HIERARCHAL INDUCTIVE PROCESS MODELING AND ANALYSIS ...

set of values that respect the constraints and, using the Levenberg-Marquardt
gradient descent method, finds a local optimum. To avoid entrapment in local
minima, the system will restart the parameter estimation from multiple
random points retaining only the parameters that produce the lowest error.
In our experiment we set the number of restarts to 128. This technique has
been found to produce reasonable matches to time series in multiple systems
(Langley et al. 2007).
4. Evaluates the performances of the produced model structures (predicted values)
against the data series (observed values) by calculating the root mean
square error (reMSE); models with the lowest reMSE will be considered best
fit models.
2.1.1 Measure of Fit
As mentioned above, HIPM evaluates and selects the best model structure and set
of parameters according to a fitness measure. The system currently uses the sum
of square error (SSE) to evaluate fitness (Bridewell et al. 2007), which is defined as
follow:
n∑
i=1
SSE(x i , x obs
i ) =
n∑
i=1
m∑
k=1
(x i,k − x obs
i,k ) 2
where x i , . . . , x n are the variables that are being fitted with m observed values for
each. To take into account the modeling of variables of varying scale, the system
uses a relative mean squared error that we define in the following way:
reMSE =
∑ n SSE(x i ,x obs
i )
i=1 s 2 (x obs
i )
nm
Here s 2 (x obs
i ) is the sample variance of the observation for x i . Across this paper
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