Tutorial: distribution with outliers
In this tutorial you will learn:
How to deal with outliers and non-gaussian distributions
In the “intrinsic distribution” tutorial, we fitted a gaussian distribution. But maybe we do not want to make such a strong assumption – we may not know the exact distribution. Perhaps it is skewed, has several sub-populations. This can be tested by trying out different models.
Here, we check compare three scenarios:
baseline model: a gaussian distribution
baseline model + outlier model
heavy-tailed distribution
Here is our data:
[1]:
import numpy as np
# velocity dispersions of dwarf galaxies by van Dokkum et al., Nature, 555, 629 https://arxiv.org/abs/1803.10237v1
values = np.array([15, 4, 2, 11, 1, -2, -1, -14, -39, -3])
values_lo = np.array([7, 16, 6, 3, 6, 5, 10, 6, 11, 13])
values_hi = np.array([7, 15, 8, 3, 6, 6, 10, 7, 14, 14])
n_data = len(values)
Visualise the data
Lets plot the data first to see what is going on:
[2]:
%matplotlib inline
import matplotlib.pyplot as plt
np.random.seed(42)
samples = []
for i in range(n_data):
# draw normal random points
u = np.random.normal(size=400)
v = values[i] + np.where(u < 0, u * values_lo[i], u * values_hi[i])
samples.append(v)
samples = np.array(samples)
plt.figure()
# for each galaxy, plot alittle cloud with its own colors
plt.violinplot(samples.transpose(), vert=False, showextrema=False)
xlabel = 'Velocity [km/s]'
plt.xlabel(xlabel)
plt.xlim(-50, 50);
Models
Gaussian
[3]:
parameters = ['mean', 'scatter']
def prior_transform(cube):
params = cube.copy()
params[0] = cube[0] * 200 - 100 # mean from -100 to +100
params[1] = 10**(cube[1] * 2) # scatter from 1 to 100
return params
import scipy.stats
def log_likelihood(params):
probs_samples = scipy.stats.norm(*params).pdf(samples)
probs_objects = probs_samples.mean(axis=1)
loglike = np.log(probs_objects + 1e-100).sum()
return loglike
import ultranest
sampler = ultranest.ReactiveNestedSampler(parameters, log_likelihood, prior_transform)
result = sampler.run()
sampler.print_results()
[ultranest] Sampling 400 live points from prior ...
[ultranest] Widening roots to 404 live points (have 400 already) ...
[ultranest] Sampling 4 live points from prior ...
[ultranest] Explored until L=-4e+01
[ultranest] Likelihood function evaluations: 5209
[ultranest] logZ = -43.83 +- 0.0769
[ultranest] Effective samples strategy satisfied (ESS = 1660.2, need >400)
[ultranest] Posterior uncertainty strategy is satisfied (KL: 0.45+-0.07 nat, need <0.50 nat)
[ultranest] Evidency uncertainty strategy is satisfied (dlogz=0.08, need <0.5)
[ultranest] logZ error budget: single: 0.09 bs:0.08 tail:0.01 total:0.08 required:<0.50
[ultranest] done iterating.
logZ = -43.833 +- 0.196
single instance: logZ = -43.833 +- 0.092
bootstrapped : logZ = -43.834 +- 0.196
tail : logZ = +- 0.010
insert order U test : converged: True correlation: inf iterations
mean : -26.5 │ ▁▁ ▁▁▁ ▁▁▁▁▁▁▁▂▃▄▆▇▇▇▆▄▃▂▁▁▁▁▁▁ ▁▁▁▁ │22.4 0.0 +- 4.1
scatter : 1.0 │▁▂▃▆▇▇▇▆▅▅▄▃▂▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁ ▁▁ │46.9 9.5 +- 4.2
Gaussian + Outliers
[4]:
parameters2 = ['mean', 'scatter', 'w_outlier']
def prior_transform2(cube):
params = cube.copy()
params[0] = cube[0] * 200 - 100 # mean from -100 to +100
params[1] = 10**(cube[1] * 2) # scatter from 1 to 100
params[2] = (cube[2] * 0.2) # fraction of outliers: allow 0-20%
return params
def log_likelihood2(params):
mean, scatter, weight = params
# compute probability if following distribution
probs_samples = scipy.stats.norm(mean, scatter).pdf(samples)
# compute probability density, if outlier:
prob_outlier = scipy.stats.uniform(-100, 200).pdf(samples)
# combine the two populations with weight:
probs_samples = probs_samples * (1 - weight) + weight * prob_outlier
# average within each object (logical OR)
probs_objects = probs_samples.mean(axis=1)
# multiply across object (logical AND)
loglike = np.log(probs_objects + 1e-100).sum()
return loglike
sampler2 = ultranest.ReactiveNestedSampler(parameters2, log_likelihood2, prior_transform2)
result2 = sampler2.run()
sampler2.print_results()
[ultranest] Sampling 400 live points from prior ...
[ultranest] Explored until L=-4e+01
[ultranest] Likelihood function evaluations: 7220
[ultranest] logZ = -44.39 +- 0.0641
[ultranest] Effective samples strategy satisfied (ESS = 1797.4, need >400)
[ultranest] Posterior uncertainty strategy is satisfied (KL: 0.46+-0.07 nat, need <0.50 nat)
[ultranest] Evidency uncertainty strategy is satisfied (dlogz=0.06, need <0.5)
[ultranest] logZ error budget: single: 0.09 bs:0.06 tail:0.01 total:0.06 required:<0.50
[ultranest] done iterating.
logZ = -44.390 +- 0.175
single instance: logZ = -44.390 +- 0.092
bootstrapped : logZ = -44.391 +- 0.175
tail : logZ = +- 0.010
insert order U test : converged: True correlation: inf iterations
mean : -25.0 │ ▁ ▁▁▁▁▁▁▁▁▂▃▅▆▇▇▇▅▅▃▁▁▁▁ ▁▁ ▁ │33.6 0.8 +- 4.3
scatter : 1.0 │▃▃▅▆▇▇▇▆▅▃▂▂▁▁▁▁▁▁▁▁▁▁▁▁▁ ▁▁▁▁ ▁ │49.2 8.7 +- 4.4
w_outlier : 0.000 │▆▇▇▇▆▆▇▆▆▇▇▇▅▆▅▆▅▅▆▅▆▄▅▄▄▄▆▄▄▃▃▅▅▄▄▄▃▄▃│0.200 0.086 +- 0.056
Heavy-tailed distribution
[5]:
parameters3 = ['mean', 'scatter', 'dof']
def prior_transform3(cube):
params = cube.copy()
params[0] = cube[0] * 200 - 100 # mean from -100 to +100
params[1] = 10**(cube[1] * 2) # scatter from 1 to 100
params[2] = 10**(cube[2]) # degrees of freedom in student-t: 1 (heavy tails)..10 (gaussian)
return params
def log_likelihood3(params):
mean, scatter, dof = params
# compute student-t probability
probs_samples = scipy.stats.t(loc=mean, scale=scatter, df=dof).pdf(samples)
# average within each object (logical OR)
probs_objects = probs_samples.mean(axis=1)
# multiply across object (logical AND)
loglike = np.log(probs_objects + 1e-100).sum()
return loglike
sampler3 = ultranest.ReactiveNestedSampler(parameters3, log_likelihood3, prior_transform3)
sampler3.run()
sampler3.print_results()
[ultranest] Sampling 400 live points from prior ...
[ultranest] Explored until L=-4e+01
[ultranest] Likelihood function evaluations: 7142
[ultranest] logZ = -44.13 +- 0.07869
[ultranest] Effective samples strategy satisfied (ESS = 1732.4, need >400)
[ultranest] Posterior uncertainty strategy is satisfied (KL: 0.46+-0.06 nat, need <0.50 nat)
[ultranest] Evidency uncertainty strategy is satisfied (dlogz=0.08, need <0.5)
[ultranest] logZ error budget: single: 0.09 bs:0.08 tail:0.01 total:0.08 required:<0.50
[ultranest] done iterating.
logZ = -44.136 +- 0.161
single instance: logZ = -44.136 +- 0.091
bootstrapped : logZ = -44.135 +- 0.161
tail : logZ = +- 0.010
insert order U test : converged: True correlation: inf iterations
mean : -19.9 │ ▁ ▁▁▁▁▁▁▁▂▂▃▄▅▅▇▇▇▇▆▅▃▂▂▁▁▁▁▁▁ ▁ │25.2 0.7 +- 4.3
scatter : 1.0 │▃▅▆▇▇▇▇▅▅▄▃▃▂▂▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ ▁ ▁ ▁ │39.4 7.5 +- 4.2
dof : 1.0 │▇▆▅▅▆▅▄▃▃▃▃▄▃▂▃▃▃▂▃▂▃▂▂▂▂▂▂▂▂▂▁▂▂▂▂▂▂▂▂│10.0 4.2 +- 2.6
Comparing the models
[6]:
plt.figure(figsize=(5, 12))
plt.xlabel(xlabel)
from ultranest.plot import PredictionBand
x = np.linspace(-50, 50, 400)
plt.subplot(3, 1, 1)
band = PredictionBand(x)
for params in sampler.results['samples'][:40]:
mean, scatter = params
band.add(scipy.stats.norm(mean, scatter).pdf(x))
band.shade(color='k', alpha=0.1, label="Gaussian")
band.line(color='k')
plt.legend(loc='upper right')
plt.title("Gaussian: lnZ=%.1f" % sampler.results['logz'])
plt.subplot(3, 1, 2)
band = PredictionBand(x)
bando = PredictionBand(x)
for params in sampler2.results['samples'][:40]:
mean, scatter, outlier_weight = params
band.add(scipy.stats.norm(mean, scatter).pdf(x) * (1 - outlier_weight))
bando.add(scipy.stats.norm(-100, 200).pdf(x) * outlier_weight)
band.shade(color='k', alpha=0.1, label="Gaussian")
band.line(color='k');
bando.shade(color='orange', alpha=0.1, label="Outliers")
bando.line(color='orange');
plt.legend(loc='upper right')
plt.title("Gaussian + outliers: lnZ=%.1f" % sampler2.results['logz'])
plt.subplot(3, 1, 3)
for params in sampler3.results['samples'][:40]:
mean, scatter, df = params
band.add(scipy.stats.t(loc=mean, scale=scatter, df=df).pdf(x))
band.shade(color='k', alpha=0.1, label='Student-t')
band.line(color='k');
plt.title("Student-t: lnZ=%.1f" % sampler3.results['logz'])
plt.legend(loc='upper right');
The models look very similar in their predictions, and their lnZ values are also quite close. So I would not have a strong preference in any direction.
Lets compare the posterior distributions with getdist:
[7]:
from getdist import MCSamples, plots
samples_g = MCSamples(samples=sampler.results['samples'],
names=sampler.results['paramnames'],
label='Gaussian',
settings=dict(smooth_scale_2D=3), sampler='nested')
samples_o = MCSamples(samples=sampler2.results['samples'][:,:2],
names=sampler2.results['paramnames'][:2],
label='Gaussian + Outlier',
settings=dict(smooth_scale_2D=3), sampler='nested')
samples_t = MCSamples(samples=sampler3.results['samples'][:,:2],
names=sampler3.results['paramnames'][:2],
label='Student-t',
settings=dict(smooth_scale_2D=3), sampler='nested')
mcsamples = [samples_g, samples_o, samples_t]
g = plots.get_subplot_plotter(width_inch=8)
g.settings.num_plot_contours = 1
g.triangle_plot(mcsamples, filled=False, contour_colors=plt.cm.Set1.colors)
Removed no burn in
Removed no burn in
Removed no burn in
Next
Try adding an outlier.
Try adjusting the number of live points (min_num_live_points) and effective sample size (min_ess) parameters above to decrease the uncertainties.
Try varying the prior ranges.
Continue with the other tutorials and explore other UltraNest features