Clinical Risk: wavelet reconstruction networks for marked point processes

1 INTRODUCTION . Clinical risk scores are commonly used analytic devices in health care . There are risk scores for predicting strep throat from sore throats ( Centor et al. , 1981 ) , mortality from vital signs ( GardnerThorpe et al. , 2006 ) , heart attacks from routine clinic visits ( D ’ Agostino et al. , 2008 ) , and many more . Policy is implemented around these risk scores , from rates of reimbursement to physician compensation ( Asch et al. , 2015 ) . When used for early warning , risk scores have been associated with reduced mortality ( Seymour et al. , 2017 ) . Underlying these approaches is the formulation of risk over time given some set of features . For example , in Cox models , a study time t0 = 0 is defined and the risk model is λ ( t ; t′ , x ) , where x are defined by features timestamped before time t′ = t0 , i.e. , a time invariant model . In Hawkes processes , the risk model is also λ ( t ; t′ , x ) but where x contains all history up to time t′ = t , i.e. , a nowcasting model . Cox and Hawkes models can be limited by their assumptions , which often are inappropriate for the health care setting , and include , for example , the assumptions of proportional hazards and summation over kernel activations . We formulate a point process model to address these limitations and develop multi-forecasting , the forecasting task across the two dimensions of time : t′ and t. To motivate our specific formulation , consider the limitations of the Hawkes process in health care . First , the Hawkes process encodes an additive relationship of change in rate from recurring precursors , i.e. , burstiness , whereas in health care , the repeated measurement of an event beyond the first , say of glucose , might be irrelevant . Second , clinical event timing may be routine , scheduled , or emergent , which suggests that kernel learning will improve model performance because changes in the rate may be time-dependent and not immediate . Third , clinical event processes are marked , with marks that could be categorical , real , or null values : e.g. , bacterial culture : staphylococcus aureus , glucose : 200 , and ketoacidosis : NA Our model addresses each of these limitations . To address the first limitation , summation , we adopt a reduction layer where we allow for reductions other than “ sum ” of kernel contributions from recurring events . To address the second and third limitations , non-specific timing and lack of marks , we propose a kernel learning method over one dimension ( time ) and two dimensions ( time-value ) using wavelet reconstructions . The motivation for wavelets is illustrated in Figure 1 , where a discrete wavelet reconstruction encodes the relationship of time-delayed events identified through cross-correlation . While cross-correlations capture all relative timings , many may be spurious or coincidental . The wavelet representation of relative timing instead is learned through likelihood optimization . To capture the effect of the value distribution of marks on events , two-dimensional wavelet reconstructions are value indexed , producing a one-dimensional reconstruction that is passed forward to the reduction layer . We show that we can encode the wavelets , the relative-to-absolute mapping , and the reduction step on a computation graph to conduct learning . We apply our model first in simulations of heart attacks and scheduled hemoglobin A1c checks . We then evaluate the model in a real cohort to forecast complications and adherence to clinical participation in patients with diabetes . Empirically , we show that our model has improved performance over baselines in single forecasts and explore its performance in multi-forecasts in terms of prediction and representation . Our results provide benefit for diabetes risk assessment . Whereas the central focus of the diabetes adherence literature is on adherence to medication and therapeutic regimen ( García-Pérez et al. , 2013 ; Edelman and Polonsky , 2017 ) , our method provides a lens to investigate adherence to continued diabetes care participation , which is important because , for one , participation is associated with better diabetes control ( Schectman et al. , 2008 ) . Additionally , forecasts of complications and adherence to participation , at different times in the care process and at different forecast distances , are central concerns in clinical decision-making . For example , many diabetes medications require regular clinical monitoring , and the risk of complications and noncompliance to clinical participation both affect regimen choice . By focusing on these outcomes , our forecasting tools provide relevant information for the clinical decision-making process . Related work . Both neural networks and Hawkes process variants are used for rate modeling in health care : a few recent examples include Choi et al . ( 2015 ) , Du et al . ( 2016 ) , Alaa et al . ( 2017 ) , and Bao et al . ( 2017 ) . To effectively model the hazard two key properties are used : ( 1 ) the ability of Hawkes processes to capture relative timing of interdependent events , and ( 2 ) the flexible functional forms of neural networks that are able to capture relative timings , albeit somewhat opaquely . The closest work is likely that of Bao et al . ( 2017 ) , where the authors adopt dyadic influence functions . However in that work marks are not used and the dyads selected are a subset of the Haar wavelet basis . Outside of health applications , related literature includes Hawkes kernel learning , e.g . Zhou et al . ( 2013 ) ; Linderman and Adams ( 2014 ) ; Lee et al . ( 2016 ) . Our method follows these approaches , instead using a wavelet representation across value and time to capture long-range dependencies . Compared to Mei and Eisner ( 2017 ) that generalizes the Hawkes process with neural networks , our method ( 1 ) allows events to have marks , ( 2 ) enables forecasting and multi-forecasting ( not just nowcasting ) , and ( 3 ) can be seen to have a generalized linear form . Our method of mapping relative timing hazard components onto absolute time possesses the advantages of the Hawkes approach and adopts a simple but well-performing neural network architecture . To achieve the mapping , a stepwise hazard approximation is made , as is done in Jing and Smola ( 2017 ) and Weiss ( 2017 ) , however , instead of LSTMs and forests that have challenging interpretations , our method remains interpretable for small data sets , where exploration with visualization similar to that of Caruana et al . ( 2015 ) can be performed . Our method uses wavelets to represent event contributions , and several survival analysis approaches have also adopted them in univariate models , e.g. , in Antoniadis et al . ( 1994 ) and Brillinger ( 1997 ) . Outside of survival analysis and point processes , wavelet-inspired neural networks have seen success , with Wave-net using wavelets to classify time series ( Bakshi and Stephanopoulos , 1993 ) , and Wavenet adopting a multi-layer hidden neural architecture to connect distant time steps ( Van Den Oord et al. , 2016 ) . Contributions . The contributions of this work are as follows : our work generalizes multivariate Hawkes processes to allow for non-additive event rate relationships . Like other works , e.g . Linderman and Adams ( 2014 ) , our work learns the kernel function that relates multivariate event histories to the rate . However , in our work we use wavelets as the kernel , akin to a multivariate development from Brillinger ( 1997 ) . We leverage the scaling property of wavelets to formulate a regularization that balances spatiotemporal generalizability with deterministic or near-deterministic event timing . Unlike many sequence models , e.g . Hochreiter and Schmidhuber ( 1997 ) , which are affected strongly by choice of time step , our work adopts an absolute and relative time frame , and therefore the granularity of the absolute time domain need not be determined a priori . Additionally , unlike some point process formulations , our work models marks that are 1-d ( event times ) and 2-d ( event times and their category or real value ) . Our work is explicit about types of forecasting tasks , and the methods are adapted for several purposes : forecast performance , interrogability , and representation . We show that our method performs as well as or better than comparison algorithms in predicting complications and forecasting adherence . 2 BACKGROUND . Let E be the set of events with target event y ∈ E the event we want to forecast . Associated with each event e is a value v ∈ V . An example consists of a sequence of ( time , event , value ) tuples and a period of interest for forecasting . For the n-th example , n ∈ { 1 . . . N } , define Tn as the number of tuples . Then the sequence can be written as ( tin , ein , vin ) for i ∈ { 1 . . . Tn } , with the period of interest denoted as τny . Let λy ( t ) be the rate functions of interest , dropping the subscript n for ease of notation . The multivariate Hawkes process can then be written as follows : λy ( t ) = λ0 ( t ) + |E|∑ e=1 βe T∑ i=1 ge ( t− ti ) 1 ( ti < t , ei = e ) where λ0 ( t ) is a baseline population rate function , ge ( · ) is a kernel function for event e relating its effect on the rate of y , βe are event-specific parameters , and 1 ( · ) is the indicator function . Typically ge ( · ) is an event-specific exponential decay function with a learnable decay parameter . Self-exciting processes are defined by gy ( · ) > 0 , bursty processes by ge ( · ) > 0 , and inhibitory processes by ge ( · ) < 0 . A few recent variations include Linderman and Adams ( 2014 ) where ge is a Bayesian graph kernel and Xu et al . ( 2017 ) where ge is an infectivity function and triggering kernel product . Given λny ( t ) , the log likelihood of the data is : LL ( X|θ ) = N∑ n=1 ( Tny∑ i=1 log λny ( tiny ) + ∫ τny λny ( t ) dt ) ( 1 ) The form of the Hawkes process is limiting , however , because ( 1 ) the effect of ge ( · ) decays over time , ( 2 ) the effect over ge ( · ) is additive , ( 3 ) the value associated with each event is not considered , and ( 4 ) the time restriction in the indicator function implies nowcasting ( 1 ( ti < t ) ) not forecasting ( 1 ( ti < t− c ) for some c > 0 ) . Making modifications to achieve these characteristics is desirable to effectively model many real-world processes . For example , a patient with new-onset diabetes schedules an appointment with a typical gap interval of 3 months , and the presence of 1 or 10 elevated readings may not affect the timing of the scheduled appointment . The former suggests the utility of kernel learning , and the latter suggests summation over ge is not the appropriate reduction . The proposed method addresses these concerns . In particular , we adopt discrete wavelet reconstructions , which both allows kernel learning and the use of marks . Additionally , by representing the point process as a neural network , we are able to ( 1 ) use maximization alongside summation in a reduction layer ( the formulation enables specification of any number of reductions ) , and ( 2 ) conduct time-dependent censoring . We formalize the model below .

Summary: The authors of this paper propose a point process model that uses wavelet reconstruction to capture complicated dependencies between events. They motivate this approach using experiments in the medical domain, which show how certain dependencies between events is better captured by their proposed model. The primary contribution of the paper is novel and could be useful in medical settings where predicting occurrence of important events such heart attacks could be challenging with alternative methods.