Bases: Kernel
GammaExponential kernel class for Gaussian Processes (GPs).
The GammaExponential class defines a kernel that generalizes the Squared Exponential kernel by introducing
a gamma parameter, allowing for greater flexibility in modeling covariance structures. It can capture processes
with varying degrees of smoothness, depending on the value of gamma.
Parameters:
-
range
(float or Variable)
–
The length scale parameter that controls how quickly the covariance decreases with distance.
-
sill
(float or Variable)
–
The variance (sill) of the kernel, representing the maximum covariance value.
-
gamma
(float or Variable)
–
The smoothness parameter. A value of 1 results in the standard exponential kernel, while a value of 2
recovers the Squared Exponential kernel. Values between 0 and 2 adjust the smoothness of the kernel.
-
scale
(optional, default:
None
)
–
An optional scale parameter that can be used to modify the metric. Default is None.
-
metric
(optional, default:
None
)
–
An optional metric used for distance calculation. Default is None.
Examples:
Creating and using a GammaExponential kernel:
from geostat.kernel import GammaExponential
# Create a GammaExponential kernel with sill=1.0, range=2.0, and gamma=1.5
gamma_exp_kernel = GammaExponential(range=2.0, sill=1.0, gamma=1.5)
locs1 = np.array([[0.0], [1.0], [2.0]])
locs2 = np.array([[0.0], [1.0], [2.0]])
covariance_matrix = gamma_exp_kernel({'locs1': locs1, 'locs2': locs2, 'sill': 1.0, 'range': 2.0, 'gamma': 1.5})
Notes:
- The
call method computes the covariance matrix using the gamma-exponential formula:
\( C(x, x') = \text{sill} \cdot \exp\left(-\left(\frac{d^2}{\text{range}^2}\right)^{\text{gamma} / 2}\right) \),
where \(d^2\) is the squared distance between locs1 and locs2.
- The
vars method returns the parameter dictionary for sill, range, and gamma using the ppp and bpp functions.
- The
GammaExponential kernel provides a more flexible covariance structure than the Squared Exponential kernel,
allowing for varying degrees of smoothness.
Source code in src/geostat/kernel.py
| class GammaExponential(Kernel):
"""
GammaExponential kernel class for Gaussian Processes (GPs).
The `GammaExponential` class defines a kernel that generalizes the Squared Exponential kernel by introducing
a gamma parameter, allowing for greater flexibility in modeling covariance structures. It can capture processes
with varying degrees of smoothness, depending on the value of `gamma`.
Parameters:
range (float or tf.Variable):
The length scale parameter that controls how quickly the covariance decreases with distance.
sill (float or tf.Variable):
The variance (sill) of the kernel, representing the maximum covariance value.
gamma (float or tf.Variable):
The smoothness parameter. A value of 1 results in the standard exponential kernel, while a value of 2
recovers the Squared Exponential kernel. Values between 0 and 2 adjust the smoothness of the kernel.
scale (optional):
An optional scale parameter that can be used to modify the metric. Default is None.
metric (optional):
An optional metric used for distance calculation. Default is None.
Examples:
Creating and using a `GammaExponential` kernel:
```python
from geostat.kernel import GammaExponential
# Create a GammaExponential kernel with sill=1.0, range=2.0, and gamma=1.5
gamma_exp_kernel = GammaExponential(range=2.0, sill=1.0, gamma=1.5)
locs1 = np.array([[0.0], [1.0], [2.0]])
locs2 = np.array([[0.0], [1.0], [2.0]])
covariance_matrix = gamma_exp_kernel({'locs1': locs1, 'locs2': locs2, 'sill': 1.0, 'range': 2.0, 'gamma': 1.5})
```
Examples: Notes:
- The `call` method computes the covariance matrix using the gamma-exponential formula:
\\( C(x, x') = \\text{sill} \cdot \exp\left(-\left(\\frac{d^2}{\\text{range}^2}\\right)^{\\text{gamma} / 2}\\right) \\),
where \\(d^2\\) is the squared distance between `locs1` and `locs2`.
- The `vars` method returns the parameter dictionary for `sill`, `range`, and `gamma` using the `ppp` and `bpp` functions.
- The `GammaExponential` kernel provides a more flexible covariance structure than the Squared Exponential kernel,
allowing for varying degrees of smoothness.
"""
def __init__(self, range, sill, gamma, scale=None, metric=None):
fa = dict(sill=sill, range=range, gamma=gamma, scale=scale)
autoinputs = scale_to_metric(scale, metric)
super().__init__(fa, dict(d2=autoinputs))
def vars(self):
return ppp(self.fa['sill']) | ppp(self.fa['range']) | bpp(self.fa['gamma'], 0., 2.)
def call(self, e):
return e['sill'] * gamma_exp(e['d2'] / tf.square(e['range']), e['gamma'])
|