dataesr/scientific-paragraphs-categorization huggingface dataset:
We then allowed fitting of all transducer parameters freely and of the actuator parameters within their 95% confidence interval. Once this is done, all common parameters (actuator transfer function and resource competition) were sampled using the same procedure and fitting on the cocaine transducer was performed. To show that parameters are well constrained (proving they minimally explain the data from Fig. 1e ), Supplementary Figs. 15  and 16  show results of sampling parameters from the final parameters distribution (without fitting at that stage) and how they compare to the data. Objective functions and model scoring. In order to evaluate and compare our models, we used the following functions. This measures the percentage of error for each point. We present the average on all experiments in Supplementary Tables  1  and 2.
Acknowledgements We are grateful to  Pau Bernado ,  Annika Urbanek , and  Anna Morato  for helpful discussions on cell-free systems and extract preparation, and to  Gottfried Otting  for the BL21 Star (DE3)::RF1-CBD 3 strain. A. P. is supported by  INRA  ( French National Institute for Agricultural Research ) and the idEx  Paris-Saclay  interdisciplinary doctoral fellowship. M. Ko. is supported by  DGA  ( French Ministry of Defense ) and Ecole Polytechnique. P. S. and J. L. F. are supported by  ANR  (grant number  ANR-18-CE33-0015 ). J. L. F. acknowledges support from  BBSRC/EPSRC  (grant number  BB/M017702/1 ).
We aim to obtain a modal decomposition of the internal geomagnetic field. In order to do so, we perform a principal component analysis of two virtual observatory datasets, with 4-month sampling time, from the CHAMP (2001)(2002)(2003)(2004)(2005)(2006)(2007)(2008)(2009) and Swarm (2014-2017) satellite records. The spatial patterns of well-resolved modes calculated from the three field components all have internal origin as expected for these datasets, except for one Swarm mode. For both datasets, we find that the modes with the shortest timescales have also the smallest length scales as expected from a physical standpoint. Also, the energy ordering of the modes is from the least to the most variable, in agreement with independent results on the main field data spectrum. This is not achieved in regularised inversions of geomagnetic field data into time-varying spherical harmonic decomposition, where the highest degree terms have also the poorest time resolution.
g. Taira et al. (2017)  for a recent overview], is a widely used statistical tool to extract a minimal number of energetic coherent structures from a time sequence of scalar maps. It has the advantage of describing the space-time evolution of a physical quantity as the superposition of a few decorrelated components (or modes), a property referred to as dimensionality reduction. It is one among different databased modal techniques used as a prospective analysis of real or simulated data. Besides, in certain cases, physically meaningful structures displaying the symmetries of underlying source mechanisms may emerge from PCA modes. By directly using records of the geomagnetic field at fixed locations (observatories) to conduct a PCA analysis, it may be possible to extract features-with small length scales and/or short timescales-left aside when constructing spherical harmonic models.
Beggan et al. (2009)  followed up the approach developed by  Mandea and Olsen (2006)  and created VO datasets from both Ørsted and CHAMP satellite measurements. They found that external fields cannot be entirely removed and produce, as a result, unrealistic secular acceleration of the main field. Saturnino et al. (2018)  illustrated an alternative methodology with 2 years of Swarm data. They neither selected the data nor removed a model for the external field. They applied instead an equivalent source dipole technique. They accounted for the recorded magnetic field by an ensemble of timevarying dipoles located at the core-mantle boundary and reconstructed VO series at a uniform altitude from these time-varying dipoles. We undertake here a PCA analysis of the VO datasets recently created by  Barrois et al.
(2018). They calculated separate datasets from CHAMP and Swarm measurements which are as free as possible from contributions of the external field. Both time series have a time resolution of 4 months with almost no gaps despite strict selection criteria of the magnetic measurements. The next section ("Data" section) consists in a description of the VO datasets. It is followed in "Methods" section by a summary of the PCA method that we employed. Our main results are presented in "Results" section, where we compare the modes calculated from CHAMP and Swarm data, respectively, and a discussion follows in "Discussion" section, focusing on the issue of time resolution. The paper ends by pointing out, in "Conclusions and perspectives" section, the main conclusions and some interesting future developments.
Virtual observatories Here, we use two sets of recent VO series, similar to the one presented by  Barrois et al. (2018)  with some small changes including the number of VOs: one of them was calculated from measurements provided by the CHAMP vector magnetometer (from July 2000 to September 2010) and the other one was computed from three Swarm vector magnetometers (from November 2013 to December 2017). For the self-consistency of this paper, let us summarise the applied procedure. To calculate the magnetic field values at each VO location, firstly a data selection needs to be applied in order to reduce the noise and aliasing due to the highfrequency currents in the magnetosphere and the ionosphere, and charges circulating near and across the satellites orbit. The initial dataset consists of CHAMP MAG-L3 and Swarm Level 1b MAG-L, version 0501 at 15-second sampling rate data.
Large outliers deviating more than 500 nT from the CHAOS-6-x3 model  (Finlay et al. 2016 ), were removed. The next step in data selection consists in considering only night-time (Sun at most 10 • above horizon) and geomagnetically quiet conditions ( K p < 3 0 , |dRC/dt| < 3 nT/h, E m ≤ 0. 8 mV/m, B z > 0 and |B y | < 10 nT, where K p is the geomagnetic activity index, RC is the ring current disturbance index, E m is the merging magnetic field at the magnetopause, B y and B z are components of the interplanetary magnetic field, see  Barrois et al. (2018)  for a precise definition of these parameters). For the CHAMP dataset, along-track sums and differences were used, while the Swarm dataset benefits from using both along-track and cross-track sums and differences, which provide a signal less perturbed by external sources.
g. Laundal and Richmond 2017). The fields due to remote magnetospheric sources are calculated in the geocentric solar magnetic coordinate system (x axis along the Earth-Sun line, dipole axis in the xz plane). In order to divide the surface of a sphere into nearly equal area regions, the Recursive Zonal Equal Area Sphere Partitioning Toolbox  (Leopardi 2006 ) was used, giving 300 VOs for the dataset provided by Chris  Finlay and Magnus Hammer (personal communication). In the vicinity of each VO, the residual field can then be represented as the gradient of a scalar potential V (determined as a linear combination of cubic polynomials in local Cartesian coordinates) inside a cylinder, separately. For each VO position and at a given epoch all data inside a cylinder of radius 700 km and within 4 months were used to build a local data vector.
Data selection for PCA The VOs data from CHAMP satellite and Swarm constellation of three satellites were used to perform a PCA analysis of the magnetic field. VO data are provided for a fixed set of spatial coordinates, which is a requirement of PCA decomposition. In order to work with the longest possible simultaneous series in a fixed number of VOs, the period under study is shorter than the satellite mission periods, from November 2001 to November 2009 for CHAMP and from March 2014 to November 2017 for Swarm. Both datasets consist of vector magnetic field values every 4 months, for a total of 25 epochs for CHAMP and 12 epochs for Swarm. The number of grid points for CHAMP is N P = 292 , since 8 of the original VOs presented recurrent gaps. For Swarm, a different set was constructed, since field values representative of the core field were systematically missing at high latitude VOs, due to strong external perturbations.
For this reason, Swarm dataset consists of only N P = 258 VOs between -60 • and 60 • latitude (sub-auroral grid). Same as in  Shore et al. (2016) , PCA is applied on a data matrix with B r , B θ and B φ components (where (r, θ , φ) are spherical polar coordinates). Although Swarm data are more precise, the longer time series from CHAMP allows to expect a better resolution of modes that have annual to decadal characteristic timescales. The equal area distribution of grid points makes it unnecessary to apply a latitudinal weighting factor, which would have been the case for the geographic regular grid spacing approach or for the existing network of ground observatories  (Shore et al. 2016). The altitude R VO of the VOs obtained from CHAMP and Swarm data is 300 km and 500 km, respectively. This slightly complicates the comparison between the modes obtained from the two datasets.
In other terms, the consecutive EOFs give a good approximation of the successive time derivatives of the field at the mean epoch. The condition that has to be met is that τ W ≪ τ c W , where τ W is the dataset win- dow width and τ c W is a critical width that depends on the relative variance of the two lowest-order time derivatives of the signal. In certain cases, but not always, τ c W may be used as an estimate for the correlation time of the signal [see  Gibson et al. (1992) ]. The degree-0 Legendre polynomial does not arise here as a principal component because the time average of the field components has been subtracted before the PCA analysis. The two lowest-order derivatives in the remaining signal are order 1 and order 2, which together allow to define the critical time width, using as an approximation to these variances the first two C X eigenval- ues.
Identification of the principal components as Legendre polynomials The (normalised) time series corresponding to the first four modes from CHAMP are represented in Fig. 1. We discuss them together with the time series for modes 1, 2 and 4 from Swarm (see Fig. 2 ). It appears that these principal components are well approximated by polynomials of degree increasing with the mode order. Using eigenvalues of the covariance matrices from CHAMP and from Swarm VO series, we obtain τ c W ∼ 30-70 year from Eq. (  4 ), much larger than each corresponding dataset window τ W , of 8 and 3. 7 year. A more precise gauge for the small-window regime is ε = (τ W /τ c W ) 2 , that should be kept such that ε ≪ 1 in order that Legendre polynomials may be used as temporal basis functions  (Gibson et al. 1992 ).
The singular spectrum and the level of noise variance The singular spectrum (the set of j values) is shown for each dataset, in Fig. 3. After a rapid decrease, not properly represented neither by an exponential nor a power law, a knee follows and then an asymptotic slightly decreasing trend. This is often referred to as a "plateau" and is attained after only a few modes at a height level that is related to the amplitude of the Gaussian noise in the data (see, e. g. Gibson et al. 1992 ). For eigenvalues above the plateau, North's rule of thumb (3) is used to locate the order in the PCA expansion where modes are no longer independently resolved and become degenerate. This occurs after order 4 for CHAMP and order 3 for Swarm datasets. Differences between the two spectra in Fig. 3  might be due to different behaviour of the field in the two periods 2001-2009 and 2014-2017, different quality of the two datasets or different sample size, M.
From results shown in Figs. 1  and 2 , the order-1 PC is a straight line, i. e. y 1 = α 1 y ′ 1 = a(tt 0 ) where t 0 is the mean epoch for each dataset. Note that, in those two figures, time t is in year/3 (4 months) and the slope is in units of (year/3) -1. The parameter a is quite similar for the two datasets. In fact, from y ′ 1 plotted in Figs. 1  and 2  and the values of α 1 , we obtain a Swarm = 1063 nT/year and a CHAMP = 1132 Fig. 3  Singular spectra for CHAMP full ( τ W = 8 year) dataset (thick blue), CHAMP shortened ( τ W = 3. 7 year) dataset (thin blue) and Swarm (thick red), in logarithmic scale. The standard error associated with each eigenvalue is given by North's rule of thumb to help identify modes that can be separated. In grey, the straight lines fitted to the noise "plateau" nT/year, a difference of only 6%.
3 ) as the number of CHAMP and Swarm VOs is almost the same. Estimated in nT, we find that the height of the noise level for Swarm data is about 0. 4 times the value for CHAMP data. Figure  3  also shows results of a test where we compare the PCA spectra using the full CHAMP dataset (8 years) with that using only the first 3. 7 years, the same length as the Swarm dataset (see the thinner solid line). This nicely supports the previous assumption that the plateau level is independent of M. It also reproduces the predicted variation in 1. Finally, this test illustrates that more and more modes are resolved as the time window is extended: the amplitude of the resolved polynomial modes increases, while the noise level remains constant. Assuming that the time series sample analytical functions,  Gibson et al.
Main PCA mode well explained as the variation of the eccentric dipole The first mode of the PCA decomposition from either CHAMP or Swarm data closely follows a straight line that corresponds exactly to the Legendre polynomial of degree 1 (Figs. 1a, 2a ). Furthermore, Fig. 6  indicates that the average harmonic degree of the spatial structure is very small, not far from 2. As a result, it does not come as a surprise that the spatial structure associated with mode 1 (in Figs. 4, 5 ), can be explained to a large extent using a simple model of a point dipole that moves with respect to the Earth's surface, at the same time as its magnetic moment is rotating. Denoting r i the Cartesian coordinates of each VO in the geographic frame, we calculate the Cartesian coordinates in the eccentric dipole frame (see "The eccentric dipole approximation" section), r f , as where R is the rotation matrix transforming the Earth's axis into the magnetic dipole axis.
The spatial signature we obtain for the moving eccentric dipole associated secular variation is clearly similar to the spatial pattern of mode 1 and to the IGRF-12 model. The spatial structures of the first mode recovered from the CHAMP and Swarm datasets are very similar. In fact, using the trend and spatial structure of PCA mode 1 computed from CHAMP dataset, we can explain 60% of the magnetic field signal in Swarm dataset. That is, where X Swarm is the data matrix from Swarm, (y 1 CHAMP , p 1 CHAMP ) are the PC and EOF for PCA mode 1 computed using CHAMP dataset and A F denotes the Frobe- nius norm of matrix A. Had we used PCA mode 1 from Swarm dataset, we would increase the explained signal in �X Swarm -y 1 CHAMP ⊗ p 1 T CHAMP � F �X Swarm � F ∼ 0. 4 Fig. 5  As in Fig. 4  but from Swarm X Swarm to 70%.
This level of predictability can meet the needs of some fields of study where only the large-scale (or long-distance) component of the main field is important. It is the case of studies on low Earth orbit satellites vulnerability to energetic particle flux  (Domingos et al. 2017). It can also be the case of magnetospheric studies where models for electrical current systems take into account the large-scale main field (e. g. Tsyganenko and Sitnov 2005). PCA mode 1 gives the average secular variation during the time interval of CHAMP and Swarm datasets. As a result, it compares well with the IGRF models for these time periods  (Thébault et al. 2015).
Higher order time derivatives The direct reading of computed PCs as Legendre polynomials of increasing degree is valid only while in the small-window-width regime, which applies to our datasets, for which τ W << τ c W. If τ W increases to the point that it gets close to or even larger than τ c W , PCs must be interpreted in a different way. Often, when using PCA on data matrices as (1), the computed PCs are expected to reveal characteristic timescales of the physical mechanisms contributing to those particular datasets. As an example, Domingos et al. (  2017 ) retrieved a quasi-periodicity of ∼ 11 year when applying PCA to the flux of high energy particles over the South Atlantic Anomaly, due to the solar forcing of the neutral atmosphere density. In Fig. 6  Average SH degree ( ℓ B ) of each mode from CHAMP and Swarm as a function of the time resolution ( T B ).
Size of symbols is proportional to the misfit to an internal field model (Table  1 ) and numbers next to symbols represent the mode ordering  the same way, if the VOs global signal includes a component due to a distinct source and with a correlation time short with respect to τ W , we expect that PCA succeeds in extracting it, if its energy rises above the noise level. This is in fact what occurs in the mode 3 from Swarm database. The second modes resolved in both CHAMP and Swarm datasets may be regarded as Secular Acceleration (SA) modes. Their importance comes from the fact that SA is seen as a window on the Earth's core dynamics, since it is mainly the result of the interaction between the flow acceleration and the field, and flow acceleration is read in terms of contributing force terms (e.
g. Aubert 2018 ). The corresponding PCs are very close to the discrete Legendre polynomial of degree-2 (see Figs. 1b, 2b ), which is precisely the required filter to calculate the second time derivative of the geomagnetic field. The spatial-time information retrieved through these modes allows for the characterisation of the components of the SA field, at the mean epoch of each dataset. In agreement with this interpretation, the EOF2 chart in Fig. 4d -f at 300 km altitude (mean epoch 2005. 8), is very similar to the chart of SA at the Earth's surface for 2006 as given by Chulliat and Maus (2014) (see their Figure  7 , also built from CHAMP data). It is dominated by spatial structures of SH degree 3 (as expected from our Fig. 6 ). The secular acceleration at the Earth's surface evolves rapidly and the whole process wherein the Atlantic equatorial patches of SA radial field change from one sign to the other takes about 4 year [see again Figure  7  of  Chulliat and Maus (2014) ].
This suggests that τ W from Swarm may be too small to properly resolve this mode and could explain why EOF2 is quite different between the two datasets , with a correlation coefficient of only 0. 47. Almost the same correlation (0. 46) can be found between mode 2 from Swarm and mode 3 from CHAMP, despite the time series corresponding to different order time derivatives. Modes 2 and 3 from CHAMP have similar average SH degree ℓ B (in contrast with the typical increase of ℓ B with the order of the mode), and they are quite similar, up to a longitude shift. The amplitude of mode 3 is higher than predicted from the power law scaling put forward by  Gibson et al. (see " The singular spectrum and the level of noise variance" section); yet, it is smaller than the amplitude of mode 2 (see Fig. 3 ) and the two modes cannot be simply combined to form a propagating wave.
It has a relatively large characteristic length scale ℓ B ∼ 3 , with two main radial flux patches over Brazil and West Africa, at southern low latitudes. It imparts a special high time variability to SA in the near-equatorial region and in particular to the SA radial flux patch over West Africa which remains equatorially symmetric over time. The fourth mode from Swarm in Fig. 5  shares the same cubic behaviour as the third mode from CHAMP  (Figs. 1c, 2d) , and some of the same symmetries can be seen in the EOF charts. In particular, the anti-symmetry over the Americas in B r  (Figs. 4g, 5j)  and the diagonally symmetric patches over Central America and South Africa in B θ  (Figs. 4h, 5k) , both stand out. In the Swarm dataset, this mode is not so well resolved and shows some degree of degeneracy (see Fig.
They did not state whether the signal would be static in the geocentric solar magnetic system used by  Barrois et al. (2018)  to calculate, in addition to the solar magnetic system, some of the external field subtracted from the VO series. Transformation between the solar magnetic system and the geographic coordinate system involves longitudinal averages that explain well the dominantly axial structure of the mode 3 that we obtained from Swarm data. Even though the CHAMP dataset covers a larger time window, we have found a well-resolved annual mode in Swarm series only. In Fig. 3 , the plateau height associated with the noise level decreases from CHAMP to Swarm from about 8. 9 × 10 2 to 1. 3 × 10 2 nT 2. The noise level in Swarm has decreased below the annual mode energy level allowing it to appear as a resolved mode.
Conclusions and perspectives This study is an effort to retrieve new information on the Earth's main magnetic field from recent geomagnetic data. We showed that PCA is an interesting alternative to the widely used SH by decomposing the signal into decorrelated modes. PCA seems to more easily retrieve a behaviour where the most rapid oscillations come from the smallest scale features, which is expected on a physical ground. The PCA decomposition is particularly useful in cases where it allows to distinguish among different physical source contributions  (Shore et al. 2016). The fit of internal and external potential field models to PCA modes guided us in a plausible separation between internal and mixed modes. We identified three clearly non-degenerate modes of internal origin from CHAMP data and two from Swarm data.
The time series of these modes (PCs) can be represented by Legendre polynomials of increasing degree. The correspondence between the principal components of rank 4 with Legendre polynomials and the partial success of the modelling of the associate EOFs as internal potential fields give us also some confidence in the interpretation of the modes of rank 4 obtained from CHAMP and Swarm. Finally, from Swarm data only, we observe a mode with an annual period and quadrupolar geometry of the radial component (mode 3). We found some interesting features in our results that can be further exploited. We also identified new problems to be tackled. • The spatial structure of an eccentric dipole moving towards Asia with amplitude increasing linearly in time is a good approximation to the SV of the main field at first order.
After an interval of about 10 years between ∼ 2005 and ∼ 2015 (from mean epoch of CHAMP to mean epoch of Swarm), mode 1 from CHAMP still explains 60% of the time variability of the field during the Swarm period. • We found that the mode 3 obtained from CHAMP data is well resolved. It corresponds to the thirdorder time derivative at the middle epoch of the CHAMP series (November 2005). Once fitted into a spherical harmonic model, the mode can be downward continued to the CMB taking care to minimise the amplification of small scale noise. Interpreting the magnetic field changes with the radial induction equation at the core surface, we can undertake the estimation not only of the surface flow velocity and of the flow acceleration but also of the second-order time derivative of the velocity. This is all the more interesting as recent studies differ in their estimation of the core flow acceleration over the period 2000-2016.