Probabilistic Partial Canonical Correlation Analysis

Partial canonical correlation analysis (partial CCA) is a statistical method that estimates a pair of linear projections onto a low dimensional space, where the correlation between two multi-dimensional variables is maximized after elimi-nating the influence of a third variable. Partial CCA is known to be closely related to a causal-ity measure between two time series. However, partial CCA requires the inverses of covariance matrices, so the calculation is not stable. This is particularly the case for high-dimensional data or small sample sizes. Additionally, we can-not estimate the optimal dimension of the sub-space in the model. In this paper, we have ad-dressed these problems by proposing a proba-bilistic interpretation of partial CCA and deriv-ing a Bayesian estimation method based on the probabilistic model. Our numerical experiments demonstrated that our methods can stably esti-mate the model parameters, even in high dimen-sions or when there are a small number of sam-ples. 1