LDA推导 + python代码
参考blog:https://www.cnblogs.com/simon-c/p/5023726.html 只有代码和注释
https://blog.csdn.net/feilong_csdn/article/details/60964027 LDA线性判别原理解析<数学推导>
https://www.cnblogs.com/viviancc/p/4133630.html 这个推导的很详细了
LDA的原理是，将带上标签的数据（点），通过投影的方法投影到维度更低的空间中，使得投影后的点形成按类别区分的情形，即相同类别的点，在投影后的空间中距离较近，不同类别的点，在投影后的空间中距离较远。
要说明LDA，首先得弄明白线性分类器(Linear Classifier)：因为LDA是一种线性分类器。对于K-分类的一个分类问题，会有K个线性函数 y=wx+b
get_ipython().magic('matplotlib inline')
import matplotlib.pyplot as plt
import numpy as np
from __future__ import print_function
from scipy import linalg
from sklearn.utils.multiclass import unique_labels
from sklearn.utils import check_array, check_X_y
from sklearn.utils.validation import check_is_fitted
__all__=['LinearDiscriminantAnalysis']
# In[427]:
#总共两个类 0 1
# 计算类内的均值
def _class_means(X,y):
means = []
# classes:[0,1]
classes = np.unique(y)
for group in classes:
# print(X)
Xg = X[y==group,:]
# print(y)
# print(Xg)
means.append(Xg.mean(0))
return np.asarray(means)
#协方差
def _cov(x):
# print("x:",x)
s = np.cov(x, rowvar=0, bias=1)
#print("s:",s)
return s
#类内的协方差
def _class_cov(X,y):
classes = np.unique(y)
covs = []
for group in classes:
# print(X)
Xg = X[y == group,:]
# print(Xg)
# print("_cov(Xg):",_cov(Xg))
covs.append(np.atleast_2d(_cov(Xg)))
#print("covs:",covs)
#print("np.:",np.average(covs, axis=0))
return np.average(covs, axis=0)
# In[428]:
class LinearDiscriminantAnalysis:
def __init__(self, n_components=None, within_between_ratio=10.0, nearest_neighbor_ratio=1.2):
self.n_components = n_components
self.within_between_ratio = within_between_ratio
self.nearest_neighbor_ratio = nearest_neighbor_ratio
# print(self.n_components)
def _solve_eigen(self,X,y):
#print("y1=",y)
#print("x1=",x)
self.means_ = _class_means(X,y)
#print("y2=",y)
self.covariance_ = _class_cov(X,y)
#print(self.means_)
#Sw:类内的散度矩阵
#Sb:类间的散度矩阵
#St:总体散度矩阵
Sw = self.covariance_ #within scatter
St = _cov(X) #total scatter
Sb = St - Sw #between scatter
# print(Sw)
# print(St)
#linalg.eigh() 计算矩阵特征值和特征向量
evals, evecs = linalg.eigh(Sb, Sw)
#排序特征向量
evecs = evecs[:, np.argsort(evals)[::-1]] #sort eigenvectors
self.scalings_ = np.asarray(evecs)
#适合局部成对的LDA训练
def fit(self, X, y):
#asarray将元组列表转换成矩阵
#reshape(-1)将y矩阵转置
X, y = check_X_y(np.asarray(X), np.asarray(y.reshape(-1)), ensure_min_samples=2)
self.classes_ = unique_labels(y)
#得到类的最大数
#self.classes_ = [0,1]
if self.n_components is None:
self.n_components = len(self.classes_) - 1
else:
self.n_components = min(len(self.classes_) - 1, self.n_components)
# print(self.classes_)
self._solve_eigen(np.asarray(X), np.asarray(y))
return self
def transform(self, X):
check_is_fitted(self, ['scalings_'], all_or_any=any)
X = check_array(X)
# print(X)
#np.dot 矩阵乘
X_new = np.dot(X, self.scalings_)
#print(self.scalings_)
print(X_new)
return X_new[:, :self.n_components]
# In[429]:
if __name__ == '__main__':
samples =200
dim = 6
lda_dim =3
data = np.random.random((samples, dim))
label = np.random.randint(0,2,size=(samples,1))
# label1 = np.random.random_integers(0,2,size=(samples,1))
# print("label:",label)
# print("label1:",label1)
# print(data)
lda = LinearDiscriminantAnalysis(lda_dim)
lda.fit(data, label)
#print(data)
lda_data = lda.transform(data)
# print ("data = ")
# print(data)
#print ("y = ")
# print(y)
# print(y.reshape(-1))
# print ("lda_data = ")
print(lda_data)
plt.scatter(data[:,0],data[:,1],c=label)
# In[409]:
get_ipython().magic('matplotlib inline')
import matplotlib.pyplot as plt
import numpy as np
x = np.arange(20)
y = x**2
plt.plot(x,y)
x=np.linspace(0,1,10) y=np.sinx plt.scatter(x,y)