This Jupyter notebook can be downloaded from understanding_fitters.ipynb, or viewed as a python script at understanding_fitters.py.

Understanding Fitters

[1]:
from IPython.display import display_markdown

import pint.toa
import pint.models
import pint.fitter
import pint.config
import pint.logging

pint.logging.setup(level="INFO")
[1]:
1
[2]:
%matplotlib inline
import matplotlib.pyplot as plt

# Turn on quantity support for plotting. This is very helpful!
from astropy.visualization import quantity_support

quantity_support()
[2]:
<astropy.visualization.units.quantity_support.<locals>.MplQuantityConverter at 0x7f776d523f10>
[3]:
# Load some TOAs and a model to fit
m, t = pint.models.get_model_and_toas(
    pint.config.examplefile("NGC6440E.par"), pint.config.examplefile("NGC6440E.tim")
)
[4]:
# You can check if a model includes a noise model with correlated errors (e.g. ECORR or TNRED) by checking the has_correlated_errors property
m.has_correlated_errors
[4]:
False

There are several fitters in PINT, each of which is a subclass of Fitter

  • DownhillWLSFitter - PINT’s workhorse fitter, which does a basic weighted least-squares minimization of the residuals.

  • DownhillGLSFitter - A generalized least squares fitter, like “tempo -G”, that can handle noise processes like ECORR and red noise that are specified by their correlation function properties.

  • WidebandDownhillFitter - A fitter that uses DM estimates associated with each TOA. Also supports generalized least squares.

  • PowellFitter - A very simple example fitter that uses the Powell method implemented in scipy. One notable feature is that it does not require evaluating derivatives w.r.t the model parameters.

  • MCMCFitter - A fitter that does an MCMC fit using the emcee package. This can be very slow, but accomodates Priors on the parameter values and can produce corner plots and other analyses of the posterior distributions of the parameters.

  • WLSFitter, GLSFitter, WidebandFitter - Simpler fitters that make no attempt to ensure convergence.

You can normally use the function pint.fitter.Fitter.auto(toas, model) to construct an appropriate fitter for your model and data.

Weighted Least Squares Fitter

[5]:
# Instantiate a fitter
wlsfit = pint.fitter.DownhillWLSFitter(toas=t, model=m)

A fit is performed by calling fit_toas()

For most fitters, multiple iterations can be limited by setting the maxiter keyword argument.

Downhill fitters will raise the pint.fitter.MaxiterReached exception if they stop before detecting convergence; you can capture this exception and continue if you don’t mind not having the best-fit answer.

[6]:
try:
    wlsfit.fit_toas(maxiter=1)
except pint.fitter.MaxiterReached:
    print("Fitter has not fully converged.")
Fitter has not fully converged.
[7]:
# A summary of the fit and resulting model parameters can easily be printed
# Only free parameters will have values and uncertainties in the Postfit column
wlsfit.print_summary()
Fitted model using downhill_wls method with 5 free parameters to 62 TOAs
Prefit residuals Wrms = 1090.5801805746107 us, Postfit residuals Wrms = 21.18210874551586 us
Chisq = 59.575 for 56 d.o.f. for reduced Chisq of 1.064

PAR                        Prefit                  Postfit            Units
=================== ==================== ============================ =====
PSR                           1748-2021E 1748-2021E                   None
EPHEM                              DE421 DE421                        None
CLOCK                       TT(BIPM2019) TT(BIPM2019)                 None
UNITS                                TDB TDB                          None
START                                                         53478.3 d
FINISH                                                        54187.6 d
TIMEEPH                             FB90 FB90                         None
T2CMETHOD                       IAU2000B IAU2000B                     None
DILATEFREQ                             N                              None
DMDATA                                 N                              None
NTOA                                   0                              None
CHI2                                                          59.5747
CHI2R                                                         1.06383
TRES                                                          21.1821 us
POSEPOCH                           53750                              d
PX                                     0                              mas
RAJ                         17h48m52.75s  17h48m52.80034691s +/- 0.00014 hourangle_second
DECJ                          -20d21m29s  -20d21m29.38331083s +/- 0.033 arcsec
PMRA                                   0                              mas / yr
PMDEC                                  0                              mas / yr
F0                               61.4855          61.485476554372(18) Hz
PEPOCH                             53750                              d
F1                            -1.181e-15            -1.1813(14)×10⁻¹⁵ Hz / s
CORRECT_TROPOSPHERE                    N                              None
PLANET_SHAPIRO                         N                              None
NE_SW                                  0                              1 / cm3
SWP                                    2
SWM                                    0                              None
DM                                 223.9                  224.114(35) pc / cm3
TZRMJD                           53801.4                              d
TZRSITE                                1 1                            None
TZRFRQ                           1949.61                              MHz

Derived Parameters:
Period = 0.016264003404376±0.000000000000005 s
Pdot = (3.125±0.004)×10⁻¹⁹
Characteristic age = 8.246e+08 yr (braking index = 3)
Surface magnetic field = 2.28e+09 G
Magnetic field at light cylinder = 4806 G
Spindown Edot = 2.868e+33 erg / s (I=1e+45 cm2 g)

[8]:
# The WLS fitter doesn't handle correlated errors
wlsfit.resids.model.has_correlated_errors
[8]:
False
[9]:
# You can request a pretty-printed covariance matrix
cov = wlsfit.get_parameter_covariance_matrix(pretty_print=True)

Parameter covariance matrix:
               RAJ       DECJ        F0         F1         DM
    RAJ      1.411e-15
   DECJ     -2.477e-14  8.328e-11
    F0      -5.932e-20  4.079e-17  3.271e-22
    F1       1.594e-26 -4.525e-24 -2.080e-29  2.079e-36
    DM      -6.523e-12  2.067e-08  4.261e-15  2.900e-21  1.221e-03


[10]:
# plot() will make a plot of the post-fit residuals
wlsfit.plot()
../_images/examples_understanding_fitters_13_0.png

Comparing models

There also a convenience function for pretty printing a comparison of two models with the differences measured in sigma.

[11]:
display_markdown(wlsfit.model.compare(wlsfit.model_init, format="markdown"), raw=True)

PARAMETER

NGC6440E.par

NGC6440E.par

Diff_Sigma1

Diff_Sigma2

PSR

1748-2021E

1748-2021E

EPHEM

DE421

DE421

CLOCK

TT(BIPM2019)

TT(BIPM2019)

UNITS

TDB

TDB

START

53478.285871419538264

Missing

FINISH

54187.58732417023191

Missing

TIMEEPH

FB90

FB90

T2CMETHOD

IAU2000B

IAU2000B

DILATEFREQ

False

False

DMDATA

False

False

NTOA

62

0

CHI2

59.574713494278086

Missing

CHI2R

1.06383416954068

Missing

TRES

21.182108745515856993

Missing

POSEPOCH

53750.0

53750.0

PX

0.0

0.0

RAJ

17h48m52.80034691s +/- 0.00014

17h48m52.75s +/- 0.05

-372.26

-1.01

DECJ

-20d21m29.38331083s +/- 0.033

-20d21m29s +/- 0.4

11.67

0.96

PMRA

0.0

0.0

PMDEC

0.0

0.0

F0

61.485476554372(18)

61.4854765540(5)

-20.60

-0.74

PEPOCH

53750.0

53750.0

F1

-1.1813(14)×10⁻¹⁵

-1.1810(10)×10⁻¹⁵

0.23

0.33

CORRECT_TROPOSPHERE

False

False

PLANET_SHAPIRO

False

False

NE_SW

0.0

0.0

SWP

2.0

2.0

SWM

0

0

DM

224.114(35)

223.90(30)

-6.12

-0.71

TZRMJD

53801.386051200748497

53801.386051200748497

TZRSITE

1

1

TZRFRQ

1949.609

1949.609

SEPARATION

0.805131 arcsec

You can see just how much F1 changed. Let’s compare the \(\chi^2\) values:

[12]:
print(f"Pre-fit chi-squared value: {wlsfit.resids_init.chi2}")
print(f"Post-fit chi-squared value: {wlsfit.resids.chi2}")
Pre-fit chi-squared value: 157920.59715077005
Post-fit chi-squared value: 59.574713494278086

Generalized Least Squares fitter

The GLS fitter is capable of handling correlated noise models.

It has some more complex options using the maxiter, threshold, and full_cov keyword arguments to fit_toas().

If maxiter is less than one, no fitting is done, just the chi-squared computation. In this case, you must provide the residuals argument.

If maxiter is one or more, so fitting is actually done, the chi-squared value returned is only approximately the chi-squared of the improved(?) model. In fact it is the chi-squared of the solution to the linear fitting problem, and the full non-linear model should be evaluated and new residuals produced if an accurate chi-squared is desired.

A first attempt is made to solve the fitting problem by Cholesky decomposition, but if this fails singular value decomposition is used instead. In this case singular values below threshold are removed.

full_cov determines which calculation is used. If True, the full covariance matrix is constructed and the calculation is relatively straightforward but the full covariance matrix may be enormous. If False, an algorithm is used that takes advantage of the structure of the covariance matrix, based on information provided by the noise model. The two algorithms should give the same result up to numerical accuracy where they both can be applied.

To test this fitter properly, we need a model that includes correlated noise components, so we will load one from NANOGrav 9yr data release.

[13]:
m1855 = pint.models.get_model(pint.config.examplefile("B1855+09_NANOGrav_9yv1.gls.par"))
[14]:
# You can check if a model includes a noise model with correlated errors (e.g. ECORR or TNRED) by checking the has_correlated_errors property
m1855.has_correlated_errors
[14]:
True
[15]:
print(m1855)
# Created: 2024-06-05T07:34:14.807453
# PINT_version: 1.0+259.g224e5f1
# User: docs
# Host: build-24596653-project-85767-nanograv-pint
# OS: Linux-5.19.0-1028-aws-x86_64-with-glibc2.35
# Python: 3.11.6 (main, Feb  1 2024, 16:47:41) [GCC 11.4.0]
# Format: pint
PSR                              B1855+09
EPHEM                               DE421
CLK                          TT(BIPM2019)
UNITS                                 TDB
START              53358.7260000000000000
FINISH             56598.8730000000000000
INFO                                   -f
TIMEEPH                              FB90
DILATEFREQ                              N
DMDATA                                  N
NTOA                                 4005
TRES                                 5.52
LAMBDA                286.863489330115613 1 0.00000001658590000000
BETA                   32.321487755503703 1 0.00000002735260000000
PMLAMBDA                          -3.2701 1 0.0141
PMBETA                            -5.0982 1 0.0291
PX                                 0.2929 1 0.2186
ECL                              IERS2003
POSEPOCH           54978.0000000000000000
F0                   186.4940812707752116 1 3.28468e-11
F1                    -6.205147513395e-16 1 1.379566413719e-19
PEPOCH             54978.0000000000000000
ECORR           -f 430_PUPPI                   0.00601
ECORR           -f L-wide_PUPPI                   0.31843
ECORR           -f L-wide_ASP                   0.79618
ECORR           -f 430_ASP                   0.01117
RNAMP                            0.017173
RNIDX                            -4.91353
TNREDAMP              -14.227505410948254
TNREDGAM                          4.91353
TNREDC                               45.0
T2EFAC          -f L-wide_PUPPI                     1.507
T2EQUAD         -f L-wide_PUPPI                   0.25518
T2EFAC          -f 430_ASP                     1.147
T2EFAC          -f L-wide_ASP                      1.15
T2EFAC          -f 430_PUPPI                     1.117
T2EQUAD         -f 430_ASP                    0.0141
T2EQUAD         -f L-wide_ASP                   0.42504
T2EQUAD         -f 430_PUPPI                    0.0264
CORRECT_TROPOSPHERE                         N
PLANET_SHAPIRO                          N
SOLARN0                               0.0
SWM                                     0
DM                              13.299393
DMX                                  14.0
DMX_0001                      0.015161863 1 0.00351684846
DMXR1_0001         53358.7274600000000001
DMXR2_0001         53358.7784100000000000
DMX_0002                     0.0152370685 1 0.00351683449
DMXR1_0002         53420.5489300000000000
DMXR2_0002         53420.5862000000000001
DMX_0003                     0.0151895956 1 0.00351649738
DMXR1_0003         53448.4736900000000000
DMXR2_0003         53457.4929400000000000
DMX_0004                     0.0151322502 1 0.00351653508
DMXR1_0004         53477.3962800000000000
DMXR2_0004         53477.4345300000000000
DMX_0005                     0.0151076504 1 0.00351662711
DMXR1_0005         53532.2328100000000000
DMXR2_0005         53532.2765600000000000
DMX_0006                      0.015263814 1 0.00351647013
DMXR1_0006         53603.0363200000000000
DMXR2_0006         53603.0897100000000000
DMX_0007                     0.0151897641 1 0.003516599
DMXR1_0007         53628.9652500000000001
DMXR2_0007         53628.9833500000000001
DMX_0008                     0.0152890326 1 0.00351651389
DMXR1_0008         53686.7976000000000000
DMXR2_0008         53686.8420300000000001
DMX_0009                     0.0152484643 1 0.0035165631
DMXR1_0009         53715.7306400000000001
DMXR2_0009         53715.7667800000000000
DMX_0010                     0.0153422398 1 0.00351652136
DMXR1_0010         53750.6227300000000000
DMXR2_0010         53750.6672300000000000
DMX_0011                      0.015354092 1 0.00351660466
DMXR1_0011         53798.5010800000000000
DMXR2_0011         53798.5391500000000000
DMX_0012                     0.0154295455 1 0.0035165139
DMXR1_0012         53851.3718100000000000
DMXR2_0012         53851.4042800000000000
DMX_0013                     0.0154693407 1 0.00351653286
DMXR1_0013         53891.2472700000000000
DMXR2_0013         53891.2851400000000000
DMX_0014                     0.0156001615 1 0.00351689266
DMXR1_0014         53926.1632600000000000
DMXR2_0014         53926.1922800000000000
DMX_0015                     0.0157477908 1 0.00351642748
DMXR1_0015         53968.0572500000000000
DMXR2_0015         53968.0951700000000000
DMX_0016                     0.0159397058 1 0.0035163291
DMXR1_0016         54008.9537200000000000
DMXR2_0016         54008.9864400000000001
DMX_0017                     0.0159157339 1 0.00351644531
DMXR1_0017         54043.8368200000000000
DMXR2_0017         54043.8724500000000001
DMX_0018                      0.016023921 1 0.00351654202
DMXR1_0018         54092.7126500000000000
DMXR2_0018         54092.7424700000000001
DMX_0019                     0.0161119039 1 0.00351660022
DMXR1_0019         54135.5828200000000000
DMXR2_0019         54135.6117100000000000
DMX_0020                     0.0163381983 1 0.00351653637
DMXR1_0020         54177.4729700000000000
DMXR2_0020         54177.5136300000000000
DMX_0021                     0.0162696012 1 0.00351647068
DMXR1_0021         54472.6627600000000000
DMXR2_0021         54472.6939300000000000
DMX_0022                     0.0161627196 1 0.00351654829
DMXR1_0022         54519.5241900000000000
DMXR2_0022         54519.5612300000000000
DMX_0023                     0.0162621197 1 0.00351655413
DMXR1_0023         54569.4095400000000000
DMXR2_0023         54569.4667500000000000
DMX_0024                     0.0164484026 1 0.00351660433
DMXR1_0024         54819.7148100000000001
DMXR2_0024         54819.7512900000000000
DMX_0025                     0.0164104711 1 0.00351663094
DMXR1_0025         54862.5833800000000000
DMXR2_0025         54862.6195900000000001
DMX_0026                      0.016569036 1 0.00351659233
DMXR1_0026         54925.4273400000000001
DMXR2_0026         54925.4664099999999999
DMX_0027                     0.0167679894 1 0.00351680449
DMXR1_0027         54981.2808400000000000
DMXR2_0027         54981.3303200000000001
DMX_0028                     0.0167977392 1 0.00351700298
DMXR1_0028         54998.2094000000000000
DMXR2_0028         54998.2266600000000000
DMX_0029                     0.0169021454 1 0.00351669195
DMXR1_0029         55108.9040200000000000
DMXR2_0029         55108.9219600000000000
DMX_0030                     0.0170268231 1 0.00351673492
DMXR1_0030         55135.8396000000000000
DMXR2_0030         55135.8607900000000001
DMX_0031                     0.0170193727 1 0.0035168257
DMXR1_0031         55170.7389400000000000
DMXR2_0031         55170.7489500000000000
DMX_0032                      0.017073879 1 0.0035166332
DMXR1_0032         55205.6495700000000000
DMXR2_0032         55205.6660600000000000
DMX_0033                     0.0171315137 1 0.00351779101
DMXR1_0033         55298.4053900000000000
DMXR2_0033         55298.4194000000000001
DMX_0034                     0.0172665141 1 0.0035171169
DMXR1_0034         55358.2410600000000000
DMXR2_0034         55358.2606200000000000
DMX_0035                     0.0173188687 1 0.00351729708
DMXR1_0035         55391.1726800000000000
DMXR2_0035         55391.1865400000000000
DMX_0036                     0.0172177568 1 0.00351711273
DMXR1_0036         55424.0694700000000000
DMXR2_0036         55424.0844000000000000
DMX_0037                     0.0172926679 1 0.00351670777
DMXR1_0037         55500.8336900000000000
DMXR2_0037         55500.8483800000000000
DMX_0038                     0.0172819014 1 0.00351683726
DMXR1_0038         55528.7626100000000000
DMXR2_0038         55528.7760200000000000
DMX_0039                     0.0172309563 1 0.00351654536
DMXR1_0039         55638.4592000000000000
DMXR2_0039         55638.4747300000000000
DMX_0040                     0.0172343241 1 0.00351659552
DMXR1_0040         55677.3593500000000000
DMXR2_0040         55677.3764100000000000
DMX_0041                     0.0172846259 1 0.00351685651
DMXR1_0041         55701.2862700000000000
DMXR2_0041         55701.3045500000000000
DMX_0042                     0.0174295333 1 0.00351754173
DMXR1_0042         55731.2369800000000000
DMXR2_0042         55731.2557500000000000
DMX_0043                     0.0174135398 1 0.00351701302
DMXR1_0043         55758.1616300000000000
DMXR2_0043         55758.1828600000000000
DMX_0044                     0.0174411084 1 0.00351698583
DMXR1_0044         55789.0676000000000000
DMXR2_0044         55789.0858500000000000
DMX_0045                     0.0176716738 1 0.00351656249
DMXR1_0045         55843.9164899999999999
DMXR2_0045         55843.9356300000000001
DMX_0046                     0.0178457703 1 0.0035165082
DMXR1_0046         55912.7231500000000000
DMXR2_0046         55912.7409100000000001
DMX_0047                     0.0179235129 1 0.00351641019
DMXR1_0047         55989.5168300000000000
DMXR2_0047         55989.5376200000000000
DMX_0048                     0.0179551569 1 0.00351643393
DMXR1_0048         56023.4263700000000000
DMXR2_0048         56023.4461900000000000
DMX_0049                     0.0180214558 1 0.00351830253
DMXR1_0049         56057.3536400000000000
DMXR2_0049         56057.3537799999999999
DMX_0050                     0.0182147041 1 0.00352479233
DMXR1_0050         56106.2205600000000000
DMXR2_0050         56106.2206100000000000
DMX_0051                     0.0181602407 1 0.00352162978
DMXR1_0051         56122.1298500000000000
DMXR2_0051         56122.1771700000000000
DMX_0052                     0.0180430586 1 0.0035266043
DMXR1_0052         56140.1376700000000000
DMXR2_0052         56140.1377500000000000
DMX_0053                     0.0181527986 1 0.00352069768
DMXR1_0053         56166.0571200000000000
DMXR2_0053         56166.0573300000000000
DMX_0054                     0.0184943818 1 0.00351633835
DMXR1_0054         56201.9569899999999999
DMXR2_0054         56212.9240400000000000
DMX_0055                     0.0182191151 1 0.00351594287
DMXR1_0055         56235.8428200000000000
DMXR2_0055         56235.8681000000000000
DMX_0056                     0.0182791281 1 0.00351620206
DMXR1_0056         56254.8200499999999999
DMXR2_0056         56254.8445300000000000
DMX_0057                     0.0183448743 1 0.00351757322
DMXR1_0057         56277.7525799999999999
DMXR2_0057         56277.7526300000000000
DMX_0058                     0.0182456031 1 0.00351582953
DMXR1_0058         56294.6690800000000000
DMXR2_0058         56306.6485600000000000
DMX_0059                     0.0180520318 1 0.00351721613
DMXR1_0059         56319.6395299999999999
DMXR2_0059         56319.6395800000000000
DMX_0060                     0.0181223112 1 0.00351582826
DMXR1_0060         56341.5587100000000000
DMXR2_0060         56341.5768099999999999
DMX_0061                      0.018162022 1 0.00351580352
DMXR1_0061         56360.5302200000000000
DMXR2_0061         56374.5115599999999999
DMX_0062                     0.0181719503 1 0.00351571012
DMXR1_0062         56380.4505300000000000
DMXR2_0062         56380.4684000000000000
DMX_0063                     0.0181438391 1 0.00351612171
DMXR1_0063         56411.3969800000000000
DMXR2_0063         56417.3732900000000000
DMX_0064                     0.0181367452 1 0.00351601523
DMXR1_0064         56432.3219500000000000
DMXR2_0064         56438.3251700000000000
DMX_0065                     0.0181205204 1 0.00351651911
DMXR1_0065         56458.2358500000000000
DMXR2_0065         56458.2537600000000000
DMX_0066                     0.0181608231 1 0.00351659645
DMXR1_0066         56479.1780800000000000
DMXR2_0066         56479.1960700000000000
DMX_0067                     0.0181467585 1 0.00351643552
DMXR1_0067         56498.1243500000000000
DMXR2_0067         56498.1420100000000000
DMX_0068                     0.0182940538 1 0.00351645118
DMXR1_0068         56519.0817400000000000
DMXR2_0068         56519.0981400000000000
DMX_0069                     0.0182906491 1 0.00351609445
DMXR1_0069         56538.0092400000000000
DMXR2_0069         56538.0256100000000000
DMX_0070                     0.0179611744 1 0.00352468823
DMXR1_0070         56557.9709500000000000
DMXR2_0070         56557.9710100000000000
DMX_0071                     0.0183625085 1 0.00351656154
DMXR1_0071         56577.9065599999999999
DMXR2_0071         56577.9228499999999999
DMX_0072                     0.0183864669 1 0.003515863
DMXR1_0072         56598.8557200000000000
DMXR2_0072         56598.8720400000000000
BINARY                                 DD
PB                      12.32717119132762 1 1.9722e-10
PBDOT                                 0.0
A1                             9.23078048 1 2.03e-07
A1DOT                                 0.0
E                              2.1634e-05 1 2.36e-08
EDOT                                  0.0
T0                 54975.5128660817000000 1 0.0019286695
OM                       276.536118059963 1 0.056323656112
OMDOT                                 0.0
M2                               0.233837 1 0.011278
SINI                             0.999461 1 0.000178
A0                                    0.0
B0                                    0.0
GAMMA                                 0.0
DR                                    0.0
DTH                                   0.0
FD1                        0.000161666384 1 3.38650356e-05
FD2                        -0.00018821003 1 4.13173074e-05
FD3                        0.000107526915 1 2.50177766e-05
TZRMJD             54981.2808461648844676
TZRSITE                                 3
TZRFRQ                              424.0
JUMP            -fe L-wide                -9.449e-06 1 9.439e-06

[16]:
ts1855 = pint.toa.get_TOAs(
    pint.config.examplefile("B1855+09_NANOGrav_9yv1.tim"), model=m1855
)
ts1855.print_summary()
Number of TOAs:  4005
Number of commands:  1
Number of observatories: 1 ['arecibo']
MJD span:  53358.727 to 56598.872
Date span: 2004-12-19 17:27:32.961266179 to 2013-11-02 20:55:40.399171358
arecibo TOAs (4005):
  Min freq:      422.187 MHz
  Max freq:      1760.728 MHz
  Min error:     0.05 us
  Max error:     17.8 us
  Median error:  1.19 us

There is currently a problem with DownhillGLSFitter: it doesn’t record appropriate noise parameters.

[17]:
glsfit = pint.fitter.GLSFitter(toas=ts1855, model=m1855)
[18]:
m1855.DMX_0001.prefix
[18]:
'DMX_'
[19]:
glsfit.fit_toas(maxiter=1)
[19]:
3900.0981539895524086
[20]:
glsfit.print_summary()
Fitted model using generalized_least_square method with 90 free parameters to 4005 TOAs
Prefit residuals Wrms = 5.5386698946489625 us, Postfit residuals Wrms = 1.3467836852341102 us
Chisq = 3900.098 for 3914 d.o.f. for reduced Chisq of 0.996

PAR                        Prefit                  Postfit            Units
=================== ==================== ============================ =====
PSR                             B1855+09 B1855+09                     None
EPHEM                              DE421 DE421                        None
CLOCK                       TT(BIPM2019) TT(BIPM2019)                 None
UNITS                                TDB TDB                          None
START                            53358.7                      53358.7 d
FINISH                           56598.9                      56598.9 d
INFO                                  -f -f                           None
TIMEEPH                             FB90 FB90                         None
BINARY                                DD DD                           None
DILATEFREQ                             N                              None
DMDATA                                 N                              None
NTOA                                4005                              None
CHI2                                                           3900.1
CHI2R                                                        0.996448
TRES                                5.52                      1.34678 us
POSEPOCH                           54978                              d
PX                                0.2929                     0.30(22) mas
ELONG                286d51m48.56158842s  286d51m48.56156705s +/- 5.9e-05 arcsec
ELAT                  32d19m17.35591981s  32d19m17.35582847s +/- 9.8e-05 arcsec
PMELONG                          -3.2701                   -3.270(14) mas / yr
PMELAT                           -5.0982                   -5.099(29) mas / yr
ECL                             IERS2003 IERS2003                     None
F0                               186.494       186.49408127078522(27) Hz
PEPOCH                             54978                              d
F1                          -6.20515e-16           -6.20497(25)×10⁻¹⁶ Hz / s
ECORR1                           0.00601                              us
ECORR2                           0.31843                              us
ECORR3                           0.79618                              us
ECORR4                           0.01117                              us
RNAMP                           0.017173
RNIDX                           -4.91353
TNREDAMP                        -14.2275
TNREDGAM                         4.91353
TNREDC                                45
EFAC1                              1.507
EQUAD1                           0.25518                              us
EFAC2                              1.147
EFAC3                               1.15
EFAC4                              1.117
EQUAD2                            0.0141                              us
EQUAD3                           0.42504                              us
EQUAD4                            0.0264                              us
CORRECT_TROPOSPHERE                    N                              None
PLANET_SHAPIRO                         N                              None
NE_SW                                  0                              1 / cm3
SWP                                    2
SWM                                    0                              None
DM                               13.2994                              pc / cm3
DMX                                   14                              pc / cm3
DMX_0001                       0.0151619                   0.0152(35) pc / cm3
DMXR1_0001                       53358.7                              d
DMXR2_0001                       53358.8                              d
DMX_0002                       0.0152371                   0.0152(35) pc / cm3
DMXR1_0002                       53420.5                              d
DMXR2_0002                       53420.6                              d
DMX_0003                       0.0151896                   0.0152(35) pc / cm3
DMXR1_0003                       53448.5                              d
DMXR2_0003                       53457.5                              d
DMX_0004                       0.0151323                   0.0151(35) pc / cm3
DMXR1_0004                       53477.4                              d
DMXR2_0004                       53477.4                              d
DMX_0005                       0.0151077                   0.0151(35) pc / cm3
DMXR1_0005                       53532.2                              d
DMXR2_0005                       53532.3                              d
DMX_0006                       0.0152638                   0.0153(35) pc / cm3
DMXR1_0006                         53603                              d
DMXR2_0006                       53603.1                              d
DMX_0007                       0.0151898                   0.0152(35) pc / cm3
DMXR1_0007                         53629                              d
DMXR2_0007                         53629                              d
DMX_0008                        0.015289                   0.0153(35) pc / cm3
DMXR1_0008                       53686.8                              d
DMXR2_0008                       53686.8                              d
DMX_0009                       0.0152485                   0.0153(35) pc / cm3
DMXR1_0009                       53715.7                              d
DMXR2_0009                       53715.8                              d
DMX_0010                       0.0153422                   0.0153(35) pc / cm3
DMXR1_0010                       53750.6                              d
DMXR2_0010                       53750.7                              d
DMX_0011                       0.0153541                   0.0154(35) pc / cm3
DMXR1_0011                       53798.5                              d
DMXR2_0011                       53798.5                              d
DMX_0012                       0.0154295                   0.0154(35) pc / cm3
DMXR1_0012                       53851.4                              d
DMXR2_0012                       53851.4                              d
DMX_0013                       0.0154693                   0.0155(35) pc / cm3
DMXR1_0013                       53891.2                              d
DMXR2_0013                       53891.3                              d
DMX_0014                       0.0156002                   0.0156(35) pc / cm3
DMXR1_0014                       53926.2                              d
DMXR2_0014                       53926.2                              d
DMX_0015                       0.0157478                   0.0158(35) pc / cm3
DMXR1_0015                       53968.1                              d
DMXR2_0015                       53968.1                              d
DMX_0016                       0.0159397                   0.0159(35) pc / cm3
DMXR1_0016                         54009                              d
DMXR2_0016                         54009                              d
DMX_0017                       0.0159157                   0.0159(35) pc / cm3
DMXR1_0017                       54043.8                              d
DMXR2_0017                       54043.9                              d
DMX_0018                       0.0160239                   0.0160(35) pc / cm3
DMXR1_0018                       54092.7                              d
DMXR2_0018                       54092.7                              d
DMX_0019                       0.0161119                   0.0161(35) pc / cm3
DMXR1_0019                       54135.6                              d
DMXR2_0019                       54135.6                              d
DMX_0020                       0.0163382                   0.0163(35) pc / cm3
DMXR1_0020                       54177.5                              d
DMXR2_0020                       54177.5                              d
DMX_0021                       0.0162696                   0.0163(35) pc / cm3
DMXR1_0021                       54472.7                              d
DMXR2_0021                       54472.7                              d
DMX_0022                       0.0161627                   0.0162(35) pc / cm3
DMXR1_0022                       54519.5                              d
DMXR2_0022                       54519.6                              d
DMX_0023                       0.0162621                   0.0163(35) pc / cm3
DMXR1_0023                       54569.4                              d
DMXR2_0023                       54569.5                              d
DMX_0024                       0.0164484                   0.0165(35) pc / cm3
DMXR1_0024                       54819.7                              d
DMXR2_0024                       54819.8                              d
DMX_0025                       0.0164105                   0.0164(35) pc / cm3
DMXR1_0025                       54862.6                              d
DMXR2_0025                       54862.6                              d
DMX_0026                        0.016569                   0.0166(35) pc / cm3
DMXR1_0026                       54925.4                              d
DMXR2_0026                       54925.5                              d
DMX_0027                        0.016768                   0.0168(35) pc / cm3
DMXR1_0027                       54981.3                              d
DMXR2_0027                       54981.3                              d
DMX_0028                       0.0167977                   0.0168(35) pc / cm3
DMXR1_0028                       54998.2                              d
DMXR2_0028                       54998.2                              d
DMX_0029                       0.0169021                   0.0169(35) pc / cm3
DMXR1_0029                       55108.9                              d
DMXR2_0029                       55108.9                              d
DMX_0030                       0.0170268                   0.0170(35) pc / cm3
DMXR1_0030                       55135.8                              d
DMXR2_0030                       55135.9                              d
DMX_0031                       0.0170194                   0.0170(35) pc / cm3
DMXR1_0031                       55170.7                              d
DMXR2_0031                       55170.7                              d
DMX_0032                       0.0170739                   0.0171(35) pc / cm3
DMXR1_0032                       55205.6                              d
DMXR2_0032                       55205.7                              d
DMX_0033                       0.0171315                   0.0171(35) pc / cm3
DMXR1_0033                       55298.4                              d
DMXR2_0033                       55298.4                              d
DMX_0034                       0.0172665                   0.0173(35) pc / cm3
DMXR1_0034                       55358.2                              d
DMXR2_0034                       55358.3                              d
DMX_0035                       0.0173189                   0.0173(35) pc / cm3
DMXR1_0035                       55391.2                              d
DMXR2_0035                       55391.2                              d
DMX_0036                       0.0172178                   0.0172(35) pc / cm3
DMXR1_0036                       55424.1                              d
DMXR2_0036                       55424.1                              d
DMX_0037                       0.0172927                   0.0173(35) pc / cm3
DMXR1_0037                       55500.8                              d
DMXR2_0037                       55500.8                              d
DMX_0038                       0.0172819                   0.0173(35) pc / cm3
DMXR1_0038                       55528.8                              d
DMXR2_0038                       55528.8                              d
DMX_0039                        0.017231                   0.0172(35) pc / cm3
DMXR1_0039                       55638.5                              d
DMXR2_0039                       55638.5                              d
DMX_0040                       0.0172343                   0.0172(35) pc / cm3
DMXR1_0040                       55677.4                              d
DMXR2_0040                       55677.4                              d
DMX_0041                       0.0172846                   0.0173(35) pc / cm3
DMXR1_0041                       55701.3                              d
DMXR2_0041                       55701.3                              d
DMX_0042                       0.0174295                   0.0174(35) pc / cm3
DMXR1_0042                       55731.2                              d
DMXR2_0042                       55731.3                              d
DMX_0043                       0.0174135                   0.0174(35) pc / cm3
DMXR1_0043                       55758.2                              d
DMXR2_0043                       55758.2                              d
DMX_0044                       0.0174411                   0.0174(35) pc / cm3
DMXR1_0044                       55789.1                              d
DMXR2_0044                       55789.1                              d
DMX_0045                       0.0176717                   0.0177(35) pc / cm3
DMXR1_0045                       55843.9                              d
DMXR2_0045                       55843.9                              d
DMX_0046                       0.0178458                   0.0178(35) pc / cm3
DMXR1_0046                       55912.7                              d
DMXR2_0046                       55912.7                              d
DMX_0047                       0.0179235                   0.0179(35) pc / cm3
DMXR1_0047                       55989.5                              d
DMXR2_0047                       55989.5                              d
DMX_0048                       0.0179552                   0.0180(35) pc / cm3
DMXR1_0048                       56023.4                              d
DMXR2_0048                       56023.4                              d
DMX_0049                       0.0180215                   0.0180(35) pc / cm3
DMXR1_0049                       56057.4                              d
DMXR2_0049                       56057.4                              d
DMX_0050                       0.0182147                   0.0182(35) pc / cm3
DMXR1_0050                       56106.2                              d
DMXR2_0050                       56106.2                              d
DMX_0051                       0.0181602                   0.0182(35) pc / cm3
DMXR1_0051                       56122.1                              d
DMXR2_0051                       56122.2                              d
DMX_0052                       0.0180431                   0.0180(35) pc / cm3
DMXR1_0052                       56140.1                              d
DMXR2_0052                       56140.1                              d
DMX_0053                       0.0181528                   0.0182(35) pc / cm3
DMXR1_0053                       56166.1                              d
DMXR2_0053                       56166.1                              d
DMX_0054                       0.0184944                   0.0185(35) pc / cm3
DMXR1_0054                         56202                              d
DMXR2_0054                       56212.9                              d
DMX_0055                       0.0182191                   0.0182(35) pc / cm3
DMXR1_0055                       56235.8                              d
DMXR2_0055                       56235.9                              d
DMX_0056                       0.0182791                   0.0183(35) pc / cm3
DMXR1_0056                       56254.8                              d
DMXR2_0056                       56254.8                              d
DMX_0057                       0.0183449                   0.0183(35) pc / cm3
DMXR1_0057                       56277.8                              d
DMXR2_0057                       56277.8                              d
DMX_0058                       0.0182456                   0.0182(35) pc / cm3
DMXR1_0058                       56294.7                              d
DMXR2_0058                       56306.6                              d
DMX_0059                        0.018052                   0.0181(35) pc / cm3
DMXR1_0059                       56319.6                              d
DMXR2_0059                       56319.6                              d
DMX_0060                       0.0181223                   0.0181(35) pc / cm3
DMXR1_0060                       56341.6                              d
DMXR2_0060                       56341.6                              d
DMX_0061                        0.018162                   0.0182(35) pc / cm3
DMXR1_0061                       56360.5                              d
DMXR2_0061                       56374.5                              d
DMX_0062                        0.018172                   0.0182(35) pc / cm3
DMXR1_0062                       56380.5                              d
DMXR2_0062                       56380.5                              d
DMX_0063                       0.0181438                   0.0181(35) pc / cm3
DMXR1_0063                       56411.4                              d
DMXR2_0063                       56417.4                              d
DMX_0064                       0.0181367                   0.0181(35) pc / cm3
DMXR1_0064                       56432.3                              d
DMXR2_0064                       56438.3                              d
DMX_0065                       0.0181205                   0.0181(35) pc / cm3
DMXR1_0065                       56458.2                              d
DMXR2_0065                       56458.3                              d
DMX_0066                       0.0181608                   0.0182(35) pc / cm3
DMXR1_0066                       56479.2                              d
DMXR2_0066                       56479.2                              d
DMX_0067                       0.0181468                   0.0181(35) pc / cm3
DMXR1_0067                       56498.1                              d
DMXR2_0067                       56498.1                              d
DMX_0068                       0.0182941                   0.0183(35) pc / cm3
DMXR1_0068                       56519.1                              d
DMXR2_0068                       56519.1                              d
DMX_0069                       0.0182906                   0.0183(35) pc / cm3
DMXR1_0069                         56538                              d
DMXR2_0069                         56538                              d
DMX_0070                       0.0179612                   0.0180(35) pc / cm3
DMXR1_0070                         56558                              d
DMXR2_0070                         56558                              d
DMX_0071                       0.0183625                   0.0184(35) pc / cm3
DMXR1_0071                       56577.9                              d
DMXR2_0071                       56577.9                              d
DMX_0072                       0.0183865                   0.0184(35) pc / cm3
DMXR1_0072                       56598.9                              d
DMXR2_0072                       56598.9                              d
PB                               12.3272           12.32717119133(20) d
PBDOT                                  0
A1                               9.23078               9.23078048(20) ls
A1DOT                                  0                              ls / s
ECC                           2.1634e-05              2.1634(24)×10⁻⁵
EDOT                                   0                              1 / s
T0                               54975.5               54975.5128(19) d
OM                               276.536                    276.53(6) deg
OMDOT                                  0                              deg / yr
M2                              0.233837                    0.234(11) solMass
SINI                            0.999461                  0.99946(18)
A0                                     0                              s
B0                                     0                              s
GAMMA                                  0                              s
DR                                     0
DTH                                    0
FD1                          0.000161666                 0.000162(34) s
FD2                          -0.00018821                  -0.00019(4) s
FD3                          0.000107527                 0.000108(25) s
TZRMJD                           54981.3                              d
TZRSITE                                3 3                            None
TZRFRQ                               424                              MHz
JUMP1                         -9.449e-06                   -9(9)×10⁻⁶ s

Derived Parameters:
Period = 0.005362100465526423±0.000000000000000008 s
Pdot = (1.78406±0.00007)×10⁻²⁰
Characteristic age = 4.762e+09 yr (braking index = 3)
Surface magnetic field = 3.13e+08 G
Magnetic field at light cylinder = 1.84e+04 G
Spindown Edot = 4.568e+33 erg / s (I=1e+45 cm2 g)

Parallax distance = (3.35±2.45)×10³ pc

Binary model BinaryDD
Mass function = 0.0055573934(4) Msun
Min / Median Companion mass (assuming Mpsr = 1.4 Msun) = 0.2470 / 0.2902 Msun
From SINI in model:
    cos(i) = 0.033(5)
    i = 88.12(31) deg
Pulsar mass (Shapiro Delay) = 1.2804782086551219 solMass

The GLS fitter produces two types of residuals, the normal residuals to the deterministic model and those from the noise model.

[21]:
glsfit.resids.time_resids
[21]:
$[-2.6044388 \times 10^{-6},~-1.2472415 \times 10^{-6},~4.1189369 \times 10^{-7},~\dots,~1.2631874 \times 10^{-6},~-3.2699274 \times 10^{-7},~4.3562553 \times 10^{-6}] \; \mathrm{s}$
[22]:
glsfit.resids.noise_resids
[22]:
{'ecorr_noise': <Quantity [-7.68053637e-11, -7.68053637e-11, -7.68053637e-11, ...,
             3.75218951e-07,  3.75218951e-07,  3.75218951e-07] s>,
 'pl_red_noise': <Quantity [-1.27090740e-06, -1.27090740e-06, -1.27090740e-06, ...,
            -1.27096267e-06, -1.27096267e-06, -1.27096267e-06] s>}
[23]:
# Here we can plot both the residuals to the deterministic model as well as the realization of the noise model residuals
# The difference will be the "whitened" residuals
fig, ax = plt.subplots(figsize=(16, 9))
mjds = glsfit.toas.get_mjds()
ax.plot(mjds, glsfit.resids.time_resids, ".")
ax.plot(mjds, glsfit.resids.noise_resids["pl_red_noise"], ".")
[23]:
[<matplotlib.lines.Line2D at 0x7f7749e3b710>]
../_images/examples_understanding_fitters_32_1.png

Choosing fitters

You can use the automatic fitter selection to help you choose between WLSFitter, GLSFitter, and their wideband variants. The default Downhill fitters generally have better performance than the plain variants.

[24]:
autofit = pint.fitter.Fitter.auto(toas=ts1855, model=m1855)
[25]:
autofit.fit_toas()
[25]:
True
[26]:
display_markdown(autofit.model.compare(glsfit.model, format="markdown"), raw=True)

PARAMETER

B1855+09_NANOGrav_9yv1.gls.par

B1855+09_NANOGrav_9yv1.gls.par

Diff_Sigma1

Diff_Sigma2

PSR

B1855+09

B1855+09

EPHEM

DE421

DE421

CLOCK

TT(BIPM2019)

TT(BIPM2019)

UNITS

TDB

TDB

START

53358.727464829469664

53358.727464829469664

FINISH

56598.871995360779607

56598.871995360779607

INFO

-f

-f

TIMEEPH

FB90

FB90

BINARY

DD

DD

DILATEFREQ

False

False

DMDATA

False

False

NTOA

4005

4005

CHI2

3900.098433923222

3900.0981539895524

CHI2R

0.9964482457647476

0.9964481742436261

TRES

1.3466265089228378848

1.3467836852341102443

POSEPOCH

54978.0

54978.0

PX

0.30(22)

0.30(22)

0.00

0.00

ELONG

286d51m48.56156713s +/- 5.9e-05

286d51m48.56156705s +/- 5.9e-05

-0.00

-0.00

ELAT

32d19m17.35582884s +/- 9.7e-05

32d19m17.35582847s +/- 9.8e-05

-0.00

-0.00

PMELONG

-3.270(14)

-3.270(14)

0.00

0.00

PMELAT

-5.099(29)

-5.099(29)

-0.00

-0.00

ECL

IERS2003

IERS2003

F0

186.49408127078522(27)

186.49408127078522(27)

0.00

0.00

PEPOCH

54978.0

54978.0

F1

-6.20497(25)×10⁻¹⁶

-6.20497(25)×10⁻¹⁶

-0.00

-0.00

ECORR1

0.00601

0.00601

ECORR2

0.31843

0.31843

ECORR3

0.79618

0.79618

ECORR4

0.01117

0.01117

RNAMP

0.017173

0.017173

RNIDX

-4.91353

-4.91353

TNREDAMP

-14.227505410948254

-14.227505410948254

TNREDGAM

4.91353

4.91353

TNREDC

45.0

45.0

EFAC1

1.507

1.507

EQUAD1

0.25518

0.25518

EFAC2

1.147

1.147

EFAC3

1.15

1.15

EFAC4

1.117

1.117

EQUAD2

0.0141

0.0141

EQUAD3

0.42504

0.42504

EQUAD4

0.0264

0.0264

CORRECT_TROPOSPHERE

False

False

PLANET_SHAPIRO

False

False

NE_SW

0.0

0.0

SWP

2.0

2.0

SWM

0

0

DM

13.299393

13.299393

PB

12.32717119133(20)

12.32717119133(20)

-0.00

-0.00

PBDOT

0.0

0.0

A1

9.23078048(20)

9.23078048(20)

-0.00

-0.00

A1DOT

0.0

0.0

ECC

2.1634(24)×10⁻⁵

2.1634(24)×10⁻⁵

0.00

0.00

EDOT

0.0

0.0

T0

54975.5128(8)

54975.5128(19)

-0.07

-0.03

OM

276.535(23)

276.53(6)

-0.07

-0.03

OMDOT

0.0

0.0

M2

0.234(11)

0.234(11)

0.00

0.00

SINI

0.99946(18)

0.99946(18)

-0.00

-0.00

A0

0.0

0.0

B0

0.0

0.0

GAMMA

0.0

0.0

DR

0.0

0.0

DTH

0.0

0.0

FD1

0.000162(34)

0.000162(34)

0.00

0.00

FD2

-0.00019(4)

-0.00019(4)

-0.00

-0.00

FD3

0.000108(25)

0.000108(25)

0.00

0.00

TZRMJD

54981.28084616488447

54981.28084616488447

TZRSITE

3

3

TZRFRQ

424.0

424.0

JUMP1

-9(9)×10⁻⁶

-9(9)×10⁻⁶

0.00

0.00

SEPARATION

0.000000 arcsec

The results are (thankfully) identical.

The MCMC fitter is considerably more complicated, so it has its own dedicated walkthroughs in MCMC_walkthrough.ipynb (for photon data) and examples/fit_NGC6440E_MCMC.py (for fitting TOAs).