Traditional and Driven IFS Correlations

In their project for the autumn, 2000, fractal geometry course, Simo Kalla and Nader Sobhan analyzed the differences in daily closing prices of six stocks for the 251 trading days in 1998. They selected Coca-Cola and Kellogg (food industry), Nokia and Motorola (telecommunications), Microsoft (software), and Chase Manhattan (banking). Dow Jones Interactive was the source of the information. Here are the driven IFS plots.

Coca-Cola Kellogg

Nokia Motorola

Microsoft Chase Manhattan

The standard statistical method of comparing two sequences {x_i} and {y_i} is to compute the correlation

rho = E((x_i - mu(x))*(y_i - mu(y)))/(sigma(x)*sigma(y))

where mu(x) is the mean value of the x_i, sigma(x) is the standard deviation of the x_i, and E(z_i) is the expected value of the z_i. (So for example, mu(x) = E(x_i).) If rho = 1, the x and y behave exactly alike; if rho = -1 they behave in the opposite way. Here are the rho values for these stocks.

The highest rho value is between Nokia and Microsoft, yet their driven IFS have some significant differences. For example, they look a bit like the reflections of one another across the diagonal. Here is an animation of these pictures. Visually, the Microsoft plot looks most like that of Coca-Cola. Here is an animation of these pictures.Can this visual comparison be quantified?

Kalla and Sobhan divided each driven IFS plot up into address length 2 subsquares, and counted the number of data points in each subsquare. Recall the notion of addresses.

Here are the values for these driven IFS, in this format:

{{n₁₁,n₁₂,n₁₃,n₁₄}, {n₂₁,n₂₂,n₂₃,n₂₄}, {n₃₁,n₃₂,n₃₃,n₃₄}, {n₄₁,n₄₂,n₄₃,n₄₄}}

Coca-Cola
{{0,4,0,0},{3,22,44,4},{0,44,109,8},{1,2,9,0}}
Kellogg
{{0,0,1,1},{0,1,12,9},{2,16,137,28},{0,6,32,5}}
Nokia
{{0,9,3,1},{6,105,44,4},{5,43,21,2},{2,2,3,0}}
Motorola
{{0,0,1,0},{0,7,33,5},{1,35,129,16},{0,3,18,2}}
Microsoft
{{0,2,1,1},{0,21,38,9},{1,37,100,13},{3,8,12,4}}
Chase
{{1,4,1,1},{2,62,55,2},{4,51,58,3},{0,3,3,0}}

As a simple measure of the differences between the driven IFS of stocks X and Y, Kalla and Sobhan proposed counting the number of points N(X,Y) in different bins between X and Y. That is, N(X,Y) = |n₁₁(X) - n₁₁(Y)| + |n₁₂(X) - n₁₂(Y)| + ... + |n₄₄(X) - n₄₄(Y)|

and then "normalizing" this count. That is, define

rho^~ = ((total number of points in X) - N(X,Y))/(total number of points in X)

Here is how Kalla and Sobhan justify rho^~ as a measure of the similarity of the driven IFS. Suppose the total number of points is 250. If n₁₁(X) =250 and n₂₂(Y) = 250, then N(X,Y) = 500 and rho^~ = (250 - 500)/250 = -1. This is reasonable, because there is no similarity at all between the two driven IFS. On the other hand, if both X and Y have the same number of points in each bin, then N(X,Y) = 0 and rho^~ = (250 - 0)/250 = +1. Again, this is sensible because at the level of length 2 addresses, the driven IFS are identical.

Here are the rho^~ values for these stocks.

Using rho^~, the highest correlation is between Coca-Cola and Microsoft, as suggested by the driven IFS plots. The highest rho value was between Nokia and Microsoft, yet their rho^~ is the fourth lowest. Motivated by these and similar observations, Kalla and Sobhan computed the correlation between rho and rho^~, obtaining a small negative value.

This was a small test, a small number of stocks and only one year's data. Moreover, comparing only the length two addresses is fairly crude. With more data, longer address squares could be compared, giving more refined measurements of the relative motions of the stocks. Kalla and Sobhan conclude, "Since fractal geometry is a relatively new field, it might open doors to exciting new ways of measuring correlations of two random variables."

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