What is a basis function expansion?
A basis function expansion (BFE) uses relatively simple equations to represent a more complex
distribution. Each term in the expansion is given a weight such that the sum of the functions
optimally represents the given distribution. In our case, we use BFEs to represent the
gravitational potential and mass distribution of the dark matter (and/or baryons) in a galaxy.
We tailor the expansion such that the zeroth order term represents the equilibrium state of the galaxy.
The BFE framework is a powerful tool for analyses of galactic dynamics and disequilibrium dynamics, in
particular. The EXP collaboration is built around the shared vision of developing these tools for galactic
dynamics, combining the varied expertise of its members to learn about disequilibrium dynamics in galaxies.
The publicly-available EXP code was developed to help achieve
this task. As evidenced in the following pages, there are a number of use cases for BFE analysis framework,
and members of our collaboration have used BFEs to provide profound insight into a variety of problems.
Interested in learning more? Check out the other tabs of this website,
the Github, and the theory page of the
readthedocs
Why should I use basis function expansions?
Basis function expansions (BFEs) provide a mathematical framework for interrogating and understanding
complex systems. Through this framework, it is possible to discover the underlying dynamics within
simulations that span the full gamut of complexity, ranging from idealized periodic boxes to large
cosmological simulations. This framework provides natural ties to analytic theory as well as new
supervised and unsupervised machine learning tools. One such tool is multi-channel Single Spectrum
Analysis (mSSA), and examples of using mSSA + BFE for dynamical discovery can be seen in
here and
here (see also the
EXP readthedocs).
A particularly powerful use of BFE is as a universal language to succinctly summarize the relevant
dynamical information in galaxies for comparison across and between different simulations.
Basis Function Expansions for Cosmological Simulations
Basis function expansions (BFEs) can be used as a post-processing analysis framework for cosmological simulations.
With BFEs, you can analyze the dark matter halo of your favorite galaxy from any cosmological
simulation to find wakes, dipoles, and more. You can also succinctly describe the stellar disk of
a galaxy, use the expansions to integrate orbits in the gravitational potential, and more. See some highlights
of recent cosmological analyses below
Add a blurb of your paper here!
Basis Function Expansions for Analytic Theorists
Basis function expansions allow us to seamlessly connect analytic theory, such as linear response, to more complex N-body simulations. Basis function
expansions enable the study of coupled modes, for example, such as those from a baryonic disk and a dark matter halo.
not sure what to say here
your paper blurb here!
Basis Function Expansions for N-body Dynamical Simulations
Basis function expansions (BFEs) can be used to both run and analyze dynamical N-body simulations. The EXP
collaboration - spearheaded by Martin Weinberg - has developed eponymous code to perform both of these
functions. EXP uses BFEs to represent the potential and mass distributions of the star and dark
matter particles of a galaxy to run simulations significantly faster than alternate techniques. The
theory underpinning BFE simulations and the implementation are discussed in
more detail in the ,
as well as these papers (,
).
The resulting simulations have both particle-based snapshot data and basis function information, including
the basis and time-evolving coefficients. These data can be used together to provide unique insight into
the underlying dynamics. EXP can also be run on simulations that were produced with different software,
including cosmological simulations, to provide BFEs at each time step. See below for
examples that use either or both of these functionalities of EXP.
Your paper links and blurbs here!
Basis Function Expansions for Observational Insight
Two dimensional basis function expansions can also be performed on observational data. Such 2D expansions
on image data describe the light (stellar) distribution in a galaxy, and provide a language for succinctly,
quantitatively summarizing the morphological features. We adopt a Fourier-Laguerre basis for image data,
which captures both the angular (Fourier) and radial (Laguerre) information. These expansions are also
how we map an image of a galaxy to a sound via sonification. We are currently developing a framework for
expansions of integral field spectrograph data, which will allow for analyses of both velocity and chemical
information.
Your paper links and blurbs here!
Basis Function Expansions for Sonification
The light profile of a galaxy image can be described with a Fourier-Laguerre basis function expansion.
The resulting expansion has both angular (Fourier, m) terms and radial (Laguerre, n) terms and a series
of coefficient weights. While we typically plot these terms and weights for a visual representation, we
can also present these same data with sounds. In plotting these data, we might decide that the n-terms
are along the x-axis and the m-terms are along the y-axis, with the coefficient values making a heatmap.
Similarly, we could choose to map the n-terms to notes on a given scale, the m-terms to octaves, and the
coefficient amplitudes to volume. This mapping of data to sound is called sonification.
We are pioneering the use of basis function expansions for sonfication. As part of this work, we have
created a GalaxyZoo project to determine the efficacy of classifying galaxy morphology through sounds,
or aural classification. Check out the project here (LINK COMING SOON) to test your own aural
classification skills!
How to get started
We have built and compiled a variety of resources to help you get started with EXP and basis function expansions!
Check out our GitHub page and accompanying
for how to install EXP.
If you want to experiment with EXP, try out the Docker image and
documentation. If you want to run pyEXP and EXP examples,
be sure to clone the respective repositories to wherever you are working with the Docker image
If you want to want to learn more about mSSA, check out this webpage and these papers:
1 and
2 (see also the
EXP readthedocs for more information)
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