lintsampler is a pure Python bundle that may simply and effectively generate random samples from any chance distribution.
Full disclosure: I’m one of many authors of lintsampler.
We frequently discover ourselves in conditions the place we’ve a chance distribution (PDF) and we have to draw random samples it. For instance, we’d wish to estimate some abstract statistics or to create a inhabitants of particles for a simulation.
If the chance distribution is a typical one, similar to a uniform distribution or a Gaussian (regular) distribution, then the numpy/scipy ecosystem supplies us with some simple methods to attract these samples, by way of the numpy.random or scipy.stats modules.
Nevertheless, out within the wild, we frequently encounter chance distributions that aren’t Gaussian. Typically, they’re very not Gaussian. For instance:
How would we draw samples from this distribution?
There are a number of widely-used strategies to attract samples from arbitrary distributions like this, similar to rejection sampling or Markov chain Monte Carlo (MCMC). These are glorious and dependable strategies, with some useful Python implementations. For instance, emcee is an MCMC sampler extensively utilized in scientific functions.
The issue with these present strategies is that they require a good quantity of setup and tuning. With rejection sampling, one has to decide on a proposal distribution, and a poor selection could make the process very inefficient. With MCMC one has to fret about whether or not the samples are converged, which generally requires some post-hoc testing to gauge.
Enter lintsampler. It’s as simple as:
from lintsampler import LintSampler
import numpy as npx = np.linspace(xmin, xmax, ngrid)
y = np.linspace(ymin, ymax, ngrid)
sampler = LintSampler((x, y), pdf)
pts = sampler.pattern(N=100000)
On this code snippet, we constructed 1D arrays alongside every of the 2 dimensions, then we fed them to the LintSampler object (imported from the lintsampler bundle) together with a pdf operate representing the chance distribution we wish to draw samples from. We didn’t spell out the pdf operate on this snippet, however there are some absolutely self-contained examples within the docs.
Now, pts is an array containing 100000 samples from the PDF. Right here they’re in a scatter plot:
The purpose of this instance was to show how simple it’s to arrange and use lintsampler. In sure instances, it’s also a lot sooner and extra environment friendly than MCMC and/or rejection sampling. In case you’re to learn how lintsampler works underneath the hood, learn on. In any other case, go to the docs, the place there are directions describing methods to set up and use lintsampler, together with instance notebooks with 1D, 2D, and 3D use instances, in addition to descriptions of a few of lintsampler’s further options: quasi Monte Carlo sampling (a.ok.a. low discrepancy sequencing), and sampling on an adaptive tree construction. There’s additionally a paper revealed within the Journal of Open Supply Software program (JOSS) describing lintsampler.
Underlying lintsampler is an algorithm we name linear interpolant sampling. The concept part of the docs provides a extra detailed and extra mathematical description of how the algorithm works, however right here it’s briefly.
The instance under illustrates what occurs underneath the hood in lintsampler while you feed a PDF and a grid to the LintSampler class. We’ll take a simple instance of a 2D Gaussian, however this technique applies in any variety of dimensions, and with a lot much less pleasant PDFs.
- First, the PDF will get evaluated on the grid. Within the instance under, the grid has uneven spacings, only for enjoyable.
- Having evaluated the PDF on the grid on this manner, we are able to estimate the full chance of every grid cell in accordance with the trapezium rule (i.e., quantity of the cell multiplied by the common of its nook densities).
- Inside every grid cell, we are able to approximate the PDF with the bilinear interpolant between the cell corners:
- This linear approximation to the PDF can then be sampled very effectively. Drawing a single pattern is a two step course of, illustrated within the determine under. First, select a random cell from the probability-weighted checklist of cells (left-hand panel). Subsequent, pattern a degree throughout the cell by way of inverse remodel sampling (right-hand panel).
It’s value understanding that the important thing step right here is the linear approximation: we describe this, in addition to extra particulars of the inverse remodel sampling course of, within the lintsampler docs. Approximating the PDF to a linear operate inside grid every cell means it has a closed, analytic type for its quantile operate (i.e., its inverse CDF), which implies doing inverse remodel sampling basically boils right down to drawing uniform samples and making use of an algebraic operate to them.
The primary factor the person wants to fret about is getting a good grid decision, in order that the linear approximation is adequate. What an excellent decision is will fluctuate from use case to make use of case, as demonstrated in a number of the instance notebooks within the lintsampler docs.
Comfortable sampling!
