## Interactive magics
%matplotlib inline
%qtconsole --colors=linux

import sys
import warnings
warnings.filterwarnings('ignore')

import regex as re
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns
import patsy as pt
from scipy import optimize

# pymc3 libraries
import pymc3 as pm
import theano as thno
import theano.tensor as T 
# import pystan

# get watermark
%install_ext https://raw.githubusercontent.com/rasbt/watermark/master/watermark.py

sns.set(style="darkgrid", palette="muted")
pd.set_option('display.mpl_style', 'default')
plt.rcParams['figure.figsize'] = 14, 6
np.random.seed(0)

"""
Explanation: GLM: Poisson Regression
A minimal reproducable 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:

Basic info on Wikipedia
GLMs: Poisson regression, exposure, and overdispersion in Chapter 6.2 of ARM, Gelmann & Hill 2006
This worked example from ARM 6.2 by Clay Ford

This very basic model is insipired by a project by Ian Osvald, which is concerend with understanding the various effects of external environmental factors upon the allergic sneezing of a test subject.
Contents


Setup

Local Functions
Generate Data



Poisson Regression

Create Design Matrices
Create Model
Sample Model
View Diagnostics and Outputs



Package Requirements (shown as a conda-env YAML):
```
$> less conda_env_pymc3_examples.yml
name: pymc3_examples
channels:
  - defaults
dependencies:
    - python=3.5
    - jupyter
    - ipywidgets
    - numpy
    - scipy
    - matplotlib
    - pandas
    - pytables
    - scikit-learn
    - statsmodels
    - seaborn
    - patsy
    - requests
    - pip
    - pip:
        - regex        
$> conda env create --file conda_env_pymc3_examples.yml
$> source activate pymc3_examples
$> pip install --process-dependency-links git+https://github.com/pymc-devs/pymc3
```
Setup
End of explanation
"""


%load_ext watermark
%watermark -dmvgp numpy,scipy,pandas,matplotlib,pymc3,theano
print('Recursion limit {}'.format(sys.getrecursionlimit()))

"""
Explanation: What versions are we running?
End of explanation
"""


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.df_summary(trcs,varnames=varnames).iterrows()})

    for i, mn in enumerate(pm.df_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') 

"""
Explanation: Local Functions
End of explanation
"""


# 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()

"""
Explanation: 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 aggegated 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
End of explanation
"""


df.groupby(['alcohol','nomeds']).mean().unstack()

"""
Explanation: View means of the various combinations (poisson mean values)
End of explanation
"""


g = sns.factorplot(x='nsneeze', row='nomeds', col='alcohol', data=df,
               kind='count', size=4, aspect=1.5)

"""
Explanation: Briefly Describe Dataset
End of explanation
"""


fml = 'nsneeze ~ alcohol + antihist + alcohol:antihist'  # full patsy formulation

fml = 'nsneeze ~ alcohol * nomeds'  # lazy, alternative patsy formulation

"""
Explanation: 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
End of explanation
"""


(mx_en, mx_ex) = pt.dmatrices(fml, df, return_type='dataframe', NA_action='raise')

pd.concat((mx_ex.head(3),mx_ex.tail(3)))

"""
Explanation: 1. Manual method, create design matrices and manually specify model
Create Design Matrices
End of explanation
"""


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)
    

"""
Explanation: Create Model
End of explanation
"""


with mdl_fish:
    
    start_MAP = pm.find_MAP(fmin=optimize.fmin_powell)
    trc_fish = pm.sample(2000, start=start_MAP, njobs=2, step=pm.NUTS())

"""
Explanation: Sample Model
End of explanation
"""


rvs_fish = [rv.name for rv in strip_derived_rvs(mdl_fish.unobserved_RVs)]
plot_traces_pymc(trc_fish[-1000:], varnames=rvs_fish)

"""
Explanation: View Diagnostics
End of explanation
"""


np.exp(pm.df_summary(trc_fish[-1000:], varnames=rvs_fish)[['mean','hpd_2.5','hpd_97.5']])

"""
Explanation: Observe:

The model converges quickly and traceplots looks pretty well mixed

Transform coeffs and recover theta values
End of explanation
"""


with pm.Model() as mdl_fish_alt:
    
    pm.glm.glm(fml, df, family=pm.glm.families.Poisson())

"""
Explanation: 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





2. Alternative method, using pymc.glm
Create Model
Alternative automatic formulation using pmyc.glm
End of explanation
"""


with mdl_fish_alt:
    
    start_MAP_alt = pm.find_MAP(fmin=optimize.fmin_powell)
    trc_fish_alt = pm.sample(4000, start=start_MAP_alt, njobs=1, step=pm.NUTS())

"""
Explanation: Sample Model
End of explanation
"""


rvs_fish_alt = [rv.name for rv in strip_derived_rvs(mdl_fish_alt.unobserved_RVs)]
plot_traces_pymc(trc_fish_alt[-1000:], varnames=rvs_fish_alt)

"""
Explanation: View Traces
End of explanation
"""


np.exp(pm.df_summary(trc_fish_alt[-1000:], varnames=rvs_fish_alt)[['mean','hpd_2.5','hpd_97.5']])

"""
Explanation: Transform coeffs
End of explanation
"""


np.percentile(trc_fish_alt[-1000:]['mu'], [25,50,75])

"""
Explanation: 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 ...
End of explanation
"""


df['nsneeze'].mean()

"""
Explanation: ... of 9.45 with a range [25%, 75%] of [4.17, 24.18], we see this is pretty close to the overall mean of:
End of explanation
"""


