Pruscha: Semiparametric Point Process and Time Series Models for Series of Events

Transcription

1 Pruscha: Semiparametric Point Process and ime Series Models for Series of Events Sonderforschungsbereich 386, Paper 4 (998) Online unter: Projektpartner

2 Semiparametric Point Process and ime Series Models for Series of Events Helmut Pruscha University of Munich, Mathematical Institute, heresienstr. 39 D Munich, Summary: We are dealing with series of events occurring at random times n and carrying further quantitive information n. Examples are sequences of extrasystoles in ECG-records. We will present two approaches for analyzing such (typically long) sequences ( n n ), n = :::. (i) A point process model is based on an intensity of the form (t) b t (), t 0, with b t astochastic intensity of the self-exciting type. (ii) A time series approach is based on a transitional GLM. he conditional expectation of the waiting time n+ = n+ ; n is set to be ( n ) h( n ()), with h a response function and n a regression term. he deterministic functions and, respectively, describe the long-term trend of the process. Keywords: Semiparametric estimation rend function Penalized least squares Point process Intensity process GLM-based time series ECG-data Introduction Series of events can be analyzed in a dual way, namely by establishing point process models for the sequence ::: of occurrence times time series models for the sequence ::: of the lengths of time intervals between events ( n = n ; n;, 0 =0) (see Cox & Lewis, 966). In the present paper, we employ intensity based point process models of the self-exciting type (Hawkes, 97) GLM-based time series models of the transitional type (Zeger & Qaqish, 988 Fahrmeir & utz, 994) In both cases the model equation is factorized into a deterministic trend function and into a stochastic part describing the short-term oscillation. As in many semiparametric problems a major task is to separate the two factors by the estimation procedure. In our situation of one single realization over a long time interval we will need a kind of ergodic behaviour of the process. he trend function is estimated by penalized least squares (l.s.) methods based on aggregated data, gained byanaveraging procedure over longer time intervals (preserving enough short-term information). For the parameter of the stochastic component we will then employ likelihood methods, thus reversing the

3 order of estimation known from partial likelihood approaches in survival time analysis. In addition to the occurrence times n metrically scaled marks n are observed, forming p-dimensional vectors and playing the role of covariates. he results will show that both approaches, the time series and the point process approach, yield satisfactory trend estimations and give insight into the inner dynamics of the process. Semiparametric Point Process Model he sequence n, n, of occurrence times is described by anintensity process t, t 0, in the multiplicative form t = (t)b t () IR d () with a deterministic and smooth function and a stochastic intensity process b. he latter is modelled in a self-exciting form by b t = b (n) (t ) =f(w n t; n ) for t ( n n+ ] () with a regression term w n being iteratively dened by w n = u(w n; ( n n )): Ex.. f(w s) =w u(w (s x)) = e ;s w + e > x we are faced with a piecewise constant intensity function b t (Pruscha, 983). With w 0 = equation () can be written in the closed form b (n) (t ) = nx i=0 n;i e ;(n; i) e > i 0 =0 0 =0: Ex.. f(w s) = + e ;s w u(w (s x)) = e ;s w + e > x the intensity function b t is of the form of Hawkes's self-exciting point process (Hawkes, 97). In fact, with w 0 =0we can write equation () as b (n) (t ) = + nx i= e ;(t; i) e > i : We will close this section by deriving the conditional expectation of the waiting times n = n ; n;. In terms of the intensity, the transition probabilities for n+, given the information F n =( ::: n n ) of the past, can be written by IP( n+ s jf n )=; exp(; n n+s) where s t = R t s u du. Conseqently, the conditional expectation of n+ is IE( n+ jf n )= Z 0 exp(; n n+s)ds : (3) Assuming (for the moment) a piecewise constant intensityon( n n+ ], say (n),weget = (n) on the ride hand side of (3): the factors in the following model equation (4) will stand in a reciprocal relation to those in ().

4 3 Semiparametric ime Series Model he sequence n, n, of the waiting times between events is now analyzed within the framework of time series. Let F n =( ::: n n ) comprise the information up to time n as above. hen the conditional expectation of n+, given F n,ismodelled by a transitional GLM of the form IE( n+ jf n )=( n )h( n ()) (4) with a smooth, deterministic trend component, a suitable response function h, and a regression term n iteratively dened by n = n; + n + G( > n ): Ex.. h() = or h() ==, with 0 =, and nonnegative, G(x) = exp(x) or G(x) =x, in the latter case with a side condition > >0 Ex.. h() =e, with 0 =0,G(x) =x. Observe that in both cases positivity ofh() is guaranteed. Zeger and Qaqish (988,.(iv)) used h() == and put = n (instead n ) in the regression term n. hese types of response functions h are in fact suggested by the two examples in sec.. Assuming for simplicity = then in the case of Ex. in sec. equation (3) yields =w n, leading to the choice h() == in the case of Ex. we getc exp(;c w n ) after some calculations, proposing the response function h() = exp(). 4 Nonparametric Estimation of the rend In order to estimate or a method of rough averaging is suggested. o this end, we divide the observation period (0 ]into K subintervals I j =(u j; u j ] of length j = u j ; u j;, j = ::: K, where K is assumed to be considerably smaller than the observed number N of events in (0 ]. We are going to base the l.s. function on the dierence Y j ; IEY j, where Y j is the rate of occurrence of events or the averaged length of time intervals between events, respectively, in the interval I j. Let for the following N j = N uj ; N uj; be the number of events in I j 4. Point process model Putting Y j = N j j wehave on the basis of model () IEY j = j Z uj u j; IE s ds = (t j) Z uj IEb s () ds j u j; for some t j [u j; u j ]. We assume, that the approximation (t j) (t j ) t j = (u j; + u j ) (5) 3

5 is possible, as well as, for larger j = u j ; u j;, Z uj IEb s () ds b () () (6) j u j; where b () () is independent ofj. Putting a(t) =(t) b () (), and using the approximations (5) and (6), we can write IEY j a(t j ). In Ex. of sec. the limit intensity value is identied as (Pruscha, 997) b () ( ) = ; () where () = a.s.- lim( n P n i= i ). In Ex. of sec. the limit intensity is ; exp(; () ). 4. ime series model We divide the interval (0 ], = N,asabove and dene Y j = j N j. Assuming for the moment that I j =( k k+ ], i.e. N j =, for some k = k(j), we have on the basis of model (4) IEY j = X i= IE( k+i )=(t j) X i= IEh( k+i; ()) for some t j Ij. By means of approximations similar to (5) and (6) above, namely (t ) (t j j), t j = (u j; + u j ), and, for larger, X i= IEh( k+i; ()) h () () (7) we arrive atiey j a(t j ), where we nowhave a(t) =(t) h () (). For the regression term n and the examples given in sec. 3, explicit formulas for h () () can be given, assuming the existence of the Cesaro limits (), (), (), G () () of the sequences n, n, n, G( > n ). Indeed, we rst have forjj < () () = ; () + G () () (8) where G () () = > () in the case G(x) =x. hen for the examples in sec. 3 Ex. (with h() =): h () () = () () Ex. (with h() =e ): If centered variables i and i are used, we get () =0 from (8), such thatwehave h () () + () = for all i.e. we can assume that the side condition (9) below is automatically fullled. 4.3 An asymptotic set-up justifying the various approximations will be given below in sec 6. We now dene for both models the penalized l.s. criterion (a) = SSE(a)+ K H () (a), with >0 a smoothing parameter and with SSE(a) =(Y ; a) > W (Y ; a) H () (a) = 4 Z 0 (a 00 (s)) ds

6 where (Y ; a) isak {vector having components Y j ; a(t j ) and where W is an appropriate K K{weight matrix. It is well known, that (a) = min is solved by natural cubic spline functions a (Green and Silverman, 994). Note that or can be estimated only in the form a = b () or a = h (), where the second factor does not depend on t, buton. 5 Parametric Estimation For the parametric part of the problem we apply likelihood methods. Maximization of the log-likelihood function l (), IR d, = N,isdoneby plugging into l () the spline solution ^a from sec. 4, leading to a function ^l (), and then solving ^l () = max under b () () =orh () () =: (9) he side condition should guarantee that the two factors and b in () or and h in (4) can be separated by the estimation procedure. Point process model: he log-likelihood function of a realization ( ::: N )ofthe point process has the form l () = NX i= log(( i )b (i;) ( i )) ; where we have to insert ^(t) =^a(t)=b () (). Z i i;! (s)b (i;) (s ) ds ime series model: Assuming that the i 's are (conditionally) exponentially distributed, the log-likelihood function of a time series realization ( ::: N ) is given by P N i= log( i; exp(; i; i )), where i ==(( i ) h( i )) cf (4). We arrive at l () = NX i= ; log ( i; )h( i; ()) ; i ( i; )h( i; ) Plugging in the solution ^(t) =^a(t)=h () () from sec. 4 and neglecting irrelevant terms we areledtoafunction^l () which can (under the side condition h () () =) be interpreted as the log-likelihood function of a realization (^ ::: ^ N ), with ^ n = n =^a( n; ), fullling the (purely parametric) model equation IE(^ n+ jf n )=h( n ()): (0) For the transitional GLM (0) quasi-likelihood methods from GLM software can be employed.! : 6 Asymptotic Set-up 6. rend function he following set-up is known from nonparametric regression (Eubank, 988) and from non-stationary time series analysis (Dahlhaus, 996). Let (t), t [0 ], be a positive 5

7 function with a continous derivative dene for >0 t (t) = t [0 ]: Let further I j =(u j; u j ], j = :::, be a division of (0 ]into subintervals of length j. Putting (0) =min j j, () =max j j,thenwe consider limits of the kind! (0) ()!! 0: () Since for s s 0 I j j (s) ; (s 0 )j j max j 0 (t)j () I j max j 0 (t)j [0 ] which tends to 0 under (), the setting (5) is established. An analogous device is used for the trend function (t) = (t= ), = N, in the time series case. 6. Ergodic properties Point process model: For each >0, the counting process N t, t [0 ], with intensity process t = (t) b t () t [0 ], may full the ergodic law Z IE b s () ds! b () () I denotes an interval of length. hen the ap- for limits of the kind (), where I proximation (6) is justied. ime series model: For each N IN thevariables n, n f ::: Ng, satisfying the equation IE N ( n+ jf n )= ( n )h( n ()) ( = N ), may full the ergodic law n N X ii N IE N h( i ())! h () () where I N comprises n N adjacent integers. hen the approximation (7) is justied. A more detailed analysis will be given elsewhere. 7 Application Data sets on long-term ECGs of patients suering from heart arrhythmias consist of the occurrence times n of ventricular extrasystoles. he covariates n are the strength of these events measured as the relative deviation from the normal beat (see Pruscha, Ulm & Schmid, 997). As an example wechoose a patient with 74 extrasystoles within 6

8 a 0-hours observation period. he reciprocal relationship of the trend functions (t) and (t) becomes apparent in the plot of Fig.. Parameters were estimated from the data of 48 patients 5 persons died of a later sudden heart attack, while 33 survived. We used point process model (), Ex., with = and normalized i 's the mean cluster size m c is also calculated from the estimated parameters, according to a cluster process representation by Hawkes&Oakes (974) ( will be denoted by PP ) time series model (4), Ex.., employing S+,glm,quasi(link=log),dispersion estimated for equation (0), with standardized i 's, ( will here be denoted by S ) he able shows the tendency, that larger positive PP -values and smaller mean clustering sizes m c correspond to smaller S -values and smaller autocorrelations r^ (), where ^ i = i =^a( i; ) as in (0). Patient N PP m c S r^ () q^ ^h() ODJFZ OVXSBH PFQWGU OEMQL he data of the 48 patients, labelled by their status =survival or =death, were submitted to a binary logistic regression, with x = Sz, y = p m c ; andz = VES/h as regressors. Hereby, VES/h is the average rate of extrasystoles per hour, and Sz is the standard deviation of the time interval lengths between those two heart beats being qualied as normal beats (not being designated as extrasystoles), vgl. ab. of Pruscha, Ulm & Schmid (997). hen in the scatter diagram of Fig., the(x 0 y 0 ) values of the 48 patients were plotted, with x 0 y 0 being the x and y values, respectively, corrected by thevariable z = VES/h on the basis of the logistic regression equation. We have three distinct outlier cases from the group lying in the "area" of group. he two most extremes of them are also outliers in the evaluation system of nonlinear dynamics (Schmid et al, 996), where the variables x = sin and y = VES were used. With respect to the quality of prediction both systems perform nearly equally well. Acknowledgments his work is supported by the Deutsche Forschungsgemeinschaft (SFB 386). he ECG data were kindly placed at my disposal by K.Ulm(echnical University,Munich). he help of dipl. math. S. Siegle with respect to Fortran programming and S+ data analysis is thankfully acknowledged. 7

9 References Cox, D. R. & Lewis, P.A.W. (966). Methuen, London. he Statistical Analysis of Series of Events. Dahlhaus, R. (996). Maximum likelihood estimation and model selection for locally stationary processes. Nonparametric Statistics, 6, 7-9. Eubank, R. L. (988). Spline Smoothing and Nonparametric Regression. M. Decker, N.Y. Fahrmeir, L. & utz, G. (994). Multivariate Statistical Modelling Based on Generalized Linear Models. Springer-Verlag, N.Y. Green, P. J. & Silverman, B. W. (994). Nonparametric Regression and Generalized Linear Models. Chapman and Hall, London. Hawkes, A. G. (97). Spectra of some self-exciting point processes. Biometrika, 58, Hawkes, A.G. & Oakes, D. (974). A cluster process representation of a self-exciting process. J. Appl. Prob.,, Pruscha, H. (983). Learning models with continuous time parameter and multivariate point processes. J. Appl. Prob., 0, Pruscha, H. (997). Semiparametric estimation in regression models for point processes based on one realization. Proceedings of the th International Workshop on Statistical Modelling, Biel 997, Pruscha, H., Ulm, K. & Schmid, G. (997). Statistische Analyse des Einusses von Herzrythmus-Storungen auf das Mortalitatsrisiko. Discussion papers, 56, SFB 386 Munchen. Schmid, G., Morll, G. E. et al. (996). Variabilityof ventricular premature complexes and mortality risk. PACE, 9, Zeger, S. L. & Qaqish, B. (988). Markov regression models for time series: a quasi likelihood approach. Biometrics, 44, Legends to Figs. Fig. (above) Extrasystoles within a 0-hours ECG record of a patient (OVXSBH). he n = 74 time points of occurrence are marked on the time scale. he occurrence frequencies (triangles) and the mean waiting times (circles) were plotted over the j = ::: 0 hours, together with the estimated trend functions and, respectively. Fig. (below) Scatterplot of the variable Cluster Size over the variable Standard Deviation, both corrected by VES/h on the basis of a logistic regression equation, for 48 patients. 8

10 Extrasystoles in a 0-Hours ECG Interval Frequency (.0) (53) Inverse of Interval Frequency ime [h] Plot of 48 Patients Suffering from Cardiac Arrythmias ClusterSize corrected by VES/h = Survived (33) = Sudden Cardiac Death (5) StandDev corrected by VES/h 9