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A one-sided sequential test
An~ Nuc1. ~e~y ~ 1 . 2 ~ N~ 1~ p~ ~ 1 0 0 9 , 1~6 030~4~4~9~)00081-X
C0~f1~t • 1996 H ~ v ~ 5c1en~ ~ d Pdnt~ ~ 6reat 8dta1n. ~1 d ~ re5erv~ 03~4~9/96 ~ ~ + ~
A 0NE-51DED 5EQUEN71AL 7E57
A. R6c2 and 1. L ~ KFK1-At0m1c En~9y Re5ear~ 1n5t1~ Appf1ed React0r P h y ~ La60r~0ry H-1525 8udape5t, E 0 . 8 0 x 49, H u n 9 ~ y e-m~1: r a c 2 ~ e r v J d ~ L h u ~m~1: 1ux~f~.k1k1.hu (Rece1ved 9 5eptem6er 1995)
A 6 ~ r a c t - 7he app~ca61~ty 0f the da5~ca1 5PR7 f0r ear1y f ~ h r e de~ct10n pr0~em5 ~ ~m1ted 6y the fact t h ~ there 15 an extra t1me de1ay 6etween the 0ccurrence 0f the f~1ure and ~5 f1r5t rec09a1t10n. C ~ e n and Adam5 deve10ped a meth0d t0 m1~m12e t~5 t1me ~ r the ca5e when the pr061em can 6e ~ r m ~ a t e d a5 te5t1n9 the mean va1ue 0f a 6au5~an ~9na1. 1n 0ur paper we pr0p0~ a pr0cedure t h ~ can 6e app~ed f0r 60th mean and var1ance te5t1n9 and that m1~m12e5 the t1me d~ay. 7he meth0d 15 6a5ed 0n a 5peda1 param~r12~10n 0f the da5~ca1 5PR7. 7he 0 n ~ d e d 5e4uent1~ te5t ( 0 5 5 7 ) can repr0duce the re5~t5 0f the C~en-Adam5 te5t when app~ed f0r mean va1ue5. C0pYr19ht • 1 ~ 6 ~ f 1 ~
~n~
L~
1. 1 N 7 R 0 D U C 7 1 0 N 7 h e 5e4uent1a1 pr06a61~ty rat10 te5t1n9 (5PR7) meth0d 15 a we11-kn0wn and w1de1y u5ed t001 f0r hyp0the515 te5t1n9 pr061em5, when the ta5k 15 t0 dec1de 6etween n0rm~1 and ~6~0rm~1 (0r de9enerate) m0de 0f the 1nve5t19ated 5t0cha5t1c 5y5tem. 7he app11ca6111ty 0f the 5PR7 re4u1re5 that the d1fference 6etween the a1ternat1ve m0de5 can 6e fu11y de5cd6ed
997
998
A. Rhc2 and 1. Lux
6y a ~ n ~ e parameter 0f a rand0m pr0ce55. Pr0ce55e5 w1th 5uch pr0pert1e5 are ca11ed re~du~5.1n 5uch a ca5e, when 1) the r e ~ d u ~ 15 a wh1te 6an5~an pr0ce55 w1th 2er0 mean and kn0wn var1ance r(k) E A/~{0,a 2} (k 15 the ~ 5 c r e ~ t1me) and 2) the n 0 r m ~ / a 6 n 0 r m ~ ~an51ent can 6e ~pre5ented, dther a5 a 5h1~ 1n the mean m 0r a ma9n1f1cat~n 0f the var1ance ~, and 3) the 0ther parameter 0f the r e ~ d u ~ rem~n5 the 5ame then the 5 P R 7 ~ a we11-def1ned M90r1thm (Hanc0ck and W1nt2, 1966) t0 dedde 6etween the M~rnat1v~5. 1n 5p1~ 0f K5 ~mphdty, the 5 P R 7 5ucce55fu1~ dcte~5 50ft chan9e5. H0wever, the m ~ h 0 d ha5 a ~ w 5h0rtc0m1n95. 1n fa~, when ~eat1n9 re~1y 50K chan9e5, very ~n9 5amp1e ~2e5 have t0 6e pr0ce55ed 1n 0rder t0 ach~ve a re11a6~ c0ndu~0n, 0r, ff the ~ m 15 a 4u1ck (pr0mpt) an5wer, then the a pr06a6111ty 0f wr0n9 ded~0n 15 1ar9e. F 0 ~ u n a t d y a n ~ y t ~ M f0rmu1~ e~5t (the avera9e 5amp1e num6er 0r A5N funct10n) t0 c0nnect the 5amp~ ~2e5 t0 the ded5~n err0r pr06a6~t~5.7hu5 K can 6e dedded 1n advance whether the pre5cr16ed re11a6~ty and the re4u1red 5amp~ ~2e5 are 60th accepta61e. 50me attempt5 have 6een made (and 5ucceeded, 1n fa~) t0 5ave the re11a6111ty 0f the te5t and t0 reduce the num6er 0f the pr0ce~ed 5amp1e5 at the 5ame t1me. 0 n e 0f them ~ W~d~5 truncated 5 P R 7 (Hanc0ck and W1nt2, 1966), wh1ch cu~ 0ff the te5t after p r 0 c e ~ 9 a ~ven num6er 0f 5amp1e5. 7he 0ther appr0ach can 6e c~1ed a5 ~he pre5umpf10n 0f 1nn0cence••. 1n fa~, 1n ear1y f~1ure 0r chan9e detect10n we are m ~ n ~ 1ntere5ted 1n the rec09n1t10n 0f the de9radat~n (~ happen5) 6ut we d0 n0t want t0 6e 60thered 6y a p e r p ~ u ~ c0nf1rmat~n that everyth1n9 ~ 0.K. Keep1n9 th15 re4~rement 1n m1nd, the 5 0 ~ 1 e d m0d1f1ed 5e4uent1M te5t ha5 6een pr0p05ed 6y Ch1en and Adam5, wh1ch detect5 0n~ a6n0rma11t~ and m1~m12e5 the d ~ e ~ n t1me (Ch1en and Adam5, 1976). A n ~ y t ~ 5dut~n5 f0r the m0~f1ed (0r Ch1en-Adam~ ~ e ~ 0n1y t0 the ca5e when the parameter t0 6e t e ~ e d 15 the mean 0f the r ~ u ~ . (Hencef0~h the m0~f1ed 5PR7, Ch1en-Adam5 0r C-A te5t refer t0 the 5ame pr0cedure.) H0wever, 1n many apphcat~n5 the d e 9 r a d ~ n can 6e m0de11ed a5 an 1n~ement 0f the var1ance 0f the re~du~. 7h15 happen5 when the r e ~ d u ~ 15 9enerated 6y an AR m0dd (R~c2 and K155, 1995). An0ther 5h0rtc0m~9 0f the m0~f1ed te5t 15 that the num6er 0f 5amp~5 nece55ary t0 make a ded~0n 15 a rand0m var1a6~, thu5 ~ can 6e very 1ar9e 1n cert~n ca5e5 (6ut 1t5 expected va1ue can 6e c~cu1ated 1n advance). 7h1rd, the pr0cedure d0e5 n0t a v e any h1nt a60ut the 5y5~m~5 5tate 6etween ~w0 ded~0n5. F u ~ h ~ m 0 r e the detecf10n t1me can 0n1y 6e decrea5ed 0n the c05t 0f 1ncrea51n9 the ~e4uency 0f wr0n9 ded5~n5. 1n 0ur paper we try t0 5dve 50me 0f the a60ve pr06~m5 v1a rec0n~der1n9 the 0r1~n~ 5 P R 7 6ut n0t f0r9~t1n9 the 6a51c m0f1vat~n 0f the Ch1en-Adam5 te5t, n a m d y that the de9raded m0de 15 the 1ntere5~n9 0ne. F1r5t we w1~ 5h0w h0w t0 5et the ded~0n err0r pr06a6111t~5 0f the 5 P R 7 1n 0rder t0 ach~ve the 5ame 0utput a5 the Ch1en-Adam5 te5t. We w1H cM1 the pr0¢edure a 0ne-~ded 8e4uent1~ 7e5t ( 0 5 5 7 ) 51nce, ~m1~f1y t0 the C-A te5t, re11a6~ ded~0n5 are made 0n~ a60ut the a6n0rm~ m0de 0f the 5y~em. 51nce 0ur pr0p0~t10n 15 n0t re5tr1cted t0 the ca5e 0f 1nve5t19at1n9 the mean va1ue5 0f the re~du~, the pr0cedure 15 ~ r ~ 9 h t ~ r w a r d ~ extended t0ward5 the te5t 0f var1ance5. 51nce there 15 a meth0d (R~c2, 1995) 0n h0w t0 6e perp~ua11y 1nf0rmed a60ut the 5tate 0f the 5y5~m
A ~
~e~f1
te5t
999
under 1nve5t19at10n ~ the hyp0the515 te5t1n9 m ~ h 0 d 15 the 5PR7, th15 a190r1thm can 6e ea51~ app11ed f0r the recent pr061em. 1n 5ummary, the 0n~f1ded 5e4uent1a1 te5t ut1112e5 the ad~anta9~ 60th 0f the da5f1ca1 and the m0~f1ed te5t5 w1~0ut ~he1r draw6aek~. 1n fact, the 0 5 5 7 1) m1~m1~5 t~n, 2) ~ c ~ e 3) ~ u ~ y ca5e 0f 5 ~
the detect10n t1me ~ a 6~n9 re5tr1cted 0 ~ y t0 t ~ ~ c 0 9 ~ t ~ n ~ d e ~ a d v 0n t e ~ n 9 d t h e r the mean ~ u e 0r the ~ 1 a n ~ 0f the M9na1 pr~d~ 1 ~ ~ (6~ ~d~) ~ 0 u t the ~ t ~ 5 5t~e e ~ n h ~h~
7he 0r9an12at10n 0f the paper 15 the f0110w1n9. 5ect10n 2 5ummar12e5 the 5PR7 and the Ch1en-Adam5 te5t. 5ec~0n 3 91ve5 0ur pr0p0~t10n h0w t0 ca1cu1ate the pr0per parameter5 0f the 5PR7 1n 0rder t0 5ave the advanta9e5 0f the C-A te5t and app1y 1t f0r the ca5e 0f var1ance te5t5.
2.7HE
5PR7 AND 7HE CH1EN-ADAM5 7E57
1n t~5 5ect10n we 0verv1ew the pr0cedure5 t0 6e u5ed ~ the 5e4u~.1n ca5e 0f hyp0the515 t e ~ 9 the ta5k ~ t0 te5t a nu11hyp0the515 ~0 a9Mn~ the a ~ e r n ~ e hyp0the~5 ~ a60ut an unkn0wn mean va1ue m 0f a n0rma1 6 a u ~ n p ~ ¢ e A/•(m, a~). 7 h e var1ance a~ 15 5upp05ed t0 6e kn0wn and c0n5tant. We have the f0110w1n9 tw0 hyp0the5e5 f0r rn: ~ 0 : m = ~0~
~1
: m = ~1,
(1)
where ~0 and ~1 are c0n5tant5. We a55ume that dur1n9 the hyp0the5e5 te5t1n9 pr0cedure the pr0pert1e5 0f the ~0urce, wh1ch 9enerate5 the ~9na15 ~ 5 , d0 n0t chan9~ 7 h e pr06a61~f1e5 0f err0r5 cau5ed 6y an 1nc0rrect ded~0n are the f0H0w1n9. 7ype 1 err0r (0r the 50-ca11ed fa15e a1arm) 0ccur5 when 0ne accept5 7~1 6ut •/0 ~ true. 1t5 pr06a61f1ty ~ den0ted 6y a. 51mf1ar1~ accept1n9 7~0 when 7~1 15 true cau5e5 7ype 11 err0r (0r m155ed a1arm). 1t5 pr06a6111ty 15 den0ted 6y ~. A5 1t wa5 ment10ned 1n the 1ntr0duct10n, 1t ~ a150 p05~61e t0 1nve5t19ate 0n1~ that the a1ternat1ve hyp0the515 15 5at15f1ed (m0d1f1ed 5PR7). 1n 5uch a ca5e the c0ncept 0f m~5ed a1arm 15 n0t app11ca61e (Ch1en and Adam5, 1976). 2.1 5e4uent1a1 pr06a61f1ty rat10 te5t (5PR7) 7 h e ta5k 15 the f0H0w1n~ Hav1n9 rec~ved the 519na15 {•, ~2,..., ~k,...} ~ 15 t0 6e dec1ded wh1ch hyp0the515 (7-/0 0r :H1) 15 true. 7he a190r1thm ~ a5 f0110w5: Fr0m the f1xed de51red err0r pr06a6111t1e5 (a a~d f1) the f0110w1n9 tw0 thre5h01d va1ue5 ( dec~10n 60unda~e5) A and 8 have t0 6e ca1cu1ated:
1000
A. R~c2 and 1. Lux
Let u5 c~cu1ate the 109-hke11h00drat~ At, a5
where P5,¢2,...,¢~1~f0r 1 = 0, 1 repre5ent the j~nt pr06a61hty denf1ty funct10n 0f the 065ervat10n 0 up t0 (,, c0nd1t10ned up0n 7~1.7hen the dec610n ma~n# pr0cedure 15 the f0110w1n9.1n the k th 5ta9e At mu5t 6e c0mpared t0 A and 8 and ~ Ak ~ A
then accept 7~1 and 5et Ak = 0,
A~ ~ 8 8
then accept ~0 and 5et A, = 0, then take an0ther 5amp1e.
(4)
A5 6n9 a5 m 15 the mean v~ue 0f a wh1te, 6au5f1an pr0ce55 ( ~ A:(m,~), the 69~ke~h00d rat~ can 6e cMcu1ated ~c~e1p:
7he a~era#e aamp~ num6er (A5N f u n ~ n 5 ) nece55ary t0 m~ke 9 dcc1a10nread5
A5N~0=
(0, 2d~00)~ (a 1 n ( ~ - )
+ (1 - a) 1n(1 ~
a)).
7he n0tat~n A 5 N ~ 0 15 u5ed when the mean take5 the va1ue m = 00 wh1~ A 5 N ~ t0 m = 0 1 .
(6) refer5
7he 5 P R 7 pr0cedure ~ M50 app11ca61e when the var1ance 15 te5ted 1n~ead 0f the mean va1ue 0f the f19nM. 1.e.the hyp0thef15 te5t1n9ta5k 15 t0 dedde 6etween the hyp0the5e5 ~0 and 7~ a60ut an unkn0wn var1ance c 0f a n0rm~ 6a~5~an pr0ceaa ( ~ .A/•(m,c~). 7he mean va~ue 0f the f19nM 15 5upp05ed t0 6e 2er0 m = 0. (Hav1n9 a n0n-2er0 6ut c0n5tant mean va1ue can 6e ~eated f1m11ar1y.H0wever, the f0rmu1ae are f1mp1er 1n ca5e 0f 2er0 mean.) 7hu5 the f0~0w1n9 hyp0the5e5 are u5ed: ~0 : ~2 = ~ ,
7~ : c 2 = 4c~,
where
4 > 1.
(8)
7he dedf10n mak1n9 pr0cedure (4) rem~n5 the 5ame 6ut the f0rmu1ae 1n E45.(5-7) chan9e a5 f0110w5: A1+, = Ak + (~+~2~94 4- 1
~ 1n(4),
(9)
.5~0 =~(~-~-~ ~ ~0~)-~(~.(~-~) + <~- 0~1.(~~ ~),
(~0~
,~,
(~1~
= ~(,- ~- ~)-~((~- ~1~(~-~ ~ ~,~(~~ ~).
A 0n~f1ded ~ 4 ~
te5t
1~1
7he m ~ n 5h0rtc0m1n9 0f the 5PR7 15 t h ~ the nece~ary 5amp~ ~2e5 c0~d 6e very 1~9e when the ded~0n err0r pr06a6111t1e5 are 5mM1. 1n ad~f10n, the f1r5t ~ c ~ n 0f a6n0rmM1~ ( u ~ 5 1t happen5 at the very 6 e ~ n ~ n ~ need5 50me extra t1me, ~nce the 1am6da ~nct10n 15 pr06a61y ~ the ~terva1 [ 8 , . . . , ~. 7~5 5h0rtc0m1n9 can 6e 0verc0me ~ we 0~y 1ntend t0 r e c 0 9 ~ a6n0rma~y ~ 5u99e~ed 6y C~en and Adam5 ~ thek m0d~ed ~4uent1~ te5t. We menf10n 6y p ~ n 9 t h ~ there 15 a n0ve1 m~h0d t0 1nterpret the 5PR7 0 u ~ 0 m ~ ~ ~ d u M e~denc~ a60ut the te5ted 5y5tem. 5~ce the meth0d (w~ch 15 ca11ed ~ u~da~9 pr0cedure) 15 0ut 0f the 5c0pe ~ the recent w0rk, 1t5 m ~ n 1dea 5ummar1~d ~ the Appen~x ~av1n9 the det~15 t0 the f 1 ~ r ~ u ~ (P~c2, 1995). 2.2 M0~f1ed 5PR7 m~h0d A5 1t wa5 ment~ned a60ve, the m £ n 5h0~c0m1n9 ~ the 0r1~n~ 5PR7 pr0cedure e m ~ 9 ~ ~0m the fact, t h ~ the 5PR7 w ~ 1ntr0du~d ~ r de4~n9 6etween tw0 (~m05t e4uM1y pr06a61e) hyp~he~5 (n0rmM 0r a6n0rm~) r ~ h e r than t0 rec09n12e a6n0rm~1ty. 1n ~ct, u~n9 any chan9e de~ct10n 5y~em, u5u~1y ~ 15 ~ 0 n a 6 ~ t0 ~5ume t h ~ the 5y5tem under 1nve5t19at10n ~ r ~ h e r ~ n0rmM than 1n a6n0rm~ ~ e . 7 h ~ e f 0 ~ the m ~ n t ~ k can 6e reduced t0 ~c09n12e ( ~ 500n and ~ ~ a 6 ~ ~ p05~61e) a6n0rmM1ty. 1n 5uch a c~e, 6y def1n1t10n, the 5y~em 15 n0rm~ unt~ R 15 d e d ~ e d t0 6e a6n0rm~. 1t ~ n0t a taut0109Y ~ R can 6e 5een ~0m the fact t h ~ the 5PR7 d0e5 n0t 5t~e anyt~n9 a60ut the 5y~em~5 5t~e ~ ~n9 ~ 8 < A~ < A ~ee. E%~) ~ r the ded~0n mak1n9 r~e5]. 1.e., the 5y~em~5 5tate ~ unde~rm1ned ~ r u n c e ~ n , n0t def1ned) except the t1me p~nt5 when 60undary ~ n 9 event5 happen. C~en and Adam5 pr0p05ed a ~fferent te5t w~ch ~ c 0 9 ~ 5 0~y a6n0rm~1~ ~ a 1nve5t19at1n9 the mean va1ue m ~ a 6au551an ~9n~ w1th kn0wn var1an~ a 9 . 7 h e pr0~dum 90e5 ~ f0110w5. 7he ~f1ke~h00d r ~ mu5t 6e c M c ~ e d ~ the 5ame way ~ ~ c~e 0f 5PR7. A ~ w ~ d 5 the 5 5 9 ~ m0~f1ed ded5~n mak1n9 p r 0 ~ d u ~ read5 A~ ~ 0
then 5et Ak = 0,
A~ ~ AcA 0
(7he 5u65c~pt cA ~ c ~
then ac~pt ~
and 5et A~ = 0,
(12)
then take an~her 5am~e.
the Ch1emAdam5 te5t). 7he 60und~y AcA can 6e c ~ c ~ e d
where 7 15 the mean t1me 6etween tw0 c0n5ecut1ve fM5e M~m5.7he rda~0n 6etween 7 and the fM5e M~m pr06a6111ty a 15 ~ven 6y the f0110w1n9 e4u~10n: ~ = 7 -~. 7he f1r5t ~me 0f d ~ e ~ 9 take5 the f01~w1n9 va~e:
(14)
a6n0rma11~ (the 50-cM~d perf0rmance 1ndex, den0ted 6y v)
, =
1002
A . Rf1c2 and 1. Lux
F0r m0re deta115 a60ut the C-A te5t 5ee the 0r191na1 art1e1e 0f Ch1en and Adam5 (1976).
3. 7 H E 0 N ~ 5 1 D E D
7E57
C0mpar1n9 the ~ n a r y 5 P R 7 t0 1t5 m0~f1ed ver5~n the f0110w1n9 6a1ance can 6e drawn: Advanta9e5 0f the m~h0d5 - 7he 5 P R 7 15 app11ca61e f0r te5t1n9 ~ther the mean 0r the var1ance 0f the ~9na1. - A n0ve1 6e11ef-updat1n9 pr0cedure can 6e app11ed 0n the 5PR7•5 0utc0me5 t0 perpe~ ua11y m 0 ~ t 0 r ~he 5y~em~5 ~ate. - 7he C-A te5t m1~m12e5 the t1me dday 6etween the ~an51t~n event 1ntended t0 rec09n12e and 1t5 f1r5t rec09n1t10n. D~w~5
~ t~ m ~
- 7 h e C-A ~5t 15 n0t app11ca6~ ~ r ~ 9 var1ance5. - 5~ the C-A t ~ t ~ c ~ 0 ~ y a6n0rma11ty, the 5ame 6 e 1 1 e h u p d ~ 9 p r 0 ~ d u m ~ n0t app11ca61e ~ r 1t5 0utput5. C0n~def1n9 the a60ve f15t5 ~ 15 ea5y t0 art1cu1ate 0ur w15h; t0 perf0rm a te5t wh1ch p055e55e5 0n1y the advanta9e5 0f the 5 P R 7 and C-A te5t5 w1th0ut th~r draw6ack5. 3.1 P r 0 p 0 ~ t ~ n 7he 6a51c 1dea 6 e ~ n d the C-A te~ 15 n0t t0 te5t the n0rma1 m0de 6ut c 0 n ~ n t r ~ 9 0 ~ y 0n the ~ ~ % ff ~ 1n 0ther w0rd5, 1t 15 a55umed that a5 ~ n 9 a5 the 5y~em ~ n0t j u ~ e d t0 6e ~ a 1 1t mu5t 6e n~ma1. 5~ce 1n ca5e 0f ~ r m ~ the A~ ~n~n ~ ~ea5~9 ~0m ~) then n0 ~ m ~ ~d h ~ p e n ff ~ e A~ ~ n ~ n decrea5e5, ~ ~ndenc~ ~ ~ ff A1 < 0 (w~ch ~ e ~ 15 ~ e ~ t0 a d ~ a 5 1 n 9 ~m6dv~n~n, Mnce the ~t1a1 ~ u e ~ r A~ 15 2er0), then 1t5 ~ u e 15 re5et t0 0. ~ ad0pt C ~ e n ~ ~ d A d ~ 5 ~ d 0 v e 0f p ~ ~ the da5~ca1 5 P ~ 1n 5 u ~ a w ~ ~h~ 1t 5ha11 ~ ~ 6enef1t5 60th 0f the C-A te5t ~ d ¢he 5 P ~ . 0 u r pr0p0~f10n ~ t0 p~ t~ ~r~e 5 ~ ~ A5~0 ~ ~6~ ~ ~ ~ . ~ ~, A5~0 = 1 15 ~ m m ~ d ~ ~ e . A5 a ~ e ~ , w~n ~ p ~ ~ 15 ~ 1 d t ~ 5 P ~ ~ 0n ~0 ~ every 5tep (1n ~ Q , ~ d 6y c 0 n ~ r u ~ n , the A~ ~ n ~ n 15 re5et t0 2er0. 7h~e~ t~5 ~ 0 ~ d u ~ ~ ~ e ~ ~ t~ m0~d 5P~ m~h0d ~ ~ C~ ~d Ad~. N0w, there ~ m ~ n e d 0ne m0re p ~ r t0 6e 5et 6y the u5er ~ d t~5 15 the ~ 5 e a1~m ~ f 1 ~ a, 0r, ~ e ~ the m e ~ t1me 7 6~ween tw0 ~ 1 ~ ~ e a1~m5 1n the C-A te5~. H 0 ~ r , the 5 P ~ need5 60~h d e ~ 0 n err0r pr06a~f1t1~ 1n 0rder t0 d~m~e the de6~0n ~ d ~ A and 8. H ~ n 9 f1xed the a ~ u e , E4.(6) f1dd5 the
A 0n~f1ded 5e4uenf1f1 te~
1003
de~red m ~ d a1~m pr06a61~y f1, ~nce 1t5 ~K hand ~de 15 kn0wn, t0 6e 1. 7hu5 the 1mp11dt e 4 u ~ n have ~ 6e ~1ved ~ r f1 15 the f01~w1n9:
(0
-.)
(16)
A~h0u9h E%(16) can a1way5 6e 5~ved 6y numer1ca1 m~h0d5, 1n Append~ 8 the reader w1~ f1nd a n a ~ c 5 0 ~ 0 n 5 ~ r 1t. 3.2 An e x a m ~ e 8e~re pr0~e~n9 ~h~ an e x ~ # e 15 ~ ~ d ~ t0 5 h ~ h0w the m ~ h 0 d w0rk5 and h0w 1t 15 c0nnected t0 the C ~ m ~ m 5 te5t. A c0mpu~r ~ m ~ n w ~ p ~ r m e d t0 9enerate a ~9na1 ~ t h an ~ r u p t chan9e ~ 1t5 mean ~ u e ~ ~ . 7he actua1 ~1ue5 ~ e the ~ ~ . 7he ~ r e 0ccurred at ~ = 800. 7 h e mean t1me 6~ween tw0 ~ 5 e a1~m 15 7 = 500 (0r a = 0.002). ~=1. ~0 = 0. 01 =
0.4.
7 h e ca1cu1ated m ~ d M ~ m ~ ~ ~ r the ~ 0 v e p ~ ~ 15 ~ = 0.91. N ~ 1t 15 p~e t0 ~ p ~ 60th the C-A and the 0 n ~ d e d te5t ~ r the ~ t ~ a 1 ~9na1. ~ 9 u r e 1 5h0w5 the 1 ~ 6 d a ~ n ~ ~ r 60th te~5. 7he de~5~n ~ d ~ (n~ ~ d her~ h ~ the ~ 0 ~ n 9 ~ u e 5 ; AcA = 3.68 ~ r the C-A te5t, wM~ A = 3.76 ~ r the 0 5 5 7 . 7he 1 ~ 6 d a ~ n ~ ~ e a1m~t the 5ame, ~ 1t w ~ ~ p e ~ e ~ (7he ~ 9 ~ ~ n ~ 5 ~e cau5ed 6y the m ~ 0 r ~ ~ 6etween the actua1 va1ue5 ~ the d e ~ 0 n ~ d ~ . ) ~r 6ett~ ~ e ~ n t~ 1~da ~t0~ ~ e t r a n 5 ~ r m ~ 1~0 t ~ 5 ~ e ~ t ~ ~1 [-1024, 102~ and ~ p ~ e d a5 ~ e 9 e r ~ . 7he ~ c a 1 ~ p ~ ~ were p ~ r m e d 6y a p m ~ a m p a ~ 9 e F ~ W (~55, 1991). 7he f1r~ detect10n 0f ~ r m ~ happened at ~ 1 ~ = 829. 5~ce the ~ r m ~ wa5 1ntr0duced at ~ = 800, the t1me 1~5e 15 2 9 . 7 h e ana1yt~a1 ~ e ~n 6y E4.(15) 15 v = 27, ~ t h ~ e 15 a 900d a ~ m e ~ 6~ween the ana1yt1ca1 and ~ m ~ c a 1 5~ut0n5. 3.3 ~
~ ~
Certa1n1y t0 repr0duce the 0utput 0f the Ch1en-Adam5 te5t 1n a d1fferent way can a150 6e re9arded a5 a re5u1t. H0wever, 0ur a1m 15 t0 90 further. 1n fact, the C-A te5t 1ack5 the a6111ty t0 te5t va~ance5. H0wever, 1n many app11cat10n5 (e.9. UAR-9enerated re~dua1~ the va~ance 5h0u1d 6e te5ted 1n5tead 0f the mean va1ue. Enc0ura9ed 6y the pre~m1nary re5u1t5 0f the 0ne-51ded pr0cedure f0r te5t1n9 mean va1ue5, we app1y the 0ne-51ded te5t f0r te5t1n9 va~ance5, t00. F0r 5uch a ca5e the pr061em 15 f0rmu1ated 6y E%(8). P~ndpa11y there 15 n0 d1fference 6etween the ta5k 0f te5t1n9 mean 0r var1ance5 v1a da5~ca1 5 P R 7 meth0d. 7he a190~thm 15 the 5ame ju5t the 1am6da funct10n and the avera9e 5amp1e num6er5 are
1004
A. R ~ ~d 1. L~ ~t~:
1am6d~ ca 1
1
1
1
1
1
1
1
2~
1
-2
X:
2
fX:
~:
0
~ata:2048
2 ~tep:
1a~6d~ 15d ~
~
~
~
~
•
•
~
2~
•
-2
X:
1
fX :
1
~:
0
1~ata : 2 0 4 8
~9ure 1: 7 h e 1am6da ~ n ~ n (~m6da.c~ ~ r the C-A te5t (upper 9raph) and ~ r the 0 5 5 7 (~m6da.15d, ~wer 9raph) chan91n9. 51nce the 0ne-~ded te5t 15 6a5ed 0n the c1a5~ca1 5PR7 w1th 5peda11y adju5ted 7ype 11 (0r m155ed a1arm) pr06a615ty, the 0n1y ta5k rema1ned t0 501ve ~ the determ1nat10n 0f th15 parameter f0r var1ance te5t1n9. H0wever, 51nce the pr061em 15 1ed 6ack t0 501ve an 1mp5dt e4uat10n f0r f1, we 0n1y have t0 1nve5t19ate, h0w th15 e4uat10n 15 affected 6y the type 0f the te5t (1.e. ~ the mean 0r the var1ance 15 t0 6e te5ted). C0mpar1n9 E45.(6) and (10) revea15 that the 1mp11dt e4uat10n 6a51ca11y 15 the 5ame f0r 60th ca5e5. 1n fact, 0n1y the f1-1ndependent part 15 d1fferent. F0r c1ar1ty, the e4uat10n5 are d15p1ayed here a9a1n: 1=
(0~ 2~a0-200)2(~ 1n(~--~) + (1 - a) 1n( ~1
a) )
1 = 2(4 ~ 1 - 1n4)-~ (a 1n(~--~) + (1 - a)1n(1~
f0r mean, ~))
f0r var1ance.
80th e4uat10n5 can 6e rewr1tten a5 E4.(8.3) 1n Append1x 8. Afterward5 the 9enera1501ut10n pre5ented 1n the Append1x can 6e app11ed f0r 60th ca5e5. 7heref0re there ~ n0 d1fference 6e~ween mean ~e~1n9 0r var1ance ~e~n9 pr061em5 an 10n9 a5 ~he dec~10n mak1n9 pr0cedure 1n the 0ne-n1ded ~e~t. F0r 111u5trat10n, a re5u1t 0f a c0mputer ~mu1at10n run 15 pre5ented 1n F19ure 2 . 7 h e var1ance 0f the n015e ~ 1ncrea5ed at the 1nterva1 [71,72] fr0m a~ t0 4a~ w1th a parameter 4 > 1. 7he actua1 va1ue5 are: 71 = 800, 72 = 1500 and 4 = 1.5. 7he 0ther va1ue5 are the 5ame a5 1n the prev10u5 examp1e. F19ure 2 5h0w5 that part 0f the 1am6da funct10n wh1ch c0nta1n5 the a6n0rma1 per10d. (7he ca1cu1ated m~5ed a1arm pr06a61f1ty 15 f1 = 0.95.) 1t can 6e 5een fr0m the f19ure, that the f1r5t detect10n 0f the a6n0rma11ty happened at 711r,t = 834. Add1t10na11y, when the 519na1 rec0vered (1.e. a9ter the tran~f10n 1nterva1 [800, 1500]), n0 further a1arm happened. 1n fact, the 1a5t a1arm wa5 at 7ta5t = 1506.
A 0ne-~d~ 5e4uent1a1te5t
1~5
50 we c0ndude, that the 0 ~ d te5t 15 a~e t0 detect (w1th m1n1mM t1me d d ~ and ~ t h 5mM1 fM5e Marm ~06a61Hty) 5mM1 chan9e5 1n t ~ var1ance5 ~ a 6au5f1an f19nM. $te~: 1
1
1
1
• X:
1
1
034
1
1
2~
1
~ata :20~8
F19ure 2: 7he 1am6da funct10n and the dedf10n 60undary f0r var1ance te5t1n9 pr061em. 7he a6n0rma11ty 0ccurred 1n the 1ntervM [800,150~ w1th a ma9n1tude 4 = 1.5.
4. 5 U M M A R Y 1n t~5 paper we pr0p0~d a hyp0the~5 ~ 5 t n 9 m ~ h 0 d t0 rec09n12e a6n0rmM 6ehav~ur w1th m1~mM t1me dday. 7he pr0cedure ~ 6a5ed 0n a 5pedM param~f12at10n 0f WMd~5 da5~cM 61nary 5PR7. ~ n c e WMd~5 0 r ~ n M w0rk dedde5 6etween tw0 hyp0th~e5, 1~ ~ rec0mmended t0 u5e m M ~ y ~ r 5uch pr06~m5 when ndther Mterna~ve 15 prefe~ed e ~ 0 r ~ . H0wever, 1n ca5e 0f ear1y fM~re chan9e d ~ e ~ n ta5k5 R ~ rea50na6~ t0 5upp05e t h ~ the 5y5tem under 1nve5t19at10n ~ 1n a n0rmM m0de. 7hu5 0ne 0f the 5y~em~ 5tate (1.e. the n0rmM 0 n ~ 15 m0re Hke1y then the an0ther (1.e. the a6n0rmM 0n~. 7heref0re the pr06~m can 6e reduced t0 rec09~2e 0~y a6n0rma11ty (1.e t~c ~ m 1~ ~0rma1 unt11 t~e ~6n0rma111~ 1~ pr0¢e~. Hav1n9 ~ m ~ d the a60ve f1mp~f1cat10n we pr0p05ed a way t0 5et the m~5ed Mama pr06a6~ty ~ 0f the 5e4uent1M te5t ~ 5uch a way that the ded5~n5 are c 0 n c e n ~ e d 0~y 0n ded~n9 ffhyp0the515 ~1 15 true. Unt~ ~ 15 dedded, the M ~ r n M ~ e hyp~he515 ~0 15 Va11d. (7~5 15 the rea50n Why We Ca11 the m ~ h 0 d a 0ne-51ded 5e4Uent1M te5t.) Hadn9 pararnetr12ed the 5 P R 7 1n the pr0p05ed Way 1t Wa5 p05f1~e t0 e11m1nMe the extra t1me de1ay 6etWeen the 0CCU~enCe 0f a6n0rmM1~ and R5 f1r5t de~Ct0n. 7he 0~y t1me dday ~ the ~ m e neC~5ary t0 9Mher en0U9h 1nf0rmat10n f0r the ded~0n mak1n9 W1th the pre5Cr16ed re11a6~ty. 51nCe the da5~CM 5 P R 7 15 Capa61e t0 te5t 60th mean and Var1anCe 0f a 6aU5f1an ~9nM, the 0n~f1ded t~t keep5 the 5ame a6~t~ (7here e~5t5 a m ~ h 0 d ~r t~t1n9 mean W1th0Ut t1me dday, name1y the C~en-Adam5 ~90r1thm. H0WeVer, ~ many praCtCM app11Cat10n5 the fM1Ure 0r chan9e d ~ e ~ n Can 6e d~Cf16ed a5 t~tn9 the Var1anCe rather than the mean Va1Ue.) Apart fr0m m1~m1f1n9 thC de~Ct10n t1me, the 0ne-51ded te5t ha5 an0ther advanta9e C0mpared t0 the da5f1CM 5 P R 7 m~h0d. 7he 1atter re4U1re5 t~0 U5er-def1ned parameter5, narndy the m15~d M a m a and fM5e M a r m pr06a6111t1e5.7he 0 5 5 7 need5 0 ~
1~6
A. Rf1~ and 1. Lux
0ne parameter, the fa15e a1arm pr06a61~ty, and the 0ther 0ne 15 ca1cu1ated dur1n9 the pr0cedure. Fr0m the v1ewp~nt 0f an expert, 1t 15 rather c0nven1ent t0 pref1x 0n1y 0ne 4uant~y than tw0, thcref0re the 0 n ~ d e d 5e4uent1a1 te5t need5 1e55 50ph15t1cated pre-c0n51derat10n5 than the 5PR7. A5 a c0n5e4uence, the 1n~rpreta~0n 0f the re5u~5 15 a~0 ea~1e~ 1n 5ummary, the 0 n ~ d e d te5t ~ rec0mmended t0 u5e whenever the da5~ca1 5 P R 7 15 appf1ca6~ t0 te5t 0n1y a6n0rma11ty. Furtherm0re, the 0 5 5 7 m1n1m~e5 the extra f1me detect1n9 a6n0rma1 6ehav10ur w1th0ut decrea51n9 the re11a6111ty 0f the c0nc1uded ded~0n.
A P P E N D 1 X A: 7 H E 8 E L 1 E F U P D A 7 1 N 6
PR0CEDURE
7he a190r1thm 15 capa61e t0 eva1uate the 0utput5 0f the da5~ca1 5 P R 7 meth0d 1n 5uch a way that the u5er w1H 6e perpetua11y 1nf0rmed a60ut the 1nve5t19ated 5y5tem•5 5tate. 1n 0rder t0 d0 1t, the 60undar1e5 wh1ch axe cr055ed dur1n9 the te5t are 5t0red 1n a 5er1e5 d = {..., A, A , . . . , 8 , A , . . . 8 , 8 , . . . } . 7h15 5er1e5 d repre5ent5 the h1~0ry 0f the 5y~em wh1ch 15 eva1uated w1th the h~p 0f the 8aye5 meth0d. 7he re5u~ 0f the eva1uat10n 15 the 6e5ef funct10n 8(7/1[d), 1 = 0,1 mea5ur1n9 the 5ke~h00d 0f that the 5y~em~5 5tate c0rre5p0nd5 t0 hyp0the5~ 7/1 c0nd1t10ned up0n the h1~0r1ca1 kn0w1ed9e repre5ented 6y the 5er1e5 d. 7he funct10n 15 re-ca1cu1ated at every new 60undary-cr05~n9 event5, 1.e. when a new ~em ~ em6edded 1nt0 the 5er1e5 d. 7he f0110w1n9 f0rmu1ae axe ~a60rated f0r 1 = 1, 1.e. when the te5ted 5tate c0rre5p0nd5 t0 the a6n0rma1 0ne. (H0wever, the re5u1t5 are a150 app5ca6~ f0r 1 = 0.) 7he actua1 f0rm 0f the 6e11ef funct10n 15 def1ned a5
8(7/1[d) ~ L(d[7/1)L(d[7/1)+ 1•
(A.1)
where the 5ke11h00d rat10 L(d[~H1) read5 a5
L(d17/~) ~ p(d17/0) P(d1~/1) .
(A.2)
7he 4uantRy P(d[7/1) 15 the c0nd1t10na1 pr06a6111ty 0f 06ta1n1n9 5er1e5 d f f the actua1 5tate 0f the 5y5tem c0rre5p0nd5 t0 the a6n0rma1 tn0de, 1.e. t0 hyp0the515 7/1. 51nce the e1ement5 0f 5e~e5 d are 1ndependent, P(d[7/1) can 6e w~tten a5 a pr0duct 0f the ~n91e event5 P(A]7"10), P(A[7/1 ), P(8[7/0) and P(8[7/1 ). 51nce the a60ve pr06a6111t1e5 are kn0wn, name1y P( 81~0 ) = 1 - ~
c0rrect dec1~0n a60ut 5tate 7/0,
P(A1~ ) = 1 - ~
c0rrect ded510n a60ut 5tate 7/1, fa15e a1arm, m~5ed a1arm,
P ( A1~0 ) = ~ P( 81~ ) = ~
(A.3)
A 0n~5Med ~4uent1f1 te5t
1007
E%(A.2) can 6e eva1uated a5
8(~H1~d): 1-- [1.-[-(~)N~(~-~)Nn] ~1, where the 5er1e5 d c0nta1n5 the 8 event N~ t1me5 and the A event f u n ~ n 8(~H~ 1d) ha5 the f01~w1n9 p r 0 p e ~ 5 :
(A.4)
NA t1me5. 7he 6e11ef
1) 1t 151ncrea51n9 when a new A dement 15 taken 1nt0 acc0unt. 2) 1t 151ncrea51n9 when a new 8 dement 15 taken 1nt0 acc0unt. 1f the 5y~em 151n a6n0rmM m0de and the 5er1e5 c0nta1n5 N 5ampM5 (1.e. N then 3) 7he 6e11ef f u n ~ n
= NA + N8)
w1H dedde 0n hyp0the515 •H1 f0r 5ure, 1.e. N~005m8(7~1]d) = 1.
(A.5)
4) A1th0u9h the 6e~ef funct10n 15 n0t 5en~t1ve t0 the 0rder 0f the ~ement50f d, the 5peed 0f 6 ~ f updat1n9 def1ned a5 ~(~H, 1d) ~ ~ N 8(~H~ 1d)
(A.6)
15 5en~t1ve t0 them. Nam~y ~(~H, 1d) ~ exp{
~(1w~ a ) N }
f0r 1ar9e N
and ~(~H11d) ~
a(1w•a) exp{a(1w•a)N}
f0r 5mM1N.
where w ~ a n 0 r m a 1 1 2 ~ n ~ 0 r . 1n ~ , ~ the 6 e # n M n 9 0 f the ~ r e n c e pr0cedure every new #ece 0f e~dence ha5 re1at1ve~ m0re we~ht (0r m0re u5efu1 c0ntr16ut10~ t0 the 0vera11 c0ndu~0n than 1ater, when the 6e11~ ~ n c t ~ n 5tart5 t0 appr0~m~e 1t5 ~ y m p t ~ va1ue. F0r m0re deta1155ee Re£ 4.
APPEND1X 8 We ~ e ~ 0 ~ n 9 ~ r 5~ut~n50f the f01~w1n9 e 4 u ~ n P - - (012~0)2 (a 1 n ( ~ - )
+ (1 - a) 1n(1~ ) )
•
(8.1)
1~8
A. R ~ 2 and 1. Lux
w h e ~ E4.(8.1) ~ve5 E4.~.1) ~ r p = 1 . 7 h e c 0 n ~ r ~ 5 a<1
and
~ m ~ n the 5ame, n a m e ~
~<1.
1t 15 5 ~ t a 6 ~ ~ a55ume, t h ~ we ~ e 100~n9 f0r 5~uf10n5 ~ r 50ft f~1~e5, 1.e. when a9 ~ (~ - ~ ~ t h 5m~1 dec1510n e~0r pm6a61hty, 1.e. a ~ 1. W1th the 5u65t1mt~n
p ~ A5N~
E4.~.1) ~ 5
0
= 1
and
k ~ (0~ 2~ 00) 2
(8.2)
-M~.
(8.3)
a5 (~-~)~f1~(1-~)~-~e
7he f19ht hand ~de 0f E4.(8~) 15 c0n5tant5 wh1ch ~ den0ted 6y C hencef0rth, 1.e
c ~ :(~ ~ ~)1-~:.~.
(~.4)
7hen E%(8.3) take5 the f0~0w1n9 f0rm:
(~-)"~
= ~
(8.5)
9
(8.6)
1ntr0dudn9 tw0 new vaf1a61~ 5 and 9 a5 def 1 - - ~ $
=
~
and
E4.(8.5) can 6e wr1tten a5 95 = C(1 - C9) -1.
(8.7)
51nce a << 1, the r00t 0f E4.(8.7) can 6e appr0x1mated a5
9 = 90(1 + ~),
(8.5)
where 90 ~ the 5d~10n ~ E % ( 8 ~ ) ~ r a = 0, 1.e. ~ = C and ~0m E%(8.6) (8.9) 5u6~1tut1n9 E4.(8.8) 1nt0 E4.(8.7) ~ d 5
C 9~(1%
e) 5 =
1 -
C90 - C ~ e ~
(8.1~
A ~d~
w ~ c h can 6e ~ r a n 9 e d
~
1009
m~
~ C
1
~(~ +~)~= 1-0~ 1 - ~
(~.11)
N0w E%(8.11) can 6e expanded 1nt0 7ay10r 5er1~ ~ 0 u n d e = 0 upt0 the f1r~ 0 r d ~
~(1+ 5~)-1~-C-C~ (1~ ~ e ) ,
(8.12)
w ~ c h can 6e 501ved ~ r e: e = C ~ (~-C(~1c~)~91 ~C~)-~59~. Hav1n9 06t~ned e, the 50u9ht m15~d ~ m
pr06a6~
(8.13)
~ can 6e e x p m ~ e d ~
~ = ~ - (1 + 9•+•)-•,
(~.1~)
App1y1n9 E%(8.6) ~ r the term 9~+~, the a60ve ~rmu1a take5 0n the ~ r m , = 1 - (1 + 0(1 -
+
51nce 1) the p ~ a m e t e r C 15 6 ~ up fr0m kn0wn va1ue5 ~0,~1,a~ and a and 2) e 15 expre55ed ~ a ~ n c ~ 0 n 0f the5e 4uant111e5, ~ 15 d ~ m ~ e d u ~ 4 u d ~ 06v10u51y the t ~ m ~ parenthe5~ ~ E4.(8.1~ ~ 9 r e ~ than 1 and thu5 f1 ~ ~ than 0ne, a5 expected. F u ~ h ~ m 0 m ~ ~ M50 06~0u5 t h ~ dnce a << 1, ~ 15 n e ~ the u ~ t ~ a5 1t can 6e 5een 1n the examp1e5.
ACKN0WLED6MEN75
7 h e w0rk w ~ p ~ t ~ 5upp0~ed 6y the 6rant 0 M F 8 95-97-47-0880,
REFERENCE5
1EEE 7ran~.Aut0ma~. C0n~Ac-21 0 ~ 0 6 e r 750-757 (1966) 519n~D ~ 0 n 7he0r~ Mc6raw-H1f1 800k C0m-
7.H. Ch1en and M.8 Adam5 (1976) J.C. Hanc0ck and P.A. w1nt2 pany, New Y0rk
7he FV1EW50~wa~ Packa9e,KFKLAEK1 App1. React0r A. R£c2 (1995) Anna~ Nuc~ Ener9y0n pre55) A. 1~c2 and 5. K155 (1995) Pr09. Nuc~ Ener9y 29~ N~3/4, 299-320
5. K155 (1991)
Phy~c5
C0~f1~t • 1996 H ~ v ~ 5c1en~ ~ d Pdnt~ ~ 6reat 8dta1n. ~1 d ~ re5erv~ 03~4~9/96 ~ ~ + ~
A 0NE-51DED 5EQUEN71AL 7E57
A. R6c2 and 1. L ~ KFK1-At0m1c En~9y Re5ear~ 1n5t1~ Appf1ed React0r P h y ~ La60r~0ry H-1525 8udape5t, E 0 . 8 0 x 49, H u n 9 ~ y e-m~1: r a c 2 ~ e r v J d ~ L h u ~m~1: 1ux~f~.k1k1.hu (Rece1ved 9 5eptem6er 1995)
A 6 ~ r a c t - 7he app~ca61~ty 0f the da5~ca1 5PR7 f0r ear1y f ~ h r e de~ct10n pr0~em5 ~ ~m1ted 6y the fact t h ~ there 15 an extra t1me de1ay 6etween the 0ccurrence 0f the f~1ure and ~5 f1r5t rec09a1t10n. C ~ e n and Adam5 deve10ped a meth0d t0 m1~m12e t~5 t1me ~ r the ca5e when the pr061em can 6e ~ r m ~ a t e d a5 te5t1n9 the mean va1ue 0f a 6au5~an ~9na1. 1n 0ur paper we pr0p0~ a pr0cedure t h ~ can 6e app~ed f0r 60th mean and var1ance te5t1n9 and that m1~m12e5 the t1me d~ay. 7he meth0d 15 6a5ed 0n a 5peda1 param~r12~10n 0f the da5~ca1 5PR7. 7he 0 n ~ d e d 5e4uent1~ te5t ( 0 5 5 7 ) can repr0duce the re5~t5 0f the C~en-Adam5 te5t when app~ed f0r mean va1ue5. C0pYr19ht • 1 ~ 6 ~ f 1 ~
~n~
L~
1. 1 N 7 R 0 D U C 7 1 0 N 7 h e 5e4uent1a1 pr06a61~ty rat10 te5t1n9 (5PR7) meth0d 15 a we11-kn0wn and w1de1y u5ed t001 f0r hyp0the515 te5t1n9 pr061em5, when the ta5k 15 t0 dec1de 6etween n0rm~1 and ~6~0rm~1 (0r de9enerate) m0de 0f the 1nve5t19ated 5t0cha5t1c 5y5tem. 7he app11ca6111ty 0f the 5PR7 re4u1re5 that the d1fference 6etween the a1ternat1ve m0de5 can 6e fu11y de5cd6ed
997
998
A. Rhc2 and 1. Lux
6y a ~ n ~ e parameter 0f a rand0m pr0ce55. Pr0ce55e5 w1th 5uch pr0pert1e5 are ca11ed re~du~5.1n 5uch a ca5e, when 1) the r e ~ d u ~ 15 a wh1te 6an5~an pr0ce55 w1th 2er0 mean and kn0wn var1ance r(k) E A/~{0,a 2} (k 15 the ~ 5 c r e ~ t1me) and 2) the n 0 r m ~ / a 6 n 0 r m ~ ~an51ent can 6e ~pre5ented, dther a5 a 5h1~ 1n the mean m 0r a ma9n1f1cat~n 0f the var1ance ~, and 3) the 0ther parameter 0f the r e ~ d u ~ rem~n5 the 5ame then the 5 P R 7 ~ a we11-def1ned M90r1thm (Hanc0ck and W1nt2, 1966) t0 dedde 6etween the M~rnat1v~5. 1n 5p1~ 0f K5 ~mphdty, the 5 P R 7 5ucce55fu1~ dcte~5 50ft chan9e5. H0wever, the m ~ h 0 d ha5 a ~ w 5h0rtc0m1n95. 1n fa~, when ~eat1n9 re~1y 50K chan9e5, very ~n9 5amp1e ~2e5 have t0 6e pr0ce55ed 1n 0rder t0 ach~ve a re11a6~ c0ndu~0n, 0r, ff the ~ m 15 a 4u1ck (pr0mpt) an5wer, then the a pr06a6111ty 0f wr0n9 ded~0n 15 1ar9e. F 0 ~ u n a t d y a n ~ y t ~ M f0rmu1~ e~5t (the avera9e 5amp1e num6er 0r A5N funct10n) t0 c0nnect the 5amp~ ~2e5 t0 the ded5~n err0r pr06a6~t~5.7hu5 K can 6e dedded 1n advance whether the pre5cr16ed re11a6~ty and the re4u1red 5amp~ ~2e5 are 60th accepta61e. 50me attempt5 have 6een made (and 5ucceeded, 1n fa~) t0 5ave the re11a6111ty 0f the te5t and t0 reduce the num6er 0f the pr0ce~ed 5amp1e5 at the 5ame t1me. 0 n e 0f them ~ W~d~5 truncated 5 P R 7 (Hanc0ck and W1nt2, 1966), wh1ch cu~ 0ff the te5t after p r 0 c e ~ 9 a ~ven num6er 0f 5amp1e5. 7he 0ther appr0ach can 6e c~1ed a5 ~he pre5umpf10n 0f 1nn0cence••. 1n fa~, 1n ear1y f~1ure 0r chan9e detect10n we are m ~ n ~ 1ntere5ted 1n the rec09n1t10n 0f the de9radat~n (~ happen5) 6ut we d0 n0t want t0 6e 60thered 6y a p e r p ~ u ~ c0nf1rmat~n that everyth1n9 ~ 0.K. Keep1n9 th15 re4~rement 1n m1nd, the 5 0 ~ 1 e d m0d1f1ed 5e4uent1M te5t ha5 6een pr0p05ed 6y Ch1en and Adam5, wh1ch detect5 0n~ a6n0rma11t~ and m1~m12e5 the d ~ e ~ n t1me (Ch1en and Adam5, 1976). A n ~ y t ~ 5dut~n5 f0r the m0~f1ed (0r Ch1en-Adam~ ~ e ~ 0n1y t0 the ca5e when the parameter t0 6e t e ~ e d 15 the mean 0f the r ~ u ~ . (Hencef0~h the m0~f1ed 5PR7, Ch1en-Adam5 0r C-A te5t refer t0 the 5ame pr0cedure.) H0wever, 1n many apphcat~n5 the d e 9 r a d ~ n can 6e m0de11ed a5 an 1n~ement 0f the var1ance 0f the re~du~. 7h15 happen5 when the r e ~ d u ~ 15 9enerated 6y an AR m0dd (R~c2 and K155, 1995). An0ther 5h0rtc0m~9 0f the m0~f1ed te5t 15 that the num6er 0f 5amp~5 nece55ary t0 make a ded~0n 15 a rand0m var1a6~, thu5 ~ can 6e very 1ar9e 1n cert~n ca5e5 (6ut 1t5 expected va1ue can 6e c~cu1ated 1n advance). 7h1rd, the pr0cedure d0e5 n0t a v e any h1nt a60ut the 5y5~m~5 5tate 6etween ~w0 ded~0n5. F u ~ h ~ m 0 r e the detecf10n t1me can 0n1y 6e decrea5ed 0n the c05t 0f 1ncrea51n9 the ~e4uency 0f wr0n9 ded5~n5. 1n 0ur paper we try t0 5dve 50me 0f the a60ve pr06~m5 v1a rec0n~der1n9 the 0r1~n~ 5 P R 7 6ut n0t f0r9~t1n9 the 6a51c m0f1vat~n 0f the Ch1en-Adam5 te5t, n a m d y that the de9raded m0de 15 the 1ntere5~n9 0ne. F1r5t we w1~ 5h0w h0w t0 5et the ded~0n err0r pr06a6111t~5 0f the 5 P R 7 1n 0rder t0 ach~ve the 5ame 0utput a5 the Ch1en-Adam5 te5t. We w1H cM1 the pr0¢edure a 0ne-~ded 8e4uent1~ 7e5t ( 0 5 5 7 ) 51nce, ~m1~f1y t0 the C-A te5t, re11a6~ ded~0n5 are made 0n~ a60ut the a6n0rm~ m0de 0f the 5y~em. 51nce 0ur pr0p0~t10n 15 n0t re5tr1cted t0 the ca5e 0f 1nve5t19at1n9 the mean va1ue5 0f the re~du~, the pr0cedure 15 ~ r ~ 9 h t ~ r w a r d ~ extended t0ward5 the te5t 0f var1ance5. 51nce there 15 a meth0d (R~c2, 1995) 0n h0w t0 6e perp~ua11y 1nf0rmed a60ut the 5tate 0f the 5y5~m
A ~
~e~f1
te5t
999
under 1nve5t19at10n ~ the hyp0the515 te5t1n9 m ~ h 0 d 15 the 5PR7, th15 a190r1thm can 6e ea51~ app11ed f0r the recent pr061em. 1n 5ummary, the 0n~f1ded 5e4uent1a1 te5t ut1112e5 the ad~anta9~ 60th 0f the da5f1ca1 and the m0~f1ed te5t5 w1~0ut ~he1r draw6aek~. 1n fact, the 0 5 5 7 1) m1~m1~5 t~n, 2) ~ c ~ e 3) ~ u ~ y ca5e 0f 5 ~
the detect10n t1me ~ a 6~n9 re5tr1cted 0 ~ y t0 t ~ ~ c 0 9 ~ t ~ n ~ d e ~ a d v 0n t e ~ n 9 d t h e r the mean ~ u e 0r the ~ 1 a n ~ 0f the M9na1 pr~d~ 1 ~ ~ (6~ ~d~) ~ 0 u t the ~ t ~ 5 5t~e e ~ n h ~h~
7he 0r9an12at10n 0f the paper 15 the f0110w1n9. 5ect10n 2 5ummar12e5 the 5PR7 and the Ch1en-Adam5 te5t. 5ec~0n 3 91ve5 0ur pr0p0~t10n h0w t0 ca1cu1ate the pr0per parameter5 0f the 5PR7 1n 0rder t0 5ave the advanta9e5 0f the C-A te5t and app1y 1t f0r the ca5e 0f var1ance te5t5.
2.7HE
5PR7 AND 7HE CH1EN-ADAM5 7E57
1n t~5 5ect10n we 0verv1ew the pr0cedure5 t0 6e u5ed ~ the 5e4u~.1n ca5e 0f hyp0the515 t e ~ 9 the ta5k ~ t0 te5t a nu11hyp0the515 ~0 a9Mn~ the a ~ e r n ~ e hyp0the~5 ~ a60ut an unkn0wn mean va1ue m 0f a n0rma1 6 a u ~ n p ~ ¢ e A/•(m, a~). 7 h e var1ance a~ 15 5upp05ed t0 6e kn0wn and c0n5tant. We have the f0110w1n9 tw0 hyp0the5e5 f0r rn: ~ 0 : m = ~0~
~1
: m = ~1,
(1)
where ~0 and ~1 are c0n5tant5. We a55ume that dur1n9 the hyp0the5e5 te5t1n9 pr0cedure the pr0pert1e5 0f the ~0urce, wh1ch 9enerate5 the ~9na15 ~ 5 , d0 n0t chan9~ 7 h e pr06a61~f1e5 0f err0r5 cau5ed 6y an 1nc0rrect ded~0n are the f0H0w1n9. 7ype 1 err0r (0r the 50-ca11ed fa15e a1arm) 0ccur5 when 0ne accept5 7~1 6ut •/0 ~ true. 1t5 pr06a61f1ty ~ den0ted 6y a. 51mf1ar1~ accept1n9 7~0 when 7~1 15 true cau5e5 7ype 11 err0r (0r m155ed a1arm). 1t5 pr06a6111ty 15 den0ted 6y ~. A5 1t wa5 ment10ned 1n the 1ntr0duct10n, 1t ~ a150 p05~61e t0 1nve5t19ate 0n1~ that the a1ternat1ve hyp0the515 15 5at15f1ed (m0d1f1ed 5PR7). 1n 5uch a ca5e the c0ncept 0f m~5ed a1arm 15 n0t app11ca61e (Ch1en and Adam5, 1976). 2.1 5e4uent1a1 pr06a61f1ty rat10 te5t (5PR7) 7 h e ta5k 15 the f0H0w1n~ Hav1n9 rec~ved the 519na15 {•, ~2,..., ~k,...} ~ 15 t0 6e dec1ded wh1ch hyp0the515 (7-/0 0r :H1) 15 true. 7he a190r1thm ~ a5 f0110w5: Fr0m the f1xed de51red err0r pr06a6111t1e5 (a a~d f1) the f0110w1n9 tw0 thre5h01d va1ue5 ( dec~10n 60unda~e5) A and 8 have t0 6e ca1cu1ated:
1000
A. R~c2 and 1. Lux
Let u5 c~cu1ate the 109-hke11h00drat~ At, a5
where P5,¢2,...,¢~1~f0r 1 = 0, 1 repre5ent the j~nt pr06a61hty denf1ty funct10n 0f the 065ervat10n 0 up t0 (,, c0nd1t10ned up0n 7~1.7hen the dec610n ma~n# pr0cedure 15 the f0110w1n9.1n the k th 5ta9e At mu5t 6e c0mpared t0 A and 8 and ~ Ak ~ A
then accept 7~1 and 5et Ak = 0,
A~ ~ 8 8
then accept ~0 and 5et A, = 0, then take an0ther 5amp1e.
(4)
A5 6n9 a5 m 15 the mean v~ue 0f a wh1te, 6au5f1an pr0ce55 ( ~ A:(m,~), the 69~ke~h00d rat~ can 6e cMcu1ated ~c~e1p:
7he a~era#e aamp~ num6er (A5N f u n ~ n 5 ) nece55ary t0 m~ke 9 dcc1a10nread5
A5N~0=
(0, 2d~00)~ (a 1 n ( ~ - )
+ (1 - a) 1n(1 ~
a)).
7he n0tat~n A 5 N ~ 0 15 u5ed when the mean take5 the va1ue m = 00 wh1~ A 5 N ~ t0 m = 0 1 .
(6) refer5
7he 5 P R 7 pr0cedure ~ M50 app11ca61e when the var1ance 15 te5ted 1n~ead 0f the mean va1ue 0f the f19nM. 1.e.the hyp0thef15 te5t1n9ta5k 15 t0 dedde 6etween the hyp0the5e5 ~0 and 7~ a60ut an unkn0wn var1ance c 0f a n0rm~ 6a~5~an pr0ceaa ( ~ .A/•(m,c~). 7he mean va~ue 0f the f19nM 15 5upp05ed t0 6e 2er0 m = 0. (Hav1n9 a n0n-2er0 6ut c0n5tant mean va1ue can 6e ~eated f1m11ar1y.H0wever, the f0rmu1ae are f1mp1er 1n ca5e 0f 2er0 mean.) 7hu5 the f0~0w1n9 hyp0the5e5 are u5ed: ~0 : ~2 = ~ ,
7~ : c 2 = 4c~,
where
4 > 1.
(8)
7he dedf10n mak1n9 pr0cedure (4) rem~n5 the 5ame 6ut the f0rmu1ae 1n E45.(5-7) chan9e a5 f0110w5: A1+, = Ak + (~+~2~94 4- 1
~ 1n(4),
(9)
.5~0 =~(~-~-~ ~ ~0~)-~(~.(~-~) + <~- 0~1.(~~ ~),
(~0~
,~,
(~1~
= ~(,- ~- ~)-~((~- ~1~(~-~ ~ ~,~(~~ ~).
A 0n~f1ded ~ 4 ~
te5t
1~1
7he m ~ n 5h0rtc0m1n9 0f the 5PR7 15 t h ~ the nece~ary 5amp~ ~2e5 c0~d 6e very 1~9e when the ded~0n err0r pr06a6111t1e5 are 5mM1. 1n ad~f10n, the f1r5t ~ c ~ n 0f a6n0rmM1~ ( u ~ 5 1t happen5 at the very 6 e ~ n ~ n ~ need5 50me extra t1me, ~nce the 1am6da ~nct10n 15 pr06a61y ~ the ~terva1 [ 8 , . . . , ~. 7~5 5h0rtc0m1n9 can 6e 0verc0me ~ we 0~y 1ntend t0 r e c 0 9 ~ a6n0rma~y ~ 5u99e~ed 6y C~en and Adam5 ~ thek m0d~ed ~4uent1~ te5t. We menf10n 6y p ~ n 9 t h ~ there 15 a n0ve1 m~h0d t0 1nterpret the 5PR7 0 u ~ 0 m ~ ~ ~ d u M e~denc~ a60ut the te5ted 5y5tem. 5~ce the meth0d (w~ch 15 ca11ed ~ u~da~9 pr0cedure) 15 0ut 0f the 5c0pe ~ the recent w0rk, 1t5 m ~ n 1dea 5ummar1~d ~ the Appen~x ~av1n9 the det~15 t0 the f 1 ~ r ~ u ~ (P~c2, 1995). 2.2 M0~f1ed 5PR7 m~h0d A5 1t wa5 ment~ned a60ve, the m £ n 5h0~c0m1n9 ~ the 0r1~n~ 5PR7 pr0cedure e m ~ 9 ~ ~0m the fact, t h ~ the 5PR7 w ~ 1ntr0du~d ~ r de4~n9 6etween tw0 (~m05t e4uM1y pr06a61e) hyp~he~5 (n0rmM 0r a6n0rm~) r ~ h e r than t0 rec09n12e a6n0rm~1ty. 1n ~ct, u~n9 any chan9e de~ct10n 5y~em, u5u~1y ~ 15 ~ 0 n a 6 ~ t0 ~5ume t h ~ the 5y5tem under 1nve5t19at10n ~ r ~ h e r ~ n0rmM than 1n a6n0rm~ ~ e . 7 h ~ e f 0 ~ the m ~ n t ~ k can 6e reduced t0 ~c09n12e ( ~ 500n and ~ ~ a 6 ~ ~ p05~61e) a6n0rmM1ty. 1n 5uch a c~e, 6y def1n1t10n, the 5y~em 15 n0rm~ unt~ R 15 d e d ~ e d t0 6e a6n0rm~. 1t ~ n0t a taut0109Y ~ R can 6e 5een ~0m the fact t h ~ the 5PR7 d0e5 n0t 5t~e anyt~n9 a60ut the 5y~em~5 5t~e ~ ~n9 ~ 8 < A~ < A ~ee. E%~) ~ r the ded~0n mak1n9 r~e5]. 1.e., the 5y~em~5 5tate ~ unde~rm1ned ~ r u n c e ~ n , n0t def1ned) except the t1me p~nt5 when 60undary ~ n 9 event5 happen. C~en and Adam5 pr0p05ed a ~fferent te5t w~ch ~ c 0 9 ~ 5 0~y a6n0rm~1~ ~ a 1nve5t19at1n9 the mean va1ue m ~ a 6au551an ~9n~ w1th kn0wn var1an~ a 9 . 7 h e pr0~dum 90e5 ~ f0110w5. 7he ~f1ke~h00d r ~ mu5t 6e c M c ~ e d ~ the 5ame way ~ ~ c~e 0f 5PR7. A ~ w ~ d 5 the 5 5 9 ~ m0~f1ed ded5~n mak1n9 p r 0 ~ d u ~ read5 A~ ~ 0
then 5et Ak = 0,
A~ ~ AcA 0
(7he 5u65c~pt cA ~ c ~
then ac~pt ~
and 5et A~ = 0,
(12)
then take an~her 5am~e.
the Ch1emAdam5 te5t). 7he 60und~y AcA can 6e c ~ c ~ e d
where 7 15 the mean t1me 6etween tw0 c0n5ecut1ve fM5e M~m5.7he rda~0n 6etween 7 and the fM5e M~m pr06a6111ty a 15 ~ven 6y the f0110w1n9 e4u~10n: ~ = 7 -~. 7he f1r5t ~me 0f d ~ e ~ 9 take5 the f01~w1n9 va~e:
(14)
a6n0rma11~ (the 50-cM~d perf0rmance 1ndex, den0ted 6y v)
, =
1002
A . Rf1c2 and 1. Lux
F0r m0re deta115 a60ut the C-A te5t 5ee the 0r191na1 art1e1e 0f Ch1en and Adam5 (1976).
3. 7 H E 0 N ~ 5 1 D E D
7E57
C0mpar1n9 the ~ n a r y 5 P R 7 t0 1t5 m0~f1ed ver5~n the f0110w1n9 6a1ance can 6e drawn: Advanta9e5 0f the m~h0d5 - 7he 5 P R 7 15 app11ca61e f0r te5t1n9 ~ther the mean 0r the var1ance 0f the ~9na1. - A n0ve1 6e11ef-updat1n9 pr0cedure can 6e app11ed 0n the 5PR7•5 0utc0me5 t0 perpe~ ua11y m 0 ~ t 0 r ~he 5y~em~5 ~ate. - 7he C-A te5t m1~m12e5 the t1me dday 6etween the ~an51t~n event 1ntended t0 rec09n12e and 1t5 f1r5t rec09n1t10n. D~w~5
~ t~ m ~
- 7 h e C-A ~5t 15 n0t app11ca6~ ~ r ~ 9 var1ance5. - 5~ the C-A t ~ t ~ c ~ 0 ~ y a6n0rma11ty, the 5ame 6 e 1 1 e h u p d ~ 9 p r 0 ~ d u m ~ n0t app11ca61e ~ r 1t5 0utput5. C0n~def1n9 the a60ve f15t5 ~ 15 ea5y t0 art1cu1ate 0ur w15h; t0 perf0rm a te5t wh1ch p055e55e5 0n1y the advanta9e5 0f the 5 P R 7 and C-A te5t5 w1th0ut th~r draw6ack5. 3.1 P r 0 p 0 ~ t ~ n 7he 6a51c 1dea 6 e ~ n d the C-A te~ 15 n0t t0 te5t the n0rma1 m0de 6ut c 0 n ~ n t r ~ 9 0 ~ y 0n the ~ ~ % ff ~ 1n 0ther w0rd5, 1t 15 a55umed that a5 ~ n 9 a5 the 5y~em ~ n0t j u ~ e d t0 6e ~ a 1 1t mu5t 6e n~ma1. 5~ce 1n ca5e 0f ~ r m ~ the A~ ~n~n ~ ~ea5~9 ~0m ~) then n0 ~ m ~ ~d h ~ p e n ff ~ e A~ ~ n ~ n decrea5e5, ~ ~ndenc~ ~ ~ ff A1 < 0 (w~ch ~ e ~ 15 ~ e ~ t0 a d ~ a 5 1 n 9 ~m6dv~n~n, Mnce the ~t1a1 ~ u e ~ r A~ 15 2er0), then 1t5 ~ u e 15 re5et t0 0. ~ ad0pt C ~ e n ~ ~ d A d ~ 5 ~ d 0 v e 0f p ~ ~ the da5~ca1 5 P ~ 1n 5 u ~ a w ~ ~h~ 1t 5ha11 ~ ~ 6enef1t5 60th 0f the C-A te5t ~ d ¢he 5 P ~ . 0 u r pr0p0~f10n ~ t0 p~ t~ ~r~e 5 ~ ~ A5~0 ~ ~6~ ~ ~ ~ . ~ ~, A5~0 = 1 15 ~ m m ~ d ~ ~ e . A5 a ~ e ~ , w~n ~ p ~ ~ 15 ~ 1 d t ~ 5 P ~ ~ 0n ~0 ~ every 5tep (1n ~ Q , ~ d 6y c 0 n ~ r u ~ n , the A~ ~ n ~ n 15 re5et t0 2er0. 7h~e~ t~5 ~ 0 ~ d u ~ ~ ~ e ~ ~ t~ m0~d 5P~ m~h0d ~ ~ C~ ~d Ad~. N0w, there ~ m ~ n e d 0ne m0re p ~ r t0 6e 5et 6y the u5er ~ d t~5 15 the ~ 5 e a1~m ~ f 1 ~ a, 0r, ~ e ~ the m e ~ t1me 7 6~ween tw0 ~ 1 ~ ~ e a1~m5 1n the C-A te5~. H 0 ~ r , the 5 P ~ need5 60~h d e ~ 0 n err0r pr06a~f1t1~ 1n 0rder t0 d~m~e the de6~0n ~ d ~ A and 8. H ~ n 9 f1xed the a ~ u e , E4.(6) f1dd5 the
A 0n~f1ded 5e4uenf1f1 te~
1003
de~red m ~ d a1~m pr06a61~y f1, ~nce 1t5 ~K hand ~de 15 kn0wn, t0 6e 1. 7hu5 the 1mp11dt e 4 u ~ n have ~ 6e ~1ved ~ r f1 15 the f01~w1n9:
(0
-.)
(16)
A~h0u9h E%(16) can a1way5 6e 5~ved 6y numer1ca1 m~h0d5, 1n Append~ 8 the reader w1~ f1nd a n a ~ c 5 0 ~ 0 n 5 ~ r 1t. 3.2 An e x a m ~ e 8e~re pr0~e~n9 ~h~ an e x ~ # e 15 ~ ~ d ~ t0 5 h ~ h0w the m ~ h 0 d w0rk5 and h0w 1t 15 c0nnected t0 the C ~ m ~ m 5 te5t. A c0mpu~r ~ m ~ n w ~ p ~ r m e d t0 9enerate a ~9na1 ~ t h an ~ r u p t chan9e ~ 1t5 mean ~ u e ~ ~ . 7he actua1 ~1ue5 ~ e the ~ ~ . 7he ~ r e 0ccurred at ~ = 800. 7 h e mean t1me 6~ween tw0 ~ 5 e a1~m 15 7 = 500 (0r a = 0.002). ~=1. ~0 = 0. 01 =
0.4.
7 h e ca1cu1ated m ~ d M ~ m ~ ~ ~ r the ~ 0 v e p ~ ~ 15 ~ = 0.91. N ~ 1t 15 p~e t0 ~ p ~ 60th the C-A and the 0 n ~ d e d te5t ~ r the ~ t ~ a 1 ~9na1. ~ 9 u r e 1 5h0w5 the 1 ~ 6 d a ~ n ~ ~ r 60th te~5. 7he de~5~n ~ d ~ (n~ ~ d her~ h ~ the ~ 0 ~ n 9 ~ u e 5 ; AcA = 3.68 ~ r the C-A te5t, wM~ A = 3.76 ~ r the 0 5 5 7 . 7he 1 ~ 6 d a ~ n ~ ~ e a1m~t the 5ame, ~ 1t w ~ ~ p e ~ e ~ (7he ~ 9 ~ ~ n ~ 5 ~e cau5ed 6y the m ~ 0 r ~ ~ 6etween the actua1 va1ue5 ~ the d e ~ 0 n ~ d ~ . ) ~r 6ett~ ~ e ~ n t~ 1~da ~t0~ ~ e t r a n 5 ~ r m ~ 1~0 t ~ 5 ~ e ~ t ~ ~1 [-1024, 102~ and ~ p ~ e d a5 ~ e 9 e r ~ . 7he ~ c a 1 ~ p ~ ~ were p ~ r m e d 6y a p m ~ a m p a ~ 9 e F ~ W (~55, 1991). 7he f1r~ detect10n 0f ~ r m ~ happened at ~ 1 ~ = 829. 5~ce the ~ r m ~ wa5 1ntr0duced at ~ = 800, the t1me 1~5e 15 2 9 . 7 h e ana1yt~a1 ~ e ~n 6y E4.(15) 15 v = 27, ~ t h ~ e 15 a 900d a ~ m e ~ 6~ween the ana1yt1ca1 and ~ m ~ c a 1 5~ut0n5. 3.3 ~
~ ~
Certa1n1y t0 repr0duce the 0utput 0f the Ch1en-Adam5 te5t 1n a d1fferent way can a150 6e re9arded a5 a re5u1t. H0wever, 0ur a1m 15 t0 90 further. 1n fact, the C-A te5t 1ack5 the a6111ty t0 te5t va~ance5. H0wever, 1n many app11cat10n5 (e.9. UAR-9enerated re~dua1~ the va~ance 5h0u1d 6e te5ted 1n5tead 0f the mean va1ue. Enc0ura9ed 6y the pre~m1nary re5u1t5 0f the 0ne-51ded pr0cedure f0r te5t1n9 mean va1ue5, we app1y the 0ne-51ded te5t f0r te5t1n9 va~ance5, t00. F0r 5uch a ca5e the pr061em 15 f0rmu1ated 6y E%(8). P~ndpa11y there 15 n0 d1fference 6etween the ta5k 0f te5t1n9 mean 0r var1ance5 v1a da5~ca1 5 P R 7 meth0d. 7he a190~thm 15 the 5ame ju5t the 1am6da funct10n and the avera9e 5amp1e num6er5 are
1004
A. R ~ ~d 1. L~ ~t~:
1am6d~ ca 1
1
1
1
1
1
1
1
2~
1
-2
X:
2
fX:
~:
0
~ata:2048
2 ~tep:
1a~6d~ 15d ~
~
~
~
~
•
•
~
2~
•
-2
X:
1
fX :
1
~:
0
1~ata : 2 0 4 8
~9ure 1: 7 h e 1am6da ~ n ~ n (~m6da.c~ ~ r the C-A te5t (upper 9raph) and ~ r the 0 5 5 7 (~m6da.15d, ~wer 9raph) chan91n9. 51nce the 0ne-~ded te5t 15 6a5ed 0n the c1a5~ca1 5PR7 w1th 5peda11y adju5ted 7ype 11 (0r m155ed a1arm) pr06a615ty, the 0n1y ta5k rema1ned t0 501ve ~ the determ1nat10n 0f th15 parameter f0r var1ance te5t1n9. H0wever, 51nce the pr061em 15 1ed 6ack t0 501ve an 1mp5dt e4uat10n f0r f1, we 0n1y have t0 1nve5t19ate, h0w th15 e4uat10n 15 affected 6y the type 0f the te5t (1.e. ~ the mean 0r the var1ance 15 t0 6e te5ted). C0mpar1n9 E45.(6) and (10) revea15 that the 1mp11dt e4uat10n 6a51ca11y 15 the 5ame f0r 60th ca5e5. 1n fact, 0n1y the f1-1ndependent part 15 d1fferent. F0r c1ar1ty, the e4uat10n5 are d15p1ayed here a9a1n: 1=
(0~ 2~a0-200)2(~ 1n(~--~) + (1 - a) 1n( ~1
a) )
1 = 2(4 ~ 1 - 1n4)-~ (a 1n(~--~) + (1 - a)1n(1~
f0r mean, ~))
f0r var1ance.
80th e4uat10n5 can 6e rewr1tten a5 E4.(8.3) 1n Append1x 8. Afterward5 the 9enera1501ut10n pre5ented 1n the Append1x can 6e app11ed f0r 60th ca5e5. 7heref0re there ~ n0 d1fference 6e~ween mean ~e~1n9 0r var1ance ~e~n9 pr061em5 an 10n9 a5 ~he dec~10n mak1n9 pr0cedure 1n the 0ne-n1ded ~e~t. F0r 111u5trat10n, a re5u1t 0f a c0mputer ~mu1at10n run 15 pre5ented 1n F19ure 2 . 7 h e var1ance 0f the n015e ~ 1ncrea5ed at the 1nterva1 [71,72] fr0m a~ t0 4a~ w1th a parameter 4 > 1. 7he actua1 va1ue5 are: 71 = 800, 72 = 1500 and 4 = 1.5. 7he 0ther va1ue5 are the 5ame a5 1n the prev10u5 examp1e. F19ure 2 5h0w5 that part 0f the 1am6da funct10n wh1ch c0nta1n5 the a6n0rma1 per10d. (7he ca1cu1ated m~5ed a1arm pr06a61f1ty 15 f1 = 0.95.) 1t can 6e 5een fr0m the f19ure, that the f1r5t detect10n 0f the a6n0rma11ty happened at 711r,t = 834. Add1t10na11y, when the 519na1 rec0vered (1.e. a9ter the tran~f10n 1nterva1 [800, 1500]), n0 further a1arm happened. 1n fact, the 1a5t a1arm wa5 at 7ta5t = 1506.
A 0ne-~d~ 5e4uent1a1te5t
1~5
50 we c0ndude, that the 0 ~ d te5t 15 a~e t0 detect (w1th m1n1mM t1me d d ~ and ~ t h 5mM1 fM5e Marm ~06a61Hty) 5mM1 chan9e5 1n t ~ var1ance5 ~ a 6au5f1an f19nM. $te~: 1
1
1
1
• X:
1
1
034
1
1
2~
1
~ata :20~8
F19ure 2: 7he 1am6da funct10n and the dedf10n 60undary f0r var1ance te5t1n9 pr061em. 7he a6n0rma11ty 0ccurred 1n the 1ntervM [800,150~ w1th a ma9n1tude 4 = 1.5.
4. 5 U M M A R Y 1n t~5 paper we pr0p0~d a hyp0the~5 ~ 5 t n 9 m ~ h 0 d t0 rec09n12e a6n0rmM 6ehav~ur w1th m1~mM t1me dday. 7he pr0cedure ~ 6a5ed 0n a 5pedM param~f12at10n 0f WMd~5 da5~cM 61nary 5PR7. ~ n c e WMd~5 0 r ~ n M w0rk dedde5 6etween tw0 hyp0th~e5, 1~ ~ rec0mmended t0 u5e m M ~ y ~ r 5uch pr06~m5 when ndther Mterna~ve 15 prefe~ed e ~ 0 r ~ . H0wever, 1n ca5e 0f ear1y fM~re chan9e d ~ e ~ n ta5k5 R ~ rea50na6~ t0 5upp05e t h ~ the 5y5tem under 1nve5t19at10n ~ 1n a n0rmM m0de. 7hu5 0ne 0f the 5y~em~ 5tate (1.e. the n0rmM 0 n ~ 15 m0re Hke1y then the an0ther (1.e. the a6n0rmM 0n~. 7heref0re the pr06~m can 6e reduced t0 rec09~2e 0~y a6n0rma11ty (1.e t~c ~ m 1~ ~0rma1 unt11 t~e ~6n0rma111~ 1~ pr0¢e~. Hav1n9 ~ m ~ d the a60ve f1mp~f1cat10n we pr0p05ed a way t0 5et the m~5ed Mama pr06a6~ty ~ 0f the 5e4uent1M te5t ~ 5uch a way that the ded5~n5 are c 0 n c e n ~ e d 0~y 0n ded~n9 ffhyp0the515 ~1 15 true. Unt~ ~ 15 dedded, the M ~ r n M ~ e hyp~he515 ~0 15 Va11d. (7~5 15 the rea50n Why We Ca11 the m ~ h 0 d a 0ne-51ded 5e4Uent1M te5t.) Hadn9 pararnetr12ed the 5 P R 7 1n the pr0p05ed Way 1t Wa5 p05f1~e t0 e11m1nMe the extra t1me de1ay 6etWeen the 0CCU~enCe 0f a6n0rmM1~ and R5 f1r5t de~Ct0n. 7he 0~y t1me dday ~ the ~ m e neC~5ary t0 9Mher en0U9h 1nf0rmat10n f0r the ded~0n mak1n9 W1th the pre5Cr16ed re11a6~ty. 51nCe the da5~CM 5 P R 7 15 Capa61e t0 te5t 60th mean and Var1anCe 0f a 6aU5f1an ~9nM, the 0n~f1ded t~t keep5 the 5ame a6~t~ (7here e~5t5 a m ~ h 0 d ~r t~t1n9 mean W1th0Ut t1me dday, name1y the C~en-Adam5 ~90r1thm. H0WeVer, ~ many praCtCM app11Cat10n5 the fM1Ure 0r chan9e d ~ e ~ n Can 6e d~Cf16ed a5 t~tn9 the Var1anCe rather than the mean Va1Ue.) Apart fr0m m1~m1f1n9 thC de~Ct10n t1me, the 0ne-51ded te5t ha5 an0ther advanta9e C0mpared t0 the da5f1CM 5 P R 7 m~h0d. 7he 1atter re4U1re5 t~0 U5er-def1ned parameter5, narndy the m15~d M a m a and fM5e M a r m pr06a6111t1e5.7he 0 5 5 7 need5 0 ~
1~6
A. Rf1~ and 1. Lux
0ne parameter, the fa15e a1arm pr06a61~ty, and the 0ther 0ne 15 ca1cu1ated dur1n9 the pr0cedure. Fr0m the v1ewp~nt 0f an expert, 1t 15 rather c0nven1ent t0 pref1x 0n1y 0ne 4uant~y than tw0, thcref0re the 0 n ~ d e d 5e4uent1a1 te5t need5 1e55 50ph15t1cated pre-c0n51derat10n5 than the 5PR7. A5 a c0n5e4uence, the 1n~rpreta~0n 0f the re5u~5 15 a~0 ea~1e~ 1n 5ummary, the 0 n ~ d e d te5t ~ rec0mmended t0 u5e whenever the da5~ca1 5 P R 7 15 appf1ca6~ t0 te5t 0n1y a6n0rma11ty. Furtherm0re, the 0 5 5 7 m1n1m~e5 the extra f1me detect1n9 a6n0rma1 6ehav10ur w1th0ut decrea51n9 the re11a6111ty 0f the c0nc1uded ded~0n.
A P P E N D 1 X A: 7 H E 8 E L 1 E F U P D A 7 1 N 6
PR0CEDURE
7he a190r1thm 15 capa61e t0 eva1uate the 0utput5 0f the da5~ca1 5 P R 7 meth0d 1n 5uch a way that the u5er w1H 6e perpetua11y 1nf0rmed a60ut the 1nve5t19ated 5y5tem•5 5tate. 1n 0rder t0 d0 1t, the 60undar1e5 wh1ch axe cr055ed dur1n9 the te5t are 5t0red 1n a 5er1e5 d = {..., A, A , . . . , 8 , A , . . . 8 , 8 , . . . } . 7h15 5er1e5 d repre5ent5 the h1~0ry 0f the 5y~em wh1ch 15 eva1uated w1th the h~p 0f the 8aye5 meth0d. 7he re5u~ 0f the eva1uat10n 15 the 6e5ef funct10n 8(7/1[d), 1 = 0,1 mea5ur1n9 the 5ke~h00d 0f that the 5y~em~5 5tate c0rre5p0nd5 t0 hyp0the5~ 7/1 c0nd1t10ned up0n the h1~0r1ca1 kn0w1ed9e repre5ented 6y the 5er1e5 d. 7he funct10n 15 re-ca1cu1ated at every new 60undary-cr05~n9 event5, 1.e. when a new ~em ~ em6edded 1nt0 the 5er1e5 d. 7he f0110w1n9 f0rmu1ae axe ~a60rated f0r 1 = 1, 1.e. when the te5ted 5tate c0rre5p0nd5 t0 the a6n0rma1 0ne. (H0wever, the re5u1t5 are a150 app5ca6~ f0r 1 = 0.) 7he actua1 f0rm 0f the 6e11ef funct10n 15 def1ned a5
8(7/1[d) ~ L(d[7/1)L(d[7/1)+ 1•
(A.1)
where the 5ke11h00d rat10 L(d[~H1) read5 a5
L(d17/~) ~ p(d17/0) P(d1~/1) .
(A.2)
7he 4uantRy P(d[7/1) 15 the c0nd1t10na1 pr06a6111ty 0f 06ta1n1n9 5er1e5 d f f the actua1 5tate 0f the 5y5tem c0rre5p0nd5 t0 the a6n0rma1 tn0de, 1.e. t0 hyp0the515 7/1. 51nce the e1ement5 0f 5e~e5 d are 1ndependent, P(d[7/1) can 6e w~tten a5 a pr0duct 0f the ~n91e event5 P(A]7"10), P(A[7/1 ), P(8[7/0) and P(8[7/1 ). 51nce the a60ve pr06a6111t1e5 are kn0wn, name1y P( 81~0 ) = 1 - ~
c0rrect dec1~0n a60ut 5tate 7/0,
P(A1~ ) = 1 - ~
c0rrect ded510n a60ut 5tate 7/1, fa15e a1arm, m~5ed a1arm,
P ( A1~0 ) = ~ P( 81~ ) = ~
(A.3)
A 0n~5Med ~4uent1f1 te5t
1007
E%(A.2) can 6e eva1uated a5
8(~H1~d): 1-- [1.-[-(~)N~(~-~)Nn] ~1, where the 5er1e5 d c0nta1n5 the 8 event N~ t1me5 and the A event f u n ~ n 8(~H~ 1d) ha5 the f01~w1n9 p r 0 p e ~ 5 :
(A.4)
NA t1me5. 7he 6e11ef
1) 1t 151ncrea51n9 when a new A dement 15 taken 1nt0 acc0unt. 2) 1t 151ncrea51n9 when a new 8 dement 15 taken 1nt0 acc0unt. 1f the 5y~em 151n a6n0rmM m0de and the 5er1e5 c0nta1n5 N 5ampM5 (1.e. N then 3) 7he 6e11ef f u n ~ n
= NA + N8)
w1H dedde 0n hyp0the515 •H1 f0r 5ure, 1.e. N~005m8(7~1]d) = 1.
(A.5)
4) A1th0u9h the 6e~ef funct10n 15 n0t 5en~t1ve t0 the 0rder 0f the ~ement50f d, the 5peed 0f 6 ~ f updat1n9 def1ned a5 ~(~H, 1d) ~ ~ N 8(~H~ 1d)
(A.6)
15 5en~t1ve t0 them. Nam~y ~(~H, 1d) ~ exp{
~(1w~ a ) N }
f0r 1ar9e N
and ~(~H11d) ~
a(1w•a) exp{a(1w•a)N}
f0r 5mM1N.
where w ~ a n 0 r m a 1 1 2 ~ n ~ 0 r . 1n ~ , ~ the 6 e # n M n 9 0 f the ~ r e n c e pr0cedure every new #ece 0f e~dence ha5 re1at1ve~ m0re we~ht (0r m0re u5efu1 c0ntr16ut10~ t0 the 0vera11 c0ndu~0n than 1ater, when the 6e11~ ~ n c t ~ n 5tart5 t0 appr0~m~e 1t5 ~ y m p t ~ va1ue. F0r m0re deta1155ee Re£ 4.
APPEND1X 8 We ~ e ~ 0 ~ n 9 ~ r 5~ut~n50f the f01~w1n9 e 4 u ~ n P - - (012~0)2 (a 1 n ( ~ - )
+ (1 - a) 1n(1~ ) )
•
(8.1)
1~8
A. R ~ 2 and 1. Lux
w h e ~ E4.(8.1) ~ve5 E4.~.1) ~ r p = 1 . 7 h e c 0 n ~ r ~ 5 a<1
and
~ m ~ n the 5ame, n a m e ~
~<1.
1t 15 5 ~ t a 6 ~ ~ a55ume, t h ~ we ~ e 100~n9 f0r 5~uf10n5 ~ r 50ft f~1~e5, 1.e. when a9 ~ (~ - ~ ~ t h 5m~1 dec1510n e~0r pm6a61hty, 1.e. a ~ 1. W1th the 5u65t1mt~n
p ~ A5N~
E4.~.1) ~ 5
0
= 1
and
k ~ (0~ 2~ 00) 2
(8.2)
-M~.
(8.3)
a5 (~-~)~f1~(1-~)~-~e
7he f19ht hand ~de 0f E4.(8~) 15 c0n5tant5 wh1ch ~ den0ted 6y C hencef0rth, 1.e
c ~ :(~ ~ ~)1-~:.~.
(~.4)
7hen E%(8.3) take5 the f0~0w1n9 f0rm:
(~-)"~
= ~
(8.5)
9
(8.6)
1ntr0dudn9 tw0 new vaf1a61~ 5 and 9 a5 def 1 - - ~ $
=
~
and
E4.(8.5) can 6e wr1tten a5 95 = C(1 - C9) -1.
(8.7)
51nce a << 1, the r00t 0f E4.(8.7) can 6e appr0x1mated a5
9 = 90(1 + ~),
(8.5)
where 90 ~ the 5d~10n ~ E % ( 8 ~ ) ~ r a = 0, 1.e. ~ = C and ~0m E%(8.6) (8.9) 5u6~1tut1n9 E4.(8.8) 1nt0 E4.(8.7) ~ d 5
C 9~(1%
e) 5 =
1 -
C90 - C ~ e ~
(8.1~
A ~d~
w ~ c h can 6e ~ r a n 9 e d
~
1009
m~
~ C
1
~(~ +~)~= 1-0~ 1 - ~
(~.11)
N0w E%(8.11) can 6e expanded 1nt0 7ay10r 5er1~ ~ 0 u n d e = 0 upt0 the f1r~ 0 r d ~
~(1+ 5~)-1~-C-C~ (1~ ~ e ) ,
(8.12)
w ~ c h can 6e 501ved ~ r e: e = C ~ (~-C(~1c~)~91 ~C~)-~59~. Hav1n9 06t~ned e, the 50u9ht m15~d ~ m
pr06a6~
(8.13)
~ can 6e e x p m ~ e d ~
~ = ~ - (1 + 9•+•)-•,
(~.1~)
App1y1n9 E%(8.6) ~ r the term 9~+~, the a60ve ~rmu1a take5 0n the ~ r m , = 1 - (1 + 0(1 -
+
51nce 1) the p ~ a m e t e r C 15 6 ~ up fr0m kn0wn va1ue5 ~0,~1,a~ and a and 2) e 15 expre55ed ~ a ~ n c ~ 0 n 0f the5e 4uant111e5, ~ 15 d ~ m ~ e d u ~ 4 u d ~ 06v10u51y the t ~ m ~ parenthe5~ ~ E4.(8.1~ ~ 9 r e ~ than 1 and thu5 f1 ~ ~ than 0ne, a5 expected. F u ~ h ~ m 0 m ~ ~ M50 06~0u5 t h ~ dnce a << 1, ~ 15 n e ~ the u ~ t ~ a5 1t can 6e 5een 1n the examp1e5.
ACKN0WLED6MEN75
7 h e w0rk w ~ p ~ t ~ 5upp0~ed 6y the 6rant 0 M F 8 95-97-47-0880,
REFERENCE5
1EEE 7ran~.Aut0ma~. C0n~Ac-21 0 ~ 0 6 e r 750-757 (1966) 519n~D ~ 0 n 7he0r~ Mc6raw-H1f1 800k C0m-
7.H. Ch1en and M.8 Adam5 (1976) J.C. Hanc0ck and P.A. w1nt2 pany, New Y0rk
7he FV1EW50~wa~ Packa9e,KFKLAEK1 App1. React0r A. R£c2 (1995) Anna~ Nuc~ Ener9y0n pre55) A. 1~c2 and 5. K155 (1995) Pr09. Nuc~ Ener9y 29~ N~3/4, 299-320
5. K155 (1991)
Phy~c5