A few functions and classes for applying the Maximum Likelihood Estimation and the Bayesian approach using the Grid Search method (i.e., uniform sampling of the parameter space).
For more info about these methods, see the notebook Inference_Notes.
The following code is saved as the Python module gridsearch so that it can be re-used in other notebooks.
CAUTION: This simple, vectorized Numpy implementation store all grid values in RAM. High resolution grids may consume a huge amount of memory!! Other solutions may greatly reduce the memory used (e.g., brute force implemented in the Scipy's optimize package: brute)
Author: B. Bovy
In [2]:
%%writefile gridsearch.py
"""
An implementation of MLE and the Bayesian approach
using the Grid-Search method.
"""
import math
import inspect
from collections import OrderedDict
import numpy as np
import pandas as pd
from scipy import stats
def chi2(mprofile, oprofile, ostd):
"""
Compute the chi-square given a measured
concentration profile (with known measurement
error) and predicted profile(s).
Parameters
----------
mprofile : 1-d or n-d array_like
the modelled concentration profile
oprofile : 1-d array_like
the measured concentration profile
ostd : 1-d array_like
the standard deviation values associated
to each profile measurements
"""
return np.sum(np.power(oprofile - mprofile, 2) /
np.power(ostd, 2),
axis=0)
def likelihood(mprofile, oprofile, ostd, log=True):
"""
Compute the (log)likelihood function
given a measured concentration profile
(with known measurement error) and predicted
profile(s).
Parameters
----------
mprofile : 1-d or n-d array_like
the modelled concentration profile
oprofile : 1-d array_like
the measured concentration profile
ostd : 1-d array_like
the standard deviation values associated
to each profile measurements
log : bool
if True, returns the log likelihood
"""
std_square = np.power(ostd, 2)
loglike = -0.5 * (
np.sum(np.power(oprofile - mprofile, 2)
/ std_square
- np.log(2 * np.pi * std_square),
axis=0)
)
if log:
return loglike
else:
return np.exp(loglike)
def ppd(likelihood, prior, log=True):
"""
Compute the (non-normalized) (log)posterior
probability distribution given the (log)likelihood
and the (log)prior probability distribution.
Parameters
----------
likelihood : float or array_like
the (log)likelihood function
prior : float or array_like
the (log)prior probability distribution
log : bool
Must be True if log-likelihood and
log-prior are given
"""
if log:
return prior + likelihood
else:
return prior * likelihood
def create_regular_grid(*ranges):
"""
Returns a regular grid for uniform sampling
in the multidimensional parameter space.
Parameters
----------
*ranges : range, range, ...
parameters ranges.
range can be either a `slice` object
or a (start, end, step) 3-tuple.
if a complex number is be given as step,
its real part will be then interpreted as
the number of points to sample within the
range.
Returns
-------
[n-d array, n-d array, ...]
an array of grid coordinates for each parameter.
all arrays can be broadcasted to the regular
grid formed by all parameters.
See Also
--------
:func:`numpy.ogrid`
"""
p_slices = [r if type(r) is slice else slice(*r)
for r in ranges]
return np.ogrid[p_slices]
def integrate_over_grid(grid_step, F, axis=None):
"""
Integrate a function over a regular grid.
Parameters
----------
grid_step : 1-d array_like
the resolution (step length) of one
(`axis`) or each (`axis=None`)
dimension of the regular grid
F : array_like
values of any function to integrate that
have been evaluated on the nodes of the
regular grid
axis : int or None
if None, integrate over the entire grid,
otherwise integrate only over the
specified axis
Returns
-------
float or n-d array
depending on `axis`, one or several
integrals
"""
if axis is None:
V = np.prod(grid_step)
else:
V = grid_step
return V * np.sum(F, axis=axis)
def normalize_ppd(ppd, grid_steps):
"""
Normalize the PPD values given on
a regular grid, so that the integral
over the grid equals 1.
"""
norm = integrate_over_grid(grid_steps, ppd)
return norm, ppd / norm
def ppd_mean(ppd, grid, grid_steps):
"""
Compute the mean of the PPD.
Parameters
----------
ppd : array_like
the sampled (and normalized) PPD
grid : array_like
grid coordinates, as returned by
:func:`create_regular_grid`
grid_steps : 1-d array_like
the resolution (step length) of each
dimension of the regular grid
Returns
-------
1-d array_like
mean values for each parameter
"""
ppd_mean = [integrate_over_grid(grid_steps, ppd * grid[dim])
for dim in range(len(grid))]
return np.array(ppd_mean)
def ppd_covmat(ppd, grid, grid_steps):
"""
Compute the covariance matrix of PPD.
"""
dimensions = range(len(grid))
ppd_mean = compute_ppd_mean(ppd, grid, grid_steps)
CM = [[integrate_over_grid(grid_steps, ppd *
grid[idim] * grid[jdim])
- ppd_mean[idim] * ppd_mean[jdim]
for jdim in dimensions]
for idim in dimensions]
return np.array(CM)
def ppd_corrmat(covmat):
"""
Compute the correlation matrix given
the covariance matrix `covmat`.
"""
CrM = [[covmat[i][j] / np.sqrt(covmat[i][i] * covmat[j][j])
for j in range(covmat.shape[1])]
for i in range(covmat.shape[0])]
return np.array(CrM)
def marginal_ppd(ppd, grid_steps, *dims):
"""
Compute the (joint) marginal PPD for one or
more parameters.
Parameters
----------
ppd : n-d array_like
the sampled (and normalized) PPD
grid_steps : 1-d array_like
the resolution (step length) of each
dimension of the regular grid
*dims : int, int, ...
parameters (grid dimensions) for which
to compute the (joint) marginal PPD
Returns
-------
1-d or n-d array
values of the (joint) marginal PPD
on the regular grid. The number of
dimensions depends on the number
of `*dims` arguments given.
"""
M = ppd.copy()
ax = 0
for d in range(len(grid_steps)):
if d in dims:
ax += 1
continue
M = integrate_over_grid(grid_steps[d], M, axis=ax)
return M
def profile_likelihood(likelihood, *dims):
"""
Compute the profile (log)likelihood for
one or more parameters
Parameters
----------
likelihood : n-d array_like
(log)likelihood
grid_steps : 1-d array_like
the resolution (step length) of each
dimension of the regular grid
*dims : int, int, ...
parameters (grid dimensions) for which
to compute the profile (log)likelihood
Returns
-------
1-d or n-d array
values of the profile (log)likelihood
on the regular grid. The number of
dimensions depends on the number
of `*dims` arguments given.
"""
Lp = likelihood.copy()
ax = 0
for d in range(likelihood.ndim):
if d in dims:
ax += 1
continue
Lp = Lp.max(axis=ax)
return Lp
def profile_likelihood_crit(profile_likelihood,
max_likelihood,
clevels=[0.674, 0.95, 0.997],
log=True):
"""
Return the critical values of the profile
likelihood that correspond to the given confidence
levels (based on the likelihood ratio test).
Useful for the calculation of confidence intervals.
Parameters
----------
profile_likelihood : n-d array_like
the profile (log)likelihood
max_likelihood : float
maximized value of the (log) likelihood
clevels : list
confidence levels
log : bool
must be True if log-likelihoods are
provided
"""
df = profile_likelihood.ndim
lambda_crit = [stats.chi2(df).ppf(cl)
for cl in clevels]
ploglike_crit = (2. * max_likelihood - lambda_crit) / 2.
if log:
return ploglike_crit
else:
return np.exp(ploglike_crit)
class CosmogenicInferenceGC():
"""
Infer a set of parameters from measured cosmogenic
profile(s) using either MLE or Bayesian inference
with the grid search sampling method.
Parameters
----------
description : string
brief description
"""
def __init__(self, description=''):
self.description = description
self.oprofile = dict()
self.parameters = OrderedDict()
self.grid = None
self.grid_size = None
self.grid_steps = None
self.mprofiles = None
self.chisq = None
self.likelihood = None
self.loglike = None
self.maxlike = None
self.mle = None
self.ppd = None
self.ppd_norm = None
self.ppd_mean = None
self.ppd_mean_f = None
self.ppd_max = None
self.ppd_max_i = None
self.ppd_max_f = None
self.ppd_covmat = None
self.ppd_corrmat = None
self.M_ppds_1d = None
def set_profile_measured(self, depth, C, std, nuclide,
**kwargs):
"""
Set the measured nuclide concentration profile.
Parameters
----------
depth : 1-d array_like
the depth values
C : 1-d array_like
the measured nuclide concentration
values
std : 1-d array_like
the standard deviation of the measured
concentrations
nuclide : 1-d array_like
allow to distinguish concatenated profiles
of multiple nuclides
**kwargs : name=value, name=value...
any other information to provide about
the profile
"""
self.oprofile['depth'] = np.array(depth)
self.oprofile['C'] = np.array(C)
self.oprofile['std'] = np.array(std)
self.oprofile['nuclide'] = np.array(nuclide)
self.oprofile.update(kwargs)
def set_profile_model(self, func):
"""
Set the mathematical model for predicting
the comsogenic concentration profiles.
Parameters
----------
func : callable
must accept depth values as the first
argument and parameter value(s) as the
other arguments for each parameter to fit,
defined in the SAME ORDER than
:attr:`CosmogenicProfileBayesGC.parameters` !
"""
self.profile_model = func
def set_parameter(self, name, srange, prior=None,
**kwargs):
"""
Set a model parameter to fit.
Parameters
----------
name : string
name of the parameter
srange : (start, stop, step)
parameter search range used to compute
the sampling regular grid. if a complex
number is given for `step`, its real part
will be the number of samples to generate
instead of a step length.
prior : callable
the prior density probability function
for the parameter (must accept a 1-d
array_like as unique argument)
**kwargs : name=value, name=value...
any other information to provide about
the parameter
"""
p = dict()
p['range'] = srange
p['prior'] = prior
p.update(kwargs)
self.parameters[name] = p
@property
def deg_freedom(self):
try:
return self.oprofile['C'].size - len(self.parameters)
except Exception:
return None
def _set_sampling_grid(self):
"""
Create the sampling regular grid.
"""
ranges = [p['range'] for p in self.parameters.values()]
self.grid = create_regular_grid(*ranges)
self.grid_sizes = [a.size for a in self.grid]
self.grid_total_size = np.prod(self.grid_sizes)
self.grid_steps = [1. * (stop - start) / step
if isinstance(step, complex)
else step
for start, stop, step in ranges]
def compute_mprofiles(self):
"""
Calculate the predicted nuclide concentration
vs. depth profiles at every node of the
sampling grid.
"""
if self.grid is None:
self._set_sampling_grid()
# array broadcasting...
depth = self.oprofile['depth'].copy()
for dim in range(len(self.grid)):
depth = np.expand_dims(depth, axis=-1)
grid = [np.expand_dims(p, axis=0) for p in self.grid]
self.mprofiles = self.profile_model(depth, *grid)
def compute_like(self, f='loglike'):
"""
Calculate the loglikelihood (`f`='loglike'),
likelihood (`f`='likelihood') or chi-square
(`f`='chisq') values at every node
of the sampling grid.
"""
oprofile = self.oprofile['C'].copy()
ostd = self.oprofile['std'].copy()
for dim in range(len(self.grid)):
oprofile = np.expand_dims(oprofile, axis=-1)
ostd = np.expand_dims(ostd, axis=-1)
if f == 'loglike':
self.likelihood = likelihood(self.mprofiles,
oprofile, ostd)
elif f == 'likelihood':
self.likelihood = likelihood(self.mprofiles,
oprofile, ostd,
log=False)
elif f == 'chisq':
self.chisq = chi2(self.mprofile,
oprofile, ostd)
def compute_from_data_model(self, data_model):
"""
Get modelled profiles, chi2_r, likelihood, prior
and ppd from the given `data_model`.
Returns a dictionary with the computed values.
"""
f_names = ['mprofile', 'chisq', 'chisq_r', 'loglike',
'prior', 'ppd']
mprofile = self.profile_model(self.oprofile['depth'],
*data_model)
chisq = chi2(mprofile, self.oprofile['C'],
self.oprofile['std'])
chisq_r = chisq / self.deg_freedom
loglike = likelihood(mprofile, self.oprofile['C'],
self.oprofile['std'])
prior_funcs = [p['prior'] for p in self.parameters.values()]
prior = np.prod([pf(dm)
for pf, dm in zip(prior_funcs, data_model)])
ppd = compute_ppd(prior, math.exp(loglike))
ppd /= self.ppd_norm
results = dict(
zip(f_names, [mprofile, chisq, chisq_r,
loglike, prior, ppd])
)
return results
def compute_mle(self, log=True,
save_mprofiles=False,
save_likelihood=False):
"""
Compute the (log)likelihood, find its
maximum, and compute 1d and 2d profile
(log)likelihoods.
"""
# compute (log)likelihood
self.compute_mprofiles()
if log:
f = 'loglike'
else:
f = 'likelihood'
self.compute_like(f=f)
# find maximum
self.maxlike = self.likelihood.max()
mle_ind = np.nonzero(self.likelihood >= self.maxlike)
self.mle = [p.flatten()[mi]
for p, mi in zip(self.grid, mle_ind)]
# profile likelihoods
self.proflike1d = [profile_likelihood(self.likelihood,
dim)
for dim in range(len(self.grid))]
self.proflike2d = [[profile_likelihood(self.likelihood,
idim, jdim)
for jdim in range(len(self.grid))]
for idim in range(len(self.grid))]
# keep or delete intermediate results
if not save_mprofiles:
self.mprofiles = None
if not save_likelihood:
self.likelihood = None
def compute_bayes(self, save_mprofiles=False,
save_likelihood=False):
"""
Compute the normalized PPD, its mean,
its covariance matrix and all the 1-d and
2-d marginal PPDs (may take a while to compute
and may consume a lot of memory, depending
on the size of the sampling grid!!).
The specified keyword arguments can be used to save
the intermediate results in the corresponding
attributes
"""
# compute the prior distribution
prior_funcs = [p['prior'] for p in self.parameters.values()]
prior = np.prod([pf(pg) for pf, pg in zip(prior_funcs, self.grid)],
axis=0)
# compute the likelihood function
self.compute_mprofiles()
self.compute_like(f='likelihood')
# compute and normalize the PPD
ppd = ppd(self.likelihood, prior)
self.ppd_norm, self.ppd = normalize_ppd(ppd, self.grid_steps)
# keep or delete intermediate results
if not save_mprofiles:
self.mprofiles = None
if not save_likelihood:
self.likelihood = None
del ppd
del grid
# compute PPD mean and mode (+ functions values)
self.ppd_mean = ppd_mean(self.ppd, self.grid,
self.grid_steps)
self.ppd_mean_f = self.compute_from_data_model(self.ppd_mean)
self.ppd_max_i = np.nonzero(self.ppd >= self.ppd.max())
self.ppd_max = [p.flatten()[mi]
for p, mi in zip(self.grid,
self.ppd_max_i)]
self.ppd_max_f = self.compute_from_data_model(self.ppd_max)
# compute PPD covavriance and correlation matrices
self.ppd_covmat = ppd_covmat(self.ppd,
self.grid,
self.grid_steps)
self.ppd_corrmat = corrmat(self.ppd_covmat)
# compute 1D marginal PPDs and find maximums
self.M_ppds_1d = [marginal_ppd(self.ppd,
self.grid_steps,
dim)
for dim in range(len(self.grid))]
M_ppds_1d_max_i = [M.argmax()
for M in self.M_ppds_1d]
self.M_ppds_1d_max = [p.flatten()[mi]
for p, mi in zip(self.grid,
M_ppds_1d_max_i)]
self.M_ppds_1d_max_f = self.compute_from_data_model(
self.M_ppds_1d_max
)
# compute 2D marginal PPDs
self.M_ppds_2d = [[marginal_ppd(self.ppd,
self.grid_steps,
idim, jdim)
for jdim in range(len(self.grid))]
for idim in range(len(self.grid))]
def setup_summary(self):
if self.grid is None:
self._set_sampling_grid()
summary = "Modelling C profile (Bayes, Grid-Search)\n\n"
summary += "DESCRIPTION:\n{desc}\n\n".format(
desc=self.description
)
summary += "MEASURED PROFILE ({N} samples):\n".format(
N=len(self.oprofile['C'])
)
summary += str(pd.DataFrame(self.oprofile))
summary += "\n\n"
summary += "PROFILE MODEL:\n{fname}\n{fdoc}\n\n".format(
fname=self.profile_model.__name__,
fdoc=inspect.getdoc(self.profile_model)
)
summary += "'UNKNOWN' PARAMETERS ({n}):\n".format(
n=len(self.parameters)
)
summary += "\n".join([
name + ":\n" +
"\n".join(["\t{0}: {1}".format(k, v)
for k, v in p.items()])
for name, p in self.parameters.items()
])
summary += "\n\ndegrees of freedom: {dof}\n\n".format(
dof=self.deg_freedom
)
summary += "GRID SEARCH:\n"
summary += "nb. of nodes per parameter: {np}\n".format(
np=self.grid_sizes
)
summary += "total nb. of nodes: {ng}\n\n".format(
ng=self.grid_total_size
)
return summary
def results_summary(self):
summary = "RESULTS:\n\n"
if self.ppd is None:
return summary + "no result yet"
summary += "parameter names in order:\n{0}\n\n".format(
self.parameters.keys()
)
summary += "PPD max:\n{0}\n\n".format(self.ppd_max)
summary += "Values at PPD max:\n"
summary += "\n".join(["{0}:\n {1}".format(k, v)
for k, v in self.ppd_max_f.items()])
summary += "\n\n"
summary += "PPD mean:\n{0}\n\n".format(self.ppd_mean)
summary += "Values at PPD mean:\n"
summary += "\n".join(["{0}:\n {1}".format(k, v)
for k, v in self.ppd_mean_f.items()])
summary += "\n\n"
summary += "1D Marginal PPD maxs:\n{0}\n\n".format(
self.M_ppds_1d_max
)
summary += "Values at 1D Marginal PPD maxs:\n"
summary += "\n".join(["{0}:\n {1}".format(k, v)
for k, v in self.M_ppds_1d_max_f.items()])
summary += "\n\n"
summary += "PPD covmat:\n{0}\n\n".format(self.ppd_covmat)
summary += "PPD corrmat:\n{0}\n\n".format(self.ppd_corrmat)
return summary
def __str__(self):
return self.setup_summary() + self.results_summary()
In [ ]: