Adaptive Mean-AutoRegressive-Moving-Average model estimation
 [z,e,ESU,REV,V,Z,SPUR] = amarma(y, mode, MOP, UC, z0, Z0, V0, W); 

 Estimates model parameters (mean and AR) with Kalman filter algorithm
 	y(t) = sum_i(a(i,t)*y(t-i)) + mu(t) + e(t)
 or the more general adaptive mean-autoregressive-moving-avarage parameters
 	y(t) = sum_i(a(i,t)*y(t-i)) + mu(t) + e(t) + sum_i(b(i,t)*e(t-i)) 

 State space model:
	z(t)=G*z(t-1) + w(t)      w(t)=N(0,W) 
	y(t)=H*z(t)   + v(t)	  v(t)=N(0,V)	

 G = I, (identity matrix)
 z = [mu(t)/(1-sum_i(a(i,t))),a_1(t-1),..,a_p(t-p),b_1(t-1),...,b_q(t-q)];
 H = [1,y(t-1),..,y(t-p),e(t-1),...,e(t-q)];
 W = E{(z(t)-G*z(t-1))*(z(t)-G*z(t-1))'}
 V = E{(y(t)-H*z(t-1))*(y(t)-H*z(t-1))'}
 v = e

 Input:
       y	Signal (AR-Process)
       Mode
	    [0,0] uses V0 and W  

       MOP     Model order [m,p,q], default [0,10,0] 
		   m=1 includes the mean term, m=0 does not. 
		   p and q must be positive integers
		   it is recommended to set q=0 (i.e. no moving average part)
		   because the optimization problem for ARMA models is 
		   non-linear and can have local optima. 
	UC	Update Coefficient, default 0
	z0	Initial state vector
	Z0	Initial Covariance matrix
      
 Output:
	z	mean-autoregressive-moving-average-parameter
               mu(t)  = z(t,1:m)	adaptive mean
               a(t,:) = z(t,m+[1:p])	adaptive autoregressive parameters
               b(t,:) = z(t,m+p+[1:q]) adaptive moving average parameters
	e	error process (Adaptively filtered process)
       REV     relative error variance MSE/MSY


 see also: AAR

 REFERENCE(S): 
 [1] A. Schlögl (2000), The electroencephalogram and the adaptive autoregressive model: theory and applications. 
     ISBN 3-8265-7640-3 Shaker Verlag, Aachen, Germany. 
 [2] Schlögl A, Lee FY, Bischof H, Pfurtscheller G
     Characterization of Four-Class Motor Imagery EEG Data for the BCI-Competition 2005.
     Journal of neural engineering 2 (2005) 4, S. L14-L22
 [3] A. Schlögl , J. Fortin, W. Habenbacher, M. Akay.
     Adaptive mean and trend removal of heart rate variability using Kalman filtering
     Proceedings of the 23rd Annual International Conference of the IEEE Engineering in Medicine and Biology Society, 
     25-28 Oct. 2001, Paper #1383, ISBN 0-7803-7213-1.

 More references can be found at 
     http://pub.ist.ac.at/~schloegl/publications/

Package: tsa