PCA与梯度上升法

PAC

主成分分析（Principal Component Analysis）

• 一个非监督的机器学习算法
• 主要用于数据的降维
• 通过降维，可以发现更便于人类理解的特征
• 其他应用：可视化；去噪

如何找到这个让样本间间距最大的轴？
如何定义样本间间距？

找到一个轴，使得样本空间的所有点映射到这个轴后，方差最大

梯度上升法解决主成分分析问题

主成分分析



代码实现
生成测试用例

import numpy as np
import matplotlib.pyplot as plt
X = np.empty((100, 2))
X[:,0] = np.random.uniform(0., 100., size=100)
X[:,1] = 0.75 * X[:,0] + 3. + np.random.normal(0, 10., size=100)
plt.scatter(X[:,0], X[:,1])
plt.show()
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demean

def demean(X):
    return X - np.mean(X, axis=0)
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X_demean = demean(X)
plt.scatter(X_demean[:,0], X_demean[:,1])
plt.show()
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梯度上升法

def f(w, X):
    return np.sum((X.dot(w)**2)) / len(X)

def df_math(w, X):
    return X.T.dot(X.dot(w)) * 2. / len(X)

def df_debug(w, X, epsilon=0.0001):
    res = np.empty(len(w))
    for i in range(len(w)):
        w_1 = w.copy()
        w_1[i] += epsilon
        w_2 = w.copy()
        w_2[i] -= epsilon
        res[i] = (f(w_1, X) - f(w_2, X)) / (2 * epsilon)
    return res

def direction(w):
    return w / np.linalg.norm(w)

def gradient_ascent(df, X, initial_w, eta, n_iters = 1e4, epsilon=1e-8):
    
    w = direction(initial_w) 
    cur_iter = 0

    while cur_iter < n_iters:
        gradient = df(w, X)
        last_w = w
        w = w + eta * gradient
        w = direction(w) # 注意1：每次求一个单位方向
        if(abs(f(w, X) - f(last_w, X)) < epsilon):
            break
            
        cur_iter += 1

    return w
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initial_w = np.random.random(X.shape[1]) # 注意2：不能用0向量开始
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eta = 0.001
# 注意3： 不能使用StandardScaler标准化数据
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gradient_ascent(df_debug, X_demean, initial_w, eta)
gradient_ascent(df_math, X_demean, initial_w, eta)
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w = gradient_ascent(df_math, X_demean, initial_w, eta)
plt.scatter(X_demean[:,0], X_demean[:,1])
plt.plot([0, w[0]*30], [0, w[1]*30], color='r')
plt.show()
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使用极端数据集测试

X2 = np.empty((100, 2))
X2[:,0] = np.random.uniform(0., 100., size=100)
X2[:,1] = 0.75 * X2[:,0] + 3.
plt.scatter(X2[:,0], X2[:,1])
plt.show()
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X2_demean = demean(X2)
w2 = gradient_ascent(df_math, X2_demean, initial_w, eta)
plt.scatter(X2_demean[:,0], X2_demean[:,1])
plt.plot([0, w2[0]*30], [0, w2[1]*30], color='r')
plt.show()
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求数据的前n个主成分

求出第一主成分以后，如何求出下一个主成分?
数据进行改变，将数据在第一个主成分上的分量去掉

在新的数据上求第一主成分
代码
生成数据

import numpy as np
import matplotlib.pyplot as plt
X = np.empty((100, 2))
X[:,0] = np.random.uniform(0., 100., size=100)
X[:,1] = 0.75 * X[:,0] + 3. + np.random.normal(0, 10., size=100)
def demean(X):
    return X - np.mean(X, axis=0)
X = demean(X)
plt.scatter(X[:,0], X[:,1])
plt.show()
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求出第一主成分

def f(w, X):
    return np.sum((X.dot(w)**2)) / len(X)

def df(w, X):
    return X.T.dot(X.dot(w)) * 2. / len(X)

def direction(w):
    return w / np.linalg.norm(w)

def first_component(X, initial_w, eta, n_iters = 1e4, epsilon=1e-8):
    
    w = direction(initial_w) 
    cur_iter = 0

    while cur_iter < n_iters:
        gradient = df(w, X)
        last_w = w
        w = w + eta * gradient
        w = direction(w) 
        if(abs(f(w, X) - f(last_w, X)) < epsilon):
            break
            
        cur_iter += 1

    return w
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initial_w = np.random.random(X.shape[1])
eta = 0.01
w = first_component(X, initial_w, eta)
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求第二主成分

X2 = np.empty(X.shape)
for i in range(len(X)):
    X2[i] = X[i] - X[i].dot(w) * w
plt.scatter(X2[:,0], X2[:,1])
plt.show()
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X2 = X - X.dot(w).reshape(-1, 1) * w
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plt.scatter(X2[:,0], X2[:,1])
plt.show()
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w2 = first_component(X2, initial_w, eta)
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w.dot(w2)
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def first_n_components(n, X, eta=0.01, n_iters = 1e4, epsilon=1e-8):
    X_pca = X.copy()
    X_pca = demean(X_pca)
    res = []
    for i in range(n):
        initial_w = np.random.random(X_pca.shape[1])
        w = first_component(X_pca, initial_w, eta)
        res.append(w)
        
        X_pca = X_pca - X_pca.dot(w).reshape(-1, 1) * w
        
    return res
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first_n_components(2, X)
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高维数据向低维数据映射

代码实现

import numpy as np


class PCA:

    def __init__(self, n_components):
        """初始化PCA"""
        assert n_components >= 1, "n_components must be valid"
        self.n_components = n_components
        self.components_ = None

    def fit(self, X, eta=0.01, n_iters=1e4):
        """获得数据集X的前n个主成分"""
        assert self.n_components <= X.shape[1], \
            "n_components must not be greater than the feature number of X"

        def demean(X):
            return X - np.mean(X, axis=0)

        def f(w, X):
            return np.sum((X.dot(w) ** 2)) / len(X)

        def df(w, X):
            return X.T.dot(X.dot(w)) * 2. / len(X)

        def direction(w):
            return w / np.linalg.norm(w)

        def first_component(X, initial_w, eta=0.01, n_iters=1e4, epsilon=1e-8):

            w = direction(initial_w)
            cur_iter = 0

            while cur_iter < n_iters:
                gradient = df(w, X)
                last_w = w
                w = w + eta * gradient
                w = direction(w)
                if (abs(f(w, X) - f(last_w, X)) < epsilon):
                    break

                cur_iter += 1

            return w

        X_pca = demean(X)
        self.components_ = np.empty(shape=(self.n_components, X.shape[1]))
        for i in range(self.n_components):
            initial_w = np.random.random(X_pca.shape[1])
            w = first_component(X_pca, initial_w, eta, n_iters)
            self.components_[i,:] = w

            X_pca = X_pca - X_pca.dot(w).reshape(-1, 1) * w

        return self

    def transform(self, X):
        """将给定的X，映射到各个主成分分量中"""
        assert X.shape[1] == self.components_.shape[1]

        return X.dot(self.components_.T)

    def inverse_transform(self, X):
        """将给定的X，反向映射回原来的特征空间"""
        assert X.shape[1] == self.components_.shape[0]

        return X.dot(self.components_)

    def __repr__(self):
        return "PCA(n_components=%d)" % self.n_components


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使用
生成数据

import numpy as np
import matplotlib.pyplot as plt
X = np.empty((100, 2))
X[:,0] = np.random.uniform(0., 100., size=100)
X[:,1] = 0.75 * X[:,0] + 3. + np.random.normal(0, 10., size=100)
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使用

from playML.PCA import PCA

pca = PCA(n_components=2)
pca.fit(X)
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pca = PCA(n_components=1)
pca.fit(X)
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降维

X_reduction = pca.transform(X)
X_reduction.shape
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恢复降维

X_restore = pca.inverse_transform(X_reduction)
X_restore.shape
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可视化

plt.scatter(X[:,0], X[:,1], color='b', alpha=0.5)
plt.scatter(X_restore[:,0], X_restore[:,1], color='r', alpha=0.5)
plt.show()
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scikit-learn中的PCA

from sklearn.decomposition import PCA
pca = PCA(n_components=1)
pca.fit(X)
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pca.components_
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X_reduction = pca.transform(X)
X_restore = pca.inverse_transform(X_reduction)
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plt.scatter(X[:,0], X[:,1], color='b', alpha=0.5)
plt.scatter(X_restore[:,0], X_restore[:,1], color='r', alpha=0.5)
plt.show()
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真实数据测试

import numpy as np
import matplotlib.pyplot as plt
from sklearn import datasets
digits = datasets.load_digits()
X = digits.data
y = digits.target
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分数据集

from sklearn.model_selection import train_test_split

X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=666)
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训练

from sklearn.neighbors import KNeighborsClassifier

knn_clf = KNeighborsClassifier()
knn_clf.fit(X_train, y_train)
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knn_clf.score(X_test, y_test)
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PCA降维

from sklearn.decomposition import PCA

pca = PCA(n_components=2)
pca.fit(X_train)
X_train_reduction = pca.transform(X_train)
X_test_reduction = pca.transform(X_test)
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knn_clf = KNeighborsClassifier()
knn_clf.fit(X_train_reduction, y_train)
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knn_clf.score(X_test_reduction, y_test)
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速度提高了 精度降低了

主成分所解释的方差

pca.explained_variance_ratio_
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from sklearn.decomposition import PCA

pca = PCA(n_components=X_train.shape[1])
pca.fit(X_train)
pca.explained_variance_ratio_
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可视化

plt.plot([i for i in range(X_train.shape[1])], 
         [np.sum(pca.explained_variance_ratio_[:i+1]) for i in range(X_train.shape[1])])
plt.show()
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主成分个数可解释95%+的方差

pca = PCA(0.95)
pca.fit(X_train)
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pca.n_components_
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说明需要28维

X_train_reduction = pca.transform(X_train)
X_test_reduction = pca.transform(X_test)
knn_clf = KNeighborsClassifier()
knn_clf.fit(X_train_reduction, y_train)
knn_clf.score(X_test_reduction, y_test)
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使用PCA对数据进行降维可视化

pca = PCA(n_components=2)
pca.fit(X)
X_reduction = pca.transform(X)
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for i in range(10):
    plt.scatter(X_reduction[y==i,0], X_reduction[y==i,1], alpha=0.8)
plt.show()
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MNIST手写数据集 训练

import numpy as np 

from sklearn.datasets import fetch_openml
 
mnist = fetch_openml('mnist_784')

X, y = mnist['data'], mnist['target']
X_train = np.array(X[:60000], dtype=float)
y_train = np.array(y[:60000], dtype=float)
X_test = np.array(X[60000:], dtype=float)
y_test = np.array(y[60000:], dtype=float)
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使用KNN

from sklearn.neighbors import KNeighborsClassifier

knn_clf = KNeighborsClassifier()
knn_clf.fit(X_train, y_train)
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%time knn_clf.score(X_test, y_test)
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PCA进行降维

from sklearn.decomposition import PCA 
pca = PCA(0.90)
pca.fit(X_train)
X_train_reduction = pca.transform(X_train)
X_test_reduction = pca.transform(X_test)
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X_train_reduction.shape
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knn_clf = KNeighborsClassifier()
knn_clf.fit(X_train_reduction, y_train)
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%time knn_clf.score(X_test_reduction, y_test)
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使用PCA降噪

例子

import numpy as np
import matplotlib.pyplot as plt
X = np.empty((100, 2))
X[:,0] = np.random.uniform(0., 100., size=100)
X[:,1] = 0.75 * X[:,0] + 3. + np.random.normal(0, 5, size=100)
plt.scatter(X[:,0], X[:,1])
plt.show()
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from sklearn.decomposition import PCA

pca = PCA(n_components=1)
pca.fit(X)
X_reduction = pca.transform(X)
X_restore = pca.inverse_transform(X_reduction)
plt.scatter(X_restore[:,0], X_restore[:,1])
plt.show()
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手写识别例子

from sklearn import datasets

digits = datasets.load_digits()
X = digits.data
y = digits.target
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制造噪音

noisy_digits = X + np.random.normal(0, 4, size=X.shape)
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取100个

example_digits = noisy_digits[y==0,:][:10]
for num in range(1,10):
    example_digits = np.vstack([example_digits, noisy_digits[y==num,:][:10]])
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绘制

def plot_digits(data):
    fig, axes = plt.subplots(10, 10, figsize=(10, 10),
                             subplot_kw={'xticks':[], 'yticks':[]},
    gridspec_kw=dict(hspace=0.1, wspace=0.1)) 
    for i, ax in enumerate(axes.flat):
        ax.imshow(data[i].reshape(8, 8),
                  cmap='binary', interpolation='nearest',
                  clim=(0, 16))

    plt.show()
    
plot_digits(example_digits)
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降噪

pca = PCA(0.5).fit(noisy_digits)
pca.n_components_
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components = pca.transform(example_digits)
filtered_digits = pca.inverse_transform(components)
plot_digits(filtered_digits)
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特征脸

加载数据

import numpy as np
import matplotlib.pyplot as plt
from sklearn.datasets import fetch_lfw_people
faces = fetch_lfw_people()
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查看属性

faces.keys()
faces.data.shape
faces.target_names
faces.images.shape
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获取36个数据

random_indexes = np.random.permutation(len(faces.data))
X = faces.data[random_indexes]
example_faces = X[:36,:]
example_faces.shape
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绘制

def plot_faces(faces):
    
    fig, axes = plt.subplots(6, 6, figsize=(10, 10),
                         subplot_kw={'xticks':[], 'yticks':[]},
    gridspec_kw=dict(hspace=0.1, wspace=0.1)) 
    for i, ax in enumerate(axes.flat):
        ax.imshow(faces[i].reshape(62, 47), cmap='bone')
    plt.show()
    
plot_faces(example_faces)
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特征脸

%%time
from sklearn.decomposition import PCA 
pca = PCA(svd_solver='randomized')# 随机方式
pca.fit(X)
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pca.components_.shape
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绘制

plot_faces(pca.components_[:36,:])
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