#!/usr/bin/env python3
import math
from typing import Optional
import torch
from ..constraints import Interval, Positive
from ..priors import Prior
from .kernel import Kernel
[docs]class PeriodicKernel(Kernel):
r"""Computes a covariance matrix based on the periodic kernel
between inputs :math:`\mathbf{x_1}` and :math:`\mathbf{x_2}`:
.. math::
\begin{equation*}
k_{\text{Periodic}}(\mathbf{x}, \mathbf{x'}) = \exp \left(
-2 \sum_i
\frac{\sin ^2 \left( \frac{\pi}{p} ({x_{i}} - {x_{i}'} ) \right)}{\lambda}
\right)
\end{equation*}
where
* :math:`p` is the period length parameter.
* :math:`\lambda` is a lengthscale parameter.
Equation is based on `David Mackay's Introduction to Gaussian Processes equation 47`_
(albeit without feature-specific lengthscales and period lengths). The exponential
coefficient was changed and lengthscale is not squared to maintain backwards compatibility
.. note::
This kernel does not have an `outputscale` parameter. To add a scaling parameter,
decorate this kernel with a :class:`gpytorch.kernels.ScaleKernel`.
:param ard_num_dims: (Default: `None`) Set this if you want a separate lengthscale for each
input dimension. It should be `d` if x1 is a `... x n x d` matrix.
:type ard_num_dims: int, optional
:param batch_shape: (Default: `None`) Set this if you want a separate lengthscale for each
batch of input data. It should be `torch.Size([b1, b2])` for a `b1 x b2 x n x m` kernel output.
:type batch_shape: torch.Size, optional
:param active_dims: (Default: `None`) Set this if you want to
compute the covariance of only a few input dimensions. The ints
corresponds to the indices of the dimensions.
:type active_dims: Tuple(int)
:param period_length_prior: (Default: `None`)
Set this if you want to apply a prior to the period length parameter.
:type period_length_prior: ~gpytorch.priors.Prior, optional
:param period_length_constraint: (Default: `Positive`) Set this if you want
to apply a constraint to the period length parameter.
:type period_length_constraint: ~gpytorch.constraints.Interval, optional
:param lengthscale_prior: (Default: `None`)
Set this if you want to apply a prior to the lengthscale parameter.
:type lengthscale_prior: ~gpytorch.priors.Prior, optional
:param lengthscale_constraint: (Default: `Positive`) Set this if you want
to apply a constraint to the lengthscale parameter.
:type lengthscale_constraint: ~gpytorch.constraints.Interval, optional
:param eps: (Default: 1e-6) The minimum value that the lengthscale can take (prevents divide by zero errors).
:type eps: float, optional
:var torch.Tensor period_length: The period length parameter. Size/shape of parameter depends on the
ard_num_dims and batch_shape arguments.
Example:
>>> x = torch.randn(10, 5)
>>> # Non-batch: Simple option
>>> covar_module = gpytorch.kernels.ScaleKernel(gpytorch.kernels.PeriodicKernel())
>>>
>>> batch_x = torch.randn(2, 10, 5)
>>> # Batch: Simple option
>>> covar_module = gpytorch.kernels.ScaleKernel(gpytorch.kernels.PeriodicKernel())
>>> # Batch: different lengthscale for each batch
>>> covar_module = gpytorch.kernels.ScaleKernel(gpytorch.kernels.PeriodicKernel(batch_size=2))
>>> covar = covar_module(x) # Output: LazyVariable of size (2 x 10 x 10)
.. _David Mackay's Introduction to Gaussian Processes equation 47:
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.81.1927&rep=rep1&type=pdf
"""
has_lengthscale = True
def __init__(
self,
period_length_prior: Optional[Prior] = None,
period_length_constraint: Optional[Interval] = None,
**kwargs,
):
super(PeriodicKernel, self).__init__(**kwargs)
if period_length_constraint is None:
period_length_constraint = Positive()
ard_num_dims = kwargs.get("ard_num_dims", 1)
self.register_parameter(
name="raw_period_length", parameter=torch.nn.Parameter(torch.zeros(*self.batch_shape, 1, ard_num_dims))
)
if period_length_prior is not None:
if not isinstance(period_length_prior, Prior):
raise TypeError("Expected gpytorch.priors.Prior but got " + type(period_length_prior).__name__)
self.register_prior(
"period_length_prior",
period_length_prior,
lambda m: m.period_length,
lambda m, v: m._set_period_length(v),
)
self.register_constraint("raw_period_length", period_length_constraint)
@property
def period_length(self):
return self.raw_period_length_constraint.transform(self.raw_period_length)
@period_length.setter
def period_length(self, value):
self._set_period_length(value)
def _set_period_length(self, value):
if not torch.is_tensor(value):
value = torch.as_tensor(value).to(self.raw_period_length)
self.initialize(raw_period_length=self.raw_period_length_constraint.inverse_transform(value))
def forward(self, x1, x2, diag=False, **params):
# Pop this argument so that we can manually sum over dimensions
last_dim_is_batch = params.pop("last_dim_is_batch", False)
# Get lengthscale
lengthscale = self.lengthscale
x1_ = x1.div(self.period_length / math.pi)
x2_ = x2.div(self.period_length / math.pi)
# We are automatically overriding last_dim_is_batch here so that we can manually sum over dimensions.
diff = self.covar_dist(x1_, x2_, diag=diag, last_dim_is_batch=True, **params)
if diag:
lengthscale = lengthscale[..., 0, :, None]
else:
lengthscale = lengthscale[..., 0, :, None, None]
exp_term = diff.sin().pow(2.0).div(lengthscale).mul(-2.0)
if not last_dim_is_batch:
exp_term = exp_term.sum(dim=(-2 if diag else -3))
return exp_term.exp()