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本文链接:https://blog.csdn.net/github_39655029/article/details/89425193
来近似y,wk为模型的权重参数,b为偏差参数,多项式函数拟合也像线性回归一样使用平方损失函数。特别地,一阶多项式函数拟合又叫做线性函数拟合。之所以选用K阶函数,主要是因为高阶多项式函数模型参数多,模型函数的选择空间更大,故高阶多项式函数比低阶多项式函数的复杂度更高。高阶多项式函数比低阶多项式函数更易在相同训练数据集上得到更低训练误差。在训练数据集给定的情况下,模型复杂度与误差间关系如下图:
若模型复杂度过低,容易出现欠拟合现象;若模型复杂度过高,易出现过拟合。应对过拟合与欠拟合的一个方法就是针对数据集选择合适复杂度的模型;
#!/usr/bin/env python
# -*- coding: utf-8 -*-
# @Time : 2019/4/22 1:21
# @Author : cunyu
# @Site : cunyu1943.github.io
# @File : PolyFuncFitting.py
# @Software: PyCharm
import d2lzh as d2l
from mxnet import autograd, gluon, nd
from mxnet.gluon import data as gdata, loss as gloss, nn
# 生成数据集
n_train, n_test, true_w, true_b = 100, 100, [1.2, -3.4, 5.6], 5
features = nd.random.normal(shape=(n_train + n_test, 1))
poly_features = nd.concat(features, nd.power(features, 2), nd.power(features, 3))
labels = (true_w[0] * poly_features[:, 0] + true_w[1] * poly_features[:, 1] + true_w[2] * poly_features[:, 2] + true_b)
labels += nd.random.normal(scale=0.1, shape=labels.shape)
print(features[:2], poly_features[:2], labels[:2])
# 定义、训练与测试模型
def semilogy(x_vals, y_vals, x_label, y_label, x2_vals=None, y2_vals=None, legend=None, figsize=(3.5, 2.5)):
d2l.set_figsize(figsize)
d2l.plt.xlabel(x_label)
d2l.plt.ylabel(y_label)
d2l.plt.semilogy(x_vals, y_vals)
if x2_vals and y2_vals:
d2l.plt.semilogy(x2_vals, y2_vals, linestyle=':')
d2l.plt.legend(legend)
d2l.plt.show()
num_epochs, loss = 100, gloss.L2Loss()
def fit_and_plot(train_features, test_features, train_labels, test_labels):
net = nn.Sequential()
net.add(nn.Dense(1))
net.initialize()
batch_size = min(10, train_labels.shape[0])
train_iter = gdata.DataLoader(gdata.ArrayDataset(train_features, train_labels), batch_size, shuffle=True)
trainer = gluon.Trainer(net.collect_params(), 'sgd', {'learning_rate': 0.01})
train_ls, test_ls = [], []
for _ in range(num_epochs):
for X, y in train_iter:
with autograd.record():
l = loss(net(X), y)
l.backward()
trainer.step(batch_size)
train_ls.append(loss(net(train_features),train_labels).mean().asscalar())
test_ls.append(loss(net(test_features),test_labels).mean().asscalar())
print('final epoch: train loss', train_ls[-1], 'test loss', test_ls[-1])
semilogy(range(1, num_epochs + 1), train_ls, 'epochs', 'loss',range(1, num_epochs + 1), test_ls, ['train', 'test'])
print('weight:', net[0].weight.data().asnumpy(),'\nbias:', net[0].bias.data().asnumpy())
# 三阶多项式函数拟合(正常)
fit_and_plot(poly_features[:n_train, :], poly_features[n_train:, :],labels[:n_train], labels[n_train:])
# 线性函数拟合(欠拟合)
fit_and_plot(features[:n_train, :], features[n_train:, :], labels[:n_train],labels[n_train:])
# 训练样本不足(过拟合)
fit_and_plot(poly_features[0:2, :], poly_features[n_train:, :], labels[0:2],labels[n_train:])