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For linear stochastic time-varying state space models with Gaussian noises, this paper investigates state estimation for the scenario where the input variables of the state equation are not fully observed but rather the input data is available only at an aggregate level. Unlike the existing filters for unknown inputs that are based on the approach of minimum-variance unbiased estimation, this paper does not impose the unbiasedness condition for state estimation; instead it incorporates a Bayesian approach to derive a modified Kalman filter by pooling the prior knowledge about the state vector at the aggregate level with the measurements on the output variables at the original level of interest. The estimated state vector is shown to be a minimum-mean-square-error estimator. The developed filter provides a unified approach to state estimation: it includes the existing filters obtained under two extreme scenarios as its special cases, i.e., the classical Kalman filter where all the inputs are observed and the filter for unknown inputs.

a b s t r a c tFor linear stochastic time-varying systems, we investigate the properties of the Kalman filter with partially observed inputs. We first establish the existence condition of a general linear filter when the unknown inputs are partially observed. Then we examine the optimality of the Kalman filter with partially observed inputs. Finally, on the basis of the established existence condition and optimality result, we investigate asymptotic stability of the filter for the corresponding time-invariant systems. It is shown that the results on existence and asymptotic stability obtained in this paper provide a unified approach to accommodating a variety of filtering scenarios as its special cases, including the classical Kalman filter and state estimation with unknown inputs.

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For the purposes of both traffic-light control and the design of roadway layouts, it is important to understand pedestrian street-crossing behavior because it is not only crucial for improving pedestrian safety but also helps to optimize vehicle flow. This paper explores the mechanism of pedestrian street crossings during the red-man phase of traffic light signals and proposes a model for pedestrians' waiting times at signalized intersections. We start from a simplified scenario for a particular pedestrian under specific traffic conditions. Then we take into account the interaction between vehicles and pedestrians via statistical unconditioning. We show that this in general leads to a U-shaped distribution of the pedestrians' intended waiting time. This U-shaped distribution characterizes the nature of pedestrian street-crossing behavior, showing that in general there are a large proportion of pedestrians who cross the street immediately after arriving at the crossing point, and a large proportion of pedestrians who are willing to wait for the entire red-man phase. The U-shaped distribution is shown to reduce to a J-shaped or Lshaped distribution for certain traffic scenarios. The proposed statistical model was applied to analyze real field data.

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• ABSTRACTThe multinomial logit model in discrete choice analysis is widely used in transport research.It has long been known that the Gumbel distribution forms the basis of the multinomial logit model. Although the Gumbel distribution is a good approximation in some applications such as route choice problems, it is chosen mainly for mathematical convenience. This can be restrictive in many other scenarios in practice. In this paper we show that the assumption of the Gumbel distribution can be substantially relaxed to include a large class of distributions that is stable with respect to the minimum operation. The distributions in the class allow heteroscedastic variances. We then seek a transformation that stabilizes the heteroscedastic variances. We show that this leads to a semiparametric choice model which links the linear combination of travel-related attributes to the choice probabilities via an unknown sensitivity function. This sensitivity function reflects the degree of travelers" sensitivity to the changes in the combined travel cost. The estimation of the semiparametric choice model is also investigated and empirical studies are used to illustrate the developed method.