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  def _extend_prior_with_posterior_data(self, x,y):
        self.x_init = np.append(self.x_init, np.array([x]), axis = 0)
        self.y_init = np.append(self.y_init, np.array(y), axis = 0)
  
  def optimize(self):
        y_max_ind = np.argmax(self.y_init)
        y_max = self.y_init[y_max_ind]
        optimal_x = self.x_init[y_max_ind]
        optimal_ei = None
        for i in range(self.n_iter):

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    def _acquisition_function(self, x):
        return -self._get_expected_improvement(x)
        
    def _get_next_probable_point(self):
            min_ei = float(sys.maxsize)
            x_optimal = None 
            
            # Trial with an array of random data points
            
            for x_start in (np.random.random((self.batch_size,self.x_init.shape[1])) * self.scale):

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    def _get_expected_improvement(self, x_new):

        # Using estimate from Gaussian surrogate instead of actual function for 
        # a new trial data point to avoid cost 
 
        mean_y_new, sigma_y_new = self.gauss_pr.predict(np.array([x_new]), return_std=True)
        sigma_y_new = sigma_y_new.reshape(-1,1)
        if sigma_y_new == 0.0:
            return 0.0
        

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from sklearn.gaussian_process import GaussianProcessRegressor
from scipy.stats import norm
from scipy.optimize import minimize
import sys
import pandas as pd

class BayesianOptimizer():
      
    def __init__(self, target_func, x_init, y_init, n_iter, scale, batch_size):
        self.x_init = x_init

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import numpy as np

def costly_function(x):
    total = np.array([])
    for x_i in x:
        total = np.append(total, np.sum(np.exp(-(x_i - 5) ** 2)))
    
    return total + np.random.randn()

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import seaborn as sns

plt.figure(figsize=(18,5))
sns.lineplot(data=pd.DataFrame({'Predicted':result,'Actual':Y_test.to_numpy()}))

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result = model.predict(start=1, end=125, exog=X_test.to_numpy().astype(float))

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from statsmodels.tsa.statespace.sarimax import SARIMAX

model = SARIMAX(endog=Y_train.to_numpy(), exog=X_train.to_numpy().astype(float), 
                order=(0, 0, 0),seasonal_order=(2, 0, 1, 2))
model = model.fit(disp=False)
model.summary()

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from pmdarima.arima import auto_arima

auto_model = auto_arima(Y_train.to_numpy(), exogenous=X_train.to_numpy(), m=2, seasonal=True,
                        suppress_warnings = True,  
                        step_wise=True, trace=True)           
auto_model.summary()

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from sklearn.model_selection import TimeSeriesSplit

tss = TimeSeriesSplit()
for train_index, test_index in tss.split(X,Y):
  Y_train, Y_test = Y.iloc[train_index], Y.iloc[test_index] 
  X_train, X_test = X.iloc[train_index], X.iloc[test_index]