Fig 3. Bayesian repertoire overlap consistently estimates true overlap

## Figure 3:

Repertoires with true overlaps ranging from 0 to 60 were subsampled in simulations. As sampling rates increase from $n_a = n_b = 30$ (left) to 40 (middle) and to 50 (right), the estimates of BRO (colored circles) approach the true values (dotted lines) symmetrically. Estimates from $\overset{\circ}{S}$ (crosses) approach the true values from below, systematically underestimating the true overlap. This bias is worse with lower sampling rates . Similar results are found when $n_a \ne n_b$, and when the total repertoire sizes are different from each other (S1 Fig).

Link to the published figure: https://doi.org/10.1371/journal.pcbi.1006898.g003

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
%matplotlib inline
from scipy.stats import hypergeom
from scipy.stats import binned_statistic as binsta
from scipy.special import logsumexp
from util import *
import palettable as pal
clrx = pal.cartocolors.qualitative.Prism_10.mpl_colors
clr = tuple(x for n,x in enumerate(clrx) if n in [1,2,4,5,6])
clr2 = pal.cartocolors.sequential.agSunset_7.mpl_colors
import matplotlib.gridspec as gridspec
import matplotlib.patches as patches

import plotly.graph_objects as go
import plotly.tools as tls
from plotly.offline import plot, iplot, init_notebook_mode
from IPython.core.display import display, HTML
init_notebook_mode(connected = True)
config={'showLink': False, 'displayModeBar': False}

# CCP, the Coupon Collector's Problem
def ccp_sample(c,pool=60):
return len(set(np.random.choice(pool,c)))

# Draw overlap
def nab_sample(s,na,nb,pool=60):
sa = np.random.hypergeometric(s,pool-s,na)
nab = np.random.hypergeometric(sa,pool-sa,nb)
return nab

# Overlap between two PCRs of depth c and overlap s
def pcr_sample(c,s):
na = ccp_sample(c)
nb = ccp_sample(c)
return nab_sample(s,na,nb),na,nb

# Draw na and nb samples from two populations of size pool_a and pool_b, with true overlap s
# and return empirical overlap between na and nb
# note that this is basically the same as nab_sample, but with two different size pools!
def nab_sample_unequal(s,na,nb,pool_a,pool_b):
sa = np.random.hypergeometric(s,pool_a-s,na)
nab = np.random.hypergeometric(sa,pool_b-sa,nb)
return nab

def p_ccp(c, pool=60):
p = np.zeros([c+1,pool+1])
p[0,0] = 1;
for row in range(1,c+1):
for k in range(1,np.min([row+2,pool+1])):
p[row,k] = p[row-1,k]*k/pool + p[row-1,k-1]*(1-(k-1)/pool)
return p[-1,:]

def p_overlap(na,nb,nab,pool=60):
p_s = np.zeros(pool+1)
# reference: hypergeom.pmf(outcome, Total, hits, Draws, loc=0)
for s in np.arange(pool+1):
# p_sa is the probability that we'd get sa from the overlap (s), just in na draws of a
p_sa = hypergeom.pmf(np.arange(pool+1),pool,s,na)
# p_nab_given_sa is the probability of getting that nab, given sa
p_nab_given_sa = hypergeom.pmf(nab,pool,np.arange(pool+1),nb)
p_s[s] = np.dot(p_sa,p_nab_given_sa)
return p_s/np.sum(p_s)

def e_overlap(na,nb,nab,pool=60):
p_s = p_overlap(na,nb,nab,pool=pool)
return np.dot(np.arange(pool+1),p_s)

def credible_interval(na,nb,nab,pct=90,pool=60):
p_s = p_overlap(na,nb,nab,pool=pool)
cdf = np.cumsum(p_s)
ccdf = np.flipud(np.cumsum(np.flipud(p_s)))
# adjust for fractions vs percents; put everything as a fraction
if pct > 1:
pct = pct/100
cutoff = (1-pct)/2
# get the lower bound.
# it's the first index at which cdf ≥ cutoff
try:
lower = np.where(cdf >= cutoff)
except IndexError:
lower = 0
# get the upper bound
# it's the first index at which ccdf ≥ 0.05
try:
upper = np.where(ccdf >= cutoff)[-1]
except IndexError:
upper=pool
expectation = np.dot(np.arange(pool+1),p_s)
# Sanity and indexing check: uncomment this line to see true tail probability ≤ 0.05
# print([cdf[lower-1],(1-ccdf[upper+1])])
return lower,expectation,upper

def p_nab_given_c(s,c,pool=60):
pna = p_ccp(c)
pnb = p_ccp(c)
nas = np.arange(1,len(pna))
nbs = np.arange(1,len(pnb))
p_gen = np.zeros([pool+1,pool+1,pool+1])
for na in nas:
p_sa = hypergeom.pmf(np.arange(pool+1),pool,s,na)
for nb in nbs:
pna_pnb = pna[na] * pnb[nb]
for nab in range(0,np.minimum(na,nb)):
p_nab_given_sa = hypergeom.pmf(nab,pool,np.arange(pool+1),nb)
p_nab_given_s = np.dot(p_sa,p_nab_given_sa)
p_gen[na,nb,nab] = p_nab_given_s * pna_pnb
return p_gen

def p_shat_given_sc(s,c,shat,pool=60):
masses = p_nab_given_c(s,c,pool=pool)
if np.sum(masses)<0.99:
print('Swapping to Monte Carlo')
return p_shat_given_sc_montecarlo(s,c,shat,pool=pool)
hist = binsta(np.ravel(shat),np.ravel(masses),statistic='sum',bins=(np.arange(pool+2)-0.5))
return hist

def p_shat_given_sc_montecarlo(s,c,shat,pool=60,n_mc=int(1e5)):
masses = np.zeros([pool+1,pool+1,pool+1])
for ii in range(n_mc):
nab,na,nb = pcr_sample(c,s)
masses[na,nb,nab] += 1
hist = binsta(np.ravel(shat),np.ravel(masses/n_mc),statistic='sum',bins=(np.arange(pool+2)-0.5))
return hist

def compute_all_estimates(pool=60):
shat = np.zeros([pool+1,pool+1,pool+1])
for na in range(1,pool+1):
for nb in range(1,pool+1):
for nab in range(0,np.minimum(na+1,nb+1)):
shat[na,nb,nab] = e_overlap(na,nb,nab,pool=pool)
return shat

def p_overlap_unequal(na,nb,nab,pool_a,pool_b):
# all loops are in terms of pool_a, which is assumed to be ≤ pool_b.
p_s = np.zeros(pool_a+1)
# reference: hypergeom.pmf(outcome, Total, hits, Draws, loc=0)
for s in np.arange(pool_a+1):
# p_sa is the probability that we'd get sa from the overlap (s), just in na draws of a
p_sa = hypergeom.pmf(np.arange(pool_a+1),pool_a,s,na)
# p_nab_given_sa is the probability of getting that nab, given sa
p_nab_given_sa = hypergeom.pmf(nab,pool_b,np.arange(pool_a+1),nb)
p_s[s] = np.dot(p_sa,p_nab_given_sa)
return p_s/np.sum(p_s)

def e_overlap_unequal(na,nb,nab,pool_a,pool_b):
# TODO. Code expects that pool_b > pool_a...
p_s = p_overlap_unequal(na,nb,nab,pool_a,pool_b)
return np.dot(np.arange(pool_a+1),p_s)

# shat = compute_all_estimates(pool=60)
# np.save('shat_60.npy',shat)
shat = np.load('shat_60.npy')

pool=60
shat = np.load('shat_60.npy')
reps=4
recovered = np.zeros([6,reps,pool+1])
planted = np.zeros([6,reps,pool+1])
recovered_pts = np.zeros([6,reps,pool+1])
nas = [30,40,50]
for idxa,na in enumerate(nas):
nb = na
for s in np.arange(0,pool+1,1):
for rep in range(3):
nab = nab_sample(s,na,nb)
planted[idxa,rep,s] = s
recovered[idxa,rep,s] = e_overlap(na,nb,nab)
recovered_pts[idxa,rep,s] = 2*nab/(na+nb)

z = np.zeros(60)
d = np.arange(1,60)
fig3 = go.Figure()

colors = ['rgba(129,105,177,0.85)', 'rgba(172,110,191,0.85)','rgba(215,128,193,0.85)']

planted1 = np.zeros((4,244))
recovered1 = np.zeros((4,244))
recovered_pts1 = np.zeros((4,244))
for index in range(0, 3):
tmp1 = planted[index,:,:]
tmp2 = recovered[index,:,:]
tmp3 = recovered_pts[index,:,:]

planted1[index] = tmp1.flatten()
recovered1[index] = tmp2.flatten()
recovered_pts1[index] = tmp3.flatten()

fig3 = go.Figure()

fig3.add_trace(go.Scatter(x = d , y = d, line = dict(width=1, dash='dash', color = 'rgb(0,0,0)'), name = 'Reference', showlegend = False))

fig3.add_trace(go.Scatter(x=planted1, y=recovered1,
mode='markers',
name = 'BRO',
marker=dict(color='rgba(129,105,177,0.85)')))
fig3.add_trace(go.Scatter(x=planted1, y=pool*recovered_pts1,
mode='markers',
name = 'S',
marker_symbol = "x",
marker=dict(color='gray')))
for idxa in [1,2]:
fig3.add_trace(go.Scatter(x=planted1[idxa], y=recovered1[idxa],
mode='markers',
name = 'BRO',
visible = False,
marker=dict(color=colors[idxa])))
fig3.add_trace(go.Scatter(x=planted1[idxa], y=pool*recovered_pts1[idxa],
mode='markers',
name = 'S',
visible = False,
marker_symbol = "x",
marker=dict(color='gray')))

fig3.update_layout(
updatemenus=[
dict(
active=0,
buttons=list([
dict(label="30 samples",
method="update",
args=[{"visible": [True,True, True, False, False, False, False]},
{"title": ""}]),
dict(label="40 samples",
method="update",
args=[{"visible": [True,False, False, True, True, False, False]},
{"title": ""
}]),
dict(label="50 samples",
method="update",
args=[{"visible": [True,False, False,False,False, True, True]},
{"title": ""}]),

]),
x=0.2,
y = 1.2
)
])

fig3.update_layout(plot_bgcolor='rgb(255,255,255)',xaxis_title="True overlap s",
yaxis_title='Estimated overlap \u015D', legend=dict(x=.85, y=.1))
fig3.update_xaxes(ticks = 'outside', showline=True, linecolor='black')
fig3.update_yaxes(ticks = 'outside', showline=True, linecolor='black')

# Plot figure
plot(fig3, filename = 'plotly_figures/fig3.html', config = config)
display(HTML('plotly_figures/fig3.html'))