带有要优化的附加变量的回归

Posted 2023-03-12

【中文标题】带有要优化的附加变量的回归

【英文标题】：Regression with additional variables to optimize

【发布时间】：2019-08-31 16:17:17

【问题描述】：

我得到了由 X 和 Y 点组成的数据(x_1,...x_n; y1,...y_n)。

我想使用两个基函数将 X 拟合到 Y：max(x,mu_1) 和 min(x,mu_2)

换句话说，我想估计以下等式：

y_i = a_1*max(x_i,mu_1)+a_2*min(x_i,mu_2)

我想找到mu_1 和mu_2 以便最适合上面的情况。我的意思是 mu_1 和 mu_2 这样当我拟合 Y 到 X 时，残差平方和最小化。

或者我可以说我需要a_1、a_2、mu_1、mu_2 以使上述拟合的残差平方和最小化。

我尝试执行以下操作：

我创建了两个参数 (mu_1 and mu_2) 的函数，它返回 Y 与 X 的拟合质量。然后我尝试使用 scipy.optimize.minimize 优化此函数。这是代码：

import numpy as np
from scipy.optimize import minimize
from sklearn.linear_model import LinearRegression

###Create X and Y
X = np.random.normal(10,1,size = 10000)
Y = np.random.normal(20,1,size = 10000)

###Create function that estimates quality of fit

def func(mu_1,mu_2):
   ### basis functions
   regressor_1 = np.maximum(X,mu_1).reshape(-1,1)
   regressor_2 = np.minimum(X,mu_2).reshape(-1,1)
   x_train = np.hstack((regressor_1,regressor_2))

   model = LinearRegression().fit(x_train,Y)

   ###I didnt find how to extract sum of squared residual, but I can get R 
   squared, so I thought that minimizing s-s-r is the same as maximizing R 
   squared and it is the same as minimizing -R^2

   objective = model.score(x_train,Y)
   return -1*objective

### Now I want to find such mu_1 and mu_2 that minimize "func"
minimum = minimize(func,0,0)
minimum.x

它不起作用。我将非常感谢任何帮助。

【参考方案1】：

这个图形拟合器使用您的功能，它似乎可以满足您的要求。

import numpy, scipy, matplotlib
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit
import warnings

xData = numpy.array([1.1, 2.2, 3.3, 4.4, 5.0, 6.6, 7.7])
yData = numpy.array([1.1, 20.2, 30.3, 60.4, 50.0, 60.6, 70.7])


def func(x, a_1, a_2, mu_1, mu_2):
    retArray = []
    for x_i in x: # process data points individually
        val = a_1*max(x_i,mu_1) + a_2*min(x_i,mu_2)
        retArray.append(val)
    return retArray


# turn off the curve_fit() "covariance estimation" warning
warnings.filterwarnings("ignore")

# these are the same as the scipy defaults
initialParameters = numpy.array([1.0, 1.0, 1.0, 1.0])

# curve fit the test data
fittedParameters, pcov = curve_fit(func, xData, yData, initialParameters)

modelPredictions = func(xData, *fittedParameters) 

absError = modelPredictions - yData

SE = numpy.square(absError) # squared errors
MSE = numpy.mean(SE) # mean squared errors
RMSE = numpy.sqrt(MSE) # Root Mean Squared Error, RMSE
Rsquared = 1.0 - (numpy.var(absError) / numpy.var(yData))

print('Parameters:', fittedParameters)
print('RMSE:', RMSE)
print('R-squared:', Rsquared)

print()


##########################################################
# graphics output section
def ModelAndScatterPlot(graphWidth, graphHeight):
    f = plt.figure(figsize=(graphWidth/100.0, graphHeight/100.0), dpi=100)
    axes = f.add_subplot(111)

    # first the raw data as a scatter plot
    axes.plot(xData, yData,  'D')

    # create data for the fitted equation plot
    xModel = numpy.linspace(min(xData), max(xData))
    yModel = func(xModel, *fittedParameters)

    # now the model as a line plot
    axes.plot(xModel, yModel)

    axes.set_xlabel('X Data') # X axis data label
    axes.set_ylabel('Y Data') # Y axis data label

    plt.show()
    plt.close('all') # clean up after using pyplot

graphWidth = 800
graphHeight = 600
ModelAndScatterPlot(graphWidth, graphHeight)