tests/test_kernels.py

"""Correctness checks for the analytic kernels."""

import jax
import jax.numpy as jnp
import numpy as np
import pytest

jax.config.update("jax_enable_x64", True)

from ballast.kernels import (
    HelmParams,
    k_ext_full,
    k_helm_mat,
    k_thelm_mat,
    matern32_blocks,
)

P = HelmParams(
    phi_ls=0.8, phi_var=0.5, psi_ls=0.5, psi_var=0.5, time_ls=2.5, time_var=1.0
)


def _m32(t, t2, ls, var):
    """Reference Matern-3/2 written so autodiff is valid (no clipped distance)."""
    r = jnp.abs(t - t2)
    lam = jnp.sqrt(3.0) / ls
    return var * (1 + lam * r) * jnp.exp(-lam * r)


def test_matern32_derivatives_match_autodiff():
    """Analytic d_t, d_t', d^2_tt' blocks agree with autodiff away from t=t'."""
    ts = jnp.array([0.0, 0.3, 1.7, 4.2])
    ts2 = jnp.array([0.5, 2.1, 3.3])
    M = matern32_blocks(ts, ts2, P.time_ls, P.time_var)

    k = jax.vmap(lambda a: jax.vmap(lambda b: _m32(a, b, P.time_ls, P.time_var))(ts2))(ts)
    dt = jax.grad(_m32, argnums=0)
    dt2 = jax.grad(_m32, argnums=1)
    d2 = jax.grad(jax.grad(_m32, argnums=0), argnums=1)

    f = lambda g: jax.vmap(lambda a: jax.vmap(lambda b: g(a, b, P.time_ls, P.time_var))(ts2))(ts)
    np.testing.assert_allclose(M[..., 0, 0], k, rtol=1e-10)
    np.testing.assert_allclose(M[..., 1, 0], f(dt), rtol=1e-8)
    np.testing.assert_allclose(M[..., 0, 1], f(dt2), rtol=1e-8)
    np.testing.assert_allclose(M[..., 1, 1], f(d2), rtol=1e-8)


def test_matern32_second_derivative_at_zero():
    """Paper Sec. H.2: d^2_{tt'} k at t=t' must be 3 sigma^2/l^2, not 0.

    This is the value GPJax-style clipped-distance autodiff silently returns as
    zero; it is also exactly P_inf[1,1] in the SPDE formulation, i.e. Var(d_t f).
    """
    t = jnp.array([1.0])
    M = matern32_blocks(t, t, 1.0, 1.0)
    assert np.isclose(M[0, 0, 1, 1], 3.0)
    M2 = matern32_blocks(t, t, P.time_ls, P.time_var)
    assert np.isclose(M2[0, 0, 1, 1], 3.0 * P.time_var / P.time_ls**2)


def test_matern32_finite_difference():
    """Independent finite-difference check of the mixed second derivative."""
    # h chosen to balance the O(h^2) truncation error of the 4-point stencil
    # against O(eps/h^2) round-off; 1e-5 leaves ~1e-4 relative truncation error
    # on a value of ~5e-3 and would fail a 1e-4 tolerance for benign reasons.
    h = 1e-3
    a, b = 1.0, 2.4
    num = (
        _m32(a + h, b + h, P.time_ls, P.time_var)
        - _m32(a + h, b - h, P.time_ls, P.time_var)
        - _m32(a - h, b + h, P.time_ls, P.time_var)
        + _m32(a - h, b - h, P.time_ls, P.time_var)
    ) / (4 * h * h)
    ana = matern32_blocks(jnp.array([a]), jnp.array([b]), P.time_ls, P.time_var)[0, 0, 1, 1]
    assert np.isclose(num, ana, rtol=1e-4)


def _helm_field_cov_reference(S, S2, p):
    """Helmholtz kernel rebuilt by autodiff of the potential/stream construction.

    F_1 = d_1 Phi + d_2 Psi,  F_2 = d_2 Phi - d_1 Psi, so
    Cov(F_a(x), F_b(x')) is assembled from mixed second derivatives of k_Phi, k_Psi.
    """
    kphi = lambda x, y: p.phi_var * jnp.exp(-0.5 * jnp.sum((x - y) ** 2) / p.phi_ls**2)
    kpsi = lambda x, y: p.psi_var * jnp.exp(-0.5 * jnp.sum((x - y) ** 2) / p.psi_ls**2)
    Hphi = jax.jacfwd(jax.jacrev(kphi, argnums=0), argnums=1)
    Hpsi = jax.jacfwd(jax.jacrev(kpsi, argnums=0), argnums=1)

    def cov(x, y):
        A, B = Hphi(x, y), Hpsi(x, y)
        return jnp.array(
            [
                [A[0, 0] + B[1, 1], A[0, 1] - B[1, 0]],
                [A[1, 0] - B[0, 1], A[1, 1] + B[0, 0]],
            ]
        )

    return jax.vmap(lambda x: jax.vmap(lambda y: cov(x, y))(S2))(S)


def test_helmholtz_matches_autodiff_construction():
    S = jnp.array([[0.0, 0.0], [0.4, -0.7], [1.2, 0.3]])
    S2 = jnp.array([[0.1, 0.2], [-0.5, 0.9]])
    ref = _helm_field_cov_reference(S, S2, P)  # (3, 2, 2, 2)
    got = k_helm_mat(S, S2, P).reshape(3, 2, 2, 2).transpose(0, 2, 1, 3)
    # got is (i, j, c, c') after the transpose of the (i,c,j,c') flattening
    np.testing.assert_allclose(got.transpose(0, 1, 2, 3), ref, rtol=1e-8, atol=1e-12)


def test_gram_matrices_are_psd():
    key = jax.random.PRNGKey(0)
    S = jax.random.uniform(key, (12, 2), minval=-2, maxval=2)
    t = jnp.linspace(0, 5, 12)
    for K in (k_thelm_mat(S, t, S, t, P), k_ext_full(S, t, P), k_helm_mat(S, S, P)):
        w = jnp.linalg.eigvalsh(0.5 * (K + K.T))
        assert w.min() > -1e-8, f"min eigenvalue {w.min()}"


def test_extended_kernel_blocks_are_consistent():
    """The f-block of the extended kernel is exactly the plain kernel."""
    key = jax.random.PRNGKey(1)
    S = jax.random.uniform(key, (5, 2), minval=-1, maxval=1)
    t = jnp.linspace(0, 3, 5)
    E = k_ext_full(S, t, P).reshape(5, 2, 2, 5, 2, 2)
    plain = k_thelm_mat(S, t, S, t, P).reshape(5, 2, 5, 2)
    np.testing.assert_allclose(E[:, :, 0, :, :, 0], plain, rtol=1e-10)