For the mathematical optimisation of the single model-free models, the
local tm model-free models, the diffusion tensor, or the global model
of the diffusion tensor together with all model-free models, it is the
chi-squared value which is minimised. For these situations the number
of parameters remains constant. Hence the result of optimising the
chi-squared value or the AIC value is identical.
Essentially, the chi-squared value is used to compare two instances of
the same model (with different parameter values) while the AIC value
is used to compare two different models!
The iterative procedure used in the new model-free optimisation
protocol, which is implemented in 'full_analysis.py', is a very
different concept! That is because at each iteration the global model
is different (well until convergence that is). Hence in this
situation, chi-squared values between iterations cannot be compared as
the models are different. But AIC can be used to compare these
different models. As I discussed in my JBNMR, 2003 publication on
model-free model selection, AIC is an attempt to approximate what is
known as the Kullback-Leibler discrepancy. This discrepancy is a
powerful concept - it is a measure of how close the model fits the
data independent of the noise associated with that data. This is an
embodiment of parsimony - the model with the lowest discrepancy value
is that which is closest to Occam's razor.
Therefore what this new protocol does is optimise the chi-squared
value within one iteration and then optimise the Kullback-Leibler
discrepancy between iterations. I have used set theory to
mathematically express these concepts and mathematically encapsulate
the entire problem, but that explanation will have to wait until my
papers are published!
The answer to your question is simply that there is no difference
between the two in finding the correct model-free parameters for a
single model. However because the model-free parameter values are
intricately linked to the diffusion tensor parameters and vice versa,
you have a chicken-and-egg scenario in which the new protocol
optimises the AIC value between iterations. Have I now completely
confused you?
Edward
On 10/6/06, Alexandar Hansen <viochemist@xxxxxxxxx> wrote:
I have a question about the AIC statistic. Let's say we're optimizing model
2 (S2 and te). As it searches for a minimum, wouldn't the difference
between chi2 and AIC just be AIC = chi2 + 4? If so, I guess I don't
understand the difference between the two in finding the correct model free
parameters.
Alex
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Re: Diffusion Tensor - Global correlation time