机器学习理论：k近邻算法

来源：古月居

前言

KNN(k-Nearest Neighbors)思想简单，应用的数学知识几乎为0，所以作为机器学习的入门非常实用、可以解释机器学习算法使用过程中的很多细节问题。能够更加完整地刻画机器学习应用的流程。

首先大致介绍一下KNN的思想，假设我们现在有两类数据集，一类是红色的点表示，另一类用蓝色的点表示，这两类点就作为我们的训练数据集，当有一个新的数据绿色的点，那么我们该怎么给这个绿色的点进行分类呢？

一般情况下，我们需要先指定一个k，当一个新的数据集来临时，我们首先计算这个新的数据跟训练集中的每一个数据的距离，一般使用欧氏距离。

然后从中选出距离最近的k个点，这个k一般选取为奇数，方便后面投票决策。在k个点中根据最多的确定新的数据属于哪一类。

KNN基础

1.先创建好数据集x_train, y_train，和一个新的数据x_new, 并使用matplot将其可视化出来。

import numpy as np
import matplotlib.pyplot as plt


raw_data_x = [[3.3935, 2.3313],
       [3.1101, 1.7815],
       [1.3438, 3.3684],
       [3.5823, 4.6792],
       [2.2804, 2.8670],
       [7.4234, 4.6965],
       [5.7451, 3.5340],
       [9.1722, 2.5111],
       [7.7928, 3.4241],
       [7.9398, 0.7916]]
raw_data_y = [0, 0, 0, 0, 0, 1, 1, 1, 1, 1]
x_train = np.array(raw_data_x)
y_train = np.array(raw_data_y)


x_new = np.array([8.0936, 3.3657])


plt.scatter(x_train[y_train==0,0], x_train[y_train==0,1], color='g')
plt.scatter(x_train[y_train==1,0], x_train[y_train==1,1], color='r')
plt.scatter(x_new[0], x_new[1], color='b')
plt.show()

1.knn过程

2.计算距离

from math import sqrt
distance = []
for x in x_train:
  d = sqrt(np.sum((x_new - x) ** 2))
  distance.append(d)


# 其实上面这些代码用一行就可以搞定
# distances = [sqrt(np.sum((x_new - x) ** 2)) for x in x_train]

输出结果：

[10.888422144185997,
 11.825242797930196,
 15.18734646375067,
 11.660703691887552,
 12.89974598548359,
 12.707715895864213,
 9.398411207752083,
 15.62480440229573,
 12.345673749536719,
 14.394770082568183]

将距离进行排序,返回的是排序之后的索引位置

nearsest = np.argsort(distances)

输出结果：array([6, 0, 3, 1, 8, 5, 4, 9, 2, 7], dtype=int64)

取k个点，假设k=5

k = 5
topk_y = [y_train[i] for i in nearest[:k]]
topk_y

输出结果：[1, 0, 0, 0, 1]

根据输出结果我们可以发现，新来的数据距离最近的5个点，有三个点属于第一类，有两个点属于第二类，根据少数服从多数原则，新来的数据就属于第一类！

投票

from collections import Counter
Counter(topk_y)

输出结果：Counter({1: 2, 0: 3})

votes = Counter(topk_y)
votes.most_common(1)
y_new = votes.most_common(1)[0][0]

输出结果：0

这样，我们就完成了一个基本的knn！

自己写一个knn函数

knn是一个不需要训练过程的机器学习算法。其数据集可以近似看成一个模型。

import numpy as np
from math import sqrt
from collections import Counter


def kNN_classifier(k, x_train, y_train, x_new):


  assert 1 <= k <= x_train.shape[0], "k must be valid"
    assert x_train.shape[0] == y_train.shape[0], "the size of x_train must be equal to the size of y_train"
    assert x_train.shape[1] == x_new.shape[0], "the feature number of x_new must be equal to x_train"


    distances = [sqrt(np.sum((x_new - x) ** 2)) for x in x_train]
    nearest = np.argsort(distances)


    topk_y = [y_train[i] for i in nearest[:k]]
    votes = Counter(topk_y)


    return votes.most_common(1)[0][0]

测试一下：

raw_data_x = [[3.3935, 2.3313],
       [3.1101, 1.7815],
       [1.3438, 3.3684],
       [3.5823, 4.6792],
       [2.2804, 2.8670],
       [7.4234, 4.6965],
       [5.7451, 3.5340],
       [9.1722, 2.5111],
       [7.7928, 3.4241],
       [7.9398, 0.7916]]
raw_data_y = [0, 0, 0, 0, 0, 1, 1, 1, 1, 1]
x_train = np.array(raw_data_x)
y_train = np.array(raw_data_y)


x_new = np.array([8.0936, 3.3657])


y_new = kNN_classifier(5, x_train, y_train, x_new)
print(y_new)

使用sklearn中的KNN

from sklearn.neighbors import KNeighborsClassifier
import numpy as np


raw_data_x = [[3.3935, 2.3313],
       [3.1101, 1.7815],
       [1.3438, 3.3684],
       [3.5823, 4.6792],
       [2.2804, 2.8670],
       [7.4234, 4.6965],
       [5.7451, 3.5340],
       [9.1722, 2.5111],
       [7.7928, 3.4241],
       [7.9398, 0.7916]]
raw_data_y = [0, 0, 0, 0, 0, 1, 1, 1, 1, 1]
x_train = np.array(raw_data_x)
y_train = np.array(raw_data_y)


x_new = np.array([8.0936, 3.3657])


knn_classifier = KNeighborsClassifier(n_neighbors=5)
knn_classifier.fit(x_train, y_train)


y_new = knn_classifier.predict(x_new.reshape(1, -1))
print(y_new[0])

自己写一个面向对象的KNN

import numpy as np
from math import sqrt
from collections import Counter


class KNNClassifier():


  def __init__(self, k):
    assert 1 <= k, "k must be valid"
        self.k = k
        self._x_train = None
        self._y_train = None


    def fit(self, x_train, y_train):
        assert x_train.shape[0] == y_train.shape[0],  
        "the size of x_train must be equal to the size of y_train"
        assert self.k <= x_train.shape[0],  
         "the size of x_train must be at least k"


        self._x_train = x_train
        self._y_train = y_train
        return self


    def predict(self, x_new):
        x_new = x_new.reshape(1, -1)
        assert self._x_train is not None and self._y_train is not None, 
        "must fit before predict"
        assert x_new.shape[1] == self._x_train.shape[1], 
        "the feature number of x must be equal to x_train"


        y_new = [self._predict(x) for x in x_new]
        return np.array(y_new)


    def _predict(self, x):
        assert x.shape[0] == self._x_train.shape[1], 
        "the feature number of x must be equal to x_train"


        distances = [sqrt(np.sum((x_train - x) ** 2)) for x_train in self._x_train]
        nearest = np.argsort(distances)


        topk_y = [self._y_train[i] for i in nearest[:self.k]]
        votes = Counter(topk_y)


        return votes.most_common(1)[0][0]


    def __repr__(self):
        return "KNN(k=%d)" % self.k

测试一下：

raw_data_x = [[3.3935, 2.3313],
       [3.1101, 1.7815],
       [1.3438, 3.3684],
       [3.5823, 4.6792],
       [2.2804, 2.8670],
       [7.4234, 4.6965],
       [5.7451, 3.5340],
       [9.1722, 2.5111],
       [7.7928, 3.4241],
       [7.9398, 0.7916]]
raw_data_y = [0, 0, 0, 0, 0, 1, 1, 1, 1, 1]
x_train = np.array(raw_data_x)
y_train = np.array(raw_data_y)


x_new = np.array([8.0936, 3.3657])


knn_clf = KNNClassifier(6)
knn_clf.fit(x_train, y_train)
y_new = knn_clf.predict(x_new)
print(y_new[0])

分割数据集

import numpy as np
from sklearn import datasets




def train_test_split(x, y, test_ratio=0.2, seed=None):


  assert x.shape[0] == y.shape[0], "the size of x must be equal to the size of y"
  assert 0.0 <= test_ratio <= 1.0, "test_ratio must be valid"


    if seed:
        np.random.seed(seed)


    shuffle_idx = np.random.permutation(len(x))


    test_size = int(len(x) * test_ratio)
    test_idx = shuffle_idx[:test_size]
    train_idx = shuffle_idx[test_size:]


    x_train = x[train_idx]
    y_train = y[train_idx]


    x_test = x[test_idx]
    y_test = y[test_idx]


    return x_train, y_train, x_test, y_test

sklearn中鸢尾花数据测试KNN

import numpy as np
from sklearn import datasets
from knn_clf import KNNClassifier


iris = datasets.load_iris()
x = iris.data
y = iris.target


x_train, y_train, x_test, y_test = train_test_split(x, y)
my_knn_clf = KNNClassifier(k=3)
my_knn_clf.fit(x_train, y_train)


y_predict = my_knn_clf.predict(x_test)
print(sum(y_predict == y_test))
print(sum(y_predict == y_test) / len(y_test))
# 也可以使用sklearn中自带的数据集拆分方法
from sklearn.model_selection import train_test_split
import numpy as np
from sklearn import datasets
from knn_clf import KNNClassifier


iris = datasets.load_iris()
x = iris.data
y = iris.target
x_train, y_train, x_test, y_test = train_test_split(x, y, 
                          test_size=0.2, random_state=666)
my_knn_clf = KNNClassifier(k=3)
my_knn_clf.fit(x_train, y_train)
y_predict = my_knn_clf.predict(x_test)
print(sum(y_predict == y_test))
print(sum(y_predict == y_test) / len(y_test))

sklearn中手写数字数据集测试KNN

首先，先来了解一下手写数字数据集。

import numpy as np
import matplotlib.pyplot as plt
from sklearn import datasets


digits = datasets.load_digits()
digits.keys()
print(digits.DESCR)
y.shape
digits.target_names
y[:100]
x[:10]
some_digit = x[666]
y[666]
some_digit_image = some_digit.reshape(8, 8)
plt.imshow(some_digit_image, cmap=plt.cm.binary)
plt.show()

接下来，就开始动手试试。

from sklearn import datasets
from shuffle_dataset import train_test_split
from knn_clf import KNNClassifier


digits = datasets.load_digits()
x = digits.data
y = digits.target


x_train, y_train, x_test, y_test = train_test_split(x, y, test_ratio=0.2)
my_knn_clf = KNNClassifier(k=3)
my_knn_clf.fit(x_train, y_train)
y_predict = my_knn_clf.predict(x_test)


print(sum(y_predict == y_test) / len(y_test))

把求acc封装成一个函数，方便调用。

def accuracy_score(y_true, y_predict):
  assert y_true.shape[0] == y_predict.shape[0],  
  "the size of y_true must be equal to the size of y_predict"


  return sum(y_true == y_predict) / len(y_true)

接下来把它封装到KNNClassifier的类中。

import numpy as np
from math import sqrt
from collections import Counter
from metrics import accuracy_score


class KNNClassifier():


  def __init__(self, k):
    assert 1 <= k, "k must be valid"
        self.k = k
        self._x_train = None
        self._y_train = None


    def fit(self, x_train, y_train):
        assert x_train.shape[0] == y_train.shape[0], 
        "the size of x_train must be equal to the size of y_train"
        assert self.k <= x_train.shape[0], 
        "the size of x_train must be at least k"


        self._x_train = x_train
        self._y_train = y_train
        return self


    def predict(self, x_new):
        # x_new = x_new.reshape(1, -1)
        assert self._x_train is not None and self._y_train is not None, 
        "must fit before predict"
        assert x_new.shape[1] == self._x_train.shape[1], 
        "the feature number of x must be equal to x_train"


        y_new = [self._predict(x) for x in x_new]
        return np.array(y_new)


    def _predict(self, x):
        assert x.shape[0] == self._x_train.shape[1], 
        "the feature number of x must be equal to x_train"


        distances = [sqrt(np.sum((x_train - x) ** 2)) for x_train in self._x_train]
        nearest = np.argsort(distances)


        topk_y = [self._y_train[i] for i in nearest[:self.k]]
        votes = Counter(topk_y)


        return votes.most_common(1)[0][0]


    def score(self, x_test, y_test):
        y_predict = self.predict(x_test)
        return accuracy_score(y_test, y_predict)


    def __repr__(self):
        return "KNN(k=%d)" % self.k

其实，在sklearn中这些都已经封装好了。

from sklearn.metrics import accuracy_score
from sklearn.model_selection import train_test_split
from sklearn.neighbors import KNeighborsClassifier


digits = datasets.load_digits()
x = digits.data
y = digits.target


x_train, x_test, y_train, y_test = train_test_split(x, y, test_size=0.2)
knn_classifier = KNeighborsClassifier(n_neighbors=3)
knn_classifier.fit(x_train, y_train)
knn_classifier.score(x_test, y_test)

超参数

k

在knn中的超参数k何时最优？

from sklearn.metrics import accuracy_score 
from sklearn.model_selection import train_test_split 
from sklearn.neighbors import KNeighborsClassifier 


digits = datasets.load_digits() 
x = digits.data 
y = digits.target 
x_train, x_test, y_train, y_test = train_test_split(x, y, test_size=0.2) 


best_score = 0.0 
best_k = -1 
for k in range(1, 11): 
  knn_clf = KNeighborsClassifier(n_neighbors=k) 
  knn_clf.fit(x_train, y_train) 
  score = knn_clf.score(x_test, y_test) 
  if score > best_score: 
    best_k = k 
    best_score = score 
print("best k=", best_k) print("best score=", best_score)

投票方式

上面这张图，绿色的球最近的三颗球分别是红色的1号，紫色的3号和蓝色的4号。如果只考虑绿色的k个近邻中多数服从少数，目前来说就是平票。

即使不是平票，红色也是距离绿色最近。此时我们就可以考虑给他们加个权重。一般使用距离的倒数作为权重。假设距离分别为1、 3、 4

红球：1 紫+蓝：1/3 + 1/4 = 7/12

这两者加起来都没有红色的权重大，因此最终将这颗绿球归为红色类别。这样能有效解决平票问题。 因此，这也算knn的一个超参数。

其实这个在sklearn封装的knn中已经考虑到了这个问题。在KNeighborsClassifier(n_neighbors=k，weights=？)

还有一个参数weights，一般有两种：uniform、distance。

from sklearn.metrics import accuracy_score
from sklearn.model_selection import train_test_split
from sklearn.neighbors import KNeighborsClassifier


digits = datasets.load_digits()
x = digits.data
y = digits.target


x_train, x_test, y_train, y_test = train_test_split(x, y, test_size=0.2)


best_method = ""
best_score = 0.0
best_k = -1
for method in["uniform", "distance"]: 
  for k in range(1, 11):
    knn_clf = KNeighborsClassifier(n_neighbors=k, weights=method)
    knn_clf.fit(x_train, y_train)
    score = knn_clf.score(x_test, y_test)
    if score > best_score:
      best_method = method
      best_k = k
      best_score = score
print("best_method=", best_method)
print("best k=", best_k)
print("best score=", best_score)

p

如果使用距离，那么有很多种距离可以使用，欧氏距离、曼哈顿距离、明可夫斯基距离。

from sklearn.metrics import accuracy_score
from sklearn.model_selection import train_test_split
from sklearn.neighbors import KNeighborsClassifier


digits = datasets.load_digits()
x = digits.data
y = digits.target


x_train, x_test, y_train, y_test = train_test_split(x, y, test_size=0.2)


best_p = -1
best_score = 0.0
best_k = -1
for p in range(1, 6):
  for k in range(1, 11):
    knn_clf = KNeighborsClassifier(n_neighbors=k, weights="distance", p=p)
    knn_clf.fit(x_train, y_train)
    score = knn_clf.score(x_test, y_test)
    if score > best_score:
      best_p = p
      best_k = k
      best_score = score
print("best_p=", best_p)
print("best k=", best_k)
print("best score=", best_score)

审核编辑：汤梓红