dear all,
i would like to write a detailed scheme about the uQ they but one dubt of mine is if the kriging is a spectral method or a general surrogate method. What do you think about it?
Thank you very much
Hi @SimoneRomani12,
I believe Kriging is not generally considered a spectral surrogate model. Kriging is based on the idea that the underlying function is a realization of a stochastic process. It uses a covariance function to quantify the correlation between different points in the input space. Spectral models decompose the model response into a series of basis functions which is not the case for Kriging.
Would you like to elaborate on why you believe Kriging could still fall into the category of spectral methods?
Best regards,
Styfen
Hi Simone,
Kriging is definitely not a spectral representation.
Without oversimplifying too much, a spectral method generally represents a function/operator as a linear superimposition of orthogonal eigenfunctions/basis elements \psi_i(\boldsymbol{x}) of some suitable functional Hilbert space \mathcal{H}, e.g. \mathcal{L}_2, of the form:
h(\boldsymbol{x}) = \sum\limits_{i = 1}^{\infty} c_i \psi_i (\boldsymbol{x}) ,
where the c_i are real coefficients.
Common examples of spectral representations include Fourier transforms for periodic signals, polynomial chaos expansions for finite-variance functions, Karounen-Loève expansions for random fields, etc. Those methods are interesting, because the coefficients themselves are often representative of some sort of interpretable property, typically related to some variation of the Parseval equality, e.g. frequency content, conditional variances/Sobol’ indices, etc.
Kriging belongs instead to the class of kernel-based interpolation, similarly to support vector machines for regression, radial basis functions, etc. While some of these can make use indeed of linear superimpositions of functions, such functions are not the basis of the same space the approximated function belongs to. Rather, they locally approximate the behavior of the function based on the distance to some form of support reference points. So they rather model the “roughness” of the function interpolating between experimental design point. When and if they have coefficients, they are often difficult to interpret, and don’t relate to global properties of the function they represent.
TL;DR => spectral representation implies global and orthogonal basis, Kriging is a locally smooth interpolant.
Hope it makes sense,
Cheers
Stefano
dear all,
thank you very much for your answers. I wanted to confirm that a Kriging was a general surrogate model. I was enough sure that it cannot be classified as spectral method but a comparison with other people has been very usefull.
Thank you so much for this clear explanation. Are there any resource (textbook) one can further read about the spectral representation method and the kernel-based interpolation approach ( as mentioned). Any recommendations?