kNN近鄰算法

算法原理

樣本點(diǎn)的特性與該鄰居點(diǎn)的特性類似，可以簡(jiǎn)單理解為“物以類聚”。因此可以使用目標(biāo)點(diǎn)的多個(gè)鄰近點(diǎn)的特性表示當(dāng)前點(diǎn)的特性。

k近鄰算法是非常特殊的，可以被認(rèn)為是沒(méi)有模型的算法，為了和其他算法統(tǒng)一，可以認(rèn)為訓(xùn)練數(shù)據(jù)集就是模型本身。

KNN分類算法：“投票法”，選擇這k 個(gè)樣本中出現(xiàn)最多的類別標(biāo)記作為預(yù)測(cè)結(jié)果。

KNN回歸算法：“平均法”，將這k 個(gè)樣本的實(shí)值輸出標(biāo)記的平均值作為預(yù)測(cè)結(jié)果。

歐拉距離公式

歐拉距離公式

化簡(jiǎn)公式

KNN算法的核心要素

1.K值的選擇：K是超參（需要給定），K值過(guò)小容易導(dǎo)致過(guò)擬合（比如噪音點(diǎn)的數(shù)據(jù)會(huì)對(duì)結(jié)果造成影響），K值過(guò)大訓(xùn)練誤差會(huì)增大，同時(shí)會(huì)使模型變得簡(jiǎn)單，容易導(dǎo)致欠擬合。

2.距離的度量：采用歐式距離。

3.決策規(guī)則：在分類模型中，主要使用多數(shù)表決法或者加權(quán)多數(shù)表決法；在回歸模型中，主要使用平均值法或者加權(quán)平均值法。（基于距離遠(yuǎn)近進(jìn)行加權(quán)，，距離越近的樣本權(quán)重越大.）。

kNN算法源碼

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分類器"""
        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):
        """根據(jù)訓(xùn)練數(shù)據(jù)集X_train和y_train訓(xùn)練kNN分類器"""
        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):
        """給定待預(yù)測(cè)數(shù)據(jù)集X_predict，返回表示X_predict的結(jié)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):
        """給定單個(gè)待預(yù)測(cè)數(shù)據(jù)x，返回x的預(yù)測(cè)結(jié)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):
        """根據(jù)測(cè)試數(shù)據(jù)集 X_test 和 y_test 確定當(dāng)前模型的準(zhǔn)確度"""

        y_predict = self.predict(X_test)
        return accuracy_score(y_test, y_predict)

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

kNN原生代碼

import numpy as np
from math import sqrt
import matplotlib.pyplot as plt
#數(shù)據(jù)處理
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])  #要判斷的新的點(diǎn)歸于屬于0還是1
X_train = np.array(raw_data_X)
y_train = np.array(raw_data_y)
#近鄰算法：計(jì)算距離
distances = []
for x_train in X_train:
    d = sqrt(np.sum((x_train-x)**2))  #求歐拉距離的公式
    distances.append(d)

print(distances)
nearest = np.argsort(distances)#按索引排序，默認(rèn)從小到大
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()