## Interactive magics
%matplotlib inline
import sys
import re
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
plt.style.use('seaborn-darkgrid')
import seaborn as sns
import patsy as pt
import pymc3 as pm
plt.rcParams['figure.figsize'] = 14, 6
np.random.seed(0)
print('Running on PyMC3 v{}'.format(pm.__version__))
Running on PyMC3 v3.4.1
This is a minimal reproducible example of Poisson regression to predict counts using dummy data.
This Notebook is basically an excuse to demo Poisson regression using PyMC3, both manually and using the glm
library to demo interactions using the patsy
library. We will create some dummy data, Poisson distributed according to a linear model, and try to recover the coefficients of that linear model through inference.
For more statistical detail see:
This very basic model is inspired by a project by Ian Osvald, which is concerned with understanding the various effects of external environmental factors upon the allergic sneezing of a test subject.
def strip_derived_rvs(rvs):
'''Convenience fn: remove PyMC3-generated RVs from a list'''
ret_rvs = []
for rv in rvs:
if not (re.search('_log',rv.name) or re.search('_interval',rv.name)):
ret_rvs.append(rv)
return ret_rvs
def plot_traces_pymc(trcs, varnames=None):
''' Convenience fn: plot traces with overlaid means and values '''
nrows = len(trcs.varnames)
if varnames is not None:
nrows = len(varnames)
ax = pm.traceplot(trcs, varnames=varnames, figsize=(12,nrows*1.4),
lines={k: v['mean'] for k, v in
pm.summary(trcs,varnames=varnames).iterrows()})
for i, mn in enumerate(pm.summary(trcs, varnames=varnames)['mean']):
ax[i,0].annotate('{:.2f}'.format(mn), xy=(mn,0), xycoords='data',
xytext=(5,10), textcoords='offset points', rotation=90,
va='bottom', fontsize='large', color='#AA0022')
This dummy dataset is created to emulate some data created as part of a study into quantified self, and the real data is more complicated than this. Ask Ian Osvald if you'd like to know more https://twitter.com/ianozsvald
nsneeze (int)
alcohol (boolean)
nomeds (boolean)
Create 4000 days of data: daily counts of sneezes which are Poisson distributed w.r.t alcohol consumption and antihistamine usage
# decide poisson theta values
theta_noalcohol_meds = 1 # no alcohol, took an antihist
theta_alcohol_meds = 3 # alcohol, took an antihist
theta_noalcohol_nomeds = 6 # no alcohol, no antihist
theta_alcohol_nomeds = 36 # alcohol, no antihist
# create samples
q = 1000
df = pd.DataFrame({
'nsneeze': np.concatenate((np.random.poisson(theta_noalcohol_meds, q),
np.random.poisson(theta_alcohol_meds, q),
np.random.poisson(theta_noalcohol_nomeds, q),
np.random.poisson(theta_alcohol_nomeds, q))),
'alcohol': np.concatenate((np.repeat(False, q),
np.repeat(True, q),
np.repeat(False, q),
np.repeat(True, q))),
'nomeds': np.concatenate((np.repeat(False, q),
np.repeat(False, q),
np.repeat(True, q),
np.repeat(True, q)))})
df.tail()
nsneeze | alcohol | nomeds | |
---|---|---|---|
3995 | 38 | True | True |
3996 | 31 | True | True |
3997 | 30 | True | True |
3998 | 34 | True | True |
3999 | 36 | True | True |
df.groupby(['alcohol','nomeds']).mean().unstack()
nsneeze | ||
---|---|---|
nomeds | False | True |
alcohol | ||
False | 1.018 | 5.866 |
True | 2.938 | 35.889 |
g = sns.factorplot(x='nsneeze', row='nomeds', col='alcohol', data=df,
kind='count', size=4, aspect=1.5)
Observe:
nomeds == False
and alcohol == False
(top-left, akak antihistamines WERE used, alcohol was NOT drunk) the mean of the poisson distribution of sneeze counts is low.alcohol == True
(top-right) increases the sneeze count nsneeze
slightlynomeds == True
(lower-left) increases the sneeze count nsneeze
furtheralcohol == True and nomeds == True
(lower-right) increases the sneeze count nsneeze
a lot, increasing both the mean and variance.Our model here is a very simple Poisson regression, allowing for interaction of terms:
$$ \theta = exp(\beta X)$$$$ Y_{sneeze\_count} ~ Poisson(\theta)$$Create linear model for interaction of terms
fml = 'nsneeze ~ alcohol + antihist + alcohol:antihist' # full patsy formulation
fml = 'nsneeze ~ alcohol * nomeds' # lazy, alternative patsy formulation
Create Design Matrices
(mx_en, mx_ex) = pt.dmatrices(fml, df, return_type='dataframe', NA_action='raise')
pd.concat((mx_ex.head(3),mx_ex.tail(3)))
Intercept | alcohol[T.True] | nomeds[T.True] | alcohol[T.True]:nomeds[T.True] | |
---|---|---|---|---|
0 | 1.0 | 0.0 | 0.0 | 0.0 |
1 | 1.0 | 0.0 | 0.0 | 0.0 |
2 | 1.0 | 0.0 | 0.0 | 0.0 |
3997 | 1.0 | 1.0 | 1.0 | 1.0 |
3998 | 1.0 | 1.0 | 1.0 | 1.0 |
3999 | 1.0 | 1.0 | 1.0 | 1.0 |
Create Model
with pm.Model() as mdl_fish:
# define priors, weakly informative Normal
b0 = pm.Normal('b0_intercept', mu=0, sd=10)
b1 = pm.Normal('b1_alcohol[T.True]', mu=0, sd=10)
b2 = pm.Normal('b2_nomeds[T.True]', mu=0, sd=10)
b3 = pm.Normal('b3_alcohol[T.True]:nomeds[T.True]', mu=0, sd=10)
# define linear model and exp link function
theta = (b0 +
b1 * mx_ex['alcohol[T.True]'] +
b2 * mx_ex['nomeds[T.True]'] +
b3 * mx_ex['alcohol[T.True]:nomeds[T.True]'])
## Define Poisson likelihood
y = pm.Poisson('y', mu=np.exp(theta), observed=mx_en['nsneeze'].values)
Sample Model
with mdl_fish:
trc_fish = pm.sample(1000, tune=1000, cores=4)
Auto-assigning NUTS sampler... Initializing NUTS using jitter+adapt_diag... Multiprocess sampling (4 chains in 4 jobs) NUTS: [b3_alcohol[T.True]:nomeds[T.True], b2_nomeds[T.True], b1_alcohol[T.True], b0_intercept] Sampling 4 chains: 100%|██████████| 8000/8000 [01:25<00:00, 93.34draws/s] The number of effective samples is smaller than 25% for some parameters.
View Diagnostics
rvs_fish = [rv.name for rv in strip_derived_rvs(mdl_fish.unobserved_RVs)]
plot_traces_pymc(trc_fish, varnames=rvs_fish)
Observe:
np.exp(pm.summary(trc_fish, varnames=rvs_fish)[['mean','hpd_2.5','hpd_97.5']])
mean | hpd_2.5 | hpd_97.5 | |
---|---|---|---|
b0_intercept | 1.017067 | 0.954463 | 1.078788 |
b1_alcohol[T.True] | 2.887893 | 2.701185 | 3.119513 |
b2_nomeds[T.True] | 5.767743 | 5.425194 | 6.175512 |
b3_alcohol[T.True]:nomeds[T.True] | 2.118607 | 1.970967 | 2.284170 |
Observe:
The contributions from each feature as a multiplier of the baseline sneezecount appear to be as per the data generation:
exp(b0_intercept): mean=1.02 cr=[0.96, 1.08]
Roughly linear baseline count when no alcohol and meds, as per the generated data:
theta_noalcohol_meds = 1 (as set above) theta_noalcohol_meds = exp(b0_intercept) = 1
exp(b1_alcohol): mean=2.88 cr=[2.69, 3.09]
non-zero positive effect of adding alcohol, a ~3x multiplier of baseline sneeze count, as per the generated data:
theta_alcohol_meds = 3 (as set above) theta_alcohol_meds = exp(b0_intercept + b1_alcohol) = exp(b0_intercept) * exp(b1_alcohol) = 1 * 3 = 3
exp(b2_nomeds[T.True]): mean=5.76 cr=[5.40, 6.17]
larger, non-zero positive effect of adding nomeds, a ~6x multiplier of baseline sneeze count, as per the generated data:
theta_noalcohol_nomeds = 6 (as set above) theta_noalcohol_nomeds = exp(b0_intercept + b2_nomeds) = exp(b0_intercept) * exp(b2_nomeds) = 1 * 6 = 6
exp(b3_alcohol[T.True]:nomeds[T.True]): mean=2.12 cr=[1.98, 2.30]
small, positive interaction effect of alcohol and meds, a ~2x multiplier of baseline sneeze count, as per the generated data:
theta_alcohol_nomeds = 36 (as set above) theta_alcohol_nomeds = exp(b0_intercept + b1_alcohol + b2_nomeds + b3_alcohol:nomeds) = exp(b0_intercept) * exp(b1_alcohol) * exp(b2_nomeds * b3_alcohol:nomeds) = 1 * 3 * 6 * 2 = 36
pymc.glm
¶Create Model
Alternative automatic formulation using pmyc.glm
with pm.Model() as mdl_fish_alt:
pm.glm.GLM.from_formula(fml, df, family=pm.glm.families.Poisson())
Sample Model
with mdl_fish_alt:
trc_fish_alt = pm.sample(2000, tune=2000)
Auto-assigning NUTS sampler... Initializing NUTS using jitter+adapt_diag... Sequential sampling (2 chains in 1 job) NUTS: [mu, alcohol[T.True]:nomeds[T.True], nomeds[T.True], alcohol[T.True], Intercept] 100%|██████████| 4000/4000 [02:08<00:00, 31.19it/s] 100%|██████████| 4000/4000 [01:11<00:00, 55.68it/s] The number of effective samples is smaller than 25% for some parameters.
View Traces
rvs_fish_alt = [rv.name for rv in strip_derived_rvs(mdl_fish_alt.unobserved_RVs)]
plot_traces_pymc(trc_fish_alt, varnames=rvs_fish_alt)
np.exp(pm.summary(trc_fish_alt, varnames=rvs_fish_alt)[['mean','hpd_2.5','hpd_97.5']])
mean | hpd_2.5 | hpd_97.5 | |
---|---|---|---|
Intercept | 1.016885e+00 | 0.955207 | 1.079243e+00 |
alcohol[T.True] | 2.889020e+00 | 2.687437 | 3.096185e+00 |
nomeds[T.True] | 5.767096e+00 | 5.384501 | 6.150425e+00 |
alcohol[T.True]:nomeds[T.True] | 2.118421e+00 | 1.958007 | 2.283186e+00 |
mu | 1.195462e+18 | 1.004549 | 1.305999e+52 |
Observe:
mu
coeff is for the overall mean of the dataset and has an extreme skew, if we look at the median value ...np.percentile(trc_fish_alt['mu'], [25,50,75])
array([ 4.06581711, 9.79920004, 24.21303451])
... of 9.45 with a range [25%, 75%] of [4.17, 24.18], we see this is pretty close to the overall mean of:
df['nsneeze'].mean()
11.42775
Example originally contributed by Jonathan Sedar 2016-05-15 github.com/jonsedar