Linear Regression

线性回归方法是机器学习里面最基础的一种方法，相关的理论方面的知识有很多，这里就不介绍了，博客主要从scikit-learn库的使用方面来探讨算法。

首先，我们使用随机生成一组数据，然后加入一些随机噪声。

import numpy as np
from sklearn.cross_validation import train_test_split

def f(x):
    return np.sin(2 * np.pi * x)

x_plot = np.linspace(0, 1, 100)

n_samples = 100
X = np.random.uniform(0, 1, size=n_samples)[:, np.newaxis]
y = f(X) + np.random.normal(scale=0.3, size=n_samples)[:, np.newaxis] ##add random noise to the dataset

X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.8)

首先，不添加正则项

fig, axes = plt.subplots(5, 2, figsize=(8, 5))
train_error = np.empty(10)
test_error = np.empty(10)
#
for ax, degree in zip(axes.ravel(), range(10)):
    est = make_pipeline(PolynomialFeatures(degree), LinearRegression())
    est.fit(X_train, y_train)
    train_error[degree] = mean_squared_error(y_train, est.predict(X_train))
    test_error[degree] = mean_squared_error(y_test, est.predict(X_test))
    plot_approximation(est, ax, label='degree=%d' %degree)
plt.show(fig)

plt.plot(np.arange(10), train_error, color='green', label='train')
plt.plot(np.arange(10), test_error, color='red', label='test')
plt.ylim(0.0, 1e0)
plt.ylabel('log(mean squared error)')
plt.xlabel('degree')
plt.legend(loc="upper left")
plt.show()

 

误差为：

当多项式的最高次幂超过6之后，训练样本的误差小，测试样本的误差过大，出现了过拟合，下面加入L2正则项：

alphas = [0.0, 1e-8, 1e-5, 1e-1]
degree = 9
fig, ax_rows = plt.subplots(3, 4, figsize=(8, 5))
for degree, ax_row in zip(range(7, 10), ax_rows):
    for alpha, ax in zip(alphas, ax_row):
        est = make_pipeline(PolynomialFeatures(degree), Ridge(alpha=alpha))
        est.fit(X_train, y_train)
        plot_approximation(est, ax, xlabel="degree=%d alpha=%r" %(degree, alpha))
#plt.tight_layout()
plt.show(fig)

具体看看不同的alpha大小对多项式系数的影响：

def plot_coefficients(est, ax, label=None, yscale='log'):
    coef = est.steps[-1][1].coef_.ravel()
    if yscale == 'log':
        ax.semilogy(np.abs(coef), marker='o', label=label)
        ax.set_ylim((1e-1, 1e8))
    else:
        ax.plot(np.abs(coef), marker='o', label=label)
    ax.set_ylabel('abs(coefficient)')
    ax.set_xlabel('coefficients')
    ax.set_xlim((1, 9))

fig, ax_rows = plt.subplots(4, 2, figsize=(8, 5))
alphas = [0.0, 1e-8, 1e-5, 1e-1]
for alpha, ax_row in zip(alphas, ax_rows):
    ax_left, ax_right = ax_row
    est = make_pipeline(PolynomialFeatures(degree), Ridge(alpha=alpha))
    est.fit(X_train, y_train)
    plot_approximation(est, ax_left, label='alpha=%r'%alpha)
    plot_coefficients(est, ax_right, label='Ridge(alpha=%r) coefficients' % alpha )

plt.show(fig)

alpha越大，因子越小，而曲线也越来越平滑。使用Ridge，可以加入L2正则项，还可以通过使用Lasso，加入L1正则项

fig, ax_rows = plt.subplots(2, 2, figsize=(8, 5))

degree = 9
alphas = [1e-3, 1e-2]

for alpha, ax_row in zip(alphas, ax_rows):
    ax_left, ax_right = ax_row
    est = make_pipeline(PolynomialFeatures(degree), Lasso(alpha=alpha))
    est.fit(X_train, y_train)
    plot_approximation(est, ax_left, label='alpha=%r' % alpha)
    plot_coefficients(est, ax_right, label='Lasso(alpha=%r) coefficients' % alpha, yscale=None)

plt.tight_layout()
plt.show(fig)

除了上述两种方式外，scikit-learn还支持同时加入L1和L2正则，需要使用ElasticNet进行训练

fig, ax_rows = plt.subplots(8, 2, figsize=(8, 5))
alphas = [1e-2, 1e-2, 1e-2, 1e-3, 1e-3, 1e-3, 1e-4, 1e-4]
ratios = [0.05, 0.85, 0.50, 0.05, 0.85, 0.50, 0.03, 0.95]
for alpha, ratio, ax_row in zip(alphas, ratios, ax_rows):
    ax_left, ax_right = ax_row
    est = make_pipeline(PolynomialFeatures(degree), ElasticNet(alpha=alpha, l1_ratio=ratio))
    est.fit(X_train, y_train)
    plot_approximation(est, ax_left, label='alpha=%r ratio=%r' % (alpha, ratio))
    plot_coefficients(est, ax_right, label="Lasso(alpah=%r ratio=%r) coefficients" % (alpha, ratio), yscale=None)

plt.show()

当alpha一定时，曲线形状并未发生明显变化，alpha限定了参数范围，alpha越小，参数取值范围越大，这与只使用L2、L1正则时相似。ratio决定了参数的取值情况，当ratio比较大时，则参数相对稀疏(只有少数几个参数的值比较大，而其余的值比较小者趋近于0)，

而ratio比较小时，参数之间差异相对较小，分布较为均匀。

数据集1：Test Scores for General Psychology

 每组数据是一个四元组，<x1, x2, x3, x4>其中x1, x2, x3表示前3次的成绩，x4表示最终成绩。现在需要有(x1, x2, x3)来预测x4.数据集总共有25条记录。其中第一行是标题。下面对比不使用正则项，使用L2正则项和使用L1正则项来做一个简单的线性回归模型。

# -*-encoding:utf-8-*-
'''
Created on 
author: dstarer
copyright: dstarer
'''

import numpy as np
from sklearn.cross_validation import train_test_split
from sklearn.linear_model import LinearRegression
from sklearn.metrics import mean_squared_error
from sklearn.linear_model import Ridge
from sklearn.linear_model import Lasso
from plot import *

def readData(filename, ignoreFirstLine=True, separtor='\t'):
    dataSet = []
    fp = open(filename, "r")
    if ignoreFirstLine:
        fp.readline()
    for line in fp.readlines():
        elements = map(int, line.strip().split(separtor))
        dataSet.append(elements)
    fp.close()
    return dataSet


def Print(message, train_error, test_error, coef):
    print "%s--------------" % message
    print "train error: %.3f" % train_error
    print "test error: %.3f" % test_error
    print coef
    print "sum of coef: ", np.sum(coef)

def process(X, y, show=True):
    error = np.empty(3)
    X_train, X_test, y_train, y_test = train_test_split(X, y)
    est = LinearRegression()
    est.fit(X_train, y_train)
    train_error = mean_squared_error(y_train, est.predict(X_train))
    test_error = mean_squared_error(y_test, est.predict(X_test))
    error[0] = test_error

    if show:
        Print(message="train without regularization", train_error=train_error, test_error=test_error, coef=est.coef_)

    ridge = Ridge()
    ridge.fit(X_train, y_train)
    train_error = mean_squared_error(y_train, ridge.predict(X_train))
    test_error = mean_squared_error(y_test, ridge.predict(X_test))
    error[1] = test_error

    if show:
        Print(message="train using L2 regularization", train_error=train_error, test_error=test_error, coef=est.coef_)

    lasso = Lasso()
    lasso.fit(X_train, y_train)
    train_error = mean_squared_error(y_train, lasso.predict(X_train))
    test_error = mean_squared_error(y_test, lasso.predict(X_test))
    error[2] = test_error

    if show:
        Print(message="train using L1 regularization", train_error=train_error, test_error=test_error, coef=est.coef_)

    if show:
        print "Data ------------"
        print "[x1  x2  x3 ] [y] [\t est \t] [\t ridge \t] [\t lasso \t]"
        for X, y, est_v, ridge_v, lasso_v in zip(X_test, y_test, est.predict(X_test), ridge.predict(X_test), lasso.predict(X_test)):
            print X, y, est_v, ridge_v, lasso_v

    return error


def error_estimate(X, y):
    error = np.empty(3)
    Iters = 20

    for i in range(Iters):
        tmp = process(X, y, show=False)
        error = error + tmp
    error /= Iters
    print "normal error: %.3f" % error[0]
    print "L2 error: %.3f" % error[1]
    print "L1 error: %.3f" % error[2]


def extract_data(filename):
    dataset = np.mat(readData(filename))

    y = dataset[:, -1]
    X = dataset[:, :-1]

    process(X, y, show=True)

    # original data set
    print "original data set:"
    error_estimate(X, y)

    print "using the first two dimensions"
    X = dataset[:, :-2]
    error_estimate(X, y)

    print "use the first and third dimensions"
    X = dataset[:, ::2]
    error_estimate(X, y)

    print "only use the third dimension"
    X = dataset[:, 2]
    error_estimate(X, y)

    print "use the second and third dimensions"
    X = dataset[:, 1:-1]
    error_estimate(X, y)

    #plot the data
    ax = plt.gca()
    X1 = dataset[:, 0]
    X2 = dataset[:, 1]
    X3 = dataset[:, 2]
    plotScatter2D(ax=ax, X=X1, y=y, color="red")
    plotScatter2D(ax=ax, X=X2, y=y, color="blue")
    plotScatter2D(ax=ax, X=X3, y=y, color="green")
    plt.show()


if '__main__' == __name__:
    extract_data("E:\\dataset\\mldata\\test_score.csv")

train without regularization--------------
train error: 2.457
test error: 16.153
[[ 0.39285405  0.5157764   1.16694498]]
sum of coef:  2.07557543476
train using L2 regularization--------------
train error: 2.457
test error: 16.159
[[ 0.39285405  0.5157764   1.16694498]]
sum of coef:  2.07557543476
train using L1 regularization--------------
train error: 2.466
test error: 16.038
[[ 0.39285405  0.5157764   1.16694498]]
sum of coef:  2.07557543476
Data ------------
[x1  x2  x3 ] [y] [     est     ] [     ridge     ] [     lasso     ]
[70 73 78] [148] [ 150.36924252] [ 150.36061654] 150.447200166
[78 75 68] [147] [ 142.87417786] [ 142.89774266] 142.910175633
[93 89 96] [192] [ 188.66231778] [ 188.65674942] 188.514072928
[93 88 93] [185] [ 184.64570642] [ 184.64589888] 184.502840202
[47 56 60] [115] [ 111.56039086] [ 111.54876013] 111.860472713
[87 79 90] [175] [ 174.14575956] [ 174.14218324] 174.036875299
[78 83 85] [175] [ 166.83845382] [ 166.82925063] 166.853488692
original data set:
normal error: 9.255
L2 error: 11.200
L1 error: 12.574
using the first two dimensions
normal error: 63.057
L2 error: 64.947
L1 error: 66.151
use the first and third dimensions
normal error: 23.051
L2 error: 23.057
L1 error: 23.230
only use the third dimension
normal error: 39.893
L2 error: 39.890
L1 error: 39.899
use the second and third dimensions
normal error: 12.268
L2 error: 12.265
L1 error: 12.260

上面是一些测试的结果，为了具体的看一下线性回归的效果，测试20次，每次数据随机划分，将测试误差绘制出来，如下图：

其中红色线是表示不加正则项的结果，不同划分下测试误差也有很大偏差。使用三种特征的组合，得到的效果总体上来说还是可以的，是不是还有其他方法会取得更好的效果呢？为了找到一种更好的预测方法，我分别从这三个特征中任选两个用于测试。测试结果已经显示在上面了，当然为了更严谨一些，我也分别测试了20次，每次也同样是随机划分数据，误差曲线如下：

第一幅图是(x1, x2),第二幅图是(x1, x3),第三幅图是(x2, x3)。很容易发现，只用(x2, x3)与同时使用(X1, X2, X3)的效果很相似！！！前面我们将训练的系数输出来了，其实不难从系数上发现，a3>a2>a1， a3>a2 + a1, 也就是说x3这个特征是最重要的，所以在只考虑x1,x2时，测试误差很大，而考虑了x3之后，误差就减小了，而同时用x2, x3时，数据集的主要特征基本被表征出来，所以此时效果基本与(x1, x2, x3)的结果相同。为了测试一下x3特征的重要性，我只使用x3特征，效果如下：

对比只使用x3,和同时使用(x1,x2)，x3的效果要比较好。当然这里我的测试方式可能欠妥，但是从平均情况来看，还是可以反映出上面的结论，最后我们在分别看一下(xi, y)的分布情况：

总体上，x3的分布相对集中一些，而x2,x1相对较为离散，波动幅度较大。未完待续，下一数据集。。。。。。