Mathematics

The Air Health Trend Indicator (AHTI) is a joint Health Canada / Environment and Climate Change Canada initiative that seeks to model the Canadian national population health risk due to acute exposure to ambient air pollution. The common model in the field uses averages of local ambient air pollution monitors to produce a population-level exposure proxy variable. This method is applied to ozone, nitrogen dioxide, particulate matter, and other similar air pollutants.

We examine the representative nature of these proxy averages on a large-scale Canadian data set, representing hundreds of monitors and dozens of city-level populations. The careful determination of temporal and spatial correlations between the disparate monitors allows for more precise estimation of population-level exposure, taking inspiration from the land-use regression models commonly used in geography. We conclude this work with an examination of the risk estimation differences between the original, simplistic population exposure metric and our new, revised metric.

Author Keywords: Air Pollution, Population Health Risk, Spatial Process, Spatio-Temporal, Temporal Process, Time Series

Population aging has become a worldwide concern since the nineteenth century. The decrease in birth rate and the increase in life expectancy will make China's population age rapidly. If the growth rate of the number of workers is less than that of the number of retirees, in the long run, there will be fewer workers per retiree. This will apply great pressure to China's public pension system in the next several decades. This is a global problem known as the "pension crisis". In this thesis, a long-term vision for China's urban pension system is presented. Based on the mathematical models and the projections for demographic variables, economic variables and pension scheme variables, we test how the changes in key variables affect the balances of the pension fund in the next 27 years. This thesis applies methods of deterministic and stochastic modeling as well as sensitivity analysis to the problem. Using sensitivity analysis, we find that the pension fund balance is highly sensitive to the changes in retirement age compared with other key variables. Monte Carlo simulations are also used to find the possible distributions of the pension fund balance by the end of the projection period. Finally, according to my analysis, several changes in retirement age are recommended in order to maintain the sustainability of China's urban basic pension scheme.

Author Keywords: China, demographic changes, Monte Carlo simulation, pension fund, sensitivity tests, sustainability

This thesis is intended to support dependent-on-crops farmers to hedge the price risks of their crops. Firstly, we applied one-factor model, which incorporated a deterministic function and a stochastic process, to predict the future prices of crops (soybean). A discrete form was employed for one-month-ahead prediction. For general prediction, de-trending and de-cyclicality were used to remove the deterministic function. Three candidate stochastic differential equations (SDEs) were chosen to simulate the stochastic process; they are mean-reverting Ornstein-Uhlenbeck (OU) process, OU process with zero mean, and Brownian motion with a drift. Least squares methods and maximum likelihood were used to estimate the parameters. Results indicated that one-factor model worked well for soybean prices. Meanwhile, we provided a two-factor model as an alternative model and it also performed well in this case. In the second main part, a zero-cost option package was introduced and we theoretically analyzed the process of hedging. In the last part, option premiums obtained based on one-factor model could be compared to those obtained from Black-Scholes model, thus we could see the differences and similarities which suggested that the deterministic function especially the cyclicality played an essential role for the soybean price, thus the one-factor model in this case was more suitable than Black-Scholes model for the underlying asset.

Author Keywords: Brownian motion, Least Squares Method, Maximum Likelihood Method, One-factor Model, Option Pricing, Ornstein-Uhlenbeck Process

Artificial Neural Networks (ANN) have been applied to many areas of research. These techniques use a series of object attributes and can be trained to recognize different classes of objects. The Self-Organizing Map (SOM) is an unsupervised machine learning technique which has been shown to be successful in the mapping of high-dimensional data into a 2D representation referred to as a map. These maps are easier to interpret and aid in the classification of data. In this work, the existing algorithms for the SOM have been extended to generate 3D maps. The higher dimensionality of the map provides for more information to be made available to the interpretation of classifications. The effectiveness of the implementation was verified using three separate standard datasets. Results from these investigations supported the expectation that a 3D SOM would result in a more effective classifier.

The 3D SOM algorithm was then applied to an analysis of galaxy morphology classifications. It is postulated that the morphology of a galaxy relates directly to how it will evolve over time. In this work, the Spectral Energy Distribution (SED) will be used as a source for galaxy attributes. The SED data was extracted from the NASA Extragalactic Database (NED). The data was grouped into sample sets of matching frequencies and the 3D SOM application was applied as a morphological classifier. It was shown that the SOMs created were effective as an unsupervised machine learning technique to classify galaxies based solely on their SED. Morphological predictions for a number of galaxies were shown to be in agreement with classifications obtained from new observations in NED.

Author Keywords: Galaxy Morphology, Multi-wavelength, parallel, Self-Organizing Maps