Python建模是什么？这个案例教会你

 import random
 import matplotlib.pyplot as plt
 import numpy as np
 ​
 def random_walk_simulation(num_balls=100, minutes=60):
     # 初始化小球位置
     positions = [(0, 0) for _ in range(num_balls)]
     
     # 记录所有时刻的位置
     all_positions = [[(0, 0)] for _ in range(num_balls)]
     
     # 每分钟的移动
     for minute in range(minutes):
         for i in range(num_balls):
             x, y = positions[i]
             # 随机选择方向：0-上，1-下，2-左，3-右
             direction = random.randint(0, 3)
             
             if direction == 0:  # 上
                 y += 1
             elif direction == 1:  # 下
                 y -= 1
             elif direction == 2:  # 左
                 x -= 1
             else:  # 右
                 x += 1
                 
             positions[i] = (x, y)
             all_positions[i].append((x, y))
     
     return positions, all_positions
 ​
 # 执行模拟
 final_positions, history = random_walk_simulation()

 # 分析最终位置
 x_coords = [pos[0] for pos in final_positions]
 y_coords = [pos[1] for pos in final_positions]
 ​
 # 计算统计信息
 mean_x = np.mean(x_coords)
 mean_y = np.mean(y_coords)
 distance_from_origin = [np.sqrt(x**2 + y**2) for x, y in final_positions]
 mean_distance = np.mean(distance_from_origin)
 ​
 print(f"平均X坐标: {mean_x:.2f}")
 print(f"平均Y坐标: {mean_y:.2f}") 
 print(f"平均距离原点: {mean_distance:.2f}")

 # 设置中文字体支持
 plt.rcParams['font.sans-serif'] = ['SimHei', 'Microsoft YaHei', 'Arial Unicode MS']
 plt.rcParams['axes.unicode_minus'] = False  # 解决负号显示问题
 ​
 # 创建三维波形图可视化概率分布
 fig = plt.figure(figsize=(16, 12))
 ​
 # 子图1: 二维原始散点图（对照）
 ax1 = fig.add_subplot(221)
 ax1.scatter(x_coords, y_coords, c='blue', alpha=0.6, s=20)
 ax1.set_xlabel('X')
 ax1.set_ylabel('Y')
 ax1.set_title('二维散点图 - 最终位置')
 ax1.grid(True)
 ax1.axhline(y=0, color='k', linestyle='--', alpha=0.3)
 ax1.axvline(x=0, color='k', linestyle='--', alpha=0.3)
 ​
 # 子图2: 三维概率密度波形图
 ax2 = fig.add_subplot(222, projection='3d')
 ​
 # 创建概率密度网格
 # 确定网格范围（基于数据范围，留有余量）
 x_min, x_max = min(x_coords)-5, max(x_coords)+5
 y_min, y_max = min(y_coords)-5, max(y_coords)+5
 ​
 # 创建网格
 grid_size = 40
 x_grid = np.linspace(x_min, x_max, grid_size)
 y_grid = np.linspace(y_min, y_max, grid_size)
 X, Y = np.meshgrid(x_grid, y_grid)
 Z = np.zeros_like(X)
 ​
 # 计算每个网格点的概率密度（二维高斯核密度估计）
 # 使用小球位置计算局部密度
 bandwidth = 2.0  # 带宽参数
 num_sim_balls = len(final_positions)  # 修正变量名
 ​
 for i in range(grid_size):
     for j in range(grid_size):
         # 计算当前网格点到所有小球的距离
         distances = np.sqrt((X[i,j] - np.array(x_coords))**2 + (Y[i,j] - np.array(y_coords))**2)
         # 高斯核密度估计
         density = np.sum(np.exp(-distances**2 / (2 * bandwidth**2))) / (num_sim_balls * bandwidth * np.sqrt(2 * np.pi))
         Z[i,j] = density
 ​
 # 平滑处理
 Z_smooth = gaussian_filter(Z, sigma=1.5)
 ​
 # 绘制三维波形表面
 surf = ax2.plot_surface(X, Y, Z_smooth, cmap='viridis', alpha=0.9, edgecolor='none', antialiased=True)
 ax2.set_xlabel('X')
 ax2.set_ylabel('Y')
 ax2.set_zlabel('概率密度')
 ax2.set_title('三维概率密度波形')
 fig.colorbar(surf, ax=ax2, shrink=0.6, label='密度')
 ax2.grid(True)
 ​
 # 子图3: 三维概率分布曲面（不同视角）
 ax3 = fig.add_subplot(223, projection='3d')
 ​
 # 使用对数变换增强低密度区域的可见性
 Z_log = np.log1p(Z_smooth)  # log(1+x) 避免log(0)
 ​
 # 绘制平滑的概率分布曲面
 surf3 = ax3.plot_surface(X, Y, Z_log, cmap='plasma', alpha=0.8, 
                         edgecolor='none', antialiased=True)
 ax3.set_xlabel('X')
 ax3.set_ylabel('Y')
 ax3.set_zlabel('log(密度)')
 ax3.set_title('三维概率曲面')
 fig.colorbar(surf3, ax=ax3, shrink=0.6, label='log(密度)')
 ax3.grid(True)
 ​
 # 子图4: 概率密度热图（俯视图）
 ax4 = fig.add_subplot(224)
 im = ax4.imshow(Z_smooth, extent=[x_min, x_max, y_min, y_max], 
                 origin='lower', cmap='hot', alpha=0.8)
 ax4.set_xlabel('X')
 ax4.set_ylabel('Y')
 ax4.set_title('概率密度热俯视图')
 fig.colorbar(im, ax=ax4, shrink=0.7, label='密度')
 ax4.grid(True, alpha=0.3)
 ax4.axhline(y=0, color='white', linestyle='--', alpha=0.5)
 ax4.axvline(x=0, color='white', linestyle='--', alpha=0.5)
 ​
 plt.tight_layout()
 plt.show()
 ​
 # 概率分布统计分析
 print("\n=== 概率分布分析 ===")
 print(f"最大概率密度: {np.max(Z_smooth):.4f}")
 print(f"平均概率密度: {np.mean(Z_smooth):.4f}")
 print(f"密度分布范围: [{np.min(Z_smooth):.4f}, {np.max(Z_smooth):.4f}]")
 ​
 # 计算概率质量中心
 total_mass = np.sum(Z_smooth)
 center_x = np.sum(X * Z_smooth) / total_mass
 center_y = np.sum(Y * Z_smooth) / total_mass
 print(f"概率分布中心: ({center_x:.2f}, {center_y:.2f})")
 ​
 # 计算分布的不确定性（香农熵近似）
 prob_normalized = Z_smooth / total_mass
 entropy = -np.sum(prob_normalized * np.log(prob_normalized + 1e-10))
 print(f"分布熵（不确定性）: {entropy:.2f}")