Diapositivo 1

1
ARMA estimation a comparison study for fMRI
David Afonso (dafonso_at_isr.ist.utl.pt) and João
Sanches (jmrs_at_isr.ist.utl.pt) Institute for
Systems and Robotics / Instituto Superior
Técnico1049-001 Lisbon, Portugal
Experimental Results Data The performance of each
method is assessed with Monte Carlo tests where
the Euclidean norm of the error, and the variance
of the difference between the signal generated by
the estimated model and the noiseless output of
the true model. Also processing time is evaluated.
Abstract This work investigates the
effectiveness of four projection methods of
finite response function (FIR) to infinite
response function (IIR) filters. This is done in
the scope of a linear physiological model
developed for functional Magnetic Resonance
Imaging (fMRI). Of the four methods the
Steiglitz-Mcbride algorithm provided the best
fit. A new method is also proposed where a
regularization term is used to keep the poles of
the IIR filter inside the unit circle in order to
make it stable. Preliminary results are promising.

• Data
• Generated from several ARMA(2,3) filters
  obtained from real functional Magnetic Resonance
  Imaging data
• Corrupted with zero mean Additive White Gaussian
  Noise (AWGN)

Problem Formulation Without loss of generality,
the transfer function of an ARMA filter is given
by also called the transfer function of a
causal stationary ARMA model of order (p,q). In
the case of real data, corrupted with noise or
not perfectly following an ARMA model the
following difference equation holds

• Methods tested
• Pronys 1
• Shankss 2
• Steiglitz-McBride 3
• Stable ARMA Estimation with Regularization in
  the Poles Positions (SAERPP)

In development
Problem The estimation of
and defining
the ARMA model that best describes the output
sequence
given the input sequence is not an easy task
where the stability of the model is a central
issue. Non linear optimization techniques are
required when computational time is relevant.
Table 1 Comparative table results with the MSE
and variance percentual values for the ARMA(3,2)
estimation for each algorithm.

• Conclusions
• With the present results Steiglitz-Mcbride 10
  provided the overall better fit, but at the
  expense of significant computing time. This was
  somewhat expected as it is the only iterative
  method on test, while the rest are one-step
  algorithms.
• Although SAERPP method proved the worst results,
  it is still in development and it is the only
  method restricting instability of the estimated
  ARMA filter. This is not shown in the example
  presented but is of crucial relevance in the
  aimed fMRI application.

Specific Problem Find a method that present the
best results in estimating our Physiologically
Based Hemodinamic (PBH) response function
ARMA(3,2) model 5.
Referências 1 Parks, T.W., and C.S. Burrus,
Digital Filter Design John Wiley Sons, 1987,
pp.226-228. 2 J. L. Shanks, Recursion Filters
For Digital Processing, Geophysics, vol.32
(1967), Nº1, pp. 33-51. 3 K. Steiglitz, L.E.
McBride, "A Technique for the Identification of
Linear Systems," IEEE Trans. Automatic Control,
Vol. AC-10 (1965), pp.461-464. 4 D. M. Afonso,
J. M. Sanches and M. H. Lauterbach, Robust Brain
Activation Detection In Functional MRI, 2008 IEEE
Int. Conf. on Image Processing, San Diego U.S.A,
October 1215, 2008 5 D. M. Afonso, J. M.
Sanches and M. H. Lauterbach (MD), Neural
physiological modeling towards a hemodynamic
response function for fMRI, 29th Annual Int.
Conf. of the IEEE Eng. in Medicine and Biology
Society, Lyon, France, August 23-26, 2007