kNN近邻算法
算法原理
样本点的特性与该邻居点的特性类似，可以简单理解为“物以类聚”。因此可以使用目标点的多个邻近点的特性表示当前点的特性。
k近邻算法是非常特殊的，可以被认为是没有模型的算法，为了和其他算法统一，可以认为训练数据集就是模型本身。
KNN分类算法：“投票法”，选择这k 个样本中出现最多的类别标记作为预测结果。
KNN回归算法：“平均法”，将这k 个样本的实值输出标记的平均值作为预测结果。
欧拉距离公式
欧拉距离公式
化简公式
KNN算法的核心要素
1.K值的选择：K是超参（需要给定），K值过小容易导致过拟合（比如噪音点的数据会对结果造成影响），K值过大训练误差会增大，同时会使模型变得简单，容易导致欠拟合。
2.距离的度量：采用欧式距离。
3.决策规则：在分类模型中，主要使用多数表决法或者加权多数表决法；在回归模型中，主要使用平均值法或者加权平均值法。（基于距离远近进行加权，，距离越近的样本权重越大.）。
kNN算法源码
import numpy as np
from math import sqrt
from collections import Counter
from sklearn.metrics import accuracy_score
class KNNClassifier:
def __init__(self, k):
"""初始化kNN分类器"""
assert k >= 1, "k must be valid"
self.k = k
self._X_train = None
self._y_train = None
def fit(self, X_train, y_train):
"""根据训练数据集X_train和y_train训练kNN分类器"""
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_predict):
"""给定待预测数据集X_predict，返回表示X_predict的结果向量"""
assert self._X_train is not None and self._y_train is not None, \
"must fit before predict!"
assert X_predict.shape[1] == self._X_train.shape[1], \
"the feature number of X_predict must be equal to X_train"
y_predict = [self._predict(x) for x in X_predict]
return np.array(y_predict)
def _predict(self, x):
"""给定单个待预测数据x，返回x的预测结果值"""
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):
"""根据测试数据集 X_test 和 y_test 确定当前模型的准确度"""
y_predict = self.predict(X_test)
return accuracy_score(y_test, y_predict)
def __repr__(self):
return "KNN(k=%d)" % self.k
kNN原生代码
import numpy as np
from math import sqrt
import matplotlib.pyplot as plt
#数据处理
raw_data_X = [[3.393533211, 2.331273381],
[3.110073483, 1.781539638],
[1.343808831, 3.368360954],
[3.582294042, 4.679179110],
[2.280362439, 2.866990263],
[7.423436942, 4.696522875],
[5.745051997, 3.533989803],
[9.172168622, 2.511101045],
[7.792783481, 3.424088941],
[7.939820817, 0.791637231]
]
raw_data_y = [0, 0, 0, 0, 0, 1, 1, 1, 1, 1]
x = np.array([8.093607318, 3.365731514]) #要判断的新的点归于属于0还是1
X_train = np.array(raw_data_X)
y_train = np.array(raw_data_y)
#近邻算法：计算距离
distances = []
for x_train in X_train:
d = sqrt(np.sum((x_train-x)**2)) #求欧拉距离的公式
distances.append(d)
print(distances)
nearest = np.argsort(distances)#按索引排序，默认从小到大
print(nearest)
k = 6 #knn算法：取6
num = [y_train[neighbors] for neighbors in nearest[:k]]
print(num)
from collections import Counter
votes = Counter(num)
print(votes.most_common(1)[0][0])
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[0],x[1],color='b')
plt.show()