name | developer | date |
---|---|---|
WEPP | USDA | 1990 |
Agro-IBIS | Wisconsin | 1990 |
APSIM | CSIRO (Australia) | 2007 |
Cycles | Penn State University | 2020 |
CyclesL | Penn State University | 2023 |
HydroGeoSphere | Iowa State University | 2023 |
2023-06-15
name | developer | date |
---|---|---|
WEPP | USDA | 1990 |
Agro-IBIS | Wisconsin | 1990 |
APSIM | CSIRO (Australia) | 2007 |
Cycles | Penn State University | 2020 |
CyclesL | Penn State University | 2023 |
HydroGeoSphere | Iowa State University | 2023 |
Inputs (\(X\)):
Outputs (\(Y\)):
Currently (Iowa and some surrounding regions)
Expansion
Deterministic computer model
\[Y = f(X)\]
for
An emulator is an estimate of \(f\), i.e. \(\widehat{f}\).
For \(f:\) 𝒳 \(\to\) 𝒴, assume \[f \sim GP(\mu, k)\]
for
For simplicity, \(\mu(x) = 0\).
For a collection of data \((Y_i, X_i)\) for \(i=1,\ldots,n\), we have
\[Y = (Y_1,\ldots,Y_n)^\top \sim N(0, \Sigma)\]
where
\[\Sigma_{i,j} = k(x_i,x_j).\]
For prediction at a new location \(\widetilde{x}\), we have the conditional distribution \(\widetilde{Y}|y\) which involves covariances \(k(x_i, \widetilde{x})\) for all \(i\).
For any 𝒳, let \[k(x_i,x_j) = \sigma^2 e^{-d(x_i,x_j)/2}\] for spatial variance \(\sigma^2\) and some distance function \(d(x_i,x_j)\).
For example,
If \(x \in \mathbb{R}\),
the squared-exponential (Gaussian) covariance kernel is \[d(x_i,x_j) = w (x_i-x_j)^2\]
where \(w\) is the weight (\(1/w\) is the length-scale/range).
If \(x \in \mathbb{R}^P\),
the automatic relevance determination kernel is \[d(x_i,x_j) = \sum_{p=1}^P w_p [x_{i,p}-x_{j,p}]^2\]
where \(w_p\) controls the strength of the relationship in the \(p\)th dimension.
If \(x \in\) ℋ (Hilbert space), the automatic dynamic relevance determination kernel is \[d(x_i,x_j) = \int w(t) [x_{i}(t)-x_{j}(t)]^2 dt\]
for some weight function \(w:\) 𝒯 \(\to \mathbb{R}^+\).
WLOG 𝒯 = [0,1].
Combine hillslope profile, length, and mean slope into a single correlation function:
Calculate scaled-integrated weight:
Introduced
Results