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

# # Shannon entropy vs Spatial entropy
# 
# Currently, there are two spatial entropy have been implemented:
# 
# - Leibovici's entropy
# - Altieri's entropy
# 
# Here we are going to compare the difference between Shannon's entropy and the spatial entropy

# ## Shannon entropy
# 
# Given a random variable $X$, with possible outcomes $x_{i}$, each with probability $P(x_{i})$, the entropy $H(X)$ of $X$ is as follows:
#     
# $$H(X) = - \sum_{i=1}^n P(x_i)log(P(x_i))$$

# In[1]:


get_ipython().run_line_magic('config', "InlineBackend.figure_format = 'retina'")


# In[2]:


import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from spatialentropy import leibovici_entropy, altieri_entropy

from utils import plot_points, random_data, cluster_data


# Define a window where we want to put our points, it's a 100*100 square

# In[3]:


window = [(0,0),(100,0),(100,100),(0,100)]


# In[4]:


# generate random types
types = np.random.choice(["lion", "tiger", "rabbit"], 200)

# complete random data
pp = random_data(window, 200)

# cluster data
cpp1, cpp1_types = cluster_data(window, 200, 10, 10, types)

# more cluster data
cpp2, cpp2_types = cluster_data(window, 200, 3, 5, types)


# In[5]:


fig, (ax1, ax2, ax3) = plt.subplots(1,3, figsize=(15,5))
plot_points(pp, types, ax=ax1, title="Random")
plot_points(cpp1, cpp1_types, ax=ax2, title="Cluster")
plot_points(cpp2, cpp2_types, ax=ax3, title="More cluster")


# Calculate shannon's entropy

# In[6]:


from scipy.stats import entropy
from collections import Counter
c = np.asarray(list(Counter(types).values()))
c = c / c.sum()
shan_ent = entropy(c)


# Calculate leibovici's entropy

# In[7]:


lb_ent1 = leibovici_entropy(pp, types, d=10)
lb_ent2 = leibovici_entropy(cpp1, cpp1_types, d=10)
lb_ent3 = leibovici_entropy(cpp2, cpp2_types, d=10)
print("Entropy:", lb_ent1.entropy, lb_ent2.entropy, lb_ent3.entropy)


# Calculate altieri's entropy

# In[8]:


ae1 = altieri_entropy(pp, types, cut=10)
ae2 = altieri_entropy(cpp1, cpp1_types, cut=10)
ae3 = altieri_entropy(cpp2, cpp2_types, cut=10)
print("Entropy:", ae1.entropy, ae2.entropy, ae3.entropy)
print("Mutual_info:", ae1.mutual_info, ae2.mutual_info, ae3.mutual_info)
print("Residue:", ae1.residue, ae2.residue, ae3.residue)


# In[9]:


fig, (ax1, ax2, ax3) = plt.subplots(1,3, figsize=(15,5))
plot_points(pp, types, ax=ax1, 
            title=f"Shannon entropy: {shan_ent:.3f}\nLeibovici entropy: {lb_ent1.entropy:.3f}\nAltieri entropy: {ae1.entropy:.3f}")
plot_points(cpp1, cpp1_types, ax=ax2, 
            title=f"Shannon entropy: {shan_ent:.3f}\nLeibovici entropy: {lb_ent2.entropy:.3f}\nAltieri entropy: {ae2.entropy:.3f}")
plot_points(cpp2, cpp2_types, ax=ax3, 
            title=f"Shannon entropy: {shan_ent:.3f}\nLeibovici entropy: {lb_ent3.entropy:.3f}\nAltieri entropy: {ae3.entropy:.3f}")


# If you are interested in the details, feel free to read what's below.
# 
# I try to explain it easier than what's in the paper, but you should refer to the paper for more details

# # Leibovici’s entropy
# 
# A new variable $Z$ is introduced. $Z$ is defined as co-occurrences across the space.
# 
# For example, we have $I$ types of cells. The combination of any two type of cells is $(x_i, x_{i'})$, the number of all combinations is denoted as $R$.
# 
# If order is preserved, $R = P_I^2 = I^2$; If the combinations are unordered, $R = C_I^2= (I^2+I)/2$
# 
# To a more general situation, consider the combination of $m$ types of cell, $R = P_I^m = I^m$ or $R=C_I^m$
# 
# At a user defined distance $d$, only co-occurrences with the distance $d$ will take into consideration.
# 
# $$H(Z|d) = \sum_{r=1}^{I^m}{p(z_r|d)}log(\frac{1}{p(z_r|d)})$$
# 
# Reference:
# 
# Leibovici, D. G., Claramunt, C., Le Guyader, D., & Brosset, D. (2014). Local and global spatio-temporal entropy indices based on distance-ratios and co-occurrences distributions. International Journal of Geographical Information Science, 28(5), 1061-1084. [link](https://www.tandfonline.com/doi/full/10.1080/13658816.2013.871284)

# # Altieri's entropy
# 
# This introduce another new vairable $W$. $w_k$ represents a series of sample window, i.e. $[0,2][2,4][4,10],[10,...]$ while $k=1,...,K$
# 
# The purpose of this entropy is to decompose it into **Spatial mutual information** $MI(Z,W)$ and **Spatial residual entropy** $H(Z)_W$.
# 
# $$H(Z)=\sum_{r=1}^Rp(z_r)log(\frac{1}{p(z_r)})=MI(Z,W)+H(Z)_W$$
# 
# $$H(Z)_W = \sum_{k=1}^Kp(w_k)H(Z|w_k)$$
# 
# $$H(Z|w_k) = \sum_{r=1}^Rp(z_r|w_k)log(\frac{1}{p(z_r|w_k)})$$
# 
# $$MI(Z,W)=\sum_{k=1}^Kp(w_k)PI(Z|w_k)$$
# 
# $$PI(Z|w_k)=\sum_{r=1}^Rp(z_r|w_k)log(\frac{p(z_r|w_k)}{p(z_r)})$$
# 
# Reference:
# 
# Altieri, L., Cocchi, D., & Roli, G. (2018). A new approach to spatial entropy measures. Environmental and ecological statistics, 25(1), 95-110. [link](https://link.springer.com/article/10.1007/s10651-017-0383-1)