[机器学习]线性回归(Linear regression)及代码实现

MLE + 高斯分布(误差满足均值为0,方差为斯塔平方的正太分布)能推出目标函数为误差平方和. 具体推到如下图.对数似然取对数仅仅是一种求解手段而已.

线性回归可以对样本非线性.只要对参数斯塔线性就行.例如$Y={\Theta }_0+{\Theta }_1{X}_1+{\Theta }_2{X}_{2}X$,可以用poly把X1 变成X1的平方或者X2的平方,或者X1X2,也就是另一种方式增加特征即增加了数据.对于同样的数据,就可以画出弧线来做回归.也就是不同阶.

样本个数有限叫做有参,样本个数无限叫做无参.

公式推导:

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线性回归广告预测:

import csv
import pandas as pd
import numpy as np
from sklearn.model_selection import train_test_split
from sklearn.linear_model import LinearRegression
import matplotlib.pyplot as plt
import matplotlib as mpl
from sklearn.metrics import accuracy_score

if __name__ == '__main__':
    #数据读取的几种方式
    path = './Advertising.csv'
    
 #     #numpy读入
#     data = np.loadtxt(path, delimiter=',', skiprows=1)#直接到第一行,因为第0行是标题,不要
#     x = data[:, 1:-1]
#     y = data[:, -1]
#     print(data)
#     print(x)
#     print(y)

#     #python自带库
#     f = file(path, 'rb')
#     print(f)
#     d = csv.reader(f)
#     for line in d:
#         print(line)
#     f.close()

#     #手写读取数据
#     f = file(path)
#     x = []
#     y = []
#     for i, d in enumerate(f):
#         if i == 0:
#             continue
#         d = d.strip()
#         if not d:
#             continue
#         d = map(float, d.split(','))
#         x.append(d[1:-1])
#         y.append(d[-1])
#     print(x)
#     print(y)


   #pandas读入
    data = pd.read_csv(path)
    data.head(5)
   # x, y = data.values[:, 1:-1], data.values[:, -1]
  #   x = data[['TV', 'Radio', 'Newspaper']]
 #     y = data['Sales']
    print(x)
    print(y)

数据如下所示(只显示前5行):

    # 数据可视化
    plt.figure(figsize=(9,12), facecolor='w')
    plt.subplot(311)
    plt.plot(data['TV'], y, 'ro')
    plt.title('TV')
    plt.grid(True)
    plt.subplot(312)
    plt.plot(data['Radio'], y, 'g^')
    plt.title('Radio')
    plt.grid(True)
    plt.subplot(313)
    plt.plot(data['Newspaper'], y, 'b*')
    plt.title('Newspaper')
    plt.grid(True)
    plt.tight_layout()
    plt.show()

    x_train, x_test, y_train, y_test = train_test_split(x, y, train_size = 0.8, random_state = 1)
    clf = LinearRegression()
    model = clf.fit(x_train, y_train)
    print('模型为:', model)
    print('sita参数 =', clf.coef_)
    print('截距项 = ', clf.intercept_)
    y_hat = clf.predict(x_test)
    mse = np.average((y_hat - np.array(y_test)) ** 2)  # Mean Squared Error
    rmse = np.sqrt(mse)  # Root Mean Squared Error
    print('MSE = ', mse)
    print('RMSE = ', rmse)
    t = np.arange(len(x_test))
    plt.plot(t, y_test, 'r-', linewidth=2, label='real data')
    plt.plot(t, y_hat, 'g-', linewidth=2, label='predict data')
    plt.legend(loc='upper right')
    plt.title('LinearRegression Predict', fontsize=18)
    plt.grid(True)
    plt.show()

交叉验证:

运用Lasso(L1正则)或者Ridge(L2正则)回归进行交叉验证选取最合适的超参数值:

import numpy as np
import matplotlib as mpl
import matplotlib.pyplot as plt
import pandas as pd
from sklearn.model_selection import train_test_split
from sklearn.linear_model import Lasso, Ridge
from sklearn.model_selection import GridSearchCV


if __name__ == "__main__":
    # pandas读入
    data = pd.read_csv('./Advertising.csv')    # TV、Radio、Newspaper、Sales
    x = data[['TV', 'Radio', 'Newspaper']]
    y = data['Sales']
    

    x_train, x_test, y_train, y_test = train_test_split(x, y, train_size=0.8, random_state=1)
    model = Lasso()
    # model = Ridge()
    alpha_can = np.logspace(-3, 2, 10)#第三个参数是个数.默认是10的多少次方,要是换就的加 base=2.这就是2的多少次方
    print('alpha参数范围:', alpha_can)
    lasso_model = GridSearchCV(model, param_grid={'alpha': alpha_can}, cv=10)
    lasso_model.fit(x_train, y_train)
    print('超参数：', lasso_model.best_params_)

    y_hat = lasso_model.predict(np.array(x_test))
    print('模型得分:',lasso_model.score(x_test, y_test))
    mse = np.average((y_hat - y_test) ** 2)  # Mean Squared Error
    rmse = np.sqrt(mse)  # Root Mean Squared Error
    print('mse:' , mse)
    print('rmse:' , rmse)
    t = np.arange(len(x_test))
    mpl.rcParams['font.sans-serif'] = [u'simHei']
    mpl.rcParams['axes.unicode_minus'] = False
    plt.plot(t, y_test, 'r-', linewidth=2, label=u'real data')
    plt.plot(t, y_hat, 'g-', linewidth=2, label=u'predict data')
    plt.title('predict', fontsize=18)
    plt.legend(loc='upper right')
    plt.grid()

ElasticNet(波士顿房价预测):

建立pipeline和poly多项式:

import numpy as np
import matplotlib.pyplot as plt
from sklearn.model_selection import train_test_split
from sklearn.linear_model import ElasticNetCV
from sklearn.preprocessing import PolynomialFeatures, StandardScaler #标准化  (x-均值)/方差
from sklearn.pipeline import Pipeline
from sklearn.metrics import mean_squared_error



def not_empty(s):
    return s != ''


if __name__ == "__main__":
#     warnings.filterwarnings(action='ignore') #不报warning
#     np.set_printoptions(suppress=True)
#     file_data = pd.read_csv('./housing.data', header=None)
#     # a = np.array([float(s) for s in str if s != ''])
#     data = np.empty((len(file_data), 14))
#     for i, d in enumerate(file_data.values):
#         d = map(float, filter(not_empty, d[0].split(' ')))
#         data[i] = d

    path = './housing.data'
    data = np.loadtxt(path)
    x, y = np.split(data, (13, ), axis=1)
    y = y.ravel()
    # data = sklearn.datasets.load_boston()
    # x = np.array(data.data)
    # y = np.array(data.target)
    print(u'样本个数：%d, 特征个数：%d' % x.shape)
    print(y.shape)
    
    # l1_ratio = np.logspace(-3, 2, 1000)
    x_train, x_test, y_train, y_test = train_test_split(x, y, train_size=0.7, random_state=0)
    model = Pipeline([
        ('ss', StandardScaler()),
        ('poly', PolynomialFeatures(degree=3, include_bias=True)),
        ('linear', ElasticNetCV(l1_ratio=np.logspace(0, 1, 50), alphas=np.logspace(-3, 2, 5),
                                fit_intercept=False, max_iter=1e3, cv=3))
    ])
    print(u'开始建模................................')
    model.fit(x_train, y_train.ravel())
    linear = model.get_params('linear')['linear']
    print(u'超参数：', linear.alpha_)
    print(u'L1 ratio：', linear.l1_ratio_)
    # print(u'系数：', linear.coef_.ravel())
    y_pred = model.predict(x_test)
    mse = mean_squared_error(y_test, y_pred)
    print(u'均方误差：', mse)

    t = np.arange(len(y_pred))
    
    plt.figure(facecolor='w')
    plt.plot(t, y_test.ravel(), 'r-', lw=2, label=u'real')
    plt.plot(t, y_pred, 'g-', lw=2, label=u'predict')
    plt.legend(loc='best')
    plt.title(u'booston house price', fontsize=18)
    plt.xlabel(u'sample numble', fontsize=15)
    plt.ylabel(u'house price', fontsize=15)
    plt.grid()
    plt.show()