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大数据与机器学习 基础篇 聚类
大数据与机器学习 基础篇 聚类

聚类(Clustering)指的是一种学习方式,即把物理或抽象对象的集合分组为由彼此类似的对象组成的多个类的分析过程。

注:本文中用到的Python及其模块安装教程参见


K-Means算法

在聚类中K-Means算法是很常用的一个算法,也是基于向量距离来做聚类。

  • 算法步骤如下:
    1. 从n个向量对象中选择任意k个向量作为初始聚类中心。
    2. 根据在步骤1中设置的k个向量(中心对象向量),计算每个对象与这k个中心对象各自的距离。
    3. 对于步骤2中的计算,任何一个向量与这k个向量都有一个距离,有的远有的近,把这个向量的距离它最近的中心向量对象归在一个类簇中。
    4. 重新计算每个类簇的中心对象向量的位置。
    5. 重复3,4两个步骤,直到类簇聚类方案中的向量归类变化极少为止。例如:一次迭代后,只有少于1%的向量还在发生类簇之间的归类漂移,那么就可以认为分类完成。
  • 这里要注意的是:
    1. 需要事先指定类簇的数量。
    2. 需要事先给定初始的类中心。

我们看这个例子:

准备一个中国城市经纬表,一共2259个向量,维度分别为(经度,维度),城市信息已经略去,数据参见附录。

设定类簇数量为5,开始聚类。

import numpy as np
import matplotlib.pyplot as plt
from sklearn.cluster import KMeans

X=[]
f=open('city.txt')
for v in f:
    X.append([float(v.split(',')[0]),float(v.split(',')[1])])

X=np.array(X)

n_clusters=5

cls=KMeans(n_clusters).fit(X)

cls.labels_

markers=['^','x','o','*','+']
for i in range(n_clusters):
    members=cls.labels_==i
    plt.scatter(X[members,0],X[members,1],s=60,marker=markers[i],c='b',alpha=0.5)

plt.title('China City Area Distribution')
plt.show()
  • 聚类效果如图:

从图片可以清楚地看出,中国的城市被划分成了东北,华中,华南,西部,西北5个区域。

  • K-Means算法所使用的距离计算公式是可以选取的,一般用欧式距离比较简单,也可以用曼哈顿距离。常用的距离度量方法还有余弦相似度。欧式距离度量会受指标不同单位刻度的影响,所以一般需要先进行标准化或者归一化,同时距离越大,个体间差异越大;空间向量余弦夹角的相似度度量不会受指标刻度的影响,余弦值落于区间[-1,1]上,值越大,差异越小。有关余弦相似度的例子在之后的文章中会具体介绍。

层次聚类

层次聚类有两种思路,一种是“凝聚的层次聚类方法”,一种是“分裂的层次聚类方法”。

  • 凝聚的层次聚类方法:在大量的样本中自底而上找那些距离比较近的样本先聚合成小的群,聚合到一定程度再由小的群聚合成更大的群。
  • 分裂的层次聚类方法:先把所有的样本分成若干个大群,再在每个群里各自重新进行聚类划分。

分裂的层次聚类方法可以通过迭代使用K-Means算法来实现,下面重点介绍凝聚的层次聚类方法。

在Scikit-learn库中,提供了一种叫做AgglomerativeClustering的分类算法。首先设置几个观察点散布在整个训练样本中,让这些观察点自下而上不断地进行类簇的合并。这种聚类合并遵循一定的原则,即基于连接度的度量来判断是否要向上继续合并两个类簇。度量有以下3种不同的策略原则:

  1. Ward策略:让所有类簇中的方差最小化。
  2. Maximum策略:力求将类簇之间的距离最大值最小化。
  3. Average linkage策略:力求将类簇之间的距离的平均值最小化。

例如,采用Ward策略:

import numpy as np
import matplotlib.pyplot as plt
from sklearn.cluster import AgglomerativeClustering

X=[]
f=open('city.txt')
for v in f:
    X.append([float(v.split(',')[0]),float(v.split(',')[1])])

X=np.array(X)

n_clusters=5

cls=AgglomerativeClustering(linkage='ward',n_clusters=n_clusters).fit(X)

cls.labels_

markers=['^','x','o','*','+']
for i in range(n_clusters):
    members=cls.labels_==i
    plt.scatter(X[members,0],X[members,1],s=60,marker=markers[i],c='b',alpha=0.5)

plt.title('China City Area Distribution')
plt.show()
  • 聚类效果如图:


密度聚类

在聚类形状不规则的情况下,K-Means这种基于欧式距离半径的类簇划分方法就不那么好用了,K-Means对于带拐弯,狭长的,不规则形状的类簇的效果就不如对圆形类簇的效果好。

这里使用sklearn中专门用来做密度分类的算法库——sklearn.cluster.DBSCAN。

例如对下表中的国家面积和人口的二维向量进行聚类。

  • 12个国家的面积,人口和GDP报表:
国家面积km^2人口GDP亿美元人均GDP美元
中国96702501392358258999607179
印度29800001247923065187071505
美国962909131740801516799753101
巴西85148772010327142242911311
日本3778731272700004901538491
澳大利亚7692024235405171505364863
加拿大9984670345910001825151990
俄罗斯170982001435512892118014819
泰国5131156704100038725674
柬埔寨181035148053581571016
韩国99600504000001221824329
朝鲜120538240522313551476
import numpy as np
from sklearn.cluster import DBSCAN
import matplotlib.pyplot as plt

X=[]
f=open('country.txt')
for v in f:
    X.append([float(v.split(',')[0]),float(v.split(',')[1])])

X=np.array(X)

#做归一化
a=X[:,:1]/max(X[:,:1])[0]*10000
b=X[:,1:]/max(X[:,1:])[0]*10000
X=np.concatenate((a,b),axis=1)

cls=DBSCAN(eps=2000,min_samples=1).fit(X)

cls.labels_

markers=['^','x','o','*','+']
for i in range(n_clusters):
    members=cls.labels_==i
    plt.scatter(X[members,0],X[members,1],s=60,marker=markers[i],c='b',alpha=0.5)

plt.title("Countrys' Area and population")
plt.show()
  • 聚类效果如图:

  • 需要注意的是:
  1. 归一化:是为了解决由于量纲或单位不同所产生的距离计算问题而进行的权重调整,目的是把两个不同维度的数据都投影到以10000为最大值的正方形区域里。
  2. DBSCAN的参数:
    • eps:设置一个阈值,再根据密度向外扩展的过程中如果发现在这个阈值距离范围内找不到向量,那么就认为这个聚簇已经查找完毕。因为归一化以后的向量都落在10000×10000的区间内,所以这里设置2000作为阈值。
    • min_samples:含义是设置聚簇最小应该拥有多少个向量。例如:如果本题中设置min_samples为3,那么右下角的俄罗斯,上面的中国和印度都将作为噪声点被消除而不会显示。

聚类评估

聚类的质量评估包括以下3个方面:

  1. 估计聚类的趋势。对于给定的数据集,评估该数据集是否存在非随机结构,也就是分布不均匀的情况。如果直接使用各种分类算法套用在样本上,然后返回一些类簇,这些类簇的分布很可能是一种错误的分类,会对人们产生误导。数据中必须存在非随机结构,聚类分析才是有意义的。
  2. 确定数据集中的簇数。上述K-Means算法在一开始就需要确定类簇的数量,并作为参数传递给算法。这里容易让人觉得有点矛盾,即人主观来决定一个类簇的数量的方法是不是可取。
  3. 测量聚类的质量。可以用量化的方法来测量聚类的质量。

聚类趋势

如果样本空间里的样本是随机出现的,本身没有聚类的趋势,那么使用聚类肯定是有问题的。对于聚类的趋势,我们常用霍普金斯统计量(Hopkins Statistics)来进行量化评估。

  1. 从所有的样本向量中随机找到n个向量,把他们成为p向量,每一个向量分别是\(p_1,p_2,……,p_n\)。对每一个向量都在样本空间里找到一个离其最近的向量,然后求距离(欧氏距离),然后用\(x_1,x_2,……,x_n\)来表示这个距离。
  2. 从样本向量的一个子集中随机找到n个向量,把他们成为q向量,每一个向量分别是\(q_1,q_2,……,q_n\)。对每一个向量都在样本空间里找到一个离其最近的向量,然后求距离(欧氏距离),然后用\(y_1,y_2,……,y_n\)来表示这个距离。
  3. 求出霍普金斯统计量H:
\(H=\frac{\sum_{i=1}^ny_i}{\sum_{i=1}^nx_i+\sum_{i=1}^ny_i}\)

如果整个样本空间是一个均匀的,没有聚类趋势(聚簇不明显)的空间,那么H应该为0.5左右。反之,如果是有聚类趋势(聚簇明显)的空间,那么H应该接近于0。

计算霍普金斯统计量:

计算中国城市经纬度的霍布金斯统计量,因为类簇数为5,所以取\(X[np.random.randint(X.shape[0],size=int(X.shape[0]/5))]\)为子集,即随机地选取\(\frac{1}{5}\)个向量作为子集。

代码如下:

import numpy as np

X=[]
f=open('city.txt')
for v in f:
    X.append([float(v.split(',')[0]),float(v.split(',')[1])])

X=np.array(X)

#做归一化
a=X[:,:1]/max(X[:,:1])[0]*100
b=X[:,1:]/max(X[:,1:])[0]*100
X=np.concatenate((a,b),axis=1)

HH=[]

for count in range(10):
    #随机选出10个
    pn=X[np.random.choice(X.shape[0],10,replace=False),:]
    xn=[]
    for i in pn:
        distance_min=1000000
        for j in X:
            if np.array_equal(j,i):
                continue
            distance=np.linalg.norm(j-i)
            if distance_min>distance:
                distance_min=distance
            xn.append(distance_min)

    #在子集X[np.random.randint(X.shape[0],size=int(X.shape[0]/5))]随机选出10个
    XX=X[np.random.randint(X.shape[0],size=int(X.shape[0]/5))]
    qn=XX[np.random.choice(XX.shape[0],10,replace=False),:]
    yn=[]
    for i in qn:
        distance_min=1000000
        for j in XX:
            if np.array_equal(j,i):
                continue
            distance=np.linalg.norm(j-i)
            if distance_min>distance:
                distance_min=distance
            yn.append(distance_min)

    Htmp=float(np.sum(yn))/(np.sum(xn)+np.sum(yn))
    HH.append(Htmp)

print(HH)
H=np.average(HH)
print("H: ",H)
输出结果为:
keyboard_arrow_down


0.11722419404269789,
0.10871923835253638,
0.09662900489410153,
0.18076604758829162,
0.12209136642523832,
0.16461772084853724,
0.09426555293455162,
0.1924580732471169,
0.14596993096952454,
0.13714731037623953

平均值为:0.13598884396788355

由此可见,中国城市的经纬度具有明显的聚类趋势。

簇数确定

确定一个样本空间里有多少簇数是很重要的,尤其是在K-Means这种算法里一开始就要求给定要被分成的簇数。况且,簇数的猜测也会影响聚类的结果,簇数太多,样本被分成很多小簇,簇数太少,样本基本没有被分开,都没有意义。

有一种经验的方法就是对于n个样本的空间,设置簇数\(p\)为\(\sqrt{\frac{n}{2}}\),在期望状态下每个簇大约有\(\sqrt{2n}\)个点。但是这种说法只能作为参考。

还有一种方法叫“肘方法”(The Elbow Method),被认为是一种更科学的方式。思路如下:

  • 尝试把样本空间划分为1个类,2个类,……,n个类,要确定哪种分法最为科学,在分成m个类簇的时候会有一个划分方法,在这种划分方法下,每个类簇的内部都有若干个向量,计算这些向量的空间中心点,即计算着m个类簇各自的空间重心在哪里。在计算每个类簇中每个向量和该类簇重心的距离(大于等于0)的和。最后把m个类簇各自的距离和相加得到一个函数var(n),n就是类簇数。

可以想象,这个平方和最大的时刻应该是分1个类——也就是不分类的时候,所有的向量到重心的距离都非常大,这样的距离的和是最大的。那么尝试着划分为2个类,3个类,4个类……随着分类的增多,第m次划分时,每个向量到自己簇的重心的距离会比上一次(m-1)次临近的机会更大,那么这个距离总和就会总体上缩小。极限情况就是最后被分成了n个类簇,n整个空间向量的数量,也就是一个向量一个类簇,每个类簇一个成员。这种情况最后距离的和就变成了0,因为每个向量距离自己(自己就是重心)的距离都是0。

当m从1,2,3……逐步向上增加的过程中,整个曲线的斜率会逐步降低,而且一开始是快速下降的。下降的过程中有一个拐点,这一个点会让人感觉曲线从立陡变成平滑,那么这个点就是要找的点。

这个样本空间被分成m个类簇之后,再分成更多的类簇时,每次的“收获”没有前面每次“收获”那么大,此时的m值就是被认为最合适的类簇数量。这个点在曲线上给人的感觉就像是人的胳膊肘一样,所以被形象地命名为“肘方法”。

例如,对中国城市的经纬表做簇数确定(这里使用曼哈顿距离):

import numpy as np
from sklearn.cluster import KMeans
import matplotlib.pyplot as plt

X=[]
f=open('city.txt')
for v in f:
    X.append([float(v.split(',')[0]),float(v.split(',')[1])])

X=np.array(X)

#做归一化
a=X[:,:1]/max(X[:,:1])[0]*100
b=X[:,1:]/max(X[:,1:])[0]*100
X=np.concatenate((a,b),axis=1)

#曼哈顿距离
def manhattan_distance(x,y):
    return np.sum(abs(x-y))

distance_sum=[]

for N in range(1,10):
    n_clusters=N

    cls=KMeans(n_clusters).fit(X)

    cls.cluster_centers_

    cls.labels_

    distance=0
    for i in range(n_clusters):
        group=cls.labels_==i
        members=X[group,:]
        for v in members:
            distance+=manhattan_distance(np.array(v),cls.cluster_centers_[i])

    distance_sum.append(distance)

for N in range(1,10):
    plt.scatter(N,distance_sum[N-1],s=60,marker='o',c='b',alpha=0.5)

plt.plot(range(1,10),distance_sum,'r',label='Fitted line',)

plt.title("Cluster Number and Distance_sum")
plt.show()
  • 输出结果如图:

这里可以发现选择5个类簇和6个类簇对中国城市经纬表进行聚类都是可以的。

  • 6个类簇的中国城市经纬图(将西北地区又分成了两部分):

有的时候会发现类簇数量可能不是很好确定,这个拐点不清晰。这种时候适当地在计算效率和收益程度上做一个平衡即可,不必太纠结,毕竟聚类是一个无监督的学习。

测定聚类质量

测定聚类质量的方法有很多,一般分为“外在方法”和“内在方法”两种。

所谓外在方法是一种依靠类别基准的方法,即已经有比较严格的类别定义时在讨论聚类是不是足够准确。这里通常使用“BCubed精度”和“BCubed召回率”来进行衡量。但是外在方法适用于有明确的外在类别基准的情况,而聚类是一种无监督的学习,更多的是在不知道基准的状况下进行的,所以我们更倾向于使用“内在方法”。

“内在方法”不会去参考类簇的标准,而是使用轮廓系数(Silhouette Coefficient)进行度量。

计算轮廓系数的思路如下:

  • 对于有n个向量的样本空间,假设它被划分成k个类簇,即\(C_1,C_2,……,C_k\)。对于任何一个样本空间中的向量v来说,可以求一个v到本类簇中其他各点的距离的平均值a(v),还可以求一个v到其他所有各类簇的最小平均距离(即从每个类簇里挑选一个离v最近的向量,然后计算距离),求这些距离的平均值,得到b(v),轮廓系数的定义为:
\(s(v)=\frac{b(v)-a(v)}{max[a(v),b(v)]}\)

一般来说,这个函数的结果在-1和1之间,a(v)表示的是类簇内部的紧凑型,越小越紧凑,而b(v)表示该类簇的其他类簇之间的分离程度。如果函数值接近1,即a(v)比较小而b(v)比较大时,说明包含v的类簇非常紧凑,而且远离其他的类簇。相反,如果函数值为负数,则说明a(v)>b(v),v距离其他类簇比距离自己所在类簇的其他对象更近,那么这种情况就不太好,应该尽可能避免。

为了让聚类中的类簇划分更合理,可以计算类簇中所有对象的轮廓系数的平均值。在一种方案里,如果轮廓系数是负数那么可以直接淘汰,如果是正数则可以在多个方案里进行比较,选择一种轮廓系数接近1的方案。

计算中国城市经纬表聚类后的轮廓系数:

#计算轮廓系数
import numpy as np
from sklearn.cluster import KMeans
import matplotlib.pyplot as plt

X=[]
f=open('city.txt')
for v in f:
    X.append([float(v.split(',')[0]),float(v.split(',')[1])])

X=np.array(X)

#做归一化
a=X[:,:1]/max(X[:,:1])[0]*100
b=X[:,1:]/max(X[:,1:])[0]*100
X=np.concatenate((a,b),axis=1)

#曼哈顿距离
def manhattan_distance(x,y):
    return np.sum(abs(x-y))

n_clusters=6

cls=KMeans(n_clusters).fit(X)

cls.labels_

index=np.random.randint(len(X))

#a(v),X[index]到类簇中其他点的距离的平均值
distance_sum=0
for v in X[cls.labels_==cls.labels_[index]]:
    distance_sum+=manhattan_distance(np.array(X[index]),np.array(v))

av=distance_sum/len(X[cls.labels_==cls.labels_[index]])
print(av)

#b(v),X[index]
distance_sum=0
distance_min=1000000
for i in range(n_clusters):
    if i==cls.labels_[index]:
        continue
    group=cls.labels_==i
    members=X[group]
    for v in members:
        distance=manhattan_distance(np.array(v),X[index])
        if distance_min>distance:
            distance_min=distance
    distance_sum+=distance

bv=distance_sum/(n_clusters-1)
print(bv)

sv=float(bv-av)/max(av,bv)
print(sv)
输出结果为:
keyboard_arrow_down


5个类簇时的轮廓系数:0.5292298922519052
6个类簇时的轮廓系数:0.6012776282140894


想了解更多关于大数据和机器学习:

中国城市经纬表
keyboard_arrow_down


经度,纬度
121.48,31.22
121.24,31.4
121.48,31.41
121.7,31.19
121.76,31.05
121.46,30.92
121.24,31
121.16,30.89
121.1,31.15
121.4,31.73
102.73,25.04
102.48,25.21
102.58,24.68
102.79,24.9
102.44,24.95
103.7,29.32
103.63,28.22
103.91,27.74
104.06,27.61
103.54,27.21
103.97,28.58
104.28,28.08
105.05,27.85
104.86,27.42
102.92,26.9
104.38,28.62
103.79,25.51
104.09,26.24
104.24,25.67
103.97,24.85
103.03,25.35
103.27,26.41
103.82,25.62
104.3,24.88
104.64,25.04
103.12,24.9
103.61,25.41
103.24,24.77
103.25,25.56
102.52,24.35
102.93,24.26
102.75,24.09
102.91,24.68
102.73,24.27
102.15,24.67
102,23.59
101.98,24.06
102.38,24.16
101,22.79
101.03,23.07
100.88,23.9
100.82,24.42
100.71,23.5
101.71,23.4
99.97,22.55
99.47,22.73
101.88,22.58
99.55,22.32
100.09,23.88
100.12,24.44
99.02,23.92
99.25,24.03
99.92,24.58
99.85,23.45
99.24,23.15
99.41,23.56
99.18,25.12
99.15,24.69
98.51,25.01
99.61,24.82
98.7,24.58
100.25,26.86
101.24,26.63
100.76,26.71
100.82,27.29
104.24,23.37
105.09,24.05
104.68,23.42
104.71,23.12
104.4,23.01
104.19,24.03
104.35,23.62
105.6,23.62
102.43,23.35
103.43,24.41
103.41,23.36
102.81,23.17
102.42,23.35
102.48,23.73
103.76,24.52
103.24,22.77
103.23,23.7
102.42,23.01
102.79,23.64
103.98,22.52
103.67,22.68
100.79,22
100.5,21.95
101.56,21.48
101.54,25.01
101.85,25.7
102.36,25.55
102.08,25.15
101.26,25.21
101.34,25.73
101.7,26.07
102.45,25.58
101.58,25.32
101.67,24.68
101.24,25.4
100.24,25.45
99.88,26.53
99.94,26.1
100.55,25.82
100.52,25.34
99.52,25.45
100.18,26.55
100.19,25.69
99.98,25.68
99.39,25.9
100.56,25.48
100.33,25.23
100.51,25.04
98.6,24.41
97.96,24.33
97.93,24.69
98.08,24.08
97.83,24
98.3,24.78
98.82,25.97
98.95,26.55
98.92,26.89
99.29,26.49
98.65,27.73
99.72,27.78
98.93,28.49
99.27,27.15
123.33,25
117.51,15.07
111.65,40.82
111.13,40.72
111.15,40.28
110,40.58
110.52,40.55
110.03,41.03
106.82,39.67
113.08,41.03
113.97,40.88
111.65,39.92
111.42,41.12
112.52,40.93
113.53,41.58
113.15,40.45
112.48,40.52
111.8,40.4
114,41.9
111.68,41.37
111.96,43.65
116.08,43.95
116.48,42.18
114.97,44.03
117.58,44.6
116.97,45.53
113.7,43.85
112.95,42.47
115.3,41.9
115,42.32
116.02,42.25
113.83,42.25
119.73,29.22
117.47,49.58
119.45,49.33
120.08,50.45
121.52,50.8
120.73,49.3
123.5,48.13
122.78,47.98
116.82,48.67
118.23,48.22
123.7,50.58
122.28,43.63
121.32,43.62
121.75,42.72
120.65,42.85
120.87,44.55
118.87,42.28
119.32,41.62
118.02,43.62
118.67,41.95
119.87,42.3
119,42.97
118.65,43.52
119.35,43.98
120.05,43.97
117.48,43.28
110,39.83
110,39.83
111.13,39.68
109.03,38.38
109.77,39.25
107.97,39.12
107.43,38.18
108.7,39.83
110.02,40.42
107.37,40.78
108.28,41.12
106.98,40.33
107.12,40.88
108.52,41.55
108.65,40.75
108.52,40.88
105.68,38.85
101.68,39.2
100.88,41.9
122.08,46.07
121.5,45.4
116.46,39.92
117.1,40.13
116.85,40.37
116.65,40.13
116.67,39.92
116.62,40.32
116.33,39.73
115.98,39.72
125.35,43.88
126.57,43.87
125.15,44.45
125.68,44.52
126.55,44.83
126.83,44.15
125.68,43.53
126.57,43.87
126.97,44.4
127.33,43.75
126.72,42.97
126.03,42.93
129.52,42.93
129.75,43.32
130.35,42.85
129.83,42.98
129,42.52
128.3,42.58
128.18,43.35
125.92,41.49
125.7,40.88
125.65,42.53
126.03,42.68
126.8,42.38
126.4,41.97
127.27,42.33
126.17,41.15
128.17,41.43
124.37,43.17
124.33,43.32
124.82,43.5
125.32,43.33
125.15,42.97
125.5,42.68
123.5,43.52
122.82,45.63
124.18,45.5
124.82,45.2
124.02,45
123.97,44.3
123.13,44.82
122.75,45.35
104.06,30.67
104.32,30.88
104.94,30.57
103.29,30.2
103.86,30.8
104.13,30.82
102.15,26.9
101.56,26.9
103.81,30.97
103.61,31.04
103.94,30.99
104.16,31.13
104.25,30.99
103.78,30.42
103.47,30.42
103.53,30.58
103.69,30.63
104.73,31.48
104.7,31.8
105.21,32.59
104.52,32.42
105.86,32.44
106.33,32.25
105.45,32.03
105.16,31.64
105.06,31.1
105.35,31.23
105.31,30.9
105.58,30.52
105.74,30.78
104.68,31.06
104.37,31.13
104.19,31.32
104.41,31.64
104.44,31.89
105.04,29.59
105.02,30.3
105.3,30.12
104.7,29.57
104.85,29.81
104.6,30.19
104.53,30.38
105.25,29.64
104.56,29.77
104.97,29.24
104.96,28.87
105.06,28.71
105.38,28.77
105.46,28.96
105.78,28.79
105.39,28.91
105.79,28.03
105.44,28.19
104.91,28.6
105.06,28.36
104.81,28.38
104.52,28.4
104.53,28.16
104.15,28.68
103.73,29.59
103.59,29.75
103.38,29.95
103.53,30.04
103.81,29.86
103.81,30.05
103.83,30.22
104.06,29.67
104.09,30
103.93,29.21
103.98,28.96
103.5,29.62
103.53,28.87
103.25,29.23
103.13,29.24
107.36,29.7
107.34,30.36
107.7,29.89
108.13,29.98
108.97,28.47
108.75,28.85
108.81,29.53
108.19,29.29
108.72,29.29
107.13,29.15
108.35,30.83
108.39,31.23
108.67,31.98
109.6,31.42
109.86,31.1
109.52,31.06
108.89,30.99
108.03,30.33
107.78,30.66
106.06,30.8
105.96,31.75
105.97,31.75
106.38,31.52
106.03,31.34
105.84,31.01
106.57,31.07
106.44,31.04
106.61,30.48
106.43,30.55
106.3,30.38
106.74,30.41
107.49,31.23
108.06,32.07
107.71,31.39
107.87,31.1
106.91,30.36
107.21,30.75
106.94,30.85
106.83,32.36
106.73,31.86
107.11,31.59
108.24,31.95
108.18,32
102.97,29.97
102.91,30.17
103.06,30.09
102.81,29.79
102.66,29.4
102.38,29.21
102.78,30.09
102.84,30.36
102.22,31.92
102.55,31.79
101.72,31.93
102.94,33.62
102.95,32.06
103.61,32.64
104.19,33.23
103.61,31.46
103.16,31.42
102.34,30.97
102.03,31.48
100.97,32.3
103.89,31.67
101.95,30.04
100.65,31.38
99.96,31.64
100.28,30.96
98.83,32.23
98.57,31.81
98.06,33.01
100.35,32.3
102.25,29.92
101.87,30.85
101.53,29.01
101,30.03
101.14,30.99
100.28,30.03
99.78,28.93
100.31,29.04
99,30
99.25,28.71
102.29,27.92
102.83,28.03
102.74,28.96
103.62,28.21
102.76,27.07
102.55,26.74
102.21,26.67
102.15,27.4
103.14,28.33
103.22,27.73
102.8,27.7
102.52,27.38
102.42,28.33
102.49,28.66
101.51,27.42
102.15,28.58
101.25,27.9
117.2,39.13
117.83,39.33
116.92,38.93
117.4,40.05
117.3,39.75
117.05,39.4
106.27,38.47
106.24,38.28
106.35,38.55
106.39,39.04
106.54,38.91
106.69,38.82
106.21,37.99
105.94,36.97
106.34,38.1
105.66,37.48
107.41,37.78
105.18,37.51
106.07,38.02
106.28,36.01
105.7,35.97
106.33,35.5
105.64,36.56
106.11,35.63
117.27,31.86
117.16,32.47
116.98,32.62
116.71,32.68
116.77,33.97
116.76,33.92
118.38,31.33
117.82,30.93
117.34,32.93
118.48,31.56
117.03,30.52
116.97,33.63
116.97,33.63
116.34,34.42
116.93,34.19
117.55,33.55
117.89,33.49
117.87,33.14
117.32,33.33
117.19,32.95
118.31,32.33
117.98,32.78
119,32.68
118.44,32.44
118.27,32.1
117.68,32.52
117.4,32.86
117.87,31.62
117.87,31.62
117.47,31.89
118.11,31.7
118.37,31.7
117.75,31.3
117.29,31.23
118.73,31.95
118.49,31.55
119.17,31.14
119.41,30.89
118.41,30.68
118.32,30.91
118.21,31.07
118.95,30.62
117.84,30.64
118.31,29.72
118.19,29.81
118.53,30.28
118.57,30.07
118.44,29.88
117.7,29.86
117.92,29.93
118.13,30.28
117.48,30.19
116.94,31.04
117.21,30.69
116.63,30.41
116.69,30.12
116.13,30.15
116.27,30.42
116.36,30.84
116.53,30.62
116.99,30.08
117.48,30.66
116.49,31.73
116.27,32.32
116.78,32.57
117.15,31.7
116.94,31.45
116.32,31.38
115.87,31.67
115.81,32.89
116.76,33.86
116.21,33.49
116.55,33.25
116.19,33.12
116.26,32.62
115.6,32.63
115.24,33.06
115.34,33.24
115.61,33.16
117,36.65
117.07,36.69
116.73,36.55
117.53,36.72
120.33,36.07
120.42,36.15
119.97,35.88
120.45,36.38
120,36.28
118.05,36.78
117.57,34.86
117.17,35.09
118.49,37.46
118.54,37.59
118.25,37.49
116.29,37.45
116.8,37.64
117.22,37.74
117.15,37.31
117.2,36.97
116.66,36.95
116,36.95
116.58,37.34
117.37,37.37
116.86,37.2
116.76,36.79
116.44,37.16
116.08,37.2
118.03,37.36
117.97,37.47
118.41,37.04
118.12,36.95
117.75,36.89
117.58,37.65
118.14,37.7
118.12,37.12
117.66,37.18
117.51,17.49
117.58,37.73
119.1,36.62
119.22,36.77
119.97,36.77
119.42,35.99
119.2,36.42
118.53,36.5
118.73,36.86
119.41,36.86
119.75,36.38
119.2,35.74
118.83,36.69
118.47,36.69
121.39,37.52
121.59,37.38
122.05,37.2
121.17,36.76
120.71,36.97
120.83,37.28
119.93,37.18
120.73,37.91
122.1,37.5
121.27,37.49
122.41,37.16
121.52,36.89
120.53,36.86
120.38,37.35
120.51,37.64
120.75,37.8
118.35,35.05
118.64,35.78
119.46,35.42
118.73,34.89
118.03,34.84
117.63,35.49
118.17,36.18
118.47,35.54
118.83,35.57
118.83,35.17
118.35,34.61
117.97,35.26
117.95,35.7
117.13,36.18
117.67,36.19
116.76,36.24
116.46,36.29
117.67,35.86
117.76,35.91
116.8,35.76
116.3,35.91
116.59,35.38
116.83,35.54
117.27,35.65
116.65,35
116.34,35.41
116.49,35.71
116.98,35.59
116.97,35.39
117.12,34.8
116.32,35.07
115.43,35.24
115.94,35.59
116.08,35.38
116.07,34.82
115.53,34.83
115.5,35.57
116.1,35.8
115.88,34.97
115.57,35.07
115.08,35.31
115.97,36.45
116.23,36.86
116.23,36.32
115.67,36.24
115.72,36.68
116.27,36.58
115.78,36.11
115.45,35.47
112.53,37.87
112.65,38.05
111.78,38.05
112.33,37.62
113.3,40.12
113.57,37.85
113.08,36.18
114.08,40.42
114.2,39.47
113.1,39.82
112.82,39.52
112.12,39.53
112.33,40.18
113.72,40.38
113.27,39.75
113.68,39.7
113.18,39.58
112.42,39.32
112.67,40.02
112.7,38.38
112.97,39.07
113.32,38.72
111.9,38.37
111.09,38.01
111.17,39.38
112.17,39.1
112.7,38.73
113.28,39.2
112.95,38.5
111.58,38.7
111.82,38.93
111.47,39.45
112.28,39
112.72,37.68
113.37,38.01
113.68,37.62
113.35,37.07
112.53,37.42
112.18,37.2
111.77,36.83
113.17,37.88
113.62,37.79
113.55,37.33
112.97,37.08
112.33,37.36
111.88,37.03
111.13,37.53
111.22,38.47
111.24,37.86
111.62,38.28
112.14,37.55
112.02,37.42
111.75,37.27
111.8,37.12
111.2,36.97
110.83,37
111.17,37.37
110.95,37.95
110.85,37.45
113.02,36.55
113.4,36.56
113.23,35.11
112.88,35.48
112.38,35.84
112.87,36.13
112.32,36.5
113.22,36.33
112.83,36.83
113.43,36.19
113.27,35.78
112.83,35.52
112.15,35.67
112.87,36.32
112.68,36.75
111.5,36.08
111.53,36.63
112.2,36.15
111.9,36.29
111.68,35.73
111.33,35.63
110.65,36.12
110.72,36.47
111.45,35.03
110.64,36.62
111.68,36.25
111.72,36.57
111.83,35.97
111.43,35.86
110.8,35.97
111.07,36.42
110.97,35.03
111.2,35.37
111.63,35.3
110.68,34.71
110.78,35.15
111.22,35.62
110.7,35.58
111.22,35.12
111.58,35.48
111.2,34.12
110.42,34.88
110.83,35.42
110.97,35.6
113.23,23.16
113.19,23.4
114.2,24.09
113.81,23.13
113.55,23.57
114.25,23.75
113.36,22.95
110.35,20.02
116.69,23.39
110.38,21.2
110.88,21.68
113.11,23.05
113.06,22.61
114.07,22.62
113.85,22.58
113.52,22.3
113.62,24.84
113.58,24.68
113.35,25.14
113.73,25.11
114.33,25.14
114.08,24.78
114.13,24.36
113.52,23.86
113.38,24.17
113.01,23.7
112.65,24.48
112.4,24.77
112.07,24.59
112.28,24.77
114.4,23.09
114.4,23.09
114.28,23.18
114.68,23.73
114.48,24.39
114.89,24.45
115.25,24.09
115.18,23.64
114.7,22.97
113.75,23.04
116.1,24.55
116.1,24.55
117.9,24.59
116.18,24.66
116.7,24.34
116.18,23.78
115.75,23.93
115.75,24.15
116.63,23.68
116.8,23.48
116.63,23.68
117.01,23.7
117.03,23.44
116.61,23.27
116.29,23.07
117.64,22.95
117.33,22.98
116.17,23.29
115.82,23.45
116.35,23.55
113.11,23.05
112.89,23.18
113.24,22.84
113.38,22.52
113.25,22.2
113.02,22.52
112.94,22.76
112.68,22.36
112.78,22.27
112.29,22.21
112.76,21.71
110.27,21.63
110.59,21.64
110.83,21.95
110.9,22.36
111.78,22.16
111.95,21.85
110.99,21.52
110.78,21.43
110.17,20.34
110.07,20.91
110.24,21.39
112.44,23.05
112.44,23.05
112.18,23.93
112.43,23.14
112.68,23.36
112.2,22.68
112.02,22.93
111.56,22.77
111.51,23.23
111.75,23.15
111.48,23.45
108.33,22.84
109.4,24.33
110.28,25.29
111.34,23.51
106.75,22.11
108.49,22.74
108.27,23.17
108.2,23.73
108.59,23.44
108.8,23.22
109.2,22.69
107.92,22.65
107.37,22.42
107.08,22.12
106.84,22.36
107.21,22.85
107.12,23.08
107.68,23.18
108.06,24.7
108.26,24.83
108.9,24.79
108.64,24.47
107.36,24.53
107.05,24.55
107.16,25.01
107.54,24.98
108.09,23.94
107.25,24.15
108.89,23.82
109.24,24.67
109.37,24.24
109.74,24.49
109.7,23.98
109.66,23.6
109.34,24.27
109.24,23.76
108.66,24.07
109.24,25.07
109.58,25.8
110.18,24.14
110.22,25.22
110.33,25.42
110.66,25.6
110.66,26.03
111.06,25.96
111.14,25.49
110.81,24.85
110.66,24.64
110.38,24.51
109.98,24.99
110.02,25.78
111.22,23.51
111.3,24.53
110.26,24.83
111.54,24.44
111,22.95
110.9,23.36
110.54,24.22
110.8,24.18
110.14,22.64
110.07,23.38
110.4,23.55
110.53,22.87
110.33,22.71
110.25,22.33
109.98,22.27
109.6,23.11
109.12,21.49
108.61,21.96
109.29,22.44
109.56,22.27
109.2,21.33
107.98,22.16
108.35,21.78
106.62,23.91
106.55,24.35
106.56,24.78
106.9,23.75
107.12,23.62
107.59,23.33
106.6,23.34
106.41,23.15
105.85,23.42
105.08,24.51
106.24,24.31
105.34,24.8
87.68,43.77
84.77,45.59
85.94,44.27
89.19,42.91
88.63,42.77
90.25,42.82
93.44,42.78
94.65,43.28
93,43.6
86.06,41.68
86.35,42.31
86.84,42.23
86.53,41.95
86.24,41.36
84.25,41.77
86.55,42.05
79.94,37.12
82.63,37.07
80.78,37.04
81.63,36.86
80.17,37.12
78.29,37.06
79.74,37.31
80.29,41.15
80.24,41.29
81.84,41.82
82.97,41.68
82.63,41.55
82.9,41.25
80.34,40.64
79.06,40.55
79.25,41.22
75.94,39.52
78.59,39.78
76.78,39.46
76.67,39.23
77.62,38.95
77.25,38.45
77.26,38.2
77.42,37.89
76.05,39.41
76.17,38.91
75.83,39.42
75.22,37.76
76.12,39.73
78.42,41.91
75.94,39.14
75.18,39.7
87.31,44.05
87.94,44.14
89.52,44.02
89.14,44
87.68,43.97
86.22,44.28
86.92,44.18
90.34,43.8
82.1,44.93
82.92,44.67
81.08,44.95
81.33,43.91
82.53,43.82
83.27,43.41
82.23,43.35
84.89,44.45
81.81,43.23
81.08,43.15
80.87,44.07
81.12,43.82
82.96,46.74
83.62,46.52
84.62,44.45
83.59,45.92
82.94,46.21
85.56,44.29
85.13,46.78
88.14,47.86
90.37,46.71
89.44,47.05
87.51,47.15
85.84,47.42
86.92,47.7
86.41,48.05
118.78,32.04
118.83,31.95
118.83,32.36
118.62,32.07
117.2,34.26
119.16,34.59
120.86,32.01
120.62,31.32
120.29,31.59
119.95,31.79
116.57,34.79
116.93,34.73
119.11,34.83
118.75,34.54
118.33,34.38
117.97,34.3
117.94,33.89
117.2,34.26
119.02,33.59
119.23,34.3
119.36,34.09
118.79,34.12
118.3,33.96
118.68,33.73
118.05,33
119.26,33.77
119.02,33.62
119.15,33.5
118.85,33.28
118.23,33.46
119.02,33.01
120.13,33.38
119.84,34.01
119.79,33.78
120.26,33.77
119.77,33.46
119.56,34.2
120.45,33.19
120.31,32.84
120.45,32.57
120.56,32.39
121.18,32.33
121.66,31.8
121.15,31.89
120.86,32.01
119.42,32.39
119.32,33.23
119.82,32.93
119.45,32.78
120.02,32.16
120.15,32.51
119.9,32.49
120.26,32.03
119.55,32.43
119.42,32.39
119.16,32.27
119.44,32.2
119.44,32.2
119.81,32.24
119.55,32
119.95,31.78
119.82,31.36
119.56,31.74
119.48,31.43
119.16,31.95
119.02,31.65
118.87,31.32
120.26,31.91
120.55,31.86
120.74,31.64
121.1,31.45
120.95,31.39
120.62,31.32
120.63,31.16
115.89,28.68
115.8,28.69
117.22,29.3
113.85,27.6
115.97,29.71
116.56,29.9
116.23,29.75
116.19,29.29
116.03,29.47
115.82,29.04
115.75,29.33
115.65,29.68
115.09,29.26
114.55,29.04
117.97,28.47
117.83,29.25
117.58,28.96
118.25,28.68
118.2,28.45
117.71,28.32
117.62,28.42
117.02,28.23
117.2,28.3
116.82,28.22
117.08,28.7
117.12,28.97
116.68,29
116.69,28.7
117.43,28.42
114.38,27.81
114.44,28.11
114.37,28.53
114.78,28.4
114.91,28.25
115.55,28.86
115.38,28.71
115.38,28.42
115.7,28.19
115.54,28.07
114.92,27.81
114.68,27.82
115.37,28.88
116.34,28
116.29,27.95
116.77,27.92
117.06,27.7
116.91,27.3
116.52,27.22
116.62,27.56
116.2,27.55
116.05,27.75
115.82,27.44
116.61,28.23
116.26,28.37
114.97,27.12
115.4,27.77
115.15,27.56
115.14,27.22
115.42,27.33
114.88,26.81
114.77,26.47
114.5,26.33
113.97,26.71
114.23,26.96
113.94,27.14
114.62,27.39
114.17,26.57
114.92,25.85
116.32,26.84
116.32,26.34
116,26.46
115.33,26.32
115.39,25.96
116.02,25.89
115.79,25.58
115.41,25.15
115.64,24.96
115.02,24.7
114.79,24.91
114.53,24.76
114.94,25.39
114.02,25.85
114.75,25.66
114.55,25.8
114.31,25.69
114.36,25.39
114.48,38.03
118.02,39.63
114.54,38.42
114.38,38.31
115.18,37.94
115.03,38.03
114.84,38.03
114.58,37.62
114.78,37.76
114.13,38.03
114.03,38.08
114.67,38.33
114.56,38.13
115.2,38.2
114.96,38.16
114.35,37.65
114.5,37.74
114.64,38.87
114.24,38.2
114.47,36.6
114.5,36.77
114.92,36.78
115.4,36.47
114.94,36.37
114.68,36.43
115.14,36.28
113.67,36.57
113.85,36.95
115.18,36.84
114.94,36.49
114.8,36.56
114.62,36.35
114.37,36.37
114.2,36.7
114.48,37.05
114.68,37.49
114.9,37.62
114.75,37.35
115.5,36.87
115.37,37.37
115.03,37.22
114.68,37.11
114.52,36.94
114.5,37.43
114.5,37.28
115.22,37.53
115.67,37.07
115.08,36.97
115.14,37.06
115.02,37.06
114.71,37
115.48,38.85
115.71,39.39
115.98,39.48
115.78,39.28
115.86,39.06
115.92,38.92
115.58,38.49
115.46,38.46
114.02,38.52
114.18,38.85
114.97,38.75
114.67,39.37
115.49,39.35
115.84,39.34
116.1,38.98
115.65,39.02
115.78,38.68
115.3,38.41
115.47,38.76
115.14,38.71
114.68,38.62
115.12,38.84
115.45,38.95
114.87,40.82
114.6,41.87
115.82,40.92
115.54,40.4
114.53,39.83
115.03,40.63
114.7,41.15
115.68,41.68
115.25,40.98
115.2,40.37
114.15,40.12
114.38,40.67
113.95,41.05
114.73,40.84
117.93,40.97
117.72,41.95
118.68,41.02
118.47,40.62
117.48,40.42
117.53,40.95
117.7,41.32
118.93,40.43
116.63,41.2
119.57,39.95
118.3,40.15
118.69,40.02
119.15,39.72
118.85,39.89
118.67,39.49
117.9,39.9
118.54,39.31
117.97,40.2
119.22,39.88
118.9,39.43
118.73,39.74
118.1,39.58
118.13,39.82
116.7,39.53
116.69,39.52
117.06,39.97
117,39.76
116.38,39.12
116.29,39.44
116.63,38.7
116.45,38.87
116.48,39.32
116.98,39.98
116.83,38.33
117.33,38.4
117.22,38.07
116.37,37.65
116.52,37.89
115.82,38.43
116.07,38.45
116.56,38.08
116.27,38.02
116.8,38.58
117.85,38.17
116.7,38.05
116.08,38.72
116.12,38.2
117.1,38.06
115.72,37.72
115.74,38.24
116.14,37.87
116.26,37.69
115.72,37.52
115.56,38.02
115.5,38.22
115.96,38.03
115.9,37.81
115.96,37.36
115.56,37.59
113.65,34.76
113.35,34.79
114.35,34.79
113.29,33.75
112.44,34.7
113.21,35.24
114.17,35.9
114.77,34.56
114.17,34.41
113.71,34.4
113.02,34.46
114.46,34.48
114,34.73
113.35,34.51
112.96,34.76
114.81,34.69
113.85,35.31
114.05,35.44
114.04,35.03
113.63,35.27
113.06,34.94
112.57,35.08
113.05,35.16
113.77,35.46
114.19,35.14
113.96,35.05
113.38,35.1
112.77,34.92
112.92,35.08
113.42,35.24
114.35,36.1
115.21,36.08
115.46,35.9
115.83,36
114.52,35.57
114.54,35.67
114.17,35.6
114.88,35.95
115.1,35.89
114.98,35.71
114.67,35.19
114.35,35.92
113.81,36.06
115.65,34.44
116.13,34.22
115.29,34.08
115.04,34.46
115.87,34.4
116.37,33.94
115.31,34.44
115.13,34.65
114.63,33.63
114.59,33.54
114.38,34.05
115.48,33.86
114.88,33.74
115.06,33.41
114.5,33.79
114.85,34.06
115.17,33.63
114.9,33.44
113.81,34.02
114.17,34.11
113.98,33.6
113.46,33.86
112.88,33.74
113.19,33.98
114.02,33.56
113.77,34.22
113.94,33.81
113.58,33.44
113.35,33.62
113.04,33.86
113.47,34.16
114.02,32.98
114.02,32.83
114,33.38
114.35,33
114.97,32.75
113.31,32.72
113.98,33.15
114.26,33.25
114.62,32.97
114.38,32.62
114.08,32.13
114.72,32.35
115.68,32.17
115.04,32.13
114.83,31.62
114.53,32.21
115.41,32.44
115.42,31.81
114.91,32.02
112.53,33.01
112.98,33.25
112.83,32.7
112.36,32.51
112.08,32.68
111.47,33.14
112.4,33.49
112.92,33.05
113.4,32.37
112.23,33.03
111.83,33.05
111.5,33.31
111.19,34.76
112.42,34.84
112.83,34.17
112.46,34.16
112.07,34.14
111.6,33.81
110.85,34.52
111.75,34.76
111.92,34.73
112.77,34.73
112.42,34.43
112.15,34.51
111.65,34.39
111.03,34.06
111.19,34.76
112.14,34.75
120.19,30.26
120.3,30.43
119.95,30.07
119.27,29.49
119.72,30.23
120.25,30.16
119.64,29.8
119.05,29.61
121.56,29.86
121.72,29.96
120.65,28.01
120.65,28.01
120.68,28.16
121.12,27.84
120.55,27.68
119.7,27.57
120.94,28.14
120.62,27.8
120.08,27.08
120.36,27.53
120.1,30.86
121.02,30.7
120.54,30.64
119.68,30.68
120.92,30.84
120.76,30.77
120.92,30.53
120.69,30.53
120.08,30.54
119.91,30.01
122.11,30.03
122.2,30.26
122.45,30.72
122.3,29.97
121.56,29.86
121.8,29.48
121.41,29.66
121.23,30.18
121.42,29.3
121.16,30.04
120.58,30.01
120.89,29.49
120.23,29.71
120.87,30.03
120.81,29.6
121.44,28.67
121.13,28.8
121.38,29.11
121.36,28.36
120.73,28.85
121.03,29.15
121.27,28.64
121.23,28.14
119.92,28.45
120.28,28.45
119.06,27.61
119.25,28.59
120.6,28.66
119.56,28.12
119.13,28.08
119.48,28.46
119.64,29.12
119.88,29.46
120.23,29.27
119.81,28.9
118.61,28.74
118.39,29.15
118.88,28.97
119.48,29.19
120.06,29.32
120.02,28.92
118.5,28.9
110.35,20.02
110.33,19.98
110.72,19.61
110.31,19.68
110.46,19.25
110.39,18.8
110.1,19.36
110,19.75
109.57,19.52
109.69,19.91
109.7,18.64
109.44,19.23
109.83,19.05
110.02,18.48
109.5,18.25
109.17,18.73
108.64,19.09
109.03,19.25
114.1,22.2
113.33,22.13
121.5,25.05
120.37,22.64
121.73,25.14
120.67,24.15
120.19,22.98
121.75,24.75
121.3,25
120.96,24.81
114.31,30.52
114.33,30.35
114.02,30.57
115.09,30.2
110.79,32.65
112.24,30.32
111.3,30.7
112.14,30.02
113.91,31.92
114.36,30.88
113.59,30.63
113.73,31.02
113.81,31.62
114.09,31.56
113.6,30.94
113.69,31.25
114.87,30.38
114.87,30.44
114.8,31.84
114.61,31.29
115,31.17
115.37,30.79
115.22,30.46
115.3,30.24
115.93,30.09
115.56,29.85
115.57,30.75
114.28,29.87
115.22,29.83
114.52,29.6
113.8,29.23
113.91,29.97
114.04,29.54
113.85,29.71
112.19,31.02
112.18,30.35
112.58,31.17
113.11,31.03
112.9,29.83
112.41,29.73
113,28.21
112.8,28.37
113.16,27.83
112.91,27.87
112.61,26.89
111.5,27.22
113.09,29.37
113.42,29.48
113.56,29.71
113.05,28.8
112.87,28.68
112.55,29.52
113.63,28.16
113.5,27.67
113.32,27.01
113.54,26.79
113.77,26.49
112.5,27.75
113,25.79
113,25.79
113.27,26.71
113.11,26.13
113.39,25.95
113.91,25.08
113.68,25.54
113.96,25.41
112.55,25.27
112.35,25.56
112.72,25.73
112.84,26.41
112.61,26.89
112.86,27.25
112.95,27.1
112.39,26.38
111.85,26.59
112.14,26.8
112.61,26.89
111.63,26.22
111.63,26.22
112.21,25.91
111.95,25.6
112.16,25.37
111.64,25.96
111.33,25.41
111.57,25.52
111.28,26.41
111.79,24.97
110.84,26.44
110.61,26.73
111.04,27.13
110.14,25.59
110.57,27.06
110.3,26.37
111.96,27.71
111.66,27.68
111.46,27.33
112.18,27.44
111.41,27.68
111.73,27.25
111.29,27.73
109.95,27.52
110.14,27.33
110.18,28.02
110.39,28.46
110.57,27.92
109.71,26.86
109.68,26.57
109.96,27.1
109.78,27.44
109.79,27.87
109.77,26.16
109.16,27.37
109.71,28.3
109.84,29
110.16,29.38
110.48,29.13
109.91,28.62
110.73,28.29
109.43,27.92
109.46,28.59
109.64,28.7
109.42,29.64
111.69,29.05
111.64,29.44
111.75,29.65
112.16,29.41
111.87,29.64
111.97,28.9
111.47,28.9
111.09,29.41
111.35,29.59
112.33,28.6
112.39,29.37
112.36,28.83
112.55,28.27
111.2,28.38
112.11,28.51
103.73,36.03
103.25,36.73
104.09,35.87
101.94,38.23
103.97,36.32
104.57,35.57
105.08,35.72
104.61,34.98
103.88,35.39
104.71,36.54
105.27,35.24
104.19,35.17
106.68,35.51
107.61,35.1
106.65,35.21
105.73,35.51
107.38,35.31
107.05,35.27
106.06,35.2
107.88,36.03
108,36.44
108.43,35.5
107.22,35.7
107.33,36.57
108.02,35.81
107.94,35.17
105.69,34.6
106.11,33.78
105.15,34.22
104.88,34.69
105.69,34.89
106.12,34.73
106.28,33.9
105.28,34.02
105.35,34.7
104.48,34.87
106.23,35
104.94,33.43
104.38,34.06
105.58,33.33
105.7,33.75
104.7,32.95
103.35,34.69
104.38,33.81
102.04,33.97
102.46,35.21
103.54,34.61
103.23,34.08
102.5,34.6
103.22,35.62
103.34,35.97
103.31,35.43
103.68,35.39
103.54,35.46
103.39,35.68
104.04,34.41
102.85,35.74
102.61,37.94
103.08,38.62
102.86,37.43
104.05,37.14
102.84,37.24
100.46,38.93
100.85,38.43
100.17,39.14
101.19,38.79
99.84,39.14
99.57,38.86
97.58,39.81
98.5,39.71
94.71,40.13
98.92,39.97
95.77,40.51
94.25,38.46
94.89,39.49
119.3,26.08
119.14,26.16
118.1,24.46
118.15,24.74
118.16,26.65
118.16,26.65
118.32,27.05
118.55,27.92
117.48,27.34
117.8,26.8
118.02,27.76
117.34,27.54
118.77,27.53
118.85,27.38
119.52,26.65
119.65,27.09
119.53,26.2
120.2,27.34
120,26.89
118.74,26.59
119.55,26.49
119.5,27.47
119.36,27.12
118.98,26.92
119.89,27.25
119,25.44
118.7,25.37
119.39,25.73
119.52,25.96
118.95,25.88
119.78,25.51
118.86,26.21
118.58,24.93
118.57,24.82
118.39,24.96
118.78,25.04
118.18,25.07
118.3,25.34
118.24,25.5
118.34,24.43
117.35,24.52
117.79,24.44
117.61,24.12
117.16,23.73
117.3,24.38
117.34,23.99
117.35,24.51
117.75,24.62
117.4,23.72
117.53,25
117.01,25.12
116.41,25.43
116.81,24.76
116.37,25.85
116.1,25.11
116.75,25.72
117.4,25.3
117.61,26.23
118.17,26.18
116.64,26.26
117.83,25.69
117.37,25.97
117.77,26.41
117.45,26.73
116.81,26.12
116.82,26.85
117.15,26.92
117.18,26.36
91.11,29.97
91.24,30.2
91.05,30.51
91.77,29.77
90.14,29.44
94.13,29.18
95.26,29.22
91.39,29.63
90.7,29.39
90.96,29.67
94.25,29.59
93.25,29.92
92.1,31.47
94.1,31.96
93.68,31.53
90.05,31.35
93.46,30.63
92.3,31.08
93.71,31.92
91.68,32.29
88.7,30.94
97.14,31.18
98.29,30.86
97.9,29.68
97.49,28.62
95.76,30.81
95.63,31.42
95.75,29.92
89.19,31.53
97.56,30.69
98.68,29.64
96.95,30.04
94.69,30.94
96.57,31.2
91.76,29.18
92.6,29.09
92.11,29.08
91.91,27.98
91.65,29.04
90.96,29.25
90.33,29.96
92,29.26
93.11,29.06
92.42,28.46
91.4,28.49
90.83,28.42
91.26,29.22
88.82,29.28
87.77,28.38
87.62,29.1
85.94,28.19
88.25,29.43
84.15,29.66
89.67,28.57
88.93,27.55
88.5,28.29
89.02,29.71
88,28.87
87.11,28.57
85.29,28.94
87.22,29.3
89.63,28.94
89.77,29.21
89.16,29.11
85.3,29.38
80,32.08
81.13,32.45
79.76,31.47
85.16,31.06
79.61,33.44
84.1,32.33
81.18,30.37
106.71,26.57
104.82,26.58
104.82,26.58
104.64,25.81
105.47,26.21
106.9,27.7
107.19,27.95
107.6,28.89
107.72,27.97
107.88,27.22
105.69,28.57
106.8,28.16
107.43,28.56
107.87,28.54
107.5,27.76
106.41,27.81
106.2,28.33
109.21,27.73
108.91,27.24
108.23,27.94
108.13,28.27
109.2,27.52
108.82,27.68
108.24,27.52
108.41,28.02
108.48,28.57
109.18,28.17
105.29,27.32
106.04,27.03
105.76,26.66
104.71,27.13
105.61,27.16
106.22,27.46
105.38,26.77
104.28,26.87
105.92,26.25
106.73,27.1
106.46,26.56
105.75,26.32
106.95,27.06
106.59,26.84
106.26,26.42
105.75,26.08
106.06,25.75
105.62,25.94
104.91,25.1
104.96,25.79
105.63,25.39
106.09,25.17
105.79,25
105.49,25.11
105.18,25.44
105.21,25.83
107.97,26.59
108.11,27.03
108.41,27.06
109.2,26.89
108.58,26.64
109.14,26.24
108.9,25.76
107.58,26.49
107.89,26.89
108.68,26.98
108.72,27.21
109.18,26.7
108.32,26.68
108.5,25.94
108.07,26.38
107.79,26.21
107.53,26.72
107.22,26.58
107.48,27.08
107.55,25.83
106.45,26.03
106.66,26.14
107.88,25.42
107.51,26.7
107.54,25.84
106.74,25.43
106.98,26.46
107.86,26
123.38,41.8
122.83,42
122.7,41.52
121.62,38.92
121.7,39.13
121.97,39.63
121.95,39.55
122.58,39.28
122.85,41.12
122.75,40.85
122.4,41.4
123.97,41.97
125.02,41.72
124.9,42.13
123.73,41.3
125.33,41.28
121.15,41.13
121.35,41.17
121.22,41.55
122.12,41.7
121.8,41.6
120.83,40.77
120.68,40.63
120.32,40.35
124.37,40.13
124.13,39.97
123.25,40.3
124.05,40.47
124.77,40.75
121.65,42
122.52,42.42
122.18,40.65
122.37,40.42
122.03,41.02
122.06,41
123.17,41.28
123.34,41.43
123.85,42.32
124.03,42.53
124.13,42.8
123.5,42.48
123.33,42.75
123.37,42.52
124.7,42.77
120.42,41.58
119.78,40.82
120.75,41.82
119.37,41.27
119.63,41.38
106.54,29.59
106.56,29.41
106.64,29.01
107.04,29.86
106.28,29.26
106.22,30.03
105.8,30.16
106.03,29.86
106.21,29.62
105.59,29.4
105.71,29.75
105.91,29.38
108.95,34.27
108.97,34.18
109.11,35.09
108.98,34.91
107.15,34.38
107.39,34.53
107.13,34.65
106.86,34.91
107.8,34.69
107.63,34.46
107.87,34.38
108.22,34.28
107.76,34.29
107.3,34.09
106.51,33.93
109.77,38.3
110.51,38.83
111.07,39.05
110.48,38.04
110.23,37.78
110.73,37.49
110.24,37.49
110.15,37.11
110.05,37.45
109.32,37.97
108.79,37.61
107.59,37.6
109.47,36.6
109.34,36.88
109.65,37.16
110.18,36.87
110.02,36.59
110.15,36.04
109.86,35.6
109.42,35.76
109.11,35.43
109.27,35.6
109.37,36
109.37,36.29
108.78,36.84
108.22,36.93
108.72,34.36
108.43,34.5
108.14,34.71
108.09,35.04
107.8,35.22
108.33,35.13
108.57,34.81
108.84,34.53
108.94,34.62
109.1,34.55
108.61,34.12
108.22,34.18
108.49,34.32
108.25,34.54
109.5,34.52
109.59,34.97
109.6,35.18
109.93,35.2
110.45,35.47
110.15,35.24
109.96,34.82
110.25,34.56
110.09,34.58
109.77,34.53
109.32,34.17
109.22,34.38
109.17,34.76
109.96,33.88
110.15,34.11
110.35,33.71
110.88,33.54
109.91,33.55
109.16,33.45
109.14,33.69
109.02,32.7
109.35,32.83
110.06,32.83
109.37,32.41
109.51,31.91
108.89,32.3
108.55,32.56
108.53,32.9
108.26,33.05
108.33,33.34
108.04,33.07
106.95,33.65
107.32,33.16
107.56,33.23
108,33.55
107.77,33
107.91,32.56
106.93,33
106.25,32.82
106.68,33.16
106.16,33.34
101.74,36.56
101.67,36.92
102.09,36.47
101.57,36.49
102.38,36.49
102.8,36.3
101.28,36.72
101.95,36.84
102.3,36.11
102.46,35.84
101.62,37.37
100.99,36.89
100.17,37.32
100.22,38.2
102,35.54
102,35.92
101.5,35.03
101.62,34.75
100.61,36.27
101.47,36.02
100.75,35.57
100.63,35.24
99.99,35.6
100.26,34.49
99.89,33.95
101.47,33.46
100.73,32.92
99.68,33.74
98.26,34.92
96.97,33.03
97.12,33.35
96.47,32.23
95.3,32.92
95.6,33.86
95.5,34.52
94.9,36.41
98.46,36.9
98.13,36.3
99.03,37.28
126.63,45.75
123.97,47.33
130.3,47.33
131.17,46.65
130.97,45.3
125.03,46.58
128.92,47.73
130,48.93
128.08,47.98
127,46.63
127.12,47.22
126.97,47.47
127.5,46.87
126.3,46.28
125.98,46.07
125.25,45.72
125.07,45.53
125.33,46.42
125.88,47.18
126.13,46.68
126.5,46.83
127.53,50.22
127.53,50.22
126.17,48.5
126.8,49.76
126.5,48.22
127.5,49.22
128.42,49.57
125.2,49.17
130.35,46.83
130.68,47.02
130.83,47.58
131.83,47.3
132.02,47.23
132.5,47.67
134.15,48.33
134,46.78
130.83,45.82
132.17,46.33
131.13,46.7
130.53,45.75
130.53,46.25
129.55,46.33
129.92,46.73
129.58,44.6
130.23,45.3
131.04,45.27
131.85,45.53
133.97,45.75
131.17,44.38
131.12,44.07
130.5,44.9
129.47,44.35
129.35,44.57
126.95,45.52
126.58,46
127.38,46.08
127.48,45.75
128.03,45.95
128.7,45.98
128.8,45.83
128.35,45.47
127.95,45.22
127.17,44.93
126.32,45.53
124.4,47.8
124.85,48.48
125.87,48.03
126.22,48.03
126.07,47.62
125.3,47.92
124.87,47.18
123.45,46.4
123.18,47.35
123.48,47.9
124.44,46.86
124.07,50.42
126.6,51.72
124.7,52.32
122.37,53.48

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大数据与机器学习 基础篇 聚类
聚类(Clustering)指的是一种学习方式,即把物理或抽象对象的集合分组为由彼此类似的对象组成的多个类的分析过程。 注:本文中用到的Python及其模块安装教程参见 K-Means算法 在聚…
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2018-05-23