Large Deviations Analysis for the Detection of 2D Hidden Gauss-Markov Random Fields Using Sensor Networks

LARGE DEVIA TIONS AN AL YSIS FOR THE DETECTION OF 2D HIDDEN GA USS -MARK O V RANDOM FIE LDS USING SENSOR NETWORKS Y oungchul Sung † , H. V incent P oor and Heejung Y u ABSTRA C T The dete ction of hidd en two-di mensional Gauss-Marko v random fields us- ing sensor netwo rks is conside red. Under a condi tional autore gressi ve model, t he error expon ent for the Ne yman-Pearson detecto r satisfyi ng a fixed le vel constraint is obtained using the larg e de viation s principle. For a symmetric first order autoregr essiv e model, the error exponent is gi ven expl icitly in terms of the SNR and an edge dependenc e fac tor (field cor - relati on). The behavior of the error exponen t as a function of correl ation strength is seen to divide into two regions depending on the v alue of the SNR. At high SNR, u ncorrelat ed observ ations maximize the error expon ent for a giv en SNR, whe reas there is non -zero optimal c orrelati on at lo w SNR. Based on the error expo nent, the energy efficie ncy (defined as the ratio of the total information gathe red to the total energy required ) of ad hoc sen- sor network for detection is examin ed for two sensor deplo yment models: an infinit e area model and and infinite density model. For a fixed sensor density , the ene rgy effici ency diminishes to zero at rate O ( area − 1 / 2 ) as the area i s inc reased. On the o ther hand, non -zero effic ienc y is possible for increa sing density dependi ng on the be havior of the physic al correl ation as a functio n of the link length. Index T erms- Neyman- Pearson detection, error e xponent, GMRF 1. INTRODUCTION Consider the design of a sensor network for the detection of a correlated stochastic signal in a fixed area. Many questions arise in such a design: Ho w do the fi eld correlation and measurement signal-to-noise (SNR) affec t the detection performan ce? W hat is the optimal se nsor density , i.e. , the number of n odes per unit area? What is the i nformation and energy trade-of f in such a sensor net- work with a d hoc routing? T o address these issues, se veral studies based on one-d imensional (1D) spatial signal models hav e been conducted (see, e.g., [1] and [2]). Ho wev er , there is an important differe nce between 1D signal models and actual spatial signals. Suppose that we take observ ations from sensors placed equidis- tantly along a line transect laid over a gi ven area. The o bserv ations may then be vie wed as samples generated by a one-dimensional process and the results from time series analysis could be applied to inv estigate their stat istical properties. Howe v er , t here is no real notion of ‘signal flow’ or dependenc e direction along the transect as t here is in a more traditionally obtained time series. For sam- ples from sensors deployed over a two-dimensional (2D) area, it is necessary t o consid er the signal depen dence in all direction in the plane, and as a consequence, answering the abo ve questions becomes more dif ficult. T o address the above questions in a 2D setting, in this pa- per , we consider the detection of 2D Gauss-Marko v random fields † Y . Sung and H. Y u are with the D ept. of Elec trical Engineering, Korea Ad- vanced Institute of Science and T echnology (KAIST), Daejeon 305-701, South Ko- rea. Email:ysung@ee.kaist. ac.kr a nd hjyu@s tein.kaist.ac.kr. H. V . Poor is with the Dept. of Electrical Engineering, Princeton Univ ersity , Princeton, NJ 08544. Email: poor@princeton.edu. The work of Y . Sung was s upported in part by Brain Korea 21 Project, the School of Informati on T echnology , KAIST . The work of H. V . Poor was supported in part by the U. S. National Science Foundation under Grants ANI-03- 38807 and CNS-06-25637. (GMRFs) using no isy obs ervation s. In p articular we consider Sen- sors i j located on a 2D lattice I . On den oting the (noisy) mea sure- ments of Sensor ij as Y ij and adopting a Ne yman-Pearson formu- lation, we can model the detection problem via n ull and alternativ e hypotheses giv en by H 0 : Y ij = W ij , ij ∈ I vs . H 1 : Y ij = X ij + W ij , ij ∈ I , (1) where { W ij } represents independent and identically distri buted (i.i.d.) N (0 , σ 2 ) noise with a known varianc e σ 2 , and { X ij } is a stationary GMRF on the 2D lattice I independen t of the mea- surement noise { W ij } . Thus, the observ ation samples form a 2D hidden GMRF under H 1 . P S f r a g r e p l a c e m e n t s ( i, j ) X ij X ij W ij Y ij Y ij Sensor ij r r Fig. 1 . Sensors on a 2D Lattice I : Hidden Markov Structure 1.1. Summary of Results The exact error probability of the detection of t he Neyman-Pearson test is not ava ilable in closed-form in the general correlated case, including the hypo theses (1). Hence, we in vok e the the lar ge de vi- ations principle and use the err or exp onent of the detection prob- ability (or, more con venien tly , its complement, t he miss probabil- ity) as an alternativ e performa nce measure. For a fi xed false-alarm lev el, the miss probability P M decays e xponentially as the sample size n increases, and the error expo nent is defined as the rate of expo nential decay , i.e., K ∆ = lim n →∞ − 1 n log P M (2) under the giv en constraint (i.e., the false alarm probability P F ≤ α ). The error exponent is a goo d performance criterion in the large sample regime since it allo ws the designer to estimate the number of samples required for a giv en detection performance. Hence, efficien t design can be ex amined through the error exp onent for large scale senso r networks. Here, we adopt the conditional autor egr ession (CA R) model for t he signal, and deri ve a closed-fo rm expression for the error expo nent K of the miss proba bility ( which is indepen dent of α ) i n the spectral domain. W e do so by exploiting the spectral str ucture of the CAR signal and the relationship between the eigen v alues of the block circulant approximation t o a block T oeplitz matrix de- scribing the 2D correlation structure. In particular , it is sh own tha t the error exponent for the detection of 2D hidden GMRF is an ex- tension of that in the 1D case obtain ed by Sung et al.[3]. As in the 1D case, it is sho wn that i.i.d. (and, t hus, uncorrelated) observ a- tions maximize the error ex ponent for a gi ve n SNR when th e SNR is high. On the other hand, there is an optimal non-zero degree of correlation at low SNR. I nterestingly , it is seen that t here i s a dis- continuity in the o ptimal c orrelation streng th as a function of SNR. In t he perfectly correlated case, the error expon ent is zero as ex- pected. For the e rror expo nent as a fun ction of SNR, we will sho w that the error expon ent increas es as log SNR for a given correlation strength at high SNR. W e conside r two asymptotic re gimes modelling the senso r de- ployme nt in 2D: an infinite area model with a fixed dens ity and an infinite density model wi th a fi xed area. Applying the results, we obtain the asym ptotic behavior of the energy e fficienc y , defined as the ratio of the total information ga thered to the required ener gy to obtain information fro m the area for a n ad hoc netw ork with mini- mum hop routing t o the f usion center . For the infinit e area model, the energy efficienc y decays to zero wit h r ate O ( area − 1 / 2 ) as we increase the cov erage area. For the infinite density model, on the other hand, a non-zero ef ficiency is possible if the decay rate of the error expo nent K ( density ) as a function of density i s slowe r t han O ( density (1 − δ ) / 2 ) , where δ is the propagation constant δ ≥ 2 . 1.2. Related W ork The detection of Gauss-Marko v processes in Gaussian noise is a classical problem. See [4] and references therein. Ho wev er , most work in this area considers only 1D signals or time series. A closed-form error expo nent was obtained and its properties were in vestigated for 1D hidden Gauss-Mark ov random processes [3]. Large deviations analyses were used to examine the issues of op- timal sensor density and optimal sampling were examined with a 1D signal model in [1] and [2]. An error exponent was obtained for t he detection of 2D GM- RFs in [5], where t he sensors are located random ly and t he Mark ov graph is based on the nearest neighbor dependenc y enabling a loop-free graph and further analysis. In this work, ho we ver , the measurement noise was not captured. Our work here focuses on the error expo nent for the detection of 2D hidden GMRF on a 2D infinite lattice, which allo ws for the consideration of m easurement noise. In parti cular we examine the abov e CAR model and inv es- tigate of the detection performance wi th respect ( w .r.t.) to various design parameters su ch as correlation strength, measuremen t SNR, sensor density and area. 2. D A T A MODEL Definition 1 (GMRF [6]) A r andom vector X = ( X 1 , X 2 , · · · , X n ) ∈ R n is a Gauss-Marko v random field w .r .t. a labelled graph G = ( ν, E ) wi th mean µ and pre cision matrix Q > 0 , if i ts pro b- ability density function is given by p ( X ) = (2 π ) − n/ 2 | Q | 1 / 2 exp „ − 1 2 ( X − µ ) T Q ( X − µ ) « , (3) and Q lm 6 = 0 ⇐ ⇒ { l, m } ∈ E for all l 6 = m . Her e, ν is th e set of all nodes { 1 , 2 , · · · , n } and E is the set of edges connecting pairs of nodes, which r epr esent the conditional depend ence structure . Note that the mean and the precision matrix fully characterize a GMRF . Note also t hat the covarian ce matrix Q − 1 is completely dense in general while the precision matrix Q has nonzero ele- ments Q lm only when there is an edge between nodes l and m in the Marko v random fiel d. Hence, when the graph is not fully con- nected, the precision matrix is sparse. The 2D indexing scheme ( i, j ) can be properly con verted to an 1D scheme t o apply Defi ni- tion 1. From here on, we use the 2D index ing scheme for con ve- nience. Definition 2 (Stationarity) A 2D GMRF on 2D doubly infinite lattice I ∞ is said to be stationary if the mean vector is constant and C ov ( X ij , X i ′ j ′ ) ∆ = E { X ij X i ′ j ′ } = c ( i − i ′ , j − j ′ ) f or some function c ( · , · ) . For a 2D stationary GM RF { X ij } , the cov ariance { γ ij } is defined as γ ij = E { X i ′ j ′ X i ′ + i,j ′ + j } = E { X 00 X ij } , (4) which does not depend on i ′ or j ′ due t o the stationarity . Further , the spectral density function of a zero-mean and stati onary Gaus- sian process { X ij } on I ∞ with cov ariance γ ij is defined as f ( ω 1 , ω 2 ) = 1 4 π 2 X ij ∈I ∞ γ ij exp( − ι ( iω 1 + j ω 2 )) , (5) where ι = √ − 1 and ( ω 1 , ω 2 ) ∈ ( − π , π ] 2 . Note that this is a 2D extension of the con v entional 1D discrete-time Fourier transform (DTFT). Definition 3 (The conditional autoregr ession (CAR)) A GMRF can be specified using a set of full conditional normal distribution s with mean and pr ecision: E { X ij | X − ij } = − 1 θ 00 X i ′ j ′ ∈I ∞ 6 =00 θ i ′ j ′ X i + i ′ ,j + j ′ , ( 6) Prec { X ij | X − ij } = θ 00 > 0 , (7) wher e X − ij denotes the set of all variables except X ij . It is shown that the GMRF defined by the C AR model (6) - ( 7) is a zero-mean stationary Gaussian process on I ∞ with the spectral density function [6] f ( ω 1 , ω 2 ) = 1 4 π 2 1 P ij ∈I ∞ θ ij exp( − ι ( iω 1 + j ω 2 )) (8) if |{ θ ij 6 = 0 }| < ∞ , θ ij = θ − i, − j , θ 00 > 0 , (9) { θ ij } is so that f ( ω 1 , ω 2 ) > 0 , ∀ ( ω 1 , ω 2 ) ∈ ( − π , π ] 2 . (10) W e assume that the 2D stochastic signal in (1) i s giv en by a sta- tionary GMRF defined by the CAR model (6) - (7) and (9) - (10). Then, t he observ ation spectrum under the two hypoth eses (1) are gi ven, respectiv ely , by S y 0 ( ω 1 , ω 2 ) = σ 2 4 π 2 and S y 1 ( ω 1 , ω 2 ) = σ 2 4 π 2 + f ( ω 1 , ω 2 ) . 3. PERFORMANCE MEASURE: ERRO R EXPONENT In this section, we in vestigate the performance of the Neyman - Pearson detector wit h l ev el α ∈ (0 , 1) for a 2D CAR signal in noisy observ ations. W e obtain the error exponen t in the spectral domain for this problem by exploiting the spectral structure of the CAR signal and t he relationship between the eigen value s of block circulant and block T oeplitz matrices representing 2D correlation structure. Theorem 1 (Err or Exponent) Consider Neyman -P earson detec- tion between the hypotheses (1) with the model (6) - (7) and with level α ∈ (0 , 1) . Assuming t hat conditions (9 and 10) hold, t he err or exponent of the miss proba bility is independent of α and is given by K = 1 4 π 2 Z π − π Z π − π „ 1 2 log σ 2 + 4 π 2 f ( ω 1 , ω 2 ) σ 2 + 1 2 σ 2 σ 2 + 4 π 2 f ( ω 1 , ω 2 ) − 1 2 « dω 1 dω 2 , (11) = 1 4 π 2 Z π − π Z π − π D ( N (0 , S y 0 ( ω 1 , ω 2 )) ||N (0 , S y 1 ( ω 1 , ω 2 )) d ω 1 dω 2 , wher e D ( ·||· ) denotes the K ullback-Leibler diver gence. Pr oof: K is giv en by the almost-sure l imit of the asymptotic Kullback-Leibler rate K = lim n →∞ 1 n log p 0 ,n p 1 ,n ( y n ) ev aluated under p 0 ,n [7]. Using the fact that we have Gaussian distributions under both hypotheses, we ha ve K = lim n →∞ 1 n „ 1 2 log det( Σ 1 ,n ) det( Σ 0 ,n ) + 1 2 y T n ( Σ − 1 1 ,n − Σ − 1 0 ,n ) y n « , Then approximating the block T oeplitz correlation matrix with a block circulant matrix and applying the the 2D Grenander -Szeg ¨ o theorem, we obtain the limit of each term as follo ws. 1 n log det( Σ 1 ,n ) → 1 4 π 2 Z π − π Z π − π log( σ 2 + 4 π 2 f ( ω 1 , ω 2 )) dω 1 dω 2 , 1 n log det( Σ 0 ,n ) → log σ 2 , 1 n y T n Σ − 1 1 ,n y n → 1 4 π 2 Z π − π Z π − π σ 2 σ 2 + 4 π 2 f ( ω 1 , ω 2 ) dω 1 dω 2 , 1 n y T n Σ − 1 0 ,n y n → 1 , almost surely .  This theorem is a 2D extension of the error exponent of 1D hidden Gauss-Marko v model based on state-space st ructure ob- tained in [3]. Intuitive ly , the err or exponent (11) can be explained using t he frequency binning argument. For each 2D frequenc y se g- ment dω 1 dω 2 , the spectra are flat, i.e., the signals are independ ent and Stein’ s lemma can be applied for the segment. The ov erall Kullback-Leibler di ver gence is the sum of contribu tions from each bin. 3.1. Symmetric First Order A u toregr ession T o in vestigate the behavior of the error exponent as a function of correlation and SNR , we further consider the sy mmetric first order autoreg ression (SF AR), described by the con ditions E { X ij | X − ij } = λ κ ( X i +1 ,j + X i − 1 ,j + X i,j +1 + X i,j − 1 ) , Prec { X ij | X − ij } = κ > 0 , where 0 ≤ λ ≤ κ 4 . (This i s a sufficient condition to satisfy (9) - (10).) Note here that θ 00 = κ and θ 1 , 0 = θ − 1 , 0 = θ 0 , 1 = θ 0 , − 1 = − λ . In this model, the correlation is symmetric for each set of four neighboring nodes. T he SF AR model is a simple yet meaningful extension o f the 1D Gau ss-Marko v random process, which has the conditional causal dependency only on the previou s sample. Here in the 2D case we have four neighborin g nodes in the four (p lanar) directions. T he spectrum of the SF AR is gi ven by f ( ω 1 , ω 2 ) = 1 4 π 2 κ (1 − 2 ζ cos ω 1 − 2 ζ cos ω 2 ) . (12) W e define the edge dependen ce f actor ζ by ζ ∆ = λ κ , 0 ≤ ζ ≤ 1 / 4 . (13) Note t hat ζ = 0 correspond s to the i. i.d. case whereas ζ = 1 / 4 corresponds to th e perfectly correlated case. Hence, the correlation strength can be captured in t his single quantity ζ for SF AR signals. The po wer of the S F AR is obtained using the in verse Fo urier trans- form via the relation (5), and is gi ven by P s = γ 00 = 2 K (4 ζ ) π κ , „ 0 ≤ ζ ≤ 1 4 « , (14) where K ( · ) is the complete elliptic integral of the fi rst kind [8]. The SNR is giv en by S NR = P s σ 2 = 2 K (4 ζ ) πκσ 2 . Using eq. (11) and the SNR, we obtain the err or expo nent i n the SF AR signal case, denoted by K s and giv en in the follo wing corollary . Corollary 1 The e rr or e xponent f or the Neyman -P earson de tector for the hypotheses (1) with the SF AR 2D signal model is given by K s = 1 4 π 2 Z π − π Z π − π „ 1 2 log 1 + SNR (2 /π ) K (4 ζ )(1 − 2 ζ cos ω 1 − 2 ζ cos ω 2 ) ! + 1 2 1 1 + SNR (2 /π ) K (4 ζ )(1 − 2 ζ cos ω 1 − 2 ζ cos ω 2 ) − 1 2 « dω 1 dω 2 . (15) Note that the S NR and correlation are separated in (15), which enables us to in vestigate the ef fects of each term separately . 3.2. Properties of the Err or Exponent K s First, it is readily seen f rom Corollary 1 that K s is a continuous function of the edge dependenc e factor ζ ( 0 ≤ ζ ≤ 1 / 4 ) for a gi ven SNR. The v alues of K s at the extreme correlations are gi v en by noting that K (0) = π 2 and K (1) = ∞ . Therefore, in the i.i.d. case (i.e., ζ = 0 ), t he corollary reduces to Stein’ s l emma as expec ted, and K s is giv en by K s = 1 2 log(1 + SNR ) + 1 2(1 + SNR ) − 1 2 = D ( N (0 , σ 2 ) ||N (0 , σ 2 + P s )) . For the perfectly correlated case ( ζ = 1 / 4 ), on the other hand, K s = 0 . In fact, in this case as well as in the i.i. d. case, the two-dimension ality is irrelev ant. The kno wn r esult that P M ∼ Θ( n − 1 / 2 ) for the perfectly correlated case is applicable. For intermediate va lues of correlation, we e valua te (15) for se veral different SNR values, as sho wn in Fig. 2. It is seen that at 0 0.05 0.1 0.15 0.2 0.25 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 ζ Error exponent, E SNR = 10 dB 0 0.05 0.1 0.15 0.2 0.25 0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.1 ζ Error exponent, E SNR = 0 dB (a) (b) 0 0.05 0.1 0.15 0.2 0.25 0.027 0.028 0.029 0.03 0.031 0.032 0.033 0.034 0.035 0.036 0.037 ζ Error exponent, E SNR = −3 dB 0 0.05 0.1 0.15 0.2 0.25 0.015 0.0155 0.016 0.0165 0.017 0.0175 0.018 0.0185 0.019 ζ Error exponent, E SNR = −5 dB (c) (d) Fig. 2 . K s as a function of ζ : (a) SNR = 10 dB, ( b) SN R = 0 dB, (c) SNR = -3 dB, (d) SNR = -5 dB high S NR K s is monotonically decreasing as ζ increases. Hence, i.i.d. observ ations gi ve t he best error performan ce for a gi ven value of SNR w hen SNR is l arge, as in the 1D case [3]. As we decrease the SNR, it is observed that a second mode grows near ζ = 1 / 4 . As we further decrease the SNR , the value of ζ of the second mode shifts tow ard 1 / 4 , and the value of the second mode exceeds that of t he i.i.d. case. Hence, there is a discontinuity in the optimal correlation as a function of SNR in the 2D case e ven if the maximal K s itself is continuo us. This is not the case in 1D. W ith regard to K s as a function of SNR, it is straightforward to see t hat it is c ontinuous and increases at the rate log SNR at high SNR for a giv en va lue of ζ . 4. AD HOC NETWORKING: INFORMA TION-ENERGY TRADE-OFF The analytical results in the prev ious section can be applied to an- swer some fundamen tal questions in the design of sensor netwo rks for d etection ap plications. W e consider a planar ad hoc sen sor net- work with minimum hop routing. T o simplify the analysis, we as- sume that (2 n + 1) 2 sensors are located on t he grid [ − n : 1 : n ] 2 with spacing r n , as shown in Fig. 1, and a fusion center is located at the center (0 , 0) . W e assume that the measurement Y ij is de- liv ered to the fusion center using the minimum h op routing, which requires a hop count of | i | + | j | . 4.1. Physical cor relation model The actual physical correlation in this model can be obtained by solving a proper continuou s index 2 D stochastic differential equa - tion (SDE), e.g., " „ ∂ ∂ x « 2 + „ ∂ ∂ y « 2 − ξ 2 # X ( x, y ) = u ( x, y ) , where u ( x, y ) is the process noise and ξ is a parameter determin- ing the correlation st rength of the fi eld. By solving a proper SDE , the edge correla tion factor ρ is give n, as a function of t he edge length r n , by ρ = f ( r n ) . T ypically , f ( · ) is a positiv e and monotonically decreasing function of r n . Further , we hav e a monotone mapping g : ρ → ζ from the edge correlation factor ρ to the edge dependen ce factor ζ , which maps zero and one to zero and 1/4, respecti vely . Thus, we ha ve ζ = g ( f ( r n )) , and for giv en phy sical parameters (with a slight abuse o f notation), K s ( SNR , ζ ) = K s ( SNR , g ( f ( r n ))) = K s ( SNR , r n ) . W e will use the arg uments SNR and ζ for K s properly if ne cessary . 4.2. Energy efficiency W e no w consider the energy efficiency of the ad hoc sensor net- work as the network size grows. T he energy ef ficiency η can be defined as η = total gathered information I t total required energ y E t , (16) where I t is giv en by the product of the number of sensors and the information K s per each sensor . W e consider two asymptotic regimes for the increase in netw ork size: an infinite area model with fixed density and an infinite density model with fixed area. The behav ior of the ener gy efficienc y as we increase the network size is summarized in the follo wing theorems. Theorem 2 (Infinite ar ea model) F or an ad hoc sensor network with incr easing ar ea and a fixed node density , the ener gy efficiency decays to zer o as we incr ease the ar ea with rate η = O “ ar ea − 1 / 2 ” . (17) Pr oof: The total energy required for data gathering is giv en b y E t = E link ( r n ) n X i = − n n X j = − n ( | i | + | j | ) = 2 n ( n + 1)(2 n + 1) E link ( r n ) , where the transmission energy per link E link ( r n ) = r δ n and δ is the propagation l oss factor . W e hav e I t = (2 n + 1) 2 K s ( r n ) , and area = Θ ( n 2 ) . The energy efficienc y is gi ven by η = (2 n + 1) 2 K s ( r n ) 2 n ( n + 1)(2 n + 1) E link ( r n ) . (18) Since r n is fixed, K s and E link do not change with n , and (17) follo ws.  Theorem 3 (Infinite density model) F or the infinite density model, a non -zer o effic iency is possible if the deca y rate of the err or exp o- nent K s as a function of density is slower than O “ density (1 − δ ) / 2 ” . (19) Pr oof: For the infinite density model, we hav e r n = Θ( n − 1 ) , r δ n = Θ( n − δ ) , density = Θ( n 2 ) . From (18), w e have η = K s ( r n ) /n 1 − δ . If K s as a function of r n decays slo wer than n 1 − δ , η does not diminish to zero.  The non-zero efficiency i n the asymptotic regime depend s on the decay rate of K s as a function of r n . Since K s ( ζ ) is gi ve n, this depends on the functions f and g in Section 4.1 and the propaga- tion loss factor δ . 5. CONCLUSIONS W e have considered the detection of 2D GMRFs from noisy ob- serv ations. W e hav e adopted the CAR model for the si gnal, and hav e used a spectral domain approach to derive the error expo- nent for the Ne yman-Pearson detector satisfying a fixe d lev el con- straint. Under the symmetric fi rst order autoregressi ve model, we hav e obtained the error expo nent exp licitly in terms of the SNR and the edge dependence factor . W e hav e in vestigated the proper- ties of the error expo nent a s a function of SNR and co rrelation. W e hav e seen that the behavior of the error exponent w .r . t. correlation strength is divided into two regions depending on S NR. At high SNR, i.i.d. (and, thus, uncorrelated) observ ations maximize the error exponen t for a given SNR, whereas there i s non-zero opti- mal v alue of correlation at low SNR. Further , it ha s been seen that there is a discontinuity for the optimal correlation as a function of SNR. Based on the error ex ponent, we have also in vestigated the energy efficiency of ad hoc sensor netwo rk for detection applica- tions. For a fix ed node den sity , the energy ef ficiency decays to zero with rate O ( area − 1 / 2 ) as we increase the area. On the other hand, non-zero efficienc y is possible with increasing density depending on physical correlation strength as a function of the link length. 6. REFERENCES [1] Y . Sung, L. T ong and H. V . Poor, “Sensor configuration and ac tiv ation for field detection in lar ge sensor arrays,” in Pro c. 2005 Information Processing in Sensor Networks (IPSN) , Los Angeles, CA, Apr . 2005. [2] J.-F . Chamberland and V . V . V eeravalli, “Ho w dense should a sensor network be for detection with correlated obs ervations?, ” IEEE T rans. Inform. Theory , vol. 52, pp. 5099 - 5106, Nov . 2006. [3] Y . Sung, L. T ong and H. V . Poor , “Neyman-Pearson detection of Gauss-Markov signals in noise: Closed-form error exponent and properties, ” IEEE T rans. In- form. Theory , vol. 52, pp. 1354 - 1365, Apr . 2006. [4] T . Kailath and H. V . Poor , “Detection of stochastic processes,” IEEE T rans. In- form. Theory , vol. 44, pp. 2230 - 2259, Oct. 1998. [5] A. Anandkumar, L. T ong and A. Swami, “De tection of Gauss-Ma rko v random field on nea rest-neighbor graph, ” in Proc. 2007 IEE E International Conference on Ac oustics, Spee ch, and Signal Processing (ICASSP’07) , Hawaii, USA, Apr . 2007. [6] H. Rue and L. Held, Gaussian Ma rkov Random F ields: Theory and Applicatons , Chapman & Hall/CRC, New Y ork, 2005. [7] I. V ajda, Theory of Statistical Infer ence and Information , Kluwer Academic Pub- lishers, Dordrecht, 1989. [8] J. Besag, “On a sys tem of two -dimensional recurrence equations, ” J ournal o f the Royal Statistical Society . Series B , vol. 43, no. 3 , pp. 302 - 309, 1981.