Linearly combines multiple Gaussian Processes (GPs) into a single GP using specified weights.
Parameters:
-
inputs
(list of GPs)
–
A list of GP objects to be combined.
-
weights
(matrix, default:
None
)
–
A matrix specifying how the inputs are to be combined. If not provided,
an identity matrix is assumed, meaning the GPs are combined without weighting.
Returns:
-
GP ( GP
) –
A new GP object representing the linear combination of the input GPs with the specified weights.
Examples:
Combining two GPs into a new multi-output GP:
Suppose you have two GPs:
\(f_1(x) \sim \mathrm{GP}(\mu_1, K_1)\) and \(f_2(x) \sim \mathrm{GP}(\mu_2, K_2)\),
and you want to create a new multi-output GP \(\mathbf{g}(x)\) defined as:
\[
\mathbf{g}(x) = \begin{pmatrix}
g_1(x) \\
g_2(x) \\
g_3(x)
\end{pmatrix} = A \begin{pmatrix}
f_1(x) \\
f_2(x)
\end{pmatrix},
\]
where \(A\) is the weights matrix. This can be implemented as:
g = Mix([f1, f2], [[a11, a12], [a21, a22], [a31, a32]])
The resulting GP \(\mathbf{g}(x)\) can then be used for fitting, generating, or predicting
with methods such as g.fit(), g.generate(), or g.predict() and its components are specified using the cats parameter.
Notes:
- The
weights parameter defines how the input GPs are linearly combined. If omitted,
each GP is assumed to be independent, and the identity matrix is used.
- The resulting GP supports all standard operations (e.g.,
fit, generate, predict).
Source code in src/geostat/model.py
| def Mix(inputs, weights=None):
"""
Linearly combines multiple Gaussian Processes (GPs) into a single GP using specified weights.
Parameters:
inputs (list of GPs):
A list of GP objects to be combined.
weights (matrix, optional):
A matrix specifying how the inputs are to be combined. If not provided,
an identity matrix is assumed, meaning the GPs are combined without weighting.
Returns:
GP (GP):
A new GP object representing the linear combination of the input GPs with the specified weights.
Examples:
Combining two GPs into a new multi-output GP:
Suppose you have two GPs:
\\(f_1(x) \sim \mathrm{GP}(\mu_1, K_1)\\) and \\(f_2(x) \sim \mathrm{GP}(\mu_2, K_2)\\),
and you want to create a new multi-output GP \\(\mathbf{g}(x)\\) defined as:
$$
\mathbf{g}(x) = \\begin{pmatrix}
g_1(x) \\\\
g_2(x) \\\\
g_3(x)
\end{pmatrix} = A \\begin{pmatrix}
f_1(x) \\\\
f_2(x)
\end{pmatrix},
$$
where \\(A\\) is the weights matrix. This can be implemented as:
```python
g = Mix([f1, f2], [[a11, a12], [a21, a22], [a31, a32]])
```
The resulting GP \\(\mathbf{g}(x)\\) can then be used for fitting, generating, or predicting
with methods such as `g.fit()`, `g.generate()`, or `g.predict()` and its components are specified using the `cats` parameter.
Examples: Notes:
- The `weights` parameter defines how the input GPs are linearly combined. If omitted,
each GP is assumed to be independent, and the identity matrix is used.
- The resulting GP supports all standard operations (e.g., `fit`, `generate`, `predict`).
"""
return GP(
mn.Mix([i.mean for i in inputs], weights),
krn.Mix([i.kernel for i in inputs], weights))
|