of-the-art technques wdely used for hyperspectral anoaly detecton [7]. Unlke generatve odelbased technques n [8] and [9], SVDD learns the support or boundary of the gven noralcy data by buldng a nal enclosng hypersphere contanng the data. The use of nonlnear kernels allows SVDD to accurately odel the nonlnear support/boundary of nontrval ultodal dstrbutons [6]. The kernel-based SVDD frst nonlnearly aps the nput data to a hgh-densonal feature space called reproducng kernel Hlbert space (RKHS) and then fnds the enclosng hypersphere of the data. Slar to SKEL, n SKAD, the SVDD-based subclassfers (one-class classfers) are optally desgned n the respectve randoly selected feature subspaces pror to optzng the enseble. A new RKHS assocated wth a kernel that s a convex cobnaton of these ndvdual kernels can be deterned, and a hypersphere enclosng the noralcy spectra can be bult. The ost copact enclosng hypersphere contanng the local noralcy spectra n ths RKHS correspondng to the sparsely cobned kernel s obtaned by nzng the radus of the hypersphere. The optal sparse weghts of the ndvdual kernels are estated by usng the gradent descent optzaton of the kernel weghts wth a l1 constrant appled on the. SKAD uses a two-step teratve process to do ths: a) the nzaton of the radus of the jont hypersphere by obtanng optal support vectors fro a cobned kernel atrx that defnes the boundary of the hypersphere and b) the gradent descent optzaton of the l1-contraned weghts of the ndvdual subclassfers to further nze the radus of the jont hypersphere. The optal jont hypersphere defnes the cobned support of the local background data across the randoly selected spectral subspaces. The jont hypersphere provdes a ore copact support of the noralcy data than the ndvdual hyperspheres bult n the feature subspaces and the hypersphere bult n the orgnal nput feature space usng regular SVDD. The spectral sgnatures that le outsde the jont hypersphere are consdered outlers. To apply SKAD on hyperspectral age data sets for anoaly detecton, for each test pxel of a hyperspectral age, a sldng dual rectangular wndow [9] s used, where the spectra between the nner and the outer X/$ IEEE

2 944 IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 9, NO. 5, SEPTEMBER 2012 wndows represent the noralcy data. The sze of the nner wndow s set to enclose the potental objects of nterest, and the sze of the outer wndow s deterned to nclude an enough nuber of spectral sgnatures to estate a copact hypersphere surroundng the local noralcy data n the RKHS. The spectral sgnature of the test pxel fro the center of the nner wndow s tested aganst the jont optal hypersphere to fnd whether t belongs to the background or s an outler and, hence, a target pxel. Both qualtatve and quanttatve perforance coparsons between SKAD and regular SVDD are provded usng hyperspectral age data sets as well as ultvarate data sets. The rest of ths letter s organzed as follows. In Secton II, the concept of SVDD s ntroduced. Secton III descrbes the SKEL usng SVDD anoaly detectors for hyperspectral target detecton and explans the pleentaton ssues. Sulaton results are presented n Secton IV. Fnally, Secton V concludes ths letter wth soe rearks about the proposed ethod. II. SVDD SVDD, ntroduced n [6], characterzes the background data set by contanng only the relevant data and excludng the superfluous space around the data set. The boundary of the data set s defned by the vectors or saples n the background data, whch are called support vectors. The pxels that le outsde ths boundary are detected as outlers or anoales. Consder a data set contanng saples represented as {x }, where x R d s a d-densonal feature vector of each data saple. Let Φ be a functon that transfors the nput feature vector to a hgh-densonal (possbly nfnte) RKHS assocated wth the kernel functon k(x, x j )= Φ(x ), Φ(x j ). The kernel-based SVDD algorth tres to fnd the sallest hypersphere n the RKHS that encloses the gven background data set n order to exclude the superfluous space around the background data set as uch as possble. Ths sphere s defned by ts center a and radus R. So, the functonal R 2 s nzed wth a constrant that the hypersphere contans all the background data ponts. If there s a possblty of outlers exstng n the background data, then slack varables are used to allow for the outlers. It can be expressed as n L(a,R)=R 2 + C ξ subject to Φ(x ) a 2 R 2 + ξ =1, 2,...,N, ξ 0 =1, 2,...,N (1) where paraeter C controls the tradeoff between the volue of the hypersphere and the errors. After applyng the Lagrange ultplers {α,=1, 2,...,N} and Karush Kuhn Tucker (KKT) condtons (see [7] for detals), the dual proble can be wrtten as ax L(α )= subject to 0 α C α k(x, x ),j =1, 2,...,N, α α j k(x, x j ) α =1. (2) Ths s a convex quadratc prograng proble and can be easly solved to obtan the optal Lagrangan ultplers {α }. The center of the hypersphere (although cannot be deterned explctly) after solvng (2) s gven by a = αφ(x ). (3) The vectors wth α =0 le nsde the hypersphere and are consdered to be a part of the background data. The vectors wth the correspondng Lagrange ultplers 0 <α <C are the support vectors that actually le on the boundary of the hypersphere. The vectors wth the correspondng Lagrange ultplers α = C are the outlers (stll support vectors) that are allowed by the ntroducton of slack varables. These vectors le outsde the hypersphere. Then, the radus of the hypersphere s gven by R 2 = 1 N b { Φ(xk ) a 2} N b k=1{ } = 1 N b k(x k, x k ) 2 α k(x k, x )+ α αj k(x, x j ) N b k=1 (4) where Φ(x k ), k =1, 2,...,N b, denotes the support vectors that le on the boundary of the background data set, and N b s the total nuber of such boundary support vectors. III. SPARSE KERNEL-BASED SVDD ENSEMBLE LEARNING Recently, t has been shown that SKEL provdes a better generalzaton perforance over a sngle classfer n the case of bnary classfcaton [2], [3]. In SKEL, the fnal kernel k(x, x j ) s consdered as a convex cobnaton of bass kernels k (x, x j ), =1, 2,...,M. Each of the bass kernels k s assocated wth ts correspondng RKHS. Each kernel and RKHS s generated fro a randoly selected feature subspace of the nput data. Each feature subspace s obtaned by the rando selecton of features usng a unfor dstrbuton. Matheatcally, t can be wrtten as M k(x, x j )= d k (P x, P x j ) subject to d 0 =1, 2,...,M, M d =1 (5) where M s the total nuber of kernels and d denotes the weghts of the ndvdual kernels. P s an F d projecton atrx whch defnes the nput features randoly selected to for the th subspace. It s defned by the eleents P f,n =1 f the nth nput feature s selected as the fth feature of the th subspace and zero, otherwse, for all n =1, 2,...,d and f =1, 2,...,F.Thel1 nor constrant s appled on the weghts of the bass kernels to proote sparsty aong the and to select only the best feature subsets that help n provng the generalzaton perforance of the classfer. In ths letter, we apply the sae prncple to SKAD usng SVDD-based one-class classfers as the subclassfers. For ths purpose, the pral SVDD proble fro (1) can be rewrtten as n L(a,R,)=R 2 + C ξ subject to a 2 2 a, Φ(x ) R 2 1+ξ =1, 2,...,N, ξ 0 =1, 2,...,N. (6),j

3 GURRAM et al.: SPARSE KERNEL-BASED HYPERSPECTRAL ANOMALY DETECTION 945 Followng the fraework of the MKL presented n [5], the pral SVDD-based enseble proble can be wrtten as n L(a,R)=R 2 + C ξ M M 1 subject to a 2 2 a, Φ (P x ) d R 2 1+ξ =1, 2,...,N, ξ 0 =1, 2,...,N, d 0 =1, 2,...,M, M d =1 (7) where each d controls the effect of the hypersphere n the RKHS assocated wth each kernel k on the jont hypersphere n the RKHS assocated wth the cobned kernel k. Φ s the nonlnear functon that aps the nput feature vectors nto the RKHS assocated wth the kernel k. One can observe that the l1 constrant appled on these weghts d leads to the selecton of the best feature subsets based on the fact that the jont enclosng hypersphere n the RKHS assocated wth the cobned kernel fro these feature subsets has a nu radus. Introducng the Lagrange ultplers α 0, γ 0, η 0, and λ 0, the Lagrangan of (7) s obtaned as L p = R 2 + C ξ + α { } 1 a 2 2 a, Φ (P x ) +1 R 2 ξ d { } γ ξ + λ d 1 η d. (8) After applyng the KKT condtons by settng the dervatves of L p wth respect to R 2, a, ξ, and d to zero and substtutng these condtons n (8), the dual proble of (7) s obtaned as ax L d =1 λ subject to α =1, 0 α C =1, 2,...,N, α α j k (P x, P x j ) λ,j =1, 2,...,M. (9) Ths dual proble s slar to the dual proble of MKL usng the SVM for bnary classfcaton, whch s explaned n [5] and s very dffcult to optze due to the last constrant n (9). Instead, slar to [5], an alternate for of the constraned optzaton proble s consdered here n d subject to J(d) M d =1, d 0 =1, 2,...,M. (10) where J(d) s gven n (11), shown at the botto of the page. The proble n (11) s the pral regular SVDD proble expressed n (1) and (6). Hence, t can be solved by usng a standard SVDD quadratc prograng solver. The dual proble for the pral proble expressed n (10) and (11) can be fored by substtutng the cobned kernel (5) nto the dual proble of the standard SVDD (2) as follows: ax J(d, α)= M α d k (P x, P x ) α α j d k (P x, P x j ),j subject to 0 α C =1, 2,...,N, α =1. (12) If the Gaussan radal bass functon (RBF) kernel s used, k(x, x )=1. Then, the dual proble can be wrtten as ax J(d, α)=1 α α j d k (P x, P x j ),j subject to 0 α C =1, 2,...,N, α =1. (13) Usng the cobned kernel k(x, x j )= M d k (P x, P x j ), (13) can be solved as a standard SVDD proble. Once the optal Lagrange ultplers α are obtaned, the objectve value of (13) s gong to be J(d) =1 M αα j d k (P x, P x j ). (14),j Ths s solved usng the gradent descent algorth, where the gradent of J(d) wth respect to each weght d s gven by J = α d α jk (P x, P x j ). (15),j Once the gradent of J(d) s coputed, d s updated by usng a descent drecton calculated usng a reduced gradent ethod descrbed n [5]. The reduced gradent ethod ensures that the weghts d are updated n such a way that the equalty constrant and the nonnegatvty constrants on d [shown n (10)] are satsfed. Optal weghts of the subclassfers d are obtaned when the gradent descent algorth converges, and the optal Lagrange ultplers α are obtaned for the fnal cobned kernel-based SVDD. Now, the radus of the jont hypersphere s gven by { R 2 = 1 N b 1 2 α d N k (P x k, P x ) b k=1 + } αα j d k (P x, P x j ) (16),j where Φ(x k ), k =1, 2,...,N b, denotes the support vectors that le on the boundary of the background data set n the M M n J(d)= s.t. R 2,{a },ξ R 2 + C ξ M M 1 d a 2 2 a, Φ (P x ) R 2 1+ξ =1, 2,...,N, ξ 0 =1, 2,...,N (11)

4 946 IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 9, NO. 5, SEPTEMBER 2012 cobned RKHS, and N b s the total nuber of such boundary support vectors. A. Algorth Ipleentaton The algorth s ntated by selectng rando feature subspaces fro the nput data to for weak classfers. The weghts of all the weak classfers or kernels are set to the sae value,.e., 1/M. The fnal kernel s obtaned by cobnng all the weghted ndvdual kernels. Then, the optzaton proble n (13) s solved to obtan the best soluton of α.theyare plugged nto (15) to be used n the gradent descent algorth for updatng the weghts of the ndvdual kernels. These two steps are contnued untl the algorth convergence crteron s et. In ths letter, the total change n the weghts s used as the convergence crteron. The algorth ternates when the change n the weghts of the ndvdual kernels s below a certan threshold. The fnal optzed sparse weghts are used to cobne the kernels and obtan the hypersphere n the RKHS assocated wth the cobned kernel as shown n (16). In ths letter, the dual-wndow technque descrbed n [9] and [7] s used to obtan the local background nforaton for every pxel n the hyperspectral age. The center of the nner wndow represents the test pxel x T. The pxels n between the nner wndow and the outer wndow consttute the background data set for that partcular test pxel. The test statstc that can be used to deterne f the test pxel s an anoaly or not usng SKAD s gven by F SKAD (x T )=1 2 α d k (P x T, P x ) + αα j d k (P x, P x j ) R 2. (17),j Ths statstc provdes a easure of slarty between the test pxel and ts background. Snce the rad of the hyperspheres enclosng the local background data sets at dfferent test pxels n a hyperspectral age are not equal, a ROC curve cannot be generated by drectly applyng varyng thresholds on the statstcs n (17). So, we use the noralzed SKAD statstc (slar to [8]) to generate ROC curves and quantfy the perforance of the detector. The noralzed SKAD test statstc s expressed n (18), shown at the botto of the page. IV. SIMULATION RESULTS A. Hyperspectral Iage Data In ths secton, we apply the kernel-rx (KRX) [9], regular SVDD, and SKAD algorths on two hyperspectral dgtal agery collecton experent (HYDICE) ages the Forest Radance I (FR-I) and the Desert Radance II (DR-II) ages. The szes of the nner wndow and the outer wndow used for the dual-wndow technque n both the test ages are 5 5 and 15 15, respectvely. The sze of the nner wndow was set to enclose the targets whose approxate sze was predeterned through a pror knowledge of the range of the Fg. 1. (a) ROC curves obtaned on the FR-I age usng KRX, regular SVDD, and SKAD. (b) ROC curves obtaned on the DR-II age usng KRX, regular SVDD, and SKAD. scene and feld of vew of the hyperspectral sensor. The outer wndow sze was deterned to provde an enough nuber of spectral saples to odel the support of the local background. The value of C for SVDD and SKAD algorths s set to ake the lower bound on the nuber of support vectors or the upper bound on the nuber of outlers allowed to be 20% of the total nuber of local background pxels beng used fro the dualwndow technque [10]. The Gaussan RBF kernel paraeter σ for SVDD and SKAD s deterned by pleentng the approxate nax technque on the randoly selected ten regons of the age to represent the background as done n [8]. The sae value s used over all the test pxels n the age. The kernel paraeter for KRX algorth fro [9] s used n ths letter as well. Each test pxel s then tested f t falls outsde or nsde of the hypersphere assocated wth the local background regon to deterne f t s an anoaly or a background pxel. Snce there s randoness nvolved n the selecton of the feature subspaces for each of the kernels, we have run the SKAD algorth fve tes and plotted error bars on the ROC curves for all the data sets. However, each run generates dfferent values of probablty of detecton and false alar rate. In order to generate the error bars, all the ROC curves are nterpolated onto the sae scale of false alar rate. Fg. 1(a) and (b) show the anoaly detecton results of KRX, regular SVDD, and the proposed sparse kernel-based SVDD (SKAD) on the FR-I age and the DR-II age. Fro Fg. 1(a) and (b), one can see that the perforance of SKAD s generally better than that of regular SVDD and KRX over the entre range of false alars for both the ages FR-I and DR-II. The densonalty of the feature subspaces s set to 50 for both the ages. The densonalty of 50 was deterned based on the perforance analyss of SKEL, whch shares the slar enseble concept wth SKAD. It turned out that ths nuber provded a near-optal perforance for SKEL. SKAD has been appled of FR-I and DR-II ages wth dfferent densonaltes of feature subspaces, and the ROC curves are plotted n Fg. 2. It can be observed that SKAD wth any densonalty of the subspace outperfors SVDD n ters of anoaly detecton. The algorth s ntalzed wth 100 kernels. The average nuber of kernels that s selected (wth nonzero weghts) after runnng the SKAD algorth on the DR-II ages s 72. The average nuber of nonzero weght kernels selected on the FR-I age s 71. F SKADN (x T )= 1 2 α d k (P x T, P x )+,j α α j d k (P x, P x j ) R 2 (18)

5 GURRAM et al.: SPARSE KERNEL-BASED HYPERSPECTRAL ANOMALY DETECTION 947 tranng saples. The fnal nuber of kernels selected after the optzaton for the two data sets are 42% and 12%, respectvely. One can, agan, observe fro the ROC curves that SKAD provdes a sgnfcantly proved probablty of detecton at a low false alar rate over regular SVDD and KRX for sonar nes data set and over KRX for onosphere data set. The confdence ntervals of the SKAD ROC curves for the ultvarate data sets are larger than those for the hyperspectral data sets due to the sall nuber of saples n the test set. Fg. 2. ROC curves obtaned on the FR-I and DR-II ages usng the dfferent nubers of features for the subspaces. (a) FR-I. (b) DR-II. Fg. 3. (a) ROC curves obtaned on the sonar nes data set usng regular SVDD and SKAD. (b) ROC curves obtaned on the onosphere data set usng regular SVDD and SKAD. One ght wonder that the proveent ade by SKAD over regular SVDD s not sgnfcant. However, SVDD s one of the state-of-the-art anoaly detecton technques, and t s hard to overcoe ts perforance [7]. In fact, no fndngs have yet been reported n lterature on anoaly detecton technques wth a superor perforance to SVDD. Moreover, the enseble structure tself can be successfully used for other applcatons, such as ultodal sensor fuson for anoaly detecton, each subclassfer recevng the nput fro the dfferent sensor odaltes. B. Multvarate Data The KRX, regular SVDD, and SKAD algorths are also copared based on ther perforance on ultvarate data sets. In ths letter, two ultvarate one-class data sets fro [11] have been used. The frst data set s the sonar nes data set, n whch the sonar sgnals that bounced off of cylndrcal etal nes represent the noralcy and the sonar sgnals that bounced off of roughly cylndrcal stones represent the outlers. The second data set s the onosphere data set whch ncludes good radar returns wth soe structure as the noralcy data and the returns wth no evdence of structure as the outlers. For both the data sets, 50% of the noralcy data ponts are randoly selected and used as the tranng set of the noralcy class. The other 50% of the noralcy data ponts along wth the outlers are used n test sets to generate the ROC curves presented n Fg. 3(a) and (b). The ntal nuber of kernels used s 50 for both the data sets. The sonar nes data set has 60 features, and the densonalty of the randoly selected feature subspaces s three. The onosphere data set has 34 features, and 5-D feature subspaces have been used. The densonalty of the feature subspaces for the two data sets was deterned by choosng the one that provdes the best proveent n the detecton perforance. The value of C s set to ake the lower bound on the nuber of support vectors or the upper bound on the nuber of outlers allowed to be 10% of the total nuber of V. C ONCLUSION In ths letter, a SKAD has been proposed for hyperspectral ages as well as soe ultvarate data sets. SKAD s presed on the prncple of enseble learnng that an optal cobnaton of ndependently estated subdecsons fro the respectve feature subspaces can sgnfcantly prove the predcton perforance of a ultvarate vector for detecton/classfcaton. In ths ethod, the weghts of ndvdual SVDD-based one-class classfers bult n randoly selected feature subspaces are optzed by nzng the radus of the hypersphere enclosng the noralcy/background data n the jont RKHS. Moreover, the weghts are sparse n nature due to the l1 constrant appled on the. The sparsty of the weghts allows the algorth to choose the best subsets of features that generate a jont RKHS, n whch the enclosng hypersphere has the ost copact support. It s well known that enseble-learnng technques have to use uch ore coputatonal resources than regular algorths to perfor ultple learnng as opposed to sngle learnng, and hence, coputatonal te requred for SKAD s uch longer than that requred for SVDD. However, ths s true only n the case of hyperspectral ages because of the dualsldng-wndow technque used. In the case of ultvarate data sets, the speed of SKAD s coparable to that of SVDD. The future work nvolves reducng the coputatonal coplexty by optzng the pleentaton of the dual-sldng-wndow SVDD used n the proposed work, whch eventually leads to a coputatonally splfed SKAD. REFERENCES [1] R. E. Schapre, The strength of weak learnablty, Mach. Learn., vol.5, no. 2, pp , Jun [2] P. Gurra and H. Kwon, Enseble learnng based on ultple kernel learnng for hyperspectral checal plue detecton, n Proc. SPIE Defense, Securty Sens. Syp., Orlando, FL, Apr. 5 9, [3] P. Gurra and H. 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