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SUMMARY:What is the dimension of a stochastic process? - Victor Panaretos 
 (EPFL)
DTSTART:20180607T150000Z
DTEND:20180607T160000Z
UID:TALK101968@talks.cam.ac.uk
CONTACT:Quentin Berthet
DESCRIPTION:How can we determine whether a mean-square continuous stochast
 ic process is\, in fact\, finite-dimensional\, and if so\, what its actual
  dimension is? And how can we do so at a given level of confidence? This q
 uestion is central to a great deal of methods for functional data analysis
 \, which require low-dimensional representations whether by functional PCA
  or other methods. The difficulty is that the determination is to be made 
 on the basis of iid replications of the process observed discretely and wi
 th measurement error contamination. This adds a ridge to the empirical cov
 ariance\, obfuscating the underlying dimension. We build a matrix-completi
 on-inspired test procedure that circumvents this issue by measuring the be
 st possible least square fit of the empirical covariance's off-diagonal el
 ements\, optimised over covariances of given finite rank. For a fixed grid
  of sufficient size\, we determine the statistic's asymptotic null distrib
 ution as the number of replications grows. We then use it to construct a b
 ootstrap implementation of a stepwise testing procedure for the collection
  of hypotheses formalising the question at hand. The procedure involves no
  tuning parameters or pre-smoothing\, is indifferent to the homoskedastici
 ty or lack of it in the measurement errors\, and does not assume a low-noi
 se regime. Based on joint work with Anirvan Chakraborty (EPFL).
LOCATION:MR11
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