Python 实现 距离公式 欧式距离、余弦距离、曼哈顿距离
距离公式 python
1、欧式距离（Euclidean Distance）
计算公式：
( x 1 − x 2 ) 2 + ( y 1 − y 2 ) 2 \sqrt{(x_1-x_2)^2+(y_1-y_2)^2} (x1−x2)2+(y1−y2)2
Python
# 计算欧氏距离
def distEclud(vecA, vecB):
return np.sqrt(np.sum(np.power((vecA - vecB), 2)))
2、曼哈顿距离（Manhattan Distance）
计算公式：
∣ x 1 − x 2 ∣ + ∣ y 1 − y 2 ∣ |x_1-x_2|+|y_1-y_2| ∣x1−x2∣+∣y1−y2∣
优点：计算速度快
Python
def manhattan(rating1, rating2):
"""Computes the Manhattan distance. Both rating1 and rating2 are dictionaries
of the form {'The Strokes': 3.0, 'Slightly Stoopid': 2.5}"""
distance = 0
commonRatings = False
for key in rating1:
if key in rating2:
distance += abs(rating1[key] - rating2[key])
commonRatings = True
if commonRatings:
return distance
else:
return -1 #Indicates no ratings in common
3、一般化
曼哈顿距离和偶是距离可以一般化为如下的明氏距离（Minkowski Distance）：
d ( x , y ) = ( ∑ k = 1 n ∣ x k − y k ∣ r ) 1 r d(x,y)=(\sum_{k=1}^{n}|x_k-y_k|^r)^{\frac{1}{r}} d(x,y)=(k=1∑n∣xk−yk∣r)r1
其中：
r = 1 r=1 r=1 时，上述公式计算的就是曼哈顿距离。
r = 2 r=2 r=2 时，上述公式计算的就是欧式距离。
r = ∞ r=\infty r=∞ 时，上述公式计算的是上确界距离（Supermum Distance）
r r r 越大，某一维上的较大差异对最终差值的影响也就越大。
Python
def minkowski(rating1, rating2, r):
"""Computes the Minkowski distance. Both rating1 and rating2 are dictionaries
of the form {'The Strokes': 3.0, 'Slightly Stoopid': 2.5}"""
distance = 0
commonRatings = False
for key in rating1:
if key in rating2:
distance += pow(abs(rating1[key] - rating2[key]), r)
commonRatings = True
if commonRatings:
return pow(distance, 1/r)
else:
return -1 #Indicates no ratings in common
4、余弦距离（Cosine Distance）
计算公式：
d = ∑ i = 1 N x i y i ∑ i N x i 2 ∑ i N y i 2 d=\frac{\sum_{i=1}^Nx_iy_i}{\sqrt{\sum_{i}^Nx_i^2}\sqrt{\sum_i^Ny_i^2}} d=∑iNxi2∑iNyi2∑i=1Nxiyi
优点：余弦距离根据向量方向来判断向量相似度，与向量各个维度的相对大小有关，不受各个维度直接数值影响。
某种程度上，归一化后的欧氏距离和余弦相似性表征能力相同
Python
# 余弦距离
def distCos(vecA, vecB):
# print(vecA)
# print(vecB)
return float(np.sum(np.array(vecA) * np.array(vecB))) / (distEclud(vecA,np.mat(np.zeros(len(vecA[0])))) * distEclud(vecB,np.mat(np.zeros(len(vecB[0])))))
5、相关测试代码
#
# FILTERINGDATA.py
#
from math import sqrt
users = {"Angelica": {"Blues Traveler": 3.5, "Broken Bells": 2.0, "Norah Jones": 4.5, "Phoenix": 5.0, "Slightly Stoopid": 1.5, "The Strokes": 2.5, "Vampire Weekend": 2.0},
"Bill":{"Blues Traveler": 2.0, "Broken Bells": 3.5, "Deadmau5": 4.0, "Phoenix": 2.0, "Slightly Stoopid": 3.5, "Vampire Weekend": 3.0},
"Chan": {"Blues Traveler": 5.0, "Broken Bells": 1.0, "Deadmau5": 1.0, "Norah Jones": 3.0, "Phoenix": 5, "Slightly Stoopid": 1.0},
"Dan": {"Blues Traveler": 3.0, "Broken Bells": 4.0, "Deadmau5": 4.5, "Phoenix": 3.0, "Slightly Stoopid": 4.5, "The Strokes": 4.0, "Vampire Weekend": 2.0},
"Hailey": {"Broken Bells": 4.0, "Deadmau5": 1.0, "Norah Jones": 4.0, "The Strokes": 4.0, "Vampire Weekend": 1.0},
"Jordyn": {"Broken Bells": 4.5, "Deadmau5": 4.0, "Norah Jones": 5.0, "Phoenix": 5.0, "Slightly Stoopid": 4.5, "The Strokes": 4.0, "Vampire Weekend": 4.0},
"Sam": {"Blues Traveler": 5.0, "Broken Bells": 2.0, "Norah Jones": 3.0, "Phoenix": 5.0, "Slightly Stoopid": 4.0, "The Strokes": 5.0},
"Veronica": {"Blues Traveler": 3.0, "Norah Jones": 5.0, "Phoenix": 4.0, "Slightly Stoopid": 2.5, "The Strokes": 3.0}
}
def manhattan(rating1, rating2):
"""Computes the Manhattan distance. Both rating1 and rating2 are dictionaries
of the form {'The Strokes': 3.0, 'Slightly Stoopid': 2.5}"""
distance = 0
commonRatings = False
for key in rating1:
if key in rating2:
distance += abs(rating1[key] - rating2[key])
commonRatings = True
if commonRatings:
return distance
else:
return -1 #Indicates no ratings in common
def computeNearestNeighbor(username, users):
"""creates a sorted list of users based on their distance to username"""
distances = []
for user in users:
if user != username:
distance = manhattan(users[user], users[username])
distances.append((distance, user))
# sort based on distance -- closest first
distances.sort()
return distances
def recommend(username, users):
"""Give list of recommendations"""
# first find nearest neighbor
nearest = computeNearestNeighbor(username, users)[0][1]
recommendations = []
# now find bands neighbor rated that user didn't
neighborRatings = users[nearest]
userRatings = users[username]
for artist in neighborRatings:
if not artist in userRatings:
recommendations.append((artist, neighborRatings[artist]))
# using the fn sorted for variety - sort is more efficient
return sorted(recommendations, key=lambda artistTuple: artistTuple[1], reverse = True)
# examples - uncomment to run
print( recommend('Hailey', users))
#print( recommend('Chan', users))