HSGP periodic (#6877)

* implement periodic psd and pass test * simplify hsgp test, set up for mark other cov_func * attempt at if-else logic in HSGP class API * np.pi in w0 * check arguments for period and nonperiod case * tests run but failing * fix one of the mypy errors * periodic cov pytest uses J not m because mpyp * avoid AttributeError by init parameterization * get rid of mypy "None" error * self.prior_linearized inside if avoids double call * need to add the mean function in prior * run pre-commit locally * add docs to test about why test_prior fails * freshen up the docs * improve coverage of args * mypy ignore type * fix parameterization test * use power_spectral_density_approx because cant sum * tbe -> the typo docs * expand warning and change the triggering clause * separate into HSGP and HSGPPeriodic * remove commented periodic in HSGP test * appease mypy * add scale parameter for HSGP

Author: theorashid

pymc/gp/__init__.py

@@ -22,4 +22,4 @@
  MarginalKron,
  MarginalSparse,
 )
-from pymc.gp.hsgp_approx import HSGP
+from pymc.gp.hsgp_approx import HSGP, HSGPPeriodic

pymc/gp/cov.py

@@ -790,6 +790,28 @@ def full_from_distance(self, dist: TensorLike, squared: bool = False) -> TensorV
  r2 = dist if squared else dist**2
  return pt.exp(-0.5 * r2)
 
+ def power_spectral_density_approx(self, J: TensorLike) -> TensorVariable:
+ """
+ Technically, this is not a spectral density but these are the first `m` coefficients of
+ the low rank approximation for the periodic kernel, which are used in the same way.
+ `J` is a vector of `np.arange(m)`.
+
+ The coefficients of the HSGP approximation for the `Periodic` kernel are given by:
+
+ .. math::
+
+ \tilde{q}_j^2 = \frac{2 \text{I}_j (\\ell^{-2})}{\\exp(\\ell^{-2})} \\
+ \tilde{q}_0^2 = \frac{\text{I}_0 (\\ell^{-2})}{\\exp(\\ell^{-2})}
+
+ where $\text{I}_j$ is the modified Bessel function of the first kind.
+ """
+ a = 1 / pt.square(self.ls)
+ c = pt.where(J > 0, 2, 1)
+
+ q2 = c * pt.iv(J, a) / pt.exp(a)
+
+ return q2
+
 
 class Linear(Covariance):
  r"""