线性回归有解析解为啥还要用梯度下降

Posted 2023-05-13

线性回归是一种用于预测真实值的方法。让人困惑的是，这些需要预测真实值的问题被称为回归问题（regression problems）。线性回归是一种用直线对输入输出值进行建模的方法。在超过二维的空间里，这条直线被想象成一个平面或者超平面（hyperplane）。预测即是通过对输入值的组合对输出值进行预判。 参考技术A 因为不能保证所选的特征是完全线性无关的，所以系数增广矩阵的秩小于n(n表示特征数),这导致虽然线性回归有解析解但是它有无穷多解（线性代数的知识），你还是无法求得一个准确的解，所以这个时候就用梯度下降法来逼近一个解。

02_有监督学习--简单线性回归模型（梯度下降法代码实现）

有监督学习--简单线性回归模型（梯度下降法代码实现）0.引入依赖1.导入数据（data.csv）2.定义损失函数3.定义模型的超参数4.定义核心梯度下降模型函数5.测试：运行梯度下降算法，计算最优的 w 和 b6.画出拟合曲线7.附录-测试数据

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有监督学习--简单线性回归模型（梯度下降法代码实现）

0.引入依赖

import numpy as np
import matplotlib.pyplot as plt

1.导入数据（data.csv）

points = np.genfromtxt(‘data.csv‘, delimiter=‘,‘)

# points
# 提取 points 中的两对数据，分别作为 x, y
# points[0][0]  等价于
# points[0,0]  # 第一行第一列数据
# points[0,0:1] # array([32.50234527])
# points[0,0:2] # 第一行数据 array([32.50234527, 31.70700585])
# points[0,0:] # 第一行数据 array([32.50234527, 31.70700585])
x = points[:,0] # 第一列数据
y = points[:,1] # 第二列数据

# 用 scatter 画出散点图
plt.scatter(x, y)
plt.show()

作图如下：

2.定义损失函数

# 损失函数是模型系数的函数，还需要传入数据的 x,y
def compute_cost(w, b, points):
    total_cost = 0
    M = len(points)
    # 逐点计算【实际数据 yi 与 模型数据 f(xi) 的差值】的平方，然后求平均
    for i in range(M):
        x = points[i, 0]
        y = points[i, 1]
        total_cost += (y - w * x - b) ** 2

    return total_cost / M

3.定义模型的超参数

alpha = 0.0001
initial_w = 0
initial_b = 0
num_iter = 10

4.定义核心梯度下降模型函数

def grad_desc(points, initial_w, initial_b, alpha, num_iter):
    w = initial_w
    b = initial_b
    # 定义一个list保存所有的损失函数值，用来显示下降的过程
    cost_list = []

    for i in range(num_iter):
        # 先计算初始值的损失函数的值
        cost_list.append(compute_cost(w, b, points))
        w, b = step_grad_desc(w, b, alpha, points)

    return [w, b, cost_list]

def step_grad_desc(current_w, current_b, alpha, points):
    sum_grad_w = 0
    sum_grad_b = 0
    M = len(points)

    # 对每一个点带入公式求和
    for i in range(M):
        x = points[i, 0]
        y = points[i, 1]
        sum_grad_w += (current_w * x + current_b - y) * x
        sum_grad_b += (current_w * x + current_b - y)

    # 用公式求当前梯度
    grad_w = 2 / M * sum_grad_w
    grad_b = 2 / M * sum_grad_b

    # 梯度下降，更新当前的 w 和 b
    updated_w = current_w - alpha * grad_w
    updated_b = current_b - alpha * grad_b

    return updated_w, updated_b

5.测试：运行梯度下降算法，计算最优的 w 和 b

w, b, cost_list = grad_desc(points, initial_w, initial_b, alpha, num_iter)

print(‘w is:‘, w)
print(‘b is:‘, b)


cost = compute_cost(w, b, points)
print(‘cost is:‘, cost)

plt.plot(cost_list)
plt.show()

输出结果如下：

w is: 1.4774173755483797
b is: 0.02963934787473238
cost is: 112.65585181499748

作图如下：

6.画出拟合曲线

# 先用 scatter 画出2维散点图
plt.scatter(x, y)

# 针对每一个x，计算出预测的值
pred_y = w * x + b
# 再用 plot 画出2维直线图
plt.plot(x, pred_y, c=‘r‘)
plt.show()

作图如下：

7.附录-测试数据

测试数据 data.csv 如下：

32.502345269453031,31.70700584656992
53.426804033275019,68.77759598163891
61.530358025636438,62.562382297945803
47.475639634786098,71.546632233567777
59.813207869512318,87.230925133687393
55.142188413943821,78.211518270799232
52.211796692214001,79.64197304980874
39.299566694317065,59.171489321869508
48.10504169176825,75.331242297063056
52.550014442733818,71.300879886850353
45.419730144973755,55.165677145959123
54.351634881228918,82.478846757497919
44.164049496773352,62.008923245725825
58.16847071685779,75.392870425994957
56.727208057096611,81.43619215887864
48.955888566093719,60.723602440673965
44.687196231480904,82.892503731453715
60.297326851333466,97.379896862166078
45.618643772955828,48.847153317355072
38.816817537445637,56.877213186268506
66.189816606752601,83.878564664602763
65.41605174513407,118.59121730252249
47.48120860786787,57.251819462268969
41.57564261748702,51.391744079832307
51.84518690563943,75.380651665312357
59.370822011089523,74.765564032151374
57.31000343834809,95.455052922574737
63.615561251453308,95.229366017555307
46.737619407976972,79.052406169565586
50.556760148547767,83.432071421323712
52.223996085553047,63.358790317497878
35.567830047746632,41.412885303700563
42.436476944055642,76.617341280074044
58.16454011019286,96.769566426108199
57.504447615341789,74.084130116602523
45.440530725319981,66.588144414228594
61.89622268029126,77.768482417793024
33.093831736163963,50.719588912312084
36.436009511386871,62.124570818071781
37.675654860850742,60.810246649902211
44.555608383275356,52.682983366387781
43.318282631865721,58.569824717692867
50.073145632289034,82.905981485070512
43.870612645218372,61.424709804339123
62.997480747553091,115.24415280079529
32.669043763467187,45.570588823376085
40.166899008703702,54.084054796223612
53.575077531673656,87.994452758110413
33.864214971778239,52.725494375900425
64.707138666121296,93.576118692658241
38.119824026822805,80.166275447370964
44.502538064645101,65.101711570560326
40.599538384552318,65.562301260400375
41.720676356341293,65.280886920822823
51.088634678336796,73.434641546324301
55.078095904923202,71.13972785861894
41.377726534895203,79.102829683549857
62.494697427269791,86.520538440347153
49.203887540826003,84.742697807826218
41.102685187349664,59.358850248624933
41.182016105169822,61.684037524833627
50.186389494880601,69.847604158249183
52.378446219236217,86.098291205774103
50.135485486286122,59.108839267699643
33.644706006191782,69.89968164362763
39.557901222906828,44.862490711164398
56.130388816875467,85.498067778840223
57.362052133238237,95.536686846467219
60.269214393997906,70.251934419771587
35.678093889410732,52.721734964774988
31.588116998132829,50.392670135079896
53.66093226167304,63.642398775657753
46.682228649471917,72.247251068662365
43.107820219102464,57.812512976181402
70.34607561504933,104.25710158543822
44.492855880854073,86.642020318822006
57.50453330326841,91.486778000110135
36.930076609191808,55.231660886212836
55.805733357942742,79.550436678507609
38.954769073377065,44.847124242467601
56.901214702247074,80.207523139682763
56.868900661384046,83.14274979204346
34.33312470421609,55.723489260543914
59.04974121466681,77.634182511677864
57.788223993230673,99.051414841748269
54.282328705967409,79.120646274680027
51.088719898979143,69.588897851118475
50.282836348230731,69.510503311494389
44.211741752090113,73.687564318317285
38.005488008060688,61.366904537240131
32.940479942618296,67.170655768995118
53.691639571070056,85.668203145001542
68.76573426962166,114.85387123391394
46.230966498310252,90.123572069967423
68.319360818255362,97.919821035242848
50.030174340312143,81.536990783015028
49.239765342753763,72.111832469615663
50.039575939875988,85.232007342325673
48.149858891028863,66.224957888054632
25.128484647772304,53.454394214850524