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BIBLIOGRAPHY
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Atkinson, R.e. and Paulson, J.A., 'An Approach to the Psychology of Instruction', Psychological Bulletin 78 (1972), 49-61.
Barry, D.M., 'Payoff Effects in the Detection of Change', Unpublished M.A. thesis, University of Southern Illinois, 1971.
Bartholomew, D.J., Stochastic Modelsfor Social Processes, Wiley, New York, 1967. Bather, J.A., 'On a Quickest Detection Problem', Annals of Mathematical Statistics 38
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view of Psychology 18 (1967), 239-286. Bellman, R., Adaptive Control Processes, Princeton University Press, Princeton, 1961. Bellman, R. and Zadeh, L.A., 'Decision-making in a Fuzzy Environment', Management
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de Finetti, B., 'Foresight: Its Logical Laws, Its Subjective Sources', Translated and reprinted in H.E. K)lkburg, Jr. and H.E. Smokier (eds.), Studies in Subjective Probability, Wiley, New York, 1964.
DeGroot, M.H., Optimal Statistical Decisions, McGraw Hill, New York, 1970. Dorien, P. and Hummon, N.P., Modeling Social Processes, Elsevier, New York, 1976. Eckles, J.E., 'Optimum Maintenance with Incomplete Information', Journal of Opera-
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Edwards, W., 'Dynamic Decision Theory and Probabilistic Information Processing', Human Factors 4 (1962),59-73.
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ApPENDIX
SOLUTION PROGRAM FOR OPTIMAL POLICY
In Chapter 7 we discussed several models for the multi-state extension of the original two-state detection of change problems, TDC and DC. Examples for the three-state case have also been presented. In this appendix we present the computer program that was used to generate the numerical results for all the examples presented in Chapter 7. The program is designed to solve numerically the recursive equations for Rn(Pn) and RnCPn) for the finite-horizon cases of problems MTDC, MDC, STDC, and SDC. As explained in Chapter 7, the program will also yield solutions to the two-state problems TDC and DC in Chapter 2. It uses, however, a coarser grid, and the results are therefore somewhat less accurate. With a sufficient number of iterations in n, the solution for the infinite-horizon values of RnCP) and RnCP) may be obtained to specified accuracy, within computational limitations.
The appropriate formulas for Rn(Pn) and RnCPn) are presented in Chapter 7, and will not be repeated here. The same applies to the Bayes' formula for revising the probability state vector. Information about the values of p~j) and x is documented in the program and will also not be repeated.
For a given three-state problem, the user must specify the: (1) Transition matrix, (2) Loss matrix, (3) Discount factor (if desired), (4) Mean and variance of the normal distributions.fj(x), j = 0, 1,2,.··, (5) N, the finite horizon, (6) The type of problem (i.e., MTDC, MDC, STDC, or SDC). A listing of the main program MSTATE is followed by the only sub
routine, NORDEN, which generates the desired normal densities. The program was written in FORTRAN to run on a PDP-llf45.
184
185
C PROGRAM ~ST~TE FTN C C C SOLUTION Of 3~STATE PARTIALLY OBSE~VA6LE MARKOV C DECISION MODEL WITH ARBITRAR, (STATIONARY) C TRANSITION MATRIX w~ICH Is UNAFFECTED BY DECISIONS AND C A~BITRAR¥ (STATIONARY) LOSS MATRI~. DISCOUNT FACTOR C IS OPTIONAL. C C PROGRAM USES NORMAL DEhSITIES WITH SPECIFIED C PARAMETtRS FOR THE OBSERVATIONS OR wILL GENERATE C "RA~DOM" DENSITIES. If SPECIFIC (NON-NORMAL) DENSITIES C A~E DESIRED, ONLY S0dROUTINE "~ORCEN" NEED BE CHANGED. C THERE IS NO RtSTR!CTION ON THE FORM OF THE DENSITIES C UStO, NOR ON THE RESULT!NG LIKlLIHOOD RATIOS. e NOTE THAT SIMPSON'S RULE REQUIRES THAT THE OENSITIES BE C EVALUAT~D AT EQUALLY SPACtD POINTS. C C PRJGRAM ASSUMtS JX3 TRA~SITJO~ MATRIX AND A 4X3 LOSS C MATRIX. IF LESS TH4~ 4 ACTIONS APE AVAILABLE, SET C CG~RESPONoINr. ROWS IN THE LOSS MATRIX TO LARGE CONSTANTS. C THE (I,J) ENTRY IN THE LO~S MATRIX IS THE C IM~EOIATE LOSS INC0~HED If ACTION I IS TA~EN IN STATE J. C PROGRAM ~AY EASILY ~E EXTtND~D TO AN ARBITRARY NU~BER C OF ACTIONS. NOT EASY 10 EXTEND PPOGRA" MUCH BEYOND C 3 STATES, ruE TO SPACE RECUIRFMENTS. C C THIs PROGRAM IS DIMENSIONED FOR A 51 By 51 DISCRETE PROBABILITY C GRID AND III VALUES ALONG THE X AXIS, BUT CAN BE C MODIFIED 8Y CMANGINr, THE OIM~NSION STATEMENT. C PROBABILITY GRIO CAN BE MORE EFrICIENTLY STORED IN A C VECTOR, AS ONLY ~ALF IS USED. C C Ir DISCOUNT FACTOR IS NOT WANTED T~EN SET DBETA"1. IN THIS C CASE, THE T~ANSITION MATRIX MUST HAVE AN ABSORBING STATE C IN ORDER 10 KEEP THE COST INCURREO OVER A~ UN-C BOUNDED ~ORIZO~ FINITE. (~N ABSORBING STATE IS NOT C NECESSA~Y IF ONLY INTERESTED IN T~t fINITE STATE SOL'N.) C
186
c C C C 17~
176 177 17A 179 1799 200
2~1
5~0
~J0
C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C l C
OIME~S!ON VOLD(51,511,'NE.(51,~ll,PRTAB(3.111),PTAB(~I), 1 Zp(e),SI~p(lll)'MINM(~1,51),TRANS(3,3),XLOSS(4,3),COST(4), 2 PARAH(2),PARAM2(2)'~ARAM3(21,XRTA8(!11) DATA NPMAX/~I/,NX/lll/,Ke~/6/,KBR/8/,LP/51
KBRaS FOR BATCH; uo OTHERWISE.
FORMAT('I',!~(/),10X, '~ULTISTATE PA~TIALLV OBSERVABLE MARKOV', I 'OECISION ?R06LEM',III,' TWANSITIoN ~ATRIX',I, 2 J(8X,3F9.4,/ll FOR~AT(II,2X,'DISCOUNT FACTOR',F6.3) FORM.T(II, I LOSS MATRII',1,1~(8X,3F9.4./l) FOR~AT(II,2X, 'F~: ',2F~,l,I,2X, If I 1 ',2F6,I,I,2X, 'F21 ',2F6,1) FORMAT('0',6II) FORMAT('0INTEGRATION INTERV'L',2F~,2) FaRHAT ('I',2~X,'EXPECTED COSTS FOR TABLED PROBABILITY VALUES',
1 1,2IX, 'HORIZON ",14, 3 II, 1~',~X,M(3X,F~ •• ).III) FORHAT(15. I STAGE ~RU8LEM DELETED".PR08. TOO SMALL') FORMAT (1~,"4.2,2~,8(2~,F7,4),1,7X,100(8(2X,F7,4) ,I)) FORMAT('~I)
PRTAB
PTAB XLO~,
~HIGH
PARAM PARAM2 PARAHj NMAX
~Lass
TRA~S
CI VNEw VOLa OBETA ICOD~ NTER MSMOl MSOI MSMD2 MSD2
N\!M~E~ CF DI5CPETf POINTS DN THE X-AXIS Te BE CONS!DE~E~ THfSt POINTS MUST 8E EQlIDISTANT AND NX MUST AGREE w:T~ T~E OlME~SIJN OF THE SECOND INDEX ~F PRTae, THIS NUH~E~ M!JST ~E ODD TO ACCOMMODATE SI~PSON'5 RULE. NU~BER OF nISCRETt PRO"AB!LITV VALUE~ CDN5IDER~D, T~ESE
~ILL BE EJU!D1STANT AND WILL INCLUDE ~.e~ AND 1,00 THIS VALUE HJ5T AG~EE WITH THE DI~ENSIONS CF vnLo,vNE~. PTAb If I~CREAStO 8EyeND 51, CHA~GE OUTPUT FOqMAT,
CONTAINS T~E DENSITIES OF THE OBSERVATIONS ,OR EAC~ STATE ,EVALUATED AT NX PO!~TS A DISTANCE Ox .PA~~. STORES THE DISCRETE V'L~ES OF P T~E INT~RVAL [XLOw,XHIGHI ON ~HE ~ AXIS WILL BE DIVIDED I~Ta NX_I EC~Al PARTS, USED TO DETE~M!NE ox. uSED ONLY IF DENSITI~S ARE GENERATED ~Y A FQRMU~A, MEAN ANO VA~IAhCE OF NORMAL DENSITY, STATE I. MEAN AND VARIANCE OF N[RMAL D~NSITY. ST.TE 2. MoAN AND VARIANCE OF NORrAl DEhSITV. ST~TE ~.
THIS PGM. ~I~L SOLVE ALL FINITE STAGE PPo~LE~5 CON~lSTIN~ OF UP TO NMAX STAGES, LOSS MATRIX, TRAN5ITIO~ ~ATKIX.
CONSTANT USED IN SIMPSGNIS RULE, CURRENT VALUE O~ OPTIMAL RETURN FUNCTION. OLD VALUE OF OPTI~AL RFTUPN FUNCTION, OISCOU~T FACTOR (0.LF.OB~TA.LE,l) al FOR NORMAL DENSITIES'.~ FOR ~ANDOM DENSITIES, SUPPRESSES SOMf ITERATIONS ON PRINTOUT. =: If Mr, VERSION 1(3-STAT~) DESIRED, .~ OiHER, -I IF 0 v~RSION I (~.STATE) DESIRED, .m OTHfR. '1 IF MO VtRSION 2C3-STATf) DESIRED, -0 STHER. '1 IF D VERSION 2 (3-STATf) DESIREO, 12 OT~ER.
1270
1~91
1292
c c c
"'RlTEC~t:lW,275)
FOR~ATCII,' N~AX NTER leooE DBETA',III) RtAO(~8R,27a) N~AX,NT!~,ICOOE,DBETA rORMAT(JI4,FI~.0) wRlTECKth,! 275) FURMAT('0MS~02 M5D2 MS~D\ MSOI ',II) READ(~B~,1276) MS"02,MSD2,MS~DI,MSDI FORMAT (81A) ~RITECKB", 1291) fORMAT('0PARA~ P~A'M2 PARAM3',II) READ(KBR,1292) PA~A~,PARAM2,PARAM3 FDRMAT(IIFI0.0J IF(MS~D2.MS02·MSMD1+MSD1 •• GT.l) GO TO 99 IF(ICODt.N[.~J GO TO 151
~RITE(~>jW,149)
149 FC;R"'AT(lI, , E"TUl SEED',III) RtAO(~BR'\5\l) 12
15~ fO~MAT(I~J 1100
151 .RI'ECK~W,375J
375 FO~"AT(II.' J 8' 3 T~ANSITION "ATPI~.-B' ROWS',III) READ(~BR,375) (CTRANSC!,J),Jol,3),I'I,3)
378 FORMAT(12F6,0) wRITUK5w,377J
317 FO>l~AT(II,' 4 AV 3 l.OSS MATRlx.-BY ROIolS',III) ~EAiJ(K'3R,37") ((~l.'JSSCI,J),J.I,3),I'I,4) X"I=J*SQIlT(PAfIAM(21J'
c
XW2'l.SJRT(PARA~2(2»)
xw3=3·S0RT(PA.AH3(2J) XlOw'f\~I~l (PAHA'" (I J.~w I, PARlM2 (Il-HI2, PARAM3 (1) -XW;)l XHIG~'AMAXI CPARA~(I).Xwl,PARA~2(11+X~2,PARAM3(ll.xW31 Gx: Cl("'rGH-XlUW) I U'X-:)
C GE~fRATt n~N~ITY F0NCTIO~S. c
CAl~ ',JRDt:~J (,"~I'VJ"QX,PAIlAM,XRTAE\,ICaDE,II,12) :)[J ~ 3 .1 I. t • N X
3 ~ ~~ ?" TA Il C 1 .J J • X ~ T I>. B ( J ) CALL NONnE~(NX,XLr~.Ox,p'~A~2,.PTA~,ICODE,II,I2J JrJ ::H31 l~l,N~
J,lI P~TABC2.I)n~TAfl(TJ
CALL NORDE~CNI,X~~',CX,PAR~H3.XRTAB,ICODE,II,12J DO 3~2 1",NX
~ ~ , P k T A ~ 0 , I ) "x II TA '\ cr )
187
188
c C PRINT PARAMETERS C
C
WRITECLP,175) «TRANS(I,Jl,J.I,~l,I.l,~) WRlTE(LP,!771 «(XLOSS(I,Jl,J"I,;Jl,ID\,41 IoI RlTECLP,17610BETA ~RITECLP,178) PARA~,PARAM2,PA~AM3
WRITE(LP,\791 MSM02,MS02,~SMn\,MSDI WRlTECLP,t799l XLOW,XHIG"
C EsTABLISH CONSTANTS T1 BE USED l~ PROGRAM C
MSI"MSMDI·~SOI MS2.MSMD2.~S02
TRI\oTRAN5(!,I) TR2PTRAiliS (2,1) TR3!.HlAN5(~,l)
TR !2"T"AN3 (l,~) TR2?TRANS (2,2) TR32'TRANS (3,2) TR13.TRANS (1 ,3) T R 2 ;\ 0 T R A ~! 5 ( 2 , ~ ) TR33'TRANS(;J,j) XLll o XLOSS (\,1) XLl2'~LOSS(\,2)
XLl3'XLOSS (1,3) XL2!'XLOSS(2, I) XL22 o XLOS,(;?,2) xL23'~LOS5 (2,3) XL3\,xLOSS(3, \) XL32 o XLOSSD,2) XL33 o XLOSS(3,3) XL4\'XLOSS(4,!) XL42·XLOSS(4,2) XL43 o XLOSS(A,3) TRXLI·TR\\*XLI1*TRI2·XLI2+TRI3*XL\;J TRXL2 o TR21*XL21+TR22*XL22+TR23*XL23 TRXL3·TR31·xL01+TR32.xL32+TR33.XL~3 IOEND~PMH-!
DEN.Il1EN
LWa l,/DEN UPhDP/2. tl P ~ A ~ 1 • N P M A 1. • 1
ISKIP.5 NSKIP.~
lJ02~8J'I,8 ;21')8 zP(J)o(J-l).NSK1P.OP
CI·O~I3.
C
C ESTAaLIS~ THE ACTuAL VALuES OF P CO~RESpaNDING TO T~E
C I~OtX Or PT~B, INITIALIZE t~PECTEr. COST FUNCTION t
DO 10 I'I.NPMA)( PaCI-l)/DEN
10 PTASeI)-p C
C I~ITIALIZt VNE~, SINCE ~E PREPARE FOR THE ~EXT
C ITERATION AT THE ~EGINNING OF EACH LOOP (SEE
C STATEMENT b24J WE INITIALIZE VNE~ INSTEAD OF VOLO, C
DO 11 !=I,NPMAX uo 11 J=I.~IPMAX
11 V>:EW(I,J)=0. C
189
C THE APP~OPRIATE SIMPSONS ~uLE MULTIPLE IS STO~ED !N SIMP, C
lC~2
DO 20 K.l.N~
lFC(~dl)"(~.NX1J 18,17,17 17 HLDI4=I.
GO TO 19 18 lCOIi-IC
HLD\4=IC 19 SIMP(K)"HLDI4 2~ CONTINUE C C SCALf P~TA8 I~ CASE ITS EhTRIfS A~E NOT fROM A
C TRUE PROBABILITY DENSITY. C
DO ~~ J-l,~
GL.OOP-". DO 36 K-l,,,"X
38 GLO~P.GLOOP.SI~peK)*PRTAB(J,K)
GLOOP.GLOOP*Cl DO 39 K'\.Nx
39 PRTAB(J.KJ"PRTAM(J,~)/GLaOp
J~ CONTI~UE C C OUTER LOQP IS ov~~ THE BOUND ON THE ~ORIZON. C
C C PREpANE fn~ THE ~EXT ITERaTIGN, C
DO 24 I"I,NPMAX DO 24 J"I,"PMAX
24 VOLn(I,J)'V~E'(I,JJ
190
c C LOOP i~RU PR08ABILITY GRID. C C C P~PROB(STATE'l) C a I PROB(STATEa2) C PG.PROB(ST~TE.3) C
C
DO 70 MPll,NPMAX
P.PHS (MP) NUPP3NPMAX!_MF' 00 70 NPll,NUPP a·PTA8(NPl IF(N.GT.ll GO TO 224
C IF N.EO.l ~E KNOW TNAT THE FUTURE COSTS ARE 0. C
XINT2~1il.
XINT3 a0, XHH4.0, GLOOP=0. GO TO 2tl
224 IF«(NP.NE,I',OR.(MP.iIIE.l" GO TO 26 IF(MSMOI.MSM02.EQ,l) GO TO 26
C' C WE FIRST'COMPUTE T~E DISCOUNTED EXpECTED COST OF C CONTINUlNG, GIVEN STARTING I~ EAC~ STATE(I,2, OR 3)1 C (",Cl,PQ)=C1"',0) CALLED XINT2 C (0,1,0) CALLED XINT3 C (~,0,1) I CALLED XINT4 C
lSTARTo2 PII, 0.0. GO TO 2l'J
27 XINT2.0dETA.GLOQP lSTAf(TlJ P'Ii!. Qa l. GO TO 21)
28 ~INT3'D8ETA.GLOOp ISTART.4 Pill. a • .,. GO TO 20
29 klNT4.0BETA'GLOOP 1STARTo!
21> PIj'l.-P-fl PSTAR,P'TRll+Q'TR21'PQ'TR31 QSTAR.P'T~12+~·lR22+PO'TRJ2 PQSol.-PSTAR-QSTAR IF('J.EQ.l1 GO TO 81l
c C ~UMERJCAL INTtGRATIO~ c
c
GLOOP.~.
DO 4~ ~=l ,N~ G!·?srA~.p~TAB(1 ,K) G2'QSTA~*?RTAB(2,K) Gx o G\+G2+ PQS*PRTAS(3,K) QPR.G2/G~
PPR.G1/GX INPPR:PP~.DEN + I. INQPR'Q~R'DEN.l, IF (I~~PH,GT.IDtN) INPFH'IDE~ If (!N~PR,GT,IU~~) IN~P~'ICE~ J,,"P=INPPR.l JNO.I'IQ?R"l
C INTERPOLATE fOR YOLO(?'] C FIT A PLANE TMRU TrlE j "~tA~EST" CCR~ERS OF T~E S~JARE C CO~JHINING (PPR,QP'<),
c
c
Vll'YO~O(INPP~,INQPR)
VI2=VOLD(JNP,INQPR) V21.VOLD(INPPR,JNO) V22.VO~D(JNP,JNQJ
C CH~CK TO SEc IF wE '~t O~ T"E OJ~GcNAL, c
IF(JNP+JNO.LE.~PMAX1) GO TO 2~e 244 eBa(V2\-Vtl)/UP
H"CV\2-YlIJ/UP 245 HLD.(PP~.PTA8II~PPA)J.A&+[QPA.PTAf(I"QP~)J*q3.V!!
GO TO 4~
25~ IF (QPI<.GE, (PHS (J~QPR) .D P2l) GC ,_ 2~5 IF(P?R,LT.(PT~B(P.?Pf\)+DP2J) tit' rc 2'4 AA=(VI2-Vll)/0f' t>6'(V22-VI2)/lJP GO TO 245
255 H(PP~,(iE,(PH6(I,.?p")+GPn) GO TC 258 AAoC V22-V21)/UP eS'(V2\-V11)/OP GO TO 245
258 AAo (V22-Y211 lUi> HB' (V22-V\ 2) IO~ HLD'(PPR.PTAB(J~P)J.AA.[QP~-PT.q(J~CJ1.8b.V22
40 GLOOP • GleOp + SI~P(~J.GX.HLD GLOOP·G~DOF·r.l·D9ETA GOTa (82,27.2h,2~1,ISTART
191
192
C 8~ IF(MS\.~Q.\) GO TO 8~1
IF(~S2.tC.ll GO TO 811 c C RECURSIvE fORMULAE ~OR EXPECTED LOSSES "OR EACH ACTION. C I~SERT E~PRES5IoNS FUR COST(I). I.l ••••• ~COR NUM8~R OF ACTIONS), C wHIcH ARE THE E~P~CTEo COSTS QF TAKING ACTION I. C THIS CONSISTS OF T~E t~PECTtD CuRRE~T LOSS PLU~ THE C (DISCOUNTtD1 EXPECTED FUTlIRE COST. I" • PARTICUlAR C PROBLEM HAS LtSS T~AN 4 POSSIHLE ACTIONS. SET THE CORRES· C ~ONDI~G COST(rl TO A L.RGt PGSIT!VE NUMBER. COP CHANGE XLOSS.) C C GLDOP'OIS:. EIP. FUTURE cnST. STARTING FROM CURRENT ~TATE.
C XI~T2.XI~T3.XINT4.0ISC. EXP, FUTLRE COST. GIVEN THAT C THE STATE IS nfiESET" TO A KNOWN VALUE. (SEE ABOH) C C DAMPLE:
c
Co5T(1)·GLOO~+?XL11·r.·XL12+pr,·xL\3
COS T ( 2) : G L 0 0 p "p ... ~. 2 1 + G. X L 2 2 + P C * x L 2 3 COST(3)·P.XI~T2+J*XI~T~+PG.Xl~T •• P.XL~I+a.XL~2.pa.XL33 [OST(':)-999. GO TO 13~~
811 COST(1)=P~S.XL13.GLQOP
c
CoST(2)'~SO~.(~.X!NT2.(II~TJ+TR2~*xL13).a)+p.xL21.Q.xL22
COST (j) .099, C05T(4)=999. GO TO 8~~
6~! COST(\)·GLonp+PQS.Xl\3+QSTAR.XlI2
C
COST«)=poXL21+PQ.XL23+HSD\*P.CX!NT2+TRXL1) CQST(~)'P.XL31+Q.XlJ2+~SOl·P.[XINT2+TRXL1)
COST(4)=9Q9.
C SELE[TTHE REST ACtION. C
~ 5 \' C C 2. A.~ I N 1 (C 051 (1) , C J 5 T ( 2) , COS T (3) • COS T (4 J J VNE~ ("1P. NP).CC2 DO 64 Jl-t.4 IF(ABS(CC2-CIJST(JL)) ,LE. •• V~~~l) GO TO 6~
6~ CONTINUE c C IN THE PRI~TOUT OF THE QPTIM.L ~OL!CY. THE ~TATES C WILL ~E .SSIJ~fO TO BE NU'18E~EO ~d.2,... INSTEAD C OF 1.2.3 .... C 6!s MINM(MP,NPloJl-\ 7~ CONTINUE
C C C C
4BJ
91
513 C C C C C
P~INT COSTS fOR SELECT£n (P,G) VALUES IN H~7RIX FG~H,
ALSn PRINT THE OPTIMAl ACTION FOR E~CH STATE,
IF(MOQ(N,~TERJ.~E.0) GO TC 1<!;J
"RITE(I.P,20~) ".ZP DO 91 11'1,~PMA~,IS~IP
DECO.(NPMAX-II)*0P IINPMAO_II ~RIH (LP, ~3\l) '"~!TE(LP,5Vl~) OECO.(V~E~(I,NU~Ei),Nu~B'1,II,N5~IP)
w R I T E (l. P , 480) (~ IN r\ ( J , N L; H ill , ~ II ~ e • ! , I I , N 5 q P)
FOR~AT(7x,8I9J
CONTINLIE "'RITE(l.P,~!8)
fOR~AT('1 OPTIMAL peLleY',II, '?ST~TE e',lll
PRINT OPTIMAL ?OLICY fOR EACH (P,C) VALUE (p.G.LE,I)
IN THt LO"ER HALF OF 51x51 GRID, UCH Raw OR COLlJ~N
CORRESPONOS TO • PROHABILITV I~CRE~ENT 0' ,m2
00 519 II'!,NPMAX I=NPMAXlpII ~RrTE(LP,52;l) (M!NMCI,Jl,J'I,II)
FOR"AT(5X,5111) -PITE (LP,522) 'QR~AT(~7X,ISTATE 1') COI'<TI'lUE. CAll txIi ENO
193
194
C
SUBROUTIN~ NORDEN (~X,XLow,D~,PARlH,XRTAB,ICODE,Il,I2) DIHENSION PARlM(2),XRT~B(III)
C COMPUTES AND STORES I~ XRTAB THE VALUES OF A NORMAL C DtNSITY. PARAM(l) AND PARAM(2) ARE THE MEAN AND VARIANt! C HE5PECTIV~LY OF THE DENSITY. C THE DENSITIES ARE EVALuATED AT THE POINTS t XLOw,XLO~.OX, ••• ,XLO~.(NX.l)*O~. C IF ICODE.EO,0 THEN "RAhOOM" DE~SITIE5 ARE GENERATED. C
LPJ5 KB'6 IF(PARAM(2),GT.~.) GO 10 I WRITE(e,Ip.0)
1~~ FORMAT (' VARIAhCE NOT POSITIVE IN SUB. NORDEN') STOP
1 IF (ICOOE,EO.0) GO TO 8~ C C GENERATE NORMAL DENSITY C
PM'PARAM(I) VM.PARAM(2) C2 o .398942/SQRT(VM) 00 5 Iol,NX XoXLow+(I-l1*DX y.(pM-Xl*rX.PH1/VM
~ XRTAB(Il.C2*EXP(V/2.) RETURN
c C GENERATE "RANDOM" OENSITITV C 8~ 00 ~0 JII,NX ~~ XRTAB(J).I./(RAN(Il,I2)+.0a5)
WRITE(O,55) XRTAB 55 ~ORMAT(6~9.3)
RETURN END
a, 65 b,65 C);,u,/,m,k,g,94 DM,1 DC, 17 d',30
di ,/, k' 86 d';,/,k,87 D .. , 121 DC3,149 DCIO,149 f(t, 11), 17 F,17 fo(x), h(x), 17 FPB,34 FNOB, 64 FNSOB,81 F j , 124 ID(P), Iw(P), 128 k, 14,64 L(x), 18 M,42, 120 MDC, 123 MTDC, 123 N,17 Po, 16 Pm 19 P*,21 Po,44 Pj> 46 Pic (m, P), 52 Ph PO, 56 Pc, 58
GLOSSARY OF SYMBOLS
195
Pic' 58 Pj (m, k), 66 p~j(m, k), 67 po(m, k), 68 POj(m, k), 72 Pilv, k), 73 Pe(v, k,P), 75 Pl(m, k, r), 87 p~(m, k, r), 91 p~l(m, k, r), 94 po(m, k, r), 94 Poim, k, r), 97 pjlV, k, r), 98 Pe(v, k, r), 102 pJjJ, 121
Pm 125 P~ilr*, 126 P(~J, 127 Qj> 35 Q,39 Qo,55 r, 81 R(P),22 R(P),22 RN (P),23 R(P, r), 62 R(P, r), 63 R', R, 107 REL,107 R', R, 107 So, SI> I, 16 s, 35, 87 SOC, 112
196 GLOSSARY OF SYMBOLS
STDC, 120 SDC, 122 t, 16 TDC, 16 T(P, x), 21 T j ,35 T, 39, 126
tOI> 51 To, TI> T, 55 TDC3, TDClO, TDC50, 149 W, 18 Wb 35 W,39 W,121 wb 121 Xno 17 xc, 64 Yb 66 Yoi k ),72 z,35 z',37 0,46 a, 16 ai' 124 (3j' 159 rN,fN,25 r*, f*, 27 r, 34 ao,53 a, 54
60,59 0,59 OJ (k), 74 O(k, P), 75 OJ (k, r), lOO O(k, r, P), lO2 e, 38 1r, 121
+ib 45 +h 46 dJ,46 +0, +10 56 (p, 56 ¢o(k), ¢l(k), 68 ¢oik) , 73 ¢l(k, r), 89 ¢o(k, r), 94 dJ(k, r, P), 96 Wo,41 <Po(i), 41 chU'), 42 1J!1o 42 1Jf(P), 42 iii, 57 </;0, 57 </;10 57 <po(k),70 1Jf(k, P), 71 rpo(k, r), rpl(k, r), 82 1Jf(k, r, P), 83
INDEX OF NAMES
Allais, M., 4 Atkinson, R. c., 12, 162, 165
Barry, D. M., 150,179 Bartholomew, D. J., 12 Bather, J. A., 12 Becker, G. B., 5 Bellman, R., 4, 23 Bernbach, H. A., 35 Birdsall, T. G., 3,9,61 Bogartz, R. S., 162, 163 Burkheimer, G. J., 9, 10, 18, 61, 80,
105, 148, 149, 150. 151, 154, 156, 177, 179
Calder, B. J., 118 Cinlar, E., 159
de Finetti, B., 4 DeGroot, M., 6,7 Dorien, P., 6
Eckles, J. E., 146, 168 Edwards, W., 5, 7, 8, 9, 103, 104, 105,
106 Egan, J. P., 113 Estes, W. K., 147, 148
Feller, W., 85, 125
Ginsberg, R., 118 Grant, D. A., 162, 163, 166 Green, D. M., 2,9,12,13,15,18,113,
119,157,158,161
Hoekstra, D. J., 1, 13, 23. 24, 25, 129, 146, 168
Howell, W. C, 9 Hummon, N, P., 6
197
Jones, L. V., 105
Kahall J. P., 105 Kemel't'Y, J. G., 37,46,47 Kozielecki, J., 4 Krantz, D. R., 5, 12
Lee, W., 4 Lichtenstein, S., 5, 32 Luce, R. D., 2,4,5,12, 13, 15, 18, 119,
157, 158, 161
MacCrimmon, K. R., 4 McClintock, C. G., 5 Millward, R. B., 35
Nemhauser, G. L., 8
Paulson, J. A., 165 Pitz, G. F., 9,12 Pollock, S. M., 1, 12,26,65, 113, 114 Polson, P. G., 35 Pratt, J. W., 7
Ramsey, F. P., 4 Rapoport, Amnon, 5,9, 10, 14, 18, 27,
31,33,34,61,68, 104, 105, 118 Roberts, R. A., 3, 9, 61 Runnels, L. K., 163
Savage, L. J., 4 Shiryaev, A. N., 12, 19, 31, 167, 170 Shtraucher, Z., 150, 156, 179 Slovic, P., 4, 5, 32 Smallwood, R. D., 13,146,168 Snell, J. L., 37, 46, 47 Sondik, E. J., 13, 146, 168 Stein, W. E., 34,171,178 Suppes, P., 4, 5, 13, 118
198 INDEX OF NAMES
Swets, J. A., 9,12,13,113 von Winterfeldt, D., 103, 104, 105, 106
Theios, J., 162 Wallsten, T. S., 5,9 Tversky, A., 4
Zadeh, L. A., 4
INDEX OF SUBJECTS
Asymptotic results for model FPB, 158
Bayes'theorem, 21, 35, 72, 127 Binary sequences, 84
Deferred decision problems, 3,9,10,179 Delay loss, 17,18,19,31,52 Descriptive models, 3, 4, 14, 32, 33 Difference equations, 68, 81
Experimental tests of the models, 147, 160
Heuristics, 32, 118
Insensitivity of the model, 104, 106, 107
Machine maintenance, I, 18, 120, 123 Markov chain approximation, 34 Markov decision processes, 13, 37, 146, 168 Medical diagnosis, 1 Multi-state detection of change problems, 14, 119, 184
Normative models, 3, 4
Optimal decision boundaries, 26,27,28,33,62, 171 Optimal policy, 13, 14, 16, 19, 21, 25, 33, 62, 65, 125, 184
Parameter estimation, 79 Performance evaluation in repeated-trial learning, 162 Posterior probability, 21, 35 Probability distribution of
number of incorrect stop decisions, 56, 77 number of observations after the change, 33, 52, 58, 73, 78, 98. total number of observations, 33,44,69, 77, 85 trial of change, 16, 18, 31, 169
Reaction time experiments, 1, 12, 18, 119, 123, 157 Receiver operating characteristics, 14, 113
199
200 INDEX OF SUBJECTS
REL function, 106 Response models, 5, 13, 15, 34, 64, 147, 165
Sensitivity analysis, 14, 33 Signal detection, 2,9,12, 13, 14, 103, 105, 113 Stochastic kernels, 35 Sufficient statistics, 19, 125, 170 System operating characteristic, 14, 112 Systems theory, 6
Validation, 103
THEORY AND DECISION LIBRARY
An International Series in the Philosophy and Methodology
of the Social and Behavioral Sciences
Editors:
Gerald Eberlein, University of Technology, Munich
Werner Leinfellner, University of Nebraska
I. Gunther Menges (ed.),1nformation, Inference, and Decision. 1974, viii + 195 pp. 2. Anatol Rapoport (ed.), Game Theory as a Theory of Conflict Resolution. 1974,
v + 283 pp. 3. Mario Bunge (ed.), The Methodological Unity of Science. 1973, viii + 264 pp. 4. Colin Cherry (ed.), Pragmatic Aspects of Human Communication. 1974, ix + 178 pp. 5. Friedrich Rapp (ed.), Contributions to a Philosophy of Technology. Studies in the
Structure of Thinking in the Technological Sciences. 1974, xv + 228 pp. 6. Werner Leinfellner and Eckehart Kohler (eds.), Developments in the Methodology of
Social &ience. 1974, x + 430 pp. 7. Jacob Marschak, Economic Information, Decision and Prediction. Selected Essays.
1974, three volumes, xviii + 389 pp.; xii + 362 pp.; x + 399 pp. 8. Carl-Axel S. Stael von Holstein (ed.), The Concept of Probability in Psychological
Experiments. 1974, xi + 153 pp. 9. Heinz J. Skala, Non-Archimedean Utility Theory. 1975, xii + 138 pp. 10. Karin D. Knorr, Hermann Strasser, and Hans Georg Zilian (eds.), Determinants and
Controls of Scientific Developments. 1975, ix + 460 pp. 11. Dirk Wendt, and Charles Vlek (eds.), Utility, Probability, and Human Decision Making.
Selected Proceedings of an Interdisciplinary Research Conference, Rome, 3-6 September, 1973. 1975, viii + 418 pp.
12. John C. Harsanyi, Essays on Ethics, Social Behavior, and Scientific Explanation. 1976, xvi + 262 pp.
13. Gerhard Schwodiauer (ed.), Equilibrium and Disequilibrium in Economic Theory. Proceedings of a Conference Organized by the Institute for Advanced Studies, Vienna, Austria, July 3-5,1974. 1978, 1+ 736 pp.
14. V. V. Kolbin, Stochastic Programming. 1977, xii + 195 pp. 15. R. Mattessich, Instrumental Reasoning and Systems Methodology. 1978, xxii + 396 pp. 16 H. Jungermann and G. de Zeeuw (eds.), Decision Making and Change in Human
Affairs. 1977, xv + 526 pp. 18. A. Rapoport, W. E. Stein, and G. J. Burkheimer, Response Models for Detection of
Change. 1978 19. H. J. Johnson, J. J. Leach, and R. G. Muhlmann (eds.), Revolutions, Systems, and
Theories, Essays in Political Philosophy. 1978 20. Stephen Gale and Gunnar Olsson (eds.), Philosophy in Geography. 1979, xxii +
470 pp. 21. Maurice Allais and Ole Hagen (eds.), Expected Utility Hypotheses and the A/lois
Paradox: Contemporary Discussions of Decisions Under Uncertainty With A//ais' Rejoinder. 1979, vii + 681 pp. + indexes.
22. Teddy Seidenfeld, Philosophical Prob/ems of Statistical Inference: Learning from R.A. Fisher. 1979, xiv + 246 pp.
