Stochastic extension and functional restrictions of ill-posed estimation problems

  • Edoardo Mosca
System Modeling And Identification
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3)


In Theorem 1 it is shown that under mild conditions the minimum-variance smoothed-estimation of a Gaussian process y(t) with covariance Ry(t,τ) from noise-corrupted measurements z(t)=y(t)+n(t), t∈T, with n(t) Gaussian with covariance Rn(t,τ), is equivalent to a purely deterministic optimization problem, namely, findingy ∈ H (Ry) that minimizes the functional
$$L \left( {y\left| z \right.} \right) = \left\| y \right\|^2 _{R_n } - 2 Re\left[ {\left\langle {z, y} \right\rangle _{R_n } } \right] + \left\| y \right\|^2 _{Ry} .$$
Here 〈·,·〉R denotes inner-product of the reproducing kernel Hilbert space (RKHS)H(R). Theorems 2 and 3 deal with the more general situation where y is a set of linear measurements on an unknown function w(α), α ∈ A.

The above stochastic-deterministic equivalence provides valuable insight and important consequences for modelling. An example shows how this equivalence allows one to make use of Kalman-Bucy filtering in a purely deterministic problem of smoothing.


Probability Measure Reproduce Kernel Hilbert Space Sample Function Deterministic Problem Covariance Kernel 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 1973

Authors and Affiliations

  • Edoardo Mosca
    • 1
  1. 1.Facoltà di IngegneriaUniversità di FirenzeItaly

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