数据归一化
将所有的数据映射到同一尺度。
首先，为什么需要数据归一化？举个简答的例子。样本间的距离时间所主导，这样在样本1以[1, 200]输入到模型中去的时候，由于200可能会直接忽略到1的存在，因此我们需要将数据进行归一化。比如将天数转换为占比1年的比例，200/365=0.5479, 100/365=0.2740。
一、最值归一化
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 outliner影响较大。
xscale=(x-xmin)/(xmax-xmin)
import numpy as np
import matplotlib.pyplot as plt
x = np.random.randint(0, 100, size=100)
x
输出结果：
array([84, 18, 75, 75, 78, 30, 39, 33, 29, 30, 48, 77, 54, 30, 1, 32, 91,
60, 73, 78, 89, 16, 71, 47, 87, 43, 24, 67, 70, 50, 58, 56, 69, 11,
19, 97, 64, 53, 37, 18, 84, 77, 6, 3, 91, 48, 14, 6, 70, 36, 93,
43, 78, 78, 73, 18, 96, 58, 77, 78, 29, 96, 75, 59, 58, 19, 65, 90,
67, 73, 72, 1, 89, 70, 59, 96, 42, 73, 58, 8, 61, 65, 78, 86, 98,
94, 52, 1, 59, 86, 44, 28, 87, 2, 91, 75, 19, 91, 46, 92])
(x-np.min(x)) / (np.max(x) - np.min(x))
输出结果：
array([0.8556701 , 0.17525773, 0.7628866 , 0.7628866 , 0.79381443,
0.29896907, 0.39175258, 0.32989691, 0.28865979, 0.29896907,
0.48453608, 0.78350515, 0.54639175, 0.29896907, 0. ,
0.31958763, 0.92783505, 0.60824742, 0.74226804, 0.79381443,
0.90721649, 0.15463918, 0.72164948, 0.4742268 , 0.88659794,
0.43298969, 0.2371134 , 0.68041237, 0.71134021, 0.50515464,
0.58762887, 0.56701031, 0.70103093, 0.10309278, 0.18556701,
0.98969072, 0.64948454, 0.53608247, 0.37113402, 0.17525773,
0.8556701 , 0.78350515, 0.05154639, 0.02061856, 0.92783505,
0.48453608, 0.13402062, 0.05154639, 0.71134021, 0.36082474,
0.94845361, 0.43298969, 0.79381443, 0.79381443, 0.74226804,
0.17525773, 0.97938144, 0.58762887, 0.78350515, 0.79381443,
0.28865979, 0.97938144, 0.7628866 , 0.59793814, 0.58762887,
0.18556701, 0.65979381, 0.91752577, 0.68041237, 0.74226804,
0.73195876, 0. , 0.90721649, 0.71134021, 0.59793814,
0.97938144, 0.42268041, 0.74226804, 0.58762887, 0.07216495,
0.6185567 , 0.65979381, 0.79381443, 0.87628866, 1. ,
0.95876289, 0.5257732 , 0. , 0.59793814, 0.87628866,
0.44329897, 0.27835052, 0.88659794, 0.01030928, 0.92783505,
0.7628866 , 0.18556701, 0.92783505, 0.46391753, 0.93814433])
X = np.random.randint(0, 100, (50, 2))
X[:10, :]
X = np.array(X, dtype=float)
X[:, 0] = (X[:, 0] - np.min(X[:, 0])) / (np.max(X[:, 0]) - np.min(X[:, 0]))
X[:, 0]
X[:, 1] = (X[:, 1] - np.min(X[:, 1])) / (np.max(X[:, 1]) - np.min(X[:, 1]))
X[:, 1]
X[:10, :]
plt.scatter(X[:,0], X[:,1])
plt.show()
np.mean(X[:,0])
np.std(X[:, 0])
np.mean(X[:,1])
np.std(X[:, 1])
二、均值方差归一化
均值方差归一化（standardization）：把所有数据归一化到均值为0方差为1的分布中。适用于数据分 布没有明显的边界，有可能存在极端的数据值。
xscale=(x-xmean)/s
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()
np.mean(x2[:,0])
np.std(x2[:,0])
np.mean(x2[:,1])
np.std(x2[:,1])
三、对训练集和测试集都进行归一化？
我们得到数据集训练模型之前，首先会把数据集进行切分，分成训练集和测试集，如果需要对数据进行归一化，我们可以很容易地通过训练集得到其均值和方差，最大值最小值。但是测试集呢？如何对测试集进行数据归一化呢？
正常情况下，测试数据集是模拟真实环境的，既然是真实环境，我们就很可能无法得到所有的测试集。因此当有一个新的数据需要进行预测时，我们需要使用训练集的均值方差，最大值最小值对测试集数据进行归一化。在scikit-learn中封装了Scaler保存训练数据集中的均值和方差等关键信息。
import numpy as np
from sklearn import datasets
from sklearn.model_selection import train_test_split
from sklearn.preprocessing import StandardScaler
iris = datasets.load_iris()
x = iris.data
y = iris.target
x[:10, :]
x_train, x_test, y_train, y_test = train_test_split(x, y, test_size=0.2, random_state=666)
standarscaler = StandardScaler()
standarscaler.fit(x_train)
standarscaler.mean_
standarscaler.scale_
standarscaler.transform(x_train)
x_train = standarscaler.transform(x_train)
x_train
x_test_standard = standarscaler.transform(x_test)
x_test_standard
接下来测试一下数据归一化之后KNN的性能：
from sklearn.neighbors import KNeighborsClassifier
knn_clf = KNeighborsClassifier()
knn_clf.fit(x_train, y_train)
knn_clf.fit(x_test_standard, y_test)
knn_clf.score(x_test_standard, y_test)
输出结果：1.0
如果训练集进行了归一化，测试集不做归一化试试？
knn_clf.score(x_test, y_test)
输出结果：0.3333333333333333
四、使用面向对象自己编写均值方差归一化
from sklearn.preprocessing import StandardScaler # 在sklearn中
import numpy as np
class StandardScale(object):
def __init__(self):
self.mean_ = None
self.scale_ = None
def fit(self, x):
"根据训练集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进行均值方差归一化"
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 scale_"
res_x = np.empty(shape=x.shape, dtype=float)
for col in range(x.shape[1]):
res_x[:, col] = (x[:, col] - self.mean_[col]) / self.scale_[col]
return res_x
五、使用面向对象自己编写最值归一化
from sklearn.preprocessing import MinMaxScaler # 在sklearn中
import numpy as np
class MinMaxScale(object):
def __init__(self):
self.mean_ = None
self.scale_ = None
def fit(self, x):
"根据训练集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])])
self.min_ = np.array([np.min(x[:, i]) for i in range(x.shape[1])])
self.max_ = np.array([np.max(x[:, i]) for i in range(x.shape[1])])
return self
def transform(self, x):
"将x进行均值方差归一化"
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 scale_"
res_x = np.empty(shape=x.shape, dtype=float)
for col in range(x.shape[1]):
res_x[:, col] = (x[:, col] - self.min_[col]) / (self.max_[col] - self.min_[col])
return res_x
其实，还有更多的数据归一化的方式，后续再进行完善！