列蜜瘘 发表于 2025-5-29 09:03:42

《Machine Learning in Action》—— 浅谈线性回归的那些事

tags:机器学习
《Machine Learning in Action》—— 浅谈线性回归的那些事

手撕机器学习算法系列文章已经肝了不少,自我感觉质量都挺不错的。目前已经更新了支持向量机SVM、决策树、K-近邻(KNN)、贝叶斯分类,读者可根据以下内容自行“充电”(持续更新中):

[*]《Machine Learning in Action》—— 剖析支持向量机,单手狂撕线性SVM: https://www.zybuluo.com/tianxingjian/note/1755051
[*]《Machine Learning in Action》—— 剖析支持向量机,优化SMO: https://www.zybuluo.com/tianxingjian/note/1756832
[*]《Machine Learning in Action》—— 懂的都懂,不懂的也能懂。非线性支持向量机: https://www.zybuluo.com/tianxingjian/note/1758624
[*]《Machine Learning in Action》—— Taoye给你讲讲决策树到底是支什么“鬼”: https://www.zybuluo.com/tianxingjian/note/1757662
[*]《Machine Learning in Action》—— 小朋友,快来玩啊,决策树呦: https://www.zybuluo.com/tianxingjian/note/1758236
[*]《Machine Learning in Action》—— 女同学问Taoye,KNN应该怎么玩才能通关: https://www.zybuluo.com/tianxingjian/note/1759263
[*]《Machine Learning in Action》—— 白话贝叶斯,“恰瓜群众”应该恰好瓜还是恰坏瓜: https://www.zybuluo.com/tianxingjian/note/1758540
[*]《Machine Learning in Action》—— 浅谈线性回归的那些事: https://www.zybuluo.com/tianxingjian/note/1761762
阅读过上述那些文章的读者应该都知道,上述内容都属于分类算法,分类的目标变量都是标称型数据(关于标称型和数据型数据的意思的可参考上文)。而本文主要讲述的内容是线性回归,它是一种回归拟合问题,会对连续性数据做出预测,而非判别某个样本属于哪一类。
本文主要包括的内容有如下几部分:

[*]线性回归,我们来谈谈小姐姐如何相亲
[*]揭开梯度下降算法的神秘面纱
[*]基于梯度下降算法实现线性回归拟合
一、线性回归,我们来谈谈小姐姐如何相亲

回归预测,回归预测,说到底就包括两个部分。
一个是回归(拟合),另一个是预测。回归是为预测做准备或者说是铺垫,只有基于已有的数据集我们才能构建一个的回归模型,然后根据这个回归模型来处理新样本数据的过程就是预测。
而线性回归就是我们的回归模型属于线性的,也就是说我们样本的每个属性特征都最多是一次的(进来的读者 应该都知道吧)
为了让读者对线性回归有个基本的了解,我们来聊聊小姐姐的相亲故事。
故事是这样的。
在很久很久以前,有位小姐姐打算去相亲,她比较在意对象的薪资情况,但这种事情也不太好意思直入主题,你说是吧?所以呢,她就想着通过相亲对象本身的属性特征来达到一个预测薪资的目的。假如说这位小姐姐认为对象的薪资主要有两个部分的数据的组成,分别是对象的年龄和头发量。对此,小姐姐想要构建出这么一个关于薪资的线性模型:
https://www.cnblogs.com/data:image/png;base64,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
中文形式的描述就是:
https://www.cnblogs.com/data:image/png;base64,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
所以呢,小姐姐现在的目的就是想要得到这么一个https://www.cnblogs.com/data:image/png;base64,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的值,然后观察和询问相亲对象的发量以及年龄,就可以根据这个线性模型计算得出其相亲对象的薪资情况。
那么,如何得到https://www.cnblogs.com/data:image/png;base64,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的值呢???就在小姐姐脑阔疼的厉害之时,Taoye是这么手握手教小姐姐的:“小姐姐,你可以先相亲1000个对象,观察并询问对象的发量和年龄之后,然后通过社会工程学来得到他的薪资情况。有了这1000组对象数据之后,你就能训练出https://www.cnblogs.com/data:image/png;base64,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的值,从而得到误差达到最小时候的这个线性模型”
小姐姐听完Taoye的讲述之后,真的是一语惊醒梦中人啊,心想:妙啊,就这么干!!!

以上例子中的内容纯属Taoye胡扯,只为描述线性回归的过程,不代表任何观点。
二、揭开梯度下降算法的神秘面纱

通过上述小姐姐的相亲故事,相信各位看官都已经对线性回归的过程有了一个基本的认识,而要具体操作线性回归,我们还需明白一个在机器学习领域中比较重要的算法,也就是梯度下降算法。
要理解梯度下降算法,我们可以将其类比成一个人下山的过程,这也是我们理解梯度下降算法时候最常用的一个例子,也就是这么一个场景:
有个人被困在山上,他现在要做的就是下山,也就是到达山的最低点。但是呢,现在整座山烟雾缭绕,可见度非常的低,所以下山的路径暂时无法确定,他必须通过自己此刻所在地的一些信息来一步步找到下山的路径。此时,就是梯度下降算法大显身手的时候了。具体怎么做呢?
是这样的,首先会以他当前所在地为基准,寻找此刻所处位置的最陡峭的地方,然后朝着下降的方向按照自己的设定走一步。走一步之后,就来到了一个新的位置,然后将这个新的位置作为基准,再找到最陡峭的地方,沿着这个方向再走一步,如此循环往复,知道走到最低点。这就是梯度下降算法的类别过程,上山同理也是一样,只不过变成了梯度上升算法。

梯度下降算法的基本过程就类似于上述下山的场景。
首先,我们会有一个可微分的函数。这个函数就类似于上述的一座山。我们的目标就是找到这个函数的最小值,也就是上述中山的最低点。根据之前的场景假设,最快的下山的方式就是找到当前位置最陡峭的方向,然后沿着此方向向下走,在这个可微分函数中,梯度反方向就代表这此山最陡峭的方向,也就是函数下降最快的方向。因为梯度的方向就是函数变化最快的方向(在后面会详细解释)
所以,我们重复利用这个方法,在达到一个新的位置之后,反复求取梯度,最后就能到达局部的最小值,这就类似于我们下山的过程。而求取梯度就确定了最陡峭的方向,也就是场景中测量方向的手段。那么为什么梯度的方向就是最陡峭的方向呢?接下来,我们从微分开始讲起:

[*]单变量
https://www.cnblogs.com/data:image/png;base64,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
对于单变量微分来讲,由于函数只有一个变量,所以此时的梯度就是函数的微分,所代表的意义就是在该点所对应的斜率。

[*]多变量(以三个变量为例)
https://www.cnblogs.com/data:image/png;base64,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
对于多变量函数来讲,此时的梯度就不再是一个具体的值,而是一个向量。我们都知道,向量是有方向的,而该梯度的方向就代表着函数在指定点中上升最快的方向。
这也就说明了为什么我们需要千方百计的求取梯度!我们需要到达山底,就需要在每一步观测到此时最陡峭的地方,梯度就恰巧告诉了我们这个方向。梯度的方向是函数在给定点上升最快的方向,那么梯度的反方向就是函数在给定点下降最快的方向,这正是我们所需要的。所以我们只要沿着梯度的方向一直走,就能走到局部的最低点!
现在,我们不妨通过代码来模拟实现这个过程。假如说,我们现在的目标函数是:
https://www.cnblogs.com/data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIQAAAAhCAYAAAAYlQP6AAAFj0lEQVR4Xu1abSiebRg+ESH5iBSiRPnIt1bkD0bCry2EybcSQhY2ah8xhR8o+aGMTKQUm5mQ8tFq2CQ1IuUzJPIRZsy8HVfv8/Z43z3b/dzP7enlua6fc9/XdZ7HedzHeVznM62rq6sr4osj8DcCWpwQnAvyCHBCcD5cQ4ATghOCE4JzQDECt14hvn37Ru/evaPt7W3a2dmhxMREcnJy4jUXicCtJ8T79+/Jx8eHrK2taXJyklpbW+nFixdkYWEhEhLNfu1WE+Ls7Iyqq6vJ0dGR4uLiaH9/n4qLiykrK4u8vLw0u7Iis7/VhMAI5dOnT2RqakouLi60urpKZWVl9PTpU3JwcBAJiWa/dqsJIV86kKO9vZ2+f/9OSUlJpKOjo9mVFZn9nSHEly9f6OvXr6x16OrqioRDuddOTk6ou7ubJiYmaHd3l6lScnIya2H/x4UY4bGCg4MVttQ7QYiVlRX6/PkzPXjwgOArLi8vydjYWHBN8PzCwgK7nQhVFpzT1NREERERZG9vTxcXFwxsEKSqqopcXV0Fny/0QTFxYm8Y77GxMdLT06Px8XGqq6u7u4TY29uj4eFhCgkJYcWESlhZWbEiCV0obl9fH0VGRpK+vr6g15aXl6moqIju379PGRkZ7Gx4mMLCQrp37x7l5+dLrlRi4pRPZmZmhvLy8oQTAn14enqaXr9+zYApKCggGxsbtufIyAiNjo4yFy8UNEHIKngIXwMKjULBF2C+cHx8TD09PexKicSgAvX19ezfZAtfeWVlJZmZmQk+XgzQm5ubVFJSQp6enpSTk8OKf3h4SE+ePGFXYBADOAnJw9zcXFCsYuJUiRBra2v08eNHCg8PZ24d7IckQg5ra2vJyMiIMjMzSUtLS1ACqjwEMpiYmJCvry8tLS0x8FNSUhjYL1++pOfPn0t2tRQL9Pn5OVMGWZsBSaAaIIlMIaTMQ2ycsjoorRADAwPk7OxMSLS0tJSePXtGbm5udHBwwAoSExNDgYGBqtRZ0LtHR0esF8fHx7MvD4YRXz1mDjBy/f39lJqaqpQK/O5gVYGW7f3hwwdqaGj4x0NInYeqcSpNCCSGttHW1kaLi4uMBAYGBjQ3N0cVFRVMNRT1ZoyO8R5GyUIW9gHBYHT+tLDv/Pw8I6mhoeGfHlf676oCjQPhKcrLy5mKBQQE/FJFVc1D1ThFEQJ9GnLs7+9PUVFRDFwwH9IHqUbbUOc6PT2lV69escFTQkKCykdjkDU0NHRtn58/f9L6+jrZ2tqStrb2tb+FhoaSn5/fb8+Fsa2pqWF4oV38qqUqm8dNxCmKEDKnDHXA+FfmHzANTE9PV4t/gMr09vaSh4cHM47oy9nZ2YykMGkgaFBQkGTkVOXLQ6zNzc0UFhbG5hDAa3Z2lsX+48cPSfNQJU4wWhQhMLyQ/R4AQ4f+DcWAXKMgipaULQOBw6XDw6APt7S0sL6MNoN48EXiy5XK3IoFGuTEHd/b25vs7OwYNPg95e3bt/To0SN2BZYyD7FxijaVMg+BK+abN29Yv8b1bWNjg8k2HL46Fgre2NjI5BsDHlzfBgcHydLSkkkybj9CB0hC4hUDNMgAFevs7GS3IdmCAQdZ09LSGHGlzENMnPL5K60QSBIGEgmC8TKDif9nkJubK/mgRUix1PGMGKCB0+PHjwne4N8LCovrutRLTJyIoaOjg82QQNatrS02uIMFiI2N/c+t8droGv4BgxWMgKOjo2lqaoq6urpYCxE6PJEaBHXsJxZodcQmf4Y64rxGCBgkXI2gCLhtuLu708OHD9UymVQ3uPLnQQkh7yC9VL7kJvJRR5x34setmwBfU/fkhNDUyivImxOCE+IaApwQnBCcEJwDihHgCsHZwRWCc4ArBOeAQAR4yxAIlKY89hfLPb/L/0UUXAAAAABJRU5ErkJggg==
则其对对应的梯度为:
https://www.cnblogs.com/data:image/png;base64,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
对此,我们可以通过如下代码来模拟梯度下降的过程,以寻找出到达最低点时候的值:
通过上述代码,我们可以发现,当函数值达到最低点的时候,此时我们的https://www.cnblogs.com/data:image/png;base64,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,与我们手动计算的基本可以划等号,这就是梯度下降所解决的问题。
针对上述代码,这里我们主要说两个点:
①:x_new = x_old - learning_rate * gradient(x_old)
在前进过程中更新x_new的时候,是通过x_old来进行的,之所以在梯度前加一个负号,主要是为了朝着梯度相反的方向前进。在前文我们也要提到,梯度的反方向就是函数在此点下降最快的方向。那么如果是上坡,也就是梯度上升算法,此时x_new的更新过程也就不需要加上负号了。
至于什么时候是梯度上升,什么时候是梯度下降,这个是根据我们实际情况是求最小值,还是最大值来决定的。
②:learning_rate
learning_rate在梯度下降算法中被称作为学习率或者说是步长,意味着我们可以通过learning_rate来控制每一步走的距离,其实就是不要走太快,从而错过了最低点。同时也要保证不要走的太慢,导致我们打到最低点需要花费大量的时间,也就是效率太低了,需要迭代很多次才能满足我们的需求。所以learning_rate的选择在梯度下降法中往往是很重要的!
需要合理的选择learning_rate值,一般来讲可取0.01,具体问题还需具体分析。
总而言之,梯度下降算法主要是根据函数的梯度来对x的值进行不断的更新迭代,以求得函数到达最小值时候的x值。
当然了,以上是该算法的一般形式,同时各位研究者也是提出了一些梯度下降算法的变种形式,主要有以下三种


[*]随机梯度下降算法(SGD,根据时间复杂度)
[*]批量梯度下降算法(BGD,根据数据量的大小)
[*]小批量梯度下降算法(MBGD,算法准确性)
关于上述三种梯度下降算法的变种形式,我们在这里挖个坑,后面有机会再来慢慢把这个坑个填上。
三、基于梯度下降算法实现线性回归拟合

这里代码实战的话,其实是牵涉到对神经网络的理解,不过我们在这里不着重讲解神经网络的内容,只简单的提一下,待手撕机器学习系列文章完成之后再来详细看看。
参考资料:《TensorFlow深度学习》——龙龙老师
我们都知道,人的大脑中包含了大量的神经元细胞,每个神经元都通过树突来获取输入的信号,然后通过轴突传递并输出信号,而神经元与神经元之间相互连接从而构成了巨大的神经网络。生物神经元的结构如下图所示:
图片来源:维基百科1943年,心理学家沃伦·麦卡洛克(Warren McCulloch)和数理逻辑学家沃尔特·皮茨(Walter Pitts)通过对生物神经元的研究,提出了模拟生物神经元机制的人工神经网络的数学模型,这一成果被美国神经学家弗兰克·罗森布拉特(Frank Rosenblatt)进一步发展成感知机(Perceptron)模型,这也是现代深度学习的基石。
我们从神经元的结构出发,来模拟这个神经元的信号处理过程。
如下图a所示,神经元输入向量https://www.cnblogs.com/data:image/png;base64,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,经过函数映射:https://www.cnblogs.com/data:image/png;base64,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后得到y,其中为函数f的自身参数。在这里,我们考虑一种简化的情况,也就是线性变换:https://www.cnblogs.com/data:image/png;base64,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,因为其中的w和x都是向量,所以我们将其展开为标量形式可表示为:
https://www.cnblogs.com/data:image/png;base64,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
上述计算的逻辑过程可通过下图b直观展现
以上神经元含有多个输入,为了方便理解,我们不妨进一步简化模型,此时的n=1,即我们假设该线性模型为:
https://www.cnblogs.com/data:image/png;base64,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
其中体现的是模型的斜率,而b体现的是截距,或者在这里我们说是偏置。
我们知道,对于一条直线来讲,我们只需要已知两个点,求解二元一次方程组,就能得到该直线的具体表达式,也就是求解出的值。
理想状态下的确是这样的,但是现实总是残酷的,我们所获取到的数据可能存在一定的误差,此时我们就根本无法构建出这么一条完美的直线来切合这些数据点。我们不妨用来表示该观测误差,假设该误差满足https://www.cnblogs.com/data:image/png;base64,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的高斯分布,则我们的模型可转换为:
https://www.cnblogs.com/data:image/png;base64,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
虽然我们不可能通过一条直线来完美的通过这些数据点,所以我们现在的需求就是尽可能找到这么一条直线,使得所有数据点到这条直线的“距离”最小,那么得到的这条直线就是我们比较满意的模型。
那么如何衡量所有数据点到达这条直线的“距离”最小?如何衡量这个模型的“好”与“不好”呢?这个时候就需要引出我们的损失函数了,一个很自然的想法就是求出当前模型的所有采样点上的预测值与真实值之间差的平方和的均值来作为这个模型的损失函数,也就是我们常常所提到的均方误差,损失函数表达如下:
https://www.cnblogs.com/data:image/png;base64,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
当我们的损失函数计算的值比较大的时候,此时说明该直线的拟合效果并不好,当我们的损失函数计算的值比较小的时候,说明此时的拟合效果达到了一个不错的程度。所以,我们不妨令损失函数值达到最小时,此时的模型参数为https://www.cnblogs.com/data:image/png;base64,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,则为:
读到这里,各位看官是不是知道下文如何走笔的了。没错,接下来就是通过梯度下降算法来求解该损失函数的最小值,
对此,我们需要求解出损失函数分别对的偏导,求解过程如下:
即:
https://www.cnblogs.com/data:image/png;base64,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
得到偏导之后,我们就可以根据旧的更新得到新的,这就是一次更新迭代过程。更新之后,我们再次重新计算偏导并更新参数,如此不断循环往复,知道我们计算的损失函数的值得到一个我们可接受的范围,即达到了我们的目的。
下面,我们通过代码来模拟实现这个过程。

[*]NumPy随机生成数据集,并通过Matplotlib初步观察数据的分布
"""
    Author: Taoye
    微信公众号: 玩世不恭的Coder
    Explain: 用于生成样本数据
    Return:
        x_data: 数据样本的一个属性
        y_data: 数据样本的另一个属性
"""
def establish_data(data_number):
    x_data = np.random.uniform(-10, 10, data_number)
    eps = np.random.normal(0, 1, data_number)
    y_data = x_data * 1.474 + 0.86 + eps
    return x_data, y_data

if __name__ == "__main__":
    x_data, y_data = establish_data(100)
    from matplotlib import pyplot as plt
    plt.scatter(x_data, y_data)
    plt.show()运行结果如下,可以看出数据分布大致可通过一条直线来进行拟合:


[*]根据数据和当前的w、b值计算均方误差
"""
    Author: Taoye
    微信公众号: 玩世不恭的Coder
    Explain: 计算均方误差
    Parameters:
        x_data: 数据样本的一个属性
        y_data: 数据样本的另一个属性
        w_now: 当前的w参数
        b_now: 当前的b参数
    Return:
        mse_value: 均方误差值
"""
def calc_mse(x_data, y_data, w_now, b_now):
    x_data, y_data = np.mat(x_data), np.mat(y_data)
    _, data_number = x_data.shape
    return np.power(w_now * x_data + b_now - y_data, 2).sum() / float(data_number)

[*]单次对w、b参数进行更新迭代
"""
    Author: Taoye
    微信公众号: 玩世不恭的Coder
    Explain: 更新迭代一次w、b
    Parameters:
        x_data: 数据样本的一个属性
        y_data: 数据样本的另一个属性
        w_now: 当前的w参数
        b_now: 当前的b参数
        learning_rate: 学习率
    Return:
        w_new: 更新迭代之后的w
        b_new: 更新迭代之后的b
"""
def step_gradient(x_data, y_data, w_now, b_now, learning_rate):
    x_data, y_data = np.mat(x_data), np.mat(y_data)
    w = (w_now * x_data + b_now - y_data) * x_data.T * 2 / x_data.shape
    b = (w_now * x_data + b_now - y_data).sum() * 2 / x_data.shape
    return w_now - w * learning_rate, b_now - b * learning_rate

[*]多次迭代更新w、b(外循环)
"""
    Author: Taoye
    微信公众号: 玩世不恭的Coder
    Explain: 多次迭代更新w、b(外循环)
    Parameters:
        x_data: 数据样本的一个属性
        y_data: 数据样本的另一个属性
        starting_w: 初始的w参数
        starting_b: 初试的b参数
        learning_rate: 学习率
        max_iter:最大迭代次数
    Return:
        w:得到的最终w
        b: 得到的最终b
        loss_list: 每次迭代计算的损失值
"""
def gradient_descent(x_data, y_data, starting_b, starting_w, learning_rate, max_iter):
    b, w = starting_b, starting_w
    loss_list = list()
    for step in range(max_iter):
        w, b = step_gradient(x_data, y_data, w, b, learning_rate)
        loss = calc_mse(x_data, y_data, w, b)
        loss_list.append(loss)
    return w, b, np.array(loss_list)

[*]拟合结果的可视化
"""
    Author: Taoye
    微信公众号: 玩世不恭的Coder
    Explain: 拟合结果的可视化
    Parameters:
        x_data: 数据样本的一个属性
        y_data: 数据样本的另一个属性
        w: 拟合得到的模型w参数
        b: 拟合得到的模型b参数
        loss_list: 每次更新迭代得到的损失函数的值
"""
def plot_result(x_data, y_data, w, b, loss_list):
    from matplotlib import pyplot as plt
    %matplotlib inline

    plt.subplot(2, 1, 1)
    plt.scatter(x_data, y_data)
    x_line_data = np.linspace(-10, 10, 1000)
    y_line_data = x_line_data * w + b
    plt.plot(x_line_data, y_line_data, "--", color = "red")

    plt.subplot(2, 1, 2)
    plt.plot(np.arange(loss_list.shape), loss_list)
    plt.show()程序运行结果如下:
从上方的运行结果来看,我们可以分析得到线性回归模型的拟合效果还不错,完全能够体现出数据的分布规律。另外,通过损失函数的变化图以及具体数值,我们可以观察到,前期损失值的变化非常的大,到了后期基本居于平缓,看比如说第一次到后面计算的损失值分别为14.215、4.0139、1.941188…..,这就是梯度下降法所体现出来的效果,也就是说我们的损失函数值越大,我们梯度下降法优化的效果也就越明显。
完整代码:
import numpy as np

"""
    Author: Taoye
    微信公众号: 玩世不恭的Coder
    Explain: 用于生成样本数据
    Return:
        x_data: 数据样本的一个属性
        y_data: 数据样本的另一个属性
"""
def establish_data(data_number):
    x_data = np.random.uniform(-10, 10, data_number)
    eps = np.random.normal(0, 1, data_number)
    y_data = x_data * 1.474 + 0.86 + eps
    return x_data, y_data

"""
    Author: Taoye
    微信公众号: 玩世不恭的Coder
    Explain: 计算均方误差
    Parameters:
        x_data: 数据样本的一个属性
        y_data: 数据样本的另一个属性
        w_now: 当前的w参数
        b_now: 当前的b参数
    Return:
        mse_value: 均方误差值
"""
def calc_mse(x_data, y_data, w_now, b_now):
    x_data, y_data = np.mat(x_data), np.mat(y_data)
    _, data_number = x_data.shape
    return np.power(w_now * x_data + b_now - y_data, 2).sum() / float(data_number)
"""
    Author: Taoye
    微信公众号: 玩世不恭的Coder
    Explain: 更新迭代一次w、b
    Parameters:
        x_data: 数据样本的一个属性
        y_data: 数据样本的另一个属性
        w_now: 当前的w参数
        b_now: 当前的b参数
        learning_rate: 学习率
    Return:
        w_new: 更新迭代之后的w
        b_new: 更新迭代之后的b
"""
def step_gradient(x_data, y_data, w_now, b_now, learning_rate):
    x_data, y_data = np.mat(x_data), np.mat(y_data)
    w = (w_now * x_data + b_now - y_data) * x_data.T * 2 / x_data.shape
    b = (w_now * x_data + b_now - y_data).sum() * 2 / x_data.shape
    return w_now - w * learning_rate, b_now - b * learning_rate
"""
    Author: Taoye
    微信公众号: 玩世不恭的Coder
    Explain: 多次迭代更新w、b(外循环)
    Parameters:
        x_data: 数据样本的一个属性
        y_data: 数据样本的另一个属性
        starting_w: 初始的w参数
        starting_b: 初试的b参数
        learning_rate: 学习率
        max_iter:最大迭代次数
    Return:
        w:得到的最终w
        b: 得到的最终b
        loss_list: 每次迭代计算的损失值
"""
def gradient_descent(x_data, y_data, starting_b, starting_w, learning_rate, max_iter):
    b, w = starting_b, starting_w
    loss_list = list()
    for step in range(max_iter):
        w, b = step_gradient(x_data, y_data, w, b, learning_rate)
        loss = calc_mse(x_data, y_data, w, b)
        loss_list.append(loss)
    return w, b, np.array(loss_list)
"""
    Author: Taoye
    微信公众号: 玩世不恭的Coder
    Explain: 拟合结果的可视化
    Parameters:
        x_data: 数据样本的一个属性
        y_data: 数据样本的另一个属性
        w: 拟合得到的模型w参数
        b: 拟合得到的模型b参数
        loss_list: 每次更新迭代得到的损失函数的值
"""
def plot_result(x_data, y_data, w, b, loss_list):
    from matplotlib import pyplot as plt
    %matplotlib inline

    plt.subplot(2, 1, 1)
    plt.scatter(x_data, y_data)
    x_line_data = np.linspace(-10, 10, 1000)
    y_line_data = x_line_data * w + b
    plt.plot(x_line_data, y_line_data, "--", color = "red")

    plt.subplot(2, 1, 2)
    plt.plot(np.arange(loss_list.shape), loss_list)
    plt.show()
if __name__ == "__main__":
    x_data, y_data = establish_data(100)
    w, b, loss_list = gradient_descent(x_data, y_data, 0, 0, 0.01, 1000)
    plot_result(x_data, y_data, w, b, loss_list)
以上就是本文线性回归的全部内容了,总体上来讲还是挺简单的,难度系数也没那么大,更多关于线性回归的内容,我们后面再来讲解。
这里我们对线性回归做一个简单的总结:
优点:结果比较容易理解,计算上并不复杂,没有太多复杂的公式和花里胡哨的内容
缺点:对非线性的数据拟合不好,时间复杂度还有一定的优化空间
适用数据类型:数值型和标称型数据
我是Taoye,爱专研,爱分享,热衷于各种技术,学习之余喜欢下象棋、听音乐、聊动漫,希望借此一亩三分地记录自己的成长过程以及生活点滴,也希望能结实更多志同道合的圈内朋友,更多内容欢迎来访微信公主号:玩世不恭的Coder。
我们下期再见,拜拜~~~
参考资料:
《机器学习实战》:Peter Harrington 人民邮电出版社
《TensorFlow深度学习》:龙龙老师
梯度下降算法原理讲解:https://blog.csdn.net/qq_41800366/article/details/86583789
推荐阅读
《Machine Learning in Action》—— 白话贝叶斯,“恰瓜群众”应该恰好瓜还是恰坏瓜
《Machine Learning in Action》—— 女同学问Taoye,KNN应该怎么玩才能通关
《Machine Learning in Action》—— 懂的都懂,不懂的也能懂。非线性支持向量机
《Machine Learning in Action》—— hao朋友,快来玩啊,决策树呦
《Machine Learning in Action》—— Taoye给你讲讲决策树到底是支什么“鬼”
《Machine Learning in Action》—— 剖析支持向量机,优化SMO
《Machine Learning in Action》—— 剖析支持向量机,单手狂撕线性SVM
print( "Hello,NumPy!" )
干啥啥不行,吃饭第一名
Taoye渗透到一家黑平台总部,背后的真相细思极恐
《大话数据库》-SQL语句执行时,底层究竟做了什么小动作?

来源:程序园用户自行投稿发布,如果侵权,请联系站长删除
免责声明:如果侵犯了您的权益,请联系站长,我们会及时删除侵权内容,谢谢合作!
页: [1]
查看完整版本: 《Machine Learning in Action》—— 浅谈线性回归的那些事