Basic Examples (2) 

Get a time series of closing prices and visualize it:

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Compute the half difference of the time series and visualize it:

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Scope (2) 

Get a time series of closing prices:

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With order 1, FractionalDTimeSeries gives the usual first difference:

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However, applying the half-difference twice on the original time series yields an equivalent first difference, but with fewer data points and on a slightly different scale:

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Get a time series of closing prices:

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Apply a fractional difference of order 0.3531 to the original time series with the default tolerance:

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For the same time series, a fractional difference of order 0.3531 and a larger tolerance will generate a time series with more data points:

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In contrast, a fractional difference of the same order 0.3531, but with a smaller tolerance, will generate a time series with fewer data points:

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Applications (3) 

Simulate an ARIMAProcess with linear trend and apply a fractional difference of order 0.5:

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Simulate an FractionalBrownianMotionProcess with Hurst index 0.65 and apply a fractional difference of order 0.45 to it:

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Get a time series of the frequencies of the word "Peru" in typical published text, and apply a fractional difference of order 0.35 and a tolerance of 0.01 to it:

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Possible Issues (4) 

FractionalDTimeSeries requires the order to be a positive real number. Negative orders (integration) are currently not implemented:

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FractionalDTimeSeries requires the tolerance to be positive:

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If the tolerance is too small, no transformation will be performed:

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If the tolerance is set too high, the transformed time series will be essentially the same as the original one, but in a different scale:

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Neat Examples (4) 

A logarithmically-transformed time series:

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Find the relationship between the statistical value of a UnitRootTest and the CorrelationFunction of a transformed time series, by using several orders between 0 and 1:

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Apply a UnitRootTest (red axis) on each differentiated time series to see up from which level of d is possible to reject H₀. Also, compute its CorrelationFunction at lag 1 (blue axis) to compare:

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Plot the results:

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