Data Analysis by Driven IFS

Background

We use the four transformations

T1(x, y) = (x/2, y/2),	the midpoint of (x, y) and (0, 0)
T2(x, y) = (x/2, y/2) + (1/2, 0),	the midpoint of (x, y) and (1, 0)
T3(x, y) = (x/2, y/2) + (0, 1/2),	the midpoint of (x, y) and (0, 1)
T4(x, y) = (x/2, y/2) + (1/2, 1/2),	the midpoint of (x, y) and (1, 1)

Starting from a point (x₀, y₀) inside the unit square, standard iterated function system (IFS) theory shows the sequence (x_k, y_k) = T_ik(x_k-1, y_k-1) uniformly fills up the unit square if the i_k are independent and uniformly distributed in {1,2,3,4}. To the extent that the square does not fill in uniformly, we can make deductions about the nonrandomness of the sequence {i_k}.

Two issues remain:

how to interpret any departures from uniform fill of the square, and
how to convert a time series {x1, x2, ... , xn} into a symbol string {i1, i2, ... , in} of 1s, 2s, 3s, and 4s.

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