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When fitting a generalised least squares (GLS) model to a spatial dataset, I was surprised to find how correlated the estimators for the range and nugget are. I was assuming a covariance matrix $\boldsymbol\Sigma$ with elements

$\Sigma_{ij} = \sigma^2\left(\pi I(i=j) + (1-\pi)\exp\left[-\left(\frac{d_{ij}}{r}\right)^2)\right]\right),$

with $\pi$ the nugget, r the range and $d_{ij}$ the distance between two observations. The estimators $\hat\pi$ and $\hat r$ turned out to be strongly correlated, meaning that quite different combinations $(\pi, r)$ can lead to similar covariance matrices. This makes it hard to do inference on these parameters, or improve their estimators. Is this problem then not ill-posed, and is their a better way to parametrize it or overcome this non-orthogonality?

Knarpie
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    Can I just check something? I'm not really sure how to interpret this covariance structure. I come from a Gaussian processes background and if when we say "nugget" we usually mean $\lambda^2$ where $\Sigma_{ij} = \sigma^2 \exp \{ - (d_{ij}/r)^2 \} + \lambda^2 \delta_{ij}$ where $\delta_{ij}$ is Kronecker's delta. In this case, it is well known that $r$ and $\lambda$ have a complex relationship – jcken Dec 02 '21 at 11:29
  • You are right, I changed the question: now the descriptions are equivalent after reparametrizing. But comforting to hear that it's a difficult story anyhow – Knarpie Dec 02 '21 at 11:46
  • The problem ultimately stems from using a Gaussian variogram with nugget. You need to be *absolutely sure* this is the right model and even then it is notoriously difficult to work with, especially in two dimensions. – whuber Dec 02 '21 at 13:55

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