Goodness of Fit and Model Comparison

Goodness of fit

It is often that we we need to know how well our model fits our data. While in linear, Gaussian regimes and under certain regularity conditions, the reduced \(\chi^2\) provides a measure of fit quality, most of the time it is unreliable and incorrect to use. For more on this, read The Do’s and Don’ts of reduced chi2.

Instead, we can almost always use the bootstrap method to estimate the quality of an MLE analysis. In 3ML, we can do this with the quite simply after a fit.

[2]:
%%capture
import matplotlib.pyplot as plt
import scipy.stats as stats
from threeML import *

Let’s go back to simulations. We will simulate a straight line.

[4]:
gen_function = Line(a=1, b=0)

x = np.linspace(0, 2, 50)

xyl_generator = XYLike.from_function(
    "sim_data", function=gen_function, x=x, yerr=0.3 * gen_function(x)
)

y = xyl_generator.y
y_err = xyl_generator.yerr

fig = xyl_generator.plot()
../_images/notebooks_gof_lrt_5_0.png

So, now we simply need to fit the data.

[5]:
fit_function = Line()

xyl = XYLike("data", x, y, y_err)

datalist = DataList(xyl)

model = Model(PointSource("xyl", 0, 0, spectral_shape=fit_function))

jl = JointLikelihood(model, datalist)

jl.fit()

fig = xyl.plot()
Best fit values:

result unit
parameter
xyl.spectrum.main.Line.a 1.08 +/- 0.08 1 / (cm2 keV s)
xyl.spectrum.main.Line.b (-7 +/- 7) x 10^-2 1 / (cm2 keV2 s)

Correlation matrix:

1.00-0.86
-0.861.00

Values of -log(likelihood) at the minimum:

-log(likelihood)
data 26.427747
total 26.427747

Values of statistical measures:

statistical measures
AIC 57.110814
BIC 60.679541
../_images/notebooks_gof_lrt_7_8.png

Now that the data are fit, we can assess the goodness of fit via simulating synthetic data sets and seeing how often these datasets have a similar likelihood. To do this, pass the JointLikelihood object to the GoodnessOfFit class.

[6]:
gof_obj = GoodnessOfFit(jl)

Now we will monte carlo some datasets. This can be computationally expensive, so we will use 3ML’s built in context manager for accessing ipython clusters. If we have a profile that is connected to a super computer, then we can simulate and fit all the datasets very quickly. Just use with parallel_computation():

[7]:
gof, data_frame, like_data_frame = gof_obj.by_mc(
    n_iterations=1000, continue_on_failure=True
)

Three things are returned, the GOF for each plugin (in our case one) as well as the total GOF, a data frame with the fitted values for each synthetic dataset, and the likelihoods for all the fits. We can see that the data have a reasonable GOF:

[8]:
gof
[8]:
OrderedDict([('total', 0.295), ('data', 0.295)])

Likelihood Ratio Tests

An essential part of MLE analysis is the likelihood ratio test (LRT) for comparing models. For nested models (those where one is a special case of the other), Wilks’ theorem posits that the LRT is \(\chi^2\) distributed, and thus the null model can be rejected with a probability read from a \(\chi^2\) table.

In a perfect world, this would always hold, but there are many regualrity conditions on Wilks’ theorem that are often violated in astromonical data. For a review, see Protassov et al and keep it close at heart whenever wanting to use the LRT.

For these reasons, in 3ML we provide a method for computing the LRT via profiling the null model via bootstrap samples. This is valid for nested models and avoids the dangers of asymmptotics and parameters defined on the extreme boundries of their distributions (spectral line normalizations, extra spectral components, etc.). This method does not avoid other problems which may arise from systmatics present in the data. As with any analysis, it is important to doubt and try and prove the result wrong as well as understanding the data/instrument.

Let’s start by simulating some data from a power law with an exponential cutoff on top of a background.

[9]:
energies = np.logspace(1, 3, 51)

low_edge = energies[:-1]
high_edge = energies[1:]

# get a blackbody source function
source_function = Cutoff_powerlaw(K=1, index=-1, xc=300, piv=100)

# power law background function
background_function = Powerlaw(K=1, index=-2.0, piv=100.0)

spectrum_generator = SpectrumLike.from_function(
    "fake",
    source_function=source_function,
    background_function=background_function,
    energy_min=low_edge,
    energy_max=high_edge,
)
[10]:
fig = spectrum_generator.view_count_spectrum()
../_images/notebooks_gof_lrt_16_0.png

We simulated a weak cutoff powerlaw. But if this was real data, we wouldn’t know that there was a cutoff. So we would fit both a power law (the null model) and a cutoff power law (the alternative model).

Let’s setup two models to fit the data via MLE in the standard 3ML way.

[11]:
powerlaw = Powerlaw(piv=100)
cutoff_powerlaw = Cutoff_powerlaw(piv=100)

ps_powerlaw = PointSource("test_pl", 0, 0, spectral_shape=powerlaw)
ps_cutoff_powerlaw = PointSource("test_cpl", 0, 0, spectral_shape=cutoff_powerlaw)

model_null = Model(ps_powerlaw)
model_alternative = Model(ps_cutoff_powerlaw)
[12]:
datalist = DataList(spectrum_generator)
[13]:
jl_null = JointLikelihood(model_null, datalist)
_ = jl_null.fit()
Best fit values:

result unit
parameter
test_pl.spectrum.main.Powerlaw.K (4.5 -0.6 +0.7) x 10^-1 1 / (cm2 keV s)
test_pl.spectrum.main.Powerlaw.index -1.38 +/- 0.09

Correlation matrix:

1.000.05
0.051.00

Values of -log(likelihood) at the minimum:

-log(likelihood)
fake 212.21517
total 212.21517

Values of statistical measures:

statistical measures
AIC 428.685660
BIC 432.254387
[14]:
jl_alternative = JointLikelihood(model_alternative, datalist)
_ = jl_alternative.fit()
Best fit values:

result unit
parameter
test_cpl.spectrum.main.Cutoff_powerlaw.K 1.26 -0.32 +0.4 1 / (cm2 keV s)
test_cpl.spectrum.main.Cutoff_powerlaw.index (-5 +/- 4) x 10^-1
test_cpl.spectrum.main.Cutoff_powerlaw.xc (1.7 -0.5 +0.7) x 10^2 keV

Correlation matrix:

1.000.75-0.90
0.751.00-0.89
-0.90-0.891.00

Values of -log(likelihood) at the minimum:

-log(likelihood)
fake 204.529337
total 204.529337

Values of statistical measures:

statistical measures
AIC 415.580413
BIC 420.794742

Ok, we now have our log(likelihoods) from each model. If we took Wilks’ theorem to heart, then we would compute:

\[\Lambda(x)=\frac{\sup\{\,\mathcal L(\theta\mid x):\theta\in\Theta_0\,\}}{\sup\{\,\mathcal L(\theta\mid x) : \theta\in\Theta\,\}}\]

or \(-2 \log(\Lambda)\) which would be \(\chi^2_{\nu}\) distributed where \(\nu\) is the number of extra parameters in the alternative model. In our case:

[15]:
# calculate the test statistic
TS = 2 * (
    jl_null.results.get_statistic_frame()["-log(likelihood)"]["total"]
    - jl_alternative.results.get_statistic_frame()["-log(likelihood)"]["total"]
)

print(f"null hyp. prob.: {stats.chi2.pdf(TS,1)}")
null hyp. prob.: 4.6734152370393636e-05

But lets check this by simulating the null distribution.

We create a LRT object by passing the null model and the alternative model (in that order).

[16]:
lrt = LikelihoodRatioTest(jl_null, jl_alternative)

Now we MC synthetic datasets again.

[17]:
lrt_results = lrt.by_mc(1000, continue_on_failure=True)

This returns three things, the null hypothesis probability, the test statistics for all the data sets, and the fitted values. We see that our null hyp. prob is:

[18]:
lrt.null_hypothesis_probability
[18]:
0.0

which is slightly different from what we obtained analytically.

We can visualize why by plotting the distributions of TS and seeing if it follows a \(\chi^2_{1}\) distribution/

[19]:
lrt.plot_TS_distribution(bins=100, ec="k", fc="white", lw=1.2)
_ = plt.legend()
../_images/notebooks_gof_lrt_31_0.png

The curve is slightly higher than we expect. Let’s rescale the curve by 1/2:

[20]:
lrt.plot_TS_distribution(scale=0.5, bins=100, ec="k", fc="white", lw=1.2)
_ = plt.legend()
../_images/notebooks_gof_lrt_33_0.png

Thus, we see that 3ML provides an automatic, and possibly efficient way to avoid the nasty problems of the LRT.

Both the GoodnessOfFit and LikelihoodRatioTest classes internally handle the generation of synthetic datasets. All current plugins have the ability to generate synthetic datasets based off their internal properties such as their background spectra and instrument responses.