正準相関分析の例

• ２つの変数群に共通の隠れ変数群を見つける
• https://cmdlinetips.com/2020/12/canonical-correlation-analysis-in-python/ より

import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns
import numpy as np

link2data = "https://raw.githubusercontent.com/mwaskom/seaborn-data/master/penguins.csv"
df = pd.read_csv(link2data)
df = df.dropna()
df.head()

	species	island	bill_length_mm	bill_depth_mm	flipper_length_mm	body_mass_g	sex
0	Adelie	Torgersen	39.1	18.7	181.0	3750.0	MALE
1	Adelie	Torgersen	39.5	17.4	186.0	3800.0	FEMALE
2	Adelie	Torgersen	40.3	18.0	195.0	3250.0	FEMALE
4	Adelie	Torgersen	36.7	19.3	193.0	3450.0	FEMALE
5	Adelie	Torgersen	39.3	20.6	190.0	3650.0	MALE

df.shape

(333, 7)

• 上記データセットを２つの変数群(bill* と fipper & body_)に分ける

x = df[['bill_length_mm', 'bill_depth_mm']]
x.head(3)

	bill_length_mm	bill_depth_mm
0	39.1	18.7
1	39.5	17.4
2	40.3	18.0

y = df[['flipper_length_mm', 'body_mass_g']]
y.head(3)

	flipper_length_mm	body_mass_g
0	181.0	3750.0
1	186.0	3800.0
2	195.0	3250.0

• どちらも標準化する

x_mc = (x - x.mean()) / (x.std())
x_mc.head(3)

	bill_length_mm	bill_depth_mm
0	-0.894695	0.779559
1	-0.821552	0.119404
2	-0.675264	0.424091

y_mc = (y - y.mean()) / (y.std())
y_mc.head(3)

	flipper_length_mm	body_mass_g
0	-1.424608	-0.567621
1	-1.067867	-0.505525
2	-0.425733	-1.188572

• 身長、体重、羽の長さなどは当然、ペンギンの種類により異なる
• なので、この２つのデータセットに共通の隠れ変数は、種類と考えられる

sklearnにより、正準相関係数と正準相関変数を求める

• 固有値の数だけ、正準相関係数と正準相関変数がある（第一正準相関変数、第二正準相関変数・・・）

from sklearn.cross_decomposition import CCA

ca = CCA()
ca.fit(x_mc, y_mc)
x_c, y_c = ca.transform(x_mc, y_mc)

x_c[:3]

array([[-1.18625232, -0.01036701],
       [-0.70957262, -0.4560358 ],
       [-0.79073194, -0.13080943],
       [-1.7186634 , -0.07362316],
       [-1.77229457,  0.73624799]])

y_c[:3]

array([[-1.40879506,  0.68286617],
       [-1.05385671,  0.42987851],
       [-0.3935502 , -0.83961988]])

print(x_c.shape)
print(y_c.shape)

(333, 2)
(333, 2)

• x_c、y_cはそれぞれ、正準相関により射影された各データ

正準相関分析結果の理解

• 正準相関分析により射影した結果をDF化

cc_res = pd.DataFrame({"Ccx_1": x_c[:,0],
                       "Ccy_1": y_c[:,0],
                       "Ccx_2": x_c[:,1],
                       "Ccy_2": y_c[:,1],
                       "Species": df.species.to_list(),
                       "Island": df.island.to_list(),
                       "Sex": df.sex.to_list()})
cc_res.head()

	Ccx_1	Ccy_1	Ccx_2	Ccy_2	Species	Island	Sex
0	-1.186252	-1.408795	-0.010367	0.682866	Adelie	Torgersen	MALE
1	-0.709573	-1.053857	-0.456036	0.429879	Adelie	Torgersen	FEMALE
2	-0.790732	-0.393550	-0.130809	-0.839620	Adelie	Torgersen	FEMALE
3	-1.718663	-0.542888	-0.073623	-0.458571	Adelie	Torgersen	FEMALE
4	-1.772295	-0.763548	0.736248	-0.014204	Adelie	Torgersen	MALE

• 一番目の正準相関変数同士（Ccx_1 , Ccy_1）の相関を求める

np.corrcoef(x_c[:,0], y_c[:,0])

array([[1.        , 0.78763151],
       [0.78763151, 1.        ]])

• 二番目の正準相関変数同士（Ccx_2 , Ccy_2）の相関を求める

np.corrcoef(x_c[:,1], y_c[:,1])

array([[1.        , 0.08638695],
       [0.08638695, 1.        ]])

• Ccx_1とCcy_1をプロットして相関具合を確認

sns.set_context("talk", font_scale=1.2)
plt.figure(figsize=(10,8))
sns.scatterplot(x="Ccx_1", y="Ccy_1", data=cc_res)
plt.title("Comp. 1, corr= %.2f" % np.corrcoef(x_c[:,0], y_c[:,0])[0,1])

Text(0.5, 1.0, 'Comp. 1, corr= 0.79')

隠れ変数であるSpecies(種類)と第一の相関変数群の相関具合を確認する

• 隠れ変数Speciesと各第一相関変数(Ccx_1とCcy_1)の関係をboxplotで可視化

plt.figure(figsize=(10,8))
plt.title('Boxplot of Canonical Correlate from Ccx_1 and latent variable(species)')
sns.boxplot(x="Species", y="Ccx_1", data=cc_res)
sns.stripplot(x="Species", y="Ccx_1", data=cc_res)

<AxesSubplot:title={'center':'Boxplot of Canonical Correlate from Ccx_1 and latent variable(species)'}, xlabel='Species', ylabel='Ccx_1'>

plt.figure(figsize=(8,6))
plt.title('Boxplot of relation before CCA')
sns.boxplot(x="species", y="bill_length_mm", data=df)
sns.stripplot(x="species", y="bill_length_mm", data=df)

<AxesSubplot:title={'center':'Boxplot of relation before CCA'}, xlabel='species', ylabel='bill_length_mm'>

plt.figure(figsize=(8,6))
plt.title('Boxplot of relation before CCA')
sns.boxplot(x="species", y="bill_depth_mm", data=df)
sns.stripplot(x="species", y="bill_depth_mm", data=df)

<AxesSubplot:title={'center':'Boxplot of relation before CCA'}, xlabel='species', ylabel='bill_depth_mm'>

• CCA前では、bill_length_mm変数と、bill_depth_mm変数それぞれ、speciesの特徴を捉えていないが、CCAを行うことで、新変数Ccx_1がspeciesの特徴をとらえている

plt.figure(figsize=(10,8))
plt.title('Boxplot of Canonical Correlate from Ccy_1 and latent variable(species)')
sns.boxplot(x="Species", y="Ccy_1", data=cc_res)
sns.stripplot(x="Species", y="Ccy_1", data=cc_res)

<AxesSubplot:title={'center':'Boxplot of Canonical Correlate from Ccy_1 and latent variable(species)'}, xlabel='Species', ylabel='Ccy_1'>

plt.figure(figsize=(8,6))
plt.title('Boxplot of relation before CCA')
sns.boxplot(x="species", y="flipper_length_mm", data=df)
sns.stripplot(x="species", y="flipper_length_mm", data=df)

<AxesSubplot:title={'center':'Boxplot of relation before CCA'}, xlabel='species', ylabel='flipper_length_mm'>

plt.figure(figsize=(8,6))
plt.title('Boxplot of relation before CCA')
sns.boxplot(x="species", y="body_mass_g", data=df)
sns.stripplot(x="species", y="body_mass_g", data=df)

<AxesSubplot:title={'center':'Boxplot of relation before CCA'}, xlabel='species', ylabel='body_mass_g'>

• CCA前では、flipper_length_mm変数と、body_mass_g変数それぞれ、speciesの特徴を捉えていないが、CCAを行うことで、新変数Ccy_1がspeciesの特徴をとらえている

plt.figure(figsize=(10,8))
sns.scatterplot(x="Ccx_1", y="Ccy_1", hue="Species", data=cc_res)
plt.title('First Pair of Canonical Covariate, corr = %.2f' % np.corrcoef(x_c[:,0], y_c[:,0])[0,1])

Text(0.5, 1.0, 'First Pair of Canonical Covariate, corr = 0.79')

• それぞれの第一相関変数同士は、相関関係を持ちつつ、共通の隠れ変数(species)の特徴を捉えている

第一相関変数、第二相関変数とオリジナルの変数の相関具合をヒートマップで確認

ccX_df = pd.DataFrame({"CCX_1":x_c[:, 0],
                       "CCX_2":x_c[:, 1],
                       "Species":df.species.astype('category').cat.codes,
                      "Island":df.island.astype('category').cat.codes,
                      "sex":df.sex.astype('category').cat.codes,
                       "bill_length":x_mc.bill_length_mm,
                      "bill_depth":x_mc.bill_depth_mm})

corr_X_df= ccX_df.corr(method='pearson') 
corr_X_df.head()

	CCX_1	CCX_2	Species	Island	sex	bill_length	bill_depth
CCX_1	1.000000e+00	7.728804e-18	0.935057	-0.561781	0.025383	0.828437	-0.734650
CCX_2	7.728804e-18	1.000000e+00	-0.078719	0.228933	0.576790	0.560082	0.678447
Species	9.350575e-01	-7.871884e-02	1.000000	-0.622428	0.010964	0.730548	-0.740346
Island	-5.617810e-01	2.289327e-01	-0.622428	1.000000	-0.012435	-0.337179	0.568031
sex	2.538332e-02	5.767897e-01	0.010964	-0.012435	1.000000	0.344078	0.372673

plt.figure(figsize=(10,8))
X_df_lt = corr_X_df.where(np.tril(np.ones(corr_X_df.shape)).astype(np.bool))

C:\Users\yoshi\AppData\Local\Temp/ipykernel_2152/3334907121.py:2: DeprecationWarning: `np.bool` is a deprecated alias for the builtin `bool`. To silence this warning, use `bool` by itself. Doing this will not modify any behavior and is safe. If you specifically wanted the numpy scalar type, use `np.bool_` here.
Deprecated in NumPy 1.20; for more details and guidance: https://numpy.org/devdocs/release/1.20.0-notes.html#deprecations
  X_df_lt = corr_X_df.where(np.tril(np.ones(corr_X_df.shape)).astype(np.bool))

<Figure size 720x576 with 0 Axes>

plt.figure(figsize=(10,8))
sns.heatmap(X_df_lt,cmap="coolwarm",annot=True,fmt='.1g')

<AxesSubplot:>

# second pair of canonical covariates with the dataset 
ccY_df = pd.DataFrame({"CCY_1":y_c[:, 0],
                       "CCY_2":y_c[:, 1],
                       "Species":df.species.astype('category').cat.codes,
                      "Island":df.island.astype('category').cat.codes,
                      "sex":df.sex.astype('category').cat.codes,
                       "flipper_length":y_mc.flipper_length_mm,
                      "body_mass":y_mc.body_mass_g})
# compute correlation with Pandas corr()
corr_Y_df= ccY_df.corr(method='pearson') 
# Get lower triangular correlation matrix
Y_df_lt = corr_Y_df.where(np.tril(np.ones(corr_Y_df.shape)).astype(np.bool))
# make a lower triangular correlation heatmap with Seaborn
plt.figure(figsize=(10,8))
sns.heatmap(Y_df_lt,cmap="coolwarm",annot=True,fmt='.1g')

C:\Users\yoshi\AppData\Local\Temp/ipykernel_2152/3193900153.py:12: DeprecationWarning: `np.bool` is a deprecated alias for the builtin `bool`. To silence this warning, use `bool` by itself. Doing this will not modify any behavior and is safe. If you specifically wanted the numpy scalar type, use `np.bool_` here.
Deprecated in NumPy 1.20; for more details and guidance: https://numpy.org/devdocs/release/1.20.0-notes.html#deprecations
  Y_df_lt = corr_Y_df.where(np.tril(np.ones(corr_Y_df.shape)).astype(np.bool))

<AxesSubplot:>

• ccx_1, ccy_1ともに、Speciesとの相関が高い
• ccx_2, ccy_2ともに、sexとの相関が高い
• よって、この２つの変数群(['bill_length_mm', 'bill_depth_mm']と['flipper_length_mm', 'body_mass_g'])それぞれに共通に影響を与えている因子として、Speciesとsexが働いていることがわかる