Mentor: Lajos Horvath
Fall 2007 project description
Fall 2007 final report
Spring 2008 project description:
Statistical Analysis of High Frequency Data
In several applications (magnetic storms, stock prices, foreign exchange rates) observations are coming in at a very high frequency. Thus, we have an extremely large amount of observations and standard statistical methods cannot be used. We use the model when the observations are treated like a discretized version of a continuous curve. We propose to continue the study of the data set containing the average monthly temperature in Prague for the last 215 years. We hope to be able to analyze the data and the progress made last semester to see if the average monthly temperature in Prague has changed over time.
The basic idea is that stochastic processes can be written as infinite sums using an orthonormal system in L2. There are several choices to use orthogonal systems in these decompositions. According to the general principle component analysis, the orthonormal system determined by the correlations between the observations should be used. In this case the first ten or twelve terms in the infinite representation can be used and a very large percentage of randomness in the data can be explained. In the first semester of this project we were able to find an orthonormal system and find the twelve terms for each year of the data set. We can now use this information to explain the data. Also, we can analyze the data to see if it follows Brownian Motion.
The mathematical theory of the suggested method is based on the L2 of stochastic processes and orthogonal functions. This theory was researched in the first semester and now we can move on to study more theory in Brownian Motion.
Spring 2008 final report