在有了相关性理论基础之后，我们即可创建一个可用于回归模型实验的数据生成器。
# 科学计算模块
import numpy as np
import pandas as pd
# 绘图模块
import matplotlib as mpl
import matplotlib.pyplot as plt
为了方便后续练习的展开，我们尝试自己创建一个数据生成器，用于自主生成一些符合某些条件、难度可控、具备某些特性的数据集。机器学习发展至今，在追求模型效果提升的过程中，模型本身可解释性逐渐变弱，对于很多集成类算法，很多模型构建思路也在朝向深度学习靠拢。这使得很多模型内部逐渐呈现灰箱甚至是黑箱的状态。但是，在初学过程中，我们仍然需要通过类似控制变量的方法、通过设计一些实验，去深入理解算法运行原理及一些优化方法的实际作用，这就需要我们自己动手，创建一些数据用于实验的原材料，通过一些实验深入了解模型原理，从“炼丹师”朝着“化学家”更进一步。
和Lesson 1中介绍的一样，课程中案例分为三类，分别是基础阶段的手动创建数据集、实战阶段的竞赛数据集和企业应用实战阶段的企业数据集。
我们先尝试生成两个特征、存在偏差，自变量和因变量存在线性关系的数据集
num_inputs = 2 # 两个特征
num_examples = 1000 # 总共一千条数据
然后尝试通过线性方程，确定自变量和因变量的真实关系
np.random.seed(24) # 设置随机数种子
np.random.randn(2, 2)
array([[ 1.32921217, -0.77003345],
[-0.31628036, -0.99081039]])
# 线性方程系数
w_true = np.array([2, -1]).reshape(-1, 1)
b_true = np.array(1)
# 扰动项相关
delta = 0.01
# 创建数据集的特征和标签取值
features = np.random.randn(num_examples, num_inputs)
labels_true = features.dot(w_true) + b_true
labels = labels_true + np.random.normal(size = labels_true.shape) * delta
features
array([[-1.07081626, -1.43871328],
[ 0.56441685, 0.29572189],
[-1.62640423, 0.2195652 ],
...,
[-1.44029131, 0.50439425],
[ 0.96604603, 1.76623359],
[ 0.57852053, -1.34028424]])
labels_true
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[ 1.83311182e+00],
[-2.47237367e+00],
[ 4.68336867e-01],
[ 2.81906560e+00],
[-8.12559166e-01],
[ 2.84911190e+00],
[ 8.16104836e-01],
[-2.23673251e+00],
[ 3.84903763e+00],
[ 3.57341849e-02],
[-6.31876174e-01],
[ 6.71457939e-01],
[-2.91185026e-01],
[ 5.69912892e+00],
[ 5.94732168e+00],
[ 1.08837176e-01],
[-4.63796210e-01],
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[-2.68017804e-01],
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[-1.86902545e-01],
[ 4.59659037e+00],
[ 2.91058551e+00],
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[ 1.88353970e+00],
[ 4.21970513e+00],
[-2.42388274e+00],
[-4.45801817e-01],
[ 7.09539251e-01],
[ 8.42884225e-01],
[ 1.66446119e+00],
[ 1.79378819e+00],
[ 3.79466825e+00],
[ 4.28262610e-01],
[-2.47199822e-02],
[-6.14811073e-01],
[ 6.59037492e-01],
[ 3.65450106e-02],
[ 2.66413132e+00],
[-1.61813369e+00],
[-1.67832099e+00],
[ 3.60926477e+00],
[ 3.74660261e+00],
[ 4.08402095e+00],
[ 3.90033163e+00],
[ 1.88854392e+00],
[ 1.63930928e+00],
[-5.48631262e-01],
[-3.06707095e+00],
[ 2.27559884e+00],
[ 2.48563735e+00],
[ 1.01023250e+00],
[-1.76039190e+00],
[-2.51795291e+00],
[ 2.26999709e+00],
[ 8.07294015e+00],
[ 3.01612410e-01],
[-1.18445060e+00],
[-6.66219484e-01],
[-9.20169796e-01],
[ 1.40495318e-01],
[ 7.18067702e-01],
[ 2.26010349e+00],
[-2.47306482e-01],
[ 1.35788517e+00],
[ 5.79860656e-01],
[ 9.93134661e-01],
[ 2.88895417e+00],
[-2.51038664e-01],
[ 3.53970607e+00],
[ 3.10975153e+00],
[-6.54386757e-01],
[ 3.30016782e+00],
[ 4.46137703e+00],
[-1.11180760e+00],
[-2.38497688e+00],
[ 1.16585846e+00],
[ 3.49732531e+00]])
注意，此时labels_true和features满足严格意义上的线性方程关系
但我们实际使用的标签labels，则是在labels_true的基础上增添了一个扰动项，np.random.normal(size = labels_true.shape) * delta，这其实也符合我们一般获取数据的情况：真实客观世界或许存在某个规律，但我们搜集到的数据往往会因为各种原因存在一定的误差，无法完全描述真实世界的客观规律，这其实也是模型误差的来源之一（另一个误差来源是模型本身捕获规律的能力）。这其中，$y=2x_1-x_2+1$相当于我们从上帝视角创建的数据真实服从的规律，而扰动项，则相当于人为创造的获取数据时的误差。
简单查看我们创建的数据集
features[: 10]
array([[-1.07081626, -1.43871328],
[ 0.56441685, 0.29572189],
[-1.62640423, 0.2195652 ],
[ 0.6788048 , 1.88927273],
[ 0.9615384 , 0.1040112 ],
[-0.48116532, 0.85022853],
[ 1.45342467, 1.05773744],
[ 0.16556161, 0.51501838],
[-1.33693569, 0.56286114],
[ 1.39285483, -0.06332798]])
labels[: 10]
array([[ 0.29161817],
[ 1.83851813],
[-2.46058022],
[ 0.44394438],
[ 2.8133944 ],
[-0.8109745 ],
[ 2.85143778],
[ 0.83156296],
[-2.22624102],
[ 3.84279053]])
plt.subplot(121)
plt.scatter(features[:, 0], labels) # 第一个特征和标签的关系
plt.subplot(122)
plt.scatter(features[:,, 1] labels) # 第二个特征和标签的关系
<AxesSubplot:>
<matplotlib.collections.PathCollection at 0x285fb4a3448>
<AxesSubplot:>
<matplotlib.collections.PathCollection at 0x285fb5cd248>
不难看出，两个特征和标签都存在一定的线性关系，并且跟特征的系数绝对值有很大关系。当然，若要增加线性模型的建模难度，可以增加扰动项的数值比例，从而削弱线性关系。
# 设置随机数种子
np.random.seed(24)
# 修改因变量
labels1 = labels_true + np.random.normal(size = labels_true.shape) * 2
# 可视化展示
# 扰动较小的情况
plt.subplot(221)
plt.scatter(features[:, 0], labels) # 第一个特征和标签的关系
plt.subplot(222)
plt.plot(features[:, 1], labels, 'ro') # 第二个特征和标签的关系
# 扰动较大的情况
plt.subplot(223)
plt.scatter(features[:, 0], labels1) # 第一个特征和标签的关系
plt.subplot(224)
plt.plot(features[:, 1], labels1, 'yo') # 第二个特征和标签的关系
<AxesSubplot:>
<matplotlib.collections.PathCollection at 0x285f92baec8>
<AxesSubplot:>
[<matplotlib.lines.Line2D at 0x285fb6f9fc8>]
<AxesSubplot:>
<matplotlib.collections.PathCollection at 0x285fb6e5388>
<AxesSubplot:>
[<matplotlib.lines.Line2D at 0x285fb75ea88>]
当然，我们也能生成非线性关系的数据集，此处我们创建满足$y=2x^2+1$规律的数据集。
np.power([2, 3], 2)
array([4, 9], dtype=int32)
# 设置随机数种子
np.random.seed(24)
num_inputs = 1 # 一个特征
num_examples = 1000 # 总共一千条数据
# 线性方程系数
w_true = np.array(2)
b_true = np.array(1)
# 特征和标签取值
features = np.random.randn(num_examples, num_inputs)
labels_true = np.power(features, 2) * w_true + b_true
labels = labels_true + np.random.normal(size = labels_true.shape) * 2
# 可视化展示
plt.scatter(features, labels)
<matplotlib.collections.PathCollection at 0x267c29d1588>
关于曲线相关，其实也可将其转化为线性相关，例如上例中，我们只需要新增一列$x_2=x$即可构建一个线性模型$y=x_2+1$，该方法也是特征工程中的一种特征衍生的方法。
此时需要注意的是，无论是创建了曲线规律的数据集，还是增加了扰动项绝对数值，都会对建模造成困难。但二者的造成困难的方式是不同的。如果是曲线规律的数据集，则代表规律本身更加复杂，此时需要使用更加复杂的模型来进行规律提取，该部分属于模型选取和参数调优的技术内容；而如果是扰动项比较大，则代表规律被掩盖的比较深，此时需要采用其他技术手段进行白噪声的处理，该部分属于特征工程技术内容。但本节内，无论是曲线规律还是白噪声数值过大，都会对线性方程建模造成困难。
这也就是控制数据集建模难度的最基础的抓手。
为了方便后续使用，我们将上述过程封装在一个函数内
A = np.arange(4).reshape(2, 2)
A
array([[0, 1],
[2, 3]])
np.power(A, 3)
array([[ 0, 1],
[ 8, 27]], dtype=int32)
A.dot(2)
array([[0, 2],
[4, 6]])
np.ones_like(A)
array([[1, 1],
[1, 1]])
def arrayGenReg(num_examples = 1000, w = [2, -1, 1], bias = True, delta = 0.01, deg = 1):
"""回归类数据集创建函数。
:param num_examples: 创建数据集的数据量
:param w: 包括截距的（如果存在）特征系数向量
:param bias：是否需要截距
:param delta：扰动项取值
:param deg：方程最高项次数
:return: 生成的特征数组和标签数组
"""
if bias == True:
num_inputs = len(w)-1 # 数据集特征个数
features_true = np.random.randn(num_examples, num_inputs) # 原始特征
w_true = np.array(w[:-1]).reshape(-1, 1) # 自变量系数
b_true = np.array(w[-1]) # 截距
labels_true = np.power(features_true, deg).dot(w_true) + b_true # 严格满足人造规律的标签
features = np.concatenate((features_true, np.ones_like(labels_true)), axis=1) # 加上全为1的一列之后的特征
else:
num_inputs = len(w)
features = np.random.randn(num_examples, num_inputs)
w_true = np.array(w).reshape(-1, 1)
labels_true = np.power(features, deg).dot(w_true)
labels = labels_true + np.random.normal(size = labels_true.shape) * delta
return features, labels
注：上述函数默认创建的是一个满足$y=2x_1-x_2+1$的数据集。
首先查看扰动项较小的时候的数据情况
# 设置随机数种子
np.random.seed(24)
# 扰动项取值为0.01
f, l = arrayGenReg(delta=0.01)
f
array([[ 1.32921217, -0.77003345, 1. ],
[-0.31628036, -0.99081039, 1. ],
[-1.07081626, -1.43871328, 1. ],
...,
[ 1.5507578 , -0.35986144, 1. ],
[-1.36267161, -0.61353562, 1. ],
[-1.44029131, 0.50439425, 1. ]])
# 绘制图像查看结果
plt.subplot(121)
plt.scatter(f[:, 0], l) # 第一个特征和标签的关系
plt.subplot(122)
plt.scatter(f[:, 1], l) # 第二个特征和标签的关系
<AxesSubplot:>
<matplotlib.collections.PathCollection at 0x267c2ac7948>
<AxesSubplot:>
<matplotlib.collections.PathCollection at 0x267c2afdf48>
然后查看扰动项较大时数据情况
# 设置随机数种子
np.random.seed(24)
# 扰动项取值为2
f, l = arrayGenReg(delta=2)
# 绘制图像查看结果
plt.subplot(121)
plt.scatter(f[:, 0], l) # 第一个特征和标签的关系
plt.subplot(122)
plt.scatter(f[:, 1], l) # 第二个特征和标签的关系
<AxesSubplot:>
<matplotlib.collections.PathCollection at 0x285fbef0048>
<AxesSubplot:>
<matplotlib.collections.PathCollection at 0x285fbd1ba48>
当特征和标签满足二阶关系时候数据表现
# 设置随机数种子
np.random.seed(24)
# 最高次项为2
f, l = arrayGenReg(deg=2)
# 绘制图像查看结果
plt.subplot(121)
plt.scatter(f[:, 0], l) # 第一个特征和标签的关系
plt.subplot(122)
plt.scatter(f[:, 1], l) # 第二个特征和标签的关系
<AxesSubplot:>
<matplotlib.collections.PathCollection at 0x285fc080f08>
<AxesSubplot:>
<matplotlib.collections.PathCollection at 0x285fc15cb48>
在定义好数据创建函数之后，即可开始尝试手动实现线性回归模型。
该数据生成器并非为线性回归模型量身定制，而是通用与此后我们学习的所有回归类模型。关于分类模型数据集的创建，我们将在介绍逻辑回归时讲解。
此外，对于机器学习模型，我们将更多的从数据规律和模型提取规律角度出发，并不会太多顾及模型可解释性等问题。上述数据本身也是从规律角度出发构建的数据，并不具备“解释性”。
此处我们使用了一个随机数种子，以确保随机结果的可重复性。当我们设置某个随机数种子之后，每次随机过程都是可重复的：我们只需要再次调用相同随机种子，就可以重复此前随机过程。
np.random.randn(9)
array([ 1.62779574, 0.35449279, 1.03752763, -0.38568351, 0.519818 ,
1.68658289, -1.32596315, 1.4289837 , -2.08935428])
np.random.seed(24)
np.random.randn(9)
array([ 1.32921217, -0.77003345, -0.31628036, -0.99081039, -1.07081626,
-1.43871328, 0.56441685, 0.29572189, -1.62640423])
np.random.randn(9)
array([ 0.2195652 , 0.6788048 , 1.88927273, 0.9615384 , 0.1040112 ,
-0.48116532, 0.85022853, 1.45342467, 1.05773744])
np.random.seed(24)
np.random.randn(9)
array([ 1.32921217, -0.77003345, -0.31628036, -0.99081039, -1.07081626,
-1.43871328, 0.56441685, 0.29572189, -1.62640423])
np.random.randn(9)
array([ 0.2195652 , 0.6788048 , 1.88927273, 0.9615384 , 0.1040112 ,
-0.48116532, 0.85022853, 1.45342467, 1.05773744])
当然，不同随机数种子所诞生的随机过程是不一样的，这也是“种子”的由来。此外，不同库中的随机过程需要用的不同随机种子也是不同的，比如上述我们用np.random.seed来规定numpy中相关随机过程，但如果是其他第三方库，如random库所定义的随机过程，就需要使用random库中的随机种子来确定。
import random
l = list(range(5))
l
[0, 1, 2, 3, 4]
random.seed(24)
random.shuffle(l)
l
[4, 2, 0, 1, 3]
random.shuffle(l)
l
[1, 4, 0, 3, 2]
l = list(range(5))
l
[0, 1, 2, 3, 4]
random.seed(24)
random.shuffle(l)
l
[4, 2, 0, 1, 3]
random.shuffle(l)
l
[1, 4, 0, 3, 2]
l = list(range(5))
l
[0, 1, 2, 3, 4]
np.random.seed(24)
random.shuffle(l)
l
[4, 0, 3, 2, 1]
random.shuffle(l)
l
[0, 4, 3, 1, 2]
l = list(range(5))
l
[0, 1, 2, 3, 4]
np.random.seed(24)
random.shuffle(l)
l
[4, 2, 3, 1, 0]
根据课程安排，本节定义的函数将后续课程中将经常使用，因此需要将其封装为一个模块方便后续调用。封装为模块有以下几种基本方法：
然后将文件保存在jupyter主目录下，并取名为ML_basic_function，后续即可通过import ML_basic_function进行调用。如果是jupyterlab用户，也可按照如下方式进行编写：
Step 1.打开左侧文件管理栏页，点击新建
Step 2.在新建目录中，选择Test File，并通过重命名将其命名为ML_basic_function.py
Step 3.在打开的文本编辑器中输入代码
并且需要注意，需要在py文件内导入相关的包。
Step 4.测试能否调用
首先重启当前jupyter
然后尝试导入自定义模块
from ML_basic_function import *
查看相关函数说明并使用
arrayGenReg?
Signature: arrayGenReg(num_examples=1000, w=[2, -1, 1], bias=True, delta=0.01, deg=1) Docstring: 回归类数据集创建函数。 :param num_examples: 创建数据集的数据量 :param w: 包括截距的（如果存在）特征系数向量 :param bias：是否需要截距 :param delta：扰动项取值 :param deg：方程最高项次数 :return: 生成的特征张和标签张量 File: f:\code file\机器学习实战\ml_basic_function.py Type: function
arrayGenReg()
(array([[-0.02288559, 0.71327691, 1. ],
[ 1.28499182, -1.56530647, 1. ],
[ 0.49971435, 0.67030438, 1. ],
...,
[-0.28042436, 0.67200216, 1. ],
[ 1.36287814, 1.02653665, 1. ],
[-1.07159586, -0.01419314, 1. ]]),
array([[ 2.39827879e-01],
[ 5.12912037e+00],
[ 1.33103149e+00],
[ 1.05298983e+00],
[ 4.68984787e+00],
[-8.31814680e-01],
[ 1.21287761e+00],
[ 1.61414139e+00],
[ 7.36961630e-01],
[ 4.35536832e-01],
[ 5.66268172e+00],
[ 1.91769533e+00],
[ 1.47292540e+00],
[ 1.78401415e-01],
[ 4.39015628e+00],
[-3.09484999e+00],
[-1.63199019e+00],
[ 2.97309215e-01],
[ 1.87154712e+00],
[ 2.69122838e+00],
[ 4.69491048e+00],
[ 2.92370059e+00],
[-1.39516732e+00],
[-3.60360214e+00],
[ 2.51687694e+00],
[ 4.89787404e+00],
[ 7.99627248e-01],
[ 2.34579580e+00],
[ 1.61243843e+00],
[-3.42711478e+00],
[ 1.97015844e+00],
[-1.29401638e+00],
[-6.13607906e-01],
[ 9.15959629e-01],
[ 1.61507913e+00],
[-7.73493234e-01],
[-3.04178989e-01],
[ 5.54798304e+00],
[ 1.51008541e-01],
[ 4.08553512e+00],
[-8.62090212e-01],
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[ 2.33028404e-01],
[-2.23458097e-01],
[ 2.69561611e+00],
[-1.11481705e+00]]))
至此就完成了自定义模块的编写与保存，后续我们还将陆续写入其他函数。