Bike Sharing Case Study
13 Sep 2024A bike-sharing system is a service in which bikes are made available for shared use to individuals on a short term basis for a price or free. Many bike share systems allow people to borrow a bike from a “dock” which is usually computer-controlled wherein the user enters the payment information, and the system unlocks it. This bike can then be returned to another dock belonging to the same system.
A US bike-sharing provider BoomBikes has recently suffered considerable dips in their revenues due to the ongoing Corona pandemic. The company is finding it very difficult to sustain in the current market scenario. So, it has decided to come up with a mindful business plan to be able to accelerate its revenue as soon as the ongoing lockdown comes to an end, and the economy restores to a healthy state.
In such an attempt, BoomBikes aspires to understand the demand for shared bikes among the people after this ongoing quarantine situation ends across the nation due to Covid-19. They have planned this to prepare themselves to cater to the people’s needs once the situation gets better all around and stand out from other service providers and make huge profits.
Business Goal
You are required to model the demand for shared bikes with the available independent variables. It will be used by the management to understand how exactly the demands vary with different features. They can accordingly manipulate the business strategy to meet the demand levels and meet the customer’s expectations. Further, the model will be a good way for management to understand the demand dynamics of a new market.
Multiple Linear Regression¶
Bike Sharing - Demand Prediction Case Study¶
1. Importing and Understanding the Data¶
In [1]:
# Import all required libraries (Numpy, Pandas, Seaborn Matplotlib) for performing multiple linear regression
import numpy as np
import pandas as pd
import seaborn as sns; sns.set_theme(color_codes=True)
import matplotlib.pyplot as plt
%matplotlib inline
# Import Statistical and Machine learning modules (Statsmodel, SKLearn) for Python
import statsmodels.api as sm
from statsmodels.stats.outliers_influence import variance_inflation_factor
from sklearn.feature_selection import RFE
from sklearn.linear_model import LinearRegression
from sklearn.metrics import r2_score, mean_absolute_error, mean_absolute_percentage_error, mean_squared_error
from sklearn.model_selection import train_test_split
from sklearn.preprocessing import MinMaxScaler
import warnings # Suppress Warnings
warnings.filterwarnings('ignore')
from IPython.core.interactiveshell import InteractiveShell
InteractiveShell.ast_node_interactivity = 'all'
# Set custom configurations For pandas.
pd.set_option("display.max_rows", 900)
pd.set_option("display.max_columns", 900)
pd.set_option('display.float_format', lambda x: '%.6f' % x)
Custom Functions¶
In [2]:
# To distinguish numerical columns either as categorical/discrete or non categorical and return as dict
def classify_feature_dtype(df,cols):
d_categories = {'int_cat': [], "float_ts":[] }
for col in cols:
if (len(df[col].unique()) < 20):
d_categories['int_cat'].append(col)
else:
d_categories['float_ts'].append(col)
return d_categories
# Print all statistical information for a given set of columns
def show_stats(df, cols):
for col in list(cols):
print("Total Nulls: {0},\nMode: {1}".format(df[col].isna().sum(), df[col].mode()[0]))
if len(df[col].unique()) < 50:
print("\nUnique: {0}\n".format(df[col].unique()))
if (df[col].dtype == int) or (df[col].dtype == float):
print("Median : {0}, \nVariance: {1}, \n\nDescribe: {2} \n".format(df[col].median(), df[col].var(), df[col].describe()))
print("ValueCounts: {0} \n\n\n".format((df[col].value_counts(normalize=True) * 100).head(5)))
print("------------------------------------------------------------------")
# Return the percentage of null values in each columns in a dataframe
def check_cols_null_pct(df):
df_non_na = df.count() / len(df) # Ratio of non null values
df_na_pct = (1 - df_non_na) * 100 # Find the Percentage of null values
return df_na_pct.sort_values(ascending=False) # Sort the resulting values in descending order
# Generates charts based on the data type of the cols, as part of the univariate analysis
# it takes dataframe, columns, train data 0,1, and feature type as args.
def univariate_plots(df, cols, target=None, ftype=None, l_dict = None):
for col in cols:
#generate plots and graphs for category type. (generates piechart, countplot, boxplot / if training data is provided it generates bar chart instead)
if ftype == "category":
fig, axs = plt.subplots(1, 3, figsize=(20, 6))
col_idx = 0
axs[col_idx].pie(x=df[col].value_counts().head(15), labels=df[col].value_counts().head(15).index, autopct="%1.1f%%",
radius=1, textprops={"fontsize": 10, "color": "Black"}, startangle=90, rotatelabels=False)
axs[col_idx].set_title("PieChart of {0}".format(col), y=1); plt.xticks(rotation=45); plt.ylabel("Percentage")
fig.subplots_adjust(wspace=0.5, hspace=0.3)
col_idx += 1
sns.countplot(data=df, y=col, order=df[col].value_counts().index, palette="viridis", ax=axs[col_idx])
if (l_dict is not None) and (l_dict.get(col) is not None):
axs[col_idx].legend([ f'{k} - {v}' for k,v in l_dict[col].items()])
axs[col_idx].set_title("Countplot of {0}".format(col)); plt.xticks(rotation=45); plt.xlabel(col); plt.ylabel("Count")
fig.subplots_adjust(wspace=0.5, hspace=0.3)
col_idx += 1
ax = sns.barplot(data=df, x=col, y=target, palette="viridis", ax=axs[col_idx], errwidth=0)
for i in ax.containers:
ax.bar_label(i,)
axs[col_idx].set_title('Barplot against target'); plt.xticks(rotation=45); plt.xlabel(col)
fig.subplots_adjust(wspace=0.5, hspace=0.3)
plt.suptitle("Univariate analysis of {0}".format(col), fontsize=12, y=0.95)
plt.tight_layout()
plt.subplots_adjust(top=0.85)
plt.show()
plt.clf()
#generate plots and graphs for numerical types. (generates boxplot, histplot, kdeplot, scatterplot)
elif ftype == "non_categorical":
fig, axs = plt.subplots(1, 4, figsize=(20, 6))
col_idx = 0
sns.boxplot(data=df, y=col, palette="viridis", flierprops=dict( marker="o", markersize=6, markerfacecolor="red", markeredgecolor="black"),
medianprops=dict(linestyle="-", linewidth=3, color="#FF9900"), whiskerprops=dict(linestyle="-", linewidth=2, color="black"),
capprops=dict(linestyle="-", linewidth=2, color="black"), ax=axs[col_idx])
axs[col_idx].set_title("Boxplot of {0}".format(col)); plt.xticks(rotation=45); plt.xlabel(col)
fig.subplots_adjust(wspace=0.5, hspace=0.3)
col_idx += 1
axs[col_idx].hist(data=df, x=col, label=col)
axs[col_idx].set_title("Histogram of {0}".format(col)); plt.xticks(rotation=45); plt.xlabel(col)
fig.subplots_adjust(wspace=0.5, hspace=0.3)
col_idx += 1
sns.kdeplot(df[col], shade=True, ax=axs[col_idx])
axs[col_idx].set_title("KDE plot of {0}".format(col)); plt.xticks(rotation=45); plt.xlabel(col)
fig.subplots_adjust(wspace=0.5, hspace=0.3)
col_idx += 1
sns.scatterplot(df[col], ax=axs[col_idx])
axs[col_idx].set_title("Scatterplot of {0}".format(col)); plt.xticks(rotation=45); plt.xlabel(col)
fig.subplots_adjust(wspace=0.5, hspace=0.3)
plt.suptitle("Univariate analysis of {0}".format(col), fontsize=12, y=0.95)
plt.tight_layout()
plt.subplots_adjust(top=0.85)
plt.show()
plt.clf()
# Perform Outlier analysis on the given dataframe.
# Find Lower threshold, Upper threshold and IQR values.
# Return the Result as a dataframe.
# find_outlier = True argument: restricts the output df to outlier columns. whereas find_outlier = False: returns results for all columns
def get_extremeval_threshld(df, find_outlier=False):
outlier_df = pd.DataFrame(columns=[i for i in df.columns if find_outlier == True], data=None)
for col in df.columns:
thirdq, firstq = df[col].quantile(0.75), df[col].quantile(0.25)
iqr = 1.5 * (thirdq - firstq)
extvalhigh, extvallow = iqr + thirdq, firstq - iqr
if find_outlier == True:
dfout = df.loc[(df[col] > extvalhigh) | (df[col] < extvallow)]
dfout = dfout.assign(name=col, thresh_low=extvallow, thresh_high=extvalhigh)
else:
dfout = pd.DataFrame([[col, extvallow, extvalhigh]], columns=['name', 'thresh_low', 'thresh_high'])
outlier_df = pd.concat([outlier_df, dfout])
# outlier_df = outlier_df.reset_index(drop=True)
outlier_df = outlier_df.set_index('name',drop=True)
return outlier_df
In [3]:
# import and read the dataset
bike_share_df = pd.read_csv('day.csv', sep=",", header=0)
bike_share_df.head()
Out[3]:
| 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 | 18.181250 | 80.583300 | 10.749882 | 331 | 654 | 985 |
| 1 | 2 | 02-01-2018 | 1 | 0 | 1 | 0 | 2 | 1 | 2 | 14.902598 | 17.686950 | 69.608700 | 16.652113 | 131 | 670 | 801 |
| 2 | 3 | 03-01-2018 | 1 | 0 | 1 | 0 | 3 | 1 | 1 | 8.050924 | 9.470250 | 43.727300 | 16.636703 | 120 | 1229 | 1349 |
| 3 | 4 | 04-01-2018 | 1 | 0 | 1 | 0 | 4 | 1 | 1 | 8.200000 | 10.606100 | 59.043500 | 10.739832 | 108 | 1454 | 1562 |
| 4 | 5 | 05-01-2018 | 1 | 0 | 1 | 0 | 5 | 1 | 1 | 9.305237 | 11.463500 | 43.695700 | 12.522300 | 82 | 1518 | 1600 |
2. EDA - Data Cleaning and Data Visualization¶
In [4]:
# Perform renaming of inconsistent columns and drop the unnecessary
bike_share_df = bike_share_df.drop(columns=['instant '], errors='ignore')
bike_share_df = bike_share_df.rename(str.strip, axis='columns')
bike_share_df = bike_share_df.rename({'dteday':'date', 'yr':'year', 'mnth':'month', 'atemp':'actual_temp', 'hum':'humidity', 'casual':'casual_users', 'registered':'registered_users', 'cnt':'total_cnt'}, axis='columns')
bike_share_df.columns # get the list of columns
bike_share_df.info() # shows column information
Out[4]:
Index(['date', 'season', 'year', 'month', 'holiday', 'weekday', 'workingday',
'weathersit', 'temp', 'actual_temp', 'humidity', 'windspeed',
'casual_users', 'registered_users', 'total_cnt'],
dtype='object')
<class 'pandas.core.frame.DataFrame'> RangeIndex: 730 entries, 0 to 729 Data columns (total 15 columns): # Column Non-Null Count Dtype --- ------ -------------- ----- 0 date 730 non-null object 1 season 730 non-null int64 2 year 730 non-null int64 3 month 730 non-null int64 4 holiday 730 non-null int64 5 weekday 730 non-null int64 6 workingday 730 non-null int64 7 weathersit 730 non-null int64 8 temp 730 non-null float64 9 actual_temp 730 non-null float64 10 humidity 730 non-null float64 11 windspeed 730 non-null float64 12 casual_users 730 non-null int64 13 registered_users 730 non-null int64 14 total_cnt 730 non-null int64 dtypes: float64(4), int64(10), object(1) memory usage: 85.7+ KB
In [5]:
# Return the percentage of null values in each columns in a dataframe
check_cols_null_pct(bike_share_df)
# Print all statistical information for a given set of columns
show_stats(bike_share_df, bike_share_df.columns)
Out[5]:
date 0.000000 season 0.000000 year 0.000000 month 0.000000 holiday 0.000000 weekday 0.000000 workingday 0.000000 weathersit 0.000000 temp 0.000000 actual_temp 0.000000 humidity 0.000000 windspeed 0.000000 casual_users 0.000000 registered_users 0.000000 total_cnt 0.000000 dtype: float64
Total Nulls: 0, Mode: 01-01-2018 ValueCounts: date 01-01-2018 0.136986 25-04-2019 0.136986 27-04-2019 0.136986 28-04-2019 0.136986 29-04-2019 0.136986 Name: proportion, dtype: float64 ------------------------------------------------------------------ Total Nulls: 0, Mode: 3 Unique: [1 2 3 4] Median : 3.0, Variance: 1.232508408967052, Describe: count 730.000000 mean 2.498630 std 1.110184 min 1.000000 25% 2.000000 50% 3.000000 75% 3.000000 max 4.000000 Name: season, dtype: float64 ValueCounts: season 3 25.753425 2 25.205479 1 24.657534 4 24.383562 Name: proportion, dtype: float64 ------------------------------------------------------------------ Total Nulls: 0, Mode: 0 Unique: [0 1] Median : 0.5, Variance: 0.2503429355281207, Describe: count 730.000000 mean 0.500000 std 0.500343 min 0.000000 25% 0.000000 50% 0.500000 75% 1.000000 max 1.000000 Name: year, dtype: float64 ValueCounts: year 0 50.000000 1 50.000000 Name: proportion, dtype: float64 ------------------------------------------------------------------ Total Nulls: 0, Mode: 1 Unique: [ 1 2 3 4 5 6 7 8 9 10 11 12] Median : 7.0, Variance: 11.903985568521309, Describe: count 730.000000 mean 6.526027 std 3.450215 min 1.000000 25% 4.000000 50% 7.000000 75% 10.000000 max 12.000000 Name: month, dtype: float64 ValueCounts: month 1 8.493151 3 8.493151 5 8.493151 7 8.493151 8 8.493151 Name: proportion, dtype: float64 ------------------------------------------------------------------ Total Nulls: 0, Mode: 0 Unique: [0 1] Median : 0.0, Variance: 0.027977901798297095, Describe: count 730.000000 mean 0.028767 std 0.167266 min 0.000000 25% 0.000000 50% 0.000000 75% 0.000000 max 1.000000 Name: holiday, dtype: float64 ValueCounts: holiday 0 97.123288 1 2.876712 Name: proportion, dtype: float64 ------------------------------------------------------------------ Total Nulls: 0, Mode: 1 Unique: [1 2 3 4 5 6 0] Median : 3.0, Variance: 4.001354830223422, Describe: count 730.000000 mean 2.995890 std 2.000339 min 0.000000 25% 1.000000 50% 3.000000 75% 5.000000 max 6.000000 Name: weekday, dtype: float64 ValueCounts: weekday 1 14.383562 2 14.383562 3 14.246575 4 14.246575 5 14.246575 Name: proportion, dtype: float64 ------------------------------------------------------------------ Total Nulls: 0, Mode: 1 Unique: [1 0] Median : 1.0, Variance: 0.21403686791814577, Describe: count 730.000000 mean 0.690411 std 0.462641 min 0.000000 25% 0.000000 50% 1.000000 75% 1.000000 max 1.000000 Name: workingday, dtype: float64 ValueCounts: workingday 1 69.041096 0 30.958904 Name: proportion, dtype: float64 ------------------------------------------------------------------ Total Nulls: 0, Mode: 1 Unique: [2 1 3] Median : 1.0, Variance: 0.29681492756073846, Describe: count 730.000000 mean 1.394521 std 0.544807 min 1.000000 25% 1.000000 50% 1.000000 75% 2.000000 max 3.000000 Name: weathersit, dtype: float64 ValueCounts: weathersit 1 63.424658 2 33.698630 3 2.876712 Name: proportion, dtype: float64 ------------------------------------------------------------------ Total Nulls: 0, Mode: 10.899153 Median : 20.4658265, Variance: 56.35097933285952, Describe: count 730.000000 mean 20.319259 std 7.506729 min 2.424346 25% 13.811885 50% 20.465826 75% 26.880615 max 35.328347 Name: temp, dtype: float64 ValueCounts: temp 26.035000 0.684932 10.899153 0.684932 27.880000 0.547945 28.563347 0.547945 23.130847 0.547945 Name: proportion, dtype: float64 ------------------------------------------------------------------ Total Nulls: 0, Mode: 32.7344 Median : 24.368225, Variance: 66.42751661885134, Describe: count 730.000000 mean 23.726322 std 8.150308 min 3.953480 25% 16.889713 50% 24.368225 75% 30.445775 max 42.044800 Name: actual_temp, dtype: float64 ValueCounts: actual_temp 32.734400 0.547945 18.781050 0.410959 31.850400 0.410959 28.598750 0.273973 23.326250 0.273973 Name: proportion, dtype: float64 ------------------------------------------------------------------ Total Nulls: 0, Mode: 61.3333 Median : 62.625, Variance: 202.70894176890425, Describe: count 730.000000 mean 62.765175 std 14.237589 min 0.000000 25% 52.000000 50% 62.625000 75% 72.989575 max 97.250000 Name: humidity, dtype: float64 ValueCounts: humidity 61.333300 0.547945 63.083300 0.410959 55.208300 0.410959 60.500000 0.410959 56.833300 0.410959 Name: proportion, dtype: float64 ------------------------------------------------------------------ Total Nulls: 0, Mode: 7.12545 Median : 12.125325, Variance: 26.996760622179345, Describe: count 730.000000 mean 12.763620 std 5.195841 min 1.500244 25% 9.041650 50% 12.125325 75% 15.625589 max 34.000021 Name: windspeed, dtype: float64 ValueCounts: windspeed 9.041918 0.410959 11.166689 0.410959 11.250104 0.410959 15.333486 0.410959 7.959064 0.410959 Name: proportion, dtype: float64 ------------------------------------------------------------------ Total Nulls: 0, Mode: 120 Median : 717.0, Variance: 471254.6181408198, Describe: count 730.000000 mean 849.249315 std 686.479875 min 2.000000 25% 316.250000 50% 717.000000 75% 1096.500000 max 3410.000000 Name: casual_users, dtype: float64 ValueCounts: casual_users 120 0.547945 968 0.547945 639 0.410959 163 0.410959 775 0.410959 Name: proportion, dtype: float64 ------------------------------------------------------------------ Total Nulls: 0, Mode: 1707 Median : 3664.5, Variance: 2432847.2895522094, Describe: count 730.000000 mean 3658.757534 std 1559.758728 min 20.000000 25% 2502.250000 50% 3664.500000 75% 4783.250000 max 6946.000000 Name: registered_users, dtype: float64 ValueCounts: registered_users 4841 0.410959 6248 0.410959 1707 0.410959 3461 0.273973 2713 0.273973 Name: proportion, dtype: float64 ------------------------------------------------------------------ Total Nulls: 0, Mode: 1096 Median : 4548.5, Variance: 3748141.098718458, Describe: count 730.000000 mean 4508.006849 std 1936.011647 min 22.000000 25% 3169.750000 50% 4548.500000 75% 5966.000000 max 8714.000000 Name: total_cnt, dtype: float64 ValueCounts: total_cnt 5409 0.273973 2424 0.273973 5698 0.273973 4459 0.273973 5119 0.273973 Name: proportion, dtype: float64 ------------------------------------------------------------------
Replace column values¶
In [6]:
# Replace columns with nominal values into strings
bike_share_df['season'] = bike_share_df['season'].replace(to_replace = [1, 2, 3, 4], value = ['Spring', 'Summer', 'Fall', 'Winter'])
bike_share_df['year'] = bike_share_df['year'].replace(to_replace = [0, 1], value = [2018, 2019])
bike_share_df['month'] = bike_share_df['month'].replace(to_replace = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], value = ['Jan', 'Feb', 'Mar', 'Apr', 'May', 'Jun', 'Jul', 'Aug', 'Sep', 'Oct', 'Nov', 'Dec'])
bike_share_df['holiday'] = bike_share_df['holiday'].replace(to_replace = [0, 1], value = ['NotHoliday', 'Holiday'])
bike_share_df['weekday'] = bike_share_df['weekday'].replace(to_replace = [0, 1, 2, 3, 4, 5, 6], value = ['Sun', 'Mon', 'Tue', 'Wed', 'Thur', 'Fri', 'Sat'])
bike_share_df['workingday'] = bike_share_df['workingday'].replace(to_replace = [0, 1], value = ['NotWorkingday', 'Workingday'])
bike_share_df['weathersit'] = bike_share_df['weathersit'].replace(to_replace = [1, 2, 3, 4], value = ['Cloud', 'Mist', 'LightRain', 'HeavyRain'])
# Convert dtype of columns to category for categorical columns
bike_share_df[['season', 'year', 'month', 'holiday', 'weekday', 'workingday', 'weathersit']] = bike_share_df[['season', 'year', 'month', 'holiday', 'weekday', 'workingday', 'weathersit']].astype('category')
# Convert date to datetime format
bike_share_df['date'] = pd.to_datetime(bike_share_df['date'], dayfirst=True, format='mixed')
# Distinguish numerical columns either as categorical/discrete or non-categorical/ continuous and return as dict
dtype_dict = classify_feature_dtype(bike_share_df, bike_share_df.columns)
Outlier Analysis¶
In [7]:
mask = bike_share_df.columns.isin(dtype_dict['int_cat']) # retrieve columns of categorical type for masking
# function to retrieve lower threshold and higher threshold for all continuous type columns
col_threshld = get_extremeval_threshld(bike_share_df.loc[:,~mask], find_outlier = False)
col_threshld
# function to retrieve lower threshold and higher threshold for Outlier columns and find the count
extreme_val_df = get_extremeval_threshld(bike_share_df.loc[:,~mask], find_outlier = True)
outlier_cols = extreme_val_df.groupby(by='name')[['thresh_low','thresh_high']].value_counts()
outlier_cols
bike_share_df[['humidity', 'windspeed', 'casual_users']].describe([.05,.10,.75,.95,.99])
Out[7]:
| thresh_low | thresh_high | |
|---|---|---|
| name | ||
| date | 2017-01-01 12:00:00 | 2020-12-29 12:00:00 |
| temp | -5.791209 | 46.483709 |
| actual_temp | -3.444381 | 50.779869 |
| humidity | 20.515637 | 104.473938 |
| windspeed | -0.834259 | 25.501498 |
| casual_users | -854.125000 | 2266.875000 |
| registered_users | -919.250000 | 8204.750000 |
| total_cnt | -1024.625000 | 10160.375000 |
Out[7]:
name thresh_low thresh_high casual_users -854.125000 2266.875000 44 humidity 20.515637 104.473938 2 windspeed -0.834259 25.501498 13 Name: count, dtype: int64
Out[7]:
| humidity | windspeed | casual_users | |
|---|---|---|---|
| count | 730.000000 | 730.000000 | 730.000000 |
| mean | 62.765175 | 12.763620 | 849.249315 |
| std | 14.237589 | 5.195841 | 686.479875 |
| min | 0.000000 | 1.500244 | 2.000000 |
| 5% | 40.741735 | 5.326052 | 88.450000 |
| 10% | 45.000000 | 6.704754 | 139.900000 |
| 50% | 62.625000 | 12.125325 | 717.000000 |
| 75% | 72.989575 | 15.625589 | 1096.500000 |
| 95% | 86.868735 | 22.999988 | 2355.000000 |
| 99% | 92.795857 | 27.380948 | 2931.680000 |
| max | 97.250000 | 34.000021 | 3410.000000 |
Univariate and Bivariate Analysis¶
In [8]:
label_dict = {'year':{0: 2018, 1:2019},'holiday':{0:'no',1:'yes'},'workingday':{0:'no',1:'yes'}}
# Plot Univariate graphs for both categorical and continuous variables. as well as perform bivariate analysis against total_cnt column
print("Univariate Analysis of categorical Variables")
univariate_plots(bike_share_df, dtype_dict['int_cat'], target='total_cnt', ftype = "category", l_dict=label_dict)
print("--------------------------------------------------------------------------")
print("Univariate Analysis of Continuous Variables")
univariate_plots(bike_share_df, dtype_dict['float_ts'], target='total_cnt', ftype = "non_categorical")
print("--------------------------------------------------------------------------")
Univariate Analysis of categorical Variables
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-------------------------------------------------------------------------- Univariate Analysis of Continuous Variables
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Observations:¶
Categorical Features¶
#### Season
• Total_cnt of users is very low for the Spring Season • Total_cnt of users is in the fall is very high • Seasonal changes have an impact on the user decision
#### Year
• there is a significant growth in total_cnt in 2019 compared to 2018. • This can be a positive sign that there is a positive growth projection
#### Weekday
• The total_cnt of users remains in a similar range for all the days of the week
• Highest on Thursday
• Lowest on Monday
• End of the week sees a slight surge in demand
#### month
• Jun saw the most user count
• whereas the Jan month saw least users count
• overall, there is a trend that the usage drops at the start and end of the year months
#### Holiday
- Total non holiday days are higher compared to holidays
#### Weathersit
- Most days in the year are cloudy
• There is a significant increase in total_cnt on Cloudy days
• Whereas there is significant decrease in LightRain days
• Users prefer to use bike platforms on a cloudy day
### Numerical Features
- There are Outliers in Windspeed, Humidty, Casual_Users features
- Windspeed, casual users are positively skewed , while humidity is negatively skewed
- registerd users, total_cnt are normally distributed
Fix Outliers¶
In [9]:
# Fix Outliers by setting either thresh low or thresh low for both extremes
lower_cutoff = col_threshld.loc['humidity','thresh_low']
bike_share_df['humidity'] = np.where((bike_share_df['humidity'] < lower_cutoff), lower_cutoff, bike_share_df['humidity'])
upper_cutoff = col_threshld.loc['windspeed','thresh_high']
bike_share_df['windspeed'] = np.where((bike_share_df['windspeed'] > upper_cutoff), upper_cutoff, bike_share_df['windspeed'])
Bivariate Analysis¶
There is a significant growth in User count for 2019, comparison to the first year 2018, also it had a downward trend at the of 2019
In [10]:
# line graph of date vs total_cnt
plt.figure(figsize=(22,4));
sns.lineplot(x='date',y='total_cnt', data=bike_share_df, palette="flare");
plt.show();
In [11]:
# Pairplot analysis of all the continuous feature columns in the dataframe.
sns.pairplot(bike_share_df[['temp','actual_temp','humidity','windspeed','casual_users','registered_users', 'total_cnt']])
plt.show();
There are high correlation between :
registered users vs total_cnt , casual_users vs total _cnt, temp vs total_cnt, actual_temp vs total_cnt
In [12]:
# Regression line analysis of all the continuous feature type columns.
axs = 331
plt.figure(figsize=(22,15))
for i in list(set(dtype_dict['float_ts']) - set(['date','total_cnt'])):
plt.subplot(axs)
sns.regplot(x=i, y='total_cnt' ,data=bike_share_df,color= 'green', line_kws={"color": "red"})
axs += 1
plt.show();
(temp, actual_temp, resitered_users, casual_users) vs total_cnt are positively correlated whereas the windspeed, humidity does not show high correlation but the regression lines show a negative trend
In [13]:
# Generate Bivariate Boxplots combinations for all the categorical and continuous columns
x_lst = list(dtype_dict['int_cat']) # x_list variable contains all the categorical Columns
y_lst = list(dtype_dict['float_ts'])# y_list contains all Continuous feature type columns
axs = 1
for x_col in x_lst:
plt.figure(figsize=(22,72))
for y_col in y_lst:
plt.subplot(18,4,axs)
sns.boxplot(x=x_col, y=y_col, data=bike_share_df, palette='tab10')
axs += 1
plt.show();
plt.show();
Multivariate Analysis¶
!!!!! Deliberately commented to avoid consuming CPU resources. and it may run for a long time. (output is already generated)
In [14]:
# # Multivariate Boxplots -
# z_lst = x_lst = list(dtype_dict['int_cat'])
# y_lst = list(dtype_dict['float_ts'])
# z_lst = x_lst.copy()
# z_lst.remove('month')
# z_lst.remove('weekday')
# axs = 1
# for x_col in x_lst:
# plt.figure(figsize=(25,650))
# for y_col in y_lst:
# for z_col in z_lst:
# plt.subplot(200,4,axs)
# sns.boxplot(x=x_col, y=y_col, hue=z_col, data=bike_share_df, palette='tab10')
# plt.legend()
# axs += 1
# plt.show();
# # axs = 1
# print("--------------------------------------------------------------------------")
# plt.show();
In [15]:
# Find the Top 10 Correlations amongst the columns using df.corr()
corr_0 = bike_share_df.corr(numeric_only=True).abs()
corr_0 = corr_0.unstack()
correlation_0 = corr_0.sort_values()
correlation_0 = corr_0.dropna()
correlation_0 = correlation_0[correlation_0 != 1.0]
correlation_target_zero = correlation_0.reset_index()
correlation_target_zero.sort_values(by=0, ascending=False).head(10)
Out[15]:
| level_0 | level_1 | 0 | |
|---|---|---|---|
| 0 | temp | actual_temp | 0.991696 |
| 6 | actual_temp | temp | 0.991696 |
| 35 | registered_users | total_cnt | 0.945411 |
| 41 | total_cnt | registered_users | 0.945411 |
| 40 | total_cnt | casual_users | 0.672123 |
| 29 | casual_users | total_cnt | 0.672123 |
| 11 | actual_temp | total_cnt | 0.630685 |
| 37 | total_cnt | actual_temp | 0.630685 |
| 36 | total_cnt | temp | 0.627044 |
| 5 | temp | total_cnt | 0.627044 |
In [16]:
# Display the Correlation values Using the SNS. Heatmap.
plt.figure(figsize=(22,8))
sns.heatmap(bike_share_df.select_dtypes(exclude=[object,'category']).corr(), annot=True)
plt.show();
Observation:
total_cnt is positively correlated with casual and registered
due to high positive correlation of total_cnt with 'registered' and 'casual', we can assume that there is relation among these as total_cnt is based of registered and casual and hence registered and casual can be removed before model building
total_cnt is negatively correlated to windspeed.
gives an impression that the shared bikes demand will be somewhat less on windy days as compared to normal days. atemp and temp are highly (positively) correlated. there we can remove one among these two before model building
3. MLR - Data Preparation (One Hot Encoding)¶
In [17]:
# Validate the dataframe Content and the column datatypes
bike_share_df.head(2)
bike_share_df.dtypes
Out[17]:
| date | season | year | month | holiday | weekday | workingday | weathersit | temp | actual_temp | humidity | windspeed | casual_users | registered_users | total_cnt | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 2018-01-01 | Spring | 2018 | Jan | NotHoliday | Mon | Workingday | Mist | 14.110847 | 18.181250 | 80.583300 | 10.749882 | 331 | 654 | 985 |
| 1 | 2018-01-02 | Spring | 2018 | Jan | NotHoliday | Tue | Workingday | Mist | 14.902598 | 17.686950 | 69.608700 | 16.652113 | 131 | 670 | 801 |
Out[17]:
date datetime64[ns] season category year category month category holiday category weekday category workingday category weathersit category temp float64 actual_temp float64 humidity float64 windspeed float64 casual_users int64 registered_users int64 total_cnt int64 dtype: object
Dummy Variables Creation¶
In [18]:
# Convert all the categorical columns to numerical values (in dummy variables format), Using get_dummies
bike_share_mod_df = bike_share_df.copy()
dummy_df = pd.get_dummies(bike_share_mod_df, columns=['season', 'year', 'month', 'holiday', 'weekday', 'workingday', 'weathersit'], dtype=int, drop_first=True)
bike_share_mod_df = dummy_df.copy()
bike_share_mod_df.head(2)
Out[18]:
| date | temp | actual_temp | humidity | windspeed | casual_users | registered_users | total_cnt | season_Spring | season_Summer | season_Winter | year_2019 | month_Aug | month_Dec | month_Feb | month_Jan | month_Jul | month_Jun | month_Mar | month_May | month_Nov | month_Oct | month_Sep | holiday_NotHoliday | weekday_Mon | weekday_Sat | weekday_Sun | weekday_Thur | weekday_Tue | weekday_Wed | workingday_Workingday | weathersit_LightRain | weathersit_Mist | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 2018-01-01 | 14.110847 | 18.181250 | 80.583300 | 10.749882 | 331 | 654 | 985 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 |
| 1 | 2018-01-02 | 14.902598 | 17.686950 | 69.608700 | 16.652113 | 131 | 670 | 801 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 |
In [19]:
bike_share_df_bk = bike_share_mod_df.copy()
#-- bike_share_mod_df = bike_share_df_bk
# Drop all the unwanted features from the dataframe.
bike_share_mod_df = bike_share_mod_df.drop(['date', 'casual_users', 'registered_users','actual_temp'], axis=1)
bike_share_mod_df = bike_share_mod_df.astype(float)
bike_share_mod_df.head(2)
Out[19]:
| temp | humidity | windspeed | total_cnt | season_Spring | season_Summer | season_Winter | year_2019 | month_Aug | month_Dec | month_Feb | month_Jan | month_Jul | month_Jun | month_Mar | month_May | month_Nov | month_Oct | month_Sep | holiday_NotHoliday | weekday_Mon | weekday_Sat | weekday_Sun | weekday_Thur | weekday_Tue | weekday_Wed | workingday_Workingday | weathersit_LightRain | weathersit_Mist | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 14.110847 | 80.583300 | 10.749882 | 985.000000 | 1.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 1.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 1.000000 | 1.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 1.000000 | 0.000000 | 1.000000 |
| 1 | 14.902598 | 69.608700 | 16.652113 | 801.000000 | 1.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 1.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 1.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 1.000000 | 0.000000 | 1.000000 | 0.000000 | 1.000000 |
4. MLR - Data Splitting for Train,Test Sets and Rescaling¶
In [20]:
# To perform multiple linear regression, We now convert the given dataset into train and test subsets, at a ratio of 80:20
df_train, df_test = train_test_split(bike_share_mod_df, train_size=0.8,test_size=0.2, random_state=0)
df_train.shape, df_test.shape
Out[20]:
((584, 29), (146, 29))
scaling using MinMaxScaler¶
In [21]:
# In order to avoid scaling issues during model training, We use the MinMaxscalar to scale all Variables Between zero and one
scaler = MinMaxScaler()
num_vars = ['temp', 'humidity', 'windspeed','total_cnt']
# A method on transformers which fits the estimator and returns the transformed training data
df_train[num_vars] = scaler.fit_transform(df_train[num_vars])
df_train.head(2)
Out[21]:
| temp | humidity | windspeed | total_cnt | season_Spring | season_Summer | season_Winter | year_2019 | month_Aug | month_Dec | month_Feb | month_Jan | month_Jul | month_Jun | month_Mar | month_May | month_Nov | month_Oct | month_Sep | holiday_NotHoliday | weekday_Mon | weekday_Sat | weekday_Sun | weekday_Thur | weekday_Tue | weekday_Wed | workingday_Workingday | weathersit_LightRain | weathersit_Mist | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 239 | 0.807351 | 0.428211 | 0.775475 | 0.496088 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 1.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 1.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 1.000000 | 0.000000 | 1.000000 | 0.000000 | 0.000000 |
| 97 | 0.344785 | 0.810325 | 0.545905 | 0.166705 | 0.000000 | 1.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 1.000000 | 0.000000 | 0.000000 | 1.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 1.000000 |
Visualizing the correlations¶
In [22]:
# Linear correlation between temp and total_cnt
plt.figure(figsize=(6,6))
sns.scatterplot(x='temp', y='total_cnt', data=df_train)
plt.show();
In [23]:
# Heatmap to check the correlation amongst the variables
plt.figure(figsize=(30,15))
sns.heatmap(df_train.corr(), annot=True)
plt.show();
5. MLR - Model Building & Analysis using SKlearn on Train data¶
In [24]:
# To build the model, We now Split the Train data set Into X_train with predictor variables and Y_train with target variable.
Y_train = df_train.pop('total_cnt')
X_train = df_train
X_train.head(2)
Y_train.head(2)
X_train.shape, Y_train.shape
Out[24]:
| temp | humidity | windspeed | season_Spring | season_Summer | season_Winter | year_2019 | month_Aug | month_Dec | month_Feb | month_Jan | month_Jul | month_Jun | month_Mar | month_May | month_Nov | month_Oct | month_Sep | holiday_NotHoliday | weekday_Mon | weekday_Sat | weekday_Sun | weekday_Thur | weekday_Tue | weekday_Wed | workingday_Workingday | weathersit_LightRain | weathersit_Mist | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 239 | 0.807351 | 0.428211 | 0.775475 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 1.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 1.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 1.000000 | 0.000000 | 1.000000 | 0.000000 | 0.000000 |
| 97 | 0.344785 | 0.810325 | 0.545905 | 0.000000 | 1.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 1.000000 | 0.000000 | 0.000000 | 1.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 1.000000 |
Out[24]:
239 0.496088 97 0.166705 Name: total_cnt, dtype: float64
Out[24]:
((584, 28), (584,))
In [25]:
# Perform Ordinary least squares Linear Regression.
lr_model = LinearRegression() # Create object lr_model
lr_model.fit(X_train, Y_train) # Fit linear model using X_train and Y_train
Out[25]:
LinearRegression()In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook.
On GitHub, the HTML representation is unable to render, please try loading this page with nbviewer.org.
LinearRegression()
In [26]:
# Get Coefficients of all the trained columns of lr_model using coef_ return it as dataframe
coef_df = pd.DataFrame(lr_model.coef_, X_train.columns, columns=["Coefficient"])
coef_df.head()
print('---------------------------------------------------')
print(f'Model_Score: {lr_model.score(X_train, Y_train)}')
print(f'Model_Intercept: {lr_model.intercept_}')
Out[26]:
| Coefficient | |
|---|---|
| temp | 0.438665 |
| humidity | -0.147242 |
| windspeed | -0.121544 |
| season_Spring | -0.093910 |
| season_Summer | 0.006138 |
--------------------------------------------------- Model_Score: 0.8370930015017464 Model_Intercept: 0.30300351026871525
Predicting on Training dataset¶
In [27]:
Y_train_pred = lr_model.predict(X_train)
Y_comp_df = pd.DataFrame({"Actual" : Y_train, "Predicted" : Y_train_pred})
Y_comp_df.head()
Out[27]:
| Actual | Predicted | |
|---|---|---|
| 239 | 0.496088 | 0.499726 |
| 97 | 0.166705 | 0.286831 |
| 503 | 0.951680 | 0.855537 |
| 642 | 0.935803 | 0.861560 |
| 498 | 0.324551 | 0.633351 |
In [28]:
def adj_r2_score(xfe, yt, ypred):
SS_Reg = np.sum((yt - ypred)**2)
SS_Total = np.sum((yt - np.mean(yt))**2)
r2 = 1-SS_Reg/SS_Total
n = len(xfe)
p = len(xfe.columns)
adj_r2 = 1-((1-r2)*(n-1))/(n-p-1)
return adj_r2
print('---------------------------------------------------')
print(f'Model_Score: {lr_model.score(X_train, Y_train)}')
print(f'Model_Intercept: {lr_model.intercept_}')
print(f'Mean Absolute Error: {mean_absolute_error(Y_train, Y_train_pred)}')
print(f'Mean Squared Error: {mean_squared_error(Y_train, Y_train_pred)}')
print(f'Root Mean Squared Error: {np.sqrt(mean_squared_error(Y_train, Y_train_pred))}')
print(f'R2_score: {r2_score(Y_train, Y_train_pred)}')
print(f'Adjusted_R2_score: {adj_r2_score(X_train, Y_train, Y_train_pred)}')
--------------------------------------------------- Model_Score: 0.8370930015017464 Model_Intercept: 0.30300351026871525 Mean Absolute Error: 0.0641989884673339 Mean Squared Error: 0.007699439599146163 Root Mean Squared Error: 0.08774645063560214 R2_score: 0.8370930015017464 Adjusted_R2_score: 0.8288742700459786
Model Analysis using SKlearn on Test data¶
In [29]:
# For testing model, We now Split the Test data set into X_test with predictor variables and Y_test with target variable.
num_vars = ['temp', 'humidity', 'windspeed','total_cnt']
df_test[num_vars] = scaler.transform(df_test[num_vars])
X_test = df_test.copy()
Y_test = X_test.pop('total_cnt')
X_test.head(5)
Y_test.head(5)
X_test.shape, Y_test.shape
Out[29]:
| temp | humidity | windspeed | season_Spring | season_Summer | season_Winter | year_2019 | month_Aug | month_Dec | month_Feb | month_Jan | month_Jul | month_Jun | month_Mar | month_May | month_Nov | month_Oct | month_Sep | holiday_NotHoliday | weekday_Mon | weekday_Sat | weekday_Sun | weekday_Thur | weekday_Tue | weekday_Wed | workingday_Workingday | weathersit_LightRain | weathersit_Mist | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 196 | 0.781941 | 0.460557 | 0.490778 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 1.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 1.000000 | 1.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 1.000000 | 0.000000 | 0.000000 |
| 187 | 0.860857 | 0.552784 | 0.345523 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 1.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 1.000000 | 0.000000 | 1.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |
| 14 | 0.217065 | 0.340487 | 0.341867 | 1.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 1.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 1.000000 | 1.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 1.000000 | 0.000000 | 1.000000 |
| 31 | 0.165779 | 0.801018 | 0.032244 | 1.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 1.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 1.000000 | 0.000000 | 0.000000 | 0.000000 | 1.000000 | 0.000000 | 0.000000 | 1.000000 | 0.000000 | 1.000000 |
| 390 | 0.352054 | 0.717517 | 0.091901 | 1.000000 | 0.000000 | 0.000000 | 1.000000 | 0.000000 | 0.000000 | 0.000000 | 1.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 1.000000 | 0.000000 | 1.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 1.000000 |
Out[29]:
196 0.678900 187 0.525771 14 0.141049 31 0.153935 390 0.466291 Name: total_cnt, dtype: float64
Out[29]:
((146, 28), (146,))
In [30]:
# Predict the y_pred values using lr_model.predict()
# Compare the actual values vs predicted values.
Y_pred = lr_model.predict(X_test)
Y_comp_df = pd.DataFrame({"Actual" : Y_test, "Predicted" : Y_pred})
Y_comp_df.head()
Out[30]:
| Actual | Predicted | |
|---|---|---|
| 196 | 0.678900 | 0.463670 |
| 187 | 0.525771 | 0.538209 |
| 14 | 0.141049 | 0.126886 |
| 31 | 0.153935 | 0.116398 |
| 390 | 0.466291 | 0.418788 |
In [31]:
# Get Coefficients of all the test columns of lr_model using lr_model.coef_ return it as dataframe
coef_df = pd.DataFrame(lr_model.coef_, X_test.columns, columns=["Coefficient"])
coef_df.head()
Out[31]:
| Coefficient | |
|---|---|
| temp | 0.438665 |
| humidity | -0.147242 |
| windspeed | -0.121544 |
| season_Spring | -0.093910 |
| season_Summer | 0.006138 |
In [32]:
print('---------------------------------------------------')
print(f'Model_Score: {lr_model.score(X_test, Y_test)}')
print(f'Model_Intercept: {lr_model.intercept_}')
print(f'Mean Absolute Error: {mean_absolute_error(Y_test, Y_pred)}')
print(f'Mean Squared Error: {mean_squared_error(Y_test, Y_pred)}')
print(f'Root Mean Squared Error: {np.sqrt(mean_squared_error(Y_test, Y_pred))}')
print(f'R2_score: {r2_score(Y_test, Y_pred)}')
print(f'Adjusted_R2_score: {adj_r2_score(X_test, Y_test, Y_pred)}')
--------------------------------------------------- Model_Score: 0.8712402002824886 Model_Intercept: 0.30300351026871525 Mean Absolute Error: 0.06804551577481996 Mean Squared Error: 0.007545283334061818 Root Mean Squared Error: 0.08686359038205718 R2_score: 0.8712402002824886 Adjusted_R2_score: 0.8404258892389816
This model Adj R2 score is 84%. We need to see if we can reduce the number of features and exclude those which are not much relevant in explaining the target variable.
6. MLR - Recursive Feature Elimination (RFE)¶
In [33]:
# Since the coefficient of determination of the prediction, is not in Recommended range,
# We now perform Recursive feature elimination, to eliminate Unnecessary features from the Training data set
rfe = RFE(estimator=lr_model, n_features_to_select= 15, step=1) # RFE rank feature and remove one feature at a time till features count = 15
rfe = rfe.fit(X_train, Y_train) # fit the RFE model on the Training data set
In [34]:
#Now combines columns and rank and return as table,
# best features are assigned rank 1.
a = list(zip(X_train.columns,rfe.support_,rfe.ranking_))
sorted(a,key=lambda x:x[2])
Out[34]:
[('temp', True, 1),
('humidity', True, 1),
('windspeed', True, 1),
('season_Spring', True, 1),
('season_Winter', True, 1),
('year_2019', True, 1),
('month_Dec', True, 1),
('month_Feb', True, 1),
('month_Jan', True, 1),
('month_Jul', True, 1),
('month_Nov', True, 1),
('month_Sep', True, 1),
('holiday_NotHoliday', True, 1),
('weathersit_LightRain', True, 1),
('weathersit_Mist', True, 1),
('workingday_Workingday', False, 2),
('weekday_Sat', False, 3),
('weekday_Sun', False, 4),
('month_May', False, 5),
('weekday_Mon', False, 6),
('weekday_Tue', False, 7),
('month_Aug', False, 8),
('month_Jun', False, 9),
('weekday_Wed', False, 10),
('month_Mar', False, 11),
('month_Oct', False, 12),
('season_Summer', False, 13),
('weekday_Thur', False, 14)]
In [35]:
# Top 15 features from RFE
rfe_top_features = X_train.columns[rfe.support_]
rfe_top_features
Out[35]:
Index(['temp', 'humidity', 'windspeed', 'season_Spring', 'season_Winter',
'year_2019', 'month_Dec', 'month_Feb', 'month_Jan', 'month_Jul',
'month_Nov', 'month_Sep', 'holiday_NotHoliday', 'weathersit_LightRain',
'weathersit_Mist'],
dtype='object')
In [36]:
# lowest ranked features from RFE
rfe_bottom_ranked = X_train.columns[~rfe.support_]
rfe_bottom_ranked
Out[36]:
Index(['season_Summer', 'month_Aug', 'month_Jun', 'month_Mar', 'month_May',
'month_Oct', 'weekday_Mon', 'weekday_Sat', 'weekday_Sun',
'weekday_Thur', 'weekday_Tue', 'weekday_Wed', 'workingday_Workingday'],
dtype='object')
7. MLR - Model Building and Analysis on top features(RFE)¶
In [37]:
# Perform Ordinary least squares Linear Regression.
X_train_rfe = X_train[rfe_top_features]
lr_model = LinearRegression()
lr_model.fit(X_train_rfe,Y_train)
# Get Coefficients of all the trained columns of lr_model using coef_ return it as dataframe
coef_df = pd.DataFrame(lr_model.coef_, X_train_rfe.columns, columns=["Coefficient"])
coef_df
print('---------------------------------------------------')
print(f'Model_Score: {lr_model.score(X_train_rfe, Y_train)}')
print(f'Model_Intercept: {lr_model.intercept_}')
Out[37]:
LinearRegression()In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook.
On GitHub, the HTML representation is unable to render, please try loading this page with nbviewer.org.
LinearRegression()
Out[37]:
| Coefficient | |
|---|---|
| temp | 0.406779 |
| humidity | -0.142953 |
| windspeed | -0.124305 |
| season_Spring | -0.098555 |
| season_Winter | 0.069679 |
| year_2019 | 0.224385 |
| month_Dec | -0.064447 |
| month_Feb | -0.049548 |
| month_Jan | -0.066233 |
| month_Jul | -0.068706 |
| month_Nov | -0.072105 |
| month_Sep | 0.053923 |
| holiday_NotHoliday | 0.051792 |
| weathersit_LightRain | -0.201380 |
| weathersit_Mist | -0.046398 |
--------------------------------------------------- Model_Score: 0.8293483572406961 Model_Intercept: 0.31058533723271714
8. MLR - Model building and Variance Inflation factor analysis using Statsmodel¶
In [38]:
def train_sm(Xtr, Ytr): # This function creates a new lr_model using statsmodel upon called.
Xtr_sm = sm.add_constant(Xtr) # Adding a constant variable , a column of ones to an array.
lr = sm.OLS(Ytr, Xtr_sm) # Create Ordinary Least Squares model
lrn_model = lr.fit() # fit the model
l_param = lrn_model.params # params
l_summary = lrn_model.summary() # summary
return l_summary, l_param, lrn_model, Xtr_sm
# Measure of VIF, Multicollinearity among features and returns the VIF score for each feature.
def find_VIF(Xtr):
vif = pd.DataFrame()
vif['Features'] = Xtr.columns
vif['VIF'] = [variance_inflation_factor(Xtr.values, i) for i in range(Xtr.shape[1])]
vif['VIF'] = round(vif['VIF'], 2)
vif = vif.sort_values(by = "VIF", ascending = False)
return vif
Base Model¶
In [39]:
# train_sm function creates a new model whenever invoked
summary, param, sm_model, X_train_sm = train_sm(X_train_rfe, Y_train)
summary, param # summary of the linear model
# find_VIF function return the VIF score for all the features
find_VIF(X_train_rfe)
Out[39]:
(<class 'statsmodels.iolib.summary.Summary'>
"""
OLS Regression Results
==============================================================================
Dep. Variable: total_cnt R-squared: 0.829
Model: OLS Adj. R-squared: 0.825
Method: Least Squares F-statistic: 184.0
Date: Mon, 23 Dec 2024 Prob (F-statistic): 1.27e-206
Time: 22:19:02 Log-Likelihood: 578.83
No. Observations: 584 AIC: -1126.
Df Residuals: 568 BIC: -1056.
Df Model: 15
Covariance Type: nonrobust
========================================================================================
coef std err t P>|t| [0.025 0.975]
----------------------------------------------------------------------------------------
const 0.3106 0.035 8.808 0.000 0.241 0.380
temp 0.4068 0.032 12.858 0.000 0.345 0.469
humidity -0.1430 0.028 -5.133 0.000 -0.198 -0.088
windspeed -0.1243 0.019 -6.697 0.000 -0.161 -0.088
season_Spring -0.0986 0.017 -5.783 0.000 -0.132 -0.065
season_Winter 0.0697 0.013 5.254 0.000 0.044 0.096
year_2019 0.2244 0.008 29.037 0.000 0.209 0.240
month_Dec -0.0644 0.017 -3.823 0.000 -0.098 -0.031
month_Feb -0.0495 0.020 -2.454 0.014 -0.089 -0.010
month_Jan -0.0662 0.020 -3.265 0.001 -0.106 -0.026
month_Jul -0.0687 0.016 -4.238 0.000 -0.101 -0.037
month_Nov -0.0721 0.018 -4.045 0.000 -0.107 -0.037
month_Sep 0.0539 0.016 3.473 0.001 0.023 0.084
holiday_NotHoliday 0.0518 0.023 2.225 0.026 0.006 0.098
weathersit_LightRain -0.2014 0.029 -7.025 0.000 -0.258 -0.145
weathersit_Mist -0.0464 0.010 -4.533 0.000 -0.067 -0.026
==============================================================================
Omnibus: 93.598 Durbin-Watson: 2.028
Prob(Omnibus): 0.000 Jarque-Bera (JB): 243.452
Skew: -0.812 Prob(JB): 1.36e-53
Kurtosis: 5.714 Cond. No. 22.0
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
""",
const 0.310585
temp 0.406779
humidity -0.142953
windspeed -0.124305
season_Spring -0.098555
season_Winter 0.069679
year_2019 0.224385
month_Dec -0.064447
month_Feb -0.049548
month_Jan -0.066233
month_Jul -0.068706
month_Nov -0.072105
month_Sep 0.053923
holiday_NotHoliday 0.051792
weathersit_LightRain -0.201380
weathersit_Mist -0.046398
dtype: float64)
Out[39]:
| Features | VIF | |
|---|---|---|
| 12 | holiday_NotHoliday | 21.000000 |
| 0 | temp | 17.040000 |
| 1 | humidity | 14.990000 |
| 2 | windspeed | 4.980000 |
| 3 | season_Spring | 4.380000 |
| 4 | season_Winter | 2.900000 |
| 14 | weathersit_Mist | 2.450000 |
| 8 | month_Jan | 2.270000 |
| 5 | year_2019 | 2.070000 |
| 7 | month_Feb | 2.030000 |
| 10 | month_Nov | 1.820000 |
| 6 | month_Dec | 1.680000 |
| 9 | month_Jul | 1.450000 |
| 13 | weathersit_LightRain | 1.350000 |
| 11 | month_Sep | 1.190000 |
Model 1¶
In [40]:
# 'holiday_NotHoliday' has very high VIF. Hence dropped
X_train_rfe = X_train_rfe.drop(["holiday_NotHoliday"], axis = 1)
summary, param, sm_model, X_train_sm = train_sm(X_train_rfe, Y_train)
summary, param
find_VIF(X_train_rfe)
Out[40]:
(<class 'statsmodels.iolib.summary.Summary'>
"""
OLS Regression Results
==============================================================================
Dep. Variable: total_cnt R-squared: 0.828
Model: OLS Adj. R-squared: 0.824
Method: Least Squares F-statistic: 195.5
Date: Mon, 23 Dec 2024 Prob (F-statistic): 1.04e-206
Time: 22:19:02 Log-Likelihood: 576.29
No. Observations: 584 AIC: -1123.
Df Residuals: 569 BIC: -1057.
Df Model: 14
Covariance Type: nonrobust
========================================================================================
coef std err t P>|t| [0.025 0.975]
----------------------------------------------------------------------------------------
const 0.3623 0.027 13.608 0.000 0.310 0.415
temp 0.4057 0.032 12.782 0.000 0.343 0.468
humidity -0.1441 0.028 -5.158 0.000 -0.199 -0.089
windspeed -0.1238 0.019 -6.648 0.000 -0.160 -0.087
season_Spring -0.0993 0.017 -5.805 0.000 -0.133 -0.066
season_Winter 0.0692 0.013 5.201 0.000 0.043 0.095
year_2019 0.2248 0.008 28.991 0.000 0.210 0.240
month_Dec -0.0646 0.017 -3.817 0.000 -0.098 -0.031
month_Feb -0.0498 0.020 -2.460 0.014 -0.090 -0.010
month_Jan -0.0675 0.020 -3.318 0.001 -0.107 -0.028
month_Jul -0.0699 0.016 -4.298 0.000 -0.102 -0.038
month_Nov -0.0753 0.018 -4.225 0.000 -0.110 -0.040
month_Sep 0.0536 0.016 3.438 0.001 0.023 0.084
weathersit_LightRain -0.1991 0.029 -6.926 0.000 -0.256 -0.143
weathersit_Mist -0.0457 0.010 -4.456 0.000 -0.066 -0.026
==============================================================================
Omnibus: 99.586 Durbin-Watson: 2.030
Prob(Omnibus): 0.000 Jarque-Bera (JB): 261.798
Skew: -0.858 Prob(JB): 1.42e-57
Kurtosis: 5.796 Cond. No. 16.9
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
""",
const 0.362280
temp 0.405727
humidity -0.144122
windspeed -0.123822
season_Spring -0.099266
season_Winter 0.069209
year_2019 0.224756
month_Dec -0.064581
month_Feb -0.049825
month_Jan -0.067528
month_Jul -0.069878
month_Nov -0.075318
month_Sep 0.053552
weathersit_LightRain -0.199087
weathersit_Mist -0.045743
dtype: float64)
Out[40]:
| Features | VIF | |
|---|---|---|
| 1 | humidity | 13.880000 |
| 0 | temp | 11.990000 |
| 2 | windspeed | 4.270000 |
| 3 | season_Spring | 4.080000 |
| 4 | season_Winter | 2.770000 |
| 13 | weathersit_Mist | 2.450000 |
| 8 | month_Jan | 2.220000 |
| 5 | year_2019 | 2.060000 |
| 7 | month_Feb | 2.010000 |
| 10 | month_Nov | 1.810000 |
| 6 | month_Dec | 1.630000 |
| 9 | month_Jul | 1.430000 |
| 12 | weathersit_LightRain | 1.330000 |
| 11 | month_Sep | 1.190000 |
Model 2¶
In [41]:
# 'humidity' is insignificant in presence of other variables. Hence dropped.
X_train_rfe = X_train_rfe.drop(["humidity"], axis = 1)
summary, param, sm_model, X_train_sm = train_sm(X_train_rfe, Y_train)
summary, param, # summary of the linear model
find_VIF(X_train_rfe)
Out[41]:
(<class 'statsmodels.iolib.summary.Summary'>
"""
OLS Regression Results
==============================================================================
Dep. Variable: total_cnt R-squared: 0.820
Model: OLS Adj. R-squared: 0.816
Method: Least Squares F-statistic: 199.5
Date: Mon, 23 Dec 2024 Prob (F-statistic): 3.02e-202
Time: 22:19:02 Log-Likelihood: 562.95
No. Observations: 584 AIC: -1098.
Df Residuals: 570 BIC: -1037.
Df Model: 13
Covariance Type: nonrobust
========================================================================================
coef std err t P>|t| [0.025 0.975]
----------------------------------------------------------------------------------------
const 0.3057 0.025 12.328 0.000 0.257 0.354
temp 0.3721 0.032 11.718 0.000 0.310 0.435
windspeed -0.0961 0.018 -5.271 0.000 -0.132 -0.060
season_Spring -0.1008 0.017 -5.769 0.000 -0.135 -0.066
season_Winter 0.0614 0.014 4.545 0.000 0.035 0.088
year_2019 0.2309 0.008 29.482 0.000 0.215 0.246
month_Dec -0.0695 0.017 -4.024 0.000 -0.103 -0.036
month_Feb -0.0483 0.021 -2.334 0.020 -0.089 -0.008
month_Jan -0.0706 0.021 -3.396 0.001 -0.111 -0.030
month_Jul -0.0592 0.016 -3.590 0.000 -0.092 -0.027
month_Nov -0.0755 0.018 -4.141 0.000 -0.111 -0.040
month_Sep 0.0442 0.016 2.794 0.005 0.013 0.075
weathersit_LightRain -0.2675 0.026 -10.260 0.000 -0.319 -0.216
weathersit_Mist -0.0775 0.008 -9.242 0.000 -0.094 -0.061
==============================================================================
Omnibus: 96.811 Durbin-Watson: 2.034
Prob(Omnibus): 0.000 Jarque-Bera (JB): 254.529
Skew: -0.835 Prob(JB): 5.37e-56
Kurtosis: 5.770 Cond. No. 15.8
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
""",
const 0.305669
temp 0.372128
windspeed -0.096087
season_Spring -0.100812
season_Winter 0.061423
year_2019 0.230880
month_Dec -0.069472
month_Feb -0.048319
month_Jan -0.070613
month_Jul -0.059188
month_Nov -0.075467
month_Sep 0.044193
weathersit_LightRain -0.267501
weathersit_Mist -0.077549
dtype: float64)
Out[41]:
| Features | VIF | |
|---|---|---|
| 0 | temp | 4.760000 |
| 1 | windspeed | 4.230000 |
| 2 | season_Spring | 3.880000 |
| 3 | season_Winter | 2.540000 |
| 7 | month_Jan | 2.170000 |
| 4 | year_2019 | 2.030000 |
| 6 | month_Feb | 2.000000 |
| 9 | month_Nov | 1.790000 |
| 5 | month_Dec | 1.570000 |
| 12 | weathersit_Mist | 1.520000 |
| 8 | month_Jul | 1.370000 |
| 10 | month_Sep | 1.170000 |
| 11 | weathersit_LightRain | 1.090000 |
Model 3¶
In [42]:
# 'windspeed' is insignificant in presence of other variables. Hence dropped.
X_train_rfe = X_train_rfe.drop(["windspeed"], axis = 1)
summary, param, sm_model, X_train_sm = train_sm(X_train_rfe, Y_train)
summary, param, # summary of the linear model
find_VIF(X_train_rfe)
Out[42]:
(<class 'statsmodels.iolib.summary.Summary'>
"""
OLS Regression Results
==============================================================================
Dep. Variable: total_cnt R-squared: 0.811
Model: OLS Adj. R-squared: 0.807
Method: Least Squares F-statistic: 204.2
Date: Mon, 23 Dec 2024 Prob (F-statistic): 1.52e-197
Time: 22:19:02 Log-Likelihood: 549.05
No. Observations: 584 AIC: -1072.
Df Residuals: 571 BIC: -1015.
Df Model: 12
Covariance Type: nonrobust
========================================================================================
coef std err t P>|t| [0.025 0.975]
----------------------------------------------------------------------------------------
const 0.2465 0.023 10.897 0.000 0.202 0.291
temp 0.3962 0.032 12.322 0.000 0.333 0.459
season_Spring -0.1029 0.018 -5.757 0.000 -0.138 -0.068
season_Winter 0.0722 0.014 5.286 0.000 0.045 0.099
year_2019 0.2308 0.008 28.807 0.000 0.215 0.247
month_Dec -0.0591 0.018 -3.370 0.001 -0.094 -0.025
month_Feb -0.0424 0.021 -2.002 0.046 -0.084 -0.001
month_Jan -0.0604 0.021 -2.849 0.005 -0.102 -0.019
month_Jul -0.0571 0.017 -3.388 0.001 -0.090 -0.024
month_Nov -0.0756 0.019 -4.055 0.000 -0.112 -0.039
month_Sep 0.0472 0.016 2.920 0.004 0.015 0.079
weathersit_LightRain -0.2863 0.026 -10.835 0.000 -0.338 -0.234
weathersit_Mist -0.0773 0.009 -9.009 0.000 -0.094 -0.060
==============================================================================
Omnibus: 96.575 Durbin-Watson: 2.075
Prob(Omnibus): 0.000 Jarque-Bera (JB): 272.387
Skew: -0.809 Prob(JB): 7.11e-60
Kurtosis: 5.928 Cond. No. 14.6
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
""",
const 0.246488
temp 0.396206
season_Spring -0.102911
season_Winter 0.072246
year_2019 0.230825
month_Dec -0.059143
month_Feb -0.042356
month_Jan -0.060353
month_Jul -0.057134
month_Nov -0.075606
month_Sep 0.047227
weathersit_LightRain -0.286307
weathersit_Mist -0.077342
dtype: float64)
Out[42]:
| Features | VIF | |
|---|---|---|
| 1 | season_Spring | 3.570000 |
| 0 | temp | 2.800000 |
| 2 | season_Winter | 2.540000 |
| 6 | month_Jan | 2.160000 |
| 3 | year_2019 | 2.020000 |
| 5 | month_Feb | 2.000000 |
| 8 | month_Nov | 1.750000 |
| 4 | month_Dec | 1.570000 |
| 11 | weathersit_Mist | 1.500000 |
| 7 | month_Jul | 1.340000 |
| 9 | month_Sep | 1.170000 |
| 10 | weathersit_LightRain | 1.070000 |
Model 4 - Alternate Model¶
- This model very much satisfies the criteria, hence can be used as final model
- The VIFs and p-values both are within an acceptable range.
- We will see if we can still reduce the features to avoid overfitting
In [43]:
# 'month_Feb' is insignificant in presence of other variables. Hence dropped.
X_train_rfe = X_train_rfe.drop(["month_Feb"], axis = 1)
summary, param, sm_model, X_train_sm = train_sm(X_train_rfe, Y_train)
summary, param, # summary of the linear model
find_VIF(X_train_rfe)
sm_model_tmp, X_train_sm_tmp = sm_model, X_train_sm
Out[43]:
(<class 'statsmodels.iolib.summary.Summary'>
"""
OLS Regression Results
==============================================================================
Dep. Variable: total_cnt R-squared: 0.810
Model: OLS Adj. R-squared: 0.806
Method: Least Squares F-statistic: 221.3
Date: Mon, 23 Dec 2024 Prob (F-statistic): 7.29e-198
Time: 22:19:02 Log-Likelihood: 547.01
No. Observations: 584 AIC: -1070.
Df Residuals: 572 BIC: -1018.
Df Model: 11
Covariance Type: nonrobust
========================================================================================
coef std err t P>|t| [0.025 0.975]
----------------------------------------------------------------------------------------
const 0.2390 0.022 10.685 0.000 0.195 0.283
temp 0.4075 0.032 12.836 0.000 0.345 0.470
season_Spring -0.1205 0.016 -7.712 0.000 -0.151 -0.090
season_Winter 0.0705 0.014 5.157 0.000 0.044 0.097
year_2019 0.2305 0.008 28.693 0.000 0.215 0.246
month_Dec -0.0487 0.017 -2.897 0.004 -0.082 -0.016
month_Jan -0.0376 0.018 -2.099 0.036 -0.073 -0.002
month_Jul -0.0593 0.017 -3.517 0.000 -0.092 -0.026
month_Nov -0.0706 0.019 -3.810 0.000 -0.107 -0.034
month_Sep 0.0472 0.016 2.909 0.004 0.015 0.079
weathersit_LightRain -0.2852 0.026 -10.766 0.000 -0.337 -0.233
weathersit_Mist -0.0770 0.009 -8.945 0.000 -0.094 -0.060
==============================================================================
Omnibus: 90.429 Durbin-Watson: 2.061
Prob(Omnibus): 0.000 Jarque-Bera (JB): 256.637
Skew: -0.757 Prob(JB): 1.87e-56
Kurtosis: 5.873 Cond. No. 14.2
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
""",
const 0.238972
temp 0.407463
season_Spring -0.120462
season_Winter 0.070526
year_2019 0.230463
month_Dec -0.048662
month_Jan -0.037632
month_Jul -0.059342
month_Nov -0.070581
month_Sep 0.047165
weathersit_LightRain -0.285170
weathersit_Mist -0.076977
dtype: float64)
Out[43]:
| Features | VIF | |
|---|---|---|
| 0 | temp | 2.790000 |
| 2 | season_Winter | 2.490000 |
| 3 | year_2019 | 2.020000 |
| 1 | season_Spring | 1.910000 |
| 7 | month_Nov | 1.740000 |
| 5 | month_Jan | 1.580000 |
| 10 | weathersit_Mist | 1.500000 |
| 4 | month_Dec | 1.460000 |
| 6 | month_Jul | 1.340000 |
| 8 | month_Sep | 1.170000 |
| 9 | weathersit_LightRain | 1.070000 |
Model 5¶
In [44]:
# 'month_Jan' is insignificant in presence of other variables. Hence dropped.
X_train_rfe = X_train_rfe.drop(["month_Jan"], axis = 1)
summary, param, sm_model, X_train_sm = train_sm(X_train_rfe, Y_train)
summary, param, # summary of the linear model
find_VIF(X_train_rfe)
Out[44]:
(<class 'statsmodels.iolib.summary.Summary'>
"""
OLS Regression Results
==============================================================================
Dep. Variable: total_cnt R-squared: 0.808
Model: OLS Adj. R-squared: 0.805
Method: Least Squares F-statistic: 241.5
Date: Mon, 23 Dec 2024 Prob (F-statistic): 4.06e-198
Time: 22:19:02 Log-Likelihood: 544.77
No. Observations: 584 AIC: -1068.
Df Residuals: 573 BIC: -1019.
Df Model: 10
Covariance Type: nonrobust
========================================================================================
coef std err t P>|t| [0.025 0.975]
----------------------------------------------------------------------------------------
const 0.2277 0.022 10.457 0.000 0.185 0.270
temp 0.4245 0.031 13.792 0.000 0.364 0.485
season_Spring -0.1279 0.015 -8.387 0.000 -0.158 -0.098
season_Winter 0.0710 0.014 5.174 0.000 0.044 0.098
year_2019 0.2302 0.008 28.583 0.000 0.214 0.246
month_Dec -0.0408 0.016 -2.485 0.013 -0.073 -0.009
month_Jul -0.0628 0.017 -3.732 0.000 -0.096 -0.030
month_Nov -0.0661 0.018 -3.580 0.000 -0.102 -0.030
month_Sep 0.0465 0.016 2.858 0.004 0.015 0.078
weathersit_LightRain -0.2860 0.027 -10.768 0.000 -0.338 -0.234
weathersit_Mist -0.0767 0.009 -8.889 0.000 -0.094 -0.060
==============================================================================
Omnibus: 86.217 Durbin-Watson: 2.072
Prob(Omnibus): 0.000 Jarque-Bera (JB): 239.004
Skew: -0.730 Prob(JB): 1.26e-52
Kurtosis: 5.773 Cond. No. 13.8
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
""",
const 0.227663
temp 0.424493
season_Spring -0.127939
season_Winter 0.070961
year_2019 0.230237
month_Dec -0.040814
month_Jul -0.062838
month_Nov -0.066071
month_Sep 0.046478
weathersit_LightRain -0.286031
weathersit_Mist -0.076718
dtype: float64)
Out[44]:
| Features | VIF | |
|---|---|---|
| 0 | temp | 2.770000 |
| 2 | season_Winter | 2.470000 |
| 3 | year_2019 | 2.020000 |
| 6 | month_Nov | 1.730000 |
| 9 | weathersit_Mist | 1.500000 |
| 4 | month_Dec | 1.420000 |
| 5 | month_Jul | 1.340000 |
| 1 | season_Spring | 1.290000 |
| 7 | month_Sep | 1.170000 |
| 8 | weathersit_LightRain | 1.060000 |
Model 6¶
In [45]:
# 'month_Nov' is insignificant in presence of other variables. Hence dropped.
X_train_rfe = X_train_rfe.drop(["month_Nov"], axis = 1)
summary, param, sm_model, X_train_sm = train_sm(X_train_rfe, Y_train)
summary, param, # summary of the linear model
find_VIF(X_train_rfe)
Out[45]:
(<class 'statsmodels.iolib.summary.Summary'>
"""
OLS Regression Results
==============================================================================
Dep. Variable: total_cnt R-squared: 0.804
Model: OLS Adj. R-squared: 0.801
Method: Least Squares F-statistic: 261.5
Date: Mon, 23 Dec 2024 Prob (F-statistic): 1.35e-196
Time: 22:19:02 Log-Likelihood: 538.31
No. Observations: 584 AIC: -1057.
Df Residuals: 574 BIC: -1013.
Df Model: 9
Covariance Type: nonrobust
========================================================================================
coef std err t P>|t| [0.025 0.975]
----------------------------------------------------------------------------------------
const 0.2086 0.021 9.782 0.000 0.167 0.251
temp 0.4508 0.030 14.932 0.000 0.392 0.510
season_Spring -0.1199 0.015 -7.865 0.000 -0.150 -0.090
season_Winter 0.0493 0.012 3.964 0.000 0.025 0.074
year_2019 0.2294 0.008 28.200 0.000 0.213 0.245
month_Dec -0.0194 0.015 -1.254 0.210 -0.050 0.011
month_Jul -0.0671 0.017 -3.955 0.000 -0.100 -0.034
month_Sep 0.0500 0.016 3.051 0.002 0.018 0.082
weathersit_LightRain -0.2827 0.027 -10.540 0.000 -0.335 -0.230
weathersit_Mist -0.0730 0.009 -8.437 0.000 -0.090 -0.056
==============================================================================
Omnibus: 80.491 Durbin-Watson: 2.075
Prob(Omnibus): 0.000 Jarque-Bera (JB): 203.056
Skew: -0.713 Prob(JB): 8.07e-45
Kurtosis: 5.513 Cond. No. 13.4
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
""",
const 0.208629
temp 0.450834
season_Spring -0.119877
season_Winter 0.049264
year_2019 0.229375
month_Dec -0.019381
month_Jul -0.067103
month_Sep 0.050030
weathersit_LightRain -0.282668
weathersit_Mist -0.073034
dtype: float64)
Out[45]:
| Features | VIF | |
|---|---|---|
| 0 | temp | 2.760000 |
| 3 | year_2019 | 2.020000 |
| 2 | season_Winter | 1.540000 |
| 8 | weathersit_Mist | 1.490000 |
| 5 | month_Jul | 1.340000 |
| 1 | season_Spring | 1.290000 |
| 4 | month_Dec | 1.270000 |
| 6 | month_Sep | 1.170000 |
| 7 | weathersit_LightRain | 1.060000 |
Model 7 Final Model¶
In [46]:
# 'month_Dec' is insignificant in presence of other variables. Hence dropped.
X_train_rfe = X_train_rfe.drop(["month_Dec"], axis = 1)
summary, param, sm_model, X_train_sm = train_sm(X_train_rfe, Y_train)
summary, param, # summary of the linear model
find_VIF(X_train_rfe)
Out[46]:
(<class 'statsmodels.iolib.summary.Summary'>
"""
OLS Regression Results
==============================================================================
Dep. Variable: total_cnt R-squared: 0.803
Model: OLS Adj. R-squared: 0.801
Method: Least Squares F-statistic: 293.7
Date: Mon, 23 Dec 2024 Prob (F-statistic): 1.67e-197
Time: 22:19:02 Log-Likelihood: 537.51
No. Observations: 584 AIC: -1057.
Df Residuals: 575 BIC: -1018.
Df Model: 8
Covariance Type: nonrobust
========================================================================================
coef std err t P>|t| [0.025 0.975]
----------------------------------------------------------------------------------------
const 0.2049 0.021 9.697 0.000 0.163 0.246
temp 0.4567 0.030 15.304 0.000 0.398 0.515
season_Spring -0.1198 0.015 -7.855 0.000 -0.150 -0.090
season_Winter 0.0461 0.012 3.785 0.000 0.022 0.070
year_2019 0.2292 0.008 28.170 0.000 0.213 0.245
month_Jul -0.0683 0.017 -4.030 0.000 -0.102 -0.035
month_Sep 0.0506 0.016 3.089 0.002 0.018 0.083
weathersit_LightRain -0.2841 0.027 -10.598 0.000 -0.337 -0.231
weathersit_Mist -0.0735 0.009 -8.500 0.000 -0.091 -0.057
==============================================================================
Omnibus: 77.442 Durbin-Watson: 2.075
Prob(Omnibus): 0.000 Jarque-Bera (JB): 189.872
Skew: -0.696 Prob(JB): 5.89e-42
Kurtosis: 5.422 Cond. No. 13.2
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
""",
const 0.204875
temp 0.456706
season_Spring -0.119780
season_Winter 0.046074
year_2019 0.229219
month_Jul -0.068302
month_Sep 0.050646
weathersit_LightRain -0.284113
weathersit_Mist -0.073536
dtype: float64)
Out[46]:
| Features | VIF | |
|---|---|---|
| 0 | temp | 2.750000 |
| 3 | year_2019 | 2.020000 |
| 7 | weathersit_Mist | 1.480000 |
| 2 | season_Winter | 1.340000 |
| 4 | month_Jul | 1.340000 |
| 1 | season_Spring | 1.250000 |
| 5 | month_Sep | 1.160000 |
| 6 | weathersit_LightRain | 1.060000 |
the VIFs and p-values both are within an acceptable range. So we go ahead and make our predictions using this model only.
9. MLR - Residual Analysis of the training data¶
In [47]:
y_train_pred = sm_model.predict(X_train_sm)
In [48]:
# Plot the histogram of the error terms
fig = plt.figure()
sns.histplot((Y_train - y_train_pred), bins = 20,kde=True)
fig.suptitle('Error Terms', fontsize = 20) # Plot heading
plt.xlabel('Errors', fontsize = 18)
plt.show();
the error terms are also normally distributed
10. MLR - Predictions on the Test Data Using the Final Model¶
In [49]:
# Dividing into X_test and Y_test
Y_test = df_test.pop('total_cnt')
X_test = df_test
In [50]:
# using our trained model to make predictions.
# Creating X_test_new dataframe by dropping variables from X_test
final_features = X_train_rfe.columns
X_test_new = X_test[final_features]
X_test_new = sm.add_constant(X_test_new) # Adding a constant variable
Y_test_pred = sm_model.predict(X_test_new) # Making predictions
Y_comp_df = pd.DataFrame({"Actual" : Y_test, "Predicted" : Y_test_pred})
Y_comp_df.head()
Out[50]:
| Actual | Predicted | |
|---|---|---|
| 196 | 0.678900 | 0.493691 |
| 187 | 0.525771 | 0.529732 |
| 14 | 0.141049 | 0.110694 |
| 31 | 0.153935 | 0.087272 |
| 390 | 0.466291 | 0.401564 |
In [51]:
# To find top 3 features of the model
coefficients = sm_model.params
# Get the feature names
feature_names = final_features
# Create a dataframe to store the top_features
top_features = pd.DataFrame({'feature': feature_names, 'coefficient': coefficients[1:]})
# Sort the dataframe by coefficient absolute value
top_features = top_features.sort_values(by='coefficient', key=abs, ascending=False)
print(top_features.reset_index(drop=True).head(5))
feature coefficient 0 temp 0.456706 1 weathersit_LightRain -0.284113 2 year_2019 0.229219 3 season_Spring -0.119780 4 weathersit_Mist -0.073536
In [52]:
print('---------------------------------------------------')
print(f'Mean Absolute Error: {mean_absolute_error(Y_test, Y_test_pred)}')
print(f'Mean Absolute Percentage Error: {mean_absolute_percentage_error(Y_test, Y_test_pred)}')
print(f'Mean Squared Error: {mean_squared_error(Y_test, Y_test_pred)}')
print(f'Root Mean Squared Error: {np.sqrt(mean_squared_error(Y_test, Y_test_pred))}')
print(f'R2_score: {r2_score(Y_test, Y_test_pred)}')
print(f'Adjusted_R2_score: {adj_r2_score(X_test, Y_test, Y_test_pred)}')
print(f'Adjusted_R2_score using sm_model_rsq_adj: {sm_model.rsquared_adj}')
--------------------------------------------------- Mean Absolute Error: 0.07332379465688707 Mean Absolute Percentage Error: 0.21722130847518956 Mean Squared Error: 0.008736601751308767 Root Mean Squared Error: 0.09346979058128228 R2_score: 0.8509104241809033 Adjusted_R2_score: 0.8152308675746238 Adjusted_R2_score using sm_model_rsq_adj: 0.8006724734729809
The R2 and Adj R2 scores are 85% and 81% respectively, hence we can conclude that the model seems to performs well on Untrained dataset
In [53]:
y_test = Y_test
y_pred = Y_test_pred
In [54]:
from sklearn.metrics import r2_score
r2_score(y_test, y_pred)
Out[54]:
0.8509104241809033
11. MLR - Model Evaluation¶
In [55]:
# Plotting y_test and y_pred to understand the spread.
fig = plt.figure()
sns.scatterplot(x=Y_test,y=Y_test_pred)
fig.suptitle('y_test vs y_pred', fontsize=20)
plt.xlabel('y_test', fontsize=18)
plt.ylabel('y_pred', fontsize=16)
plt.show();
We can see that the equation of our best fitted line is:
$ total cnt = 0.204875C - 0.456706 \times temp - 0.119780 \times season Spring + 0.046074 \times season Winter + 0.229219 \times year 2019 - 0.068302 \times month Jul + 0.050646 \times month Sep - 0.284113 \times weathersit LightRain - 0.073536 \times weathersitMist $
The Bike Sharing demand prediction: Inference:
People usually prefer bikes during Cloudy, non-rainy/non-snowy days
There is a positive growth in 2019, therefore even though pandemic has made things worse there is higher chance of getting positive growth once things get back to normalcy.
people prefer to ride bikes when the temperature is moderate.
Most people prefer to use bikes in the fall.
Therefore it is advised to Bike sharing company to prepare/plan/forecast ahead of these situations.
to cater the needs/demand of the users, which ultimately result in more users on the platform and a positive growth.
- Thus enabling much needed return on investment.
In [ ]:
Note:
When you perform dummy encoding (also known as one-hot encoding) in machine learning, the first category is often implicit, meaning it's not explicitly represented as a column. This is done to avoid multicollinearity, which can lead to unstable estimates of the regression coefficients.
Notice that the
cloudycategory is not explicitly represented as a column. This is because the first category (cloudy) is implicit, and the coefficients for the other categories (LightRainandMist) are estimated relative to this implicit category.Now, let's say we want to infer the final model. We can write the model equation as:
y = β0 + β1 * LightRain + β2 * Mist + ε
where y is the response variable, β0 is the intercept, β1 and β2 are the coefficients for the
LightRainandMistcategories, respectively, and ε is the error term.To infer the effect of the
cloudycategory, we need to consider the implicit category. We can do this by setting theLightRainandMistvariables to 0, which effectively selects thecloudycategory.The predicted value for the
cloudycategory is then:y = β0 + 0 * β1 + 0 * β2 = β0
So, the intercept (β0) represents the expected value of the response variable when the
weathersitvariable iscloudy.To illustrate this with an example, let's say we have the following estimated coefficients:
- β0 = 10 , β1 = -2, β2 = -3
The predicted values for each category would be:
cloudy: y = 10 (since β0 represents the expected value forCloudy)Lightrain: y = 10 - 2 = 8Mist: y = 10 - 3 = 7
In this example, the implicit cloudy category has an expected value of 10, while the Lightrain and Mist categories have expected values of 8 and 7, respectively, relative to the cloudy category.