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import inspect
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from tkinter import *
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import tkinter as tk
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import mpl_toolkits.axisartist as axisartist
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import matplotlib.pyplot as plt
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from scipy.optimize import curve_fit
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from scipy.optimize import approx_fprime
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import data as gl_data
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import numpy as np
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from X1 import *
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#------------------借用setphoto（）
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set_img1 = None
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set_img2 = None
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# 这里有个将图像输出到主界面的函数
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def set_phtot(index1):
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global set_img1,set_img2
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# 显示图像
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Canvas2 = gl_data.Canvas2
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if index1 == 1:
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if set_img2 != None:
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Canvas2.delete(set_img2)
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gl_data.Img1=PhotoImage(file=r"dot.png")
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# print(gl_data.Img1)
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gl_data.Img1 = gl_data.Img1.zoom(2, 2)
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set_img1=Canvas2.create_image(2, 10, image=gl_data.Img1, anchor=NW)
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# Canvas2.itemconfigure(set_img1,int(420 * gl_data.MAGNIDICATION), int(300 * gl_data.MAGNIDICATION))
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Canvas2.update()
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print("已输出数据点")
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elif index1 == 2:
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if set_img1 != None:
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Canvas2.delete(set_img1)
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gl_data.Img2=PhotoImage(file=r"line.png")
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# print(gl_data.Img2)
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gl_data.Img2 = gl_data.Img2.zoom(2, 2)
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set_img2=Canvas2.create_image(2, 10, image=gl_data.Img2, anchor=NW)
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# Canvas2.itemconfigure(set_img2, 0,0,int(420*gl_data.MAGNIDICATION), int(300*gl_data.MAGNIDICATION))
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Canvas2.update()
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print("已输出数据点和曲线")
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def window():
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root_window = Tk()
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root_window.title('函数拟合')
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root_window.geometry('900x600') # 设置窗口大小:宽x高,注,此处不能为 "*",必须使用 "x"
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# 设置主窗口的背景颜色,颜色值可以是英文单词，或者颜色值的16进制数,除此之外还可以使用Tk内置的颜色常量
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root_window["background"] = "white"
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root_window.resizable(0, 0) # 防止用户调整尺寸
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label1 = tk.Label(root_window, text="样本数据\n集文件", font=('Times', 8), bg="white",
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width=13, height=3, # 设置标签内容区大小
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padx=0, pady=0, borderwidth=0, )
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label1.place(x=10, y=4) # 设置填充区距离、边框宽度和其样式（凹陷式）
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label3 = tk.Label(root_window, text="", font=('Times', 8), bg="white", fg="black",
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width=39, height=2, padx=0, pady=0, borderwidth=0, relief="ridge", highlightcolor="blue")
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label3.place(x=122, y=10)
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return root_window
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# if __name__ == '__main__':
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# root_window = window()
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# root_window.mainloop()
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def draw_axis(low, high, step=250):
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fig = plt.figure(figsize=(4.4, 3.2)) # 设置显示大小
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ax = axisartist.Subplot(fig, 111) # 使用axisartist.Subplot方法创建一个绘图区对象ax
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fig.add_axes(ax) # 将绘图区对象添加到画布中
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ax.axis[:].set_visible(False)# 通过set_visible方法设置绘图区所有坐标轴隐藏
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ax.axis["x"] = ax.new_floating_axis(0, 0) # 添加新的x坐标轴
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ax.axis["x"].set_axisline_style("-|>", size=1.0) # 给x坐标轴加上箭头
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ax.axis["y"] = ax.new_floating_axis(1, 0) # 添加新的y坐标轴
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ax.axis["y"].set_axisline_style("-|>", size=1.0) # y坐标轴加上箭头
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ax.axis["x"].set_axis_direction("bottom") # 设置x、y轴上刻度显示方向
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ax.axis["y"].set_axis_direction("left") # 设置x、y轴上刻度显示方向
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# plt.xlim(low, high) # 把x轴的刻度范围设置
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# plt.ylim(low, high) # 把y轴的刻度范围设置
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# 把x轴和y轴的刻度范围设置
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ax.set_xlim(low, high)
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ax.set_ylim(low, high)
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ax.set_xticks(np.arange(low, high + 5, step)) # 把x轴的刻度间隔设置
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ax.set_yticks(np.arange(low, high + 5, step)) # 把y轴的刻度间隔设置
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# plt.show()
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return ax
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# if __name__ == '__main__':
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# draw_axis(gl_data.LOW, gl_data.HIGH)
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def selfdata_show(sampleData, low, high):
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x = sampleData[:,0]
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y = sampleData[:,1]
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ax = draw_axis(low, high) # 绘制x、y轴
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ax.scatter(x, y, c='red') # 绘制样本数据点
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# plt.show() #显示图片
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plt.savefig(r"dot.png", facecolor='w', bbox_inches='tight') # 保存为png文件
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plt.clf()
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set_phtot(1) # 后续函数将图像显示在主界面中
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# if __name__ == '__main__':
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# sampleData = random_points(20, 0, 1000) # 生成样本点
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# selfdata_show(sampleData, 0, 1000) # 绘制样本数据点
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# 编程4.5-----------------------------------------
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# m次多项式的最小二乘法
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# 这里是没有对数据进行了标准化处理的
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def least_square_method_UnNormalized(m, sampleData):
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x = sampleData[:, 0]
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y = sampleData[:, 1]
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# 使用标准化的x构造A矩阵
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A = np.vander(x, m + 1, increasing=True)
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# 计算拟合系数theta，使用最小二乘法的正规方程
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theta = np.linalg.inv(A.T @ A) @ A.T @ y
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# 计算残差平方和和其他统计量
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residuals = y - A @ theta
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residuals_squared_sum = residuals.T @ residuals
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degrees_of_freedom = len(y) - (m + 1)
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mean_squared_error = residuals_squared_sum / degrees_of_freedom
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covariance_matrix = np.linalg.inv(A.T @ A) * mean_squared_error
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return theta, covariance_matrix
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# 这里是对数据进行了标准化处理的
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def least_square_method(m, sampleData):
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x = sampleData[:, 0]
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y = sampleData[:, 1]
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# 标准化x
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x_mean = x.mean()
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x_std = x.std()
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x_normalized = (x - x_mean) / x_std
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# 使用标准化的x构造A矩阵
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A = np.vander(x_normalized, m + 1, increasing=True)
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# 计算拟合系数theta，使用最小二乘法的正规方程
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theta = np.linalg.inv(A.T @ A) @ A.T @ y
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# 计算残差平方和和其他统计量
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residuals = y - A @ theta
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residuals_squared_sum = residuals.T @ residuals
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degrees_of_freedom = len(y) - (m + 1)
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mean_squared_error = residuals_squared_sum / degrees_of_freedom
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covariance_matrix = np.linalg.inv(A.T @ A) * mean_squared_error
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return theta, covariance_matrix
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# def curve_draw_line(curveData,sampleData):
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# x_fit = curveData[:,0]
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# y_fit = curveData[:,1]
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# x = sampleData[:,0]
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# y = sampleData[:,1]
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# # 创建图形对象并绘制拟合曲线和样本点
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# ax = draw_axis(0, 1000,step=250)
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# ax.plot(x_fit, y_fit, label=f'FitCurve')
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# ax.scatter(x, y, color='red', label='SampleData')
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# # ax.set_xlabel('X')
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# # ax.set_ylabel('Y')
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# plt.legend()
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# plt.show()
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def draw_dots_and_line(curveData, sampleData, low,high):
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x_fit, y_fit = curveData[:, 0], curveData[:, 1]
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x, y = sampleData[:, 0], sampleData[:, 1]
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# 创建图形对象并绘制拟合曲线和样本点
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ax = draw_axis(low,high,step=250)
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ax.plot(x_fit, y_fit, label=f'FitCurve')
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ax.scatter(x, y, color='red', label='SampleData')
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# ax.set_xlabel('X')
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# ax.set_ylabel('Y')
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plt.legend()
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plt.show()
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# 重写compute_curveData以适应标准化的需要
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def compute_curveData2(low, high, step, theta, m, x_mean, x_std):
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x_fit = np.arange(low, high, step)
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# 将x_fit标准化
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x_fit_normalized = (x_fit - x_mean) / x_std
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A_fit = np.vander(x_fit_normalized, m + 1, increasing=True)
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y_fit = A_fit @ theta
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return np.column_stack((x_fit, y_fit))
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if __name__ == '__main__':
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# sampleData = random_points(20,0, 1000)
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# x = sampleData[:,0]
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# y = sampleData[:,1]
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# 这里为了便于观察使用设计好的数据
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x = np.array([0, 100, 200, 300, 400, 500, 600, 700, 800, 900])
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sy = np.array([10, 20, 10, 50, 80, 130, 210, 340, 550, 890])
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sampleData = np.array(list(zip(x, sy)))# 将两个一维数组拼接成二维数组
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m = 3 # 使用三次函数拟合
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theta, covariance_matrix = least_square_method(m, sampleData)
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curveData = compute_curveData2(0,1000,1,theta,m, x.mean(), x.std())
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draw_dots_and_line(curveData,sampleData,0,1000)
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# 编程4.5 END-----------------------------------------
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# 编程 4.12 -----------------------------------------
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def gradient_descent_method(sampleData, degree, learning_rate=0.001, iterations=50000):
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def error_function(y_pred, y): # 定义误差函数error_function()
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return y_pred - y
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def gradient(degree,error,x_normalized,sampleNum,learning_rate): # 定义梯度函数gradient()
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# 对每个参数计算梯度
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for i in range(degree + 1):
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# 计算损失函数对参数的偏导数（梯度）
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gradient = np.dot(error, x_normalized**i) * 2 / sampleNum
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# 梯度下降更新参数
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theta[i] -= learning_rate * gradient
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return theta
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sx, sy = sampleData[:, 0], sampleData[:, 1]
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# 初始化参数（多项式系数）为0
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theta = np.zeros(degree + 1)
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sampleNum = len(sx) # 样本数量
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# 归一化x以改善数值稳定性
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x_normalized = (sx - sx.mean()) / sx.std()
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# 迭代梯度下降
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for _ in range(iterations):
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# 通过当前参数计算多项式的值
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y_pred = np.polyval(theta[::-1], x_normalized)
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# 计算预测值与真实值之间的误差
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error = error_function(y_pred, sy)
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# 计算梯度并更新theta
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theta = gradient(degree, error, x_normalized, sampleNum, learning_rate)
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# --------------------
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# 设计矩阵
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A = np.vander(x_normalized, N=degree + 1, increasing=True)
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# 计算残差方差
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residuals = sy - np.polyval(theta[::-1], x_normalized)
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residuals_squared_sum = np.sum(residuals ** 2)
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variance = residuals_squared_sum / (sampleNum - (degree + 1))
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# 估计协方差矩阵
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# 这里使用A的伪逆来近似最小二乘法中的(A^T A)^-1项
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covariance_matrix = np.linalg.pinv(A.T @ A) * variance
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# --------------------
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return theta, covariance_matrix
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def compute_curveData2(low, high, step, theta, m, x_mean, x_std):
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x_fit = np.arange(low, high, step)
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# 将x_fit标准化
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x_fit_normalized = (x_fit - x_mean) / x_std
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A_fit = np.vander(x_fit_normalized, m + 1, increasing=True)
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y_fit = A_fit @ theta
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return np.column_stack((x_fit, y_fit))
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if __name__ == '__main__':
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# sampleData = random_points(20,0, 1000)
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# sx, sy = sampleData[:, 0], sampleData[:, 1]
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# 这里为了便于观察使用设计好的数据
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sx = np.array([0, 100, 200, 300, 400, 500, 600, 700, 800, 900])
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sy = np.array([10, 20, 10, 50, 80, 130, 210, 340, 550, 890])
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sampleData = np.array(list(zip(sx, sy)))
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# 设置一个较小的学习率和较多的迭代次数
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learning_rate = 0.001
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iterations = 50000
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degree = 4
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# 梯度下降算法求解
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theta, covariance_matrix = gradient_descent_method(sampleData, degree, learning_rate, iterations)
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# 计算画图所需要的拟合数据
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curveData = compute_curveData2(0, 1000, 1, theta, degree, sx.mean(), sx.std())
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# 画拟合结果
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draw_dots_and_line(curveData, sampleData, 0, 1000)
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# 这里是为了后续fit函数中调用梯度下降法
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# 这里将梯度下降法的输入变为了与curve_fit相同，由于协方差矩阵只对于多项式函数适用，因此这里返回的第二个值为False
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# 这里搞好了，就是函数的梯度不能够确定，因此调用了包approx_fprime
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def gradient_descent_method_X5(func, sx, sy, learning_rate=0.001, iterations=5000):
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def error_function(y_pred, y): # 定义误差函数error_function()
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return (y_pred - y) ** 2
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# 定义损失函数和梯度的计算
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def loss_grad(theta):
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y_pred = func(sx_normalized, *theta)
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# 计算预测值与真实值之间的误差
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error = error_function(y_pred, sy)
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# 调用包来实现对任意函数计算梯度
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grad = approx_fprime(theta, lambda t: np.mean((func(sx_normalized, *t) - sy) ** 2), epsilon=1e-6)
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return error, grad
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# 因为X4和X5中的函数方式不一样，X5中不是function类，没有self参数，因此要处理不同的参数要求
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sig = inspect.signature(func) # 获取所有参数的名称
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params = [param.name for param in sig.parameters.values()]
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# print(params)
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# 动态确定func需要的参数数量（减去self, sx或者只有sx）
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if 'self' in params:
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params_count = func.__code__.co_argcount - 2
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else:
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params_count = func.__code__.co_argcount - 1
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# print(params_count)
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# 基于参数数量初始化theta
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theta = np.random.randn(params_count) * 0.01
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# 标准化sx
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mean_sx = np.mean(sx)
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std_sx = np.std(sx)
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sx_normalized = (sx - mean_sx) / std_sx
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# 梯度下降循环
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for _ in range(iterations):
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err, grad = loss_grad(theta)
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theta -= learning_rate * grad
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# 由于不再特定于多项式函数，以下部分相关于残差方差和协方差矩阵的计算需要根据实际情况调整
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# 但是实际没有使用到这个矩阵，因此返回False
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return theta, False
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