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from typing import Optional, Sequence
import tensorflow as tf
from ..base import Parameter, TensorType
from ..utilities import positive
from .base import Combination, Kernel
[docs]class ChangePoints(Combination):
r"""
The ChangePoints kernel defines a fixed number of change-points along a 1d
input space where different kernels govern different parts of the space.
The kernel is by multiplication and addition of the base kernels with
sigmoid functions (σ). A single change-point kernel is defined as::
K₁(x, x') * (1 - σ(x)) * (1 - σ(x')) + K₂(x, x') * σ(x) * σ(x')
where K₁ is deactivated around the change-point and K₂ is activated. The
single change-point version can be found in :cite:t:`lloyd2014`. Each sigmoid
is a logistic function defined as::
σ(x) = 1 / (1 + exp{-s(x - x₀)})
parameterized by location "x₀" and steepness "s".
The key reference is :cite:t:`lloyd2014`.
"""
def __init__(
self,
kernels: Sequence[Kernel],
locations: TensorType,
steepness: TensorType = 1.0,
name: Optional[str] = None,
):
"""
:param kernels: list of kernels defining the different regimes
:param locations: list of change-point locations in the 1d input space
:param steepness: the steepness parameter(s) of the sigmoids, this can be
common between them or decoupled
"""
if len(kernels) != len(locations) + 1:
raise ValueError(
"Number of kernels ({nk}) must be one more than the number of "
"changepoint locations ({nl})".format(nk=len(kernels), nl=len(locations))
)
if isinstance(steepness, Sequence) and len(steepness) != len(locations):
raise ValueError(
"Dimension of steepness ({ns}) does not match number of changepoint "
"locations ({nl})".format(ns=len(steepness), nl=len(locations))
)
super().__init__(kernels, name=name)
self.locations = Parameter(locations)
self.steepness = Parameter(steepness, transform=positive())
def _set_kernels(self, kernels: Sequence[Kernel]) -> None:
# it is not clear how to flatten out nested change-points
self.kernels = list(kernels)
def K(self, X: tf.Tensor, X2: Optional[tf.Tensor] = None) -> tf.Tensor:
sig_X = self._sigmoids(X) # N1 x 1 x Ncp
sig_X2 = self._sigmoids(X2) if X2 is not None else sig_X # N2 x 1 x Ncp
# `starters` are the sigmoids going from 0 -> 1, whilst `stoppers` go
# from 1 -> 0, dimensions are N1 x N2 x Ncp
starters = sig_X * tf.transpose(sig_X2, perm=(1, 0, 2))
stoppers = (1 - sig_X) * tf.transpose((1 - sig_X2), perm=(1, 0, 2))
# prepend `starters` with ones and append ones to `stoppers` since the
# first kernel has no start and the last kernel has no end
N1 = tf.shape(X)[0]
N2 = tf.shape(X2)[0] if X2 is not None else N1
ones = tf.ones((N1, N2, 1), dtype=X.dtype)
starters = tf.concat([ones, starters], axis=2)
stoppers = tf.concat([stoppers, ones], axis=2)
# now combine with the underlying kernels
kernel_stack = tf.stack([k(X, X2) for k in self.kernels], axis=2)
return tf.reduce_sum(kernel_stack * starters * stoppers, axis=2)
def K_diag(self, X: tf.Tensor) -> tf.Tensor:
N = tf.shape(X)[0]
sig_X = tf.reshape(self._sigmoids(X), (N, -1)) # N x Ncp
ones = tf.ones((N, 1), dtype=X.dtype)
starters = tf.concat([ones, sig_X * sig_X], axis=1) # N x Ncp
stoppers = tf.concat([(1 - sig_X) * (1 - sig_X), ones], axis=1)
kernel_stack = tf.stack([k(X, full_cov=False) for k in self.kernels], axis=1)
return tf.reduce_sum(kernel_stack * starters * stoppers, axis=1)
def _sigmoids(self, X: tf.Tensor) -> tf.Tensor:
locations = tf.sort(self.locations) # ensure locations are ordered
locations = tf.reshape(locations, (1, 1, -1))
steepness = tf.reshape(self.steepness, (1, 1, -1))
return tf.sigmoid(steepness * (X[:, :, None] - locations))