From b0e624bf1d953a9eb1c652c910d26eb72ea8bd34 Mon Sep 17 00:00:00 2001 From: NiPreps Bot Date: Wed, 30 Oct 2024 02:09:04 +0000 Subject: [PATCH] docs(main): Update docs of development line --- main/.buildinfo | 2 +- main/_modules/eddymotion/cli/parser.html | 11 +- main/_modules/eddymotion/cli/run.html | 11 +- main/_modules/eddymotion/data/dmri.html | 11 +- main/_modules/eddymotion/data/filtering.html | 11 +- main/_modules/eddymotion/data/pet.html | 11 +- main/_modules/eddymotion/data/splitting.html | 11 +- main/_modules/eddymotion/data/utils.html | 11 +- main/_modules/eddymotion/estimator.html | 13 +- main/_modules/eddymotion/exceptions.html | 11 +- main/_modules/eddymotion/math/utils.html | 11 +- main/_modules/eddymotion/model/base.html | 11 +- main/_modules/eddymotion/model/dmri.html | 11 +- main/_modules/eddymotion/model/gpr.html | 955 ++++++++++++++++++ main/_modules/eddymotion/model/pet.html | 11 +- .../eddymotion/registration/ants.html | 11 +- .../eddymotion/registration/utils.html | 11 +- .../eddymotion/testing/simulations.html | 11 +- main/_modules/eddymotion/utils.html | 11 +- main/_modules/eddymotion/viz/signals.html | 11 +- main/_modules/index.html | 12 +- .../_sources/api/eddymotion.model.gpr.rst.txt | 7 + main/_sources/api/eddymotion.model.rst.txt | 1 + main/_static/documentation_options.js | 2 +- main/api/eddymotion.cli.html | 11 +- main/api/eddymotion.cli.parser.html | 11 +- main/api/eddymotion.cli.run.html | 11 +- main/api/eddymotion.data.dmri.html | 11 +- main/api/eddymotion.data.filtering.html | 11 +- main/api/eddymotion.data.html | 11 +- main/api/eddymotion.data.pet.html | 11 +- main/api/eddymotion.data.splitting.html | 11 +- main/api/eddymotion.data.utils.html | 11 +- main/api/eddymotion.estimator.html | 11 +- main/api/eddymotion.exceptions.html | 11 +- main/api/eddymotion.math.html | 11 +- main/api/eddymotion.math.utils.html | 11 +- main/api/eddymotion.model.base.html | 11 +- main/api/eddymotion.model.dmri.html | 17 +- main/api/eddymotion.model.gpr.html | 741 ++++++++++++++ main/api/eddymotion.model.html | 12 +- main/api/eddymotion.model.pet.html | 17 +- main/api/eddymotion.registration.ants.html | 11 +- main/api/eddymotion.registration.html | 11 +- main/api/eddymotion.registration.utils.html | 11 +- main/api/eddymotion.testing.html | 11 +- main/api/eddymotion.testing.simulations.html | 23 +- main/api/eddymotion.utils.html | 11 +- main/api/eddymotion.viz.html | 11 +- main/api/eddymotion.viz.signals.html | 13 +- main/changes.html | 11 +- main/developers.html | 39 +- main/genindex.html | 114 ++- main/index.html | 11 +- main/installation.html | 11 +- main/links.html | 11 +- main/objects.inv | Bin 2151 -> 2421 bytes main/py-modindex.html | 17 +- main/running.html | 11 +- main/search.html | 11 +- main/searchindex.js | 2 +- main/usage.html | 11 +- 62 files changed, 2171 insertions(+), 300 deletions(-) create mode 100644 main/_modules/eddymotion/model/gpr.html create mode 100644 main/_sources/api/eddymotion.model.gpr.rst.txt create mode 100644 main/api/eddymotion.model.gpr.html diff --git a/main/.buildinfo b/main/.buildinfo index 590a8b6f..c2f187f0 100644 --- a/main/.buildinfo +++ b/main/.buildinfo @@ -1,4 +1,4 @@ # Sphinx build info version 1 # This file hashes the configuration used when building these files. When it is not found, a full rebuild will be done. -config: b653a8fbb9dbb05add28ed0cd449ad69 +config: 05dbe88b606d77b21e3f378bdfa07057 tags: 645f666f9bcd5a90fca523b33c5a78b7 diff --git a/main/_modules/eddymotion/cli/parser.html b/main/_modules/eddymotion/cli/parser.html index e3b5e8a1..a10edc26 100644 --- a/main/_modules/eddymotion/cli/parser.html +++ b/main/_modules/eddymotion/cli/parser.html @@ -5,7 +5,7 @@ - eddymotion.cli.parser - eddymotion 24.0.0.dev207 documentation + eddymotion.cli.parser - eddymotion 24.0.0.dev211 documentation @@ -166,7 +166,7 @@
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diff --git a/main/_modules/eddymotion/data/splitting.html b/main/_modules/eddymotion/data/splitting.html index 8d438f8b..d77a448b 100644 --- a/main/_modules/eddymotion/data/splitting.html +++ b/main/_modules/eddymotion/data/splitting.html @@ -5,7 +5,7 @@ - eddymotion.data.splitting - eddymotion 24.0.0.dev207 documentation + eddymotion.data.splitting - eddymotion 24.0.0.dev211 documentation @@ -166,7 +166,7 @@
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diff --git a/main/_modules/eddymotion/data/utils.html b/main/_modules/eddymotion/data/utils.html index b1c59c3f..f290927c 100644 --- a/main/_modules/eddymotion/data/utils.html +++ b/main/_modules/eddymotion/data/utils.html @@ -5,7 +5,7 @@ - eddymotion.data.utils - eddymotion 24.0.0.dev207 documentation + eddymotion.data.utils - eddymotion 24.0.0.dev211 documentation @@ -166,7 +166,7 @@
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diff --git a/main/_modules/eddymotion/estimator.html b/main/_modules/eddymotion/estimator.html index 56a7d998..b2ac900b 100644 --- a/main/_modules/eddymotion/estimator.html +++ b/main/_modules/eddymotion/estimator.html @@ -5,7 +5,7 @@ - eddymotion.estimator - eddymotion 24.0.0.dev207 documentation + eddymotion.estimator - eddymotion 24.0.0.dev211 documentation @@ -166,7 +166,7 @@
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diff --git a/main/_modules/eddymotion/math/utils.html b/main/_modules/eddymotion/math/utils.html index 49ff16d2..aafe20a4 100644 --- a/main/_modules/eddymotion/math/utils.html +++ b/main/_modules/eddymotion/math/utils.html @@ -5,7 +5,7 @@ - eddymotion.math.utils - eddymotion 24.0.0.dev207 documentation + eddymotion.math.utils - eddymotion 24.0.0.dev211 documentation @@ -166,7 +166,7 @@
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diff --git a/main/_modules/eddymotion/model/base.html b/main/_modules/eddymotion/model/base.html index 3a3f294d..43fcf6a3 100644 --- a/main/_modules/eddymotion/model/base.html +++ b/main/_modules/eddymotion/model/base.html @@ -5,7 +5,7 @@ - eddymotion.model.base - eddymotion 24.0.0.dev207 documentation + eddymotion.model.base - eddymotion 24.0.0.dev211 documentation @@ -166,7 +166,7 @@
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diff --git a/main/_modules/eddymotion/model/dmri.html b/main/_modules/eddymotion/model/dmri.html index 0c04b08c..3180dee9 100644 --- a/main/_modules/eddymotion/model/dmri.html +++ b/main/_modules/eddymotion/model/dmri.html @@ -5,7 +5,7 @@ - eddymotion.model.dmri - eddymotion 24.0.0.dev207 documentation + eddymotion.model.dmri - eddymotion 24.0.0.dev211 documentation @@ -166,7 +166,7 @@
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diff --git a/main/_modules/eddymotion/model/gpr.html b/main/_modules/eddymotion/model/gpr.html new file mode 100644 index 00000000..e4049eeb --- /dev/null +++ b/main/_modules/eddymotion/model/gpr.html @@ -0,0 +1,955 @@ + + + + + + + + eddymotion.model.gpr - eddymotion 24.0.0.dev211 documentation + + + + + + + + + + + + + + + + + Contents + + + + + + Menu + + + + + + + + Expand + + + + + + Light mode + + + + + + + + + + + + + + Dark mode + + + + + + + Auto light/dark, in light mode + + + + + + + + + + + + + + + Auto light/dark, in dark mode + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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Source code for eddymotion.model.gpr

+# emacs: -*- mode: python; py-indent-offset: 4; indent-tabs-mode: nil -*-
+# vi: set ft=python sts=4 ts=4 sw=4 et:
+#
+# Copyright The NiPreps Developers <nipreps@gmail.com>
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+# You may obtain a copy of the License at
+#
+#     http://www.apache.org/licenses/LICENSE-2.0
+#
+# Unless required by applicable law or agreed to in writing, software
+# distributed under the License is distributed on an "AS IS" BASIS,
+# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+# See the License for the specific language governing permissions and
+# limitations under the License.
+#
+# We support and encourage derived works from this project, please read
+# about our expectations at
+#
+#     https://www.nipreps.org/community/licensing/
+#
+"""Derivations from scikit-learn for Gaussian Processes."""
+
+from __future__ import annotations
+
+from numbers import Integral, Real
+from typing import Callable, Mapping, Sequence
+
+import numpy as np
+from scipy import optimize
+from scipy.optimize._minimize import Bounds
+from sklearn.gaussian_process import GaussianProcessRegressor
+from sklearn.gaussian_process.kernels import (
+    Hyperparameter,
+    Kernel,
+)
+from sklearn.metrics.pairwise import cosine_similarity
+from sklearn.utils._param_validation import Interval, StrOptions
+
+BOUNDS_A: tuple[float, float] = (0.1, 2.35)
+"""The limits for the parameter *a* (angular distance in rad)."""
+BOUNDS_LAMBDA: tuple[float, float] = (1e-3, 1000)
+"""The limits for the parameter λ (signal scaling factor)."""
+THETA_EPSILON: float = 1e-5
+"""Minimum nonzero angle."""
+LBFGS_CONFIGURABLE_OPTIONS = {"disp", "maxiter", "ftol", "gtol"}
+"""The set of extended options that can be set on the default BFGS."""
+CONFIGURABLE_OPTIONS: Mapping[str, set] = {
+    "Nelder-Mead": {"disp", "maxiter", "adaptive", "fatol"},
+    "CG": {"disp", "maxiter", "gtol"},
+}
+"""
+A mapping from optimizer names to the option set they allow.
+
+Add new optimizers to this list, including what options may be
+configured.
+"""
+NONGRADIENT_METHODS = {"Nelder-Mead"}
+"""A set of gradients that do not allow analytical gradients."""
+SUPPORTED_OPTIMIZERS = set(CONFIGURABLE_OPTIONS.keys()) | {"fmin_l_bfgs_b"}
+"""A set of supported optimizers (automatically created)."""
+
+
+

+[docs]
+class EddyMotionGPR(GaussianProcessRegressor):
+    r"""
+    A GP regressor specialized for eddymotion.
+
+    This specialization of the default GP regressor is created to allow
+    the following extended behaviors:
+
+    * Pacify Scikit-learn's estimator parameter checker to allow optimizers
+      given by name (as a string) other than the default BFGS.
+    * Enable custom options of optimizers.
+      See :obj:`~scipy.optimize.minimize` for the available options.
+      Please note that only a few of them are currently supported.
+
+    In the future, this specialization would be the right place for hyperparameter
+    optimization using cross-validation and such.
+
+    In principle, Scikit-Learn's implementation normalizes the training data
+    as in [Andersson15]_ (see
+    `FSL's souce code <https://git.fmrib.ox.ac.uk/fsl/eddy/-/blob/2480dda293d4cec83014454db3a193b87921f6b0/DiffusionGP.cpp#L218>`__).
+    From their paper (p. 167, end of first column):
+
+        Typically one just substracts the mean (:math:`\bar{\mathbf{f}}`)
+        from :math:`\mathbf{f}` and then add it back to
+        :math:`f^{*}`, which is analogous to what is often done in
+        "traditional" regression.
+
+    Finally, the parameter :math:`\sigma^2` maps on to Scikit-learn's ``alpha``
+    of the regressor.
+    Because it is not a parameter of the kernel, hyperparameter selection
+    through gradient-descent with analytical gradient calculations
+    would not work (the derivative of the kernel w.r.t. alpha is zero).
+
+    I believe this is overlooked in [Andersson15]_, or they actually did not
+    use analytical gradient-descent:
+
+        *A note on optimisation*
+
+        It is suggested, for example in Rasmussen and Williams (2006), that
+        an optimisation method that uses derivative information should be
+        used when finding the hyperparameters that maximise Eq. (12).
+        The reason for that is that such methods typically use fewer steps, and
+        when the cost of calculating the derivatives is small/moderate compared
+        to calculating the functions itself (as is the case for Eq. (12)) then
+        execution time can be much shorter.
+        However, we found that for the multi-shell case a heuristic optimisation
+        method such as the Nelder-Mead simplex method (Nelder and Mead, 1965) was
+        frequently better at avoiding local maxima.
+        Hence, that was the method we used for all optimisations in the present
+        paper.
+    
+    **Multi-shell regression (TODO).**
+    For multi-shell modeling, the kernel :math:`k(\textbf{x}, \textbf{x'})`
+    is updated following Eq. (14) in [Andersson15]_.
+
+    .. math::
+        k(\textbf{x}, \textbf{x'}) = C_{\theta}(\mathbf{g}, \mathbf{g'}; a) C_{b}(|b - b'|; \ell)
+
+    and :math:`C_{b}` is based the log of the b-values ratio, a measure of distance along the
+    b-direction, according to Eq. (15) given by:
+
+    .. math::
+        C_{b}(b, b'; \ell) = \exp\left( - \frac{(\log b - \log b')^2}{2 \ell^2} \right),
+
+    :math:`b` and :math:`b'` being the b-values, and :math:`\mathbf{g}` and
+    :math:`\mathbf{g'}` the unit diffusion-encoding gradient unit vectors of the
+    shells; and :math:`{a, \ell}` some hyperparameters.
+
+    The full GP regression kernel :math:`\mathbf{K}` is then updated for a 2-shell case as
+    follows (Eq. (16) in [Andersson15]_):
+
+    .. math::
+        \begin{equation}
+        \mathbf{K} = \left[
+        \begin{matrix}
+            \lambda C_{\theta}(\theta (\mathbf{G}_{1}); a) + \sigma_{1}^{2} \mathbf{I} &
+            \lambda C_{\theta}(\theta (\mathbf{G}_{2}, \mathbf{G}_{1}); a) C_{b}(b_{2}, b_{1}; \ell) \\
+            \lambda C_{\theta}(\theta (\mathbf{G}_{1}, \mathbf{G}_{2}); a) C_{b}(b_{1}, b_{2}; \ell) &
+            \lambda C_{\theta}(\theta (\mathbf{G}_{2}); a) + \sigma_{2}^{2} \mathbf{I} \\
+        \end{matrix}
+        \right]
+        \end{equation}
+
+    References
+    ----------
+    .. [Andersson15] J. L. R. Andersson. et al., An integrated approach to
+        correction for off-resonance effects and subject movement in diffusion MR
+        imaging, NeuroImage 125 (2016) 1063-11078
+
+    """
+
+    _parameter_constraints: dict = {
+        "kernel": [None, Kernel],
+        "alpha": [Interval(Real, 0, None, closed="left"), np.ndarray],
+        "optimizer": [StrOptions(SUPPORTED_OPTIMIZERS), callable, None],
+        "n_restarts_optimizer": [Interval(Integral, 0, None, closed="left")],
+        "copy_X_train": ["boolean"],
+        "normalize_y": ["boolean"],
+        "n_targets": [Interval(Integral, 1, None, closed="left"), None],
+        "random_state": ["random_state"],
+    }
+
+    def __init__(
+        self,
+        kernel: Kernel | None = None,
+        *,
+        alpha: float = 0.5,
+        optimizer: str | Callable | None = "fmin_l_bfgs_b",
+        n_restarts_optimizer: int = 0,
+        copy_X_train: bool = True,
+        normalize_y: bool = True,
+        n_targets: int | None = None,
+        random_state: int | None = None,
+        eval_gradient: bool = True,
+        tol: float | None = None,
+        disp: bool | int | None = None,
+        maxiter: int | None = None,
+        ftol: float | None = None,
+        gtol: float | None = None,
+        adaptive: bool | int | None = None,
+        fatol: float | None = None,
+    ):
+        super().__init__(
+            kernel,
+            alpha=alpha,
+            optimizer=optimizer,
+            n_restarts_optimizer=n_restarts_optimizer,
+            normalize_y=normalize_y,
+            copy_X_train=copy_X_train,
+            n_targets=n_targets,
+            random_state=random_state,
+        )
+
+        self.tol = tol
+        self.eval_gradient = eval_gradient if optimizer not in NONGRADIENT_METHODS else False
+        self.maxiter = maxiter
+        self.disp = disp
+        self.ftol = ftol
+        self.gtol = gtol
+        self.adaptive = adaptive
+        self.fatol = fatol
+
+    def _constrained_optimization(
+        self,
+        obj_func: Callable,
+        initial_theta: np.ndarray,
+        bounds: Sequence[tuple[float, float]] | Bounds,
+    ) -> tuple[float, float]:
+        options = {}
+        if self.optimizer == "fmin_l_bfgs_b":
+            from sklearn.utils.optimize import _check_optimize_result
+
+            for name in LBFGS_CONFIGURABLE_OPTIONS:
+                if (value := getattr(self, name, None)) is not None:
+                    options[name] = value
+
+            opt_res = optimize.minimize(
+                obj_func,
+                initial_theta,
+                method="L-BFGS-B",
+                bounds=bounds,
+                jac=self.eval_gradient,
+                options=options,
+                args=(self.eval_gradient,),
+                tol=self.tol,
+            )
+            _check_optimize_result("lbfgs", opt_res)
+            return opt_res.x, opt_res.fun
+
+        if isinstance(self.optimizer, str) and self.optimizer in CONFIGURABLE_OPTIONS:
+            for name in CONFIGURABLE_OPTIONS[self.optimizer]:
+                if (value := getattr(self, name, None)) is not None:
+                    options[name] = value
+
+            opt_res = optimize.minimize(
+                obj_func,
+                initial_theta,
+                method=self.optimizer,
+                bounds=bounds,
+                jac=self.eval_gradient,
+                options=options,
+                args=(self.eval_gradient,),
+                tol=self.tol,
+            )
+            return opt_res.x, opt_res.fun
+
+        if callable(self.optimizer):
+            return self.optimizer(obj_func, initial_theta, bounds=bounds)
+
+        raise ValueError(f"Unknown optimizer {self.optimizer}.")

+
+
+
+

+[docs]
+class ExponentialKriging(Kernel):
+    """A scikit-learn's kernel for DWI signals."""
+
+    def __init__(
+        self,
+        beta_a: float = 0.01,
+        beta_l: float = 2.0,
+        a_bounds: tuple[float, float] = BOUNDS_A,
+        l_bounds: tuple[float, float] = BOUNDS_LAMBDA,
+    ):
+        r"""
+        Initialize an exponential Kriging kernel.
+
+        Parameters
+        ----------
+        beta_a : :obj:`float`, optional
+            Minimum angle in rads.
+        beta_l : :obj:`float`, optional
+            The :math:`\lambda` hyperparameter.
+        a_bounds : :obj:`tuple`, optional
+            Bounds for the a parameter.
+        l_bounds : :obj:`tuple`, optional
+            Bounds for the :math:`\lambda` hyperparameter.
+
+        """
+        self.beta_a = beta_a
+        self.beta_l = beta_l
+        self.a_bounds = a_bounds
+        self.l_bounds = l_bounds
+
+    @property
+    def hyperparameter_a(self) -> Hyperparameter:
+        return Hyperparameter("beta_a", "numeric", self.a_bounds)
+
+    @property
+    def hyperparameter_beta_l(self) -> Hyperparameter:
+        return Hyperparameter("beta_l", "numeric", self.l_bounds)
+
+    def __call__(
+        self, X: np.ndarray, Y: np.ndarray | None = None, eval_gradient: bool = False
+    ) -> np.ndarray | tuple[np.ndarray, np.ndarray]:
+        """
+        Return the kernel K(X, Y) and optionally its gradient.
+
+        Parameters
+        ----------
+        X : :obj:`~numpy.ndarray`
+            Gradient table (X)
+        Y : :obj:`~numpy.ndarray`, optional
+            Gradient table (Y, optional)
+        eval_gradient : :obj:`bool`, optional
+            Determines whether the gradient with respect to the log of
+            the kernel hyperparameter is computed.
+            Only supported when Y is ``None``.
+
+        Returns
+        -------
+        K : ndarray of shape (n_samples_X, n_samples_Y)
+            Kernel k(X, Y)
+
+        K_gradient : ndarray of shape (n_samples_X, n_samples_X, n_dims),\
+                optional
+            The gradient of the kernel k(X, X) with respect to the log of the
+            hyperparameter of the kernel. Only returned when `eval_gradient`
+            is True.
+
+        """
+        thetas = compute_pairwise_angles(X, Y)
+        C_theta = exponential_covariance(thetas, self.beta_a)
+
+        if not eval_gradient:
+            return self.beta_l * C_theta
+
+        K_gradient = np.zeros((*thetas.shape, 2))
+        K_gradient[..., 0] = self.beta_l * C_theta * thetas / self.beta_a**2  # Derivative w.r.t. a
+        K_gradient[..., 1] = C_theta
+
+        return self.beta_l * C_theta, K_gradient
+
+

+[docs]
+    def diag(self, X: np.ndarray) -> np.ndarray:
+        """Returns the diagonal of the kernel k(X, X).
+
+        The result of this method is identical to np.diag(self(X)); however,
+        it can be evaluated more efficiently since only the diagonal is
+        evaluated.
+
+        Parameters
+        ----------
+        X : ndarray of shape (n_samples_X, n_features)
+            Left argument of the returned kernel k(X, Y)
+
+        Returns
+        -------
+        K_diag : ndarray of shape (n_samples_X,)
+            Diagonal of kernel k(X, X)
+        """
+        return self.beta_l * np.ones(X.shape[0])

+
+
+

+[docs]
+    def is_stationary(self) -> bool:
+        """Returns whether the kernel is stationary."""
+        return True

+
+
+    def __repr__(self) -> str:
+        return f"ExponentialKriging (a={self.beta_a}, λ={self.beta_l})"

+
+
+
+

+[docs]
+class SphericalKriging(Kernel):
+    """A scikit-learn's kernel for DWI signals."""
+
+    def __init__(
+        self,
+        beta_a: float = 1.38,
+        beta_l: float = 0.5,
+        a_bounds: tuple[float, float] = BOUNDS_A,
+        l_bounds: tuple[float, float] = BOUNDS_LAMBDA,
+    ):
+        r"""
+        Initialize a spherical Kriging kernel.
+
+        Parameters
+        ----------
+        beta_a : :obj:`float`, optional
+            Minimum angle in rads.
+        beta_l : :obj:`float`, optional
+            The :math:`\lambda` hyperparameter.
+        a_bounds : :obj:`tuple`, optional
+            Bounds for the ``a`` parameter.
+        l_bounds : :obj:`tuple`, optional
+            Bounds for the :math:`\lambda` hyperparameter.
+
+        """
+        self.beta_a = beta_a
+        self.beta_l = beta_l
+        self.a_bounds = a_bounds
+        self.l_bounds = l_bounds
+
+    @property
+    def hyperparameter_a(self) -> Hyperparameter:
+        return Hyperparameter("beta_a", "numeric", self.a_bounds)
+
+    @property
+    def hyperparameter_beta_l(self) -> Hyperparameter:
+        return Hyperparameter("beta_l", "numeric", self.l_bounds)
+
+    def __call__(
+        self, X: np.ndarray, Y: np.ndarray | None = None, eval_gradient: bool = False
+    ) -> np.ndarray | tuple[np.ndarray, np.ndarray]:
+        """
+        Return the kernel K(X, Y) and optionally its gradient.
+
+        Parameters
+        ----------
+        X : :obj:`~numpy.ndarray`
+            Gradient table (X)
+        Y : :obj:`~numpy.ndarray`, optional
+            Gradient table (Y, optional)
+        eval_gradient : :obj:`bool`, optional
+            Determines whether the gradient with respect to the log of
+            the kernel hyperparameter is computed.
+            Only supported when Y is ``None``.
+
+        Returns
+        -------
+        K : ndarray of shape (n_samples_X, n_samples_Y)
+            Kernel k(X, Y)
+
+        K_gradient : ndarray of shape (n_samples_X, n_samples_X, n_dims),\
+                optional
+            The gradient of the kernel k(X, X) with respect to the log of the
+            hyperparameter of the kernel. Only returned when ``eval_gradient``
+            is True.
+
+        """
+        thetas = compute_pairwise_angles(X, Y)
+        C_theta = spherical_covariance(thetas, self.beta_a)
+
+        if not eval_gradient:
+            return self.beta_l * C_theta
+
+        deriv_a = np.zeros_like(thetas)
+        nonzero = thetas <= self.beta_a
+        deriv_a[nonzero] = (
+            1.5
+            * self.beta_l
+            * (thetas[nonzero] / self.beta_a**2 - thetas[nonzero] ** 3 / self.beta_a**4)
+        )
+        K_gradient = np.dstack((deriv_a, C_theta))
+
+        return self.beta_l * C_theta, K_gradient
+
+

+[docs]
+    def diag(self, X: np.ndarray) -> np.ndarray:
+        """Returns the diagonal of the kernel k(X, X).
+
+        The result of this method is identical to np.diag(self(X)); however,
+        it can be evaluated more efficiently since only the diagonal is
+        evaluated.
+
+        Parameters
+        ----------
+        X : ndarray of shape (n_samples_X, n_features)
+            Left argument of the returned kernel k(X, Y)
+
+        Returns
+        -------
+        K_diag : ndarray of shape (n_samples_X,)
+            Diagonal of kernel k(X, X)
+        """
+        return self.beta_l * np.ones(X.shape[0])

+
+
+

+[docs]
+    def is_stationary(self) -> bool:
+        """Returns whether the kernel is stationary."""
+        return True

+
+
+    def __repr__(self) -> str:
+        return f"SphericalKriging (a={self.beta_a}, λ={self.beta_l})"

+
+
+
+

+[docs]
+def exponential_covariance(theta: np.ndarray, a: float) -> np.ndarray:
+    r"""
+    Compute the exponential covariance for given distances and scale parameter.
+
+    Implements :math:`C_{\theta}`, following Eq. (9) in [Andersson15]_:
+
+    .. math::
+        \begin{equation}
+        C(\theta) = e^{-\theta/a} \,\, \text{for} \, 0 \leq \theta \leq \pi,
+        \end{equation}
+
+    :math:`\theta` being computed as:
+
+    .. math::
+        \theta(\mathbf{g}, \mathbf{g'}) = \arccos(|\langle \mathbf{g}, \mathbf{g'} \rangle|)
+
+    Parameters
+    ----------
+    theta : :obj:`~numpy.ndarray`
+        Array of distances between points.
+    a : :obj:`float`
+        Scale parameter that controls the range of the covariance function.
+
+    Returns
+    -------
+    :obj:`~numpy.ndarray`
+        Exponential covariance values for the input distances.
+
+    """
+    return np.exp(-theta / a)

+
+
+
+

+[docs]
+def spherical_covariance(theta: np.ndarray, a: float) -> np.ndarray:
+    r"""
+    Compute the spherical covariance for given distances and scale parameter.
+
+    Implements :math:`C_{\theta}`, following Eq. (10) in [Andersson15]_:
+
+    .. math::
+        \begin{equation}
+        C(\theta) =
+        \begin{cases}
+        1 - \frac{3 \theta}{2 a} + \frac{\theta^3}{2 a^3} & \textnormal{if} \; \theta \leq a \\
+        0 & \textnormal{if} \; \theta > a
+        \end{cases}
+        \end{equation}
+
+    :math:`\theta` being computed as:
+
+    .. math::
+        \theta(\mathbf{g}, \mathbf{g'}) = \arccos(|\langle \mathbf{g}, \mathbf{g'} \rangle|)
+
+    Parameters
+    ----------
+    theta : :obj:`~numpy.ndarray`
+        Array of distances between points.
+    a : :obj:`float`
+        Scale parameter that controls the range of the covariance function.
+
+    Returns
+    -------
+    :obj:`~numpy.ndarray`
+        Spherical covariance values for the input distances.
+
+    """
+    return np.where(theta <= a, 1 - 1.5 * theta / a + 0.5 * (theta**3) / (a**3), 0.0)

+
+
+
+

+[docs]
+def compute_pairwise_angles(
+    X: np.ndarray,
+    Y: np.ndarray | None = None,
+    closest_polarity: bool = True,
+    dense_output: bool = True,
+) -> np.ndarray:
+    r"""Compute pairwise angles across diffusion gradient encoding directions.
+
+    Following [Andersson15]_, it computes the smallest of the angles between
+    each pair if ``closest_polarity`` is ``True``, i.e.,
+
+    .. math::
+
+        \theta(\mathbf{g}, \mathbf{g'}) = \arccos(|\langle \mathbf{g}, \mathbf{g'} \rangle|)
+
+    Parameters
+    ----------
+    X : {array-like, sparse matrix} of shape (n_samples_X, n_features)
+        Input data.
+    Y : {array-like, sparse matrix} of shape (n_samples_Y, n_features), optional
+        Input data. If ``None``, the output will be the pairwise
+        similarities between all samples in ``X``.
+    dense_output : :obj:`bool`, optional
+        Whether to return dense output even when the input is sparse. If
+        ``False``, the output is sparse if both input arrays are sparse.
+    closest_polarity : :obj:`bool`, optional
+        ``True`` to consider the smallest of the two angles between the crossing
+         lines resulting from reversing each vector pair.
+
+    Returns
+    -------
+    :obj:`~numpy.ndarray`
+        Pairwise angles across diffusion gradient encoding directions.
+
+    Examples
+    --------
+    >>> X = np.asarray([(1.0, -1.0), (0.0, 0.0), (0.0, 0.0)]).T
+    >>> compute_pairwise_angles(X, closest_polarity=False)[0, 1]  # doctest: +ELLIPSIS
+    3.1415...
+    >>> X = np.asarray([(1.0, -1.0), (0.0, 0.0), (0.0, 0.0)]).T
+    >>> Y = np.asarray([(1.0, -1.0), (0.0, 0.0), (0.0, 0.0)]).T
+    >>> compute_pairwise_angles(X, Y, closest_polarity=False)[0, 1]  # doctest: +ELLIPSIS
+    3.1415...
+    >>> X = np.asarray([(1.0, -1.0), (0.0, 0.0), (0.0, 0.0)]).T
+    >>> compute_pairwise_angles(X)[0, 1]
+    0.0
+
+    """
+
+    cosines = np.clip(cosine_similarity(X, Y, dense_output=dense_output), -1.0, 1.0)
+    thetas = np.arccos(np.abs(cosines)) if closest_polarity else np.arccos(cosines)
+    thetas[np.abs(thetas) < THETA_EPSILON] = 0.0
+    return thetas

+
+
+
+
+
+ +
+ + +
+
+
+
+ Copyright © Copyright 2021, The eddymotion developers +
+ Made with Sphinx and @pradyunsg's + + Furo + +
+
+ +
+
+ +
+
+ +
+
+ + + + + + \ No newline at end of file diff --git a/main/_modules/eddymotion/model/pet.html b/main/_modules/eddymotion/model/pet.html index a7f8dbcb..2ccb7b1a 100644 --- a/main/_modules/eddymotion/model/pet.html +++ b/main/_modules/eddymotion/model/pet.html @@ -5,7 +5,7 @@ - eddymotion.model.pet - eddymotion 24.0.0.dev207 documentation + eddymotion.model.pet - eddymotion 24.0.0.dev211 documentation @@ -166,7 +166,7 @@
-
eddymotion 24.0.0.dev207 documentation
 +
eddymotion 24.0.0.dev211 documentation
@@ -189,14 +189,14 @@
eddymotion - 24.0.0.dev207 + 24.0.0.dev211
+
diff --git a/main/_modules/eddymotion/registration/ants.html b/main/_modules/eddymotion/registration/ants.html index 92907e49..546f0b57 100644 --- a/main/_modules/eddymotion/registration/ants.html +++ b/main/_modules/eddymotion/registration/ants.html @@ -5,7 +5,7 @@ - eddymotion.registration.ants - eddymotion 24.0.0.dev207 documentation + eddymotion.registration.ants - eddymotion 24.0.0.dev211 documentation @@ -166,7 +166,7 @@
-
eddymotion 24.0.0.dev207 documentation
 +
eddymotion 24.0.0.dev211 documentation
@@ -189,14 +189,14 @@
eddymotion - 24.0.0.dev207 + 24.0.0.dev211
+
diff --git a/main/_modules/eddymotion/registration/utils.html b/main/_modules/eddymotion/registration/utils.html index fbb6949f..a877cd4f 100644 --- a/main/_modules/eddymotion/registration/utils.html +++ b/main/_modules/eddymotion/registration/utils.html @@ -5,7 +5,7 @@ - eddymotion.registration.utils - eddymotion 24.0.0.dev207 documentation + eddymotion.registration.utils - eddymotion 24.0.0.dev211 documentation @@ -166,7 +166,7 @@
-
eddymotion 24.0.0.dev207 documentation
 +
eddymotion 24.0.0.dev211 documentation
@@ -189,14 +189,14 @@
eddymotion - 24.0.0.dev207 + 24.0.0.dev211
+
diff --git a/main/_modules/eddymotion/testing/simulations.html b/main/_modules/eddymotion/testing/simulations.html index 1c7de10d..3924ef8a 100644 --- a/main/_modules/eddymotion/testing/simulations.html +++ b/main/_modules/eddymotion/testing/simulations.html @@ -5,7 +5,7 @@ - eddymotion.testing.simulations - eddymotion 24.0.0.dev207 documentation + eddymotion.testing.simulations - eddymotion 24.0.0.dev211 documentation @@ -166,7 +166,7 @@
-
eddymotion 24.0.0.dev207 documentation
 +
eddymotion 24.0.0.dev211 documentation
@@ -189,14 +189,14 @@
eddymotion - 24.0.0.dev207 + 24.0.0.dev211
+
diff --git a/main/_modules/eddymotion/utils.html b/main/_modules/eddymotion/utils.html index 0616ccf0..db68ea54 100644 --- a/main/_modules/eddymotion/utils.html +++ b/main/_modules/eddymotion/utils.html @@ -5,7 +5,7 @@ - eddymotion.utils - eddymotion 24.0.0.dev207 documentation + eddymotion.utils - eddymotion 24.0.0.dev211 documentation @@ -166,7 +166,7 @@
-
eddymotion 24.0.0.dev207 documentation
 +
eddymotion 24.0.0.dev211 documentation
@@ -189,14 +189,14 @@
eddymotion - 24.0.0.dev207 + 24.0.0.dev211
+
diff --git a/main/_modules/eddymotion/viz/signals.html b/main/_modules/eddymotion/viz/signals.html index 5af5253e..bb7c8ed0 100644 --- a/main/_modules/eddymotion/viz/signals.html +++ b/main/_modules/eddymotion/viz/signals.html @@ -5,7 +5,7 @@ - eddymotion.viz.signals - eddymotion 24.0.0.dev207 documentation + eddymotion.viz.signals - eddymotion 24.0.0.dev211 documentation @@ -166,7 +166,7 @@
-
eddymotion 24.0.0.dev207 documentation
 +
eddymotion 24.0.0.dev211 documentation
@@ -189,14 +189,14 @@
eddymotion - 24.0.0.dev207 + 24.0.0.dev211
+
diff --git a/main/_modules/index.html b/main/_modules/index.html index 449639a2..a4b7c1a4 100644 --- a/main/_modules/index.html +++ b/main/_modules/index.html @@ -5,7 +5,7 @@ - Overview: module code - eddymotion 24.0.0.dev207 documentation + Overview: module code - eddymotion 24.0.0.dev211 documentation @@ -166,7 +166,7 @@
-
eddymotion 24.0.0.dev207 documentation
 +
eddymotion 24.0.0.dev211 documentation
@@ -189,14 +189,14 @@
eddymotion - 24.0.0.dev207 + 24.0.0.dev211
-
+
diff --git a/main/_sources/api/eddymotion.model.gpr.rst.txt b/main/_sources/api/eddymotion.model.gpr.rst.txt new file mode 100644 index 00000000..c1d92c2c --- /dev/null +++ b/main/_sources/api/eddymotion.model.gpr.rst.txt @@ -0,0 +1,7 @@ +eddymotion.model.gpr module +=========================== + +.. automodule:: eddymotion.model.gpr + :members: + :undoc-members: + :show-inheritance: diff --git a/main/_sources/api/eddymotion.model.rst.txt b/main/_sources/api/eddymotion.model.rst.txt index cc3c6fae..b759fb58 100644 --- a/main/_sources/api/eddymotion.model.rst.txt +++ b/main/_sources/api/eddymotion.model.rst.txt @@ -14,4 +14,5 @@ Submodules eddymotion.model.base eddymotion.model.dmri + eddymotion.model.gpr eddymotion.model.pet diff --git a/main/_static/documentation_options.js b/main/_static/documentation_options.js index 4ccc2535..df920f96 100644 --- a/main/_static/documentation_options.js +++ b/main/_static/documentation_options.js @@ -1,5 +1,5 @@ const DOCUMENTATION_OPTIONS = { - VERSION: '24.0.0.dev207', + VERSION: '24.0.0.dev211', LANGUAGE: 'en', COLLAPSE_INDEX: false, BUILDER: 'html', diff --git a/main/api/eddymotion.cli.html b/main/api/eddymotion.cli.html index d6fde614..30fec92c 100644 --- a/main/api/eddymotion.cli.html +++ b/main/api/eddymotion.cli.html @@ -6,7 +6,7 @@ - eddymotion.cli package - eddymotion 24.0.0.dev207 documentation + eddymotion.cli package - eddymotion 24.0.0.dev211 documentation @@ -167,7 +167,7 @@
-
eddymotion 24.0.0.dev207 documentation
 +
eddymotion 24.0.0.dev211 documentation
@@ -190,14 +190,14 @@
eddymotion - 24.0.0.dev207 + 24.0.0.dev211
diff --git a/main/api/eddymotion.cli.parser.html b/main/api/eddymotion.cli.parser.html index 42a6e808..b044e06d 100644 --- a/main/api/eddymotion.cli.parser.html +++ b/main/api/eddymotion.cli.parser.html @@ -6,7 +6,7 @@ - eddymotion.cli.parser module - eddymotion 24.0.0.dev207 documentation + eddymotion.cli.parser module - eddymotion 24.0.0.dev211 documentation @@ -167,7 +167,7 @@
-
eddymotion 24.0.0.dev207 documentation
 +
eddymotion 24.0.0.dev211 documentation
@@ -190,14 +190,14 @@
eddymotion - 24.0.0.dev207 + 24.0.0.dev211
-
+
diff --git a/main/api/eddymotion.cli.run.html b/main/api/eddymotion.cli.run.html index 9611c947..011266d7 100644 --- a/main/api/eddymotion.cli.run.html +++ b/main/api/eddymotion.cli.run.html @@ -6,7 +6,7 @@ - eddymotion.cli.run module - eddymotion 24.0.0.dev207 documentation + eddymotion.cli.run module - eddymotion 24.0.0.dev211 documentation @@ -167,7 +167,7 @@
-
eddymotion 24.0.0.dev207 documentation
 +
eddymotion 24.0.0.dev211 documentation
@@ -190,14 +190,14 @@
eddymotion - 24.0.0.dev207 + 24.0.0.dev211
-
+
diff --git a/main/api/eddymotion.data.dmri.html b/main/api/eddymotion.data.dmri.html index 04a544b3..53175faf 100644 --- a/main/api/eddymotion.data.dmri.html +++ b/main/api/eddymotion.data.dmri.html @@ -6,7 +6,7 @@ - eddymotion.data.dmri module - eddymotion 24.0.0.dev207 documentation + eddymotion.data.dmri module - eddymotion 24.0.0.dev211 documentation @@ -167,7 +167,7 @@
-
eddymotion 24.0.0.dev207 documentation
 +
eddymotion 24.0.0.dev211 documentation
@@ -190,14 +190,14 @@
eddymotion - 24.0.0.dev207 + 24.0.0.dev211
-
+
diff --git a/main/api/eddymotion.data.filtering.html b/main/api/eddymotion.data.filtering.html index 3d3bd740..fc34489a 100644 --- a/main/api/eddymotion.data.filtering.html +++ b/main/api/eddymotion.data.filtering.html @@ -6,7 +6,7 @@ - eddymotion.data.filtering module - eddymotion 24.0.0.dev207 documentation + eddymotion.data.filtering module - eddymotion 24.0.0.dev211 documentation @@ -167,7 +167,7 @@
-
eddymotion 24.0.0.dev207 documentation
 +
eddymotion 24.0.0.dev211 documentation
@@ -190,14 +190,14 @@
eddymotion - 24.0.0.dev207 + 24.0.0.dev211
-
+
diff --git a/main/api/eddymotion.data.html b/main/api/eddymotion.data.html index 924dcaaa..6d4499f3 100644 --- a/main/api/eddymotion.data.html +++ b/main/api/eddymotion.data.html @@ -6,7 +6,7 @@ - eddymotion.data package - eddymotion 24.0.0.dev207 documentation + eddymotion.data package - eddymotion 24.0.0.dev211 documentation @@ -167,7 +167,7 @@
-
eddymotion 24.0.0.dev207 documentation
 +
eddymotion 24.0.0.dev211 documentation
@@ -190,14 +190,14 @@
eddymotion - 24.0.0.dev207 + 24.0.0.dev211
diff --git a/main/api/eddymotion.data.pet.html b/main/api/eddymotion.data.pet.html index b9ebdf91..40573b4c 100644 --- a/main/api/eddymotion.data.pet.html +++ b/main/api/eddymotion.data.pet.html @@ -6,7 +6,7 @@ - eddymotion.data.pet module - eddymotion 24.0.0.dev207 documentation + eddymotion.data.pet module - eddymotion 24.0.0.dev211 documentation @@ -167,7 +167,7 @@
-
eddymotion 24.0.0.dev207 documentation
 +
eddymotion 24.0.0.dev211 documentation
@@ -190,14 +190,14 @@
eddymotion - 24.0.0.dev207 + 24.0.0.dev211
-
+
diff --git a/main/api/eddymotion.data.splitting.html b/main/api/eddymotion.data.splitting.html index 57cb9ab9..3c64950c 100644 --- a/main/api/eddymotion.data.splitting.html +++ b/main/api/eddymotion.data.splitting.html @@ -6,7 +6,7 @@ - eddymotion.data.splitting module - eddymotion 24.0.0.dev207 documentation + eddymotion.data.splitting module - eddymotion 24.0.0.dev211 documentation @@ -167,7 +167,7 @@
-
eddymotion 24.0.0.dev207 documentation
 +
eddymotion 24.0.0.dev211 documentation
@@ -190,14 +190,14 @@
eddymotion - 24.0.0.dev207 + 24.0.0.dev211
-
+
diff --git a/main/api/eddymotion.data.utils.html b/main/api/eddymotion.data.utils.html index 5e84073a..506de1ff 100644 --- a/main/api/eddymotion.data.utils.html +++ b/main/api/eddymotion.data.utils.html @@ -6,7 +6,7 @@ - eddymotion.data.utils module - eddymotion 24.0.0.dev207 documentation + eddymotion.data.utils module - eddymotion 24.0.0.dev211 documentation @@ -167,7 +167,7 @@
-
eddymotion 24.0.0.dev207 documentation
 +
eddymotion 24.0.0.dev211 documentation
@@ -190,14 +190,14 @@
eddymotion - 24.0.0.dev207 + 24.0.0.dev211
-
+
diff --git a/main/api/eddymotion.estimator.html b/main/api/eddymotion.estimator.html index 50b71f59..2559ba4a 100644 --- a/main/api/eddymotion.estimator.html +++ b/main/api/eddymotion.estimator.html @@ -6,7 +6,7 @@ - eddymotion.estimator module - eddymotion 24.0.0.dev207 documentation + eddymotion.estimator module - eddymotion 24.0.0.dev211 documentation @@ -167,7 +167,7 @@
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diff --git a/main/api/eddymotion.exceptions.html b/main/api/eddymotion.exceptions.html index b5a737d5..55899d5f 100644 --- a/main/api/eddymotion.exceptions.html +++ b/main/api/eddymotion.exceptions.html @@ -6,7 +6,7 @@ - eddymotion.exceptions module - eddymotion 24.0.0.dev207 documentation + eddymotion.exceptions module - eddymotion 24.0.0.dev211 documentation @@ -167,7 +167,7 @@
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diff --git a/main/api/eddymotion.math.html b/main/api/eddymotion.math.html index 0f806739..872c6792 100644 --- a/main/api/eddymotion.math.html +++ b/main/api/eddymotion.math.html @@ -6,7 +6,7 @@ - eddymotion.math package - eddymotion 24.0.0.dev207 documentation + eddymotion.math package - eddymotion 24.0.0.dev211 documentation @@ -167,7 +167,7 @@
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diff --git a/main/api/eddymotion.model.base.html b/main/api/eddymotion.model.base.html index b96dffbd..8a198835 100644 --- a/main/api/eddymotion.model.base.html +++ b/main/api/eddymotion.model.base.html @@ -6,7 +6,7 @@ - eddymotion.model.base module - eddymotion 24.0.0.dev207 documentation + eddymotion.model.base module - eddymotion 24.0.0.dev211 documentation @@ -167,7 +167,7 @@
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diff --git a/main/api/eddymotion.model.dmri.html b/main/api/eddymotion.model.dmri.html index 357fccec..c44bfe4a 100644 --- a/main/api/eddymotion.model.dmri.html +++ b/main/api/eddymotion.model.dmri.html @@ -3,10 +3,10 @@ - + - eddymotion.model.dmri module - eddymotion 24.0.0.dev207 documentation + eddymotion.model.dmri module - eddymotion 24.0.0.dev211 documentation @@ -167,7 +167,7 @@
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diff --git a/main/api/eddymotion.model.gpr.html b/main/api/eddymotion.model.gpr.html new file mode 100644 index 00000000..fa392792 --- /dev/null +++ b/main/api/eddymotion.model.gpr.html @@ -0,0 +1,741 @@ + + + + + + + + + eddymotion.model.gpr module - eddymotion 24.0.0.dev211 documentation + + + + + + + + + + + + + + + + + Contents + + + + + + Menu + + + + + + + + Expand + + + + + + Light mode + + + + + + + + + + + + + + Dark mode + + + + + + + Auto light/dark, in light mode + + + + + + + + + + + + + + + Auto light/dark, in dark mode + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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+

eddymotion.model.gpr module

+

Derivations from scikit-learn for Gaussian Processes.

+
+
+eddymotion.model.gpr.BOUNDS_A: tuple[float, float] = (0.1, 2.35)
+

The limits for the parameter a (angular distance in rad).

+
+ +
+
+eddymotion.model.gpr.BOUNDS_LAMBDA: tuple[float, float] = (0.001, 1000)
+

The limits for the parameter λ (signal scaling factor).

+
+ +
+
+eddymotion.model.gpr.CONFIGURABLE_OPTIONS: Mapping[str, set] = {'CG': {'disp', 'gtol', 'maxiter'}, 'Nelder-Mead': {'adaptive', 'disp', 'fatol', 'maxiter'}}
+

A mapping from optimizer names to the option set they allow.

+

Add new optimizers to this list, including what options may be +configured.

+
+ +
+
+class eddymotion.model.gpr.EddyMotionGPR(*args: Any, **kwargs: Any)[source]
+

Bases: GaussianProcessRegressor

+

A GP regressor specialized for eddymotion.

+

This specialization of the default GP regressor is created to allow +the following extended behaviors:

+
    +

  • Pacify Scikit-learn’s estimator parameter checker to allow optimizers +given by name (as a string) other than the default BFGS.

    • +

  • Enable custom options of optimizers. +See minimize for the available options. +Please note that only a few of them are currently supported.

    • +
+

In the future, this specialization would be the right place for hyperparameter +optimization using cross-validation and such.

+

In principle, Scikit-Learn’s implementation normalizes the training data +as in [Andersson15] (see +FSL’s souce code). +From their paper (p. 167, end of first column):

+
+

Typically one just substracts the mean (\(\bar{\mathbf{f}}\)) +from \(\mathbf{f}\) and then add it back to +\(f^{*}\), which is analogous to what is often done in +“traditional” regression.

+
+

Finally, the parameter \(\sigma^2\) maps on to Scikit-learn’s alpha +of the regressor. +Because it is not a parameter of the kernel, hyperparameter selection +through gradient-descent with analytical gradient calculations +would not work (the derivative of the kernel w.r.t. alpha is zero).

+

I believe this is overlooked in [Andersson15], or they actually did not +use analytical gradient-descent:

+
+

A note on optimisation

+

It is suggested, for example in Rasmussen and Williams (2006), that +an optimisation method that uses derivative information should be +used when finding the hyperparameters that maximise Eq. (12). +The reason for that is that such methods typically use fewer steps, and +when the cost of calculating the derivatives is small/moderate compared +to calculating the functions itself (as is the case for Eq. (12)) then +execution time can be much shorter. +However, we found that for the multi-shell case a heuristic optimisation +method such as the Nelder-Mead simplex method (Nelder and Mead, 1965) was +frequently better at avoiding local maxima. +Hence, that was the method we used for all optimisations in the present +paper.

+
+

Multi-shell regression (TODO). +For multi-shell modeling, the kernel \(k(\textbf{x}, \textbf{x'})\) +is updated following Eq. (14) in [Andersson15].

+
+
+\[k(\textbf{x}, \textbf{x'}) = C_{\theta}(\mathbf{g}, \mathbf{g'}; a) C_{b}(|b - b'|; \ell)\]
+
+

and \(C_{b}\) is based the log of the b-values ratio, a measure of distance along the +b-direction, according to Eq. (15) given by:

+
+
+\[C_{b}(b, b'; \ell) = \exp\left( - \frac{(\log b - \log b')^2}{2 \ell^2} \right),\]
+
+

\(b\) and \(b'\) being the b-values, and \(\mathbf{g}\) and +\(\mathbf{g'}\) the unit diffusion-encoding gradient unit vectors of the +shells; and \({a, \ell}\) some hyperparameters.

+

The full GP regression kernel \(\mathbf{K}\) is then updated for a 2-shell case as +follows (Eq. (16) in [Andersson15]):

+
+
+\[\begin{split}\begin{equation} +\mathbf{K} = \left[ +\begin{matrix} + \lambda C_{\theta}(\theta (\mathbf{G}_{1}); a) + \sigma_{1}^{2} \mathbf{I} & + \lambda C_{\theta}(\theta (\mathbf{G}_{2}, \mathbf{G}_{1}); a) C_{b}(b_{2}, b_{1}; \ell) \\ + \lambda C_{\theta}(\theta (\mathbf{G}_{1}, \mathbf{G}_{2}); a) C_{b}(b_{1}, b_{2}; \ell) & + \lambda C_{\theta}(\theta (\mathbf{G}_{2}); a) + \sigma_{2}^{2} \mathbf{I} \\ +\end{matrix} +\right] +\end{equation}\end{split}\]
+
+

References

+
+
+[Andersson15] +(1,2,3,4,5,6,7) +

J. L. R. Andersson. et al., An integrated approach to +correction for off-resonance effects and subject movement in diffusion MR +imaging, NeuroImage 125 (2016) 1063-11078

+
+
+
+ +
+
+class eddymotion.model.gpr.ExponentialKriging(*args: Any, **kwargs: Any)[source]
+

Bases: Kernel

+

A scikit-learn’s kernel for DWI signals.

+
+
+diag(X: numpy.ndarray) → numpy.ndarray[source]
+

Returns the diagonal of the kernel k(X, X).

+

The result of this method is identical to np.diag(self(X)); however, +it can be evaluated more efficiently since only the diagonal is +evaluated.

+
+
Parameters:
+

X (ndarray of shape (n_samples_X, n_features)) – Left argument of the returned kernel k(X, Y)

+
+
Returns:
+

K_diag – Diagonal of kernel k(X, X)

+
+
Return type:
+

ndarray of shape (n_samples_X,)

+
+
+
+ +
+
+property hyperparameter_a: sklearn.gaussian_process.kernels.Hyperparameter
+
+ +
+
+property hyperparameter_beta_l: sklearn.gaussian_process.kernels.Hyperparameter
+
+ +
+
+is_stationary() → bool[source]
+

Returns whether the kernel is stationary.

+
+ +
+ +
+
+eddymotion.model.gpr.LBFGS_CONFIGURABLE_OPTIONS = {'disp', 'ftol', 'gtol', 'maxiter'}
+

The set of extended options that can be set on the default BFGS.

+
+ +
+
+eddymotion.model.gpr.NONGRADIENT_METHODS = {'Nelder-Mead'}
+

A set of gradients that do not allow analytical gradients.

+
+ +
+
+eddymotion.model.gpr.SUPPORTED_OPTIMIZERS = {'CG', 'Nelder-Mead', 'fmin_l_bfgs_b'}
+

A set of supported optimizers (automatically created).

+
+ +
+
+class eddymotion.model.gpr.SphericalKriging(*args: Any, **kwargs: Any)[source]
+

Bases: Kernel

+

A scikit-learn’s kernel for DWI signals.

+
+
+diag(X: numpy.ndarray) → numpy.ndarray[source]
+

Returns the diagonal of the kernel k(X, X).

+

The result of this method is identical to np.diag(self(X)); however, +it can be evaluated more efficiently since only the diagonal is +evaluated.

+
+
Parameters:
+

X (ndarray of shape (n_samples_X, n_features)) – Left argument of the returned kernel k(X, Y)

+
+
Returns:
+

K_diag – Diagonal of kernel k(X, X)

+
+
Return type:
+

ndarray of shape (n_samples_X,)

+
+
+
+ +
+
+property hyperparameter_a: sklearn.gaussian_process.kernels.Hyperparameter
+
+ +
+
+property hyperparameter_beta_l: sklearn.gaussian_process.kernels.Hyperparameter
+
+ +
+
+is_stationary() → bool[source]
+

Returns whether the kernel is stationary.

+
+ +
+ +
+
+eddymotion.model.gpr.THETA_EPSILON: float = 1e-05
+

Minimum nonzero angle.

+
+ +
+
+eddymotion.model.gpr.compute_pairwise_angles(X: np.ndarray, Y: np.ndarray | None = None, closest_polarity: bool = True, dense_output: bool = True) → np.ndarray[source]
+

Compute pairwise angles across diffusion gradient encoding directions.

+

Following [Andersson15], it computes the smallest of the angles between +each pair if closest_polarity is True, i.e.,

+
+
+\[\theta(\mathbf{g}, \mathbf{g'}) = \arccos(|\langle \mathbf{g}, \mathbf{g'} \rangle|)\]
+
+
+
Parameters:
+
    +

  • X ({array-like, sparse matrix} of shape (n_samples_X, n_features)) – Input data.

    • +

  • Y ({array-like, sparse matrix} of shape (n_samples_Y, n_features), optional) – Input data. If None, the output will be the pairwise +similarities between all samples in X.

    • +

  • dense_output (bool, optional) – Whether to return dense output even when the input is sparse. If +False, the output is sparse if both input arrays are sparse.

    • +

  • closest_polarity (bool, optional) –

    +
    +
    True to consider the smallest of the two angles between the crossing

    lines resulting from reversing each vector pair.

    +
    +
    +
    • +
+
+
Returns:
+

Pairwise angles across diffusion gradient encoding directions.

+
+
Return type:
+

ndarray

+
+
+

Examples

+
>>> X = np.asarray([(1.0, -1.0), (0.0, 0.0), (0.0, 0.0)]).T
+>>> compute_pairwise_angles(X, closest_polarity=False)[0, 1]  
+3.1415...
+>>> X = np.asarray([(1.0, -1.0), (0.0, 0.0), (0.0, 0.0)]).T
+>>> Y = np.asarray([(1.0, -1.0), (0.0, 0.0), (0.0, 0.0)]).T
+>>> compute_pairwise_angles(X, Y, closest_polarity=False)[0, 1]  
+3.1415...
+>>> X = np.asarray([(1.0, -1.0), (0.0, 0.0), (0.0, 0.0)]).T
+>>> compute_pairwise_angles(X)[0, 1]
+0.0
+
+
+
+ +
+
+eddymotion.model.gpr.exponential_covariance(theta: numpy.ndarray, a: float) → numpy.ndarray[source]
+

Compute the exponential covariance for given distances and scale parameter.

+

Implements \(C_{\theta}\), following Eq. (9) in [Andersson15]:

+
+
+\[\begin{equation} +C(\theta) = e^{-\theta/a} \,\, \text{for} \, 0 \leq \theta \leq \pi, +\end{equation}\]
+
+

\(\theta\) being computed as:

+
+
+\[\theta(\mathbf{g}, \mathbf{g'}) = \arccos(|\langle \mathbf{g}, \mathbf{g'} \rangle|)\]
+
+
+
Parameters:
+
    +

  • theta (ndarray) – Array of distances between points.

    • +

  • a (float) – Scale parameter that controls the range of the covariance function.

    • +
+
+
Returns:
+

Exponential covariance values for the input distances.

+
+
Return type:
+

ndarray

+
+
+
+ +
+
+eddymotion.model.gpr.spherical_covariance(theta: numpy.ndarray, a: float) → numpy.ndarray[source]
+

Compute the spherical covariance for given distances and scale parameter.

+

Implements \(C_{\theta}\), following Eq. (10) in [Andersson15]:

+
+
+\[\begin{split}\begin{equation} +C(\theta) = +\begin{cases} +1 - \frac{3 \theta}{2 a} + \frac{\theta^3}{2 a^3} & \textnormal{if} \; \theta \leq a \\ +0 & \textnormal{if} \; \theta > a +\end{cases} +\end{equation}\end{split}\]
+
+

\(\theta\) being computed as:

+
+
+\[\theta(\mathbf{g}, \mathbf{g'}) = \arccos(|\langle \mathbf{g}, \mathbf{g'} \rangle|)\]
+
+
+
Parameters:
+
    +

  • theta (ndarray) – Array of distances between points.

    • +

  • a (float) – Scale parameter that controls the range of the covariance function.

    • +
+
+
Returns:
+

Spherical covariance values for the input distances.

+
+
Return type:
+

ndarray

+
+
+
+ +
+ +
+
+ +
+ +
+
+ + + + + + + \ No newline at end of file diff --git a/main/api/eddymotion.model.html b/main/api/eddymotion.model.html index f917baef..f3d23b59 100644 --- a/main/api/eddymotion.model.html +++ b/main/api/eddymotion.model.html @@ -6,7 +6,7 @@ - eddymotion.model package - eddymotion 24.0.0.dev207 documentation + eddymotion.model package - eddymotion 24.0.0.dev211 documentation @@ -167,7 +167,7 @@
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@@ -546,7 +548,7 @@

Submodules +

diff --git a/main/api/eddymotion.model.pet.html b/main/api/eddymotion.model.pet.html index 12b5ebec..6cfa5e04 100644 --- a/main/api/eddymotion.model.pet.html +++ b/main/api/eddymotion.model.pet.html @@ -3,10 +3,10 @@ - + - eddymotion.model.pet module - eddymotion 24.0.0.dev207 documentation + eddymotion.model.pet module - eddymotion 24.0.0.dev211 documentation @@ -167,7 +167,7 @@
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Previous
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+
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@@ -190,14 +190,14 @@
eddymotion - 24.0.0.dev207 + 24.0.0.dev211
diff --git a/main/developers.html b/main/developers.html index 43740ef1..509959bd 100644 --- a/main/developers.html +++ b/main/developers.html @@ -6,7 +6,7 @@ - For developers - eddymotion 24.0.0.dev207 documentation + For developers - eddymotion 24.0.0.dev211 documentation @@ -167,7 +167,7 @@
-
eddymotion 24.0.0.dev207 documentation
 +
eddymotion 24.0.0.dev211 documentation
@@ -190,14 +190,14 @@
eddymotion - 24.0.0.dev207 + 24.0.0.dev211 diff --git a/main/genindex.html b/main/genindex.html index bf360f5b..35c58fde 100644 --- a/main/genindex.html +++ b/main/genindex.html @@ -4,7 +4,7 @@ - Index - eddymotion 24.0.0.dev207 documentation + Index - eddymotion 24.0.0.dev211 documentation @@ -165,7 +165,7 @@
-
eddymotion 24.0.0.dev207 documentation
 +
eddymotion 24.0.0.dev211 documentation
@@ -188,14 +188,14 @@
eddymotion - 24.0.0.dev207 + 24.0.0.dev211
+ - - + - -
  • eddymotion.model.base @@ -554,11 +567,20 @@

    E

  • module
+ +
  • eddymotion.model.dmri
    • +
  • + eddymotion.model.gpr + +
  • @@ -625,6 +647,8 @@

    E

  • EddyMotionEstimator (class in eddymotion.estimator) +
    • +
  • EddyMotionGPR (class in eddymotion.model.gpr)
  • em_affines (eddymotion.data.dmri.DWI attribute) @@ -633,6 +657,10 @@

    E

  • estimate() (eddymotion.estimator.EddyMotionEstimator static method) +
    • +
  • exponential_covariance() (in module eddymotion.model.gpr) +
    • +
  • ExponentialKriging (class in eddymotion.model.gpr)
    • @@ -708,6 +736,28 @@

    G

    +
    +

    H

    + + + +
    +
    +

    I

    @@ -738,6 +788,12 @@

    I

    @@ -746,16 +802,18 @@

    I

    L

    + - @@ -811,6 +869,8 @@

    M

  • eddymotion.model.base
  • eddymotion.model.dmri +
    • +
  • eddymotion.model.gpr
  • eddymotion.model.pet
    • @@ -835,6 +895,16 @@

    M

    +
    +

    N

    + + +
    +
    +

    P

    @@ -919,12 +989,12 @@

    S

  • (eddymotion.data.pet.PET method)
    • - - +
    @@ -941,6 +1017,8 @@

    S

    T

    + + + +
        + eddymotion.model.gpr +
        @@ -490,7 +497,7 @@

    Python Module Index

    - + diff --git a/main/running.html b/main/running.html index a8472dcc..7b7dd389 100644 --- a/main/running.html +++ b/main/running.html @@ -6,7 +6,7 @@ - Running Eddymotion - eddymotion 24.0.0.dev207 documentation + Running Eddymotion - eddymotion 24.0.0.dev211 documentation @@ -167,7 +167,7 @@
    -
    eddymotion 24.0.0.dev207 documentation
     +
    eddymotion 24.0.0.dev211 documentation
    @@ -190,14 +190,14 @@
    eddymotion - 24.0.0.dev207 + 24.0.0.dev211
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    -
    eddymotion 24.0.0.dev207 documentation
     +
    eddymotion 24.0.0.dev211 documentation
    @@ -190,14 +190,14 @@
    eddymotion - 24.0.0.dev207 + 24.0.0.dev211
    -
    +
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    eddymotion 24.0.0.dev207 documentation
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    eddymotion 24.0.0.dev211 documentation
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