Near the Diagonal

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The covariance kernel of a Gaussian process encapsulates all its fluctuation properties. Particularly its behaviour near the diagonal can reveal valuable information on the behaviour of the process, which in turn can be important as vehicle for or a target of statistical inference. Well-known examples include sample path regularity and effective dimensionality, and a more recent example pertains to Markov properties, via positive-definite extension from the vicinity of the diagonal.

Alas, the covariance itself is seldom available. Rather, it often needs to be estimated from noise-corrupted discrete observations on sample paths -- and the effects of noise are most perturbative precisely near the diagonal. In the tradition of functional data analysis, these effects are mitigated by smoothing the "raw covariance" corresponding to the noise-corrupted discrete data. Yet smoothing over the diagonal introduces biases that obfuscate properties such as those listed above. Path regularity will now be dictated by the choice of smoother; dimensionality will be influenced by the smoother's effective degrees of freedom; and Markov properties will be distorted by the smoother's bandwidth.

We will see how completion-inspired methods can be effective in circumventing the noise problem and allowing progress on some long-standing problems in functional data analysis, that ultimately hinge on the covariance structure near the diagonal. Time permitting, we will touch on statistical problems including inferring the dimensionality of a random process, analysing rough functional data, continuum graphical modelling, extrapolating correlations, and analysing non-separable random fields.