Random Fractals and the Stock Market

Surrogates of the Stock Market - Cartoon Driven IFS

Mean-Centered Bins - Example 1

Here we use the cartoon with generator having turning points (1/3, 2/3) and (5/9, 1/3).
Here are the successive differences of the cartoon data, where B1 and B3 are 0.1σ from the mean. Note most differences lie in bin1 and bin4, with few in bin2 and bin3. Here is the IFS driven by these differences as data points. The most visible features are (i) most IFS points lie on the line connecting the lower left and upper right corners of the square, and (ii) in the squares with address 2 and 3, almost all of the points connect the lower left and upper right corners of these squares. Feature (i) results from the preponderance of data points lying in bin1 and bin4. Moreover, to the limit of our vision, data points land in these bins in all combinations. Feature (ii) is the result of data points in bin1 and bin4 being followed occasionally by data points in bin2 and in bin3. Note consecutive data points rarely land in bin2 (the square with address 22 contains few IFS points), and even fewer consecutive data points land in bin3 (the square with address 33 contains only 3 IFS points).
To illustrate the extent to which the details of this driven IFS depend on the distance A from B1 and B3 to B2, in this animation we let A change from 0.25 standard deviations (σ) to 2.0σ, in steps of 0.25σ.
Click the picture to animate.

Return to Equal-size bins.