迴歸分析
迴歸分析（Regression Analysis)是確定兩種或兩種以上變量間相互依賴的定量關係的一種統計分析方法 ，是一種預測性的建模技術。
線性迴歸：
簡單而言，就是將輸入項分別乘以一些常量，再將結果加起來得到輸出。線性迴歸包括一元線性迴歸和多元線性迴歸。
一元線性迴歸
線型迴歸分析中，如果僅有一個自變量與一個因變量，且其關係大致上可用一條直線表示，則稱之爲簡單迴歸分析（一元線性迴歸）。
如果發現因變量Y和自變量X之間存在高度的正相關，可以確定一條直線的方程，使得所有的數據點儘可能接近這條擬合的直線。簡單迴歸分析的模型可以用以下方程表示：Y=a+bx。其中：Y爲因變量，a爲截距，b爲相關係數，x爲自變量。
用python實現一元線性迴歸：
一個簡單的線性迴歸例子：預測房價，通過房子面積預測房子價值
假設收集到數據如下表：square_feet:平方英尺、price：價格（元/平方英尺）
| square_feet | price | |
| 1 | 150 | 6450 |
| 2 | 200 | 7450 |
| 3 | 250 | 8450 |
| 4 | 300 | 9450 |
| 5 | 350 | 11450 |
| 6 | 400 | 15450 |
| 7 | 600 | 18450 |
（1）在一元線性迴歸中，必須在數據中找出一種線性關係y(X)=a+bX。
其中y(X)是關於特定平方英尺的價格值（需要預測的值），a是一個常數，b是迴歸係數
（2）將文件保存爲CSV文件，命名爲input_data.csv（可以用Excel來做，要加上列名，要和代碼.py文件在一個目錄下）
import matplotlib.pyplot as plt
import numpy as np
import pandas as pd
from sklearn import datasets, linear_model
#讀取數據
def get_date(file_name):
data = pd.read_csv(file_name)
X_parameter = []
Y_parameter = []
#遍歷數據
for single_square_feet,single_price_feet in zip(data['square_feet'],data['price']):
X_parameter.append([float(single_square_feet)])
Y_parameter.append([float(single_price_feet)])
return X_parameter,Y_parameter
X,Y = get_date('input_data.csv')
print(X)
print(Y)
#輸出如下：
[[150.0], [200.0], [250.0], [300.0], [350.0], [400.0], [600.0]]
[[6450.0], [7450.0], [8450.0], [9450.0], [11450.0], [15450.0], [18450.0]]
（3）把X_parameter,Y_parameter擬合爲線性迴歸模型。需要寫一個函數輸入X_parameter,Y_parameter和需要進行預測的房子面積值（平方英尺值：square_feet），返回a（常數），b（迴歸係數）和預測的價格。這裏使用scikit-learn機器學習算法。
import matplotlib.pyplot as plt
import numpy as np
import pandas as pd
from sklearn import datasets, linear_model
#讀取數據
def get_date(file_name):
data = pd.read_csv(file_name)
X_parameter = []
Y_parameter = []
# 遍歷數據
for single_square_feet,single_price_feet in zip(data['square_feet'],data['price']):
X_parameter.append([float(single_square_feet)])
Y_parameter.append([float(single_price_feet)])
return X_parameter,Y_parameter
# 將數據擬合到線性模型
def linear_model_main (X_parameter,Y_parameter,predict_value):
# 創建線性迴歸對象
regr = linear_model.LinearRegression()
regr.fit(X_parameter,Y_parameter) # 用X_parameter,Y_parameter訓練模型
predice_outcome = regr.predict(predict_value) # 用predict_value（房屋面積）預測房價
predictions = {}
predictions['intercept'] = regr.intercept_ # 存儲a（截距）的值
predictions['coefficient'] = regr.coef_ # 存儲b（迴歸係數）的值
predictions['predicted_value'] = predice_outcome # 存儲y（預測的房價）的值
return predictions
X,Y = get_date('input_data.csv')
predictvalue = [[700]] # 書中此處代碼是“predictvalue = 700”，但是會報錯：ValueError: Expected 2D array, got scalar array instead:
result = linear_model_main(X,Y,predictvalue)
print(result)
print("輸出intercept的值：",result['intercept'])
print("輸出coefficient的值：",result['coefficient'])
print("輸出predicted_value的值：",result['predicted_value'])
# 輸出結果如下：
{'intercept': array([1771.80851064]), 'coefficient': array([[28.77659574]]), 'predicted_value': array([[21915.42553191]])}
輸出intercept的值： [1771.80851064] # a（截距）
輸出coefficient的值： [[28.77659574]] # b（迴歸係數）
輸出predicted_value的值： [[21915.42553191]] # 當房屋面積(x)爲700時，預測的房價(y)爲21915
# 即模型是y(x)= 1771 + 28 * X
（4）爲了驗證，需要查看數據是否擬合線性迴歸，所以需要寫一個函數，輸入X_parameter,Y_parameter，顯示數據擬合的直線
#顯示線性擬合模型結果
def show_linear_line (X_parameter,Y_parameter):
#創建線性迴歸對象
regr = linear_model.LinearRegression()
regr.fit(X_parameter, Y_parameter)
plt.scatter(X_parameter, Y_parameter,color='blue')
plt.plot(X_parameter,regr.predict(X_parameter),color='red',linewidth=4)
plt.xticks(())
plt.yticks(())
plt.show()
X,Y = get_date('input_data.csv')
show_linear_line(X,Y)
從下圖可以看出：直線基本可以擬合所有數據點
多元線性迴歸
多元線性迴歸是簡單線性迴歸的推廣，指的是多個因變量對多個自變量的迴歸。其中最常用的是只限於一個因變量但有多個自變量的情況，也叫多重回歸。多重回歸的一般形式如下：a代表截距，b1，b2，...，bk爲迴歸係數。
用python實現多元線性迴歸
當結果值的影響因素有多個時，可以採用多元線性迴歸模型
例如：商品銷售額可能與電視廣告投入、收音機廣告投入、報紙廣告投入有關係，則：
數據使用Advertising.csv（文末有）
import pandas as pd
data = pd.read_csv('Advertising.csv')
print(data.head()) # 顯示前5行數據
print(data.shape) # 顯示數據維度
# 輸出如下：
TV radio newspaper sales
0 230.1 37.8 69.2 22.1
1 44.5 39.3 45.1 10.4
2 17.2 45.9 69.3 9.3
3 151.5 41.3 58.5 18.5
4 180.8 10.8 58.4 12.9
(200, 4) # 即200行*4列
在這個案例中，通過在電視、廣播、報紙上不同的廣告投入，預測產品銷量。
使用散點圖可視化顯示特徵與響應之間的關係
import pandas as pd
import seaborn as sns
import matplotlib.pyplot as plt
# 使用散點圖可視化顯示特徵與響應之間的關係
data = pd.read_csv('Advertising.csv')
# x_vars和y_vars裏的內容要和Advertising.csv裏的列名完全一樣，否則會出現KeyError
sns.pairplot(data,x_vars=['TV','radio','newspaper'],y_vars='sales',size=7,aspect=0.8,kind='reg')
# 通過假如參數kind='reg',seaborn可以添加一條最佳擬合直線和95%的置信帶
plt.show()
運行後得到下圖：
x軸都爲0-300時，圖示應該是這樣的
可以看出，銷量和TV投入有較強的線性關係，而raido、newspaper和銷量的線性關係更弱。
線性迴歸模型
優點：快速、沒有調節參數、可輕易解釋、可理解
缺點：相比以他模型，預測準確率不高
（1）使用pandas構建X（特徵向量）和y（標籤列）
scikit-learn要求X是一個特徵矩陣，y是一個Numpy向量，pandas構建在Numpy之上。
import pandas as pd
import seaborn as sns
import matplotlib.pyplot as plt
data = pd.read_csv('Advertising.csv')
# 創建特徵向量
feature_clos = ['TV','radio','newspaper']
# 使用列表選擇原始數據DataFrame的子集
X = data[feature_clos]
X = data[['TV','radio','newspaper']]
# 從DataFrame中選擇一個Series
y = data['sales']
y = data.sales
print(X)
print(y)
#輸出X如下：
TV radio newspaper
0 230.1 37.8 69.2
1 44.5 39.3 45.1
2 17.2 45.9 69.3
3 151.5 41.3 58.5
4 180.8 10.8 58.4
.. ... ... ...
195 38.2 3.7 13.8
196 94.2 4.9 8.1
197 177.0 9.3 6.4
198 283.6 42.0 66.2
199 232.1 8.6 8.7
[200 rows x 3 columns]
#輸出y如下：
0 22.1
1 10.4
2 9.3
3 18.5
4 12.9
...
195 7.6
196 9.7
197 12.8
198 25.5
199 13.4
Name: sales, Length: 200, dtype: float64
2.構建訓練集與測試集
構建訓練集與測試集，分別保存在X_train、y_train、X_test、y_test中
import pandas as pd
import seaborn as sns
import matplotlib.pyplot as plt
data = pd.read_csv('Advertising.csv')
# 創建特徵向量
feature_clos = ['TV','radio','newspaper']
# 使用列表選擇原始數據DataFrame的子集
X = data[feature_clos]
X = data[['TV','radio','newspaper']]
# 從DataFrame中選擇一個Series
y = data['sales']
y = data.sales
from sklearn.model_selection import train_test_split
# cross_validation模塊在0.18版本中被棄用，現在已經被model_selection代替。
# 所以在導入的時候把"sklearn.cross_validation import train_test_split "
# 更改爲"from sklearn.model_selection import train_test_split"
X_train,X_test,y_train,y_test = train_test_split(X,y,random_state=1)
# 75%用於訓練，25%用於測試
print(X_train.shape)
print(X_test.shape)
print(y_train.shape)
print(y_test.shape)
# 輸出結果如下：
(150, 3)
(50, 3)
(150,)
(50,)
3.sklearn的線性迴歸
使用sklearn做線性迴歸
import pandas as pd
import seaborn as sns
import matplotlib.pyplot as plt
data = pd.read_csv('Advertising.csv')
# 創建特徵向量
feature_clos = ['TV','radio','newspaper']
# 使用列表選擇原始數據DataFrame的子集
X = data[feature_clos]
X = data[['TV','radio','newspaper']]
# 從DataFrame中選擇一個Series
y = data['sales']
y = data.sales
from sklearn.model_selection import train_test_split
# cross_validation模塊在0.18版本中被棄用，現在已經被model_selection代替。
# 所以在導入的時候把"sklearn.cross_validation import train_test_split "
# 更改爲"from sklearn.model_selection import train_test_split"
X_train,X_test,y_train,y_test = train_test_split(X,y,random_state=1)
from sklearn.linear_model import LinearRegression
linreg = LinearRegression()
model = linreg.fit(X_train,y_train) # 線性迴歸
print(model)
print(linreg.intercept_) # 截距
print(linreg.coef_) # 係數
# 輸出結果如下：
LinearRegression(copy_X=True, fit_intercept=True, n_jobs=None, normalize=False)
2.8769666223179335
[0.04656457 0.17915812 0.00345046]
所以線性迴歸結果如下：
y=2.8769666223179335+0.04656457*TV+0.17915812*radio+0.00345046*newspaper
4.預測
通過線性模擬求出迴歸模型後，可通過模型預測數據，通過predict函數即可求出預測結果
import pandas as pd
import seaborn as sns
import matplotlib.pyplot as plt
data = pd.read_csv('Advertising.csv')
# 創建特徵向量
feature_clos = ['TV','radio','newspaper']
# 使用列表選擇原始數據DataFrame的子集
X = data[feature_clos]
X = data[['TV','radio','newspaper']]
# 從DataFrame中選擇一個Series
y = data['sales']
y = data.sales
from sklearn.model_selection import train_test_split
# cross_validation模塊在0.18版本中被棄用，現在已經被model_selection代替。
# 所以在導入的時候把"sklearn.cross_validation import train_test_split "
# 更改爲"from sklearn.model_selection import train_test_split"
X_train,X_test,y_train,y_test = train_test_split(X,y,random_state=1)
from sklearn.linear_model import LinearRegression
linreg = LinearRegression()
model = linreg.fit(X_train,y_train) # 線性迴歸
y_pred = linreg.predict(X_test)
print(y_pred)
#輸出結果如下：
[21.70910292 16.41055243 7.60955058 17.80769552 18.6146359 23.83573998
16.32488681 13.43225536 9.17173403 17.333853 14.44479482 9.83511973
17.18797614 16.73086831 15.05529391 15.61434433 12.42541574 17.17716376
11.08827566 18.00537501 9.28438889 12.98458458 8.79950614 10.42382499
11.3846456 14.98082512 9.78853268 19.39643187 18.18099936 17.12807566
21.54670213 14.69809481 16.24641438 12.32114579 19.92422501 15.32498602
13.88726522 10.03162255 20.93105915 7.44936831 3.64695761 7.22020178
5.9962782 18.43381853 8.39408045 14.08371047 15.02195699 20.35836418
20.57036347 19.60636679]
5.評價測度
對於分類問題，評價測度是準確率，但不適用於迴歸問題，因此使用針對連續數值的評價測度
這裏介紹3中常用針對線性迴歸的評價測度：
平均絕對誤差（Mean Absolute Error ，MAE）
即平均絕對值誤差，它表示預測值和觀測值之間絕對誤差的平均值
（絕對誤差：測量值與真實值之差）
平均絕對誤差MAD=1/(N)*(|x1-xm|+|x2-xm|+..+|xN-xm|)
平均值：xm
均方誤差（mean-square error, MSE）
是指參數估計值與參數真值之差平方的期望值
均方誤差可以評價數據的變化程度，MSE的值越小，說明預測模型描述實驗數據具有更好的精確度。
均方根誤差（Root Mean Square Error，RMSE）
均方根誤差是均方誤差的算術平方根
案例1：真實值= [2,4,6,8]，預測值= [4,6,8,10]
案例2：真實值= [2,4,6,8]，預測值= [4,6,8,12]
案例1的MAE = 2.0，RMSE = 2.0
MAE =1/(4)*[|(2-4)|+|(4-6)|+|(6-8)|+|(8-10)|]
RMSE =sqrt{1/(4)*[sqr(2-4)+sqr(4-6)+sqr(6-8)+sqr(8-10)}
案例2的MAE = 2.5，RMSE = 2.65
MAE =1/(4)*[|(2-4)|+|(4-6)|+|(6-8)|+|(8-12)|]
RMSE =sqrt{1/(4)*[sqr(2-4)+sqr(4-6)+sqr(6-8)+sqr(8-12)}
這裏使用RMES進行評價測度
import pandas as pd
import seaborn as sns
import matplotlib.pyplot as plt
data = pd.read_csv('Advertising.csv')
# 創建特徵向量
feature_clos = ['TV','radio','newspaper']
# 使用列表選擇原始數據DataFrame的子集
X = data[feature_clos]
X = data[['TV','radio','newspaper']]
# 從DataFrame中選擇一個Series
y = data['sales']
y = data.sales
from sklearn.model_selection import train_test_split
# cross_validation模塊在0.18版本中被棄用，現在已經被model_selection代替。
# 所以在導入的時候把"sklearn.cross_validation import train_test_split "
# 更改爲"from sklearn.model_selection import train_test_split"
X_train,X_test,y_train,y_test = train_test_split(X,y,random_state=1)
from sklearn.linear_model import LinearRegression
linreg = LinearRegression()
model = linreg.fit(X_train,y_train) # 線性迴歸
y_pred = linreg.predict(X_test)
# 計算sales預測的RMSE
from sklearn import metrics
import numpy as np
sum_mean = 0
for i in range(len(y_pred)):
sum_mean += (y_pred[i] - y_test.iloc[i])**2 # 應用y_test.iloc[i]替換y_test.values[i]
sum_erro = np.sqrt(sum_mean/50)
# 輸出RMSE
print("輸出RMSE:",sum_erro)
# 輸出結果如下：
輸出RMSE: 1.4046514230328953
接下來繪製ROC曲線，代碼如下：
import pandas as pd
import seaborn as sns
import matplotlib.pyplot as plt
data = pd.read_csv('Advertising.csv')
# 創建特徵向量
feature_clos = ['TV','radio','newspaper']
# 使用列表選擇原始數據DataFrame的子集
X = data[feature_clos]
X = data[['TV','radio','newspaper']]
# 從DataFrame中選擇一個Series
y = data['sales']
y = data.sales
from sklearn.model_selection import train_test_split
# cross_validation模塊在0.18版本中被棄用，現在已經被model_selection代替。
# 所以在導入的時候把"sklearn.cross_validation import train_test_split "
# 更改爲"from sklearn.model_selection import train_test_split"
X_train,X_test,y_train,y_test = train_test_split(X,y,random_state=1)
from sklearn.linear_model import LinearRegression
linreg = LinearRegression()
model = linreg.fit(X_train,y_train) # 線性迴歸
y_pred = linreg.predict(X_test)
# 計算sales預測的RMSE
from sklearn import metrics
import numpy as np
sum_mean = 0
for i in range(len(y_pred)):
sum_mean += (y_pred[i] - y_test.iloc[i])**2 # 應用y_test.iloc[i]替換y_test.values[i]
sum_erro = np.sqrt(sum_mean/50)
import matplotlib.pyplot as plt
plt.figure()
plt.plot(range(len(y_pred)),y_pred,'b',label="predict")
plt.plot(range(len(y_pred)),y_test,'r',label="test")
plt.legend(loc="upper right")
plt.xlabel("the number of sales")
plt.ylabel("value of sales")
plt.show()
上面的曲線是真是的曲線，下面的曲線是預測的曲線
至此，一次多元線性迴歸預測結束
可以看下這個實例：
基於線性迴歸的股票預測
#附 advertising.csv（數據本身沒有第一列（序號），CSV文件裏應當刪除第一列）
TV,radio,newspaper,sales
1,230.1,37.8,69.2,22.1
2,44.5,39.3,45.1,10.4
3,17.2,45.9,69.3,9.3
4,151.5,41.3,58.5,18.5
5,180.8,10.8,58.4,12.9
6,8.7,48.9,75,7.2
7,57.5,32.8,23.5,11.8
8,120.2,19.6,11.6,13.2
9,8.6,2.1,1,4.8
10,199.8,2.6,21.2,10.6
11,66.1,5.8,24.2,8.6
12,214.7,24,4,17.4
13,23.8,35.1,65.9,9.2
14,97.5,7.6,7.2,9.7
15,204.1,32.9,46,19
16,195.4,47.7,52.9,22.4
17,67.8,36.6,114,12.5
18,281.4,39.6,55.8,24.4
19,69.2,20.5,18.3,11.3
20,147.3,23.9,19.1,14.6
21,218.4,27.7,53.4,18
22,237.4,5.1,23.5,12.5
23,13.2,15.9,49.6,5.6
24,228.3,16.9,26.2,15.5
25,62.3,12.6,18.3,9.7
26,262.9,3.5,19.5,12
27,142.9,29.3,12.6,15
28,240.1,16.7,22.9,15.9
29,248.8,27.1,22.9,18.9
30,70.6,16,40.8,10.5
31,292.9,28.3,43.2,21.4
32,112.9,17.4,38.6,11.9
33,97.2,1.5,30,9.6
34,265.6,20,0.3,17.4
35,95.7,1.4,7.4,9.5
36,290.7,4.1,8.5,12.8
37,266.9,43.8,5,25.4
38,74.7,49.4,45.7,14.7
39,43.1,26.7,35.1,10.1
40,228,37.7,32,21.5
41,202.5,22.3,31.6,16.6
42,177,33.4,38.7,17.1
43,293.6,27.7,1.8,20.7
44,206.9,8.4,26.4,12.9
45,25.1,25.7,43.3,8.5
46,175.1,22.5,31.5,14.9
47,89.7,9.9,35.7,10.6
48,239.9,41.5,18.5,23.2
49,227.2,15.8,49.9,14.8
50,66.9,11.7,36.8,9.7
51,199.8,3.1,34.6,11.4
52,100.4,9.6,3.6,10.7
53,216.4,41.7,39.6,22.6
54,182.6,46.2,58.7,21.2
55,262.7,28.8,15.9,20.2
56,198.9,49.4,60,23.7
57,7.3,28.1,41.4,5.5
58,136.2,19.2,16.6,13.2
59,210.8,49.6,37.7,23.8
60,210.7,29.5,9.3,18.4
61,53.5,2,21.4,8.1
62,261.3,42.7,54.7,24.2
63,239.3,15.5,27.3,15.7
64,102.7,29.6,8.4,14
65,131.1,42.8,28.9,18
66,69,9.3,0.9,9.3
67,31.5,24.6,2.2,9.5
68,139.3,14.5,10.2,13.4
69,237.4,27.5,11,18.9
70,216.8,43.9,27.2,22.3
71,199.1,30.6,38.7,18.3
72,109.8,14.3,31.7,12.4
73,26.8,33,19.3,8.8
74,129.4,5.7,31.3,11
75,213.4,24.6,13.1,17
76,16.9,43.7,89.4,8.7
77,27.5,1.6,20.7,6.9
78,120.5,28.5,14.2,14.2
79,5.4,29.9,9.4,5.3
80,116,7.7,23.1,11
81,76.4,26.7,22.3,11.8
82,239.8,4.1,36.9,12.3
83,75.3,20.3,32.5,11.3
84,68.4,44.5,35.6,13.6
85,213.5,43,33.8,21.7
86,193.2,18.4,65.7,15.2
87,76.3,27.5,16,12
88,110.7,40.6,63.2,16
89,88.3,25.5,73.4,12.9
90,109.8,47.8,51.4,16.7
91,134.3,4.9,9.3,11.2
92,28.6,1.5,33,7.3
93,217.7,33.5,59,19.4
94,250.9,36.5,72.3,22.2
95,107.4,14,10.9,11.5
96,163.3,31.6,52.9,16.9
97,197.6,3.5,5.9,11.7
98,184.9,21,22,15.5
99,289.7,42.3,51.2,25.4
100,135.2,41.7,45.9,17.2
101,222.4,4.3,49.8,11.7
102,296.4,36.3,100.9,23.8
103,280.2,10.1,21.4,14.8
104,187.9,17.2,17.9,14.7
105,238.2,34.3,5.3,20.7
106,137.9,46.4,59,19.2
107,25,11,29.7,7.2
108,90.4,0.3,23.2,8.7
109,13.1,0.4,25.6,5.3
110,255.4,26.9,5.5,19.8
111,225.8,8.2,56.5,13.4
112,241.7,38,23.2,21.8
113,175.7,15.4,2.4,14.1
114,209.6,20.6,10.7,15.9
115,78.2,46.8,34.5,14.6
116,75.1,35,52.7,12.6
117,139.2,14.3,25.6,12.2
118,76.4,0.8,14.8,9.4
119,125.7,36.9,79.2,15.9
120,19.4,16,22.3,6.6
121,141.3,26.8,46.2,15.5
122,18.8,21.7,50.4,7
123,224,2.4,15.6,11.6
124,123.1,34.6,12.4,15.2
125,229.5,32.3,74.2,19.7
126,87.2,11.8,25.9,10.6
127,7.8,38.9,50.6,6.6
128,80.2,0,9.2,8.8
129,220.3,49,3.2,24.7
130,59.6,12,43.1,9.7
131,0.7,39.6,8.7,1.6
132,265.2,2.9,43,12.7
133,8.4,27.2,2.1,5.7
134,219.8,33.5,45.1,19.6
135,36.9,38.6,65.6,10.8
136,48.3,47,8.5,11.6
137,25.6,39,9.3,9.5
138,273.7,28.9,59.7,20.8
139,43,25.9,20.5,9.6
140,184.9,43.9,1.7,20.7
141,73.4,17,12.9,10.9
142,193.7,35.4,75.6,19.2
143,220.5,33.2,37.9,20.1
144,104.6,5.7,34.4,10.4
145,96.2,14.8,38.9,11.4
146,140.3,1.9,9,10.3
147,240.1,7.3,8.7,13.2
148,243.2,49,44.3,25.4
149,38,40.3,11.9,10.9
150,44.7,25.8,20.6,10.1
151,280.7,13.9,37,16.1
152,121,8.4,48.7,11.6
153,197.6,23.3,14.2,16.6
154,171.3,39.7,37.7,19
155,187.8,21.1,9.5,15.6
156,4.1,11.6,5.7,3.2
157,93.9,43.5,50.5,15.3
158,149.8,1.3,24.3,10.1
159,11.7,36.9,45.2,7.3
160,131.7,18.4,34.6,12.9
161,172.5,18.1,30.7,14.4
162,85.7,35.8,49.3,13.3
163,188.4,18.1,25.6,14.9
164,163.5,36.8,7.4,18
165,117.2,14.7,5.4,11.9
166,234.5,3.4,84.8,11.9
167,17.9,37.6,21.6,8
168,206.8,5.2,19.4,12.2
169,215.4,23.6,57.6,17.1
170,284.3,10.6,6.4,15
171,50,11.6,18.4,8.4
172,164.5,20.9,47.4,14.5
173,19.6,20.1,17,7.6
174,168.4,7.1,12.8,11.7
175,222.4,3.4,13.1,11.5
176,276.9,48.9,41.8,27
177,248.4,30.2,20.3,20.2
178,170.2,7.8,35.2,11.7
179,276.7,2.3,23.7,11.8
180,165.6,10,17.6,12.6
181,156.6,2.6,8.3,10.5
182,218.5,5.4,27.4,12.2
183,56.2,5.7,29.7,8.7
184,287.6,43,71.8,26.2
185,253.8,21.3,30,17.6
186,205,45.1,19.6,22.6
187,139.5,2.1,26.6,10.3
188,191.1,28.7,18.2,17.3
189,286,13.9,3.7,15.9
190,18.7,12.1,23.4,6.7
191,39.5,41.1,5.8,10.8
192,75.5,10.8,6,9.9
193,17.2,4.1,31.6,5.9
194,166.8,42,3.6,19.6
195,149.7,35.6,6,17.3
196,38.2,3.7,13.8,7.6
197,94.2,4.9,8.1,9.7
198,177,9.3,6.4,12.8
199,283.6,42,66.2,25.5
200,232.1,8.6,8.7,13.4