In [1]:
!python -m pip install statsmodels
Requirement already satisfied: statsmodels in /usr/local/lib/python3.10/dist-packages (0.14.1)
Requirement already satisfied: numpy<2,>=1.18 in /usr/local/lib/python3.10/dist-packages (from statsmodels) (1.25.2)
Requirement already satisfied: scipy!=1.9.2,>=1.4 in /usr/local/lib/python3.10/dist-packages (from statsmodels) (1.11.4)
Requirement already satisfied: pandas!=2.1.0,>=1.0 in /usr/local/lib/python3.10/dist-packages (from statsmodels) (1.5.3)
Requirement already satisfied: patsy>=0.5.4 in /usr/local/lib/python3.10/dist-packages (from statsmodels) (0.5.6)
Requirement already satisfied: packaging>=21.3 in /usr/local/lib/python3.10/dist-packages (from statsmodels) (23.2)
Requirement already satisfied: python-dateutil>=2.8.1 in /usr/local/lib/python3.10/dist-packages (from pandas!=2.1.0,>=1.0->statsmodels) (2.8.2)
Requirement already satisfied: pytz>=2020.1 in /usr/local/lib/python3.10/dist-packages (from pandas!=2.1.0,>=1.0->statsmodels) (2023.4)
Requirement already satisfied: six in /usr/local/lib/python3.10/dist-packages (from patsy>=0.5.4->statsmodels) (1.16.0)
In [2]:
import numpy as np
from sklearn.utils import shuffle
import scipy.stats as ss
import scipy.sparse as sparse
from itertools import *
import pandas as pd
import matplotlib.pyplot as plt
import statsmodels.stats.multitest as smm

Задача 1¶

Задача 1.1 Генерация данных¶

In [3]:
def generate(mu1, sigma1, n1, mu2, sigma2, n2): # функция генерации смеси двух нормальных распределений
  sample1 = np.random.normal(mu1, sigma1, n1)
  sample2 = np.random.normal(mu2, sigma2, n2)
  X = np.concatenate([sample1, sample2])
  labels = np.concatenate([np.zeros(n1, dtype=int), np.ones(n2, dtype=int)])
  return shuffle(X, labels, random_state=666)

Задача 1.2 EM¶

In [4]:
class Model:
  def __init__(self, X): # запоминаем данные
    self.X = X
    self.N = len(self.X)

  def fit_predict(self, max_iter=15, seed=42):
    #np.random.seed(seed)

    # разбиваем множество значений на два равных промежутка и из каждого выбираем случайное значение матожидания
    self.mu1, self.mu2 = np.random.choice(sorted(self.X)[0:self.N//2]), np.random.choice(sorted(self.X)[self.N//2+1:])
    # задаём начальные стандартные отклонения
    self.sigma1, self.sigma2 = 1, 1
    # задаём долю значений из класса
    self.q1 = np.random.random()
    self.q2 = 1 - self.q1

    def _estimate_params(X, y_prob): # оценка параметров распределения по известной разметке (M-step)
      q1_ = np.mean(y_prob)
      q2_ = 1 - q1_
      mu1_ = np.sum(X * y_prob) / np.sum(y_prob)
      sigma1_ = np.sum((X-mu1_)**2 * y_prob) / np.sum(y_prob)
      mu2_ = np.sum(X * (1-y_prob)) / np.sum(1-y_prob)
      sigma2_ = np.sum((X-mu2_)**2 * (1-y_prob)) / np.sum(1-y_prob)
      return q1_, q2_, mu1_, sigma1_, mu2_, sigma2_

    self.y_pred_old = None

    for i in range(max_iter): # итерируемся до сходимости, но не более max_iter раз
      # получение новой разметки по заданным параметрам (E-step)
      p1 = self.q1 * ss.norm.pdf(self.X, self.mu1, self.sigma1)
      p2 = self.q2 * ss.norm.pdf(self.X, self.mu2, self.sigma2)
      self.y_pred = np.array(p1 - p2 < 0, dtype=int)
      self.y_prob = p1 / (p1 + p2)
      # оценка параметров распределения по известной разметке (M-step)
      self.q1, self.q2, self.mu1, self.sigma1, self.mu2, self.sigma2 = _estimate_params(self.X, self.y_prob)
      # остановка при сходимости (разметка не изменилась после очередной итерации)
      if (self.y_pred == self.y_pred_old).all():
        break
      else:
        self.y_pred_old = np.copy(self.y_pred)

    # финальная оценка параметров (гарантирует наличие по крайней мере одной итерации алгоритма)
    self.q1, self.q2, self.mu1, self.sigma1, self.mu2, self.sigma2 = _estimate_params(self.X, self.y_prob)
    p1 = self.q1 * ss.norm.pdf(self.X, self.mu1, self.sigma1)
    p2 = self.q2 * ss.norm.pdf(self.X, self.mu2, self.sigma2)
    self.y_pred = np.array(p1 - p2 < 0, dtype=int)
    self.y_prob = p1 / (p1 + p2)
    self.n1, self.n2 = np.sum(1-self.y_pred), np.sum(self.y_pred)
    return self.y_pred


  def get_params(self):
    return self.mu1, self.sigma1, self.n1, self.mu2, self.sigma2, self.n2

  def get_probs(self):
    return self.y_prob

  def log_liklihood(self):
    return np.sum(np.log(self.q1 * ss.norm.pdf(self.X, self.mu1, self.sigma1) + self.q2 * ss.norm.pdf(self.X, self.mu2, self.sigma2)))

  def get_treshhold(self):
    a = self.sigma2**2-self.sigma1**2
    b = 2*self.mu2*self.sigma1**2 - 2*self.mu1*self.sigma2**2
    c = self.mu1**2 * self.sigma2**2 - self.mu2**2 * self.sigma1**2 - 2 * self.sigma1**2 * self.sigma2**2 * np.log( (self.q1*self.sigma2) / (self.q2*self.sigma1) )
    D = b**2 - 4 * a * c
    return (-b + np.sqrt(D)) / (2*a)

  def estimate_qual(self, y_true): # оценка качества
    eps = 1e-6
    TP = np.sum(self.y_pred * y_true)
    FN = np.sum(y_true) - TP
    FP = np.sum(self.y_pred) - TP
    TN = len(y_true) - TP - FN - FP
    SEN = TP / (TP + FN + eps)
    SPC = TN / (FP + TN + eps)
    PPV = TP / (TP + FP + eps)
    FPR = FP / (FP + TN + eps)
    F1 = 2 * PPV * SEN / (PPV + SEN + eps)
    ACC = (TP + TN) / (TP + FN + FP + TN + eps)
    return {"sensitivity" : SEN,
            "specifity" : SPC,
            "precision" : PPV,
            "F1-score" : F1,
            "accuracy" : ACC}
In [5]:
np.random.seed(666)
iters = [
    (0, 1, 10, 1, 1, 5),
    (0, 1, 100, 1, 1, 50),
    (0, 1, 1000, 1, 1, 500)
]
In [6]:
for iter in iters:
  params = pd.DataFrame({
    "mu1" : [],
    "sigma1" : [],
    "n1" : [],
    "mu2" : [],
    "sigma2" : [],
    "n2" : []})
  params.loc["real"] = iter
  np.random.seed(42)
  X, y_true = generate(*iter)
  model = Model(X)
  y_pred = model.fit_predict()
  colours = np.array(["orange"] * len(y_pred))
  colours[y_true == 0] = "blue"
  colours[y_true == 1] = "orange"
  params.loc["estimated"] = model.get_params()
  y_probs = model.get_probs()
  log_liklihood = model.log_liklihood()
  quality = model.estimate_qual(y_true)
  description = f'''
_______________Model description_______________
Gaussian Mixture Models based on EM-algorythm

_______________Model parameters_______________
{params}
logLiklihood = {log_liklihood}

________________Model quality________________
{pd.Series(quality)}
  '''
  # пара костылей, чтобы графики выглядели так, как хочется
  fig, (ax1, ax2) = plt.subplots(nrows=1, ncols=2, figsize = (12,3), tight_layout = True)
  ax1.axis('off')
  ax1.text(0, 0.5, description, ha="left", va="center", fontfamily="monospace")
  ax2.scatter([], [], c=colours[y_true == 0][0], s=3, label=f"x from class N({params['mu1'].loc['real']:.02f}, {params['sigma1'].loc['real']:.02f})")
  ax2.scatter([], [], c=colours[y_true == 1][1], s=3, label=f"x from class N({params['mu2'].loc['real']:.02f}, {params['sigma2'].loc['real']:.02f})")
  ax2.vlines(model.get_treshhold(), 0, 1, linestyle="dashed", colors="black", alpha=0.5, label="Treshhold")
  ax2.scatter(X, y_probs, c=colours, s=3)
  plt.ylabel(f"Probability of class N({params['mu1'].loc['estimated']:.02f}, {params['sigma1'].loc['estimated']:.02f})")
  ax2.legend()
  plt.show()

Выводы¶

Хочется сразу отметить, что в процессе реализации EM-алгоритма выяснилось, что он очень чувствителен к начальным предположениям и легко сваливается в локальный минимум. В частности, если выбирать оба начальных матожидания из всего диапазона X или оценивать их по случайной разметке, то в конце алгоритма все элементы оказываются в одном классе. Поэтому пришлось схитрить и выбирать начальные матожидания, разбив значения X на два равных промежутка. Тем не менее смесь всё равно не очень хорошо разделяется. В случае маленькой выборки данных недостаточно, чтобы успешно оценить параметры, в случае большой выборки исходные распределения перекрываются сильно, поэтому разбиение на классы элементов, которые находятся между математическими ожиданиями происходит почти случайно. Также отмечу, что распределение данной смеси очень похоже на сумму нормальных распределений (хотя, естественно им не является), что также может объяснять, почему EM-алгоритм плохо разделяет смесь и склонен относить все элементы к одному распределению.

In [7]:
X = np.linspace(-4, 4, 101)
Y1 = 2/3 * ss.norm.pdf(X, 0, 1) + 1/3 * ss.norm.pdf(X, 1, 1)
Y2 = ss.norm.pdf(X, 2/3 * 0 + 1/3 * 1, np.sqrt(2/3 * 1 + 1/3 * 1))
plt.plot(X, Y1, label="Mixture of N(0, 1) and N(1, 1)")
plt.plot(X, Y2, label="N($\\frac{1}{3}$, 1) = N(0, 1) + N(1, 1)")
plt.title("Density of different distributions")
plt.legend(loc='center left', bbox_to_anchor=(1, 0.5))
Out[7]:
<matplotlib.legend.Legend at 0x7b8991839f60>

Задача 2¶

In [8]:
class MarkovChain:
  def __init__(self, alphabet, order):
    self.alphabet = np.array(list(alphabet))
    self.A = len(self.alphabet)
    self.order = order+1
    self._generate_Ps()

  def _generate_Ps(self): # функция, создающая матрицы перехода
    self.Ps = list()
    for i in range(self.order):
      P = np.random.random(tuple([self.A] * (i+1)))
      P /= np.sum(P, axis=-1, keepdims=True)
      self.Ps.append(P)

  def generate_seqs(self, length, N):
    seqs = list()
    for _ in range(N):
      i = 0
      seq = ''
      for s in range(length):
        pre_string = seq[-i:]
        pre = []
        for _ in pre_string:
          pre.append(np.where(self.alphabet == _)[0][0])
        seq += np.random.choice(self.alphabet, p=self.Ps[i][tuple(pre)])
        if i < self.order-1:
          i += 1
      seqs.append(seq)
    return seqs

  def fit(self, seqs, eps=1e-6, erg=False):

    self.N = len(seqs)
    self.length = min([len(seq) for seq in seqs])
    self.Kmers = [None for _ in range(self.order)]


    i = self.order-1
    while i > -1: #+ erg*(self.order-2):
      imers = np.ones(tuple([self.A] * (i+1))) * eps / (self.A ** (i+1))
      for seq in seqs:
        if erg:
          js = self.length-i-1
        else:
          js = 1
        for j in range(js):
          i_mer_string = seq[j:j+i+1]
          i_mer = []
          for _ in i_mer_string:
            i_mer.append(np.where(self.alphabet == _)[0][0])
          imers[tuple(i_mer)] += 1
      self.Kmers[i] = imers
      i -= 1

    for i in range(self.order-1, -1, -1):
      if erg:
        if i == self.order-1 and i != 0:
          self.Ps[i] = self.Kmers[i] / self.Kmers[i-1].reshape(tuple([self.A] * (i) + [1]))
        else:
          if i != 0:
            self.Ps[i] = self.Kmers[i] / self.Kmers[i-1].reshape(tuple([self.A] * (i) + [1]))
          else:
            self.Ps[i] = self.Kmers[i] / (self.N * self.length)


          '''
          жалкие потуги постичь эргодичность

          rows = np.array([[j for k in range(self.A)] for j in range(self.A ** (i+1))]).ravel()
          cols = np.array([[[l*(self.A)**(i)+j for l in range(self.A)] for k in range(self.A)] for j in range(self.A ** i)]).ravel()
          data = self.Ps[i+1].ravel()
          matrix = sparse.coo_matrix((data, (rows, cols)))
          eigenvalues, eigenvectors = sparse.linalg.eigs(matrix)
          P = eigenvectors[eigenvalues == 1]
          P /= np.sum(P, axis=-1, keepdims=True)
          self.Ps[i] = P
          '''

      else:
        if i != 0:
          self.Ps[i] = self.Kmers[i] / self.Kmers[i-1].reshape(tuple([self.A] * (i) + [1]))
        else:
          self.Ps[i] = self.Kmers[i] / self.N

    if erg:
      self.k = self.A ** self.order - self.A ** (self.order-1)
    else:
      self.k = self.A ** self.order - 1
    self._log_L(seqs)

    return self.Ps

  def BIC(self):
    return self.k * np.log(self.N) - self.log_L

  def AIC(self):
    return self.k * 2 - self.log_L

  def _log_L(self, seqs):
    self.log_Ps = [np.log(_) for _ in self.Ps]
    self.log_L = 0
    for seq in seqs:
      i = 0
      for j in range(self.length):
        i_mer_string = seq[j-i:j+1]
        i_mer = []
        for _ in i_mer_string:
          i_mer.append(np.where(self.alphabet == _)[0][0])
        self.log_L += self.log_Ps[i][tuple(i_mer)]
        if i < self.order-1:
          i += 1
In [9]:
def print_Ps(Ps):
  for i in range(len(Ps)):
    print(f"""
P{i}=
{Ps[i]}""")

Задача 2.1 Генерация последовательностей¶

In [10]:
alphabet = "ab"
order = 1
N = 1000
length = 3000
generation = MarkovChain(alphabet, order)
real_Ps = generation.Ps
seqs = generation.generate_seqs(length, N)
estimation = MarkovChain(alphabet, order)
estimated_Ps = estimation.fit(seqs, erg=False)
print("Parameters of model")
print("Real transition matrices")
print_Ps(real_Ps)
print("Estimated transition matrices")
print_Ps(estimated_Ps)
Parameters of model
Real transition matrices

P0=
[0.86605874 0.13394126]

P1=
[[0.14906449 0.85093551]
 [0.52586667 0.47413333]]
Estimated transition matrices

P0=
[0.879 0.121]

P1=
[[0.16040956 0.83959044]
 [0.56198347 0.43801653]]

Задача 2.2 Сравнение моделей¶

In [11]:
print("Real transition matrices of 1-order Markov chain model")
print_Ps(real_Ps)
AICs, BICs = [], []
orders = [_ for _ in range(4)]
for order in orders:
  estimation = MarkovChain(alphabet, order)
  estimated_Ps = estimation.fit(seqs, erg=False)
  AIC = estimation.AIC()
  BIC = estimation.BIC()
  AICs.append(AIC)
  BICs.append(BIC)
  print(f"""
{order}-order Markov chain model

Estimated transition matrices""")
  print_Ps(estimated_Ps)
  print(f"""
AIC = {AIC}\tBIC = {BIC}
  """)
fig, (ax1, ax2) = plt.subplots(nrows=1, ncols=2, figsize = (12,3), tight_layout = True)
ax1.plot(orders, AICs)
ax1.title.set_text("AIC")
ax1.set_xlabel("Order of model")
ax2.plot(orders, BICs)
ax2.title.set_text("BIC")
ax2.set_xlabel("Order of model")
Real transition matrices of 1-order Markov chain model

P0=
[0.86605874 0.13394126]

P1=
[[0.14906449 0.85093551]
 [0.52586667 0.47413333]]

0-order Markov chain model

Estimated transition matrices

P0=
[0.879 0.121]

AIC = 4062922.6265882384	BIC = 4062927.5343435174
  

1-order Markov chain model

Estimated transition matrices

P0=
[0.879 0.121]

P1=
[[0.16040956 0.83959044]
 [0.56198347 0.43801653]]

AIC = 1770470.1698345244	BIC = 1770484.8931003613
  

2-order Markov chain model

Estimated transition matrices

P0=
[0.879 0.121]

P1=
[[0.16040956 0.83959044]
 [0.56198347 0.43801653]]

P2=
[[[0.15602837 0.84397163]
  [0.51897019 0.48102981]]

 [[0.14705882 0.85294118]
  [0.45283019 0.54716981]]]

AIC = 1774592.5031855742	BIC = 1774626.857472527
  

3-order Markov chain model

Estimated transition matrices

P0=
[0.879 0.121]

P1=
[[0.16040956 0.83959044]
 [0.56198347 0.43801653]]

P2=
[[[0.15602837 0.84397163]
  [0.51897019 0.48102981]]

 [[0.14705882 0.85294118]
  [0.45283019 0.54716981]]]

P3=
[[[[9.09090932e-02 9.09090907e-01]
   [4.87394958e-01 5.12605042e-01]]

  [[1.64490862e-01 8.35509138e-01]
   [5.40845070e-01 4.59154930e-01]]]


 [[[6.24999992e-09 9.99999994e-01]
   [5.17241379e-01 4.82758621e-01]]

  [[1.66666668e-01 8.33333332e-01]
   [5.17241379e-01 4.82758621e-01]]]]

AIC = 2116049.4443266215	BIC = 2116123.060655806
  
Out[11]:
Text(0.5, 0, 'Order of model')

Выводы¶

Параметры, полученные в результате оценки близки к реальным параметрам, причём наилучшее качество у модели первого порядка (т.е. того же порядка, что был у генерации). При этом стоит отметить, что модель второго порядка описывает данные почти также хорошо. При этом для данной модели выполняется

$P_2[i, j, k] = P(S_n = \alpha_k | S_{n-1} = \alpha_j, S_{n-2} = \alpha_i) \approx P_1[j, k] = P(S_n = \alpha_k | S_{n-1} = \alpha_j)$

Это указывает на то, что данная модель почти не учитывает влияние $(n-2)$-ой буквы. При этом бернуллиевская модель (цепь Маркова нулевого порядка) описывет данные крайне плохо, в то время как модель третьего порядка является слишком сложной и содержит избыточные параметры, поэтому для неё критерии $AIC$ и $BIC$ имеют большие значения.

Задача 2.3 Реальные данные¶

In [12]:
!wget -O  B_subtilis.fasta.gz https://ftp.ncbi.nlm.nih.gov/genomes/all/GCA/000/009/045/GCA_000009045.1_ASM904v1/GCA_000009045.1_ASM904v1_genomic.fna.gz
!gunzip -d  B_subtilis.fasta.gz
--2024-03-10 20:23:47--  https://ftp.ncbi.nlm.nih.gov/genomes/all/GCA/000/009/045/GCA_000009045.1_ASM904v1/GCA_000009045.1_ASM904v1_genomic.fna.gz
Resolving ftp.ncbi.nlm.nih.gov (ftp.ncbi.nlm.nih.gov)... 130.14.250.11, 130.14.250.12, 2607:f220:41e:250::10, ...
Connecting to ftp.ncbi.nlm.nih.gov (ftp.ncbi.nlm.nih.gov)|130.14.250.11|:443... connected.
HTTP request sent, awaiting response... 200 OK
Length: 1248394 (1.2M) [application/x-gzip]
Saving to: ‘B_subtilis.fasta.gz’

B_subtilis.fasta.gz 100%[===================>]   1.19M  --.-KB/s    in 0.09s   

2024-03-10 20:23:47 (12.9 MB/s) - ‘B_subtilis.fasta.gz’ saved [1248394/1248394]

In [13]:
with open("B_subtilis.fasta") as inp:
    header = inp.readline().strip()
    name = header.split()[0][1:]
    genome = []
    for line in inp:
        line = line.strip()
        genome.append(line)
    genome = "".join(genome).upper() # на всякий случай

i, j, G = 0, 1000, len(genome)
B_seqs = []
while j < G:
  seq = genome[i:j]
  B_seqs.append(seq)
  i += 1000
  j += 1000
In [20]:
AICs, BICs = [], []
orders = [_ for _ in range(6)]
alphabet = "ATGC"
full_est_Ps = []
for order in orders:
  estimation = MarkovChain(alphabet, order)
  estimated_Ps = estimation.fit(B_seqs, erg=False)
  full_est_Ps.append(estimated_Ps)
  AIC = estimation.AIC()
  BIC = estimation.BIC()
  AICs.append(AIC)
  BICs.append(BIC)
  print(f"""
{order}-order Markov chain model

Estimated transition matrices""")
  if order < max(orders):
    print("Same as for the highest-order model")
  else:
    print_Ps(estimated_Ps)
  print(f"""
AIC = {AIC}\tBIC = {BIC}
  """)
fig, (ax1, ax2) = plt.subplots(nrows=1, ncols=2, figsize = (12,3), tight_layout = True)
ax1.plot(orders, AICs)
ax1.title.set_text("AIC")
ax1.set_xlabel("Order of model")
ax2.plot(orders, BICs)
ax2.title.set_text("BIC")
ax2.set_xlabel("Order of model")
0-order Markov chain model

Estimated transition matrices
Same as for the highest-order model

AIC = 5810167.532098773	BIC = 5810186.571313384
  

1-order Markov chain model

Estimated transition matrices
Same as for the highest-order model

AIC = 5741972.615781027	BIC = 5742067.811854083
  

2-order Markov chain model

Estimated transition matrices
Same as for the highest-order model

AIC = 5719849.615952403	BIC = 5720249.439459241
  

3-order Markov chain model

Estimated transition matrices
Same as for the highest-order model

AIC = 5782479.94242272	BIC = 5784098.275664681
  

4-order Markov chain model

Estimated transition matrices
Same as for the highest-order model

AIC = 9147755.11509343	BIC = 9154247.487275887
  

5-order Markov chain model

Estimated transition matrices

P0=
[0.28232503 0.28493476 0.20474496 0.22799526]

P1=
[[0.35882353 0.28235294 0.18991597 0.16890756]
 [0.15237302 0.36636137 0.24229808 0.23896753]
 [0.28852839 0.21552723 0.21784473 0.27809965]
 [0.31009365 0.26430801 0.21540062 0.21019771]]

P2=
[[[0.38173302 0.28337237 0.16393443 0.17096019]
  [0.2202381  0.32142857 0.24404762 0.21428571]
  [0.33185841 0.21238938 0.19911504 0.25663717]
  [0.32338308 0.23383085 0.22885572 0.21393035]]

 [[0.30601093 0.42076503 0.13114754 0.1420765 ]
  [0.15909091 0.41363636 0.19090909 0.23636364]
  [0.38831615 0.20618557 0.19243986 0.21305842]
  [0.32055749 0.28571429 0.17421603 0.2195122 ]]

 [[0.36144578 0.28915663 0.1686747  0.18072289]
  [0.26344086 0.3172043  0.19354839 0.22580645]
  [0.33510638 0.21276596 0.19148936 0.2606383 ]
  [0.2875     0.30416667 0.19166667 0.21666667]]

 [[0.31543624 0.2885906  0.24832215 0.14765101]
  [0.13779528 0.33464567 0.27559055 0.2519685 ]
  [0.24637681 0.26570048 0.22222222 0.26570048]
  [0.24752475 0.23762376 0.32673267 0.18811881]]]

P3=
[[[[0.42944785 0.21472393 0.1595092  0.19631902]
   [0.27272727 0.23140496 0.28099174 0.21487603]
   [0.4        0.08571429 0.31428571 0.2       ]
   [0.47945205 0.1369863  0.24657534 0.1369863 ]]

  [[0.40540541 0.33783784 0.08108108 0.17567568]
   [0.19444444 0.31481481 0.26851852 0.22222222]
   [0.30487805 0.2804878  0.17073171 0.24390244]
   [0.375      0.18055556 0.23611111 0.20833333]]

  [[0.34666667 0.24       0.17333333 0.24      ]
   [0.10416667 0.29166667 0.375      0.22916667]
   [0.33333333 0.24444444 0.22222222 0.2       ]
   [0.34482759 0.24137931 0.24137931 0.17241379]]

  [[0.44615385 0.2        0.23076923 0.12307692]
   [0.08510638 0.44680851 0.36170213 0.10638298]
   [0.26086957 0.2173913  0.36956522 0.15217391]
   [0.30232558 0.20930233 0.30232558 0.18604651]]]


 [[[0.44642857 0.28571429 0.10714286 0.16071429]
   [0.2987013  0.33766234 0.20779221 0.15584416]
   [0.41666667 0.20833333 0.125      0.25      ]
   [0.34615385 0.15384615 0.23076923 0.26923077]]

  [[0.2        0.45714286 0.15714286 0.18571429]
   [0.21428571 0.41208791 0.13736264 0.23626374]
   [0.38095238 0.22619048 0.20238095 0.19047619]
   [0.34615385 0.24038462 0.22115385 0.19230769]]

  [[0.30973451 0.30088496 0.15929204 0.2300885 ]
   [0.25       0.28333333 0.21666667 0.25      ]
   [0.41071429 0.19642857 0.17857143 0.21428571]
   [0.24193548 0.33870968 0.19354839 0.22580645]]

  [[0.32608696 0.42391304 0.16304348 0.08695652]
   [0.13414634 0.34146341 0.2804878  0.24390244]
   [0.26       0.34       0.2        0.2       ]
   [0.31746032 0.22222222 0.26984127 0.19047619]]]


 [[[0.35555556 0.16666667 0.3        0.17777778]
   [0.25       0.25       0.25       0.25      ]
   [0.33333333 0.16666667 0.21428571 0.28571429]
   [0.31111111 0.22222222 0.24444444 0.22222222]]

  [[0.28571429 0.32653061 0.24489796 0.14285714]
   [0.20338983 0.3220339  0.20338983 0.27118644]
   [0.41666667 0.22222222 0.25       0.11111111]
   [0.30952381 0.33333333 0.19047619 0.16666667]]

  [[0.42857143 0.36507937 0.06349206 0.14285714]
   [0.125      0.35       0.375      0.15      ]
   [0.25       0.30555556 0.19444444 0.25      ]
   [0.24489796 0.30612245 0.26530612 0.18367347]]

  [[0.28985507 0.24637681 0.26086957 0.20289855]
   [0.02739726 0.34246575 0.46575342 0.16438356]
   [0.23913043 0.17391304 0.36956522 0.2173913 ]
   [0.23076923 0.32692308 0.28846154 0.15384615]]]


 [[[0.34042553 0.22340426 0.26595745 0.17021277]
   [0.24418605 0.26744186 0.12790698 0.36046512]
   [0.22972973 0.17567568 0.17567568 0.41891892]
   [0.31818182 0.25       0.15909091 0.27272727]]

  [[0.34285714 0.31428571 0.14285714 0.2       ]
   [0.15294118 0.31764706 0.2        0.32941176]
   [0.38571429 0.28571429 0.1        0.22857143]
   [0.296875   0.265625   0.171875   0.265625  ]]

  [[0.21568627 0.45098039 0.19607843 0.1372549 ]
   [0.2        0.45454545 0.16363636 0.18181818]
   [0.26086957 0.32608696 0.2173913  0.19565217]
   [0.18181818 0.45454545 0.16363636 0.2       ]]

  [[0.36       0.26       0.22       0.16      ]
   [0.10416667 0.25       0.35416667 0.29166667]
   [0.25757576 0.1969697  0.1969697  0.34848485]
   [0.18421053 0.23684211 0.26315789 0.31578947]]]]

P4=
[[[[[4.14285714e-01 1.85714286e-01 2.57142857e-01 1.42857143e-01]
    [2.00000000e-01 2.00000000e-01 2.85714286e-01 3.14285714e-01]
    [3.46153846e-01 7.69230769e-02 2.30769231e-01 3.46153846e-01]
    [2.81250000e-01 6.25000000e-02 4.37500000e-01 2.18750000e-01]]

   [[4.84848485e-01 2.72727273e-01 1.21212121e-01 1.21212121e-01]
    [2.14285714e-01 3.21428571e-01 2.14285714e-01 2.50000000e-01]
    [5.88235294e-01 1.47058824e-01 1.17647059e-01 1.47058824e-01]
    [1.53846154e-01 1.92307692e-01 3.07692308e-01 3.46153846e-01]]

   [[3.92857143e-01 2.85714286e-01 1.78571429e-01 1.42857143e-01]
    [1.62760417e-10 1.66666667e-01 8.33333333e-01 1.62760417e-10]
    [3.18181818e-01 1.81818182e-01 3.18181818e-01 1.81818182e-01]
    [5.00000000e-01 7.14285715e-02 2.85714286e-01 1.42857143e-01]]

   [[2.85714286e-01 2.28571429e-01 2.85714286e-01 2.00000000e-01]
    [1.00000000e-01 4.00000000e-01 4.00000000e-01 1.00000000e-01]
    [3.33333333e-01 1.66666667e-01 3.88888889e-01 1.11111111e-01]
    [1.00000000e-01 9.76562500e-11 6.00000000e-01 3.00000000e-01]]]


  [[[5.33333333e-01 2.33333333e-01 1.66666667e-01 6.66666667e-02]
    [1.60000000e-01 4.40000000e-01 2.00000000e-01 2.00000000e-01]
    [6.66666666e-01 1.66666667e-01 1.62760417e-10 1.66666667e-01]
    [2.30769231e-01 2.30769231e-01 2.30769231e-01 3.07692308e-01]]

   [[3.33333333e-01 5.23809524e-01 9.52380953e-02 4.76190477e-02]
    [8.82352941e-02 3.82352941e-01 2.35294118e-01 2.94117647e-01]
    [5.17241379e-01 1.03448276e-01 1.72413793e-01 2.06896552e-01]
    [4.16666667e-01 1.25000000e-01 4.16666667e-02 4.16666667e-01]]

   [[4.40000000e-01 2.80000000e-01 1.20000000e-01 1.60000000e-01]
    [8.69565218e-02 3.47826087e-01 2.17391304e-01 3.47826087e-01]
    [5.71428571e-01 7.14285715e-02 2.14285714e-01 1.42857143e-01]
    [2.50000000e-01 3.50000000e-01 2.00000000e-01 2.00000000e-01]]

   [[3.33333333e-01 3.33333333e-01 2.22222222e-01 1.11111111e-01]
    [2.30769231e-01 2.30769231e-01 3.84615385e-01 1.53846154e-01]
    [5.29411765e-01 1.17647059e-01 2.94117647e-01 5.88235295e-02]
    [1.33333333e-01 2.66666667e-01 2.66666667e-01 3.33333333e-01]]]


  [[[3.84615385e-01 1.92307692e-01 2.30769231e-01 1.92307692e-01]
    [2.77777778e-01 1.66666667e-01 2.22222222e-01 3.33333333e-01]
    [7.69230770e-02 7.51201923e-11 6.15384615e-01 3.07692308e-01]
    [4.44444444e-01 1.11111111e-01 3.88888889e-01 5.55555556e-02]]

   [[2.00000000e-01 6.00000000e-01 1.95312500e-10 2.00000000e-01]
    [6.97544643e-11 6.42857143e-01 2.14285714e-01 1.42857143e-01]
    [4.44444444e-01 3.33333333e-01 5.42534722e-11 2.22222222e-01]
    [5.45454545e-01 2.72727273e-01 9.09090910e-02 9.09090910e-02]]

   [[3.33333333e-01 2.66666667e-01 3.33333333e-01 6.66666667e-02]
    [9.09090910e-02 3.63636364e-01 4.54545454e-01 9.09090910e-02]
    [5.00000000e-01 1.00000000e-01 2.00000000e-01 2.00000000e-01]
    [3.33333333e-01 2.22222222e-01 3.33333333e-01 1.11111111e-01]]

   [[2.50000000e-01 3.00000000e-01 3.50000000e-01 1.00000000e-01]
    [7.14285715e-02 4.28571429e-01 3.57142857e-01 1.42857143e-01]
    [1.42857143e-01 1.42857143e-01 2.14285714e-01 5.00000000e-01]
    [1.00000000e-01 1.00000000e-01 6.00000000e-01 2.00000000e-01]]]


  [[[2.75862069e-01 3.10344828e-01 2.41379310e-01 1.72413793e-01]
    [3.07692308e-01 3.07692308e-01 1.53846154e-01 2.30769231e-01]
    [2.66666667e-01 2.66666667e-01 2.66666667e-01 2.00000000e-01]
    [1.25000000e-01 1.25000000e-01 3.75000000e-01 3.75000000e-01]]

   [[2.44140625e-10 2.50000000e-01 5.00000000e-01 2.50000000e-01]
    [9.52380953e-02 3.33333333e-01 2.85714286e-01 2.85714286e-01]
    [2.35294118e-01 2.94117647e-01 5.74448529e-11 4.70588235e-01]
    [8.00000000e-01 2.00000000e-01 1.95312500e-10 1.95312500e-10]]

   [[5.83333333e-01 2.50000000e-01 1.66666667e-01 8.13802083e-11]
    [2.00000000e-01 3.00000000e-01 1.00000000e-01 4.00000000e-01]
    [4.11764706e-01 1.17647059e-01 1.76470588e-01 2.94117647e-01]
    [1.42857143e-01 1.42857143e-01 1.39508928e-10 7.14285714e-01]]

   [[3.07692308e-01 4.61538461e-01 1.53846154e-01 7.69230770e-02]
    [1.11111111e-01 3.33333333e-01 3.33333333e-01 2.22222222e-01]
    [7.69230770e-02 3.84615385e-01 3.07692308e-01 2.30769231e-01]
    [3.75000000e-01 2.50000000e-01 2.50000000e-01 1.25000000e-01]]]]



 [[[[4.00000000e-01 4.00000000e-02 3.60000000e-01 2.00000000e-01]
    [6.25000000e-02 4.37500000e-01 3.12500000e-01 1.87500000e-01]
    [6.66666666e-01 1.62760417e-10 1.62760417e-10 3.33333333e-01]
    [1.11111111e-01 2.22222222e-01 3.33333333e-01 3.33333333e-01]]

   [[1.30434783e-01 4.78260870e-01 1.73913043e-01 2.17391304e-01]
    [1.15384615e-01 4.23076923e-01 1.92307692e-01 2.69230769e-01]
    [3.75000000e-01 2.50000000e-01 1.87500000e-01 1.87500000e-01]
    [2.50000000e-01 1.66666667e-01 3.33333333e-01 2.50000000e-01]]

   [[6.00000000e-01 4.00000000e-01 9.76562500e-11 9.76562500e-11]
    [4.00000000e-01 2.00000000e-01 4.00000000e-01 1.95312500e-10]
    [3.25520833e-10 3.25520833e-10 6.66666666e-01 3.33333333e-01]
    [1.62760417e-10 1.66666667e-01 6.66666666e-01 1.66666667e-01]]

   [[4.44444444e-01 2.22222222e-01 3.33333333e-01 1.08506944e-10]
    [2.44140625e-10 5.00000000e-01 2.50000000e-01 2.50000000e-01]
    [1.66666667e-01 1.66666667e-01 5.00000000e-01 1.66666667e-01]
    [2.85714286e-01 1.42857143e-01 2.85714286e-01 2.85714286e-01]]]


  [[[3.57142857e-01 2.85714286e-01 1.42857143e-01 2.14285714e-01]
    [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
    [9.09090910e-02 2.72727273e-01 1.81818182e-01 4.54545454e-01]
    [3.84615385e-01 3.07692308e-01 1.53846154e-01 1.53846154e-01]]

   [[3.07692308e-01 4.61538462e-01 7.69230769e-02 1.53846154e-01]
    [1.86666667e-01 4.13333333e-01 1.20000000e-01 2.80000000e-01]
    [2.80000000e-01 3.20000000e-01 2.00000000e-01 2.00000000e-01]
    [3.72093023e-01 3.02325581e-01 1.62790698e-01 1.62790698e-01]]

   [[4.06250000e-01 3.43750000e-01 1.25000000e-01 1.25000000e-01]
    [2.10526316e-01 4.21052632e-01 5.26315790e-02 3.15789474e-01]
    [3.52941176e-01 5.88235295e-02 1.76470588e-01 4.11764706e-01]
    [2.50000000e-01 3.12500000e-01 1.87500000e-01 2.50000000e-01]]

   [[3.33333333e-01 2.22222222e-01 2.77777778e-01 1.66666667e-01]
    [2.00000000e-01 3.20000000e-01 2.80000000e-01 2.00000000e-01]
    [1.30434783e-01 3.47826087e-01 3.04347826e-01 2.17391304e-01]
    [3.00000000e-01 4.00000000e-01 2.00000000e-01 1.00000000e-01]]]


  [[[2.85714286e-01 3.71428571e-01 1.71428571e-01 1.71428571e-01]
    [2.35294118e-01 3.23529412e-01 1.47058824e-01 2.94117647e-01]
    [1.66666667e-01 1.66666667e-01 2.77777778e-01 3.88888889e-01]
    [2.30769231e-01 1.92307692e-01 3.84615385e-01 1.92307692e-01]]

   [[5.33333333e-01 2.66666667e-01 1.33333333e-01 6.66666667e-02]
    [5.88235295e-02 7.05882353e-01 1.76470588e-01 5.88235295e-02]
    [3.84615385e-01 7.51201923e-11 1.53846154e-01 4.61538461e-01]
    [4.00000000e-01 2.66666667e-01 6.66666667e-02 2.66666667e-01]]

   [[3.04347826e-01 3.04347826e-01 8.69565218e-02 3.04347826e-01]
    [3.63636364e-01 3.63636364e-01 2.72727273e-01 8.87784091e-11]
    [3.00000000e-01 3.00000000e-01 4.00000000e-01 9.76562500e-11]
    [3.33333333e-01 8.33333334e-02 2.50000000e-01 3.33333333e-01]]

   [[4.00000000e-01 2.00000000e-01 1.33333333e-01 2.66666667e-01]
    [9.52380953e-02 1.90476190e-01 5.23809524e-01 1.90476190e-01]
    [3.33333333e-01 1.66666667e-01 4.16666667e-01 8.33333334e-02]
    [4.28571429e-01 7.14285715e-02 4.28571429e-01 7.14285715e-02]]]


  [[[3.33333333e-01 2.33333333e-01 2.33333333e-01 2.00000000e-01]
    [1.79487179e-01 3.58974359e-01 1.53846154e-01 3.07692308e-01]
    [4.00000000e-01 2.00000000e-01 6.51041666e-11 4.00000000e-01]
    [3.75000000e-01 1.25000000e-01 3.75000000e-01 1.25000000e-01]]

   [[5.45454545e-01 2.72727273e-01 8.87784091e-11 1.81818182e-01]
    [7.14285715e-02 2.50000000e-01 2.50000000e-01 4.28571429e-01]
    [2.60869565e-01 2.60869565e-01 1.73913043e-01 3.04347826e-01]
    [2.50000000e-01 3.00000000e-01 2.00000000e-01 2.50000000e-01]]

   [[7.69230770e-02 5.38461538e-01 3.07692308e-01 7.69230770e-02]
    [1.76470588e-01 4.70588235e-01 2.35294118e-01 1.17647059e-01]
    [3.00000000e-01 2.00000000e-01 1.00000000e-01 4.00000000e-01]
    [2.00000000e-01 4.00000000e-01 3.00000000e-01 1.00000000e-01]]

   [[4.00000000e-01 3.00000000e-01 3.00000000e-01 4.88281250e-11]
    [2.14285714e-01 5.00000000e-01 2.85714286e-01 6.97544643e-11]
    [1.17647059e-01 1.76470588e-01 2.35294118e-01 4.70588235e-01]
    [1.66666667e-01 2.50000000e-01 2.50000000e-01 3.33333333e-01]]]]



 [[[[4.06250000e-01 2.18750000e-01 2.50000000e-01 1.25000000e-01]
    [1.33333333e-01 4.00000000e-01 2.66666667e-01 2.00000000e-01]
    [2.96296296e-01 7.40740741e-02 2.59259259e-01 3.70370370e-01]
    [3.12500000e-01 2.50000000e-01 1.25000000e-01 3.12500000e-01]]

   [[3.88888889e-01 3.88888889e-01 5.42534722e-11 2.22222222e-01]
    [5.42534722e-11 6.11111111e-01 3.33333333e-01 5.55555556e-02]
    [5.55555555e-01 5.55555556e-02 1.66666667e-01 2.22222222e-01]
    [2.22222222e-01 3.33333333e-01 2.22222222e-01 2.22222222e-01]]

   [[4.28571429e-01 2.85714286e-01 2.14285714e-01 7.14285715e-02]
    [1.39508928e-10 4.28571428e-01 4.28571428e-01 1.42857143e-01]
    [3.33333333e-01 4.44444444e-01 1.11111111e-01 1.11111111e-01]
    [5.00000000e-01 2.50000000e-01 8.13802083e-11 2.50000000e-01]]

   [[2.85714286e-01 5.00000000e-01 7.14285715e-02 1.42857143e-01]
    [9.76562500e-11 5.00000000e-01 4.00000000e-01 1.00000000e-01]
    [3.63636364e-01 9.09090910e-02 3.63636364e-01 1.81818182e-01]
    [4.00000000e-01 1.00000000e-01 4.00000000e-01 1.00000000e-01]]]


  [[[5.00000000e-01 7.14285715e-02 2.14285714e-01 2.14285714e-01]
    [6.25000000e-02 3.75000000e-01 3.75000000e-01 1.87500000e-01]
    [5.00000000e-01 1.66666667e-01 8.33333334e-02 2.50000000e-01]
    [2.85714286e-01 4.28571428e-01 2.85714286e-01 1.39508928e-10]]

   [[6.66666667e-01 8.33333334e-02 1.66666667e-01 8.33333334e-02]
    [1.05263158e-01 3.15789474e-01 2.63157895e-01 3.15789474e-01]
    [3.33333333e-01 3.33333333e-01 2.50000000e-01 8.33333334e-02]
    [3.75000000e-01 4.37500000e-01 6.25000000e-02 1.25000000e-01]]

   [[4.00000000e-01 2.66666667e-01 2.00000000e-01 1.33333333e-01]
    [2.50000000e-01 3.75000000e-01 1.25000000e-01 2.50000000e-01]
    [2.22222222e-01 2.22222222e-01 1.08506944e-10 5.55555555e-01]
    [5.00000000e-01 5.00000000e-01 2.44140625e-10 2.44140625e-10]]

   [[2.30769231e-01 1.53846154e-01 1.53846154e-01 4.61538461e-01]
    [1.42857143e-01 5.00000000e-01 1.42857143e-01 2.14285714e-01]
    [3.75000000e-01 1.25000000e-01 3.75000000e-01 1.25000000e-01]
    [1.42857143e-01 4.28571428e-01 1.42857143e-01 2.85714286e-01]]]


  [[[2.59259259e-01 3.33333333e-01 2.96296296e-01 1.11111111e-01]
    [1.73913043e-01 2.60869565e-01 4.34782609e-01 1.30434783e-01]
    [5.00000000e-01 2.50000000e-01 2.44140625e-10 2.50000000e-01]
    [2.22222222e-01 1.11111111e-01 4.44444444e-01 2.22222222e-01]]

   [[2.00000000e-01 1.95312500e-10 4.00000000e-01 4.00000000e-01]
    [1.42857143e-01 5.00000000e-01 1.42857143e-01 2.14285714e-01]
    [4.66666667e-01 1.33333333e-01 1.33333333e-01 2.66666667e-01]
    [1.66666667e-01 6.66666666e-01 1.66666667e-01 1.62760417e-10]]

   [[3.33333333e-01 3.33333333e-01 1.11111111e-01 2.22222222e-01]
    [2.72727273e-01 1.81818182e-01 4.54545454e-01 9.09090910e-02]
    [2.85714286e-01 2.85714286e-01 1.42857143e-01 2.85714286e-01]
    [3.33333333e-01 3.33333333e-01 2.22222222e-01 1.11111111e-01]]

   [[4.16666667e-01 2.50000000e-01 8.33333334e-02 2.50000000e-01]
    [6.51041666e-11 2.00000000e-01 5.33333333e-01 2.66666667e-01]
    [4.61538461e-01 1.53846154e-01 2.30769231e-01 1.53846154e-01]
    [4.44444444e-01 2.22222222e-01 3.33333333e-01 1.08506944e-10]]]


  [[[5.00000000e-01 2.50000000e-01 2.00000000e-01 5.00000000e-02]
    [1.76470588e-01 2.94117647e-01 4.11764706e-01 1.17647059e-01]
    [2.77777778e-01 5.55555556e-02 3.33333333e-01 3.33333333e-01]
    [3.57142857e-01 2.14285714e-01 2.85714286e-01 1.42857143e-01]]

   [[4.88281249e-10 4.88281249e-10 4.88281249e-10 9.99999999e-01]
    [1.20000000e-01 3.20000000e-01 3.20000000e-01 2.40000000e-01]
    [4.41176471e-01 2.64705882e-01 8.82352941e-02 2.05882353e-01]
    [3.33333333e-01 3.33333333e-01 1.66666667e-01 1.66666667e-01]]

   [[2.72727273e-01 2.72727273e-01 9.09090910e-02 3.63636364e-01]
    [1.22070312e-10 8.75000000e-01 1.25000000e-01 1.22070312e-10]
    [4.11764706e-01 5.74448529e-11 1.76470588e-01 4.11764706e-01]
    [1.00000000e-01 4.00000000e-01 9.76562500e-11 5.00000000e-01]]

   [[3.33333333e-01 4.16666667e-01 1.66666667e-01 8.33333334e-02]
    [1.76470588e-01 3.52941176e-01 1.76470588e-01 2.94117647e-01]
    [2.66666667e-01 2.66666667e-01 2.66666667e-01 2.00000000e-01]
    [3.75000000e-01 1.25000000e-01 5.00000000e-01 1.22070312e-10]]]]



 [[[[3.43750000e-01 1.56250000e-01 3.75000000e-01 1.25000000e-01]
    [2.38095238e-01 2.85714286e-01 2.85714286e-01 1.90476190e-01]
    [8.00000000e-02 4.00000000e-02 2.40000000e-01 6.40000000e-01]
    [5.00000000e-01 1.87500000e-01 2.50000000e-01 6.25000000e-02]]

   [[3.33333333e-01 4.76190476e-01 1.42857143e-01 4.76190477e-02]
    [1.30434783e-01 4.78260870e-01 1.73913043e-01 2.17391304e-01]
    [4.54545454e-01 2.72727273e-01 9.09090910e-02 1.81818182e-01]
    [3.22580645e-01 2.90322581e-01 2.25806452e-01 1.61290323e-01]]

   [[4.70588235e-01 4.11764706e-01 5.88235295e-02 5.88235295e-02]
    [3.07692308e-01 3.07692308e-01 3.07692308e-01 7.69230770e-02]
    [1.53846154e-01 3.07692308e-01 7.69230770e-02 4.61538461e-01]
    [2.90322581e-01 3.54838710e-01 1.61290323e-01 1.93548387e-01]]

   [[1.42857143e-01 2.14285714e-01 4.28571429e-01 2.14285714e-01]
    [1.81818182e-01 4.54545454e-01 2.72727273e-01 9.09090910e-02]
    [4.28571428e-01 2.85714286e-01 1.42857143e-01 1.42857143e-01]
    [2.50000000e-01 2.50000000e-01 3.33333333e-01 1.66666667e-01]]]


  [[[5.83333333e-01 8.33333334e-02 8.33333334e-02 2.50000000e-01]
    [9.09090910e-02 7.27272727e-01 1.81818182e-01 8.87784091e-11]
    [1.95312500e-10 2.00000000e-01 1.95312500e-10 8.00000000e-01]
    [5.71428571e-01 1.39508928e-10 2.85714286e-01 1.42857143e-01]]

   [[3.07692308e-01 2.30769231e-01 2.30769231e-01 2.30769231e-01]
    [1.11111111e-01 4.44444444e-01 2.22222222e-01 2.22222222e-01]
    [4.11764706e-01 2.94117647e-01 1.76470588e-01 1.17647059e-01]
    [3.21428571e-01 3.92857143e-01 1.42857143e-01 1.42857143e-01]]

   [[4.07407407e-01 2.96296296e-01 1.48148148e-01 1.48148148e-01]
    [4.50000000e-01 2.50000000e-01 1.50000000e-01 1.50000000e-01]
    [4.28571428e-01 5.71428571e-01 1.39508928e-10 1.39508928e-10]
    [1.25000000e-01 4.37500000e-01 1.87500000e-01 2.50000000e-01]]

   [[4.73684210e-01 2.63157895e-01 1.57894737e-01 1.05263158e-01]
    [2.94117647e-01 4.11764706e-01 1.17647059e-01 1.76470588e-01]
    [2.72727273e-01 2.72727273e-01 2.72727273e-01 1.81818182e-01]
    [2.35294118e-01 3.52941176e-01 1.76470588e-01 2.35294118e-01]]]


  [[[3.63636364e-01 2.72727273e-01 9.09090910e-02 2.72727273e-01]
    [3.04347826e-01 2.17391304e-01 2.17391304e-01 2.60869565e-01]
    [4.00000000e-01 1.00000000e-01 2.00000000e-01 3.00000000e-01]
    [1.39508928e-10 1.42857143e-01 5.71428571e-01 2.85714286e-01]]

   [[4.54545454e-01 2.72727273e-01 1.81818182e-01 9.09090910e-02]
    [8.00000000e-02 5.60000000e-01 2.00000000e-01 1.60000000e-01]
    [2.22222222e-01 4.44444444e-01 1.08506944e-10 3.33333333e-01]
    [2.00000000e-01 3.00000000e-01 9.76562500e-11 5.00000000e-01]]

   [[2.50000000e-01 4.16666667e-01 8.33333334e-02 2.50000000e-01]
    [4.00000000e-01 3.33333333e-01 6.66666667e-02 2.00000000e-01]
    [7.00000000e-01 9.76562500e-11 3.00000000e-01 9.76562500e-11]
    [3.33333333e-01 1.11111111e-01 2.22222222e-01 3.33333333e-01]]

   [[3.00000000e-01 4.00000000e-01 3.00000000e-01 9.76562500e-11]
    [3.90625000e-11 5.20000000e-01 3.20000000e-01 1.60000000e-01]
    [1.11111111e-01 3.33333333e-01 3.33333333e-01 2.22222222e-01]
    [1.81818182e-01 4.54545454e-01 1.81818182e-01 1.81818182e-01]]]


  [[[1.11111111e-01 4.44444444e-01 3.33333333e-01 1.11111111e-01]
    [7.51201923e-11 1.53846154e-01 1.53846154e-01 6.92307692e-01]
    [2.72727273e-01 9.09090910e-02 1.81818182e-01 4.54545454e-01]
    [5.00000000e-01 1.25000000e-01 3.75000000e-01 1.22070312e-10]]

   [[8.00000000e-01 1.95312500e-10 2.00000000e-01 1.95312500e-10]
    [2.50000000e-01 1.66666667e-01 1.66666667e-01 4.16666667e-01]
    [2.94117647e-01 1.17647059e-01 3.52941176e-01 2.35294118e-01]
    [2.14285714e-01 1.42857143e-01 3.57142857e-01 2.85714286e-01]]

   [[1.76470588e-01 5.29411765e-01 2.35294118e-01 5.88235295e-02]
    [7.69230770e-02 5.38461538e-01 7.51201923e-11 3.84615385e-01]
    [2.30769231e-01 2.30769231e-01 1.53846154e-01 3.84615385e-01]
    [2.60869565e-01 1.73913043e-01 3.47826087e-01 2.17391304e-01]]

   [[1.42857143e-01 5.71428571e-01 2.85714286e-01 1.39508928e-10]
    [1.08506944e-10 5.55555555e-01 3.33333333e-01 1.11111111e-01]
    [4.00000000e-01 2.00000000e-01 1.00000000e-01 3.00000000e-01]
    [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]]]]]

P5=
[[[[[[3.44827586e-01 2.06896552e-01 2.75862069e-01 1.72413793e-01]
     [3.84615385e-01 2.30769231e-01 2.30769231e-01 1.53846154e-01]
     [3.88888889e-01 1.66666667e-01 1.11111111e-01 3.33333333e-01]
     [4.00000000e-01 1.00000000e-01 3.00000000e-01 2.00000000e-01]]

    [[2.85714286e-01 2.85714286e-01 4.28571429e-01 3.48772321e-11]
     [3.48772321e-11 1.42857143e-01 4.28571429e-01 4.28571429e-01]
     [3.00000000e-01 1.00000000e-01 3.00000000e-01 3.00000000e-01]
     [4.54545455e-01 2.72727273e-01 2.72727273e-01 2.21946023e-11]]

    [[2.22222222e-01 3.33333333e-01 3.33333333e-01 1.11111111e-01]
     [1.22070312e-10 1.22070312e-10 1.22070312e-10 1.00000000e+00]
     [3.33333333e-01 3.33333333e-01 1.66666667e-01 1.66666667e-01]
     [3.33333333e-01 2.22222222e-01 2.22222222e-01 2.22222222e-01]]

    [[2.71267361e-11 4.44444444e-01 4.44444444e-01 1.11111111e-01]
     [1.22070312e-10 5.00000000e-01 1.22070312e-10 5.00000000e-01]
     [3.57142857e-01 1.42857143e-01 3.57142857e-01 1.42857143e-01]
     [1.42857143e-01 4.28571429e-01 2.85714286e-01 1.42857143e-01]]]


   [[[4.37500000e-01 1.87500000e-01 1.87500000e-01 1.87500000e-01]
     [2.22222222e-01 2.22222222e-01 2.22222222e-01 3.33333333e-01]
     [5.00000000e-01 6.10351562e-11 5.00000000e-01 6.10351562e-11]
     [5.00000000e-01 2.50000000e-01 2.50000000e-01 6.10351562e-11]]

    [[1.66666667e-01 3.33333333e-01 1.66666667e-01 3.33333333e-01]
     [1.11111111e-01 4.44444444e-01 1.11111111e-01 3.33333333e-01]
     [5.00000000e-01 1.66666667e-01 4.06901042e-11 3.33333333e-01]
     [1.42857143e-01 2.85714286e-01 1.42857143e-01 4.28571429e-01]]

    [[4.00000000e-01 2.50000000e-01 1.50000000e-01 2.00000000e-01]
     [4.88281250e-11 6.00000000e-01 2.00000000e-01 2.00000000e-01]
     [6.10351562e-11 2.50000000e-01 2.50000000e-01 5.00000000e-01]
     [2.00000000e-01 2.00000000e-01 4.00000000e-01 2.00000000e-01]]

    [[6.10351562e-11 6.10351562e-11 2.50000000e-01 7.50000000e-01]
     [2.00000000e-01 2.00000000e-01 2.00000000e-01 4.00000000e-01]
     [2.50000000e-01 1.25000000e-01 3.75000000e-01 2.50000000e-01]
     [3.33333333e-01 1.11111111e-01 4.44444444e-01 1.11111111e-01]]]


   [[[5.45454545e-01 2.72727273e-01 1.81818182e-01 2.21946023e-11]
     [3.05175781e-11 2.50000000e-01 3.75000000e-01 3.75000000e-01]
     [2.00000000e-01 2.00000000e-01 2.00000000e-01 4.00000000e-01]
     [2.50000000e-01 5.00000000e-01 2.50000000e-01 6.10351562e-11]]

    [[2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [2.44140625e-10 2.44140625e-10 2.44140625e-10 9.99999999e-01]
     [4.88281250e-11 2.00000000e-01 2.00000000e-01 6.00000000e-01]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]]

    [[5.71428571e-01 2.85714286e-01 1.42857143e-01 3.48772321e-11]
     [5.00000000e-01 2.50000000e-01 2.50000000e-01 6.10351562e-11]
     [1.42857143e-01 3.48772321e-11 5.71428571e-01 2.85714286e-01]
     [2.50000000e-01 2.50000000e-01 6.10351562e-11 5.00000000e-01]]

    [[2.85714286e-01 2.85714286e-01 2.85714286e-01 1.42857143e-01]
     [2.44140625e-10 9.99999999e-01 2.44140625e-10 2.44140625e-10]
     [6.10351562e-11 6.10351562e-11 5.00000000e-01 5.00000000e-01]
     [5.00000000e-01 1.22070312e-10 5.00000000e-01 1.22070312e-10]]]


   [[[7.00000000e-01 2.00000000e-01 2.44140625e-11 1.00000000e-01]
     [5.00000000e-01 3.05175781e-11 5.00000000e-01 3.05175781e-11]
     [6.00000000e-01 1.00000000e-01 2.44140625e-11 3.00000000e-01]
     [5.71428571e-01 1.42857143e-01 3.48772321e-11 2.85714286e-01]]

    [[2.44140625e-10 2.44140625e-10 9.99999999e-01 2.44140625e-10]
     [6.10351562e-11 5.00000000e-01 6.10351562e-11 5.00000000e-01]
     [6.10351562e-11 5.00000000e-01 2.50000000e-01 2.50000000e-01]
     [9.99999999e-01 2.44140625e-10 2.44140625e-10 2.44140625e-10]]

    [[3.33333333e-01 3.33333333e-01 3.33333333e-01 4.06901042e-11]
     [8.13802083e-11 3.33333333e-01 6.66666667e-01 8.13802083e-11]
     [2.85714286e-01 4.28571429e-01 3.48772321e-11 2.85714286e-01]
     [1.22070312e-10 5.00000000e-01 5.00000000e-01 1.22070312e-10]]

    [[2.44140625e-10 9.99999999e-01 2.44140625e-10 2.44140625e-10]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [3.33333333e-01 4.06901042e-11 3.33333333e-01 3.33333333e-01]
     [3.33333333e-01 6.66666667e-01 8.13802083e-11 8.13802083e-11]]]]



  [[[[3.75000000e-01 1.87500000e-01 2.50000000e-01 1.87500000e-01]
     [3.48772321e-11 2.85714286e-01 2.85714286e-01 4.28571429e-01]
     [2.00000000e-01 4.00000000e-01 4.88281250e-11 4.00000000e-01]
     [5.00000000e-01 5.00000000e-01 1.22070312e-10 1.22070312e-10]]

    [[2.50000000e-01 6.10351562e-11 2.50000000e-01 5.00000000e-01]
     [9.09090909e-02 5.45454545e-01 1.81818182e-01 1.81818182e-01]
     [8.00000000e-01 4.88281250e-11 2.00000000e-01 4.88281250e-11]
     [4.00000000e-01 4.00000000e-01 2.00000000e-01 4.88281250e-11]]

    [[1.00000000e+00 6.10351562e-11 6.10351562e-11 6.10351562e-11]
     [2.44140625e-10 9.99999999e-01 2.44140625e-10 2.44140625e-10]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [2.44140625e-10 2.44140625e-10 9.99999999e-01 2.44140625e-10]]

    [[3.33333333e-01 3.33333333e-01 8.13802083e-11 3.33333333e-01]
     [8.13802083e-11 6.66666667e-01 8.13802083e-11 3.33333333e-01]
     [6.66666667e-01 8.13802083e-11 3.33333333e-01 8.13802083e-11]
     [6.10351562e-11 2.50000000e-01 5.00000000e-01 2.50000000e-01]]]


   [[[1.42857143e-01 4.28571429e-01 2.85714286e-01 1.42857143e-01]
     [2.72727273e-01 4.54545455e-01 1.81818182e-01 9.09090909e-02]
     [1.22070312e-10 1.22070312e-10 5.00000000e-01 5.00000000e-01]
     [2.44140625e-10 2.44140625e-10 2.44140625e-10 9.99999999e-01]]

    [[3.33333333e-01 3.33333333e-01 8.13802083e-11 3.33333333e-01]
     [3.84615385e-01 3.84615385e-01 1.87800481e-11 2.30769231e-01]
     [6.25000000e-01 1.25000000e-01 1.25000000e-01 1.25000000e-01]
     [5.00000000e-01 3.00000000e-01 2.44140625e-11 2.00000000e-01]]

    [[5.33333333e-01 4.00000000e-01 6.66666667e-02 1.62760417e-11]
     [3.33333333e-01 6.66666667e-01 8.13802083e-11 8.13802083e-11]
     [2.00000000e-01 4.00000000e-01 2.00000000e-01 2.00000000e-01]
     [1.66666667e-01 1.66666667e-01 3.33333333e-01 3.33333333e-01]]

    [[4.00000000e-01 3.00000000e-01 3.00000000e-01 2.44140625e-11]
     [8.13802083e-11 3.33333333e-01 3.33333333e-01 3.33333333e-01]
     [2.44140625e-10 9.99999999e-01 2.44140625e-10 2.44140625e-10]
     [3.00000000e-01 2.00000000e-01 4.00000000e-01 1.00000000e-01]]]


   [[[1.81818182e-01 5.45454545e-01 1.81818182e-01 9.09090909e-02]
     [2.85714286e-01 1.42857143e-01 4.28571429e-01 1.42857143e-01]
     [3.33333333e-01 3.33333333e-01 8.13802083e-11 3.33333333e-01]
     [5.00000000e-01 6.10351562e-11 2.50000000e-01 2.50000000e-01]]

    [[1.22070312e-10 5.00000000e-01 5.00000000e-01 1.22070312e-10]
     [3.05175781e-11 6.25000000e-01 2.50000000e-01 1.25000000e-01]
     [4.00000000e-01 4.00000000e-01 4.88281250e-11 2.00000000e-01]
     [3.75000000e-01 2.50000000e-01 2.50000000e-01 1.25000000e-01]]

    [[2.50000000e-01 3.75000000e-01 2.50000000e-01 1.25000000e-01]
     [2.44140625e-10 9.99999999e-01 2.44140625e-10 2.44140625e-10]
     [6.66666667e-01 8.13802083e-11 3.33333333e-01 8.13802083e-11]
     [1.00000000e+00 1.22070312e-10 1.22070312e-10 1.22070312e-10]]

    [[4.00000000e-01 4.88281250e-11 6.00000000e-01 4.88281250e-11]
     [1.42857143e-01 5.71428571e-01 1.42857143e-01 1.42857143e-01]
     [2.50000000e-01 6.10351562e-11 7.50000000e-01 6.10351562e-11]
     [2.50000000e-01 5.00000000e-01 6.10351562e-11 2.50000000e-01]]]


   [[[2.22222222e-01 3.33333333e-01 3.33333333e-01 1.11111111e-01]
     [2.22222222e-01 4.44444444e-01 3.33333333e-01 2.71267361e-11]
     [1.66666667e-01 3.33333333e-01 4.06901042e-11 5.00000000e-01]
     [6.66666667e-01 8.13802083e-11 3.33333333e-01 8.13802083e-11]]

    [[3.33333333e-01 3.33333333e-01 8.13802083e-11 3.33333333e-01]
     [8.13802083e-11 6.66666667e-01 3.33333333e-01 8.13802083e-11]
     [2.00000000e-01 4.00000000e-01 2.00000000e-01 2.00000000e-01]
     [5.00000000e-01 1.22070312e-10 1.22070312e-10 5.00000000e-01]]

    [[4.44444444e-01 2.22222222e-01 2.22222222e-01 1.11111111e-01]
     [1.22070312e-10 1.22070312e-10 1.00000000e+00 1.22070312e-10]
     [2.00000000e-01 2.00000000e-01 2.00000000e-01 4.00000000e-01]
     [9.99999999e-01 2.44140625e-10 2.44140625e-10 2.44140625e-10]]

    [[5.00000000e-01 1.22070312e-10 1.22070312e-10 5.00000000e-01]
     [6.10351562e-11 7.50000000e-01 6.10351562e-11 2.50000000e-01]
     [6.10351562e-11 2.50000000e-01 2.50000000e-01 5.00000000e-01]
     [4.00000000e-01 4.00000000e-01 2.00000000e-01 4.88281250e-11]]]]



  [[[[2.00000000e-01 4.00000000e-01 3.00000000e-01 1.00000000e-01]
     [2.00000000e-01 2.00000000e-01 4.00000000e-01 2.00000000e-01]
     [5.00000000e-01 1.66666667e-01 1.66666667e-01 1.66666667e-01]
     [2.00000000e-01 2.00000000e-01 2.00000000e-01 4.00000000e-01]]

    [[4.00000000e-01 4.88281250e-11 4.00000000e-01 2.00000000e-01]
     [6.66666667e-01 8.13802083e-11 3.33333333e-01 8.13802083e-11]
     [7.50000000e-01 2.50000000e-01 6.10351562e-11 6.10351562e-11]
     [8.33333333e-01 4.06901042e-11 1.66666667e-01 4.06901042e-11]]

    [[9.99999999e-01 2.44140625e-10 2.44140625e-10 2.44140625e-10]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [1.25000000e-01 3.75000000e-01 2.50000000e-01 2.50000000e-01]
     [6.10351562e-11 2.50000000e-01 7.50000000e-01 6.10351562e-11]]

    [[1.25000000e-01 3.75000000e-01 3.75000000e-01 1.25000000e-01]
     [1.22070312e-10 5.00000000e-01 5.00000000e-01 1.22070312e-10]
     [1.42857143e-01 3.48772321e-11 2.85714286e-01 5.71428571e-01]
     [2.44140625e-10 9.99999999e-01 2.44140625e-10 2.44140625e-10]]]


   [[[2.44140625e-10 9.99999999e-01 2.44140625e-10 2.44140625e-10]
     [8.13802083e-11 6.66666667e-01 8.13802083e-11 3.33333333e-01]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [2.44140625e-10 9.99999999e-01 2.44140625e-10 2.44140625e-10]]

    [[2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [2.71267361e-11 5.55555556e-01 3.33333333e-01 1.11111111e-01]
     [8.13802083e-11 6.66666667e-01 8.13802083e-11 3.33333333e-01]
     [1.00000000e+00 1.22070312e-10 1.22070312e-10 1.22070312e-10]]

    [[2.50000000e-01 6.25000000e-01 3.05175781e-11 1.25000000e-01]
     [1.66666667e-01 1.66666667e-01 4.06901042e-11 6.66666667e-01]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]]

    [[3.33333333e-01 3.33333333e-01 1.66666667e-01 1.66666667e-01]
     [8.13802083e-11 3.33333333e-01 6.66666667e-01 8.13802083e-11]
     [2.44140625e-10 9.99999999e-01 2.44140625e-10 2.44140625e-10]
     [2.44140625e-10 9.99999999e-01 2.44140625e-10 2.44140625e-10]]]


   [[[4.88281250e-11 6.00000000e-01 2.00000000e-01 2.00000000e-01]
     [2.50000000e-01 5.00000000e-01 6.10351562e-11 2.50000000e-01]
     [2.00000000e-01 4.88281250e-11 4.00000000e-01 4.00000000e-01]
     [9.99999999e-01 2.44140625e-10 2.44140625e-10 2.44140625e-10]]

    [[2.44140625e-10 2.44140625e-10 2.44140625e-10 9.99999999e-01]
     [6.10351562e-11 5.00000000e-01 2.50000000e-01 2.50000000e-01]
     [4.88281250e-11 4.00000000e-01 4.88281250e-11 6.00000000e-01]
     [2.44140625e-10 2.44140625e-10 9.99999999e-01 2.44140625e-10]]

    [[2.00000000e-01 6.00000000e-01 2.00000000e-01 4.88281250e-11]
     [2.44140625e-10 9.99999999e-01 2.44140625e-10 2.44140625e-10]
     [1.22070312e-10 1.22070312e-10 1.00000000e+00 1.22070312e-10]
     [5.00000000e-01 5.00000000e-01 1.22070312e-10 1.22070312e-10]]

    [[6.66666667e-01 3.33333333e-01 8.13802083e-11 8.13802083e-11]
     [1.22070312e-10 1.22070312e-10 1.00000000e+00 1.22070312e-10]
     [3.33333333e-01 3.33333333e-01 8.13802083e-11 3.33333333e-01]
     [2.44140625e-10 2.44140625e-10 9.99999999e-01 2.44140625e-10]]]


   [[[2.00000000e-01 4.00000000e-01 2.00000000e-01 2.00000000e-01]
     [1.66666667e-01 1.66666667e-01 1.66666667e-01 5.00000000e-01]
     [2.85714286e-01 1.42857143e-01 1.42857143e-01 4.28571429e-01]
     [1.22070312e-10 1.00000000e+00 1.22070312e-10 1.22070312e-10]]

    [[9.99999999e-01 2.44140625e-10 2.44140625e-10 2.44140625e-10]
     [4.06901042e-11 5.00000000e-01 3.33333333e-01 1.66666667e-01]
     [2.00000000e-01 4.00000000e-01 4.00000000e-01 4.88281250e-11]
     [5.00000000e-01 1.22070312e-10 5.00000000e-01 1.22070312e-10]]

    [[1.22070312e-10 5.00000000e-01 5.00000000e-01 1.22070312e-10]
     [5.00000000e-01 5.00000000e-01 1.22070312e-10 1.22070312e-10]
     [8.13802083e-11 8.13802083e-11 1.00000000e+00 8.13802083e-11]
     [3.48772321e-11 1.42857143e-01 1.42857143e-01 7.14285714e-01]]

    [[2.44140625e-10 2.44140625e-10 2.44140625e-10 9.99999999e-01]
     [2.44140625e-10 9.99999999e-01 2.44140625e-10 2.44140625e-10]
     [1.66666667e-01 4.06901042e-11 6.66666667e-01 1.66666667e-01]
     [5.00000000e-01 1.22070312e-10 5.00000000e-01 1.22070312e-10]]]]



  [[[[5.00000000e-01 1.25000000e-01 2.50000000e-01 1.25000000e-01]
     [3.33333333e-01 2.22222222e-01 1.11111111e-01 3.33333333e-01]
     [2.85714286e-01 4.28571429e-01 1.42857143e-01 1.42857143e-01]
     [4.00000000e-01 2.00000000e-01 2.00000000e-01 2.00000000e-01]]

    [[2.50000000e-01 2.50000000e-01 6.10351562e-11 5.00000000e-01]
     [5.00000000e-01 5.00000000e-01 6.10351562e-11 6.10351562e-11]
     [1.22070312e-10 1.22070312e-10 5.00000000e-01 5.00000000e-01]
     [6.66666667e-01 8.13802083e-11 8.13802083e-11 3.33333333e-01]]

    [[2.50000000e-01 6.10351562e-11 2.50000000e-01 5.00000000e-01]
     [6.10351562e-11 5.00000000e-01 6.10351562e-11 5.00000000e-01]
     [5.00000000e-01 2.50000000e-01 6.10351562e-11 2.50000000e-01]
     [3.33333333e-01 6.66666667e-01 8.13802083e-11 8.13802083e-11]]

    [[2.44140625e-10 2.44140625e-10 2.44140625e-10 9.99999999e-01]
     [2.44140625e-10 2.44140625e-10 2.44140625e-10 9.99999999e-01]
     [8.13802083e-11 8.13802083e-11 3.33333333e-01 6.66666667e-01]
     [8.13802083e-11 3.33333333e-01 6.66666667e-01 8.13802083e-11]]]


   [[[2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [2.44140625e-10 9.99999999e-01 2.44140625e-10 2.44140625e-10]
     [1.22070312e-10 1.22070312e-10 1.00000000e+00 1.22070312e-10]
     [2.44140625e-10 9.99999999e-01 2.44140625e-10 2.44140625e-10]]

    [[5.00000000e-01 1.22070312e-10 5.00000000e-01 1.22070312e-10]
     [3.48772321e-11 5.71428571e-01 2.85714286e-01 1.42857143e-01]
     [3.33333333e-01 1.66666667e-01 1.66666667e-01 3.33333333e-01]
     [1.66666667e-01 3.33333333e-01 3.33333333e-01 1.66666667e-01]]

    [[5.00000000e-01 5.00000000e-01 6.10351562e-11 6.10351562e-11]
     [2.00000000e-01 4.88281250e-11 4.00000000e-01 4.00000000e-01]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [1.25000000e-01 3.75000000e-01 3.75000000e-01 1.25000000e-01]]

    [[5.00000000e-01 2.50000000e-01 6.10351562e-11 2.50000000e-01]
     [2.44140625e-10 2.44140625e-10 9.99999999e-01 2.44140625e-10]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]]]


   [[[1.42857143e-01 1.42857143e-01 5.71428571e-01 1.42857143e-01]
     [8.13802083e-11 3.33333333e-01 6.66666667e-01 8.13802083e-11]
     [1.22070312e-10 1.22070312e-10 1.22070312e-10 1.00000000e+00]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]]

    [[1.22070312e-10 5.00000000e-01 1.22070312e-10 5.00000000e-01]
     [3.33333333e-01 8.13802083e-11 8.13802083e-11 6.66666667e-01]
     [2.44140625e-10 9.99999999e-01 2.44140625e-10 2.44140625e-10]
     [6.10351562e-11 6.10351562e-11 2.50000000e-01 7.50000000e-01]]

    [[3.48772321e-11 4.28571429e-01 4.28571429e-01 1.42857143e-01]
     [1.22070312e-10 5.00000000e-01 5.00000000e-01 1.22070312e-10]
     [6.66666667e-01 3.33333333e-01 8.13802083e-11 8.13802083e-11]
     [2.00000000e-01 4.00000000e-01 2.00000000e-01 2.00000000e-01]]

    [[2.44140625e-10 9.99999999e-01 2.44140625e-10 2.44140625e-10]
     [2.44140625e-10 9.99999999e-01 2.44140625e-10 2.44140625e-10]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [4.88281250e-11 4.00000000e-01 6.00000000e-01 4.88281250e-11]]]


   [[[5.00000000e-01 2.50000000e-01 2.50000000e-01 6.10351562e-11]
     [4.06901042e-11 6.66666667e-01 4.06901042e-11 3.33333333e-01]
     [5.00000000e-01 5.00000000e-01 1.22070312e-10 1.22070312e-10]
     [2.44140625e-10 2.44140625e-10 2.44140625e-10 9.99999999e-01]]

    [[2.44140625e-10 9.99999999e-01 2.44140625e-10 2.44140625e-10]
     [6.66666667e-01 8.13802083e-11 3.33333333e-01 8.13802083e-11]
     [3.33333333e-01 8.13802083e-11 8.13802083e-11 6.66666667e-01]
     [1.22070312e-10 5.00000000e-01 1.22070312e-10 5.00000000e-01]]

    [[2.44140625e-10 2.44140625e-10 2.44140625e-10 9.99999999e-01]
     [4.88281250e-11 2.00000000e-01 2.00000000e-01 6.00000000e-01]
     [5.00000000e-01 6.10351562e-11 2.50000000e-01 2.50000000e-01]
     [3.33333333e-01 6.66666667e-01 8.13802083e-11 8.13802083e-11]]

    [[8.13802083e-11 6.66666667e-01 8.13802083e-11 3.33333333e-01]
     [1.22070312e-10 1.00000000e+00 1.22070312e-10 1.22070312e-10]
     [1.22070312e-10 5.00000000e-01 5.00000000e-01 1.22070312e-10]
     [2.44140625e-10 2.44140625e-10 2.44140625e-10 9.99999999e-01]]]]]




 [[[[[2.00000000e-01 4.00000000e-01 2.00000000e-01 2.00000000e-01]
     [2.44140625e-10 2.44140625e-10 2.44140625e-10 9.99999999e-01]
     [4.44444444e-01 1.11111111e-01 1.11111111e-01 3.33333333e-01]
     [8.00000000e-01 2.00000000e-01 4.88281250e-11 4.88281250e-11]]

    [[2.44140625e-10 2.44140625e-10 2.44140625e-10 9.99999999e-01]
     [4.28571429e-01 2.85714286e-01 1.42857143e-01 1.42857143e-01]
     [2.00000000e-01 4.88281250e-11 6.00000000e-01 2.00000000e-01]
     [3.33333333e-01 3.33333333e-01 8.13802083e-11 3.33333333e-01]]

    [[2.50000000e-01 2.50000000e-01 5.00000000e-01 6.10351562e-11]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [1.00000000e+00 1.22070312e-10 1.22070312e-10 1.22070312e-10]]

    [[9.99999999e-01 2.44140625e-10 2.44140625e-10 2.44140625e-10]
     [1.22070312e-10 1.22070312e-10 5.00000000e-01 5.00000000e-01]
     [6.66666667e-01 8.13802083e-11 3.33333333e-01 8.13802083e-11]
     [6.66666667e-01 3.33333333e-01 8.13802083e-11 8.13802083e-11]]]


   [[[1.00000000e+00 8.13802083e-11 8.13802083e-11 8.13802083e-11]
     [9.09090909e-02 4.54545455e-01 1.81818182e-01 2.72727273e-01]
     [2.50000000e-01 6.10351562e-11 5.00000000e-01 2.50000000e-01]
     [6.00000000e-01 4.00000000e-01 4.88281250e-11 4.88281250e-11]]

    [[8.13802083e-11 1.00000000e+00 8.13802083e-11 8.13802083e-11]
     [2.21946023e-11 4.54545455e-01 2.72727273e-01 2.72727273e-01]
     [4.88281250e-11 2.00000000e-01 4.00000000e-01 4.00000000e-01]
     [5.71428571e-01 2.85714286e-01 3.48772321e-11 1.42857143e-01]]

    [[5.00000000e-01 1.66666667e-01 4.06901042e-11 3.33333333e-01]
     [2.50000000e-01 5.00000000e-01 6.10351562e-11 2.50000000e-01]
     [6.66666667e-01 3.33333333e-01 8.13802083e-11 8.13802083e-11]
     [3.33333333e-01 3.33333333e-01 3.33333333e-01 8.13802083e-11]]

    [[3.33333333e-01 6.66666667e-01 8.13802083e-11 8.13802083e-11]
     [1.22070312e-10 1.00000000e+00 1.22070312e-10 1.22070312e-10]
     [5.00000000e-01 6.10351562e-11 5.00000000e-01 6.10351562e-11]
     [8.13802083e-11 6.66666667e-01 3.33333333e-01 8.13802083e-11]]]


   [[[8.33333333e-01 4.06901042e-11 1.66666667e-01 4.06901042e-11]
     [7.50000000e-01 6.10351562e-11 2.50000000e-01 6.10351562e-11]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]]

    [[1.22070312e-10 5.00000000e-01 5.00000000e-01 1.22070312e-10]
     [9.99999999e-01 2.44140625e-10 2.44140625e-10 2.44140625e-10]
     [1.22070312e-10 1.22070312e-10 1.22070312e-10 1.00000000e+00]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]]

    [[2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [5.00000000e-01 1.22070312e-10 1.22070312e-10 5.00000000e-01]
     [9.99999999e-01 2.44140625e-10 2.44140625e-10 2.44140625e-10]]

    [[2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [2.44140625e-10 2.44140625e-10 9.99999999e-01 2.44140625e-10]
     [7.50000000e-01 6.10351562e-11 2.50000000e-01 6.10351562e-11]
     [2.44140625e-10 9.99999999e-01 2.44140625e-10 2.44140625e-10]]]


   [[[7.50000000e-01 6.10351562e-11 6.10351562e-11 2.50000000e-01]
     [5.00000000e-01 5.00000000e-01 1.22070312e-10 1.22070312e-10]
     [8.13802083e-11 8.13802083e-11 6.66666667e-01 3.33333333e-01]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]]

    [[2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [1.22070312e-10 5.00000000e-01 1.22070312e-10 5.00000000e-01]
     [9.99999999e-01 2.44140625e-10 2.44140625e-10 2.44140625e-10]
     [2.44140625e-10 2.44140625e-10 2.44140625e-10 9.99999999e-01]]

    [[2.44140625e-10 2.44140625e-10 9.99999999e-01 2.44140625e-10]
     [2.44140625e-10 9.99999999e-01 2.44140625e-10 2.44140625e-10]
     [3.33333333e-01 3.33333333e-01 8.13802083e-11 3.33333333e-01]
     [9.99999999e-01 2.44140625e-10 2.44140625e-10 2.44140625e-10]]

    [[1.22070312e-10 5.00000000e-01 5.00000000e-01 1.22070312e-10]
     [9.99999999e-01 2.44140625e-10 2.44140625e-10 2.44140625e-10]
     [1.22070312e-10 1.00000000e+00 1.22070312e-10 1.22070312e-10]
     [1.22070312e-10 1.22070312e-10 5.00000000e-01 5.00000000e-01]]]]



  [[[[8.00000000e-01 4.88281250e-11 2.00000000e-01 4.88281250e-11]
     [1.00000000e+00 6.10351562e-11 6.10351562e-11 6.10351562e-11]
     [5.00000000e-01 1.22070312e-10 1.22070312e-10 5.00000000e-01]
     [3.33333333e-01 8.13802083e-11 3.33333333e-01 3.33333333e-01]]

    [[2.50000000e-01 3.75000000e-01 2.50000000e-01 1.25000000e-01]
     [2.50000000e-01 3.75000000e-01 2.50000000e-01 1.25000000e-01]
     [2.50000000e-01 3.05175781e-11 2.50000000e-01 5.00000000e-01]
     [5.00000000e-01 2.50000000e-01 2.50000000e-01 3.05175781e-11]]

    [[2.44140625e-10 2.44140625e-10 9.99999999e-01 2.44140625e-10]
     [6.66666667e-01 3.33333333e-01 8.13802083e-11 8.13802083e-11]
     [5.00000000e-01 1.22070312e-10 1.22070312e-10 5.00000000e-01]
     [6.00000000e-01 2.00000000e-01 2.00000000e-01 4.88281250e-11]]

    [[6.00000000e-01 4.00000000e-01 4.88281250e-11 4.88281250e-11]
     [2.50000000e-01 2.50000000e-01 5.00000000e-01 6.10351562e-11]
     [5.00000000e-01 1.22070312e-10 5.00000000e-01 1.22070312e-10]
     [5.00000000e-01 1.22070312e-10 5.00000000e-01 1.22070312e-10]]]


   [[[3.33333333e-01 1.66666667e-01 2.50000000e-01 2.50000000e-01]
     [1.66666667e-01 4.44444444e-01 1.11111111e-01 2.77777778e-01]
     [3.33333333e-01 8.13802083e-11 3.33333333e-01 3.33333333e-01]
     [3.33333333e-01 4.06901042e-11 3.33333333e-01 3.33333333e-01]]

    [[2.85714286e-01 3.57142857e-01 1.42857143e-01 2.14285714e-01]
     [1.29032258e-01 2.25806452e-01 2.58064516e-01 3.87096774e-01]
     [4.44444444e-01 2.22222222e-01 1.11111111e-01 2.22222222e-01]
     [2.38095238e-01 2.38095238e-01 2.85714286e-01 2.38095238e-01]]

    [[2.85714286e-01 2.85714286e-01 1.42857143e-01 2.85714286e-01]
     [1.25000000e-01 3.75000000e-01 1.25000000e-01 3.75000000e-01]
     [4.88281250e-11 2.00000000e-01 4.00000000e-01 4.00000000e-01]
     [4.00000000e-01 2.00000000e-01 4.88281250e-11 4.00000000e-01]]

    [[2.50000000e-01 3.75000000e-01 1.87500000e-01 1.87500000e-01]
     [1.53846154e-01 3.84615385e-01 1.87800481e-11 4.61538462e-01]
     [1.42857143e-01 4.28571429e-01 3.48772321e-11 4.28571429e-01]
     [5.71428571e-01 2.85714286e-01 3.48772321e-11 1.42857143e-01]]]


   [[[4.61538462e-01 1.53846154e-01 3.84615385e-01 1.87800481e-11]
     [9.09090909e-02 4.54545455e-01 1.81818182e-01 2.72727273e-01]
     [2.50000000e-01 2.50000000e-01 6.10351562e-11 5.00000000e-01]
     [5.00000000e-01 2.50000000e-01 2.50000000e-01 6.10351562e-11]]

    [[6.10351562e-11 7.50000000e-01 2.50000000e-01 6.10351562e-11]
     [1.25000000e-01 5.00000000e-01 1.25000000e-01 2.50000000e-01]
     [2.44140625e-10 2.44140625e-10 2.44140625e-10 9.99999999e-01]
     [6.66666667e-01 1.66666667e-01 4.06901042e-11 1.66666667e-01]]

    [[3.33333333e-01 4.06901042e-11 1.66666667e-01 5.00000000e-01]
     [2.44140625e-10 9.99999999e-01 2.44140625e-10 2.44140625e-10]
     [3.33333333e-01 6.66666667e-01 8.13802083e-11 8.13802083e-11]
     [3.48772321e-11 4.28571429e-01 4.28571429e-01 1.42857143e-01]]

    [[6.10351562e-11 2.50000000e-01 6.10351562e-11 7.50000000e-01]
     [4.88281250e-11 4.00000000e-01 4.88281250e-11 6.00000000e-01]
     [3.33333333e-01 8.13802083e-11 3.33333333e-01 3.33333333e-01]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]]]


   [[[2.50000000e-01 3.33333333e-01 3.33333333e-01 8.33333333e-02]
     [1.25000000e-01 5.00000000e-01 1.25000000e-01 2.50000000e-01]
     [4.00000000e-01 2.44140625e-11 3.00000000e-01 3.00000000e-01]
     [5.00000000e-01 1.66666667e-01 1.66666667e-01 1.66666667e-01]]

    [[2.00000000e-01 6.00000000e-01 2.00000000e-01 4.88281250e-11]
     [2.50000000e-01 2.50000000e-01 3.75000000e-01 1.25000000e-01]
     [1.42857143e-01 4.28571429e-01 1.42857143e-01 2.85714286e-01]
     [4.00000000e-01 4.00000000e-01 4.88281250e-11 2.00000000e-01]]

    [[3.33333333e-01 8.13802083e-11 3.33333333e-01 3.33333333e-01]
     [3.05175781e-11 5.00000000e-01 2.50000000e-01 2.50000000e-01]
     [2.85714286e-01 3.48772321e-11 1.42857143e-01 5.71428571e-01]
     [4.00000000e-01 2.00000000e-01 4.88281250e-11 4.00000000e-01]]

    [[4.06901042e-11 1.66666667e-01 6.66666667e-01 1.66666667e-01]
     [3.05175781e-11 3.75000000e-01 1.25000000e-01 5.00000000e-01]
     [5.00000000e-01 2.50000000e-01 6.10351562e-11 2.50000000e-01]
     [1.22070312e-10 5.00000000e-01 5.00000000e-01 1.22070312e-10]]]]



  [[[[5.00000000e-01 1.00000000e-01 2.00000000e-01 2.00000000e-01]
     [3.07692308e-01 1.53846154e-01 3.84615385e-01 1.53846154e-01]
     [1.66666667e-01 3.33333333e-01 4.06901042e-11 5.00000000e-01]
     [5.00000000e-01 1.66666667e-01 4.06901042e-11 3.33333333e-01]]

    [[2.50000000e-01 3.75000000e-01 2.50000000e-01 1.25000000e-01]
     [9.09090909e-02 4.54545455e-01 9.09090909e-02 3.63636364e-01]
     [4.00000000e-01 4.88281250e-11 6.00000000e-01 4.88281250e-11]
     [6.00000000e-01 2.00000000e-01 1.00000000e-01 1.00000000e-01]]

    [[8.13802083e-11 3.33333333e-01 3.33333333e-01 3.33333333e-01]
     [3.33333333e-01 8.13802083e-11 6.66666667e-01 8.13802083e-11]
     [2.00000000e-01 4.00000000e-01 2.00000000e-01 2.00000000e-01]
     [2.85714286e-01 1.42857143e-01 5.71428571e-01 3.48772321e-11]]

    [[8.33333333e-01 4.06901042e-11 4.06901042e-11 1.66666667e-01]
     [4.88281250e-11 4.00000000e-01 6.00000000e-01 4.88281250e-11]
     [2.00000000e-01 3.00000000e-01 1.00000000e-01 4.00000000e-01]
     [2.00000000e-01 4.88281250e-11 6.00000000e-01 2.00000000e-01]]]


   [[[2.50000000e-01 5.00000000e-01 2.50000000e-01 3.05175781e-11]
     [7.50000000e-01 6.10351562e-11 2.50000000e-01 6.10351562e-11]
     [1.22070312e-10 5.00000000e-01 1.22070312e-10 5.00000000e-01]
     [9.99999999e-01 2.44140625e-10 2.44140625e-10 2.44140625e-10]]

    [[9.99999999e-01 2.44140625e-10 2.44140625e-10 2.44140625e-10]
     [1.66666667e-01 5.00000000e-01 8.33333333e-02 2.50000000e-01]
     [3.33333333e-01 3.33333333e-01 8.13802083e-11 3.33333333e-01]
     [2.44140625e-10 2.44140625e-10 2.44140625e-10 9.99999999e-01]]

    [[4.00000000e-01 6.00000000e-01 4.88281250e-11 4.88281250e-11]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [5.00000000e-01 1.22070312e-10 1.22070312e-10 5.00000000e-01]
     [3.33333333e-01 1.66666667e-01 4.06901042e-11 5.00000000e-01]]

    [[1.66666667e-01 5.00000000e-01 3.33333333e-01 4.06901042e-11]
     [6.10351562e-11 5.00000000e-01 5.00000000e-01 6.10351562e-11]
     [9.99999999e-01 2.44140625e-10 2.44140625e-10 2.44140625e-10]
     [2.50000000e-01 5.00000000e-01 6.10351562e-11 2.50000000e-01]]]


   [[[4.28571429e-01 2.85714286e-01 1.42857143e-01 1.42857143e-01]
     [3.48772321e-11 1.42857143e-01 5.71428571e-01 2.85714286e-01]
     [5.00000000e-01 1.22070312e-10 5.00000000e-01 1.22070312e-10]
     [4.28571429e-01 3.48772321e-11 2.85714286e-01 2.85714286e-01]]

    [[5.00000000e-01 5.00000000e-01 6.10351562e-11 6.10351562e-11]
     [6.10351562e-11 5.00000000e-01 2.50000000e-01 2.50000000e-01]
     [3.33333333e-01 8.13802083e-11 6.66666667e-01 8.13802083e-11]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]]

    [[8.13802083e-11 6.66666667e-01 3.33333333e-01 8.13802083e-11]
     [3.33333333e-01 6.66666667e-01 8.13802083e-11 8.13802083e-11]
     [7.50000000e-01 2.50000000e-01 6.10351562e-11 6.10351562e-11]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]]

    [[5.00000000e-01 2.50000000e-01 2.50000000e-01 6.10351562e-11]
     [2.44140625e-10 2.44140625e-10 9.99999999e-01 2.44140625e-10]
     [3.33333333e-01 8.13802083e-11 3.33333333e-01 3.33333333e-01]
     [5.00000000e-01 2.50000000e-01 2.50000000e-01 6.10351562e-11]]]


   [[[6.66666667e-01 1.66666667e-01 1.66666667e-01 4.06901042e-11]
     [3.33333333e-01 3.33333333e-01 8.13802083e-11 3.33333333e-01]
     [5.00000000e-01 1.22070312e-10 1.22070312e-10 5.00000000e-01]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]]

    [[1.22070312e-10 5.00000000e-01 5.00000000e-01 1.22070312e-10]
     [6.10351562e-11 2.50000000e-01 5.00000000e-01 2.50000000e-01]
     [3.63636364e-01 2.72727273e-01 9.09090909e-02 2.72727273e-01]
     [2.50000000e-01 5.00000000e-01 2.50000000e-01 6.10351562e-11]]

    [[5.00000000e-01 5.00000000e-01 6.10351562e-11 6.10351562e-11]
     [1.22070312e-10 5.00000000e-01 1.22070312e-10 5.00000000e-01]
     [6.00000000e-01 2.00000000e-01 4.88281250e-11 2.00000000e-01]
     [2.44140625e-10 9.99999999e-01 2.44140625e-10 2.44140625e-10]]

    [[5.00000000e-01 4.06901042e-11 3.33333333e-01 1.66666667e-01]
     [2.44140625e-10 2.44140625e-10 9.99999999e-01 2.44140625e-10]
     [3.33333333e-01 1.66666667e-01 1.66666667e-01 3.33333333e-01]
     [2.44140625e-10 2.44140625e-10 9.99999999e-01 2.44140625e-10]]]]



  [[[[3.00000000e-01 2.00000000e-01 1.00000000e-01 4.00000000e-01]
     [1.42857143e-01 2.85714286e-01 5.71428571e-01 3.48772321e-11]
     [3.48772321e-11 1.42857143e-01 1.42857143e-01 7.14285714e-01]
     [8.33333333e-01 4.06901042e-11 4.06901042e-11 1.66666667e-01]]

    [[5.71428571e-01 1.42857143e-01 3.48772321e-11 2.85714286e-01]
     [2.14285714e-01 3.57142857e-01 2.85714286e-01 1.42857143e-01]
     [5.00000000e-01 1.66666667e-01 1.66666667e-01 1.66666667e-01]
     [5.00000000e-01 8.33333333e-02 2.50000000e-01 1.66666667e-01]]

    [[1.66666667e-01 3.33333333e-01 4.06901042e-11 5.00000000e-01]
     [8.13802083e-11 6.66666667e-01 8.13802083e-11 3.33333333e-01]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [3.33333333e-01 1.66666667e-01 1.66666667e-01 3.33333333e-01]]

    [[8.13802083e-11 6.66666667e-01 3.33333333e-01 8.13802083e-11]
     [2.44140625e-10 9.99999999e-01 2.44140625e-10 2.44140625e-10]
     [3.33333333e-01 8.13802083e-11 8.13802083e-11 6.66666667e-01]
     [9.99999999e-01 2.44140625e-10 2.44140625e-10 2.44140625e-10]]]


   [[[5.00000000e-01 1.66666667e-01 1.66666667e-01 1.66666667e-01]
     [3.33333333e-01 8.13802083e-11 8.13802083e-11 6.66666667e-01]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [5.00000000e-01 1.22070312e-10 1.22070312e-10 5.00000000e-01]]

    [[5.00000000e-01 5.00000000e-01 1.22070312e-10 1.22070312e-10]
     [2.85714286e-01 2.85714286e-01 4.28571429e-01 3.48772321e-11]
     [1.42857143e-01 5.71428571e-01 1.42857143e-01 1.42857143e-01]
     [3.33333333e-01 8.33333333e-02 1.66666667e-01 4.16666667e-01]]

    [[3.33333333e-01 1.66666667e-01 4.06901042e-11 5.00000000e-01]
     [4.06901042e-11 3.33333333e-01 3.33333333e-01 3.33333333e-01]
     [2.50000000e-01 2.50000000e-01 6.10351562e-11 5.00000000e-01]
     [1.42857143e-01 2.85714286e-01 5.71428571e-01 3.48772321e-11]]

    [[4.00000000e-01 4.00000000e-01 2.00000000e-01 4.88281250e-11]
     [4.06901042e-11 3.33333333e-01 5.00000000e-01 1.66666667e-01]
     [6.10351562e-11 5.00000000e-01 2.50000000e-01 2.50000000e-01]
     [2.00000000e-01 6.00000000e-01 2.00000000e-01 4.88281250e-11]]]


   [[[2.44140625e-10 2.44140625e-10 9.99999999e-01 2.44140625e-10]
     [3.48772321e-11 4.28571429e-01 4.28571429e-01 1.42857143e-01]
     [2.50000000e-01 2.50000000e-01 6.10351562e-11 5.00000000e-01]
     [2.44140625e-10 2.44140625e-10 2.44140625e-10 9.99999999e-01]]

    [[3.33333333e-01 3.33333333e-01 3.33333333e-01 8.13802083e-11]
     [3.05175781e-11 5.00000000e-01 2.50000000e-01 2.50000000e-01]
     [6.10351562e-11 7.50000000e-01 6.10351562e-11 2.50000000e-01]
     [1.22070312e-10 1.22070312e-10 5.00000000e-01 5.00000000e-01]]

    [[6.66666667e-01 8.13802083e-11 3.33333333e-01 8.13802083e-11]
     [5.00000000e-01 5.00000000e-01 1.22070312e-10 1.22070312e-10]
     [2.44140625e-10 2.44140625e-10 2.44140625e-10 9.99999999e-01]
     [6.10351562e-11 5.00000000e-01 5.00000000e-01 6.10351562e-11]]

    [[1.22070312e-10 5.00000000e-01 5.00000000e-01 1.22070312e-10]
     [6.10351562e-11 2.50000000e-01 5.00000000e-01 2.50000000e-01]
     [3.33333333e-01 6.66666667e-01 8.13802083e-11 8.13802083e-11]
     [2.44140625e-10 2.44140625e-10 2.44140625e-10 9.99999999e-01]]]


   [[[5.00000000e-01 2.50000000e-01 1.25000000e-01 1.25000000e-01]
     [1.66666667e-01 1.66666667e-01 1.66666667e-01 5.00000000e-01]
     [1.66666667e-01 1.66666667e-01 3.33333333e-01 3.33333333e-01]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]]

    [[8.13802083e-11 6.66666667e-01 3.33333333e-01 8.13802083e-11]
     [1.42857143e-01 2.85714286e-01 1.42857143e-01 4.28571429e-01]
     [2.50000000e-01 5.00000000e-01 6.10351562e-11 2.50000000e-01]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]]

    [[5.00000000e-01 1.22070312e-10 1.22070312e-10 5.00000000e-01]
     [3.33333333e-01 8.13802083e-11 6.66666667e-01 8.13802083e-11]
     [6.10351562e-11 2.50000000e-01 6.10351562e-11 7.50000000e-01]
     [2.50000000e-01 3.75000000e-01 1.25000000e-01 2.50000000e-01]]

    [[5.00000000e-01 5.00000000e-01 1.22070312e-10 1.22070312e-10]
     [8.13802083e-11 3.33333333e-01 3.33333333e-01 3.33333333e-01]
     [8.13802083e-11 8.13802083e-11 8.13802083e-11 1.00000000e+00]
     [6.10351562e-11 5.00000000e-01 5.00000000e-01 6.10351562e-11]]]]]




 [[[[[7.69230769e-01 1.87800481e-11 7.69230769e-02 1.53846154e-01]
     [1.42857143e-01 3.48772321e-11 1.42857143e-01 7.14285714e-01]
     [2.50000000e-01 3.05175781e-11 3.05175781e-11 7.50000000e-01]
     [7.50000000e-01 6.10351562e-11 6.10351562e-11 2.50000000e-01]]

    [[1.22070312e-10 1.00000000e+00 1.22070312e-10 1.22070312e-10]
     [3.33333333e-01 6.66666667e-01 4.06901042e-11 4.06901042e-11]
     [2.50000000e-01 2.50000000e-01 5.00000000e-01 6.10351562e-11]
     [8.13802083e-11 8.13802083e-11 8.13802083e-11 1.00000000e+00]]

    [[7.50000000e-01 1.25000000e-01 3.05175781e-11 1.25000000e-01]
     [5.00000000e-01 5.00000000e-01 1.22070312e-10 1.22070312e-10]
     [4.28571429e-01 3.48772321e-11 1.42857143e-01 4.28571429e-01]
     [3.00000000e-01 1.00000000e-01 5.00000000e-01 1.00000000e-01]]

    [[2.00000000e-01 4.88281250e-11 4.00000000e-01 4.00000000e-01]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [1.22070312e-10 1.22070312e-10 1.22070312e-10 1.00000000e+00]
     [4.88281250e-11 2.00000000e-01 2.00000000e-01 6.00000000e-01]]]


   [[[5.71428571e-01 2.85714286e-01 3.48772321e-11 1.42857143e-01]
     [1.42857143e-01 5.71428571e-01 1.42857143e-01 1.42857143e-01]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [6.10351562e-11 5.00000000e-01 6.10351562e-11 5.00000000e-01]]

    [[2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [9.09090909e-02 7.27272727e-01 9.09090909e-02 9.09090909e-02]
     [3.33333333e-01 3.33333333e-01 1.66666667e-01 1.66666667e-01]
     [9.99999999e-01 2.44140625e-10 2.44140625e-10 2.44140625e-10]]

    [[5.00000000e-01 3.00000000e-01 2.00000000e-01 2.44140625e-11]
     [2.44140625e-10 9.99999999e-01 2.44140625e-10 2.44140625e-10]
     [3.33333333e-01 8.13802083e-11 6.66666667e-01 8.13802083e-11]
     [5.00000000e-01 6.10351562e-11 6.10351562e-11 5.00000000e-01]]

    [[2.50000000e-01 6.10351562e-11 7.50000000e-01 6.10351562e-11]
     [1.66666667e-01 5.00000000e-01 1.66666667e-01 1.66666667e-01]
     [5.00000000e-01 2.50000000e-01 6.10351562e-11 2.50000000e-01]
     [6.10351562e-11 2.50000000e-01 5.00000000e-01 2.50000000e-01]]]


   [[[4.06901042e-11 3.33333333e-01 3.33333333e-01 3.33333333e-01]
     [2.50000000e-01 5.00000000e-01 6.10351562e-11 2.50000000e-01]
     [1.00000000e+00 8.13802083e-11 8.13802083e-11 8.13802083e-11]
     [9.99999999e-01 2.44140625e-10 2.44140625e-10 2.44140625e-10]]

    [[2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [8.13802083e-11 1.00000000e+00 8.13802083e-11 8.13802083e-11]
     [3.33333333e-01 3.33333333e-01 8.13802083e-11 3.33333333e-01]
     [2.44140625e-10 2.44140625e-10 2.44140625e-10 9.99999999e-01]]

    [[3.33333333e-01 3.33333333e-01 3.33333333e-01 8.13802083e-11]
     [6.10351562e-11 2.50000000e-01 5.00000000e-01 2.50000000e-01]
     [2.44140625e-10 9.99999999e-01 2.44140625e-10 2.44140625e-10]
     [9.99999999e-01 2.44140625e-10 2.44140625e-10 2.44140625e-10]]

    [[1.66666667e-01 4.06901042e-11 3.33333333e-01 5.00000000e-01]
     [8.13802083e-11 3.33333333e-01 3.33333333e-01 3.33333333e-01]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [6.66666667e-01 8.13802083e-11 8.13802083e-11 3.33333333e-01]]]


   [[[2.50000000e-01 6.10351562e-11 7.50000000e-01 6.10351562e-11]
     [3.48772321e-11 2.85714286e-01 2.85714286e-01 4.28571429e-01]
     [2.44140625e-10 9.99999999e-01 2.44140625e-10 2.44140625e-10]
     [1.22070312e-10 1.00000000e+00 1.22070312e-10 1.22070312e-10]]

    [[2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [4.88281250e-11 6.00000000e-01 2.00000000e-01 2.00000000e-01]
     [2.50000000e-01 2.50000000e-01 5.00000000e-01 6.10351562e-11]
     [2.44140625e-10 9.99999999e-01 2.44140625e-10 2.44140625e-10]]

    [[2.50000000e-01 6.10351562e-11 5.00000000e-01 2.50000000e-01]
     [2.44140625e-10 9.99999999e-01 2.44140625e-10 2.44140625e-10]
     [5.00000000e-01 2.50000000e-01 2.50000000e-01 6.10351562e-11]
     [1.22070312e-10 1.22070312e-10 1.00000000e+00 1.22070312e-10]]

    [[2.50000000e-01 2.50000000e-01 5.00000000e-01 6.10351562e-11]
     [2.44140625e-10 9.99999999e-01 2.44140625e-10 2.44140625e-10]
     [6.10351562e-11 2.50000000e-01 5.00000000e-01 2.50000000e-01]
     [2.44140625e-10 2.44140625e-10 2.44140625e-10 9.99999999e-01]]]]



  [[[[2.85714286e-01 2.85714286e-01 1.42857143e-01 2.85714286e-01]
     [9.99999999e-01 2.44140625e-10 2.44140625e-10 2.44140625e-10]
     [3.33333333e-01 8.13802083e-11 3.33333333e-01 3.33333333e-01]
     [6.66666667e-01 8.13802083e-11 3.33333333e-01 8.13802083e-11]]

    [[2.44140625e-10 9.99999999e-01 2.44140625e-10 2.44140625e-10]
     [4.06901042e-11 1.66666667e-01 5.00000000e-01 3.33333333e-01]
     [3.33333333e-01 3.33333333e-01 1.66666667e-01 1.66666667e-01]
     [3.33333333e-01 3.33333333e-01 3.33333333e-01 8.13802083e-11]]

    [[6.66666667e-01 4.06901042e-11 1.66666667e-01 1.66666667e-01]
     [1.22070312e-10 5.00000000e-01 1.22070312e-10 5.00000000e-01]
     [9.99999999e-01 2.44140625e-10 2.44140625e-10 2.44140625e-10]
     [3.33333333e-01 3.33333333e-01 8.13802083e-11 3.33333333e-01]]

    [[5.00000000e-01 5.00000000e-01 1.22070312e-10 1.22070312e-10]
     [8.13802083e-11 3.33333333e-01 6.66666667e-01 8.13802083e-11]
     [5.00000000e-01 5.00000000e-01 1.22070312e-10 1.22070312e-10]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]]]


   [[[2.50000000e-01 5.00000000e-01 3.05175781e-11 2.50000000e-01]
     [9.99999999e-01 2.44140625e-10 2.44140625e-10 2.44140625e-10]
     [1.22070312e-10 1.22070312e-10 1.22070312e-10 1.00000000e+00]
     [2.44140625e-10 2.44140625e-10 2.44140625e-10 9.99999999e-01]]

    [[1.22070312e-10 5.00000000e-01 1.22070312e-10 5.00000000e-01]
     [4.06901042e-11 5.00000000e-01 1.66666667e-01 3.33333333e-01]
     [2.00000000e-01 4.88281250e-11 2.00000000e-01 6.00000000e-01]
     [6.66666667e-01 4.06901042e-11 4.06901042e-11 3.33333333e-01]]

    [[7.50000000e-01 6.10351562e-11 2.50000000e-01 6.10351562e-11]
     [5.00000000e-01 5.00000000e-01 6.10351562e-11 6.10351562e-11]
     [8.13802083e-11 8.13802083e-11 1.00000000e+00 8.13802083e-11]
     [2.44140625e-10 9.99999999e-01 2.44140625e-10 2.44140625e-10]]

    [[3.33333333e-01 3.33333333e-01 1.66666667e-01 1.66666667e-01]
     [3.48772321e-11 7.14285714e-01 3.48772321e-11 2.85714286e-01]
     [9.99999999e-01 2.44140625e-10 2.44140625e-10 2.44140625e-10]
     [1.22070312e-10 1.22070312e-10 5.00000000e-01 5.00000000e-01]]]


   [[[6.66666667e-01 1.66666667e-01 4.06901042e-11 1.66666667e-01]
     [2.50000000e-01 2.50000000e-01 5.00000000e-01 6.10351562e-11]
     [8.13802083e-11 3.33333333e-01 3.33333333e-01 3.33333333e-01]
     [5.00000000e-01 1.22070312e-10 5.00000000e-01 1.22070312e-10]]

    [[1.22070312e-10 1.22070312e-10 1.00000000e+00 1.22070312e-10]
     [8.13802083e-11 6.66666667e-01 3.33333333e-01 8.13802083e-11]
     [9.99999999e-01 2.44140625e-10 2.44140625e-10 2.44140625e-10]
     [1.22070312e-10 1.22070312e-10 5.00000000e-01 5.00000000e-01]]

    [[5.00000000e-01 5.00000000e-01 1.22070312e-10 1.22070312e-10]
     [1.22070312e-10 5.00000000e-01 1.22070312e-10 5.00000000e-01]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [2.00000000e-01 4.00000000e-01 4.88281250e-11 4.00000000e-01]]

    [[1.22070312e-10 5.00000000e-01 5.00000000e-01 1.22070312e-10]
     [1.22070312e-10 1.00000000e+00 1.22070312e-10 1.22070312e-10]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]]]


   [[[3.33333333e-01 3.33333333e-01 8.13802083e-11 3.33333333e-01]
     [1.22070312e-10 1.00000000e+00 1.22070312e-10 1.22070312e-10]
     [1.22070312e-10 1.22070312e-10 5.00000000e-01 5.00000000e-01]
     [1.66666667e-01 3.33333333e-01 3.33333333e-01 1.66666667e-01]]

    [[1.22070312e-10 5.00000000e-01 1.22070312e-10 5.00000000e-01]
     [1.42857143e-01 4.28571429e-01 1.42857143e-01 2.85714286e-01]
     [5.00000000e-01 1.22070312e-10 1.22070312e-10 5.00000000e-01]
     [6.66666667e-01 8.13802083e-11 3.33333333e-01 8.13802083e-11]]

    [[3.33333333e-01 3.33333333e-01 3.33333333e-01 8.13802083e-11]
     [9.99999999e-01 2.44140625e-10 2.44140625e-10 2.44140625e-10]
     [3.33333333e-01 8.13802083e-11 8.13802083e-11 6.66666667e-01]
     [9.99999999e-01 2.44140625e-10 2.44140625e-10 2.44140625e-10]]

    [[2.44140625e-10 9.99999999e-01 2.44140625e-10 2.44140625e-10]
     [6.66666667e-01 3.33333333e-01 8.13802083e-11 8.13802083e-11]
     [2.44140625e-10 2.44140625e-10 9.99999999e-01 2.44140625e-10]
     [5.00000000e-01 1.22070312e-10 5.00000000e-01 1.22070312e-10]]]]



  [[[[4.28571429e-01 1.42857143e-01 1.42857143e-01 2.85714286e-01]
     [1.11111111e-01 5.55555556e-01 1.11111111e-01 2.22222222e-01]
     [1.25000000e-01 2.50000000e-01 1.25000000e-01 5.00000000e-01]
     [8.13802083e-11 6.66666667e-01 8.13802083e-11 3.33333333e-01]]

    [[6.10351562e-11 2.50000000e-01 2.50000000e-01 5.00000000e-01]
     [1.66666667e-01 3.33333333e-01 1.66666667e-01 3.33333333e-01]
     [3.00000000e-01 2.00000000e-01 2.00000000e-01 3.00000000e-01]
     [3.33333333e-01 8.13802083e-11 6.66666667e-01 8.13802083e-11]]

    [[1.22070312e-10 5.00000000e-01 5.00000000e-01 1.22070312e-10]
     [2.44140625e-10 9.99999999e-01 2.44140625e-10 2.44140625e-10]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [2.44140625e-10 2.44140625e-10 2.44140625e-10 9.99999999e-01]]

    [[5.00000000e-01 1.22070312e-10 5.00000000e-01 1.22070312e-10]
     [2.44140625e-10 2.44140625e-10 9.99999999e-01 2.44140625e-10]
     [2.50000000e-01 2.50000000e-01 5.00000000e-01 6.10351562e-11]
     [1.22070312e-10 1.00000000e+00 1.22070312e-10 1.22070312e-10]]]


   [[[2.44140625e-10 9.99999999e-01 2.44140625e-10 2.44140625e-10]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [1.22070312e-10 5.00000000e-01 5.00000000e-01 1.22070312e-10]
     [1.22070312e-10 1.00000000e+00 1.22070312e-10 1.22070312e-10]]

    [[1.22070312e-10 5.00000000e-01 5.00000000e-01 1.22070312e-10]
     [5.71428571e-01 1.42857143e-01 1.42857143e-01 1.42857143e-01]
     [1.22070312e-10 5.00000000e-01 1.22070312e-10 5.00000000e-01]
     [3.33333333e-01 3.33333333e-01 3.33333333e-01 8.13802083e-11]]

    [[2.85714286e-01 4.28571429e-01 1.42857143e-01 1.42857143e-01]
     [1.22070312e-10 1.22070312e-10 5.00000000e-01 5.00000000e-01]
     [1.22070312e-10 5.00000000e-01 1.22070312e-10 5.00000000e-01]
     [6.10351562e-11 5.00000000e-01 5.00000000e-01 6.10351562e-11]]

    [[2.44140625e-10 2.44140625e-10 9.99999999e-01 2.44140625e-10]
     [2.50000000e-01 5.00000000e-01 6.10351562e-11 2.50000000e-01]
     [2.44140625e-10 2.44140625e-10 9.99999999e-01 2.44140625e-10]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]]]


   [[[6.66666667e-01 8.13802083e-11 3.33333333e-01 8.13802083e-11]
     [3.33333333e-01 8.13802083e-11 3.33333333e-01 3.33333333e-01]
     [2.44140625e-10 2.44140625e-10 2.44140625e-10 9.99999999e-01]
     [1.22070312e-10 1.22070312e-10 1.22070312e-10 1.00000000e+00]]

    [[3.33333333e-01 3.33333333e-01 8.13802083e-11 3.33333333e-01]
     [5.00000000e-01 5.00000000e-01 1.22070312e-10 1.22070312e-10]
     [2.00000000e-01 4.00000000e-01 4.88281250e-11 4.00000000e-01]
     [2.44140625e-10 2.44140625e-10 9.99999999e-01 2.44140625e-10]]

    [[5.00000000e-01 1.22070312e-10 5.00000000e-01 1.22070312e-10]
     [5.00000000e-01 1.22070312e-10 5.00000000e-01 1.22070312e-10]
     [2.44140625e-10 2.44140625e-10 2.44140625e-10 9.99999999e-01]
     [5.00000000e-01 1.22070312e-10 5.00000000e-01 1.22070312e-10]]

    [[3.33333333e-01 3.33333333e-01 8.13802083e-11 3.33333333e-01]
     [8.13802083e-11 6.66666667e-01 8.13802083e-11 3.33333333e-01]
     [1.00000000e+00 1.22070312e-10 1.22070312e-10 1.22070312e-10]
     [2.44140625e-10 2.44140625e-10 2.44140625e-10 9.99999999e-01]]]


   [[[4.00000000e-01 6.00000000e-01 4.88281250e-11 4.88281250e-11]
     [3.33333333e-01 8.13802083e-11 3.33333333e-01 3.33333333e-01]
     [2.44140625e-10 2.44140625e-10 9.99999999e-01 2.44140625e-10]
     [3.33333333e-01 6.66666667e-01 8.13802083e-11 8.13802083e-11]]

    [[2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [8.13802083e-11 6.66666667e-01 3.33333333e-01 8.13802083e-11]
     [5.00000000e-01 1.25000000e-01 1.25000000e-01 2.50000000e-01]
     [2.50000000e-01 2.50000000e-01 6.10351562e-11 5.00000000e-01]]

    [[5.00000000e-01 3.33333333e-01 4.06901042e-11 1.66666667e-01]
     [1.22070312e-10 5.00000000e-01 5.00000000e-01 1.22070312e-10]
     [3.33333333e-01 8.13802083e-11 6.66666667e-01 8.13802083e-11]
     [1.22070312e-10 1.22070312e-10 5.00000000e-01 5.00000000e-01]]

    [[6.10351562e-11 1.00000000e+00 6.10351562e-11 6.10351562e-11]
     [1.22070312e-10 5.00000000e-01 1.22070312e-10 5.00000000e-01]
     [6.66666667e-01 8.13802083e-11 8.13802083e-11 3.33333333e-01]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]]]]



  [[[[6.00000000e-01 2.00000000e-01 1.00000000e-01 1.00000000e-01]
     [4.88281250e-11 4.88281250e-11 2.00000000e-01 8.00000000e-01]
     [5.00000000e-01 2.50000000e-01 6.10351562e-11 2.50000000e-01]
     [9.99999999e-01 2.44140625e-10 2.44140625e-10 2.44140625e-10]]

    [[6.66666667e-01 3.33333333e-01 8.13802083e-11 8.13802083e-11]
     [2.00000000e-01 4.88281250e-11 6.00000000e-01 2.00000000e-01]
     [5.71428571e-01 3.48772321e-11 1.42857143e-01 2.85714286e-01]
     [1.22070312e-10 5.00000000e-01 1.22070312e-10 5.00000000e-01]]

    [[8.00000000e-01 2.00000000e-01 4.88281250e-11 4.88281250e-11]
     [2.44140625e-10 9.99999999e-01 2.44140625e-10 2.44140625e-10]
     [1.66666667e-01 4.06901042e-11 3.33333333e-01 5.00000000e-01]
     [8.33333333e-01 4.06901042e-11 4.06901042e-11 1.66666667e-01]]

    [[2.00000000e-01 2.00000000e-01 6.00000000e-01 4.88281250e-11]
     [8.13802083e-11 3.33333333e-01 6.66666667e-01 8.13802083e-11]
     [6.10351562e-11 2.50000000e-01 6.10351562e-11 7.50000000e-01]
     [1.22070312e-10 5.00000000e-01 5.00000000e-01 1.22070312e-10]]]


   [[[2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [1.22070312e-10 5.00000000e-01 5.00000000e-01 1.22070312e-10]]

    [[3.33333333e-01 8.13802083e-11 8.13802083e-11 6.66666667e-01]
     [1.25000000e-01 5.00000000e-01 3.75000000e-01 3.05175781e-11]
     [3.75000000e-01 3.75000000e-01 3.05175781e-11 2.50000000e-01]
     [3.33333333e-01 1.66666667e-01 1.66666667e-01 3.33333333e-01]]

    [[5.33333333e-01 2.00000000e-01 1.33333333e-01 1.33333333e-01]
     [2.22222222e-01 5.55555556e-01 2.71267361e-11 2.22222222e-01]
     [8.13802083e-11 3.33333333e-01 3.33333333e-01 3.33333333e-01]
     [1.42857143e-01 4.28571429e-01 2.85714286e-01 1.42857143e-01]]

    [[2.50000000e-01 2.50000000e-01 5.00000000e-01 6.10351562e-11]
     [6.10351562e-11 5.00000000e-01 2.50000000e-01 2.50000000e-01]
     [1.22070312e-10 5.00000000e-01 1.22070312e-10 5.00000000e-01]
     [1.22070312e-10 1.22070312e-10 1.00000000e+00 1.22070312e-10]]]


   [[[3.33333333e-01 6.66666667e-01 8.13802083e-11 8.13802083e-11]
     [8.13802083e-11 3.33333333e-01 6.66666667e-01 8.13802083e-11]
     [9.99999999e-01 2.44140625e-10 2.44140625e-10 2.44140625e-10]
     [2.50000000e-01 2.50000000e-01 5.00000000e-01 6.10351562e-11]]

    [[2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [3.48772321e-11 4.28571429e-01 2.85714286e-01 2.85714286e-01]
     [2.44140625e-10 2.44140625e-10 2.44140625e-10 9.99999999e-01]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]]

    [[5.71428571e-01 3.48772321e-11 4.28571429e-01 3.48772321e-11]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [8.13802083e-11 8.13802083e-11 3.33333333e-01 6.66666667e-01]
     [2.85714286e-01 4.28571429e-01 2.85714286e-01 3.48772321e-11]]

    [[9.99999999e-01 2.44140625e-10 2.44140625e-10 2.44140625e-10]
     [6.10351562e-11 2.50000000e-01 7.50000000e-01 6.10351562e-11]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [2.00000000e-01 2.00000000e-01 4.00000000e-01 2.00000000e-01]]]


   [[[6.10351562e-11 5.00000000e-01 6.10351562e-11 5.00000000e-01]
     [4.88281250e-11 8.00000000e-01 2.00000000e-01 4.88281250e-11]
     [1.22070312e-10 5.00000000e-01 1.22070312e-10 5.00000000e-01]
     [9.99999999e-01 2.44140625e-10 2.44140625e-10 2.44140625e-10]]

    [[3.33333333e-01 3.33333333e-01 3.33333333e-01 8.13802083e-11]
     [1.66666667e-01 5.00000000e-01 4.06901042e-11 3.33333333e-01]
     [3.33333333e-01 6.66666667e-01 8.13802083e-11 8.13802083e-11]
     [2.00000000e-01 4.88281250e-11 4.88281250e-11 8.00000000e-01]]

    [[6.10351562e-11 7.50000000e-01 2.50000000e-01 6.10351562e-11]
     [6.10351562e-11 5.00000000e-01 6.10351562e-11 5.00000000e-01]
     [5.00000000e-01 6.10351562e-11 2.50000000e-01 2.50000000e-01]
     [8.13802083e-11 6.66666667e-01 8.13802083e-11 3.33333333e-01]]

    [[6.66666667e-01 8.13802083e-11 8.13802083e-11 3.33333333e-01]
     [2.44140625e-10 2.44140625e-10 9.99999999e-01 2.44140625e-10]
     [2.50000000e-01 6.10351562e-11 2.50000000e-01 5.00000000e-01]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]]]]]




 [[[[[5.45454545e-01 9.09090909e-02 2.72727273e-01 9.09090909e-02]
     [2.00000000e-01 4.00000000e-01 2.00000000e-01 2.00000000e-01]
     [4.16666667e-01 1.66666667e-01 1.66666667e-01 2.50000000e-01]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]]

    [[2.00000000e-01 4.88281250e-11 2.00000000e-01 6.00000000e-01]
     [4.06901042e-11 8.33333333e-01 4.06901042e-11 1.66666667e-01]
     [3.33333333e-01 3.33333333e-01 4.06901042e-11 3.33333333e-01]
     [6.10351562e-11 7.50000000e-01 6.10351562e-11 2.50000000e-01]]

    [[5.00000000e-01 1.22070312e-10 1.22070312e-10 5.00000000e-01]
     [2.44140625e-10 2.44140625e-10 9.99999999e-01 2.44140625e-10]
     [5.00000000e-01 1.66666667e-01 4.06901042e-11 3.33333333e-01]
     [3.12500000e-01 3.75000000e-01 1.25000000e-01 1.87500000e-01]]

    [[2.50000000e-01 2.50000000e-01 5.00000000e-01 3.05175781e-11]
     [8.13802083e-11 3.33333333e-01 3.33333333e-01 3.33333333e-01]
     [6.10351562e-11 6.10351562e-11 5.00000000e-01 5.00000000e-01]
     [2.44140625e-10 9.99999999e-01 2.44140625e-10 2.44140625e-10]]]


   [[[1.42857143e-01 1.42857143e-01 7.14285714e-01 3.48772321e-11]
     [1.00000000e-01 3.00000000e-01 5.00000000e-01 1.00000000e-01]
     [8.13802083e-11 6.66666667e-01 3.33333333e-01 8.13802083e-11]
     [2.44140625e-10 9.99999999e-01 2.44140625e-10 2.44140625e-10]]

    [[8.13802083e-11 6.66666667e-01 8.13802083e-11 3.33333333e-01]
     [9.09090909e-02 6.36363636e-01 1.81818182e-01 9.09090909e-02]
     [7.50000000e-01 6.10351562e-11 2.50000000e-01 6.10351562e-11]
     [2.00000000e-01 2.00000000e-01 4.00000000e-01 2.00000000e-01]]

    [[4.00000000e-01 2.00000000e-01 4.00000000e-01 4.88281250e-11]
     [3.33333333e-01 8.13802083e-11 6.66666667e-01 8.13802083e-11]
     [2.44140625e-10 2.44140625e-10 2.44140625e-10 9.99999999e-01]
     [1.22070312e-10 5.00000000e-01 5.00000000e-01 1.22070312e-10]]

    [[4.00000000e-01 4.00000000e-01 1.00000000e-01 1.00000000e-01]
     [1.11111111e-01 1.11111111e-01 4.44444444e-01 3.33333333e-01]
     [1.42857143e-01 3.48772321e-11 4.28571429e-01 4.28571429e-01]
     [4.00000000e-01 2.00000000e-01 2.00000000e-01 2.00000000e-01]]]


   [[[6.25000000e-01 3.75000000e-01 3.05175781e-11 3.05175781e-11]
     [1.42857143e-01 3.48772321e-11 2.85714286e-01 5.71428571e-01]
     [9.99999999e-01 2.44140625e-10 2.44140625e-10 2.44140625e-10]
     [9.99999999e-01 2.44140625e-10 2.44140625e-10 2.44140625e-10]]

    [[6.10351562e-11 7.50000000e-01 2.50000000e-01 6.10351562e-11]
     [6.10351562e-11 2.50000000e-01 2.50000000e-01 5.00000000e-01]
     [2.50000000e-01 6.10351562e-11 2.50000000e-01 5.00000000e-01]
     [2.44140625e-10 2.44140625e-10 9.99999999e-01 2.44140625e-10]]

    [[1.22070312e-10 1.00000000e+00 1.22070312e-10 1.22070312e-10]
     [6.10351562e-11 2.50000000e-01 5.00000000e-01 2.50000000e-01]
     [2.44140625e-10 9.99999999e-01 2.44140625e-10 2.44140625e-10]
     [4.06901042e-11 3.33333333e-01 3.33333333e-01 3.33333333e-01]]

    [[3.33333333e-01 3.33333333e-01 1.11111111e-01 2.22222222e-01]
     [2.21946023e-11 1.81818182e-01 8.18181818e-01 2.21946023e-11]
     [2.00000000e-01 4.88281250e-11 4.00000000e-01 4.00000000e-01]
     [1.66666667e-01 3.33333333e-01 3.33333333e-01 1.66666667e-01]]]


   [[[1.00000000e+00 1.22070312e-10 1.22070312e-10 1.22070312e-10]
     [8.13802083e-11 3.33333333e-01 3.33333333e-01 3.33333333e-01]
     [3.33333333e-01 4.06901042e-11 3.33333333e-01 3.33333333e-01]
     [3.33333333e-01 3.33333333e-01 8.13802083e-11 3.33333333e-01]]

    [[1.22070312e-10 1.22070312e-10 1.00000000e+00 1.22070312e-10]
     [2.00000000e-01 4.00000000e-01 4.88281250e-11 4.00000000e-01]
     [8.13802083e-11 8.13802083e-11 8.13802083e-11 1.00000000e+00]
     [2.44140625e-10 2.44140625e-10 9.99999999e-01 2.44140625e-10]]

    [[8.13802083e-11 8.13802083e-11 6.66666667e-01 3.33333333e-01]
     [1.22070312e-10 5.00000000e-01 5.00000000e-01 1.22070312e-10]
     [9.99999999e-01 2.44140625e-10 2.44140625e-10 2.44140625e-10]
     [2.44140625e-10 2.44140625e-10 9.99999999e-01 2.44140625e-10]]

    [[8.13802083e-11 3.33333333e-01 3.33333333e-01 3.33333333e-01]
     [8.13802083e-11 3.33333333e-01 3.33333333e-01 3.33333333e-01]
     [5.00000000e-01 6.10351562e-11 2.50000000e-01 2.50000000e-01]
     [1.00000000e+00 1.22070312e-10 1.22070312e-10 1.22070312e-10]]]]



  [[[[4.28571429e-01 4.28571429e-01 1.42857143e-01 3.48772321e-11]
     [2.44140625e-10 2.44140625e-10 9.99999999e-01 2.44140625e-10]
     [9.99999999e-01 2.44140625e-10 2.44140625e-10 2.44140625e-10]
     [6.66666667e-01 8.13802083e-11 3.33333333e-01 8.13802083e-11]]

    [[2.44140625e-10 2.44140625e-10 9.99999999e-01 2.44140625e-10]
     [2.50000000e-01 3.75000000e-01 1.25000000e-01 2.50000000e-01]
     [1.22070312e-10 5.00000000e-01 1.22070312e-10 5.00000000e-01]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]]

    [[2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [2.44140625e-10 9.99999999e-01 2.44140625e-10 2.44140625e-10]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [6.10351562e-11 2.50000000e-01 5.00000000e-01 2.50000000e-01]]

    [[2.50000000e-01 2.50000000e-01 5.00000000e-01 6.10351562e-11]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [1.22070312e-10 5.00000000e-01 5.00000000e-01 1.22070312e-10]
     [9.99999999e-01 2.44140625e-10 2.44140625e-10 2.44140625e-10]]]


   [[[1.00000000e+00 6.10351562e-11 6.10351562e-11 6.10351562e-11]
     [3.33333333e-01 8.13802083e-11 3.33333333e-01 3.33333333e-01]
     [3.33333333e-01 8.13802083e-11 3.33333333e-01 3.33333333e-01]
     [3.33333333e-01 3.33333333e-01 8.13802083e-11 3.33333333e-01]]

    [[3.33333333e-01 6.66666667e-01 8.13802083e-11 8.13802083e-11]
     [4.16666667e-01 4.16666667e-01 1.66666667e-01 2.03450521e-11]
     [4.06901042e-11 6.66666667e-01 1.66666667e-01 1.66666667e-01]
     [8.33333333e-01 1.66666667e-01 4.06901042e-11 4.06901042e-11]]

    [[4.28571429e-01 2.85714286e-01 3.48772321e-11 2.85714286e-01]
     [4.88281250e-11 2.00000000e-01 2.00000000e-01 6.00000000e-01]
     [1.00000000e+00 8.13802083e-11 8.13802083e-11 8.13802083e-11]
     [5.00000000e-01 1.22070312e-10 5.00000000e-01 1.22070312e-10]]

    [[3.33333333e-01 4.44444444e-01 1.11111111e-01 1.11111111e-01]
     [9.09090909e-02 3.63636364e-01 1.81818182e-01 3.63636364e-01]
     [2.50000000e-01 5.00000000e-01 2.50000000e-01 6.10351562e-11]
     [6.10351562e-11 7.50000000e-01 2.50000000e-01 6.10351562e-11]]]


   [[[5.45454545e-01 2.72727273e-01 2.21946023e-11 1.81818182e-01]
     [2.50000000e-01 1.25000000e-01 2.50000000e-01 3.75000000e-01]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [5.00000000e-01 2.50000000e-01 2.50000000e-01 6.10351562e-11]]

    [[2.22222222e-01 6.66666667e-01 2.71267361e-11 1.11111111e-01]
     [2.00000000e-01 4.00000000e-01 2.00000000e-01 2.00000000e-01]
     [8.13802083e-11 6.66666667e-01 3.33333333e-01 8.13802083e-11]
     [6.66666667e-01 8.13802083e-11 3.33333333e-01 8.13802083e-11]]

    [[8.13802083e-11 1.00000000e+00 8.13802083e-11 8.13802083e-11]
     [2.50000000e-01 5.00000000e-01 2.50000000e-01 6.10351562e-11]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]]

    [[1.00000000e+00 1.22070312e-10 1.22070312e-10 1.22070312e-10]
     [3.48772321e-11 4.28571429e-01 4.28571429e-01 1.42857143e-01]
     [3.33333333e-01 3.33333333e-01 3.33333333e-01 8.13802083e-11]
     [7.50000000e-01 6.10351562e-11 2.50000000e-01 6.10351562e-11]]]


   [[[2.71267361e-11 5.55555556e-01 1.11111111e-01 3.33333333e-01]
     [4.00000000e-01 2.00000000e-01 4.88281250e-11 4.00000000e-01]
     [8.13802083e-11 8.13802083e-11 6.66666667e-01 3.33333333e-01]
     [1.22070312e-10 5.00000000e-01 1.22070312e-10 5.00000000e-01]]

    [[4.00000000e-01 4.00000000e-01 2.00000000e-01 4.88281250e-11]
     [1.42857143e-01 2.85714286e-01 2.85714286e-01 2.85714286e-01]
     [5.00000000e-01 5.00000000e-01 1.22070312e-10 1.22070312e-10]
     [3.33333333e-01 3.33333333e-01 3.33333333e-01 8.13802083e-11]]

    [[3.33333333e-01 6.66666667e-01 8.13802083e-11 8.13802083e-11]
     [8.13802083e-11 1.00000000e+00 8.13802083e-11 8.13802083e-11]
     [3.33333333e-01 8.13802083e-11 8.13802083e-11 6.66666667e-01]
     [1.22070312e-10 1.22070312e-10 1.00000000e+00 1.22070312e-10]]

    [[6.10351562e-11 5.00000000e-01 6.10351562e-11 5.00000000e-01]
     [1.66666667e-01 6.66666667e-01 4.06901042e-11 1.66666667e-01]
     [3.33333333e-01 3.33333333e-01 8.13802083e-11 3.33333333e-01]
     [6.10351562e-11 6.10351562e-11 2.50000000e-01 7.50000000e-01]]]]



  [[[[1.00000000e+00 6.10351562e-11 6.10351562e-11 6.10351562e-11]
     [6.66666667e-01 3.33333333e-01 8.13802083e-11 8.13802083e-11]
     [2.44140625e-10 2.44140625e-10 9.99999999e-01 2.44140625e-10]
     [3.33333333e-01 8.13802083e-11 6.66666667e-01 8.13802083e-11]]

    [[1.42857143e-01 2.85714286e-01 4.28571429e-01 1.42857143e-01]
     [2.00000000e-01 6.00000000e-01 2.00000000e-01 4.88281250e-11]
     [4.88281250e-11 4.00000000e-01 2.00000000e-01 4.00000000e-01]
     [1.66666667e-01 1.66666667e-01 3.33333333e-01 3.33333333e-01]]

    [[5.00000000e-01 6.10351562e-11 2.50000000e-01 2.50000000e-01]
     [2.44140625e-10 2.44140625e-10 2.44140625e-10 9.99999999e-01]
     [5.00000000e-01 5.00000000e-01 1.22070312e-10 1.22070312e-10]
     [3.33333333e-01 3.33333333e-01 3.33333333e-01 8.13802083e-11]]

    [[2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [2.44140625e-10 2.44140625e-10 2.44140625e-10 9.99999999e-01]
     [6.10351562e-11 2.50000000e-01 6.10351562e-11 7.50000000e-01]
     [5.00000000e-01 1.22070312e-10 5.00000000e-01 1.22070312e-10]]]


   [[[4.00000000e-01 2.00000000e-01 4.00000000e-01 4.88281250e-11]
     [8.13802083e-11 1.00000000e+00 8.13802083e-11 8.13802083e-11]
     [5.00000000e-01 1.22070312e-10 1.22070312e-10 5.00000000e-01]
     [2.44140625e-10 9.99999999e-01 2.44140625e-10 2.44140625e-10]]

    [[1.22070312e-10 1.00000000e+00 1.22070312e-10 1.22070312e-10]
     [1.42857143e-01 7.85714286e-01 1.74386161e-11 7.14285714e-02]
     [6.00000000e-01 2.00000000e-01 4.88281250e-11 2.00000000e-01]
     [5.00000000e-01 6.10351562e-11 6.10351562e-11 5.00000000e-01]]

    [[5.00000000e-01 1.22070312e-10 5.00000000e-01 1.22070312e-10]
     [2.50000000e-01 5.00000000e-01 2.50000000e-01 6.10351562e-11]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [3.33333333e-01 8.13802083e-11 3.33333333e-01 3.33333333e-01]]

    [[1.00000000e+00 1.22070312e-10 1.22070312e-10 1.22070312e-10]
     [6.66666667e-01 8.13802083e-11 3.33333333e-01 8.13802083e-11]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [4.88281250e-11 2.00000000e-01 6.00000000e-01 2.00000000e-01]]]


   [[[8.13802083e-11 3.33333333e-01 3.33333333e-01 3.33333333e-01]
     [4.88281250e-11 2.00000000e-01 2.00000000e-01 6.00000000e-01]
     [2.44140625e-10 9.99999999e-01 2.44140625e-10 2.44140625e-10]
     [8.13802083e-11 6.66666667e-01 3.33333333e-01 8.13802083e-11]]

    [[5.00000000e-01 5.00000000e-01 4.06901042e-11 4.06901042e-11]
     [2.00000000e-01 4.00000000e-01 2.00000000e-01 2.00000000e-01]
     [9.99999999e-01 2.44140625e-10 2.44140625e-10 2.44140625e-10]
     [8.13802083e-11 6.66666667e-01 8.13802083e-11 3.33333333e-01]]

    [[4.28571429e-01 2.85714286e-01 1.42857143e-01 1.42857143e-01]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [3.33333333e-01 8.13802083e-11 8.13802083e-11 6.66666667e-01]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]]

    [[6.66666667e-01 3.33333333e-01 8.13802083e-11 8.13802083e-11]
     [2.44140625e-10 2.44140625e-10 9.99999999e-01 2.44140625e-10]
     [1.22070312e-10 1.00000000e+00 1.22070312e-10 1.22070312e-10]
     [8.13802083e-11 1.00000000e+00 8.13802083e-11 8.13802083e-11]]]


   [[[8.13802083e-11 6.66666667e-01 8.13802083e-11 3.33333333e-01]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [8.13802083e-11 3.33333333e-01 8.13802083e-11 6.66666667e-01]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]]

    [[2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [1.53846154e-01 3.07692308e-01 3.07692308e-01 2.30769231e-01]
     [1.25000000e-01 2.50000000e-01 3.05175781e-11 6.25000000e-01]
     [6.10351562e-11 2.50000000e-01 6.10351562e-11 7.50000000e-01]]

    [[9.99999999e-01 2.44140625e-10 2.44140625e-10 2.44140625e-10]
     [6.66666667e-01 3.33333333e-01 8.13802083e-11 8.13802083e-11]
     [8.13802083e-11 3.33333333e-01 8.13802083e-11 6.66666667e-01]
     [1.22070312e-10 5.00000000e-01 1.22070312e-10 5.00000000e-01]]

    [[1.00000000e+00 1.22070312e-10 1.22070312e-10 1.22070312e-10]
     [4.88281250e-11 6.00000000e-01 4.88281250e-11 4.00000000e-01]
     [1.22070312e-10 1.22070312e-10 5.00000000e-01 5.00000000e-01]
     [5.00000000e-01 1.22070312e-10 5.00000000e-01 1.22070312e-10]]]]



  [[[[5.00000000e-01 1.22070312e-10 5.00000000e-01 1.22070312e-10]
     [1.25000000e-01 3.05175781e-11 5.00000000e-01 3.75000000e-01]
     [1.66666667e-01 3.33333333e-01 4.06901042e-11 5.00000000e-01]
     [5.00000000e-01 1.22070312e-10 1.22070312e-10 5.00000000e-01]]

    [[2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [1.22070312e-10 5.00000000e-01 1.22070312e-10 5.00000000e-01]
     [5.00000000e-01 1.22070312e-10 1.22070312e-10 5.00000000e-01]
     [3.33333333e-01 1.11111111e-01 3.33333333e-01 2.22222222e-01]]

    [[6.66666667e-01 8.13802083e-11 8.13802083e-11 3.33333333e-01]
     [2.44140625e-10 2.44140625e-10 2.44140625e-10 9.99999999e-01]
     [5.00000000e-01 1.22070312e-10 1.22070312e-10 5.00000000e-01]
     [4.88281250e-11 4.00000000e-01 4.00000000e-01 2.00000000e-01]]

    [[2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [2.44140625e-10 2.44140625e-10 9.99999999e-01 2.44140625e-10]
     [8.13802083e-11 8.13802083e-11 3.33333333e-01 6.66666667e-01]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]]]


   [[[5.00000000e-01 2.50000000e-01 2.50000000e-01 6.10351562e-11]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [2.44140625e-10 2.44140625e-10 2.44140625e-10 9.99999999e-01]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]]

    [[3.33333333e-01 6.66666667e-01 8.13802083e-11 8.13802083e-11]
     [1.22070312e-10 5.00000000e-01 1.22070312e-10 5.00000000e-01]
     [1.22070312e-10 1.22070312e-10 5.00000000e-01 5.00000000e-01]
     [4.00000000e-01 4.00000000e-01 2.00000000e-01 4.88281250e-11]]

    [[4.00000000e-01 4.88281250e-11 4.00000000e-01 2.00000000e-01]
     [1.22070312e-10 1.00000000e+00 1.22070312e-10 1.22070312e-10]
     [1.66666667e-01 1.66666667e-01 1.66666667e-01 5.00000000e-01]
     [6.10351562e-11 5.00000000e-01 6.10351562e-11 5.00000000e-01]]

    [[3.33333333e-01 6.66666667e-01 8.13802083e-11 8.13802083e-11]
     [1.22070312e-10 1.00000000e+00 1.22070312e-10 1.22070312e-10]
     [2.00000000e-01 4.00000000e-01 2.00000000e-01 2.00000000e-01]
     [6.10351562e-11 6.10351562e-11 6.10351562e-11 1.00000000e+00]]]


   [[[3.33333333e-01 8.13802083e-11 6.66666667e-01 8.13802083e-11]
     [1.11111111e-01 2.22222222e-01 4.44444444e-01 2.22222222e-01]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [9.99999999e-01 2.44140625e-10 2.44140625e-10 2.44140625e-10]]

    [[2.44140625e-10 9.99999999e-01 2.44140625e-10 2.44140625e-10]
     [1.42857143e-01 5.71428571e-01 3.48772321e-11 2.85714286e-01]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [2.00000000e-01 4.00000000e-01 2.00000000e-01 2.00000000e-01]]

    [[3.33333333e-01 6.66666667e-01 8.13802083e-11 8.13802083e-11]
     [3.33333333e-01 6.66666667e-01 8.13802083e-11 8.13802083e-11]
     [1.22070312e-10 1.22070312e-10 1.22070312e-10 1.00000000e+00]
     [4.00000000e-01 2.00000000e-01 2.00000000e-01 2.00000000e-01]]

    [[5.00000000e-01 1.66666667e-01 3.33333333e-01 4.06901042e-11]
     [6.10351562e-11 2.50000000e-01 7.50000000e-01 6.10351562e-11]
     [2.50000000e-01 3.05175781e-11 3.75000000e-01 3.75000000e-01]
     [4.00000000e-01 2.00000000e-01 4.00000000e-01 4.88281250e-11]]]


   [[[2.44140625e-10 9.99999999e-01 2.44140625e-10 2.44140625e-10]
     [5.00000000e-01 2.50000000e-01 6.10351562e-11 2.50000000e-01]
     [5.00000000e-01 1.22070312e-10 1.22070312e-10 5.00000000e-01]
     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]]

    [[2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
     [4.88281250e-11 4.00000000e-01 4.88281250e-11 6.00000000e-01]
     [6.66666667e-01 8.13802083e-11 8.13802083e-11 3.33333333e-01]
     [2.44140625e-10 2.44140625e-10 2.44140625e-10 9.99999999e-01]]

    [[5.00000000e-01 6.10351562e-11 6.10351562e-11 5.00000000e-01]
     [1.22070312e-10 1.22070312e-10 1.22070312e-10 1.00000000e+00]
     [2.44140625e-10 2.44140625e-10 9.99999999e-01 2.44140625e-10]
     [3.33333333e-01 6.66666667e-01 8.13802083e-11 8.13802083e-11]]

    [[3.33333333e-01 6.66666667e-01 8.13802083e-11 8.13802083e-11]
     [8.13802083e-11 6.66666667e-01 8.13802083e-11 3.33333333e-01]
     [3.33333333e-01 8.13802083e-11 8.13802083e-11 6.66666667e-01]
     [3.33333333e-01 3.33333333e-01 3.33333333e-01 8.13802083e-11]]]]]]

AIC = 30287748.393798966	BIC = 30313736.9217434
  
Out[20]:
Text(0.5, 0, 'Order of model')

Выводы¶

Реальные данные, представленные геномом бактерии лучше всего описываются Марковской моделью второго порядка, т.е. на букву в каждой позиции влияют буквы в двух предыдущих позициях. Это в целом соотносится с триплетностью генетического кода, т.к. употребление букв внутри триплета обуславливается частотатами кодируемых аминокислот и предпочтением определённых кодонов, вызванным неодинаковой представленностью синонимичных тРНК, а также влиянием используемых триплетов на устойчивость мРНК. При этом четвёртая буква уже будет относиться к следующему кодону и не так сильно зависть от предыдущего кодона. Конечно для аминокислотных последовательностей (а следовательно и для кодонов) тоже имеются предпочтения в использовании, однако пологаю, что влияние этого явления будет меньше. Также стоит отметить, что помимо последовательностей кодирующих белок в геноме имеются участки, к которым применима другая логика. В частности для генома бктерий верно избегание CpG, сайтов эндонуклеаз рестрикции, наличие последовательности Шайна — Дальгарно. Поэтому глобально описать геном бактерии одной Марковской цепью фиксированного порядка не удастся.

Задача 2.4 Эргодичность*¶

Задача 2.4.1 Праметры в предположении эргодичности¶

К сожалению,трёх семестров математического анализа, одного семестра дифференциальных уравнений, одного семестра линейной алгебры, одного семестра комбинаторики и одного семестра теории вероятностей на ФББ недостаточно, чтобы понять хотя бы малую часть биоалгоритмов. На сколько я понял в предположении эргодичности нужно оценить матрицу переходов k-того порядка по количеству (k+1)-меров и k-меров во всём наборе данных, в то время как остальные матрицы переходов более низких порядков получаются из неё в силу определённых предельных соотношений и должны являться соответствующими собственными векторами полученного тензора. Но я так и не понял, как это решается математически, а тем паче не смог реализовать в коде. Посему в предположении эргодичности все тензоры переходов оцениваю по частотам букв на всём протяжении каждой строки, в то время как без предположения эргодичности оценка проводилась только по первым k буквам каждой строки.

Статья про эргодичность:

Fasino, Dario & Tudisco, Francesco. (2020). Ergodicity coefficients for higher-order stochastic processes.

Статья про поиск собственных векторов (вообще ничего не понял):

Benson, Austin & Gleich, David. (2019). Computing Tensor $Z$-Eigenvectors with Dynamical Systems. SIAM Journal on Matrix Analysis and Applications. 40. 1311-1324. 10.1137/18M1229584.

In [15]:
print("Real transition matrices of 1-order Markov chain model")
alphabet = "ab"
print_Ps(real_Ps)
AICs, BICs = [], []
orders = [_ for _ in range(4)]
for order in orders:
  estimation = MarkovChain(alphabet, order)
  estimated_Ps = estimation.fit(seqs, erg=True)
  AIC = estimation.AIC()
  BIC = estimation.BIC()
  AICs.append(AIC)
  BICs.append(BIC)
  print(f"""
{order}-order Markov chain model with ergodicity

Estimated transition matrices""")
  print_Ps(estimated_Ps)
  print(f"""
AIC = {AIC}\tBIC = {BIC}
  """)
fig, (ax1, ax2) = plt.subplots(nrows=1, ncols=2, figsize = (12,3), tight_layout = True)
ax1.plot(orders, AICs)
ax1.title.set_text("AIC")
ax1.set_xlabel("Order of model")
ax2.plot(orders, BICs)
ax2.title.set_text("BIC")
ax2.set_xlabel("Order of model")
Real transition matrices of 1-order Markov chain model

P0=
[0.86605874 0.13394126]

P1=
[[0.14906449 0.85093551]
 [0.52586667 0.47413333]]

0-order Markov chain model with ergodicity

Estimated transition matrices

P0=
[0.38195333 0.61771333]

AIC = 1996218.4365915784	BIC = 1996223.3443468574
  

1-order Markov chain model with ergodicity

Estimated transition matrices

P0=
[0.38195333 0.61771333]

P1=
[[0.14907929 0.85059955]
 [0.52567912 0.47397984]]

AIC = 1766562.3018151168	BIC = 1766572.1173256747
  

2-order Markov chain model with ergodicity

Estimated transition matrices

P0=
[0.38195333 0.61771333]

P1=
[[0.14907929 0.85059955]
 [0.52567912 0.47397984]]

P2=
[[[0.14904229 0.85068843]
  [0.52562308 0.47405578]]

 [[0.14907556 0.8505939 ]
  [0.52573629 0.47390052]]]

AIC = 1766566.6025238943	BIC = 1766586.2335450102
  

3-order Markov chain model with ergodicity

Estimated transition matrices

P0=
[0.38195333 0.61771333]

P1=
[[0.14907929 0.85059955]
 [0.52567912 0.47397984]]

P2=
[[[0.14904229 0.85068843]
  [0.52562308 0.47405578]]

 [[0.14907556 0.8505939 ]
  [0.52573629 0.47390052]]]

P3=
[[[[0.14697565 0.85271013]
   [0.52688586 0.47281135]]

  [[0.1489885  0.85070309]
   [0.52589022 0.473757  ]]]


 [[[0.14939782 0.85034051]
   [0.52540755 0.47426781]]

  [[0.14917244 0.85047241]
   [0.52557471 0.47405051]]]]

AIC = 1766573.802805225	BIC = 1766613.0648474568
  
Out[15]:
Text(0.5, 0, 'Order of model')

Выводы¶

Если оценивать параметры модели вдоль всей последовательности, а не только в начале (надеюсь, что это правильное понимание эргодичности), то качество моделей повышается. Формально лучше всего данные всё ещё описываются моделью первого порядка, однако модели более высоких порядков всё ещё хороши и переобучение ненаблюдается. Бернуллиевская модель всё ещё не достаточна для описания данных.

Задача 2.4.2 Проверка эргодичности¶

In [16]:
def make_suseqs(seqs, min_length=6): #функция нарезки подпоследовательностей
  subseqs = []
  for seq in seqs:
    start = np.random.randint(len(seq)-min_length)
    subseq = seq[start:]
    subseqs.append(subseq)
  return subseqs
In [17]:
AICs, BICs = [], []
orders = [_ for _ in range(6)]
alphabet = "ATGC"
for order in orders:
  print(f"Ergodicity for {order}-order Markov chain")

  # из-за ограничения полной вероятности, количество независимых вероятностей меньше
  # (вероятность одной буквы из алфавита выражается через вероятности остальных букв),
  # поэтому оставим для тестирования только независимые параметры
  full_P = []
  for param in full_est_Ps[order]:
    param = np.atleast_2d(np.stack(param, axis=-1))
    for tup in np.ndindex((*param.shape[:-1],param.shape[-1]-1)):
      tup1=(*tup[:-1],tup[-1])
      full_P.append(param[tup1])

  Pss = [[] for _ in range(order+1)]
  for i in range(20):
    B_subseqs = make_suseqs(B_seqs)
    estimation = MarkovChain(alphabet, order)
    estimated_Ps = estimation.fit(B_subseqs, erg=False)
    for est_P, all_P in zip(estimated_Ps, Pss):
      all_P.append(est_P)

  # оставим для тестирования только независимые параметры
  samples = []
  for param in Pss:
    param = np.stack(param, axis=-1)
    for tup in np.ndindex((*param.shape[:-2],param.shape[-2]-1)):
      tup1=(*tup,slice(None))
      samples.append(param[tup1])

  pvalues = []
  for sample, p in zip(samples, full_P):
    statistic, pvalue = ss.ttest_1samp(sample, p)
    pvalues.append(pvalue)

  reject, pvalues_corr, alphacSidak, alphacBonf = smm.multipletests(pvalues, alpha=0.05, method="bonferroni")
  print(f"Does any probability differ?\t{(reject == True).any()}")
Ergodicity for 0-order Markov chain
Does any probability differ?	True
Ergodicity for 1-order Markov chain
Does any probability differ?	True
Ergodicity for 2-order Markov chain
Does any probability differ?	True
Ergodicity for 3-order Markov chain
Does any probability differ?	True
Ergodicity for 4-order Markov chain
Does any probability differ?	True
Ergodicity for 5-order Markov chain
Does any probability differ?	True

Выводы¶

Во первых я не уверен в корректности применения t-теста, хотя бы потому, что данный тест предполагает нормальность данность, а распределение вероятностей точно не нормально, т.к. $0 \le p \le 1$. Так же оценка вероятностей производилась по доле соответствующих k-меров в наборе последовательностей. Для них можно было бы применить одновыборочный t-тест (или даже z-тест) пропорций, однако тогда не ясно зачем проводить 20 итераций и как делать поправку на множественное тестирование. Поэтому был поведён t-тест на равенство среднего значения параметра в 20 повторах значению соответсвующего параметра в оценки из задачи 2.2. При этом было учтено, что из всех $\sum_{i=1}^{order+1}|A|^i$ параметров модели независимыми являются только $|A|^{order+1}-1$. Также была применена поправка на множественное тестирование гипотез (FWER: Бонферрони), чтобы отвергнуть как можно меньшее число нулевых гипотез, тем не менее доказать эргодичность не удалось. На реальных данных модель любого порядка даёт различные параметры для подпоследовательностей и исходных последовательностей. Это грустно, но, возможно, что так и должно быть.