1. RECPAD - 14ª Conferência Portuguesa de Reconhecimento de Padrões, Aveiro, 23 de Outubro de 2009 David Afonso (dafonso@isr.ist.utl.pt) and João Sanches (jmrs@isr.ist.utl.pt) Institute for Systems and Robotics / Instituto Superior Técnico 1049-001 Lisbon, Portugaldafonso@isr.ist.utl.ptjmrs@isr.ist.utl.pt ARMA estimation – a comparison study for fMRI 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. 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. 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 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 12–15, 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 Output sequence Input sequence Residue Autoregressive parameters Moving Average parameters Brain Tissues Input stimuli signal x (n) Oxygen Metabolism Vascular Demand Output signal y (n) - Vascular Controller Vascular response - 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. FIR space x (n) y (n) ARMA space a (p) b (q) ? Specific Problem: Find a method that present the best results in estimating our Physiologically Based Hemodinamic (PBH) response function ARMA(3,2) model [5]. 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. Methods tested  Prony’s [1]  Shanks’s [2]  Steiglitz-McBride [3]  Stable ARMA Estimation with Regularization in the Poles Positions (SAERPP) 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) In development Table 1 – Comparative table results with the MSE and variance percentual values for the ARMA(3,2) estimation for each algorithm.