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1# Frankenstein: 1D disc brightness profile reconstruction from Fourier data
2# using non-parametric Gaussian Processes
3#
4# Copyright (C) 2019-2020 R. Booth, J. Jennings, M. Tazzari
5#
6# This program is free software: you can redistribute it and/or modify
7# it under the terms of the GNU General Public License as published by
9# the Free Software Foundation, either version 3 of the License, or
10# (at your option) any later version.
11#
12# This program is distributed in the hope that it will be useful,
13# but WITHOUT ANY WARRANTY; without even the implied warranty of
14# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
15# GNU General Public License for more details.
16#
17# You should have received a copy of the GNU General Public License
18# along with this program. If not, see <https://www.gnu.org/licenses/>
19#
20"""This module contains methods for fitting a radial brightness profile to a set
21 of deprojected visibities.
22"""
23import abc
24from collections import defaultdict
25import logging
26import numpy as np
28from frank.constants import rad_to_arcsec
29from frank.filter import CriticalFilter
30from frank.hankel import DiscreteHankelTransform
31from frank.statistical_models import (
32 GaussianModel, LogNormalMAPModel, VisibilityMapping
33)
35class FrankRadialFit(metaclass=abc.ABCMeta):
36 """
37 Base class for results of frank fits.
39 Parameters
40 ----------
41 vis_map : VisibilityMapping object
42 Mapping between image and visibility plane.
43 info: dict
44 Dictionary containing useful quantities for reproducing a fit
45 (such as the hyperparameters used)
46 geometry: SourceGeometry object, optional
47 Geometry used to correct the visibilities for the source
48 inclination. If not provided, the geometry determined during the
49 fit will be used.
50 """
51 def __init__(self, vis_map, info, geometry):
52 self._vis_map = vis_map
53 self._geometry = geometry
54 self._info = info
56 def predict(self, u, v, I=None, geometry=None):
57 r"""
58 Predict the visibilities in the sky-plane
60 Parameters
61 ----------
62 u, v : array, unit = :math:`\lambda`
63 uv-points to predict the visibilities at
64 I : array, optional, unit = Jy
65 Intensity points to predict the vibilities of. If not specified,
66 the mean will be used. The intensity should be specified at the
67 collocation points, I[k] = :math:`I(r_k)`
68 geometry: SourceGeometry object, optional
69 Geometry used to correct the visibilities for the source
70 inclination. If not provided, the geometry determined during the
71 fit will be used
73 Returns
74 -------
75 V(u,v) : array, unit = Jy
76 Predicted visibilties of a source with a radial flux distribution
77 given by :math:`I` and the position angle, inclination and phase
78 centre determined by the geometry object
79 """
80 if geometry is None:
81 geometry = self._geometry
83 if I is None:
84 I = self.I
86 if geometry is not None:
87 u, v, wz = geometry.deproject(u, v, use3D=True)
88 else:
89 wz = np.zeros_like(u)
91 q = np.hypot(u, v)
92 V = self._vis_map.predict_visibilities(I, q, wz, geometry=geometry)
94 # Undo phase centering
95 if geometry is not None:
96 _, _, V = geometry.undo_correction(u, v, V)
98 return V
100 def predict_deprojected(self, q=None, I=None, geometry=None,
101 block_size=10**5, assume_optically_thick=True):
102 r"""
103 Predict the visibilities in the deprojected-plane
105 Parameters
106 ----------
107 q : array, default = self.q, unit = :math:`\lambda`
108 1D uv-points to predict the visibilities at
109 I : array, optional, unit = Jy / sr
110 Intensity points to predict the vibilities of. If not specified,
111 the mean will be used. The intensity should be specified at the
112 collocation points, I[k] = I(r_k)
113 geometry: SourceGeometry object, optional
114 Geometry used to correct the visibilities for the source
115 inclination. If not provided, the geometry determined during the
116 fit will be used
117 block_size : int, default = 10**5
118 Maximum matrix size used in the visibility calculation
119 assume_optically_thick : bool, default = True
120 Whether to correct the visibility amplitudes for the source
121 inclination
123 Returns
124 -------
125 V(q) : array, unit = Jy
126 Predicted visibilties of a source with a radial flux distribution
127 given by :math:`I`. The amplitude of the visibilities are reduced
128 according to the inclination of the source, for consistency with
129 `uvplot`
131 Notes
132 -----
133 The visibility amplitudes are still reduced due to the projection,
134 for consistency with `uvplot`
135 """
136 if geometry is None:
137 geometry = self._geometry
139 if I is None:
140 I = self.I
142 V = self._vis_map.predict_visibilities(I, q, q*0, geometry=geometry)
144 return V
146 def interpolate_brightness(self, Rpts, I=None):
147 r"""
148 Interpolate the brightness profile to the requested set of points.
150 The interpolation is done using the Fourier-Bessel series.
152 Parameters
153 ----------
154 Rpts : array, unit = arcsec
155 The points to interpolate the brightness to.
156 I : array, optional, unit = Jy / Sr
157 Intensity points to interpolate. If not specified, the MAP/mean
158 will be used. The intensity should be specified at the collocation
159 points, I[k] = I(r_k).
161 Returns
162 -------
163 I(Rpts) : array, unit = Jy / Sr
164 Brightness at the radial points.
166 Notes
167 -----
168 The resulting brightness will be consistent with higher-resolution fits
169 as long as the original fit has sufficient resolution. By sufficient
170 resolution we simply mean that the missing terms in the Fourier-Bessel
171 series are negligible, which will typically be the case if the
172 brightness was obtained from a frank fit with 100 points or more.
173 """
174 Rpts = np.array(Rpts)
176 return self._vis_map.interpolate(I, Rpts, space='Real')
179 @abc.abstractproperty
180 def MAP(self):
181 pass
183 @property
184 def I(self):
185 return self.MAP
187 @property
188 def r(self):
189 """Radius points, unit = arcsec"""
190 return self._vis_map.r
192 @property
193 def Rmax(self):
194 """Maximum radius, unit = arcsec"""
195 return self._vis_map.Rmax
196 @property
197 def q(self):
198 r"""Frequency points, unit = :math:`\lambda`"""
199 return self._vis_map.q
201 @property
202 def Qmax(self):
203 r"""Maximum frequency, unit = :math:`\lambda`"""
204 return self._vis_map.Qmax
206 @property
207 def size(self):
208 """Number of points in reconstruction"""
209 return self._vis_map.size
211 @property
212 def geometry(self):
213 """SourceGeometry object"""
214 return self._geometry
216 @property
217 def info(self):
218 """Fit quantities for reference"""
219 return self._info
222class FrankGaussianFit(FrankRadialFit):
223 """
224 Result of a frank fit with a Gaussian brightness model.
226 Parameters
227 ----------
228 DHT : DiscreteHankelTransform
229 A DHT object with N bins that defines H(p). The DHT is used to compute
230 :math:`S(p)`
231 fit : GaussianModel object
232 Result of fitting with MAP power spectrum.
233 info: dict, optional
234 Dictionary containing useful quantities for reproducing a fit
235 (such as the hyperparameters used)
236 geometry: SourceGeometry object, optional
237 Geometry used to correct the visibilities for the source
238 inclination. If not provided, the geometry determined during the
239 fit will be used.
240 """
242 def __init__(self, DHT, fit, info={}, geometry=None):
243 FrankRadialFit.__init__(self, DHT, info, geometry)
244 self._fit = fit
246 def draw(self, N):
247 """Compute N draws from the posterior"""
248 return np.random.multivariate_normal(self.mean, self.covariance, N)
250 def log_likelihood(self, I=None):
251 r"""
252 Compute one of two types of likelihood.
254 If :math:`I` is provided, this computes
256 .. math:
257 \log[P(I,V|S)].
259 Otherwise the marginalized likelihood is computed,
261 .. math:
262 \log[P(V|S)].
265 Parameters
266 ----------
267 I : array, size = N, optional, unit = Jy / sr
268 Intensity :math:`I(r)` to compute the likelihood of
270 Returns
271 -------
272 log_P : float
273 Log likelihood, :math:`\log[P(I,V|p)]` or :math:`\log[P(V|p)]`
275 Notes
276 -----
277 1. The prior probability P(S) is not included.
278 2. The likelihoods take the form:
280 .. math::
281 \log[P(I,V|p)] = \frac{1}{2} j^T I - \frac{1}{2} I^T D^{-1} I
282 - \frac{1}{2} \log[\det(2 \pi S)] + H_0
284 and
286 .. math::
287 \log[P(V|p)] = \frac{1}{2} j^T D j
288 + \frac{1}{2} \log[\det(D)/\det(S)] + H_0
290 where
292 .. math::
293 H_0 = -\frac{1}{2} V^T w V + \frac{1}{2} \sum \log(w /2 \pi)
295 is the noise likelihood.
296 """
297 return self._fit.log_likelihood(I)
300 def solve_non_negative(self):
301 """Compute the best fit solution with non-negative intensities"""
302 return self._fit.solve_non_negative()
304 @property
305 def mean(self):
306 """Posterior mean, unit = Jy / sr"""
307 return self._fit.mean
309 @property
310 def MAP(self):
311 """Posterior maximum, unit = Jy / sr"""
312 return self.mean
314 @property
315 def covariance(self):
316 """Posterior covariance, unit = (Jy / sr)**2"""
317 return self._fit.covariance
319 @property
320 def power_spectrum(self):
321 """Power spectrum coefficients"""
322 return self._fit.power_spectrum
326class FrankLogNormalFit(FrankRadialFit):
327 """
328 Result of a frank fit with a Gaussian brightness model.
330 Parameters
331 ----------
332 DHT : DiscreteHankelTransform
333 A DHT object with N bins that defines H(p). The DHT is used to compute
334 :math:`S(p)`
335 fit : LogNormalMAPModel object
336 Result of fitting with MAP power spectrum.
337 info: dict, optional
338 Dictionary containing useful quantities for reproducing a fit
339 (such as the hyperparameters used)
340 geometry: SourceGeometry object, optional
341 Geometry used to correct the visibilities for the source
342 inclination. If not provided, the geometry determined during the
343 fit will be used.
344 """
346 def __init__(self, DHT, fit, info={}, geometry=None):
347 FrankRadialFit.__init__(self, DHT, info, geometry)
348 self._fit = fit
350 def log_likelihood(self, I=None):
351 r"""
352 Compute the likelihood,
354 .. math:
355 \log[P(I,V|S)].
357 Parameters
358 ----------
359 I : array, size = N, optional
360 Intensity :math:`I(r)=exp(s0*s)` to compute the likelihood of
362 Returns
363 -------
364 log_P : float
365 Log likelihood, :math:`\log[P(I,V|p)]`
367 Notes
368 -----
369 1. The prior probability P(S) is not included.
370 2. The likelihood takes the form:
372 .. math::
373 \log[P(I,V|p)] = j^T I - \frac{1}{2} I^T D^{-1} I
374 - \frac{1}{2} \log[\det(2 \pi S)] + H_0
376 where
378 .. math::
379 H_0 = -\frac{1}{2} V^T w V + \frac{1}{2} \sum \log(w /2 \pi)
381 is the noise likelihood.
382 """
384 if I is not None:
385 return self._fit.log_likelihood(np.log(I))
386 else:
387 return self._fit.log_likelihood()
389 @property
390 def MAP(self):
391 """Posterior maximum, unit = Jy / sr"""
392 return np.exp((self._fit.MAP + self._fit.s_0) * self._fit.scale)
394 @property
395 def covariance(self):
396 """Posterior covariance, unit = log[(Jy / sr)**2]"""
397 return self._fit.covariance
399 @property
400 def power_spectrum(self):
401 """Power spectrum coefficients"""
402 return self._fit.power_spectrum
405class FourierBesselFitter(object):
406 """
407 Fourier-Bessel series model for fitting visibilities
409 Parameters
410 ----------
411 Rmax : float, unit = arcsec
412 Radius of support for the functions to transform, i.e.,
413 f(r) = 0 for R >= Rmax
414 N : int
415 Number of collocation points
416 geometry : SourceGeometry object
417 Geometry used to deproject the visibilities before fitting
418 nu : int, default = 0
419 Order of the discrete Hankel transform (DHT)
420 block_data : bool, default = True
421 Large temporary matrices are needed to set up the data. If block_data
422 is True, we avoid this, limiting the memory requirement to block_size
423 elements.
424 assume_optically_thick : bool, default = True
425 Whether to correct the visibility amplitudes by a factor of
426 1 / cos(inclination); see frank.geometry.rescale_total_flux
427 scale_height : function R --> H, optional
428 Specifies the vertical thickness of disc as a function of radius. Both
429 R and H should be in arcsec. Assumes a Gaussian vertical structure.
430 Only works with assume_optically_thick=False
431 block_size : int, default = 10**5
432 Size of the matrices if blocking is used
433 verbose : bool, default = False
434 Whether to print notification messages
435 """
437 def __init__(self, Rmax, N, geometry, nu=0, block_data=True,
438 assume_optically_thick=True, scale_height=None,
439 block_size=10 ** 5, verbose=True):
441 Rmax /= rad_to_arcsec
443 self._geometry = geometry
445 self._DHT = DiscreteHankelTransform(Rmax, N, nu)
447 if assume_optically_thick:
448 if scale_height is not None:
449 raise ValueError("Optically thick models must have zero "
450 "scale-height")
451 model = 'opt_thick'
452 elif scale_height is not None:
453 model = 'debris'
454 else:
455 model = 'opt_thin'
457 self._vis_map = VisibilityMapping(self._DHT, geometry,
458 model, scale_height=scale_height,
459 block_data=block_data, block_size=block_size,
460 check_qbounds=False, verbose=verbose)
462 self._info = {'Rmax' : self._DHT.Rmax * rad_to_arcsec,
463 'N' : self._DHT.size
464 }
466 self._verbose = verbose
468 def preprocess_visibilities(self, u, v, V, weights=1):
469 r"""Prepare the visibilities for fitting.
471 This step will be done by the automatically fit method, but it can be
472 expensive. This method is provided to enable the pre-processing to be
473 done once when conducting multiple fits with different hyperprior
474 parameters.
476 Parameters
477 ----------
478 u,v : 1D array, unit = :math:`\lambda`
479 uv-points of the visibilies
480 V : 1D array, unit = Jy
481 Visibility amplitudes at q
482 weights : 1D array, optional, unit = Jy^-2
483 Weights of the visibilities, weight = 1 / sigma^2, where sigma is
484 the standard deviation
486 Returns
487 -------
488 processed_visibilities : dict
489 Processed visibilities needed in subsequent steps of the fit
491 Notes
492 -----
493 Re-using the processed visibilities is only valid with certain parameter
494 changes. For example N, Rmax, geometry, nu, assume_optically_thick, and
495 scale_height cannot be changed. This will be checked before fits are
496 conducted.
497 """
498 return self._vis_map.map_visibilities(u, v, V, weights)
500 def _build_matrices(self, mapping):
501 r"""
502 Compute the matrices M and j from the visibility data.
504 Also compute
505 .. math:
506 `H0 = 0.5*\log[det(weights/(2*np.pi))]
507 - 0.5*np.sum(V * weights * V):math:`
508 """
509 self._vis_map.check_hash(mapping['hash'])
511 self._M = mapping['M']
512 self._j = mapping['j']
514 self._H0 = mapping['null_likelihood']
516 def fit_method(self):
517 """Name of the fit method"""
518 return type(self).__name__
520 def fit_preprocessed(self, preproc_vis):
521 r"""
522 Fit the pre-processed visibilties. The last step in the fitting
523 procedure.
525 Parameters
526 ----------
527 preproc_vis : pre-processed visibilities
528 Visibilities to fit that have been processed by
529 `self.preprocess_visibilities`.
531 Returns
532 -------
533 sol : FrankRadialFit
534 Least-squares Fourier-Bessel series fit
535 """
536 if self._verbose:
537 logging.info(' Fitting pre-processed visibilities for brightness'
538 ' profile using {}'.format(self.fit_method()))
540 self._build_matrices(preproc_vis)
542 return self._fit()
544 def fit(self, u, v, V, weights=1):
545 r"""
546 Fit the visibilties
548 Parameters
549 ----------
550 u,v : 1D array, unit = :math:`\lambda`
551 uv-points of the visibilies
552 V : 1D array, unit = Jy
553 Visibility amplitudes at q
554 weights : 1D array, optional, unit = J^-2
555 Weights of the visibilities, weight = 1 / sigma^2, where sigma is
556 the standard deviation
558 Returns
559 -------
560 sol : FrankRadialFit
561 Least-squares Fourier-Bessel series fit
562 """
563 if self._verbose:
564 logging.info(' Fitting for brightness profile using'
565 ' {}'.format(self.fit_method()))
567 self._geometry.fit(u, v, V, weights)
569 mapping = self.preprocess_visibilities(u, v, V, weights)
570 self._build_matrices(mapping)
572 return self._fit()
574 def _fit(self):
575 """Fit step. Computes the best fit given the pre-processed data"""
576 fit = GaussianModel(self._DHT, self._M, self._j,
577 noise_likelihood=self._H0)
579 self._sol = FrankGaussianFit(self._vis_map, fit, self._info,
580 geometry=self._geometry.clone())
582 return self._sol
585 @property
586 def r(self):
587 """Radius points, unit = arcsec"""
588 return self._DHT.r * rad_to_arcsec
590 @property
591 def Rmax(self):
592 """Maximum radius, unit = arcsec"""
593 return self._DHT.Rmax * rad_to_arcsec
595 @property
596 def q(self):
597 r"""Frequency points, unit = :math:`\lambda`"""
598 return self._DHT.q
600 @property
601 def Qmax(self):
602 r"""Maximum frequency, unit = :math:`\lambda`"""
603 return self._DHT.Qmax
605 @property
606 def size(self):
607 """Number of points in reconstruction"""
608 return self._DHT.size
610 @property
611 def geometry(self):
612 """Geometry object"""
613 return self._geometry
616class FrankFitter(FourierBesselFitter):
617 """
618 Fit a Gaussian process model using the Discrete Hankel Transform of
619 Baddour & Chouinard (2015).
621 The GP model is based upon Oppermann et al. (2013), which use a maximum
622 a posteriori estimate for the power spectrum as the GP prior for the
623 real-space coefficients
625 Parameters
626 ----------
627 Rmax : float, unit = arcsec
628 Radius of support for the functions to transform, i.e., f(r) = 0 for
629 R >= Rmax.
630 N : int
631 Number of collaction points
632 geometry : SourceGeometry object
633 Geometry used to deproject the visibilities before fitting
634 nu : int, default = 0
635 Order of the discrete Hankel transform, given by J_nu(r)
636 block_data : bool, default = True
637 Large temporary matrices are needed to set up the data. If block_data
638 is True, we avoid this, limiting the memory requirement to block_size
639 elements
640 block_size : int, default = 10**5
641 Size of the matrices if blocking is used
642 alpha : float >= 1, default = 1.05
643 Order parameter of the inverse gamma prior for the power spectrum
644 coefficients
645 p_0 : float >= 0, default = None, unit=Jy^2
646 Scale parameter of the inverse gamma prior for the power spectrum
647 coefficients. If not provided p_0 = 1e-15 (method="Normal") or
648 1e-35 (method="LogNormal") will be used.
649 weights_smooth : float >= 0, default = 1e-4
650 Spectral smoothness prior parameter. Zero is no smoothness prior
651 tol : float > 0, default = 1e-3
652 Tolerence for convergence of the power spectrum iteration
653 method : string, default="Normal"
654 Model used for the brightness reconstrution. This must be one of
655 "Normal" of "LogNormal".
656 I_scale : float, default = 1e5, unit= Jy/Sr
657 Brightness scale. Only used in the LogNormal model. Note the
658 LogNormal model produces I(Rmax) = I_scale.
659 max_iter: int, default = 2000
660 Maximum number of fit iterations
661 check_qbounds: bool, default = True
662 Whether to check if the first (last) collocation point is smaller
663 (larger) than the shortest (longest) deprojected baseline in the dataset
664 store_iteration_diagnostics: bool, default = False
665 Whether to store the power spectrum parameters and brightness profile
666 for each fit iteration
667 assume_optically_thick : bool, default = True
668 Whether to correct the visibility amplitudes by a factor of
669 1 / cos(inclination); see frank.geometry.rescale_total_flux
670 scale_height : function R --> H, optional
671 Specifies the vertical thickness of disc as a function of radius. Both
672 R and H should be in arcsec. Assumes a Gaussian vertical structure.
673 Only works with assume_optically_thick=False
674 verbose: bool
675 Whether to print notification messages
676 convergence_failure: string, default = 'raise'
677 Decide what to do when the frank model does not converge within max_iter.
678 Should be one of:
679 'raise' : raise an error
680 'warn' : print a warning message and continue
681 'ignore' : Ignore the error.
683 References
684 ----------
685 Baddour & Chouinard (2015)
686 DOI: https://doi.org/10.1364/JOSAA.32.000611
687 Oppermann et al. (2013)
688 DOI: https://doi.org/10.1103/PhysRevE.87.032136
689 """
691 def __init__(self, Rmax, N, geometry, nu=0, block_data=True,
692 block_size=10 ** 5, alpha=1.05, p_0=None, weights_smooth=1e-4,
693 tol=1e-3, method='Normal', I_scale=1e5, max_iter=2000, check_qbounds=True,
694 store_iteration_diagnostics=False, assume_optically_thick=True,
695 scale_height=None, verbose=True, convergence_failure='raise'):
697 if method not in {'Normal', 'LogNormal'}:
698 raise ValueError('FrankFitter supports following mehods:\n\t'
699 '{ "Normal", "LogNormal"}"')
700 self._method = method
702 super(FrankFitter, self).__init__(Rmax, N, geometry, nu, block_data,
703 assume_optically_thick, scale_height,
704 block_size, verbose)
706 # Reinstate the bounds check: FourierBesselFitter does not check bounds
707 self._vis_map.check_qbounds=check_qbounds
709 if p_0 is None:
710 if method == 'Normal':
711 p_0 = 1e-15
712 else:
713 p_0 = 1e-35
715 self._s_scale = np.log(I_scale)
717 self._filter = CriticalFilter(
718 self._DHT, alpha, p_0, weights_smooth, tol
719 )
721 self._max_iter = max_iter
723 self._store_iteration_diagnostics = store_iteration_diagnostics
725 self._info.update({'alpha' : alpha, 'wsmooth' : weights_smooth,
726 'p0' : p_0, 'method': method})
728 if convergence_failure not in {'raise', 'warn', 'ignore'}:
729 raise ValueError("convergence_failure must be one of 'raise',"
730 f"'warn', or 'ignore', nor {convergence_failure}")
731 self._convergence_failure = convergence_failure
733 def fit_method(self):
734 """Name of the fit method"""
735 return '{}: {} method'.format(type(self).__name__, self._method)
737 def _fit(self):
738 """Fit step. Computes the best fit given the pre-processed data"""
740 if self._store_iteration_diagnostics:
741 self._iteration_diagnostics = defaultdict(list)
743 # Inital guess for power spectrum
744 pI = np.ones([self.size])
746 # Do an extra iteration based on a power-law guess
747 fit = self._perform_fit(pI, guess=np.ones_like(pI), fit_method='Normal')
749 pI = np.max(self._DHT.transform(fit.MAP)**2)
750 pI *= (self.q / self.q[0])**-2
752 fit = self._perform_fit(pI, fit_method='Normal')
754 # Now that we've got a reasonable initial brightness, setup the
755 # log-normal power spectrum estimate
756 if self._method == 'LogNormal':
757 s = np.log(np.maximum(fit.MAP, 1e-3 * fit.MAP.max()))
758 s -= self._s_scale
760 pI = np.max(self._DHT.transform(s)**2)
761 pI *= (self.q / self.q[0])**-4
763 fit = self._perform_fit(pI, guess=s)
767 count = 0
768 pi_old = 0
769 while (not self._filter.check_convergence(pI, pi_old) and
770 count <= self._max_iter):
772 if self._verbose and logging.getLogger().isEnabledFor(logging.INFO):
773 print('\r {} iteration {}'.format(type(self).__name__, count),
774 end='', flush=True)
776 pi_old = pI.copy()
777 pI = self._filter.update_power_spectrum(fit)
779 fit = self._perform_fit(pI, guess=fit.MAP)
781 if self._store_iteration_diagnostics:
782 self._iteration_diagnostics['power_spectrum'].append(pI)
783 self._iteration_diagnostics['MAP'].append(fit.MAP)
785 count += 1
787 # Check / report convergence
788 if count < self._max_iter:
789 if self._verbose and logging.getLogger().isEnabledFor(logging.INFO):
790 print()
791 logging.info(' Convergence criterion met at iteration'
792 ' {}'.format(count-1))
793 else:
794 if self._verbose and logging.getLogger().isEnabledFor(logging.INFO):
795 print()
796 logging.info(' Convergence criterion not met; fit stopped at'
797 ' max_iter specified in your parameter file,'
798 ' {}'.format(self._max_iter))
800 msg = f'Convergence not met within {self._max_iter} '
801 msg += f'iterations.\nTry increasing max_iter, or '
802 msg += 'try increasing alpha since convergence can '
803 msg += 'be very slow for alpha close to 1.'
804 if self._convergence_failure == 'raise':
805 msg += '\nAlternatively set convergence_failure to'
806 msg += "'warn' or 'ignore' to continue despite the"
807 msg += 'failure.'
808 raise RuntimeError(msg)
809 elif self._convergence_failure == 'warn':
810 if logging.getLogger().isEnabledFor(logging.INFO):
811 logging.info(msg)
812 else:
813 print(msg)
814 elif self._convergence_failure == 'ignore':
815 pass
817 if self._store_iteration_diagnostics:
818 self._iteration_diagnostics['num_iterations'] = count
820 # Save the best fit
821 if self._method == "Normal":
822 self._sol = FrankGaussianFit(self._vis_map, fit, self._info,
823 geometry=self._geometry.clone())
824 else:
825 self._sol = FrankLogNormalFit(self._vis_map, fit, self._info,
826 geometry=self._geometry.clone())
828 # Compute the power spectrum covariance at the maximum
829 self._ps = pI
830 self._ps_cov = None
832 return self._sol
834 def draw_powerspectrum(self, Ndraw=1):
835 """
836 Draw N sets of power-spectrum parameters.
838 The draw is taken from the Laplace-approximated (Gaussian) posterior
839 distribution for p,
840 :math:`P(p) ~ G(p - p_MAP, p_cov)`
842 Parameters
843 ----------
844 Ndraw : int, default = 1
845 Number of draws
847 Returns
848 -------
849 p : array, size = (N, Ndraw)
850 Power spectrum draws
851 """
853 log_p = np.random.multivariate_normal(np.log(self._ps),
854 self._ps_cov, Ndraw)
856 return np.exp(log_p)
858 def _perform_fit(self, p, guess=None, fit_method=None):
859 """
860 Find the posterior mean and covariance given p
862 Parameters
863 ----------
864 p : array, size = N
865 Power spectrum parameters
866 guess : array, size = N, option
867 Initial guess for brightness used in the fit.
868 fit_method : string, optional.
869 One of {"Normal", "LogNormal"}. Brightness profile
870 method used in fit.
872 Returns
873 -------
874 sol : FrankRadialFit
875 Posterior solution object for P(I|V,p)
876 """
877 if fit_method is None:
878 fit_method = self._method
880 if fit_method == 'Normal':
881 return GaussianModel(self._DHT, self._M, self._j, p,
882 guess=guess,
883 noise_likelihood=self._H0)
884 elif fit_method == 'LogNormal':
885 return LogNormalMAPModel(self._DHT, self._M, self._j, p,
886 guess=guess, s0=self._s_scale,
887 noise_likelihood=self._H0)
888 else:
889 raise ValueError('fit_method must be one of the following:\n\t'
890 '{"Normal", "LogNormal"}')
892 def log_prior(self, p=None):
893 """
894 Compute the log prior probability, log(P(p)),
896 .. math:
897 `log[P(p)] ~ np.sum(p0/pi - alpha*np.log(p0/pi))
898 - 0.5*np.log(p) (weights_smooth*T) np.log(p)`
900 Parameters
901 ----------
902 p : array, size = N, optional
903 Power spectrum coefficients. If not provided, the MAP values are
904 used
906 Returns
907 -------
908 log[P(p)] : float
909 Log prior probability
911 Notes
912 -----
913 Computed up to a normalizing constant that depends on alpha and p0
914 """
916 if p is None:
917 p = self._ps
919 return self._filter.log_prior(p)
922 def log_likelihood(self, sol=None):
923 r"""
924 Compute the log likelihood :math:`log[P(p, V)]`,
926 .. math:
927 \log[P(p, V)] ~ \log[P(V|p)] + \log[P(p)]
930 Parameters
931 ----------
932 sol : FrankRadialFit object, optional
933 Posterior solution given a set power spectrum parameters, :math:`p`.
934 If not provided, the MAP solution will be used
936 Returns
937 -------
938 log[P(p, V)] : float
939 Log prior probability
941 Notes
942 -----
943 Computed up to a normalizing constant that depends on alpha and p0
944 """
946 if sol is None:
947 sol = self.MAP_solution
949 return self.log_prior(sol.power_spectrum) + sol.log_likelihood()
951 def log_evidence_laplace(self):
952 r"""Compute the evidence, p(V), for the best fit model.
954 Uses the Laplace approximation, I.e. we model the posterior p(p, V) as
955 a Gaussian in \tau = log(p) centered on p_MAP with the covariance
956 determined by the curvate of log P(p,V) at p_MAP:
957 P_Laplace(\tau) ~ N(\tau-\tau_MAP, \Sigma) * p(V)
958 where p(V) is the approximated evidence. Here
959 \Sigma_i,j = - d^2 \log(P) / d\tau_i d\tau_j
960 and p(V) is determined such that P_Lapace(\tau_MAP) = P(p_MAP, V).
961 """
962 Sigma_inv = self._filter.covariance_MAP(self._sol, ret_inv=True)
964 sign, logdet = np.linalg.slogdet(Sigma_inv/(2*np.pi))
965 laplace = self.log_likelihood() - 0.5*logdet
967 return laplace
969 @property
970 def MAP_solution(self):
971 """Reconstruction for the maximum a posteriori power spectrum"""
972 return self._sol
974 @property
975 def MAP_spectrum(self):
976 """Maximum a posteriori power spectrum"""
977 return self._ps
979 @property
980 def MAP_spectrum_covariance(self):
981 """Covariance matrix of the maximum a posteriori power spectrum"""
982 if self._ps_cov is None:
983 self._ps_cov = self._filter.covariance_MAP(self._sol)
985 return self._ps_cov
987 @property
988 def iteration_diagnostics(self):
989 """Dict containing power spectrum parameters and posterior mean
990 brightness profile at each fit iteration, and number of iterations"""
991 return self._iteration_diagnostics