Machine Learning Examples

We describe two scenarioes below. The first one assumes a parametric lens model.
The second one uses the roulette formalism as a parameter-free model.

Parameter Recovery

The vanilla problem is to assume particular lens and source models, and aim to estimate the relevant parameters. For instance,

• The SIS lens model has a single parameter $R_E$ (Einstein radius),
• The spherical source model has three parameters, the size $\sigma$ and the position $(x,y)$.
• Additionally, there is a distance parameter $\chi$ which is the distance to the lens relative to the distance to the source (plane).

A simple script to generate a random dataset could be the following, but the ranges for the random parameters should be reviewed. As given, it will output a csv file with 300 datapoints.

#! /usr/bin/env python3

import numpy as np
from random import randint

n = 300
fn = "dataset.csv"

def getline(idx):

    # Source
    sigma = randint(1,60)

    # Lens
    einsteinR = randint(10,50)
    chi = randint(30,70)

    # Polar Source Co-ordinates
    phi = randint(0,359)
    R = randint(einsteinR,100)

    # Cartesian Co-ordinates
    x = R*np.cos(np.pi*phi/180)
    y = R*np.sin(np.pi*phi/180)

    srcmode = "Spherical"
    lensmode = "SIS"
    simmode = "Raytrace"
    return f'"{idx:04}","image-{idx:04}.png",{srcmode},{lensmode},{simmode},{chi},' \
         + f'{einsteinR},{sigma},{x},{y}'

header = ( "index,filename,source,lens,model,chi,"
         + "einsteinR,sigma,x,y\n"
         )

with open(fn, 'w') as f:
    f.write(header)
    for i in range(n):
        l = getline(i+1)
        f.write(l)
        f.write("\n")

The command to generate the $400\times400$ images from these datapoints is

python3 CosmoSimPy/datagen.py -C -Z 400 --csvfile dataset.csv --directory images

The -C flag is important to avoid leaking information to the machine learning system. It centres the image around the centre of mass, which is a well-definece reference point for any image. Without the flag, the lens is at the centre of the image, and the machine learning model would learn to take advantage of this fact.

The machine learning problem is to estimate the columns chi, einsteinR, sigma, x, and y in the CSV file from the corresponding image files.

Roulette Amplitude Recovery

The roulette amplitudes can be thought of as coefficients of a Taylor expansion of the lens potential. Thus they give a local description of the lens potential in a single point. The hope is to use this as a parameter-free model, but the research is still in early stages.

The basic dataset can be generated as above, or more parameters can be varied, including the choice of lens and source models. However, when we generate the images, we need to generate the roulette amplitudes as well. This can be done as follows:

python3 CosmoSimPy/datagen.py \
   -C -Z 800 -z 400 -D images \
   --nterms 5 --outfile roulette.csv --csvfile dataset.csv --xireference

This generates a new file roulette.csv wgucg includes the roulette amplitudes instead of lens parameters.

The columns we want to estimate in this scenario are

• The roulette amplitudes alpha[$m$][$s$] and beta[$m$][$s$] up to a chosen maximum $m$. This is the local description of the lens potential $\psi$.
• The position of the reference point xiX and xiY relative to the centre of mass. This is the point where we have the local description of $\psi$.
• Possibly sigma if we want to resimulate.
  The position of the source is not required, as it is inferred from the roulette amplitudes.

The --xireference option tells the simulator to calculate the amplitudes in the image of the centre of the actual source. Without the option, the amplitudes are calculated in the apparent centre of mass.

Source information, including the position (x, y) are copied from the original dataset. Other colums of the CSV file are described under Roulette Formalism.