梯度下降法解决简单线性回归问题

逃离我推掉我的手 2022-01-23 06:11 292阅读 0赞

如下数据保存在data.csv文件中，由一元线性回归模型y = 1.477 ∗ x + 0.089 + ε（ε为随机噪声）生成，第一列为x,第二列为y。

data.csv

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

上面的数据，二维图像如下：

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaWNhbzE5OTA_size_16_color_FFFFFF_t_70][]

定义损失函数如下：

# loss = 1/N ![20190602205048714.png][]      #

代码如下：

#coding=utf-8
    
    import numpy as np
    
    # 在当前w、b参数下计算数据集的均方误差
    def compute_error_for_line_given_points(w, b, points):
        totalError = 0
        for i in range(0, len(points)):
            x = points[i, 0]
            y = points[i, 1]
            # 计算误差和
            totalError = totalError + (y - (w * x + b)) **2
        # 返回均方误差
        return totalError / float(len(points))
    
    # 计算梯度，更新w、b
    def step_gradient(w_current, b_current, points, learningRate):
        w_gradient = 0
        b_gradient = 0
        N = float(len(points))
        for i in range(0, len(points)):
            x = points[i, 0]
            y = points[i, 1]
            # 计算数据集上梯度和
            w_gradient = w_gradient + 2 * x * ((w_current * x + b_current) - y)
            b_gradient = b_gradient + 2 * ((w_current * x + b_current) - y)
    
        # 计算数据集上平均梯度
        average_w_gradient = w_gradient / N
        average_b_gradient = b_gradient / N
    
        # 更新w、b
        new_w = w_current - (learningRate * average_w_gradient)
        new_b = b_current - (learningRate * average_b_gradient)
    
        return [new_w, new_b]
    
    
    # 梯度下降迭代
    def gradient_descent_runner(points, starting_w, starting_b, learning_rate, num_iterations):
        w = starting_w
        b = starting_b
        for i in range(num_iterations):
            w, b = step_gradient(w, b, np.array(points), learning_rate)
    
        # 取最终的w、b
        return [w, b]
    
    
    def run():
        # 读取csv文件中的数据
        points = np.genfromtxt("./data.csv", delimiter=",")
    
        # 学习率
        learning_rate = 0.0001
    
        # 初始化w,b值
        initial_w = 0
        initial_b = 0
    
        # 迭代次数
        num_iterations = 1000
    
        # 当前w、b参数下在数据集上的均方误差
        print("Starting gradient descent at w = {0}, b = {1}, error = {2}"
              .format(initial_w,
                      initial_b,
                      compute_error_for_line_given_points(initial_w, initial_b, points)))
    
        print("Running...")
        [w, b] = gradient_descent_runner(points, initial_w, initial_b, learning_rate, num_iterations)
        print("After {0} iterations w = {1}, b = {2}, error = {3}"
              .format(num_iterations,
                      w,
                      b,
                      compute_error_for_line_given_points(w, b, points)))
    
    if __name__ == "__main__":
        run()

执行结果如下：

![20190602210020859.png][]

拟合结果为 w = 1.4777440851894448, b = 0.08893651993741342 极为符合生成该数据集的线性模型。

[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaWNhbzE5OTA_size_16_color_FFFFFF_t_70]: /images/20220123/8e0190fe07a24aa2a5ef5386579b37ec.png
[20190602205048714.png]: /images/20220123/08fdbe3f6a1d4d27acdd74938fd0d25f.png
[20190602210020859.png]: /images/20220123/813314b011c444158a1962a3e2cf7951.png