Fractal dimension standard deviations and autocorrelations in currency pairs

As you may have noticed I have gone quite heavily into chaos based analysis during the past few weeks using different types of statistics. On my last post about the fractal dimension I talked about how the fractal dimension could finally be a clue as to why it is easier to find inefficiencies on some instruments compared to others and today I want to go deeper into some interesting statistical behavior I have found pertaining to the fractal dimension within Forex instruments, mainly about the standard deviations and auto-correlations of rolling window fractal dimension measurements within these series. Today I am going to show you the data I have gathered on this subject across 16 currency pairs, how it relates to previous findings and why this further reinforces the relationship between the number of inefficiencies that can be found on symbols and their fractal dimension.


The fractal dimension gives us an idea about the character of a financial time series (see previous posts on the subject for more details). With this in mind it is reasonable to assume that if we slice the data into shorter timeframes we might be able to relate the amount of change in the fractal dimension to whether or not we are able to find inefficiencies within a symbol. If the character of a series changes slowly through time and is different from randomness then we should expect the fractal dimension to have a narrower variation while if variations are large we might expect to be able to find less inefficiencies because the behavior of the instrument would inherently become less predictable. Trying to explore this hypothesis I divided the 1H data for 16 Forex symbols from 1986 to 2016 into 200 bar sections and studied the properties of the distribution of fractal dimension values calculated using the “rodogram” technique using the fractaldim R package on each symbol.

The standard deviation values for the fractal dimension distributions for each symbol are shown on the graph above. As you can see the standard deviation of the fractal dimension is lowest for the symbols where I have empirically been able to find the most inefficiencies, mainly the EUR/USD, USD/CHF and USD/JPY, in good agreement with my last blog post attempting to relate the fractal dimension with this fact. However the EUR/JPY continues to be expected to yield few systems within this test while experimentally we have always found that it yields more systems than symbols like the AUD/USD. This highlights that although the findings I have made within the past few weeks are part of the puzzle they still don’t paint a crystal clear picture of what actually leads to a prediction of predictability.


To continue my study I decided to look at the 100 lag auto-correlations for all the different symbols and actually found something quite surprising. The symbols that traditionally generate more systems, like the EUR/USD and the USD/JPY, have almost no statistically relevant auto-correlations of the fractal dimension while the symbols that are very difficult to exploit, such as the AUD/USD and the USD/CAD have high auto-correlations. This means that for these symbols the next value of the fractal dimension within the series is bound to look more like the last while their standard deviation suggests that the range of variation can be very large through time. This may seem contradictory at first so let me take some time to better explain what is going on here.

On these symbols each 200 bar period is usually followed by another period where the fractal dimension is similar, however in time the fractal dimension oscillates through very extreme values, reason why the standard deviation of the fractal dimension is much higher than for symbols like the EUR/USD. So while in the EUR/USD you have rather uncorrelated changes in the fractal dimension over a narrow range in symbols like the AUD/USD you have highly correlated changes in the fractal dimension across significantly much larger ranges. This is perfectly coherent with the finding of long term inefficiencies because you want the character of the series to remain rather constant which implies that it’s more important to have a narrow range in the fractal dimension than a high auto-correlation.


The fractal dimension variations define such huge differences in character between series – some maintaining their character in the long term while others slowly oscillating between widely different character values – that this requires much further study. Perhaps there are ways to exploit series with very high auto-correlations in the fractal dimension – like the USD/CAD – which are very different from the finding of simple long term inefficiencies. If you would like to learn more about financial time series and how you can analyse them using tools such as R please consider joining, a website filled with educational videos, trading systems, development and a sound, honest and transparent approach towards automated trading.strategies

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2 Responses to “Fractal dimension standard deviations and autocorrelations in currency pairs”

  1. Jonathan Shaw says:

    I think you’ve got to an interesting place Daniel.

    It makes sense that the eurusd, the biggest volume instrument, has the most stable characteristics.

    Your methodology for finding inefficiencies has required them to be long-term stable. So the issue may not be whether certain instruments have inefficiencies or not, but rather whether they have stable persistent inefficiencies.

    Seems like a fruitful place to have reached.

    • admin says:

      Hi Jonathan,

      Thanks for commenting. Yes, I think that this leads to some very interesting conclusions as well. I have submitted an article to AutomatedTrader magazine which further explores this with some additional calculations (varying slice size, difference between the fractal dimension in the real data vs random series, etc). Hopefully we can discuss that article when it comes out! Thanks again for posting,

      Best Regards,


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