<SYSTEM_TASK:>
Given the following text description, write Python code to implement the functionality described below step by step
<END_TASK>
<USER_TASK:>
Description:
 "Spatial Clustering" - the Galaxy Correlation Function
The degree to which objects positions are correlated with each other - "clustered" - is of great interest in astronomy. 
We expect galaxies to appear in groups and clusters, as they fall together under gravity
Step1:  The Correlation Function
The 2-point correlation function $\xi(\theta)$ is defined as "the probability of finding two galaxies separated by an angular distance $\theta$ with respect to that expected for a random distribution" (Peebles 1980), and is an excellent summary statistic for quantifying the clustering of galaxies.
The simplest possible estimator for this excess probability is just 
$\hat{\xi}(\theta) = \frac{DD - RR}{RR}$, 
where $DD(\theta) = N_{\rm pairs}(\theta) / N_D(N_D-1)/2$. Here, $N_D$ is the total number of galaxies in the dataset, and $N_{\rm pairs}(\theta)$ is the number of galaxy pairs with separation lying in a bin centered on $\theta$. $RR(\theta)$ is the same quantity computed in a "random catalog," covering the same field of view but with uniformly randomly distributed positions.
Correlations between mock galaxies distributed uniformly randomly over the survey "footprint" helps account for spurious effects in the correlation function that might arise from weird survey area design.
We'll use Mike Jarvis' TreeCorr code (Jarvis et al 2004) to compute this correlation function estimator efficiently. You can read more about better estimators starting from the TreeCorr wiki.
Step2:  Random Catalogs
First we'll need a random catalog. Let's make it the same size as the data one.
Step3:  Now let's plot both catalogs, and compare.
Step4:  Estimating $\xi(\theta)$
<ASSISTANT_TASK:>
Python Code:
%load_ext autoreload
%autoreload 2
import numpy as np
import SDSS
import pandas as pd
import matplotlib.pyplot as plt
%matplotlib inline
import copy
# We want to select galaxies, and then are only interested in their positions on the sky.
data = pd.read_csv("downloads/SDSSobjects.csv",usecols=['ra','dec','u','g',\
                                                'r','i','size'])
# Filter out objects with bad magnitude or size measurements:
data = data[(data['u'] > 0) & (data['g'] > 0) & (data['r'] > 0) & (data['i'] > 0) & (data['size'] > 0)]
# Make size cuts, to exclude stars and nearby galaxies, and magnitude cuts, to get good galaxy detections:
data = data[(data['size'] > 0.8) & (data['size'] < 10.0) & (data['i'] > 17) & (data['i'] < 22)]
# Drop the things we're not so interested in:
del data['u'], data['g'], data['r'], data['i'],data['size']
data.head()
Ngals = len(data)
ramin,ramax = np.min(data['ra']),np.max(data['ra'])
decmin,decmax = np.min(data['dec']),np.max(data['dec'])
print Ngals,"galaxy-like objects in (ra,dec) range (",ramin,":",ramax,",",decmin,":",decmax,")"
Explanation: "Spatial Clustering" - the Galaxy Correlation Function
The degree to which objects positions are correlated with each other - "clustered" - is of great interest in astronomy. 
We expect galaxies to appear in groups and clusters, as they fall together under gravity: the statistics of galaxy clustering should contain information about galaxy evolution during hierarchical structure formation.
Let's try and measure a clustering signal in our SDSS photometric object catalog.
End of explanation
# !pip install --upgrade TreeCorr
Explanation: The Correlation Function
The 2-point correlation function $\xi(\theta)$ is defined as "the probability of finding two galaxies separated by an angular distance $\theta$ with respect to that expected for a random distribution" (Peebles 1980), and is an excellent summary statistic for quantifying the clustering of galaxies.
The simplest possible estimator for this excess probability is just 
$\hat{\xi}(\theta) = \frac{DD - RR}{RR}$, 
where $DD(\theta) = N_{\rm pairs}(\theta) / N_D(N_D-1)/2$. Here, $N_D$ is the total number of galaxies in the dataset, and $N_{\rm pairs}(\theta)$ is the number of galaxy pairs with separation lying in a bin centered on $\theta$. $RR(\theta)$ is the same quantity computed in a "random catalog," covering the same field of view but with uniformly randomly distributed positions.
Correlations between mock galaxies distributed uniformly randomly over the survey "footprint" helps account for spurious effects in the correlation function that might arise from weird survey area design.
We'll use Mike Jarvis' TreeCorr code (Jarvis et al 2004) to compute this correlation function estimator efficiently. You can read more about better estimators starting from the TreeCorr wiki.
End of explanation
random = pd.DataFrame({'ra' : ramin + (ramax-ramin)*np.random.rand(Ngals), 'dec' : decmin + (decmax-decmin)*np.random.rand(Ngals)})
print len(random), type(random)
Explanation: Random Catalogs
First we'll need a random catalog. Let's make it the same size as the data one.
End of explanation
fig, ax = plt.subplots(nrows=1, ncols=2)
fig.set_size_inches(15, 6)
plt.subplots_adjust(wspace=0.2)
    
random.plot(kind='scatter', x='ra', y='dec', ax=ax[0], title='Random')
ax[0].set_xlabel('RA / deg')
ax[0].set_ylabel('Dec. / deg')
data.plot(kind='scatter', x='ra', y='dec', ax=ax[1], title='Data')
ax[1].set_xlabel('RA / deg')
ax[1].set_ylabel('Dec. / deg')
Explanation: Now let's plot both catalogs, and compare.
End of explanation
import treecorr
random_cat = treecorr.Catalog(ra=random['ra'], dec=random['dec'], ra_units='deg', dec_units='deg')
data_cat = treecorr.Catalog(ra=data['ra'], dec=data['dec'], ra_units='deg', dec_units='deg')
# Set up some correlation function estimator objects:
sep_units='arcmin'
min_sep=0.5
max_sep=10.0
N = 7
bin_size = np.log10(1.0*max_sep/min_sep)/(1.0*N)
dd = treecorr.NNCorrelation(bin_size=bin_size, min_sep=min_sep, max_sep=max_sep, sep_units=sep_units, bin_slop=0.05/bin_size)
rr = treecorr.NNCorrelation(bin_size=bin_size, min_sep=min_sep, max_sep=max_sep, sep_units=sep_units, bin_slop=0.05/bin_size)
# Process the data:
dd.process(data_cat)
rr.process(random_cat)
# Combine into a correlation function and its variance:
xi, varxi = dd.calculateXi(rr)
plt.figure(figsize=(15,8))
plt.rc('xtick', labelsize=16) 
plt.rc('ytick', labelsize=16)
plt.errorbar(np.exp(dd.logr),xi,np.sqrt(varxi),c='blue',linewidth=2)
# plt.xscale('log')
plt.xlabel('$\\theta / {\\rm arcmin}$',fontsize=20)
plt.ylabel('$\\xi(\\theta)$',fontsize=20)
plt.ylim([-0.1,0.2])
plt.grid(True)
Explanation: Estimating $\xi(\theta)$
End of explanation
<END_TASK>