Explanation

This section is to explain pulsar timing, how PINT works, and why it is built the way it is.

PINT is used for pulsar timing and related activities. Some of it may make a lot more sense if you know something about the science it is used for. You can find an excellent introduction in the Handbook of Pulsar Astronomy, by Lorimer and Kramer. This document is aimed at using PINT specifically, and may also be more understandable if you have used other pulsar timing software, TEMPO or TEMPO2, though we hope that you will find PINT sufficient for all your needs!

Time

With modern instrumentation, we are able to measure time - both time intervals and an absolute time scale - to stupendous accuracy. Pulsar timing is a powerful tool in large part because it takes advantage of that accuracy. Getting time measurements and calculations right to this level of accuracy does require a certain amount of care, in general and while using (and writing) PINT.

Precision

The first challenge that arises is numerical precision. Computers necessarily represent real numbers to finite precision. Python, in particular, uses floating-point numbers that occupy 64 bits, 11 of which encode the exponent and 53 of which encode the mantissa. This means that numbers are represented with a little less than 16 decimal digits of precision:

>>> import numpy as np
>>> np.finfo(float).eps
2.220446049250313e-16
>>> 1 + np.finfo(float).eps
1.0000000000000002
>>> 1 + np.finfo(float).eps/2
1.0
>>> 1 + np.finfo(float).eps/2 == 1
True

Unfortunately, we have observations spanning decades and we would often like to work with time measurements at the nanosecond level. It turns out that python’s floating-point numbers simply don’t have the precision we need for this:

>>> import astropy.units as u
>>> (10*u.year*np.finfo(float).eps).to(u.ns)
<Quantity 70.07194824 ns>

That is, if I want to represent a ten-year span, the smallest increment python’s floating-point can cope with is about 70 nanoseconds - not enough for accurate pulsar timing work! There are a number of ways to approach this problem, all somewhat awkward in python. One approach of interest is that numpy provides floating-point types, for example, numpy.longdouble, with more precision:

>>> np.finfo(np.longdouble).eps
1.084202172485504434e-19
>>> (10*u.year*np.finfo(np.longdouble).eps).to(u.ns)
<Quantity 0.03421482 ns>

These numbers are represented with 80 bits, and most desktop and server machines have hardware for computing with these numbers, so they are not much slower than ordinary (“double-precision”, 64-bit) floating-point numbers. Let me warn you about one point of possible confusion: modern computers have very complicated cache setups that prefer data to be aligned just so in memory, so numpy generally pads these numbers out with zeroes and stores them in larger memory spaces. Thus you will often see np.float96 and np.float128 types; these contain only numbers with 80-bit precision. Actual 128-bit precision is not currently available in numpy, in part because on almost all current machines all calculations must be carried out in software, which takes 20-50 times as long.

An alternative approach to dealing with more precision than your machine’s floating-point numbers natively support is to represent numbers as a pair of double-precision values, with the second providing additional digits of precision to the first. These are generically called double-double numbers, and can be faster than “proper” 128-bit floating-point numbers. Sadly these are not implemented in numpy either. But because it is primarily time that requires such precision, astropy provides a type astropy.time.Time (and astropy.time.TimeDelta) that uses a similar representation internally: two floating-point numbers, one of which is the integer number of days (in the Julian-Day system) and one of which is the fractional day. This allows very satisfactory precision:

>>> (1*u.day*np.finfo(float).eps).to(u.ns)
<Quantity 0.01918465 ns>
>>> t = astropy.time.Time("2019-08-19", format="iso")
>>> t
<Time object: scale='utc' format='iso' value=2019-08-19 00:00:00.000>
>>> (t + 0.1*u.ns) - t
<TimeDelta object: scale='tai' format='jd' value=1.1102230246251565e-15>
>>> ((t + 0.1*u.ns) - t).to(u.ns)
<Quantity 0.09592327 ns>

Thus it is important when dealing with a time to ensure that it is stored in either an astropy time object or a np.longdouble. Because python’s default is to use less precision, it is easy to lose digits:

>>> 1+np.finfo(np.longdouble).eps
1.0000000000000000001
>>> print("Number: {}".format(1+np.finfo(np.longdouble).eps))
Number: 1.0
>>> print("Number: {}".format(1+np.finfo(float).eps))
Number: 1.0000000000000002
>>> print("Number: {}".format(str(1+np.finfo(np.longdouble).eps)))
Number: 1.0000000000000000001

Time Scales

A second concern when dealing with times at this level of precision is that Einstein’s theory of relativity becomes relevant: Clocks at the surface of the Earth advance more slowly than clocks in space nearby because of the slowing of time by the Earth’s gravity. As the Earth moves around the Sun, its changing velocity affects clock rates, and so does its movement deeper and shallower in the Sun’s gravity well. Of course none of these things affect a pulsar’s rotation, so we need some way to compensate for that.

On a more human scale, observations are recorded in convenient time units, often UTC; but UTC has leap seconds, so some days have one more second (or one fewer) than others!

The upshot of all this is that if you care about accuracy, you need to be quite careful about how you measure your time. Fortunately, there is a well-defined system of time scales, and astropy.time.Time automatically keeps track of which one your time is in and does the appropriate conversions - as long as you tell it what kind of time you’re putting in, and what kind of time you’re asking for:

>>> t = astropy.time.Time("2019-08-19", format="iso", scale="utc")
>>> t
<Time object: scale='utc' format='iso' value=2019-08-19 00:00:00.000>
>>> t.tdb
<Time object: scale='tdb' format='iso' value=2019-08-19 00:01:09.183>

The conventional time scale for working with pulsars, and the one PINT uses, is Barycentric Dynamical Time (TDB). You should be aware that there is another time scale, not yet fully supported in PINT, called Barycentric Coordinate Time (TCB). Because of different handling of relativistic corrections, the TCB timescale does not advance at the same rate as TDB (there is also a many-second offset). TEMPO2 uses TCB by default, so you may encounter pulsar timing models or even measurements that use TCB. PINT provides a command line tool tcb2tdb to approximately convert TCB timing models to TDB. PINT can also optionally convert TCB timing models to TDB (approximately) upon read.

Note that the need for leap seconds is because the Earth’s rotation is somewhat erratic - no, we’re not about to be thrown off, but its unpredictability can get as large as a second after a few years. So the International_Earth_Rotation_Service announces leap seconds about six months in advance. This means that astropy and pint need to keep their lists of leap seconds up-to-date by checking the IERS website from time to time.

It is also conventional to record pulsar data with reference to an observatory clock, usually a maser, that may drift with respect to International Atomic Time (TAI). Usually, GPS is used to track the deviations of this observatory clock and these deviations are recorded in a file known as a clock file. PINT also needs up-to-date versions of these observatory clock correction files to produce accurate results.

Even more detail about how PINT handles time scales is available on the github wiki.

Specifically, there is a complexity in using MJDs to specify times in the UTC time scale, which is the customary way observatories work. PINT attempts to handle this correctly by default, but if you see timing anomalies on days with leap seconds, this may be the problem. Alternatively, you may not be using up-to-date leap-second data files, or the process that generated the MJDs may not (this is a particular concern when working with X-ray or gamma-ray data).

Dispersion Measure (DM)

Radio waves emitted by the pulsar experience dispersion as they travel through the ionized interstellar medium (ISM). The time delay due to the interstellar dispersion is given by \(\frac{K\times DM}{\nu^2}\), where \(\nu\) is the frequency of the radio signal. The dominant source of this dispersion is the presence of free electrons in the ISM, and to a first approximation, the DM can be interpreted as the electron column density along the line of sight to the pulsar. \(K\) is known as the DM constant, and should be equal to \(\frac{e^2}{8 \pi ^2 c \epsilon _0 m_e} \approx 1.3445365918(9)\times 10^{-7}\; \text{m}^2/\text{s}\) based on the latest measurements of the physical constants. However, pulsar astronomers have traditionally used a fixed value \(1.3447217\times 10^{-7}\; \text{m}^2/\text{s}\) for \(K\) over the years. For example, the Handbook of Pulsar Astronomy by Lorimer & Kramer (Chapter 5) provides the value \(2.41\times 10^{-4}\; \text{MHz}^{-2} \text{pc}\, \text{cm}^{-3} s^{-1}\) for the reciprocal of \(K\). PINT follows this convention to be compatible with older pulsar ephemerides and with other pulsar timing packages. The value of \(K\) used by PINT can be accessed as pint.DMconst.

It should also be noted that there are other effects contributing to the dispersion delay than the free electrons, such as ions in the ISM, interstellar magnetic fields, and the ISM temperature. Hence, it has been argued (see Kulkarni 2020 https://arxiv.org/abs/2007.02886) that the dispersion slope \(K\times DM\) should be treated as the primary observable rather than the DM, which is usually interpreted as the electron column density. The dispersion slope corresponding to a DM value can be computed using pint.derived_quantities.dispersion_slope(). A DM value measured based on the conventional value of \(K\) can be converted to a value based on the latest physical constant values using pint.utils.convert_dispersion_measure().

The total DM and dispersion slope predicted by a given timing model (pint.models.timing_model.TimingModel) for a given set of TOAs (pint.toa.TOAs) can be computed using pint.models.timing_model.TimingModel.total_dm() and pint.models.timing_model.TimingModel.total_dispersion_slope() methods respectively.

Offsets in pulsar timing

Offsets arise in pulsar timing models for a variety of reasons. The different types of offsets are listed below:

Overall phase offset (PHOFF)

The pulse phase corresponding to the TOAs are usually computed in reference to an arbitrary fiducial TOA known as the TZR TOA (see pint.models.absolute_phase.AbsPhase). Since the choice of the TZR TOA is arbitrary, there can be an overall phase offset between the TZR TOA and the measured TOAs. There are three ways to account for this offset: (1) subtract the weighted mean from the timing residuals, (2) make the TZR TOA (given by the TZRMJD parameter) fittable, or (3) introduce a fittable phase offset parameter between measured TOAs and the TZR TOA. Traditionally, pulsar timing packages have opted to implicitly subtract the residual mean, and this is the default behavior of PINT. Option (2) is hard to implement because the TZR TOA may be specified at any observatory, and computing the TZR phase requires the application of the clock corrections. The explicit phase offset (option 3) can be invoked by adding the PHOFF parameter, (implemented in pint.models.phase_offset.PhaseOffset). If the explicit offset PHOFF is given, the implicit residual mean subtraction behavior will be disabled.

In the pulsar ephemeris (par) file, an example PHOFF parameter looks like this:

PHOFF 0.1 1 0.001

System-dependent delays (`JUMP`s)

It is very common to have TOAs for the same pulsar obtained using different observatories, telescope receivers, backend systems, and data processing pipelines, especially in long-running campaigns. Delays can arise between the TOAs measured using such different systems due to, among other reasons, instrumental delays, differences in algorithms used for RFI mitigation, folding, TOA measurement etc., and the choice of different template profiles used for TOA measurement. Such offsets are usually modeled using phase jumps (the JUMP parameter, see pint.models.jump.PhaseJump) between TOAs generated from different systems.

Here are some examples for JUMP parameters in a par file:

JUMP -f 430_PUPPI 0.01 1 1e-5 JUMP tel ao 0.01 1 1e-5 JUMP mjd 55000 55100 0.01 1 1e-5 JUMP freq 1000 1400 0.01 1 1e-5

System-dependent DM offsets (`DMJUMP`s and `FDJUMPDM`s)

Similar to system-dependent delays, offsets can arise between wideband DM values measured using different systems due to the choice of template portraits with different fiducial DMs. This is usually modeled using DM jumps (the DMJUMP parameter, see pint.models.dispersion_model.DispersionJump). This type of offset only applies to the wideband DM values and not to the wideband TOAs.

Here are some examples for DMJUMP parameters in a par file:

DMJUMP -f 430_PUPPI 1e-4 1 1e-5 DMJUMP tel ao 1e-4 1 1e-5 DMJUMP mjd 55000 55100 1e-4 1 1e-5 DMJUMP freq 1000 1400 1e-4 1 1e-5

Similar offsets also arise in the case of narrowband TOAs. Unlike the wideband case, these offsets manifest as system-dependent corrections to the DM delay. They are modeled using the FDJUMPDM parameters (see see pint.models.dispersion_model.FDJumpDM)

Here are some examples for FDJUMPDM parameters in a par file:

FDJUMPDM -f 430_PUPPI 1e-4 1 1e-5 FDJUMPDM -f L-wide_PUPPI 1e-4 1 1e-5

System- and frequency-dependent offsets (`FDJUMP`s)

In narrowband datasets, the template profiles often do not adequately model the frequency-dependent evolution of pulse profiles, resulting in a frequency-dependent artefact in the timing residuals. This systematic effect is usually modeled phenomenologically as a log-polynomial function of frequency whose coefficients are the so-called FD parameters (see pint.models.frequency_dependent.FD). Sometimes, this effect needs to be modeled separately for different systems since different template profiles will be used for each system. This is achieved through system-dependent FD parameters or FDJUMP`s (see :class:`pint.models.fdjump.FDJump).

Here are some examples for FDJUMP parameters in a par file:

FD1JUMP -f L-wide_PUPPI 1e-4 1 1e-5 FD2JUMP -f L-wide_PUPPI 1e-4 1 1e-5 FD1JUMP -f 430_PUPPI 1e-4 1 1e-5 FD2JUMP -f 430_PUPPI 1e-4 1 1e-5

Observatories

PINT comes with a number of defined observatories. Those on the surface of the Earth are TopoObs instances. It can also pull in observatories from astropy, and you can define your own. Observatories are generally referenced when reading TOA files, but can also be accessed directly:

import pint.observatory
gbt = pint.observatory.get_observatory("gbt")

Observatory definitions

Observatory definitions are included in pint.config.runtimefile("observatories.json"). To see the existing names, pint.observatory.Observatory.names_and_aliases() will return a dictionary giving all of the names (primary keys) and potential aliases (values). You can also find the full list at Observatory List.

The observatory data are stored in JSON format. A simple example is:

"gbt": {
    "tempo_code": "1",
    "itoa_code": "GB",
    "clock_file": "time_gbt.dat",
    "apply_gps2utc": true,
    "itrf_xyz": [
        882589.289,
        -4924872.368,
        3943729.418
    ],
    "fullname": "The Robert C. Byrd Green Bank Telescope",
    "origin": "This data was obtained by Joe Swiggum from Ryan Lynch in 2021 September.\n"
},

The observatory is defined by its name (gbt) and its position. This can be given as geocentric coordinates in the International_Terrestrial_Reference_System (ITRF) through the itrf_xyz triple (units as m), or geodetic coordinates (WGS84 assumed) through lat, lon, alt (units are deg and m). Conversion is done through Astropy_EarthLocation.

The time corrections are specified by the clock_file parameter, which gives the time corrections to be applied to site arrival times to get to UTC. Usually this is done by reference to an observatory time standard that is tied to GPS, and so the times are in UTC(GPS). The apply_gps2utc parameter is a boolean that selects whether to apply the correction from UTC(GPS) to UTC that is derived from BIPM Circular T.

Other attributes are optional. Here we have also specified the tempo_code and itoa_code, and a human-readable origin string.

A more complex/complete example is:

"jbroach": {
      "clock_file": [
          {
              "name": "jbroach2jb.clk",
              "valid_beyond_ends": true
          },
          "jb2gps.clk"
      ],
      "clock_fmt": "tempo2",
      "apply_gps2utc" : true,
      "aliases": [
          "jboroach"
      ],
      "bogus_last_correction": true,
      "itrf_xyz": [
          3822625.769,
          -154105.255,
          5086486.256
      ],
      "origin": [
          "The Lovell telescope at Jodrell Bank.",
          "These are the coordinates used for VLBI as of March 2020 (MJD 58919). They are based on",
          "a fiducial position at MJD 50449 plus a (continental) drift velocity of",
          "[-0.0117, 0.0170, 0.0093] m/yr. This data was obtained from Ben Perera in September 2021.",
          "This data is for the Roach instrument - a different clock file is required for this instrument to accommodate recorded instrumental delays."
      ]
  }

Here we have included additional explicit aliases, specified the clock format via clock_fmt, and specified that the last entry in the clock file is bogus (bogus_last_correction). There are two clock files included in clock_file:

  • jbroach2jb.clk (where we also specify that it is valid_beyond_ends)

  • jb2gps.clk

These are combined to reference this particular telescope/instrument combination. For the full set of options, see TopoObs.

Adding New Observatories

In addition to modifying pint.config.runtimefile("observatories.json"), there are other ways to add new observatories. Make sure you define any new observatory before you load any TOAs.

1. You can define them pythonically:

import pint.observatory.topo_obs
import astropy.coordinates
newobs = pint.observatory.topo_obs.TopoObs("newobs", location=astropy.coordinates.EarthLocation.of_site("keck"), origin="another way to get Keck")

This can be done by specifying the ITRF coordinates, (lat, lon, alt), or a EarthLocation instance.

2. You can include them just for the duration of your python session:

import io
from pint.observatory.topo_obs import load_observatories
# GBT but no clock file
fakeGBT = r"""{
    "gbt": {
        "tempo_code": "1",
        "itoa_code": "GB",
        "clock_file": "",
        "apply_gps2utc": false,
        "itrf_xyz": [
            882589.289,
            -4924872.368,
            3943729.418
        ],
        "origin": "The Robert C. Byrd Green Bank Telescope.\nThis data was obtained by Joe Swiggum from Ryan Lynch in 2021 September.\nHowever this has no clock correction"
    }
    }"""
load_observatories(io.StringIO(fakeGBT), overwrite=True)

Note that since we are overwriting an existing observatory (rather than defining a completely new one) we specify overwrite=True.

3. You can define them in a different file on disk. If you took the JSON above and put it into a file /home/user/anothergbt.json, you could then do:

export $PINT_OBS_OVERRIDE=/home/user/anothergbt.json

(or the equivalent in your shell of choice) before you start any PINT scripts. By default this will overwrite any existing definitions.

4. You can rely on astropy. For instance:

import pint.observatory
keck = pint.observatory.Observatory.get("keck")

will find Keck. astropy.coordinates.EarthLocation.get_site_names() will return a list of potential observatories.

External Data

In order to provide sub-microsecond accuracy, PINT needs a certain number of data files, for example Solar System ephemerides, that would be cumbersome to include in the package itself. Further, some of this external data needs to be kept up-to-date - precise measurements of the Earth’s rotation, for example, or observatory clock corrections.

Most of this external data is obtained through astropy’s data downloading mechanism (see astropy.utils.data). This will result in the data being downloaded the first time it is required on your machine but thereafter stored in a “cache” in your home directory. If you plan to operate offline, you may want to run some commands before disconnecting to ensure that this data has been downloaded. Data that must be up-to-date is generally in the form of a time series, and “up-to-date” generally means that it must cover the times that occur in your data. This can be an issue for simulation and forecasting; there should always be a mechanism to allow out-of-date data if you can accept lower accuracy.

Clock corrections

Not all the data that PINT uses is easily accessible for programs to download. Observatory clock corrections, for example, may need to be obtained from the observatory through various means (often talking to a support scientist). PINT uses a global repository, https://ipta.github.io/pulsar-clock-corrections/ to retrieve up-to-date clock corrections for all telescopes it knows about. PINT should notify you when your clock files are out of date for the data you are using; be aware that you may obtain reduced accuracy if you have old clock correction files.

Normally, if you try to do some operation that requires unavailable clock corrections, PINT will emit a warning but continue. If you want to be stricter, you can specify limit="error" to various functions like pint.toa.get_TOAs().

If you need to check how up to date your clock corrections are, you can use something like get_observatory("gbt").last_clock_correction_mjd(): the function pint.observatory.Observatory.last_clock_correction_mjd() checks when clock corrections are valid for. For most telescopes, this combines the per-telescope clock correction with PINT’s global GPS and BIPM clock corrections (both of which cannot be reliably extrapolated too far into the future). PINT provides two convenience functions, pint.observatory.list_last_correction_mjds() and pint.observatory.check_for_new_clock_files_in_tempo12_repos(), that will help you check the state of your clock corrections.

If you need clock files that are not in the global repository, perhaps more recent versions or clock files for telescopes not included in the global repository or specific versions for reproducibility, you have several options:

  1. Set the environment variable PINT_CLOCK_OVERRIDE to point to a directory that contains clock files. Any clock file found there will supersede the version found in the global repository. You can also use pint.observatory.export_clock_files() to export the clock files you are currently using to a directory for use in this way later.

  2. Modify src/pint/data/runtime/observatories.json so that the observatory you are interested in points to the correct clock file. (You may have to redo pip install for PINT to make this take effect.) If you set clock_dir="TEMPO" or clock_dir="TEMPO2" then PINT will look in the clock directory referenced by your environment variables $TEMPO or $TEMPO2 (and nowhere else; it will no longer find clock corrections for this observatory that are included with PINT). You can also specify a specific directory as clock_dir="/home/burnell/clock-files/". Editing this file also allows you to choose between TEMPO- and TEMPO2-format clock corrections with the clock_fmt argument.

  3. Create a new observatory in your own code. This involves creating a new pint.observatory.topo_obs.TopoObs object like those in src/pint/data/runtime/observatories.json. As long as this object is created before you read in any TOAs that need it, and as long as its name does not overlap with any existing observatory, you should be able to create your custom observatory and point the clock correction files to the right place as above.

Structure of Pulsar Timing Data Formats

Pulsar timing data has traditionally been divided into two parts: a list of pulse arrival times, with sufficient metadata to work with (a .tim file), and a description of the timing model, with parameter values, metadata, and some fitting instructions (a .par file). These have been ad-hoc formats, created to be easy to work with (originally) using 1980s FORTRAN code (specifically TEMPO). The advent of a second tool that works with these files (TEMPO2) did not, unfortunately, come with a standardization effort, and so files varied further in structure and were not necessarily interpreted in the same way by both tools. As PINT is a third tool, we would prefer to avoid introducing our own, incompatible (obviously or subtly) file formats. We therefore formalize them here.

We are aware that not every set of timing data or parameters “in the wild” will follow these rules. We hope to be able to lay out a clear and specific description of these files and how they are interpreted, then elaborate on how non-conforming files are handled, as well as how TEMPO and TEMPO2 interpret these same files. Where possible we have tried to ensure that our description agrees with both TEMPO and TEMPO2, but as they disagree for some existing files, it may be necessary to offer PINT some guidance on how to interpret some files.

Parameter files (.par)

Parameter files are text files, consisting of a collection of lines whose order is irrelevant. Lines generally begin with an all-uppercase parameter name, then a space-separated list of values whose interpretation depends on the parameter.

We separate parsing such a file into two steps: determining the structure of the timing model, that is, which components make up the timing model and how many parameters they have, then extracting the values and settings from the par file into the model. It is the intention that in PINT these two steps can be carried out separately, for example manually constructing a timing model from a collection of components then feeding it parameter values from a parameter file. It is also the intent that, unlike TEMPO and TEMPO2, PINT should be able to clearly indicate when anomalies have occurred, for example if some parameter was present in the parameter file but not used by any model.

Selecting timing model components

We describe a simple procedure for selecting the relevant timing model components.

  • If the BINARY line is present in the parameter file, its value determines which binary model to use; if not, no binary model is used.

  • Each model component has one or more “special parameters” or families of parameters identified by a common prefix. If a par file contains a special parameter, or a known alias of one, then the timing model uses the corresponding component.

  • Components are organized into categories. No more than one component from each category may be present; some categories may be required but in others no component is necessary: - Solar system dispersion - Astrometry - Interstellar dispersion - Binary - Spin-down - Timing noise

  • Each component may indicate that it supersedes one or more others, that is, that its parameters are a superset of the previous model. In this case, if both are suggested by the parameter file, the component that is superseded is discarded. If applying this rule does not reduce the number of components in the category down to one, then the model is ambiguous.

We note that many parameters have “aliases”, alternative names used in certain par files. For these purposes, aliases are treated as equivalent to the special parameters they are aliases for. Also note that not all parameters need to be special for any component; the intent is for each component to identify a parameter that is unique to it (or models that supersede it) and will always be present.

We intend that PINT have facilities for managing parameter files that are ambiguous by this definition, whether by applying heuristics or by allowing users to clarify their intent.

This scheme as it stands has a problem: some parameter files found “in the wild” specify equatorial coordinates for the pulsar but ecliptic values for the proper motion. These files should certainly use ecliptic coordinates for fitting.

Timing files (.tim)

There are several commonly-used timing file formats. These are collections of lines, but in some cases they can contain structure in the form of blocks that are meant to be omitted from reading or have their time adjusted. We recommend use of the most flexible format, that defined by TEMPO2 and now also supported (to the extent that the engine permits) by TEMPO.

Fitting

A very common operation with PINT is fitting a timing model to timing data. Fundamentally this operation tries to adjust the model parameters to minimize the residuals produced when the model is applied to a set of TOAs. The result of this process is a set of best-fit model parameters, uncertainties on (and correlations between) these, and residuals from this best-fit model. This is carried out by constructing a pint.fitter.Fitter object from a pint.toa.TOAs object and a pint.models.timing_model.TimingModel object and then running the pint.fitter.Fitter.fit_toas() method; there are several example notebooks that demonstrate this. Nevertheless there are some subtleties to how fitting works in PINT that we explain here.

Timing noise and correlated errors

Precision pulsar timing requires a quite sophisticated model of the errors that appear in our measurement. While each TOA has an associated uncertainty estimate, in reality these can need to be adjusted to reflect unmodelled sources of error; PINT (and TEMPO and TEMPO2) provide two adjustments, EFAC and EQUAD. If these are set, and the claimed uncertainty is U, PINT will treat the uncertainty on a data point as \(\textrm{EFAC}\sqrt{U^2+\textrm{EQUAD}^2}\). We also expect a certain amount of correlation between measurements that were taken simultaneously but at different frequencies; this is parametrized by ECORR. More, the way we choose to handle “red” timing noise in pulsars is to treat it as a noise component that introduces long-term correlations in the timing measurements, where the amount of those correlations depends on the time between measurements and the spectrum of the timing noise. The introduction of correlations between the errors on TOAs requires a somewhat more complicated procedure for fitting models to TOAs, and even to simply measuring the goodness of fit of a model to TOAs.

The most direct way of handling correlated errors between TOAs is by constructing a covariance matrix describing all the correlations between the measurements; a square root of this matrix can be computed using the Cholesky decomposition, and this square root can be used to transform the fitting problem into a conventional least-squares problem. This procedure is described in Coles_et_al_2011 and implemented in PINT (via the full_cov=True option to fitters). Unfortunately this method requires a decomposition of a matrix that is the size of the number of TOAs by the number of TOAs; this can be very expensive in terms of memory and computation.

Fortunately, Lentati_et_al_2013 and van_Haasteren_and_Vallisneri_2015 describe a method for using a low-rank approximation to the covariance matrix to remove the need to ever construct these very large matrices; the implementation in PINT follows the mathematics in the NANOGrav_9-year data analysis paper, Appendix C.

The idea of this reduced-rank approach is to represent the correlations using basis functions - blocks of 1s for each set of residuals grouped by ECORR, or sinusoids for a red noise model - whose coefficients are added to the list of parameters to be fit. The linear least-squares fitting problem is then adjusted based on the prior estimates of the amplitudes of these basis functions (for example the ECORR value or the amplitude of sinusoids of that frequency in the timing model), and this modified least-squares fit is carried out. The best-fit combinations of these noise basis functions can be subtracted from the residuals to produce “whitened” residuals, and the goodness of fit can be described by taking the usual chi-squared of these whitened residuals and adding a term based on the sizes of the noise basis coefficients.

Specifically the mathematics takes an approximate solution and models the residuals as

\[\delta t = M\epsilon + Fa + Uj + n\]

where \(M\) is the Jacobian matrix of the model (the derivative of each predicted TOA with respect to each model parameter, \(\epsilon\) is an error in the model parameters, \(F\) is a “Fourier design matrix”, a set of sine and cosine functions at each of a range of frequencies, \(a\) is the amplitudes of these basis functions in the red noise contribution, \(U\) is a matrix of basis functions representing the ECORR blocks, \(j\) is their coefficients, and \(n\) is a vector of uncorrelated noise of amplitude coming from the adjusted TOA uncertainties. The NANOGrav_9-year paper gives expressions for the likelihood of such a representation, suitable for use in Bayesian fitting methods, but for PINT’s fitters the goal is to find the maximum-likelihood values for \(\epsilon\), a corresponding set of residuals \(n\), and a goodness-of-fit statistic distributed as a \(\chi^2\) distribution for some number of degrees of freedom.

The paper develops this, constructing additional matrices

\[ \begin{align}\begin{aligned}N_{ij} = E_i^2(\sigma_i^2+Q_i^2)\delta_{ij}\\T = \begin{bmatrix} M & F & U \end{bmatrix}\\\begin{split}b = \begin{bmatrix} \epsilon \\ a \\ j \end{bmatrix}\end{split}\\\begin{split}B = \begin{bmatrix} \infty & 0 & 0 \\ 0 & \phi & 0 \\ 0 & 0 & J \end{bmatrix}\end{split}\end{aligned}\end{align} \]

where \(N\) is a diagonal matrix of the adjusted TOA uncertainties, and \(B\) is a block matrix with diagonal matrices on the blocks; the \(\infty\) is a diagonal matrix of infinities (we will be using \(B^{-1]\)), while \(\phi\) and \(J\) are “weights” corresponding to the noise basis functions’ expected amplitudes.

They then construct the objects \(d = T^T N^{-1} \delta t\) and \(\Sigma = (B^{-1} + T^T N^{-1} T)\). Then they say that the maximum likelihood values of \(b\) and its uncertainties are given by

\[ \begin{align}\begin{aligned}b = \Sigma^{-1} d\\\textrm{cov}(b) = \Sigma^{-1}\end{aligned}\end{align} \]

This is what is implemented in PINT’s fitters, both the generalized least-squares fitter for narrowband data, and the fitter used for all wideband data (whether it has correlated errors or not).

It is perhaps worth noting that if \(B^{-1}\) were zero or omitted, these would be the equations for a linear least squares fit for \(b\) to match \(\delta t\) with variances represented in \(N\). The addition of \(B^{-1}\) in \(\Sigma\) is where our knowledge about the amplitudes of the noise basis functions is applied.

The formula is not worked out in the NANOGrav_9-year data set paper, but if we want a goodness-of-fit statistic for a set of model parameters that correctly reflects both the mis-fit of the data and also the penalization of the noise components, we need to fix all the model parameters we care about, reducing \(M\) to almost nothing (just a constant offset). So we compute residuals at the model parameters of interest, then we then do the fit as above, obtaining a maximum-likelihood \(b\) and a set of whitened residuals \(n\). We then report, as our goodness of fit,

\[\chi_G^2 = n^T N n + b^T B^{-1} b\]

Fitting algorithms

PINT is designed to be able to offer several alternative algorithms to arrive at the best-fit model. This both because fitting can be a time-consuming process if a suboptimal algorithm is chosen, and because different kinds of model and data require different calculations - narrowband (TOA-only) versus wideband (TOA and DM measurements) and uncorrelated errors versus correlated errors.

The TEMPO/TEMPO2 and default PINT fitting algorithms (pint.fitter.WidebandTOAFitter, for example), leaving aside the rank-reduced case, proceed like:

  1. Evaluate the model and its derivatives at the starting point \(x\), producing a set of residuals \(\delta y\) and a Jacobian M.

  2. Compute \(\delta x\) to minimize \(\left| M\delta x - \delta y \right|_C\), where \(\left| \cdot \right|_C\) is the squared amplitude of a vector with respect to the data uncertainties/covariance \(C\).

  3. Update the starting point by \(\delta x\).

TEMPO and TEMPO2 can check whether the predicted improvement of chi-squared, assuming the linear model is correct, is enough to warrant continuing; if so, they jump back to step 1 unless the maximum number of iterations is reached. PINT does not contain this check.

This algorithm is the Gauss-Newton_algorithm for solving nonlinear least-squares problems, and even in one-complex-dimensional cases can exhibit convergence behavior that is literally chaotic. For TEMPO/TEMPO2 and PINT, the problem is that the model is never actually evaluated at the updated starting point before committing to it; it can be invalid (ECC > 1) or the step can be large enough that the derivative does not match the function and thus the chi-squared value after the step can be worse than the initial chi-squared. These issues particularly arise with poorly constrained parameters like M2 or SINI. Users experienced with pulsar timing are frequently all too familiar with this phenomenon and have a collection of tricks for evading it.

PINT contains a slightly more sophisticated algorithm, implemented in pint.fitter.DownhillFitter, that takes more careful steps:

  1. Evaluate the model and its derivatives at the starting point \(x\), producing a set of residuals \(\delta y\) and a Jacobian M.

  2. Compute \(\delta x\) to minimize \(\left| M\delta x - \delta y \right|_C\), where \(\left| \cdot \right|_C\) is the squared amplitude of a vector with respect to the data uncertainties/covariance \(C\).

  3. Set \(\lambda\) to 1.

  4. Evaluate the model at the starting point plus \(\lambda \delta x\). If this is invalid or worse than the starting point, divide \(\lambda\) by two and repeat this step. If \(\lambda\) is too small, accept the best point seen to date and exit without convergence.

  5. If the model improved but only slightly with \(\lambda=1\), exit with convergence. If the maximum number of iterations was reached, exit without convergence. Otherwise update the starting point and return to step 1.

This ensures that PINT tries taking smaller steps if problems arise, and claims convergence only if a normal step worked. It does not solve the problems that arise if some parameters are nearly degenerate, enough to cause problems with the numerical linear algebra.

As a rule, this kind of problem is addressed with the Levenberg-Marquardt algorithm, which operates on the same principle of taking reduced steps when the derivative appears not to match the function, but does so in a way that also reduces issues with degenerate parameters; unfortunately it is not clear how to adapt this problem to the rank-reduced case. Nevertheless, PINT contains an implementation in pint.fitter.WidebandLMFitter, but it does not perform as well as one might hope in practice and must be considered experimental.

Coding Style

We would like PINT to be easy to use and easy to contribute to. To this end we’d like to ask that if you’re going to contribute code or documentation that you try to follow the below style advice. We know that not all of the existing code does this, and it’s something we’d like to change.

For a specific listing of the rules we try to write PINT code by, please see PINT coding style.

More general rules and explanations:

  • Think about how someone might want to use your code in various ways. Is it called something helpful so that they will be able to find it? Will they be able to do something different with it than you wrote it for? How will it respond if they give it incorrect values?

  • Code should follow PEP8. Most importantly, if at all possible, class names should be in CamelCase, while function names should be in snake_case. There is also advice there on line length and whitespace. You can check your code with the tool flake8, but I’m afraid much of PINT’s existing code emits a blizzard of warnings.

  • Files should be formatted according to the much more specific rules enforced by the tool black. This is as simple as pip install black and then running black on a python file. If an existing file does not follow this style please don’t convert it unless you are modifying almost all the file anyway; it will mix in formatting changes with the actual substantive changes you are making when it comes time for us to review your pull request.

  • Functions, modules, and classes should have docstrings. These should start with a short one-line description of what the function (or module or class) does. Then, if you want to say more than fits in a line, a blank line and a longer description. If you can, if it’s something that will be used widely, please follow the numpy docstring guidelines - these result in very helpful usage descriptions in both the interpreter and online docs. Check the HTML documentation for the thing you are modifying to see if it looks okay.

  • Tests are great! When there is a good test suite, you can make changes without fear you’re going to break something. Unit tests are a special kind of test, that isolate the functionality of a small piece of code and test it rigorously.

    • When you write a new function, write a few tests for it. You will never have a clearer idea of how it’s supposed to work than right after you wrote it. And anyway you probably used some code to see if it works, right? Make that into a test, it’s not hard. Feed it some bogus data, make sure it raises an exception. Make sure it does the right thing on empty lists, multidimensional arrays, and NaNs as input - even if that’s to raise an exception. We use pytest. You can easily run just your new tests.

    • Give tests names that describe what property of what thing they are testing. We don’t call test functions ourselves so there is no advantage to them having short names. It is perfectly reasonable to have a function called test_download_parallel_fills_cache or test_cache_size_changes_correctly_when_files_are_added_and_removed.

    • If your function depends on complicated other functions or data, consider using something like unittest.Mock to replace that complexity with mock functions that return specific values. This is designed to let you test your function specifically in isolation from potential bugs in other parts of the code.

    • When you find a bug, you presumably have some code that triggers it. You’ll want to narrow that down as much as possible for debugging purposes, so please turn that bug test case into a test - before you fix the bug! That way you know the bug stays fixed.

    • If you’re trying to track down a tricky bug and you have a test case that triggers it, running pytest tests/test_my_buggy_code.py --pdb will drop you into the python debugger pdb at the moment failure occurs so you can inspect local variables and generally poke around.

  • When you’re working with a physical quantity or an array of these, something that has units, please use Quantity to keep track of what these units are. If you need a plain floating-point number out of one, use .to(u.m).value, where u.m should be replaced by the units you want the number to be in. This will raise an exception (good!) if the units can’t be converted (u.kg for example) and convert if it’s in a compatible unit (u.cm, say). Adding units to a number when you know what they are is as simple as multiplying.

  • When you want to let the user know some information from deep inside PINT, remember that they might be running a GUI application where they can’t see what comes out of print. Please use logger. Conveniently, this has levels debug, info, warning, and error; the end user can decide which levels of severity they want to see.

  • When something goes wrong and your code can’t continue and still produce a sensible result, please raise an exception. Usually you will want to raise a ValueError with a description of what went wrong, but if you want users to be able to do something with the specific thing that went wrong (for example, they might want to use an exception to know that they have emptied a container), you can quickly create a new exception class (no more than class PulsarProblem(ValueError): pass) that the user can specifically catch and distinguish from other exceptions. Similarly, if you’re catching an exception some code might raise, use except PulsarProblem: to catch just the kind you can deal with.

There are a number of tools out there that can help with the mechanical aspects of cleaning up your code and catching some obvious bugs. Most of these are installed through PINT’s requirements_dev.txt.

  • flake8 reads through code and warns about style issues, things like confusing indentation, unused variable names, un-initialized variables (usually a typo), and names that don’t follow python conventions. Unfortunately a lot of existing PINT code has some or all of these problems. flake8-diff checks only the code that you have touched - for the most part this pushes you to clean up functions and modules you work on as you go.

  • isort sorts your module’s import section into conventional order.

  • black is a draconian code formatter that completely rearranges the whitespace in your code to standardize the appearance of your formatting. blackcellmagic allows you to have black format the cells in a Jupyter notebook.

  • pre-commit allows git to automatically run some checks before you check in your code. It may require an additional installation step.

  • make coverage can show you if your tests aren’t even exercising certain parts of your code.

  • editorconfig allows PINT to specify how your editor should format PINT files in a way that many editors can understand (though some, including vim and emacs, require a plugin to notice).

Your editor, whether it is emacs, vim, JupyterLab, Spyder, or some more graphical tool, can probably be made to understand that you are editing python and do things like highlight syntax, offer tab completion on identifiers, automatically indent text, automatically strip trailing white space, and possibly integrate some of the above tools.

The Zen of Python

by Tim Peters:

>>> import this
The Zen of Python, by Tim Peters

Beautiful is better than ugly.
Explicit is better than implicit.
Simple is better than complex.
Complex is better than complicated.
Flat is better than nested.
Sparse is better than dense.
Readability counts.
Special cases aren't special enough to break the rules.
Although practicality beats purity.
Errors should never pass silently.
Unless explicitly silenced.
In the face of ambiguity, refuse the temptation to guess.
There should be one-- and preferably only one --obvious way to do it.
Although that way may not be obvious at first unless you're Dutch.
Now is better than never.
Although never is often better than *right* now.
If the implementation is hard to explain, it's a bad idea.
If the implementation is easy to explain, it may be a good idea.
Namespaces are one honking great idea -- let's do more of those!