Integration of micro- and macroscopic models for pedestrian evacuation simulation

Simulation of pedestrian evacuations of smart buildings in emergency is a powerful tool for building analysis, dynamic evacuation planning and real-time response to the evolving state of evacuations. Macroscopic pedestrian models are low-complexity models that are and well suited to algorithmic analysis and planning, but are quite abstract. Microscopic simulation models allow for a high level of simulation detail but can be computationally intensive. By combining micro- and macro- models we can use each to overcome the shortcomings of the other and enable new capability and applications for pedestrian evacuation simulation that would not be possible with either alone. We develop the EvacSim multi-agent pedestrian simulator and procedurally generate macroscopic flow graph models of building space, integrating micro- and macroscopic approaches to simulation of the same emergency space. By “coupling” flow graph parameters to microscopic simulation results, the graph model captures some of the higher detail and fidelity of the complex microscopic simulation model. The coupled flow graph is used for analysis and prediction of the movement of pedestrians in the microscopic simulation, and investigate the performance of dynamic evacuation planning in simulated emergencies using a variety of strategies for allocation of macroscopic evacuation routes to microscopic pedestrian agents. The predictive capability of the coupled flow graph is exploited for the decomposition of microscopic simulation space into multiple future states in a scalable manner. By simulating multiple future states of the emergency in short time frames, this enables sensing strategy based on simulation scenario pattern matching which we show to achieve fast scenario matching, enabling rich, real-time feedback in emergencies in buildings with meagre sensing capabilities.

Financial modelling with 2-EPT probability density functions

The class of all Exponential-Polynomial-Trigonometric (EPT) functions is classical and equal to the Euler-d’Alembert class of solutions of linear differential equations with constant coefficients. The class of non-negative EPT functions defined on [0;1) was discussed in Hanzon and Holland (2010) of which EPT probability density functions are an important subclass. EPT functions can be represented as ceAxb, where A is a square matrix, b a column vector and c a row vector where the triple (A; b; c) is the minimal realization of the EPT function. The minimal triple is only unique up to a basis transformation. Here the class of 2-EPT probability density functions on R is defined and shown to be closed under a variety of operations. The class is also generalised to include mixtures with the pointmass at zero. This class coincides with the class of probability density functions with rational characteristic functions. It is illustrated that the Variance Gamma density is a 2-EPT density under a parameter restriction. A discrete 2-EPT process is a process which has stochastically independent 2-EPT random variables as increments. It is shown that the distribution of the minimum and maximum of such a process is an EPT density mixed with a pointmass at zero. The Laplace Transform of these distributions correspond to the discrete time Wiener-Hopf factors of the discrete time 2-EPT process. A distribution of daily log-returns, observed over the period 1931-2011 from a prominent US index, is approximated with a 2-EPT density function. Without the non-negativity condition, it is illustrated how this problem is transformed into a discrete time rational approximation problem. The rational approximation software RARL2 is used to carry out this approximation. The non-negativity constraint is then imposed via a convex optimisation procedure after the unconstrained approximation. Sufficient and necessary conditions are derived to characterise infinitely divisible EPT and 2-EPT functions. Infinitely divisible 2-EPT density functions generate 2-EPT Lévy processes. An assets log returns can be modelled as a 2-EPT Lévy process. Closed form pricing formulae are then derived for European Options with specific times to maturity. Formulae for discretely monitored Lookback Options and 2-Period Bermudan Options are also provided. Certain Greeks, including Delta and Gamma, of these options are also computed analytically. MATLAB scripts are provided for calculations involving 2-EPT functions. Numerical option pricing examples illustrate the effectiveness of the 2-EPT approach to financial modelling.