机器学习（1）——K近邻算法

KNN的函数写法

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] #返回类型

使用scikit-learn中的KNN

from sklearn.neighbors import KNeighborsClassifier
import numpy as np
KNN_classifier=KNeighborsClassifier(n_neighbors=6) #传入k的值
#这里我随便搞的训练数据
x_train=np.arange(0,100).reshape(-1,2) #x是矩阵
y_train=np.random.randint(0,2,50) #y是数组
KNN_classifier.fit(x_train,y_train) #传入训练数据集
x=np.array([1,3]) #测试数据
x=x.reshape(1,-1) #测试数据只能传矩阵为参数
y=KNN_classifier.predict(x)[0] #因为我只测试了一组数据，所以取[0]即可

KNN的类写法

KNN.py

import numpy as np
from math import sqrt
from collections import Counter
from K近邻算法包.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 #私有变量
        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): #自我描述，在创建对象时打印
        return "KNN(k=%d)" % self.k

测试算法正确率

model_selection.py:

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: #固定随机种子，好调试
        np.random.seed(seed)

    shuffled_indexes = np.random.permutation(len(X)) #len(矩阵)是行数

    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

可以在jupyter notebook中调用一下试试：

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

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

%run F:/python3玩转机器学习/K近邻算法/model_selection.py

X_train, X_test, y_train, y_test=train_test_split(X,y,test_ratio=0.2)

%run F:/python3玩转机器学习/K近邻算法/KNN.py
    
my_knn_clf.fit(X_train,y_train)

y_predict=my_knn_clf.predict(X_test)

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

scikit-learn中的model_selection:

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)

分类准确度

自己编写一个包

编写metrics.py:

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)

在KNN.py中调用：

import numpy as np
from math import sqrt
from collections import Counter
from K近邻算法包.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 #私有变量
        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): #自我描述，在创建对象时打印
        return "KNN(k=%d)" % self.k

用sklearn测试准确度

data=datasets.load_digits()

X=data.data

y=data.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)

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) ，也可以这样：KNN_clf.score(X_test,y_test)

超参数

超参数：在算法运行前需要决定的参数

模型参数：算法过程中运行的参数

KNN没有模型参数，KNN中的k是典型的超参数

根据上面的手写数字的数据集，暴力查找最好的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)

有时候，距离的权重可能有影响，比如：

最近的是红色，那么我们按权重来比就是对距离取倒数求和。

明科夫斯基距离：

获得了一个超参数p

查找最好的p和k(网格搜索)：

%%time

best_p=-1
best_score=0.0
best_k=-1
for k in range(1,11):
    for p in range(1,6):
        knn_clf=KNeighborsClassifier(n_neighbors=k,weights="distance")
        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_p=",best_p)
print("best_k=",best_k)
print("best_score=",best_score)

使用scikit-learn中的网格搜索：

定义网格参数：

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)]
}
]

初始化一个分类器对象：

knn_clf=KNeighborsClassifier()

导入网格搜索：

from sklearn.model_selection import GridSearchCV

实例化：

grid_search = GridSearchCV(knn_clf,param_grid)

拟合：

%%time
grid_search.fit(X_train,y_train)

最优分类器：

grid_search.best_estimator_

grid_search.best_score_

grid_search.best_params_

将knn_clf赋为最优参数的分类器：

knn_clf=grid_search.best_estimator_

knn_clf.score(X_test,y_test)

加速、显示具体信息：

grid_search=GridSearchCV(knn_clf,param_grid,n_jobs=-1，verbose=2)#-1是将所有核并行 verbose是输出信息的详细情况

%%time
grid_search.fit(X_train,y_train)

数据归一化

将所有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

对向量：

x1=np.random.randint(0,100,size=100)

(x1-np.min(x1))/(np.max(x1)-np.min(x1))

对矩阵：

X=np.random.randint(0,100,(50,2))

X=np.array(X,dtype=float)

对每一列特征值归一化：

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]))

绘制散点图：

plt.scatter(X[:,0],X[:,1])
plt.show()

均值方差归一化（standardization）

把所有数据归一到均值为0，方差为1的分布

适用于数据没有明显的边界，不受极端值影响

S是标准差。

实例：

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])

plt.scatter(x2[:,0],x2[:,1])
plt.show()

对测试数据集如何归一化？

不能简单的只对测试数据集归一化，应该用（x_test-x_mean_train）/std_train

使用scikit-learn归一化

from sklearn import datasets
import numpy

iris=datasets.load_iris()

X=iris.data
y=iris.target

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)

from sklearn.preprocessing import StandardScaler

standardScaler=StandardScaler()

standardScaler.fit(X_train)

standardScaler.mean_ #均值

standardScaler.scale_ #标准差，std已经不能再用

X_train=standardScaler.transform(X_train) #返回归一化的矩阵

X_test_standard=standardScaler.transform(X_test) #测试数据集要用训练数据集来归一化

测试归一化后的准确率：

from sklearn.neighbors import KNeighborsClassifier

knn_clf = KNeighborsClassifier(n_neighbors=3)

knn_clf.fit(X_train,y_train) #用归一化的训练数据X训练

knn_clf.score(X_test_standard,y_test) #用归一化的测试数据X测试

返回1.0，因为鸢尾花的数据比较少，准确高也是自然地。

sklearn.preprocessing中还有MinMaxScaler（最值归一化），用法类似。

自己写StandardScaler类

preprocessing.py:

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

关于K近邻算法

最大的缺点：效率低下

如果训练集有m个样本，n个特征，那么每预测一个数据，要O（m*n）

可以使用KD-Tree、Ball-Tree优化，但依然低效

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会越来越大”

解决方法：降维，如PCA

机器学习流程：