我们把均值向量替换到均值函数，协方差矩阵替换到核函数。假设有一组观测值，它们的时刻对应一个向量 #高斯核函数 def gaussian_kernel ( x1 , x2 , l = 0.5 , sigma_f = 0.2 ) : m , n = x1 . shape [ 0 ] , x2 . shape [ 0 ] dist_matrix = np . zeros ( ( m , n ) , dtype = float ) for i in range ( m ) : for j in range ( n ) : dist_matrix [ i ] [ j ] = np . sum ( ( x1 [ i ] - x2 [ j ] ) ** 2 ) return sigma_f ** 2 * np . exp ( - 0.5 / l ** 2 * dist_matrix ) #生成观测值，取sin函数没有别的用意，单纯就是为了计算出Y def getY ( X ) : X = np . asarray ( X ) Y = np . sin ( X ) * 0.4 + np . random . normal ( 0 , 0.05 , size = X . shape ) return Y . tolist ( ) #根据观察点X，修正生成高斯过程新的均值和协方差 def update ( X , X_star ) : X = np . asarray ( X ) X_star = np . asarray ( X_star ) K_YY = gaussian_kernel ( X , X ) # K(X,X) K_ff = gaussian_kernel ( X_star , X_star ) # K(X*, X*) K_Yf = gaussian_kernel ( X , X_star ) # K(X, X*) K_fY = K_Yf . T # K(X*, X) 协方差矩阵是对称的，因此分块互为转置 K_YY_inv = np . linalg . inv ( K_YY + 1e - 8 * np . eye ( len ( X ) ) ) # (N, N) mu_star = K_fY . dot ( K_YY_inv ) . dot ( Y ) cov_star = K_ff - K_fY . dot ( K_YY_inv ) . dot ( K_Yf ) return mu_star , cov_star f , ax = plt . subplots ( 2 , 1 , sharex = True , sharey = True ) #绘制高斯过程的先验 X_pre = np . arange ( 0 , 10 , 0.1 ) mu_pre = np . array ( [ 0 ] * len ( X_pre ) ) Y_pre = mu_pre cov_pre = gaussian_kernel ( X_pre , X_pre ) uncertainty = 1.96 * np . sqrt ( np . diag ( cov_pre ) ) #取95%置信区间 """95%可能性,该区间包含了这个时刻下的高斯分布均值""" ax [ 0 ] . fill_between ( X_pre , Y_pre + uncertainty , Y_pre - uncertainty , alpha = 0.1 ) ax [ 0 ] . plot ( X_pre , Y_pre , label = "expection" ) ax [ 0 ] . legend ( ) #绘制基于观测值的高斯过程后验 X = np . array ( [ 1 , 3 , 7 , 9 ] ) . reshape ( - 1 , 1 ) #4*1矩阵 Y = getY ( X ) X_star = np . arange ( 0 , 10 , 0.1 ) . reshape ( - 1 , 1 ) mu_star , cov_star = update ( X , X_star ) print ( mu_star . shape ) # (100,1) print ( cov_star . shape ) # (100,100) 反映100个时刻各自之间的方差 # 将高维数组展开成一维 Y_star = mu_star . ravel ( ) # (100,) # 时刻9的值(后验，真实) print ( Y_star [ 90 ] , Y [ 3 ] ) # 0.10295047719514798 0.10295050291616215 uncertainty = 1.96 * np . sqrt ( np . diag ( cov_star ) ) #取95%置信区间 ax [ 1 ] . fill_between ( X_star . ravel ( ) , Y_star + uncertainty , Y_star - uncertainty , alpha = 0.1 ) ax [ 1 ] . plot ( X_star , Y_star , label = "expection" ) ax [ 1 ] . scatter ( X , Y , label = "observation point" , c = "red" , marker = "x" ) ax [ 1 ] . legend ( ) plt . show ( )

图中半透明区域表示在该时刻的均值附近取95%置信区间的区域，可以看出，由于观测值的引入，连续域上各个时刻的均值发生了变化，并且95%置信区间的空间变窄，证明确定性正在增强。

补充内容：关于置信区间