Biomedical Engineering Reference
In-Depth Information
Figure 6
. Inability of linear regression on log-log plots to correctly identify power law distri-
butions. Simulation data (circles) and resulting least-squares fit (line) for the 5,112 points in
Figure 5 for which
x
1. The
R
2
of the regression line is 0.962.
Maximum likelihood fitting of a power law distribution gave B = -1.30
0.006, with a negative log-likelihood of 18481.51. Similarly fitting a log-normal
distribution gave
E
[log
X
] = 2.60 0.02 and T(log
X
) = 1.48 0.02, with a
negative log-likelihood of 17,218.22. As one can see from Figure 8, the log-
normal provides a very good fit to almost all of the data, whereas even the best
fitting power-law distribution is not very good at all.
31
A rigorous application of the logic of error testing (50) would now consider
the probability of getting at least this good a fit to a log-normal if the data were
actually generated by a power law. However, since in this case the data were
e
18481.51-17218.22
13 million times more likely under the log-normal distribution,
any sane test would reject the power-law hypothesis.
8.5. Other Measures of Complexity
Considerations of space preclude an adequate discussion of further
complexity measures. It will have to suffice to point to some of the leading ones.
The
thermodynamic depth
of Lloyd and Pagels (182) measures the amount
of information required to specify a trajectory leading to a final state, and
is related both to departure from thermodynamic equilibrium and to retrodiction
(209). Huberman and Hogg (210), and later Wolpert and Macready (211),
proposed to measure complexity as the
dissimilarity
between different levels
of a given system, on the grounds that self-similar structures are actually very
