The authors consider the pth-order rational discrete-time linear model for a process whose value at time t is

y(t) = hx(t) + v(t),

where x(t + 1) = Fx(t) + Tv(t), v(t) is a white noise input sequence of given variance, x(t) is a p × 1 state vector, and F, T, and h are constant matrices of size p × p, p × 1 and 1 × p, respectively. The modeling problem involves predicting the future output vector

Y⁺ = (y(t), y(t + 1), . . . )

from the past output vector

Y⁻ = (y(t − 1), y(t − 2), . . . ).

The authors discuss three methods: canonical correlation, which bases the partial state for future prediction on the canonical components of Y⁻, which maximize mutual information with respect to Y⁺; principal components of the covariance Hankel matrix, which picks orthonormal components of the past maximally correlated with the future; and predictive efficiency, where Y⁻ is compressed into a partial state with the smallest error in predicting Y⁺. The authors favor the last method and present an algorithm for its realization, the unweighted principal components (UPC) algorithm. They describe no implementations or numerical experiments, but they do give a number of contexts for which their models are appropriate.