Research Report Statistical Research Unit Department of Economics University of Gothenburg

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1 Research Repor Saisical Research Uni Deparmen of Economics Universiy of Gohenburg Sweden Hoelling s T Mehod in Mulivariae On-Line Surveillance. On he Delay of an Alarm E. Andersson Research Repor 008:3 ISSN Mailing address: Fax Phone Home Page: Saisical Research Uni Na: Na: hp:// P.O. Box 640 In: In: SE Göeborg Sweden

2 HOTELLING S T METHOD IN MULTIVARIATE ON-LINE SURVEILLANCE. ON THE DELAY OF AN ALARM E. Andersson Saisical Research Uni Goeborg Universiy Sweden Absrac A sysem for deecing changes in an on-going process is needed in many siuaions. Online monioring (surveillance) is used in early deecion of disease oubreaks, of paiens a risk and of financial insabiliy. By coninually monioring one or several indicaors, we can, early, deec a change in he processes of ineres. There are several suggesed mehods for mulivariae surveillance, one of which is he Hoelling s T. Since one aim in surveillance is quick deecion of a change, i is imporan o use evaluaion measures ha reflec he imeliness of an alarm. One suggesed measure is he expeced delay of an alarm, in relaion o he ime of change (τ) in he process. Here we invesigae a delay measure for he bivariae siuaion. Generally, he measure depends on boh change imes (i.e. τ and τ). We show ha, for a bivariae siuaion using he T mehod, he delay only depends on τ and τ hrough he disance τ-τ. Key words: Monioring; On-line; Surveillance; T; Timeliness.

3 . Inroducion In many siuaions i is imporan o monior one or several processes in order o deec imporan changes as soon as possible. In on-line monioring we have repeaed decisions: a each new ime poin, a new observaion becomes available and a new decision has o be made in order o decide wheher he process has changed or nor. In his siuaion he mehodology of saisical surveillance is appropriae. Examples of differen areas where saisical surveillance has been used is urning poin deecion (see (Nefci 98), (Hamilon 989), (Royson 99), (Baron 00), (Bock, Andersson and Frisén 007), (Andersson 004) and (Andersson, Bock and Frisén 004). Anoher area is deecion of growh reardaion of foeuses, see (Pezold, Sonesson, Bergman and Kieler 004). Ye anoher is deecion of an increased level, emerging from a source and spreading spaially ((Järpe 999)). In indusrial qualiy conrol, saisical surveillance has been used in he form of conrol chars, such as xbar-chars, s-chars and R-chars. These chars were developed for he univariae case and ofen only he las observaion is used (he Shewhar mehod ((Shewhar 93)). The ime scale can differ from one applicaion o anoher (daily, weekly, monhly), bu common o all surveillance is he repeaed decisions: a each new ime poin anoher observaion becomes available and once more we have o decide wheher he change has occurred or no. There is always a risk of a false alarm. A good alarm sysem should no have oo many false alarms. Bu we mus also consider ha we wan a sysem wih high deecion abiliy if here really has been a change we wan o deec a change quickly. When evaluaing surveillance sysems, we ofen use oher measures han size and power, insead here is a rade off beween false alarms and delay of moivaed alarms (see e.g. (Frisén 99) and (Frisén 007)). The false alarms are ofen measured by he average run lengh, ARL0. For he moivaed alarms i is imporan o consider how long i ook unil a signal was called, for example measured by he expeced delay ime. In many siuaions we monior several processes, which can change a he same ime or a differen imes. There are several approaches o mulivariae surveillance, see (Sonesson and Frisén 005). One approach is o reduce he mulidimensional daa a each ime poin, o a scalar. The Hoelling s T is an example of his approach ((Hoelling 947)). In his paper we presen a delay measure for he mulivariae siuaion. We also show a resul regarding he delay of he T mehod, when monioring a bivariae process, where he change imes are no equal.. Model and mehod Firs we sudy he univariae surveillance siuaion. A some unknown ime au here is a change in μ (he expeced value of X). In he simples case he change can be a shif in he mean, i.e. 0 μ(s)μ, s < τ μ(s)μ, s τ exemplified in Figure.

4 Figure : The vecor μ, as a funcion of ime, for τ (lef) and τ3 (righ). Anoher example is a change from a consan level o an increasing, unspecified funcion so ha he vecor μ is 0 μ()...μ(s-)μ(s)μ, s < τ 0 0 μ()...μ( τ-)μ and μ <μ( τ)<...<μ(s), s τ which is exemplified in Figure. Figure : The vecor μ, as a funcion of ime, for τ (lef) and τ3 (righ). Now we urn o he bivariae case. A each decision ime, a new bivariae observaion becomes available, and a decision ime s we have he observaions (X,Y). A an unknown ime τ X here is a change in μ X and correspondingly for he process Y. The wo processes X and Y have he same variance (i.e. Var[Y()] Var[X()] σ ) and have covariance ρ σ (i.e. E[(X()- μ X ())(Y()- μ Y ())] ρ σ ). The variables X(s) and Y(s) are (possibly) dependen bu no X(s) and X(s-j) or Y(s) and Y(s-j) or X(s) and Y(sj). A an unknown ime τ X here is a change in μ X and correspondingly for he process Y. Thus, for he same value of τ (τ X τ Y ), X and Y have he same disribuion. When ρ0, X and Y are independen, condiional on τ. The aim is o deec he firs change in eiher μ X 3

5 or μ Y, when hese processes may change a differen ime poins τ X and τ Y. We sudy he siuaion when he τ values are no idenical or have known lags). An early mulivariae surveillance mehod is he T mehod of (Hoelling 947). The covariance marix is assumed o be known, see e.g. (Al 985) ( x( s) μ ( s)) ( y( s) μ ( s)) ρ( x( s) μ ( s))( y( s) μ ( s)) T(s) + >k. ( ρ ) σ ( ρ ) σ ( ρ ) σ The alarm limi, k, is chosen o give a specified false alarm propery, e.g. a specific ARL 0. The ime of alarm, A, is defined as A min{s: T(s)>k}.. A measure of delay in mulivariae surveillance For an on-line sysem, he abiliy o deec a change quickly is imporan, i.e. we wan a shor delay of a moivaed alarm. For mos surveillance mehods, he delay of an alarm depends on when he change did occur, in relaion o he sar of he surveillance. In he univariae siuaion, he delay can be measured by he condiional expeced delay, defined as CED() E[ τ τ, τ ]. () A A Many evaluaions are made using only τ, e.g CED() which is equivalen o ARL -. However i is imporan o consider oher change poin imes also. In he mulivariae siuaion where we wan o deec he firs change, τ (), he delay depends on boh change poins, τ X and τ Y. In (Wessman 999) and (Andersson 007) he following delay measure was suggesed CED(, ) E[ A -τ () A τ (), τ X, τ Y ]. ().3 Resuls A simulaion sudy reveal he following, regarding he CED(, ) of he T mehod (he complee sudy is presened in (Andersson 007)). Below, CED curves for ρ{0, 0.5} are presened. 4

6 CED(, ) CED Figure 3: CED(, ) for T when {, 5, 0}. Lef: ρ0.0, righ: ρ0.5. The graphs above indicae ha, for T, he CED(, ) only depends on he disance ( - ). This will be generally proven below..3. The delay of he T mehod for a bivariae process The CED(, ) in () is based on he moivaed alarms, i.e. alarms afer ime τ (), so ha P ( ) CED(, ) ( i τ() ) P( A i A τ() ) ( τ() ) A i i. P ( τ τ () ) τ A.3.. Simulaneous change poins () Firs we consider he siuaion wih simulaneous changes, τ X τ Y. Then τ () and he condiional expeced delay equals CED(,) (( ) P( A ) + ( + ) P( A + ) +...) P ( A ) ( ) ( ) ( ) i P A i. P A The probabiliy in he denominaor, P ( A ), equals P ( A ) P ( A > ) PT ( () < k... T ( ) < k) PT ( ( i) < k). () 5

7 The probabiliy is independen of ime, so we denoe PT ( ( i) < k ) by p 0. Thus 0 P ( A ) PT ( ( i) < kμx( i) μy( i) μ ) (p 0 ) -. For he probabiliy in he nominaor, P ( A i), we have P ( A i) i PT ( ( j) < k) P( T ( i) > k). j We are ineresed in CED(,). For a value i, he probabiliy is divided ino ime poins before and ime poins afer 0 0 PT ( ( j) < kμ, μ ) * j i P( T ( j) < k μ ( j + ), μ ( j + )) * j P( T ( i) > k μ ( i + ), μ ( i + )). We denoe P( T ( j) < k μ ( j + ), μ ( j + )) by p (j-+) and P( T ( j) > k μ ( j + ), μ ( j + )) by q (j-+). Then P ( A i) ( ) 0 i p p( j + ) q ( i + ). j Thus, he condiional expeced delay equals CED(,) ( i ) P( A i) P ( A ) ( ) ( 0) ( ) ( ) i i p p j + q i + ( p0) i j i i p( j) q( i+ ). 0 j 6

8 Hence CED(,) is independen of. If μ is consan (i.e. μ (s-τ+) μ for s τ), he probabiliies are consan over ime PT ( ( j) < kμ, μ ) p, ( ( ) >, ) q, PT i kμ μ and for his siuaion he CED(,) equals i i ( p) q Differen change poins Second, we look a differen change imes, τ X and τ Y, where > and hence τ (), τ (). The condiional expeced delay equals CED(, ) ( i τ() ) P( A i A τ() ) τ () ( i ) P( A i) P ( A ) ( i ) P( A i) + ( i ) P( A i). P ( A ) Using he same noaion as above, we have P ( A ) (p 0 ) -. For he probabiliy P( A i), we have, for i< P ( A i) 0 0 PT ( ( j) < k μ, μ )* j i 0 PT ( ( j) < kμ ( j + ), μ )* j 0 PT ( ( i) > kμ ( i + ), μ ), 7

9 and for i P ( A i) 0 0 PT ( ( j) < k μ, μ )* j 0 PT ( ( j) < kμ ( j + ), μ )* j i P( T ( j) < k μ ( j + ), μ ( j + )) * j P( T ( i) > k μ ( i + ), μ ( i + )). Denoe he probabiliies by P( T ( j) < k μ ( j + ), μ ( j + )) p (j- +, j- +), 0 PT ( ( j) < kμ ( j + ), μ ) p 0 (j- +), P( T ( j) > k μ ( j + ), μ ( j + )) q (j- +, j- +), 0 PT ( ( j) > kμ ( j + ), μ ) q 0 (j- +). Thus, for i< and for i we have he wo following expressions P( A i) ( 0) i p p0( j + ) q 0( i + ) j and P ( A i) ( 0) i p p0( j + ) p( j +, j + ) q ( i +, i + ). j j 8

10 The expression for CED(, ) can be pariioned ino one sum for i< and anoher sum for i. The firs sum can be expressed as ( i ) P( A i) i ( i ) ( p0) p0( j + ) q 0( i + ) j i i ( p0) p0( j) q0( i + ). 0 j The second sum can be expressed as ( i ) P( A i) ( ) ( 0) i p p0( j + ) * j i p( j +, j + ) q ( i +, i + ) j ( 0) 0( ) i i p p j p( + j, j) q( i+, i ( ) + ). j j The complee expression for he condiional expeced delay is CED(, ) ( i ) P( A i) + ( i ) P( A i) P ( A ) 9

11 i i ( p0) p0( j) q0( i+ ) 0 j + ( p0 ) ( 0) 0( ) i i p p j p( + j, j) q( i+, i ( ) + ) j j ( p0 ) which equals i i p0( j) q0( i + ) + 0 j 0( ) i i p j p( + j, j) q( i+, i ( ) + ). j j Denoe - by c. Then he condiional expeced delay equals CED(, ) c () i i p0( j) q0( i + ) + 0 j () c 0( ) i i p j p( c+ j, j) q( i+, i c + ), c j j which is independen of and, and only depends on he disance ( - )c. For a consan μ (i.e. μ (s-τ+) μ for s τ), he probabiliies are consan over ime P( T ( j) < k μ ( j + ), μ ( j + )) p, 0 PT ( ( j) < kμ ( j + ), μ ) p 0, P( T ( j) > k μ ( j + ), μ ( j + )) q, 0 PT ( ( i) > kμ ( i + ), μ ) q 0, 0

12 and hen CED(, ) c i () i ( p0 ) q0 0 i i p0 p q. c c + () ( ) ( ) Thus when X is independen over ime and likewise wih Y, he CED(, ) for T only depends on he disance beween he change imes, - c.. Summary On-line monioring of mulivariae daa is considered and he siuaion when he processes under surveillance change a differen ime poins is sudied. In on-line monioring, he delay of an alarm is an imporan evaluaion measure. A measure of he expeced delay is suggesed, for he mulivariae siuaion. One approach o mulivariae surveillance is o reduce he daa a each ime poin, o a scalar and hen monior his scalar by univariae surveillance. Here we sudy one reducion, namely he Hoellings T. We prove ha he condiional expeced delay for T, in he siuaion wih wo processes, only depends on he disance beween he change imes. By using he T a each ime poin, we only include he informaion from he curren ime poin. A univariae correspondence is he Shewhar mehod (for ime independen daa), where only he curren observaion is used. I has been shown, e.g. in (Frisén and Wessman 999), ha he condiional expeced delay for he Shewhar mehod, is independen of he ime of change (i.e. CED(i) in () is consan over differen values of i). Acknowledgemen This work was suppored by he Swedish Emergency Managemen Agency.

13 References Al, F. B. (985), Mulivariae Qualiy Conrol, ed. K. S. Johnson N L, Wiley. Andersson, E. (004), "The Impac of Inensiy in Surveillance of Cyclical Processes," Communicaions in Saisics-Simulaion and Compuaion, 33, Andersson, E. (007), "Effec of Dependency in Sysems for Mulivariae Surveillance," Research Repor 007:, Saisical Research Uni, Göeborg Universiy, Sweden. Andersson, E., Bock, D., and Frisén, M. (004), "Deecion of Turning Poins in Business Cycles," Journal of Business Cycle Measuremen and Analysis,, Baron, M. (00), "Bayes and Asympoically Poinwise Opimal Sopping Rules for he Deecion of Influenza Epidemics.," in Case Sudies in Bayesian Saisics (Vol. 6), eds. C. Gasonis, R. E. Kass, A. Carriquiry, A. Gelman, D. Higdon, D. K. Pauler and I. Verdinelli, New York: Springer-Verlag, pp Bock, D., Andersson, E., and Frisén, M. (007), "The Relaion beween Saisical Surveillance and Technical Analysis in Finance," in Financial Surveillance, ed. M. Frisén, Chicheser: Wiley, pp Frisén, M. (99), "Evaluaions of Mehods for Saisical Surveillance," Saisics in Medicine,, Frisén, M. (007), "Properies and Use of he Shewhar Mehod and Followers," Sequenial analysis, 6, Frisén, M., and Wessman, P. (999), "Evaluaions of Likelihood Raio Mehods for Surveillance. Differences and Robusness," Communicaions in Saisics-Simulaions and Compuaions, 8, Hamilon, J. D. (989), "A New Approach o he Economic Analysis of Nonsaionary Time Series and he Business Cycle," Economerica, 57, Hoelling, H. (947), "Mulivariae Qualiy Conrol," in Techniques of Saisical Analysis, eds. C. Eisenhar, M. W. Hasay and W. A. Wallis, New York: McGraw-Hill. Järpe, E. (999), "Surveillance of Spaial Paerns.," Communicaions in Saisics. Theory and Mehods, 8, Nefci, S. (98), "Opimal Predicion of Cyclical Downurns," Journal of Economic Dynamics and Conrol, 4, 5-4.

14 Pezold, M., Sonesson, C., Bergman, E., and Kieler, H. (004), "Surveillance in Longiudinal Models. Deecion of Inra-Uerine Growh Resricion," Biomerics, 60, Royson, P. (99), "Idenifying he Ferile Phase of he Human Mensrual Cycle," Saisics in Medicine, 0, -40. Shewhar, W. A. (93), Economic Conrol of Qualiy of Manufacured Produc, London: MacMillan and Co. Sonesson, C., and Frisén, M. (005), "Mulivariae Surveillance," in Spaial Surveillance for Public Healh, eds. A. Lawson and K. Kleinman, Wiley. Wessman, P. (999), "The Surveillance of Several Processes wih Differen Change Poins," Research Repor 999:, Deparmen of Saisics, Göeborg Universiy, Sweden. 3

15 Research Repor 007:3 Bock, D.: Consequences of using he probabiliy of a false alarm as he false alarm measure. 007:4 Frisén, M.: Principles for Mulivariae Surveillance. 007:5 Andersson, E., Bock, D. & Frisén, M.: 007:6 Bock, D., Andersson, E. & Frisén, M.: 007:7 Andersson, E., Kühlmann-Berenzon, S., Linde, A., Schiöler, L., Rubinova, S. & Frisén, M.: 007:8 Bock, D., Andersson, E. & Frisén, M.: Modeling influenza incidence for he purpose of on-line monioring. Saisical Surveillance of Epidemics: Peak Deecion of Influenza in Sweden. Predicions by early indicaors of he ime and heigh of yearly influenza oubreaks in Sweden. Similariies and differences beween saisical surveillance and cerain decision rules in finance. 007:9 Bock, D.: Evaluaions of likelihood based surveillance of volailiy. 007:0 Bock, D. & Peersson, K. 007: Frisén, M. & Andersson, E. 007: Frisén, M., Andersson, E. & Schiöler, L. 007:3 Frisén, M., Andersson, E. & Peersson, K. Exploraive analysis of spaial aspecs on he Swedish influenza daa. Semiparameric surveillance of oubreaks. Robus oubreak surveillance of epidemics in Sweden. Semiparameric esimaion of oubreak regression. 007:4 Peersson, K. Unimodal regression in he wo-parameer exponenial family wih consan or known dispersion parameer 007:5 Peersson, K. On curve esimaion under order resricions 008: Frisén, M. Inroducion o financial surveillance 008: Jonsson, R When does Heckman s wo-sep procedure for censored daa work and when does i no?