Fitting quasi-normal modes in GW150914

We demonstrate how to use ringdown to fit damped sinusoids in data from the first LIGO detection, GW150914. We will make a number of pedagogical plots along the way, the actual fit should take under 2 minutes in a modern desktop computer.

NOTE: this example is inteneded to demonstrate the functionality of the ringdown package using GW150914 as an example; it is not meant to reproduce the result from Isi, et al. (2019): we use a more general polarization model, assume different priors, and apply slightly different data conditioning; therefore, the results should not be expected to be identical Isi et al (2019).

We begin with some standard imports and global settings.

Preliminaries

%matplotlib inline
%config InlineBackend.figure_format = 'retina'

# disable numpy multithreading to avoid conflicts
# with jax multiprocessing in numpyro
import os
os.environ["OMP_NUM_THREADS"] = "1"
import numpy as np

# import jax and set it up to use double precision
from jax import config
config.update("jax_enable_x64", False)

# import numpyro and set it up to use 4 CPU devices
import numpyro
numpyro.set_host_device_count(4)
numpyro.set_platform('cpu')

# we will use matplotlib, arviz and seaborn for some of the plotting
import matplotlib.pyplot as plt
import arviz as az
import seaborn as sns

# disable some warning shown by importing LALSuite from a notebook
import warnings
warnings.filterwarnings("ignore", "Wswiglal-redir-stdio")

# import ringdown package
import ringdown as rd

# set plotting context
sns.set_context('notebook')
sns.set_palette('colorblind')

Tip

ringdown can run on GPUs! You can activate this through numpyro.set_platform. If running on a GPU, you will likely see significant performance improvements by using single precision, i.e., config.update("jax_enable_x64", False); if doing so, ringdown will automatically attempt to scale quantities so that single precision is safe—but you should always check that this works for your data.

Data

In this example, we will load GW150914 directly from disk. To do this, we first download it from the Gravitational-Wave Open Science Center here. We grab 32s of data in each IFO, sampled at 16 kHz. Ringdown supports reading both frame files and HDF5 files from GWOSC.

Verify data integrity by checking that the MD5 sums are:

MD5(H-H1_GWOSC_16KHZ_R1-1126259447-32.hdf5)= b345645c0488287301824fe617f58dc2
MD5(L-L1_GWOSC_16KHZ_R1-1126259447-32.hdf5)= 5eb246ff21364d38471c95452f012853

!wget -nc https://www.gw-openscience.org/eventapi/html/GWTC-1-confident/GW150914/v3/H-H1_GWOSC_16KHZ_R1-1126259447-32.hdf5
!wget -nc https://www.gw-openscience.org/eventapi/html/GWTC-1-confident/GW150914/v3/L-L1_GWOSC_16KHZ_R1-1126259447-32.hdf5

!openssl md5 H-H1_GWOSC_16KHZ_R1-1126259447-32.hdf5
!openssl md5 L-L1_GWOSC_16KHZ_R1-1126259447-32.hdf5

Tip

Besides custom GWOSC files, ringdown can load any data that is readable by the pandas package; additionally, with the optional dependency GWpy, ringdown can directly read in LIGO-Virgo-KAGRA frame files (*.gwf), as well as discover and load public or proprietary strain data. See docs for rd.Fit.load_data for further info.

Fitting Kerr modes

The first step in a ringdown analysis is creating a Fit object. This object will contain the data and prior information, and will perform the necessary operations to run the model (including data conditioning, if requested).

Broadly speaking, ringdown currently allows for two types of fits based on how the frequencies (\(f\)) and damping rates (\(\gamma\)) of the damped sinusoids are parametrized:

1. Kerr fits in which the frequencies and damping rates of all modes are set as a function of black hole mass (\(M\)) and spin (\(\chi\)), assuming some mode labeling (e.g., \(\ell=|m|=2,\, n=0\));

2. Free fits in which the frequencies and damping rates are allowed to vary freely.

An intermediate type of fit can be obtained by allowing for small deviations away from Kerr through the introduction of deviation parameters \(\delta f\) and \(\delta \gamma\), but we will consider this type of beyond-Kerr fit a functional extension of the Kerr fits above.

We will initialize our Fit object by specifying which modes we would like to fit. We will first carry out a Kerr fit containing both the fundamental quadrupolar mode and an overtone.

The fit object expects to receive a list of modes to include in the fit; the most explicit way to specify them is to give tuples of (p, s, l, m, n), where p = +/- 1 indicates prograde/retrograde, s = -2 is the spin weight, l and m are the usual angular quantum numbers, and n is the tone index. The modes are elliptically polarized, so there is no need to specify both + and - m azimuthal numbers. (See Isi & Farr (2021).)

fit = rd.Fit(modes=[(1, -2, 2, 2, 0), (1, -2, 2, 2, 1)])

We will also set an analysis time, target sky location and analysis duration. We will use some reference values corresponding to an estimate of the peak of GW150914:

fit.set_target(1126259462.4083147, ra=1.95, dec=-1.27, psi=0.82, duration=0.2)

Tip

By default, the target time is assumed to correspond to arrival at geocenter. If you want to specify time of arrival at a specific detector, you can use the reference_ifo argument. For example, the above is the same as

fit.set_target(1126259462.423, ra=1.95, dec=-1.27, psi=0.82, duration=0.2, reference_ifo='H1')

Alternatively, you can also specify detector start times and antenna patterns directly by passing corresponding dictionaries to the set_target function.

The actual target times at each detector will be computed automatically for detectors for which data is available.

Data preparation

One of the main features of the Fit object is its ability to load and manipulate data. In particular, we will usually want to:

1. filter and downsample data to reduce the number of data points analyzed;

2. estimate or load the autocovariance function (ACF) of the data

3. subselect some relevant data based on the target time and duration.

We begin by loading some data. The fit offers a variety of ways for doing this (see TIP above), but in this example we will load them directly from files on disk.

fit.load_data('{i}-{ifo}_GWOSC_16KHZ_R1-1126259447-32.hdf5', ifos=['H1', 'L1'], kind='GWOSC')

As a sanity-check, we can plot of the raw strain we just loaded:

for ifo, data in fit.data.items():
    data[::4].plot(label=ifo)
plt.xlabel(r'$t_\mathrm{GPS} / \mathrm{s}$');
plt.ylabel(r'$h(t)$');
plt.legend(loc='best');

Now that we’ve got the strain loaded up, we want to condition it a bit. LIGO does not have very good high-frequency sensitivity; also, our modes do not have a lot of high-frequency content. Additionally, LIGO contains a lot of low-frequency noise and our templates do not have a lot of power at low frequencies. Here are estimated noise PSDs for our data:

for ifo, data in fit.data.items():
    plt.loglog(data.get_psd().iloc[1:], label=ifo)
plt.xlabel(r'$f / \mathrm{Hz}$');
plt.ylabel(r'$P\left( f \right) / \mathrm{Hz}^{-1}$');
plt.legend(loc='best');

Data conditioning

Because the cost of our analysis scales as \(N^2\) for a data segment of \(N\) samples, and because the noise in LIGO combined with the finite bandwidth of ringdown modes means that the data are uninformative at both very low and very high frequencies, we can bandpass the data, downsample considerably (to eliminate high-frequency content), and truncate to a very short data segment (to eliminate low frequency content). The condition_data method below accomplishes this.

When we downsample, we want to be sure to preserve a sample as close as possible to the requested target time at each detector. One issue we have to deal with is that the signal arrives in the detectors at different times (the time delay between detectors depends on the source location on the sky). The Fit object knows how to handle this for us.

fit.condition_data(ds=8, f_min=20)

Warning

The conditioning function has a large number of options to manipulate the data in several ways. You should always make sure that the defaults make sense for your data!

We can visualize the segment of conditioned analysis data as another sanity check:

for i, d in fit.analysis_data.items():
    d.plot(label=i)
    plt.axvline(fit.start_times[i], ls='--', alpha=0.5, lw=3, c='gray')
plt.xlabel(r'$t_\mathrm{GPS} / \mathrm{s}$');
plt.ylabel(r'$h(t)$');
plt.legend(loc='best');

The wave arrived first at Livingston, then at Hanford, which is why the time segments are not exactly aligned.

Autocovariance function

A key ingredient in our likelihood calculation is the autocovariance function (ACF) of the noise, which is the time-domain equivalent of the power spectral density (PSD). You can load estimates of the ACF from disk, derive it from a PSD, or estimate it directly from the data. We do the latter below, with some default options.

fit.compute_acfs(f_min=20)

We can use the ACFs to whiten the analysis data and plot it once again.

wd = fit.whiten(fit.analysis_data)
for i, d in wd.items():
    d.plot(label=i)
plt.xlabel(r'$t_\mathrm{GPS} / \mathrm{s}$');
plt.ylabel(r'whitened $h(t)$');
plt.legend(loc='best');

The signal is visible towards the beginning of the whitened data.

Run fit

We are almost ready to run the fit. Before sampling over the posterior on modes, we need to set up some priors with reasonable choices. We can see all the options available by using querying the valid_model_settings argument.

print(fit.valid_model_settings)

['a_scale_max', 'marginalized', 'm_min', 'm_max', 'chi_min', 'chi_max', 'cosi_min', 'cosi_max', 'cosi', 'df_min', 'df_max', 'dg_min', 'dg_max', 'f_min', 'f_max', 'g_min', 'g_max', 'flat_amplitude_prior', 'mode_ordering', 'single_polarization']

These settings determine the prior and type of fit. For a vanilla Kerr fit with generic mode polarizations (not parametrized in terms of a source inclination), we only need to specify an amplitude scale, and a mass range.

fit.update_model(a_scale_max=5e-20, m_min=40.0, m_max=140.0, cosi=-1)

We are finally ready to run! This should only take a couple of minutes on a modern desktop computer using CPUs.

fit.run()

WARNING:root:running with float32 precision

fit.result

arviz.InferenceData

• posterior

  <xarray.Dataset> Size: 27MB
  Dimensions:     (chain: 4, draw: 1000, mode: 2, ifo: 2, time_index: 410)
  Coordinates:
    * chain       (chain) int64 32B 0 1 2 3
    * draw        (draw) int64 8kB 0 1 2 3 4 5 6 7 ... 993 994 995 996 997 998 999
    * mode        (mode) |S10 20B b'1,-2,2,2,0' b'1,-2,2,2,1'
    * ifo         (ifo) <U2 16B 'H1' 'L1'
    * time_index  (time_index) int64 3kB 0 1 2 3 4 5 6 ... 404 405 406 407 408 409
  Data variables:
      a_scale     (chain, draw, mode) float32 32kB 3.477e-21 ... 3.297e-21
      chi         (chain, draw) float32 16kB 0.1735 0.8477 ... 0.5773 0.6874
      f           (chain, draw, mode) float32 32kB 236.9 222.7 ... 237.6 232.4
      g           (chain, draw, mode) float32 32kB 330.5 1.009e+03 ... 229.9 699.0
      m           (chain, draw) float32 16kB 54.34 78.9 59.82 ... 64.21 71.66
      a           (chain, draw, mode) float32 32kB 2.994e-21 ... 1.976e-21
      ax_unit     (chain, draw, mode) float32 32kB 0.4851 -2.144 ... 0.7748 1.316
      ay_unit     (chain, draw, mode) float32 32kB 0.0456 0.6439 ... -0.04075
      ellip       (chain, draw, mode) float32 32kB -1.0 -1.0 -1.0 ... -1.0 -1.0
      h_det_mode  (chain, draw, ifo, mode, time_index) float32 26MB 1.26e-21 .....
      phi         (chain, draw, mode) float32 32kB 0.5805 -2.326 ... 0.9118 -1.535
  Attributes:
      created_at:                 2024-10-09T19:14:10.419458+00:00
      arviz_version:              0.18.0
      inference_library:          numpyro
      inference_library_version:  0.14.0

  xarray.Dataset

  • Dimensions:
    • chain: 4
    • draw: 1000
    • mode: 2
    • ifo: 2
    • time_index: 410
  • Coordinates: (5)
    • chain
      (chain)
      int64
      0 1 2 3

      array([0, 1, 2, 3])

    • draw
      (draw)
      int64
      0 1 2 3 4 5 ... 995 996 997 998 999

      array([  0,   1,   2, ..., 997, 998, 999])

    • mode
      (mode)
      |S10
      b'1,-2,2,2,0' b'1,-2,2,2,1'

      array([b'1,-2,2,2,0', b'1,-2,2,2,1'], dtype='|S10')

    • ifo
      (ifo)
      <U2
      'H1' 'L1'

      array(['H1', 'L1'], dtype='<U2')

    • time_index
      (time_index)
      int64
      0 1 2 3 4 5 ... 405 406 407 408 409

      array([  0,   1,   2, ..., 407, 408, 409])

  • Data variables: (11)
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      (chain, draw, mode)
      float32
      3.477e-21 1.293e-20 ... 3.297e-21

      array([[[3.4773978e-21, 1.2927894e-20],
              [2.8206861e-21, 2.3516822e-21],
              [1.5243102e-21, 3.0607363e-20],
              ...,
              [3.0480432e-21, 2.2881981e-21],
              [4.0521971e-21, 1.2992526e-20],
              [3.1275073e-21, 6.9806806e-21]],
      
             [[5.4427091e-22, 1.0643996e-21],
              [2.0455926e-21, 2.5647082e-21],
              [7.3843313e-22, 1.3800598e-21],
              ...,
              [1.7903628e-21, 7.8002712e-21],
              [1.4707116e-21, 1.8249669e-20],
              [2.4170070e-20, 1.6454656e-20]],
      
             [[6.6238785e-22, 1.8149667e-21],
              [1.7212556e-20, 6.1389916e-21],
              [1.4397865e-21, 7.4518076e-21],
              ...,
              [1.1247980e-21, 5.4158083e-21],
              [4.3079797e-21, 1.8897545e-21],
              [1.7087859e-21, 1.3628704e-21]],
      
             [[2.3030974e-21, 1.2723745e-20],
              [3.7294860e-21, 1.7290401e-21],
              [2.5168877e-21, 1.3922679e-20],
              ...,
              [4.0382404e-21, 5.7713086e-21],
              [2.6591803e-20, 3.1560935e-21],
              [3.0757335e-20, 3.2965211e-21]]], dtype=float32)

    • chi
      (chain, draw)
      float32
      0.1735 0.8477 ... 0.5773 0.6874

      array([[0.17351581, 0.84766716, 0.3978106 , ..., 0.74278647, 0.7147096 ,
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             [0.8614453 , 0.76676786, 0.70323986, ..., 0.7809218 , 0.8010022 ,
              0.59431285],
             [0.5278961 , 0.8009608 , 0.71480477, ..., 0.5075927 , 0.8518078 ,
              0.53907436],
             [0.5801063 , 0.8645191 , 0.34760875, ..., 0.60181135, 0.5772611 ,
              0.6874492 ]], dtype=float32)

    • f
      (chain, draw, mode)
      float32
      236.9 222.7 254.6 ... 237.6 232.4

      array([[[236.86917, 222.74187],
              [254.5521 , 251.97954],
              [236.6825 , 226.3812 ],
              ...,
              [231.90498, 227.73607],
              [249.39153, 244.38443],
              [230.84863, 224.88812]],
      
             [[237.39664, 235.24336],
              [249.3445 , 245.30995],
              [247.145  , 241.9712 ],
              ...,
              [238.15404, 234.55331],
              [246.85246, 243.49217],
              [243.51054, 236.43723]],
      
             [[238.849  , 230.73605],
              [247.72739, 244.35443],
              [246.06277, 241.12427],
              ...,
              [241.63228, 233.0626 ],
              [247.92444, 245.49605],
              [234.41692, 226.64827]],
      
             [[229.4771 , 222.56949],
              [254.7288 , 252.47716],
              [231.33054, 220.41713],
              ...,
              [258.93173, 251.55467],
              [244.43445, 237.02507],
              [237.62968, 232.3749 ]]], dtype=float32)

    • g
      (chain, draw, mode)
      float32
      330.5 1.009e+03 ... 229.9 699.0

      array([[[ 330.46085, 1009.23517],
              [ 183.71727,  552.8711 ],
              [ 295.5515 ,  903.4446 ],
              ...,
              [ 207.42654,  628.90497],
              [ 232.61372,  706.26227],
              [ 236.9491 ,  721.5601 ]],
      
             [[ 164.7366 ,  495.09192],
              [ 214.33075,  648.9493 ],
              [ 234.21008,  711.4781 ],
              ...,
              [ 199.55396,  603.6706 ],
              [ 198.8709 ,  600.77435],
              [ 261.3902 ,  796.9515 ]],
      
             [[ 272.01202,  830.4513 ],
              [ 199.59276,  602.9569 ],
              [ 229.47804,  696.7386 ],
              ...,
              [ 279.68256,  854.13025],
              [ 176.90523,  532.1638 ],
              [ 264.49954,  807.3642 ]],
      
             [[ 249.67961,  761.50244],
              [ 175.12485,  526.14856],
              [ 297.4342 ,  909.28125],
              ...,
              [ 275.90872,  841.0582 ],
              [ 266.65872,  813.3394 ],
              [ 229.93625,  698.9609 ]]], dtype=float32)

    • m
      (chain, draw)
      float32
      54.34 78.9 59.82 ... 64.21 71.66

      array([[54.33652 , 78.903984, 59.82206 , ..., 77.09888 , 69.88179 ,
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             [86.21115 , 73.40285 , 69.818115, ..., 77.9786  , 76.88435 ,
              65.1985  ],
             [63.679035, 76.60927 , 70.833084, ..., 62.183983, 81.4627  ,
              65.33093 ],
             [68.52415 , 80.69554 , 59.751102, ..., 61.633476, 64.209274,
              71.65532 ]], dtype=float32)

    • a
      (chain, draw, mode)
      float32
      2.994e-21 3.861e-21 ... 1.976e-21

      array([[[2.9937298e-21, 3.8605424e-21],
              [1.4161864e-21, 2.1244864e-21],
              [2.0004844e-21, 2.4813991e-21],
              ...,
              [1.4508679e-21, 1.4121182e-21],
              [2.0901739e-21, 2.5408052e-21],
              [2.0666665e-21, 2.3076513e-21]],
      
             [[8.5101365e-22, 7.8831322e-22],
              [1.6242530e-21, 1.9596727e-21],
              [1.4344238e-21, 1.6872953e-21],
              ...,
              [1.6242563e-21, 1.3556039e-21],
              [1.4453322e-21, 1.9409288e-21],
              [2.0110501e-21, 2.4174299e-21]],
      
             [[2.0481765e-21, 2.6848609e-21],
              [1.7675846e-21, 2.2144443e-21],
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              [2.5929176e-21, 3.8089749e-21],
              [1.4545283e-21, 1.8309887e-21],
              [1.7470048e-21, 1.9090212e-21]],
      
             [[1.9287754e-21, 2.3085373e-21],
              [1.6940046e-21, 2.2406804e-21],
              [1.9546714e-21, 2.5214545e-21],
              ...,
              [2.0012550e-21, 2.9671835e-21],
              [2.3626213e-21, 2.4927104e-21],
              [1.8012754e-21, 1.9757526e-21]]], dtype=float32)

    • ax_unit
      (chain, draw, mode)
      float32
      0.4851 -2.144 ... 0.7748 1.316

      array([[[ 0.48514748, -2.1441562 ],
              [ 0.23285832, -0.63512695],
              [-1.9047188 ,  1.0096748 ],
              ...,
              [-0.5967601 ,  0.51618433],
              [ 0.4364355 , -0.448423  ],
              [ 1.1458021 ,  1.1227202 ]],
      
             [[-0.05973805,  0.6742787 ],
              [-0.27318636, -0.6825636 ],
              [-1.893921  ,  0.7743336 ],
              ...,
              [ 1.7690477 ,  0.2671678 ],
              [-0.0632791 ,  1.7895939 ],
              [-0.5950781 , -0.43692905]],
      
             [[ 0.46109566,  0.11992762],
              [-0.8902265 , -0.7373623 ],
              [-0.8839796 ,  1.5368804 ],
              ...,
              [ 0.51267964,  0.83566463],
              [ 0.54544955,  1.2810864 ],
              [-1.7680807 ,  0.5134899 ]],
      
             [[-2.0081127 , -0.339901  ],
              [ 1.3810637 , -0.09298101],
              [-0.6475459 ,  2.3640652 ],
              ...,
              [ 0.01932434,  0.07286919],
              [ 1.100048  , -2.6414356 ],
              [ 0.7747906 ,  1.3160509 ]]], dtype=float32)

    • ay_unit
      (chain, draw, mode)
      float32
      0.0456 0.6439 ... 0.5118 -0.04075

      array([[[ 0.04560344,  0.64386237],
              [-2.0102537 , -0.62472504],
              [ 0.6682435 ,  0.45586178],
              ...,
              [-1.0895063 , -0.29514876],
              [ 0.7078754 , -0.22546363],
              [ 1.3717451 , -1.1256288 ]],
      
             [[ 0.73998266,  1.809628  ],
              [-0.9391397 ,  0.27766412],
              [ 0.10780344,  0.74245054],
              ...,
              [ 0.46543923,  0.41493052],
              [ 0.89748746,  0.93028814],
              [-0.52060074,  0.59505767]],
      
             [[-0.6184641 , -1.3067467 ],
              [-0.05283857, -0.57828313],
              [ 2.097068  , -1.6894375 ],
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              [ 0.95613545, -1.8844889 ],
              [ 1.6083623 ,  0.51687473],
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             [[ 0.3580762 , -0.37850082],
              [ 1.0159644 , -1.0074627 ],
              [ 0.39721525,  0.04988404],
              ...,
              [-1.2495695 , -0.1323228 ],
              [-0.56009036,  1.062371  ],
              [ 0.5117768 , -0.04075171]]], dtype=float32)

    • ellip
      (chain, draw, mode)
      float32
      -1.0 -1.0 -1.0 ... -1.0 -1.0 -1.0

      array([[[-1., -1.],
              [-1., -1.],
              [-1., -1.],
              ...,
              [-1., -1.],
              [-1., -1.],
              [-1., -1.]],
      
             [[-1., -1.],
              [-1., -1.],
              [-1., -1.],
              ...,
              [-1., -1.],
              [-1., -1.],
              [-1., -1.]],
      
             [[-1., -1.],
              [-1., -1.],
              [-1., -1.],
              ...,
              [-1., -1.],
              [-1., -1.],
              [-1., -1.]],
      
             [[-1., -1.],
              [-1., -1.],
              [-1., -1.],
              ...,
              [-1., -1.],
              [-1., -1.],
              [-1., -1.]]], dtype=float32)

    • h_det_mode
      (chain, draw, ifo, mode, time_index)
      float32
      1.26e-21 3.198e-21 ... 0.0 0.0

      array([[[[[ 1.2603652e-21,  3.1980493e-21,  3.1551892e-21, ...,
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                [-3.2640494e-22, -2.3905186e-21, -2.1431436e-21, ...,
                  0.0000000e+00,  0.0000000e+00,  0.0000000e+00]],
      
               [[-1.9626714e-21, -2.8049961e-21, -2.1466300e-21, ...,
                 -0.0000000e+00, -0.0000000e+00, -0.0000000e+00],
                [ 1.6341367e-21,  2.3334793e-21,  1.6010258e-21, ...,
                  0.0000000e+00,  0.0000000e+00,  0.0000000e+00]]],
      
      
              [[[-1.0967488e-21,  4.2805829e-22,  1.4724995e-21, ...,
                  1.7535639e-39, -1.8559938e-37, -2.4248089e-37],
                [ 2.8728486e-21,  8.7262314e-22, -7.2063517e-22, ...,
                  0.0000000e+00,  0.0000000e+00,  0.0000000e+00]],
      
               [[ 3.5078913e-22, -7.7654412e-22, -1.3016001e-21, ...,
                  7.1577154e-38,  1.8221992e-37,  1.7680730e-37],
                [-1.7393094e-21, -6.9188863e-23,  9.3805361e-22, ...,
                  0.0000000e+00,  0.0000000e+00,  0.0000000e+00]]],
      ...
                 -2.1019477e-44, -2.9427268e-44, -2.1019477e-44],
                [ 3.9358823e-22, -1.4595636e-21, -1.6438434e-21, ...,
                  0.0000000e+00,  0.0000000e+00,  0.0000000e+00]],
      
               [[-1.2913647e-21, -2.2268696e-21, -1.8658331e-21, ...,
                  2.1019477e-44,  2.2420775e-44,  1.2611686e-44],
                [ 6.2687421e-22,  1.5413751e-21,  1.2648593e-21, ...,
                  0.0000000e+00,  0.0000000e+00,  0.0000000e+00]]],
      
      
              [[[-1.1251283e-22,  1.5034973e-21,  2.0943741e-21, ...,
                  3.7109186e-41,  2.9310960e-41,  9.4321400e-42],
                [ 1.9799214e-21,  4.9873942e-23, -9.4684079e-22, ...,
                  0.0000000e+00,  0.0000000e+00,  0.0000000e+00]],
      
               [[-5.7778965e-22, -1.5465989e-21, -1.6003709e-21, ...,
                 -2.8823308e-41, -1.7035585e-41,  3.1389086e-43],
                [-8.8380038e-22,  4.7740125e-22,  9.6002105e-22, ...,
                  0.0000000e+00,  0.0000000e+00,  0.0000000e+00]]]]],
            dtype=float32)

    • phi
      (chain, draw, mode)
      float32
      0.5805 -2.326 ... 0.9118 -1.535

      array([[[ 0.58045393, -2.3260062 ],
              [ 1.420855  , -1.1310031 ],
              [ 0.91910726, -1.8008652 ],
              ...,
              [ 0.62855434, -1.8194796 ],
              [ 1.1680415 , -1.513524  ],
              [ 0.7674691 , -1.798354  ]],
      
             [[ 0.79691285, -0.07896552],
              [ 1.1519281 , -1.4279199 ],
              [ 1.3026539 , -1.0794383 ],
              ...,
              [ 0.80304027, -1.5010616 ],
              [ 1.3954115 , -0.8274952 ],
              [ 0.9156429 , -1.8192612 ]],
      
             [[ 0.8252706 , -1.8209163 ],
              [ 1.20132   , -1.5001959 ],
              [ 1.3467891 , -1.3923522 ],
              ...,
              [ 1.0280203 , -1.7483218 ],
              [ 1.3985995 , -0.80545884],
              [ 0.78142536, -1.875503  ]],
      
             [[ 0.64200133, -2.1586194 ],
              [ 1.465612  , -1.0207753 ],
              [ 0.8044823 , -1.7561287 ],
              ...,
              [ 1.3980746 , -1.2992423 ],
              [ 0.694595  , -2.1663902 ],
              [ 0.9117774 , -1.5346278 ]]], dtype=float32)

  • Indexes: (5)
    • chain
      PandasIndex

      PandasIndex(Index([0, 1, 2, 3], dtype='int64', name='chain'))

    • draw
      PandasIndex

      PandasIndex(Index([  0,   1,   2,   3,   4,   5,   6,   7,   8,   9,
             ...
             990, 991, 992, 993, 994, 995, 996, 997, 998, 999],
            dtype='int64', name='draw', length=1000))

    • mode
      PandasIndex

      PandasIndex(Index([b'1,-2,2,2,0', b'1,-2,2,2,1'], dtype='object', name='mode'))

    • ifo
      PandasIndex

      PandasIndex(Index(['H1', 'L1'], dtype='object', name='ifo'))

    • time_index
      PandasIndex

      PandasIndex(Index([  0,   1,   2,   3,   4,   5,   6,   7,   8,   9,
             ...
             400, 401, 402, 403, 404, 405, 406, 407, 408, 409],
            dtype='int64', name='time_index', length=410))

  • Attributes: (4)

    created_at :

    2024-10-09T19:14:10.419458+00:00

    arviz_version :

    0.18.0

    inference_library :

    numpyro

    inference_library_version :

    0.14.0

  
  
• log_likelihood

  <xarray.Dataset> Size: 40kB
  Dimensions:  (chain: 4, draw: 1000)
  Coordinates:
    * chain    (chain) int64 32B 0 1 2 3
    * draw     (draw) int64 8kB 0 1 2 3 4 5 6 7 ... 993 994 995 996 997 998 999
  Data variables:
      logl_0   (chain, draw) float32 16kB 1.494e+03 1.499e+03 ... 1.493e+03
      logl_1   (chain, draw) float32 16kB 1.441e+03 1.44e+03 ... 1.442e+03
  Attributes:
      created_at:                 2024-10-09T19:14:17.538225+00:00
      arviz_version:              0.18.0
      inference_library:          numpyro
      inference_library_version:  0.14.0

  xarray.Dataset

  • Dimensions:
    • chain: 4
    • draw: 1000
  • Coordinates: (2)
    • chain
      (chain)
      int64
      0 1 2 3

      array([0, 1, 2, 3])

    • draw
      (draw)
      int64
      0 1 2 3 4 5 ... 995 996 997 998 999

      array([  0,   1,   2, ..., 997, 998, 999])

  • Data variables: (2)
    • logl_0
      (chain, draw)
      float32
      1.494e+03 1.499e+03 ... 1.493e+03

      array([[1493.5675, 1499.1731, 1493.4001, ..., 1497.129 , 1495.4211,
              1495.2665],
             [1499.6652, 1499.5714, 1499.6493, ..., 1496.7065, 1496.3275,
              1490.9393],
             [1497.8118, 1494.1586, 1497.876 , ..., 1497.2014, 1498.4344,
              1498.8124],
             [1494.4321, 1498.7954, 1494.1819, ..., 1495.2463, 1493.6338,
              1493.2018]], dtype=float32)

    • logl_1
      (chain, draw)
      float32
      1.441e+03 1.44e+03 ... 1.442e+03

      array([[1441.1221, 1440.2253, 1441.3783, ..., 1441.4905, 1441.1123,
              1441.6765],
             [1440.8134, 1441.0652, 1440.7178, ..., 1441.5568, 1441.1821,
              1441.5983],
             [1440.3643, 1441.1818, 1441.332 , ..., 1441.1575, 1441.0171,
              1441.3025],
             [1441.609 , 1440.1702, 1441.4915, ..., 1439.5232, 1441.4421,
              1441.8235]], dtype=float32)

  • Indexes: (2)
    • chain
      PandasIndex

      PandasIndex(Index([0, 1, 2, 3], dtype='int64', name='chain'))

    • draw
      PandasIndex

      PandasIndex(Index([  0,   1,   2,   3,   4,   5,   6,   7,   8,   9,
             ...
             990, 991, 992, 993, 994, 995, 996, 997, 998, 999],
            dtype='int64', name='draw', length=1000))

  • Attributes: (4)

    created_at :

    2024-10-09T19:14:17.538225+00:00

    arviz_version :

    0.18.0

    inference_library :

    numpyro

    inference_library_version :

    0.14.0

  
  
• sample_stats

  <xarray.Dataset> Size: 12kB
  Dimensions:    (chain: 4, draw: 1000)
  Coordinates:
    * chain      (chain) int64 32B 0 1 2 3
    * draw       (draw) int64 8kB 0 1 2 3 4 5 6 7 ... 993 994 995 996 997 998 999
  Data variables:
      diverging  (chain, draw) bool 4kB False False False ... False False False
  Attributes:
      created_at:                 2024-10-09T19:14:10.449647+00:00
      arviz_version:              0.18.0
      inference_library:          numpyro
      inference_library_version:  0.14.0

  xarray.Dataset

  • Dimensions:
    • chain: 4
    • draw: 1000
  • Coordinates: (2)
    • chain
      (chain)
      int64
      0 1 2 3

      array([0, 1, 2, 3])

    • draw
      (draw)
      int64
      0 1 2 3 4 5 ... 995 996 997 998 999

      array([  0,   1,   2, ..., 997, 998, 999])

  • Data variables: (1)
    • diverging
      (chain, draw)
      bool
      False False False ... False False

      array([[False, False, False, ..., False, False, False],
             [False, False, False, ..., False, False, False],
             [False, False, False, ..., False, False, False],
             [False, False, False, ..., False, False, False]])

  • Indexes: (2)
    • chain
      PandasIndex

      PandasIndex(Index([0, 1, 2, 3], dtype='int64', name='chain'))

    • draw
      PandasIndex

      PandasIndex(Index([  0,   1,   2,   3,   4,   5,   6,   7,   8,   9,
             ...
             990, 991, 992, 993, 994, 995, 996, 997, 998, 999],
            dtype='int64', name='draw', length=1000))

  • Attributes: (4)

    created_at :

    2024-10-09T19:14:10.449647+00:00

    arviz_version :

    0.18.0

    inference_library :

    numpyro

    inference_library_version :

    0.14.0

  
  
• observed_data

  <xarray.Dataset> Size: 13kB
  Dimensions:       (logl_0_dim_0: 0, logl_1_dim_0: 0, ifo: 2, mode: 2,
                     time_index: 410, time_index_1: 410)
  Coordinates:
    * logl_0_dim_0  (logl_0_dim_0) int64 0B
    * logl_1_dim_0  (logl_1_dim_0) int64 0B
    * ifo           (ifo) <U2 16B 'H1' 'L1'
    * mode          (mode) |S10 20B b'1,-2,2,2,0' b'1,-2,2,2,1'
    * time_index    (time_index) int64 3kB 0 1 2 3 4 5 ... 404 405 406 407 408 409
    * time_index_1  (time_index_1) int64 3kB 0 1 2 3 4 5 ... 405 406 407 408 409
  Data variables:
      logl_0        (logl_0_dim_0) float32 0B
      logl_1        (logl_1_dim_0) float32 0B
      strain        (ifo, time_index) float64 7kB 1.768e-20 ... -3.06e-20
  Attributes:
      created_at:                 2024-10-09T19:14:17.542262+00:00
      arviz_version:              0.18.0
      inference_library:          numpyro
      inference_library_version:  0.14.0

  xarray.Dataset

  • Dimensions:
    • logl_0_dim_0: 0
    • logl_1_dim_0: 0
    • ifo: 2
    • mode: 2
    • time_index: 410
    • time_index_1: 410
  • Coordinates: (6)
    • logl_0_dim_0
      (logl_0_dim_0)
      int64

      array([], dtype=int64)

    • logl_1_dim_0
      (logl_1_dim_0)
      int64

      array([], dtype=int64)

    • ifo
      (ifo)
      <U2
      'H1' 'L1'

      array(['H1', 'L1'], dtype='<U2')

    • mode
      (mode)
      |S10
      b'1,-2,2,2,0' b'1,-2,2,2,1'

      array([b'1,-2,2,2,0', b'1,-2,2,2,1'], dtype='|S10')

    • time_index
      (time_index)
      int64
      0 1 2 3 4 5 ... 405 406 407 408 409

      array([  0,   1,   2, ..., 407, 408, 409])

    • time_index_1
      (time_index_1)
      int64
      0 1 2 3 4 5 ... 405 406 407 408 409

      array([  0,   1,   2, ..., 407, 408, 409])

  • Data variables: (3)
    • logl_0
      (logl_0_dim_0)
      float32

      array([], dtype=float32)

    • logl_1
      (logl_1_dim_0)
      float32

      array([], dtype=float32)

    • strain
      (ifo, time_index)
      float64
      1.768e-20 -1.541e-20 ... -3.06e-20

      array([[ 1.76796321e-20, -1.54054700e-20,  1.83856314e-20,
              -1.75991326e-20,  1.67560019e-20, -1.89603520e-20,
               1.75309124e-20, -1.89963585e-20,  1.52483263e-20,
              -1.84297766e-20,  1.73012983e-20, -1.55967387e-20,
               1.50888913e-20, -1.64252408e-20,  1.20238979e-20,
              -1.41381311e-20,  9.37125325e-21, -1.14791431e-20,
               8.80106106e-21, -7.73420053e-21,  4.70309027e-21,
              -8.43071824e-21,  3.47149081e-21, -3.11562122e-21,
              -4.79650539e-22, -6.20086111e-22, -3.80075396e-21,
               2.30999848e-21, -5.86063126e-21,  4.15074369e-21,
              -7.67497159e-21,  8.09969183e-21, -1.15661467e-20,
               8.36396944e-21, -1.30562785e-20,  1.28105578e-20,
              -1.48227396e-20,  1.34615083e-20, -1.67260657e-20,
               1.52344252e-20, -1.70097146e-20,  1.48689093e-20,
              -1.68276867e-20,  1.89558073e-20, -1.75091350e-20,
               1.70652771e-20, -1.84044602e-20,  1.94442395e-20,
              -1.64234380e-20,  1.71399647e-20, -1.55740358e-20,
               1.71264595e-20, -1.31808794e-20,  1.45784664e-20,
              -1.18050350e-20,  1.52974014e-20, -1.00384832e-20,
               1.06373214e-20, -8.81369079e-21,  9.69845431e-21,
      ...
              -4.66016192e-22,  2.27805218e-20, -1.85416866e-20,
              -1.32716769e-20, -1.24932661e-21,  2.73000500e-20,
              -2.18083534e-20, -1.35511533e-20,  4.38414648e-22,
               3.23624849e-20, -2.63860706e-20, -1.51965805e-20,
              -1.02756396e-21,  3.76378293e-20, -2.68843605e-20,
              -1.65706092e-20, -7.59215183e-22,  4.27899095e-20,
              -2.95328023e-20, -1.68005781e-20,  7.74792524e-22,
               4.68167290e-20, -3.08458047e-20, -1.93251203e-20,
              -2.20019443e-22,  5.14817587e-20, -3.07707609e-20,
              -2.00164156e-20,  7.41604409e-22,  5.41088021e-20,
              -3.18080695e-20, -2.13626729e-20,  1.77819957e-21,
               5.69557960e-20, -3.23507740e-20, -2.51323025e-20,
               9.51361439e-22,  5.91603185e-20, -3.24274750e-20,
              -2.67000369e-20,  9.92805449e-22,  5.96055987e-20,
              -3.34311216e-20, -2.92891324e-20,  5.04086797e-22,
               6.00871271e-20, -3.33574359e-20, -3.34435150e-20,
              -9.10971583e-22,  5.99175389e-20, -3.18699463e-20,
              -3.49478787e-20, -1.54582941e-22,  5.92217267e-20,
              -3.28705903e-20, -3.76802127e-20,  8.08347159e-22,
               5.95473104e-20, -3.05980188e-20]])

  • Indexes: (6)
    • logl_0_dim_0
      PandasIndex

      PandasIndex(Index([], dtype='int64', name='logl_0_dim_0'))

    • logl_1_dim_0
      PandasIndex

      PandasIndex(Index([], dtype='int64', name='logl_1_dim_0'))

    • ifo
      PandasIndex

      PandasIndex(Index(['H1', 'L1'], dtype='object', name='ifo'))

    • mode
      PandasIndex

      PandasIndex(Index([b'1,-2,2,2,0', b'1,-2,2,2,1'], dtype='object', name='mode'))

    • time_index
      PandasIndex

      PandasIndex(Index([  0,   1,   2,   3,   4,   5,   6,   7,   8,   9,
             ...
             400, 401, 402, 403, 404, 405, 406, 407, 408, 409],
            dtype='int64', name='time_index', length=410))

    • time_index_1
      PandasIndex

      PandasIndex(Index([  0,   1,   2,   3,   4,   5,   6,   7,   8,   9,
             ...
             400, 401, 402, 403, 404, 405, 406, 407, 408, 409],
            dtype='int64', name='time_index_1', length=410))

  • Attributes: (4)

    created_at :

    2024-10-09T19:14:17.542262+00:00

    arviz_version :

    0.18.0

    inference_library :

    numpyro

    inference_library_version :

    0.14.0

  
  
• constant_data

  <xarray.Dataset> Size: 3MB
  Dimensions:          (ifo: 2, time_index: 410, time_index_1: 410, scale_dim_0: 1)
  Coordinates:
    * ifo              (ifo) <U2 16B 'H1' 'L1'
    * time_index       (time_index) int64 3kB 0 1 2 3 4 5 ... 405 406 407 408 409
    * time_index_1     (time_index_1) int64 3kB 0 1 2 3 4 ... 405 406 407 408 409
    * scale_dim_0      (scale_dim_0) int64 8B 0
  Data variables:
      cholesky_factor  (ifo, time_index, time_index_1) float64 3MB 1.702e-20 .....
      epoch            (ifo) float64 16B 1.126e+09 1.126e+09
      fc               (ifo) float64 16B -0.4509 0.2052
      fp               (ifo) float64 16B 0.5787 -0.5274
      scale            (scale_dim_0) float64 8B 1.0
      time             (ifo, time_index) float64 7kB -2.646e-05 ... 0.1997
  Attributes:
      created_at:                 2024-10-09T19:14:17.548972+00:00
      arviz_version:              0.18.0
      inference_library:          numpyro
      inference_library_version:  0.14.0

  xarray.Dataset

  • Dimensions:
    • ifo: 2
    • time_index: 410
    • time_index_1: 410
    • scale_dim_0: 1
  • Coordinates: (4)
    • ifo
      (ifo)
      <U2
      'H1' 'L1'

      array(['H1', 'L1'], dtype='<U2')

    • time_index
      (time_index)
      int64
      0 1 2 3 4 5 ... 405 406 407 408 409

      array([  0,   1,   2, ..., 407, 408, 409])

    • time_index_1
      (time_index_1)
      int64
      0 1 2 3 4 5 ... 405 406 407 408 409

      array([  0,   1,   2, ..., 407, 408, 409])

    • scale_dim_0
      (scale_dim_0)
      int64
      0

      array([0])

  • Data variables: (6)
    • cholesky_factor
      (ifo, time_index, time_index_1)
      float64
      1.702e-20 0.0 ... 5.414e-22

      array([[[ 1.70162852e-20,  0.00000000e+00,  0.00000000e+00, ...,
                0.00000000e+00,  0.00000000e+00,  0.00000000e+00],
              [-1.21921275e-20,  1.18703829e-20,  0.00000000e+00, ...,
                0.00000000e+00,  0.00000000e+00,  0.00000000e+00],
              [ 1.67041647e-20, -3.20580538e-22,  3.22832372e-21, ...,
                0.00000000e+00,  0.00000000e+00,  0.00000000e+00],
              ...,
              [ 1.47732792e-22, -1.76856341e-22,  9.11966762e-22, ...,
                4.89149199e-22,  0.00000000e+00,  0.00000000e+00],
              [-1.13814628e-23,  2.00086138e-22, -1.12946169e-21, ...,
               -6.99247883e-23,  4.88967326e-22,  0.00000000e+00],
              [-5.74181550e-23, -7.52898779e-23,  1.06830958e-21, ...,
                2.43216452e-22, -7.00434232e-23,  4.88938554e-22]],
      
             [[ 4.96780263e-20,  0.00000000e+00,  0.00000000e+00, ...,
                0.00000000e+00,  0.00000000e+00,  0.00000000e+00],
              [ 1.59178059e-20,  4.70587903e-20,  0.00000000e+00, ...,
                0.00000000e+00,  0.00000000e+00,  0.00000000e+00],
              [-8.27058689e-21,  1.96013279e-20,  4.48919996e-20, ...,
                0.00000000e+00,  0.00000000e+00,  0.00000000e+00],
              ...,
              [-1.92059527e-20,  6.95933882e-21,  1.51630981e-20, ...,
                5.42071687e-22,  0.00000000e+00,  0.00000000e+00],
              [-9.89763117e-22, -1.99401421e-20,  9.00937322e-21, ...,
               -1.09589372e-23,  5.41737935e-22,  0.00000000e+00],
              [ 1.94466981e-20, -7.62276806e-21, -1.43424629e-20, ...,
                2.29957342e-22, -1.02647805e-23,  5.41383538e-22]]])

    • epoch
      (ifo)
      float64
      1.126e+09 1.126e+09

      array([1.12625946e+09, 1.12625946e+09])

    • fc
      (ifo)
      float64
      -0.4509 0.2052

      array([-0.45094782,  0.20520961])

    • fp
      (ifo)
      float64
      0.5787 -0.5274

      array([ 0.57874241, -0.52743343])

    • scale
      (scale_dim_0)
      float64
      1.0

      array([1.])

    • time
      (ifo, time_index)
      float64
      -2.646e-05 0.0004618 ... 0.1997

      array([[-2.64644623e-05,  4.61816788e-04,  9.50098038e-04,
               1.43837929e-03,  1.92666054e-03,  2.41494179e-03,
               2.90322304e-03,  3.39150429e-03,  3.87978554e-03,
               4.36806679e-03,  4.85634804e-03,  5.34462929e-03,
               5.83291054e-03,  6.32119179e-03,  6.80947304e-03,
               7.29775429e-03,  7.78603554e-03,  8.27431679e-03,
               8.76259804e-03,  9.25087929e-03,  9.73916054e-03,
               1.02274418e-02,  1.07157230e-02,  1.12040043e-02,
               1.16922855e-02,  1.21805668e-02,  1.26688480e-02,
               1.31571293e-02,  1.36454105e-02,  1.41336918e-02,
               1.46219730e-02,  1.51102543e-02,  1.55985355e-02,
               1.60868168e-02,  1.65750980e-02,  1.70633793e-02,
               1.75516605e-02,  1.80399418e-02,  1.85282230e-02,
               1.90165043e-02,  1.95047855e-02,  1.99930668e-02,
               2.04813480e-02,  2.09696293e-02,  2.14579105e-02,
               2.19461918e-02,  2.24344730e-02,  2.29227543e-02,
               2.34110355e-02,  2.38993168e-02,  2.43875980e-02,
               2.48758793e-02,  2.53641605e-02,  2.58524418e-02,
               2.63407230e-02,  2.68290043e-02,  2.73172855e-02,
               2.78055668e-02,  2.82938480e-02,  2.87821293e-02,
      ...
               1.71386719e-01,  1.71875000e-01,  1.72363281e-01,
               1.72851562e-01,  1.73339844e-01,  1.73828125e-01,
               1.74316406e-01,  1.74804688e-01,  1.75292969e-01,
               1.75781250e-01,  1.76269531e-01,  1.76757812e-01,
               1.77246094e-01,  1.77734375e-01,  1.78222656e-01,
               1.78710938e-01,  1.79199219e-01,  1.79687500e-01,
               1.80175781e-01,  1.80664062e-01,  1.81152344e-01,
               1.81640625e-01,  1.82128906e-01,  1.82617188e-01,
               1.83105469e-01,  1.83593750e-01,  1.84082031e-01,
               1.84570312e-01,  1.85058594e-01,  1.85546875e-01,
               1.86035156e-01,  1.86523438e-01,  1.87011719e-01,
               1.87500000e-01,  1.87988281e-01,  1.88476562e-01,
               1.88964844e-01,  1.89453125e-01,  1.89941406e-01,
               1.90429688e-01,  1.90917969e-01,  1.91406250e-01,
               1.91894531e-01,  1.92382812e-01,  1.92871094e-01,
               1.93359375e-01,  1.93847656e-01,  1.94335938e-01,
               1.94824219e-01,  1.95312500e-01,  1.95800781e-01,
               1.96289062e-01,  1.96777344e-01,  1.97265625e-01,
               1.97753906e-01,  1.98242188e-01,  1.98730469e-01,
               1.99218750e-01,  1.99707031e-01]])

  • Indexes: (4)
    • ifo
      PandasIndex

      PandasIndex(Index(['H1', 'L1'], dtype='object', name='ifo'))

    • time_index
      PandasIndex

      PandasIndex(Index([  0,   1,   2,   3,   4,   5,   6,   7,   8,   9,
             ...
             400, 401, 402, 403, 404, 405, 406, 407, 408, 409],
            dtype='int64', name='time_index', length=410))

    • time_index_1
      PandasIndex

      PandasIndex(Index([  0,   1,   2,   3,   4,   5,   6,   7,   8,   9,
             ...
             400, 401, 402, 403, 404, 405, 406, 407, 408, 409],
            dtype='int64', name='time_index_1', length=410))

    • scale_dim_0
      PandasIndex

      PandasIndex(Index([0], dtype='int64', name='scale_dim_0'))

  • Attributes: (4)

    created_at :

    2024-10-09T19:14:17.548972+00:00

    arviz_version :

    0.18.0

    inference_library :

    numpyro

    inference_library_version :

    0.14.0

  
  

The result of our fit is held in fit.result. This is just an arviz.InferenceData object, with some additional ringdown-specific functionality. As such we can use all tools in ArViz easily.

az.summary(fit.result, var_names=['a', 'm', 'chi', 'f', 'g'])

	mean	sd	hdi_3%	hdi_97%	mcse_mean	mcse_sd	ess_bulk	ess_tail	r_hat
a[b'1,-2,2,2,0']	0.000	0.000	0.000	0.000	0.000	0.000	2590.0	2327.0	1.0
a[b'1,-2,2,2,1']	0.000	0.000	0.000	0.000	0.000	0.000	2751.0	2486.0	1.0
m	72.383	8.122	56.953	86.946	0.185	0.131	1855.0	1101.0	1.0
chi	0.683	0.164	0.367	0.910	0.005	0.003	1762.0	929.0	1.0
f[b'1,-2,2,2,0']	241.321	8.398	226.042	257.261	0.165	0.117	2730.0	2114.0	1.0
f[b'1,-2,2,2,1']	235.969	9.784	218.280	254.070	0.215	0.152	2333.0	1157.0	1.0
g[b'1,-2,2,2,0']	226.454	41.814	149.710	304.645	0.996	0.753	1863.0	1026.0	1.0
g[b'1,-2,2,2,1']	687.530	130.421	448.097	931.308	3.093	2.339	1866.0	1026.0	1.0

We can visualize the result of the fit using a trace plot. This allows us to visualize the chains and diagnose any potential divergences.

az.plot_trace(fit.result, var_names=['m', 'chi', 'a']);

We can access the posterior samples directly in fit.result.posterior. We can also obtain a Pandas DataFrame with all chains stacked together. This can be really convenient for plotting and other manipulations.

# draw 500 samples from the posterior
fit.result.get_parameter_dataframe(500, latex=True)

	$M / M_\odot$	$\chi$	$f_{220} / \mathrm{Hz}$	$f_{221} / \mathrm{Hz}$	$\gamma_{220} / \mathrm{Hz}$	$\gamma_{221} / \mathrm{Hz}$	$A_{220}$	$A_{221}$	$\phi_{220}$	$\phi_{221}$	$\epsilon_{220}$	$\epsilon_{221}$
1975	63.744255	0.540083	240.402634	232.453522	271.023010	827.261841	2.028834e-21	1.978116e-21	0.798366	-1.844549	-1.0	-1.0
3331	53.371269	0.003806	228.270950	211.972427	337.275848	1027.417725	2.578548e-21	2.752907e-21	0.477533	-2.331038	-1.0	-1.0
3241	73.982391	0.701359	232.859207	227.951736	221.233810	672.114441	1.674814e-21	1.410554e-21	0.476132	-2.101680	-1.0	-1.0
1453	63.589539	0.471834	231.500549	222.680450	275.225342	840.881165	1.630561e-21	1.525404e-21	0.492855	-1.945385	-1.0	-1.0
1212	69.349838	0.704752	249.137009	243.949631	235.612534	715.690735	1.714517e-21	2.247939e-21	1.348513	-0.894499	-1.0	-1.0
...	...	...	...	...	...	...	...	...	...	...	...	...
782	67.635185	0.553754	228.531662	221.206268	254.636353	777.048218	1.895225e-21	1.964108e-21	0.629840	-1.903309	-1.0	-1.0
229	63.197891	0.504437	237.314835	228.843048	275.359772	840.965393	1.794975e-21	1.633226e-21	0.822161	-1.845080	-1.0	-1.0
2854	84.826508	0.839987	234.419861	231.915436	172.660767	519.961731	1.472406e-21	1.449929e-21	0.786228	-1.568629	-1.0	-1.0
2540	65.534164	0.606795	244.368256	237.496628	259.099792	789.716858	1.888778e-21	2.443565e-21	1.024894	-1.650668	-1.0	-1.0
1686	79.150085	0.791397	237.265060	233.864227	194.870026	589.087463	1.303603e-21	7.788810e-22	0.690618	-1.612360	-1.0	-1.0

500 rows × 12 columns

Plot results

We can make some quick plots of the results using arviz and seaborn. Below we demonstrate a few possible ways of doing this.

Remnant mass and spin

az.plot_pair(fit.result, var_names=['m', 'chi'], marginals=True,
             kind='kde', reference_values={'m': 70, 'chi': 0.7});

Mode amplitudes

df = fit.result.get_parameter_dataframe()
g = sns.jointplot(x=df['a_220'], y=df['a_221'], kind='kde', fill=True);
g.ax_joint.set_xlabel('$A_{220}$');
g.ax_joint.set_ylabel('$A_{221}$');

Mode frequencies

df = fit.result.get_mode_parameter_dataframe(500, latex=True)
key_map = fit.result.get_parameter_key_map(modes=False)

pg = sns.PairGrid(df, vars=[key_map[k] for k in ['f', 'g']],
                  diag_sharey=False, hue='mode')
pg.map_diag(sns.kdeplot);
pg.map_upper(sns.scatterplot);
pg.map_lower(rd.utils.kdeplot, levels=[0.9, 0.5, 0.1]);

Waveform reconstructions

Plot 90% credible interval for the reconstructions at each detector.

fig, axs = plt.subplots(1, len(fit.ifos), figsize=(12, 4))
axs = np.atleast_1d(axs)
for mode in fit.result.modes:
    hdet_mode_cls = [fit.result.get_strain_quantile(q, mode=mode) for q in [0.5, 0.05, 0.95]]
    for i, ax in zip(fit.ifos, axs):
        m, l, h = [hcl[i] for hcl in hdet_mode_cls]
        ax.plot(m, label=mode.get_label())
        ax.fill_between(m.index, l, h, alpha=0.5)

hdet_cls = [fit.result.get_strain_quantile(q) for q in [0.5, 0.05, 0.95]]
for i, ax in zip(fit.ifos, axs):
    m, l, h = [hcl[i] for hcl in hdet_cls]

    ax.plot(m, label='total')
    ax.fill_between(m.index, l, h, alpha=0.5)

    ax.set_xlim(fit.start_times[i]- 0.001,
                fit.start_times[i] + 0.025)
    ax.set_xlabel(r'$t_\mathrm{GPS} / \mathrm{s}$')
    ax.set_ylabel(f'$h(t)$ {i}');
    ax.legend();

Signal to noise ratio

snrs = fit.result.compute_posterior_snrs(optimal=False, network=True)
sns.kdeplot(snrs);
plt.axvline(np.median(snrs), ls='--')
plt.xlabel('matched-filter network SNR');

Note

This page was generated from a Jupyter notebook that can be downloaded here.