To infer the hidden states from the noisy observations and make predictions based on a set of input states and output observations are two challenging problems in many research areas. Examples of applications many include position estimation from various measurable radio signals in indoor environments, self-navigation for autonomous cars, modeling and predicting of the traffic flows, and flow pattern analysis for crowds of people. In this thesis, we mainly use the Bayesian inference framework for position estimation in an indoor environment, where the radio propagation is uncertain. In Bayesian inference framework, it is usually hard to get analytical solutions. In such cases, we resort to Monte Carlo methods to solve the problem numerically. In addition, we apply Bayesian nonparametric modeling for trajectory learning in sport analytics.

The main contribution of this thesis is to propose sequential Monte Carlo methods, namely particle filtering and smoothing, for a novel indoor positioning framework based on proximity reports. The experiment results have been further compared with theoretical bounds derived for this proximity based positioning system. To improve the performance, Bayesian non-parametric modeling, namely Gaussian process, has been applied to better indicate the radio propagation conditions. Then, the position estimates obtained sequentially using filtering and smoothing are further compared with a static solution, which is known as fingerprinting.

Moreover, we propose a trajectory learning framework for flow estimation in sport analytics based on Gaussian processes. To mitigate the computation deficiency of Gaussian process, a grid-based on-line algorithm has been adopted for real-time applications. The resulting trajectory modeling for individual athlete can be used for many purposes, such as performance prediction and analysis, health condition monitoring, etc. Furthermore, we aim at modeling the flow of groups of athletes, which could be potentially used for flow pattern recognition, strategy planning, etc.

Estimation of unknown parameters is considered as one of the major research areas in statistical signal processing. In the most recent decades, approaches in estimation theory have become more and more attractive in practical applications. Examples of such applications may include, but are not limited to, positioning using various measurable radio signals in indoor environments, self-navigation for autonomous cars, image processing, radar tracking and so on. One issue that is usually encountered when solving an estimation problem is to identify a good system model, which may have great impacts on the estimation performance. In this thesis, we are interested in studying estimation problems particularly in inferring the unknown positions from noisy radio signal measurements. In addition, the modeling of the system is studied by investigating the relationship between positions and radio signal strength measurements.

One of the main contributions of this thesis is to propose a novel indoor positioning framework based on proximity measurements, which are obtained by quantizing the received signal strength measurements. Sequential Monte Carlo methods, to be more specific particle filter and smoother, are utilized for estimating unknown positions from proximity measurements. The Cramér-Rao bounds for proximity-based positioning are further derived as a benchmark for the positioning accuracy in this framework.

Secondly, to improve the estimation performance, Bayesian non-parametric modeling, namely Gaussian processes, have been adopted to provide more accurate and flexible models for both dynamic motions and radio signal strength measurements. Then, the Cramér-Rao bounds for Gaussian process based system models are derived and evaluated in an indoor positioning scenario.

In addition, we estimate the positions of stationary devices by comparing the individual signal strength measurements with a pre-constructed fingerprinting database. The positioning accuracy is further compared to the case where a moving device is positioned using a time series of radio signal strength measurements.

Moreover, Gaussian processes have been applied to sports analytics, where trajectory modeling for athletes is studied. The proposed framework can be further utilized to carry out, for instance, performance prediction and analysis, health condition monitoring, etc. Finally, a grey-box modeling is proposed to analyze the forces, particularly in cross-country skiing races, by combining a deterministic kinetic model with Gaussian process.