Classification of oximetry signals using Bayesian neural networks to assist in the detection of the obstructive sleep apnoea syndrome

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1 Classificatio of oximetry sigals usig Bayesia eural etworks to assist i the detectio of the obstructive sleep apoea sydrome JV Marcos 1, R Horero 1, D Álvarez 1, IT Nabey 2, F del Campo 3, C Zamarró 4 1 Biomedical Egieerig Group, E.T.S.I. de Telecomuicació, Uiversity of Valladolid, Camio del Cemeterio s/, 47011, Valladolid, Spai 2 Neural Computigo-liearity ad Complexity Research Group, School of Egieerig ad Applied Scieces, Asto Uiversity, Asto Triagle, B4 7ET, Birmigham, Uited Kigdom 3 Hospital Uiversitario del Río Hortega, Servicio de Neumología, c/ Dulzaia 2, 47012, Valladolid, Spai 4 Hospital Clíico Uiversitario, Servicio de Neumología, Travesía de la Choupaa s/, 15706, Satiago de Compostela, Spai jvmarcos@gmail.com, robhor@tel.uva.es, dalvgo@ribera.tel.uva.es, i.t.abey@asto.ac.uk, fsas@telefoica.et, carlos.zamarro.saz@sergas.es Abstract Nowadays, polysomography (PSG) is the gold-stadard to diagose obstructive sleep apoea sydrome (OSAS). However, it is complex, time-cosumig ad expesive. Noctural pulse oximetry, which provides oxyge saturatio (SaO 2 ) recordigs, allows us to overcome these difficulties ad could be a alterative to PSG. I the preset study, multilayer perceptro (MLP) eural etworks were applied to help i OSAS diagosis usig iformatio from SaO 2 sigals. We performed time ad spectral aalysis of these recordigs to extract 14 features related to OSAS. Accordig to the priciple used for etwork optimisatio, wwe compared the performace of two differet MLP classifiers: maximum likelihood (ML) ad Bayesia (BY) MLP etworks. A total of 187 subjects suspected of sufferig from OSAS took part i the study. Their SaO 2 sigals were divided ito a traiig set with 74 recordigs ad a test set with 113 recordigs to develop ad validate the classifiers. BY-MLP etworks achieved the best performace o the test set with 85.58% accuracy (87.76% sesitivity ad 82.39% specificity). These results were substatially better tha those provided by ML-MLP etworks, which were affected by overfittig ad achieved a accuracy of 76.81% (86.42% sesitivity ad 62.83% specificity). Our results suggest that the Bayesia framework is preferred to implemet our MLP classifiers. The proposed BY- MLP etworks could be used for early OSAS detectio, cotributig to ad thus reduce the umber of required PSGs. Keywords: obstructive sleep apoea sydrome (OSAS), octural pulse oximetry, multilayer perceptro (MLP), maximum likelihood, Bayesia iferece

2 1 Itroductio Feedforward eural etworks represet a powerful tool for patter recogitio tasks. Several advatages ca be foud oare associated with these algorithms. Firstly, eural etworks ca lear from the eviromet i which they operate without the requiremet of ay previous assumptio (Hayki 1996, Zhag 2000). I additio, eural etworks are capable of uiversal approximatio, i.e. they ca approximate ay cotiuous mappig, provided that sufficietly may hidde uits are available (Horik 1991). Fially, eural etworks are oliear models. Thus, they ca be applied to model real-world complex relatioships (Zhag 2000). The multilayer perceptro (MLP) is the most commoly studied ad used feedforward eural etwork. It is usually used for classificatio purposes sice MLP etworks ca estimate posterior probabilities (Bishop 1995, Hayki 1999): see equatio (6). These eural etworks have provided promisig results i a wide variety of patter recogitio problems such as hadwritte digit recogitio, fault diagosis, bakruptcity predictio ad medical diagosis (Zhag 2000). Traditioally, MLP etworks were based o the maximum likelihood (ML) approach were applied to solve these problems. Accordig to this priciple, etwork weights (w (which are the parameters of the model i a statistical sese)) are determied by miimisig a objective fuctio that represets the error betwee the desired output ad the etwork output. As a result, a sigle set of weights is obtaied as optimum. I this cotext, more complex models are typically better able to fit the traiig data. However, this does ot ivolve ecessarily imply good geeralisatio capability (Bishop 1995). The Bayesia (BY) framework provides a alterative approach to implemet MLP algorithms. Twhich accouts for the ucertaity o i the etwork weights is take ito accout by cosiderig a prior probability distributio fuctio p(w) over weight space. The prior assumptio is that etworks with small weights are preferred: these give rise to smoother mappigs ad hece the etwork is likely to overfit. Oce the traiig data D is observed, the posterior probability p(w D) ca be obtaied (Bishop 1995). This distributio is cocetrated o weight values that are more cosistet with data i the traiig setca be used to itegrate over all possible parameters, weighted by their probability. Moreover, it ca be used to evaluate the relevace of each iput variable for the etwork predictios (Neal 1996, Nabey 2002). The aim of this study is to aalyse the performace of Bayesia MLP etworks to assist i the diagosis of the obstructive sleep apoea sydrome (OSAS). Moreover, we compared these algorithms with maximum likelihood MLP etworks. Both etwork models were applied i OSAS detectio usig iformatio from octural oxyge saturatio sigals. Patiets affected by OSAS suffer repetitive occlusio of the upper airway durig sleep, leadig to a complete (apoea) or partial (hypopoea) cessatio of the airflow (Qureshi ad Ballard 2003). The recurrece of these episodes has severe implicatios o for the health of the patiet. Ideed: for example, OSAS is cosidered a risk factor for the developmet of cardiovascular diseases such as hypertesio, cardiac failure, arrhythmias ad atherosclerosis (Lattimore et al 2003). Additioally, excessive daytime sleepiess due to sleep fragmetatio has bee poited out as a major cause of traffic ad idustrial accidets (George 2001). As a result, early diagosis of OSAS is required i order to apply a effective treatmet. NowadaysCurretly, octural polysomography (PSG) is the gold stadard i for OSAS diagosis (Qureshi ad Ballard 2003). This test is performed i a special sleep uit ad must be supervised by a traied techicia. Usually, the followig recordigs are moitored durig PSG: electroecephalogram (EEG), electrocardiogram (ECG), electromyiogram (EMG), electro-oculogram (EOG), oximetry, asal airflow ad respiratory effort. Subsequetly, a medical expert must aalyse these this data to provide a fial diagosis. Despite its diagostic reliability, PSG is complex, time-cosumig ad expesive (Beet ad Kiear 1999). Therefore, simplified diagostic techiques would be of great practical iterest. Arterial oxyge saturatio (SaO 2 ) recordigs from octural pulse oximetry represet a alterative to PSG. These could be acquired i the home of the patiet, resultig i reduced complexity ad cost. Pulse oximetry is widely kow i pulmoary medicie (Netzer et al 2001). It provides useful iformatio about respiratory dyamics durig sleep. Subjects sufferig from OSAS are usually characterised by SaO 2 sigals with high istability. The Commet [1]: Ca these be quatified here?

3 repetitio of apoeas is reflected by frequet drops ad the correspodig restoratios of the saturatio value due to the lack of oxyge durig each apoeic evet. I cotrast, cotrol subjects ted to preset a costat SaO 2 value aroud 97% (Netzer et al 2001). Thus, oximetry data could be usefulis useful forto detectig OSAS. Differet methodologies have bee previously proposed to perform OSAS diagosis from SaO 2 recordigs. Visual ispectio represets the easiest aalysis techique (Rodríguez et al 1996). However, automated aalysis would allow to reduce the time required for a fial diagosis. Covetioal oximetry idices were suggested for this purpose. These idices are usually provided by the oximetry equipmet. They iclude the oxyge desaturatio idex over 2% (ODI2), 3% (ODI3) ad 4% (ODI4), ad the cumulative time spet below a give level of saturatio. Typically, a saturatio level of 90% (CT90) is applied (Lévy et al 1996, Roche et al 2002, Vázquez et al 2000, Netzer et al 2001, Magalag et al 2003). Recetly, sigal processig techiques have bee also used for automated aalysis of oximetry recordigs through spectral ad oliear methods (Zamarró et al 2003, Álvarez et al 2006, Del Campo et al 2006, Álvarez et al 2007, Horero et al 2007). MoreoverI additio, patter classificatio techiques such as eural etworks have bee applied for the developmet of diagostic algorithms (El- Solh et al 2003, Marcos et al 2008, Polat et al 2008). I the preset study, we modelled the OSAS diagosis problem as a patter recogitio task. Subjects must be assiged to oe of two possible groups: OSAS positive or egative. Several features extracted from SaO 2 recordigs were fed ito MLP etwork classifiers to idetify subjects with OSAS. We evaluated ad compared the capability of MLP etworks developed withi two differet frameworks: the maximum likelihood ad the Bayesia approaches. 2 Subjects ad sigals A total of 187 SaO 2 recordigs from subjects suspected of sufferig from OSAS were available for the study. Usually, sleep aalysis was carried out from midight to 8:00 AM i the Sleep Uit of the Hospital Clíico Uiversitario of Satiago de Compostela (Spai). The Review Board o Huma Studies at this istitutio approved the protocol. Covetioal PSG ad octural pulse oximetry were simultaeously performed o each of the subjects. Oximetry sigals were recorded by meas of a Criticare 504 oximeter (CSI, Waukeska, U.S.A.) at a samplig frequecy of 0.2 Hz. The equipmet used to perform PSG was a polygraph (Ultrasom Network, Nicolet, Madiso, W.I., U.S.A.). Sigals obtaied i the polysomographic study were EEG, EOG, chi EMG, airflow (three-port thermistor), ECG ad measuremet of chest wall movemet. These recordigs were aalysed by a expert accordig to the system by Rechtschaffe ad Kales to obtai a diagosis for each subject (Rechtschaffe ad Kales 1968). Apoea was defied as a cessatio of airflow for 10 secods or loger. Hypopoea was defied as a reductio, without complete cessatio, i airflow of at least 50%, accompaied by a decrease of more tha 4% i the saturatio of haemoglobi. The average apoea-hypopoea idex (AHI) was calculated for hourly periods of sleep from apoeic/hypopoeic episodes captured i PSG. Fially, a threshold of AHI 10 evets/h was established to determie the presece of OSAS i a subject. A positive diagosis of OSAS was cofirmed i 111 subjects, i.e % of the populatio uder study. There were o sigificat differeces betwee OSAS OSAS-positive ad egative groups i age, body mass idex (BMI) ad recordig time. O the other had, the percetage of males was higher i the OSAS OSAS-positive group (84.68%) tha i the OSAS OSASegative (69.74%). The iitial populatio was radomly divided ito traiig ad test sets to develop both types of eural etwork classifiers. The proportio of OSAS OSAS-positive ad egative subjects was preserved i each of these sets. The traiig set (74 subjects) was used for etwork optimisatio. The test set (113 subjects) was applied to estimate the geeralisatio capability of the classifiers. Table 1 summarises the demographic ad cliical data for the whole populatio as well as for traiig ad test sets. Commet [2]: What was the methodology used? Expert judgmet? Commet [3]: Whe OSAS positive is used as a adjective, it should be hypheated. I have tried to fix all of these, but may have missed a few.

4 Table 1. Demographic ad cliical statistics of all subjects, traiig set ad test set. Data are preseted as mea stadard deviatio. : umber of subjects; BMI: body mass idex; AHI: apoea/hypopoea idex computed as evets for hourly periods. All subjects All ( = 187) OSAS Positive ( = 111) OSAS Negative ( = 76) Age (years) Males (%) BMI (kg/m 2 ) Recordig Time (h) AHI (evets/h) Traiig set All ( = 74) OSAS Positive ( = 44) OSAS Negative ( = 30) Age (years) Males (%) BMI (kg/m 2 ) Recordig Time (h) AHI (evets/h) Test set All ( = 113) OSAS Positive ( = 67) OSAS Negative ( = 46) Age (years) Males (%) BMI (kg/m 2 ) Recordig Time (h) AHI (evets/h) Methods Patter recogitio techiques were applied to model the OSAS diagosis problem usig oxymetry data. Our methodology ivolved several stages: 1) feature extractio, 2) feature preprocessig ad 3) patter classificatio. 3.1 Feature extractio The purpose of the feature extractio stage was to summarise the iformatio i SaO 2 recordigs usig a reduced set of parameters or features. Evets of apoea are accompaied by hypoxaemia, which is reflected i oximetry sigals with a marked decrease i the saturatio value. These recordigs ted to preset a differet behaviour for OSAS positive ad egative subjects. Therefore, they ca be useful for OSAS detectio. The extracted features must provide suitable measures i order to differetiate sigals from both populatios. We used sigal processig techiques to extract features from oximetry data. Iitially, covetioal statistical aalysis was applied. Moreover, we aalysed SaO 2 sigals usig spectral ad oliear methods. As stated i our previous research (Zamarró et al 2003, Álvarez et al 2006, Del Campo et al 2006, Álvarez et al 2007, Horero et al 2007), spectral ad oliear features from oximetry sigals provided sigificat statistical differeces betwee OSAS positive ad egative subjects. Fially, a total of 14 features were computed from SaO 2 recordigs. These ca be divided ito two groups: time-domai ad frequecy-domai features Time-domai aalysis of oximetry data Time represetatio of oximetry sigals reflects respiratory dyamics durig sleep. Apoea evets are characterised by a decrease i the SaO 2 value due to airway obstructio ad reduced airflow. Therefore, sigals from positive subjects are usually associated to istability. They reflect cotiuous drops ad subsequet restoratios of the saturatio value because of the repetitio of apoeas. I cotrast, sigals from OSAS egative subjects ted to preset a ear

5 costat saturatio value aroud 97% (Netzer et al 2001). To illustrate this, the oximetry recordigs correspodig to a cotrol subject (AHI = 0 evets/h), a patiet sufferig from OSAS (AHI = 44 evets/h) ad a ucertai OSAS positive patiet (AHI = 14 evets/h) from our database are depicted i figure 1. I order to quatify these dyamic differeces, we aalysed SaO 2 sigals i the time domai usig covetioal statistics ad oliear methods. We estimated the first four stadard momets for the distributio of the variable represetig the SaO 2 value. The probability desity fuctio of this variable was modelled with a discrete uiform distributio. Each stadard momet was estimated by averagig the values computed from sigal epochs of 200 samples. The followig features were extracted: Feature 1. First statistical momet i the time domai (SMT1). SMT1 represets the expected value for of the distributio of SaO 2 samples. It is supposed to beusually lower for positive patiets due to frequet drops i the saturatio value. Feature 2. Secod statistical momet i the time domai (SMT2). SMT2 represets the variace for of the distributio of SaO 2 samples. Positive patiets are expected to provide higher values of SMT2 due to istability of their recordigs. Feature 3. Third statistical momet i the time domai (SMT3). SMT3 measures the asymmetry for of the distributio of SaO 2 samples. Typically, it is egative for subjects from both populatios. However, its magitude is usually greater i positive subjects due to the higher cocetratio of samples with low values of saturatio. Feature 4. Fourth statistical momet i the time domai (SMT4). SMT4 evaluates the sharpess for of the distributio of SaO 2 samples. It is expected to be higher i cotrol subjects sice their oximetry sigals ted to be costat. I additio, we aalysed oximetry recordigs with three differet oliear methods: approximate etropy (ApE), cetral tedecy measure (CTM) ad Lempel-Ziv complexity Figure 1. Time represetatio of oximetry recordigs correspodig to (a) a OSAS egative subject (AHI = 0 evets/h), (b) a OSAS positive subject (AHI = 44 evets/h) ad (c) a ucertai OSAS positive subject (AHI = 14 evets/h).

6 (LZC). Accordig to our previous research, these methods ca be used to evaluate properties of oximetry sigals that are related to OSAS (Álvarez et al 2006, Del Campo et al 2006, Álvarez et al 2007, Horero et al 2007). Sigals were divided ito epochs of 200 samples to compute these features. For each of them, the average value from all sigal epochs represeted the fial estimate. The followig oliear methods were applied o SaO 2 sigals: Feature 5. Approximate etropy (ApE). ApE estimates irregularity of time series, with high values of ApE correspodig to irregular sigals (Picus 2001). This method requires to specifythe specificatio of two desig parameters: a ru legth m ad a tolerace widow r. These were set to 1 ad 0.25 times the stadard deviatio of the origial sequece, respectively (Horero et al 2007). ApE measures the logarithmic likelihood that rus of patters that are close (withi r) for m cotiguous observatios remai close (withi the same tolerace widow width r) o subsequet icremetal comparisos (Picus 2001). Usually, high values of ApE are associated with SaO 2 sigals from OSAS positive subjects. These ted to preset a more irregular behaviour due to frequet chages i the saturatio value (Horero et al 2007). Feature 6. Cetral tedecy measure (CTM). CTM quatifies the variability of the sigal (Cohe et al 1996), assigig low values to sigals with a high degree of chaos. It is computed from secod-order differece plots represetig (s t+2 s t+1 ) vs. (s t+1 s t ), where s = (s 1,, s t,, s T ) is the a time series of legth T. The umber of poits that fall iside a circular regio of radius cetred o the origi is obtaied. The, it is divided by the total umber of poits to compute CTM. I the preset study, a radius = 0.25 was applied to estimate CTM from our SaO 2 sigals (Álvarez et al 2006). Oximetry recordigs from OSAS positive patiets are characterised by high variability due to the recurrece of apoeas, which is reflected i low CTM values for these subjects (Álvarez et al 2006). Feature 7. Lempel-Ziv complexity (LZC). LZC estimates the complexity of the sigal (Lempel ad Ziv 1976). Series with high complexity provide high values of LZC. It is related to the umber of distict substrigs ad the rate of their recurrece alog a give sequece. The sigal is trasformed ito a fiite symbol sequece by comparig each sample with a fixed threshold. I this study, LZC was computed by covertig SaO 2 sigals ito 0-1 sequeces. The media value of the sigal samples was used as threshold (Álvarez et al 2006). The resultig sequece is scaed from left to right, icreasig the complexity couter by oe uit every time a ew subsequece of cosecutive characters is ecoutered (Lempel ad Ziv 1976). High values of LZC are expected for SaO 2 sigals from OSAS positive subjects (Álvarez et al 2006) Frequecy-domai aalysis of oximetry data Our precedig studies cocluded that spectral aalysis of oximetry sigals reflects sigificat differeces betwee positive ad egative subjects. The sigal power associated with frequecy compoets located i the bad betwee ad Hz is usually higher i subjects with OSAS tha i ormal cotrols (Zamarró et al 2003). The duratio of a apoea usually rages from 30 s to 2 mi, icludig the awakeig respose after the evet. The repetitio of apoeas durig sleep origiates phase-lagged chages i SaO 2 sigals with the same periodicity. Thus, the miimum ad maximum frequecies for the occurrece of apoeas would approximately correspod to the limits of that bad. This behaviour is illustrated i figure 2. It depicts the power spectral desity (PSD) fuctio computed from the SaO 2 recordigs show i figure 1. As it ca be observed, frequet chages i the saturatio value due to apoeas lead to a higher badwidth i SaO 2 sigals from positive subjects. I additio, a peak i the bad betwee ad Hz reflects a periodic compoet i the oximetry recordig from the patiet with severe OSAS. Differet methods ca be used to compute the PSD from o-statioary data such as our SaO 2 sigals. We applied the o-parametric Welch s method to estimate the PSD usig a

7 Figure 2. Power spectral desity (PSD) computed from oximetry recordigs correspodig to (a) a OSAS egative subject (AHI = 0 evets/h), (b) a OSAS positive subject (AHI = 44 evets/h) ad (c) a ucertai OSAS positive subject (AHI = 14 evets/h). The dashed lie idicates the limits for the bad betwee ad Hz. Haig widow with a legth of 300 samples (50% overlappig) (Welch 1967). Iitially, we aalysed the statistical properties of the variable represetig the frequecy compoet of oximetry sigals. The ormalised PSD was used as the probability desity fuctio of this variable. The followig features were computed: Feature 8. First statistical momet i the frequecy domai (SMF1). SMF1 estimates the expected value of the variable defied by the frequecy compoet. It is expected to beusually higher larger for OSA-S positive patiets sice their SaO 2 sigals teds to preset have a higher larger badwidth. Feature 9. Secod statistical momet i the frequecy domai (SMF2). SMF2 represets the variace of the variable defied by the frequecy compoet. Similarly, higher values of this feature are associated to patiets affected by OSAS. Feature 10. Third statistical momet i the frequecy domai (SMF3). SMF3 measures the asymmetry of the variable defied by the frequecy compoet. It is positive for both groups of subjects. High values are associated to OSAS- egative subjects sice the PSD of their sigals ted to be cocetrated o frequecies close to zero. Feature 11. Fourth statistical momet i the frequecy domai (SMF4). SMF4 evaluates the sharpess of the distributio defied by the frequecy compoet. It is expected to be higher for OSAS- egative subjects sice the power of the sigal is cocetrated i low frequecy compoets. I additio, other three spectral features were computed from SaO 2 sigals. These are directly related to the aalysis of the PSD i the frequecy bad betwee ad Hz (Zamarró et al 2003). We computed the followig features:

8 Feature 12. Total area uder the PSD (S T ). S T evaluates the power of the sigal uder study. High values are associated to sigals from positive subjects due to frequet chages ad variability. Feature 13. Area eclosed i the bad of iterest (S B ). S B measures the sigal power correspodig to frequecies i the bad betwee ad Hz. As idicated beforeabove, it is usually higher for OSAS- positive patiets. Feature 14. Peak amplitude of the PSD i the bad of iterest (PA). PA represets the most sigificat frequecy compoet cotaied i the bad betwee ad Hz. It is expected to be higher i OSAS- positive patiets due to periodic chages i SaO 2 sigals correspodig to these frequecies. 3.2 Feature preprocessig The preprocessig stage avoids possible differeces betwee the magitudes of the iput features (Bishop 1995). Each of them was ormalised to have zero mea ad uit variace. The followig liear trasformatio was applied: where ad i are the sample ad feature idices, respectively, i x ~ x i i i, (1) x i is the ormalised value of feature i for sample, x~ i is its correspodig raw value, i is the mea value of feature i ad is its stadard deviatio. i 3.3 Classificatio We used multilayer perceptro (MLP) eural etwork classifiers to process the ormalised features. The purpose of this stage is to classify patters from SaO 2 recordigs ito oe of two possible groups: OSAS positive or egative. MLP eural etworks suitably adapt to classificatio tasks sice they are capable of estimatig posterior probabilities (Bishop 1995, Hayki 1999). Classifiers based o MLP preset some advatages i compariso with other covetioal techiques such as discrimiat aalysis. Maily, they avoid prior assumptios about the statistical distributio of the iput data (Zhag 2000). Moreover, MLP etworks ca establish complex oliear decisio boudaries i the iput space (Bishop 1995, Hayki 1999). For a give problem, desigig eural etworks requires to take some decisios about etwork architecture ad traiig. I this study, iput patters must be labelled as OSAS positive or egative, which represets a two-class classificatio problem. A sigle ode is required i the output layer of the etwork. A logistic activatio fuctio was used for this ode i order to iterpret etwork outputs as probabilities (Bishop 1995). O the other had, we decided to evaluate MLP etworks with a sigle hidde layer. As idicated i (Horik 1991), etworks with this architecture are capable of uiversal approximatio. The hyperbolic taget activatio fuctio was used for hidde odes uits sice it provides faster covergece of the traiig algorithms (Hayki 1999). We traied our MLP etworks usig a codig scheme t = 1 if the iput patter belogs to class C 1 (OSAS- positive group) ad t = 0 if it belogs to class C 0 (OSAS- egative group), with t represetig the target value ad C j the membership variable. As a result, the etwork output ca be iterpreted as the probability of havig OSAS give the iput patter. Therefore, the Bayes decisio rule ca be applied to perform patter classificatio i order to miimise the probability of misclassificatio (Bishop 1995). It states the followig: Decide C j for x if p ( C j x) max p( C j x), (2) j 0,1 where p(c j x) represets the posterior probability of class C j for the iput patter x.

9 Traiig MLP etworks ivolves to adjustig etwork weights from based o a fiite set of data that represets the statistical properties of the problem. I this cotext, two differet techiques ca be applied: the maximum likelihood criterio ad the Bayesia iferece approach. I this study, we propose to compare the performace of MLP classifiers from both frameworks to assist i OSAS diagosis usig oximetry data Maximum likelihood I classificatio problems, the probability of observig the target value t (represeted by a 0-1 ecodig) is modelled with a Beroulli distributio (Bishop 1995). It is expressed as: p t x y t 1 t 1 y, (3) where y is the etwork output value, i.e. the estimate for the posterior probability p(c 1 x). Accordig to this, the likelihood (L) of observig the traiig set is give by the followig expressio: L N 1 y t 1 t 1 y, (4) where N is the size of the traiig set ad it has bee assumed that traiig samples are statistically idepedet. The aim of the maximum likelihood approach is to adjust etwork weights i order to maximise L. Usually, the egative logarithm of the likelihood is cosidered. The, the optimisatio process is equivalet to miimise the expressio resulted from that trasformatio. It is referred to as the cross-etropy error fuctio (E D ) ad is expressed as (Bishop 1995): E D N 1 t l y (1 t )l(1 y ). (5) As a result, the maximum likelihood approach allows to obtai the weight vector w that best fits the traiig datathe optimal weights caot be foud directly sice there is a o-liear depedece o the weights. Istead, a iterative approach is used: partial derivatives of the error fuctio with respect to the weights ca be determied aalytically, ad these are used i secod-order o-liear optimisatio algorithms. SubsequetlyAfter weight optimisatio, etwork predictios for ew iput patters are computed as: y l H ho t h 1 i 1 I ih i f ( x, w) g w g w x b b, (6) h o Formatted: Idet: First lie: 0 cm Commet [4]: I would move this equatio ad the descriptio befote equatio (2). where I is the umber of features i the iput vector, H is the umber of hidde eurosuits, w ho is the weight coectig hidde euro uit h with output euro uit o, b o is the bias associated to output euro o, w ih is the weight coectig the feature i of the iput patter with hidde euro uit h, b h is the bias associated to hidde euro uit h, g t ( ) is the activatio fuctio for euros uits i the hidde layer ad g l ( ) is the activatio fuctio for the output layer eurouit Bayesia iferece The Bayesia approach suggests to models the posterior probability desity fuctio of the weight vector rather tha determiig a optimum set of etwork weights (Bishop 1995, Nabey 2002). Whe the maximum likelihood approach is applied, ddifferet (represetative) traiig sets represetig the problem uder study lead to differet etwork weights whe the maximum likelihood approach is applied. Bayesia techiques aim to cosider accout for this ucertaity by represetig the degrees of belief i the values of the weight vector (Bishop 1995) with a probability distributio. Accordig to the Bayes theorem, the posterior distributio of the weights (w) give the traiig set (D) is expressed as:

10 p D w p w p w D, (7) p D where p(w) is the prior probability fuctio over weight space, p(d w) is the likelihood of the traiig data (computed usig equatio (4)) ad p(d) is a ormalisatio factor kow as the evidece (Nabey 2002). Oce the posterior has bee calculated, it ca be used to ifer the distributio of output values by computig the followig expressio: p t x, D p t x, w p w D dw (8) where p(t x,w) is the model for the distributio of the oise o the target for a give weight vector. I patter classificatio problems, it is give by the expressio i (3). Therefore, the probability of membership of a iput patter to the OSAS positive group is obtaied as: p C1 x, D p C1 x, w p w D dw y x, w p w D dw (9) We used the evidece procedure (Mackay 1992) to implemet Bayesia MLP etworks. A Gaussia approximatio to the posterior (Laplace approximatio) is used to solve compute itegrals such as those metioed before (Nabey 2002)i equatios (8) ad (9). I the absece of data, the prior probability distributio is chose i order to favour small weights. Smooth mappigs are preferred sice they provide better geeralisatio (Bishop 1995). Thus, the prior is modelled usig the followig zero-mea expoetial Gaussia fuctio: p w exp EW exp w, (10) Z Z 2 W where Z W is a ormalisatio factor ad, is referred to as a hyperparameter, is the iverse variace of the Gaussia. It cotrols the distributio of other the etwork parameters, i.e. etwork weights ad biases. O the other had, the likelihood fuctio for the traiig data give i (4) ca be writte as: L N p D w y 1 y exp G D w 1 t W 1 t where G(D w) = E D is the cross-etropy error fuctio as i equatio (5). Accordig to the expressio i (7), the Bayes theorem is applied to compute the posterior probability from p(w) ad p(d w). It is give by: p w D 1 exp Z S G E W 1 exp Z S (11) S w, (12) 1 2 where Z S is a ormalisatio factor for the Gaussia, E W w ad S(w) = G(D w) + E W (w). 2 I practice, this distributio is approximated by a Gaussia cetred o the maximum posterior weight vector (w MP ): p w D 1 1 T exp S wmp w A w * (13) Z 2 S where Z * S is the ormalisatio factor for the Gaussia, Δw = w w MP ad A is the Hessia matrix of the error fuctio S(w). The implemetatio of the Bayesia MLP etwork requires the use of a oliear optimisatio algorithm to fid w MP, which is foud by miimisig S(w). Additioally, the hyperparameter is periodically updated durig this optimisatio process. The maximum posterior value for the hyperparameter is foud by maximisig the likelihood p(d α) (Bishop 1995, Nabey 2002).

11 The hyperparameter cotrols the shape of the prior distributio of weights. Complex forms of this hyperparameter ca be specified. Specifically, differet values of ca be used to model the distributio of the weights associated to each iput feature. This procedure is kow as automatic relevace determiatio (ARD) (Neal 1996, Nabey 2002). Oce the etwork has bee traied, the importace of a iput feature (x i ) ca be evaluated by aalysig its associated hyperparameter ( i ). A low value of i correspods to a large variace prior, which allows weights of large magitude. Thus, the feature x i is very relevat for predictig the output. Coversely, a large value of i is iterpreted as low ifluece of the iput feature o etwork output. As a result, feature selectio is implemeted i the etwork traiig process. 4 Results We computed the proposed 14 features from each SaO 2 sigal i our database. Table 2 summarises the mea value of each feature for the complete set of sigals. Subsequetly, the preprocessig stage was applied o each feature. The obtaied patters were fed ito a MLP classifier. We developed MLP etworks usig the two optimisatio frameworks described before: maximum likelihood (ML) ad Bayesia iferece (BY). The Netlab software was used to implemet these algorithms (Nabey 2002). 4.1 Traiig The traiig set with 74 subjects was used to develop our classificatio algorithms. The scaled cojugate gradiet was used to optimise both types of MLP classifiers (Moller 1993). It was applied to miimise the cross-etropy error fuctio give by E D i order to determie the optimum weight vector for etworks correspodig to the ML criterio. I the BY framework, the optimisatio algorithm was used to fid the most probable set of weights by miimisig the error fuctio S(w); this is alterated with re-estimatio of hyperparameters usig the evidece framework (Mackay 1992).

12 Differet etwork architectures of both types were evaluated by varyig the umber of hidde odes from 2 to 20. The geeralisatio performace of each etwork cofiguratio was measured with sesitivity, specificity ad accuracy. Additioally, we used receiver operatig characteristics (ROC) aalysis. The area uder the ROC curve (AUROC) was computed as a measure of classificatio ability (Haley ad McNeil 1982). Give the radom ature of etwork iitialisatio, the performace measures were averaged for a total of 10 rus, i.e. a total of 10 differet etworks were traied for each cofiguratio. Table 3 summarises the results achieved o the traiig set for both types of MLP etworks. As it ca be observed, ML-MLP etworks ca correctly classify all the samples i the traiig set. This is achieved for etwork cofiguratios with more tha 4 odes i the hidde layer. I cotrast, BY-MLP etworks provided lower classificatio ability o the traiig data, reachig a accuracy value aroud 90% ad a AUROC close to Table 2. Time ad spectral features extracted from octural oximetry recordigs for all subjects uder study. Data are preseted as mea stadard deviatio. : umber of subjects; SMT1: first statistical momet i the time domai; SMT2: secod statistical momet i the time domai; SMT3: third statistical momet i the time domai; SMT4: fourth statistical momet i the time domai; ApE: approximate etropy; CTM: cetral tedecy measure; LZC: Lempel-Ziv complexity; SMF1: first statistical momet i the frequecy domai; SMF2: secod statistical momet i the frequecy domai; SMF3: third statistical momet i the frequecy domai; SMF4: fourth statistical momet i the frequecy domai; S T : total area uder the power spectral desity; S B : area eclosed i the bad of iterest; PA: peak amplitude of the power spectral desity i the bad of iterest. TIME FEATURES All ( = 187) OSAS Positive ( = 111) OSAS Negative ( = 76) SMT SMT SMT SMT ApE CTM LZC SPECTRAL FEATURES All ( = 187) OSAS Positive ( = 111) OSAS Negative ( = 76) SMF SMF SMF SMF S T S B PA

13 Table 3. Classificatio results achieved by maximum likelihood ad Bayesia MLP classifiers o the traiig set. Se: sesitivity; Sp: specificity; Ac: accuracy; AUROC: area uder the ROC curve. Maximum likelihood Bayesia iferece Hidde odesuits Se (%) Sp (%) Ac (%) AUROC Se (%) Sp (%) Ac (%) AUROC ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 0.00 Formatted Table 4.2 Test The traied etworks were also evaluated o previously usee data to obtai a more reliable measure of geeralisatio capability. The test set with 113 subjects was used. The results achieved by ML-MLP ad BY-MLP etworks are summarised i table 4. As expected, the results o the test set were lower tha those achieved o the traiig set by both types of MLP etworks. However, the decrease was sigificatly marked for ML-MLP etworks. This was ot very surprisig. The etworks have 16H+1 parameters (where H is the umber of hidde uits) ad hece whe H>4 there are more parameters tha traiig data samples. I additio, it was observed that there were slight differeces betwee etwork cofiguratios with a differet umber of hidde odes. It which was observed for both ML- MLP ad BY-MLP etworks. Thus, less complex algorithms such as etworks with 2 hidde odes are preferred. The ML-MLP algorithm with this cofiguratio provided a mea accuracy of 76.81% (86.42% sesitivity ad 62.83% specificity) ad a mea AUROC of The accuracy reached by BY-MLP classifiers o the test set was substatially higher. The BY-MLP etwork with 2 odes i the hidde layer achieved a mea accuracy of 85.58% (87.76% sesitivity ad % specificity) ad a mea AUROC of Formatted: Fot: Italic Formatted: Fot: Italic Formatted: Fot: Italic Commet [5]: I would also iclude results from logistic regressio as compariso. Table 4. Classificatio results achieved by maximum likelihood ad Bayesia MLP classifiers o the test set. Se: sesitivity; Sp: specificity; Ac: accuracy; AUROC: area uder the ROC curve. Maximum likelihood Bayesia iferece Hidde odesuits Se (%) Sp (%) Ac (%) AUROC Se (%) Sp (%) Ac (%) AUROC ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 0.00 Formatted Table

14 5 Discussio Classifiers based o Bayesia MLP etworks were evaluated as a assistat tool to assist i OSAS diagosis from octural oximetry data. A total of 14 features computed from timedomai ad frequecy-domai aalysis of SaO 2 sigals were used as iputs. The performace of these classifiers was compared with that achieved by commo ML-MLP etworks. We foud that the classificatio ability of these algorithms was improved by BY-MLP. Our results idicate thatusig Bayesia iferece represets a more effective optimisatio techique to implemet our MLP classifiers. It is ofte preferred i applicatios with a reduced small dataset such as that preseted i this study. BY-MLP etworks provided the best classificatio performace with a accuracy of 85.58% ad a AUROC of 0.90 o the test set. These etworks sigificatly outperformed classifiers based o ML-MLP (76.81% accuracy ad 0.86 AUROC). The opposite situatio was observed for data i the traiig set. The accuracy reached o this set by BY-MLP etworks raged from 87.70% (BY-MLP with 20 hidde odes) to 90.10% (BY-MLP with 2 hidde odes). For ML- MLP etworks, it was close or equal to 100%. The compariso of the results achieved o both sets idicates that ML-MLP etworks overfitted the traiig data. These etworks are more likely to be affected by overfittig sice their iheret complexity is greater tha that of BY- MLP etworks (Bishop 1995). For a give traiig set, the bias-variace trade-off requires the best compromise betwee a good represetatio of the data ad etwork complexity i order to achieve high geeralisatio capability. A simple or iflexible model (large bias) ca lead to uderfit the data. The model may ot have the ability to lear eough the uderlyig distributio. O the other had, a too complex or flexible model (large variace) could capture the oise preset i the traiig data, leadig to overfittig. Both situatios result i poor geeralisatio (Bishop 1995, Hayki 1999). The mai advatage of BY-MLP etworks is that the effect of the bias-variace dilemma is o relevatreduced. They implemet regularisatio by icludig the hyperparameter, which cotrols the distributio of the weights ad the etwork complexity durig traiig (Nabey 2002). May real applicatios such as the proposed OSAS diagosis problem preset a traiig set with a limited size. Thus, regularisatio techiques are required to develop effective eural etwork algorithms. I our precedig work, a MLP etwork with oliear features from oximetry data was developed to help i OSAS diagosis (Marcos et al 2008). It provided a accuracy of 85.5% ad a AUROC of 0.90 o a test set with 83 subjects usig weight decay regularisatio. These results were similar to those obtaied with BY-MLP etworks developed i this study. However, weight decay requires the user to adjust a additioal regularisatio parameter, which cotrols the trade-off betwee reducig traiig error ad favourig small weight values. A high large umber of experimetal rus are eeded to optimise this parameter. Therefore, the Bayesia approach represets a more efficiet regularisatio techique sice the hyperparameter is automatically updated i the traiig process. Moreover, it allows us to specify complex priors to implemet automatic relevace determiatio. This procedure provides a meas to perform feature selectio i the etwork traiig algorithm (Nabey 2002). From our experimets, we foud that ApE, LZC, SMT1 ad SMT4 were determied as the most relevat features to classify SaO 2 recordigs as OSAS- positive or egative. This result suggests that time-domai features from oximetry data provided the most relevat iformatio to detect OSAS. Other methodologies to aalyse oximetry sigals for OSAS diagosis were have bee previously proposed to aalyse oximetry sigals for OSAS diagosis. Visual ispectio of these recordigs provided a sesitivity of 91% ad a specificity of 69% (Rodríguez et al 1996). However, this is a time-cosumig techique ad the iterpretatio of the sigals may differ from oe expert to aother. Covetioal oximetry idices were proposed for automated aalysis of SaO 2 recordigs. Most of the commercial oximeters ca provide ODI2, ODI3, ODI4 ad the cumulative time spet below 90% of saturatio. The diagostic capability of these idices has bee evaluated i other studies. The reported results varied amog differet researchers: the sesitivity raged from 31% to 98% ad the specificity from 41% to 100% (Netzer et al 2001). Vázquez et al (2000) reported 98% sesitivity ad 88% specificity by Commet [6]: i.e. time-domai features were better tha frequecy-domai features Commet [7]: Iclude a table of fial alpha values for each feature. Ideally, it might also be good to trai some models o a reduced set of iputs to see if this improves performace. Commet [8]: It would be very helpful to iclude Accuracy (ad where posible AUROC) values i this paragraph. Perhaps a summary table of the algorithm results would also help.

15 meas of ODI4 whe a threshold of 15 evets/h was applied to defie OSAS. It represets the highest diagostic accuracy reported by these idices. However, a defiitio of arousal differet to the criteria proposed by the Atlas Task Force was applied (The Atlas Task Force 1992). Lévy et al (1996) proposed a additioal idex (Δ idex) from oximetry sigals to detect OSAS. It is a measure of the variability of the SaO 2 recordigs. This parameter achieved 98% sesitivity ad 46% specificity usig a threshold of 15 evets/h o the AHI to determie the presece of OSAS. Magalag et al (2003) achieved similar results usig the Δ idex, with a sesitivity of 91% ad a specificity of 59%. These results were improved by usig the Δ idex together with the other covetioal idices, which yielded a sesitivity of 90% ad a specificity of 70%. Fially, Roche et al (2002) suggested to combie iformatio from oximetry data with cliical features usig logistic regressio. This model achieved a accuracy of 62.1%. The preseted algorithms improved the classificatio accuracy reported i the cited studies. Moreover, we used our database of SaO 2 recordigs to compare the diagostic capability of our etworks ad covetioal oximetry idices. I order to do this, we computed the classificatio results of these idices o our sigal database. For each idex, the threshold (l) that provided the highest accuracy o the traiig set was selected. Subsequetly, it was applied o sigals i the test set to compute sesitivity, specificity ad accuracy values. I additio, we computed the AUROC from data i the test set. The obtaied results are summarised i table 5. The best diagostic performace was provided by ODI2 ad ODI3, which achieved a sesitivity of 76.12%, a specificity of 93.48% ad a accuracy of 83.19%. I additio, CT90 provided the best AUROC value with As it ca be observed, the BY-MLP classifiers clearly outperformed covetioal oximetry idexes. The proposed algorithms represet a more effective techique for automated OSAS diagosis from SaO 2 data. Some limitatios ca be foud i our methodology. A larger populatio would be desirable to obtai a better represetatio of the statistical properties of our classificatio problem. The geeralisatio capability of our etworks could be improved by usig a larger traiig set. Similarly, a more accurate estimate of our classificatio results could be obtaied through a larger test set. Moreover, our results were computed as the average value of several rus. A model selectio stage is required i order to select a optimum classificatio algorithm from those traied i our experimets. I summary, BY-MLP classifiers have show to be a useful tool to assist i OSAS diagosis from oximetry data. They achieved a accuracy of 85.58% (87.76% sesitivity ad 82.39% specificity) ad a AUROC of They outperformed ML-MLP etworks, which provided a accuracy of 76.81% ad a AUROC of Our results suggest that Bayesia techiques are preferred for the optimisatio of our MLP etworks. I additio, the proposed BY-MLP algorithms improved the diagostic capability of visual ispectio of SaO 2 recordigs ad covetioal oximetry idices. Our BY-MLP etworks could be used to assist medical experts for early detectio of OSAS oly usig oximetry sigals. These algorithms could be applied as a effective techique for OSAS screeig, cotributig to reduce the umber of required PSG Table 5. Diagostic results achieved by covetioal oximetry idices. ODI2: oxyge desaturatio idex over 2%; ODI3: oxyge desaturatio idex over 3%; ODI4: oxyge desaturatio idex over 4%; CT90: cumulative time spet below 90% of saturatio; l: threshold value determied o the traiig set; Se: sesitivity; Sp: specificity; Ac: accuracy; AUROC: area uder the ROC curve. Idex l Se (%) Sp (%) Ac (%) AUROC ODI ODI ODI CT tests. Other possible future work classificatio of segmets of dataset so that a more refied diagosis ca be give; it might also icrease performace.

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