GLM: Poisson Regression

In [1]:
## 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.

Local Functions

In [2]:
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')

Generate Data

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

Assumptions:

  • The subject sneezes N times per day, recorded as nsneeze (int)
  • The subject may or may not drink alcohol during that day, recorded as alcohol (boolean)
  • The subject may or may not take an antihistamine medication during that day, recorded as the negative action nomeds (boolean)
  • I postulate (probably incorrectly) that sneezing occurs at some baseline rate, which increases if an antihistamine is not taken, and further increased after alcohol is consumed.
  • The data is aggregated per day, to yield a total count of sneezes on that day, with a boolean flag for alcohol and antihistamine usage, with the big assumption that nsneezes have a direct causal relationship.

Create 4000 days of data: daily counts of sneezes which are Poisson distributed w.r.t alcohol consumption and antihistamine usage

In [3]:
# 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)))})
In [4]:
df.tail()
Out[4]:
nsneeze alcohol nomeds
3995 38 True True
3996 31 True True
3997 30 True True
3998 34 True True
3999 36 True True

View means of the various combinations (Poisson mean values)

In [5]:
df.groupby(['alcohol','nomeds']).mean().unstack()
Out[5]:
nsneeze
nomeds False True
alcohol
False 1.018 5.866
True 2.938 35.889

Briefly Describe Dataset

In [6]:
g = sns.factorplot(x='nsneeze', row='nomeds', col='alcohol', data=df,
               kind='count', size=4, aspect=1.5)
../_images/notebooks_GLM-poisson-regression_12_0.png

Observe:

  • This looks a lot like poisson-distributed count data (because it is)
  • With 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.
  • Changing alcohol == True (top-right) increases the sneeze count nsneeze slightly
  • Changing nomeds == True (lower-left) increases the sneeze count nsneeze further
  • Changing both alcohol == True and nomeds == True (lower-right) increases the sneeze count nsneeze a lot, increasing both the mean and variance.

Poisson Regression

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

In [7]:
fml = 'nsneeze ~ alcohol + antihist + alcohol:antihist'  # full patsy formulation
In [8]:
fml = 'nsneeze ~ alcohol * nomeds'  # lazy, alternative patsy formulation

1. Manual method, create design matrices and manually specify model

Create Design Matrices

In [9]:
(mx_en, mx_ex) = pt.dmatrices(fml, df, return_type='dataframe', NA_action='raise')
In [10]:
pd.concat((mx_ex.head(3),mx_ex.tail(3)))
Out[10]:
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

In [11]:
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

In [12]:
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

In [13]:
rvs_fish = [rv.name for rv in strip_derived_rvs(mdl_fish.unobserved_RVs)]
plot_traces_pymc(trc_fish, varnames=rvs_fish)
../_images/notebooks_GLM-poisson-regression_29_0.png

Observe:

  • The model converges quickly and traceplots looks pretty well mixed

Transform coeffs and recover theta values

In [14]:
np.exp(pm.summary(trc_fish, varnames=rvs_fish)[['mean','hpd_2.5','hpd_97.5']])
Out[14]:
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:

    1. 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

    2. 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

    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

    4. 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

2. Alternative method, using pymc.glm

Create Model

Alternative automatic formulation using ``pmyc.glm``

In [15]:
with pm.Model() as mdl_fish_alt:

    pm.glm.GLM.from_formula(fml, df, family=pm.glm.families.Poisson())

Sample Model

In [16]:
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

In [17]:
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)
../_images/notebooks_GLM-poisson-regression_41_0.png

Transform coeffs

In [18]:
np.exp(pm.summary(trc_fish_alt, varnames=rvs_fish_alt)[['mean','hpd_2.5','hpd_97.5']])
Out[18]:
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:

  • The traceplots look well mixed
  • The transformed model coeffs look moreorless the same as those generated by the manual model
  • Note also that the mu coeff is for the overall mean of the dataset and has an extreme skew, if we look at the median value …
In [19]:
np.percentile(trc_fish_alt['mu'], [25,50,75])
Out[19]:
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:

In [20]:
df['nsneeze'].mean()
Out[20]:
11.42775

Example originally contributed by Jonathan Sedar 2016-05-15 github.com/jonsedar