# Licensed under a 3-clause BSD style license - see LICENSE.rst """ 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 2012 [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] http://adsabs.harvard.edu/abs/2012arXiv1207.5578S .. [2] http://astroML.org/ https://github.com//astroML/astroML/ """ from __future__ import (absolute_import, division, print_function, unicode_literals) import warnings import numpy as np from ..utils.compat.funcsigs import signature from ..utils.exceptions import AstropyUserWarning from ..extern.six.moves import range # TODO: implement other fitness functions from appendix B of Scargle 2012 __all__ = ['FitnessFunc', 'Events', 'RegularEvents', 'PointMeasures', 'bayesian_blocks'] 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 2012 [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 -------- 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. (2012) http://adsabs.harvard.edu/abs/2012arXiv1207.5578S 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) class FitnessFunc(object): """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. (2012) http://adsabs.harvard.edu/abs/2012arXiv1207.5578S """ def __init__(self, p0=0.05, gamma=None, ncp_prior=None): self.p0 = p0 self.gamma = gamma self.ncp_prior = ncp_prior def validate_input(self, t, x=None, sigma=None): """Validate inputs to the model. Parameters ---------- t : array_like times of observations x : array_like (optional) values observed at each time sigma : float or array_like (optional) errors in values x Returns ------- t, x, sigma : array_like, float or None validated and perhaps modified versions of inputs """ # validate array input t = np.asarray(t, dtype=float) if x is not None: x = np.asarray(x) if sigma is not None: sigma = np.asarray(sigma) # find unique values of t t = np.array(t) if t.ndim != 1: raise ValueError("t must be a one-dimensional array") unq_t, unq_ind, unq_inv = np.unique(t, return_index=True, return_inverse=True) # if x is not specified, x will be counts at each time if x is None: if sigma is not None: raise ValueError("If sigma is specified, x must be specified") else: sigma = 1 if len(unq_t) == len(t): x = np.ones_like(t) else: x = np.bincount(unq_inv) t = unq_t # if x is specified, then we need to simultaneously sort t and x else: # TODO: allow broadcasted x? x = np.asarray(x) if x.shape not in [(), (1,), (t.size,)]: raise ValueError("x does not match shape of t") x += np.zeros_like(t) if len(unq_t) != len(t): raise ValueError("Repeated values in t not supported when " "x is specified") t = unq_t x = x[unq_ind] # verify the given sigma value if sigma is None: sigma = 1 else: sigma = np.asarray(sigma) if sigma.shape not in [(), (1,), (t.size,)]: raise ValueError('sigma does not match the shape of x') return t, x, sigma def fitness(self, **kwargs): raise NotImplementedError() def p0_prior(self, N): """ Empirical prior, parametrized by the false alarm probability ``p0`` See eq. 21 in Scargle (2012) 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() def compute_ncp_prior(self, N): """ If ``ncp_prior`` is not explicitly defined, compute it from ``gamma`` or ``p0``. """ if self.ncp_prior is not None: return self.ncp_prior elif 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`` is not defined, and cannot compute " "it as neither ``gamma`` nor ``p0`` is defined.") 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) # ---------------------------------------------------------------- # 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 True: i_cp -= 1 change_points[i_cp] = ind if ind == 0: break ind = last[ind - 1] change_points = change_points[i_cp:] return edges[change_points] 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 2012). 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. """ def __init__(self, p0=0.05, gamma=None, ncp_prior=None): if p0 is not None and gamma is None and ncp_prior is None: warnings.warn('p0 does not seem to accurately represent the false ' 'positive rate for event data. It is highly ' 'recommended that you run random trials on signal-' 'free noise to calibrate ncp_prior to achieve a ' 'desired false positive rate.', AstropyUserWarning) super(Events, self).__init__(p0, gamma, ncp_prior) def fitness(self, N_k, T_k): # eq. 19 from Scargle 2012 return N_k * (np.log(N_k) - np.log(T_k)) def validate_input(self, t, x, sigma): t, x, sigma = super(Events, self).validate_input(t, x, sigma) if x is not None and np.any(x % 1 > 0): raise ValueError("x must be integer counts for fitness='events'") return t, x, sigma 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 2012). 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(RegularEvents, self).__init__(p0, gamma, ncp_prior) def validate_input(self, t, x, sigma): t, x, sigma = super(RegularEvents, self).validate_input(t, x, sigma) if not np.all((x == 0) | (x == 1)): raise ValueError("Regular events must have only 0 and 1 in x") return t, x, sigma def fitness(self, T_k, N_k): # Eq. 75 of Scargle 2012 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) 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 2012). 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(PointMeasures, self).__init__(p0, gamma, ncp_prior) def fitness(self, a_k, b_k): # eq. 41 from Scargle 2012 return (b_k * b_k) / (4 * a_k) def validate_input(self, t, x, sigma): if x is None: raise ValueError("x must be specified for point measures") return super(PointMeasures, self).validate_input(t, x, sigma)