Projects | Hongjia Henry Chen

Converting foreign currency from a financial literacy and proportional reasoning perspective

Mon, 22 Feb 2021 00:00:00 +0000

Can undergraduates convert $100 of foreign currency? Can they convert 100 NZD into foreign currency? We examine foreign currency conversion from a financial literacy lens and compare it to a proportional reasoning perspective. A qualitative and quantitative analysis of 108 undergraduate students revealed that consistent proportional reasoning contributed a greater impact on accuracy compared to financial literacy. We also find no evidence of a difference in preferred currency exchange representations and accuracy, bizarrely, more experienced currency exchangers performed below average. Furthermore we take a deeper dive into the different approaches and methodology students undertook to solve each particular problem and uncover common misconceptions.

Supervised by Dr. Igor' Kontorvich.

Mathematical billiards: Periodic orbits within quadrilaterials

Fri, 27 Nov 2020 00:00:00 +0000

Mathematical billiards is a dynamical system that models billiards in an idealised environment. The billiard ball is considered to be a point mass and satisfies the law of reflection when interacting with the boundary; the shape of the mathematical billiard is arbitrary. These simple constraints lead to surprisingly deep and complex dynamics.

We focus on billiards with quadrilateral boundaries. The existence of periodic orbits in all polygons is currently one of the most resistant problems in dynamics. This dissertation makes progress on this conjecture and explores the existence of periodic orbits within squares, rectangles and parallelograms. We provide alternative proofs to classical results for square billiards with additional insights and connections to number theory.

We also take a dynamical systems approach which enables the use of bifurcation theory and parameter continuation. We introduce a novel continuation formulation which bypasses the extreme degeneracies exhibited by mathematical billiards and use it to compute branches of periodic solutions as a parameter varies the shape of the billiard from a square to a rectangle and parallelogram. The insights gained from the numerical exploration lead us to prove that there exist no period-4 orbits within the parallelogram and to show in a computer-assisted manner the existence of a bifurcation diagram that exhibits a period-adding sequence, where a periodic orbit has its period change under parameter variation in successive jumps of four each time.

Supervised by Professor Hinke M. Osinga.

Dynamics of Twitter Hashtags

Fri, 28 Feb 2020 00:00:00 +0000

In this project, we quantify the statistical properties and dynamics of the frequency of hashtag use on Twitter. Hashtags are special words used in social media to attract attention and to organize content. Looking at the collection of all hashtags used in a period of time, we identify the scaling laws underpinning the hashtag frequency distribution (Zipf’s law), the number of unique hashtags as a function of sample size (Heaps' law), and the fluctuations around expected values (Taylor’s law). While these scaling laws appear to be universal, in the sense that similar exponents are observed irrespective of when the sample is gathered, the volume and the nature of the hashtags depend strongly on time, with the appearance of bursts at the minute scale, fat-tailed noise, and long-range correlations. We quantify this dynamics by computing the Jensen-Shannon divergence between hashtag distributions obtained τ times apart and we find that the speed of change decays roughly as 1/τ. Our findings are based on the analysis of 3.5×10^9 hashtags used between 2015 and 2016.

This project culminated in the following paper published in Chaos: An Interdisciplinary Journal of Nonlinear Science.

Supervised by Professor Eduardo G. Altmann.

One thing leads to another: Modelling earthquake occurrences

Thu, 27 Feb 2020 00:00:00 +0000

The timestamps of earthquakes form a point pattern in time and the locations form a point pattern in space. We model earthquakes in the Canterbury region using temporal, spatial and spatio-temporal point processes. We also use temporal self-exciting models to describe how after an earthquake occurs, there is an immediate increase in probability of another earthquake event. In this project, we use latent Gaussian fields to explain the spatial and temporal structure behind earthquake events. We explore how both spatial and temporal information can contribute to earthquake occurrences.

Supervised by Dr. Charlotte Jones-Todd.

Non-convergent Nash equilibria on a circular market

Wed, 27 Feb 2019 00:00:00 +0000

We use the Hotelling-Downs spatial model of competition to investigate the behavior of competing firms when the location space is a circle. Unlike the standard Hotelling-Downs model we assume that customers with some probability buy from retailers other than the closest one. We characterise convergent Nash equilibria and non-convergent Nash equilibria depending on the vector of those probabilities.

Supervised by Professor Arkadii Slinko.