KNN原理
KNN 就是 k-Nearest Neighbor, 过程如下：
- train: 将数据集的数据和对应的标签读入到类中
- test: 首先计算一个矩阵（num_test, num_train）存储测试集中每一个数据与训练集中每个数据的距离，这里我们采用L2距离进行计算。
- 随后在测试第i张测试图像时，我们选取距离最小的前k个训练图片，将其对应的label存入到closest_y中，然后找出出现次数最多的label，将其作为该测试图像的预测结果。
KNN类
import numpy as np
from past.builtins import xrange
class KNearestNeighbor(object):
""" a kNN classifier with L2 distance """
def __init__(self):
pass
def train(self, X, y):
""" Train the classifier. For k-nearest neighbors this is just memorizing the training data. Inputs: - X: A numpy array of shape (num_train, D) containing the training data consisting of num_train samples each of dimension D. - y: A numpy array of shape (N,) containing the training labels, where y[i] is the label for X[i]. """
self.X_train = X
self.y_train = y
def predict(self, X, k=1, num_loops=0):
""" Predict labels for test data using this classifier. Inputs: - X: A numpy array of shape (num_test, D) containing test data consisting of num_test samples each of dimension D. - k: The number of nearest neighbors that vote for the predicted labels. - num_loops: Determines which implementation to use to compute distances between training points and testing points. Returns: - y: A numpy array of shape (num_test,) containing predicted labels for the test data, where y[i] is the predicted label for the test point X[i]. """
if num_loops == 0:
dists = self.compute_distances_no_loops(X)
elif num_loops == 1:
dists = self.compute_distances_one_loop(X)
elif num_loops == 2:
dists = self.compute_distances_two_loops(X)
else:
raise ValueError('Invalid value %d for num_loops' % num_loops)
return self.predict_labels(dists, k=k)
def compute_distances_two_loops(self, X):
""" Compute the distance between each test point in X and each training point in self.X_train using a nested loop over both the training data and the test data. Inputs: - X: A numpy array of shape (num_test, D) containing test data. Returns: - dists: A numpy array of shape (num_test, num_train) where dists[i, j] is the Euclidean distance between the ith test point and the jth training point. """
num_test = X.shape[0]
num_train = self.X_train.shape[0]
dists = np.zeros((num_test, num_train))
for i in xrange(num_test):
for j in xrange(num_train):
#####################################################################
# TODO: #
# Compute the l2 distance between the ith test point and the jth #
# training point, and store the result in dists[i, j]. You should #
# not use a loop over dimension. #
#####################################################################
dists[i][j] = np.sqrt(np.sum(np.square(X[i]-self.X_train[j])))
#####################################################################
# END OF YOUR CODE #
#####################################################################
return dists
def compute_distances_one_loop(self, X):
""" Compute the distance between each test point in X and each training point in self.X_train using a single loop over the test data. Input / Output: Same as compute_distances_two_loops """
num_test = X.shape[0]
num_train = self.X_train.shape[0]
dists = np.zeros((num_test, num_train))
for i in xrange(num_test):
#######################################################################
# TODO: #
# Compute the l2 distance between the ith test point and all training #
# points, and store the result in dists[i, :]. #
#######################################################################
dists[i] = np.sqrt(np.sum(np.square(self.X_train-X[i]),axis=1))
#######################################################################
# END OF YOUR CODE #
#######################################################################
return dists
def compute_distances_no_loops(self, X):
""" Compute the distance between each test point in X and each training point in self.X_train using no explicit loops. Input / Output: Same as compute_distances_two_loops """
num_test = X.shape[0]
num_train = self.X_train.shape[0]
dists = np.zeros((num_test, num_train))
#########################################################################
# TODO: #
# Compute the l2 distance between all test points and all training #
# points without using any explicit loops, and store the result in #
# dists. #
# #
# You should implement this function using only basic array operations; #
# in particular you should not use functions from scipy. #
# #
# HINT: Try to formulate the l2 distance using matrix multiplication #
# and two broadcast sums. #
#########################################################################
# (a-b)^2 = a^2-2a*b+b^2
dist_left = np.sum(np.square(X),axis=1,keepdims=True) #size(num_test,1)
dist_middle = np.multiply(np.dot(X,self.X_train.T),-2) # size(num_test, num_train)
dist_right = np.sum(np.square(self.X_train),axis=1) #size(num_train,)
dists = np.sqrt(dist_middle + dist_left + dist_right)
#########################################################################
# END OF YOUR CODE #
#########################################################################
return dists
def predict_labels(self, dists, k=1):
""" Given a matrix of distances between test points and training points, predict a label for each test point. Inputs: - dists: A numpy array of shape (num_test, num_train) where dists[i, j] gives the distance betwen the ith test point and the jth training point. Returns: - y: A numpy array of shape (num_test,) containing predicted labels for the test data, where y[i] is the predicted label for the test point X[i]. """
num_test = dists.shape[0]
y_pred = np.zeros(num_test)
for i in xrange(num_test):
# A list of length k storing the labels of the k nearest neighbors to
# the ith test point.
closest_y = []
#########################################################################
# TODO: #
# Use the distance matrix to find the k nearest neighbors of the ith #
# testing point, and use self.y_train to find the labels of these #
# neighbors. Store these labels in closest_y. #
# Hint: Look up the function numpy.argsort. #
#########################################################################
closest_y = self.y_train[np.argsort(dists[i])[:k]]
#########################################################################
# TODO: #
# Now that you have found the labels of the k nearest neighbors, you #
# need to find the most common label in the list closest_y of labels. #
# Store this label in y_pred[i]. Break ties by choosing the smaller #
# label. #
#########################################################################
y_pred[i] = np.argmax(np.bincount(closest_y))
# dists_k_min = np.argsort(dists[i])[:k]
# close_y = self.y_train[dists_k_min]
# y_pred[i] = np.argmax(np.bincount(close_y))
#########################################################################
# END OF YOUR CODE #
#########################################################################
return y_pred
小tips
- np.flatnonzero（array）将array平铺并得到不为零的元素的索引
- np.linalg.norm 求向量的范数
x_norm=np.linalg.norm(x, ord=None, axis=None, keepdims=False)
ord=1: 列和的最大值
ord=2: 求特征值，然后求最大特征值得算术平方根
ord=np.inf, 行和的最大值
axis: 处理类型
axis=1表示按行向量处理，求多个行向量的范数
axis=0表示按列向量处理，求多个列向量的范数
axis=None表示矩阵范数 - np.concatenate 数组拼接
np.concatenate((arr1,arr2),axis)
axis=0时表示，对数组的第0个中括号后的数组维度进行拼接
axis=1时表示，对数组的第1个中括号后的数组维度进行拼接 - 使用k折运算时，绘制误差图使用plt.errorbar
plt.errorbar(x,
y,
yerr=None,
xerr=None,
fmt='',
ecolor=None,
elinewidth=None,
capsize=None,
capthick=None
)
x,y: 数据点的位置坐标
xerr,yerr: 数据的误差范围，经常使用标准差
fmt: 数据点的标记样式以及相互之间连接线样式
ecolor: 误差棒的线条颜***r> elinewidth: 误差棒的线条粗细
capsize: 误差棒边界横杠的大小
capthick: 误差棒边界横杠的厚度
ms: 数据点的大小
mfc: 数据点的颜***r> mec: 数据点边缘的颜色
-
numpy的广播机制
两个数组的size,从后往前一致时可以进行广播，或者前面的维度一致，有一个数组的维度为1时也可以进行广播
例： （4,2，3）与（2,3）可以进行广播，在第一个维度上进行扩展
（4,3）和（4,1）在1的维度上进行扩展
（4,3）和（3，）在4的维度上进行扩展 -
numpy的随机数组
# 创建2行2列取值范围为[0,1）的数组
np.random.rand(2,2)
# 创建2行3列，取值范围为标准正态分布的数组
np.random.randn(2,3)
# 创建指定大小的数组，数组数值随机取于[low, high）之间。
np.random.randint(1,20,size=(2,2,3))
# 随机选取一个数
np.random.choice(a, size=None, replace=True, p=None)
a: 指定的一维数组或者整数。如果是整数，则该方法等同于np.arange(a)
size:数组大小
replace:生成的数组中元素是否可以重复。默认为True,即可以重复
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