径向基函数神经网络（RBFNN）

RBF 就是计算两个点之间的相似度。

真正把特征从低维空间映射到高维空间的是你选择的隐层的节点数，隐层的节点数就是高维空间的维数。

如下实现了输入层到隐层之间文随机权重的径向基函数神经网络（RBFNN）

def pairwise_distances(X, Y):
    D = -2 * X @ Y.T + np.sum(Y ** 2, axis=1) + np.sum(X ** 2, axis=1)[:, np.newaxis]
    D[D < 0] = 0
    return D

class RBFNN:
    def __init__(self):
        self.N_i = None
        self.N_o = None
        self.N_h = 200
        self.sigma = 1
        self.beta = 1e-6

        
    def train(self, X, Y):
        self.N_i = X.shape[0]
        self.N_o = Y.shape[0]
        self.W_i = np.random.uniform(-self.sigma,self.sigma, (self.N_h, self.N_i))
        
        # 隐层输出
        H = np.exp(-pairwise_distances(self.W_i, X.T))
        
        self.W_o = Y.dot(H.T.dot(np.linalg.inv(H.dot(H.T)+self.beta*np.eye(self.N_h))))
     
    def predict(self, X):
        H = np.exp(-pairwise_l2_distances(self.W_i, X.T))
        return self.W_o.dot(H)

在洛伦兹系统的动态重建实验上表现很好：

np.random.seed()

model = RBFNN()

n = 1  # 使用 n 个历史点作为输入

num_train = 3000
x_train = np.vstack([select_samples(x,0+i,num_train) for i in range(n)])
y_train = select_samples(x,n,num_train)

model.train(x_train, y_train)

num_test = 5000
test_start = 5000
P = np.empty((3,num_test))
Q = select_samples(x,test_start, num_test)
p = np.vstack([select_samples(x,test_start-n+i,1) for i in range(n)])


for i in range(num_test):
    p_next = model.predict(p)
    P[:,i] = np.squeeze(p_next)
    p = np.vstack([p,p_next])[3:]


fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax = fig.add_subplot()
plt.plot(*P, 'r')
plt.plot(*Q, 'g')

plt.figure()
dim = ['x','y','z']
for i in range(3):
    plt.subplot(3,1,i+1)
    plt.plot(P[i,:].T, label='prediction')
    plt.plot(Q[i,:].T, label='true')
    plt.ylabel(dim[i])
    plt.legend(loc='upper right')