import math
import matplotlib.pyplot as plt
import numpy as np
import scipy
import scipy.stats

"""
Explanation: Maximum Likelihood Estimates (MLEs)
By Delaney Granizo-Mackenzie and Andrei Kirilenko developed as part of the Masters of Finance curriculum at MIT Sloan.
Part of the Quantopian Lecture Series:

www.quantopian.com/lectures
github.com/quantopian/research_public

Notebook released under the Creative Commons Attribution 4.0 License.

In this tutorial notebook we'll do the following things:
1. Compute the MLE for a normal distribution.
2. Compute the MLE for an exponential distribution.
3. Fit a normal distribution to asset returns using MLE.
First we need to import some libraries
End of explanation
"""


TRUE_MEAN = 40
TRUE_STD = 10
X = np.random.normal(TRUE_MEAN, TRUE_STD, 1000)

"""
Explanation: Normal Distribution
We'll start by sampling some data from a normal distribution.
End of explanation
"""


def normal_mu_MLE(X):
    # Get the number of observations
    T = len(X)
    # Sum the observations
    s = sum(X)
    return 1.0/T * s

def normal_sigma_MLE(X):
    T = len(X)
    # Get the mu MLE
    mu = normal_mu_MLE(X)
    # Sum the square of the differences
    s = sum( np.power((X - mu), 2) )
    # Compute sigma^2
    sigma_squared = 1.0/T * s
    return math.sqrt(sigma_squared)

"""
Explanation: Now we'll define functions that given our data, will compute the MLE for the $\mu$ and $\sigma$ parameters of the normal distribution.
Recall that
$$\hat\mu = \frac{1}{T}\sum_{t=1}^{T} x_t$$
$$\hat\sigma = \sqrt{\frac{1}{T}\sum_{t=1}^{T}{(x_t - \hat\mu)^2}}$$
End of explanation
"""


print "Mean Estimation"
print normal_mu_MLE(X)
print np.mean(X)
print "Standard Deviation Estimation"
print normal_sigma_MLE(X)
print np.std(X)

"""
Explanation: Now let's try our functions out on our sample data and see how they compare to the built-in np.mean and np.std
End of explanation
"""


mu, std = scipy.stats.norm.fit(X)
print "mu estimate: " + str(mu)
print "std estimate: " + str(std)

"""
Explanation: Now let's estimate both parameters at once with scipy's built in fit() function.
End of explanation
"""


pdf = scipy.stats.norm.pdf
# We would like to plot our data along an x-axis ranging from 0-80 with 80 intervals
# (increments of 1)
x = np.linspace(0, 80, 80)
plt.hist(X, bins=x, normed='true')
plt.plot(pdf(x, loc=mu, scale=std))
plt.xlabel('Value')
plt.ylabel('Observed Frequency')
plt.legend(['Fitted Distribution PDF', 'Observed Data', ]);

"""
Explanation: Now let's plot the distribution PDF along with the data to see how well it fits. We can do that by accessing the pdf provided in scipy.stats.norm.pdf.
End of explanation
"""


TRUE_LAMBDA = 5
X = np.random.exponential(TRUE_LAMBDA, 1000)

"""
Explanation: Exponential Distribution
Let's do the same thing, but for the exponential distribution. We'll start by sampling some data.
End of explanation
"""


def exp_lamda_MLE(X):
    T = len(X)
    s = sum(X)
    return s/T

print "lambda estimate: " + str(exp_lamda_MLE(X))

# The scipy version of the exponential distribution has a location parameter
# that can skew the distribution. We ignore this by fixing the location
# parameter to 0 with floc=0
_, l = scipy.stats.expon.fit(X, floc=0)

pdf = scipy.stats.expon.pdf
x = range(0, 80)
plt.hist(X, bins=x, normed='true')
plt.plot(pdf(x, scale=l))
plt.xlabel('Value')
plt.ylabel('Observed Frequency')
plt.legend(['Fitted Distribution PDF', 'Observed Data', ]);

"""
Explanation: numpy defines the exponential distribution as
$$\frac{1}{\lambda}e^{-\frac{x}{\lambda}}$$
So we need to invert the MLE from the lecture notes. There it is
$$\hat\lambda = \frac{T}{\sum_{t=1}^{T} x_t}$$
Here it's just the reciprocal, so
$$\hat\lambda = \frac{\sum_{t=1}^{T} x_t}{T}$$
End of explanation
"""


prices = get_pricing('TSLA', fields='price', start_date='2014-01-01', end_date='2015-01-01')
# This will give us the number of dollars returned each day
absolute_returns = np.diff(prices)
# This will give us the percentage return over the last day's value
# the [:-1] notation gives us all but the last item in the array
# We do this because there are no returns on the final price in the array.
returns = absolute_returns/prices[:-1]

"""
Explanation: MLE for Asset Returns
Now we'll fetch some real returns and try to fit a normal distribution to them using MLE.
End of explanation
"""


mu, std = scipy.stats.norm.fit(returns)
pdf = scipy.stats.norm.pdf
x = np.linspace(-1,1, num=100)
h = plt.hist(returns, bins=x, normed='true')
l = plt.plot(x, pdf(x, loc=mu, scale=std))

"""
Explanation: Let's use scipy's fit function to get the $\mu$ and $\sigma$ MLEs.
End of explanation
"""


from statsmodels.stats.stattools import jarque_bera
jarque_bera(returns)

jarque_bera(np.random.normal(0, 1, 100))

"""
Explanation: Of course, this fit is meaningless unless we've tested that they obey a normal distribution first. We can test this using the Jarque-Bera normality test. The Jarque-Bera test will reject the hypothesis of a normal distribution if the p-value is under a c
End of explanation
"""


