Application of Complex Hilbert Principal Component Analysis to Financial Data

We addressed two areas of concern regarding the analysis of a financial time series with a correlation structure, coarse graining (or renormalization) and the extraction of leading and lagging structures. We introduce the complex Hilbert principal component analysis to solve these two problems, and apply them to the time series of 33 Tokyo Stock Exchange industry indices and Tokyo Stock Price Index. Our data analysis proved that the time difference between the leading indices and the lagging indices decreases when the market mode is dominant, i.e., the herd behavior of the market.

I. Introduction

We addressed two areas of concern regarding the analysis of a financial time series with a correlation structure. The first area of concern was the coarse graining or renormalization of a time series. To anticipate financial and economic crises, it is important to monitor a large amount of data recorded at different time intervals. The simplest method to tackle this situation is to match the time interval with the longest time interval. For example, let us consider a situation when we investigate the producer price index (PPI) and stock prices. PPI is reported monthly by the government, and stock exchanges record the stock prices for each transaction. Thus, we define the monthly stock price. The simplest and the most frequently adopted method is to consider the closing prices of the last trading day of the month as the monthly data. However, this method ignores the microstructure of the time series of a stock price. To overcome this problem, we introduced the concept of coarse graining or renormalization.