Temporal modeling of real-life systems, such as social networks, financial markets and medical decision-support systems, is important to understand them better, and make predictions. Temporal data of these systems have irregular time granularity; therefore continuous-time models are a natural fit. In this thesis, we focus on one of the building blocks in this process, statistical inference. Additionally, we apply continuous-time models to a medical informatics application.
In our inference algorithms, we focus on continuous-time Markov processes (CTMPs). Answering queries about the components as a CTMP evolves involves inferring the probability distributions of the state of the system at query time points. When the number of components is large, exact inference becomes intractable since the state space is exponentially large in the number of components. Structured representations provide a framework to apply inference methods in an efficient way. Such representations usually also discretize time. However, choosing the right time- width is challenging since the observations are not synchronized among components and there might be large intervals without any observations. Therefore, our inference algorithms use the structured representation of continuous-time Bayesian networks (CTBNs), but they can also be applied to other continuous-time representations. A CTBN provides a compact representation using local dependencies. Unfortunately, exploiting the structure in the dynamics does not alleviate the need to represent the full joint space, and exact inference in CTBNs is intractable.
Our approximate inference computations concentrate on the key calculation for a CTMP, the matrix exponential. We use two different expansion of the matrix exponential to derive different approximation algorithms. Our first algorithm keeps the solution in the factorized state space by using uniformization. It is the first non-sampling method to have bounded error. Also, it has better experimental results than the previous methods. Our second algorithm is built upon the sum of time-ordered products. It combines the advantages of deterministic and sampling methods as it is deterministic and anytime. It converges to the true distribution in the limit of infinite computation time, and it is not random. Random methods such as sampling methods can lead to instability when used inside parameter estimation algorithms. We show that it performs as well as or better than the current best sampling approaches on benchmark problems.
Our last work is an application of a multivariate Gaussian process (MGP) to a medical informatics problem. We estimate the blood gas values of a patient during mechanical ventilation in a pediatric intensive care unit. Frequent blood gas values allow more responsive care which can reduce the duration of ventilation and risk of lung injury. Estimating these values from non-invasive measurements can reduce the number of invasive blood tests, which are challenging in children. We estimate them by using previous values of all variables, and current values of all non-invasive variables. We develop an MGP model because the variables are naturally continuous. Our results show promising prediction accuracies, which could be used to automate the ventilation process.
