Technical Report / Universität Dortmund, SFB 475 Komplexitätsreduktion in Multivariaten Datenstrukturen, No. 2006,23

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1 econsor Der Open-Access-Publikaionsserver der ZBW Leibniz-Informaionszenrum Wirschaf The Open Access Publicaion Server of he ZBW Leibniz Informaion Cenre for Economics Schelinger, Karen; Fried, Roland; Gaher, Ursula Working Paper Robus Filers for Inensive Care Monioring: Beyond he Running Median Technical Repor / Universiä Dormund, SFB 475 Komplexiäsredukion in Mulivariaen Daensrukuren, No. 2006,23 Provided in Cooperaion wih: Collaboraive Research Cener 'Reducion of Complexiy in Mulivariae Daa Srucures' (SFB 475), Universiy of Dormund Suggesed Ciaion: Schelinger, Karen; Fried, Roland; Gaher, Ursula (2006) : Robus Filers for Inensive Care Monioring: Beyond he Running Median, Technical Repor / Universiä Dormund, SFB 475 Komplexiäsredukion in Mulivariaen Daensrukuren, No. 2006,23 This Version is available a: hp://hdl.handle.ne/10419/22666 Nuzungsbedingungen: Die ZBW räum Ihnen als Nuzerin/Nuzer das unengelliche, räumlich unbeschränke und zeilich auf die Dauer des Schuzrechs beschränke einfache Rech ein, das ausgewähle Werk im Rahmen der uner hp:// nachzulesenden vollsändigen Nuzungsbedingungen zu vervielfäligen, mi denen die Nuzerin/der Nuzer sich durch die erse Nuzung einversanden erklär. Terms of use: The ZBW grans you, he user, he non-exclusive righ o use he seleced work free of charge, erriorially unresriced and wihin he ime limi of he erm of he propery righs according o he erms specified a hp:// By he firs use of he seleced work he user agrees and declares o comply wih hese erms of use. zbw Leibniz-Informaionszenrum Wirschaf Leibniz Informaion Cenre for Economics

2 Robus Filers for Inensive Care Monioring Beyond he Running Median Karen Schelinger Roland Fried Ursula Gaher Fachbereich Saisik Universiä Dormund Dormund Absrac Curren alarm sysems on inensive care unis creae a very high rae of false posiive alarms because mos of hem simply compare he physiological measuremens o fixed hresholds. An improvemen can be expeced when he acual measuremens are replaced by smoohed esimaes of he underlying signal. However, classical filering procedures are no appropriae for signal exracion as sandard assumpions, like saionariy, do no hold here: he measured ime series ofen show long periods wihou change, bu also upward or downward rends, sudden shifs and numerous large measuremen arefacs. Alernaive approaches are needed o exrac he relevan informaion from he daa, i.e. he underlying signal of he moniored variables and he relevan paerns of change, like abrup shifs and rends. This aricle reviews recen research on filer based online signal exracion mehods which are designed for applicaion in inensive care. 1

3 1 Inroducion Monioring sysems in inensive care need o be credible ools for judging he sae of he criically ill. Apar from he acual measuremens, relevan paerns of change, like abrup shifs or monoonic rends, conain essenial informaion abou a paien's condiion. Therefore, mehods are required for he reliable exracion of his informaion from he daa. A he same ime he mehods have o be able o deal wih many arefacs and irrelevan minor flucuaions. Mos alarm sysems currenly used for he haemodynamic monioring in inensive care are essenially based on hresholds: violaions of he upper or lower conrol limi acivae an alarm someimes afer a cerain offse ime. For example, he monioring sysem we sudy here riggers an alarm for he sysolic arerial blood pressure if he measuremens exceed he upper conrol limi or fall below he lower conrol limi for a leas four seconds. This offse ime is one possibiliy o make he sysem robus agains single measuremen arefacs. However, experience wih real daa ses suggess ha in pracice such arefacs also occur in 'paches' of several consecuive values. Thus, even his proceeding does no compleely avoid false alarms (see Fig. 1) Sysolic Arerial Blood Pressure Upper and Lower Conrol Limis Alarms mmhg :00 10:05 10:10 10:15 10:20 10:25 10:30 Time Figure 1 Curren alarm sysems rigger false alarms, e.g. because of measuremen arefacs. 2

4 Pre-processing he inpu daa for an alarm sysem by robus online filering can be expeced o yield considerably less false alarms as i does no only remove single bu also shor paches of arefacs. We review some robus versions from he broad variey of filering mehods, exhibiing cerain characerisics which are desirable in he online monioring conex. Measuremens ( y ) Z from a physiological variable, recorded in shor ime inervals up o once per second, can be represened by a simple 'signal plus noise' model Here, µ y = µ + u for Z. symbolises he underlying 'rue' biosignal, which is assumed o vary smoohly wih a few sudden changes, and defines he noise componen. This componen can conain large aberran values, e.g. due o measuremen arefacs, which are called 'ouliers' in he saisical lieraure. A simple approach for exracing he signal u is o use a moving average: In ime windows of fixed lengh n he average of he observaions is calculaed for esimaion of he signal in he window cenre. Moving averages are popular since hey race rends and are very efficien for Gaussian samples. However, sudden level shifs are 'smeared' and ouliers can cause a considerable bias (see Fig. 2). A running median, as suggesed by Tukey [1], is robus agains ouliers and capable of racing level shifs. This filer can resis up o [n/2] ouliers wihin one ime window bu i deerioraes in rend periods (see also Fig. 2). In he following, we review filering echniques for signal exracion which are robus agains ouliers bu addiionally capable of racing rends, rend changes and level shifs. We compare hese mehods by applicaions o inensive care daa, discuss heir performance in siuaions which are of paricular ineres in he online monioring conex, and poin ou furher demands for fuure research. µ 3

5 1/min Pulse Measuremens Moving Average Running Median 17:15 17:17 17:19 17:21 17:23 17:25 Time Figure 2 Moving average and running median applied o a ime series of pulse measuremens from inensive care. 2 Signal Exracion Mehods 2.1 Simple Robus Regression Filers In view of he weakness of he running medians in rend periods, Davies, Fried and Gaher [2] achieve a beer adapaion o emporal rends by assuming he signal o be locally linear insead of locally consan. This means, wihin a ime window cenred a ime poin he following model is applied: y i = µ + β i + ε, i + for i = -m,...,m, where again represens he underlying signal, and β is he slope in he µ window cenre, while ε, i describes he noise componen. Sandard mehods for he esimaion of and β such as leas squares µ regression are no suiable in he presence of ouliers. I is raher advisable o apply robus regression mehods which are able o deal wih a cerain amoun of conaminaion wihou becoming srongly affeced. Denoing he residuals in a window by r ( + i = y + i µ + βi) for i = -m,...,m, 4

6 Davies, Fried and Gaher [2] survey he following echniques for esimaing and β : µ - L 1 Regression m ) = arg min r µ, β i= m L1 L1 ( µ, β + i - Leas Median of Squares (LMS) Regression [3] ( µ LMS, β LMS ) = arg min{ med µ, β - Repeaed Median (RM) Regression [4] i= m,..., m ( r 2 + i )} β RM = y + i + j med med i = m,..., m j= m,..., m; j i i j µ RM = med { y+ i= m,..., m i y β RM i} The LMS filer offers he highes robusness agains many large ouliers and is able o rack level shifs and rend changes well. The RM filer slighly smoohes such changes. Neverheless, he repeaed median is considered he bes choice for signal exracion because i does no only offer considerable robusness agains ouliers, bu i is also sable w.r.. moderae variaions in he daa. Addiionally, compuaion of he RM filer is much faser: In [5] an algorihm for he RM regression line is presened which only needs linear ime for an updae. Here he erm 'updae' means ha esimaion akes place by using he sored informaion from he las ime window only insering he new informaion given by he mos curren daa poin and deleing ha of he oldes daa poin. Thus, updae algorihms save a lo of compuaion ime as he esimaes do no have o be calculaed for each window from scrach. Esimaion of he parameers and β in he cenre of he ime window µ means a delay of m ime unis for he filer oupu. Taking ino accoun he urgency of reliable oupu on inensive care unis, only minimal delays are accepable. Thus, signal exracion as described above is raher suiable for rerospecive analyses when applying a large window widh. 5

7 For signal exracion wihou ime delay, Gaher, Schelinger and Fried [6] examine he online esimaes online = µ + β m µ + m, esimaing he signal value a he mos recen ime poin. Since boh RM and LMS regression show cerain advanages in [2], hese mehods are considered again and compared o wo furher robus regression mehods in he online siuaion: leas rimmed squares (LTS) regression [7] and deepes regression (DR) [8]. I urns ou ha he differences in he oucomes beween LMS and LTS regression are negligible, and also ha here is lile difference beween he repeaed median and deepes regression filers (see Fig. 3). In he online siuaion, LMS and LTS rack shifs wih a longer delay han heir compeiors and end o overshoo shifs, while RM and DR show more sable resuls (see also Fig. 3). Considering he compuaional speed, again he repeaed median is recommended for applicaions in inensive care. mmhg mmhg Mean Pulmonary Arery Blood Pressure LMS Filer LTS Filer Time Mean Pulmonary Arery Blood Pressure RM Filer DR Filer Time Figure 3 Online signals exraced wih four differen regression filers. 6

8 2.2 Repeaed Median Hybrid Filers As poined ou above, a simple RM filer does no preserve sudden level shifs as such bu 'smears' hem somewha [2]. Heinonen and Neuvo [9], [10] emphasise he advanages of linear median hybrid filers for preserving such signal edges. FIR median hybrid (FMH) filers are compuaionally even less demanding han running medians and preserve shifs similarly good or even beer han hese. An FMH filer is defined as he median of several linear subfilers: { Φ, Φ, Φ } FMH( y ) = med K, 1 2 M. For signal exracion from blood pressure measuremens, Heinonen, Kalli, Turjanmaa and Neuvo [11] use a simple FMH filer wih M = 3 subfilers, consising of wo one-sided moving averages and he cenral observaion as cenral subfiler: Φ m 1 1 ( y ) = m y i, Φ ( y ) = y i = 1 1 2, Φ 3 ( y ) = m y+ i. Similar o running medians, such simple FMH filers assume he signal o be locally consan. Predicive FMH (PFMH) filers use one-sided weighed averages insead of ordinary half-window averages for racking linear rends [10]. Combined FMH filers, finally, combine he srucures for a local consan and for a local linear signal. However, hese filers can only remove single isolaed ouliers and hence, hey are no sufficienly robus for applicaions in inensive care. Fried, Bernhol and Gaher [12] consruc hybrid filers based on RM regression o combine he robusness of he repeaed median wih he beer shif preservaion of FMH filers. Several filers are invesigaed, using eiher he cenral observaion y or he median of all observaions in he window µ as cenral subfiler. Insead of one-sided means hey use one sided medians m i = 1 F µ med{ y,..., y } and B med{ y,..., y }, = m 1 µ = m 7

9 and insead of he one-sided weighed averages hey apply one-sided RM filers F F F RM med{ y + mβ,..., y + β } where B RM = m 1 F β is he RM slope esimae based on he observaions y -m,...,y -1, and is defined analogously for he oher half of he window. Since hese subfilers make predicions for he cenral value, he procedures are called 'predicive' or 'combined' if boh, median and RM subfilers, are used. In general RM based filers are no affeced by rends and aenuae Gaussian as well as spiky noise well. The smoohes signal esimaions are obained by he ordinary RM filer, bu on he oher hand i also smoohes ou shifs and rend changes. In conras, he predicive RM hybrid filer PRMH ( y ) = med{ RM F, y, RM can preserve rend changes and level shifs almos exacly even wihin rends bu i aenuaes Gaussian noise less efficienly, and like he oher RM hybrid filers i is more affeced by many ouliers. Also, RM hybrid filers are designed for delayed signal exracion and hence, for online signal exracion differen subfilers had o be applied. B } 2.3 Nesed Filers An approach for combining he smoohness of he moving average wih he robusness and shif preservaion of he running median is given by modified rimmed means (MTM) [13]. The idea is o calculae he median of all observaions in he window and hen 'rim', i.e. discard, hose observaions which deviae more han a specified muliple of a robus scale esimae, e.g. he median absolue deviaion abou he median (MAD) [14] MAD σ + i i = m,..., m = med { r }. The arihmeic mean of he remaining observaions is hen aken as signal esimae in he cenre of he ime window. These MTM esimaes are boh robus agains ouliers and efficien for Gaussian noise. Also, hey can preserve large shifs in an oherwise consan level beer han ordinary running medians [15]. 8

10 Since he locaion-based MTM deerioraes in rend periods Gaher and Fried [16] exend his idea o he rimmed repeaed median (TRM): Wihin each ime window a RM regression line is fied and he MAD calculaed from is residuals for esimaing he local variabiliy [17]. Observaions deviaing more han a muliple of he residual MAD from he fied line are rimmed, and he final signal esimae is derived by a leas squares fi o he remaining observaions. This TRM filer is almos as robus as a varian applying anoher RM regression in he second sep, bu i is more efficien for Gaussian errors. To furher improve he preservaion of shifs, Bernhol, Fried, Gaher and Wegener [18] use a smaller window widh in he firs sep for he iniial RM fi. Because of he nesed design of he windows for he firs and he second regression sep, he prefix 'double window' (DW) is added o he esimaes which resuls in DWMRM and DWTRM (see Fig. 4) Iniial RM Fi Trimming Boundaries Final Leas Squares Fi o he Trimmed Daa Signal Esimae Inner window Figure 4 DWTRM fi o a single ime window of widh n=31: In he second sep only he observaions wihin he rimming boundaries around he RM line are used o calculae he leas squares fi. 9

11 Using his double window echnique considerably improves he performance of he RM filers concerning he preservaion of shifs. In general, shifs which are large relaive o he observaional noise are raced more accuraely han smaller shifs. If he applicaion allows for a relaively large ouer window widh, he signal esimaion can also be improved by using a shor inner window for he iniial RM slope esimaion and a larger ouer window for he level esimaion. Firs experiences show ha he DWTRM filer seems even more promising for delayed signal exracion keeping in mind he demands for robusness and he allowable ime delay. Ye, hese mehods have no been invesigaed carefully in full online analysis. 2.4 Weighed Repeaed Median Filers In analogy o he popular weighed median (WM) filers, Fried, Einbeck and Gaher [19] consruc weighed repeaed median (WRM) filers. While he former are based on he idea ha a consan level is more likely for close-by observaions, he laer filers assume he signal slope o be more likely o be he same in shor ime lags. Suiable symmeric bell-shaped (in delayed/rerospecive analysis) or monoonic (in full online analysis) weighing schemes allow o use longer ime windows han ordinary running medians or RM filers which correspond o uniform weighs. Considering n observaions (x i,y i ), i=1,...,n, where he x i are no necessarily equidisan, and wo ses of ineger weighs w i and repeaed median is defined by µ β WRM WRM ( x) = med w j= 1,..., n ( x) = med j= 1,..., n j yi y o med w i o i j xi x j j WRM { w o ( y ( x x) β ( x) )}. j j j w j, he weighed Here, he operaor symbolises replicaion, i.e. w i y i means ha y i is replicaed w i imes. This newly defined mehod is hen compared o L 1 and weighed L 1 filers. Among oher hings, he sudy deermines he minimal window widh which is necessary for he invesigaed mehods o resis a cerain number h of 10

12 successive ouliers, while aking hese devian values ino accoun if heir number is larger han h. The reason for his lies in he fac ha, moving a ime window hrough a series of measuremens, a some poin he ime series conains h subsequen ouliers or 'spikes' (which are sill regarded as a sequence of arefacs) while in he subsequen ime window he presence of h+1 successive ouliers wih he same size and sign may already indicae a shif [20]. In his way, window widhs are deermined which allow for racking shifs lasing a leas h+1 observaions while eliminaing a smaller number of ouliers. For he RM weighing improves he adjusmen o nonlinear rends, allows for larger window widhs, and increases he efficiency, while for he L 1 filer weighing can increase robusness and efficiency. For online signal exracion, he WRM filer racks shifs beer han he L 1 filer, which has some difficulies in disinguishing relevan from irrelevan paerns. The weighing reduces he bias of he RM, implying ha he WRM also ouperforms he sandard RM filer in racing shifs. Also, he WRM filer shows generally he smoohes signal esimaions in applicaion o ime series, while he L 1 filer overshoos shifs and is wiggly. In conclusion, a suiably designed weighed RM filer can be recommended for online signal exracion. In he rerospecive siuaion he weighed L 1 filers provide even beer resuls han he WRM filers. Paricularly for moderae ouliers, weighed L 1 filers show he leas biased resuls and furher hey race large shifs wih a smaller ime delay. However, if i is possible ha several oulier paches occur close o each oher and hus inrude ino he same ime window, he sandard RM filer may sill be he bes choice because of is maximal breakdown poin. 2.5 Exended Robus Regression Filers In conras o LMS filers, RM filers are more vulnerable o large ouliers while hey accommodae small ouliers well (see e.g. [16] and [17]). Also, large ouliers are usually easier o deec han small ones. Therefore, i is worhwhile o add auomaic rules for oulier deecion and replacemen o he repeaed median o increase he robusness of he signal esimaion [21]. Likewise we can apply auomaic rules for level shif deecion o he RM filers invesigaed in [2] and [6]. 11

13 Similarly o he nesed filers approach, an observaion is regarded as oulier if he corresponding absolue deviaion from he curren regression line is larger han a specified muliple d of a robus scale esimaion, i.e. if r i d σ + >. However, here only he nex, incoming observaion is screened for oulyingness before enering he acualised ime window by exrapolaing he previous regression line. Deeced ouliers are replaced and no longer considered in he following analysis. In his way hey lose heir influence on he esimaions. For cerain 'wors case' scenarios, replacing ouliers by he simple exrapolaion of he regression line, gives beer resuls han oher down-sizing replacemen sraegies, for he price of reduced Gaussian efficiency. For he scale esimaion, several robus esimaors are invesigaed and heir respecive advanages elaboraed. These are, in addiion o he MAD (see Secion 2.3), Rousseeuw's and Croux' S n and Q n esimaors [22] and he 'lengh of he shores half' (LSH) [23], [24]. The Q n and he LSH scale esimaor give he bes resuls in case of many large ouliers of similar size, bu he Q n provides beer efficiency, especially when idenical measuremens occur, e.g. due o rounding Pulse Measuremens Simple RM Filer Exended RM Filer 1/min :05 11:15 11:25 11:35 11:45 11:55 Time Figure 5 Comparison of he simple (delayed) RM filer wih is exended version including oulier and shif deecion. 12

14 For shif deecion, a simple majoriy rule is added o he filering procedure: Considering he mos recen m observaions in he ime window, he number of observaions wih residuals larger han a cerain bound and same sign is couned. If his number exceeds m/2, his indicaes a level shif and he procedure moves o he nex window no overlapping he curren one. This rule enables he regression filers o deec and hus preserve shifs, and hence i overcomes he bigges disadvanage of he RM filer (see Fig. 5). Also, he delay in following shifs decreases ideally o a minimal delay of m/2 +1 ime unis. In his conex, regression based filers wih addiional shif deecion rules seem preferable o oher shif preserving procedures like LMS or FMH filers. Furher, some simple rules can be added o overcome problems in he infrequen case ha oo many observaions are idenified as ouliers and hus replaced [21]. The rules for oulier reamen and shif deecion can also be applied for online signal exracion. However, he minimal delay of shif deecion canno be reduced furher because of he necessary differeniaion of shifs and oulier paches. 3 Applicaions The differen approaches described in he previous secions, produced differen filers which seem promising for applicaion o online monioring daa from inensive care. Especially filers based on he repeaed median show good resuls. However, he choice of he appropriae filer should depend on he characerisics of he underlying signal whenever known. Summarising he oucomes described above, he following recommendaions can be given: For rerospecive signal exracion he predicive RM hybrid (PRMH) filer (Sec. 2.2) seems o be he bes choice if he signal is assumed o conain many jumps and rend changes, while he simple RM filer (Sec. 2.1) yields beer resuls if many ouliers bu no abrup changes are expeced. A compromise beween hese wo mehods is given by he DWTRM filer (Sec. 2.3), while he RM filer in combinaion wih addiional rules (Sec. 2.5) works beer han he simple RM filer in he occurrence of many oulier paches and level shifs (see Fig. 5). 13

15 160 Sysolic Arerial Blood Pressure RM Filer DWTRM Filer 140 PRMH Filer mmhg :30 14:35 14:40 14:45 14:50 14:55 15:00 Time Figure 6 Comparison of a nesed RM (DWTRM) and an RM hybrid filer (PRMH) wih he simple RM filer for rerospecive applicaion. For online signal exracion, he weighed version of he RM filer (Sec. 2.4) seems he bes choice a he momen bu anoher promising approach based on he RM is currenly under research, adaping he window widh a each ime poin. To provide a comparison of he specific benefis of he proposed filers, we presen some applicaions o inensive care ime series here. The comparison of he simple RM filer wih is exended version (Sec. 2.5) in Fig. 5 shows how much a shif deecion rule can improve he RM filer. However, local exremes, i.e. sudden rend changes, canno be raced as well wih his exended RM filer. In ha case, he applicaion of he PRMH (Sec. 2.2) or he DWTRM filer (Sec. 2.3) is more recommendable (see Fig. 6). In Fig. 6 we see ha he predicive RM hybrid filer (PRMH) races he sudden shifs and local exremes very accuraely. However, he PRMH signal shows he larges variabiliy, especially in relaively consan periods, e.g. from 14:30h o 14:40h here. The simple RM filer oupu is he smoohes bu smears sudden shifs and 'cus' local exremes. As poined ou before, he DWTRM signal is a compromise beween he RM and he PRMH filer oupu: I is smooher han he PRMH signal bu races rend changes and shifs beer han he RM filer. 14

16 Real-ime applicaion of hese filers implies a ime delay of half he window widh used for he signal exracion. Therefore, filers have been examined for heir online applicaion wihou any ime delay. As displayed in Fig. 3, simple regression filers (Sec. 2.1) are suiable for his purpose bu even he online version of he RM filer sill possesses some disadvanages such as he slow reacion o level shifs. Weighed RM filers (WRM, Sec. 2.4), which are under curren research, can possibly improve upon simple online RM filers. Online signal exracion by such mehods can be used for improving monioring sysems for haemodynamic variables: Basing an alarm sysem on he signal insead of he acual measuremens would no rigger alarms a he occurrence of single measuremen arefacs or irrelevan paches of ouliers. Therefore, we claim ha such an alarm sysem will have a lower false alarm rae han alarm sysems based on raw measuremens. Fig. 7 shows ha basing he alarm sysem on he exraced RM online signal avoids false alarms due o arefacs. However, he sysem would sill reac o he sudden change of hear rae around 15:00h Hear Rae Signal from RM Filer Upper Conrol Limi 1/min :10 14:20 14:30 14:40 14:50 15:00 15:10 Time Figure 7 A filer based alarm sysem does no rigger alarms in case of measuremen arefacs. 15

17 Sysolic Arerial Blood Pressure Upper and Lower Conrol Limis 23 Alarms wih he Curren Sysem mmhg :30 12:35 12:40 12:45 12:50 12:55 13:00 Time Figure 8 Depending on he alarm seings, he overall alarm rae of curren monioring sysems can be very high even in he absence of ouliers Sysolic Arerial Blood Pressure Upper and Lower Conrol Limis 7 Alarms wih he Filer Based Sysem mmhg :30 12:35 12:40 12:45 12:50 12:55 13:00 Time Figure 9 A filer based alarm sysem can reduce he overall alarm rae drasically. 16

18 Furher, an alarm sysem based on he daa signal can also reduce he overall alarm rae: Fig. 8 shows half an hour of sysolic arerial blood pressure measuremens wih he alarm seings and alarms riggered by he currenly used alarm sysem. In Fig. 9 i is demonsraed ha alhough hese daa do no conain any measuremen arefacs an alarm sysem based on he filer oupu can reduce his high alarm rae considerably. This resul does no imply a decrease of he false alarm rae. However, a decrease of he number of oal alarms may also be considered an improvemen. One should noe ha he figures and applicaions given here are only exemplary. The general superioriy of a filer based alarm sysem has ye o be shown based on a sensible and careful definiion of he erm 'false alarm'. The full assessmen of such a new alarm sysem is one aim of a currenly running clinical sudy conduced a he Hospial of he Universiy of Regensburg and suppored by he collaboraive research cenre SFB 475 a he Universiy of Dormund. 4 Discussion The mehods recommended for univariae signal exracion here, are based on a simple linear regression approach. The ordinary repeaed median regression filer improves on running medians in rend periods bu lacks he propery of preserving sudden shifs. Differen approaches o overcome his problem have been proposed and work well for paricular siuaions, bu here is no universal procedure wihou any deficiencies. Double window TRM filers are promising for delayed signal exracion while weighed RM filers are hopeful candidaes for he online analysis. Furher invesigaions show ha median based filers are also robus agains he presence of auocorrelaions. Compared o procedures based on leas squares, robus locaion or regression filers, based on he median, rimmed means or he repeaed median respecively, gain relaive efficiency in he frequen case of posiive correlaions. In ha case, hey also ouperform filers incorporaing he auocorrelaions explicily ino he analysis (Fried [25], Fried and Gaher [26]). In he infrequen siuaion of srong negaive auocorrelaions a Prais-Winsen ransformaion of he daa is worhwhile and improves he ordinary RM filer. 17

19 Wih he linear ime RM updae algorihm developed by Bernhol and Fried [5] or is advancemen o a linear sorage algorihm [12], he RM filer is compuaionally feasible even for high frequency daa. This updae algorihm also means linear compuaion ime for all of he recommended filers. Oulier and shif deecion as described in Secion 2.5 does no add furher compuaion ime when using e.g. an O(log n) MAD updae algorihm [18] for he scale esimaion. Thus, an RM filer wih such exensions offers an accepable choice for signal exracion. Anoher approach o improve he compuaional speed is proposed by Fried and Gaher [27]. Dividing he ime window ino n 2 disjoin segmens, each of lengh n 1, he level wihin each segmen is esimaed by an ordinary median or by repeaed median regression. Then he RM, or anoher procedure, can be applied o his pre-processed oupu window of widh n2. Hence, he compuaion ime can be shrunk by a facor when using a n 1 linear ime algorihm and by 2 n 1 when using an algorihm needing quadraic ime. For rerospecive analyses, compuaion imes are no as crucial as for realime applicaions bu hey are sill imporan because of he possible magniude of he daa ses. Hence, he compuabiliy of he filers should always be aken ino accoun when choosing he 'righ' filer. For he filering procedures described above, he window widh n = 2m + 1 has been assumed o be fixed hroughou. The suiable choice of he widh is no rivial ask and depends on saisical as well as medical demands. Larger window widhs generally imply a smooher filer oupu, bu hey also increase he bias a shifs. To ackle he problem of accurae racing of level shifs Gaher and Fried [16] inroduce a procedure wih a variable, daa-adapive choice of he window widh for delayed signal exracion wih he RM filer. This adapive procedure improves he racing of rend changes and level shifs, while sill achieving smooh signal esimaions and high robusness. Therefore, i is of special ineres for fuure research o conver his mehod o he siuaion of signal exracion wihou any ime delay o make i applicable for online monioring in inensive care. 18

20 Using exraced signals for judging he paien's sae of healh may improve he work on inensive care unis noably. However, he superioriy of filer based alarm sysems is a ask ye o be compleed, and here is sill need for research on fully adapable signal exracion mehods bes suiable in he online monioring conex. Neverheless, he presened signal exracion filers provide a solid background for he developmen of enhanced alarm sysems wih lower false alarm raes. Acknowledgemens We graefully acknowledge he suppor of he German Research Foundaion DFG hrough he collaboraive research cenre SFB 475 "Reducion of Complexiy for Mulivariae Daa Srucures". 19

21 4 References [1] Tukey, J.W.: Exploraory Daa Analysis, Reading, MA, Addison- Wesley, [2] Davies, P.L.; Fried, R.; Gaher, U.: Robus Signal Exracion for Online Monioring Daa, Journal of Saisical Planning and Inference 122 (2004), pp [3] Rousseeuw, P.J.: Leas Median of Squares Regression, Journal of he American Saisical Associaion 79 (1984), No. 388, pp [4] Siegel, A.F.: Robus Regression Using Repeaed Medians, Biomerika 69 (1982), pp [5] Bernhol, T.; Fried, R.: Compuing he Updae of he Repeaed Median Regression Line in Linear Time, Informaion Processing Leers 88 (3) (2003), pp [6] Gaher, U.; Schelinger, K.; Fried, R.: Online Signal Exracion by Robus Linear Regression, o appear in: Compuaional Saisics, Issue 02/2006. [7] Rousseeuw, P.J.: Mulivariae Esimaion wih High Breakdown Poin, Proceedings of he 4h Pannonian Symposium on Mahemaical Saisics and Probabiliy Vol. B, Dordrech: Reidel, [8] Rousseeuw, P.J.; Huber, M.: Regression Deph, Journal of he American Saisical Associaion 94 (1999), No. 446, pp [9] Heinonen, P., Neuvo, Y.: FIR-Median Hybrid Filers, IEEE Transacions on Acousics, Speech, and Signal Processing 35 (1987), pp [10] Heinonen, P., Neuvo, Y.: FIR-Median Hybrid Filers wih Predicive FIR Subsrucures, IEEE Transacions on Acousics, Speech, and Signal Processing 36 (1988), pp

22 [11] Heinonen, P., Kalli, S., Turjanmaa, V., Neuvo, Y.: Generalized Median Filers for Biological Signal Processing, Proceedings of he Sevenh European Conference on Circui Theory and Design (1985), pp [12] Fried, R., Bernhol, T., Gaher, U.: Repeaed Median and Hybrid Filers, o appear in: Compuaional Saisics and Daa Analysis, Special Issue on 'Saisical Signal Exracion and Filering' (2006). [13] Lee, Y., Kassam, S.A.: Generalized Median Filering and Relaed Nonlinear Filering Techniques, IEEE Transacions on Acousics, Speech and Signal Processing 33 (1985), pp [14] Huber, P.J.: Robus Saisics, Wiley, New York, [15] Himaya, N., Kassam, S.A.: Approximae Performance Analysis of Edge Preserving Filers, IEEE Transacions on Signal Processing 4 (1993), pp [16] Gaher, U., Fried, R.: Mehods and Algorihms for Robus Filering, Proc. in Compuaional Saisics, J. Anoch (ed.), Physica, Heidelberg, 2004, pp [17] Gaher, U., Fried, R.: Robus Esimaion of Scale for Local Linear Temporal Trends, Tara Mounains Mahemaical Publicaions 26 (2003), pp [18] Bernhol, T., Fried, R., Gaher, U., Wegener, I.: Modified Repeaed Median Filers, condiionally acceped for Saisics and Compuing (2004). [19] Fried, R., Einbeck, J., Gaher, U.: Weighed Repeaed Median Smoohing and Filering, submied o Journal of he American Saisical Associaion (2005). [20] Imhoff, M., Bauer, M., Gaher, U., Fried, R.: Paern Deecion in Inensive Care Monioring Time Series wih Auoregressive Models: Influence of he Model Order, Biomerical Journal 44-6 (2002), pp [21] Fried, R.: Robus Filering of Time Series wih Trends, Journal of Nonparameric Saisics 16 (3) (2004), pp

23 [22] Rousseeuw, P.J., Croux, C.: Alernaives o he Median Absolue Deviaion, Journal of he American Saisical Associaion 88 (1993), pp [23] Grübel, R.: Lengh of he Shorh, Annals of Saisics 16 (1988), pp [24] Rousseeuw, P.J., Leroy, A.: A Robus Scale Esimaor Based on he Shores Half, Saisica Neerlandica 42 (1988), pp [25] Fried, R.: Robus Locaion Esimaion under Dependence, o appear in: Journal of Saisical Compuaion and Simulaion (2006). [26] Fried, R., Gaher, U.: Robus Trend Esimaion for AR(1) Disurbances, Ausrian Journal of Saisics, Vol. 34 (2) (2005), pp [27] Fried, R., Gaher, U.: Fas and Robus Filering of Time Series wih Trends, Proc. in Compuaional Saisics, W. Härdle and B. Rönz (eds.), Physica, Heidelberg, 2002, pp