LoginSignup
1
2

More than 5 years have passed since last update.

@maskot1977(Ikemen Mas Kot)

曲線回帰を本当はPythonで解きたいけど表計算で解けと言われたので

Posted at

線形回帰を本当はPythonで解きたいけど表計算で解けと言われたので と 線形重回帰を本当はPythonで解きたいけど表計算で解けと言われたので の続編です。前の記事を読んでもらっていれば、曲線に回帰するのは簡単。

用いるデータ

分子性物質のデータ（融点・沸点）

import pandas as pd
data = [['HF', 19.5, 20.0],
        ['HCl', -84.9, 36.5],
        ['HBr', -67.0, 80.9],
        ['HI', -35.1, 127.9],
        ['H2O', 100.0, 18.0],
        ['H2S', -60.7, 34.1],
        ['H2Se', -42, 81.0],
        ['H2Te', -1.8, 129.6],
        ['NH3', -33.4, 17.0],
        ['PH3', -87, 34.0],
        ['AsH3', -55, 77.9],
        ['SbH3', -17.1, 124.8],
        ['CH4', -161.49, 16.0],
        ['SiH4', -111.8, 32.1],
        ['GeH4', -90, 76.6],
        ['SnH4', -52, 122.7],
        ['He', -268.934, 4.0],
        ['Ne', -246.048, 20.2],
        ['Ar', -185.7, 39.9],
        ['Kr', -152.3, 83.8],
        ['Xe', -108.1, 131.3],
       ]
df = pd.DataFrame(data, columns = ['molecule', 'boiling point', 'molecular weight'])
df
molecule boiling point molecular weight
0 HF 19.500 20.0
1 HCl -84.900 36.5
2 HBr -67.000 80.9
3 HI -35.100 127.9
4 H2O 100.000 18.0
5 H2S -60.700 34.1
6 H2Se -42.000 81.0
7 H2Te -1.800 129.6
8 NH3 -33.400 17.0
9 PH3 -87.000 34.0
10 AsH3 -55.000 77.9
11 SbH3 -17.100 124.8
12 CH4 -161.490 16.0
13 SiH4 -111.800 32.1
14 GeH4 -90.000 76.6
15 SnH4 -52.000 122.7
16 He -268.934 4.0
17 Ne -246.048 20.2
18 Ar -185.700 39.9
19 Kr -152.300 83.8
20 Xe -108.100 131.3 
X = df.loc[:, ['molecular weight']].as_matrix()
X

array([[  20. ],
       [  36.5],
       [  80.9],
       [ 127.9],
       [  18. ],
       [  34.1],
       [  81. ],
       [ 129.6],
       [  17. ],
       [  34. ],
       [  77.9],
       [ 124.8],
       [  16. ],
       [  32.1],
       [  76.6],
       [ 122.7],
       [   4. ],
       [  20.2],
       [  39.9],
       [  83.8],
       [ 131.3]])

Y = df['boiling point'].as_matrix()
Y

array([  19.5  ,  -84.9  ,  -67.   ,  -35.1  ,  100.   ,  -60.7  ,
        -42.   ,   -1.8  ,  -33.4  ,  -87.   ,  -55.   ,  -17.1  ,
       -161.49 , -111.8  ,  -90.   ,  -52.   , -268.934, -246.048,
       -185.7  , -152.3  , -108.1  ])

%matplotlib inline
import matplotlib.pyplot as plt
# 散布図
plt.figure(figsize=(8,6))
plt.scatter(X, Y, alpha=0.3)
for name, x, y in zip(df.loc[:, ['molecule']].as_matrix(), X, Y):
    plt.text(x, y, name[0], size=8)
plt.xlabel('molecular weight')
plt.ylabel('boiling point')
plt.grid()
plt.show()

曲線回帰を本当はPythonで解きたいけど表計算で解けと言われたので_4_0.png

import math
import numpy as np
logX = np.array([[math.log(x)] for x in X[:,0]])

logX

array([[ 2.99573227],
       [ 3.59731226],
       [ 4.39321382],
       [ 4.85124871],
       [ 2.89037176],
       [ 3.52929738],
       [ 4.39444915],
       [ 4.86445278],
       [ 2.83321334],
       [ 3.52636052],
       [ 4.35542595],
       [ 4.82671246],
       [ 2.77258872],
       [ 3.46885603],
       [ 4.33859708],
       [ 4.80974235],
       [ 1.38629436],
       [ 3.0056826 ],
       [ 3.68637632],
       [ 4.42843301],
       [ 4.87748478]])

まずは一番便利な scikit-learn から

from sklearn import linear_model
lr = linear_model.LinearRegression()

lr.fit(logX, Y)

LinearRegression(copy_X=True, fit_intercept=True, n_jobs=1, normalize=False)

# 回帰係数
lr.coef_

array([ 33.87205496])

# 切片
lr.intercept_

-211.66384118386563

print("y = f(x) = wlogx + t; (w, t) = ({0}, {1})".format(lr.coef_[0], lr.intercept_))

y = f(x) = wlogx + t; (w, t) = (33.872054963641745, -211.66384118386563)

# 決定係数R2
lr.score(logX, Y)

0.13487033631665779

%matplotlib inline
import matplotlib.pyplot as plt
# 散布図
plt.figure(figsize=(5,4))
plt.scatter(X, Y, alpha=0.3)

# 回帰直線
plt.plot(sorted(X), sorted(lr.predict(logX)))
for name, x, y in zip(df.loc[:, ['molecule']].as_matrix(), X, Y):
    plt.text(x, y, name[0], size=8)
plt.xlabel('molecular weight')
plt.ylabel('boiling point')
plt.grid()
plt.show()

曲線回帰を本当はPythonで解きたいけど表計算で解けと言われたので_14_0.png

次は、ガチPythonで。

# 平均値を求める関数
def mean(list):
    sum = 0
    for x in list:
        sum += x
    return sum / len(list)

# 分散を求める関数
def variance(list):
    ave = mean(list)
    sum = 0
    for x in list:
        sum += (x - ave) ** 2
    return sum / len(list)

# 標準偏差を求める関数
import math
def standard_deviation(list):
    return math.sqrt(variance(list))

# 共分散 = 偏差積の平均
def covariance(list1, list2): 
    mean1 = mean(list1)
    mean2 = mean(list2)
    sum = 0
    for d1, d2 in zip(list1, list2):
        sum += (d1 - mean1) * (d2 - mean2)
    return sum / len(list1)

# 相関係数 = 共分散を list1, list2 の標準偏差で割ったもの
def correlation(list1, list2):
    return covariance(list1, list2) / (standard_deviation(list1) * standard_deviation(list2))

# 回帰直線の傾き＝相関係数＊（（yの標準偏差）／（xの標準偏差））
def w_fit(xlist, ylist):
    return correlation(xlist, ylist) * standard_deviation(ylist) / standard_deviation(xlist)

# y切片＝yの平均－（傾き＊xの平均）
def t_fit(xlist, ylist):
    return mean(ylist) - w_fit(xlist, ylist) * mean(xlist)

# 回帰直線の式を表示
w = w_fit(logX, Y)
t = t_fit(logX, Y)
print("y = f(x) = wlogx + t; (w, t) = ({0}, {1})".format(w, t))

y = f(x) = wlogx + t; (w, t) = ([ 33.87205496], [-211.66384118])

# 回帰直線の式を関数として表現
def f(x):
    return w * x + t

# 決定係数R2
def r2(xlist, ylist):
    wa1 = 0.
    wa2 = 0.
    for x, y in zip(xlist, ylist):
        wa1 += (y - f(x))**2
        wa2 += (y - mean(ylist))**2
    return 1. - wa1 / wa2

r2(logX, Y)

array([ 0.13487034])

さて、表計算で解けと言われたので pandas で書いてみましょうか。

import copy
from IPython.display import display
excel = copy.deepcopy(df)
excel
molecule boiling point molecular weight
0 HF 19.500 20.0
1 HCl -84.900 36.5
2 HBr -67.000 80.9
3 HI -35.100 127.9
4 H2O 100.000 18.0
5 H2S -60.700 34.1
6 H2Se -42.000 81.0
7 H2Te -1.800 129.6
8 NH3 -33.400 17.0
9 PH3 -87.000 34.0
10 AsH3 -55.000 77.9
11 SbH3 -17.100 124.8
12 CH4 -161.490 16.0
13 SiH4 -111.800 32.1
14 GeH4 -90.000 76.6
15 SnH4 -52.000 122.7
16 He -268.934 4.0
17 Ne -246.048 20.2
18 Ar -185.700 39.9
19 Kr -152.300 83.8
20 Xe -108.100 131.3 
excel['y'] = excel['boiling point']
excel['x'] = excel['molecular weight']
excel['log(x)'] = [math.log(x) for x in excel['x']]
mean_y = mean(excel['y'])
mean_logx = mean(excel['log(x)'])
display(excel, pd.DataFrame([[mean_y, mean_logx]], columns=['y','log(x)'], index=['mean']))
molecule boiling point molecular weight y x log(x)
0 HF 19.500 20.0 19.500 20.0 2.995732
1 HCl -84.900 36.5 -84.900 36.5 3.597312
2 HBr -67.000 80.9 -67.000 80.9 4.393214
3 HI -35.100 127.9 -35.100 127.9 4.851249
4 H2O 100.000 18.0 100.000 18.0 2.890372
5 H2S -60.700 34.1 -60.700 34.1 3.529297
6 H2Se -42.000 81.0 -42.000 81.0 4.394449
7 H2Te -1.800 129.6 -1.800 129.6 4.864453
8 NH3 -33.400 17.0 -33.400 17.0 2.833213
9 PH3 -87.000 34.0 -87.000 34.0 3.526361
10 AsH3 -55.000 77.9 -55.000 77.9 4.355426
11 SbH3 -17.100 124.8 -17.100 124.8 4.826712
12 CH4 -161.490 16.0 -161.490 16.0 2.772589
13 SiH4 -111.800 32.1 -111.800 32.1 3.468856
14 GeH4 -90.000 76.6 -90.000 76.6 4.338597
15 SnH4 -52.000 122.7 -52.000 122.7 4.809742
16 He -268.934 4.0 -268.934 4.0 1.386294
17 Ne -246.048 20.2 -246.048 20.2 3.005683
18 Ar -185.700 39.9 -185.700 39.9 3.686376
19 Kr -152.300 83.8 -152.300 83.8 4.428433
20 Xe -108.100 131.3 -108.100 131.3 4.877485
y log(x)
mean -82.898667 3.801516 
excel['y-mean(y)'] = [y - mean_y for y in excel['y']]
excel['log(x)-mean(log(x))'] = [logx - mean_logx for logx in excel['log(x)']]
display(excel, pd.DataFrame([[mean_y, mean_logx]], columns=['y','log(x)'], index=['mean']))
molecule boiling point molecular weight y x log(x) y-mean(y) log(x)-mean(log(x))
0 HF 19.500 20.0 19.500 20.0 2.995732 102.398667 -0.805784
1 HCl -84.900 36.5 -84.900 36.5 3.597312 -2.001333 -0.204204
2 HBr -67.000 80.9 -67.000 80.9 4.393214 15.898667 0.591697
3 HI -35.100 127.9 -35.100 127.9 4.851249 47.798667 1.049732
4 H2O 100.000 18.0 100.000 18.0 2.890372 182.898667 -0.911145
5 H2S -60.700 34.1 -60.700 34.1 3.529297 22.198667 -0.272219
6 H2Se -42.000 81.0 -42.000 81.0 4.394449 40.898667 0.592933
7 H2Te -1.800 129.6 -1.800 129.6 4.864453 81.098667 1.062936
8 NH3 -33.400 17.0 -33.400 17.0 2.833213 49.498667 -0.968303
9 PH3 -87.000 34.0 -87.000 34.0 3.526361 -4.101333 -0.275156
10 AsH3 -55.000 77.9 -55.000 77.9 4.355426 27.898667 0.553909
11 SbH3 -17.100 124.8 -17.100 124.8 4.826712 65.798667 1.025196
12 CH4 -161.490 16.0 -161.490 16.0 2.772589 -78.591333 -1.028928
13 SiH4 -111.800 32.1 -111.800 32.1 3.468856 -28.901333 -0.332660
14 GeH4 -90.000 76.6 -90.000 76.6 4.338597 -7.101333 0.537081
15 SnH4 -52.000 122.7 -52.000 122.7 4.809742 30.898667 1.008226
16 He -268.934 4.0 -268.934 4.0 1.386294 -186.035333 -2.415222
17 Ne -246.048 20.2 -246.048 20.2 3.005683 -163.149333 -0.795834
18 Ar -185.700 39.9 -185.700 39.9 3.686376 -102.801333 -0.115140
19 Kr -152.300 83.8 -152.300 83.8 4.428433 -69.401333 0.626917
20 Xe -108.100 131.3 -108.100 131.3 4.877485 -25.201333 1.075968
y log(x)
mean -82.898667 3.801516 
excel['(y-mean(y))**2'] = [sa ** 2 for sa in excel['y-mean(y)']]
excel['(log(x)-mean(log(x)))**2'] = [sa ** 2 for sa in excel['log(x)-mean(log(x))']]
display(excel, pd.DataFrame([[mean_y, mean_logx]], columns=['y','log(x)'], index=['mean']))
molecule boiling point molecular weight y x log(x) y-mean(y) log(x)-mean(log(x)) (y-mean(y))**2 (log(x)-mean(log(x)))**2
0 HF 19.500 20.0 19.500 20.0 2.995732 102.398667 -0.805784 10485.486935 0.649288
1 HCl -84.900 36.5 -84.900 36.5 3.597312 -2.001333 -0.204204 4.005335 0.041699
2 HBr -67.000 80.9 -67.000 80.9 4.393214 15.898667 0.591697 252.767602 0.350106
3 HI -35.100 127.9 -35.100 127.9 4.851249 47.798667 1.049732 2284.712535 1.101938
4 H2O 100.000 18.0 100.000 18.0 2.890372 182.898667 -0.911145 33451.922268 0.830185
5 H2S -60.700 34.1 -60.700 34.1 3.529297 22.198667 -0.272219 492.780802 0.074103
6 H2Se -42.000 81.0 -42.000 81.0 4.394449 40.898667 0.592933 1672.700935 0.351569
7 H2Te -1.800 129.6 -1.800 129.6 4.864453 81.098667 1.062936 6576.993735 1.129834
8 NH3 -33.400 17.0 -33.400 17.0 2.833213 49.498667 -0.968303 2450.118002 0.937611
9 PH3 -87.000 34.0 -87.000 34.0 3.526361 -4.101333 -0.275156 16.820935 0.075711
10 AsH3 -55.000 77.9 -55.000 77.9 4.355426 27.898667 0.553909 778.335602 0.306816
11 SbH3 -17.100 124.8 -17.100 124.8 4.826712 65.798667 1.025196 4329.464535 1.051027
12 CH4 -161.490 16.0 -161.490 16.0 2.772589 -78.591333 -1.028928 6176.597675 1.058692
13 SiH4 -111.800 32.1 -111.800 32.1 3.468856 -28.901333 -0.332660 835.287068 0.110663
14 GeH4 -90.000 76.6 -90.000 76.6 4.338597 -7.101333 0.537081 50.428935 0.288456
15 SnH4 -52.000 122.7 -52.000 122.7 4.809742 30.898667 1.008226 954.727602 1.016519
16 He -268.934 4.0 -268.934 4.0 1.386294 -186.035333 -2.415222 34609.145248 5.833298
17 Ne -246.048 20.2 -246.048 20.2 3.005683 -163.149333 -0.795834 26617.704967 0.633352
18 Ar -185.700 39.9 -185.700 39.9 3.686376 -102.801333 -0.115140 10568.114135 0.013257
19 Kr -152.300 83.8 -152.300 83.8 4.428433 -69.401333 0.626917 4816.545068 0.393024
20 Xe -108.100 131.3 -108.100 131.3 4.877485 -25.201333 1.075968 635.107202 1.157708
y log(x)
mean -82.898667 3.801516 
variance_y = mean(excel['(y-mean(y))**2'])
variance_logx = mean(excel['(log(x)-mean(log(x)))**2'])
sd_y = math.sqrt(variance_y)
sd_logx = math.sqrt(variance_logx)
display(excel, pd.DataFrame([[mean_y, mean_logx], [variance_y, variance_logx], [sd_y, sd_logx]], 
                            columns=['y','log(x)'], index=['mean', 'variance', 'sd']))
molecule boiling point molecular weight y x log(x) y-mean(y) log(x)-mean(log(x)) (y-mean(y))**2 (log(x)-mean(log(x)))**2
0 HF 19.500 20.0 19.500 20.0 2.995732 102.398667 -0.805784 10485.486935 0.649288
1 HCl -84.900 36.5 -84.900 36.5 3.597312 -2.001333 -0.204204 4.005335 0.041699
2 HBr -67.000 80.9 -67.000 80.9 4.393214 15.898667 0.591697 252.767602 0.350106
3 HI -35.100 127.9 -35.100 127.9 4.851249 47.798667 1.049732 2284.712535 1.101938
4 H2O 100.000 18.0 100.000 18.0 2.890372 182.898667 -0.911145 33451.922268 0.830185
5 H2S -60.700 34.1 -60.700 34.1 3.529297 22.198667 -0.272219 492.780802 0.074103
6 H2Se -42.000 81.0 -42.000 81.0 4.394449 40.898667 0.592933 1672.700935 0.351569
7 H2Te -1.800 129.6 -1.800 129.6 4.864453 81.098667 1.062936 6576.993735 1.129834
8 NH3 -33.400 17.0 -33.400 17.0 2.833213 49.498667 -0.968303 2450.118002 0.937611
9 PH3 -87.000 34.0 -87.000 34.0 3.526361 -4.101333 -0.275156 16.820935 0.075711
10 AsH3 -55.000 77.9 -55.000 77.9 4.355426 27.898667 0.553909 778.335602 0.306816
11 SbH3 -17.100 124.8 -17.100 124.8 4.826712 65.798667 1.025196 4329.464535 1.051027
12 CH4 -161.490 16.0 -161.490 16.0 2.772589 -78.591333 -1.028928 6176.597675 1.058692
13 SiH4 -111.800 32.1 -111.800 32.1 3.468856 -28.901333 -0.332660 835.287068 0.110663
14 GeH4 -90.000 76.6 -90.000 76.6 4.338597 -7.101333 0.537081 50.428935 0.288456
15 SnH4 -52.000 122.7 -52.000 122.7 4.809742 30.898667 1.008226 954.727602 1.016519
16 He -268.934 4.0 -268.934 4.0 1.386294 -186.035333 -2.415222 34609.145248 5.833298
17 Ne -246.048 20.2 -246.048 20.2 3.005683 -163.149333 -0.795834 26617.704967 0.633352
18 Ar -185.700 39.9 -185.700 39.9 3.686376 -102.801333 -0.115140 10568.114135 0.013257
19 Kr -152.300 83.8 -152.300 83.8 4.428433 -69.401333 0.626917 4816.545068 0.393024
20 Xe -108.100 131.3 -108.100 131.3 4.877485 -25.201333 1.075968 635.107202 1.157708
y log(x)
mean -82.898667 3.801516
variance 7050.465101 0.828803
sd 83.967048 0.910386 
excel['(y-mean(y)) * (log(x)-mean(log(x)))'] = excel['y-mean(y)'] * excel['log(x)-mean(log(x))']
display(excel, pd.DataFrame([[mean_y, mean_logx], [variance_y, variance_logx], [sd_y, sd_logx]], 
                            columns=['y','log(x)'], index=['mean', 'variance', 'sd']))
molecule boiling point molecular weight y x log(x) y-mean(y) log(x)-mean(log(x)) (y-mean(y))**2 (log(x)-mean(log(x)))**2 (y-mean(y)) * (log(x)-mean(log(x)))
0 HF 19.500 20.0 19.500 20.0 2.995732 102.398667 -0.805784 10485.486935 0.649288 -82.511226
1 HCl -84.900 36.5 -84.900 36.5 3.597312 -2.001333 -0.204204 4.005335 0.041699 0.408681
2 HBr -67.000 80.9 -67.000 80.9 4.393214 15.898667 0.591697 252.767602 0.350106 9.407199
3 HI -35.100 127.9 -35.100 127.9 4.851249 47.798667 1.049732 2284.712535 1.101938 50.175802
4 H2O 100.000 18.0 100.000 18.0 2.890372 182.898667 -0.911145 33451.922268 0.830185 -166.647151
5 H2S -60.700 34.1 -60.700 34.1 3.529297 22.198667 -0.272219 492.780802 0.074103 -6.042901
6 H2Se -42.000 81.0 -42.000 81.0 4.394449 40.898667 0.592933 1672.700935 0.351569 24.250157
7 H2Te -1.800 129.6 -1.800 129.6 4.864453 81.098667 1.062936 6576.993735 1.129834 86.202719
8 NH3 -33.400 17.0 -33.400 17.0 2.833213 49.498667 -0.968303 2450.118002 0.937611 -47.929713
9 PH3 -87.000 34.0 -87.000 34.0 3.526361 -4.101333 -0.275156 16.820935 0.075711 1.128506
10 AsH3 -55.000 77.9 -55.000 77.9 4.355426 27.898667 0.553909 778.335602 0.306816 15.453336
11 SbH3 -17.100 124.8 -17.100 124.8 4.826712 65.798667 1.025196 4329.464535 1.051027 67.456530
12 CH4 -161.490 16.0 -161.490 16.0 2.772589 -78.591333 -1.028928 6176.597675 1.058692 80.864803
13 SiH4 -111.800 32.1 -111.800 32.1 3.468856 -28.901333 -0.332660 835.287068 0.110663 9.614330
14 GeH4 -90.000 76.6 -90.000 76.6 4.338597 -7.101333 0.537081 50.428935 0.288456 -3.813988
15 SnH4 -52.000 122.7 -52.000 122.7 4.809742 30.898667 1.008226 954.727602 1.016519 31.152836
16 He -268.934 4.0 -268.934 4.0 1.386294 -186.035333 -2.415222 34609.145248 5.833298 449.316648
17 Ne -246.048 20.2 -246.048 20.2 3.005683 -163.149333 -0.795834 26617.704967 0.633352 129.839763
18 Ar -185.700 39.9 -185.700 39.9 3.686376 -102.801333 -0.115140 10568.114135 0.013257 11.836560
19 Kr -152.300 83.8 -152.300 83.8 4.428433 -69.401333 0.626917 4816.545068 0.393024 -43.508844
20 Xe -108.100 131.3 -108.100 131.3 4.877485 -25.201333 1.075968 635.107202 1.157708 -27.115836
y log(x)
mean -82.898667 3.801516
variance 7050.465101 0.828803
sd 83.967048 0.910386 
covar_logxy = mean(excel['(y-mean(y)) * (log(x)-mean(log(x)))'])
corr_logxy = covar_logxy / (sd_logx * sd_y)
display(excel, pd.DataFrame([[mean_y, mean_logx], [variance_y, variance_logx], [sd_y, sd_logx]], 
                            columns=['y','log(x)'], index=['mean', 'variance', 'sd']),
       pd.DataFrame([covar_logxy, corr_logxy], index=['covariance', 'correlation'], columns=['log(x),y']))
molecule boiling point molecular weight y x log(x) y-mean(y) log(x)-mean(log(x)) (y-mean(y))**2 (log(x)-mean(log(x)))**2 (y-mean(y)) * (log(x)-mean(log(x)))
0 HF 19.500 20.0 19.500 20.0 2.995732 102.398667 -0.805784 10485.486935 0.649288 -82.511226
1 HCl -84.900 36.5 -84.900 36.5 3.597312 -2.001333 -0.204204 4.005335 0.041699 0.408681
2 HBr -67.000 80.9 -67.000 80.9 4.393214 15.898667 0.591697 252.767602 0.350106 9.407199
3 HI -35.100 127.9 -35.100 127.9 4.851249 47.798667 1.049732 2284.712535 1.101938 50.175802
4 H2O 100.000 18.0 100.000 18.0 2.890372 182.898667 -0.911145 33451.922268 0.830185 -166.647151
5 H2S -60.700 34.1 -60.700 34.1 3.529297 22.198667 -0.272219 492.780802 0.074103 -6.042901
6 H2Se -42.000 81.0 -42.000 81.0 4.394449 40.898667 0.592933 1672.700935 0.351569 24.250157
7 H2Te -1.800 129.6 -1.800 129.6 4.864453 81.098667 1.062936 6576.993735 1.129834 86.202719
8 NH3 -33.400 17.0 -33.400 17.0 2.833213 49.498667 -0.968303 2450.118002 0.937611 -47.929713
9 PH3 -87.000 34.0 -87.000 34.0 3.526361 -4.101333 -0.275156 16.820935 0.075711 1.128506
10 AsH3 -55.000 77.9 -55.000 77.9 4.355426 27.898667 0.553909 778.335602 0.306816 15.453336
11 SbH3 -17.100 124.8 -17.100 124.8 4.826712 65.798667 1.025196 4329.464535 1.051027 67.456530
12 CH4 -161.490 16.0 -161.490 16.0 2.772589 -78.591333 -1.028928 6176.597675 1.058692 80.864803
13 SiH4 -111.800 32.1 -111.800 32.1 3.468856 -28.901333 -0.332660 835.287068 0.110663 9.614330
14 GeH4 -90.000 76.6 -90.000 76.6 4.338597 -7.101333 0.537081 50.428935 0.288456 -3.813988
15 SnH4 -52.000 122.7 -52.000 122.7 4.809742 30.898667 1.008226 954.727602 1.016519 31.152836
16 He -268.934 4.0 -268.934 4.0 1.386294 -186.035333 -2.415222 34609.145248 5.833298 449.316648
17 Ne -246.048 20.2 -246.048 20.2 3.005683 -163.149333 -0.795834 26617.704967 0.633352 129.839763
18 Ar -185.700 39.9 -185.700 39.9 3.686376 -102.801333 -0.115140 10568.114135 0.013257 11.836560
19 Kr -152.300 83.8 -152.300 83.8 4.428433 -69.401333 0.626917 4816.545068 0.393024 -43.508844
20 Xe -108.100 131.3 -108.100 131.3 4.877485 -25.201333 1.075968 635.107202 1.157708 -27.115836
y log(x)
mean -82.898667 3.801516
variance 7050.465101 0.828803
sd 83.967048 0.910386
log(x),y
covariance 28.073248
correlation 0.367247 
w = corr_logxy * sd_y / sd_logx
t = mean_y - w * mean_logx
display(excel, pd.DataFrame([[mean_y, mean_logx], [variance_y, variance_logx], [sd_y, sd_logx]], 
                            columns=['y', 'log(x)'], index=['mean', 'variance', 'sd']),
        pd.DataFrame([covar_logxy, corr_logxy], index=['covariance', 'correlation'], columns=['log(x),y']),
        pd.DataFrame([[w, t]], columns=["w", "t"], index=["y = f(x) = wlog(x) + t"]))
molecule boiling point molecular weight y x log(x) y-mean(y) log(x)-mean(log(x)) (y-mean(y))**2 (log(x)-mean(log(x)))**2 (y-mean(y)) * (log(x)-mean(log(x)))
0 HF 19.500 20.0 19.500 20.0 2.995732 102.398667 -0.805784 10485.486935 0.649288 -82.511226
1 HCl -84.900 36.5 -84.900 36.5 3.597312 -2.001333 -0.204204 4.005335 0.041699 0.408681
2 HBr -67.000 80.9 -67.000 80.9 4.393214 15.898667 0.591697 252.767602 0.350106 9.407199
3 HI -35.100 127.9 -35.100 127.9 4.851249 47.798667 1.049732 2284.712535 1.101938 50.175802
4 H2O 100.000 18.0 100.000 18.0 2.890372 182.898667 -0.911145 33451.922268 0.830185 -166.647151
5 H2S -60.700 34.1 -60.700 34.1 3.529297 22.198667 -0.272219 492.780802 0.074103 -6.042901
6 H2Se -42.000 81.0 -42.000 81.0 4.394449 40.898667 0.592933 1672.700935 0.351569 24.250157
7 H2Te -1.800 129.6 -1.800 129.6 4.864453 81.098667 1.062936 6576.993735 1.129834 86.202719
8 NH3 -33.400 17.0 -33.400 17.0 2.833213 49.498667 -0.968303 2450.118002 0.937611 -47.929713
9 PH3 -87.000 34.0 -87.000 34.0 3.526361 -4.101333 -0.275156 16.820935 0.075711 1.128506
10 AsH3 -55.000 77.9 -55.000 77.9 4.355426 27.898667 0.553909 778.335602 0.306816 15.453336
11 SbH3 -17.100 124.8 -17.100 124.8 4.826712 65.798667 1.025196 4329.464535 1.051027 67.456530
12 CH4 -161.490 16.0 -161.490 16.0 2.772589 -78.591333 -1.028928 6176.597675 1.058692 80.864803
13 SiH4 -111.800 32.1 -111.800 32.1 3.468856 -28.901333 -0.332660 835.287068 0.110663 9.614330
14 GeH4 -90.000 76.6 -90.000 76.6 4.338597 -7.101333 0.537081 50.428935 0.288456 -3.813988
15 SnH4 -52.000 122.7 -52.000 122.7 4.809742 30.898667 1.008226 954.727602 1.016519 31.152836
16 He -268.934 4.0 -268.934 4.0 1.386294 -186.035333 -2.415222 34609.145248 5.833298 449.316648
17 Ne -246.048 20.2 -246.048 20.2 3.005683 -163.149333 -0.795834 26617.704967 0.633352 129.839763
18 Ar -185.700 39.9 -185.700 39.9 3.686376 -102.801333 -0.115140 10568.114135 0.013257 11.836560
19 Kr -152.300 83.8 -152.300 83.8 4.428433 -69.401333 0.626917 4816.545068 0.393024 -43.508844
20 Xe -108.100 131.3 -108.100 131.3 4.877485 -25.201333 1.075968 635.107202 1.157708 -27.115836
y log(x)
mean -82.898667 3.801516
variance 7050.465101 0.828803
sd 83.967048 0.910386
log(x),y
covariance 28.073248
correlation 0.367247
w t
y = f(x) = wlog(x) + t 33.872055 -211.663841 
# 回帰直線の式を関数として表現
def f(x):
    return w * x + t

excel['f(x)'] = f(excel['log(x)'])
display(excel, pd.DataFrame([[mean_y, mean_logx], [variance_y, variance_logx], [sd_y, sd_logx]], 
                            columns=['y','log(x)'], index=['mean', 'variance', 'sd']),
        pd.DataFrame([covar_logxy, corr_logxy], index=['covariance', 'correlation'], columns=['log(x),y']),
        pd.DataFrame([[w, t]], columns=["w", "t"], index=["y = f(x) = wlog(x) + t"]))
molecule boiling point molecular weight y x log(x) y-mean(y) log(x)-mean(log(x)) (y-mean(y))**2 (log(x)-mean(log(x)))**2 (y-mean(y)) * (log(x)-mean(log(x))) f(x)
0 HF 19.500 20.0 19.500 20.0 2.995732 102.398667 -0.805784 10485.486935 0.649288 -82.511226 -110.192233
1 HCl -84.900 36.5 -84.900 36.5 3.597312 -2.001333 -0.204204 4.005335 0.041699 0.408681 -89.815483
2 HBr -67.000 80.9 -67.000 80.9 4.393214 15.898667 0.591697 252.767602 0.350106 9.407199 -62.856661
3 HI -35.100 127.9 -35.100 127.9 4.851249 47.798667 1.049732 2284.712535 1.101938 50.175802 -47.342078
4 H2O 100.000 18.0 100.000 18.0 2.890372 182.898667 -0.911145 33451.922268 0.830185 -166.647151 -113.761010
5 H2S -60.700 34.1 -60.700 34.1 3.529297 22.198667 -0.272219 492.780802 0.074103 -6.042901 -92.119286
6 H2Se -42.000 81.0 -42.000 81.0 4.394449 40.898667 0.592933 1672.700935 0.351569 24.250157 -62.814818
7 H2Te -1.800 129.6 -1.800 129.6 4.864453 81.098667 1.062936 6576.993735 1.129834 86.202719 -46.894829
8 NH3 -33.400 17.0 -33.400 17.0 2.833213 49.498667 -0.968303 2450.118002 0.937611 -47.929713 -115.697083
9 PH3 -87.000 34.0 -87.000 34.0 3.526361 -4.101333 -0.275156 16.820935 0.075711 1.128506 -92.218764
10 AsH3 -55.000 77.9 -55.000 77.9 4.355426 27.898667 0.553909 778.335602 0.306816 15.453336 -64.136614
11 SbH3 -17.100 124.8 -17.100 124.8 4.826712 65.798667 1.025196 4329.464535 1.051027 67.456530 -48.173172
12 CH4 -161.490 16.0 -161.490 16.0 2.772589 -78.591333 -1.028928 6176.597675 1.058692 80.864803 -117.750564
13 SiH4 -111.800 32.1 -111.800 32.1 3.468856 -28.901333 -0.332660 835.287068 0.110663 9.614330 -94.166559
14 GeH4 -90.000 76.6 -90.000 76.6 4.338597 -7.101333 0.537081 50.428935 0.288456 -3.813988 -64.706643
15 SnH4 -52.000 122.7 -52.000 122.7 4.809742 30.898667 1.008226 954.727602 1.016519 31.152836 -48.747984
16 He -268.934 4.0 -268.934 4.0 1.386294 -186.035333 -2.415222 34609.145248 5.833298 449.316648 -164.707202
17 Ne -246.048 20.2 -246.048 20.2 3.005683 -163.149333 -0.795834 26617.704967 0.633352 129.839763 -109.855195
18 Ar -185.700 39.9 -185.700 39.9 3.686376 -102.801333 -0.115140 10568.114135 0.013257 11.836560 -86.798700
19 Kr -152.300 83.8 -152.300 83.8 4.428433 -69.401333 0.626917 4816.545068 0.393024 -43.508844 -61.663715
20 Xe -108.100 131.3 -108.100 131.3 4.877485 -25.201333 1.075968 635.107202 1.157708 -27.115836 -46.453409
y log(x)
mean -82.898667 3.801516
variance 7050.465101 0.828803
sd 83.967048 0.910386
log(x),y
covariance 28.073248
correlation 0.367247
w t
y = f(x) = wlog(x) + t 33.872055 -211.663841 
excel['(y-f(x))**2'] = (excel['y'] - excel['f(x)'])**2
display(excel, pd.DataFrame([[mean_y, mean_logx], [variance_y, variance_logx], [sd_y, sd_logx]], 
                            columns=['y','log(x)'], index=['mean', 'variance', 'sd']),
        pd.DataFrame([covar_logxy, corr_logxy], index=['covariance', 'correlation'], columns=['log(x),y']),
        pd.DataFrame([[w, t]], columns=["w", "t"], index=["y = f(x) = wlog(x) + t"]))
molecule boiling point molecular weight y x log(x) y-mean(y) log(x)-mean(log(x)) (y-mean(y))**2 (log(x)-mean(log(x)))**2 (y-mean(y)) * (log(x)-mean(log(x))) f(x) (y-f(x))**2
0 HF 19.500 20.0 19.500 20.0 2.995732 102.398667 -0.805784 10485.486935 0.649288 -82.511226 -110.192233 16820.075290
1 HCl -84.900 36.5 -84.900 36.5 3.597312 -2.001333 -0.204204 4.005335 0.041699 0.408681 -89.815483 24.161969
2 HBr -67.000 80.9 -67.000 80.9 4.393214 15.898667 0.591697 252.767602 0.350106 9.407199 -62.856661 17.167258
3 HI -35.100 127.9 -35.100 127.9 4.851249 47.798667 1.049732 2284.712535 1.101938 50.175802 -47.342078 149.868481
4 H2O 100.000 18.0 100.000 18.0 2.890372 182.898667 -0.911145 33451.922268 0.830185 -166.647151 -113.761010 45693.769454
5 H2S -60.700 34.1 -60.700 34.1 3.529297 22.198667 -0.272219 492.780802 0.074103 -6.042901 -92.119286 987.171545
6 H2Se -42.000 81.0 -42.000 81.0 4.394449 40.898667 0.592933 1672.700935 0.351569 24.250157 -62.814818 433.256643
7 H2Te -1.800 129.6 -1.800 129.6 4.864453 81.098667 1.062936 6576.993735 1.129834 86.202719 -46.894829 2033.543613
8 NH3 -33.400 17.0 -33.400 17.0 2.833213 49.498667 -0.968303 2450.118002 0.937611 -47.929713 -115.697083 6772.809882
9 PH3 -87.000 34.0 -87.000 34.0 3.526361 -4.101333 -0.275156 16.820935 0.075711 1.128506 -92.218764 27.235494
10 AsH3 -55.000 77.9 -55.000 77.9 4.355426 27.898667 0.553909 778.335602 0.306816 15.453336 -64.136614 83.477714
11 SbH3 -17.100 124.8 -17.100 124.8 4.826712 65.798667 1.025196 4329.464535 1.051027 67.456530 -48.173172 965.541992
12 CH4 -161.490 16.0 -161.490 16.0 2.772589 -78.591333 -1.028928 6176.597675 1.058692 80.864803 -117.750564 1913.138297
13 SiH4 -111.800 32.1 -111.800 32.1 3.468856 -28.901333 -0.332660 835.287068 0.110663 9.614330 -94.166559 310.938239
14 GeH4 -90.000 76.6 -90.000 76.6 4.338597 -7.101333 0.537081 50.428935 0.288456 -3.813988 -64.706643 639.753932
15 SnH4 -52.000 122.7 -52.000 122.7 4.809742 30.898667 1.008226 954.727602 1.016519 31.152836 -48.747984 10.575609
16 He -268.934 4.0 -268.934 4.0 1.386294 -186.035333 -2.415222 34609.145248 5.833298 449.316648 -164.707202 10863.225340
17 Ne -246.048 20.2 -246.048 20.2 3.005683 -163.149333 -0.795834 26617.704967 0.633352 129.839763 -109.855195 18548.480187
18 Ar -185.700 39.9 -185.700 39.9 3.686376 -102.801333 -0.115140 10568.114135 0.013257 11.836560 -86.798700 9781.467196
19 Kr -152.300 83.8 -152.300 83.8 4.428433 -69.401333 0.626917 4816.545068 0.393024 -43.508844 -61.663715 8214.936167
20 Xe -108.100 131.3 -108.100 131.3 4.877485 -25.201333 1.075968 635.107202 1.157708 -27.115836 -46.453409 3800.302233
y log(x)
mean -82.898667 3.801516
variance 7050.465101 0.828803
sd 83.967048 0.910386
log(x),y
covariance 28.073248
correlation 0.367247
w t
y = f(x) = wlog(x) + t 33.872055 -211.663841 
r2 = 1. - sum(excel['(y-f(x))**2']) / sum(excel['(y-mean(y))**2'])
display(excel, pd.DataFrame([[mean_y, mean_logx], [variance_y, variance_logx], [sd_y, sd_logx]], 
                            columns=['y','log(x)'], index=['mean', 'variance', 'sd']),
        pd.DataFrame([covar_logxy, corr_logxy], index=['covariance', 'correlation'], columns=['log(x),y']),
        pd.DataFrame([[w, t, r2]], columns=["w", "t", "R2"], index=["y = f(x) = wlog(x) + t"]))
molecule boiling point molecular weight y x log(x) y-mean(y) log(x)-mean(log(x)) (y-mean(y))**2 (log(x)-mean(log(x)))**2 (y-mean(y)) * (log(x)-mean(log(x))) f(x) (y-f(x))**2
0 HF 19.500 20.0 19.500 20.0 2.995732 102.398667 -0.805784 10485.486935 0.649288 -82.511226 -110.192233 16820.075290
1 HCl -84.900 36.5 -84.900 36.5 3.597312 -2.001333 -0.204204 4.005335 0.041699 0.408681 -89.815483 24.161969
2 HBr -67.000 80.9 -67.000 80.9 4.393214 15.898667 0.591697 252.767602 0.350106 9.407199 -62.856661 17.167258
3 HI -35.100 127.9 -35.100 127.9 4.851249 47.798667 1.049732 2284.712535 1.101938 50.175802 -47.342078 149.868481
4 H2O 100.000 18.0 100.000 18.0 2.890372 182.898667 -0.911145 33451.922268 0.830185 -166.647151 -113.761010 45693.769454
5 H2S -60.700 34.1 -60.700 34.1 3.529297 22.198667 -0.272219 492.780802 0.074103 -6.042901 -92.119286 987.171545
6 H2Se -42.000 81.0 -42.000 81.0 4.394449 40.898667 0.592933 1672.700935 0.351569 24.250157 -62.814818 433.256643
7 H2Te -1.800 129.6 -1.800 129.6 4.864453 81.098667 1.062936 6576.993735 1.129834 86.202719 -46.894829 2033.543613
8 NH3 -33.400 17.0 -33.400 17.0 2.833213 49.498667 -0.968303 2450.118002 0.937611 -47.929713 -115.697083 6772.809882
9 PH3 -87.000 34.0 -87.000 34.0 3.526361 -4.101333 -0.275156 16.820935 0.075711 1.128506 -92.218764 27.235494
10 AsH3 -55.000 77.9 -55.000 77.9 4.355426 27.898667 0.553909 778.335602 0.306816 15.453336 -64.136614 83.477714
11 SbH3 -17.100 124.8 -17.100 124.8 4.826712 65.798667 1.025196 4329.464535 1.051027 67.456530 -48.173172 965.541992
12 CH4 -161.490 16.0 -161.490 16.0 2.772589 -78.591333 -1.028928 6176.597675 1.058692 80.864803 -117.750564 1913.138297
13 SiH4 -111.800 32.1 -111.800 32.1 3.468856 -28.901333 -0.332660 835.287068 0.110663 9.614330 -94.166559 310.938239
14 GeH4 -90.000 76.6 -90.000 76.6 4.338597 -7.101333 0.537081 50.428935 0.288456 -3.813988 -64.706643 639.753932
15 SnH4 -52.000 122.7 -52.000 122.7 4.809742 30.898667 1.008226 954.727602 1.016519 31.152836 -48.747984 10.575609
16 He -268.934 4.0 -268.934 4.0 1.386294 -186.035333 -2.415222 34609.145248 5.833298 449.316648 -164.707202 10863.225340
17 Ne -246.048 20.2 -246.048 20.2 3.005683 -163.149333 -0.795834 26617.704967 0.633352 129.839763 -109.855195 18548.480187
18 Ar -185.700 39.9 -185.700 39.9 3.686376 -102.801333 -0.115140 10568.114135 0.013257 11.836560 -86.798700 9781.467196
19 Kr -152.300 83.8 -152.300 83.8 4.428433 -69.401333 0.626917 4816.545068 0.393024 -43.508844 -61.663715 8214.936167
20 Xe -108.100 131.3 -108.100 131.3 4.877485 -25.201333 1.075968 635.107202 1.157708 -27.115836 -46.453409 3800.302233
y log(x)
mean -82.898667 3.801516
variance 7050.465101 0.828803
sd 83.967048 0.910386
log(x),y
covariance 28.073248
correlation 0.367247
w t R2
y = f(x) = wlog(x) + t 33.872055 -211.663841 0.13487

できたッ！

1
2
0

Register as a new user and use Qiita more conveniently

  1. You get articles that match your needs
  2. You can efficiently read back useful information
  3. You can use dark theme
What you can do with signing up
Sign upLogin
1
2