Business Finland 6G Visible project’s objective is the development of 6G-era service and architecture solutions utilizing autonomous and semi-autonomous driving, with both physical and logical computational elements and with use cases for real-life verification and validation. Finnish Meteorological Institute is focusing especially on weather- and safety-related services for autonomous vehicles. We are tailoring our road weather services for the special needs of autonomous driving, keeping in mind that autonomous vehicles are more sensitive to the harsh winter weather conditions and benefit from more accurate weather information considering the sensor systems of each vehicle. Employing weather radar-based nowcasting of more accurate short-term precipitation forecasting benefits autonomous traffic, especially in cases of heavy local precipitation by re-routing/route planning and avoiding heaviest precipitation. Evaluation of autonomous vehicles’ sensor systems’ sensitivity to harsh weather conditions allows for weather forecasting based on the real vulnerability of each vehicle.

A possibly time-dependent transition intensity matrix or generator $(Q(t))$ characterizes the law of a Markov jump process (MP). For a time homogeneous MP, the transition probability matrix (TPM) can be expressed as a matrix exponential of $Q$. However, when dealing with a time non-homogeneous MP, there is often no simple analytical form of the TPM in terms of $Q(t)$, unless they all commute. This poses a challenge because when a continuous MP is observed intermittently, a TPM is required to build a likelihood. In this paper, we show that the estimation of the transition intensities of a two-state nonhomogeneous Markov model can be carried out by augmenting the intermittent observations with honest random times associated with two independent driving Poisson point processes, and that sampling the full path is not required. We propose a pseudo-marginal McMC algorithm to estimate the transition rates using the augmented data. Finally, we illustrate our approach by simulating a continuous MP and by using observed (intermittent) time grids extracted from real clinical visits data.

We study discrete-time, discrete-state multistate Markov models from the perspective of algebraic statistics. These models are widely studied in event history analysis, and are characterized by the state space, the initial distribution and the transition probabilities. A finite path under the multistate Markov model is a particular set of states occupied at finite time instances $\{1, \dots, n\}$. The main goal of this paper is to establish a bridge between event history analysis and algebraic statistics. The joint probabilities of finite paths in these models have a natural monomial parametrization in terms of the initial distribution and the transition probabilities. We study the polynomial relations among joint path probabilities. When the statistical constraints on the parameters are disregarded, nonhomogeneous multistate Markov models of arbitrary order can be viewed as slices of decomposable hierarchical models. This yields a complete description of their vanishing ideals as toric ideals generated by explicit families of binomials. Moreover, the variety of this vanishing ideal equals the nonhomogeneous multistate Markov model on the probability simplex. In contrast, homogeneous multistate Markov models exhibit different algebraic behavior, as time homogeneity imposes additional polynomial relations, leading to vanishing ideals that are strictly larger than in the nonhomogeneous case. We also derive families of binomial relations that vanish on homogeneous multistate Markov models. We investigate maximum likelihood estimation from statistical and algebraic perspectives. For nonhomogeneous models, classical and algebraic formulas agree; in the homogeneous case, the algebraic approach is more complex. Lastly, we provide data applications where we demonstrate the statistical theory to obtain the maximum likelihood estimates of the parameters under specific multistate Markov models.