# Licensed under a 3-clause BSD style license - see LICENSE.rst
"""Bayesian Blocks for Time Series Analysis.
Bayesian Blocks for Time Series Analysis
========================================
Dynamic programming algorithm for solving a piecewise-constant model for
various datasets. This is based on the algorithm presented in Scargle
et al 2013 [1]_. This code was ported from the astroML project [2]_.
Applications include:
- finding an optimal histogram with adaptive bin widths
- finding optimal segmentation of time series data
- detecting inflection points in the rate of event data
The primary interface to these routines is the :func:`bayesian_blocks`
function. This module provides fitness functions suitable for three types
of data:
- Irregularly-spaced event data via the :class:`Events` class
- Regularly-spaced event data via the :class:`RegularEvents` class
- Irregularly-spaced point measurements via the :class:`PointMeasures` class
For more fine-tuned control over the fitness functions used, it is possible
to define custom :class:`FitnessFunc` classes directly and use them with
the :func:`bayesian_blocks` routine.
One common application of the Bayesian Blocks algorithm is the determination
of optimal adaptive-width histogram bins. This uses the same fitness function
as for irregularly-spaced time series events. The easiest interface for
creating Bayesian Blocks histograms is the :func:`astropy.stats.histogram`
function.
References
----------
.. [1] https://ui.adsabs.harvard.edu/abs/2013ApJ...764..167S
.. [2] https://www.astroml.org/ https://github.com//astroML/astroML/
.. [3] Bellman, R.E., Dreyfus, S.E., 1962. Applied Dynamic
Programming. Princeton University Press, Princeton.
https://press.princeton.edu/books/hardcover/9780691651873/applied-dynamic-programming
.. [4] Bellman, R., Roth, R., 1969. Curve fitting by segmented
straight lines. J. Amer. Statist. Assoc. 64, 1079–1084.
https://www.tandfonline.com/doi/abs/10.1080/01621459.1969.10501038
"""
import warnings
from inspect import signature
import numpy as np
from astropy.utils.exceptions import AstropyUserWarning
# TODO: implement other fitness functions from appendix C of Scargle 2013
__all__ = ["FitnessFunc", "Events", "RegularEvents", "PointMeasures", "bayesian_blocks"]
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def bayesian_blocks(t, x=None, sigma=None, fitness="events", **kwargs):
r"""Compute optimal segmentation of data with Scargle's Bayesian Blocks.
This is a flexible implementation of the Bayesian Blocks algorithm
described in Scargle 2013 [1]_.
Parameters
----------
t : array-like
data times (one dimensional, length N)
x : array-like, optional
data values
sigma : array-like or float, optional
data errors
fitness : str or object
the fitness function to use for the model.
If a string, the following options are supported:
- 'events' : binned or unbinned event data. Arguments are ``gamma``,
which gives the slope of the prior on the number of bins, or
``ncp_prior``, which is :math:`-\ln({\tt gamma})`.
- 'regular_events' : non-overlapping events measured at multiples of a
fundamental tick rate, ``dt``, which must be specified as an
additional argument. Extra arguments are ``p0``, which gives the
false alarm probability to compute the prior, or ``gamma``, which
gives the slope of the prior on the number of bins, or ``ncp_prior``,
which is :math:`-\ln({\tt gamma})`.
- 'measures' : fitness for a measured sequence with Gaussian errors.
Extra arguments are ``p0``, which gives the false alarm probability
to compute the prior, or ``gamma``, which gives the slope of the
prior on the number of bins, or ``ncp_prior``, which is
:math:`-\ln({\tt gamma})`.
In all three cases, if more than one of ``p0``, ``gamma``, and
``ncp_prior`` is chosen, ``ncp_prior`` takes precedence over ``gamma``
which takes precedence over ``p0``.
Alternatively, the fitness parameter can be an instance of
:class:`FitnessFunc` or a subclass thereof.
**kwargs :
any additional keyword arguments will be passed to the specified
:class:`FitnessFunc` derived class.
Returns
-------
edges : ndarray
array containing the (N+1) edges defining the N bins
Examples
--------
.. testsetup::
>>> np.random.seed(12345)
Event data:
>>> t = np.random.normal(size=100)
>>> edges = bayesian_blocks(t, fitness='events', p0=0.01)
Event data with repeats:
>>> t = np.random.normal(size=100)
>>> t[80:] = t[:20]
>>> edges = bayesian_blocks(t, fitness='events', p0=0.01)
Regular event data:
>>> dt = 0.05
>>> t = dt * np.arange(1000)
>>> x = np.zeros(len(t))
>>> x[np.random.randint(0, len(t), len(t) // 10)] = 1
>>> edges = bayesian_blocks(t, x, fitness='regular_events', dt=dt)
Measured point data with errors:
>>> t = 100 * np.random.random(100)
>>> x = np.exp(-0.5 * (t - 50) ** 2)
>>> sigma = 0.1
>>> x_obs = np.random.normal(x, sigma)
>>> edges = bayesian_blocks(t, x_obs, sigma, fitness='measures')
References
----------
.. [1] Scargle, J et al. (2013)
https://ui.adsabs.harvard.edu/abs/2013ApJ...764..167S
.. [2] Bellman, R.E., Dreyfus, S.E., 1962. Applied Dynamic
Programming. Princeton University Press, Princeton.
https://press.princeton.edu/books/hardcover/9780691651873/applied-dynamic-programming
.. [3] Bellman, R., Roth, R., 1969. Curve fitting by segmented
straight lines. J. Amer. Statist. Assoc. 64, 1079–1084.
https://www.tandfonline.com/doi/abs/10.1080/01621459.1969.10501038
See Also
--------
astropy.stats.histogram : compute a histogram using bayesian blocks
"""
FITNESS_DICT = {
"events": Events,
"regular_events": RegularEvents,
"measures": PointMeasures,
}
fitness = FITNESS_DICT.get(fitness, fitness)
if type(fitness) is type and issubclass(fitness, FitnessFunc):
fitfunc = fitness(**kwargs)
elif isinstance(fitness, FitnessFunc):
fitfunc = fitness
else:
raise ValueError("fitness parameter not understood")
return fitfunc.fit(t, x, sigma)
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class FitnessFunc:
"""Base class for bayesian blocks fitness functions.
Derived classes should overload the following method:
``fitness(self, **kwargs)``:
Compute the fitness given a set of named arguments.
Arguments accepted by fitness must be among ``[T_k, N_k, a_k, b_k, c_k]``
(See [1]_ for details on the meaning of these parameters).
Additionally, other methods may be overloaded as well:
``__init__(self, **kwargs)``:
Initialize the fitness function with any parameters beyond the normal
``p0`` and ``gamma``.
``validate_input(self, t, x, sigma)``:
Enable specific checks of the input data (``t``, ``x``, ``sigma``)
to be performed prior to the fit.
``compute_ncp_prior(self, N)``: If ``ncp_prior`` is not defined explicitly,
this function is called in order to define it before fitting. This may be
calculated from ``gamma``, ``p0``, or whatever method you choose.
``p0_prior(self, N)``:
Specify the form of the prior given the false-alarm probability ``p0``
(See [1]_ for details).
For examples of implemented fitness functions, see :class:`Events`,
:class:`RegularEvents`, and :class:`PointMeasures`.
References
----------
.. [1] Scargle, J et al. (2013)
https://ui.adsabs.harvard.edu/abs/2013ApJ...764..167S
"""
def __init__(self, p0=0.05, gamma=None, ncp_prior=None):
self.p0 = p0
self.gamma = gamma
self.ncp_prior = ncp_prior
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def fitness(self, **kwargs):
raise NotImplementedError()
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def p0_prior(self, N):
"""Empirical prior, parametrized by the false alarm probability ``p0``.
See eq. 21 in Scargle (2013).
Note that there was an error in this equation in the original Scargle
paper (the "log" was missing). The following corrected form is taken
from https://arxiv.org/abs/1304.2818
"""
return 4 - np.log(73.53 * self.p0 * (N**-0.478))
# the fitness_args property will return the list of arguments accepted by
# the method fitness(). This allows more efficient computation below.
@property
def _fitness_args(self):
return signature(self.fitness).parameters.keys()
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def compute_ncp_prior(self, N):
"""
If ``ncp_prior`` is not explicitly defined, compute it from ``gamma``
or ``p0``.
"""
if self.gamma is not None:
return -np.log(self.gamma)
elif self.p0 is not None:
return self.p0_prior(N)
else:
raise ValueError(
"``ncp_prior`` cannot be computed as neither "
"``gamma`` nor ``p0`` is defined."
)
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def fit(self, t, x=None, sigma=None):
"""Fit the Bayesian Blocks model given the specified fitness function.
Parameters
----------
t : array-like
data times (one dimensional, length N)
x : array-like, optional
data values
sigma : array-like or float, optional
data errors
Returns
-------
edges : ndarray
array containing the (M+1) edges defining the M optimal bins
"""
t, x, sigma = self.validate_input(t, x, sigma)
# compute values needed for computation, below
if "a_k" in self._fitness_args:
ak_raw = np.ones_like(x) / sigma**2
if "b_k" in self._fitness_args:
bk_raw = x / sigma**2
if "c_k" in self._fitness_args:
ck_raw = x * x / sigma**2
# create length-(N + 1) array of cell edges
edges = np.concatenate([t[:1], 0.5 * (t[1:] + t[:-1]), t[-1:]])
block_length = t[-1] - edges
# arrays to store the best configuration
N = len(t)
best = np.zeros(N, dtype=float)
last = np.zeros(N, dtype=int)
# Compute ncp_prior if not defined
if self.ncp_prior is None:
ncp_prior = self.compute_ncp_prior(N)
else:
ncp_prior = self.ncp_prior
# ----------------------------------------------------------------
# Start with first data cell; add one cell at each iteration
# ----------------------------------------------------------------
for R in range(N):
# Compute fit_vec : fitness of putative last block (end at R)
kwds = {}
# T_k: width/duration of each block
if "T_k" in self._fitness_args:
kwds["T_k"] = block_length[: (R + 1)] - block_length[R + 1]
# N_k: number of elements in each block
if "N_k" in self._fitness_args:
kwds["N_k"] = np.cumsum(x[: (R + 1)][::-1])[::-1]
# a_k: eq. 31
if "a_k" in self._fitness_args:
kwds["a_k"] = 0.5 * np.cumsum(ak_raw[: (R + 1)][::-1])[::-1]
# b_k: eq. 32
if "b_k" in self._fitness_args:
kwds["b_k"] = -np.cumsum(bk_raw[: (R + 1)][::-1])[::-1]
# c_k: eq. 33
if "c_k" in self._fitness_args:
kwds["c_k"] = 0.5 * np.cumsum(ck_raw[: (R + 1)][::-1])[::-1]
# evaluate fitness function
fit_vec = self.fitness(**kwds)
A_R = fit_vec - ncp_prior
A_R[1:] += best[:R]
i_max = np.argmax(A_R)
last[R] = i_max
best[R] = A_R[i_max]
# ----------------------------------------------------------------
# Now find changepoints by iteratively peeling off the last block
# ----------------------------------------------------------------
change_points = np.zeros(N, dtype=int)
i_cp = N
ind = N
while i_cp > 0:
i_cp -= 1
change_points[i_cp] = ind
if ind == 0:
break
ind = last[ind - 1]
if i_cp == 0:
change_points[i_cp] = 0
change_points = change_points[i_cp:]
return edges[change_points]
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class Events(FitnessFunc):
r"""Bayesian blocks fitness for binned or unbinned events.
Parameters
----------
p0 : float, optional
False alarm probability, used to compute the prior on
:math:`N_{\rm blocks}` (see eq. 21 of Scargle 2013). For the Events
type data, ``p0`` does not seem to be an accurate representation of the
actual false alarm probability. If you are using this fitness function
for a triggering type condition, it is recommended that you run
statistical trials on signal-free noise to determine an appropriate
value of ``gamma`` or ``ncp_prior`` to use for a desired false alarm
rate.
gamma : float, optional
If specified, then use this gamma to compute the general prior form,
:math:`p \sim {\tt gamma}^{N_{\rm blocks}}`. If gamma is specified, p0
is ignored.
ncp_prior : float, optional
If specified, use the value of ``ncp_prior`` to compute the prior as
above, using the definition :math:`{\tt ncp\_prior} = -\ln({\tt
gamma})`.
If ``ncp_prior`` is specified, ``gamma`` and ``p0`` is ignored.
"""
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def fitness(self, N_k, T_k):
# eq. 19 from Scargle 2013
return N_k * (np.log(N_k / T_k))
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class RegularEvents(FitnessFunc):
r"""Bayesian blocks fitness for regular events.
This is for data which has a fundamental "tick" length, so that all
measured values are multiples of this tick length. In each tick, there
are either zero or one counts.
Parameters
----------
dt : float
tick rate for data
p0 : float, optional
False alarm probability, used to compute the prior on :math:`N_{\rm
blocks}` (see eq. 21 of Scargle 2013). If gamma is specified, p0 is
ignored.
ncp_prior : float, optional
If specified, use the value of ``ncp_prior`` to compute the prior as
above, using the definition :math:`{\tt ncp\_prior} = -\ln({\tt
gamma})`. If ``ncp_prior`` is specified, ``gamma`` and ``p0`` are
ignored.
"""
def __init__(self, dt, p0=0.05, gamma=None, ncp_prior=None):
self.dt = dt
super().__init__(p0, gamma, ncp_prior)
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def fitness(self, T_k, N_k):
# Eq. C23 of Scargle 2013
M_k = T_k / self.dt
N_over_M = N_k / M_k
eps = 1e-8
if np.any(N_over_M > 1 + eps):
warnings.warn(
"regular events: N/M > 1. Is the time step correct?",
AstropyUserWarning,
)
one_m_NM = 1 - N_over_M
N_over_M[N_over_M <= 0] = 1
one_m_NM[one_m_NM <= 0] = 1
return N_k * np.log(N_over_M) + (M_k - N_k) * np.log(one_m_NM)
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class PointMeasures(FitnessFunc):
r"""Bayesian blocks fitness for point measures.
Parameters
----------
p0 : float, optional
False alarm probability, used to compute the prior on :math:`N_{\rm
blocks}` (see eq. 21 of Scargle 2013). If gamma is specified, p0 is
ignored.
ncp_prior : float, optional
If specified, use the value of ``ncp_prior`` to compute the prior as
above, using the definition :math:`{\tt ncp\_prior} = -\ln({\tt
gamma})`. If ``ncp_prior`` is specified, ``gamma`` and ``p0`` are
ignored.
"""
def __init__(self, p0=0.05, gamma=None, ncp_prior=None):
super().__init__(p0, gamma, ncp_prior)
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def fitness(self, a_k, b_k):
# eq. 41 from Scargle 2013
return (b_k * b_k) / (4 * a_k)
