machinelearning_notebook/4_logistic_regression/Logistic_regression.py

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# # Logistic Regression
#
# 逻辑回归(Logistic Regression, LR)模型其实仅在线性回归的基础上，套用了一个逻辑函数，但也就由于这个逻辑函数，使得逻辑回归模型成为了机器学习领域一颗耀眼的明星，更是计算广告学的核心。本节主要详述逻辑回归模型的基础。
#
#
# ## 1 逻辑回归模型
# 回归是一种比较容易理解的模型，就相当于$y=f(x)$，表明自变量$x$与因变量$y$的关系。最常见问题有如医生治病时的望、闻、问、切，之后判定病人是否生病或生了什么病，其中的望闻问切就是获取自变量$x$，即特征数据，判断是否生病就相当于获取因变量$y$，即预测分类。
#
# 最简单的回归是线性回归，在此借用Andrew NG的讲义，有如图所示，$X$为数据点——肿瘤的大小，$Y$为观测值——是否是恶性肿瘤。通过构建线性回归模型，如$h_\theta(x)$所示，构建线性回归模型后，即可以根据肿瘤大小，预测是否为恶性肿瘤$h_\theta(x)) \ge 0.5$为恶性，$h_\theta(x) \lt 0.5$为良性。
#
# ![LinearRegression](images/fig1.gif)
#
# 然而线性回归的鲁棒性很差，例如在上图的数据集上建立回归，因最右边噪点的存在，使回归模型在训练集上表现都很差。这主要是由于线性回归在整个实数域内敏感度一致，而分类范围，需要在$[0,1]$。
#
# 逻辑回归就是一种减小预测范围，将预测值限定为$[0,1]$间的一种回归模型，其回归方程与回归曲线如图2所示。逻辑曲线在$z=0$时，十分敏感，在$z>>0$或$z<<0$处，都不敏感，将预测值限定为$(0,1)$。
#
#

# +
# %matplotlib inline
import matplotlib.pyplot as plt
import numpy as np

plt.figure()
plt.axis([-10,10,0,1])
plt.grid(True)
X=np.arange(-10,10,0.1)
y=1/(1+np.e**(-X))
plt.plot(X,y,'b-')
plt.title("Logistic function")
plt.show()
# -

# ### 逻辑回归表达式
#
# 这个函数称为Logistic函数(logistic function)，也称为Sigmoid函数(sigmoid function)。函数公式如下：
#
# $$
# g(z) = \frac{1}{1+e^{-z}}
# $$
#
# Logistic函数当z趋近于无穷大时，g(z)趋近于1；当z趋近于无穷小时，g(z)趋近于0。Logistic函数的图形如上图所示。Logistic函数求导时有一个特性，这个特性将在下面的推导中用到，这个特性为：
# $$
# g'(z) =  \frac{d}{dz} \frac{1}{1+e^{-z}} \\
#       =  \frac{1}{(1+e^{-z})^2}(e^{-z}) \\
#       =  \frac{1}{(1+e^{-z})} (1 - \frac{1}{(1+e^{-z})}) \\
#       =  g(z)(1-g(z))
# $$
#
#

# +
# %matplotlib inline
import matplotlib.pyplot as plt
import numpy as np

plt.figure()
plt.axis([-10,10,0,1])
plt.grid(True)
X=np.arange(-10,10,0.1)
y=1/(1+np.e**(-X))
plt.plot(X,y,'b-')
plt.title("Logistic function")
plt.show()
# -

# 逻辑回归本质上是线性回归，只是在特征到结果的映射中加入了一层函数映射，即先把特征线性求和，然后使用函数$g(z)$将最为假设函数来预测。$g(z)$可以将连续值映射到0到1之间。线性回归模型的表达式带入$g(z)$，就得到逻辑回归的表达式:
#
# $$
# h_\theta(x) = g(\theta^T x) = \frac{1}{1+e^{-\theta^T x}}
# $$

# ### 逻辑回归的软分类
#
# 现在我们将y的取值$h_\theta(x)$通过Logistic函数归一化到(0,1)间，$y$的取值有特殊的含义，它表示结果取1的概率，因此对于输入$x$分类结果为类别1和类别0的概率分别为：
#
# $$
# P(y=1|x,\theta) = h_\theta(x) \\
# P(y=0|x,\theta) = 1 - h_\theta(x)
# $$
#
# 对上面的表达式合并一下就是：
#
# $$
# p(y|x,\theta) = (h_\theta(x))^y (1 - h_\theta(x))^{1-y}
# $$
#
#

# ### 梯度上升
#
# 得到了逻辑回归的表达式，下一步跟线性回归类似，构建似然函数，然后最大似然估计，最终推导出$\theta$的迭代更新表达式。只不过这里用的不是梯度下降，而是梯度上升，因为这里是最大化似然函数。
#
# 我们假设训练样本相互独立，那么似然函数表达式为：
# ![Loss](images/eq_loss.png)
#
# 同样对似然函数取log，转换为：
# ![LogLoss](images/eq_logloss.png)
#
# 转换后的似然函数对$\theta$求偏导，在这里我们以只有一个训练样本的情况为例：
# ![LogLossDiff](images/eq_logloss_diff.png)
#
# 这个求偏导过程中：
# * 第一步是对$\theta$偏导的转化，依据偏导公式：$y=lnx$, $y'=1/x$。
# * 第二步是根据g(z)求导的特性g'(z) = g(z)(1 - g(z)) 。
# * 第三步就是普通的变换。
#
# 这样我们就得到了梯度上升每次迭代的更新方向，那么$\theta$的迭代表达式为：
# $$
# \theta_j := \theta_j + \alpha(y^i - h_\theta(x^i)) x_j^i
# $$
#
#

# ## Program

# +
# %matplotlib inline

from __future__ import division
import numpy as np
import sklearn.datasets
import matplotlib.pyplot as plt

np.random.seed(0)


# +
# load sample data
data, label = sklearn.datasets.make_moons(200, noise=0.30)

print("data  = ", data[:10, :])
print("label = ", label[:10])

plt.scatter(data[:,0], data[:,1], c=label)
plt.title("Original Data")

# +
def plot_decision_boundary(predict_func, data, label):
    """画出结果图
    Args:
        pred_func (callable): 预测函数
        data (numpy.ndarray): 训练数据集合
        label (numpy.ndarray): 训练数据标签
    """
    x_min, x_max = data[:, 0].min() - .5, data[:, 0].max() + .5
    y_min, y_max = data[:, 1].min() - .5, data[:, 1].max() + .5
    h = 0.01

    xx, yy = np.meshgrid(np.arange(x_min, x_max, h), np.arange(y_min, y_max, h))

    Z = predict_func(np.c_[xx.ravel(), yy.ravel()])
    Z = Z.reshape(xx.shape)

    plt.contourf(xx, yy, Z, cmap=plt.cm.Spectral)
    plt.scatter(data[:, 0], data[:, 1], c=label, cmap=plt.cm.Spectral)
    plt.show()



# +
def sigmoid(x):
    return 1.0 / (1 + np.exp(-x))

class Logistic(object):
    """logistic回归模型"""
    def __init__(self, data, label):
        self.data = data
        self.label = label

        self.data_num, n = np.shape(data)
        self.weights = np.ones(n)
        self.b = 1

    def train(self, num_iteration=150):
        """随机梯度上升算法
        Args:
            data (numpy.ndarray): 训练数据集
            labels (numpy.ndarray): 训练标签
            num_iteration (int): 迭代次数
        """
        for j in range(num_iteration):
            data_index = list(range(self.data_num))
            for i in range(self.data_num):
                # 学习速率
                alpha = 0.01
                rand_index = int(np.random.uniform(0, len(data_index)))
                error = self.label[rand_index] - sigmoid(sum(self.data[rand_index] * self.weights + self.b))
                self.weights += alpha * error * self.data[rand_index]
                self.b += alpha * error
                del(data_index[rand_index])

    def predict(self, predict_data):
        """预测函数"""
        result = list(map(lambda x: 1 if sum(self.weights * x + self.b) > 0 else 0,
                     predict_data))
        return np.array(result)

# -

logistic = Logistic(data, label)
logistic.train(200)
plot_decision_boundary(lambda x: logistic.predict(x), data, label)

# ## How to use sklearn to resolve the problem
#

# +
from sklearn.linear_model.logistic import LogisticRegression
from sklearn.metrics import confusion_matrix
from sklearn.metrics import accuracy_score
import matplotlib.pyplot as plt

# calculate train/test data number
N = len(data)
N_train = int(N*0.8)
N_test = N - N_train

# split train/test data
x_train = data[:N_train, :]
y_train = label[:N_train]
x_test  = data[N_train:, :]
y_test  = label[N_train:]

# do logistic regression
lr=LogisticRegression()
lr.fit(x_train,y_train)

pred_train = lr.predict(x_train)
pred_test  = lr.predict(x_test)

# calculate train/test accuracy
acc_train = accuracy_score(y_train, pred_train)
acc_test = accuracy_score(y_test, pred_test)
print("accuracy train = %f" % acc_train)
print("accuracy test = %f" % acc_test)

# plot confusion matrix
cm = confusion_matrix(y_test,pred_test)

plt.matshow(cm)
plt.title(u'Confusion Matrix')
plt.colorbar()
plt.ylabel(u'Groundtruth')
plt.xlabel(u'Predict')
plt.show()
# -

# ## Multi-class recognition

# ### Load & show the data

# +
import matplotlib.pyplot as plt 
from sklearn.datasets import load_digits

# load data
digits = load_digits()

# copied from notebook 02_sklearn_data.ipynb
fig = plt.figure(figsize=(6, 6))  # figure size in inches
fig.subplots_adjust(left=0, right=1, bottom=0, top=1, hspace=0.05, wspace=0.05)

# plot the digits: each image is 8x8 pixels
for i in range(64):
    ax = fig.add_subplot(8, 8, i + 1, xticks=[], yticks=[])
    ax.imshow(digits.images[i], cmap=plt.cm.binary)
    
    # label the image with the target value
    ax.text(0, 7, str(digits.target[i]))
# -

# ### Visualizing the Data
#
# A good first-step for many problems is to visualize the data using one of the Dimensionality Reduction techniques we saw earlier. We'll start with the most straightforward one, Principal Component Analysis (PCA).
#
# PCA seeks orthogonal linear combinations of the features which show the greatest variance, and as such, can help give you a good idea of the structure of the data set. Here we'll use RandomizedPCA, because it's faster for large N.

# +
from sklearn.decomposition import PCA
pca = PCA(n_components=2, svd_solver="randomized")
proj = pca.fit_transform(digits.data)

plt.scatter(proj[:, 0], proj[:, 1], c=digits.target)
plt.colorbar()
# -

# A weakness of PCA is that it produces a linear dimensionality reduction:
# this may miss some interesting relationships in the data.  If we want to
# see a nonlinear mapping  of the data, we can use one of the several
# methods in the `manifold` module.  Here we'll use [Isomap](https://blog.csdn.net/VictoriaW/article/details/78497316) (a concatenation
# of Isometric Mapping) which is a manifold learning method based on
# graph theory:

# +
from sklearn.manifold import Isomap
iso = Isomap(n_neighbors=5, n_components=2)
proj = iso.fit_transform(digits.data)

plt.scatter(proj[:, 0], proj[:, 1], c=digits.target)
plt.colorbar()
# -

# ## Program

# +
from sklearn.datasets import load_digits
from sklearn.linear_model.logistic import LogisticRegression
from sklearn.metrics import accuracy_score

import matplotlib.pyplot as plt 

# load digital data
digits, dig_label = load_digits(return_X_y=True)
print(digits.shape)

# calculate train/test data number
N = len(digits)
N_train = int(N*0.8)
N_test = N - N_train

# split train/test data
x_train = digits[:N_train, :]
y_train = dig_label[:N_train]
x_test  = digits[N_train:, :]
y_test  = dig_label[N_train:]

# do logistic regression
lr=LogisticRegression()
lr.fit(x_train,y_train)

pred_train = lr.predict(x_train)
pred_test  = lr.predict(x_test)

# calculate train/test accuracy
acc_train = accuracy_score(y_train, pred_train)
acc_test = accuracy_score(y_test, pred_test)
print("accuracy train = %f, accuracy_test = %f" % (acc_train, acc_test))

score_train = lr.score(x_train, y_train)
score_test  = lr.score(x_test, y_test)
print("score_train = %f, score_test = %f" % (score_train, score_test))



# +
from sklearn.metrics import confusion_matrix

# plot confusion matrix
cm = confusion_matrix(y_test,pred_test)

plt.matshow(cm)
plt.title(u'Confusion Matrix')
plt.colorbar()
plt.ylabel(u'Groundtruth')
plt.xlabel(u'Predict')
plt.show()
# -

# ## Exercise - How to draw mis-classfied data?
#
# 1. How to obtain the mis-classified index?
# 2. How to draw them?

# ## References
#
# * [逻辑回归模型(Logistic Regression, LR)基础](https://www.cnblogs.com/sparkwen/p/3441197.html)
# * [逻辑回归（Logistic Regression）](http://www.cnblogs.com/BYRans/p/4713624.html)