Module cancorr 

Canonical correlations between the past and the future of a stationary series — the weighted (CCA) subspace step of Bauer (2001). The number of non-zero canonical correlations is the state dimension (McMillan degree), which is what the SVC criterion and the Gumbel-thresholded variant read off the estimated correlations.

The blocks are built from the sample autocovariances: the joint vector (x_{t+f-1}, …, x_t, x_{t-1}, …, x_{t-p}) is contiguous, so its covariance is the (f+p) × (f+p) Toeplitz matrix of γ̂(0..f+p-1), which the biased estimator keeps positive semi-definite — hence all estimated canonical correlations are guaranteed to lie in [0, 1].

Structs

Cancorr

Output of canonical_correlations.

SvcConfig

Configuration of svc_order.

SvcResult

Output of svc_order.

Enums

SvcPenalty

Penalty C(T) of the SVC criterion.

Functions

canonical_correlations

Canonical correlations between (x_t, …, x_{t+f-1}) and (x_{t-1}, …, x_{t-p}), from the biased sample autocovariances.

canonical_correlations_from_acov

Same, from an autocovariance sequence r[0..=f+p-1] (exact inputs make exact tests possible).

canonical_correlations_stacked

Canonical correlations from the stacked-sample covariances, Bauer’s (2001) convention: averages of Y⁺_{t,f} (Z⁻_{t,p})' etc. over t = p+1 … T−f (his index range, divisor 1/T — it cancels in the correlations). Differs from canonical_correlations only by edge effects O((f+p)/T); the sample joint second-moment matrix is an average of outer products, hence PSD, so correlations stay in [0, 1]. The series is demeaned first (the paper’s model is mean-zero; demeaning keeps the estimator translation-invariant).

svc_order

Bauer’s (2001, eq. 4) SVC estimator of the McMillan degree, output-only scalar case with CCA weighting: SVC(n) = σ̂²_{n+1} + C(T)·2n/T (d(n) = 2n parameters for a scalar state-space system), minimised over n = 0..=M, M = min(f, p), with the convention σ̂_{M+1} = 0. Consistency (his Theorem 3) needs C(T)/(fp log log T) → ∞.