LDA与QDA

作者:桂。

时间:2017-05-23  06:37:31

链接:http://www.cnblogs.com/xingshansi/p/6892317.html 


前言

仍然是python库函数scikit-learn的学习笔记,内容Regression-1.2Linear and Quadratic Discriminant Analysis部分,主要包括:

    1)线性分类判别(Linear discriminant analysis, LDA)

  2)二次分类判别(Quadratic discriminant analysis, QDA)

  3)Fisher判据

 一、线性分类判别

对于二分类问题,LDA针对的是:数据服从高斯分布,且均值不同,方差相同

概率密度:

p是数据的维度。

分类判别函数:

可以看出结果是关于x的一次函数:wx+w0,线性分类判别的说法由此得来。

参数计算:

 

二、二次分类判别

对于二分类问题,QDA针对的是:数据服从高斯分布,且均值不同,方差不同

数据方差相同的时候,一次判别就可以,如左图所示;但如果方差差别较大,就是一个二次问题了,像右图那样。

 从sklearn给的例子中,也容易观察到:

QDA对数据有更好的适用性,QDA判别公式:

 

三、Fisher判据

  A-Fisher理论推导

 Fisher一个总原则是:投影之后的数据,最小化类内误差,同时最大化类间误差

其中,分别对应投影后的类均值。对应投影后的类内方差。

重写类内总方差、类间距离:

 准则函数重写:

这就是泛化瑞利熵的形式了,容易求解:

其中常借助SVD求解:Sw = U∑VT,Sw-1 = U∑-1VT,借助特征值分解也是可以的。

   B-LDA与Fisher

这里的LDA指代上面提到的利用概率求解的贝叶斯最优估计器。可以证明:

二分类任务中两类数据满足高斯分布且方差相同时,线性判别分析(指Fisher方法)产生贝叶斯最优分类器(指本文的LDA)。

 对于Fisher,求J的最大值:

贝叶斯最优,即距离分类中心距离除以两个类中心距离的比值最小:

二者倒数关系,一个最大化,一个最小化,所以是等价的。

补充一句:误差为高斯分布,与最小二乘也是等价的

这样一来,求解就有了三个思路:奇异值分解SVD,最小二乘Lsqr,特征值分解eigen,这也是Sklearn的思路:

  C-多分类LDA

定义类内散度矩阵:

定义类间散度矩阵:

定义总散度矩阵:

可见ST 、SB、 SW三者任意取两个即可。

准则函数:

其中。 

最优化求解:

看作投影矩阵,其闭式解是的d'个最大非零广义特征值对应的特征向量组成的矩阵。d'通常远小于原数据属性数d,这就实现了监督的降维。

参数求解利用train data:

 

四、Sk-learn基本用法

  LDA:

lda = LinearDiscriminantAnalysis(solver="svd", store_covariance=True)
y_pred = lda.fit(X, y).predict(X)

  QDA:

qda = QuadraticDiscriminantAnalysis(store_covariances=True)
y_pred = qda.fit(X, y).predict(X)

  LDA与QDA应用实例:

"""
====================================================================
Linear and Quadratic Discriminant Analysis with confidence ellipsoid
====================================================================

Plot the confidence ellipsoids of each class and decision boundary
"""
print(__doc__)

from scipy import linalg
import numpy as np
import matplotlib.pyplot as plt
import matplotlib as mpl
from matplotlib import colors

from sklearn.discriminant_analysis import LinearDiscriminantAnalysis
from sklearn.discriminant_analysis import QuadraticDiscriminantAnalysis

###############################################################################
# colormap
cmap = colors.LinearSegmentedColormap(
    'red_blue_classes',
    {'red': [(0, 1, 1), (1, 0.7, 0.7)],
     'green': [(0, 0.7, 0.7), (1, 0.7, 0.7)],
     'blue': [(0, 0.7, 0.7), (1, 1, 1)]})
plt.cm.register_cmap(cmap=cmap)


###############################################################################
# generate datasets
def dataset_fixed_cov():
    '''Generate 2 Gaussians samples with the same covariance matrix'''
    n, dim = 300, 2
    np.random.seed(0)
    C = np.array([[0., -0.23], [0.83, .23]])
    X = np.r_[np.dot(np.random.randn(n, dim), C),
              np.dot(np.random.randn(n, dim), C) + np.array([1, 1])]
    y = np.hstack((np.zeros(n), np.ones(n)))
    return X, y


def dataset_cov():
    '''Generate 2 Gaussians samples with different covariance matrices'''
    n, dim = 300, 2
    np.random.seed(0)
    C = np.array([[0., -1.], [2.5, .7]]) * 2.
    X = np.r_[np.dot(np.random.randn(n, dim), C),
              np.dot(np.random.randn(n, dim), C.T) + np.array([1, 4])]
    y = np.hstack((np.zeros(n), np.ones(n)))
    return X, y


###############################################################################
# plot functions
def plot_data(lda, X, y, y_pred, fig_index):
    splot = plt.subplot(2, 2, fig_index)
    if fig_index == 1:
        plt.title('Linear Discriminant Analysis')
        plt.ylabel('Data with fixed covariance')
    elif fig_index == 2:
        plt.title('Quadratic Discriminant Analysis')
    elif fig_index == 3:
        plt.ylabel('Data with varying covariances')

    tp = (y == y_pred)  # True Positive
    tp0, tp1 = tp[y == 0], tp[y == 1]
    X0, X1 = X[y == 0], X[y == 1]
    X0_tp, X0_fp = X0[tp0], X0[~tp0]
    X1_tp, X1_fp = X1[tp1], X1[~tp1]

    alpha = 0.5

    # class 0: dots
    plt.plot(X0_tp[:, 0], X0_tp[:, 1], 'o', alpha=alpha,
             color='red')
    plt.plot(X0_fp[:, 0], X0_fp[:, 1], '*', alpha=alpha,
             color='#990000')  # dark red

    # class 1: dots
    plt.plot(X1_tp[:, 0], X1_tp[:, 1], 'o', alpha=alpha,
             color='blue')
    plt.plot(X1_fp[:, 0], X1_fp[:, 1], '*', alpha=alpha,
             color='#000099')  # dark blue

    # class 0 and 1 : areas
    nx, ny = 200, 100
    x_min, x_max = plt.xlim()
    y_min, y_max = plt.ylim()
    xx, yy = np.meshgrid(np.linspace(x_min, x_max, nx),
                         np.linspace(y_min, y_max, ny))
    Z = lda.predict_proba(np.c_[xx.ravel(), yy.ravel()])
    Z = Z[:, 1].reshape(xx.shape)
    plt.pcolormesh(xx, yy, Z, cmap='red_blue_classes',
                   norm=colors.Normalize(0., 1.))
    plt.contour(xx, yy, Z, [0.5], linewidths=2., colors='k')

    # means
    plt.plot(lda.means_[0][0], lda.means_[0][1],
             'o', color='black', markersize=10)
    plt.plot(lda.means_[1][0], lda.means_[1][1],
             'o', color='black', markersize=10)

    return splot


def plot_ellipse(splot, mean, cov, color):
    v, w = linalg.eigh(cov)
    u = w[0] / linalg.norm(w[0])
    angle = np.arctan(u[1] / u[0])
    angle = 180 * angle / np.pi  # convert to degrees
    # filled Gaussian at 2 standard deviation
    ell = mpl.patches.Ellipse(mean, 2 * v[0] ** 0.5, 2 * v[1] ** 0.5,
                              180 + angle, facecolor=color, edgecolor='yellow',
                              linewidth=2, zorder=2)
    ell.set_clip_box(splot.bbox)
    ell.set_alpha(0.5)
    splot.add_artist(ell)
    splot.set_xticks(())
    splot.set_yticks(())


def plot_lda_cov(lda, splot):
    plot_ellipse(splot, lda.means_[0], lda.covariance_, 'red')
    plot_ellipse(splot, lda.means_[1], lda.covariance_, 'blue')


def plot_qda_cov(qda, splot):
    plot_ellipse(splot, qda.means_[0], qda.covariances_[0], 'red')
    plot_ellipse(splot, qda.means_[1], qda.covariances_[1], 'blue')

###############################################################################
for i, (X, y) in enumerate([dataset_fixed_cov(), dataset_cov()]):
    # Linear Discriminant Analysis
    lda = LinearDiscriminantAnalysis(solver="svd", store_covariance=True)
    y_pred = lda.fit(X, y).predict(X)
    splot = plot_data(lda, X, y, y_pred, fig_index=2 * i + 1)
    plot_lda_cov(lda, splot)
    plt.axis('tight')

    # Quadratic Discriminant Analysis
    qda = QuadraticDiscriminantAnalysis(store_covariances=True)
    y_pred = qda.fit(X, y).predict(X)
    splot = plot_data(qda, X, y, y_pred, fig_index=2 * i + 2)
    plot_qda_cov(qda, splot)
    plt.axis('tight')
plt.suptitle('Linear Discriminant Analysis vs Quadratic Discriminant Analysis')
plt.show()

LDA与PCA,LDA与PCA都可以借助SVD求解,但本质是不同的:

顺便提一句之前梳理的独立成分分析(ICA)与PCA的差别,PCA立足点是相关性,是基于协方差矩阵(二阶统计量);而ICA立足点是独立性,利用概率分布(也就是高阶统计量),当然如果是正态分布,二者就等价了。

关于LDA、PCA降维的对比,Sklearn给出了IRIS数据的示例:

 The Iris dataset represents 3 kind of Iris flowers (Setosa, Versicolour and Virginica) with 4 attributes: sepal length, sepal width, petal length and petal width.即数据类别数是3,每一个样本对应的特征维度是4,现在分别用LDA、PCA降至2维。

code:

print(__doc__)

import matplotlib.pyplot as plt

from sklearn import datasets
from sklearn.decomposition import PCA
from sklearn.discriminant_analysis import LinearDiscriminantAnalysis

iris = datasets.load_iris()

X = iris.data
y = iris.target
target_names = iris.target_names

pca = PCA(n_components=2)
X_r = pca.fit(X).transform(X)

lda = LinearDiscriminantAnalysis(n_components=2)
X_r2 = lda.fit(X, y).transform(X)

# Percentage of variance explained for each components
print('explained variance ratio (first two components): %s'
      % str(pca.explained_variance_ratio_))

plt.figure()
colors = ['navy', 'turquoise', 'darkorange']
lw = 2

for color, i, target_name in zip(colors, [0, 1, 2], target_names):
    plt.scatter(X_r[y == i, 0], X_r[y == i, 1], color=color, alpha=.8, lw=lw,
                label=target_name)
plt.legend(loc='best', shadow=False, scatterpoints=1)
plt.title('PCA of IRIS dataset')

plt.figure()
for color, i, target_name in zip(colors, [0, 1, 2], target_names):
    plt.scatter(X_r2[y == i, 0], X_r2[y == i, 1], alpha=.8, color=color,
                label=target_name)
plt.legend(loc='best', shadow=False, scatterpoints=1)
plt.title('LDA of IRIS dataset')

plt.show()

 Sklearn还提供了归一化:

只在求解方法为lsqr和eigen时有效,就是前面提到的特征值分解和最小二乘啦。

分别用归一化/不归一化:

clf1 = LinearDiscriminantAnalysis(solver='lsqr', shrinkage='auto').fit(X, y)
clf2 = LinearDiscriminantAnalysis(solver='lsqr', shrinkage=None).fit(X, y)

  应用实例:

from __future__ import division

import numpy as np
import matplotlib.pyplot as plt

from sklearn.datasets import make_blobs
from sklearn.discriminant_analysis import LinearDiscriminantAnalysis


n_train = 20  # samples for training
n_test = 200  # samples for testing
n_averages = 50  # how often to repeat classification
n_features_max = 75  # maximum number of features
step = 4  # step size for the calculation


def generate_data(n_samples, n_features):
    """Generate random blob-ish data with noisy features.

    This returns an array of input data with shape `(n_samples, n_features)`
    and an array of `n_samples` target labels.

    Only one feature contains discriminative information, the other features
    contain only noise.
    """
    X, y = make_blobs(n_samples=n_samples, n_features=1, centers=[[-2], [2]])

    # add non-discriminative features
    if n_features > 1:
        X = np.hstack([X, np.random.randn(n_samples, n_features - 1)])
    return X, y

acc_clf1, acc_clf2 = [], []
n_features_range = range(1, n_features_max + 1, step)
for n_features in n_features_range:
    score_clf1, score_clf2 = 0, 0
    for _ in range(n_averages):
        X, y = generate_data(n_train, n_features)

        clf1 = LinearDiscriminantAnalysis(solver='lsqr', shrinkage='auto').fit(X, y)
        clf2 = LinearDiscriminantAnalysis(solver='lsqr', shrinkage=None).fit(X, y)

        X, y = generate_data(n_test, n_features)
        score_clf1 += clf1.score(X, y)
        score_clf2 += clf2.score(X, y)

    acc_clf1.append(score_clf1 / n_averages)
    acc_clf2.append(score_clf2 / n_averages)

features_samples_ratio = np.array(n_features_range) / n_train

plt.plot(features_samples_ratio, acc_clf1, linewidth=2,
         label="Linear Discriminant Analysis with shrinkage", color='navy')
plt.plot(features_samples_ratio, acc_clf2, linewidth=2,
         label="Linear Discriminant Analysis", color='gold')

plt.xlabel('n_features / n_samples')
plt.ylabel('Classification accuracy')

plt.legend(loc=1, prop={'size': 12})
plt.suptitle('Linear Discriminant Analysis vs. \
shrinkage Linear Discriminant Analysis (1 discriminative feature)')
plt.show()

  结果图可以看出,shrinkage更鲁棒:

参考

  • http://blog.csdn.net/daunxx/article/details/51881956
  • http://scikit-learn.org/stable/modules/lda_qda.html
posted @ 2017-05-23 10:34  LeeLIn。  阅读(8273)  评论(0编辑  收藏  举报