三维可视化

# ===== 示例1：多项式分布可视化 =====
from scipy.stats import multinomial
import numpy as np
import matplotlib.pyplot as plt

def visualize_multinomial(p_array, num=20):
    # 生成网格
    x1_array = np.arange(num + 1)
    x2_array = np.arange(num + 1)
    xx1, xx2 = np.meshgrid(x1_array, x2_array)
    
    # 计算第三维（x1 + x2 + x3 = num）
    xx3 = num - xx1 - xx2
    xx3 = np.where(xx3 >= 0.0, xx3, np.nan)
    
    # 计算PMF
    PMF_ff = multinomial.pmf(
        x=np.array(([xx1.ravel(), xx2.ravel(), xx3.ravel()])).T,
        n=num, p=p_array
    )
    PMF_ff = np.where(PMF_ff > 0.0, PMF_ff, np.nan)
    PMF_ff = np.reshape(PMF_ff, xx1.shape)
    
    # 三维散点图
    fig = plt.figure(figsize=(5, 5))
    ax = plt.axes(projection="3d")
    
    scatter_plot = ax.scatter3D(
        xx1.ravel(), xx2.ravel(), xx3.ravel(),
        s=50, marker='.',
        alpha=1,
        c=PMF_ff.ravel(),
        cmap='RdYlBu_r'
    )
    
    ax.set_proj_type('ortho')
    ax.set_xlabel('$x_1$')
    ax.set_ylabel('$x_2$')
    ax.set_zlabel('$x_3$')
    ax.set_xlim(0, num)
    ax.set_ylim(0, num)
    ax.set_zlim3d(0, num)
    ax.view_init(azim=30, elev=30)
    ax.grid()
    plt.show()

# 示例：均匀概率
p_array = [1/3, 1/3, 1/3]
visualize_multinomial(p_array)

# 示例：不均匀概率
p_array = [0.2, 0.2, 0.6]
visualize_multinomial(p_array)


# ===== 示例2：三维高斯分布概率密度 =====
def Mahal_d(Mu, Sigma, x):
    # 计算马哈距离
    x_demeaned = x - Mu  # 中心化
    inv_covmat = np.linalg.inv(Sigma)  # 协方差矩阵求逆
    mahal_sq = x_demeaned @ inv_covmat @ x_demeaned.T
    mahal_sq = np.diag(mahal_sq)  # 对角线元素
    mahal_d = np.sqrt(mahal_sq)  # 开平方
    return mahal_d

def Mahal_d_2_pdf(d, Sigma):
    # 将马氏距离转化为概率密度
    scale_1 = np.sqrt(np.linalg.det(Sigma))
    scale_2 = (2*np.pi)**(3/2)
    gaussian = np.exp(-d**2/2)
    pdf = gaussian/scale_1/scale_2
    return pdf

# 生成三维网格
x1 = np.linspace(-2, 2, 31)
x2 = np.linspace(-2, 2, 31)
x3 = np.linspace(-2, 2, 31)
xxx1, xxx2, xxx3 = np.meshgrid(x1, x2, x3)

# 定义参数
Mu = np.array([[0, 0, 0]])
Sigma = np.array([[1, 0.6, -0.4],
                  [0.6, 1.5, 1],
                  [-0.4, 1, 2]])

x_array = np.vstack([xxx1.ravel(), xxx2.ravel(), xxx3.ravel()]).T

# 计算马氏距离和概率密度
d_array = Mahal_d(Mu, Sigma, x_array)
d_array = d_array.reshape(xxx1.shape)
pdf_zz = Mahal_d_2_pdf(d_array, Sigma)

# 可视化切片（沿x3轴）
fig = plt.figure(figsize=(6, 6))
ax = fig.add_subplot(projection='3d')

x3_slice_idx = 15  # 选择x3 = 0的切片
ax.scatter(xxx1[:, :, x3_slice_idx].ravel(),
          xxx2[:, :, x3_slice_idx].ravel(),
          xxx3[:, :, x3_slice_idx].ravel(),
          c=pdf_zz[:, :, x3_slice_idx].ravel(),
          cmap='turbo', s=4)

ax.view_init(azim=-120, elev=30)
ax.set_proj_type('ortho')
ax.set_xlabel('$x_1$')
ax.set_ylabel('$x_2$')
ax.set_zlabel('$x_3$')
plt.show()


# 💡 关键学习点：
# 1. multinomial.pmf()：计算多项式分布概率
# 2. x1 + x2 + x3 = n：多项式分布约束
# 3. 马哈距离：考虑协方差的距离度量
# 4. 三维散点图：c参数映射颜色
# 5. 切片可视化：观察高维分布的局部结构