Mathematics

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