from scipy.spatial.distance import pdist, squareform
from numpy import exp
from scipy.linalg import eigh
import numpy as np 
def rbf_kernel_pca(X, gamma, n_components):
    """
    RBF kernel PCA implementation.    
    Parameters
    ------------
    X: {NumPy ndarray}, shape = [n_examples, n_features]  
    gamma: float
        Tuning parameter of the RBF kernel    
    n_components: int
        Number of principal components to return    
    Returns
    ------------
    X_pc: {NumPy ndarray}, shape = [n_examples, k_features]
        Projected dataset   
    """
    # Calculate pairwise squared Euclidean distances
    # in the MxN dimensional dataset.
    sq_dists = pdist(X, 'sqeuclidean')    
    # Convert pairwise distances into a square matrix.
    mat_sq_dists = squareform(sq_dists)    
    # Compute the symmetric kernel matrix.
    K = exp(-gamma * mat_sq_dists)    
    # Center the kernel matrix.
    N = K.shape[0]
    one_n = np.ones((N,N)) / N
    K = K - one_n.dot(K) - K.dot(one_n) + one_n.dot(K).dot(one_n)    
    # Obtaining eigenpairs from the centered kernel matrix
    # scipy.linalg.eigh returns them in ascending order
    eigvals, eigvecs = eigh(K)
    eigvals, eigvecs = eigvals[::-1], eigvecs[:, ::-1]    
    # Collect the top k eigenvectors (projected examples)
    X_pc = np.column_stack([eigvecs[:, i]
                           for i in range(n_components)])    
    return X_pc


#Example — separating half-moon shapes
"""Now, let us apply our rbf_kernel_pca on some nonlinear example datasets. We will start by creating a two-dimensional dataset of 100 example points representing two half-moon shapes:"""

from sklearn.datasets import make_moons
import matplotlib.pyplot as plt

n_examples = 100
X, y = make_moons(n_examples, random_state=123)
plt.scatter(X[y==0, 0], X[y==0, 1],color='red', marker='^', alpha=0.5)
plt.scatter(X[y==1, 0], X[y==1, 1],color='blue', marker='o', alpha=0.5)
plt.tight_layout()
plt.show()

"""For the purposes of illustration, the half-moon of triangle symbols will represent one class, and the half-moon depicted by the circle symbols will represent the examples from another class:"""