一、引言
在当今数据驱动的时代，深度学习已成为解决复杂问题的有力工具。它广泛应用于图像识别、自然语言处理和预测分析等领域。回归分析是统计学的一种基础方法，用于描述变量之间的关系。通过回归模型，我们可以预测连续的数值输出，这在经济学、工程学、医疗等领域有着至关重要的应用。
其中，线性回归是最简单且最常用的回归方法之一，它假设预测变量与目标变量之间存在线性关系。虽然线性回归在许多场景中表现良好，但当数据具有非线性特征时，它的预测能力可能会受到限制。在这些情况下，曲线回归（如多项式回归或其他非线性模型）通常可以提供更好的拟合效果。
二、线性回归
2.1、定义
线性回归是一种基本的预测分析方法，它通过拟合一条直线来描述自变量（特征）与因变量（目标）之间的关系。
2.2、设计思路
导入模块
python
import numpy as np
import torch
import random
import torch.nn as nn
import matplotlib.pyplot as plt输入数据
python
point = [[1.8, 8.5], [1.6, 8.4], [2.3, 7.2], [0.7, 8.6], [1.8, 7.6], [4.9, 4.1], [2.5, 6.6], [4.4, 4.1],
[7.4, 0.6], [1.9, 7.5],
[5.8, 2.2], [6.8, 0.6], [2.9, 6.1], [5.6, 2.1], [1.0, 8.4], [5.3, 3.1], [3.2, 6.8], [3.2, 5.1],
[6.4, 0.9], [2.9, 5.6],
[5.5, 2.9], [4.7, 4.3], [8.0, 0.9], [2.1, 6.2], [7.2, 0.7], [4.1, 4.0], [5.6, 2.4], [1.3, 7.7],
[3.2, 6.0], [4.7, 2.8],
[5.4, 2.9], [4.0, 4.6], [4.3, 4.1], [1.3, 8.5], [2.5, 7.1], [4.1, 4.2], [5.4, 3.4], [6.4, 2.8],
[7.0, 2.9], [2.4, 7.0],
[1.2, 7.6], [7.5, 0.4], [7.7, 0.7], [1.8, 8.2], [0.6, 8.9], [4.5, 4.2], [7.3, 0.3], [7.4, 1.2],
[4.0, 5.7], [7.0, 0.4],
[6.7, 2.3], [1.3, 7.9], [1.7, 7.2], [4.8, 3.7], [1.3, 7.3], [5.4, 3.8], [3.9, 5.6], [3.1, 5.7],
[3.2, 6.3], [2.5, 6.6],
[0.9, 8.9], [1.4, 7.5], [0.8, 8.1], [1.9, 8.3], [4.2, 3.6], [1.7, 7.2], [7.7, 0.5], [5.5, 2.1],
[4.2, 5.2], [3.9, 4.9],
[4.2, 4.4], [4.0, 5.9], [4.3, 3.4], [7.0, 1.0], [7.6, 0.8], [7.3, 0.1], [5.6, 3.0], [6.4, 3.0],
[-0.0, 9.1], [2.9, 6.7],
[4.4, 3.6], [6.4, 2.2], [5.3, 3.2], [5.7, 2.7], [6.5, 1.5], [7.4, 1.1], [6.2, 2.1], [5.6, 1.4],
[5.7, 2.0], [3.0, 5.2],
[4.5, 5.6], [6.8, 1.7], [6.5, 1.3], [4.2, 4.5], [3.3, 6.5], [2.7, 5.2], [5.8, 3.5], [7.8, 0.9],
[5.5, 3.0], [1.2, 8.0],
[4.2, 4.2], [0.9, 8.6], [7.0, 1.0], [0.2, 9.6], [5.9, 3.0], [2.3, 6.5], [3.3, 5.1], [5.9, 2.2],
[6.8, 1.7], [4.6, 3.8],
[6.3, 1.3], [1.2, 8.4], [6.8, 1.6], [5.0, 2.3], [7.4, 0.1], [3.1, 5.9], [4.9, 3.8], [1.8, 7.5],
[7.9, 0.3], [2.8, 5.2],
[2.4, 7.2], [4.0, 4.0], [6.8, 1.7], [6.6, 1.9], [4.9, 4.4], [6.4, 2.9], [7.3, 0.7], [2.1, 7.6],
[1.9, 7.7], [0.7, 9.2],
[3.7, 4.8], [0.5, 8.9], [4.8, 4.4], [5.7, 2.7], [4.0, 3.8], [6.1, 1.6], [6.7, 0.3], [0.3, 8.5],
[5.3, 1.7], [2.9, 5.6],
[0.9, 7.8], [2.9, 6.5], [0.2, 8.8], [8.0, 0.7], [1.8, 6.7], [3.0, 6.0], [5.0, 3.7], [2.8, 5.3],
[4.2, 5.2], [4.5, 5.2],
[8.1, 0.6], [4.4, 3.9], [7.3, 1.4], [5.7, 2.0], [1.9, 7.2], [3.5, 4.4], [4.4, 4.4], [2.6, 6.3],
[6.0, 2.9], [2.5, 7.1],
[6.0, 2.3], [6.5, 1.2], [0.3, 9.6], [2.3, 6.6], [7.6, 0.4], [0.2, 9.3], [1.1, 8.7], [3.5, 5.2],
[7.0, 2.0], [6.5, 2.1],
[7.8, 0.6], [4.1, 4.3], [1.2, 8.9], [1.0, 8.9], [5.6, 3.4], [5.6, 2.0], [4.7, 3.3], [7.7, 0.8],
[7.4, 1.4], [3.2, 4.9],
[4.8, 3.9], [5.6, 2.8], [1.4, 8.7], [2.4, 7.2], [8.0, 0.3], [4.9, 3.8], [2.3, 6.9], [5.8, 2.7],
[1.9, 7.0], [5.0, 2.9],
[2.2, 7.4], [6.1, 2.6], [6.7, 1.0], [4.6, 3.6], [7.9, 0.2], [3.1, 5.8], [4.7, 4.1], [1.5, 8.1],
[2.3, 7.0], [4.2, 5.0],
[5.6, 2.2], [5.9, 2.6], [3.3, 4.8], [2.5, 5.6], [2.1, 7.5], [0.8, 7.4], [6.2, 2.6], [4.2, 3.8],
[0.8, 8.3], [4.5, 4.1],
[6.2, 2.0], [7.8, 1.0], [2.6, 6.0], [4.2, 4.2], [1.6, 7.8], [4.1, 4.2], [5.8, 2.7], [4.0, 5.8],
[0.9, 7.8], [6.7, 1.6],
[0.2, 8.2], [1.1, 7.7], [2.1, 7.1], [6.0, 2.8], [4.0, 4.9], [7.5, 1.6], [6.1, 1.7], [3.5, 5.9],
[6.3, 1.6], [8.0, 0.3],
[5.4, 2.6], [7.6, 0.2], [5.8, 2.9], [1.9, 6.6], [0.4, 8.2], [5.7, 2.1], [3.2, 6.2], [5.2, 3.5],
[7.6, 0.2], [1.8, 7.3],
[0.5, 8.4], [5.5, 3.6], [5.2, 3.4], [6.0, 2.3], [5.0, 3.8], [3.3, 5.5], [7.4, 1.3], [4.2, 4.3],
[2.4, 7.0], [2.1, 6.5],
[7.7, 0.7], [5.6, 2.7], [6.3, 1.4], [5.3, 2.0], [0.4, 8.5], [2.0, 7.7], [5.8, 3.8], [4.3, 4.5],
[0.9, 8.9], [3.7, 4.7],
[7.0, 1.5], [6.2, 2.0], [2.5, 6.2], [3.8, 5.5], [1.8, 8.4], [3.3, 5.5], [7.9, 0.4], [1.9, 7.8],
[5.6, 3.1], [7.9, 0.6],
[4.8, 3.7], [5.1, 3.9], [6.9, 1.3], [3.3, 5.8], [3.8, 5.1], [5.3, 3.5], [1.3, 7.8], [0.8, 8.2],
[1.9, 7.8], [4.9, 3.6],
[6.8, 2.4], [7.5, 0.2], [4.8, 3.3], [3.9, 4.4], [4.3, 4.2], [6.2, 1.9], [7.2, 0.8], [2.7, 6.0],
[1.1, 7.7], [7.0, 0.1],
[0.8, 7.3], [5.6, 3.5], [0.8, 8.1], [4.7, 3.8], [3.9, 4.5], [4.7, 3.2], [1.3, 7.7], [7.2, 0.8],
[4.2, 4.0], [1.2, 8.9]]划分训练集和测试集
python
# 将 point1 分割为训练集和测试集
random.shuffle(point) # 随机打乱数据
split_index = int(0.1 * len(point)) # 取前 10% 的数据作为测试集
train_point = point[split_index:]
test_point = point[:split_index]
x_train = np.array([point[0] for point in train_point])
y_train = np.array([point[1] for point in train_point])
x_test = np.array([point[0] for point in test_point])
y_test = np.array([point[1] for point in test_point])转换为Tensor张量
python
x_train = torch.from_numpy(x_train).float()
y_train = torch.from_numpy(y_train).float()构建模型
python
class ModelClass(nn.Module):
def __init__(self):
super().__init__()
self.layer1 = nn.Linear(1, 8)
self.layer2 = nn.Linear(8, 1)
def forward(self, x):
x = torch.tanh(self.layer1(x))
x = self.layer2(x)
return x
model = ModelClass()构建损失函数和优化器
python
criterion = nn.MSELoss()
optimizer = torch.optim.Adam(model.parameters(), lr=0.005, weight_decay=0.01)模型训练
python
for n in range(1, 2001):
# 前向传播
y_pred = model(x_train.unsqueeze(1))
# 计算损失
loss = criterion(y_pred.squeeze(1), y_train)
# 反向传播和优化
optimizer.zero_grad()
loss.backward()
optimizer.step()
if n % 100 == 0 or n == 1:
print(n,loss.item())可视化
python
step_list = []
loss_list = []
test_step_list = []
test_loss_list = []
for n in range(1, 2001):
# 前向传播
y_pred = model(x_train.unsqueeze(1))
# 计算损失
loss = criterion(y_pred.squeeze(1), y_train)
# 反向传播和优化
optimizer.zero_grad()
loss.backward()
optimizer.step()
# 更新右侧的损失图数据并绘制
step_list.append(n)
loss_list.append(loss.detach())
# 显示频率设置
if n % 100 == 0 or n == 1:
print(n,loss.item())
# 绘制左侧的拟合图
ax1.clear()
ax1.set_xlim(0, 10)
ax1.set_ylim(0, 10)
ax1.set_xlabel("x axis label")
ax1.set_ylabel("y axis label")
ax1.scatter(x_train, y_train)
x_range = torch.tensor(np.arange(0, 10, 0.1)).unsqueeze(1).float()
y_range = model(x_range).detach().numpy()
ax1.plot(x_range, y_range, 'r') # 绘制拟合线
# 计算测试集损失
y_pred_test = model(torch.tensor(x_test).unsqueeze(1).float())
loss_test = criterion(y_pred_test.squeeze(1), torch.from_numpy(y_test).float())
test_step_list.append(n)
test_loss_list.append(loss_test.detach())
ax2.clear()
ax2.plot(step_list, loss_list, 'r-', label='Train Loss')
ax2.plot(test_step_list, test_loss_list, 'b-', label='Test Loss') # 绘制测试集损失
ax2.set_xlabel("Step")
ax2.set_ylabel("Loss")
ax2.legend()
plt.show()完整代码
python
import numpy as np
import torch
import random
import torch.nn as nn
import matplotlib.pyplot as plt
# 创造数据，数据集
point = [[1.8, 8.5], [1.6, 8.4], [2.3, 7.2], [0.7, 8.6], [1.8, 7.6], [4.9, 4.1], [2.5, 6.6], [4.4, 4.1],
[7.4, 0.6], [1.9, 7.5],
[5.8, 2.2], [6.8, 0.6], [2.9, 6.1], [5.6, 2.1], [1.0, 8.4], [5.3, 3.1], [3.2, 6.8], [3.2, 5.1],
[6.4, 0.9], [2.9, 5.6],
[5.5, 2.9], [4.7, 4.3], [8.0, 0.9], [2.1, 6.2], [7.2, 0.7], [4.1, 4.0], [5.6, 2.4], [1.3, 7.7],
[3.2, 6.0], [4.7, 2.8],
[5.4, 2.9], [4.0, 4.6], [4.3, 4.1], [1.3, 8.5], [2.5, 7.1], [4.1, 4.2], [5.4, 3.4], [6.4, 2.8],
[7.0, 2.9], [2.4, 7.0],
[1.2, 7.6], [7.5, 0.4], [7.7, 0.7], [1.8, 8.2], [0.6, 8.9], [4.5, 4.2], [7.3, 0.3], [7.4, 1.2],
[4.0, 5.7], [7.0, 0.4],
[6.7, 2.3], [1.3, 7.9], [1.7, 7.2], [4.8, 3.7], [1.3, 7.3], [5.4, 3.8], [3.9, 5.6], [3.1, 5.7],
[3.2, 6.3], [2.5, 6.6],
[0.9, 8.9], [1.4, 7.5], [0.8, 8.1], [1.9, 8.3], [4.2, 3.6], [1.7, 7.2], [7.7, 0.5], [5.5, 2.1],
[4.2, 5.2], [3.9, 4.9],
[4.2, 4.4], [4.0, 5.9], [4.3, 3.4], [7.0, 1.0], [7.6, 0.8], [7.3, 0.1], [5.6, 3.0], [6.4, 3.0],
[-0.0, 9.1], [2.9, 6.7],
[4.4, 3.6], [6.4, 2.2], [5.3, 3.2], [5.7, 2.7], [6.5, 1.5], [7.4, 1.1], [6.2, 2.1], [5.6, 1.4],
[5.7, 2.0], [3.0, 5.2],
[4.5, 5.6], [6.8, 1.7], [6.5, 1.3], [4.2, 4.5], [3.3, 6.5], [2.7, 5.2], [5.8, 3.5], [7.8, 0.9],
[5.5, 3.0], [1.2, 8.0],
[4.2, 4.2], [0.9, 8.6], [7.0, 1.0], [0.2, 9.6], [5.9, 3.0], [2.3, 6.5], [3.3, 5.1], [5.9, 2.2],
[6.8, 1.7], [4.6, 3.8],
[6.3, 1.3], [1.2, 8.4], [6.8, 1.6], [5.0, 2.3], [7.4, 0.1], [3.1, 5.9], [4.9, 3.8], [1.8, 7.5],
[7.9, 0.3], [2.8, 5.2],
[2.4, 7.2], [4.0, 4.0], [6.8, 1.7], [6.6, 1.9], [4.9, 4.4], [6.4, 2.9], [7.3, 0.7], [2.1, 7.6],
[1.9, 7.7], [0.7, 9.2],
[3.7, 4.8], [0.5, 8.9], [4.8, 4.4], [5.7, 2.7], [4.0, 3.8], [6.1, 1.6], [6.7, 0.3], [0.3, 8.5],
[5.3, 1.7], [2.9, 5.6],
[0.9, 7.8], [2.9, 6.5], [0.2, 8.8], [8.0, 0.7], [1.8, 6.7], [3.0, 6.0], [5.0, 3.7], [2.8, 5.3],
[4.2, 5.2], [4.5, 5.2],
[8.1, 0.6], [4.4, 3.9], [7.3, 1.4], [5.7, 2.0], [1.9, 7.2], [3.5, 4.4], [4.4, 4.4], [2.6, 6.3],
[6.0, 2.9], [2.5, 7.1],
[6.0, 2.3], [6.5, 1.2], [0.3, 9.6], [2.3, 6.6], [7.6, 0.4], [0.2, 9.3], [1.1, 8.7], [3.5, 5.2],
[7.0, 2.0], [6.5, 2.1],
[7.8, 0.6], [4.1, 4.3], [1.2, 8.9], [1.0, 8.9], [5.6, 3.4], [5.6, 2.0], [4.7, 3.3], [7.7, 0.8],
[7.4, 1.4], [3.2, 4.9],
[4.8, 3.9], [5.6, 2.8], [1.4, 8.7], [2.4, 7.2], [8.0, 0.3], [4.9, 3.8], [2.3, 6.9], [5.8, 2.7],
[1.9, 7.0], [5.0, 2.9],
[2.2, 7.4], [6.1, 2.6], [6.7, 1.0], [4.6, 3.6], [7.9, 0.2], [3.1, 5.8], [4.7, 4.1], [1.5, 8.1],
[2.3, 7.0], [4.2, 5.0],
[5.6, 2.2], [5.9, 2.6], [3.3, 4.8], [2.5, 5.6], [2.1, 7.5], [0.8, 7.4], [6.2, 2.6], [4.2, 3.8],
[0.8, 8.3], [4.5, 4.1],
[6.2, 2.0], [7.8, 1.0], [2.6, 6.0], [4.2, 4.2], [1.6, 7.8], [4.1, 4.2], [5.8, 2.7], [4.0, 5.8],
[0.9, 7.8], [6.7, 1.6],
[0.2, 8.2], [1.1, 7.7], [2.1, 7.1], [6.0, 2.8], [4.0, 4.9], [7.5, 1.6], [6.1, 1.7], [3.5, 5.9],
[6.3, 1.6], [8.0, 0.3],
[5.4, 2.6], [7.6, 0.2], [5.8, 2.9], [1.9, 6.6], [0.4, 8.2], [5.7, 2.1], [3.2, 6.2], [5.2, 3.5],
[7.6, 0.2], [1.8, 7.3],
[0.5, 8.4], [5.5, 3.6], [5.2, 3.4], [6.0, 2.3], [5.0, 3.8], [3.3, 5.5], [7.4, 1.3], [4.2, 4.3],
[2.4, 7.0], [2.1, 6.5],
[7.7, 0.7], [5.6, 2.7], [6.3, 1.4], [5.3, 2.0], [0.4, 8.5], [2.0, 7.7], [5.8, 3.8], [4.3, 4.5],
[0.9, 8.9], [3.7, 4.7],
[7.0, 1.5], [6.2, 2.0], [2.5, 6.2], [3.8, 5.5], [1.8, 8.4], [3.3, 5.5], [7.9, 0.4], [1.9, 7.8],
[5.6, 3.1], [7.9, 0.6],
[4.8, 3.7], [5.1, 3.9], [6.9, 1.3], [3.3, 5.8], [3.8, 5.1], [5.3, 3.5], [1.3, 7.8], [0.8, 8.2],
[1.9, 7.8], [4.9, 3.6],
[6.8, 2.4], [7.5, 0.2], [4.8, 3.3], [3.9, 4.4], [4.3, 4.2], [6.2, 1.9], [7.2, 0.8], [2.7, 6.0],
[1.1, 7.7], [7.0, 0.1],
[0.8, 7.3], [5.6, 3.5], [0.8, 8.1], [4.7, 3.8], [3.9, 4.5], [4.7, 3.2], [1.3, 7.7], [7.2, 0.8],
[4.2, 4.0], [1.2, 8.9]]
# 将 point1 分割为训练集和测试集
random.shuffle(point) # 随机打乱数据
split_index = int(0.1 * len(point)) # 取前 10% 的数据作为测试集
# 划分数据集
train_point = point[split_index:] # 训练集包含 90% 的数据
test_point = point[:split_index] # 测试集为前 10% 的数据
# 将训练集和测试集的数据分别提取为特征和目标
x_train = np.array([point[0] for point in train_point]) # 训练特征
y_train = np.array([point[1] for point in train_point]) # 训练目标
x_test = np.array([point[0] for point in test_point]) # 测试特征
y_test = np.array([point[1] for point in test_point]) # 测试目标
# 转换为PyTorch的张量
x_train = torch.from_numpy(x_train).float() # 将训练特征转换为浮点型张量
y_train = torch.from_numpy(y_train).float() # 将训练目标转换为浮点型张量
# 定义前向模型
class ModelClass(nn.Module):
def __init__(self):
super().__init__()
# 定义网络层
self.layer1 = nn.Linear(1, 8) # 第一个线性层，输入为1维，输出为8维
self.layer2 = nn.Linear(8, 1) # 第二个线性层，输入为8维，输出为1维
def forward(self, x):
# 前向传播函数
x = torch.tanh(self.layer1(x)) # 第一个层的输出应用tanh激活函数
x = self.layer2(x) # 经过第二个层
return x
# 实例化模型
model = ModelClass()
# 定义损失函数和优化器
criterion = nn.MSELoss() # 均方误差损失函数
optimizer = torch.optim.Adam(model.parameters(), lr=0.005, weight_decay=0.01) # Adam优化器
# 初始化绘图
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 6)) # 创建绘图窗口，包含两个子图
# 开始迭代
step_list = [] # 用于存储训练的步骤
loss_list = [] # 用于存储训练损失
test_step_list = [] # 用于存储测试的步骤
test_loss_list = [] # 用于存储测试损失
for n in range(1, 2001): # 训练迭代2000轮
# 前向传播
y_pred = model(x_train.unsqueeze(1)) # 将训练输入传入模型，reshape为合适维度
# 计算损失
loss = criterion(y_pred.squeeze(1), y_train) # 计算模型预测值与真实值之间的损失
# 反向传播和优化
optimizer.zero_grad() # 清除之前的梯度
loss.backward() # 计算当前损失的梯度
optimizer.step() # 更新模型参数
# 更新右侧的损失图数据并绘制
step_list.append(n) # 记录当前步数
loss_list.append(loss.detach()) # 记录当前损失值
# 显示频率设置
if n % 100 == 0 or n == 1: # 每100步输出一次损失值
print(n, loss.item()) # 打印当前步数和损失值
# 绘制左侧的拟合图
ax1.clear() # 清除当前图
ax1.set_xlim(0, 10) # 设置x轴范围
ax1.set_ylim(0, 10) # 设置y轴范围
ax1.set_xlabel("x axis label") # x轴标签
ax1.set_ylabel("y axis label") # y轴标签
ax1.scatter(x_train, y_train) # 绘制训练数据点
x_range = torch.tensor(np.arange(0, 10, 0.1)).unsqueeze(1).float() # 生成预测输入范围
y_range = model(x_range).detach().numpy() # 计算拟合线的预测输出
ax1.plot(x_range, y_range, 'r') # 绘制拟合线
# 计算测试集损失
y_pred_test = model(torch.tensor(x_test).unsqueeze(1).float()) # 模型对测试集进行预测
loss_test = criterion(y_pred_test.squeeze(1), torch.from_numpy(y_test).float()) # 计算测试集损失
test_step_list.append(n) # 记录测试步数
test_loss_list.append(loss_test.detach()) # 记录测试损失
ax2.clear() # 清除当前测试损失图
ax2.plot(step_list, loss_list, 'r-', label='Train Loss') # 绘制训练损失
ax2.plot(test_step_list, test_loss_list, 'b-', label='Test Loss') # 绘制测试集损失
ax2.set_xlabel("Step") # x轴标签
ax2.set_ylabel("Loss") # y轴标签
ax2.legend() # 显示图例
plt.show() # 显示绘图窗口 三、曲线回归
3.1定义
曲线回归是一种用于拟合非线性关系的回归分析方法。与线性回归不同，曲线回归允许因变量与自变量之间存在更复杂的关系。常见的曲线回归形式包括多项式回归、指数回归和对数回归等。在多项式回归中，我们可以通过引入高次项来扩展模型的灵活性
常见模型
多项式回归：用于拟合二次或更高次的曲线，例如二次曲线
逻辑回归：虽然名字中有"回归"，逻辑回归实际上是用于分类问题的模型。
指数回归与对数回归：用于处理特定类型的数据关系，比如指数增长数据或对数增长数据。
3.2、设计思路
输入数据
python
point = [[0.5, 8.6], [0.5, 9.3], [0.6, 8.9], [0.6, 8.3], [0.6, 8.0], [0.7, 7.8], [0.7, 8.9], [0.7, 9.7],
[0.7, 9.1], [0.8, 9.2],
[0.8, 8.5], [0.8, 8.4], [0.9, 8.8], [0.9, 8.6], [0.9, 8.2], [1.0, 8.2], [1.0, 6.6], [1.0, 6.3],
[1.0, 6.9], [1.1, 7.1],
[1.1, 7.7], [1.1, 6.5], [1.2, 7.0], [1.2, 7.7], [1.2, 6.1], [1.3, 7.7], [1.3, 6.5], [1.3, 6.9],
[1.3, 5.3], [1.4, 5.7],
[1.4, 5.8], [1.4, 5.6], [1.5, 6.8], [1.5, 6.7], [1.5, 6.6], [1.6, 3.6], [1.6, 5.3], [1.6, 6.9],
[1.6, 5.9], [1.7, 6.0],
[1.7, 4.7], [1.7, 5.0], [1.8, 4.5], [1.8, 5.6], [1.8, 4.2], [1.9, 3.8], [1.9, 4.5], [1.9, 5.8],
[1.9, 6.7], [2.0, 6.5],
[2.0, 6.3], [2.0, 4.9], [2.1, 5.9], [2.1, 3.6], [2.1, 3.8], [2.2, 4.8], [2.2, 4.3], [2.2, 4.6],
[2.2, 4.1], [2.3, 3.5],
[2.3, 2.9], [2.3, 4.4], [2.4, 4.5], [2.4, 3.6], [2.4, 4.3], [2.5, 5.0], [2.5, 2.3], [2.5, 4.4],
[2.5, 6.0], [2.6, 3.4],
[2.6, 3.6], [2.6, 3.6], [2.7, 4.9], [2.7, 3.6], [2.7, 5.1], [2.8, 5.1], [2.8, 3.5], [2.8, 2.0],
[2.8, 3.7], [2.9, 2.5],
[2.9, 3.3], [2.9, 2.8], [3.0, 2.5], [3.0, 1.4], [3.0, 4.1], [3.1, 2.8], [3.1, 4.1], [3.1, 2.2],
[3.1, 3.1], [3.2, 3.2],
[3.2, 3.0], [3.2, 3.7], [3.3, 3.7], [3.3, 2.9], [3.3, 4.0], [3.4, 2.7], [3.4, 3.0], [3.4, 2.3],
[3.4, 1.8], [3.5, 3.4],
[3.5, 3.9], [3.5, 3.1], [3.6, 3.1], [3.6, 2.4], [3.6, 2.1], [3.7, 2.3], [3.7, 1.3], [3.7, 2.7],
[3.8, 2.0], [3.8, 2.2],
[3.8, 3.0], [3.8, 2.0], [3.9, 3.1], [3.9, 1.9], [3.9, 0.0], [4.0, 1.6], [4.0, 1.9], [4.0, 1.8],
[4.1, 2.6], [4.1, 2.0],
[4.1, 1.2], [4.1, 2.5], [4.2, 2.0], [4.2, 0.1], [4.2, 1.7], [4.3, 1.2], [4.3, 2.4], [4.3, 2.1],
[4.4, 1.3], [4.4, 1.0],
[4.4, 1.6], [4.4, 2.8], [4.5, 2.8], [4.5, 2.1], [4.5, 1.9], [4.6, 3.0], [4.6, 2.3], [4.6, 2.3],
[4.7, 3.0], [4.7, 0.4],
[4.7, 1.6], [4.7, 1.1], [4.8, 2.6], [4.8, 2.9], [4.8, 2.9], [4.9, 2.5], [4.9, 2.4], [4.9, 1.9],
[5.0, 1.9], [5.0, 2.9],
[5.0, 1.4], [5.0, 2.0], [5.1, 3.4], [5.1, 2.5], [5.1, 1.7], [5.2, 2.7], [5.2, 2.2], [5.2, 1.9],
[5.3, 1.5], [5.3, 2.6],
[5.3, 1.9], [5.3, 1.2], [5.4, 2.2], [5.4, 2.6], [5.4, 1.2], [5.5, 1.8], [5.5, 2.4], [5.5, 3.0],
[5.6, 2.7], [5.6, 3.6],
[5.6, 2.2], [5.6, 2.4], [5.7, 2.2], [5.7, 3.3], [5.7, 2.2], [5.8, 3.0], [5.8, 0.9], [5.8, 2.6],
[5.9, 2.5], [5.9, 1.5],
[5.9, 2.4], [5.9, 2.1], [6.0, 2.2], [6.0, 1.7], [6.0, 2.8], [6.1, 1.4], [6.1, 2.5], [6.1, 2.0],
[6.2, 2.5], [6.2, 2.5],
[6.2, 1.0], [6.2, 2.4], [6.3, 1.1], [6.3, 2.9], [6.3, 3.5], [6.4, 2.3], [6.4, 5.0], [6.4, 2.8],
[6.5, 1.5], [6.5, 4.0],
[6.5, 3.6], [6.6, 3.8], [6.6, 2.7], [6.6, 2.6], [6.6, 2.1], [6.7, 3.1], [6.7, 3.6], [6.7, 3.5],
[6.8, 2.7], [6.8, 3.0],
[6.8, 2.5], [6.9, 2.9], [6.9, 3.9], [6.9, 3.6], [6.9, 3.4], [7.0, 3.4], [7.0, 3.4], [7.0, 4.5],
[7.1, 3.9], [7.1, 4.6],
[7.1, 4.4], [7.2, 4.1], [7.2, 3.2], [7.2, 2.7], [7.2, 4.2], [7.3, 4.1], [7.3, 5.7], [7.3, 3.7],
[7.4, 3.0], [7.4, 4.0],
[7.4, 3.9], [7.5, 4.3], [7.5, 3.5], [7.5, 4.4], [7.5, 6.2], [7.6, 4.0], [7.6, 5.7], [7.6, 6.6],
[7.7, 6.1], [7.7, 5.4],
[7.7, 2.5], [7.8, 5.6], [7.8, 4.1], [7.8, 5.9], [7.8, 5.1], [7.9, 4.5], [7.9, 5.1], [7.9, 5.5],
[8.0, 5.8], [8.0, 5.0],
[8.0, 6.0], [8.1, 5.8], [8.1, 5.9], [8.1, 5.6], [8.1, 5.2], [8.2, 4.0], [8.2, 6.4], [8.2, 4.5],
[8.3, 6.2], [8.3, 5.7],
[8.3, 5.3], [8.4, 4.9], [8.4, 6.9], [8.4, 5.0], [8.4, 7.4], [8.5, 5.0], [8.5, 7.5], [8.5, 7.1],
[8.6, 6.4], [8.6, 6.0],
[8.6, 7.5], [8.7, 5.8], [8.7, 7.7], [8.7, 6.2], [8.7, 6.6], [8.8, 6.2], [8.8, 8.1], [8.8, 7.7],
[8.9, 7.4], [8.9, 8.2],
[8.9, 7.4], [9.0, 7.6], [9.0, 6.7], [9.0, 7.7], [9.0, 8.2], [9.1, 7.7], [9.1, 9.2], [9.1, 9.1],
[9.2, 8.5], [9.2, 7.4],
[9.2, 8.5], [9.3, 9.2], [9.3, 8.3], [9.3, 9.7], [9.3, 8.5], [9.4, 8.2], [9.4, 9.9], [9.4, 8.5],
[9.5, 9.9], [9.5, 8.7]]划分训练集和测试集
python
# 将 point1 分割为训练集和测试集
random.shuffle(point) # 随机打乱数据
split_index = int(0.1 * len(point)) # 取前 10% 的数据作为测试集
train_point = point[split_index:]
test_point = point[:split_index]
x_train = np.array([point[0] for point in train_point])
y_train = np.array([point[1] for point in train_point])
x_test = np.array([point[0] for point in test_point])
y_test = np.array([point[1] for point in test_point])转换为Tensor张量
python
x_train = torch.from_numpy(x_train).float()
y_train = torch.from_numpy(y_train).float()构建模型
python
class ModelClass(nn.Module):
def __init__(self):
super().__init__()
self.layer1 = nn.Linear(1, 8)
self.layer2 = nn.Linear(8, 1)
def forward(self, x):
x = torch.tanh(self.layer1(x))
x = self.layer2(x)
return x
model = ModelClass()构建损失函数和优化器
python
criterion = nn.MSELoss()
optimizer = torch.optim.Adam(model.parameters(), lr=0.005, weight_decay=0.01)模型训练
python
for n in range(1, 2001):
# 前向传播
y_pred = model(x_train.unsqueeze(1))
# 计算损失
loss = criterion(y_pred.squeeze(1), y_train)
# 反向传播和优化
optimizer.zero_grad()
loss.backward()
optimizer.step()
if n % 100 == 0 or n == 1:
print(n,loss.item())可视化
python
step_list = []
loss_list = []
test_step_list = []
test_loss_list = []
for n in range(1, 2001):
# 前向传播
y_pred = model(x_train.unsqueeze(1))
# 计算损失
loss = criterion(y_pred.squeeze(1), y_train)
# 反向传播和优化
optimizer.zero_grad()
loss.backward()
optimizer.step()
# 更新右侧的损失图数据并绘制
step_list.append(n)
loss_list.append(loss.detach())
# 显示频率设置
if n % 100 == 0 or n == 1:
print(n,loss.item())
# 绘制左侧的拟合图
ax1.clear()
ax1.set_xlim(0, 10)
ax1.set_ylim(0, 10)
ax1.set_xlabel("x axis label")
ax1.set_ylabel("y axis label")
ax1.scatter(x_train, y_train)
x_range = torch.tensor(np.arange(0, 10, 0.1)).unsqueeze(1).float()
y_range = model(x_range).detach().numpy()
ax1.plot(x_range, y_range, 'r') # 绘制拟合线
# 计算测试集损失
y_pred_test = model(torch.tensor(x_test).unsqueeze(1).float())
loss_test = criterion(y_pred_test.squeeze(1), torch.from_numpy(y_test).float())
test_step_list.append(n)
test_loss_list.append(loss_test.detach())
ax2.clear()
ax2.plot(step_list, loss_list, 'r-', label='Train Loss')
ax2.plot(test_step_list, test_loss_list, 'b-', label='Test Loss') # 绘制测试集损失
ax2.set_xlabel("Step")
ax2.set_ylabel("Loss")
ax2.legend()
plt.show()完整代码
python
import numpy as np
import torch
import random
import torch.nn as nn
import matplotlib.pyplot as plt
# 创造数据，数据集
point = [[0.5, 8.6], [0.5, 9.3], [0.6, 8.9], [0.6, 8.3], [0.6, 8.0], [0.7, 7.8], [0.7, 8.9], [0.7, 9.7],
[0.7, 9.1], [0.8, 9.2],
[0.8, 8.5], [0.8, 8.4], [0.9, 8.8], [0.9, 8.6], [0.9, 8.2], [1.0, 8.2], [1.0, 6.6], [1.0, 6.3],
[1.0, 6.9], [1.1, 7.1],
[1.1, 7.7], [1.1, 6.5], [1.2, 7.0], [1.2, 7.7], [1.2, 6.1], [1.3, 7.7], [1.3, 6.5], [1.3, 6.9],
[1.3, 5.3], [1.4, 5.7],
[1.4, 5.8], [1.4, 5.6], [1.5, 6.8], [1.5, 6.7], [1.5, 6.6], [1.6, 3.6], [1.6, 5.3], [1.6, 6.9],
[1.6, 5.9], [1.7, 6.0],
[1.7, 4.7], [1.7, 5.0], [1.8, 4.5], [1.8, 5.6], [1.8, 4.2], [1.9, 3.8], [1.9, 4.5], [1.9, 5.8],
[1.9, 6.7], [2.0, 6.5],
[2.0, 6.3], [2.0, 4.9], [2.1, 5.9], [2.1, 3.6], [2.1, 3.8], [2.2, 4.8], [2.2, 4.3], [2.2, 4.6],
[2.2, 4.1], [2.3, 3.5],
[2.3, 2.9], [2.3, 4.4], [2.4, 4.5], [2.4, 3.6], [2.4, 4.3], [2.5, 5.0], [2.5, 2.3], [2.5, 4.4],
[2.5, 6.0], [2.6, 3.4],
[2.6, 3.6], [2.6, 3.6], [2.7, 4.9], [2.7, 3.6], [2.7, 5.1], [2.8, 5.1], [2.8, 3.5], [2.8, 2.0],
[2.8, 3.7], [2.9, 2.5],
[2.9, 3.3], [2.9, 2.8], [3.0, 2.5], [3.0, 1.4], [3.0, 4.1], [3.1, 2.8], [3.1, 4.1], [3.1, 2.2],
[3.1, 3.1], [3.2, 3.2],
[3.2, 3.0], [3.2, 3.7], [3.3, 3.7], [3.3, 2.9], [3.3, 4.0], [3.4, 2.7], [3.4, 3.0], [3.4, 2.3],
[3.4, 1.8], [3.5, 3.4],
[3.5, 3.9], [3.5, 3.1], [3.6, 3.1], [3.6, 2.4], [3.6, 2.1], [3.7, 2.3], [3.7, 1.3], [3.7, 2.7],
[3.8, 2.0], [3.8, 2.2],
[3.8, 3.0], [3.8, 2.0], [3.9, 3.1], [3.9, 1.9], [3.9, 0.0], [4.0, 1.6], [4.0, 1.9], [4.0, 1.8],
[4.1, 2.6], [4.1, 2.0],
[4.1, 1.2], [4.1, 2.5], [4.2, 2.0], [4.2, 0.1], [4.2, 1.7], [4.3, 1.2], [4.3, 2.4], [4.3, 2.1],
[4.4, 1.3], [4.4, 1.0],
[4.4, 1.6], [4.4, 2.8], [4.5, 2.8], [4.5, 2.1], [4.5, 1.9], [4.6, 3.0], [4.6, 2.3], [4.6, 2.3],
[4.7, 3.0], [4.7, 0.4],
[4.7, 1.6], [4.7, 1.1], [4.8, 2.6], [4.8, 2.9], [4.8, 2.9], [4.9, 2.5], [4.9, 2.4], [4.9, 1.9],
[5.0, 1.9], [5.0, 2.9],
[5.0, 1.4], [5.0, 2.0], [5.1, 3.4], [5.1, 2.5], [5.1, 1.7], [5.2, 2.7], [5.2, 2.2], [5.2, 1.9],
[5.3, 1.5], [5.3, 2.6],
[5.3, 1.9], [5.3, 1.2], [5.4, 2.2], [5.4, 2.6], [5.4, 1.2], [5.5, 1.8], [5.5, 2.4], [5.5, 3.0],
[5.6, 2.7], [5.6, 3.6],
[5.6, 2.2], [5.6, 2.4], [5.7, 2.2], [5.7, 3.3], [5.7, 2.2], [5.8, 3.0], [5.8, 0.9], [5.8, 2.6],
[5.9, 2.5], [5.9, 1.5],
[5.9, 2.4], [5.9, 2.1], [6.0, 2.2], [6.0, 1.7], [6.0, 2.8], [6.1, 1.4], [6.1, 2.5], [6.1, 2.0],
[6.2, 2.5], [6.2, 2.5],
[6.2, 1.0], [6.2, 2.4], [6.3, 1.1], [6.3, 2.9], [6.3, 3.5], [6.4, 2.3], [6.4, 5.0], [6.4, 2.8],
[6.5, 1.5], [6.5, 4.0],
[6.5, 3.6], [6.6, 3.8], [6.6, 2.7], [6.6, 2.6], [6.6, 2.1], [6.7, 3.1], [6.7, 3.6], [6.7, 3.5],
[6.8, 2.7], [6.8, 3.0],
[6.8, 2.5], [6.9, 2.9], [6.9, 3.9], [6.9, 3.6], [6.9, 3.4], [7.0, 3.4], [7.0, 3.4], [7.0, 4.5],
[7.1, 3.9], [7.1, 4.6],
[7.1, 4.4], [7.2, 4.1], [7.2, 3.2], [7.2, 2.7], [7.2, 4.2], [7.3, 4.1], [7.3, 5.7], [7.3, 3.7],
[7.4, 3.0], [7.4, 4.0],
[7.4, 3.9], [7.5, 4.3], [7.5, 3.5], [7.5, 4.4], [7.5, 6.2], [7.6, 4.0], [7.6, 5.7], [7.6, 6.6],
[7.7, 6.1], [7.7, 5.4],
[7.7, 2.5], [7.8, 5.6], [7.8, 4.1], [7.8, 5.9], [7.8, 5.1], [7.9, 4.5], [7.9, 5.1], [7.9, 5.5],
[8.0, 5.8], [8.0, 5.0],
[8.0, 6.0], [8.1, 5.8], [8.1, 5.9], [8.1, 5.6], [8.1, 5.2], [8.2, 4.0], [8.2, 6.4], [8.2, 4.5],
[8.3, 6.2], [8.3, 5.7],
[8.3, 5.3], [8.4, 4.9], [8.4, 6.9], [8.4, 5.0], [8.4, 7.4], [8.5, 5.0], [8.5, 7.5], [8.5, 7.1],
[8.6, 6.4], [8.6, 6.0],
[8.6, 7.5], [8.7, 5.8], [8.7, 7.7], [8.7, 6.2], [8.7, 6.6], [8.8, 6.2], [8.8, 8.1], [8.8, 7.7],
[8.9, 7.4], [8.9, 8.2],
[8.9, 7.4], [9.0, 7.6], [9.0, 6.7], [9.0, 7.7], [9.0, 8.2], [9.1, 7.7], [9.1, 9.2], [9.1, 9.1],
[9.2, 8.5], [9.2, 7.4],
[9.2, 8.5], [9.3, 9.2], [9.3, 8.3], [9.3, 9.7], [9.3, 8.5], [9.4, 8.2], [9.4, 9.9], [9.4, 8.5],
[9.5, 9.9], [9.5, 8.7]]
# 将 point1 分割为训练集和测试集
random.shuffle(point) # 随机打乱数据
split_index = int(0.1 * len(point)) # 取前 10% 的数据作为测试集
# 划分数据集
train_point = point[split_index:] # 训练集包含 90% 的数据
test_point = point[:split_index] # 测试集为前 10% 的数据
# 将训练集和测试集的数据分别提取为特征和目标
x_train = np.array([point[0] for point in train_point]) # 训练特征
y_train = np.array([point[1] for point in train_point]) # 训练目标
x_test = np.array([point[0] for point in test_point]) # 测试特征
y_test = np.array([point[1] for point in test_point]) # 测试目标
# 转换为PyTorch的张量
x_train = torch.from_numpy(x_train).float() # 将训练特征转换为浮点型张量
y_train = torch.from_numpy(y_train).float() # 将训练目标转换为浮点型张量
# 定义前向模型
class ModelClass(nn.Module):
def __init__(self):
super().__init__()
# 定义网络层
self.layer1 = nn.Linear(1, 8) # 第一个线性层，输入为1维，输出为8维
self.layer2 = nn.Linear(8, 1) # 第二个线性层，输入为8维，输出为1维
def forward(self, x):
# 前向传播函数
x = torch.tanh(self.layer1(x)) # 第一个层的输出应用tanh激活函数
x = self.layer2(x) # 经过第二个层
return x
# 实例化模型
model = ModelClass()
# 定义损失函数和优化器
criterion = nn.MSELoss() # 均方误差损失函数
optimizer = torch.optim.Adam(model.parameters(), lr=0.005, weight_decay=0.01) # Adam优化器
# 初始化绘图
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 6)) # 创建绘图窗口，包含两个子图
# 开始迭代
step_list = [] # 用于存储训练的步骤
loss_list = [] # 用于存储训练损失
test_step_list = [] # 用于存储测试的步骤
test_loss_list = [] # 用于存储测试损失
for n in range(1, 2001): # 训练迭代2000轮
# 前向传播
y_pred = model(x_train.unsqueeze(1)) # 将训练输入传入模型，reshape为合适维度
# 计算损失
loss = criterion(y_pred.squeeze(1), y_train) # 计算模型预测值与真实值之间的损失
# 反向传播和优化
optimizer.zero_grad() # 清除之前的梯度
loss.backward() # 计算当前损失的梯度
optimizer.step() # 更新模型参数
# 更新右侧的损失图数据并绘制
step_list.append(n) # 记录当前步数
loss_list.append(loss.detach()) # 记录当前损失值
# 显示频率设置
if n % 100 == 0 or n == 1: # 每100步输出一次损失值
print(n, loss.item()) # 打印当前步数和损失值
# 绘制左侧的拟合图
ax1.clear() # 清除当前图
ax1.set_xlim(0, 10) # 设置x轴范围
ax1.set_ylim(0, 10) # 设置y轴范围
ax1.set_xlabel("x axis label") # x轴标签
ax1.set_ylabel("y axis label") # y轴标签
ax1.scatter(x_train, y_train) # 绘制训练数据点
x_range = torch.tensor(np.arange(0, 10, 0.1)).unsqueeze(1).float() # 生成预测输入范围
y_range = model(x_range).detach().numpy() # 计算拟合线的预测输出
ax1.plot(x_range, y_range, 'r') # 绘制拟合线
# 计算测试集损失
y_pred_test = model(torch.tensor(x_test).unsqueeze(1).float()) # 模型对测试集进行预测
loss_test = criterion(y_pred_test.squeeze(1), torch.from_numpy(y_test).float()) # 计算测试集损失
test_step_list.append(n) # 记录测试步数
test_loss_list.append(loss_test.detach()) # 记录测试损失
ax2.clear() # 清除当前测试损失图
ax2.plot(step_list, loss_list, 'r-', label='Train Loss') # 绘制训练损失
ax2.plot(test_step_list, test_loss_list, 'b-', label='Test Loss') # 绘制测试集损失
ax2.set_xlabel("Step") # x轴标签
ax2.set_ylabel("Loss") # y轴标签
ax2.legend() # 显示图例
plt.show() # 显示绘图窗口