多项式拟合

 

多项式拟合

 

 

In [1]:

%matplotlib inline
import gluonbook as gb
from mxnet import nd,autograd, gluon
from mxnet.gluon import data as gdata,loss as gloss,nn

 

生成数据集 $$y=1.2x - 3.4x^2 + 5.6x^3 + 5 + \alpha$$

In [2]:

n_train = 100
n_test = 100
true_w = [1.2,-3.4,5.6]
true_b = 5

In [10]:

features = nd.random.normal(shape=(n_train+n_test,1))
features.shape

Out[10]:

(200, 1)

In [11]:

poly_features = nd.concat(features, nd.power(features, 2),nd.power(features, 3))
poly_features.shape

Out[11]:

(200, 3)

In [12]:

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)

In [13]:

features[:2], poly_features[:2], labels[:2]

Out[13]:

(
 [[-0.6474401 ]
  [-0.55203336]]
 <NDArray 2x1 @cpu(0)>, 
 [[-0.6474401   0.41917866 -0.27139306]
  [-0.55203336  0.30474085 -0.1682271 ]]
 <NDArray 2x3 @cpu(0)>, 
 [1.1710231 2.493016 ]
 <NDArray 2 @cpu(0)>)

 

定义，训练，测试模型

 

平方损失函数

In [14]:

num_epochs = 100
loss = gloss.L2Loss()

In [17]:

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])
    print('weight:',net[0].weight.data().asnumpy())
    print('bias:',net[0].bias.data().asnumpy())

In [18]:

fit_and_plot(poly_features[:n_train, :], poly_features[n_train:, :],
             labels[:n_train], labels[n_train:])

 

final epoch: train loss 0.0054066028 test loss 0.007267852
weight: [[ 1.1381239 -3.3803644  5.616167 ]]
bias: [4.9635367]

In [19]:

poly_features.shape

Out[19]:

(200, 3)

 

欠拟合

In [21]:

features.shape

Out[21]:

(200, 1)

In [20]:

fit_and_plot(features[:n_train, :], features[n_train:, :], labels[:n_train],
             labels[n_train:])

 

final epoch: train loss 91.158066 test loss 147.58788
weight: [[18.628803]]
bias: [1.0468582]

 

过拟合

In [22]:

fit_and_plot(poly_features[0:2, :], poly_features[n_train:, :], labels[0:2],
             labels[n_train:])

 

final epoch: train loss 0.3388414 test loss 481.85403
weight: [[-0.5109072   0.31804863 -0.13049412]]
bias: [0.9656807]