机器学习实验二K-近邻算法及应用

一：作业信息

博客班级	https://edu.cnblogs.com/campus/ahgc/machinelearning/
作业要求	https://edu.cnblogs.com/campus/ahgc/machinelearning/homework/12004
作业目标	理解感知器算法原理并能针对实际构建感知器模型并进行预测
学号	3180701323

二：实验目的
1.理解K-近邻算法原理，能实现算法K近邻算法；
2.掌握常见的距离度量方法；
3.掌握K近邻树实现算法；
4.针对特定应用场景及数据，能应用K近邻解决实际问题。

三：实验内容
1.实现曼哈顿距离、欧氏距离、闵式距离算法，并测试算法正确性。
2.实现K近邻树算法；
3.针对iris数据集，应用sklearn的K近邻算法进行类别预测。
4.针对iris数据集，编制程序使用K近邻树进行类别预测。

四：实验报告要求
1.对照实验内容，撰写实验过程、算法及测试结果；
2.代码规范化：命名规则、注释；
3.分析核心算法的复杂度；
4.查阅文献，讨论K近邻的优缺点；
5.举例说明K近邻的应用场景。

五:实验过程
In [1]:

import math
from itertools import combinations

·p1= 1 曼哈顿距离
·p2= 2 欧氏距离
·p3= inf 闵式距离minkowski_distance
In [2]:

def L(x, y, p=2):
    # x1 = [1, 1], x2 = [5,1]
    if len(x) == len(y) and len(x) > 1:
        sum = 0
        for i in range(len(x)):
            sum += math.pow(abs(x[i] - y[i]), p)
        return math.pow(sum, 1/p)
    else:
        return 0

In [3]:

# 课本例3.1
x1 = [1, 1]
x2 = [5, 1]
x3 = [4, 4]

In [4]:

# x1, x2
for i in range(1, 5):
    r = { '1-{}'.format(c):L(x1, c, p=i) for c in [x2, x3]}
    print(min(zip(r.values(), r.keys())))

python实现，遍历所有数据点，找出n个距离最近的点的分类情况，少数服从多数
In [5]:

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
%matplotlib inline

from sklearn.datasets import load_iris
from sklearn.model_selection import train_test_split

from collections import Counter

In [6]:

# data
iris = load_iris()
df = pd.DataFrame(iris.data, columns=iris.feature_names)
df['label'] = iris.target
df.columns = ['sepal length', 'sepal width', 'petal length', 'petal width', 'label']
# data = np.array(df.iloc[:100, [0, 1, -1]])

In [7]:

df

In [8]:

plt.scatter(df[:50]['sepal length'], df[:50]['sepal width'], label='0')
plt.scatter(df[50:100]['sepal length'], df[50:100]['sepal width'], label='1')
plt.xlabel('sepal length')
plt.ylabel('sepal width')
plt.legend()

In [9]:

data = np.array(df.iloc[:100, [0, 1, -1]])
X, y = data[:,:-1], data[:,-1]
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)

In [10]:

class KNN:
    def __init__(self, X_train, y_train, n_neighbors=3, p=2):
        """
        parameter: n_neighbors 临近点个数
        parameter: p 距离度量
        """
        self.n = n_neighbors
        self.p = p
        self.X_train = X_train
        self.y_train = y_train

    def predict(self, X):
        # 取出n个点
        knn_list = []
        for i in range(self.n):
            dist = np.linalg.norm(X - self.X_train[i], ord=self.p)
            knn_list.append((dist, self.y_train[i]))

        for i in range(self.n, len(self.X_train)):
            max_index = knn_list.index(max(knn_list, key=lambda x: x[0]))
            dist = np.linalg.norm(X - self.X_train[i], ord=self.p)
            if knn_list[max_index][0] > dist:
                knn_list[max_index] = (dist, self.y_train[i])
        # 统计
        knn = [k[-1] for k in knn_list]
        count_pairs = Counter(knn)
        max_count = sorted(count_pairs, key=lambda x:x)[-1]
        return max_count
    def score(self, X_test, y_test):
        right_count = 0
        n = 10
        for X, y in zip(X_test, y_test):
            label = self.predict(X)
            if label == y:
                right_count += 1
        return right_count / len(X_test)

In [11]:

clf = KNN(X_train, y_train)

In [12]:

clf.score(X_test, y_test)

In [13]:

test_point = [6.0, 3.0]
print('Test Point: {}'.format(clf.predict(test_point)))

In [14]:

plt.scatter(df[:50]['sepal length'], df[:50]['sepal width'], label='0')
plt.scatter(df[50:100]['sepal length'], df[50:100]['sepal width'], label='1')
plt.plot(test_point[0], test_point[1], 'bo', label='test_point')
plt.xlabel('sepal length')
plt.ylabel('sepal width')
plt.legend()

scikitlearn
In [15]:

from sklearn.neighbors import KNeighborsClassifier

In [16]:

clf_sk = KNeighborsClassifier()
clf_sk.fit(X_train, y_train)

In [17]:

clf_sk.score(X_test, y_test)

sklearn.neighbors.KNeighborsClassifier
·n_neighbors: 临近点个数
·p: 距离度量
·algorithm: 近邻算法，可选{'auto', 'ball_tree', 'kd_tree', 'brute'}
·weights: 确定近邻的权重

kd树
In [18]:

# kd-tree每个结点中主要包含的数据结构如下
class KdNode(object):
    def __init__(self, dom_elt, split, left, right):
        self.dom_elt = dom_elt # k维向量节点(k维空间中的一个样本点)
        self.split = split # 整数（进行分割维度的序号）
        self.left = left # 该结点分割超平面左子空间构成的kd-tree
        self.right = right # 该结点分割超平面右子空间构成的kd-tree
class KdTree(object):
    def __init__(self, data):
        k = len(data[0]) # 数据维度

        def CreateNode(split, data_set): # 按第split维划分数据集exset创建KdNode
            if not data_set: # 数据集为空
                return None
            # key参数的值为一个函数，此函数只有一个参数且返回一个值用来进行比较
            # operator模块提供的itemgetter函数用于获取对象的哪些维的数据，参数为需要获取的数据在对象
            #data_set.sort(key=itemgetter(split)) # 按要进行分割的那一维数据排序
            data_set.sort(key=lambda x: x[split])
            split_pos = len(data_set) // 2 # //为Python中的整数除法
            median = data_set[split_pos] # 中位数分割点
            split_next = (split + 1) % k # cycle coordinates

            # 递归的创建kd树
            return KdNode(median, split,
                         CreateNode(split_next, data_set[:split_pos]), # 创建左子树
                         CreateNode(split_next, data_set[split_pos + 1:])) # 创建右子树

        self.root = CreateNode(0, data) # 从第0维分量开始构建kd树,返回根节点

# KDTree的前序遍历
def preorder(root):
    print (root.dom_elt)
    if root.left: # 节点不为空
        preorder(root.left)
    if root.right:
        preorder(root.right)

In [19]:

# 对构建好的kd树进行搜索，寻找与目标点最近的样本点：
from math import sqrt
from collections import namedtuple

# 定义一个namedtuple,分别存放最近坐标点、最近距离和访问过的节点数
result = namedtuple("Result_tuple", "nearest_point nearest_dist nodes_visited")

def find_nearest(tree, point):
    k = len(point) # 数据维度
    def travel(kd_node, target, max_dist):
        if kd_node is None:
            return result([0] * k, float("inf"), 0) # python中用float("inf")和float("-inf")表示正负
        nodes_visited = 1

        s = kd_node.split # 进行分割的维度
        pivot = kd_node.dom_elt # 进行分割的“轴”

        if target[s] <= pivot[s]: # 如果目标点第s维小于分割轴的对应值(目标离左子树更近)
            nearer_node = kd_node.left # 下一个访问节点为左子树根节点
            further_node = kd_node.right # 同时记录下右子树
        else: # 目标离右子树更近
            nearer_node = kd_node.right # 下一个访问节点为右子树根节点
            further_node = kd_node.left

        temp1 = travel(nearer_node, target, max_dist) # 进行遍历找到包含目标点的区域

        nearest = temp1.nearest_point # 以此叶结点作为“当前最近点”
        dist = temp1.nearest_dist # 更新最近距离

        nodes_visited += temp1.nodes_visited

        if dist < max_dist:
            max_dist = dist # 最近点将在以目标点为球心，max_dist为半径的超球体内

        temp_dist = abs(pivot[s] - target[s]) # 第s维上目标点与分割超平面的距离
        if max_dist < temp_dist: # 判断超球体是否与超平面相交
            return result(nearest, dist, nodes_visited) # 不相交则可以直接返回，不用继续判断

        #----------------------------------------------------------------------
        # 计算目标点与分割点的欧氏距离
        temp_dist = sqrt(sum((p1 - p2) ** 2 for p1, p2 in zip(pivot, target)))

        if temp_dist < dist: # 如果“更近”
            nearest = pivot # 更新最近点
            dist = temp_dist # 更新最近距离
            max_dist = dist # 更新超球体半径

        # 检查另一个子结点对应的区域是否有更近的点
        temp2 = travel(further_node, target, max_dist)

        nodes_visited += temp2.nodes_visited
        if temp2.nearest_dist < dist: # 如果另一个子结点内存在更近距离
            nearest = temp2.nearest_point # 更新最近点
            dist = temp2.nearest_dist # 更新最近距离

        return result(nearest, dist, nodes_visited)

    return travel(tree.root, point, float("inf")) # 从根节点开始递归

In [20]:

data = [[2,3],[5,4],[9,6],[4,7],[8,1],[7,2]]
kd = KdTree(data)
preorder(kd.root)

In [21]:

from time import clock
from random import random

# 产生一个k维随机向量，每维分量值在0~1之间
def random_point(k):
    return [random() for _ in range(k)]

# 产生n个k维随机向量
def random_points(k, n):
    return [random_point(k) for _ in range(n)]

In [22]:

ret = find_nearest(kd, [3,4.5])
print (ret)

In [23]:

N = 400000
t0 = clock()
kd2 = KdTree(random_points(3, N)) # 构建包含四十万个3维空间样本点的kd树
ret2 = find_nearest(kd2, [0.1,0.5,0.8]) # 四十万个样本点中寻找离目标最近的点
t1 = clock()
print ("time: ",t1-t0, "s")
print (ret2)

实验结果：

六：实验小结
通过这次实验我理解了K-近邻算法原理，也能在特定的实际问题中使用K近邻算法去解决问题；也掌握了常见的距离度量方法以及K近邻树实现算法。