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]]

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

    [[2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]
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     [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]
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     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]]

    [[3.33333333e-01 3.33333333e-01 1.66666667e-01 1.66666667e-01]
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     [2.44140625e-10 9.99999999e-01 2.44140625e-10 2.44140625e-10]]]


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

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

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

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


   [[[2.00000000e-01 4.00000000e-01 2.00000000e-01 2.00000000e-01]
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    [[9.99999999e-01 2.44140625e-10 2.44140625e-10 2.44140625e-10]
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     [5.00000000e-01 1.22070312e-10 5.00000000e-01 1.22070312e-10]]

    [[1.22070312e-10 5.00000000e-01 5.00000000e-01 1.22070312e-10]
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     [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]
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     [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]
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     [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]
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     [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]
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     [1.25000000e-01 3.75000000e-01 3.75000000e-01 1.25000000e-01]]

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

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



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


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



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


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     [2.00000000e-01 6.00000000e-01 2.00000000e-01 4.88281250e-11]]]


   [[[2.44140625e-10 2.44140625e-10 9.99999999e-01 2.44140625e-10]
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    [[7.50000000e-01 1.25000000e-01 3.05175781e-11 1.25000000e-01]
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    [[2.00000000e-01 4.88281250e-11 4.00000000e-01 4.00000000e-01]
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   [[[4.06901042e-11 3.33333333e-01 3.33333333e-01 3.33333333e-01]
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    [[1.66666667e-01 4.06901042e-11 3.33333333e-01 5.00000000e-01]
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   [[[2.50000000e-01 6.10351562e-11 7.50000000e-01 6.10351562e-11]
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     [2.50000000e-01 2.50000000e-01 2.50000000e-01 2.50000000e-01]]]


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

    [[3.33333333e-01 3.33333333e-01 1.66666667e-01 1.66666667e-01]
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     [1.22070312e-10 1.22070312e-10 5.00000000e-01 5.00000000e-01]]]


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

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

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

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

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

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



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

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

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


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

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

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

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

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