Multivariate Density Estimation by Neural Networks

PRISMA: A novel approach for deriving probabilistic surrogate safety measures for risk evaluation

Surrogate Safety Measures (SSMs) are used to express road safety in terms of the safety risk in traffic conflicts. Typically, SSMs rely on assumptions regarding the future evolution of traffic participant trajectories to generate a measure of risk, restricting their applicability to scenarios where these assumptions are valid. In response to this limitation, we present the novel Probabilistic RISk Measure derivAtion (PRISMA) method. The objective of the PRISMA method is to derive SSMs that can be used to calculate in real time the probability of a specific event (e.g., a crash). The PRISMA method adopts a data-driven approach to predict the possible future traffic participant trajectories, thereby reducing the reliance on specific assumptions regarding these trajectories. Since the PRISMA is not bound to specific assumptions, the PRISMA method offers the ability to derive multiple SSMs for various scenarios. The occurrence probability of the specified event is based on simulations and combined with a regression model, this enables our derived SSMs to make real-time risk estimations. To illustrate the PRISMA method, an SSM is derived for risk evaluation during longitudinal traffic interactions. Since there is no known method to objectively estimate risk from first principles, i.e., there is no known risk ground truth, it is very difficult, if not impossible, to objectively compare the relative merits of two SSMs. Instead, we provide a method for benchmarking our derived SSM with respect to expected risk trends. The application of the benchmarking illustrates that the SSM matches the expected risk trends. Whereas the derived SSM shows the potential of the PRISMA method, future work involves applying the approach for other types of traffic conflicts, such as lateral traffic conflicts or interactions with vulnerable road users.

Multivariate Density Estimation with Deep Neural Mixture Models

Albeit worryingly underrated in the recent literature on machine learning in general (and, on deep learning in particular), multivariate density estimation is a fundamental task in many applications, at least implicitly, and still an open issue. With a few exceptions, deep neural networks (DNNs) have seldom been applied to density estimation, mostly due to the unsupervised nature of the estimation task, and (especially) due to the need for constrained training algorithms that ended up realizing proper probabilistic models that satisfy Kolmogorov’s axioms. Moreover, in spite of the well-known improvement in terms of modeling capabilities yielded by mixture models over plain single-density statistical estimators, no proper mixtures of multivariate DNN-based component densities have been investigated so far. The paper fills this gap by extending our previous work on neural mixture densities (NMMs) to multivariate DNN mixtures. A maximum-likelihood (ML) algorithm for estimating Deep NMMs (DNMMs) is handed out, which satisfies numerically a combination of hard and soft constraints aimed at ensuring satisfaction of Kolmogorov’s axioms. The class of probability density functions that can be modeled to any degree of precision via DNMMs is formally defined. A procedure for the automatic selection of the DNMM architecture, as well as of the hyperparameters for its ML training algorithm, is presented (exploiting the probabilistic nature of the DNMM). Experimental results on univariate and multivariate data are reported on, corroborating the effectiveness of the approach and its superiority to the most popular statistical estimation techniques.