Michele Cappellari Python Programs

The software below is freely available. However, if you use it for published research, you are requested to cite the paper highlighted in red where the method is described.

You are not allowed to re-distribute any of these programs (modified or not) without explicit permission from the author.

All my packages below are on the Python Package Index (PyPI).

See my review on galaxy structure and evolution

1 MgeFit: Multi-Gaussian Expansion Fitting
1.1 The MgeFit package in Python
2 JAM: Jeans Anisotropic Modelling

2.1 The JAM package in Python
2.2 The JAM source code in the C language
3 VorBin: Voronoi Binning

3.1 The VorBin package in Python
3.2 The VorBin package in Julia
4 pPXF: Penalized PiXel-Fitting

4.1 Libraries of models for stellar-population with pPXF
4.2 Libraries of stellar templates for stars/gas kinematics with pPXF
4.3 The pPXF package in Python
5 CapFit: Linearly-constrained Nonlinear Least-Squares

5.1 The CapFit package in Python
6 PaFit: Fitting Kinematic PA
6.1 The PaFit package in Python
7 LtsFit: Robust linear regression with scatter in arbitrary dimension
7.1 The LtsFit package in Python
8 LOESS: Adaptive smoothing in one or two dimensions
8.1 The loess package in Python
9 AdaMet: Adaptive Metropolis for Bayesian analysis
9.1 The AdaMet package in Python
References

1 MgeFit: Multi-Gaussian Expansion Fitting

Robust technique to perform Multi-Gaussian Expansion (MGE) fits to galaxy images

Figure 1: Illustration of the steps involved in the MGE fit to the S0 galaxy NGC 4342 using the MgeFit package. The figures were produced by the mge_fit_example.py script included in the Python distribution of the software.

This software obtains an accurate Multi-Gaussian Expansion (MGE) parametrization (Emsellem et al., 1994; Cappellari, 2002) for a galaxy surface brightness with the fitting algorithm of Cappellari (2002). Given that Gaussians are not orthogonal functions, MGE fits are in general strongly degenerate, with difficult global convergence, but the mge_fit_sectors method solves all problems, making MGE fitting an automated, reliable and robust process (Figure 1).

See Cappellari et al. (2013a) and Zhu et al. (2024) for large scale applications of this software to the study of the M/L ratio and the Fundamental Plane of early-type galaxies and Cappellari et al. (2012) for an application to the study of the stellar IMF. The MGE parametrization is useful in the construction of realistic dynamical models of galaxies (see JAM in section 2), for PSF deconvolution of images, for the correction and estimation of dust absorption effects, or for galaxy photometry.

1.1 The MgeFit package in Python

The source code of the MgeFit package, including four test galaxy images, is on the Python Package Index (PyPI). The changes are documented in the CHANGELOG.

How to install: Use pip install mgefit. Without write access to the global site-packages directory, use pip install --user mgefit.

Usage examples: are in the directory “examples” inside the main package folder inside site-packages.

MgeFit requires the scientific core packages NumPy, SciPy and Matplotlib, and the examples use Astropy to read FITS images.

NOTE: I would appreciate if you drop me an e-mail (address at the bottom) when you download the above procedures.

2 JAM: Jeans Anisotropic Modelling

Jeans Anisotropic Modelling of the dynamics, stellar kinematics or proper motions of axisymmetric or spherical galaxies

Figure 2: (a) Including both proper motions and radial velocities. Top Panels: the components of the symmetric velocity second moment tensor measured from the particles of a realistic N-body simulation. Bottom Panels: the corresponding JAM model predictions. Note the striking agreement! (Cappellari et al., 2012). (b) The JAM package includes an implementation for both the cylindrically-aligned solution JAM_cyl of Cappellari (2008) and the spherically-aligned solution JAM_sph of Cappellari (2020). Comparisons between these two extreme solutions allows for a robust assessment of possible systematic modeling uncertainties (see Cappellari 2020 for some examples). (c) Validating black hole (BH) recovery with JAM. A detailed JAM modelling accurately recovers the “benchmark” BH mass in NGC4258, as inferred from maser observations (figure taken from Drehmer et al. 2015). An equally good agreement was found with JAM for the “benchmark” BH in the Milky Way (Sec.4.1.2 of Feldmeier-Krause et al. 2017). Other successful BH comparisons, between JAM and Schwarzschild’s method, were presented e.g. in Cappellari et al. (2010), for 25 galaxies, and Krajnović et al. (2018), for 7 galaxies. (d) This is the first dynamical modelling (using JAM_sph) of the high-quality Gaia DR2 kinematics of the Milky Way, with full 6D phase-space coordinates. For the first time, there is no dynamical degeneracy and the model has no freedom to fit the data, yet all features are reproduced. This is an important test of galaxy dynamics. (figure taken from Nitschai et al. 2020)

Figure 3: Examples of JAM data-model comparisons. The bi-symmetrized SAURON stellar kinematics of 6 Elliptical (left) and 6 S0 (right) fast-rotator early-type galaxies is compared to the predictions of the anisotropic Jeans models with JAM. The kinematics varies widely for different galaxies, yet these two-parameters (!) models are able to correctly predict the shape of a pair of two-dimensional functions (\(V\) and \(V_\mathrm {rms}\)), once the observed surface brightness is given (this is fig. 10 of Cappellari 2016).

Figure 4: Left Panel: JAM works better than Schwarzschild on simulated galaxies. A detailed comparison using the Illustris simulations with the JAM and Schwarzschild's (SCH) dynamical models, found that JAM recovers the known enclosed total mass within \(1R_{e}\) more accurately than SCH. Specifically, the 68th percentile deviation (\(1\sigma \) error) over all 45 model fits is \(1.6\times \) smaller for JAM than SCH (this is fig. 4 from Jin et al. 2019). Right Panel: JAM works better than Schwarzschild on real galaxies. A detailed comparison, for 54 real disk galaxies with CALIFA data, between the circular velocity V_CO measured from CO gas and the corresponding V_JAM and V_SCH from dynamical modelling of integral-field stellar kinematics using the JAM and Schwarzschild's (SCH) methods, found that JAM recovers the densities more accurately than SCH. Specifically, between 0.8–1.6 \(R_{e}\) (inside the red boxes), where the gas is well-resolved and V_c is better determined, the mean 1\(\sigma \) error is \(1.7\times \) smaller for JAM than SCH (this is fig. 8 from Leung et al. 2018).

The JAM software in this section can be used for the dynamical modelling of galaxies or other gravitationally-bound systems of particles. It was used e.g. to measure the mass of supermassive black holes in galaxies, to infer their dark-matter content or to measure galaxy masses and density profiles.

JAM implements a solution of the Jeans equations which allows for orbital anisotropy (three-integrals distribution function) and also provides the full second moment tensor, including both proper motions and radial velocities (Figure 2a), for both axisymmetric (Cappellari et al., 2012) and spherical geometry (Cappellari, 2015). The technique was introduced in Cappellari (2008), for the cylindrically-aligned case, and in Cappellari (2020), for the spherically-aligned case (Figure 2b), and I called it the Jeans Anisotropic Modelling (JAM) method. It relies on the Multi-Gaussian Expansion parametrization (Emsellem et al., 1994; Cappellari, 2002) for the galaxy surface brightness.

With the addition of a single extra parameter \(\beta _z=1-(\sigma _z/\sigma _R)^2\) (in cylindrical alignment) or \(\beta _r=1-(\sigma _\theta /\sigma _r)^2\) (in spherical alignment), the simple and user-friendly three-integrals JAM method already provides a dramatic improvement over the classic but less general two-integrals \(f(E,L_z)\) Jeans (1922) models. However, JAM also allows for tangential anisotropy \(\gamma =1-(\sigma _\phi /\sigma _R)^2\) and/or for spatially varying anisotropy (different for every MGE Gaussian). The JAM models provide an excellent description of state-of-the-art integral-field stellar kinematics of real galaxies (Figure 3). This makes the technique well suited to measure the inclination, the dynamical M/L and angular momenta of early-type fast-rotators and spiral galaxies. JAM was found to be more accurate than Schwarzschil’d modelling when measuring the density profiles of both real and simulated galaxies (Figure 4).

The JAM routines are designed for axisymmetric or spherical geometry, (i) they can provide the proper motion dispersion tensor (Cappellari, 2012; Cappellari, 2015) (Figure 2a), (ii) allow for the inclusion of dark matter, (iii) variable stellar M/L, (iv) spatially varying anisotropy and (v) multiple kinematic components and (vi) supermassive black holes (BH; Figure 2c). The JAM package also includes a routine to compute the circular velocity from the MGE models. Some sample applications of the JAM method are given below:

Dynamical masses for a sample of 9 passive early-type galaxies at redshift \(z\approx 2\) (Cappellari et al., 2009)
Dark matter from stellar dynamics and gravitational lensing (van de Ven et al., 2010; Barnabè et al., 2012)
Total mass profiles from two-dimensional stellar kinematics up to 4 R_e (Cappellari et al., 2015)
A systematic variation of the stellar IMF in 260 early-type galaxies (Cappellari et al., 2012)
Discrete dynamical models of \(\omega \) Centauri (Watkins et al., 2013)
First Gaia DR2 dynamical model of the Milky Way with six phase-space coordinates (Nitschai et al., 2020; Nitschai et al., 2021)
Dynamical modelling of the final (DR17) MaNGA sample of 10K galaxies with integral-field spectroscopy (Zhu et al., 2023)
Time-delay cosmography from dynamical models and lensing time-delay (Shajib et al., 2023)
Dynamical masses for 3200 LEGA-C galaxies at redshift \(z\approx 0.8\) (Cappellari, 2023)

To construct dynamical models with the JAM method one needs to describe the galaxies surface brightness via the Multi-Gaussian Expansion parametrization using my MgeFit package in section 1.

2.1 The JAM package in Python

The source code of the JamPy package is on the Python Package Index (PyPI). The changes are documented in the CHANGELOG.

How to install: Use pip install jampy. Without write access to the global site-packages directory, use pip install --user jampy.

Usage examples: are in the directory ”examples” inside the main package folder in site-packages.

Also required is my plotbin package (automatically installed by pip).

The pure-python JamPy package only requires the scientific core packages NumPy, SciPy and Matplotlib.

2.2 The JAM (cylindrical-only) source code in the C language

Note that the JAM Python code is extremely well vectorized. You should not assume the C version will be significantly faster without benchmarking with identical setup. Also note that the C version only implements the cylindrically aligned JAM solution, not the spherically-aligned one. Laura Watkins has translated the JAM procedures into the C language. Laura’s code is available here.

NOTE: I would appreciate if you drop me an e-mail (address at the bottom) when you download the above procedures.

3 VorBin: Voronoi Binning

Adaptively spatial bin two-dimensional data to a constant signal-to-noise ratio per bin

Figure 5: Left: Four-coloring of Voronoi binning; Middle: Unbinned versus Voronoi binned stellar kinematics from Integral-Field Spectroscopy (Cappellari and Copin, 2003); Right: Voronoi binning of abundance from X-ray data (Sanders et al., 2004).

The Voronoi Binning method by Cappellari and Copin (2003) optimally solves the problem of preserving the maximum spatial resolution of general two-dimensional data (or higher dimensions), given a constraint on the minimum signal-to-noise ratio (Figure 5).

The Voronoi binning method has been applied to a variety of types of data. A review of the concepts and applications to (i) X-ray data, (ii) integral-field spectroscopy, (iii) Fabry-Perot interferometry, (iv) N-body simulations, (v) standard images and (vi) other regularly or irregularly sampled data is given in Cappellari (2009).

3.1 The VorBin package in Python

The source code of the VorBin package, with examples and instructions, is on the Python Package Index (PyPI). The changes are documented in the CHANGELOG.

How to install: Use pip install vorbin. Without write access to the global site-packages directory, use pip install --user vorbin.

User Manual: Is available on THIS PAGE.

Usage example: is in the package folder inside site-packages.

The VorBin package requires the scientific core packages NumPy, SciPy and Matplotlib.

My optional plotbin package contains routines to visualize Voronoi 2D-binned or unbinned data like in Figure 5.

3.2 The VorBin package in Julia

VorBin was ported to the Julia language by Michael Reefe. You can find it HERE.

NOTE: I would appreciate if you drop me an e-mail (address at the bottom) when you download the above procedures.

4 pPXF: Penalized PiXel-Fitting

Stellar or gas kinematics and stellar population from galaxy spectra via full spectrum fitting with photometry

Figure 6: Left: Stellar and gas kinematics fit with pPXF. The black line (mostly hidden by the fit) is the relative flux of the observed spectrum. The red line is the pPXF fit for the stellar component, while the orange line is a fit to the gas emission lines. The green symbols at the bottom are the fit residuals, while the blue lines is the gas-only best-fitting spectrum. The main absorption and emission features are indicated at the top of the plot (taken from Cappellari 2017). Right: Spectra and photometry with pPXF. The top panel shows a fit to the 28-bands COSMOS photometry for a galaxy in the LEGA-C survey at redshift \(z\approx 0.7\). The grey band indicates where the spectrum is also available. The middle panel is the fit to the spectrum and gas emission. The bottom panel shows the recovered star formation history (SFH) using 43(Ages)\(\times \)9([M/H])=387 FSPS population templates (taken from Cappellari 2023).

This software implements the Penalized PiXel-Fitting method (pPXF) to extract the stellar or gas kinematics (Figure 6 Left) and stellar population (Figure 6 Right) from absorption-line spectra of galaxies, using a maximum penalized likelihood approach. The method was originally described in Cappellari and Emsellem (2004) and was significantly upgraded in subsequent years and particularly in Cappellari (2017) and with the inclusion of photometry and linear constraints in Cappellari (2023). The method is very general and robust. For this reason it was applied to a variety of situations. The following key features are implemented in the current pPXF program:

Optimal template: Fitted together with the kinematics to minimize template-mismatch errors. Also, useful to extract gas kinematics or derive emission-corrected line-strengths indexes. One can use synthetic templates to study the stellar population of galaxies via ”Full-Spectrum Fitting” instead of using traditional line-strengths;
Regularization of templates weights: To reduce the noise in the recovery of the stellar population parameters and attach a physical meaning to the output weights assigned to the templates in term of the star formation history (SFH) or metallicity distribution of an individual galaxy;
Multiple gaseous/stellar components: Each template in pPXF can be assigned to a different kinematic component, with a different LOSVD (Johnston et al., 2013). This allows one to extract multiple kinematic components and to fit the gas emission lines together with the stellar kinematics and population; Penalization applies to both gas and stars and regularization can still be enforced to the stellar templates;
Fitting photometry and spectra: One can fit multiple photometric bands (SED) simultaneously with a spectrum (Cappellari, 2023);
Kinematic bulge/disk decomposition: One can enforce a ratio between the flux of two kinematic components, e.g., to try to infer the kinematic of bulge and disk components (Tabor et al., 2017);
Linear constraints on templates or kinematics: One can enforce general linear constraints on either the stellar templates (e.g. require the total weights in one kinematic component to be large than a certain fraction) or the kinematic parameters (e.g. requiring the dispersion of one kinematic component to be larger than a certain fraction of that of another component) (Cappellari, 2023);
Iterative sigma clipping: To clean the spectra from residual bad pixels or cosmic rays;
Two-sided LOSVD fitting: To reduce systematic errors in the kinematics, using two spectra taken at the opposite sides of the galaxy nucleus (Sec.3.6 of Rix et al. 1992);
Additive/multiplicative polynomials: To correct low frequency continuum variations. Also useful for calibration purposes;
Inclusion of sky spectrum: To deal with spectra heavily contaminated by the sky spectrum (Sec.3.1 of Weijmans et al. 2009);
Reddening fit: To determine the reddening of the spectrum using the extinction curve of (Calzetti et al., 2000);
Inclusion of covariance matrix: To account for correlated errors in the spectral pixels (e.g. due to rebinning and interpolations).

See Emsellem et al. (2004, SAURON), Cappellari et al. (2011, ATLAS^3D), Falcón-Barroso et al. (2017, CALIFA), van de Sande et al. (2017, SAMI) and Westfall et al. (2019, MaNGA) for some ever-increasing large-scale applications of the pPXF method to the measurement of the stellar or gas kinematics of galaxies.

4.1 Libraries of models for stellar-population with pPXF

Figure 7: pPXF is the most accurate full-spectrum fitting method. In a detailed study, pPXF was found to produce the smallest errors (compared to the known true values) and to be significantly faster than all other full-spectrum fitting codes (figure from Woo et al. 2024).

pPXF is the most efficient, reliable and flexible implementation of the “Full-Spectrum Fitting” method to study stellar populations (see Figure 7). This technique has effectively rendered the traditional line-strength indices obsolete, facilitated by the development of high-quality spectral models. pPXF was designed to be independent on any specific set of stellar-population models and has already been used with nearly every available one. Here is an incomplete list of stellar population models that were used with pPXF:

GALAXEV (Bruzual and Charlot, 2003)
Vazdekis models (Vazdekis et al., 2010)
Maraston model (Maraston and Strömbäck, 2011)
FSPS (Conroy et al., 2009)
BPASS (Stanway and Eldridge, 2018)
XSL (Verro et al., 2022a)

pPXF allows one to extract multiple kinematic components, with different stellar populations, from a spectrum. Gas emission lines can be fitted simultaneously, avoiding the need for masking them. This is particularly useful when studying the stellar population of galaxies with prominent emission lines (e.g. the Balmer lines) filling important absorption features (see Figure 6).

Please also cite the source of the stellar population models HERE if you use pPXF with any of the included libraries of spectral templates.

4.2 Libraries of stellar templates for stars/gas kinematics with pPXF

The ability of the pPXF method to fit a large set of stellar templates together with the kinematics allows the template mismatch problem to be virtually eliminated. This is particularly useful given the current availability of large stellar libraries spanning wide ranges of physical parameters and having good spectral resolution. Excellent results have been obtained by using a few hundred template stars with pPXF, from which generally about 10-20 are selected by the program to provide detailed fits to high S/N galaxy spectra. An incomplete list of libraries that have been successfully used with pPXF for the kinematics extraction is given below:

Indo-U.S. Coudé Feed Spectral Library: 1273 stars covering the region from 3460 – 9464 Å at a spectral resolution of 1.35 Å (FWHM), \(\sigma \approx 30\) km/s, \(R\approx 4200\). From Valdes et al. (2004)
Ca II triplet : 706 stars covering the region from 8348 – 9020 Å at a spectral resolution of 1.5 Å (FWHM), \(\sigma \approx 22\) km/s, \(R\approx 5700\). From Cenarro et al. (2001)
ELODIE: 1388 stars covering the region from 4000 – 6800 Å at a spectral resolution of 0.5 Å (FWHM), \(\sigma \approx 13\) km/s, \(R\approx 10,000\). From Prugniel and Soubiran (2001)
MILES Library of Stellar Spectra: 985 well-calibrated stars covering the region from 3525 – 7500 Å at a spectral resolution of 2.51 Å (FWHM), \(\sigma \approx 64\) km/s, \(R\approx 2000\). From Sánchez-Blázquez et al. (2006) and Falcón-Barroso et al. (2011)
MaStar Stellar Library: 10K well-calibrated stars covering the region from 3,622 – 10,354 Å at a spectral resolution of 3.5 Å (FWHM), \(\sigma \approx 70\) km/s, \(R\approx 1800\). From Yan et al. (2019).
X-Shooter Spectral Library: 683 stars covering three segments from 3,000 – 24,500 Å at a spectral resolution with \(\sigma \approx 13\) km/s, \(R\approx 10,000\). From Verro et al. (2022b).
GNIRS library of late spectral templates: 40 stars with a range of CO equivalent widths, covering the K-band region from 2.18 – 2.43 \(\mu \)m at a spectral resolution of 3.4 Å (FWHM), \(\sigma \approx 19\) km/s, \(R\approx 6600\). From Winge et al. (2009)
MARCS synthetic library of stellar spectra covering the wavelength region from 1300 Å to 20 \(\mu \)m at a spectral resolution of \(\sigma \approx 6.4\) km/s, \(R\approx 20,000\). This is not as reliable as an empirical stellar library but it can be very useful in special situations, due to the wide wavelength coverage and high spectral resolution. From Gustafsson et al. (2008)
SYNTHE synthetic library of stellar spectra covering the wavelength region from 2500 Å to 1.05 \(\mu \)m at a spectral resolution of \(\sigma \approx 6.4\) km/s, \(R\approx 20,000\). The same caveats and advantages apply to this synthetic library as to the previous one. From Munari et al. (2005)

4.3 The pPXF package in Python

The source code of the ppxf package, with examples and instructions, is on the Python Package Index (PyPI). The changes are documented in the CHANGELOG.

How to install: Use pip install ppxf. Without write access to the global site-packages directory, use pip install --user ppxf.

User Manual: Detailed documentation is on THIS PAGE (always up to date).

Usage examples: Jupyter Notebooks ppxf examples are available HERE. Python examples are in the directory “examples” inside the main ppxf package folder inside site-packages.

ppxf is written in pure Python and only requires the scientific core packages NumPy, SciPy and Matplotlib, and the examples use Astropy to read FITS data.

NOTE: I would appreciate if you drop me an e-mail (address at the bottom) when you download the above procedures.

5 CapFit: Linearly-constrained Nonlinear Least-Squares

A Python package for efficient constrained least-squares fitting

Figure 8: This is the kind of linearly-constrained non-linear leats-squares problems that CapFit is designed to solve. The contours show the Rosenbrock function, which is a popular benchmark for non-linear least-squares method. The magenta dot is the unconstrained minimum; the red dot is the minimum subject to the linear equality constraints (red line); the green dot is the minimum subject to the linear inequality constraints (green triangle), while the yellow dot satisfies bot the equality and inequality constraints. The white line shows the feasible steps taken by CapFit to reach the inequality constrained solution.

CapFit solves linearly-constrained non-linear least-squares optimization problems.

It supports linear inequality/equality constraints and bound constraints. Additionally, parameters can be tied (enforcing non-linear constraints) or fixed without modifying the fitting function. CapFit implements Algorithm 2 of Cappellari (2023). It combines two very successful ideas:

The Sequential Quadratic Programming (SQP);
The Levenberg-Marquardt (LM) method.

CapFit can be described as a Levenberg-Marquardt algorithm with linear constraints (which include simple bounds as a special case).

It is designed for situations where the user function is complex and computationally expensive compared to the small quadratic subproblem, CapFit generally outperforms the best unconstrained or bound-constrained least-squares algorithms in terms of robustness and number of function evaluations (Figure 8). Additionally, it supports more general constraints.

5.1 The CapFit package in Python

The pure-Python CapFit package is on the Python Package Index (PyPI).

How to install: Use pip install capfit. Without write access to the global site-packages directory, use pip install --user capfit.

Usage examples: are in the directory “examples” inside the main package folder inside site-packages. The full documentation is on the Python Package Index (PyPI)

The CapFit procedure is also used for the nonlinear optimization for both MgeFit in section 1 and pPXF in section 4 above.

6 PaFit: Fitting Kinematic PA

Fit the global kinematic position-angle of galaxies

Figure 9: Left Panel: Best fitting symmetrized kinematics at the optimal position angle determined by PaFit. Right Panel: The corresponding mock observed stellar kinematics, with overlaid the best-fitting kinematic PA. The black dots indicate the location were the kinematics was sampled.

This software implements the method presented in Appendix C of Krajnović et al. (2006) to measure the global kinematic position-angle (PA) from integral field observations of a galaxy stellar or gas kinematics (Figure 9).

See Cappellari et al. (2007), Krajnović et al. (2011) and Graham et al. (2018) for large scale applications of this software to the study of the stellar kinematic misalignment of early-type galaxies. See Davor’s Krajnović page for the related Kinemetry package.

6.1 The PaFit package in Python

The PaFit package is on the Python Package Index (PyPI).

How to install: Use pip install pafit. Without write access to the global site-packages directory, use pip install --user pafit.

Also required is my plotbin package (automatically installed by pip).

PaFit requires the scientific core packages NumPy and Matplotlib.

NOTE: I would appreciate if you drop me an e-mail (address at the bottom) when you download PaFit.

7 LtsFit: Robust linear regression with scatter in arbitrary dimension

Fit lines, planes, or hyperplanes to data with errors in all variables, possible outliers and intrinsic scatter

Figure 10: Fitting of the (M/L)-\(\sigma \) relation in the Virgo cluster using ltsfit (taken from Cappellari et al. 2013a). The green outliers above the relation are automatically removed from the fit. They turn out to be background galaxies and for this reason they do not follow the cluster relation.

This software implements the method presented in Sec. 3.2 of Cappellari et al. (2013a) to perform extremely robust fit of lines or planes to data with errors in all variables, possible large outliers (bad data) and unknown intrinsic scatter.

The code combines the Least Trimmed Squares (LTS) robust technique, proposed by Rousseeuw (1984) and speeded up in Rousseeuw & Van Driessen (2006), into a least-squares fitting algorithm which allows for intrinsic scatter and errors in all coordinates. This method makes the fit converge to the correct solution even in the presence of numerous catastrophic outliers (like in Figure 10), where the much simpler \(\sigma \)-clipping approach can converge to the wrong solution.

7.1 The LtsFit package in Python

The LtsFit package is on the Python Package Index (PyPI).

How to install: Use pip install ltsfit. Without write access to the global site-packages directory, use pip install --user ltsfit.

Usage examples: are in the directory “examples” inside the main package folder inside site-packages.

LtsFit requires the scientific core packages NumPy, SciPy and Matplotlib.

NOTE: I would appreciate if you drop me an e-mail (address at the bottom) when you download LtsFit.

8 LOESS: Adaptive smoothing in one or two dimensions

Recover mean trends from noisy data in one or two dimension

Figure 11: Application of the loess_2d routine (with frac=0.6) to the recovery of two underlying functions from a noisy distribution of 200 scattered values. For comparison, the right panels show the distribution obtained via simple averaging on a grid, but using \(100\times \) more values than the ones used for the LOESS recovery. The LOESS approach achieves a comparable result while requiring a much smaller number of input values.

This software provides an improved implementation of the one-dimensional (Cleveland 1979) and two-dimensional (Cleveland & Devlin 1988) Locally Weighted Regression (LOESS) method to recover the mean trends of the population from noisy data in one or two dimensions (Figure 11). It includes a robust approach to deal with outliers (bad data).

These programs were implemented and used to produce the two-dimensional maps and the one-dimensional mean trends in Cappellari et al. (2013b).

8.1 The loess package in Python

The loess package is on the Python Package Index (PyPI).

How to install: Use pip install loess. Without write access to the global site-packages directory, use pip install --user loess.

Usage examples: are in the directory “examples” inside the main package folder inside site-packages.

Also required to run the 2D example is my plotbin package (automatically installed by pip).

NOTE: I would appreciate if you drop me an e-mail (address at the bottom) when you download the code.

9 AdaMet: Adaptive Metropolis for Bayesian analysis

A Python package for efficient Bayesian analysis

Figure 12: Corner plot for the posterior of the model parameters obtained when fitting the black hole mass in the galaxy NGC1277 using the JAM modelling method and the AdaMet Bayesian code (taken from Krajnović et al. 2018). The Python code to produce this figure is included in the JamPy package in section 2.

This AdaMet package is the implementation of Cappellari et al. (2013a) of the Adaptive Metropolis algorithm of Haario (2001). In many real-world applications, it was found to be more efficient and robust than the emcee approach, whose warm-up phase scales linearly with the number of walkers (Figure 12).

9.1 The AdaMet package in Python

The user-friendly AdaMet package is on the Python Package Index (PyPI).

How to install: Use pip install adamet. Without write access to the global site-packages directory, use pip install --user adamet.

Usage examples: are in the directory ”examples” inside the main package folder inside site-packages.

Also required for plotting is my plotbin package (automatically installed by pip).

Comments and suggestions are welcome to the address on my Homepage.

Latest changes: September 14, 2024

Go back to Michele Cappellari Homepage

References

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