Boom Bikes Linear Regression

Sachin Shekhar
SachinShekhar@Outlook.com

# Importing essential libraries
import numpy as np
import pandas as pd

from matplotlib import pyplot as plt
import seaborn as sns

from sklearn.model_selection import train_test_split
from sklearn.preprocessing import MinMaxScaler
from sklearn.feature_selection import RFE
from sklearn.linear_model import LinearRegression
from sklearn.metrics import r2_score
import statsmodels.api as sm
from statsmodels.stats.outliers_influence import variance_inflation_factor

import warnings
warnings.filterwarnings('ignore')

Exploratory Data Analysis

Loading Data

# Loading data
df = pd.read_csv('day.csv')
df.head()

	instant	dteday	season	yr	mnth	holiday	weekday	workingday	weathersit	temp	atemp	hum	windspeed	casual	registered	cnt
0	1	01-01-2018	1	0	1	0	1	1	2	14.110847	18.18125	80.5833	10.749882	331	654	985
1	2	02-01-2018	1	0	1	0	2	1	2	14.902598	17.68695	69.6087	16.652113	131	670	801
2	3	03-01-2018	1	0	1	0	3	1	1	8.050924	9.47025	43.7273	16.636703	120	1229	1349
3	4	04-01-2018	1	0	1	0	4	1	1	8.200000	10.60610	59.0435	10.739832	108	1454	1562
4	5	05-01-2018	1	0	1	0	5	1	1	9.305237	11.46350	43.6957	12.522300	82	1518	1600

df.shape

(730, 16)

# Creating list of categorical & numerical variables for easy access
cat_vars = ['season', 'yr', 'mnth', 'holiday', 'weekday', 'workingday', 'weathersit']
num_vars = ['dteday', 'temp', 'atemp', 'hum', 'windspeed', 'casual', 'registered', 'cnt']

Cleaning Data

Missing Values

df.isnull().sum()

instant       0
dteday        0
season        0
yr            0
mnth          0
holiday       0
weekday       0
workingday    0
weathersit    0
temp          0
atemp         0
hum           0
windspeed     0
casual        0
registered    0
cnt           0
dtype: int64

Duplicate Values

df.duplicated().sum()

0

Non-essential Features

# 'instant' is an index which won't contribute to the dependent variable change
df.drop('instant', axis=1, inplace=True)

# 'cnt' variable includes counts of 'casual' and 'registered'
df.drop(['casual', 'registered'], axis=1, inplace=True)
num_vars.remove('casual')
num_vars.remove('registered')

Data Types

df.dtypes

dteday         object
season          int64
yr              int64
mnth            int64
holiday         int64
weekday         int64
workingday      int64
weathersit      int64
temp          float64
atemp         float64
hum           float64
windspeed     float64
cnt             int64
dtype: object

# Changing datatype of 'dteday' to datetime64
df['dteday'] = pd.to_datetime(df['dteday'], format='%d-%m-%Y')

# Converting values of 'dteday' to absolute unix epoch day for easy analysis
df['dteday'] = (df['dteday'] - pd.Timestamp('1970-01-01')) // pd.Timedelta('1d')

df['dteday'].dtypes

dtype('int64')

# Converting categorical variable values into strings
df['season'].replace({1:'spring', 2:'summer', 3:'fall', 4:'winter'}, inplace=True)
df['yr'].replace({0: '2018', 1:'2019'}, inplace=True)
df['mnth'].replace({1: 'january', 2: 'february', 3: 'march', 4: 'april', 5: 'may', 6: 'june', 7: 'july', 8: 'august', 9: 'september', 10: 'october', 11: 'november', 12: 'december'}, inplace=True)
df['holiday'].replace({0: 'no', 1: 'yes'}, inplace=True)
df['weekday'].replace({0: 'sunday', 1: 'monday', 2: 'tuesday', 3: 'wednesday', 4: 'thursday', 5: 'friday', 6: 'saturday'}, inplace=True)
df['workingday'].replace({0: 'no', 1: 'yes'}, inplace=True)
df['weathersit'].replace({1: 'clear', 2: 'misty', 3: 'light_weather', 4: 'heavy_weather'}, inplace=True)

Univariate Analysis

Numerical Variables

# Helper functions for Numerical Univariate Analysis
def num_uni_box_analysis(var, friendly_name):
  ax = sns.boxplot(data=df, x=var)
  ax.set_xticks(list(df[var].quantile([0,0.25,0.5,0.75,0.95,1])))
  ax.set_xlabel(friendly_name)
  ax.set_title(f'{friendly_name} Distribution Univariate Analysis', fontsize=20)
  ax.figure.set_size_inches(24,8)

def num_uni_hist_analysis(var, friendly_name):
  ax = sns.histplot(data=df ,x=var, stat='percent')
  ax.set_xlabel(friendly_name)
  ax.set_title(f'{friendly_name} Distribution Histogram', fontsize=20)
  ax.figure.set_size_inches(24,8)

df.describe()

	dteday	temp	atemp	hum	windspeed	cnt
count	730.000000	730.000000	730.000000	730.000000	730.000000	730.000000
mean	17896.500000	20.319259	23.726322	62.765175	12.763620	4508.006849
std	210.877136	7.506729	8.150308	14.237589	5.195841	1936.011647
min	17532.000000	2.424346	3.953480	0.000000	1.500244	22.000000
25%	17714.250000	13.811885	16.889713	52.000000	9.041650	3169.750000
50%	17896.500000	20.465826	24.368225	62.625000	12.125325	4548.500000
75%	18078.750000	26.880615	30.445775	72.989575	15.625589	5966.000000
max	18261.000000	35.328347	42.044800	97.250000	34.000021	8714.000000

Feature: cnt (Target Variable)

num_uni_box_analysis('cnt', 'Rider Count')
plt.show()

num_uni_hist_analysis('cnt', 'Rider Count')
plt.show()

Feature: dteday

num_uni_box_analysis('dteday', 'Date')
plt.show()

pd.to_datetime((17896.5 * pd.Timedelta('1d')) + pd.Timestamp('1970-01-01'), unit='s')

Timestamp('2018-12-31 12:00:00')

The median makes sense as the data was collected for two years.

num_uni_hist_analysis('dteday', 'Date')
plt.show()

This uniform distribution makes sense as data was recorded every day.

Feature: temp

num_uni_box_analysis('temp', 'Temperature')
plt.show()

num_uni_hist_analysis('temp', 'Temperature')
plt.show()

Feature: atemp

num_uni_box_analysis('atemp', 'Feel Temperature')
plt.show()

num_uni_hist_analysis('atemp', 'Feel Temperature')
plt.show()

Feature: hum

num_uni_box_analysis('hum', 'Humidity')
plt.show()

num_uni_hist_analysis('hum', 'Humidity')
plt.show()

Feature: windspeed

num_uni_box_analysis('windspeed', 'Wind Speed')
plt.show()

num_uni_hist_analysis('windspeed', 'Wind Speed')
plt.show()

Assessment:
Data present in numerical variables make sense.

Categorical Variables

# Helper function for Categorical Univariate Analysis
def cat_uni_analysis(var, friendly_name):
  tmp_df = df[var].value_counts(normalize=True).mul(100).rename('Percent').reset_index().rename(columns={'index': var})
  ax = sns.barplot(data=tmp_df, x=var, y='Percent')
  for p in ax.patches:
    txt = str(p.get_height().round(2)) + '%'
    txt_x = p.get_x() + (p.get_width()/2)
    txt_y = p.get_height()
    ax.annotate(txt, (txt_x, txt_y), size=11, ha='center', va='bottom')
  ax.set_title(f'{friendly_name} Univariate Analysis (Normalised)', fontsize=20)
  ax.set_xlabel(friendly_name)
  ax.figure.set_size_inches(16,8)
  return ax

cat_uni_analysis('season', 'Season')
plt.show()

cat_uni_analysis('yr', 'Year')
plt.show()

cat_uni_analysis('mnth', 'Month')
plt.show()

cat_uni_analysis('holiday', 'Holiday')
plt.show()

cat_uni_analysis('weekday', 'Weekday')
plt.show()

cat_uni_analysis('workingday', 'Working Day')
plt.show()

cat_uni_analysis('weathersit', 'Weather')
plt.show()

Assessment:
Data present in categorical variables make sense.

Bivariate Analysis

Numerical Variables

ax = sns.pairplot(df)
ax.figure.set_size_inches(24,20)

Categorical Variables

# Helper function for Categorical Bivariate Analysis
def cat_bi_analysis(var, friendly_name):
  ax = sns.boxplot(data=df, x=var, y='cnt')
  ax.figure.set_size_inches(24,12)
  ax.set_title(f'{friendly_name} vs Rental Count', fontsize=20)
  return ax

Feature: season

cat_bi_analysis('season', 'Season')
plt.show()

Assessment:

• Spring has less number of rentals than other seasons.
• Summer & fall have comparable number of highest rentals.

Feature: yr

ax = cat_bi_analysis('yr', 'Year')
ax.figure.set_size_inches(16,12)
plt.show()

Assessment:
2019 saw increase in rentals from 2018.

Feature: mnth

cat_bi_analysis('mnth', 'Month')
plt.show()

Assessment:
Starting January, no. of rentals increase until June. Then, it stays steady for several months before falling down starting October.

Feature: holiday

ax = cat_bi_analysis('holiday', 'Holiday')
ax.figure.set_size_inches(16,12)
plt.show()

Assessment:
When there is no holiday, rental count is more consistent.

Feature: weekday

cat_bi_analysis('weekday', 'Weekday')
plt.show()

Assessment:
Weekdays don't say anything special about no. of rentals. Although, thursdays see more consistency in rentals.

Feature: workingday

ax = cat_bi_analysis('workingday', 'Working Day')
ax.figure.set_size_inches(16,12)
plt.show()

Assessment:
Workingday doesn't say anything about rental counts.

Feature: weathersit

cat_bi_analysis('weathersit', 'Weather')
plt.show()

Assessment:

• When there's light snow or light rain, rental counts drop significantly.
• Clear weather sees most rental counts.
• Misty weather sees moderate rental counts.

Correlations

# Heatmap
ax = sns.heatmap(df.corr(), cmap='RdYlGn', annot=True)
ax.figure.set_size_inches(12,10)
plt.show()

Data Preparation

Dummy Variables Creation

cat_dum = pd.get_dummies(df[cat_vars], drop_first=True)
df.drop(columns=cat_vars, inplace=True)
df = pd.concat([df, cat_dum], axis=1)
df.columns

Index(['dteday', 'temp', 'atemp', 'hum', 'windspeed', 'cnt', 'season_spring',
       'season_summer', 'season_winter', 'yr_2019', 'mnth_august',
       'mnth_december', 'mnth_february', 'mnth_january', 'mnth_july',
       'mnth_june', 'mnth_march', 'mnth_may', 'mnth_november', 'mnth_october',
       'mnth_september', 'holiday_yes', 'weekday_monday', 'weekday_saturday',
       'weekday_sunday', 'weekday_thursday', 'weekday_tuesday',
       'weekday_wednesday', 'workingday_yes', 'weathersit_light_weather',
       'weathersit_misty'],
      dtype='object')

Train-Test Split

df_train, df_test = train_test_split(df, train_size=0.7, random_state=100)

df_train.shape

(510, 31)

df_test.shape

(220, 31)

Rescaling Training Set

# Learn Min & Max values
scaler = MinMaxScaler()
scaler.fit(df_train[num_vars])

MinMaxScaler()

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# Scale
df_train[num_vars] = scaler.transform(df_train[num_vars])
df_train.head()

	dteday	temp	atemp	hum	windspeed	cnt	season_spring	season_summer	season_winter	yr_2019	...	holiday_yes	weekday_monday	weekday_saturday	weekday_sunday	weekday_thursday	weekday_tuesday	weekday_wednesday	workingday_yes	weathersit_light_weather	weathersit_misty
576	0.791209	0.815169	0.766351	0.725633	0.264686	0.827658	0	0	0	1	...	0	0	0	0	0	0	1	1	0	0
426	0.585165	0.442393	0.438975	0.640189	0.255342	0.465255	1	0	0	1	...	0	0	0	1	0	0	0	0	0	1
728	1.000000	0.245101	0.200348	0.498067	0.663106	0.204096	1	0	0	1	...	0	1	0	0	0	0	0	1	0	0
482	0.662088	0.395666	0.391735	0.504508	0.188475	0.482973	0	1	0	1	...	0	0	0	1	0	0	0	0	0	1
111	0.152473	0.345824	0.318819	0.751824	0.380981	0.191095	0	1	0	0	...	0	0	0	1	0	0	0	0	0	1

5 rows × 31 columns

Model Training

# Creating function to easily calculate VIF
def calculate_vif(x_df):
  vif_df = pd.DataFrame({'Feature': x_df.columns, 'VIF': [ variance_inflation_factor(x_df.values, i) for i in range(x_df.shape[1])]})
  vif_df['VIF'] = vif_df['VIF'].round(2)
  vif_df = vif_df.sort_values(by='VIF', ascending=False)
  return vif_df

# Creating function to easily train models
def train_model(y_df, x_df):
  # Preparing for intercept
  x_df_sm = sm.add_constant(x_df)
  # Training model
  lr = sm.OLS(y_df, x_df_sm)
  lr_model = lr.fit()
  return lr_model

# Extract Target Variable
y_train = df_train.pop('cnt')
X_train = df_train

# Creating function to easily select features
def auto_select_features(n, x_df=X_train):
  lm = LinearRegression()
  lm.fit(x_df, y_train)
  selector = RFE(lm, n_features_to_select=n)
  selector = selector.fit(x_df, y_train)
  top_features_df = pd.DataFrame({'Feature': x_df.columns, 'Selected': selector.support_, 'Rank': selector.ranking_})
  selected_vars = top_features_df[top_features_df['Selected'] == True]['Feature'].to_list()
  return selected_vars

Model 1

X_train_1 = X_train[auto_select_features(15)]

train_model(y_train, X_train_1).summary()

OLS Regression Results

Dep. Variable:	cnt	R-squared:	0.845
Model:	OLS	Adj. R-squared:	0.840
Method:	Least Squares	F-statistic:	179.4
Date:	Tue, 30 Aug 2022	Prob (F-statistic):	8.57e-189
Time:	01:25:12	Log-Likelihood:	514.13
No. Observations:	510	AIC:	-996.3
Df Residuals:	494	BIC:	-928.5
Df Model:	15
Covariance Type:	nonrobust

	coef	std err	t	P>|t|	[0.025	0.975]
const	0.2876	0.033	8.733	0.000	0.223	0.352
dteday	-0.0750	0.064	-1.168	0.244	-0.201	0.051
temp	0.4920	0.028	17.658	0.000	0.437	0.547
hum	-0.1698	0.038	-4.525	0.000	-0.243	-0.096
windspeed	-0.1935	0.026	-7.541	0.000	-0.244	-0.143
season_summer	0.0875	0.013	6.684	0.000	0.062	0.113
season_winter	0.1296	0.019	6.807	0.000	0.092	0.167
yr_2019	0.2675	0.033	8.143	0.000	0.203	0.332
mnth_august	0.0586	0.017	3.500	0.001	0.026	0.092
mnth_february	-0.0392	0.023	-1.711	0.088	-0.084	0.006
mnth_january	-0.0671	0.024	-2.785	0.006	-0.114	-0.020
mnth_october	0.0373	0.018	2.104	0.036	0.002	0.072
mnth_september	0.1272	0.017	7.592	0.000	0.094	0.160
holiday_yes	-0.0897	0.025	-3.518	0.000	-0.140	-0.040
weathersit_light_weather	-0.2443	0.026	-9.291	0.000	-0.296	-0.193
weathersit_misty	-0.0541	0.010	-5.187	0.000	-0.075	-0.034

Omnibus:	67.981	Durbin-Watson:	2.036
Prob(Omnibus):	0.000	Jarque-Bera (JB):	155.260
Skew:	-0.713	Prob(JB):	1.93e-34
Kurtosis:	5.296	Cond. No.	31.9

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

calculate_vif(X_train_1)

	Feature	VIF
0	dteday	73.85
6	yr_2019	30.19
2	hum	26.58
1	temp	14.17
5	season_winter	5.67
3	windspeed	4.05
4	season_summer	2.43
9	mnth_january	2.29
14	weathersit_misty	2.26
8	mnth_february	1.72
7	mnth_august	1.69
10	mnth_october	1.67
11	mnth_september	1.41
13	weathersit_light_weather	1.24
12	holiday_yes	1.05

# Dropping 'dteday' because it has both high p-value & high VIF
X_train_1 = X_train_1.drop('dteday', axis=1)

# Re-train model
train_model(y_train, X_train_1).summary()

OLS Regression Results

Dep. Variable:	cnt	R-squared:	0.844
Model:	OLS	Adj. R-squared:	0.840
Method:	Least Squares	F-statistic:	192.0
Date:	Tue, 30 Aug 2022	Prob (F-statistic):	1.20e-189
Time:	01:25:13	Log-Likelihood:	513.43
No. Observations:	510	AIC:	-996.9
Df Residuals:	495	BIC:	-933.3
Df Model:	14
Covariance Type:	nonrobust

	coef	std err	t	P>|t|	[0.025	0.975]
const	0.2721	0.030	9.027	0.000	0.213	0.331
temp	0.4899	0.028	17.612	0.000	0.435	0.545
hum	-0.1741	0.037	-4.662	0.000	-0.247	-0.101
windspeed	-0.1922	0.026	-7.495	0.000	-0.243	-0.142
season_summer	0.0935	0.012	7.757	0.000	0.070	0.117
season_winter	0.1153	0.015	7.890	0.000	0.087	0.144
yr_2019	0.2303	0.008	28.508	0.000	0.214	0.246
mnth_august	0.0545	0.016	3.327	0.001	0.022	0.087
mnth_february	-0.0260	0.020	-1.304	0.193	-0.065	0.013
mnth_january	-0.0508	0.020	-2.586	0.010	-0.089	-0.012
mnth_october	0.0405	0.018	2.311	0.021	0.006	0.075
mnth_september	0.1232	0.016	7.508	0.000	0.091	0.155
holiday_yes	-0.0915	0.025	-3.593	0.000	-0.141	-0.041
weathersit_light_weather	-0.2429	0.026	-9.245	0.000	-0.295	-0.191
weathersit_misty	-0.0534	0.010	-5.124	0.000	-0.074	-0.033

Omnibus:	70.142	Durbin-Watson:	2.046
Prob(Omnibus):	0.000	Jarque-Bera (JB):	165.258
Skew:	-0.724	Prob(JB):	1.30e-36
Kurtosis:	5.384	Cond. No.	18.6

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

calculate_vif(X_train_1)

	Feature	VIF
1	hum	21.40
0	temp	13.18
2	windspeed	3.88
4	season_winter	2.98
3	season_summer	2.20
13	weathersit_misty	2.19
5	yr_2019	2.06
8	mnth_january	1.79
6	mnth_august	1.63
9	mnth_october	1.60
7	mnth_february	1.47
10	mnth_september	1.37
12	weathersit_light_weather	1.22
11	holiday_yes	1.05

# Dropping 'mnth_february' because it has high p-value
X_train_1 = X_train_1.drop('mnth_february', axis=1)

# Re-train model
train_model(y_train, X_train_1).summary()

OLS Regression Results

Dep. Variable:	cnt	R-squared:	0.844
Model:	OLS	Adj. R-squared:	0.840
Method:	Least Squares	F-statistic:	206.3
Date:	Tue, 30 Aug 2022	Prob (F-statistic):	1.90e-190
Time:	01:25:14	Log-Likelihood:	512.56
No. Observations:	510	AIC:	-997.1
Df Residuals:	496	BIC:	-937.8
Df Model:	13
Covariance Type:	nonrobust

	coef	std err	t	P>|t|	[0.025	0.975]
const	0.2586	0.028	9.131	0.000	0.203	0.314
temp	0.5058	0.025	20.204	0.000	0.457	0.555
hum	-0.1759	0.037	-4.709	0.000	-0.249	-0.102
windspeed	-0.1927	0.026	-7.511	0.000	-0.243	-0.142
season_summer	0.0992	0.011	8.831	0.000	0.077	0.121
season_winter	0.1237	0.013	9.433	0.000	0.098	0.150
yr_2019	0.2299	0.008	28.458	0.000	0.214	0.246
mnth_august	0.0568	0.016	3.485	0.001	0.025	0.089
mnth_january	-0.0392	0.018	-2.236	0.026	-0.074	-0.005
mnth_october	0.0388	0.017	2.220	0.027	0.004	0.073
mnth_september	0.1255	0.016	7.684	0.000	0.093	0.158
holiday_yes	-0.0932	0.025	-3.664	0.000	-0.143	-0.043
weathersit_light_weather	-0.2420	0.026	-9.208	0.000	-0.294	-0.190
weathersit_misty	-0.0532	0.010	-5.103	0.000	-0.074	-0.033

Omnibus:	66.371	Durbin-Watson:	2.046
Prob(Omnibus):	0.000	Jarque-Bera (JB):	154.759
Skew:	-0.691	Prob(JB):	2.48e-34
Kurtosis:	5.318	Cond. No.	18.5

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

calculate_vif(X_train_1)

	Feature	VIF
1	hum	18.19
0	temp	11.70
2	windspeed	3.62
4	season_winter	2.57
12	weathersit_misty	2.16
5	yr_2019	2.04
3	season_summer	1.97
6	mnth_august	1.61
8	mnth_october	1.60
7	mnth_january	1.55
9	mnth_september	1.35
11	weathersit_light_weather	1.21
10	holiday_yes	1.04

# Dropping 'hum' because it has high VIF
X_train_1 = X_train_1.drop('hum', axis=1)

# Re-train model
train_model(y_train, X_train_1).summary()

OLS Regression Results

Dep. Variable:	cnt	R-squared:	0.837
Model:	OLS	Adj. R-squared:	0.833
Method:	Least Squares	F-statistic:	212.6
Date:	Tue, 30 Aug 2022	Prob (F-statistic):	6.26e-187
Time:	01:25:14	Log-Likelihood:	501.40
No. Observations:	510	AIC:	-976.8
Df Residuals:	497	BIC:	-921.8
Df Model:	12
Covariance Type:	nonrobust

	coef	std err	t	P>|t|	[0.025	0.975]
const	0.1610	0.020	8.169	0.000	0.122	0.200
temp	0.4810	0.025	19.248	0.000	0.432	0.530
windspeed	-0.1597	0.025	-6.338	0.000	-0.209	-0.110
season_summer	0.0952	0.011	8.321	0.000	0.073	0.118
season_winter	0.1136	0.013	8.600	0.000	0.088	0.140
yr_2019	0.2339	0.008	28.522	0.000	0.218	0.250
mnth_august	0.0549	0.017	3.303	0.001	0.022	0.088
mnth_january	-0.0450	0.018	-2.524	0.012	-0.080	-0.010
mnth_october	0.0347	0.018	1.948	0.052	-0.000	0.070
mnth_september	0.1185	0.017	7.135	0.000	0.086	0.151
holiday_yes	-0.0945	0.026	-3.638	0.000	-0.146	-0.043
weathersit_light_weather	-0.2892	0.025	-11.656	0.000	-0.338	-0.240
weathersit_misty	-0.0810	0.009	-9.230	0.000	-0.098	-0.064

Omnibus:	64.117	Durbin-Watson:	2.056
Prob(Omnibus):	0.000	Jarque-Bera (JB):	148.090
Skew:	-0.672	Prob(JB):	6.96e-33
Kurtosis:	5.272	Cond. No.	12.2

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

calculate_vif(X_train_1)

	Feature	VIF
0	temp	5.15
1	windspeed	3.35
4	yr_2019	2.04
3	season_winter	1.99
2	season_summer	1.91
5	mnth_august	1.61
7	mnth_october	1.59
11	weathersit_misty	1.54
8	mnth_september	1.34
6	mnth_january	1.22
10	weathersit_light_weather	1.10
9	holiday_yes	1.04

# Dropping 'mnth_october' because it has high p-value
X_train_1 = X_train_1.drop('mnth_october', axis=1)

# Re-train model
train_model(y_train, X_train_1).summary()

OLS Regression Results

Dep. Variable:	cnt	R-squared:	0.836
Model:	OLS	Adj. R-squared:	0.832
Method:	Least Squares	F-statistic:	230.3
Date:	Tue, 30 Aug 2022	Prob (F-statistic):	2.65e-187
Time:	01:25:15	Log-Likelihood:	499.46
No. Observations:	510	AIC:	-974.9
Df Residuals:	498	BIC:	-924.1
Df Model:	11
Covariance Type:	nonrobust

	coef	std err	t	P>|t|	[0.025	0.975]
const	0.1550	0.020	7.942	0.000	0.117	0.193
temp	0.4911	0.025	20.028	0.000	0.443	0.539
windspeed	-0.1570	0.025	-6.223	0.000	-0.207	-0.107
season_summer	0.0941	0.011	8.212	0.000	0.072	0.117
season_winter	0.1261	0.012	10.869	0.000	0.103	0.149
yr_2019	0.2335	0.008	28.400	0.000	0.217	0.250
mnth_august	0.0520	0.017	3.131	0.002	0.019	0.085
mnth_january	-0.0422	0.018	-2.368	0.018	-0.077	-0.007
mnth_september	0.1137	0.016	6.902	0.000	0.081	0.146
holiday_yes	-0.0957	0.026	-3.674	0.000	-0.147	-0.045
weathersit_light_weather	-0.2834	0.025	-11.473	0.000	-0.332	-0.235
weathersit_misty	-0.0801	0.009	-9.118	0.000	-0.097	-0.063

Omnibus:	56.180	Durbin-Watson:	2.067
Prob(Omnibus):	0.000	Jarque-Bera (JB):	121.643
Skew:	-0.614	Prob(JB):	3.85e-27
Kurtosis:	5.053	Cond. No.	11.8

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

calculate_vif(X_train_1)

	Feature	VIF
0	temp	5.05
1	windspeed	3.34
4	yr_2019	2.04
2	season_summer	1.89
5	mnth_august	1.59
3	season_winter	1.55
10	weathersit_misty	1.54
7	mnth_september	1.31
6	mnth_january	1.22
9	weathersit_light_weather	1.08
8	holiday_yes	1.04

lr_model_1 = train_model(y_train, X_train_1)

Model 2

X_train_2 = X_train_1.drop('temp', axis=1)

train_model(y_train, X_train_2).summary()

OLS Regression Results

Dep. Variable:	cnt	R-squared:	0.703
Model:	OLS	Adj. R-squared:	0.697
Method:	Least Squares	F-statistic:	118.3
Date:	Tue, 30 Aug 2022	Prob (F-statistic):	8.53e-125
Time:	01:25:15	Log-Likelihood:	348.80
No. Observations:	510	AIC:	-675.6
Df Residuals:	499	BIC:	-629.0
Df Model:	10
Covariance Type:	nonrobust

	coef	std err	t	P>|t|	[0.025	0.975]
const	0.4553	0.017	27.130	0.000	0.422	0.488
windspeed	-0.2593	0.033	-7.815	0.000	-0.324	-0.194
season_summer	0.1169	0.015	7.637	0.000	0.087	0.147
season_winter	0.0686	0.015	4.548	0.000	0.039	0.098
yr_2019	0.2502	0.011	22.795	0.000	0.229	0.272
mnth_august	0.1711	0.021	8.218	0.000	0.130	0.212
mnth_january	-0.2037	0.021	-9.541	0.000	-0.246	-0.162
mnth_september	0.1936	0.021	9.028	0.000	0.151	0.236
holiday_yes	-0.1134	0.035	-3.245	0.001	-0.182	-0.045
weathersit_light_weather	-0.2900	0.033	-8.747	0.000	-0.355	-0.225
weathersit_misty	-0.0989	0.012	-8.429	0.000	-0.122	-0.076

Omnibus:	5.753	Durbin-Watson:	1.932
Prob(Omnibus):	0.056	Jarque-Bera (JB):	5.805
Skew:	-0.200	Prob(JB):	0.0549
Kurtosis:	3.337	Cond. No.	8.87

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

calculate_vif(X_train_2)

	Feature	VIF
0	windspeed	2.60
3	yr_2019	1.84
1	season_summer	1.63
9	weathersit_misty	1.53
2	season_winter	1.47
5	mnth_january	1.21
4	mnth_august	1.19
6	mnth_september	1.12
8	weathersit_light_weather	1.08
7	holiday_yes	1.04

lr_model_2 = train_model(y_train, X_train_2)

Model 3

# Starting without 'dteday'
X_train_3 = X_train.drop('dteday', axis=1)
X_train_3 = X_train_3[auto_select_features(15, X_train_3)]

# Train model
train_model(y_train, X_train_3).summary()

OLS Regression Results

Dep. Variable:	cnt	R-squared:	0.845
Model:	OLS	Adj. R-squared:	0.840
Method:	Least Squares	F-statistic:	179.4
Date:	Tue, 30 Aug 2022	Prob (F-statistic):	8.15e-189
Time:	01:25:16	Log-Likelihood:	514.19
No. Observations:	510	AIC:	-996.4
Df Residuals:	494	BIC:	-928.6
Df Model:	15
Covariance Type:	nonrobust

	coef	std err	t	P>|t|	[0.025	0.975]
const	0.3197	0.036	8.859	0.000	0.249	0.391
temp	0.4815	0.037	13.005	0.000	0.409	0.554
hum	-0.1622	0.038	-4.291	0.000	-0.236	-0.088
windspeed	-0.1887	0.026	-7.315	0.000	-0.239	-0.138
season_spring	-0.0613	0.021	-2.881	0.004	-0.103	-0.019
season_summer	0.0423	0.015	2.761	0.006	0.012	0.072
season_winter	0.1019	0.018	5.656	0.000	0.067	0.137
yr_2019	0.2304	0.008	28.487	0.000	0.215	0.246
mnth_december	-0.0355	0.018	-2.024	0.043	-0.070	-0.001
mnth_january	-0.0434	0.018	-2.393	0.017	-0.079	-0.008
mnth_july	-0.0553	0.018	-3.030	0.003	-0.091	-0.019
mnth_november	-0.0387	0.019	-2.057	0.040	-0.076	-0.002
mnth_september	0.0755	0.017	4.466	0.000	0.042	0.109
holiday_yes	-0.0911	0.026	-3.557	0.000	-0.141	-0.041
weathersit_light_weather	-0.2465	0.026	-9.331	0.000	-0.298	-0.195
weathersit_misty	-0.0543	0.010	-5.194	0.000	-0.075	-0.034

Omnibus:	66.656	Durbin-Watson:	2.025
Prob(Omnibus):	0.000	Jarque-Bera (JB):	161.040
Skew:	-0.682	Prob(JB):	1.07e-35
Kurtosis:	5.392	Cond. No.	20.8

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

calculate_vif(X_train_3)

	Feature	VIF
1	hum	30.89
0	temp	17.79
2	windspeed	4.72
3	season_spring	4.37
5	season_winter	4.06
4	season_summer	2.82
14	weathersit_misty	2.32
6	yr_2019	2.09
10	mnth_november	1.85
8	mnth_january	1.75
9	mnth_july	1.59
7	mnth_december	1.56
11	mnth_september	1.41
13	weathersit_light_weather	1.28
12	holiday_yes	1.06

# Dropping 'hum' due to high VIF
X_train_3 = X_train_3.drop('hum', axis=1)

# Re-train model
train_model(y_train, X_train_3).summary()

OLS Regression Results

Dep. Variable:	cnt	R-squared:	0.839
Model:	OLS	Adj. R-squared:	0.835
Method:	Least Squares	F-statistic:	184.5
Date:	Tue, 30 Aug 2022	Prob (F-statistic):	4.75e-186
Time:	01:25:17	Log-Likelihood:	504.85
No. Observations:	510	AIC:	-979.7
Df Residuals:	495	BIC:	-916.2
Df Model:	14
Covariance Type:	nonrobust

	coef	std err	t	P>|t|	[0.025	0.975]
const	0.2450	0.032	7.618	0.000	0.182	0.308
temp	0.4387	0.036	12.093	0.000	0.367	0.510
windspeed	-0.1585	0.025	-6.276	0.000	-0.208	-0.109
season_spring	-0.0713	0.021	-3.314	0.001	-0.113	-0.029
season_summer	0.0349	0.015	2.251	0.025	0.004	0.065
season_winter	0.0869	0.018	4.831	0.000	0.052	0.122
yr_2019	0.2345	0.008	28.687	0.000	0.218	0.251
mnth_december	-0.0428	0.018	-2.413	0.016	-0.078	-0.008
mnth_january	-0.0500	0.018	-2.719	0.007	-0.086	-0.014
mnth_july	-0.0500	0.019	-2.703	0.007	-0.086	-0.014
mnth_november	-0.0395	0.019	-2.064	0.040	-0.077	-0.002
mnth_september	0.0687	0.017	4.015	0.000	0.035	0.102
holiday_yes	-0.0918	0.026	-3.522	0.000	-0.143	-0.041
weathersit_light_weather	-0.2917	0.025	-11.840	0.000	-0.340	-0.243
weathersit_misty	-0.0801	0.009	-9.198	0.000	-0.097	-0.063

Omnibus:	69.242	Durbin-Watson:	2.024
Prob(Omnibus):	0.000	Jarque-Bera (JB):	171.476
Skew:	-0.698	Prob(JB):	5.81e-38
Kurtosis:	5.473	Cond. No.	18.9

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

calculate_vif(X_train_3)

	Feature	VIF
0	temp	5.17
1	windspeed	4.67
4	season_winter	2.95
2	season_spring	2.89
3	season_summer	2.24
5	yr_2019	2.07
9	mnth_november	1.81
7	mnth_january	1.66
8	mnth_july	1.59
13	weathersit_misty	1.57
6	mnth_december	1.47
10	mnth_september	1.35
12	weathersit_light_weather	1.09
11	holiday_yes	1.06

lr_model_3 = train_model(y_train, X_train_3)

Model 4

# Continuing further with Model 3
X_train_4 = X_train_3.drop('temp', axis=1)

# Train model
train_model(y_train, X_train_4).summary()

OLS Regression Results

Dep. Variable:	cnt	R-squared:	0.792
Model:	OLS	Adj. R-squared:	0.786
Method:	Least Squares	F-statistic:	144.9
Date:	Tue, 30 Aug 2022	Prob (F-statistic):	1.82e-159
Time:	01:25:17	Log-Likelihood:	438.84
No. Observations:	510	AIC:	-849.7
Df Residuals:	496	BIC:	-790.4
Df Model:	13
Covariance Type:	nonrobust

	coef	std err	t	P>|t|	[0.025	0.975]
const	0.5954	0.016	37.460	0.000	0.564	0.627
windspeed	-0.1902	0.029	-6.662	0.000	-0.246	-0.134
season_spring	-0.2503	0.018	-14.125	0.000	-0.285	-0.216
season_summer	-0.0497	0.016	-3.160	0.002	-0.081	-0.019
season_winter	-0.0231	0.018	-1.307	0.192	-0.058	0.012
yr_2019	0.2457	0.009	26.616	0.000	0.228	0.264
mnth_december	-0.1086	0.019	-5.660	0.000	-0.146	-0.071
mnth_january	-0.1202	0.020	-6.056	0.000	-0.159	-0.081
mnth_july	-0.0182	0.021	-0.875	0.382	-0.059	0.023
mnth_november	-0.0995	0.021	-4.731	0.000	-0.141	-0.058
mnth_september	0.0534	0.019	2.750	0.006	0.015	0.092
holiday_yes	-0.0855	0.030	-2.887	0.004	-0.144	-0.027
weathersit_light_weather	-0.3117	0.028	-11.150	0.000	-0.367	-0.257
weathersit_misty	-0.0869	0.010	-8.800	0.000	-0.106	-0.068

Omnibus:	56.274	Durbin-Watson:	1.943
Prob(Omnibus):	0.000	Jarque-Bera (JB):	132.272
Skew:	-0.588	Prob(JB):	1.89e-29
Kurtosis:	5.200	Cond. No.	9.27

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

calculate_vif(X_train_4)

	Feature	VIF
0	windspeed	3.89
1	season_spring	2.89
3	season_winter	2.85
2	season_summer	2.02
4	yr_2019	1.83
8	mnth_november	1.80
6	mnth_january	1.64
12	weathersit_misty	1.53
5	mnth_december	1.46
7	mnth_july	1.20
9	mnth_september	1.18
11	weathersit_light_weather	1.09
10	holiday_yes	1.06

# Dropping 'mnth_july' due to high p-value
X_train_4 = X_train_4.drop('mnth_july', axis=1)

# Re-train model
train_model(y_train, X_train_4).summary()

OLS Regression Results

Dep. Variable:	cnt	R-squared:	0.791
Model:	OLS	Adj. R-squared:	0.786
Method:	Least Squares	F-statistic:	157.0
Date:	Tue, 30 Aug 2022	Prob (F-statistic):	2.08e-160
Time:	01:25:18	Log-Likelihood:	438.45
No. Observations:	510	AIC:	-850.9
Df Residuals:	497	BIC:	-795.9
Df Model:	12
Covariance Type:	nonrobust

	coef	std err	t	P>|t|	[0.025	0.975]
const	0.5886	0.014	42.474	0.000	0.561	0.616
windspeed	-0.1901	0.029	-6.659	0.000	-0.246	-0.134
season_spring	-0.2439	0.016	-15.125	0.000	-0.276	-0.212
season_summer	-0.0432	0.014	-3.116	0.002	-0.070	-0.016
season_winter	-0.0173	0.016	-1.058	0.290	-0.050	0.015
yr_2019	0.2458	0.009	26.633	0.000	0.228	0.264
mnth_december	-0.1081	0.019	-5.635	0.000	-0.146	-0.070
mnth_january	-0.1201	0.020	-6.057	0.000	-0.159	-0.081
mnth_november	-0.0987	0.021	-4.699	0.000	-0.140	-0.057
mnth_september	0.0586	0.018	3.172	0.002	0.022	0.095
holiday_yes	-0.0852	0.030	-2.877	0.004	-0.143	-0.027
weathersit_light_weather	-0.3118	0.028	-11.156	0.000	-0.367	-0.257
weathersit_misty	-0.0863	0.010	-8.762	0.000	-0.106	-0.067

Omnibus:	56.410	Durbin-Watson:	1.953
Prob(Omnibus):	0.000	Jarque-Bera (JB):	131.512
Skew:	-0.592	Prob(JB):	2.77e-29
Kurtosis:	5.188	Cond. No.	8.93

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

calculate_vif(X_train_4)

	Feature	VIF
0	windspeed	3.52
3	season_winter	2.75
1	season_spring	2.71
2	season_summer	1.87
7	mnth_november	1.80
4	yr_2019	1.77
6	mnth_january	1.64
11	weathersit_misty	1.52
5	mnth_december	1.46
8	mnth_september	1.16
10	weathersit_light_weather	1.09
9	holiday_yes	1.06

# Dropping 'season_winter' due to high p-value
X_train_4 = X_train_4.drop('season_winter', axis=1)

# Re-train model
train_model(y_train, X_train_4).summary()

OLS Regression Results

Dep. Variable:	cnt	R-squared:	0.791
Model:	OLS	Adj. R-squared:	0.786
Method:	Least Squares	F-statistic:	171.2
Date:	Tue, 30 Aug 2022	Prob (F-statistic):	2.71e-161
Time:	01:25:19	Log-Likelihood:	437.88
No. Observations:	510	AIC:	-851.8
Df Residuals:	498	BIC:	-800.9
Df Model:	11
Covariance Type:	nonrobust

	coef	std err	t	P>|t|	[0.025	0.975]
const	0.5834	0.013	44.987	0.000	0.558	0.609
windspeed	-0.1910	0.029	-6.694	0.000	-0.247	-0.135
season_spring	-0.2367	0.015	-16.212	0.000	-0.265	-0.208
season_summer	-0.0374	0.013	-2.937	0.003	-0.062	-0.012
yr_2019	0.2459	0.009	26.646	0.000	0.228	0.264
mnth_december	-0.1166	0.017	-6.689	0.000	-0.151	-0.082
mnth_january	-0.1216	0.020	-6.143	0.000	-0.160	-0.083
mnth_november	-0.1101	0.018	-6.120	0.000	-0.145	-0.075
mnth_september	0.0607	0.018	3.304	0.001	0.025	0.097
holiday_yes	-0.0854	0.030	-2.883	0.004	-0.144	-0.027
weathersit_light_weather	-0.3155	0.028	-11.374	0.000	-0.370	-0.261
weathersit_misty	-0.0872	0.010	-8.883	0.000	-0.106	-0.068

Omnibus:	60.252	Durbin-Watson:	1.945
Prob(Omnibus):	0.000	Jarque-Bera (JB):	145.320
Skew:	-0.619	Prob(JB):	2.78e-32
Kurtosis:	5.303	Cond. No.	8.79

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

calculate_vif(X_train_4)

	Feature	VIF
0	windspeed	3.23
1	season_spring	2.37
3	yr_2019	1.74
2	season_summer	1.70
5	mnth_january	1.62
10	weathersit_misty	1.48
6	mnth_november	1.23
7	mnth_september	1.16
4	mnth_december	1.13
9	weathersit_light_weather	1.07
8	holiday_yes	1.06

lr_model_4 = train_model(y_train, X_train_4)

Model 5

# Continuing from Model 4 with objective to contain VIF within 2.5
X_train_5 = X_train_4.drop('windspeed', axis=1)

# Train model
train_model(y_train, X_train_5).summary()

OLS Regression Results

Dep. Variable:	cnt	R-squared:	0.772
Model:	OLS	Adj. R-squared:	0.767
Method:	Least Squares	F-statistic:	169.0
Date:	Tue, 30 Aug 2022	Prob (F-statistic):	3.79e-153
Time:	01:25:19	Log-Likelihood:	415.91
No. Observations:	510	AIC:	-809.8
Df Residuals:	499	BIC:	-763.2
Df Model:	10
Covariance Type:	nonrobust

	coef	std err	t	P>|t|	[0.025	0.975]
const	0.5309	0.011	49.274	0.000	0.510	0.552
season_spring	-0.2616	0.015	-17.771	0.000	-0.291	-0.233
season_summer	-0.0521	0.013	-3.988	0.000	-0.078	-0.026
yr_2019	0.2452	0.010	25.469	0.000	0.226	0.264
mnth_december	-0.1109	0.018	-6.108	0.000	-0.147	-0.075
mnth_january	-0.1062	0.021	-5.178	0.000	-0.146	-0.066
mnth_november	-0.1189	0.019	-6.349	0.000	-0.156	-0.082
mnth_september	0.0662	0.019	3.461	0.001	0.029	0.104
holiday_yes	-0.0879	0.031	-2.844	0.005	-0.149	-0.027
weathersit_light_weather	-0.3349	0.029	-11.639	0.000	-0.391	-0.278
weathersit_misty	-0.0861	0.010	-8.412	0.000	-0.106	-0.066

Omnibus:	68.305	Durbin-Watson:	1.986
Prob(Omnibus):	0.000	Jarque-Bera (JB):	198.368
Skew:	-0.637	Prob(JB):	8.41e-44
Kurtosis:	5.777	Cond. No.	8.28

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

calculate_vif(X_train_5)

	Feature	VIF
0	season_spring	1.84
4	mnth_january	1.61
9	weathersit_misty	1.44
1	season_summer	1.31
5	mnth_november	1.12
6	mnth_september	1.12
3	mnth_december	1.10
7	holiday_yes	1.06
8	weathersit_light_weather	1.03
2	yr_2019	0.02

lr_model_5 = train_model(y_train, X_train_5)

Model 6

X_train_6 = X_train[auto_select_features(10)]

# Train model
train_model(y_train, X_train_6).summary()

OLS Regression Results

Dep. Variable:	cnt	R-squared:	0.833
Model:	OLS	Adj. R-squared:	0.830
Method:	Least Squares	F-statistic:	248.8
Date:	Tue, 30 Aug 2022	Prob (F-statistic):	1.02e-186
Time:	01:25:20	Log-Likelihood:	495.21
No. Observations:	510	AIC:	-968.4
Df Residuals:	499	BIC:	-921.8
Df Model:	10
Covariance Type:	nonrobust

	coef	std err	t	P>|t|	[0.025	0.975]
const	0.2755	0.026	10.547	0.000	0.224	0.327
temp	0.5660	0.022	25.833	0.000	0.523	0.609
hum	-0.2848	0.032	-9.026	0.000	-0.347	-0.223
windspeed	-0.2012	0.026	-7.729	0.000	-0.252	-0.150
season_summer	0.1011	0.011	9.084	0.000	0.079	0.123
season_winter	0.1508	0.011	13.840	0.000	0.129	0.172
yr_2019	0.2264	0.008	27.262	0.000	0.210	0.243
mnth_august	0.0496	0.017	2.973	0.003	0.017	0.082
mnth_september	0.1190	0.017	7.174	0.000	0.086	0.152
holiday_yes	-0.0893	0.026	-3.409	0.001	-0.141	-0.038
weathersit_light_weather	-0.1916	0.026	-7.496	0.000	-0.242	-0.141

Omnibus:	56.891	Durbin-Watson:	2.021
Prob(Omnibus):	0.000	Jarque-Bera (JB):	105.867
Skew:	-0.673	Prob(JB):	1.03e-23
Kurtosis:	4.781	Cond. No.	15.4

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

calculate_vif(X_train_6)

	Feature	VIF
1	hum	10.02
0	temp	8.52
2	windspeed	3.57
5	yr_2019	2.01
3	season_summer	1.82
4	season_winter	1.73
6	mnth_august	1.57
7	mnth_september	1.30
9	weathersit_light_weather	1.09
8	holiday_yes	1.04

# Dropping 'hum' due to high VIF
X_train_6 = X_train_6.drop('hum', axis=1)

# Re-train model
train_model(y_train, X_train_6).summary()

OLS Regression Results

Dep. Variable:	cnt	R-squared:	0.806
Model:	OLS	Adj. R-squared:	0.802
Method:	Least Squares	F-statistic:	230.3
Date:	Tue, 30 Aug 2022	Prob (F-statistic):	1.39e-171
Time:	01:25:21	Log-Likelihood:	456.65
No. Observations:	510	AIC:	-893.3
Df Residuals:	500	BIC:	-851.0
Df Model:	9
Covariance Type:	nonrobust

	coef	std err	t	P>|t|	[0.025	0.975]
const	0.0900	0.017	5.181	0.000	0.056	0.124
temp	0.5464	0.023	23.262	0.000	0.500	0.593
windspeed	-0.1427	0.027	-5.252	0.000	-0.196	-0.089
season_summer	0.0905	0.012	7.588	0.000	0.067	0.114
season_winter	0.1316	0.012	11.430	0.000	0.109	0.154
yr_2019	0.2332	0.009	26.169	0.000	0.216	0.251
mnth_august	0.0394	0.018	2.196	0.029	0.004	0.075
mnth_september	0.1002	0.018	5.650	0.000	0.065	0.135
holiday_yes	-0.0848	0.028	-3.005	0.003	-0.140	-0.029
weathersit_light_weather	-0.2520	0.027	-9.483	0.000	-0.304	-0.200

Omnibus:	63.117	Durbin-Watson:	1.996
Prob(Omnibus):	0.000	Jarque-Bera (JB):	111.174
Skew:	-0.757	Prob(JB):	7.23e-25
Kurtosis:	4.714	Cond. No.	9.93

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

calculate_vif(X_train_6)

	Feature	VIF
0	temp	4.99
1	windspeed	3.01
4	yr_2019	2.00
2	season_summer	1.79
5	mnth_august	1.57
3	season_winter	1.44
6	mnth_september	1.29
8	weathersit_light_weather	1.06
7	holiday_yes	1.04

lr_model_6 = train_model(y_train, X_train_6)

Model 7

X_train_7 = X_train[auto_select_features(5)]

# Train model
train_model(y_train, X_train_7).summary()

OLS Regression Results

Dep. Variable:	cnt	R-squared:	0.750
Model:	OLS	Adj. R-squared:	0.748
Method:	Least Squares	F-statistic:	302.8
Date:	Tue, 30 Aug 2022	Prob (F-statistic):	2.85e-149
Time:	01:25:21	Log-Likelihood:	392.69
No. Observations:	510	AIC:	-773.4
Df Residuals:	504	BIC:	-748.0
Df Model:	5
Covariance Type:	nonrobust

	coef	std err	t	P>|t|	[0.025	0.975]
const	0.2832	0.031	9.044	0.000	0.222	0.345
temp	0.5684	0.023	24.875	0.000	0.524	0.613
hum	-0.1791	0.037	-4.785	0.000	-0.253	-0.106
windspeed	-0.2185	0.031	-6.989	0.000	-0.280	-0.157
yr_2019	0.2295	0.010	22.722	0.000	0.210	0.249
weathersit_light_weather	-0.1752	0.031	-5.661	0.000	-0.236	-0.114

Omnibus:	8.563	Durbin-Watson:	1.996
Prob(Omnibus):	0.014	Jarque-Bera (JB):	8.626
Skew:	-0.276	Prob(JB):	0.0134
Kurtosis:	3.316	Cond. No.	14.7

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

calculate_vif(X_train_7)

	Feature	VIF
1	hum	8.17
0	temp	6.40
2	windspeed	3.40
3	yr_2019	2.01
4	weathersit_light_weather	1.08

# Dropping hum due to high VIF
X_train_7 = X_train_7.drop('hum', axis=1)

# Re-train model
train_model(y_train, X_train_7).summary()

OLS Regression Results

Dep. Variable:	cnt	R-squared:	0.739
Model:	OLS	Adj. R-squared:	0.737
Method:	Least Squares	F-statistic:	357.3
Date:	Tue, 30 Aug 2022	Prob (F-statistic):	1.01e-145
Time:	01:25:22	Log-Likelihood:	381.36
No. Observations:	510	AIC:	-752.7
Df Residuals:	505	BIC:	-731.5
Df Model:	4
Covariance Type:	nonrobust

	coef	std err	t	P>|t|	[0.025	0.975]
const	0.1603	0.018	8.767	0.000	0.124	0.196
temp	0.5541	0.023	23.946	0.000	0.509	0.600
windspeed	-0.1774	0.031	-5.777	0.000	-0.238	-0.117
yr_2019	0.2337	0.010	22.744	0.000	0.214	0.254
weathersit_light_weather	-0.2165	0.030	-7.135	0.000	-0.276	-0.157

Omnibus:	10.058	Durbin-Watson:	1.979
Prob(Omnibus):	0.007	Jarque-Bera (JB):	10.298
Skew:	-0.300	Prob(JB):	0.00580
Kurtosis:	3.352	Cond. No.	8.98

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

calculate_vif(X_train_7)

	Feature	VIF
0	temp	3.06
1	windspeed	2.73
2	yr_2019	1.98
3	weathersit_light_weather	1.04

lr_model_7 = train_model(y_train, X_train_7)

Model 8

Going all manual

X_train_8 = X_train

# Train model
train_model(y_train, X_train_8).summary()

OLS Regression Results

Dep. Variable:	cnt	R-squared:	0.850
Model:	OLS	Adj. R-squared:	0.841
Method:	Least Squares	F-statistic:	90.67
Date:	Tue, 30 Aug 2022	Prob (F-statistic):	1.15e-176
Time:	01:25:22	Log-Likelihood:	523.13
No. Observations:	510	AIC:	-984.3
Df Residuals:	479	BIC:	-853.0
Df Model:	30
Covariance Type:	nonrobust

	coef	std err	t	P>|t|	[0.025	0.975]
const	0.4066	0.097	4.212	0.000	0.217	0.596
dteday	-0.1107	0.330	-0.335	0.738	-0.759	0.538
temp	0.3893	0.144	2.708	0.007	0.107	0.672
atemp	0.0523	0.140	0.374	0.708	-0.222	0.327
hum	-0.1570	0.039	-4.014	0.000	-0.234	-0.080
windspeed	-0.1800	0.027	-6.734	0.000	-0.233	-0.127
season_spring	-0.0415	0.030	-1.365	0.173	-0.101	0.018
season_summer	0.0459	0.026	1.735	0.083	-0.006	0.098
season_winter	0.1115	0.028	3.932	0.000	0.056	0.167
yr_2019	0.2867	0.166	1.729	0.084	-0.039	0.613
mnth_august	0.0478	0.066	0.729	0.466	-0.081	0.177
mnth_december	-0.0068	0.117	-0.058	0.954	-0.237	0.224
mnth_february	-0.0458	0.042	-1.092	0.275	-0.128	0.037
mnth_january	-0.0771	0.052	-1.493	0.136	-0.178	0.024
mnth_july	-0.0134	0.055	-0.244	0.808	-0.121	0.094
mnth_june	0.0192	0.038	0.506	0.613	-0.055	0.094
mnth_march	-0.0033	0.028	-0.117	0.907	-0.058	0.052
mnth_may	0.0300	0.025	1.193	0.234	-0.019	0.079
mnth_november	-0.0062	0.106	-0.059	0.953	-0.214	0.202
mnth_october	0.0347	0.094	0.371	0.711	-0.149	0.218
mnth_september	0.1149	0.077	1.498	0.135	-0.036	0.266
holiday_yes	-0.1344	0.066	-2.050	0.041	-0.263	-0.006
weekday_monday	-0.0247	0.015	-1.692	0.091	-0.053	0.004
weekday_saturday	-0.0554	0.072	-0.771	0.441	-0.197	0.086
weekday_sunday	-0.0496	0.072	-0.691	0.490	-0.191	0.091
weekday_thursday	0.0043	0.015	0.285	0.776	-0.025	0.034
weekday_tuesday	-0.0261	0.015	-1.775	0.077	-0.055	0.003
weekday_wednesday	-0.0102	0.015	-0.663	0.508	-0.040	0.020
workingday_yes	-0.0602	0.071	-0.845	0.399	-0.200	0.080
weathersit_light_weather	-0.2496	0.027	-9.321	0.000	-0.302	-0.197
weathersit_misty	-0.0582	0.011	-5.487	0.000	-0.079	-0.037

Omnibus:	77.579	Durbin-Watson:	2.009
Prob(Omnibus):	0.000	Jarque-Bera (JB):	204.012
Skew:	-0.758	Prob(JB):	5.01e-45
Kurtosis:	5.702	Cond. No.	207.

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

calculate_vif(X_train_8)

	Feature	VIF
0	dteday	1828.85
8	yr_2019	697.07
1	temp	444.57
2	atemp	383.10
27	workingday_yes	102.24
10	mnth_december	63.07
17	mnth_november	52.74
3	hum	42.29
18	mnth_october	40.63
19	mnth_september	26.66
9	mnth_august	23.76
23	weekday_sunday	19.90
22	weekday_saturday	18.97
13	mnth_july	13.64
5	season_spring	13.56
7	season_winter	12.43
12	mnth_january	11.31
6	season_summer	10.15
14	mnth_june	6.33
11	mnth_february	6.04
4	windspeed	5.92
15	mnth_march	4.26
20	holiday_yes	3.76
16	mnth_may	3.16
29	weathersit_misty	2.47
21	weekday_monday	2.09
25	weekday_tuesday	2.09
24	weekday_thursday	2.04
26	weekday_wednesday	1.99
28	weathersit_light_weather	1.34

# Dropping 'dteday' due to high p-value & high VIF
X_train_8 = X_train_8.drop('dteday', axis=1)

# Re-train model
train_model(y_train, X_train_8).summary()

OLS Regression Results

Dep. Variable:	cnt	R-squared:	0.850
Model:	OLS	Adj. R-squared:	0.841
Method:	Least Squares	F-statistic:	93.97
Date:	Tue, 30 Aug 2022	Prob (F-statistic):	1.24e-177
Time:	01:25:23	Log-Likelihood:	523.07
No. Observations:	510	AIC:	-986.1
Df Residuals:	480	BIC:	-859.1
Df Model:	29
Covariance Type:	nonrobust

	coef	std err	t	P>|t|	[0.025	0.975]
const	0.3915	0.085	4.593	0.000	0.224	0.559
temp	0.3873	0.143	2.699	0.007	0.105	0.669
atemp	0.0547	0.139	0.392	0.695	-0.219	0.329
hum	-0.1579	0.039	-4.050	0.000	-0.234	-0.081
windspeed	-0.1802	0.027	-6.751	0.000	-0.233	-0.128
season_spring	-0.0409	0.030	-1.349	0.178	-0.100	0.019
season_summer	0.0458	0.026	1.733	0.084	-0.006	0.098
season_winter	0.1121	0.028	3.969	0.000	0.057	0.168
yr_2019	0.2312	0.008	28.360	0.000	0.215	0.247
mnth_august	0.0291	0.034	0.849	0.396	-0.038	0.096
mnth_december	-0.0444	0.034	-1.308	0.191	-0.111	0.022
mnth_february	-0.0373	0.033	-1.119	0.264	-0.103	0.028
mnth_january	-0.0640	0.034	-1.887	0.060	-0.131	0.003
mnth_july	-0.0274	0.035	-0.773	0.440	-0.097	0.042
mnth_june	0.0097	0.025	0.384	0.701	-0.040	0.059
mnth_march	0.0010	0.025	0.041	0.967	-0.048	0.050
mnth_may	0.0256	0.021	1.196	0.232	-0.016	0.068
mnth_november	-0.0395	0.037	-1.076	0.282	-0.112	0.033
mnth_october	0.0058	0.036	0.159	0.874	-0.065	0.077
mnth_september	0.0916	0.032	2.837	0.005	0.028	0.155
holiday_yes	-0.1345	0.066	-2.054	0.041	-0.263	-0.006
weekday_monday	-0.0247	0.015	-1.688	0.092	-0.053	0.004
weekday_saturday	-0.0557	0.072	-0.777	0.438	-0.197	0.085
weekday_sunday	-0.0499	0.072	-0.696	0.487	-0.191	0.091
weekday_thursday	0.0042	0.015	0.280	0.780	-0.025	0.034
weekday_tuesday	-0.0261	0.015	-1.776	0.076	-0.055	0.003
weekday_wednesday	-0.0103	0.015	-0.670	0.503	-0.040	0.020
workingday_yes	-0.0605	0.071	-0.849	0.396	-0.200	0.079
weathersit_light_weather	-0.2492	0.027	-9.324	0.000	-0.302	-0.197
weathersit_misty	-0.0578	0.011	-5.486	0.000	-0.079	-0.037

Omnibus:	78.724	Durbin-Watson:	2.008
Prob(Omnibus):	0.000	Jarque-Bera (JB):	207.783
Skew:	-0.768	Prob(JB):	7.59e-46
Kurtosis:	5.724	Cond. No.	92.4

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

calculate_vif(X_train_8)

	Feature	VIF
0	temp	441.52
1	atemp	382.19
26	workingday_yes	63.90
2	hum	41.23
4	season_spring	13.52
6	season_winter	12.43
22	weekday_sunday	12.35
21	weekday_saturday	11.84
5	season_summer	9.88
16	mnth_november	7.16
8	mnth_august	6.92
17	mnth_october	6.82
11	mnth_january	6.09
12	mnth_july	5.95
3	windspeed	5.87
9	mnth_december	5.78
18	mnth_september	5.04
10	mnth_february	4.50
14	mnth_march	3.72
13	mnth_june	3.09
19	holiday_yes	2.82
15	mnth_may	2.45
28	weathersit_misty	2.43
7	yr_2019	2.14
20	weekday_monday	2.09
24	weekday_tuesday	2.09
23	weekday_thursday	2.04
25	weekday_wednesday	1.97
27	weathersit_light_weather	1.34

# Dropping 'atemp' due to high p-value & high VIF
X_train_8 = X_train_8.drop('atemp', axis=1)

# Re-train model
train_model(y_train, X_train_8).summary()

OLS Regression Results

Dep. Variable:	cnt	R-squared:	0.850
Model:	OLS	Adj. R-squared:	0.841
Method:	Least Squares	F-statistic:	97.49
Date:	Tue, 30 Aug 2022	Prob (F-statistic):	1.34e-178
Time:	01:25:24	Log-Likelihood:	522.99
No. Observations:	510	AIC:	-988.0
Df Residuals:	481	BIC:	-865.2
Df Model:	28
Covariance Type:	nonrobust

	coef	std err	t	P>|t|	[0.025	0.975]
const	0.3912	0.085	4.594	0.000	0.224	0.558
temp	0.4405	0.047	9.412	0.000	0.349	0.532
hum	-0.1574	0.039	-4.043	0.000	-0.234	-0.081
windspeed	-0.1825	0.026	-7.003	0.000	-0.234	-0.131
season_spring	-0.0404	0.030	-1.335	0.182	-0.100	0.019
season_summer	0.0464	0.026	1.761	0.079	-0.005	0.098
season_winter	0.1126	0.028	3.994	0.000	0.057	0.168
yr_2019	0.2311	0.008	28.384	0.000	0.215	0.247
mnth_august	0.0273	0.034	0.806	0.421	-0.039	0.094
mnth_december	-0.0444	0.034	-1.309	0.191	-0.111	0.022
mnth_february	-0.0374	0.033	-1.123	0.262	-0.103	0.028
mnth_january	-0.0642	0.034	-1.894	0.059	-0.131	0.002
mnth_july	-0.0285	0.035	-0.808	0.420	-0.098	0.041
mnth_june	0.0084	0.025	0.336	0.737	-0.041	0.058
mnth_march	0.0008	0.025	0.032	0.974	-0.048	0.050
mnth_may	0.0247	0.021	1.164	0.245	-0.017	0.066
mnth_november	-0.0396	0.037	-1.077	0.282	-0.112	0.033
mnth_october	0.0056	0.036	0.156	0.876	-0.066	0.077
mnth_september	0.0908	0.032	2.822	0.005	0.028	0.154
holiday_yes	-0.1348	0.065	-2.059	0.040	-0.263	-0.006
weekday_monday	-0.0246	0.015	-1.687	0.092	-0.053	0.004
weekday_saturday	-0.0557	0.072	-0.777	0.437	-0.197	0.085
weekday_sunday	-0.0495	0.072	-0.691	0.490	-0.190	0.091
weekday_thursday	0.0042	0.015	0.282	0.778	-0.025	0.034
weekday_tuesday	-0.0261	0.015	-1.774	0.077	-0.055	0.003
weekday_wednesday	-0.0102	0.015	-0.669	0.504	-0.040	0.020
workingday_yes	-0.0600	0.071	-0.844	0.399	-0.200	0.080
weathersit_light_weather	-0.2499	0.027	-9.380	0.000	-0.302	-0.198
weathersit_misty	-0.0578	0.011	-5.495	0.000	-0.079	-0.037

Omnibus:	78.047	Durbin-Watson:	2.007
Prob(Omnibus):	0.000	Jarque-Bera (JB):	205.442
Skew:	-0.762	Prob(JB):	2.45e-45
Kurtosis:	5.710	Cond. No.	70.4

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

calculate_vif(X_train_8)

	Feature	VIF
25	workingday_yes	63.89
0	temp	43.77
1	hum	41.18
3	season_spring	13.50
5	season_winter	12.40
21	weekday_sunday	12.34
20	weekday_saturday	11.83
4	season_summer	9.85
15	mnth_november	7.16
16	mnth_october	6.82
7	mnth_august	6.79
10	mnth_january	6.09
11	mnth_july	5.91
8	mnth_december	5.78
2	windspeed	5.59
17	mnth_september	5.02
9	mnth_february	4.50
13	mnth_march	3.72
12	mnth_june	3.04
18	holiday_yes	2.81
14	mnth_may	2.43
27	weathersit_misty	2.43
6	yr_2019	2.14
19	weekday_monday	2.09
23	weekday_tuesday	2.09
22	weekday_thursday	2.04
24	weekday_wednesday	1.97
26	weathersit_light_weather	1.33

# Dropping 'workingday_yes' due to high p-value & high VIF
X_train_8 = X_train_8.drop('workingday_yes', axis=1)

# Re-train model
train_model(y_train, X_train_8).summary()

OLS Regression Results

Dep. Variable:	cnt	R-squared:	0.850
Model:	OLS	Adj. R-squared:	0.842
Method:	Least Squares	F-statistic:	101.1
Date:	Tue, 30 Aug 2022	Prob (F-statistic):	1.88e-179
Time:	01:25:24	Log-Likelihood:	522.61
No. Observations:	510	AIC:	-989.2
Df Residuals:	482	BIC:	-870.7
Df Model:	27
Covariance Type:	nonrobust

	coef	std err	t	P>|t|	[0.025	0.975]
const	0.3301	0.045	7.359	0.000	0.242	0.418
temp	0.4421	0.047	9.455	0.000	0.350	0.534
hum	-0.1565	0.039	-4.025	0.000	-0.233	-0.080
windspeed	-0.1831	0.026	-7.034	0.000	-0.234	-0.132
season_spring	-0.0401	0.030	-1.326	0.185	-0.100	0.019
season_summer	0.0466	0.026	1.766	0.078	-0.005	0.098
season_winter	0.1119	0.028	3.971	0.000	0.057	0.167
yr_2019	0.2315	0.008	28.493	0.000	0.216	0.248
mnth_august	0.0266	0.034	0.785	0.433	-0.040	0.093
mnth_december	-0.0437	0.034	-1.288	0.198	-0.110	0.023
mnth_february	-0.0369	0.033	-1.109	0.268	-0.102	0.029
mnth_january	-0.0638	0.034	-1.884	0.060	-0.130	0.003
mnth_july	-0.0292	0.035	-0.827	0.409	-0.098	0.040
mnth_june	0.0077	0.025	0.309	0.757	-0.041	0.057
mnth_march	0.0007	0.025	0.027	0.979	-0.048	0.049
mnth_may	0.0241	0.021	1.134	0.257	-0.018	0.066
mnth_november	-0.0408	0.037	-1.111	0.267	-0.113	0.031
mnth_october	0.0062	0.036	0.172	0.864	-0.065	0.077
mnth_september	0.0909	0.032	2.824	0.005	0.028	0.154
holiday_yes	-0.0842	0.026	-3.202	0.001	-0.136	-0.033
weekday_monday	-0.0247	0.015	-1.692	0.091	-0.053	0.004
weekday_saturday	0.0034	0.015	0.226	0.822	-0.026	0.033
weekday_sunday	0.0096	0.015	0.640	0.522	-0.020	0.039
weekday_thursday	0.0041	0.015	0.274	0.785	-0.025	0.034
weekday_tuesday	-0.0258	0.015	-1.758	0.079	-0.055	0.003
weekday_wednesday	-0.0094	0.015	-0.618	0.537	-0.039	0.021
weathersit_light_weather	-0.2498	0.027	-9.378	0.000	-0.302	-0.197
weathersit_misty	-0.0577	0.011	-5.482	0.000	-0.078	-0.037

Omnibus:	77.558	Durbin-Watson:	2.007
Prob(Omnibus):	0.000	Jarque-Bera (JB):	200.597
Skew:	-0.764	Prob(JB):	2.76e-44
Kurtosis:	5.666	Cond. No.	35.1

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

calculate_vif(X_train_8)

	Feature	VIF
1	hum	37.70
0	temp	36.94
3	season_spring	11.85
5	season_winter	11.71
4	season_summer	7.02
16	mnth_october	6.32
15	mnth_november	6.07
7	mnth_august	5.89
11	mnth_july	5.30
10	mnth_january	5.22
2	windspeed	5.11
8	mnth_december	4.92
17	mnth_september	4.27
9	mnth_february	4.00
13	mnth_march	3.36
12	mnth_june	2.93
14	mnth_may	2.43
26	weathersit_misty	2.39
6	yr_2019	2.14
23	weekday_tuesday	2.06
19	weekday_monday	2.03
22	weekday_thursday	1.99
24	weekday_wednesday	1.94
21	weekday_sunday	1.92
20	weekday_saturday	1.85
25	weathersit_light_weather	1.32
18	holiday_yes	1.12

# Dropping 'season_spring' due to high p-value & high VIF
X_train_8 = X_train_8.drop('season_spring', axis=1)

# Re-train model
train_model(y_train, X_train_8).summary()

OLS Regression Results

Dep. Variable:	cnt	R-squared:	0.849
Model:	OLS	Adj. R-squared:	0.841
Method:	Least Squares	F-statistic:	104.8
Date:	Tue, 30 Aug 2022	Prob (F-statistic):	4.33e-180
Time:	01:25:25	Log-Likelihood:	521.68
No. Observations:	510	AIC:	-989.4
Df Residuals:	483	BIC:	-875.0
Df Model:	26
Covariance Type:	nonrobust

	coef	std err	t	P>|t|	[0.025	0.975]
const	0.3058	0.041	7.461	0.000	0.225	0.386
temp	0.4498	0.046	9.690	0.000	0.359	0.541
hum	-0.1584	0.039	-4.072	0.000	-0.235	-0.082
windspeed	-0.1859	0.026	-7.155	0.000	-0.237	-0.135
season_summer	0.0694	0.020	3.472	0.001	0.030	0.109
season_winter	0.1343	0.023	5.947	0.000	0.090	0.179
yr_2019	0.2312	0.008	28.445	0.000	0.215	0.247
mnth_august	0.0468	0.030	1.544	0.123	-0.013	0.106
mnth_december	-0.0475	0.034	-1.407	0.160	-0.114	0.019
mnth_february	-0.0527	0.031	-1.695	0.091	-0.114	0.008
mnth_january	-0.0791	0.032	-2.480	0.013	-0.142	-0.016
mnth_july	-0.0095	0.032	-0.296	0.767	-0.072	0.053
mnth_june	0.0135	0.025	0.550	0.582	-0.035	0.062
mnth_march	-0.0099	0.023	-0.423	0.673	-0.056	0.036
mnth_may	0.0227	0.021	1.068	0.286	-0.019	0.064
mnth_november	-0.0397	0.037	-1.082	0.280	-0.112	0.032
mnth_october	0.0062	0.036	0.170	0.865	-0.065	0.077
mnth_september	0.1069	0.030	3.581	0.000	0.048	0.166
holiday_yes	-0.0846	0.026	-3.217	0.001	-0.136	-0.033
weekday_monday	-0.0255	0.015	-1.749	0.081	-0.054	0.003
weekday_saturday	0.0028	0.015	0.186	0.853	-0.027	0.033
weekday_sunday	0.0098	0.015	0.649	0.516	-0.020	0.039
weekday_thursday	0.0044	0.015	0.296	0.767	-0.025	0.034
weekday_tuesday	-0.0261	0.015	-1.774	0.077	-0.055	0.003
weekday_wednesday	-0.0094	0.015	-0.618	0.537	-0.039	0.021
weathersit_light_weather	-0.2475	0.027	-9.303	0.000	-0.300	-0.195
weathersit_misty	-0.0573	0.011	-5.441	0.000	-0.078	-0.037

Omnibus:	75.649	Durbin-Watson:	2.012
Prob(Omnibus):	0.000	Jarque-Bera (JB):	192.184
Skew:	-0.751	Prob(JB):	1.85e-42
Kurtosis:	5.605	Cond. No.	34.4

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

calculate_vif(X_train_8)

	Feature	VIF
0	temp	36.69
1	hum	36.05
4	season_winter	8.03
15	mnth_october	6.16
14	mnth_november	5.87
6	mnth_august	5.21
3	season_summer	4.83
2	windspeed	4.81
10	mnth_july	4.75
7	mnth_december	4.38
16	mnth_september	4.03
9	mnth_january	3.12
11	mnth_june	2.90
8	mnth_february	2.48
13	mnth_may	2.42
25	weathersit_misty	2.37
12	mnth_march	2.35
5	yr_2019	2.13
22	weekday_tuesday	2.05
18	weekday_monday	2.00
21	weekday_thursday	1.98
23	weekday_wednesday	1.93
20	weekday_sunday	1.90
19	weekday_saturday	1.83
24	weathersit_light_weather	1.31
17	holiday_yes	1.12

# Dropping 'mnth_october' due to high p-value & high VIF
X_train_8 = X_train_8.drop('mnth_october', axis=1)

# Re-train model
train_model(y_train, X_train_8).summary()

OLS Regression Results

Dep. Variable:	cnt	R-squared:	0.849
Model:	OLS	Adj. R-squared:	0.842
Method:	Least Squares	F-statistic:	109.2
Date:	Tue, 30 Aug 2022	Prob (F-statistic):	4.16e-181
Time:	01:25:25	Log-Likelihood:	521.67
No. Observations:	510	AIC:	-991.3
Df Residuals:	484	BIC:	-881.2
Df Model:	25
Covariance Type:	nonrobust

	coef	std err	t	P>|t|	[0.025	0.975]
const	0.3083	0.038	8.062	0.000	0.233	0.383
temp	0.4496	0.046	9.699	0.000	0.358	0.541
hum	-0.1576	0.039	-4.086	0.000	-0.233	-0.082
windspeed	-0.1858	0.026	-7.161	0.000	-0.237	-0.135
season_summer	0.0675	0.017	4.067	0.000	0.035	0.100
season_winter	0.1366	0.018	7.645	0.000	0.101	0.172
yr_2019	0.2312	0.008	28.491	0.000	0.215	0.247
mnth_august	0.0440	0.025	1.732	0.084	-0.006	0.094
mnth_december	-0.0521	0.020	-2.558	0.011	-0.092	-0.012
mnth_february	-0.0556	0.026	-2.137	0.033	-0.107	-0.004
mnth_january	-0.0820	0.027	-3.060	0.002	-0.135	-0.029
mnth_july	-0.0122	0.028	-0.444	0.657	-0.066	0.042
mnth_june	0.0120	0.023	0.525	0.600	-0.033	0.057
mnth_march	-0.0121	0.020	-0.616	0.538	-0.051	0.026
mnth_may	0.0217	0.020	1.063	0.288	-0.018	0.062
mnth_november	-0.0450	0.020	-2.292	0.022	-0.084	-0.006
mnth_september	0.1035	0.022	4.677	0.000	0.060	0.147
holiday_yes	-0.0844	0.026	-3.216	0.001	-0.136	-0.033
weekday_monday	-0.0255	0.015	-1.747	0.081	-0.054	0.003
weekday_saturday	0.0029	0.015	0.189	0.850	-0.027	0.033
weekday_sunday	0.0098	0.015	0.652	0.515	-0.020	0.039
weekday_thursday	0.0045	0.015	0.298	0.766	-0.025	0.034
weekday_tuesday	-0.0261	0.015	-1.778	0.076	-0.055	0.003
weekday_wednesday	-0.0094	0.015	-0.617	0.537	-0.039	0.021
weathersit_light_weather	-0.2473	0.027	-9.311	0.000	-0.300	-0.195
weathersit_misty	-0.0573	0.011	-5.454	0.000	-0.078	-0.037

Omnibus:	75.109	Durbin-Watson:	2.011
Prob(Omnibus):	0.000	Jarque-Bera (JB):	190.219
Skew:	-0.747	Prob(JB):	4.95e-42
Kurtosis:	5.592	Cond. No.	25.3

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

calculate_vif(X_train_8)

	Feature	VIF
0	temp	35.77
1	hum	32.56
2	windspeed	4.67
4	season_winter	4.54
6	mnth_august	3.92
3	season_summer	3.77
10	mnth_july	3.71
11	mnth_june	2.55
9	mnth_january	2.53
15	mnth_september	2.46
24	weathersit_misty	2.35
13	mnth_may	2.23
5	yr_2019	2.13
21	weekday_tuesday	2.04
8	mnth_february	2.00
17	weekday_monday	1.97
20	weekday_thursday	1.96
14	mnth_november	1.92
22	weekday_wednesday	1.91
19	weekday_sunday	1.89
12	mnth_march	1.88
18	weekday_saturday	1.81
7	mnth_december	1.79
23	weathersit_light_weather	1.31
16	holiday_yes	1.12

# Dropping 'weekday_saturday' due to high p-value
X_train_8 = X_train_8.drop('weekday_saturday', axis=1)

# Re-train model
train_model(y_train, X_train_8).summary()

OLS Regression Results

Dep. Variable:	cnt	R-squared:	0.849
Model:	OLS	Adj. R-squared:	0.842
Method:	Least Squares	F-statistic:	114.0
Date:	Tue, 30 Aug 2022	Prob (F-statistic):	3.92e-182
Time:	01:25:26	Log-Likelihood:	521.65
No. Observations:	510	AIC:	-993.3
Df Residuals:	485	BIC:	-887.4
Df Model:	24
Covariance Type:	nonrobust

	coef	std err	t	P>|t|	[0.025	0.975]
const	0.3099	0.037	8.312	0.000	0.237	0.383
temp	0.4496	0.046	9.710	0.000	0.359	0.541
hum	-0.1580	0.038	-4.105	0.000	-0.234	-0.082
windspeed	-0.1859	0.026	-7.173	0.000	-0.237	-0.135
season_summer	0.0676	0.017	4.075	0.000	0.035	0.100
season_winter	0.1367	0.018	7.657	0.000	0.102	0.172
yr_2019	0.2312	0.008	28.519	0.000	0.215	0.247
mnth_august	0.0440	0.025	1.735	0.083	-0.006	0.094
mnth_december	-0.0522	0.020	-2.567	0.011	-0.092	-0.012
mnth_february	-0.0557	0.026	-2.142	0.033	-0.107	-0.005
mnth_january	-0.0821	0.027	-3.064	0.002	-0.135	-0.029
mnth_july	-0.0121	0.028	-0.440	0.660	-0.066	0.042
mnth_june	0.0118	0.023	0.518	0.605	-0.033	0.057
mnth_march	-0.0122	0.020	-0.620	0.536	-0.051	0.026
mnth_may	0.0216	0.020	1.063	0.288	-0.018	0.062
mnth_november	-0.0451	0.020	-2.300	0.022	-0.084	-0.007
mnth_september	0.1035	0.022	4.681	0.000	0.060	0.147
holiday_yes	-0.0843	0.026	-3.218	0.001	-0.136	-0.033
weekday_monday	-0.0268	0.013	-2.114	0.035	-0.052	-0.002
weekday_sunday	0.0084	0.013	0.640	0.522	-0.017	0.034
weekday_thursday	0.0031	0.013	0.237	0.813	-0.023	0.029
weekday_tuesday	-0.0275	0.013	-2.144	0.033	-0.053	-0.002
weekday_wednesday	-0.0108	0.013	-0.802	0.423	-0.037	0.016
weathersit_light_weather	-0.2474	0.027	-9.323	0.000	-0.300	-0.195
weathersit_misty	-0.0572	0.010	-5.456	0.000	-0.078	-0.037

Omnibus:	75.078	Durbin-Watson:	2.012
Prob(Omnibus):	0.000	Jarque-Bera (JB):	190.847
Skew:	-0.746	Prob(JB):	3.62e-42
Kurtosis:	5.599	Cond. No.	25.1

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

calculate_vif(X_train_8)

	Feature	VIF
0	temp	35.18
1	hum	32.43
2	windspeed	4.64
4	season_winter	4.50
6	mnth_august	3.92
3	season_summer	3.72
10	mnth_july	3.70
11	mnth_june	2.55
15	mnth_september	2.46
9	mnth_january	2.46
23	weathersit_misty	2.35
13	mnth_may	2.23
5	yr_2019	2.13
8	mnth_february	1.96
14	mnth_november	1.91
12	mnth_march	1.86
7	mnth_december	1.78
20	weekday_tuesday	1.60
17	weekday_monday	1.55
19	weekday_thursday	1.55
21	weekday_wednesday	1.53
18	weekday_sunday	1.50
22	weathersit_light_weather	1.30
16	holiday_yes	1.12

# Dropping 'weekday_thursday' due to high p-value
X_train_8 = X_train_8.drop('weekday_thursday', axis=1)

# Re-train model
train_model(y_train, X_train_8).summary()

OLS Regression Results

Dep. Variable:	cnt	R-squared:	0.849
Model:	OLS	Adj. R-squared:	0.842
Method:	Least Squares	F-statistic:	119.2
Date:	Tue, 30 Aug 2022	Prob (F-statistic):	3.65e-183
Time:	01:25:26	Log-Likelihood:	521.62
No. Observations:	510	AIC:	-995.2
Df Residuals:	486	BIC:	-893.6
Df Model:	23
Covariance Type:	nonrobust

	coef	std err	t	P>|t|	[0.025	0.975]
const	0.3109	0.037	8.405	0.000	0.238	0.384
temp	0.4497	0.046	9.723	0.000	0.359	0.541
hum	-0.1580	0.038	-4.112	0.000	-0.234	-0.083
windspeed	-0.1860	0.026	-7.183	0.000	-0.237	-0.135
season_summer	0.0678	0.017	4.097	0.000	0.035	0.100
season_winter	0.1368	0.018	7.679	0.000	0.102	0.172
yr_2019	0.2312	0.008	28.567	0.000	0.215	0.247
mnth_august	0.0440	0.025	1.738	0.083	-0.006	0.094
mnth_december	-0.0525	0.020	-2.584	0.010	-0.092	-0.013
mnth_february	-0.0555	0.026	-2.137	0.033	-0.106	-0.004
mnth_january	-0.0821	0.027	-3.067	0.002	-0.135	-0.029
mnth_july	-0.0120	0.028	-0.435	0.664	-0.066	0.042
mnth_june	0.0116	0.023	0.511	0.610	-0.033	0.056
mnth_march	-0.0122	0.020	-0.620	0.536	-0.051	0.026
mnth_may	0.0216	0.020	1.064	0.288	-0.018	0.062
mnth_november	-0.0452	0.020	-2.308	0.021	-0.084	-0.007
mnth_september	0.1035	0.022	4.686	0.000	0.060	0.147
holiday_yes	-0.0845	0.026	-3.229	0.001	-0.136	-0.033
weekday_monday	-0.0279	0.012	-2.346	0.019	-0.051	-0.005
weekday_sunday	0.0074	0.012	0.596	0.552	-0.017	0.032
weekday_tuesday	-0.0285	0.012	-2.386	0.017	-0.052	-0.005
weekday_wednesday	-0.0118	0.013	-0.932	0.352	-0.037	0.013
weathersit_light_weather	-0.2480	0.026	-9.403	0.000	-0.300	-0.196
weathersit_misty	-0.0574	0.010	-5.482	0.000	-0.078	-0.037

Omnibus:	75.371	Durbin-Watson:	2.012
Prob(Omnibus):	0.000	Jarque-Bera (JB):	191.496
Skew:	-0.749	Prob(JB):	2.61e-42
Kurtosis:	5.602	Cond. No.	25.0

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

calculate_vif(X_train_8)

	Feature	VIF
0	temp	34.95
1	hum	32.34
2	windspeed	4.63
4	season_winter	4.47
6	mnth_august	3.92
10	mnth_july	3.70
3	season_summer	3.68
11	mnth_june	2.54
15	mnth_september	2.45
9	mnth_january	2.44
22	weathersit_misty	2.33
13	mnth_may	2.23
5	yr_2019	2.13
8	mnth_february	1.93
14	mnth_november	1.91
12	mnth_march	1.85
7	mnth_december	1.78
19	weekday_tuesday	1.40
17	weekday_monday	1.38
20	weekday_wednesday	1.38
18	weekday_sunday	1.34
21	weathersit_light_weather	1.29
16	holiday_yes	1.12

# Dropping 'mnth_july' due to high p-value
X_train_8 = X_train_8.drop('mnth_july', axis=1)

# Re-train model
train_model(y_train, X_train_8).summary()

OLS Regression Results

Dep. Variable:	cnt	R-squared:	0.849
Model:	OLS	Adj. R-squared:	0.843
Method:	Least Squares	F-statistic:	124.8
Date:	Tue, 30 Aug 2022	Prob (F-statistic):	3.55e-184
Time:	01:25:27	Log-Likelihood:	521.52
No. Observations:	510	AIC:	-997.0
Df Residuals:	487	BIC:	-899.6
Df Model:	22
Covariance Type:	nonrobust

	coef	std err	t	P>|t|	[0.025	0.975]
const	0.3098	0.037	8.402	0.000	0.237	0.382
temp	0.4389	0.039	11.276	0.000	0.362	0.515
hum	-0.1559	0.038	-4.093	0.000	-0.231	-0.081
windspeed	-0.1844	0.026	-7.198	0.000	-0.235	-0.134
season_summer	0.0707	0.015	4.674	0.000	0.041	0.100
season_winter	0.1404	0.016	8.902	0.000	0.109	0.171
yr_2019	0.2315	0.008	28.716	0.000	0.216	0.247
mnth_august	0.0521	0.017	3.007	0.003	0.018	0.086
mnth_december	-0.0520	0.020	-2.567	0.011	-0.092	-0.012
mnth_february	-0.0531	0.025	-2.094	0.037	-0.103	-0.003
mnth_january	-0.0804	0.026	-3.039	0.003	-0.132	-0.028
mnth_june	0.0175	0.018	0.960	0.338	-0.018	0.053
mnth_march	-0.0094	0.019	-0.508	0.612	-0.046	0.027
mnth_may	0.0250	0.019	1.332	0.184	-0.012	0.062
mnth_november	-0.0453	0.020	-2.316	0.021	-0.084	-0.007
mnth_september	0.1095	0.017	6.341	0.000	0.076	0.143
holiday_yes	-0.0835	0.026	-3.205	0.001	-0.135	-0.032
weekday_monday	-0.0282	0.012	-2.380	0.018	-0.051	-0.005
weekday_sunday	0.0071	0.012	0.577	0.564	-0.017	0.031
weekday_tuesday	-0.0288	0.012	-2.417	0.016	-0.052	-0.005
weekday_wednesday	-0.0120	0.013	-0.948	0.344	-0.037	0.013
weathersit_light_weather	-0.2485	0.026	-9.438	0.000	-0.300	-0.197
weathersit_misty	-0.0577	0.010	-5.532	0.000	-0.078	-0.037

Omnibus:	76.510	Durbin-Watson:	2.016
Prob(Omnibus):	0.000	Jarque-Bera (JB):	196.590
Skew:	-0.756	Prob(JB):	2.05e-43
Kurtosis:	5.639	Cond. No.	24.5

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

calculate_vif(X_train_8)

	Feature	VIF
1	hum	31.94
0	temp	19.57
2	windspeed	4.56
4	season_winter	3.48
3	season_summer	3.06
9	mnth_january	2.40
21	weathersit_misty	2.32
5	yr_2019	2.11
13	mnth_november	1.91
12	mnth_may	1.90
8	mnth_february	1.84
6	mnth_august	1.84
7	mnth_december	1.78
11	mnth_march	1.65
10	mnth_june	1.63
14	mnth_september	1.51
18	weekday_tuesday	1.40
19	weekday_wednesday	1.38
16	weekday_monday	1.37
17	weekday_sunday	1.34
20	weathersit_light_weather	1.28
15	holiday_yes	1.11

# Dropping 'mnth_march' due to high p-value
X_train_8 = X_train_8.drop('mnth_march', axis=1)

# Re-train model
train_model(y_train, X_train_8).summary()

OLS Regression Results

Dep. Variable:	cnt	R-squared:	0.849
Model:	OLS	Adj. R-squared:	0.843
Method:	Least Squares	F-statistic:	130.9
Date:	Tue, 30 Aug 2022	Prob (F-statistic):	3.48e-185
Time:	01:25:27	Log-Likelihood:	521.38
No. Observations:	510	AIC:	-998.8
Df Residuals:	488	BIC:	-905.6
Df Model:	21
Covariance Type:	nonrobust

	coef	std err	t	P>|t|	[0.025	0.975]
const	0.3007	0.032	9.366	0.000	0.238	0.364
temp	0.4492	0.033	13.519	0.000	0.384	0.514
hum	-0.1575	0.038	-4.152	0.000	-0.232	-0.083
windspeed	-0.1846	0.026	-7.210	0.000	-0.235	-0.134
season_summer	0.0728	0.015	5.017	0.000	0.044	0.101
season_winter	0.1439	0.014	10.158	0.000	0.116	0.172
yr_2019	0.2313	0.008	28.759	0.000	0.215	0.247
mnth_august	0.0540	0.017	3.202	0.001	0.021	0.087
mnth_december	-0.0478	0.018	-2.591	0.010	-0.084	-0.012
mnth_february	-0.0460	0.021	-2.175	0.030	-0.088	-0.004
mnth_january	-0.0725	0.021	-3.386	0.001	-0.115	-0.030
mnth_june	0.0183	0.018	1.008	0.314	-0.017	0.054
mnth_may	0.0264	0.019	1.425	0.155	-0.010	0.063
mnth_november	-0.0427	0.019	-2.263	0.024	-0.080	-0.006
mnth_september	0.1118	0.017	6.722	0.000	0.079	0.144
holiday_yes	-0.0832	0.026	-3.199	0.001	-0.134	-0.032
weekday_monday	-0.0277	0.012	-2.346	0.019	-0.051	-0.004
weekday_sunday	0.0072	0.012	0.584	0.559	-0.017	0.032
weekday_tuesday	-0.0286	0.012	-2.400	0.017	-0.052	-0.005
weekday_wednesday	-0.0118	0.013	-0.930	0.353	-0.037	0.013
weathersit_light_weather	-0.2478	0.026	-9.431	0.000	-0.299	-0.196
weathersit_misty	-0.0575	0.010	-5.522	0.000	-0.078	-0.037

Omnibus:	74.968	Durbin-Watson:	2.019
Prob(Omnibus):	0.000	Jarque-Bera (JB):	187.078
Skew:	-0.751	Prob(JB):	2.38e-41
Kurtosis:	5.559	Cond. No.	19.8

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

calculate_vif(X_train_8)

	Feature	VIF
1	hum	26.45
0	temp	17.72
2	windspeed	4.20
4	season_winter	3.11
3	season_summer	3.03
20	weathersit_misty	2.27
5	yr_2019	2.09
9	mnth_january	1.99
12	mnth_november	1.88
11	mnth_may	1.80
6	mnth_august	1.76
7	mnth_december	1.67
10	mnth_june	1.61
8	mnth_february	1.58
13	mnth_september	1.43
17	weekday_tuesday	1.40
18	weekday_wednesday	1.38
15	weekday_monday	1.37
16	weekday_sunday	1.33
19	weathersit_light_weather	1.26
14	holiday_yes	1.11

# Dropping 'weekday_sunday' due to high p-value
X_train_8 = X_train_8.drop('weekday_sunday', axis=1)

# Re-train model
train_model(y_train, X_train_8).summary()

OLS Regression Results

Dep. Variable:	cnt	R-squared:	0.849
Model:	OLS	Adj. R-squared:	0.843
Method:	Least Squares	F-statistic:	137.6
Date:	Tue, 30 Aug 2022	Prob (F-statistic):	3.47e-186
Time:	01:25:28	Log-Likelihood:	521.21
No. Observations:	510	AIC:	-1000.
Df Residuals:	489	BIC:	-911.5
Df Model:	20
Covariance Type:	nonrobust

	coef	std err	t	P>|t|	[0.025	0.975]
const	0.3028	0.032	9.500	0.000	0.240	0.365
temp	0.4494	0.033	13.536	0.000	0.384	0.515
hum	-0.1584	0.038	-4.184	0.000	-0.233	-0.084
windspeed	-0.1843	0.026	-7.206	0.000	-0.235	-0.134
season_summer	0.0726	0.014	5.010	0.000	0.044	0.101
season_winter	0.1440	0.014	10.171	0.000	0.116	0.172
yr_2019	0.2313	0.008	28.789	0.000	0.216	0.247
mnth_august	0.0538	0.017	3.192	0.002	0.021	0.087
mnth_december	-0.0475	0.018	-2.579	0.010	-0.084	-0.011
mnth_february	-0.0461	0.021	-2.181	0.030	-0.088	-0.005
mnth_january	-0.0726	0.021	-3.393	0.001	-0.115	-0.031
mnth_june	0.0182	0.018	1.002	0.317	-0.017	0.054
mnth_may	0.0262	0.019	1.416	0.157	-0.010	0.063
mnth_november	-0.0433	0.019	-2.301	0.022	-0.080	-0.006
mnth_september	0.1116	0.017	6.718	0.000	0.079	0.144
holiday_yes	-0.0830	0.026	-3.191	0.002	-0.134	-0.032
weekday_monday	-0.0294	0.011	-2.585	0.010	-0.052	-0.007
weekday_tuesday	-0.0304	0.012	-2.640	0.009	-0.053	-0.008
weekday_wednesday	-0.0135	0.012	-1.106	0.269	-0.038	0.011
weathersit_light_weather	-0.2481	0.026	-9.450	0.000	-0.300	-0.197
weathersit_misty	-0.0570	0.010	-5.497	0.000	-0.077	-0.037

Omnibus:	73.000	Durbin-Watson:	2.014
Prob(Omnibus):	0.000	Jarque-Bera (JB):	183.029
Skew:	-0.730	Prob(JB):	1.80e-40
Kurtosis:	5.545	Cond. No.	19.7

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

calculate_vif(X_train_8)

	Feature	VIF
1	hum	26.42
0	temp	17.60
2	windspeed	4.17
4	season_winter	3.10
3	season_summer	3.03
19	weathersit_misty	2.26
5	yr_2019	2.09
9	mnth_january	1.99
12	mnth_november	1.88
11	mnth_may	1.80
6	mnth_august	1.76
7	mnth_december	1.66
10	mnth_june	1.61
8	mnth_february	1.58
13	mnth_september	1.43
16	weekday_tuesday	1.31
17	weekday_wednesday	1.30
15	weekday_monday	1.28
18	weathersit_light_weather	1.26
14	holiday_yes	1.11

# Dropping 'mnth_june' due to high p-value
X_train_8 = X_train_8.drop('mnth_june', axis=1)

# Re-train model
train_model(y_train, X_train_8).summary()

OLS Regression Results

Dep. Variable:	cnt	R-squared:	0.849
Model:	OLS	Adj. R-squared:	0.843
Method:	Least Squares	F-statistic:	144.8
Date:	Tue, 30 Aug 2022	Prob (F-statistic):	4.67e-187
Time:	01:25:29	Log-Likelihood:	520.68
No. Observations:	510	AIC:	-1001.
Df Residuals:	490	BIC:	-916.7
Df Model:	19
Covariance Type:	nonrobust

	coef	std err	t	P>|t|	[0.025	0.975]
const	0.2991	0.032	9.447	0.000	0.237	0.361
temp	0.4630	0.030	15.290	0.000	0.404	0.523
hum	-0.1618	0.038	-4.290	0.000	-0.236	-0.088
windspeed	-0.1861	0.026	-7.290	0.000	-0.236	-0.136
season_summer	0.0771	0.014	5.593	0.000	0.050	0.104
season_winter	0.1437	0.014	10.154	0.000	0.116	0.172
yr_2019	0.2309	0.008	28.780	0.000	0.215	0.247
mnth_august	0.0494	0.016	3.036	0.003	0.017	0.081
mnth_december	-0.0453	0.018	-2.478	0.014	-0.081	-0.009
mnth_february	-0.0435	0.021	-2.074	0.039	-0.085	-0.002
mnth_january	-0.0691	0.021	-3.272	0.001	-0.111	-0.028
mnth_may	0.0196	0.017	1.131	0.258	-0.014	0.054
mnth_november	-0.0418	0.019	-2.226	0.026	-0.079	-0.005
mnth_september	0.1089	0.016	6.644	0.000	0.077	0.141
holiday_yes	-0.0840	0.026	-3.231	0.001	-0.135	-0.033
weekday_monday	-0.0291	0.011	-2.559	0.011	-0.051	-0.007
weekday_tuesday	-0.0304	0.012	-2.643	0.008	-0.053	-0.008
weekday_wednesday	-0.0134	0.012	-1.093	0.275	-0.037	0.011
weathersit_light_weather	-0.2480	0.026	-9.447	0.000	-0.300	-0.196
weathersit_misty	-0.0567	0.010	-5.470	0.000	-0.077	-0.036

Omnibus:	72.757	Durbin-Watson:	2.015
Prob(Omnibus):	0.000	Jarque-Bera (JB):	177.468
Skew:	-0.737	Prob(JB):	2.91e-39
Kurtosis:	5.485	Cond. No.	19.4

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

calculate_vif(X_train_8)

	Feature	VIF
1	hum	25.37
0	temp	14.62
2	windspeed	4.08
4	season_winter	3.10
3	season_summer	2.77
18	weathersit_misty	2.25
5	yr_2019	2.08
9	mnth_january	1.96
11	mnth_november	1.87
7	mnth_december	1.65
6	mnth_august	1.64
10	mnth_may	1.58
8	mnth_february	1.57
12	mnth_september	1.39
15	weekday_tuesday	1.31
16	weekday_wednesday	1.30
14	weekday_monday	1.28
17	weathersit_light_weather	1.26
13	holiday_yes	1.11

# Dropping 'weekday_wednesday' due to high p-value
X_train_8 = X_train_8.drop('weekday_wednesday', axis=1)

# Re-train model
train_model(y_train, X_train_8).summary()

OLS Regression Results

Dep. Variable:	cnt	R-squared:	0.848
Model:	OLS	Adj. R-squared:	0.843
Method:	Least Squares	F-statistic:	152.7
Date:	Tue, 30 Aug 2022	Prob (F-statistic):	6.74e-188
Time:	01:25:29	Log-Likelihood:	520.06
No. Observations:	510	AIC:	-1002.
Df Residuals:	491	BIC:	-921.7
Df Model:	18
Covariance Type:	nonrobust

	coef	std err	t	P>|t|	[0.025	0.975]
const	0.2964	0.032	9.390	0.000	0.234	0.358
temp	0.4644	0.030	15.347	0.000	0.405	0.524
hum	-0.1629	0.038	-4.319	0.000	-0.237	-0.089
windspeed	-0.1849	0.026	-7.250	0.000	-0.235	-0.135
season_summer	0.0768	0.014	5.571	0.000	0.050	0.104
season_winter	0.1427	0.014	10.102	0.000	0.115	0.170
yr_2019	0.2307	0.008	28.758	0.000	0.215	0.246
mnth_august	0.0488	0.016	3.000	0.003	0.017	0.081
mnth_december	-0.0446	0.018	-2.441	0.015	-0.081	-0.009
mnth_february	-0.0435	0.021	-2.075	0.039	-0.085	-0.002
mnth_january	-0.0694	0.021	-3.288	0.001	-0.111	-0.028
mnth_may	0.0192	0.017	1.110	0.267	-0.015	0.053
mnth_november	-0.0406	0.019	-2.165	0.031	-0.077	-0.004
mnth_september	0.1092	0.016	6.661	0.000	0.077	0.141
holiday_yes	-0.0895	0.025	-3.510	0.000	-0.140	-0.039
weekday_monday	-0.0267	0.011	-2.389	0.017	-0.049	-0.005
weekday_tuesday	-0.0277	0.011	-2.466	0.014	-0.050	-0.006
weathersit_light_weather	-0.2477	0.026	-9.434	0.000	-0.299	-0.196
weathersit_misty	-0.0566	0.010	-5.452	0.000	-0.077	-0.036

Omnibus:	71.655	Durbin-Watson:	2.017
Prob(Omnibus):	0.000	Jarque-Bera (JB):	178.070
Skew:	-0.721	Prob(JB):	2.15e-39
Kurtosis:	5.511	Cond. No.	19.4

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

calculate_vif(X_train_8)

	Feature	VIF
1	hum	25.16
0	temp	14.62
2	windspeed	4.08
4	season_winter	3.08
3	season_summer	2.77
17	weathersit_misty	2.25
5	yr_2019	2.08
9	mnth_january	1.95
11	mnth_november	1.87
7	mnth_december	1.65
6	mnth_august	1.64
10	mnth_may	1.58
8	mnth_february	1.57
12	mnth_september	1.39
15	weekday_tuesday	1.25
16	weathersit_light_weather	1.25
14	weekday_monday	1.24
13	holiday_yes	1.07

# Dropping 'mnth_may' due to high p-value
X_train_8 = X_train_8.drop('mnth_may', axis=1)

# Re-train model
train_model(y_train, X_train_8).summary()

OLS Regression Results

Dep. Variable:	cnt	R-squared:	0.848
Model:	OLS	Adj. R-squared:	0.843
Method:	Least Squares	F-statistic:	161.6
Date:	Tue, 30 Aug 2022	Prob (F-statistic):	9.63e-189
Time:	01:25:30	Log-Likelihood:	519.42
No. Observations:	510	AIC:	-1003.
Df Residuals:	492	BIC:	-926.6
Df Model:	17
Covariance Type:	nonrobust

	coef	std err	t	P>|t|	[0.025	0.975]
const	0.2908	0.031	9.331	0.000	0.230	0.352
temp	0.4686	0.030	15.603	0.000	0.410	0.528
hum	-0.1568	0.037	-4.201	0.000	-0.230	-0.083
windspeed	-0.1859	0.025	-7.290	0.000	-0.236	-0.136
season_summer	0.0835	0.012	6.729	0.000	0.059	0.108
season_winter	0.1427	0.014	10.098	0.000	0.115	0.170
yr_2019	0.2303	0.008	28.730	0.000	0.215	0.246
mnth_august	0.0477	0.016	2.939	0.003	0.016	0.080
mnth_december	-0.0437	0.018	-2.391	0.017	-0.080	-0.008
mnth_february	-0.0417	0.021	-1.992	0.047	-0.083	-0.001
mnth_january	-0.0675	0.021	-3.209	0.001	-0.109	-0.026
mnth_november	-0.0396	0.019	-2.114	0.035	-0.076	-0.003
mnth_september	0.1084	0.016	6.616	0.000	0.076	0.141
holiday_yes	-0.0901	0.025	-3.534	0.000	-0.140	-0.040
weekday_monday	-0.0272	0.011	-2.440	0.015	-0.049	-0.005
weekday_tuesday	-0.0281	0.011	-2.501	0.013	-0.050	-0.006
weathersit_light_weather	-0.2497	0.026	-9.530	0.000	-0.301	-0.198
weathersit_misty	-0.0568	0.010	-5.479	0.000	-0.077	-0.036

Omnibus:	71.020	Durbin-Watson:	2.022
Prob(Omnibus):	0.000	Jarque-Bera (JB):	170.053
Skew:	-0.727	Prob(JB):	1.18e-37
Kurtosis:	5.427	Cond. No.	19.0

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

calculate_vif(X_train_8)

	Feature	VIF
1	hum	25.08
0	temp	14.58
2	windspeed	4.01
4	season_winter	3.08
3	season_summer	2.30
16	weathersit_misty	2.25
5	yr_2019	2.07
9	mnth_january	1.95
10	mnth_november	1.87
7	mnth_december	1.65
6	mnth_august	1.63
8	mnth_february	1.57
11	mnth_september	1.39
14	weekday_tuesday	1.25
15	weathersit_light_weather	1.25
13	weekday_monday	1.23
12	holiday_yes	1.07

# Dropping 'hum' due to high VIF
X_train_8 = X_train_8.drop('hum', axis=1)

# Re-train model
train_model(y_train, X_train_8).summary()

OLS Regression Results

Dep. Variable:	cnt	R-squared:	0.843
Model:	OLS	Adj. R-squared:	0.838
Method:	Least Squares	F-statistic:	165.0
Date:	Tue, 30 Aug 2022	Prob (F-statistic):	4.10e-186
Time:	01:25:30	Log-Likelihood:	510.43
No. Observations:	510	AIC:	-986.9
Df Residuals:	493	BIC:	-914.9
Df Model:	16
Covariance Type:	nonrobust

	coef	std err	t	P>|t|	[0.025	0.975]
const	0.2110	0.025	8.399	0.000	0.162	0.260
temp	0.4391	0.030	14.790	0.000	0.381	0.497
windspeed	-0.1576	0.025	-6.303	0.000	-0.207	-0.108
season_summer	0.0780	0.013	6.215	0.000	0.053	0.103
season_winter	0.1315	0.014	9.322	0.000	0.104	0.159
yr_2019	0.2341	0.008	28.905	0.000	0.218	0.250
mnth_august	0.0454	0.016	2.750	0.006	0.013	0.078
mnth_december	-0.0515	0.018	-2.788	0.006	-0.088	-0.015
mnth_february	-0.0475	0.021	-2.238	0.026	-0.089	-0.006
mnth_january	-0.0776	0.021	-3.649	0.000	-0.119	-0.036
mnth_november	-0.0397	0.019	-2.086	0.038	-0.077	-0.002
mnth_september	0.1014	0.017	6.119	0.000	0.069	0.134
holiday_yes	-0.0910	0.026	-3.509	0.000	-0.142	-0.040
weekday_monday	-0.0275	0.011	-2.428	0.016	-0.050	-0.005
weekday_tuesday	-0.0318	0.011	-2.791	0.005	-0.054	-0.009
weathersit_light_weather	-0.2935	0.024	-12.005	0.000	-0.342	-0.245
weathersit_misty	-0.0817	0.009	-9.434	0.000	-0.099	-0.065

Omnibus:	69.898	Durbin-Watson:	2.030
Prob(Omnibus):	0.000	Jarque-Bera (JB):	166.985
Skew:	-0.717	Prob(JB):	5.49e-37
Kurtosis:	5.409	Cond. No.	16.2

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

calculate_vif(X_train_8)

	Feature	VIF
0	temp	5.24
1	windspeed	3.98
3	season_winter	2.63
4	yr_2019	2.06
2	season_summer	2.04
9	mnth_november	1.82
5	mnth_august	1.63
15	weathersit_misty	1.57
6	mnth_december	1.42
10	mnth_september	1.35
8	mnth_january	1.30
7	mnth_february	1.27
12	weekday_monday	1.23
13	weekday_tuesday	1.23
14	weathersit_light_weather	1.09
11	holiday_yes	1.07

lr_model_8 = train_model(y_train, X_train_8)

Model 9

# Continuing from Model 8 to contain VIF within 5
X_train_9 = X_train_8.drop('temp', axis=1)

# Train model
train_model(y_train, X_train_9).summary()

OLS Regression Results

Dep. Variable:	cnt	R-squared:	0.773
Model:	OLS	Adj. R-squared:	0.766
Method:	Least Squares	F-statistic:	112.0
Date:	Tue, 30 Aug 2022	Prob (F-statistic):	4.27e-148
Time:	01:25:31	Log-Likelihood:	416.79
No. Observations:	510	AIC:	-801.6
Df Residuals:	494	BIC:	-733.8
Df Model:	15
Covariance Type:	nonrobust

	coef	std err	t	P>|t|	[0.025	0.975]
const	0.5261	0.016	32.917	0.000	0.495	0.558
windspeed	-0.2304	0.029	-7.827	0.000	-0.288	-0.173
season_summer	0.0449	0.015	3.032	0.003	0.016	0.074
season_winter	0.0841	0.016	5.097	0.000	0.052	0.116
yr_2019	0.2461	0.010	25.453	0.000	0.227	0.265
mnth_august	0.1007	0.019	5.223	0.000	0.063	0.139
mnth_december	-0.1661	0.020	-8.251	0.000	-0.206	-0.127
mnth_february	-0.2087	0.022	-9.550	0.000	-0.252	-0.166
mnth_january	-0.2760	0.020	-13.935	0.000	-0.315	-0.237
mnth_november	-0.1121	0.022	-5.075	0.000	-0.155	-0.069
mnth_september	0.1198	0.020	6.037	0.000	0.081	0.159
holiday_yes	-0.0855	0.031	-2.748	0.006	-0.147	-0.024
weekday_monday	-0.0329	0.014	-2.418	0.016	-0.060	-0.006
weekday_tuesday	-0.0280	0.014	-2.047	0.041	-0.055	-0.001
weathersit_light_weather	-0.3117	0.029	-10.634	0.000	-0.369	-0.254
weathersit_misty	-0.0934	0.010	-9.024	0.000	-0.114	-0.073

Omnibus:	55.168	Durbin-Watson:	1.933
Prob(Omnibus):	0.000	Jarque-Bera (JB):	90.368
Skew:	-0.702	Prob(JB):	2.38e-20
Kurtosis:	4.510	Cond. No.	9.34

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

calculate_vif(X_train_9)

	Feature	VIF
0	windspeed	3.21
2	season_winter	2.54
3	yr_2019	1.86
1	season_summer	1.82
8	mnth_november	1.81
14	weathersit_misty	1.56
5	mnth_december	1.41
7	mnth_january	1.28
6	mnth_february	1.26
4	mnth_august	1.24
11	weekday_monday	1.21
12	weekday_tuesday	1.21
9	mnth_september	1.17
13	weathersit_light_weather	1.09
10	holiday_yes	1.07

lr_model_9 = train_model(y_train, X_train_9)

Residual Analysis

# Creating function to plot residual distribution
def get_res(x_df, lr_model):
  X_train_sm = sm.add_constant(x_df)
  y_train_predicted = lr_model.predict(X_train_sm)
  residuals = y_train - y_train_predicted
  return (y_train_predicted, residuals)

def plot_res_dist(x_df, lr_model):
  ax = sns.distplot(get_res(x_df, lr_model)[1])
  ax.figure.set_size_inches(10, 8)
  ax.set_xlabel('Residuals')
  return ax

def plot_res_scatter(x_df, lr_model):
  ax = sns.scatterplot(get_res(x_df, lr_model)[0], get_res(x_df, lr_model)[1])
  ax.figure.set_size_inches(10, 8)
  ax.set_xlabel('Predicted')
  ax.set_ylabel('Residuals')
  sns.lineplot(y_train, [0]*len(y_train), color='red')
  return ax

Model 1

plot_res_dist(X_train_1, lr_model_1)
plt.show()

Assessment:
Residuals are normally distributed around the mean 0. So, the "Normality of Residual" assumption holds.

plot_res_scatter(X_train_1, lr_model_1)
plt.show()

Assessment:
Homoscedasticity assumption holds.

Model 2

plot_res_dist(X_train_2, lr_model_2)
plt.show()

Assessment:
Residuals are normally distributed around the mean 0. So, the "Normality of Residual" assumption holds.

plot_res_scatter(X_train_2, lr_model_2)
plt.show()

Assessment:
Homoscedasticity assumption holds.

Model 3

plot_res_dist(X_train_3, lr_model_3)
plt.show()

Assessment:
Residuals are normally distributed around the mean 0. So, the "Normality of Residual" assumption holds.

plot_res_scatter(X_train_3, lr_model_3)
plt.show()

Assessment:
Homoscedasticity assumption holds.

Model 4

plot_res_dist(X_train_4, lr_model_4)
plt.show()

Assessment:
Residuals are normally distributed around the mean 0. So, the "Normality of Residual" assumption holds.

plot_res_scatter(X_train_4, lr_model_4)
plt.show()

Assessment:
Homoscedasticity assumption holds.

Model 5

plot_res_dist(X_train_5, lr_model_5)
plt.show()

Assessment:
Residuals are normally distributed around the mean 0. So, the "Normality of Residual" assumption holds.

plot_res_scatter(X_train_5, lr_model_5)
plt.show()

Assessment:
Homoscedasticity assumption holds.

Model 6

plot_res_dist(X_train_6, lr_model_6)
plt.show()

Assessment:
Residuals are normally distributed around the mean 0. So, the "Normality of Residual" assumption holds.

plot_res_scatter(X_train_6, lr_model_6)
plt.show()

Assessment:
Homoscedasticity assumption holds.

Model 7

plot_res_dist(X_train_7, lr_model_7)
plt.show()

Assessment:
Residuals are normally distributed around the mean 0. So, the "Normality of Residual" assumption holds.

plot_res_scatter(X_train_7, lr_model_7)
plt.show()

Assessment:
Homoscedasticity assumption holds.

Model 8

plot_res_dist(X_train_8, lr_model_8)
plt.show()

Assessment:
Residuals are normally distributed around the mean 0. So, the "Normality of Residual" assumption holds.

plot_res_scatter(X_train_8, lr_model_8)
plt.show()

Assessment:
Homoscedasticity assumption holds.

Model 9

plot_res_dist(X_train_9, lr_model_9)
plt.show()

Assessment:
Residuals are normally distributed around the mean 0. So, the "Normality of Residual" assumption holds.

plot_res_scatter(X_train_9, lr_model_9)
plt.show()

Assessment:
Homoscedasticity assumption holds.

Evaluation on Test Set

Rescaling Test Set

df_test[num_vars] = scaler.transform(df_test[num_vars])

Extract Target Variable

y_test = df_test.pop('cnt')
X_test = df_test

Predict & Evaluate

# Creating function to easily calculate R-squared on test set
def r2_on_test_set(lr_model, train_df):
  X_test_df = X_test[train_df.columns]
  X_test_sm = sm.add_constant(X_test_df)
  y_test_predicted = lr_model.predict(X_test_sm)
  r2 = r2_score(y_test, y_test_predicted)
  return r2

def get_adjusted_r2(r2,p):
  N = y_test.shape[0]
  return 1 - (((1-r2)*(N-1))/(N-p-1))

model_names = ['Model 1', 'Model 2', 'Model 3', 'Model 4', 'Model 5', 'Model 6', 'Model 7', 'Model 8', 'Model 9']
lr_models = [lr_model_1, lr_model_2, lr_model_3, lr_model_4, lr_model_5, lr_model_6, lr_model_7, lr_model_8, lr_model_9]
train_sets = [X_train_1, X_train_2, X_train_3, X_train_4, X_train_5, X_train_6, X_train_7, X_train_8, X_train_9]

test_r2_scores = []
num_predictors = []
test_adjusted_r2_scores = []
train_r2_scores = []
train_adjusted_r2_scores = []

for lr_model, train_set in zip(lr_models, train_sets):
  r2 = r2_on_test_set(lr_model, train_set)
  p = len(lr_model.params)
  adjusted_r2 = get_adjusted_r2(r2,p)
  test_r2_scores.append(r2)
  num_predictors.append(p)
  test_adjusted_r2_scores.append(adjusted_r2)
  train_r2_scores.append(lr_model.rsquared)
  train_adjusted_r2_scores.append(lr_model.rsquared_adj)

result_df = pd.DataFrame({'Model': model_names, 'No. of Predictors': num_predictors, 'R-squared (Train)': train_r2_scores, 'R-squared (Test)': test_r2_scores, 'Adj. R-squared (Train)': train_adjusted_r2_scores, 'Adj. R-squared (Test)': test_adjusted_r2_scores})
result_df

	Model	No. of Predictors	R-squared (Train)	R-squared (Test)	Adj. R-squared (Train)	Adj. R-squared (Test)
0	Model 1	12	0.835710	0.791771	0.832081	0.779700
1	Model 2	11	0.703381	0.558414	0.697437	0.535061
2	Model 3	15	0.839147	0.813676	0.834598	0.799976
3	Model 4	12	0.790832	0.779084	0.786211	0.766277
4	Model 5	11	0.772009	0.791792	0.767440	0.780781
5	Model 6	10	0.805678	0.773139	0.802180	0.762284
6	Model 7	5	0.738932	0.693713	0.736865	0.686557
7	Model 8	17	0.842630	0.810699	0.837522	0.794768
8	Model 9	16	0.772802	0.730680	0.765904	0.709452

Assessment:

• Model 2 shows overfitting behaviour as it is behaving poorly on test set.
• R-squared scores of all other models on test set are within 5% of the R-squared scores on the train set.
• The best performing model based on Adjusted R-squared score on test set is Model 3. With 15 predictors, it has Adjusted R-squared score on test set is approx 80%.

Final Model

final_model = lr_model_3

final_model.summary()

OLS Regression Results

Dep. Variable:	cnt	R-squared:	0.839
Model:	OLS	Adj. R-squared:	0.835
Method:	Least Squares	F-statistic:	184.5
Date:	Tue, 30 Aug 2022	Prob (F-statistic):	4.75e-186
Time:	01:25:41	Log-Likelihood:	504.85
No. Observations:	510	AIC:	-979.7
Df Residuals:	495	BIC:	-916.2
Df Model:	14
Covariance Type:	nonrobust

	coef	std err	t	P>|t|	[0.025	0.975]
const	0.2450	0.032	7.618	0.000	0.182	0.308
temp	0.4387	0.036	12.093	0.000	0.367	0.510
windspeed	-0.1585	0.025	-6.276	0.000	-0.208	-0.109
season_spring	-0.0713	0.021	-3.314	0.001	-0.113	-0.029
season_summer	0.0349	0.015	2.251	0.025	0.004	0.065
season_winter	0.0869	0.018	4.831	0.000	0.052	0.122
yr_2019	0.2345	0.008	28.687	0.000	0.218	0.251
mnth_december	-0.0428	0.018	-2.413	0.016	-0.078	-0.008
mnth_january	-0.0500	0.018	-2.719	0.007	-0.086	-0.014
mnth_july	-0.0500	0.019	-2.703	0.007	-0.086	-0.014
mnth_november	-0.0395	0.019	-2.064	0.040	-0.077	-0.002
mnth_september	0.0687	0.017	4.015	0.000	0.035	0.102
holiday_yes	-0.0918	0.026	-3.522	0.000	-0.143	-0.041
weathersit_light_weather	-0.2917	0.025	-11.840	0.000	-0.340	-0.243
weathersit_misty	-0.0801	0.009	-9.198	0.000	-0.097	-0.063

Omnibus:	69.242	Durbin-Watson:	2.024
Prob(Omnibus):	0.000	Jarque-Bera (JB):	171.476
Skew:	-0.698	Prob(JB):	5.81e-38
Kurtosis:	5.473	Cond. No.	18.9

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.