Time-Space Scaling of Financial Time Series

Abstract

We propose a new method to describe scaling behavior of time series. We introduce an extension of extreme values: maximum and minimum. By using these extreme values determined by time-space scaling with a spread (or width), functions of this spread are defined. One is the number of these extreme values, and the other is the total variation among these extreme values. These functions are independent of time scale. In high frequency data, observations can occur at varying time intervals. In particular, on fractal analysis, interpolation influences the results. Using these functions, we can analyze non-equidistant data without interpolation. Moreover the problem of choosing the appropriate time scale to use for analyzing market data is avoided. In other words, 'time' is defined by fluctuations here. Lastly, these functions are related to a viewpoint of investor whose transaction costs coincide with the spread.