Driven IFS and Financial Cartoons

Background

For comparison with financial data, we shall use a version of the self-affine cartoons developed by Benoit Mandelbrot.

In its simplest version, we begin with a broken line segment having vertices

(0, 0), (x1, y1), (x2, y2), and (1, 1)

Call the linear pieces U1, D, and U2 (first up segment, down segment, and second up segment).

In the picture on the left we have (x1, y1) = (4/9, 2/3) and (x2, y2) = (5/9, 1/3).

selected randomly.

To generate the cartoon picture, iterate this process:

replace each segment with a scaled version of one of these six combinations

{U1, D, U2}, {U1, U2, D}, {D, U1, U2}, {D, U2, U1}, {U2, U1, D}, and {U2, D, U1},

Note in this example, U1 = U2, so there are really only three combinations.

Note the scaled version of the segments must begin and end at the endpoints of the segment being replaced.

If the segment being replaced has negative slope, so will two of the three scaled copies.

The middle picture shows the second generation, with scaled segments {D, U, U} {U, D, U}, and {U, U, D}.

The right picture shows ten equally-spaced sample points superimposed on the third-generation graph.

The final issue is extracting the sample points.

Common data are sampled at equal time intervals, so we shall do the same thing here.

By moving the points (x₁, y₁) and (x₂, y₂), always preserving the relation 0 <= x₁ <= x₂ <= 1, we can synthesize a wide range of behaviors, including long-term dependence and large jumps, both found in real data.

Examples are given in the Samples section.

Return to Background.