机器学习-基础分类算法-KNN详解

KNN-k近邻算法

k-Nearest Neighbors

• 思想极度简单
• 应用数学只是少
• 效果好
• 可以解释机器学习算法使用过程中的很多细节问题
• 更完整的刻画机器学习应用的流程

创建简单测试用例

import numpy as np
import matplotlib.pyplot as plt
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_train = np.array(raw_data_X)
y_train = np.array(raw_data_y)

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.show()

预测

x = np.array([8.093607318, 3.365731514])

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()

KNN过程

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

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

数组排序返回索引

np.argsort(distances)
nearest = np.argsort(distances)

最近k个点相近的y坐标

k = 6
topK_y = [y_train[neighbor] for neighbor in nearest[:k]]
topK_y

统计

from collections import Counter
votes = Counter(topK_y)
votes.most_common(1)
predict_y = votes.most_common(1)[0][0]

封装

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


def kNN_classify(k, X_train, y_train, x):

    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 equal to the size of y_train"
    assert X_train.shape[1] == x.shape[0], \
        "the feature number of x must be equal to X_train"

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

• 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

使用scikit-learn中的kNN

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_train = np.array(raw_data_X)
y_train = np.array(raw_data_y)

x = np.array([8.093607318, 3.365731514])
from sklearn.neighbors import KNeighborsClassifier
kNN_classifier = KNeighborsClassifier(n_neighbors=6)
kNN_classifier.fit(X_train, y_train)
#预测
kNN_classifier.predict(x)
#转化为矩阵
X_predict = x.reshape(1, -1)
kNN_classifier.predict(X_predict)
y_predict = kNN_classifier.predict(X_predict)

优化封装的KNN

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


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

判断机器学习算法的性能-训练数据分割测试数据

分一部分数据设置为测试数据
、
train test split
加载数据

import numpy as np
import matplotlib.pyplot as plt
from sklearn import datasets 
iris = datasets.load_iris()
iris.keys()
X = iris.data
y = iris.target
X.shape
y.shape

**分离出一部分数据做训练，另外一部分数据做测试 **
将索引随机排列

shuffled_indexes = np.random.permutation(len(X))
shuffled_indexes

设置测试数据百分比
获取测试数据

test_ratio = 0.2
test_size = int(len(X) * test_ratio)
test_indexes = shuffled_indexes[:test_size]
train_indexes = shuffled_indexes[test_size:]
X_train = X[train_indexes]
y_train = y[train_indexes]
X_test = X[test_indexes]
y_test = y[test_indexes]

封装算法

import numpy as np
def train_test_split(X, y, test_ratio=0.2, seed=None):
    """将数据 X 和 y 按照test_ratio分割成X_train, X_test, y_train, y_test"""
    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_ration must be valid"

    if seed:
        np.random.seed(seed)

    shuffled_indexes = np.random.permutation(len(X))

    test_size = int(len(X) * test_ratio)
    test_indexes = shuffled_indexes[:test_size]
    train_indexes = shuffled_indexes[test_size:]

    X_train = X[train_indexes]
    y_train = y[train_indexes]

    X_test = X[test_indexes]
    y_test = y[test_indexes]

    return X_train, X_test, y_train, y_test

sklearn中的train_test_split

分类准确度

加载手写数字数据

import numpy as np
import matplotlib.pyplot as plt
from sklearn import datasets
digits = datasets.load_digits()
digits.keys()
X = digits.data
X.shape
y = digits.target
y.shape

查看数据

some_digit = X[666]
some_digit_image = some_digit.reshape(8, 8)
import matplotlib
import matplotlib.pyplot as plt
plt.imshow(some_digit_image, cmap = matplotlib.cm.binary)
plt.show()

预测准确率

from playML.model_selection import train_test_split

X_train, X_test, y_train, y_test = train_test_split(X, y, test_ratio=0.2)
from playML.kNN import KNNClassifier

my_knn_clf = KNNClassifier(k=3)
my_knn_clf.fit(X_train, y_train)
y_predict = my_knn_clf.predict(X_test)
sum(y_predict == y_test) / len(y_test)

封装自己的accuracy_score

import numpy as np


def accuracy_score(y_true, y_predict):
    '''计算y_true和y_predict之间的准确率'''
    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)

scikit-learn中的accuracy_score

from sklearn.model_selection import train_test_split

X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=666)
from sklearn.neighbors import KNeighborsClassifier

knn_clf = KNeighborsClassifier(n_neighbors=3)
knn_clf.fit(X_train, y_train)
y_predict = knn_clf.predict(X_test)
from sklearn.metrics import accuracy_score
accuracy_score(y_test, y_predict)

超参数和模型参数

• 超参数：在算法运行前需要决定的参数
• 模型参数：算法过程中学习的参数

KNN算法没有模型参数
KNN算法中的K是典型的超参数
寻找好的超参数

• 领域知识
• 经验数值
• 实验搜索

加载手写数字数据集

import numpy as np
from sklearn import datasets
digits = datasets.load_digits()
X = digits.data
y = digits.target
from sklearn.model_selection import train_test_split
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=666)
from sklearn.neighbors import KNeighborsClassifier
knn_clf = KNeighborsClassifier(n_neighbors=3)
knn_clf.fit(X_train, y_train)
knn_clf.score(X_test, y_test)

寻找最好的k、

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)

加入是否考虑距离

best_score = 0.0
best_k = -1
best_method = ""
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_k = k
            best_score = score
            best_method = method
        
print("best_method =", best_method)
print("best_k =", best_k)
print("best_score =", best_score)

明可夫斯基距离

best_score = 0.0
best_k = -1
best_p = -1

for k in range(1, 11):
    for p in range(1, 6):
        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_k = k
            best_p = p
            best_score = score
        
print("best_k =", best_k)
print("best_p =", best_p)
print("best_score =", best_score)

网格搜索 Grid Search

定义搜索参数

param_grid = [
    {
        'weights': ['uniform'], 
        'n_neighbors': [i for i in range(1, 11)]
    },
    {
        'weights': ['distance'],
        'n_neighbors': [i for i in range(1, 11)], 
        'p': [i for i in range(1, 6)]
    }
]

knn_clf = KNeighborsClassifier()
from sklearn.model_selection import GridSearchCV

grid_search = GridSearchCV(knn_clf, param_grid)
grid_search.fit(X_train, y_train)

查看最佳分类器参数和准确度

grid_search.best_estimator_
grid_search.best_score_

更多的距离定义

• 向量空间余弦相似度 Cosine Similarity
• 调整余弦相似度 Adjusted Cosine Similarity
• 皮尔森相关系数 Pearson Correlation Coefficient
• Jaccard相似吸收 Jaccard Coeffcient

数据归一化 Feature Scaling

将所有的数据映射到同一尺度
最值归一化 normalization：把所有数据映射到0-1之间

适用于分布有明显边界的情况；受outlier影响较大

x = np.random.randint(0, 100, 100) 
(x - np.min(x)) / (np.max(x) - np.min(x))

X = np.random.randint(0, 100, (50, 2))
X = np.array(X, dtype=float)
X[:,0] = (X[:,0] - np.min(X[:,0])) / (np.max(X[:,0]) - np.min(X[:,0]))
X[:,1] = (X[:,1] - np.min(X[:,1])) / (np.max(X[:,1]) - np.min(X[:,1]))
X[:10,:]

均值方差归一化 standardization
数据分布没有明显的边界，有可能存在极端数据值
均值方差归一化：把所有数据归一到均值为0方差为1的分布中

X2 = np.random.randint(0, 100, (50, 2))
X2 = np.array(X2, dtype=float)
X2[:,0] = (X2[:,0] - np.mean(X2[:,0])) / np.std(X2[:,0])
X2[:,1] = (X2[:,1] - np.mean(X2[:,1])) / np.std(X2[:,1])

测试数据集的归一化

测试数据说模拟真实环境

• 真实环境很有可能无法得到所有测试数据的均值和方差
• 对数据的归一化也是算法的一部分

(x_test-mean_train)/std_train

scikit-learn中的Scaler

加载数据

import numpy as np
from sklearn import datasets
iris = datasets.load_iris()
X = iris.data
y = iris.target

数据分割

from sklearn.model_selection import train_test_split
X_train, X_test, y_train, y_test = train_test_split(iris.data, iris.target, test_size=0.2, random_state=666)

数据归一化

from sklearn.preprocessing import StandardScaler 
standardScalar = StandardScaler() 
standardScalar.fit(X_train)
standardScalar.transform(X_train)#归一化

归一化结果赋值

X_train = standardScalar.transform(X_train)
X_test_standard = standardScalar.transform(X_test) 

使用归一化后的数据进行knn分类

from sklearn.neighbors import KNeighborsClassifier
knn_clf = KNeighborsClassifier(n_neighbors=3)
knn_clf.fit(X_train, y_train)
knn_clf.score(X_test_standard, y_test)

实现自己的standardScaler

import numpy as np


class StandardScaler:

    def __init__(self):
        self.mean_ = None
        self.scale_ = None

    def fit(self, X):
        """根据训练数据集X获得数据的均值和方差"""
        assert X.ndim == 2, "The dimension of X must be 2"

        self.mean_ = np.array([np.mean(X[:,i]) for i in range(X.shape[1])])
        self.scale_ = np.array([np.std(X[:,i]) for i in range(X.shape[1])])

        return self

    def transform(self, X):
        """将X根据这个StandardScaler进行均值方差归一化处理"""
        assert X.ndim == 2, "The dimension of X must be 2"
        assert self.mean_ is not None and self.scale_ is not None, \
               "must fit before transform!"
        assert X.shape[1] == len(self.mean_), \
               "the feature number of X must be equal to mean_ and std_"

        resX = np.empty(shape=X.shape, dtype=float)
        for col in range(X.shape[1]):
            resX[:,col] = (X[:,col] - self.mean_[col]) / self.scale_[col]
        return resX

有关k近邻算法

解决分类问题
天然可以解决多分类问题
思想简单，效果强大
最大缺点：效率低下
如果训练集有m个样本，n个特征，则预测每一个新的数据，需要O(m*n)
优化 使用树结构：KD-Tree，Ball-Tree
缺点2：高度数据相关
缺点3：预测结果不具有可解释性
维数灾难
随着维度的增加，“看似相近”的两个点之间的距离越来越大
解决方法：降维、