the book provides a brief introduction to SAS, SPSS, and BMDP, along with their use in performing ANOVA. The book also has a chapter devoted to experimental designs and the corresponding ANOVA. In terms of coverage, a nice feature of the book is the inclusion of a chapter on  nite population models—typically not found in books on experimental designs and ANOVA. Several appendixes are given at the end of the book discussing some of the standard distributions, the Satterthwaite approximation, rules for computing the sums of squares, degrees of freedom, expected mean squares, and so forth. The exercises at the end of each chapter contain a number of numerical problems. Some of my quibbles about the book are the following. At times, it simply gives expressions without adequate motivation or examples. A reader who is not already familiar with ANOVA techniques will wonder as to the relevance of some of the expressions. Just to give an example, the quantity “sum of squares due to a contrast” is de ned on page 65. The algebraic property that the sums of squares due to a set of a ƒ 1 orthogonal contrasts will add up to the sum of squares due to an effect having a ƒ 1 df is then stated. Given the level of the book, discussion of such a property appears to be irrelevant. I did not see this property used anywhere in the book; neither did I see the sum of squares due to a contrast explicitly used or mentioned later in the book. Examples in which the one-way model is adequate are mentioned only after introducing the model and the assumptions, and the examples are buried inside the remarks (in small print) following the model. This is also the case with the two-way model with interaction (Chap. 4). The authors indicate in the preface that the remarks are mostly meant to include results to be kept out of the main body of the text. I believe that good examples should be the starting point for introducing ANOVA models. The authors present the analysis of  xed, random, and mixed models simultaneously. Motivating examples that distinguish between these scenarios should have been made the highlight of the presentation in each chapter rather than deferred to the later part of the chapter under “worked out examples” or buried within the remarks. The authors discuss transformations to correct lack of normality and lack of homoscedasticity (Sec. 2.22). However, these are not illustrated with any real examples. Regarding tests concerning the departure from the model assumptions, formal tests are presented in some detail; however, graphical procedures are only very brie y mentioned under a remark. I consider this to be a glaring omission. Consequently, I would be somewhat hesitant to recommend this book to anyone interested in actual data analysis using ANOVA unless the application is such that one of the standard models (along with the standard assumptions) is known to be adequate and diagnostic checks are not called for. Obviously, this is an unlikely scenario in most applications. The preceding criticisms aside, I can see myself consulting this book to refer to an ANOVA table, to look up an expected value or test statistic under a random or mixed-effects model, or to refer to the use of SAS, SPSS, or BMDP for performing ANOVA. The book is indeed an excellent source of reference for the ANOVA based on  xed, random, and mixed-effects models.

Linear Models in Statistics

Vehicle trajectory prediction is crucial for autonomous driving and advanced driver assistant systems. While existing approaches may sample from a predicted distribution of vehicle trajectories, they lack the ability to explore it – a key ability for evaluating safety from a planning and verification perspective. In this work, we devise a novel approach for generating realistic and diverse vehicle trajectories. We first extend the generative adversarial network (GAN) framework with a low-dimensional approximate semantic space, and shape that space to capture semantics such as merging and turning. We then sample from this space in a way that mimics the predicted distribution, but allows us to control coverage of semantically distinct outcomes. We validate our approach on a publicly available dataset and show results that achieve state-of-the-art prediction performance, while providing improved coverage of the space of predicted trajectory semantics.

DiversityGAN: Diversity-Aware Vehicle Motion Prediction via Latent Semantic Sampling

Agent behavior is arguably the greatest source of uncertainty in trajectory planning for autonomous vehicles. This problem has motivated significant amounts of work in the behavior prediction community on learning rich distributions of the future states and actions of agents. However, most current works on chance-constrained trajectory planning under agent or obstacle uncertainty either assume Gaussian uncertainty or linear constraints, which is limiting, or requires sampling, which can be computationally intractable to encode in an optimization problem. In this letter, we extend the state-of-the-art by presenting a methodology to upper-bound chance-constraints defined by polynomials and mixture models with potentially non-Gaussian components. Our method achieves its generality by using statistical moments of the distributions in concentration inequalities to upper-bound the probability of constraint violation. With this method, optimization-based trajectory planners can plan trajectories that are chance-constrained with respect to a wide range of distributions representing predictions of agent future positions. In experiments, we show that the resulting optimization problem can be solved with state-of-the-art nonlinear program solvers to plan trajectories fast enough for use online.

Non-Gaussian Chance-Constrained Trajectory Planning for Autonomous Vehicles Under Agent Uncertainty

Motion planning and control algorithms for autonomous vehicles need to be safe, and consider future movements of other road users to ensure collision-free trajectories. In this letter, we present a control scheme based on Model Predictive Control (MPC) with robust constraint satisfaction where the constraint uncertainty, stemming from the road users’ behavior, is multimodal. The method combines ideas from tube-based and scenario-based MPC strategies in order to approximate the expected cost and to guarantee robust state and input constraint satisfaction. In particular, we design a feedback policy that is a function of the disturbance mode and allows the controller to take less conservative actions. The effectiveness of the proposed approach is illustrated through two numerical simulations, where we compare it against a standard robust MPC formulation.

/pdf/a-robust-scenario-mpc-approach-for-uncertain-multi-modal-2p904rq5wu.pdf

A Robust Scenario MPC Approach for Uncertain Multi-Modal Obstacles

Predicting future motions of road participants is an important task for driving autonomously in urban scenes. Existing models excel at predicting marginal trajectories for single agents, yet it remains an open question to jointly predict scene compliant trajectories over multiple agents. The challenge is due to exponentially increasing prediction space as a function of the number of agents. In this work, we exploit the underlying relations between interacting agents and decouple the joint prediction problem into marginal prediction problems. Our proposed approach M2I first classifies interacting agents as pairs of influencers and reactors, and then leverages a marginal prediction model and a conditional prediction model to predict trajectories for the influencers and reactors, respectively. The predictions from interacting agents are combined and selected according to their joint likelihoods. Experiments show that our simple but effective approach achieves state-of-the-art performance on the Waymo Open Motion Dataset interactive prediction benchmark.

M2I: From Factored Marginal Trajectory Prediction to Interactive Prediction

This paper presents fast non-sampling based methods to assess the risk of trajectories for autonomous vehicles when probabilistic predictions of other agents' futures are generated by deep neural networks (DNNs). The presented methods address a wide range of representations for uncertain predictions including both Gaussian and non-Gaussian mixture models for predictions of both agent positions and controls. We show that the problem of risk assessment when Gaussian mixture models (GMMs) of agent positions are learned can be solved rapidly to arbitrary levels of accuracy with existing numerical methods. To address the problem of risk assessment for non-Gaussian mixture models of agent position, we propose finding upper bounds on risk using Chebyshev's Inequality and sums-of-squares (SOS) programming; they are both of interest as the former is much faster while the latter can be arbitrarily tight. These approaches only require statistical moments of agent positions to determine upper bounds on risk. To perform risk assessment when models are learned for agent controls as opposed to positions, we develop TreeRing, an algorithm analogous to tree search over the ring of polynomials that can be used to exactly propagate moments of control distributions into position distributions through nonlinear dynamics. The presented methods are demonstrated on realistic predictions from DNNs trained on the Argoverse and CARLA datasets and are shown to be effective for rapidly assessing the probability of low probability events.

http://www.roboticsproceedings.org/rss16/p089.pdf

Fast Risk Assessment for Autonomous Vehicles Using Learned Models of Agent Futures.

Agent behavior is arguably the greatest source of uncertainty in trajectory planning for autonomous vehicles. This problem has motivated significant amounts of work in the behavior prediction community on learning rich distributions of the future states and actions of agents. However, most current works on chance-constrained trajectory planning under agent or obstacle uncertainty either assume Gaussian uncertainty or linear constraints, which is limiting, or requires sampling, which can be computationally intractable to encode in an optimization problem. In this paper, we extend the state-of-the-art by presenting a methodology to upper-bound chance-constraints defined by polynomials and mixture models with potentially non-Gaussian components. Our method achieves its generality by using statistical moments of the distributions in concentration inequalities to upper-bound the probability of constraint violation. With this method, optimization-based trajectory planners can plan trajectories that are chance-constrained with respect to a wide range of distributions representing predictions of agent future positions. In experiments, we show that the resulting optimization problem can be solved with state-of-the-art nonlinear program solvers to plan trajectories fast enough for use online.

Non-Gaussian Chance-Constrained Trajectory Planning for Autonomous Vehicles under Agent Uncertainty.

This paper presents fast non-sampling based methods to assess the risk for trajectories of autonomous vehicles when probabilistic predictions of other agents’ futures are generated by deep neural networks (DNNs). The presented methods address a wide range of representations for uncertain predictions including both Gaussian and non-Gaussian mixture models to predict both agent positions and control inputs conditioned on the scene contexts. We show that the problem of risk assessment when Gaussian mixture models of agent positions are learned can be solved rapidly to arbitrary levels of accuracy with existing numerical methods. To address the problem of risk assessment for non-Gaussian mixture models of agent position, we propose finding upper bounds on risk using nonlinear Chebyshev’s Inequality and sums-of-squares programming; they are both of interest as the former is much faster while the latter can be arbitrarily tight. These approaches only require higher order statistical moments of agent positions to determine upper bounds on risk. To perform risk assessment when models are learned for agent control inputs as opposed to positions, we propagate the moments of uncertain control inputs through the nonlinear motion dynamics to obtain the exact moments of uncertain position over the planning horizon. To this end, we construct deterministic linear dynamical systems that govern the exact time evolution of the moments of uncertain position in the presence of uncertain control inputs. The presented methods are demonstrated on realistic predictions from DNNs trained on the Argoverse and CARLA datasets and are shown to be effective for rapidly assessing the probability of low probability events.

Allen Wang

Papers

Non-Gaussian Chance-Constrained Trajectory Planning for Autonomous Vehicles Under Agent Uncertainty

Fast Risk Assessment for Autonomous Vehicles Using Learned Models of Agent Futures.

Non-Gaussian Chance-Constrained Trajectory Planning for Autonomous Vehicles under Agent Uncertainty.

Fast nonlinear risk assessment for autonomous vehicles using learned conditional probabilistic models of agent futures

Fast Risk Assessment for Autonomous Vehicles Using Learned Models of Agent Futures