Question

我正在使用SCAD罚函数实现阈值回归，SCAD具有自己定义的导数，因此我也实现了SCAD导数的功能。我将导数传递给“优化”的jac参数。我首先使用没有SCAD惩罚功能的目标功能来进行初步猜测，并且它可以正常工作。但是，当使用带有SCAD的惩罚函数来解决此问题时，迭代器无法给出正确的结果，并返回“由于精度损失而不一定实现的期望误差”。

我该怎么办？

import pandas as pd
import scipy as sc
from scipy import stats as st
import numpy as np
from scipy import optimize as opt
from mpl_toolkits.mplot3d import Axes3D
from matplotlib import pyplot as plt

def loss_SCAD(beta_matrix):
    M_MSE = np.dot(x, beta_matrix[:tmp_lt])[:, None]
    M_c = np.array(beta_matrix[-M:], ndmin=1)
    scad = 0

    for idx, c in enumerate(M_c):
        nor_distri_x = (x[:, 1][:, None] - c) / tmp_h
        nor_distri_CDF = st.norm.cdf(nor_distri_x)  # 光滑示性函数
        M_MSE += np.dot(x, beta_matrix[tmp_lt * (idx + 1):tmp_lt * (idx + 2)])[:, None] * nor_distri_CDF
        beta_param = np.sum(np.abs(beta_matrix[tmp_lt * (idx + 1):tmp_lt * (idx + 2)]))
        if beta_param > a * tmp_lam:
            scad += 0.5 * ((a + 1) * tmp_lam ** 2)
        elif beta_param > tmp_lam:
            scad += ((a ** 2 - 1) * tmp_lam ** 2 - (beta_param - a * tmp_lam) ** 2) / (2 * (a - 1))
        else:
            scad += tmp_lam * beta_param

    MSE = 0.5 * np.average((y - M_MSE) ** 2) + scad
    return MSE


def loss_derivative(beta_matrix):
    deriv_list = []
    other_M_MSE, nor_distri_CDF = 0, 0
    first_M_MSE = y - np.dot(x, beta_matrix[:tmp_lt])[:, None]
    M_c = np.array(beta_matrix[-M:], ndmin=1)

    for idx, c in enumerate(M_c):
        nor_distri_x = (x[:, 1][:, None] - c) / tmp_h
        nor_distri_CDF = st.norm.cdf(nor_distri_x)  # 光滑示性函数
        other_M_MSE += np.dot(x, beta_matrix[tmp_lt * (idx + 1):tmp_lt * (idx + 2)])[:, None] * nor_distri_CDF

    square_part = first_M_MSE - other_M_MSE

    for idx, beta_param in enumerate(beta_matrix[:tmp_lt]):  # 更新梯度：第一段
        deriv_list.append(np.average(square_part * -x[:, idx][:, None]))

    for idx, c in enumerate(M_c):
        beta_param = np.sum(np.abs(beta_matrix[tmp_lt * (idx + 1):tmp_lt * (idx + 2)]))
        if beta_param == 0:  # 更新梯度：SCAD函数
            scad = 0
        elif beta_param > tmp_lam:
            scad = np.max(((a * tmp_lam - beta_param) / (a - 1), 0))
        else:
            scad = tmp_lam
        for i in range(tmp_lt):  # 更新梯度：分段
            deriv_list.append(np.average(square_part * -x[:, i][:, None] * nor_distri_CDF) + scad)

    for idx, c in enumerate(M_c):  # 更新梯度：门槛参数
        nor_distri_PDF = st.norm.pdf((tmp_x - c) / tmp_h)
        deriv_list.append(np.average(square_part * np.dot(x, beta_matrix[tmp_lt * (idx + 1):tmp_lt * (idx + 2)])[:, None] * nor_distri_PDF / tmp_h))

    return np.array(deriv_list)


def loss(beta_matrix):
    M_MSE = np.dot(x, beta_matrix[:tmp_lt])[:, None]
    M_c = np.array(beta_matrix[-M:], ndmin=1)

    for idx, c in enumerate(M_c):
        nor_distri_x = (x[:, 1][:, None] - c) / tmp_h
        nor_distri_CDF = st.norm.cdf(nor_distri_x)  # 光滑示性函数
        M_MSE += np.dot(x, beta_matrix[tmp_lt * (idx + 1):tmp_lt * (idx + 2)])[:, None] * nor_distri_CDF

    MSE = 0.5 * np.average((y - M_MSE) ** 2)
    return MSE


np.random.seed(1171359)
a = 3.7
tmp_x = np.random.normal(loc=0, scale=2, size=(400, 1))
tmp_beta = [2, -2, 5, 2, 9, 7]
tmp_c = [-np.sqrt(2), np.sqrt(2)]
M = np.shape(tmp_c)[0]
x1, x2, x3 = tmp_x[tmp_x < tmp_c[0]], tmp_x[(tmp_x >= tmp_c[0]) & (tmp_x < tmp_c[1])], tmp_x[tmp_x >= tmp_c[1]]
y1, y2, y3 = tmp_beta[0] + tmp_beta[1] * x1, tmp_beta[2] + tmp_beta[3] * x2, tmp_beta[4] + tmp_beta[5] * x3
x = np.reshape(np.concatenate((x1, x2, x3)), (-1, 1))
y = np.reshape(np.concatenate((y1, y2, y3)) + np.random.randn(400), (-1, 1))
x = np.concatenate((np.ones_like(x), x), axis=1)
tmp_lt = x.shape[1]
tmp_h = np.log(400) / 400 * tmp_x.std(ddof=1)
tmp_lam = 0.5

tmp_res = opt.minimize(fun=loss, x0=np.append(np.ones(7), 2))
tmp_result = opt.minimize(fun=loss_SCAD, x0=tmp_res.x, jac=loss_derivative)
# print(tmp_result.x, tmp_res.x, sep='\n')

k = tmp_result.x[:-2].reshape((2, -1), order='F')
y_hat_1 = np.dot(x[x[:, 1] < tmp_result.x[-2]], k[:, 0].reshape((-1, 1)))
y_hat_2 = np.dot(x[(x[:, 1] >= tmp_result.x[-2]) & (x[:, 1] < tmp_result.x[-1])], k[:, 0].reshape((-1, 1)) + k[:, 1].reshape((-1, 1)))
y_hat_3 = np.dot(x[x[:, 1] >= tmp_result.x[-1]], k[:, 0].reshape((-1, 1)) + k[:, 1].reshape((-1, 1)) + k[:, 2].reshape((-1, 1)))
print(k, tmp_result.fun, tmp_result, sep='\n')

fig = plt.figure(111, figsize=(6, 6))
plt.plot(x[:, 1][x[:, 1] < tmp_result.x[-2]], y_hat_1, c='orange', alpha=0.6, linewidth=4)
plt.plot(x[:, 1][(x[:, 1] >= tmp_result.x[-2]) & (x[:, 1] < tmp_result.x[-1])], y_hat_2, c='blue', alpha=0.6, linewidth=4)
plt.plot(x[:, 1][x[:, 1] >= tmp_result.x[-1]], y_hat_3, c='green', alpha=0.6, linewidth=4)
plt.plot(x1, y1, c='black', alpha=0.6, linewidth=4)
plt.plot(x2, y2, c='black', alpha=0.6, linewidth=4)
plt.plot(x3, y3, c='black', alpha=0.6, linewidth=4)
plt.scatter(x[:, 1], y, s=12, alpha=0.5, c='gray')
plt.axis([tmp_x.min() * 1.05, tmp_x.max() * 1.05, y.min() * 1.05, y.max() * 1.05])
plt.show()

fun: 1.6869804934437402
 hess_inv: array([[1, 0, 0, 0, 0, 0, 0, 0],
       [0, 1, 0, 0, 0, 0, 0, 0],
       [0, 0, 1, 0, 0, 0, 0, 0],
       [0, 0, 0, 1, 0, 0, 0, 0],
       [0, 0, 0, 0, 1, 0, 0, 0],
       [0, 0, 0, 0, 0, 1, 0, 0],
       [0, 0, 0, 0, 0, 0, 1, 0],
       [0, 0, 0, 0, 0, 0, 0, 1]])
      jac: array([ 1.98384534e-07, -2.81814249e-07, -1.51727486e-08, -2.17460279e-08,
       -1.51727486e-08, -2.17460279e-08, -1.15503569e+00, -6.99892939e-01])
  message: 'Desired error not necessarily achieved due to precision loss.'
     nfev: 84
      nit: 0
     njev: 72
   status: 2
  success: False
        x: array([ 1.93751362, -2.04506983,  3.09129046,  4.13404934,  4.32073797,
        4.87646363, -1.391239  ,  1.4099789 ])

关于“ Scipy.optimize”：消息：“由于精度损失而不一定实现所需的错误

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