Random Fractals and the Stock Market

Surrogates of the Stock Market - Cartoon Driven IFS

Equal-Size Bins - Example 1

Here we use the cartoon with generator having turning points (1/3, 2/3) and (5/9, 1/3).

Here is the time series of the cartoon data. Here is the IFS driven by the data. Note the familiar backward Z, a characteristic of consecutive data points lying in the same bin or in adjoining bins. Here are the successive differences of the cartoon data. Note most differences lie in bin2 and very few in bin4. Here is the IFS driven by these differences of data points. The main visible features are (i) most IFS points lie on the gasket (because so few data points lie in bin4), (ii) the great concentration of IFS points at corner 2 (because most data points lie in bin2), and (iii) of the three sides of the gasket, the bottom is the most densely filled in (because many more consecutive data points lie in bin1 and bin2, than in bin2 and bin3, or in bin1 and bin3).

To illustrate the extent to which the details of this driven IFS depend on the particular random sequence of up and down segments, here are two more difference and driven IFS plots with different random sequences.

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