Noise Enhancement for Weighted Sum of Type I and II Error Probabilities with Constraints

Transcription

1 etropy Article Noie Ehacemet for Weighted Sum of Type I ad II Error Probabilitie with Cotrait Shuju iu, Tig Yag ad Kui Zhag * College of Commuicatio Egieerig, Chogqig Uiverity, Chogqig 444, Chia; liuj@cqu.edu.c (S..); yadtt28@26.com (T.Y.) * Correpodece: zk@cqu.edu.c; Tel.: Received: 2 April 27; Accepted: 2 Jue 27; Publihed: 4 Jue 27 Abtract: I thi paper, the oie-ehaced detectio problem i ivetigated for the biary hypothei-tetig. The optimal additive oie i determied accordig to a criterio propoed by DeGroot ad Schervih (2), which aim to miimize the weighted um of type I ad II error probabilitie uder cotrait o type I ad II error probabilitie. Baed o a geeric compoite hypothei-tetig formulatio, the optimal additive oie i obtaied. The ufficiet coditio are alo deduced to verify whether the uage of the additive oie ca or caot improve the detectability of a give detector. I additio, ome additioal reult are obtaied accordig to the pecificity of the biary hypothei-tetig, ad a algorithm i developed for fidig the correpodig optimal oie. Fially, umerical example are give to verify the theoretical reult ad proof of the mai theorem are preeted i the Appedix. Keyword: oie ehacemet; hypothei tetig; weighted um; error probability. Itroductio I the biary hypothei tetig problem, there are uually a ull hypothei H ad a alterative hypothei H, ad the objective of tetig i to be determie truthfule of them baed o the obervatio data ad a deciio rule. Due to the preece of oie, the deciio reult obviouly caot be abolutely correct. Geerally, two erroeou deciio may occur i the igal detectio: type I error that reject a true ull hypothei ad type II error that accept a fale ull hypothei []. I the claical tatitical theory, the Neyma Pearo criterio i uually applied to obtai a deciio rule that miimize the type II error probability β with a cotrait o the type I error probability α. However, the miimum β may ot alway correpod to the optimal deciio reult. For itace, i the example of [2], a biary hypothei tetig i deiged to determie the mea of a ormal data: the mea equal to uder H ad equal to uder H. Uder the cotrait that the type I error probability i fixed to.5, the type II error probability i decreaed from.9 to.26 whe the data ize icreae from 2 to, wherea the rejectio regio of the ull hypothei i chaged from (., + ) to (.5, + ). I uch cae, more iformatio brig a wore deciio reult eve a maller type II error probability i achieved with the fixed type I error probability. Similarly, the deciio rule that miimize the type I error probability for a fixed type II error probability may ot perform well. Therefore, it could ot be appropriate to imply miimize oe of the two error probabilitie i practice. The ideal cae i to develop a deciio criterio that miimize the two type of error probabilitie imultaeouly, but it i almot impoible i practical applicatio. I order to obtai a better deciio reult to balace the type I ad II error probabilitie, DeGroot ad Schervih [] propoed a criterio to miimize a weighted um of type I ad II error probabilitie, i.e., mi[c α(φ) + c 2 β(φ)], where φ repreet the deciio rule, c ad c 2 are the φ weight coefficiet correpodig to α ad β, repectively, ad c, c 2 >. Furthermore, DeGroot Etropy 27, 9, 276; doi:.339/e

2 Etropy 27, 9, of 22 alo provided the optimal deciio procedure to miimize the weighted um. The deciio rule i give a follow. If c f (x H ) < c 2 f (x H ), the ull hypothei H i rejected, where f (x H ) ad f (x H ) are the repective probability deity fuctio (pdf) of the obervatio x uder H ad H. If c f (x H ) > c 2 f (x H ), the alterative hypothei H i rejected. I additio, if c f (x H ) = c 2 f (x H ), the hypothei H ca be either rejected or ot. The optimal detector i thi cae i cloely related to the ditributio of the obervatio. Thi implie that oce the ditributio chage, the detector hould be adjuted accordigly. But i the cae where the detector i fixed, thi weighted um rule caot be directly applied. I uch a cae, fidig a alterative method to miimize the weighted um of type I ad II error probabilitie itead of chagig the detector i importat. Fortuately, the tochatic reoace (SR) theory provide a mea to olve thi problem. The SR, firt dicovered by Bezi et al. [3] i 98, i a pheomeo where oie play a poitive role i ehacig igal ad ytem through a oliear ytem uder certai coditio. The pheomeo of SR i the igal detectio i alo called oie-ehaced detectio. Recet tudie idicate that the ytem output performace ca be improved igificatly by addig oie to the ytem iput or icreaig the backgroud oie level [4 22]. The improvemet achieved via oie ca be meaured i the form of icreaed igal-to-oie ratio (SNR) [7 ], mutual iformatio (MI) [,2] or detectio probability [3 6], or i the form of decreaed Baye rik [7,8]. For example, the SNR gai of a parallel ucoupled array of bitable ocillator, operatig i a mixture of iuoidal igal ad Gauia white oie, i maximized via extra array oie [8]. I additio, due to the added array oie, the performace of a fiite array cloely approache to a ifiite array. I [], the throughput MI of threhold euro i icreaed by icreaig the iteity of fait iput oie. The optimal additive oie to maximize the detectio probability with a cotrait o fale-alarm probability i tudied i [3], ad the ufficiet coditio for improvability ad o-improvability are deduced. I [7], the effect of additive idepedet oie o the performace of uboptimal detector are ivetigated accordig to the retricted Baye criterio, where the miimum oie modified Baye rik i explored with certai cotrait o the coditioal rik. Ipired by thi cocept, it i reaoable to cojecture that a proper oie ca decreae the weighted um of type I ad II error probabilitie for a fixed detector. I the abece of cotrait, it i obviou that the additive oie that miimize the weighted um i a cotat vector, wherea the correpodig type I or II error probability may exceed a certai value to caue a bad deciio reult. To avoid thi problem, two cotrait are eforced o type I ad II error probabilitie, repectively, to keep a balace. The aim of thi work i to fid the optimal additive oie that miimize the weighted um of type I ad II error probabilitie with the cotrait o type I ad II error probabilitie for a fixed detector. Furthermore, the work ca alo be exteded to ome applicatio, uch a the eergy detectio i eor etwork [23,24] ad the idepedet Beroulli trial [25]. The mai cotributio of thi paper are ummarized a follow: Formulatio of the optimizatio problem for miimizig the oie modified weighted um of type I ad II error probabilitie uder the cotrait o the two error probabilitie i preeted. Derivatio of the optimal oie that miimize the weighted um ad ufficiet coditio for improvability ad oimprovability for a geeral compoite hypothei tetig problem are provided. Aalyi of the characteritic of the optimal additive oie that miimize the weighted um for a imple hypothei tetig problem i tudied ad the correpodig algorithm to olve the optimizatio problem i developed. Numerical reult are preeted to verify the theoretical reult ad to demotrate the uperior performace of the propoed detector. The remaider of thi paper i orgaized a follow: i Sectio 2, a oie modified compoite hypothei tetig problem i formulated firt for miimizig the weighted um of type I ad II error probabilitie uder differet cotrait. The the ufficiet coditio for improvability ad

3 Etropy 27, 9, of 22 oimprovability are give ad the optimal additive oie i derived. I Sectio 3, additioal theoretical reult are aalyzed for a imple hypothei tetig problem. Fially, imulatio reult are how i Sectio 4 ad cocluio are made i Sectio 5. Notatio: ower-cae bold letter deote vector, with υ i deotig the i-th elemet of υ; deote the value of parameter Θ; f υ (υ ) deote the pdf of υ for a give parameter value Θ = ; Ω i deote the et of all poible parameter value of Θ uder H i ; δ( ) deote the Dirac fuctio;, ad deote iterectio, uio ad ull et, repectively;, ( ) T,, E{ }, mi, max ad arg deote covolutio, trapoe, itegral, expectatio, miimum, maximum ad argumet operator, repectively; if{ } ad up{ } deote the ifimum ad upremum operator, repectively; mea ummatio; ad H deote the repective gradiet ad Heia operator. 2. Noie Ehaced Compoite Hypothei Tetig 2.. Problem Formulatio Coider the followig biary compoite hypothei tetig problem: { H : f x (x ), Ω H : f x (x ), Ω () where x R N i the obervatio vector, H ad H are the ull ad the alterative hypotheize, repectively, deote the value of parameter Θ, f x (x ) repreet the pdf of x for a give parameter value Θ =. The parameter Θ ha multiple poible value uder each hypothei ad deote the pdf of ay parameter value Θ = uder H ad H by ϖ () ad ϖ (). I additio, Ω ad Ω deote the repective et of all poible value of Θ uder H ad H. It i true that Ω Ω = ad the uio of them form the parameter pace Ω, i.e., Ω = Ω Ω. Without lo of geerality, a deciio rule (detector) i coidered a: φ(x) = {, x Γ, x Γ (2) where Γ ad Γ form the obervatio pace Γ. Actually, the detector chooe H if x Γ, otherwie chooe H if x Γ. I order to ivetigate the performace of the detector achieved via a additive oie, a oie modified obervatio y i obtaied by addig a idepedet additive oie to the origial obervatio x, i.e., y = x +. For a give parameter value Θ =, the pdf of y i calculated by the covolutio of the pdf of x ad, give by: f y (y ) = f x (x ) p () = p () f x (y )d (3) R N where p () deote the pdf of. For a fixed detector, the oie modified type I ad II error probabilitie of the detector for give parameter value ow i expreed a: α y (φ; ) = φ(y) f y (y )d y = f y (y Γ )d y, Ω (4) Γ β y (φ; ) = ( φ(y)) f y (y )d y = f y (y )d y, Ω (5) Γ Γ Correpodigly, the average oie modified type I ad II error probabilitie are calculated by: α y (φ) = α y (φ; )ϖ ()d (6) Ω

4 Etropy 27, 9, of 22 β y (φ) = β y (φ; )ϖ ()d (7) Ω From (6) ad (7), the weighted um of the two type of average error probabilitie i obtaied a: Er y = c α y (φ) + c 2 β y (φ) = c Ω α y (φ; )ϖ ()d + c 2 Ω β y (φ; )ϖ ()d (8) where c ad c 2 are the weight aiged for the type I ad II error probabilitie, which ca be predefied accordig to the actual ituatio. For example, if the prior probabilitie are kow, the value of c ad c 2 equal the prior probabilitie correpodig to H ad H, repectively. Beide, the value of c ad c 2 ca alo be determied baed o the expected deciio reult. I thi work, the aim i to fid the optimal idepedet additive oie, which miimize the weighted um of the average error probabilitie uder the cotrait o the maximum type I ad II error probabilitie for differet parameter value. The optimizatio problem ca be formulated a below: p opt () = argmi p () Ery (9) ubject to max Ω α y (φ; ) α o max Ω β y (φ; ) β o () where α o ad β o are the upper limit for the type I ad II error probabilitie, repectively. I order to explicitly expre the optimizatio problem decribed i (9) ad (), ubtitutig (3) ito (4) produce: α y (φ; ) = Γ R N p () f x (y )d dy = R N p () Γ f x (y )d yd =, Ω () R N p ()A ()d = E{A ()} where A () = f x (y )d y = φ(y) f x (y )d y, Ω (2) Γ Γ It hould be oted that A () ca be viewed a the type I error probability obtaied by addig a cotat vector to x for Ω. Therefore, α x (φ; ) = A () = Γ φ(x) f x(x )d x deote the type I error probability for the origial obervatio x. Similarly, β y (φ; ) i (5) ca be expreed a: where β y (φ; ) = E{B ()}, Ω (3) B () = f x (y )d y = ( φ(y)) f x (y )d y, Ω (4) Γ Γ The B () ca be treated a the type II error probability obtaied by addig a cotat vector to x for Ω ad β x (φ; ) = B () = Γ ( φ(x)) f x(x )d x i the origial type II error probability without addig oie for Ω. With () ad (3), (8) become: [ ] Er y = p () c A ()ϖ ()d + c 2 B ()ϖ ()d d = E{Er()} (5) R N Ω Ω where Er() = c Ω A ()ϖ ()d + c 2 Ω B ()ϖ ()d (6)

5 Etropy 27, 9, of 22 Accordigly, Er() i the weighted um of two type of average error probabilitie achieved by addig a cotat vector to the origial obervatio x. Naturally, Er x = Er() deote the weighted um of type I ad II average error probabilitie for the origial obervatio x. Combied (), (3) ad (5), the optimizatio problem i (9) ad () ow i: p opt () = argmi E{Er()} (7) p () ubject to max Ω E{A ()} α o max Ω E{B ()} β o (8) 2.2. Sufficiet Coditio for Improvability ad No-improvability I practice, the olutio of the optimizatio problem i (7) ad (8) require a reearch over all poible oie ad thi procedure i complicated. Therefore, it i worthwhile to determie whether the detector ca or caot be improved by addig additive oie i advace. From (7) ad (8), a detector i coidered to be improvable if there exit oe oie that atifie E{Er()} < Er x = Er(), maxe{a ()} α o ad maxe{b ()} β o imultaeouly; otherwie, the detector i coidered to Ω Ω be o-improvable. The ufficiet coditio for o-improvability ca be obtaied accordig to the characteritic of A (), B () ad Er(), which are provided i Theorem. Theorem. If there exit Ω ( Ω ) uch that A () α o (B () β o ) implie Er() Er() for ay P, where P repreet the covex et of all poible additive oie, ad if A () (B ()) ad Er() are covex fuctio over P, the the detector i o-improvable. The proof i provided i Appedix A. Uder the coditio i Theorem, the detector caot be improved ad it i ueceary to olve the optimizatio problem i (7) ad (8). I other word, if the coditio i Theorem are atified, the three iequitie Er y Er x, maxe{a ()} α o ad maxe{b ()} β o caot be Ω Ω achieved imultaeouly by addig ay additive oie. I additio, eve if the coditio i Theorem are ot atified, the detector ca alo be o-improvable. Thi implie the ufficiet coditio for improvability eed to be addreed. The ufficiet coditio for improvability are dicued ow. Suppoe that A (x) ( Ω ), B (x) ( Ω ) ad Er(x) are ecod-order cotiuouly differetiable aroud x =. I order to facilitate the ubequet aalyi, ix auxiliary fuctio are predefied a follow baed o the firt ad ecod partial derivative of A (x), B (x) ad Er(x) with repect to the elemet of x. The firt three auxiliary fuctio a () g), b () g) ad er () g) are defied a the weight um of the firt partial derivative of A (x), B (x) ad Er(x), repectively, baed o the coefficiet vector g. Specifically: a () N A g) g (x) i = g T A x (x), Ω (9) i= i b () N B g) g (x) i = g T B x (x), Ω (2) i= i er () g) N Er(x) g i = g T Er(x) (2) x i= i

6 Etropy 27, 9, of 22 where g i a N-dimeioal colum vector, g T i the trapoitio of g, x i ad g i are the i-th elemet of x ad g, repectively. I additio, deote the gradiet operator, thereby A (x) ( B (x), Er(x)) i a N-dimeioal colum vector with i-th elemet A (x)/ x i ( B (x)/ x i, E r (x)/ x i ), i =,..., N. The lat three auxiliary fuctio a (2) g), b (2) g) ad er (2) g) are defied a the weight um of the ecod partial derivative of A (x), B (x) ad Er(x) baed o the N N coefficiet matrix gg T, i.e., a (2) g) b (2) g) N j= N j= er (2) g) N i= N i= N j= g j g i 2 A (x) ( x j x i ) = gt H(A (x))g, Ω (22) g j g i 2 B (x) ( x j x i ) = gt H(B (x))g, Ω (23) N i= g j g i 2 Er(x) ( x j x i ) = gt H(Er(x))g (24) where H deote the Heia operator, H(A (x)) (H(B (x)),h(er(x))) i a N N matrix with it (j, i)-th elemet deoted by 2 A (x)/( x j x i ) ( 2 B (x)/( x j x i ), 2 Er(x)/( x j x i )), where i, j =,..., N. Baed o the defiitio i (9) (24), Theorem 2 preet the ufficiet coditio for improvability. Theorem 2. Suppoe that Λ ad Λ are the et of all poible value of that maximize A () ad B (), repectively, α o = maxa () ad β o = maxb (). The detector i improvable, if there exit a N-dimeioal Ω Ω colum vector g that atifie oe of the followig coditio for all Λ ad Λ : () er () g) x= <, a () g) x= <, b () g) x= < ; (2) er () g) x= >, a () g) x= >, b () g) x= > ; (3) er (2) g) x= <, a (2) g) x= <, b (2) g) x= <. The proof i preeted i Appedix B. Theorem 2 idicate that uder the coditio (), (2) or (3), there alway exit oie that decreae the weighted um of average error probabilitie uder the cotrait o the type I ad II error probabilitie. I additio, alterative ufficiet coditio for improvability ca be obtaied by defiig the followig two fuctio, ad they are: { } I(t) = if Er() max A () = t, R N Ω { } S(t) = up maxb () Ω max A () = t, R N Ω where I(t) ad S(t) are the miimum weighted um of two type of average error probabilitie ad the maximum type II error probability for a give maximum type I error probability obtaied via addig a cotat vector, repectively. If there i a t α o uch that I(t ) Er() ad S() β o, the detector i improvable. More pecifically, there exit a cotat vector that atifie maxa ( ) = t α o, Ω Er( ) Er() ad maxb ( ) β o imultaeouly. However, i mot cae, the olutio of the Ω optimizatio problem i (7) ad (8) i ot a cotat vector. A more practical ufficiet coditio for improvability i how i Theorem 3. Theorem 3. et α = maxα x (φ; ) ad β = maxβ x (φ; ) be the repective maximum type I ad II error Ω Ω probabilitie without addig ay oie, ad uppoe that α α o, β β o ad S( α) = β. If I(t) ad S(t) (25) (26)

7 Etropy 27, 9, of 22 are ecod-order cotiuouly differetiable aroud t = α, ad I ( α) < ad S ( α) < hold at the ame time, the the detector i improvable. The proof i give i Appedix C. Additioally, the followig fuctio J(ε) ad G(ε) are defied: { } J(ε) = if Er() max B () = ε, R N Ω { } G(ε) = up maxa () Ω max B () = ε, R N Ω A imilar cocluio to the Theorem 3 ca be made a well, provided i Corollary. (27) (28) Corollary. The detector i improvable, if J ( β) ad G ( β) hold, where J(ε) ad G(ε) are ecod-order cotiuouly differetiable aroud ε = β, ad G( β) = α. The proof i imilar to that of Theorem 3 ad it i omitted here Optimal Additive Noie I geeral, it i difficult to olve the optimizatio problem i (7) ad (8) directly, becaue the olutio i obtaied baed o the earch over all poible additive oie. Hece, i order to reduce the computatioal complexity, oe ca utilize Parze widow deity etimatio to obtai a approximate olutio. Actually, the pdf of the optimal additive oie ca be approximated by: p () = η l ϑ l () (29) l= where η l ad l= η l =, while ϑ l ( ) repreet the widow fuctio that atifie ϑ l (x) for ay x ad ϑ l (x)dx = for l =,...,. The widow fuctio ca be a coie widow, rectagular widow, or Gau widow fuctio. With (29), the optimizatio problem i implified to obtai the parameter value correpodig to each widow fuctio. I uch cae, global optimizatio algorithm ca be applied uch a Particle warm optimizatio (PSO), At coloy algorithm (ACA), ad Geetic algorithm (GA) [26 28]. If the umber of parameter value i Ω ad Ω are fiite, the optimal additive oie for (7) ad (8) i a radomizatio of o more tha M + K cotat vector. I thi cae, Ω ad Ω ca be expreed by Ω = {, 2,..., M } ad Ω = {, 2,..., K }, where M ad K are fiite poitive iteger. The Theorem 4 tate thi claim. Theorem 4. Suppoe that each compoet i the optimal additive oie i fiite, amely i [a i, b i ] for i =,..., N, where a i ad b i are two fiite value. If A i ( ) ad B i ( ) are cotiuou fuctio, the pdf of the optimal additive oie for the optimizatio problem i (7) ad (8) ca be expreed a: where η l ad M+K l= η l =. p () = M+K η l δ( l ) (3) l= The proof i imilar to that of Theorem 4 i [7] ad Theorem 3 i [3], ad omitted here. I ome pecial cae, the optimal additive ca be olved directly baed o the characteritic of I(t) (H(ε)). For example, let Er mi = mii(t) = I(t m ) (Er mi = mih(ε) = I(ε m )) ad maxa ( m ) = t m t ε Ω (maxb ( m ) = ε m ). If t m α o (ε m β o ) ad maxb ( m ) β o (maxa ( m ) α o ), the optimal additive Ω Ω Ω

8 Etropy 27, 9, of 22 oie i a cotat vector with pdf of p () = δ( m ). I additio, equality of max Ω E{A ()} = α o (max Ω E{B ()} = β o ) hold if t m > α o (ε m > β o ). 3. Noie Ehaced Simple Hypothei Tetig I thi ectio, the oie ehaced biary imple hypothei tetig problem i coidered, which i a pecial cae of the optimizatio problem i (9) ad (). Therefore, the cocluio obtaied i Sectio 2 are alo applicable i thi ectio. Furthermore, due to the pecificity of imple biary hypothei tetig problem, ome additioal reult are alo obtaied. 3.. Problem Formulatio Whe Ω i = { i }, i =,, the compoite biary hypothei tetig problem decribed i () i implified to a imple biary hypothei tetig problem. I thi cae, the probability of i uder H i equal to, i.e., ϖ i () = for i =,. Therefore, the correpodig oie modified type I ad II error probabilitie i rewritte a: α y (φ) = α y (φ; ) = p () f (y )dyd = E{A ()} (3) R N Γ β y (φ) = β y (φ; ) = p () f (y )dyd = E{B ()} (32) R N Γ where f ( ) ad f ( ) repreet the pdf of x uder H ad H, repectively, ad A () ad B () are: A () = f (y )dy (33) Γ B () = f (y )dy (34) Γ Correpodigly, the weighted um of oie modified type I ad II error probabilitie i calculated by: Er y = c α y (φ) + c 2 β y (φ) = c E{A ()} + c 2 E{B ()} = R N p ()(c A () + c 2 B ())d (35) = E{Er()} where A a reult, the optimizatio problem i (9) ad () become: Er() = c A () + c 2 B () (36) p opt () = argmi E{Er()} (37) p () ubject to { E{A ()} α o E{B ()} β o (38) Baed o the defiitio i (33) ad (34), A () ad B () are viewed a the oie modified type I ad II error probabilitie obtaied by addig a cotat vector oie. Furthermore, A () ad B () are the origial type I ad II error probabilitie, repectively Algorithm for the Optimal Additive Noie Accordig to the Theorem 4 i Sectio 2.3, the optimal additive oie for the optimizatio problem i (37) ad (38) i a radomizatio of mot two cotat vector with the pdf p opt () = ηδ( ) +

9 Etropy 27, 9, of 22 ( η)δ( 2 ). I order to fid the value of η, ad 2, we firt divide each cotat vector ito four dijoit et accordig to the relatiohip of A () ad α o, B () ad β o. To be pecific, the four dijoit et are Q = { A () α o, B () β o }, Q 2 = { A () α o, B () > β o }, Q 3 = { A () > α o, B () β o }, ad Q 4 = { A () > α o, B () > β o }. The, we calculate the miimum { Er(), the } correpodig et of all poible value of i deoted by Q e = = argmier(). It hould be oted that Q e i the optimal additive oie that miimize the weighted um without cotrait. It i obviou that Q, Q 2 ad Q 3 do ot exit if all the elemet of Q e belog to Q 4. I other word, if Q e Q 4, there i o additive oie that atifie E{Er()} < Er() uder the cotrait of E{A ()} α o ad E{B ()} β o. Therefore, if the detector i improvable, the elemet of Q e mut come from Q, Q 2 ad/or Q 3. Theorem 5 i ow provided to fid the value of η, ad 2. Theorem 5. et η = α o A ( 2 ) A ( ) A ( 2 ) ad η 2 = β o B ( 2 ) B ( ) B ( 2 ). () If Q e Q =, the η = ad Q e Q uch that Er y opt = Er( ) = mier(). (2) If Q e Q 2 = ad Q e Q 3 = are true, the we have Q e Q 2, 2 Q e Q 3, η η η 2, ad Er y opt = mier(). (3) If Q e Q 2, the Er y opt i obtaied whe η = η 2, ad the correpodig E{A ()} achieve the miimum ad E{B ()} = β o. (4) If Q e Q 3, the Er y opt i achieved whe η = η, ad the correpodig E{A ()} = α o ad E{B ()} reache the miimum. The correpodig proof are provided i Appedix D. From (3) ad (4) i Theorem 5, uder the cotrait o E{A ()} α o ad E{B ()} β o, the olutio of the optimizatio problem i (37) ad (38) i idetical with the additive oie that miimize E{A ()} (E{B ()}) whe Q e Q 2 (Q e Q 3 ). I uch cae, the optimal olutio ca be obtaied eaily by referrig the algorithm provided i [4]. 4. Numerical Reult I thi ectio, a biary hypothei tetig problem i tudied to verify the theoretical aalyi, ad it i: { H : x = v (39) H : x = Θ + v where x R i a obervatio, Θ i a cotat or radom variable, ad v i the backgroud oie with pdf p v ( ). From (39), the pdf of x uder H i f (x) = p v (x), ad the pdf of x uder H for a give parameter value Θ = i deoted by f (x) = p v ( ) p ( ), where p ( ) repreet the pdf of Θ =. A oie modified obervatio y i obtaied via addig a additive idepedet oie to the obervatio x, i.e., y = x +. If the additive oie i a cotat vector, the pdf of y uder H i calculated a f (y) = f (x ), ad the pdf of y uder H for Θ = i f (y) = f (x ). I additio, a liear- quadratic detector i utilized here, give by: T(y) = d y 2 H + d y + d 2 > γ (4) < H where d, d ad d 2 are detector parameter, ad γ deote the detectio threhold. I the umerical example, α o = α x ad β o = β x, where α x ad β x are the origial type I ad II error probabilitie, repectively.

10 Etropy 27, 9, 276 of Rayleigh Ditributio Backgroud Noie Suppoe that Θ = i a cotat, the problem how i (39) repreet a imple biary hypothei tetig problem. Here, we et d = d 2 = ad d =, the the detector become T(y) = y H > γ (4) < H It i aumed that the backgroud oie v obey the mixture of Rayleigh ditributio with zero-mea uch that p v (v) = M i= m i ϕ i (v µ i ), where m i for i =,..., M, M i= m i =, ad ϕ i (x) = { x 2 i exp( x2 ), x 2i 2, x < (42) I the imulatio, the variace of all the Rayleigh compoet are aumed to be the ame, i.e., i = for i =,..., M. I additio, the parameter are pecified a M = 4, u =.2, u 2 =.4, u 3 = 2 π2.2, u 4 = 2 π2.4 ad m i =.25 for i =,..., 4. From (33) ad (34), the oie modified type I error probability A () ad type II error probability B () obtaied by addig a cotat vector i calculated a: where Φ(x) = x A () = B () = γ + γ f (y)dy = f (y)dy = 4 i= 4 i= m i Φ(γ µ i ) (43) m i Φ(γ µ i ) (44) x 2 exp( x2 2 2 )dt, whe x > ; Φ(x) =, whe x. Accordigly, α x = A () = 4 i= m iφ(γ µ i ) ad β x = B () = 4 i= m iφ(γ µ i ). et c = β x /(α x + β x ) ad c 2 = α x /(α x + β x ), the oie modified weighted um of the two type of error probabilitie obtaied via addig a cotat vector i Er() = c 4 i= m iφ(γ µ i ) + c 2 4 i= m iφ(γ µ i ). From Sectio 3.2, the pdf of the optimal additive oie that miimize weighted um of type I ad () = ηδ( ) + ( η)δ( 2 ), uder the two cotrait that α y α x ad β y β x. Moreover, the optimal additive oie for the cae without ay cotrait i a cotat vector. Figure plot the miimum oie modified weighted um of type I ad II error probabilitie obtaied uder o cotrait ad two cotrait that α y α x ad β y β x, ad the origial weighted um without addig ay oie for differet value of whe = 3 ad γ = /2. Whe, there i o oie that decreae the weighted um. With the icreae of, oie exhibit a poitive effect o the detectio performace. To be pecific, whe < < 2, the weighted um ca be decreaed by addig a cotat vector for the o cotrait cae. Whe > 2, the weighted um ca be decreaed addig the oie uder two cotrait. The oie modified weighted um obtaied without ay cotrait i le tha or equal to that obtaied uder the two cotrait, ad the differece betwee them firt decreae to zero for 3 < < 4 ad the gradually icreae whe > 4. I additio, oce exceed a certai value, o oie exit that ca decreae the weighted um for ay cae. II error probabilitie i deoted by p opt

11 weighted um ca be decreaed addig the oie uder two cotrait. The oie modified weighted um obtaied without ay cotrait i le tha or equal to that obtaied uder the two cotrait, ad the differece betwee them firt decreae to zero for 3 < < 4 ad the gradually icreae whe >. I additio, oce exceed a certai value, o oie exit that 4 ca decreae the weighted um for ay cae. Etropy 27, 9, 276 of Er.2. Origial 3 Noie Modified (two cotrait) Noie Modified (o cotrait) Figure. The miimum oie oie modified modified weighted weighted um um of the of type the Itype ad II I error ad II probabilitie error probabilitie obtaied obtaied uder ouder cotrait o cotrait ad two cotrait, ad two cotrait, ad the origial ad the weighted origial um weighted for differet um for whe differet = 3 ad whe γ = /2. = 3 ad γ = 2. Figure 2 how the the type type I ad I ad II error II error probabilitie correpodig to the to weighted the weighted um i Figure um i. Figure From both. From Figure both Figure ad 2, it iad oberved 2, it i that oberved oe ofthat the oie oe of modified the oie Type modified I ad II Type errori probabilitie ad II error probabilitie perform wore perform tha the wore origial tha oe the for origial the ooe cotrait for the cae. o cotrait Therefore, cae. though Therefore, the oie though modified the oie weighted modified um obtaied weighted with um o obtaied cotrait with io le cotrait tha thati obtaied le tha uder that obtaied the two cotrait, uder the two the cotrait, correpodig Table the. The oie correpodig optimal i actually additive ot oie oie uitable actually that tomiimize addot the uitable the obervatio. weighted to add um to Furthermore, uder obervatio. two cotrait whefurthermore, miimum ad o whe value of cotrait the themiimum oie for modified variou value weighted of where the oie um = 3 i ad modified obtaied γ = 2weighted uder. the two um cotrait, i obtaied theuder correpodig the two cotrait, type II probability the correpodig equal to the type origial II probability oe ad theequal type I probability to the origial achieve oe the ad miimum the type for I Two Cotrait No Cotrait probability 2 < < 3 achieve. Coverely, the whe miimum > for 4, the 2 correpodig < < 3. Coverely, type I probability whe > equal, the to the correpodig origial oe 4 η 2 o type ad the I probability type II probability equal to achieve the origial the miimum. oe ad the The type reult II probability are coitet achieve with the part miimum. (3) ad part The (4) reult i Theorem are coitet 5. Epecially, with.95 forpart 3 < (3) < - ad 4, the part miimum -(4) Theorem value - of5. the Epecially,.789 oie modified for weighted 3 < < 4, um the miimum obtaied uder value oof cotrait the oie.25 i modified equal.982 toweighted that obtaied.7963 um uder obtaied.695 twouder cotrait,.928 o cotrait ad thei correpodig equal to that obtaied type I aduder II error two probabilitie cotrait, are ad the the ame, correpodig which alo type agree I ad withii part error (2) probabilitie i Theorem are 5. Ithe order ame, to /3.896 which furtheralo illutrate agree the with reult part i(2) Figure i Theorem ad 2, 5. Table I order provide to further the illutrate optimal additive the reult oie i Figure added for ad the two 2, Table differet provide cae. the optimal additive oie added for the two differet cae. Etropy 27, 9, of 23 α.8.7 Origial.7.6 Noie Modified (two cotrait) Noie Modified (o cotrait) Origial. 3 Noie Modified (two cotrait). Noie Modified (o cotrait) (a) Figure Figure The The type type I (a) I (a) ad ad II II (b) (b) error error probabilitie probabilitie correpodig correpodig to to the the weighted weighted um um i i Figure Figure.. Figure 3 depict the miimum oie modified weighted um of the type I ad II error probabilitie veru for the cae of o cotrait ad two cotrait, ad the origial weighted um, whe = ad γ = 2. The correpodig type I ad II error probabilitie are depicted i Figure 4a,b, repectively. It i ee i Figure 3, the improvemet of the weighted um obtaied by addig oie firt icreae ad the decreae with the icreae of, ad fially they all coverge to the ame value. The differece for the cae with ad without cotrait are very mall i mot cae. I the mall iterval of, i.e., (, 2), the differece eve decreae to zero. O the other β (b)

12 Etropy 27, 9, of 22 Table. The optimal additive oie that miimize the weighted um uder two cotrait ad o cotrait for variou where = 3 ad γ = /2. Two Cotrait No Cotrait 2 η o / Figure 3 depict the miimum oie modified weighted um of the type I ad II error probabilitie veru for the cae of o cotrait ad two cotrait, ad the origial weighted um, whe = ad γ = /2. The correpodig type I ad II error probabilitie are depicted i Figure 4a,b, repectively. It i ee i Figure 3, the improvemet of the weighted um obtaied by addig oie firt icreae ad the decreae with the icreae of, ad fially they all coverge to the ame value. The differece for the cae with ad without cotrait are very mall i mot cae. I the mall iterval of, i.e., (, 2 ), the differece eve decreae to zero. O the other had, the oie modified type I error probability obtaied uder o cotrait i igificatly greater tha the origial oe for <, while the correpodig type II error probability i le tha that obtaied uder the two cotrait. The ituatio, however, i revered for 2 < < 3. Whe > 3, there i o oie that decreae the weighted um uder the two cotrait, while the weighted um i till decreaed by addig a cotat vector for o cotrait cae. Whe > 4, the weighted um caot be decreaed by addig ay oie for all the cae. Furthermore, Table 2 how the optimal additive oie that Etropy miimize 27, the 9, 278 weighted um uder the cae of o ad two cotrait. 3 of Er.3.25 Figure The The miimum miimum oie oie modified modified weighted weighted um um of theof type the Itype ad II I error ad II probabilitie error probabilitie obtaied obtaied uder o uder cotrait o cotrait ad two cotrait, ad two cotrait, ad the origial ad weighted the origial umweighted for differet um for whe differet = ad whe γ = /2. = ad γ = (a).2 Origial.5 Noie Modified (two cotrait) Noie Modified (o cotrait) Table 2. The optimal additive oie that miimize the.7 weighted um uder two cotrait ad o cotrait for variou Origial where = ad γ = /2. Origial Noie Modified (two cotrait).6 Noie Modified (two cotrait).8 Noie Modified (o cotrait) Noie Modified (o cotrait) Two Cotrait No Cotrait η o / α β (b)

13 Figure 3. The miimum oie modified weighted um of the type I ad II error probabilitie Etropy obtaied 27, 9, uder 276 o cotrait ad two cotrait, ad the origial weighted um for differet 3 of 22 whe = ad γ = 2..8 Origial Noie Modified (two cotrait) Noie Modified (o cotrait).7.6 Origial Noie Modified (two cotrait) Noie Modified (o cotrait) α β (a) Figure Figure4. 4. The The type type II ad ad II II error error probabilitie probabilitie correpodig correpodig to the to weighted the weighted um um i Figure i Figure 3 are how 3 are how i (a) ad i (a) (b), ad repectively. (b), repectively. Table 2. The optimal additive oie that miimize the weighted um uder two cotrait ad o Etropy 27, 9, Figure cotrait how the miimum for variou oie modified weighted um of type I ad II error probabilitie where = ad γ = 2. 4 of 23 veru γ for the cae of o cotrait ad two cotrait, ad the origial weighted um, whe = ad = 3. The correpodig type Two I adcotrait II error probabilitie are No Cotrait depicted i Figure 6a,b, repectively. decreae Afor γ illutrated 2 < γ < γ i Figure 3, ad 5, whe o improvemet γ i cloe to zero, ca the η origial be obtaied weighted whe γ > γ 2 um Er x approache 2. O the to zero. other had, o the miimum I uch cae, oie additive modified oie weighted exit to decreae um obtaied the weighteduder um. For o the cotrait cae of two cotrait, i maller the tha that improvemet of the weighted um firt icreae for γ < γ < γ 2 ad the decreae for γ 2 < γ < γ 3, obtaied uder the two cotrait for γ.6563 < γ < γ , ad the.448/.6563 differece betwee them firt icreae ad o improvemet ca be obtaied whe γ > γ 2. O the other had, the miimum oie modified ad the weighted decreae umfor obtaied both 2.5 uder γ<.652 o γ < cotrait γ2 ad.469 i maller γ < γ tha < γ 3 that. Whe obtaied.62 γ > uder γ3, there twotill cotrait exit for a cotat γ < γ < γ 3, ad the differece 3.25 betwee - them - firt icreae - ad the.5866 decreae for both γ < γ < γ vector that decreae the weighted um, but it may be ot a uitable oie i the 2 practical ad γ 2 < γ < γ 3. Whe γ > γ 3, there till exit a cotat vector that decreae the weighted um, but applicatio accordig to the type II probability depicted i Figure 6b. Furthermore, i order to it may Figure be ot 5 how a uitable the oie miimum theoie practical modified applicatio weighted accordig um of totype the type I ad II II probability error probabilitie depicted tudy the veru i Figure reult γ 6b. illutrated for Furthermore, i cae of o cotrait order Figure to tudy 5 ad ad the two reult 6, Table cotrait, illutrated 3 how ad ithe Figure the origial 5optimal ad weighted 6, Table additive um, 3 howoie whe the that miimize optimal the weighted additive oie um that for miimize the cae the of weighted o ad two um for cotrait. the cae of o ad two cotrait. = ad = 3. The correpodig type I ad II error probabilitie are depicted i Figure 6a,b, x repectively. A illutrated i Figure 5, whe γ i cloe to zero, the origial weighted um Er approache to zero. I uch cae, o additive oie Origial exit to decreae the weighted um. For the cae.3 Noie Modified (two cotrait) of two cotrait, the improvemet of the weighted um firt icreae for γ< γ < γ2 ad the.25.2 Noie Modified (o cotrait) (b) Er.5..5 γ.5 γ 2.5 γ γ Figure 5. Figure The 5. The miimum oie modified weighted um um of the type of the I adtype II error I probabilitie ad II error obtaied probabilitie obtaied uder o cotrait o cotrait ad twoad cotrait, two cotrait, ad the origial ad weighted the origial um forweighted differet γ whe um for = ad differet γ = 3. whe ad = Origial Noie Modified (two cotrait) Noie Modified (o cotrait) Origial Noie Modified (two cotrait) Noie Modified (o cotrait)

14 γ.5 γ γ γ Figure 5. The miimum oie modified weighted um of the type I ad II error probabilitie Etropy 27, obtaied 9, 276uder o cotrait ad two cotrait, ad the origial weighted um for differet 4 of γ 22 whe = ad = Origial Noie Modified (two cotrait) Noie Modified (o cotrait) Origial Noie Modified (two cotrait) Noie Modified (o cotrait).4.4 α.3 β γ 2 γ.5 γ γ γ (a) (b) Figure Figure 6. The 6. The type type I ad I ad II error II error probabilitie probabilitie correpodig correpodig to the weighted to the weighted um i Figure um i 5 Figure are how 5 are i how (a) ad i (b), (a) repectively. ad (b), repectively. Table Table 3. The 3. The optimal optimal additive additive oie oie that that miimize miimize the the weighted weighted um um uder uder two two cotrait cotrait ad ad o o cotrait cotrait for for variou variou γ where γ where = ad = ad = 3. = 3. Two Cotrait No Cotrait γ Two Cotrait η No Cotrait 2 o γ η - o Gauia Mixture Backgroud Noie Suppoe that Θ i a radom variable with followig pdf: ϖ () = ρδ( ) + ( ρ)δ( + ) (45) Therefore, we have Ω = {} ad Ω = {, }. I the imulatio, we et d =, d =, d 2 = 2 /4 ad γ =, the detector i expreed a: T(y) = y H > (46) < H Moreover, we aume that v i a zero-mea ymmetric Gauia mixture oie with pdf of p v (v) = M i= m iψ i (v µ i ), where m i, M i= m i = ad: ψ i (v) = 2πi 2 exp( v2 2i 2 ) (47) et M = 4 ad the mea value of the ymmetric Gauia compoet are et a [ ] with correpodig weight [ ]. I additio, the variace of Gauia compoet are the ame, i.e., i = for i =,..., 4. Accordig to (2) ad (4), the oie modified type I error probability obtaied by addig a cotat vector to x i calculated by:

15 Etropy 27, 9, of 22 A () = 4 i= m i (Ψ( /2 + µ i + ) + Ψ( /2 µ i )) (48) ad the correpodig type II error probabilitie for Θ = ad are repectively calculated a: where Ψ(x) = x B () = B () = 4 i= 4 i= m i (Ψ( 3/2 + µ i + m i (Ψ( /2 + µ i + 2π exp( t2 2 2 )dt. Accordigly: ) + Ψ( /2 µ i )) (49) ) + Ψ( 3/2 µ i )) (5) B () = ρb () + ( ρ)b () (5) Therefore, the origial type I ad type II error probabilitie for Θ = ad are α x (φ; ) = A (), β x (φ; ) = B () ad β x (φ; ) = B (), repectively. Due to the ymmetry property of v, oe obtai B () = B (). I thi cae, the origial average type II error probability i β x = B () = ρβ x (φ; ) + ( ρ)β x (φ; ) = B () = B (). The oie modified weighted um of type I ad average type II error probabilitie correpodig to the cotat vector i expreed by Er() = c A () + c 2 B (). The value of c ad c 2 are till pecified a β x /(α x + β x ) ad α x /(α x + β x ), repectively. From Theorem 4 i Sectio 2.3, the optimal additive oie that miimize the weighted um i a radomizatio with a pdf of p opt () = η δ( ) + η 2 δ( 2 ) + η 3 δ( 3 ), where η i for i =,..., 3, ad 3 i= η i =. Figure 7 how the detectio performace of the origial detector ad the oie ehaced detector that miimize the weighted um of type I ad average type II error probabilitie uder the cotrait that α y (φ; ) α o ad maxβ y (φ; ) β o, for differet value of where = ad Ω ρ =.6. The miimum achievable oie modified weighted um i plotted i Figure 7a, ad the correpodig type I error probability ad type II error probabilitie for Θ = ad are depicted i Figure 7b d, repectively. From Figure 7, the origial weighted um, type I error probabilitie, ad type II error probabilitie for Θ = ad icreae a decreae toward zero. I Figure 7a, whe i cloe to zero, the weighted um ca be decreaed igificatly. With the icreae of, the improvemet obtaied by addig oie i reduced gradually to zero. I other word, the pheomeo of oie-ehaced detectio performace caot occur whe exceed a certai value. I Figure 7b, the oie modified type I error probability tay at.5 for <.7 ad the icreae gradually to equal to the origial type I error probability. Moreover, the oie modified type II error probabilitie for Θ = correpodig to the miimum weighted um icreae from zero to that of origial detector, how i Figure 7c, while the type II error probabilitie for Θ = of the oie ehaced detector i equal to that of the origial detector all the time. I fact, the type II error probability for Θ = alo reache the miimum uder the cotrait that α y (φ; ) α o ad maxβ y (φ; ) β o i thi example. I additio, Ω Table 4 offer the optimal additive oie that miimize the weighted um for differet value of to explai the reult i Figure 7. It hould be oted that the optimal oie i ot uique.y.

16 Etropy 27, 9, of 22 Etropy 27, 9, of Er x Er y.35.3 α x (φ;) α y (φ;) Er.6.4 α β (a) (c) β x (φ;) β y (φ;) β (b) (d) β x (φ;-) β y (φ;-) Figure Figure The The weighted weighted um, um, type type I error I error probabilitie, probabilitie, ad ad type type II error II probabilitie error probabilitie for Θ = for ad Θ = ad of the origial of the detector origial ad detector the oie ad ehaced the oie ehaced detector for detector differet for differet where = ad where ρ =.6 = how ad i ρ = (a),.6 (b), how (c) ad i (d), (a), repectively (b), (c) ad (d), repectively. Table From 4. Figure The optimal 7, the additive origial oie weighted that miimize um, the weighted type I error um uder probabilitie, two cotrait ad for type variou II error probabilitie where for = ad Θ = ρ = ad.6. icreae a decreae toward zero. I Figure 7a, whe i cloe to zero, the weighted um ca be decreaed igificatly. With the icreae of, the improvemet obtaied by addig oie 2 i reduced 3 gradually η to zero. η 2 I other η 3 word, the pheomeo. of oie-ehaced.2286 detectio - performace - caot. occur whe - exceed - a certai value. I Figure.27b, the.2286 oie modified.2255 type I error - probability.843 tay at for < -.7 ad the icreae gradually.5 to equal.2287 to the.228 origial type.242 I error probability..53 Moreover,.3446 the.244 oie modified type II error.8 probabilitie.28 for.285 Θ = correpodig.268 to the.5943 miimum weighted.2449 um.68 icreae from zero to that of origial detector, how i Figure 7c, while the type II error probabilitie for Θ = of the Figure oie ehaced 8a demotrate detector the i equal weighted to that um of the oforigial type I ad detector average all the type time. II error I fact, probabilitie the type II y error of theprobability origial detector for Θ = ad alo thereache oie ehaced the miimum detector uder veru the cotrait, where that =.8 α (;) ad φ ρ α o = ad.6. The correpodig y type I error probability ad type II error probabilitie for Θ = ad are max β ( φ ; ) βo i thi example. I additio, Table 4 offer the optimal additive oie that depicted Ω i Figure 8b d, repectively. From Figure 8a, the weighted um caot be decreaed uder miimize the cotrait the weighted o differet um error for differet probabilitie value for of < ad to explai < 2. the Coverely, reult i there Figure exit 7. It additive hould be oie oted uder that the the cotrait optimal oie that i reduce ot uique. the weighted um for < < 2, ad the correpodig improvemet firt icreae ad the decreae with the icreae of. Comparig Figure 8b with Figure 8a, it i oted that the chage of the oie modified type I error probability i imilar to that of the oie modified weighted um. I Figure 8c, the oie modified type II error probability for Θ =

17 Etropy 27, 9, of 23 Table 4. The optimal additive oie that miimize the weighted um uder two cotrait for variou where = ad ρ = η η 2 3 Etropy 27, 9, of 22 η firt decreae to the.2 miimum.2286 ad the.2255 icreae a- icreae,.843 while.587 the type- II error probability for Θ = of the oie modified detector i alway equal to that of the origial detector, how i Figure 8d. I additio, i order to further illutrate the reult i Figure 8, Table 5 how the optimal oie that miimize.8 the weighted.28 um.285 uder the.268 cae of two.5943 cotrait Er x Er y.35.3 α x (φ;) α y (φ;).6.25 Er.4 α.2.2 β (a) (b).6 β x (φ;) β x (φ;-) β y (φ;).4 β y (φ;-).2 β (c) (d) Figure Figure The The weighted weighted um, um, type type I error I error probabilitie, probabilitie, ad ad type type II error II error probabilitie probabilitie for Θ for = ad Θ = ad of the origial of the origial detector detector ad the ad oiethe ehaced oie ehaced detector for detector differet for differet where =.8 where ad ρ =.8.6 how ad i ρ = (a),.6(b), how (c) ad i (d), (a), (b), repectively. (c) ad (d), repectively. Table Figure 5. The 8a optimal demotrate additivethe oie weighted that miimize um the of weighted type I ad umaverage uder two type cotrait II error for probabilitie variou of the origial where detector =.8 ad ad ρ =.6. the oie ehaced detector veru, where =.8 ad ρ =.6. The correpodig type I error probability ad type II error probabilitie for Θ = ad are depicted i Figure 8b d, repectively. 2 From Figure 3 8a, the ηweighted um η 2 caot ηbe 3 decreaed uder the cotrait.65o differet.63 error.63 probabilitie - for.6267 < ad.3733 < 2. Coverely, - there exit additive oie.85 uder.248 the cotrait.249that -.25 reduce the.8262 weighted.3 um for.438 < < 2, ad the correpodig.95 improvemet.295 firt.296 icreae ad -.29 the decreae.76 with.96 the icreae.78 of. Comparig Figure 8b with Figure 8a, it i oted that the chage of the oie modified type I error probability i 5. imilar Cocluio to that of the oie modified weighted um. I Figure 8c, the oie modified type II error probability for Θ = firt decreae to the miimum ad the icreae a icreae, while the I thi paper, a oie-ehaced detectio problem ha bee ivetigated for a geeral compoite hypothei tetig. Uder the cotrait of type I ad II error probabilitie, the miimizatio of the weighted um of average type I ad II error probabilitie ha bee explored by addig a additive

18 Etropy 27, 9, of 22 idepedet oie. The ufficiet coditio for improvability of the weighted um are provided, ad a imple algorithm to earch the optimal oie i developed. The ome additioal theoretical reult are made baed o the pecificity of the biary imple hypothei tetig problem. The tudie o differet oie ditributio cofirm the theoretical aalyi that the optimal additive oie ideed miimize the weighted um uder certai coditio. To be oted that, theoretical reult ca alo be exteded to a broad cla of oie ehaced optimizatio problem uder two iequality cotrait uch a the miimizatio of Baye rik uder the differet cotrait of coditio rik for a biary hypothei tetig problem. Ackowledgmet: Thi reearch i partly upported by the Baic ad Advaced Reearch Project i Chogqig (Grat No. ctc26jcyja34, No. ctc26jcyja43), the Natioal Natural Sciece Foudatio of Chia (Grat No. 6572, No , No , No , No ) ad the Project No. 627CDJQJ6887 upported by the Fudametal Reearch Fud for the Cetral Uiveritie. Author Cotributio: Shuju iu raied the idea of the framework to olve oie ehaced detectio problem. Tig Yag ad Shuju iu cotributed to the draftig of the maucript, iterpretatio of the reult, ome experimetal deig ad checked the maucript. Kui Zhag cotributed to develop the algorithm of fidig the correpodig optimal oie ad Tig Yag cotributed to the proof of the theorie developed i thi paper. All author have read ad approved the fial maucript. Coflict of Iteret: The author declare o coflict of iteret. Appedix A. Proof of Theorem Proof. Due to the covexity of A () ad accordig to the Jee iequality, the type I error probability i (4) i calculated a: α y (φ; ) = E{A ()} A (E{}) (A) The cotradictio method i utilized to prove thi theorem. Suppoe that the detector ca be improved by addig oie. The improvability mea that α y (φ; ) α o for ay Ω, ad the A (E{}) α o from (A). Sice E{} P, A (E{}) α o implie Er(E{}) Er() baed o the aumptio i Theorem, (5) ca be recalculated a: Er y = E{Er()} Er(E{}) Er() = Er x (A2) where the firt iequality hold accordig to the covexity of Er(). From (A) ad (A2), the iequality Er y < Er x caot be achieved by addig ay oie uder the coditio preeted i Theorem. Therefore, the detector i oimprovable, which cotradict the aumptio. Similarly, the alterative coditio for oimprovability tated i the parethee ca alo be proved. Appedix B. Proof of Theorem 2 Proof. Accordig to the defiitio i (9) ad (), improvability for a detector mea that there exit at leat oe pdf p () to atify three coditio, i.e., Er y = R N p ()Er()d < Er(), R N p ()A ()d α o for ay Ω ad R N p ()B ()d β o for ay Ω. Suppoe that the oie pdf p () coit of ifiiteimal oie compoet, i.e., p () = l= λ lδ( ε l ). The three coditio ca be rewritte a follow: Er y = λ l Er(ε l ) < Er() l= λ l A (ε l ) α o, Ω l= λ l B (ε l ) β o, Ω l= (A3) (A4) (A5)

19 Etropy 27, 9, of 22 Sice ε l, l =,...,, i ifiiteimal, Er(ε l ), A (ε l ) ad B (ε l ) ca be expreed approximately with Taylor erie expaio a Er() + εl TEr +.5εT l Hε l, A () + εl TA +.5εl THA ε l ad B () + εl TB +.5εl THB ε l, where Er(A,B ) ad H(H A,HB ) are the gradiet ad the Heia matrix of Er(x)(A (x),b (x)) aroud x =, repectively. Therefore, (A3) (A5) are rewritte a: λ l ε T l Er +.5 λ l εl T Hε l < l= l= (A6) l= l= λ l ε T l A +.5 λ l ε T l B +.5 l= l= λ l ε T l HA ε l α o A (), Ω λ l ε T l HB ε l β o B (), Ω et ε l be expreed by ε l = τ l g, where g i a N-dimeioal real vector ad τ l i a ifiiteimal real value, l =,...,. Accordigly, oe obtai: (A7) (A8) l= λ l τ l g T Er +.5 l= λ l τ 2 l gt Hg < (A9) l= l= λ l τ l g T A +.5 λ l τ l g T B +.5 l= l= λ l τ 2 l gt H A g α o A (), Ω λ l τ 2 l gt H B g β o B (), Ω Baed o the defiitio give i (9) (24), (A9) (A) are implified a: (k er () g) + er (2) g)) x= < (A) (A) (A2) (k a () g) + a (2) g)) x= < 2(α o A ()) l= λ lτl 2, Ω (A3) (k b () g) + b (2) g)) x= < 2(β o B ()) l= λ lτl 2, Ω (A4) where k = 2l= λ lτ l /l= λ lτl 2. A α o = A () for Λ ad α o > A () for Ω Λ, the right-had ide of (A3) approache to plu ifiity for Ω Λ. Similarly, whe β o = B () for Λ ad β o > B () for Ω Λ, the right-had ide of (A4) alo goe to plu ifiity for Ω Λ. Therefore, we oly eed to coider the cae of Λ ad Λ. I doig o, (A2) (A4) are ow: (k er () g) + er (2) g)) x= < (A5) (k a () g) + a (2) g)) x= <, Λ (A6) (k b () g) + b (2) g)) x= <, Λ (A7) It i obviou that k ca be et a ay real value by chooig appropriate λ l ad τ l. A a reult, (A5) (A7) ca be atified by electig a uitable value of k uder each coditio i Theorem 2. That i: () Iequalitie (A5) (A7) ca be atified by ettig k a a ufficietly large poitive umber, if er () g) x= <, a () g) x= <, b () g) x= < hold.

20 Etropy 27, 9, of 22 (2) Iequalitie (A5) (A7) ca be atified by ettig k a a ufficietly large egative umber, if er () g) x= >, ea () g) x= >, b () g) x= > hold. (3) Iequalitie (A5) (A7) ca be atified by ettig k a zero, if er (2) g) x= <, a (2) g) x= <, b (2) g) x= < hold. Appedix C. Proof of Theorem 3 Proof. Sice I(t) ad S(t) are ecod-order cotiuouly differetiable aroud t = α, there exit a ξ > uch that I ( ) < ad S ( ) < for = ( α ξ, α + ξ). If oe add a oie with pdf p ˆ () =.5δ( ) +.5δ( 2 ), where maxa ( ) = α + ξ ad maxa ( 2 ) = α ξ, to the Ω Ω origial obervatio x, the maximum value of correpodig oie modified type I ad II error probabilitie are: { } maxe{a ( ˆ)} E maxa ( ˆ).5( α + ξ) +.5( α ξ) = α α o Ω Ω { } maxe{b ( ˆ)} E maxb ( ˆ).5S( α + ξ) +.5S( α ξ) S( α) = β β o Ω Ω I additio: E{Er( ˆ)} = E{I(t)} =.5I( α + ξ) +.5I( α ξ) < I( α) (A8) (A9) (A2) Oe obtai E{Er( ˆ)} < Er() becaue I( α) Er() accordig to the defiitio of I(t). A a reult, the detector i improvable. Appedix D. Proof of Theorem 5 Proof. Part (): If Q e Q =, ay Q e Q atifie the cotrait of A ( ) α o ad B ( ) β o baed o the defiitio of Q ad Er y opt = Er( ) = mier() < Er() accordig to the defiitio of Q e. Part (2): If Q e Q 2 = ad Q e Q 3 = imultaeouly, there exit Q e Q 2 ad 2 Q e Q 3 uch that Er( ) = Er( 2 ) = mier() baed o the defiitio of Q e. I order to meet the cotrait that E{A ()} α o ad E{B ()} β o, the oie compoet η, ad 2 hould atify the followig two iequalitie: ηa ( ) + ( η)a ( 2 ) α o (A2) ηb ( ) + ( η)b ( 2 ) β o (A22) Coequetly, η η = α o A ( 2 ) A ( ) A ( 2 ) ad η η 2 = β o B ( 2 ) accordig to the defiitio of B ( ) B ( 2 ) Q 2 ad Q 3. If η η η 2, the oie with pdf p opt () = ηδ( ) + ( η)δ( 2 ) ca miimize E{Er()} ad atify the two iequalitie, ad Er y opt = ηer( ) + ( η)er( 2 ) = mier(). Part (3): If Q e Q 2, the optimal additive oie i ot a cotat vector, i.e., η =. Therefore, oe of ad 2 belog to Q 2 ad the ecod oe come from Q or Q 3. I additio, η, ad 2 hould alo atify the two cotrait i (A2) ad (A22). Firt, uppoe that Q 2 ad 2 Q, the (A2) hold baed o the defiitio of Q ad Q 2. We hould oly coider the cotrait i (A22), which implie η η 2. It i true that A ( 2 ) α o ad B ( 2 ) β o accordig to the defiitio of Q. If Er( ) > Er( 2 ), we have Er( 2 ) < Er y opt = ηer( ) + ( η)er( 2 ), which cotradict with the defiitio of p opt (). Hece, Er( ) < Er( 2 ) ad the miimum of E{Er()} i obtaied whe η = η 2. Next, uppoe that Q 2 ad 2 Q 3. The two iequalitie i (A2) ad (A22) require that η η η 2. If Er( ) > Er( 2 ), the miimum of E{Er()} i obtaied whe η = η. I uch

21 Etropy 27, 9, of 22 cae, there exit a oie with pdf p ˆ () = ςp opt () + ( ς)δ( e ) that atifie E{A ( ˆ)} α o ad E{B ( ˆ)} β o imultaeouly, where e Q e ad ς. Therefore, E{Er( ˆ)} = ςer y opt + ( ς)er( e ) < Er y opt ice Er( e) = mier() < Er y opt, which cotradict with the defiitio of p opt (). A a reult, Er( ) < Er( 2 ) ad the miimum of E{Er()} i obtaied whe η = η 2. Whe η = η 2, oe obtai E{B ()} = ηb ( ) + ( η)b ( 2 ) = β o. I other word, the miimum of E{Er()} i obtaied whe E{A ()} achieve the miimum ad E{A ()}. Accordigly, oe obtai Er y opt = c α y opt + c 2β o. Part (4): The proof of Part (4) i imilar to that of Part (3) ad it i omitted here. Referece. DeGroot, M.H.; Sxhervih, M.J. Probability ad Statitic, 4d ed.; Addio-Weley: Boto, MA, USA, Pericchi,.; Pereira, C. Adaptative igificace level uig optimal deciio rule: Balacig by weightig the error probabilitie. Braz. J. Probab. Stat. 26, 3, 7 9. [CroRef] 3. Bezi, R.; Sutera, A.; Vulpiai, A. The mechaim of tochatic reoace. J. Phy. A Math. 98, 4, [CroRef] 4. Patel, A.; Koko, B. Noie beefit i quatizer-array correlatio detectio ad watermark decodig. IEEE Tra. Sigal Proce. 2, 59, [CroRef] 5. Ha, D.; i, P.; A, S.; Shi, P. Multi-frequecy weak igal detectio baed o wavelet traform ad parameter compeatio bad-pa multi-table tochatic reoace. Mech. Syt. Sigal Proce. 26, 7 7, 995. [CroRef] 6. Addeo, P.; Pierro, V.; Filatrella, G. Iterplay betwee detectio trategie ad tochatic reoace propertie. Commu. Noliear Sci. Numer. Simul. 26, 3, 5 3. [CroRef] 7. Gigl, Z.; Makra, P.; Vajtai, R. High igal-to-oie ratio gai by tochatic reoace i a double well. Fluct. Noie ett. 2,, [CroRef] 8. Makra, P.; Gigl, Z. Sigal-to-oie ratio gai i o-dyamical ad dyamical bitable tochatic reoator. Fluct. Noie ett. 22, 2, [CroRef] 9. Makra, P.; Gigl, Z.; Fulei, T. Sigal-to-oie ratio gai i tochatic reoator drive by coloured oie. Phy. ett. A 23, 37, [CroRef]. Dua, F.; Chapeau-Blodeau, F.; Abbott, D. Noie-ehaced SNR gai i parallel array of bitable ocillator. Electro. ett. 26, 42, 8 9. [CroRef]. Mitaim, S.; Koko, B. Adaptive tochatic reoace i oiy euro baed o mutual iformatio. IEEE Tra. Neural Netw. 24, 5, [CroRef] [PubMed] 2. Patel, A.; Koko, B. Mutual-Iformatio Noie Beefit i Browia Model of Cotiuou ad Spikig Neuro. I Proceedig of the 26 Iteratioal Joit Coferece o Neural Network, Vacouver, BC, Caada, 6 2 July 26; pp Che, H.; Varhey, P.K.; Kay, S.M.; Michel, J.H. Theory of the tochatic reoace effect i igal detectio: Part I fixed detector. IEEE Tra. Sigal Proce. 27, 55, [CroRef] 4. Patel, A.; Koko, B. Optimal oie beefit i Neyma Pearo ad iequality cotraied igal detectio. IEEE Tra. Sigal Proce. 29, 57, Bayram, S.; Gezici, S. Stochatic reoace i biary compoite hypothei-tetig problem i the Neyma Pearo framework. Digit. Sigal Proce. 22, 22, [CroRef] 6. Bayrama, S.; Gultekib, S.; Gezici, S. Noie ehaced hypothei-tetig accordig to retricted Neyma Pearo criterio. Digit. Sigal Proce. 24, 25, [CroRef] 7. Bayram, S.; Gezici, S.; Poor, H.V. Noie ehaced hypothei-tetig i the retricted Bayeia framework. IEEE Tra. Sigal Proce. 2, 58, [CroRef] 8. Bayram, S.; Gezici, S. Noie ehaced M-ary compoite hypothei-tetig i the preece of partial prior iformatio. IEEE Tra. Sigal Proce. 2, 59, [CroRef] 9. Che, H.; Varhey,.R.; Varhey, P.K. Noie-ehaced iformatio ytem. Proc. IEEE 24, 2, [CroRef] 2. Weber, J.F.; Waldma, S.D. Stochatic Reoace i a Method to Improve the Bioythetic Repoe of Chodrocyte to Mechaical Stimulatio. J. Orthop. Re. 25, 34, [CroRef] [PubMed]

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