  In this work, we develop a novel principal component analysis (PCA) for
semimartingales by introducing a suitable spectral analysis for the quadratic
variation operator. Motivated by high-dimensional complex systems typically
found in interest rate markets, we investigate correlation in high-dimensional
high-frequency data generated by continuous semimartingales. In contrast to the
traditional PCA methodology, the directions of large variations are not
deterministic, but rather they are bounded variation adapted processes which
maximize quadratic variation almost surely. This allows us to reduce
dimensionality from high-dimensional semimartingale systems in terms of
quadratic covariation rather than the usual covariance concept.
  The proposed methodology allows us to investigate space-time data driven by
multi-dimensional latent semimartingale state processes. The theory is applied
to discretely-observed stochastic PDEs which admit finite-dimensional
realizations. In particular, we provide consistent estimators for
finite-dimensional invariant manifolds for Heath-Jarrow-Morton models. More
importantly, components of the invariant manifold associated to volatility and
drift dynamics are consistently estimated and identified. The proposed
methodology is illustrated with both simulated and real data sets.
