import numpy as np
import pandas as pd
import pymc3 as pm
from scipy import stats
from scipy import optimize
import matplotlib.pyplot as plt
import seaborn as sns
import re

%matplotlib inline

"""
Explanation: GLM: Negative Binomial Regression
This notebook demos negative binomial regression using the glm submodule. It closely follows the GLM Poisson regression example by Jonathan Sedar (which is in turn insipired by a project by Ian Osvald) except the data here is negative binomially distributed instead of Poisson distributed.
Negative binomial regression is used to model count data for which the variance is higher than the mean. The negative binomial distribution can be thought of as a Poisson distribution whose rate parameter is gamma distributed, so that rate parameter can be adjusted to account for the increased variance.
Contents


Setup

Convenience Functions
Generate Data
Poisson Data
Negative Binomial Data
Visualize the Data





Negative Binomial Regression

Create GLM Model
View Results



Setup
End of explanation
"""


def plot_traces(trcs, varnames=None):
    '''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')
        
def strip_derived_rvs(rvs):
    '''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

"""
Explanation: Convenience Functions
Taken from the Poisson regression example.
End of explanation
"""


np.random.seed(123)

# Mean Poisson 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_pois = 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_pois.groupby(['nomeds', 'alcohol'])['nsneeze'].agg(['mean', 'var'])

"""
Explanation: Generate Data
As in the Poisson regression example, we assume that sneezing occurs at some baseline rate, and that consuming alcohol, not taking antihistamines, or doing both, increase its frequency.
Poisson Data
First, let's look at some Poisson distributed data from the Poisson regression example.
End of explanation
"""


# Gamma shape parameter
alpha = 10

def get_nb_vals(mu, alpha, size):
    """Generate negative binomially distributed samples by
    drawing a sample from a gamma distribution with mean `mu` and
    shape parameter `alpha', then drawing from a Poisson
    distribution whose rate parameter is given by the sampled
    gamma variable.

    """

    g = stats.gamma.rvs(alpha, scale=mu / alpha, size=size)
    return stats.poisson.rvs(g)

# Create samples
n = 1000
df = pd.DataFrame({
        'nsneeze': np.concatenate((get_nb_vals(theta_noalcohol_meds, alpha, n),
                                   get_nb_vals(theta_alcohol_meds, alpha, n),
                                   get_nb_vals(theta_noalcohol_nomeds, alpha, n),
                                   get_nb_vals(theta_alcohol_nomeds, alpha, n))),
        'alcohol': np.concatenate((np.repeat(False, n),
                                   np.repeat(True, n),
                                   np.repeat(False, n),
                                   np.repeat(True, n))),
        'nomeds': np.concatenate((np.repeat(False, n),
                                      np.repeat(False, n),
                                      np.repeat(True, n),
                                      np.repeat(True, n)))})

df.groupby(['nomeds', 'alcohol'])['nsneeze'].agg(['mean', 'var'])

"""
Explanation: Since the mean and variance of a Poisson distributed random variable are equal, the sample means and variances are very close.
Negative Binomial Data
Now, suppose every subject in the dataset had the flu, increasing the variance of their sneezing (and causing an unfortunate few to sneeze over 70 times a day). If the mean number of sneezes stays the same but variance increases, the data might follow a negative binomial distribution.
End of explanation
"""


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

# Make x-axis ticklabels less crowded
ax = g.axes[1, 0]
labels = range(len(ax.get_xticklabels(which='both')))
ax.set_xticks(labels[::5])
ax.set_xticklabels(labels[::5]);

"""
Explanation: As in the Poisson regression example, we see that drinking alcohol and/or not taking antihistamines increase the sneezing rate to varying degrees. Unlike in that example, for each combination of alcohol and nomeds, the variance of nsneeze is higher than the mean. This suggests that a Poisson distrubution would be a poor fit for the data since the mean and variance of a Poisson distribution are equal.
Visualize the Data
End of explanation
"""


fml = 'nsneeze ~ alcohol + nomeds + alcohol:nomeds'

with pm.Model() as model:
    pm.glm.GLM.from_formula(formula=fml, data=df, family=pm.glm.families.NegativeBinomial())
    
    # Old initialization
    # start = pm.find_MAP(fmin=optimize.fmin_powell)
    # C = pm.approx_hessian(start)
    # trace = pm.sample(4000, step=pm.NUTS(scaling=C))
    
    trace = pm.sample(2000, njobs=2)

"""
Explanation: Negative Binomial Regression
Create GLM Model
End of explanation
"""


rvs = [rv.name for rv in strip_derived_rvs(model.unobserved_RVs)]
plot_traces(trace[1000:], varnames=rvs);

# Transform coefficients to recover parameter values
np.exp(pm.df_summary(trace[1000:], varnames=rvs)[['mean','hpd_2.5','hpd_97.5']])

"""
Explanation: View Results
End of explanation
"""


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

df.nsneeze.mean()

trace[1000:]['alpha'].mean()

"""
Explanation: The mean values are close to the values we specified when generating the data:
- The base rate is a constant 1.
- Drinking alcohol triples the base rate.
- Not taking antihistamines increases the base rate by 6 times.
- Drinking alcohol and not taking antihistamines doubles the rate that would be expected if their rates were independent. If they were independent, then doing both would increase the base rate by 3*6=18 times, but instead the base rate is increased by 3*6*2=16 times.
Finally, even though the sample for mu is highly skewed, its median value is close to the sample mean, and the mean of alpha is also quite close to its actual value of 10.
End of explanation
"""


