[Dacon] 인구 데이터 기반 소득 예측 경진대회
데이콘의 “인구 데이터 기반 소득 예측 경진대회”에 참여하여 작성한 글이며, 간단한 데이터 전처리 및 EDA, LightGBM과 XGBoost 모델을 활용한 Ensemble 구현 코드를 소개합니다.
코드실행은 Google Colab의 GPU, Standard RAM 환경에서 진행했습니다.
➔ 데이콘에서 읽기
0. Import Packages
- 라이브러리 불러오기
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
!pip install -U pandas-profiling
import numpy as np
import pandas as pd
import matplotlib
import matplotlib.pyplot as plt
import sklearn
import pandas_profiling
import seaborn as sns
import random as rn
import os
import scipy.stats as stats
from sklearn.preprocessing import LabelEncoder
from sklearn.preprocessing import OneHotEncoder
from sklearn.model_selection import train_test_split
from sklearn.ensemble import VotingClassifier
from sklearn import metrics
import xgboost as xgb
import lightgbm as lgb
from collections import Counter
import warnings
%matplotlib inline
warnings.filterwarnings(action='ignore')
- 주요 라이브러리 버전 확인
1
2
3
4
5
6
print("numpy version: {}". format(np.__version__))
print("pandas version: {}". format(pd.__version__))
print("matplotlib version: {}". format(matplotlib.__version__))
print("scikit-learn version: {}". format(sklearn.__version__))
print("xgboost version: {}". format(xgb.__version__))
print("lightgbm version: {}". format(lgb.__version__))
numpy version: 1.21.6
pandas version: 1.3.5
matplotlib version: 3.2.2
scikit-learn version: 1.0.2
xgboost version: 0.90
lightgbm version: 2.2.3
- 랜덤 시드 고정
1
2
3
4
5
# reproducibility
seed_num = 42
np.random.seed(seed_num)
rn.seed(seed_num)
os.environ['PYTHONHASHSEED']=str(seed_num)
1. Load and Check Dataset
Variable Description
| age | workclass | fnlwgt | education |
| 나이 | 일 유형 | CPS(Current Population Survey) 가중치 | 교육수준 |
| education.num | marital.status | occupation |
| 교육수준 번호 | 결혼 상태 | 직업 |
| relationship | race | sex | capital.gain | capital.loss | hours.per.week | native.country |
| 가족관계 | 인종 | 성별 | 자본 이익 | 자본 손실 | 주당 근무시간 | 본 국적 |
- 데이터 불러오기
1
2
3
4
5
6
7
train = pd.read_csv('/content/drive/MyDrive/Forecasting_income/dataset/train.csv')
test = pd.read_csv('/content/drive/MyDrive/Forecasting_income/dataset/test.csv')
train.columns = train.columns.str.replace('.','_')
test.columns = test.columns.str.replace('.','_')
train.head()
id age workclass fnlwgt education education_num marital_status \
0 0 32 Private 309513 Assoc-acdm 12 Married-civ-spouse
1 1 33 Private 205469 Some-college 10 Married-civ-spouse
2 2 46 Private 149949 Some-college 10 Married-civ-spouse
3 3 23 Private 193090 Bachelors 13 Never-married
4 4 55 Private 60193 HS-grad 9 Divorced
occupation relationship race sex capital_gain capital_loss \
0 Craft-repair Husband White Male 0 0
1 Exec-managerial Husband White Male 0 0
2 Craft-repair Husband White Male 0 0
3 Adm-clerical Own-child White Female 0 0
4 Adm-clerical Not-in-family White Female 0 0
hours_per_week native_country target
0 40 United-States 0
1 40 United-States 1
2 40 United-States 0
3 30 United-States 0
4 40 United-States 0
- Pandas Profiling Report 생성하기
1
2
pr=train.profile_report()
pr.to_file('/content/drive/MyDrive/Forecasting_income/pr_report.html')
- Pandas Profiling을 활용하면 아래와 같이 데이터 프레임을 쉽고 효율적으로 탐색할 수 있습니다.
Pandas Profiling Report의 Alert 활용하기
Variable Pairs with the High Correlation
relationship-sexage-marital.statusworkclass-occupationeducation-education.numrelationship-marital.statusrace-native.countrysex-occupationtarget-relationship
Data Type
- Numeric (7) :
id,age,fnlwgt,education.num,capital.gain,capital.loss,hours.per.week - Categorical (9) :
workclass,education,marital.status,occupation,relationship,race,sex,native.country,target
Note
workclass와occupation이 같은 비율 (10.5%)의 결측치(Missing Value)를 가집니다.native.country는 583(3.3%)의 결측치(Missing Value)를 가지므로 해당 행(Row)을 삭제해주겠습니다.capital.gain와capital.loss는 높은 왜도(Skewness)를 가집니다. 이상치(Outlier)를 확인하고 필요시 제거하거나 변환 함수를 적용하겠습니다.
1
train.info()
<class 'pandas.core.frame.DataFrame'>
RangeIndex: 17480 entries, 0 to 17479
Data columns (total 16 columns):
# Column Non-Null Count Dtype
--- ------ -------------- -----
0 id 17480 non-null int64
1 age 17480 non-null int64
2 workclass 15644 non-null object
3 fnlwgt 17480 non-null int64
4 education 17480 non-null object
5 education_num 17480 non-null int64
6 marital_status 17480 non-null object
7 occupation 15637 non-null object
8 relationship 17480 non-null object
9 race 17480 non-null object
10 sex 17480 non-null object
11 capital_gain 17480 non-null int64
12 capital_loss 17480 non-null int64
13 hours_per_week 17480 non-null int64
14 native_country 16897 non-null object
15 target 17480 non-null int64
dtypes: int64(8), object(8)
memory usage: 2.1+ MB
2. Data Preprocessing
(1) Missing Value
1
train.columns[train.isnull().any()]
Index(['workclass', 'occupation', 'native_country'], dtype='object')
1
train[train["workclass"].isnull()]
id age workclass fnlwgt education education_num \
15081 15081 90 NaN 77053 HS-grad 9
15082 15082 66 NaN 186061 Some-college 10
15084 15084 51 NaN 172175 Doctorate 16
15086 15086 61 NaN 135285 HS-grad 9
15087 15087 71 NaN 100820 HS-grad 9
... ... ... ... ... ... ...
17475 17475 35 NaN 320084 Bachelors 13
17476 17476 30 NaN 33811 Bachelors 13
17477 17477 71 NaN 287372 Doctorate 16
17478 17478 41 NaN 202822 HS-grad 9
17479 17479 72 NaN 129912 HS-grad 9
marital_status occupation relationship race \
15081 Widowed NaN Not-in-family White
15082 Widowed NaN Unmarried Black
15084 Never-married NaN Not-in-family White
15086 Married-civ-spouse NaN Husband White
15087 Married-civ-spouse NaN Husband White
... ... ... ... ...
17475 Married-civ-spouse NaN Wife White
17476 Never-married NaN Not-in-family Asian-Pac-Islander
17477 Married-civ-spouse NaN Husband White
17478 Separated NaN Not-in-family Black
17479 Married-civ-spouse NaN Husband White
sex capital_gain capital_loss hours_per_week native_country \
15081 Female 0 4356 40 United-States
15082 Female 0 4356 40 United-States
15084 Male 0 2824 40 United-States
15086 Male 0 2603 32 United-States
15087 Male 0 2489 15 United-States
... ... ... ... ... ...
17475 Female 0 0 55 United-States
17476 Female 0 0 99 United-States
17477 Male 0 0 10 United-States
17478 Female 0 0 32 United-States
17479 Male 0 0 25 United-States
target
15081 0
15082 0
15084 1
15086 0
15087 0
... ...
17475 1
17476 0
17477 1
17478 0
17479 0
[1836 rows x 16 columns]
1
train['workclass'].unique()
array(['Private', 'State-gov', 'Local-gov', 'Self-emp-not-inc',
'Self-emp-inc', 'Federal-gov', 'Without-pay', nan, 'Never-worked'],
dtype=object)
workclass와occupation열(Column)에서 결측치가 포함된 행은 삭제합니다.- 두 열이 동시에 결측치를 갖는 경우가 대부분이므로,
workclass의 결측치만Never-worked와 같은 이미 존재하는 특성으로 채우는 것은 의미가 없습니다. workclass와occupation에 새로운 feature을 부여하는 방법도 시도하였지만, One-hot Encoding을 했을 때 생기는 테스트 데이터와의 값 차이 때문에 다른 방법을 고려해볼 필요가 있다고 생각합니다 😔
1
2
3
4
print(sum(train['workclass'].isna()))
print(sum(train['occupation'].isna()))
fill_na = train['workclass'].isna()
1836
1843
1
2
3
4
5
df_train = train.dropna()
print(sum(df_train['workclass'].isna()))
print(sum(df_train['occupation'].isna()))
print(sum(df_train['native_country'].isna()))
0
0
0
1
df_train
id age workclass fnlwgt education education_num \
0 0 32 Private 309513 Assoc-acdm 12
1 1 33 Private 205469 Some-college 10
2 2 46 Private 149949 Some-college 10
3 3 23 Private 193090 Bachelors 13
4 4 55 Private 60193 HS-grad 9
... ... ... ... ... ... ...
15076 15076 35 Private 337286 Masters 14
15077 15077 36 Private 182074 Some-college 10
15078 15078 50 Self-emp-inc 175070 Prof-school 15
15079 15079 39 Private 202937 Some-college 10
15080 15080 33 Private 96245 Assoc-acdm 12
marital_status occupation relationship race \
0 Married-civ-spouse Craft-repair Husband White
1 Married-civ-spouse Exec-managerial Husband White
2 Married-civ-spouse Craft-repair Husband White
3 Never-married Adm-clerical Own-child White
4 Divorced Adm-clerical Not-in-family White
... ... ... ... ...
15076 Never-married Exec-managerial Not-in-family Asian-Pac-Islander
15077 Divorced Adm-clerical Not-in-family White
15078 Married-civ-spouse Prof-specialty Husband White
15079 Divorced Tech-support Not-in-family White
15080 Married-civ-spouse Prof-specialty Husband White
sex capital_gain capital_loss hours_per_week native_country \
0 Male 0 0 40 United-States
1 Male 0 0 40 United-States
2 Male 0 0 40 United-States
3 Female 0 0 30 United-States
4 Female 0 0 40 United-States
... ... ... ... ... ...
15076 Male 0 0 40 United-States
15077 Male 0 0 45 United-States
15078 Male 0 0 45 United-States
15079 Female 0 0 40 Poland
15080 Male 0 0 50 United-States
target
0 0
1 1
2 0
3 0
4 0
... ...
15076 0
15077 0
15078 1
15079 0
15080 0
[15081 rows x 16 columns]
(2) Outlier
1
2
3
4
5
6
7
fig, ax = plt.subplots(1, 2, figsize=(12,3))
g = sns.distplot(df_train['capital_gain'], color='b', label='Skewness : {:.2f}'.format(df_train['capital_gain'].skew()), ax=ax[0])
g = g.legend(loc='best')
g = sns.distplot(df_train['capital_loss'], color='b', label='Skewness : {:.2f}'.format(df_train['capital_loss'].skew()), ax=ax[1])
g = g.legend(loc='best')
plt.show()
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
numeric_fts = ['age', 'fnlwgt', 'education_num', 'capital_gain', 'capital_loss', 'hours_per_week']
outlier_ind = []
for i in numeric_fts:
Q1 = np.percentile(df_train[i],25)
Q3 = np.percentile(df_train[i],75)
IQR = Q3-Q1
outlier_list = df_train[(df_train[i] < Q1 - IQR * 1.5) | (df_train[i] > Q3 + IQR * 1.5)].index
outlier_ind.extend(outlier_list)
outlier_ind = Counter(outlier_ind)
multi_outliers = list(k for k,j in outlier_ind.items() if j > 2)
# Drop outliers
train_df = df_train.drop(multi_outliers, axis = 0).reset_index(drop = True)
train_df
id age workclass fnlwgt education education_num \
0 0 32 Private 309513 Assoc-acdm 12
1 1 33 Private 205469 Some-college 10
2 2 46 Private 149949 Some-college 10
3 3 23 Private 193090 Bachelors 13
4 4 55 Private 60193 HS-grad 9
... ... ... ... ... ... ...
15043 15076 35 Private 337286 Masters 14
15044 15077 36 Private 182074 Some-college 10
15045 15078 50 Self-emp-inc 175070 Prof-school 15
15046 15079 39 Private 202937 Some-college 10
15047 15080 33 Private 96245 Assoc-acdm 12
marital_status occupation relationship race \
0 Married-civ-spouse Craft-repair Husband White
1 Married-civ-spouse Exec-managerial Husband White
2 Married-civ-spouse Craft-repair Husband White
3 Never-married Adm-clerical Own-child White
4 Divorced Adm-clerical Not-in-family White
... ... ... ... ...
15043 Never-married Exec-managerial Not-in-family Asian-Pac-Islander
15044 Divorced Adm-clerical Not-in-family White
15045 Married-civ-spouse Prof-specialty Husband White
15046 Divorced Tech-support Not-in-family White
15047 Married-civ-spouse Prof-specialty Husband White
sex capital_gain capital_loss hours_per_week native_country \
0 Male 0 0 40 United-States
1 Male 0 0 40 United-States
2 Male 0 0 40 United-States
3 Female 0 0 30 United-States
4 Female 0 0 40 United-States
... ... ... ... ... ...
15043 Male 0 0 40 United-States
15044 Male 0 0 45 United-States
15045 Male 0 0 45 United-States
15046 Female 0 0 40 Poland
15047 Male 0 0 50 United-States
target
0 0
1 1
2 0
3 0
4 0
... ...
15043 0
15044 0
15045 1
15046 0
15047 0
[15048 rows x 16 columns]
1
print(train_df['capital_gain'].skew(), train_df['capital_loss'].skew())
12.004940559585881 4.607122286739042
- 이상치들을 제거하였음에도 두 변수는 여전히 높은 왜도를 보이고 있어, 로그 변환(Log Transformation)을 진행했습니다.
1
2
3
4
5
6
7
8
# log transformation
train_df['capital_gain'] = train_df['capital_gain'].map(lambda i: np.log(i) if i > 0 else 0)
test['capital_gain'] = test['capital_gain'].map(lambda i: np.log(i) if i > 0 else 0)
train_df['capital_loss'] = train_df['capital_loss'].map(lambda i: np.log(i) if i > 0 else 0)
test['capital_loss'] = test['capital_loss'].map(lambda i: np.log(i) if i > 0 else 0)
print(train_df['capital_gain'].skew(), train_df['capital_loss'].skew())
3.0945787119106676 4.390015583095806
3. Feature Engineering
(1) Correlation
- 범주형(Categorical) 데이터를 라벨 인코더(Label Encoder)를 통해 수치형(Numerical)으로 변환한 후 상관관계를 확인합니다.
- Categorical :
workclass,education,marital.status,occupation,relationship,race,sex,native.country
1
2
3
4
5
6
7
8
la_train = train_df.copy()
cat_fts = ['workclass', 'education', 'marital_status', 'occupation', 'relationship', 'race', 'sex', 'native_country']
for i in range(len(cat_fts)):
encoder = LabelEncoder()
la_train[cat_fts[i]] = encoder.fit_transform(la_train[cat_fts[i]])
la_train.head()
id age workclass fnlwgt education education_num marital_status \
0 0 32 2 309513 7 12 2
1 1 33 2 205469 15 10 2
2 2 46 2 149949 15 10 2
3 3 23 2 193090 9 13 4
4 4 55 2 60193 11 9 0
occupation relationship race sex capital_gain capital_loss \
0 2 0 4 1 0.0 0.0
1 3 0 4 1 0.0 0.0
2 2 0 4 1 0.0 0.0
3 0 3 4 0 0.0 0.0
4 0 1 4 0 0.0 0.0
hours_per_week native_country target
0 40 38 0
1 40 38 1
2 40 38 0
3 30 38 0
4 40 38 0
- 앞서 수행한 Pandas Profiling Report의 Alert 섹션을 참고하여 상관계수를 계산했습니다.
- 유의미한 상관관계를 가지고 있다고 생각되는 변수 Pair는
relationship-sex,occupation-workclass,education-education.num입니다.
1
2
# Pearson
la_train[['age','marital_status', 'relationship', 'sex', 'occupation', 'workclass']].corr()
age marital_status relationship sex occupation \
age 1.000000 -0.271955 -0.240331 0.087515 -0.007994
marital_status -0.271955 1.000000 0.180281 -0.124481 0.023856
relationship -0.240331 0.180281 1.000000 -0.590077 -0.052109
sex 0.087515 -0.124481 -0.590077 1.000000 0.061443
occupation -0.007994 0.023856 -0.052109 0.061443 1.000000
workclass 0.081100 -0.044000 -0.070512 0.078764 0.010194
workclass
age 0.081100
marital_status -0.044000
relationship -0.070512
sex 0.078764
occupation 0.010194
workclass 1.000000
1
la_train[['education', 'education_num', 'race', 'native_country']].corr()
education education_num race native_country
education 1.000000 0.348614 0.011236 0.079063
education_num 0.348614 1.000000 0.034686 0.097485
race 0.011236 0.034686 1.000000 0.126654
native_country 0.079063 0.097485 0.126654 1.000000
범주형인 변수는 Cramer’s V 공식을 활용하여 상관관계를 확인했습니다.
\(V = \sqrt{\frac{\chi^2}{N \cdot \min(k-1, r-1)}}\)
- $N$: 전체 관측값의 합
- $k$: 행의 개수
- $r$: 열의 개수
1
2
3
4
5
6
7
stat = stats.chi2_contingency(la_train[['race', 'native_country']].values, correction=False)[0]
obs = np.sum(la_train[['race', 'native_country']].values)
mini = min(la_train[['race', 'native_country']].values.shape)-1
# Cramer's V
V = np.sqrt((stat/(obs*mini)))
print(V)
0.11306993147326666
(2) String to numerical
- 범주형 데이터를 모델의 Input으로 사용하기 위해서는 수치형 데이터로 변환시킬 필요가 있습니다. 라벨 인코더는 불필요한 상관관계를 보일 가능성이 있기에 원핫 인코더(One-hot Encoder)를 사용했습니다.
- Categorical :
workclass,education,marital.status,occupation,relationship,race,sex,native.country
1
2
train_dataset = train_df.copy()
test_dataset = test.copy()
get_dummies를 사용하여 원핫 인코딩을 진행했습니다.
1
2
3
4
5
train_dataset = pd.get_dummies(train_dataset)
test_dataset = pd.get_dummies(test_dataset)
print(train_dataset.columns)
print(test_dataset.columns)
Index(['id', 'age', 'fnlwgt', 'education_num', 'capital_gain', 'capital_loss',
'hours_per_week', 'target', 'workclass_Federal-gov',
'workclass_Local-gov',
...
'native_country_Portugal', 'native_country_Puerto-Rico',
'native_country_Scotland', 'native_country_South',
'native_country_Taiwan', 'native_country_Thailand',
'native_country_Trinadad&Tobago', 'native_country_United-States',
'native_country_Vietnam', 'native_country_Yugoslavia'],
dtype='object', length=106)
Index(['id', 'age', 'fnlwgt', 'education_num', 'capital_gain', 'capital_loss',
'hours_per_week', 'workclass_Federal-gov', 'workclass_Local-gov',
'workclass_Private',
...
'native_country_Portugal', 'native_country_Puerto-Rico',
'native_country_Scotland', 'native_country_South',
'native_country_Taiwan', 'native_country_Thailand',
'native_country_Trinadad&Tobago', 'native_country_United-States',
'native_country_Vietnam', 'native_country_Yugoslavia'],
dtype='object', length=104)
- Train 데이터와 Test 데이터의 열 길이를 맞춰주는 작업을 합니다.
1
2
3
4
5
6
7
8
9
test_col = []
add_test = []
for i in test_dataset.columns:
test_col.append(i)
for j in train_dataset.columns:
if j not in test_col:
add_test.append(j)
add_test.remove('target')
- Test 데이터의
native.country열에는 ‘Holand-Netherlands’ 특성이 없는걸까요?
1
print(add_test)
['native_country_Holand-Netherlands']
1
2
3
4
5
for d in add_test:
test_dataset[d] = 0
print(train_dataset.columns)
print(test_dataset.columns)
Index(['id', 'age', 'fnlwgt', 'education_num', 'capital_gain', 'capital_loss',
'hours_per_week', 'target', 'workclass_Federal-gov',
'workclass_Local-gov',
...
'native_country_Portugal', 'native_country_Puerto-Rico',
'native_country_Scotland', 'native_country_South',
'native_country_Taiwan', 'native_country_Thailand',
'native_country_Trinadad&Tobago', 'native_country_United-States',
'native_country_Vietnam', 'native_country_Yugoslavia'],
dtype='object', length=106)
Index(['id', 'age', 'fnlwgt', 'education_num', 'capital_gain', 'capital_loss',
'hours_per_week', 'workclass_Federal-gov', 'workclass_Local-gov',
'workclass_Private',
...
'native_country_Puerto-Rico', 'native_country_Scotland',
'native_country_South', 'native_country_Taiwan',
'native_country_Thailand', 'native_country_Trinadad&Tobago',
'native_country_United-States', 'native_country_Vietnam',
'native_country_Yugoslavia', 'native_country_Holand-Netherlands'],
dtype='object', length=105)
- Train 데이터의 Target 열을 제외하면, 열 길이가 잘 맞춰진것을 확인할 수 있습니다.
4. Modeling
- 먼저, Train과 Validation 데이터를
train_test_split함수를 사용하여 만들어줍니다.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
test_size =0.15
train_data, val_data = train_test_split(train_dataset, test_size = test_size, random_state = seed_num)
drop_col = ['target', 'id']
train_x = train_data.drop(drop_col, axis = 1)
train_y = pd.DataFrame(train_data['target'])
val_x = val_data.drop(drop_col, axis = 1)
val_y = pd.DataFrame(val_data['target'])
print(train_x.shape, train_y.shape)
print(val_x.shape, val_y.shape)
(12790, 104) (12790, 1)
(2258, 104) (2258, 1)
- LGBM과 XGboost를 Soft Voting하여 간단한 앙상블(Ensemble) 파이프라인을 제작했습니다.
- Soft Voting은 LGBM, XGboost 모델의 예측 확률을 평균 계산하여 최종 Class를 결정합니다.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
LGBClassifier = lgb.LGBMClassifier(random_state = seed_num)
lgbm = LGBClassifier.fit(train_x.values,
train_y.values.ravel(),
eval_set = [(train_x.values, train_y), (val_x.values, val_y)],
eval_metric ='auc', early_stopping_rounds = 1000,
verbose = True)
XGBClassifier = xgb.XGBClassifier(max_depth = 6, learning_rate = 0.01, n_estimators = 10000, random_state = seed_num)
xgb = XGBClassifier.fit(train_x.values,
train_y.values.ravel(),
eval_set = [(train_x.values, train_y), (val_x.values, val_y)],
eval_metric = 'auc', early_stopping_rounds = 1000,
verbose = True)
voting = VotingClassifier(estimators=[('xgb', xgb),('lgbm', lgbm)], voting='soft')
vot = voting.fit(train_x.values, train_y.values)
5. Evaluation & Submission
1
2
3
4
5
6
7
l_val_y_pred = lgbm.predict(val_x.values)
x_val_y_pred = xgb.predict(val_x.values)
v_val_y_pred = vot.predict(val_x.values)
print(metrics.accuracy_score(l_val_y_pred, val_y))
print(metrics.accuracy_score(x_val_y_pred, val_y))
print(metrics.accuracy_score(v_val_y_pred, val_y))
0.8702391496899912
0.8680248007085917
0.8596102745792737
1
print(metrics.classification_report(v_val_y_pred, val_y))
precision recall f1-score support
0 0.93 0.89 0.91 1800
1 0.63 0.72 0.68 458
accuracy 0.86 2258
macro avg 0.78 0.81 0.79 2258
weighted avg 0.87 0.86 0.86 2258
1
2
3
4
5
6
val_xgb = pd.Series(l_val_y_pred, name="XGB")
val_lgbm = pd.Series(x_val_y_pred, name="LGBM")
ensemble_results = pd.concat([val_xgb,val_lgbm],axis=1)
sns.heatmap(ensemble_results.corr(), annot=True)
plt.show()
- Soft Voting을 진행했음에도 성능이 향상되지 않았습니다.
- 두 모델의 예측은 높은 상관관계를 가지기 때문에, 앙상블을 해도 성능이 향상되지 않는 것이라 예상해봅니다.
This post is licensed under CC BY 4.0 by the author.