Analysis Results

3ML stores the results of a fit in a container we call an “Analysis Result” (AR). The structure of this object is designed to be useable in a live sense within an active analysis (python script, ipython interactive shell, jupyter notebook) as well as storable as a FITS file for saving results for later.

The structure is nearly the same between MLE and Bayesian analyses in order to make a seamless functionality between all analyses.

%%capture
import numpy as np

np.seterr(all="ignore")
from threeML import *
from threeML.analysis_results import *
import astropy.units as u

silence_logs()
from tqdm.auto import tqdm
from jupyterthemes import jtplot

%matplotlib inline
jtplot.style(context="talk", fscale=1, ticks=True, grid=False)
import matplotlib.pyplot as plt

set_threeML_style()

Let’s take a look at what we can do with an AR. First, we will simulate some data.

gen_function = Line(a=2, b=0) + Gaussian(F=30.0, mu=25.0, sigma=1)

# Generate a dataset using the line and a gaussian.
# constant 20% error

x = np.linspace(0, 50, 50)

xy = XYLike.from_function(
    "sim_data", function=gen_function, x=x, yerr=0.2 * gen_function(x)
)

fig = xy.plot()

MLE Results

First we will demonstrate how AR’s work for an MLE analysis on our synthetic data. As we will see, most of the functionality exists in the Bayesian AR’s as well.

Let’s do a simple likelihood maximization of our data and model.

fitfun = Line() + Gaussian()

fitfun.b_1.bounds = (-10, 10.0)
fitfun.a_1.bounds = (-100, 100.0)
fitfun.F_2 = 25.0
fitfun.F_2.bounds = (1e-3, 200.0)
fitfun.mu_2 = 25.0
fitfun.mu_2.bounds = (0.0, 100.0)
fitfun.sigma_2.bounds = (1e-3, 10.0)

model = Model(PointSource("fake", 0.0, 0.0, fitfun))

data = DataList(xy)

jl = JointLikelihood(model, DataList(xy))
_ = jl.fit()

Best fit values:

	result	unit
parameter
fake.spectrum.main.composite.a_1	1.83 +/- 0.11	1 / (keV s cm2)
fake.spectrum.main.composite.b_1	(5 +/- 4) x 10^-3	1 / (s cm2 keV2)
fake.spectrum.main.composite.F_2	(2.8 +/- 0.4) x 10	1 / (s cm2)
fake.spectrum.main.composite.mu_2	(2.498 +/- 0.013) x 10	keV
fake.spectrum.main.composite.sigma_2	(9.4 +/- 1.2) x 10^-1	keV

Correlation matrix:

1.00	-0.85	-0.04	0.00	-0.09
-0.85	1.00	0.00	-0.02	-0.00
-0.04	0.00	1.00	0.01	-0.10
0.00	-0.02	0.01	1.00	0.13
-0.09	-0.00	-0.10	0.13	1.00

Values of -log(likelihood) at the minimum:

	-log(likelihood)
sim_data	22.850182
total	22.850182

Values of statistical measures:

	statistical measures
AIC	57.064001
BIC	65.260479

We can get our errors as always, but the results cannot be propagated (error propagation assumes Gaussian errors, i.e., symmetric errors) In this case though errors are pretty symmetric, so we are likely in the case where the MLE is actually normally distributed.

jl.get_errors()

	result	unit
parameter
fake.spectrum.main.composite.a_1	1.83 +/- 0.11	1 / (keV s cm2)
fake.spectrum.main.composite.b_1	(5 +/- 4) x 10^-3	1 / (s cm2 keV2)
fake.spectrum.main.composite.F_2	(2.8 +/- 0.4) x 10	1 / (s cm2)
fake.spectrum.main.composite.mu_2	(2.498 +/- 0.013) x 10	keV
fake.spectrum.main.composite.sigma_2	(9.4 -1.2 +1.3) x 10^-1	keV

	value	negative_error	positive_error	error	unit
fake.spectrum.main.composite.a_1	1.828787	-0.113679	0.113257	0.113468	1 / (keV s cm2)
fake.spectrum.main.composite.b_1	0.004730	-0.003840	0.003848	0.003844	1 / (s cm2 keV2)
fake.spectrum.main.composite.F_2	28.300406	-4.010681	4.010163	4.010422	1 / (s cm2)
fake.spectrum.main.composite.mu_2	24.984325	-0.126610	0.126626	0.126618	keV
fake.spectrum.main.composite.sigma_2	0.939472	-0.118509	0.131205	0.124857	keV

We need to get the AnalysisResults object that is created after a fit is performed. The AR object is a member of the JointLikelihood object

ar = jl.results

We can display the results of the analysis. Note, when a fit is performed, the post display is actaully from the internal AR.

ar.display()

Best fit values:

	result	unit
parameter
fake.spectrum.main.composite.a_1	1.83 +/- 0.11	1 / (keV s cm2)
fake.spectrum.main.composite.b_1	(5 +/- 4) x 10^-3	1 / (s cm2 keV2)
fake.spectrum.main.composite.F_2	(2.8 +/- 0.4) x 10	1 / (s cm2)
fake.spectrum.main.composite.mu_2	(2.498 +/- 0.013) x 10	keV
fake.spectrum.main.composite.sigma_2	(9.4 +/- 1.2) x 10^-1	keV

Correlation matrix:

1.00	-0.85	-0.04	0.00	-0.09
-0.85	1.00	0.00	-0.02	-0.00
-0.04	0.00	1.00	0.01	-0.10
0.00	-0.02	0.01	1.00	0.13
-0.09	-0.00	-0.10	0.13	1.00

Values of -log(likelihood) at the minimum:

	-log(likelihood)
sim_data	22.850182
total	22.850182

Values of statistical measures:

	statistical measures
AIC	57.064001
BIC	65.260479

By default, the equal tail intervals are displayed. We can instead display highest posterior densities (equal in the MLE case)

ar.display("hpd")

Best fit values:

	result	unit
parameter
fake.spectrum.main.composite.a_1	1.83 +/- 0.11	1 / (keV s cm2)
fake.spectrum.main.composite.b_1	(5 +/- 4) x 10^-3	1 / (s cm2 keV2)
fake.spectrum.main.composite.F_2	(2.8 +/- 0.4) x 10	1 / (s cm2)
fake.spectrum.main.composite.mu_2	(2.498 +/- 0.013) x 10	keV
fake.spectrum.main.composite.sigma_2	(9.4 +/- 1.2) x 10^-1	keV

Correlation matrix:

1.00	-0.85	-0.04	0.00	-0.09
-0.85	1.00	0.00	-0.02	-0.00
-0.04	0.00	1.00	0.01	-0.10
0.00	-0.02	0.01	1.00	0.13
-0.09	-0.00	-0.10	0.13	1.00

Values of -log(likelihood) at the minimum:

	-log(likelihood)
sim_data	22.850182
total	22.850182

Values of statistical measures:

	statistical measures
AIC	57.064001
BIC	65.260479

The AR stores several properties from the analysis:

ar.analysis_type

'MLE'

ar.covariance_matrix

array([[ 1.28746233e-02, -3.69321204e-04, -1.73338055e-02,
         4.23643660e-05, -1.24992299e-03],
       [-3.69321204e-04,  1.47745831e-05,  1.69000627e-06,
        -7.82459296e-06, -6.60149889e-07],
       [-1.73338055e-02,  1.69000627e-06,  1.60827685e+01,
         3.31026944e-03, -4.93090579e-02],
       [ 4.23643660e-05, -7.82459296e-06,  3.31026944e-03,
         1.57414866e-02,  1.98986176e-03],
       [-1.24992299e-03, -6.60149889e-07, -4.93090579e-02,
         1.98986176e-03,  1.55050159e-02]])

ar.get_point_source_flux(1 * u.keV, 0.1 * u.MeV)

	flux	low bound	hi bound
fake: total	1.8349585255912096e-05 erg / (s cm2)	1.6977267476857178e-05 erg / (s cm2)	1.9752427441516853e-05 erg / (s cm2)

ar.optimized_model

Model summary:

	N
Point sources	1
Extended sources	0
Particle sources	0

Free parameters (5):

	value	min_value	max_value	unit
fake.spectrum.main.composite.a_1	1.828787	-100.0	100.0	keV-1 s-1 cm-2
fake.spectrum.main.composite.b_1	0.00473	-10.0	10.0	s-1 cm-2 keV-2
fake.spectrum.main.composite.F_2	28.300406	0.001	200.0	s-1 cm-2
fake.spectrum.main.composite.mu_2	24.984325	0.0	100.0	keV
fake.spectrum.main.composite.sigma_2	0.939472	0.001	10.0	keV

Fixed parameters (2):
(abridged. Use complete=True to see all fixed parameters)


Properties (0):

(none)


Linked parameters (0):

(none)

Independent variables:

(none)

Linked functions (0):

(none)

Saving results to disk

The beauty of the analysis result is that all of this information can be written to disk and restored at a later time. The statistical parameters, best-fit model, etc. can all be recovered.

AR’s are stored as a structured FITS file. We write the AR like this:

ar.write_to("test_mle.fits", overwrite=True)

The FITS file can be examines with any normal FITS reader.

import astropy.io.fits as fits

ar_fits = fits.open("test_mle.fits")
ar_fits.info()

Filename: test_mle.fits
No.    Name      Ver    Type      Cards   Dimensions   Format
  0  PRIMARY       1 PrimaryHDU       6   ()
  1  ANALYSIS_RESULTS    1 BinTableHDU     38   5R x 9C   [36A, D, D, D, D, 16A, 5D, D, D]

However, to easily pull the results back into the 3ML framework, we use the \({\tt load\_analysis\_results}\) function:

ar_reloaded = load_analysis_results("test_mle.fits")

ar_reloaded.get_statistic_frame()

	-log(likelihood)
sim_data	22.850182
total	22.850182

You can get a DataFrame with the saved results:

ar_reloaded.get_data_frame()

	value	negative_error	positive_error	error	unit
fake.spectrum.main.composite.a_1	1.828787	-0.111646	0.114910	0.113278	1 / (keV s cm2)
fake.spectrum.main.composite.b_1	0.004730	-0.003890	0.003772	0.003831	1 / (s cm2 keV2)
fake.spectrum.main.composite.F_2	28.300406	-3.936790	3.971710	3.954250	1 / (s cm2)
fake.spectrum.main.composite.mu_2	24.984325	-0.120819	0.129137	0.124978	keV
fake.spectrum.main.composite.sigma_2	0.939472	-0.126708	0.123221	0.124965	keV

Analysis Result Sets

When doing time-resolved analysis or analysing a several objects, we can save several AR’s is a set. This is achieved with the analysis result set. We can pass an array of AR’s to the set and even set up descriptions for the different entries.

from threeML.analysis_results import AnalysisResultsSet

analysis_set = AnalysisResultsSet([ar, ar_reloaded])

# index as time bins
analysis_set.set_bins("testing", [-1, 1], [3, 5], unit="s")

# write to disk
analysis_set.write_to("analysis_set_test.fits", overwrite=True)

analysis_set = load_analysis_results("analysis_set_test.fits")

analysis_set[0].display()

Best fit values:

	result	unit
parameter
fake.spectrum.main.composite.a_1	1.83 +/- 0.11	1 / (keV s cm2)
fake.spectrum.main.composite.b_1	(5 +/- 4) x 10^-3	1 / (s cm2 keV2)
fake.spectrum.main.composite.F_2	(2.8 +/- 0.4) x 10	1 / (s cm2)
fake.spectrum.main.composite.mu_2	(2.498 +/- 0.013) x 10	keV
fake.spectrum.main.composite.sigma_2	(9.4 +/- 1.2) x 10^-1	keV

Correlation matrix:

1.00	-0.85	-0.04	0.00	-0.09
-0.85	1.00	0.00	-0.02	-0.00
-0.04	0.00	1.00	0.01	-0.10
0.00	-0.02	0.01	1.00	0.13
-0.09	-0.00	-0.10	0.13	1.00

Values of -log(likelihood) at the minimum:

	-log(likelihood)
sim_data	22.850182
total	22.850182

Values of statistical measures:

	statistical measures
AIC	57.064001
BIC	65.260479

Error propagation

In 3ML, we propagate errors for MLE reults via sampling of the covariance matrix instead of Taylor exanding around the maximum of the likelihood and computing a jacobain. Thus, we can achieve non-linear error propagation.

You can use the results for propagating errors non-linearly for analytical functions:

p1 = ar.get_variates("fake.spectrum.main.composite.b_1")
p2 = ar.get_variates("fake.spectrum.main.composite.a_1")

print("Propagating a+b, with a and b respectively:")
print(p1)
print(p2)

print("\nThis is the result (with errors):")
res = p1 + p2
print(res)

print(res.equal_tail_interval())

Propagating a+b, with a and b respectively:
equal-tail: (5 +/- 4) x 10^-3, hpd: (5 +/- 4) x 10^-3
equal-tail: 1.83 +/- 0.11, hpd: 1.83 +/- 0.11

This is the result (with errors):
equal-tail: 1.83 +/- 0.11, hpd: 1.83 -0.10 +0.11
(1.723750109365203, 1.9423958735368796)

The propagation accounts for covariances. For example this has error of zero (of course) since there is perfect covariance.

print("\nThis is 50 * a/a:")
print(50 * p1 / p1)

This is 50 * a/a:
equal-tail: (5.0 +/- 0) x 10, hpd: (5.0 +/- 0) x 10

You can use arbitrary (np) functions

print("\nThis is arcsinh(b + 5*) / np.log10(b) (why not?)")
print(np.arcsinh(p1 + 5 * p2) / np.log10(p2))

This is arcsinh(b + 5*) / np.log10(b) (why not?)
equal-tail: (1.11 -0.08 +0.10) x 10, hpd: (1.11 -0.09 +0.08) x 10

Errors can become asymmetric. For example, the ratio of two gaussians is asymmetric notoriously:

print("\nRatio a/b:")
print(p2 / p1)

Ratio a/b:
equal-tail: (3.1 -1.6 +5) x 10^2, hpd: (3.1 -2.0 +3.5) x 10^2

You can always use it with arbitrary functions:

def my_function(x, a, b):
    return b * x**a


print("\nPropagating using a custom function:")
print(my_function(2.3, p1, p2))

Propagating using a custom function:
equal-tail: 1.83 +/- 0.11, hpd: 1.83 -0.10 +0.11

This is an example of an error propagation to get the plot of the model with its errors (which are propagated without assuming linearity on parameters)

def go(fitfun, ar, model):
    fig, ax = plt.subplots()

    # Gather the parameter variates

    arguments = {}

    for par in fitfun.parameters.values():
        if par.free:
            this_name = par.name

            this_variate = ar.get_variates(par.path)

            # Do not use more than 1000 values (would make computation too slow for nothing)

            if len(this_variate) > 1000:
                this_variate = np.random.choice(this_variate, size=1000)

            arguments[this_name] = this_variate

    # Prepare the error propagator function

    pp = ar.propagate(
        ar.optimized_model.fake.spectrum.main.shape.evaluate_at, **arguments
    )

    # You can just use it as:

    print(pp(5.0))

    # Make the plot

    energies = np.linspace(0, 50, 100)

    low_curve = np.zeros_like(energies)
    middle_curve = np.zeros_like(energies)
    hi_curve = np.zeros_like(energies)

    free_parameters = model.free_parameters

    p = tqdm(total=len(energies), desc="Propagating errors")

    with use_astromodels_memoization(False):
        for i, e in enumerate(energies):
            this_flux = pp(e)

            low_bound, hi_bound = this_flux.equal_tail_interval()

            low_curve[i], middle_curve[i], hi_curve[i] = (
                low_bound,
                this_flux.median,
                hi_bound,
            )

            p.update(1)

    ax.plot(energies, middle_curve, "--", color="black")
    ax.fill_between(energies, low_curve, hi_curve, alpha=0.5, color="blue")

go(fitfun, ar, model)

equal-tail: 1.85 -0.12 +0.11, hpd: 1.85 +/- 0.11

Bayesian Analysis Results

Analysis Results work exactly the same under Bayesian analysis.

Let’s run the analysis first.

for parameter in ar.optimized_model:
    model[parameter.path].value = parameter.value

model.fake.spectrum.main.composite.a_1.set_uninformative_prior(Uniform_prior)
model.fake.spectrum.main.composite.b_1.set_uninformative_prior(Uniform_prior)
model.fake.spectrum.main.composite.F_2.set_uninformative_prior(Log_uniform_prior)
model.fake.spectrum.main.composite.mu_2.set_uninformative_prior(Uniform_prior)
model.fake.spectrum.main.composite.sigma_2.set_uninformative_prior(Log_uniform_prior)

bs = BayesianAnalysis(model, data)
bs.set_sampler("emcee")
bs.sampler.setup(n_iterations=1000, n_burn_in=100, n_walkers=20)
samples = bs.sample()

Maximum a posteriori probability (MAP) point:

	result	unit
parameter
fake.spectrum.main.composite.a_1	1.84 +/- 0.12	1 / (keV s cm2)
fake.spectrum.main.composite.b_1	(4.8 -3.5 +4) x 10^-3	1 / (s cm2 keV2)
fake.spectrum.main.composite.F_2	(2.73 -0.8 +0.29) x 10	1 / (s cm2)
fake.spectrum.main.composite.mu_2	(2.500 -0.015 +0.017) x 10	keV
fake.spectrum.main.composite.sigma_2	(9.3 -1.5 +1.2) x 10^-1	keV

Values of -log(posterior) at the minimum:

	-log(posterior)
sim_data	-26.135147
total	-26.135147

Values of statistical measures:

	statistical measures
AIC	63.633930
BIC	71.830409
DIC	-211.914484
PDIC	-273.939715

Again, we grab the results from the BayesianAnalysis object:

ar2 = bs.results

We can write and read the results to/from a file:

ar2.write_to("test_bayes.fits", overwrite=True)

ar2_reloaded = load_analysis_results("test_bayes.fits")

The AR holds the posterior samples from the analysis. We can see the saved and live reults are the same:

np.allclose(ar2_reloaded.samples, ar2.samples)

True

NOTE: MLE AR’s store samples as well. These are the samples from the covariance matrix

We can examine the marginal distributions of the parameters:

fig = ar2.corner_plot()

We can return pandas DataFrames with equal tail or HPD results.

ar2.get_data_frame("equal tail")

	value	negative_error	positive_error	error	unit
fake.spectrum.main.composite.a_1	1.835780	-0.117094	0.121585	0.119340	1 / (keV s cm2)
fake.spectrum.main.composite.b_1	0.004788	-0.003469	0.003726	0.003597	1 / (s cm2 keV2)
fake.spectrum.main.composite.F_2	27.272560	-7.694274	2.934516	5.314395	1 / (s cm2)
fake.spectrum.main.composite.mu_2	24.998973	-0.148410	0.174541	0.161475	keV
fake.spectrum.main.composite.sigma_2	0.928535	-0.147395	0.123220	0.135307	keV

ar2.get_data_frame("hpd")

	value	negative_error	positive_error	error	unit
fake.spectrum.main.composite.a_1	1.835780	-0.127038	0.107425	0.117231	1 / (keV s cm2)
fake.spectrum.main.composite.b_1	0.004788	-0.003212	0.003942	0.003577	1 / (s cm2 keV2)
fake.spectrum.main.composite.F_2	27.272560	-6.226229	3.792182	5.009206	1 / (s cm2)
fake.spectrum.main.composite.mu_2	24.998973	-0.177014	0.131523	0.154269	keV
fake.spectrum.main.composite.sigma_2	0.928535	-0.155726	0.097920	0.126823	keV

Error propagation operates the same way. Internally, the process is the same as the MLE results, however, the samples are those of the posterior rather than the (assumed) covariance matrix.

p1 = ar2.get_variates("fake.spectrum.main.composite.b_1")
p2 = ar2.get_variates("fake.spectrum.main.composite.a_1")

print(p1)
print(p2)

res = p1 + p2

print(res)

equal-tail: (4.5 -3.1 +4) x 10^-3, hpd: (4.5 -2.9 +4) x 10^-3
equal-tail: 1.83 -0.11 +0.13, hpd: 1.83 -0.12 +0.11
equal-tail: 1.84 -0.11 +0.12, hpd: 1.84 -0.12 +0.11

To demonstrate how the two objects (MLE and Bayes) are the same, we see that our plotting function written for the MLE result works on our Bayesian results seamlessly.

go(fitfun, ar2, model)

equal-tail: 1.86 -0.11 +0.12, hpd: 1.86 +/- 0.12