Using Fractal Dimension to Test Currency Pair Relationships
Summary
The article explains how fractal dimension can describe the structure formed by simultaneous observations of multiple variables. It introduces box counting and the Grassberger–Procaccia correlation-dimension approach, where changes in estimated dimension across scales can help distinguish underlying degrees of freedom from noise. A synthetic noisy line illustrates how dimension can rise at small scales as random variation fills more of the space.
The practical study examines closing prices and price increments for seven major forex pairs over 2000–2009, across timeframes from five minutes to daily. Price-level dimensions changed little as pairs were added, while increment dimensions rose as additional pairs entered the data. Although AUDUSD and NZDUSD increments showed stronger correlation on the daily than five-minute interval, the dimension estimates did not reveal a lower-dimensional joint structure. The authors therefore conclude that relationships are obscured by noise, leaving the question unresolved. The study is limited by its historical sample and insufficient observations at weekly or monthly frequencies; its results do not establish that currency pairs are truly independent.
Key ideas
- Fractal dimension estimates how observations fill a multidimensional space and can be interpreted as a proxy for independent variables.
- Box counting estimates dimension from how occupied cells change with cell size, while the article favors the Grassberger–Procaccia method for higher-dimensional data.
- A noisy one-dimensional structure can appear to have a higher dimension at small scales because noise fills the surrounding space.
- In the studied forex data, increment dimensions increased as more currency pairs were added across the examined timeframes.
- The findings suggest that noise obscures possible currency relationships rather than proving that those relationships are absent.
Tags
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.