# -*- coding: utf-8 -*-
"""
Created on Fri Aug 12 13:20:49 2022

@author: Rene
"""
import numpy as np
from scipy import optimize
import matplotlib.pyplot as plt
import pylab as p
import math

def CDF_T_k(t,k,n):
    num_values = np.arange(t-k+1,t+1,dtype=np.float32)
    denom_values = np.arange(n-k+1,n+1,dtype=np.float32)
    numerator = np.product(num_values)
    denominator = np.product(denom_values)
    value = numerator/denominator
    return(value)

def solve_critical_value(alpha,k,n):
    
    def f(t):
        value = alpha - CDF_T_k(t,k,n)
        return(value)
    
    root = optimize.brentq(f, 1, n)
    return(root)

n = 350
k = 5
t = 61
alpha = 0.05
t_alpha = solve_critical_value(alpha,k,n)
Crit_value = math.trunc(t_alpha)

m = 500
n_values = np.zeros(m-Crit_value+1)
Type_II_error = np.zeros(m-Crit_value+1)
for i in np.arange(1,(m-Crit_value)+2):
  n_values[i-1] = Crit_value+(i-1)
  Type_II_error[i-1] = 1-CDF_T_k(Crit_value,k,n_values[i-1])

data = [61,19,56,24,16]
t_5 = max(data)
p_5 = CDF_T_k(t_5,k,n)

# Plotting the analysis results    
LB = 190
UB = 500
off_set_x = 5
off_set_y = 0.1
max_y = 1
    
# Setting the attributes for the Figure
Figure = plt.figure()     
plt.rc('font', family='serif', size='10')
plt.figure(figsize=(9, 4.5))
Figure.subplots_adjust(hspace=0.4, wspace=0.4)
x_lims = (LB-off_set_x,UB+off_set_x)
y_lims = (0-off_set_y,max_y+off_set_y)
plt.axvline(0, lw=2, ls = '-',color = 'lightgray',alpha=0.5)
plt.axhline(0, lw=2, ls = '-',color = 'lightgray',alpha=0.5)
plt.xlim(x_lims)
plt.ylim(y_lims)
plt.xlabel('t')
y_title = 'Type II Error $= Pr(T_5 \geq $'+ f'{Crit_value:2.0f}'+'$ |N=n)$'
plt.ylabel(y_title)
plt.plot(n_values,Type_II_error,ls='-',lw=0.5,color='red')

plt.axhline(1, lw=1, ls = ':',color = 'black',alpha=0.5)

p.arrow(n, 0, 0, 0.90, head_width=5, head_length=0.05, color='green')
p.arrow(n, 0.95, 0, -0.90, head_width=5, head_length=0.05,color='green')

plt.axvline(n, lw=2, ls = ':',color = 'black',alpha=0.5)
text_str = r'$1-\alpha = $'+ f'{1-alpha:2.2f}'
plt.text(n-5,0.5,text_str, horizontalalignment = 'right', verticalalignment = 'bottom', color = 'green')

plt.axvline(t_alpha, lw=2, ls = '--',color = 'blue',alpha=0.5)
text_str = r'$t_{\alpha} = $' +  f'{Crit_value:2.2f}' 
plt.text(t_alpha+5,-0.05,text_str, horizontalalignment = 'left', verticalalignment = 'center', color = 'blue')

text_str = r"$H_0: N = $" + f'{n:2.0f}'
plt.text(450,0.2,text_str, horizontalalignment = 'left', verticalalignment = 'center', color = 'green')

text_str = r"$H_1: N < $" + f'{n:2.0f}'
plt.text(450,0.1,text_str, horizontalalignment = 'left', verticalalignment = 'center', color = 'green')

text_str = r"German Tank Example: $T_5 = max(X_1,\ldots,X_5)$"
plt.suptitle(text_str,size='14')

plt.savefig('German_Tank_Example_Type_II_350.png', dpi=300)