回归分析笔记（Live update）

回归方程拟合度评价

• $SST$：总偏差平方和：因变量实际值与因变量平均值的差的平方和，反应因变量取值的总体波动情况
• $SSR$：回归平方和：因变量回归值与其均值的差的平方和，反映了因变量偏差值由X的关系所引起的偏差，可以由回归曲线进行解释。
• $SSE$：残差平方和：因变量实际值与回归值的差的平方和，反映了因变量偏差值由X的关系以外的其他关系所引起的偏差，不可以由回归曲线进行解释。

$$SST=SSR+SSE$$

定义:

$$R2=SSR/SST$$

其取值范围为$0\sim 1$越接近于1说明拟合效果越好。

线性回归实战

sklearn通过linear_model模块（广义线性模型）来进行线性回归预测，其模型原理为：

$$y(w,x)=w_0+w_1x_1+w_2x_2+\cdots+w_px_p$$

数据结构

为了更好地刻画参数，有必要将工艺数据参数化设计成一个数组。

class MaterialProperties:
    def __init__(self, product_id, supplier, steel_type, thickness, width, temperatures, speed, fan_frequency, rolling_before_thickness, rolling_after_thickness, treatment_before_hardness, target_hardness_upper_limit, target_hardness_lower_limit):
        self.product_id = product_id  # 产品唯一编号
        self.supplier = supplier  # 原料商
        self.steel_type = steel_type  # 钢种
        self.thickness = thickness  # 厚度
        self.width = width  # 宽度
        self.temperatures = temperatures  # 温度数组
        self.speed = speed  # 速度
        self.fan_frequency = fan_frequency  # 风机频率
        self.rolling_before_thickness = rolling_before_thickness  # 轧制前厚度
        self.rolling_after_thickness = rolling_after_thickness  # 轧制后厚度
        self.treatment_before_hardness = treatment_before_hardness  # 处理前硬度
        self.target_hardness_upper_limit = target_hardness_upper_limit  # 目标硬度上限
        self.target_hardness_lower_limit = target_hardness_lower_limit  # 目标硬度下限

        # 计算TV和前制程压下率
        self.TV = self.calculate_TV()
        self.pre_process_reduction_rate = self.calculate_pre_process_reduction_rate()

    def calculate_TV(self):
        """计算TV值"""
        return self.thickness * self.speed

    def calculate_pre_process_reduction_rate(self):
        """计算前制程压下率"""
        if self.rolling_before_thickness == 0:
            # 如果轧制前厚度为0，表示一轧一退料，压下率为0
            return 0
        else:
            return (self.rolling_before_thickness - self.rolling_after_thickness) / self.rolling_before_thickness

    def get_property(self, property_name):
        """通过产品编号和属性名获取属性值"""
        return getattr(self, property_name, None)

# 示例数据
product_id = "A001FG"
supplier = "供应商A"
steel_type = "钢种B"
thickness = 10.5
width = 1.2
temperatures = [20, 30, 40, 50, 60]  # 温度区间数组
speed = 100
fan_frequency = 50
rolling_before_thickness = 0  # 轧制前厚度为0，表示一轧一退料
rolling_after_thickness = 10.0
treatment_before_hardness = [30, 40, 50, 60]  # 处理前硬度数组
target_hardness_upper_limit = 50  # 示例目标硬度上限值
target_hardness_lower_limit = 30  # 示例目标硬度下限值

# 创建实例
material = MaterialProperties(product_id, supplier, steel_type, thickness, width, temperatures, speed, fan_frequency, rolling_before_thickness, rolling_after_thickness, treatment_before_hardness, target_hardness_upper_limit, target_hardness_lower_limit)

# 打印实例数据
print(f"{material.product_id}.supplier: {material.supplier}")
print(f"{material.product_id}.steel_type: {material.steel_type}")
print(f"{material.product_id}.thickness: {material.thickness}")
print(f"{material.product_id}.width: {material.width}")
print(f"{material.product_id}.temperatures: {material.temperatures}")
print(f"{material.product_id}.speed: {material.speed}")
print(f"{material.product_id}.fan_frequency: {material.fan_frequency}")
print(f"{material.product_id}.rolling_before_thickness: {material.rolling_before_thickness}")
print(f"{material.product_id}.rolling_after_thickness: {material.rolling_after_thickness}")
print(f"{material.product_id}.treatment_before_hardness: {material.treatment_before_hardness}")
print(f"{material.product_id}.TV: {material.TV}")
print(f"{material.product_id}.pre_process_reduction_rate: {material.pre_process_reduction_rate}")
print(f"{material.product_id}.target_hardness_upper_limit: {material.target_hardness_upper_limit}")
print(f"{material.product_id}.target_hardness_lower_limit: {material.target_hardness_lower_limit}")

一元线性回归