Minimum Spanning Tree Statistics

Minimum Spanning Tree Statistics#

The minimum spanning tree (MST), a graph constructed from a distribution of points, draws lines between pairs of points so that all points are linked in a single skeletal structure. The structure/tree contains no loops and the minimal possible total edge length.

An example MST constructed from a random subset of points.

Measuring the MST with the MinimumSpanningTree class#

To MinimumSpanningTree class is imported from acm.estimators.galaxy_clustering.mst:

from acm.estimators.galaxy_clustering.mst import MinimumSpanningTree

To initialise and setup the MST class

mst = MinimumSpanningTree()
# Jitter dispersion scale, the point-process smoothing scale
sigmaJ = 3.
# Size of the box for the simulation data set
boxsize = 1000.
# Order of the maximum percolation statistics
Nthpoint = 10
# OPTIONAL: origin of the box.
origin = 0.
# OPTIONAL: Divide the box, to compute the MST in subboxes.
split = 1
# OPTIONAL: The number of iterations to marginalise over the stochasticity from the jitter dispersion.
iterations = 1
# OPTIONAL: The number of bins used for the MST distributions. 
quartiles = 50
# Setup MST
mst.setup(sigmaJ, boxsize, Nthpoint, origin=origin, split=split, iterations=iterations, quartiles=quartiles)

The class can be used to compute percolation statistics, which are a way to characterise paths eminating from a single point.

The percolation paths from a point marked by 'x'. The colormap corresponds to the number of edges percolating along the MST from a single point. The lengths of the paths and the straightline distance between end points are extracted to measure properties of the MST.

Compute the MST statistics with, where the positions are inputed as pos where x=pos[:,0], y=pos[:,1] and z=pos[:,2].

mstdict = mst.get_percolation_statistics(pos)

Where mstdict is a dictionary containing the percolation statistics:

mstdict['mst1pt'] # which gives the distribution of the MST edge lengths.
mstdict['mst2pt'] # which gives the distribution of 2 edges in the MST.
mstdict['end2pt'] # which gives the distribution of the distances between the end points of the associated 2 edges.

For the next N-points you can get the statistics for the N-th edge distributions using:

mstdict['mst%ipt'%N]
mstdict['end%ipt'%N]

and to plot all the outputs:

plot_percolation_statistics(mstdict)

The distribution of the percolation paths. The solid lines correspond to the distance along the MST, and the dashed lines the direct distances between the end points of the paths.

MST Emulator Mock Challenge with the MinimumSpanningTree class#

To construct the MST emulators we must first initiate the EMC class and set some variables

from acm.projects.emc_new.mst import MinimumSpanningTree

# define the smoothing scale used for the measurement, either:
# - '0p0' = No point-process smoothing
# - '3p0' = 3 Mpc/h point-process smoothing [DEFAULT]
sigmaJ = '3p0'

# define the rebining value, this averages consecutive bins to 
# reduce the dimensionality of the data vectors
# NOTE: rebin must be a factor of 50 (the total number of bins
# for the MST statistics)
rebin = 5

# define the maximum percolation path distance
# where 1 <= Nptmax <= 10
Nptmax = 10

mst = MinimumSpanningTree(sigmaJ=sigmaJ, rebin=rebin, Nptmax=Nptmax)

We now collect the data vectors to compute the covariance matrix for the MST statistics

cov_data = mst.create_covariance()

and now to construct the lhc file for the MST

cout = mst.create_lhc()
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