Predictive accuracy¶
We compare the predictive accuracy of convex regression with several different machine learning models for regression. The models are:
Linear regression
Ridge regression
Lasso regression
Support vector regression
We use the same data as in the previous example.
The following examples are run in python 3.9.15 with Intel(R) Core(TM) i5-1135G7 CPU @ 2.40GHz.
Example: Estimating convex function and calculate mean squared errors¶
# import packages
import numpy as np
from cvxreg.models import CR, PCR
from sklearn import linear_model, svm
# generate data
np.random.seed(0)
n, d, SNR = 100, 3, 3
x = np.random.uniform(low=-1, high=1, size=(n*2, d))
y_true = np.linalg.norm(x, axis=1)**2 + 3
sigma = np.sqrt(np.var(y_true, ddof=1, axis=0)/SNR)
nse = np.random.normal(0, sigma, n*2)
y = y_true + nse
# split data into training and testing
x_tr, y_tr = x[:n,:], y[:n]
x_te, y_te = x[-n:,:], y_true[-n:]
# fit the penalized convex regression model
pcr = PCR(c=0.01)
pcr.fit(x_tr, y_tr)
# calculate mean squared errors
y_hat = pcr.predict(x_te)
mse = np.mean((y_te - y_hat)**2)
print('Mean squared error of PCR: %.4f' % mse) # MSE = 0.024
# fit the linear regression model
lr = linear_model.LinearRegression()
lr.fit(x_tr, y_tr)
# calculate mean squared errors
y_hat = lr.predict(x_te)
mse = np.mean((y_te - y_hat)**2)
print('Mean squared error of linear regression: %.4f' % mse) # MSE = 0.310
# fit the ridge regression model
rr = linear_model.Ridge(alpha=0.01)
rr.fit(x_tr, y_tr)
# calculate mean squared errors
y_hat = rr.predict(x_te)
mse = np.mean((y_te - y_hat)**2)
print('Mean squared error of ridge regression: %.4f' % mse) # MSE = 0.310
# fit the lasso regression model
lasso = linear_model.Lasso(alpha=0.01)
lasso.fit(x_tr, y_tr)
# calculate mean squared errors
y_hat = lasso.predict(x_te)
mse = np.mean((y_te - y_hat)**2)
print('Mean squared error of lasso regression: %.4f' % mse) # MSE = 0.308
# fit the support vector regression model
svr = svm.SVR(kernel='rbf')
svr.fit(x_tr, y_tr)
# calculate mean squared errors
y_hat = svr.predict(x_te)
mse = np.mean((y_te - y_hat)**2)
print('Mean squared error of support vector regression: %.4f' % mse) # MSE = 0.042
We concude that the penalized convex regression model has the best predictive accuracy.
