#!/usr/bin/env python
# coding: utf-8

# # Part 1 
# 
#  - start with fitting data to a parametric model
#  
#  - so we need to create the log-likehood
#  - let's create some fake data first.

# In[1]:


get_ipython().run_line_magic('matplotlib', 'inline')
import numpy as np
from matplotlib import pyplot as plt


T = (np.random.exponential(size=1000)/1.5) ** 2.3

plt.hist(T, bins=50);


# Let's use the Weibull parametric model: https://en.wikipedia.org/wiki/Weibull_distribution

# In[2]:


def pdf(params, t):
    lambda_, rho_ = params
    # I'm not going to make you type this out - it's annoying and error prone. 
    return rho_ / lambda_ * (t / lambda_) ** (rho_ - 1) * np.exp(-(t/lambda_) ** rho_)

# okay, but we actually need the _log_ of the pdf
def log_pdf(params, t):
    lambda_, rho_ = params
    # I'm not going to make you type this out - it's annoying and error prone. 
    return np.log(rho_) - np.log(lambda_) + (rho_ - 1) * (np.log(t) - np.log(lambda_)) - (t/lambda_) ** rho_

# now we can define the log likehood

def log_likelihood(params, t):
    return np.sum(log_pdf(params, t)) 


# ## `scipy.optimize.minimize`

# In[3]:


from scipy.optimize import minimize

results = minimize(log_likelihood, 
        x0 = np.array([1.0, 1.0]),  # some initial guess of the parameters. 
        method=None, 
        args=(T, ))

print(results)


# In[4]:


def log_likelihood(params, t):
    return -np.sum(log_pdf(params, t)) 


# In[5]:


results = minimize(log_likelihood, 
        x0 = np.array([1.0, 1.0]),  # some initial guess of the parameters. 
        method=None, 
        args=(T, ))

print(results)


# In[6]:


# weibull parameters must be greater than 0!
# we can "nudge" the minimizer to understand this using the bounds argument
results = minimize(log_likelihood, 
        x0 = np.array([1.0, 1.0]),
        method=None, 
        args=(T, ),
        bounds=((0, None), (0, None)))

print(results)


# ## Takeaway: `minimize` is a very flexible function for small-mid size parameter optimization

# ### A few problems though:
# 
# 1. You are stuck with only using known parametric models that are easy to implement. 
# 2. Not very fast minimization routines
# 
# Let's move to Part 2.