机器学习入门
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既然咱们学Python，也肯定要讲讲机器学习。因为它的很多库都是用Python写的。开始先装上它们玩一下。
Tensorflow
安装一下。
$ pip install tensorflow
ERROR: Could not find a version that satisfies the requirement tensorflow
ERROR: No matching distribution found for tensorflow
$ type python
python is aliased to `/usr/local/Cellar/python@3.9/3.9.1_6/bin/python3'
然而 Tensorflow 2只支持 Python 3.5–3.8。我们用的是3.9 。
% type python3
python3 is /usr/bin/python3
% python3 -V
Python 3.8.2
注意到我系统里的python3是3.8.2版本。这个Python版本对应的pip安装到哪儿呢。
% python3 -m pip -V
pip 21.0.1 from /Users/lzw/Library/Python/3.8/lib/python/site-packages/pip (python 3.8)
对应的pip在这里。那我更改一下.zprofile文件里。最近我更改了我的shell。.zprofile就相当于之前的.bash_profile。加入一行。
alias pip3=/Users/lzw/Library/Python/3.8/bin/pip3
这样，我们用python3和pip3来玩Tensorflow。
% pip3 install tensorflow
...
Successfully installed absl-py-0.12.0 astunparse-1.6.3 cachetools-4.2.1 certifi-2020.12.5 chardet-4.0.0 flatbuffers-1.12 gast-0.3.3 google-auth-1.27.1 google-auth-oauthlib-0.4.3 google-pasta-0.2.0 grpcio-1.32.0 h5py-2.10.0 idna-2.10 keras-preprocessing-1.1.2 markdown-3.3.4 numpy-1.19.5 oauthlib-3.1.0 opt-einsum-3.3.0 protobuf-3.15.6 pyasn1-0.4.8 pyasn1-modules-0.2.8 requests-2.25.1 requests-oauthlib-1.3.0 rsa-4.7.2 tensorboard-2.4.1 tensorboard-plugin-wit-1.8.0 tensorflow-2.4.1 tensorflow-estimator-2.4.0 termcolor-1.1.0 typing-extensions-3.7.4.3 urllib3-1.26.3 werkzeug-1.0.1 wheel-0.36.2 wrapt-1.12.1
安装了很多库。用上官网的一个例子。
import tensorflow as tf
mnist = tf.keras.datasets.mnist
(x_train, y_train), (x_test, y_test) = mnist.load_data()
x_train, x_test = x_train / 255.0, x_test / 255.0
model = tf.keras.models.Sequential([
tf.keras.layers.Flatten(input_shape=(28, 28)),
tf.keras.layers.Dense(128, activation='relu'),
tf.keras.layers.Dropout(0.2),
tf.keras.layers.Dense(10)
])
predictions = model(x_train[:1]).numpy()
print(predictions)
运行一下。
$ /usr/bin/python3 tf.py
Downloading data from https://storage.googleapis.com/tensorflow/tf-keras-datasets/mnist.npz
11493376/11490434 [==============================] - 10s 1us/step
[[ 0.15477428 -0.3877643 0.0994779 0.07474922 -0.26219758 -0.03550266
0.32226565 -0.37141111 0.10925996 -0.0115255 ]]
可见下载了数据集，接着输出了结果。
接下来，看看图片分类的例子。
# TensorFlow and tf.keras
import tensorflow as tf
# Helper libraries
import numpy as np
import matplotlib.pyplot as plt
print(tf.__version__)
报错。
ModuleNotFoundError: No module named 'matplotlib'
安装一下。
% pip3 install matplotlib
正确了。
$ /usr/bin/python3 image.py
2.4.1
进行复制粘贴例子代码。
# TensorFlow and tf.keras
import tensorflow as tf
# Helper libraries
import numpy as np
import matplotlib.pyplot as plt
fashion_mnist = tf.keras.datasets.fashion_mnist
(train_images, train_labels), (test_images, test_labels) = fashion_mnist.load_data()
class_names = ['T-shirt/top', 'Trouser', 'Pullover', 'Dress', 'Coat',
'Sandal', 'Shirt', 'Sneaker', 'Bag', 'Ankle boot']
print(train_images.shape)
print(len(train_labels))
输出了结果。注意到这里有train_images、train_labels、test_images、test_labels。就是分为训练数据集和测试数据集。
(60000, 28, 28)
60000
接着试试打印出图片来。
print(train_images[0])
看下结果。
[[ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0]
[ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0]
[ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0]
[ 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 13 73 0
0 1 4 0 0 0 0 1 1 0]
[ 0 0 0 0 0 0 0 0 0 0 0 0 3 0 36 136 127 62
54 0 0 0 1 3 4 0 0 3]
[ 0 0 0 0 0 0 0 0 0 0 0 0 6 0 102 204 176 134
144 123 23 0 0 0 0 12 10 0]
[ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 155 236 207 178
107 156 161 109 64 23 77 130 72 15]
[ 0 0 0 0 0 0 0 0 0 0 0 1 0 69 207 223 218 216
216 163 127 121 122 146 141 88 172 66]]
....
这里节选了部分结果。
print(len(train_images[0][0]))
输出28。所以很清楚，这是一个横宽为28的矩阵。继续打印。
print(len(train_images[0][0][0])
TypeError: object of type 'numpy.uint8' has no len()
所以很明白。每张图片都是28*28*3的数组。最后一维数组保存的是rgb值。然而发现我们的想法可能是错的。
print(train_images[0][1][20])
0
print(train_images[0][1])
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
说明每张图片是28*28的数组。捣鼓了一阵。我们终于知道了秘密。
先来看看输出的图。
plt.figure()
plt.imshow(train_images[0])
plt.colorbar()
plt.grid(False)
plt.show()
看到右边的颜色条吗。0到250。原来这是在两种颜色里的渐变。可是它怎么知道是哪两种颜色。我们哪里告诉它了。
接着我们把第二张图也打印出来。
plt.imshow(train_images[1])
很有意思。这难道是pyplot依赖库默认的吗。继续运行官网给的代码。
plt.figure(figsize=(10,10))
for i in range(25):
plt.subplot(5,5,i+1)
plt.xticks([])
plt.yticks([])
plt.grid(False)
plt.imshow(train_images[i], cmap=plt.cm.binary)
plt.xlabel(class_names[train_labels[i]])
plt.show()
注意到这里显示了图片以及它们的分类。终于我们知道了cmp参数。如果cmp什么都不写，一定会是刚刚我们那种色彩的。果然。
plt.imshow(train_images[i])
这会我们搜索pyplot cmap。找到一些资料。
plt.imshow(train_images[i], cmap=plt.cm.PiYG)
改一下代码。
plt.figure(figsize=(10,10))
for i in range(25):
plt.subplot(2,5,i+1) ## 改这行
plt.xticks([])
plt.yticks([])
plt.grid(False)
plt.imshow(train_images[i], cmap=plt.cm.Blues)
plt.xlabel(class_names[train_labels[i]])
plt.show()
然而报错了。
ValueError: num must be 1 <= num <= 10, not 11
这意味着什么。之前的5,5,i+1到底什么意思。为什么改成2就不行了。尽管我们直观地知道大概是5行5列的意思。但为什么会报这个错误。11是怎么计算出来的。num又是什么意思。10是什么意思。注意到2*5=10。所以也许当i=11的时候出错了。当改成for i in range(10):时，得到了以下结果。
这会稍微看一下文档，得知subplot(nrows, ncols, index, **kwargs)。嗯，到此我们很明白了。
plt.figure(figsize=(10,10))
for i in range(25):
plt.subplot(5,5,i+1)
# plt.xticks([])
plt.yticks([])
plt.grid(False)
plt.imshow(train_images[i], cmap=plt.cm.Blues)
plt.xlabel(class_names[train_labels[i]])
plt.show()
注意到0 25这种就叫xticks。当我们放大缩小这个框的时候，会有不同的展示。
注意到放大缩小框，xticks和xlabels会有不同的显示。
model = tf.keras.Sequential([
tf.keras.layers.Flatten(input_shape=(28, 28)),
tf.keras.layers.Dense(128, activation='relu'),
tf.keras.layers.Dense(10)
])
model.compile(optimizer='adam',
loss=tf.keras.losses.SparseCategoricalCrossentropy(from_logits=True),
metrics=['accuracy'])
model.fit(train_images, train_labels, epochs=10)
test_loss, test_acc = model.evaluate(test_images, test_labels, verbose=2)
print('\nTest accuracy:', test_acc)
注意到了这里定义model的方式，用到了类Sequential。注意这些参数，28,28、128、relu、10。注意到需要compile和fit。fit是拟合的意思。注意到28,28就是图形大小。
Epoch 1/10
1875/1875 [==============================] - 2s 928us/step - loss: 0.6331 - accuracy: 0.7769
Epoch 2/10
1875/1875 [==============================] - 2s 961us/step - loss: 0.3860 - accuracy: 0.8615
Epoch 3/10
1875/1875 [==============================] - 2s 930us/step - loss: 0.3395 - accuracy: 0.8755
Epoch 4/10
1875/1875 [==============================] - 2s 1ms/step - loss: 0.3071 - accuracy: 0.8890
Epoch 5/10
1875/1875 [==============================] - 2s 1ms/step - loss: 0.2964 - accuracy: 0.8927
Epoch 6/10
1875/1875 [==============================] - 2s 985us/step - loss: 0.2764 - accuracy: 0.8955
Epoch 7/10
1875/1875 [==============================] - 2s 961us/step - loss: 0.2653 - accuracy: 0.8996
Epoch 8/10
1875/1875 [==============================] - 2s 1ms/step - loss: 0.2549 - accuracy: 0.9052
Epoch 9/10
1875/1875 [==============================] - 2s 1ms/step - loss: 0.2416 - accuracy: 0.9090
Epoch 10/10
1875/1875 [==============================] - 2s 1ms/step - loss: 0.2372 - accuracy: 0.9086
313/313 - 0s - loss: 0.3422 - accuracy: 0.8798
Test accuracy: 0.879800021648407
模型已经训练出来了。来改下参数。
model = tf.keras.Sequential([
tf.keras.layers.Flatten(input_shape=(28, 28)),
tf.keras.layers.Dense(28, activation='relu'), # 128 -> 28
tf.keras.layers.Dense(10)
])
修改一下Dense的第一个参数。
Epoch 1/10
1875/1875 [==============================] - 2s 714us/step - loss: 6.9774 - accuracy: 0.3294
Epoch 2/10
1875/1875 [==============================] - 1s 715us/step - loss: 1.3038 - accuracy: 0.4831
Epoch 3/10
1875/1875 [==============================] - 1s 747us/step - loss: 1.0160 - accuracy: 0.6197
Epoch 4/10
1875/1875 [==============================] - 1s 800us/step - loss: 0.7963 - accuracy: 0.6939
Epoch 5/10
1875/1875 [==============================] - 2s 893us/step - loss: 0.7006 - accuracy: 0.7183
Epoch 6/10
1875/1875 [==============================] - 1s 747us/step - loss: 0.6675 - accuracy: 0.7299
Epoch 7/10
1875/1875 [==============================] - 1s 694us/step - loss: 0.6681 - accuracy: 0.7330
Epoch 8/10
1875/1875 [==============================] - 1s 702us/step - loss: 0.6675 - accuracy: 0.7356
Epoch 9/10
1875/1875 [==============================] - 1s 778us/step - loss: 0.6508 - accuracy: 0.7363
Epoch 10/10
1875/1875 [==============================] - 1s 732us/step - loss: 0.6532 - accuracy: 0.7350
313/313 - 0s - loss: 0.6816 - accuracy: 0.7230
Test accuracy: 0.7229999899864197
注意到Test accuracy前后发生了变化。Epoch这样的是fit函数输出的日志。注意到当是128时，accuracy从0.7769变到0.9086。而当是28时，accuracy从0.3294变到0.7350。这会注意到，我们先是用训练集，来调优loss和accuracy。接着用测试数据集来测试。先来看看train_labels。
print(train_labels)
[9 0 0 ... 3 0 5]
print(len(train_labels))
60000
这意味着用0到9来表示这些类别。刚好class_names也是有10个。
class_names = ['T-shirt/top', 'Trouser', 'Pullover', 'Dress', 'Coat',
'Sandal', 'Shirt', 'Sneaker', 'Bag', 'Ankle boot']
再来改一改。
model = tf.keras.Sequential([
tf.keras.layers.Flatten(input_shape=(28, 28)),
tf.keras.layers.Dense(28, activation='relu'),
tf.keras.layers.Dense(5) # 10 -> 5
])
model.compile(optimizer='adam',
loss=tf.keras.losses.SparseCategoricalCrossentropy(from_logits=True),
metrics=['accuracy'])
model.fit(train_images, train_labels, epochs=10)
出错了。
tensorflow.python.framework.errors_impl.InvalidArgumentError: Received a label value of 9 which is outside the valid range of [0, 5). Label values: 4 3 2 9 4 1 6 0 7 9 1 6 5 2 3 8 6 3 8 0 3 5 6 1 2 6 3 6 8 4 8 4
[[node sparse_categorical_crossentropy/SparseSoftmaxCrossEntropyWithLogits/SparseSoftmaxCrossEntropyWithLogits (defined at /curiosity-courses/ml/tf/image.py:53) ]] [Op:__inference_train_function_538]
Function call stack:
train_function
改成把Sequential的第三个参数Dense的参数改成15就可以了。结果区别不大。试试改改Epoch。
model = tf.keras.Sequential([
tf.keras.layers.Flatten(input_shape=(28, 28)),
tf.keras.layers.Dense(28, activation='relu'),
tf.keras.layers.Dense(15)
])
model.compile(optimizer='adam',
loss=tf.keras.losses.SparseCategoricalCrossentropy(from_logits=True),
metrics=['accuracy'])
model.fit(train_images, train_labels, epochs=15) # 10 -> 15
test_loss, test_acc = model.evaluate(test_images, test_labels, verbose=2)
print('\nTest accuracy:', test_acc)
Epoch 1/15
1875/1875 [==============================] - 2s 892us/step - loss: 6.5778 - accuracy: 0.3771
Epoch 2/15
1875/1875 [==============================] - 2s 872us/step - loss: 1.3121 - accuracy: 0.4910
Epoch 3/15
1875/1875 [==============================] - 2s 909us/step - loss: 1.0900 - accuracy: 0.5389
Epoch 4/15
1875/1875 [==============================] - 1s 730us/step - loss: 1.0422 - accuracy: 0.5577
Epoch 5/15
1875/1875 [==============================] - 1s 709us/step - loss: 0.9529 - accuracy: 0.5952
Epoch 6/15
1875/1875 [==============================] - 1s 714us/step - loss: 0.9888 - accuracy: 0.5950
Epoch 7/15
1875/1875 [==============================] - 1s 767us/step - loss: 0.8678 - accuracy: 0.6355
Epoch 8/15
1875/1875 [==============================] - 1s 715us/step - loss: 0.8247 - accuracy: 0.6611
Epoch 9/15
1875/1875 [==============================] - 1s 721us/step - loss: 0.8011 - accuracy: 0.6626
Epoch 10/15
1875/1875 [==============================] - 1s 711us/step - loss: 0.8024 - accuracy: 0.6622
Epoch 11/15
1875/1875 [==============================] - 1s 781us/step - loss: 0.7777 - accuracy: 0.6696
Epoch 12/15
1875/1875 [==============================] - 1s 724us/step - loss: 0.7764 - accuracy: 0.6728
Epoch 13/15
1875/1875 [==============================] - 1s 731us/step - loss: 0.7688 - accuracy: 0.6767
Epoch 14/15
1875/1875 [==============================] - 1s 715us/step - loss: 0.7592 - accuracy: 0.6793
Epoch 15/15
1875/1875 [==============================] - 1s 786us/step - loss: 0.7526 - accuracy: 0.6792
313/313 - 0s - loss: 0.8555 - accuracy: 0.6418
Test accuracy: 0.6417999863624573
注意改成15。区别也不大。 tf.keras.layers.Dense(88, activation='relu'),是重要的。试着128改成88。得到了Test accuracy: 0.824999988079071。128时，是0.879800021648407。28时，是0.7229999899864197。是不是越大越好，然而当改成256时，是Test accuracy: 0.8409000039100647。这不禁让我们思考loss和accuracy的含义。
probability_model = tf.keras.Sequential([model,
tf.keras.layers.Softmax()])
接下来预测一下。注意到Sequential和上面的一样。注意到参数model和 tf.keras.layers.Softmax()。
probability_model = tf.keras.Sequential([model,
tf.keras.layers.Softmax()])
predictions = probability_model.predict(test_images)
def plot_image(i, predictions_array, true_label, img):
true_label, img = true_label[i], img[i]
plt.grid(False)
plt.xticks([])
plt.yticks([])
plt.imshow(img, cmap=plt.cm.binary)
predicted_label = np.argmax(predictions_array)
if predicted_label == true_label:
color = 'blue'
else:
color = 'red'
plt.xlabel("{} {:2.0f}% ({})".format(class_names[predicted_label],
100*np.max(predictions_array),
class_names[true_label]),
color=color)
def plot_value_array(i, predictions_array, true_label):
true_label = true_label[i]
plt.grid(False)
plt.xticks(range(10))
plt.yticks([])
thisplot = plt.bar(range(10), predictions_array, color="#777777")
plt.ylim([0, 1])
predicted_label = np.argmax(predictions_array)
thisplot[predicted_label].set_color('red')
thisplot[true_label].set_color('blue')
i = 0
plt.figure(figsize=(6,3))
plt.subplot(1,2,1)
plot_image(i, predictions[i], test_labels, test_images)
plt.subplot(1,2,2)
plot_value_array(i, predictions[i], test_labels)
plt.show()
这说明这个图片99%的可能是Ankle boot。注意到plot_image是显示左边的图。plot_value_array是输出右边的图。
num_rows = 5
num_cols = 3
num_images = num_rows*num_cols
plt.figure(figsize=(2*2*num_cols, 2*num_rows))
for i in range(num_images):
plt.subplot(num_rows, 2*num_cols, 2*i+1)
plot_image(i, predictions[i], test_labels, test_images)
plt.subplot(num_rows, 2*num_cols, 2*i+2)
plot_value_array(i, predictions[i], test_labels)
plt.tight_layout()
plt.show()
注意到这里只是显示更多的测试结果。所以使用流程我们大致很清楚。所以我们还不知道背后怎么计算的。但我们知道如何使用它们。它们背后是微积分。如何理解微积分呢。
比如说有个数字，1到100让你猜。每次你猜多少。我告诉你小了还是大了。你猜50。我说小了。你猜80。我说大了。你猜65。我说大了。你猜55。我说小了。你猜58。我说，嗯，猜对了。
机器学习，就是在背后模拟类似的过程。只不过复杂一些。可能是很多的1到100，要猜很多数。同时每次猜都要进行很多运算。以及每次判断是否大了还是小了，要计算很多。
PyTorch
安装一下。这个支持3.9版本的Python。
$ pip install torch torchvision
Collecting torch
Downloading torch-1.8.0-cp39-none-macosx_10_9_x86_64.whl (120.6 MB)
|████████████████████████████████| 120.6 MB 224 kB/s
Collecting torchvision
Downloading torchvision-0.9.0-cp39-cp39-macosx_10_9_x86_64.whl (13.1 MB)
|████████████████████████████████| 13.1 MB 549 kB/s
Requirement already satisfied: numpy in /usr/local/lib/python3.9/site-packages (from torch) (1.20.1)
Collecting typing-extensions
Downloading typing_extensions-3.7.4.3-py3-none-any.whl (22 kB)
Requirement already satisfied: pillow>=4.1.1 in /usr/local/lib/python3.9/site-packages (from torchvision) (8.0.1)
Installing collected packages: typing-extensions, torch, torchvision
Successfully installed torch-1.8.0 torchvision-0.9.0 typing-extensions-3.7.4.3
检验一下。
import torch
x = torch.rand(5, 3)
print(x)
出错了。
Traceback (most recent call last):
File "torch.py", line 1, in <module>
import torch
File "torch.py", line 2, in <module>
x = torch.rand(5, 3)
AttributeError: partially initialized module 'torch' has no attribute 'rand' (most likely due to a circular import)
谷歌一下这个错误信息。原来是因为我们的文件也叫torch。重名了。改一下然后就正确了。
tensor([[0.5520, 0.9446, 0.5543],
[0.6192, 0.0908, 0.8726],
[0.0223, 0.7685, 0.9814],
[0.4019, 0.5406, 0.3861],
[0.5485, 0.6040, 0.2387]])
找到一个例子。
# -*- coding: utf-8 -*-
import torch
import math
dtype = torch.float
device = torch.device("cpu")
# device = torch.device("cuda:0") # Uncomment this to run on GPU
# Create random input and output data
x = torch.linspace(-math.pi, math.pi, 2000, device=device, dtype=dtype)
y = torch.sin(x)
# Randomly initialize weights
a = torch.randn((), device=device, dtype=dtype)
b = torch.randn((), device=device, dtype=dtype)
c = torch.randn((), device=device, dtype=dtype)
d = torch.randn((), device=device, dtype=dtype)
learning_rate = 1e-6
for t in range(2000):
# Forward pass: compute predicted y
y_pred = a + b * x + c * x ** 2 + d * x ** 3
# Compute and print loss
loss = (y_pred - y).pow(2).sum().item()
if t % 100 == 99:
print(t, loss)
# Backprop to compute gradients of a, b, c, d with respect to loss
grad_y_pred = 2.0 * (y_pred - y)
grad_a = grad_y_pred.sum()
grad_b = (grad_y_pred * x).sum()
grad_c = (grad_y_pred * x ** 2).sum()
grad_d = (grad_y_pred * x ** 3).sum()
# Update weights using gradient descent
a -= learning_rate * grad_a
b -= learning_rate * grad_b
c -= learning_rate * grad_c
d -= learning_rate * grad_d
print(f'Result: y = {a.item()} + {b.item()} x + {c.item()} x^2 + {d.item()} x^3')
运行一下。
99 1273.537353515625
199 849.24853515625
299 567.4786987304688
399 380.30291748046875
499 255.92752075195312
599 173.2559814453125
699 118.2861328125
799 81.72274780273438
899 57.39331817626953
999 41.198158264160156
1099 30.41307830810547
1199 23.227672576904297
1299 18.438262939453125
1399 15.244369506835938
1499 13.113286972045898
1599 11.690631866455078
1699 10.740333557128906
1799 10.105220794677734
1899 9.6804780960083
1999 9.39621353149414
Result: y = -0.011828352697193623 + 0.8360244631767273 x + 0.002040589228272438 x^2 + -0.09038365632295609 x^3
看看只用numpy库的代码。
# -*- coding: utf-8 -*-
import numpy as np
import math
# Create random input and output data
x = np.linspace(-math.pi, math.pi, 2000)
y = np.sin(x)
# Randomly initialize weights
a = np.random.randn()
b = np.random.randn()
c = np.random.randn()
d = np.random.randn()
learning_rate = 1e-6
for t in range(2000):
# Forward pass: compute predicted y
# y = a + b x + c x^2 + d x^3
y_pred = a + b * x + c * x ** 2 + d * x ** 3
# Compute and print loss
loss = np.square(y_pred - y).sum()
if t % 100 == 99:
print(t, loss)
# Backprop to compute gradients of a, b, c, d with respect to loss
grad_y_pred = 2.0 * (y_pred - y)
grad_a = grad_y_pred.sum()
grad_b = (grad_y_pred * x).sum()
grad_c = (grad_y_pred * x ** 2).sum()
grad_d = (grad_y_pred * x ** 3).sum()
# Update weights
a -= learning_rate * grad_a
b -= learning_rate * grad_b
c -= learning_rate * grad_c
d -= learning_rate * grad_d
print(f'Result: y = {a} + {b} x + {c} x^2 + {d} x^3')
注意到这是两种方式来计算。
这两个例子，是先生成了一组x和y。接着假设是三次方程。再接着用些方法把系数迭代计算出来。这些算法是怎样的呢。注意到是循环了2000次。每次拟合地精确一些。这里先不细究。
最后
目前，我们不懂机器学习背后是怎么计算的。然而，暂时不重要。我们用上面类似的知识已经可以来干很多事情了。还可以用机器学习来处理文本、音频等的。 等我们试探了几十个例子，再学原理也不迟。
练习
- 学生像上面那样探索一遍。