Random Fractals and the Stock Market

Surrogates of the Stock Market - Cartoon Driven IFS

Zero-Centered Bins

So we take B₂, the boundary between bin₂ and bin₃, to be 0, and place B₁ and B₃ symmetrically about B₂ = 0.

Here we use zero-centered bins, symmetrically spaced, to drive the IFS with data generated by several cartoons.

For later comparison with some financial data sets, we take cartoons with 3128 data points.

The motivation for this test is Nial Neger's observation that for stocks we are interested in determining whether the differences are positive or negative.

The amount of visible structure depends on how large A, the distance from B1 and B3 to B2, is relative to the clustering of the differences about B2 = 0.

Here are some examples. We show the time series of successive differences and the driven IFS.

Here we use the cartoon with generator having turning points (1/3,2/3) and (5/9,1/3). We take B2 = 0 and B1 and B3 one-quarter the range of values on either side of B2. Here we use the cartoon with generator having turning points (1/3,2/3) and (5/9,1/3). We take B2 = 0 and B1 and B3 one-tenth the range of values on either side of B2. Here we use the cartoon with generator having turning points (2/9,2/3) and (5/9,1/3). We take B2 = 0 and B1 and B3 one-quarter the range of values on either side of B2. Here we use the cartoon with generator having turning points (2/9,2/3) and (5/9,1/3). We take B2 = 0 and B1 and B3 one-tenth the range of values on either side of B2. Here we use the cartoon with generator having turning points (1/9,2/3) and (5/9,1/3). We take B2 = 0 and B1 and B3 one-quarter the range of values on either side of B2. Here we use the cartoon with generator having turning points (2/9,2/3) and (5/9,1/3). We take B2 = 0 and B1 and B3 one-tenth the range of values on either side of B2.

From these examples, we see that as the x coordinate of the turning point (x,2/3) decreases toward 0, the range of the differences increases and the majority of the differences cluster ever more closely (relative to the range) about 0.

Consequently, in order for the IFS to reveal much structure, B1 and B3 must be placed closer to 0.

To more illustrate the effect of placement of B1 and B3, here we let the bin boundaries range from .025 of the range to .30 of the range, in steps of .025.

Click the picture to animate.

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