機器學習演算法及程式碼實現–支援向量機

1、支援向量機

SVM希望通過N-1維的分隔超平面線性分開N維的資料，距離分隔超平面最近的點被叫做支援向量，我們利用SMO（SVM實現方法之一）最大化支援向量到分隔面的距離，這樣當新樣本點進來時，其被分類正確的概率也就更大。我們計算樣本點到分隔超平面的函式間隔，如果函式間隔為正，則分類正確，函式間隔為負，則分類錯誤，函式間隔的絕對值除以||w||就是幾何間隔，幾何間隔始終為正，可以理解為樣本點到分隔超平面的幾何距離。若資料不是線性可分的，那我們引入核函式的概念，從某個特徵空間到另一個特徵空間的對映是通過核函式來實現的，我們利用核函式將資料從低維空間對映到高維空間，低維空間的非線性問題在高維空間往往會成為線性問題，再利用N-1維分割超平面對資料分類。

2、分類

線性可分、線性不可分

3、超平面公式（先考慮線性可分）

W*X+b=0
其中W={w1,w2,,,w3}，為權重向量
下面用簡單的二維向量講解（思維導圖）

4、尋找超平面

5、例子

6、線性不可分

對映到高維

演算法思路（思維導圖）

核函式舉例

程式碼

# -*- coding: utf-8 -*-
from sklearn import svm

# 資料
x = [[2, 0], [1, 1], [2, 3]]
# 標籤
y = [0, 0, 1]
# 線性可分的svm分類器，用線性的核函式
clf = svm.SVC(kernel='linear'
 
)
# 訓練
clf.fit(x, y)
print clf

# 獲得支援向量
print clf.support_vectors_

# 獲得支援向量點在原資料中的下標
print clf.support_

# 獲得每個類支援向量的個數
print clf.n_support_

# 預測
print clf.predict([2, 0])

# -*- coding: utf-8 -*-
import numpy as np
import pylab as pl
from sklearn import svm

np.random.seed(0)  # 值固定，每次隨機結果不變
# 2組20
 
個二維的隨機數，20個0，20個1的y  （20，2）20行2列
X = np.r_[np.random.randn(20, 2) - [2, 2], np.random.randn(20, 2) + [2, 2]]
Y = [0] * 20 + [1] * 20

# 訓練
clf = svm.SVC(kernel='linear')
clf.fit(X, Y)


w = clf.coef_[0]
a = -w[0] / w[1]
xx = np.linspace(-5, 5)
yy = a * xx - (clf.intercept_[0] / w[1])  # 點斜式 平分的線


b = clf.support_vectors_[0]
yy_down = a* xx +(b[1] - a*b[0])
b = clf.support_vectors_[-1]
yy_up = a* xx +(b[1] - a*b[0])  # 兩條虛線

print "w: ", w
print "a: ", a
# print " xx: ", xx
# print " yy: ", yy
print "support_vectors_: ", clf.support_vectors_
print "clf.coef_: ", clf.coef_

# In scikit-learn coef_ attribute holds the vectors of the separating hyperplanes for linear models. It has shape (n_classes, n_features) if n_classes > 1 (multi-class one-vs-all) and (1, n_features) for binary classification.
#
# In this toy binary classification example, n_features == 2, hence w = coef_[0] is the vector orthogonal to the hyperplane (the hyperplane is fully defined by it + the intercept).
#
# To plot this hyperplane in the 2D case (any hyperplane of a 2D plane is a 1D line), we want to find a f as in y = f(x) = a.x + b. In this case a is the slope of the line and can be computed by a = -w[0] / w[1].




# plot the line, the points, and the nearest vectors to the plane
pl.plot(xx, yy, 'k-')
pl.plot(xx, yy_down, 'k--')
pl.plot(xx, yy_up, 'k--')

pl.scatter(clf.support_vectors_[:, 0], clf.support_vectors_[:, 1],
           s=80, facecolors='none')
pl.scatter(X[:, 0], X[:, 1], c=Y, cmap=pl.cm.Paired)

pl.axis('tight')
pl.show()

# -*- coding: utf-8 -*-
from __future__ import print_function

from time import time
import logging  # 列印程式進展的資訊
import matplotlib.pyplot as plt

from sklearn.cross_validation import train_test_split
from sklearn.datasets import fetch_lfw_people
from sklearn.grid_search import GridSearchCV
from sklearn.metrics import classification_report
from sklearn.metrics import confusion_matrix
from sklearn.decomposition import RandomizedPCA
from sklearn.svm import SVC


print(__doc__)

# 列印程式進展的資訊
logging.basicConfig(level=logging.INFO, format='%(asctime)s %(message)s')


###############################################################################
# 下載人臉資料集,並匯入

lfw_people = fetch_lfw_people(min_faces_per_person=70, resize=0.4)

# 資料集多少，長寬多少
n_samples, h, w = lfw_people.images.shape

# x是特徵向量的矩陣，獲取矩陣列數，即緯度
X = lfw_people.data
n_features = X.shape[1]

# y是分類標籤向量
y = lfw_people.target
# 類別裡面有誰的名字
target_names = lfw_people.target_names
# 名字有多少行，即有多少人要區分
n_classes = target_names.shape[0]

# 列印
print("Total dataset size:")
print("n_samples: %d" % n_samples)
print("n_features: %d" % n_features)
print("n_classes: %d" % n_classes)


###############################################################################
# 將資料集劃分為訓練集和測試集，測試集佔0.25
X_train, X_test, y_train, y_test = train_test_split(
    X, y, test_size=0.25)


###############################################################################
# PCA降維
n_components = 150  # 組成元素數量

print("Extracting the top %d eigenfaces from %d faces"
      % (n_components, X_train.shape[0]))
t0 = time()
# 建立PCA模型
pca = RandomizedPCA(n_components=n_components, whiten=True).fit(X_train)
print("done in %0.3fs" % (time() - t0))

# 提取特徵臉
eigenfaces = pca.components_.reshape((n_components, h, w))

print("Projecting the input data on the eigenfaces orthonormal basis")
t0 = time()
# 將特徵向量轉化為低維矩陣
X_train_pca = pca.transform(X_train)
X_test_pca = pca.transform(X_test)
print("done in %0.3fs" % (time() - t0))


###############################################################################
# Train a SVM classification model

print("Fitting the classifier to the training set")
t0 = time()
# C錯誤懲罰權重 gamma 建立核函式的不同比例
param_grid = {'C': [1e3, 5e3, 1e4, 5e4, 1e5],
              'gamma': [0.0001, 0.0005, 0.001, 0.005, 0.01, 0.1], }
# 選擇核函式，建SVC，嘗試執行，獲得最好引數
clf = GridSearchCV(SVC(kernel='rbf', class_weight='auto'), param_grid)
# 訓練
clf = clf.fit(X_train_pca, y_train)
print("done in %0.3fs" % (time() - t0))
print("Best estimator found by grid search:")
print(clf.best_estimator_)  # 輸出最佳引數


###############################################################################
# Quantitative evaluation of the model quality on the test set

print("Predicting people's names on the test set")
t0 = time()
# 預測
y_pred = clf.predict(X_test_pca)
print("done in %0.3fs" % (time() - t0))

print(classification_report(y_test, y_pred, target_names=target_names))  # 與真實情況作對比求置信度
print(confusion_matrix(y_test, y_pred, labels=range(n_classes)))  # 對角線的為預測正確的，a預測為a


###############################################################################
# Qualitative evaluation of the predictions using matplotlib

def plot_gallery(images, titles, h, w, n_row=3, n_col=4):
    """Helper function to plot a gallery of portraits"""
    plt.figure(figsize=(1.8 * n_col, 2.4 * n_row))
    plt.subplots_adjust(bottom=0, left=.01, right=.99, top=.90, hspace=.35)
    for i in range(n_row * n_col):
        plt.subplot(n_row, n_col, i + 1)
        plt.imshow(images[i].reshape((h, w)), cmap=plt.cm.gray)
        plt.title(titles[i], size=12)
        plt.xticks(())
        plt.yticks(())


# plot the result of the prediction on a portion of the test set

def title(y_pred, y_test, target_names, i):
    pred_name = target_names[y_pred[i]].rsplit(' ', 1)[-1]
    true_name = target_names[y_test[i]].rsplit(' ', 1)[-1]
    return 'predicted: %s\ntrue:      %s' % (pred_name, true_name)

prediction_titles = [title(y_pred, y_test, target_names, i)
                     for i in range(y_pred.shape[0])]

plot_gallery(X_test, prediction_titles, h, w)  # 畫出測試集和它的title

# plot the gallery of the most significative eigenfaces

eigenface_titles = ["eigenface %d" % i for i in range(eigenfaces.shape[0])]
plot_gallery(eigenfaces, eigenface_titles, h, w)  # 列印特徵臉

plt.show()  # 顯示