scikit-learn：4.7. Pairwise metrics, Affinities and Kernels

參考：http://scikit-learn.org/stable/modules/metrics.html

The sklearn.metrics.pairwise submodule implements utilities to evaluate pairwise distances（样本对的距离） or affinity of sets of samples（样本集的相似度）。

Distance metrics are functions d(a, b) such that d(a, b) < d(a, c) if objects a and b are considered “more similar” than objects a and c. 

Kernels are measures of similarity, i.e. s(a, b) > s(a, c) if objects a and b are considered “more similar” than objects a and c. 

1、Cosine similarity

向量点积的L2-norm：

if  and  are row vectors, their cosine similarity  is defined as:

This kernel is a popular choice for computing the similarity of documents represented as tf-idf vectors.

2、Linear kernel

If x and y are column vectors, their linear kernel is:

(x, y) = x_transport * y

3、Polynomial kernel

Conceptually, the polynomial kernels considers not only the similarity between vectors under the same dimension, but also across dimensions. When used in machine learning algorithms, this allows to account for feature interaction.

The polynomial kernel is defined as:

4、Sigmoid kernel

defined as:

5、RBF kernel

defined as:

If  the kernel is known as the Gaussian kernel of variance .

6、Chi-squared kernel

defined as:

The chi-squared kernel is a very popular choice for training non-linear SVMs in computer vision applications. It can be computed usingchi2_kernel and then passed to an sklearn.svm.SVC with kernel="precomputed":

>>> from sklearn.svm import SVC
>>> from sklearn.metrics.pairwise import chi2_kernel
>>> X = [[0, 1], [1, 0], [.2, .8], [.7, .3]]
>>> y = [0, 1, 0, 1]
>>> K = chi2_kernel(X, gamma=.5)
>>> K                        
array([[ 1.        ,  0.36...,  0.89...,  0.58...],
       [ 0.36...,  1.        ,  0.51...,  0.83...],
       [ 0.89...,  0.51...,  1.        ,  0.77... ],
       [ 0.58...,  0.83...,  0.77... ,  1.        ]])

>>> svm = SVC(kernel='precomputed').fit(K, y)
>>> svm.predict(K)
array([0, 1, 0, 1])

It can also be directly used as the kernel argument:

>>> svm = SVC(kernel=chi2_kernel).fit(X, y)
>>> svm.predict(X)
array([0, 1, 0, 1])