dataesr/scientific-paragraphs-categorization huggingface dataset:
In Fig. 2 , we present noise models for four pulsars as an example of belonging to either of the above categories. Below, we present a summary of the noise models in the three datasets for each of the 25 pulsars. PSR J0030+0541 (Category 1). DR2full and DR2new largely agree, both with achromatic noise only. The DR2 posterior is better constrained. DR1 finds the same achromatic noise, but is unable to cleanly distinguish it from the DM noise. PSR J0613-0200 (Category 3). DR2 and DR1 both see achromatic noise of similar amplitude, but the spectral index is unconstrained for DR1, whereas DR2 finds the achromatic noise to have an index >4. DR2new is unable to detect this noise, probably due to the relatively short data (∼10 yr), and only finds DM variations. The posterior for γ DM shifts somewhat between the three datasets but are largely consistent and slightly flatter than Kolmogorov.
PSR J0751+1807 (Category 1). All three datasets are largely consistent, finding only DM variations with a spectral slope consistent with Kolmogorov. PSR J0900-3144 (Category 3). DR2 and DR2new posteriors are consistent, finding achromatic noise with a very flat spectral shape <2, and DM variations with an unconstrained spectral slope. The achromatic noise is much flatter than expected from classical power-law timing noise and may represent another noise process. The chromatic noise in DR1 data is consistent but unable to discern the achromatic noise. PSR J1012+5307 (Category 3). DR2full has a tightly constrained posterior for DM and achromatic noise. The achromatic index is very flat <2 and appears to be dominated by the highfrequency power above 1/yr. DR2new finds similar achromatic noise, and similar amplitude DM variations, but is barely able to constrain γ DM.
We speculate that this is because the longer dataset of DR2full can rule out very steep DM variations given the absence of steep achromatic noise. DR1 finds achromatic noise with a marginally steeper spectrum and finds an upper limit on DM variations just below that observed in the DR2full data. PSR J1022+1001 (Category 2). DR2full and DR2new are consistent with each other and show a very flat spectrum DM variation and an achromatic noise with γ red ∼ 4. The flatspectrum DM variation in this pulsar could likely be due to high-frequency residuals remaining from the solar wind contribution as this pulsar passes close to the ecliptic. A revised interpretation is given in Sect. 5. 1, after performing a chromatic index evaluation for each time-correlated component. DR1 showed flat spectrum achromatic noise and little variation in DM.
PSR J1730-2304 (Category 1). Generally consistent between the three datasets, with DR1 unable to distinguish between DM and achromatic noise. DR2full and DR2new find only DM variations consistent with Kolmogorov. PSR J1738+0333 (Category 2). DR1 does not find any noise. DR2full and DR2new find a flat-spectrum noise process, but the model selection prefers achromatic noise for DR2full and chromatic noise for DR2new. In both cases, the evidence is marginal for any one particular model. PSR J1744-1134 (Category 3). The three datasets are superficially similar, but paint a confusing picture when taken together. DR1 finds no variations in DM and an achromatic process with γ red ∼ 3. DR2full finds DM variations with γ DM < 1, amplitude above the upper limit for DM variations from DR1, and achromatic noise of similar amplitude but with a steeper, but largely unconstrained spectral slope, γ red > 2.
5. Indeed, the DR2full posterior for the achromatic noise is bimodal with a component that is largely consistent with DR1 and a component that is much steeper (γ red > 5). DR2new does not find evidence of achromatic noise, but has consistent DM variations with DR2full. The very flat spectrum DM variations, inconsistent with DR1, suggest that the chromatic noise is not a power-law process but instead dominated by high-frequency variations to which DR1 was insensitive because of the limited instantaneous frequency coverage. The inconsistency of the achromatic noise may indicate there is another underlying process at work in this pulsar beyond a single power-law achromatic noise and DM variations. This pulsar is observed by all members of the IPTA, and hence we suspect that the picture will become clearer when studied with a combined IPTA dataset.
PSR J1751-2857 (Category 1). Generally consistent between the three datasets. DR1 only finds upper limits, but DR2full and DR2new find a DM variation consistent with Kolmogorov. PSR J1801-1417 (Category 1). Generally consistent between the three datasets. DR2full and DR2new better constrain the DM variation, and the spectral shape is consistent with Kolmogorov. PSR J1804-2717 (Category 1). Generally consistent between the three datasets. DR2full and DR2new much better constrain the DM variations, and the spectral shape is consistent with Kolmogorov, although both prefer a flatter γ DM. PSR J1843-1113 (Category 1). Generally consistent between the three datasets. DR2full and DR2new much better constrain the DM variation, and the spectral shape is consistent with Kolmogorov. PSR J1857+0943 (Category 2).
PSR J1910+1256 (Category 1). Generally consistent between the three datasets. DR1 only finds upper limits, while DR2full and DR2new find DM variations consistent with Kolmogorov, but γ DM is not well constrained. PSR J1911+1347 (Category 2). Generally consistent between the three datasets. DR1 is not able to distinguish between DM and achromatic noise, but DR2full and DR2new find a DM variation consistent with Kolmogorov. PSR J1918-0642 (Category 2). DR1 finds achromatic noise with a preference for large γ red , and does not find significant DM variations. DR2full does not find evidence for achromatic noise, but instead finds a DM variation consistent with Kolmogorov, although also consistent with a steeper spectrum. DR2new is consistent with DR2full. This is another example of achromatic noise in DR1 being interpreted as chromatic noise when wide-band receivers are used.
Changes in noise models after the inclusion of the InPTA data Here we study the impact of including low radio frequency observations from the InPTA on the estimated noise models. The InPTA dataset complements the EPTA data with simultaneous observations at 300-500 MHz and at 1260-1460 MHz between MJDs 58235 and 59496 observed with the upgraded Giant Metrewave Radio Telescope (uGMRT). The frequency coverage at 300-500 MHz is particularly important since EPTA has a limited number of observations at this frequency band. Therefore, the inclusion of the InPTA data is of particular interest in constraining noise due to the IISM such as DM and scattering variations. To allow a quantitative comparison of the posterior noise models before and after the inclusion of InPTA data, we adapt and employ a tension metric package as detailed in Raveri & Doux (2021) 6.
We begin by introducing the probability density for parameter differences between the posteriors of two sets of parameters θ 1 and θ 2 to be P(∆θ) where ∆θ ≡ θ 1 -θ 2. Employing a joint posterior P(θ 1 , θ 2 ) ≡ P(θ 1 , θ 2 |d 1 , d 2 ), we can write P(θ 1 , ∆θ) = P(θ 1 , θ 1 -∆θ), where d 1 and d 2 refer to the two data sets. Integrating the base parameters, we obtain P(∆θ) = V π P(θ 1 , θ 1 -∆θ) dθ 1 , (9) where V π is the volume of the parameter space where the prior is non-vanishing, also known as the support of the prior. If the two datasets, specified by the parameter θ, are conditionally independent, it is possible to sample the posteriors P(θ 1 ) and P(θ 2 ) separately  (Raveri & Doux 2021). In this scenario, the expression for P(∆θ) becomes P(∆θ) = V π P(θ 1 ) P(θ 1 -∆θ) dθ 1. ( 10 ) Thereafter, we obtain the mean probability of the presence of a parameter shift using the following equation.
PSR J1744-1134 As described in Sect. 4. 1, this pulsar exhibits a bimodal posterior for achromatic noise and a very flat spectrum for DM variations with the DR2full dataset. The inclusion of the InPTA data measures DM variations with a much lower spectral index and a lower amplitude. This removes the observed bimodality in the achromatic noise and allows a tighter constraint on a single mode as shown in Fig. 5.
PSR J1909-3744 The behaviour observed in PSR J1909-3744 appears to be very similar to PSR J0613-0200, in which including the InPTA data produces a lower spectral index and a higher amplitude for the chromatic noise. This is also likely caused by the sharp change in DM variations in the last two years of the dataset (cf. Fig. 3 ). This feature was already reported in  Tarafdar et al. (2022)  and is also shown in  Curyło et al. (2023)  that used, respectively, the first data release of InPTA and the ultra-wide-bandwidth lowfrequency data from PPTA. Interestingly, we do not observe any associated impact on the RN posterior distributions.
Implications As expected the much-improved frequency coverage of DR2 has meant that the chromatic noise is much better constrained with the DR2full and DR2new datasets. Surprisingly, in some cases, signals attributed to achromatic processes in DR1 have been attributed to chromatic noise in DR2. Of particular interest, however, are pulsars such as J1713+0747 and J1744-1134, in which the choice of dataset seems to affect the noise model hyperparameters in a way that is not easily understood. Although these effects are rather subtle, these are two of the best-timed pulsars in the EPTA, and this inconsistency may be an indication that more complex noise models may be needed for the best-timed pulsars and longest datasets. We assume that the noise process is a stationary Gaussian process with a power-law spectral density, and this may suggest that one or more of these assumptions are incorrect.
Further insights are hampered by the fact that it can be hard to disentangle the effect of improvements in the observing systems, especially sensitivity and observing frequency coverage, from changes in the observable pulsar noise properties, either due to time-variable processes or from processes only detectable on long timescales. Combined IPTA datasets bring additional complications, but the combination of improved instantaneous sensitivity and frequency coverage, particularly in the latest instrumentation, will be our best chance to separate instrumental effects from those intrinsic to the pulsars. Furthermore, the direct combination of data can make use of a dropout analysis to identify any telescopeor backend-specific instrumental effects that may mask the real behaviour of the pulsars.
Pulsar DR2full+ DR2new 4. 9 2. 0 J2322+2057 X X 4. 1 2. 0 X X 3. 7 1. 9 α RN α DM B cus std B cus+RN30 cus α RN α DM B cus std B cus+RN30 cus J0030+0451 -1. 23 Notes. For each dataset, the two first columns display the medians and 95% confidence intervals of the chromatic index posterior distributions corresponding, respectively, to the RN and DM components. The values displayed in bold emphasise the numbers that are inconsistent with the expected values (0 for RN and 2 for DM). The third columns show the Bayes factor for the customised model against the standard noise model (RN and DM with resp. 30 and 100 frequency bins). The last columns display the Bayes factor evaluations for an additional red noise component with 30 frequency bins with the customised noise models. customised noise models over the standard models are shown in Col.
3 of Table  7  for each dataset. Following the  Kass & Raftery (1995)  scale, we find -weak support (B cus stand ∈ [1, 3]) for 11 and 13 pulsars, -positive support (B cus  stand ∈ [3, 20]) for five and seven pulsars, -strong support (B cus stand ∈ [20, 100]) for two and zero pulsars, -very strong support (B cus stand ≥ 100) for six and four pulsars, in DR2full+ and DR2new respectively. We notice higher Bayes factor values for DR2full+ compared to DR2new for most pulsars, which can be partially explained by the better performance of the standard noise models to describe higher PSD frequencies for DR2new as its timing baseline is approximately two times shorter than DR2full+. Furthermore, cases with "strong" and "very strong" evidence are obtained for the pulsars that include two time-correlated components (RN+DM, DM+SV), except for PSR  J1801-1417.
We also observe a very significant reduction of Bayes factors for PSR J1909-3744 with the DR2new, even if the preferred model for this pulsar includes both RN and DM for the two datasets. To assess the validity of the customised noise models, we checked the chromatic index α values used for the timecorrelated noise components fixed at 0, 2 and 4 for RN, DM and SV respectively. To do so, we used custom noise models for each pulsar and set α for each noise component as a hyperparameter to estimate its posterior distribution, as performed in recent PTA analyses (e. g. Goncharov et al. 2021; Chalumeau et al. 2022). If the pulsar has two time-correlated components, we perform two analyses to evaluate each α independently while fixing the index of the other component to its relevant value (0 if RN and 2 if DM).
Simulations We have implemented a toolkit to generate repeatable simulations of EPTA datasets to validate noise models. Simulations are generated using the toasim framework that is distributed with tempo2. In brief, the method is to generate a set of 'idealised' TOAs that produce zero residual with respect to a seed pulsar parameter file and then add perturbations for each of the simulated noise processes. For an idealised TOA t i , the final simulated TOA is given by t final,i = t i + y red (t i ) + y dm (t i ) + y scat (t i ) + y white (t i ), ( 12 ) where each of the perturbations y is discussed in the following sections. In principle, this can only be solved iteratively, but in practise we can neglect second-order corrections as long as the perturbations are small. Therefore, we typically subtract a quadratic from each y(t), which significantly reduces the general amplitude of the perturbations, especially for steep red noise processes.
Removing a quadratic in this way does not affect the noise models, but it does mean that the mean pulsar spin frequency and frequency derivative are unchanged from realisation to realisation. This is hence equivalent to fitting for F0 and F1 on each dataset, and small variations in these parameters are largely uninteresting, and in many senses arbitrary for a simulation, this does not affect the overall usefulness of the simulations. Significant improvements have been implemented to the toasim code as part of this work, which were required to ensure that the definitions of the model parameters are consistent between this work and the simulation code. The simulation code has been adapted to be consistent with the tempo2 and temponest definitions of the model parameters, although these are generally trivially related to other definitions.
P-P plots One tool to validate Bayesian analysis tools is by studying posterior quantiles from simulated datasets  (Cook et al. 2006). Our method largely follows that implemented in bilby  (Ashton et al. 2019) , where we simulate a large number of datasets, j, with 'true' model parameters Θ j drawn from a prior distribution p(Θ). We then use our Bayesian software, in this case temponest, to generate samples, θ i, j of the posterior distribution of the parameters given the prior distribution (see Table  9 ), for each of the simulated datasets, following the same approach as for the real data. We can then, for each parameter i, and each simulated dataset j, compute the quantile q i, j , within the 1-d posterior that the 'true' value of the parameter lies, that is, the weighted fraction of samples for which θ i, j < Θ i, j.
This is quite common in the pulsar datasets, as power-law red-noise processes below the detection threshold will have log amplitude unbounded on the low side and the spectral slope may simply return the prior. If the prior does not match the simulation and posterior generation, this will inevitably lead to a perceived error in the posterior computation for such cases. We choose a prior range such that most simulations have noise processes that are largely representative of what we estimated in the real data, and wide enough that the posterior of the real data would not be heavily constrained by the choice of prior. Results: For simplicity, we first consider the simulation of only achromatic red noise. The P-P plot, shown in Fig. 6a , indicates generally good consistency in the posterior of γ red , but there is an excess at low confidence intervals for log (A red ).
The investigation of individual results shows a correlation between the log (A red ) confidence interval and the injected γ red , visualised in Fig. 6b. We believe that the observed correlation suggests that the excess results are driven by cases where γ red is large, and hence we suspect that they are cases where the red noise has such a steep spectrum that it is no longer able to be expressed in a Fourier basis with lowest frequency at 1/T span. To understand this, we recall that the Fourier-basis Gaussian process models used in this work, and widely used by other similar projects are naturally periodic on T span due to the nature of the Fourier basis. This use of a Fourier basis vastly speeds up the computation compared to directly evaluating the covariance matrix for these large datasets, but it requires that the noise process can be well approximated by a periodic function.
Our hypothesis is that there is an intrinsic bias for slightly flatter spectral slopes, which then manifests itself as a corresponding bias in the amplitude at 1 yr -1. We repeat this with simulations that include achromatic noise red noise, excess white noise, DM variations, and scattering variations. The resulting P-P plots are very similar to the simulation with achromatic red noise only. Figure  6d  shows the P-P plot when the noise is periodic on 1/T span , showing a similarly small but significant bias in the posterior parameters for all three parameters. The similarity between all three components of the noise model to that obtained when only injecting achromatic noise suggests that the bias is due to something fundamental in the Fourier domain Gaussian process modelling of the noise, rather than being driven by the choice of the dataset.
Conclusions The noise models presented here are our current best estimates for stochastic noise in the 25-pulsar EPTA DR2 dataset. With the inclusion of wide-band receivers and additional low-frequency data from the InPTA collaboration, all but four pulsars show evidence for chromatic noise, specifically in the form of DM variations. However, several of these pulsars seem to have powerlaw noise with a much flatter spectrum than expected from DM variations in the turbulent IISM. This may be attributed to additional high-frequency terms, but it may also suggest other processes leaking into the model for DM variations. Compared to the EPTA DR1 and InPTA datasets, we find that the choice of preferred noise model can change, including a couple of cases A49, page 16 of 20 where chromatic noise that was ruled out in EPTA DR1 becomes detected in EPTA DR2.
We feel that this is evidence that the assumptions typically made in noise modelling are not strictly true for all pulsars. Either there are additional processes that need to be considered, such as noise that is not purely modelled by a power law, or the spectral properties of the noise vary over time, or there are systematic instrumental effects that affect particular parts of the dataset. In reality, none of these assumptions are likely completely true, and the improvements in data quality and time span of the EPTA DR2 data particularly highlight these subtle effects. The fact that the best noise models can depend on the frequency and time coverage of the dataset implies that great care is needed when comparing the noise models across different PTAs, and hence we suggest that a thorough investigation into the time and frequency stationary of the pulsar noise models, as well as exploration of additional noise terms, is best undertaken within the IPTA framework, which necessarily has better time and frequency coverage than any individual PTA.
We further saw that even if we simulate noise that has the same periodic nature as the noise model, there is a very small bias (∼2%) in the recovered spectral slope. Although these tests require large numbers of trials and, hence, large computing costs, we encourage further testing of this nature within the IPTA framework to attempt to better understand the origin and any possible effect of these biases. It is important to note that although we find several areas of investigation for a better understanding of the pulsar noise processes, the custom noise models presented here model the noise in the EPTA dataset extremely well, and the majority of the pulsar noise models are consistent between all datasets that we have compared. We also demonstrate that our customised noise models improve our overall sensitivity to GWB signals over the 'standard' models for both the complete DR2full+ dataset as well as when focussing only on the new purpose-built EPTA instrumentation in DR2new.
Acknowledgements. The  European Pulsar Timing Array  (EPTA) is a collaboration between European and partner institutes, namely  ASTRON (NL) ,  INAF/Osservatorio di Cagliari (IT) ,  Max-Planck-Institut für Radioastronomie (GER) ,  Nançay/Paris Observatory (FRA) , the  University of Manchester (UK) , the  University of Birmingham (UK) , the  University of East Anglia (UK) , the  University of Bielefeld (GER) , the  University of Paris (FRA) , the  University of Milan-Bicocca (IT) , the  Foundation for Research and Technology, Hellas  (GR), and  Peking University (CHN) , with the aim to provide high-precision pulsar timing to work towards the direct detection of low-frequency gravitational waves. An Advanced Grant of the  European Research Council  allowed to implement the Large European Array for Pulsars (LEAP) under Grant Agreement Number  227947  ( PI M.
We acknowledge financial support from " Programme National de Cosmologie and Galaxies" (PNCG) , and " Programme National Hautes Energies" (PNHE)  funded by  CNRS/INSU-IN2P3-INP ,  CEA  and  CNES, France. We acknowledge financial support from  Agence Nationale de la Recherche  ( ANR-18-CE31-0015 ),  France. The  Westerbork Synthesis Radio  Telescope is operated by the  Netherlands Institute for Radio Astronomy (ASTRON ) with support from the  Netherlands Foundation for Scientific Research (NWO). The  Sardinia Radio Telescope  (SRT) is funded by the  Department of University and Research (MIUR) , the  Italian Space Agency (ASI) , and the  Autonomous Region of Sardinia (RAS)  and is operated as a National Facility by the  National Institute for Astrophysics (INAF). The work is supported by the  National SKA programme of China  ( 2020SKA0120100 ),  Max-Planck Partner Group ,  NSFC   11690024 ,  CAS Cultivation Project for FAST Scientific.
This work is also supported as part of the " LEGACY "  MPG-CAS  collaboration on low- frequency gravitational wave astronomy. JA acknowledges support from the  European Commission  (Grant Agreement number:  101094354 ). JA and SCha were partially supported by the  Stavros Niarchos Foundation (SNF)  and the  Hellenic Foundation for Research and Innovation  (H. F. R. I. ) under the  2nd Call of  the " Science and Society -Action Always strive for excellence -Theodoros Papazoglou " (Project Number:  01431 ). AC acknowledges support from the  Paris Île-de-France Region. AC, AF, ASe, ASa, EB, DI, GMS, MBo acknowledge financial support provided under the  European Union 's  H2020   ERC Consolidator Grant  " Binary Massive Black Hole Astrophysics " ( B Massive , Grant Agreement:  818691 ). GD, KLi, RK and MK acknowledge support from  European Research Council (ERC)   Synergy Grant  " BlackHoleCam ", Grant Agreement Number  610058.