Regression with additional variables to optimize

Question

I am given data which consists of X and Y points (x_1,...x_n; y1,...y_n).

I want to fit X to Y using two basis functions: max(x,mu_1) and min(x,mu_2)

In other words I want to estimate the following equation:

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

I want to find mu_1 and mu_2 such that the fit above is best possible. I mean such mu_1 and mu_2 so that when I fit Y to X sum of squared residual is minimized.

Or I could say that I need a_1, a_2, mu_1, mu_2 such that the sum of squared residuals for the fit above is minimized.

I tried to do the following:

I created the function of two arguments (mu_1 and mu_2) that returns the quality of the fit of Y to X. And then I tried to optimize this function using scipy.optimize.minimize. Here is code:

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 SSR 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

It doesnt work. I will really appreciate any help.

score 1 · Accepted Answer · answered Apr 10 '19 at 15:10

This graphical fitter uses your function, it seems to do what you are asking for.

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)

Regression with additional variables to optimize

1 Answers1