Lattice algorithms for recursive instrumental variable methods

We develop recursive lattice algorithms for the estimation of the AR parameters of an ARMA process using an instrumental variable (IV) such as z ( n )= y 2 ( n ), which leads to AR estimates based on the third‐order cumulants, or z ( n )= y ( n‐q ), which leads to AR estimates based on the higher‐order Yule‐Walker equations. In the non‐recursive mode the resulting set of linear equations involves the cross‐correlation matrix of the observed process y ( n ), and the instrumental process z ( n ) and hence this matrix is generally not Hermitian. Our time‐ and order‐recursive algorithm leads to a pair of lattices, one excited by y ( n ) and the other by z ( n ), the lattices being coupled through order and time update equations. We show that IV‐based AR model fitting is equivalent to linear prediction with non‐conventional orthogonality conditions. Extensions to ARMA processes are discussed. The joint‐process estimation problem is also treated. Some statistical analysis is carried out and convergence results are given when y ( n ) and z ( n ) can be modelled as stationary processes. The only assumption that we make about z ( n ), the associated process, is that the cross‐correlation of y ( n ) and z ( n ) suffices for the estimation of the AR parameters, i.e. we assume that z ( n ) is an instrumental process. Particular choices of z ( n ) legal to estimates based on the higher‐order cumulants, which are useful when the non‐Gaussian signal is corrupted by additive coloured Gaussian noise.