Interval-censored data solely records the aggregated counts of events during specific time intervals - such as the number of patients admitted to the hospital or the volume of vehicles passing traffic loop detectors - and not the exact occurrence time of the events. It is currently not understood how to fit the Hawkes point processes to this kind of data. Its typical loss function (the point process log-likelihood) cannot be computed without exact event times. Furthermore, it does not have the independent increments property to use the Poisson likelihood. This work builds a novel point process, a set of tools, and approximations for fitting Hawkes processes within interval-censored data scenarios. First, we define the Mean Behavior Poisson process (MBPP), a novel Poisson process with a direct parameter correspondence to the popular self-exciting Hawkes process. We fit MBPP in the interval-censored setting using an interval-censored Poisson log-likelihood (IC-LL). We use the parameter equivalence to uncover the parameters of the associated Hawkes process. Second, we introduce two novel exogenous functions to distinguish the exogenous from the endogenous events. We propose the multi-impulse exogenous function - for when the exogenous events are observed as event time - and the latent homogeneous Poisson process exogenous function - for when the exogenous events are presented as interval-censored volumes. Third, we provide several approximation methods to estimate the intensity and compensator function of MBPP when no analytical solution exists. Fourth and finally, we connect the interval-censored loss of MBPP to a broader class of Bregman divergence-based functions. Using the connection, we show that the popularity estimation algorithm Hawkes Intensity Process (HIP) is a particular case of the MBPP. We verify our models through empirical testing on synthetic data and real-world data.
ICODEN: Ordinary Differential Equation Neural Networks for Interval-Censored Data
Department of Biostatistics and Health Data Science, University of Pittsburgh, PA, USA（匹兹堡大学生物统计与健康数据科学系）；Center for Applied Statistics and School of Statistics, Renmin University of China, Beijing, China（中国人民大学应用统计中心与统计学院）；Department of Applied Statistics, Konkuk University, Seoul, Republic of Korea（韩国康国民大学应用统计系）
Neural interval-censored survival regression with feature selection
Instituto de Ciencias Matemáticas, CSIC（数学科学研究所（西班牙高等科学研究理事会））；Universidad Autónoma de Madrid（马德里自治大学）；Harvard University（哈佛大学）；Université Paris 1 Panthéon-Sorbonne（巴黎第一大学先贤祠-索邦大学）；University of North Carolina, Chapel Hill（北卡罗来纳大学教堂山分校）
Interpretable Deep Regression Models with Interval-Censored Failure Time Data
School of Mathematics, Jilin University（吉林大学数学学院）；School of Economics and Statistics, Guangzhou University（广州大学经济与统计学院）；Department of Statistics, Chinese University of Hong Kong（香港中文大学统计系）；School of Data Science, Fudan University（复旦大学数据科学学院）
