Boom Bike Sharing¶
Problem Statement¶
A 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 v ry difficult to sustain in the current market scenario. So, it has decided to co me 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 ation due to Covid-19. They have planned this to prepare themselves to cater o the people's needs once the situation gets better all around and stand out from other service providers and make huge profits.
They have contracted a consulting company to understand the factors on w hich the demand for these shared bikes depends. Specifically, they want to unders tand the factors affecting the demand for these shared bikes in the American market. The company wants
- Which variables are significant in predicting the demand for shared bikes.
- How well those variables describe the bike demands
Based on various meteorological surveys and people's styles, the service provider firm has gathered a large dataset on daily bike demands across the American market based on some factors.
Business Goal¶
the company want 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.
Table of content¶
Reading and Understanding Data¶
Lets import numpy and pandas and read the bike sharing data set
In [1]:
# supress Warning
import warnings
warnings.filterwarnings("ignore")
In [3]:
# Importing necessarylibraries
import pandas as pd
import numpy as np
In [5]:
Bikesharing = pd.read_csv("day.csv")
In [7]:
Bikesharing.head()
Out[7]:
| 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 |
In [9]:
Bikesharing.shape
Out[9]:
(730, 16)
In [11]:
Bikesharing.info()
<class 'pandas.core.frame.DataFrame'> RangeIndex: 730 entries, 0 to 729 Data columns (total 16 columns): # Column Non-Null Count Dtype --- ------ -------------- ----- 0 instant 730 non-null int64 1 dteday 730 non-null object 2 season 730 non-null int64 3 yr 730 non-null int64 4 mnth 730 non-null int64 5 holiday 730 non-null int64 6 weekday 730 non-null int64 7 workingday 730 non-null int64 8 weathersit 730 non-null int64 9 temp 730 non-null float64 10 atemp 730 non-null float64 11 hum 730 non-null float64 12 windspeed 730 non-null float64 13 casual 730 non-null int64 14 registered 730 non-null int64 15 cnt 730 non-null int64 dtypes: float64(4), int64(11), object(1) memory usage: 91.4+ KB
- No missing Values found in data
In [14]:
Bikesharing.describe()
Out[14]:
| instant | season | yr | mnth | holiday | weekday | workingday | weathersit | temp | atemp | hum | windspeed | casual | registered | cnt | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| count | 730.000000 | 730.000000 | 730.000000 | 730.000000 | 730.000000 | 730.000000 | 730.000000 | 730.000000 | 730.000000 | 730.000000 | 730.000000 | 730.000000 | 730.000000 | 730.000000 | 730.000000 |
| mean | 365.500000 | 2.498630 | 0.500000 | 6.526027 | 0.028767 | 2.995890 | 0.690411 | 1.394521 | 20.319259 | 23.726322 | 62.765175 | 12.763620 | 849.249315 | 3658.757534 | 4508.006849 |
| std | 210.877136 | 1.110184 | 0.500343 | 3.450215 | 0.167266 | 2.000339 | 0.462641 | 0.544807 | 7.506729 | 8.150308 | 14.237589 | 5.195841 | 686.479875 | 1559.758728 | 1936.011647 |
| min | 1.000000 | 1.000000 | 0.000000 | 1.000000 | 0.000000 | 0.000000 | 0.000000 | 1.000000 | 2.424346 | 3.953480 | 0.000000 | 1.500244 | 2.000000 | 20.000000 | 22.000000 |
| 25% | 183.250000 | 2.000000 | 0.000000 | 4.000000 | 0.000000 | 1.000000 | 0.000000 | 1.000000 | 13.811885 | 16.889713 | 52.000000 | 9.041650 | 316.250000 | 2502.250000 | 3169.750000 |
| 50% | 365.500000 | 3.000000 | 0.500000 | 7.000000 | 0.000000 | 3.000000 | 1.000000 | 1.000000 | 20.465826 | 24.368225 | 62.625000 | 12.125325 | 717.000000 | 3664.500000 | 4548.500000 |
| 75% | 547.750000 | 3.000000 | 1.000000 | 10.000000 | 0.000000 | 5.000000 | 1.000000 | 2.000000 | 26.880615 | 30.445775 | 72.989575 | 15.625589 | 1096.500000 | 4783.250000 | 5966.000000 |
| max | 730.000000 | 4.000000 | 1.000000 | 12.000000 | 1.000000 | 6.000000 | 1.000000 | 3.000000 | 35.328347 | 42.044800 | 97.250000 | 34.000021 | 3410.000000 | 6946.000000 | 8714.000000 |
Data Quality Check¶
Changing some categorical variable for further analysis of data continous and categorical separately¶
- Deleting some columns casual, registered as both comes under cnt column
- No need of instant column
- no need of dteday as month and year is already there for analysis
- changing mnth as month
- changing temp : temperature
- changing hum: humidity
In [18]:
# copying the Bikesharing data set
data = Bikesharing.copy()
In [20]:
data.rename(columns={"yr" : "year", "mnth":"month", "temp":"Temperature" ,"hum":"humidity"}, inplace=True)
In [22]:
data.head()
Out[22]:
| instant | dteday | season | year | month | holiday | weekday | workingday | weathersit | Temperature | atemp | humidity | 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 |
In [24]:
drop_columns = ['instant','dteday','casual','registered']
data = data.drop(drop_columns, axis=1)
In [26]:
data.head()
Out[26]:
| season | year | month | holiday | weekday | workingday | weathersit | Temperature | atemp | humidity | windspeed | cnt | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 0 | 1 | 0 | 1 | 1 | 2 | 14.110847 | 18.18125 | 80.5833 | 10.749882 | 985 |
| 1 | 1 | 0 | 1 | 0 | 2 | 1 | 2 | 14.902598 | 17.68695 | 69.6087 | 16.652113 | 801 |
| 2 | 1 | 0 | 1 | 0 | 3 | 1 | 1 | 8.050924 | 9.47025 | 43.7273 | 16.636703 | 1349 |
| 3 | 1 | 0 | 1 | 0 | 4 | 1 | 1 | 8.200000 | 10.60610 | 59.0435 | 10.739832 | 1562 |
| 4 | 1 | 0 | 1 | 0 | 5 | 1 | 1 | 9.305237 | 11.46350 | 43.6957 | 12.522300 | 1600 |
Checking correlation between numerical variables¶
In [29]:
data[['Temperature','atemp','humidity','windspeed']].corr()
Out[29]:
| Temperature | atemp | humidity | windspeed | |
|---|---|---|---|---|
| Temperature | 1.000000 | 0.991696 | 0.128565 | -0.158186 |
| atemp | 0.991696 | 1.000000 | 0.141512 | -0.183876 |
| humidity | 0.128565 | 0.141512 | 1.000000 | -0.248506 |
| windspeed | -0.158186 | -0.183876 | -0.248506 | 1.000000 |
- A numerical value ranging from -1 to 1, which indicates the strength and direction of the relationship between them , highly correlated to each other.
- As seen
Temperatureandatempis almost 1 - Dropping feature
atemp
In [32]:
data.drop('atemp', axis =1 ,inplace= True)
In [34]:
data['season'].unique()
Out[34]:
array([1, 2, 3, 4], dtype=int64)
In [36]:
data['season'] = data['season'].replace({1 : 'spring',
2: 'summer',
3 : 'fall',
4 : 'winter'})
In [38]:
data['month'] = data['month'].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' })
In [40]:
data['weekday'].unique()
Out[40]:
array([1, 2, 3, 4, 5, 6, 0], dtype=int64)
In [42]:
data['weekday'] = data['weekday'].replace({0 :'Sunday',
1: 'Monday',
2: 'Tuesday',
3: 'Wednesday',
4: 'Thursday',
5: 'Friday',
6: 'Saturday'})
In [44]:
data['weathersit'].unique()
Out[44]:
array([2, 1, 3], dtype=int64)
In [46]:
data['weathersit'] = data['weathersit'].replace({1 : 'clear' ,2: 'cloudy' , 3: 'light_rain' , 4: 'heavy_rain'
})
Visualising the Data¶
Let's now spend some time doing what is arguably the most important step - understanding the data.
- If there is some obvious multicollinearity going on, this is the first place to catch it
- Here's where you'll also identify if some predictors directly have a strong association with the outcome variable that is cnt
We'll visualise our data using matplotlib and seaborn.
In [49]:
data.head()
Out[49]:
| season | year | month | holiday | weekday | workingday | weathersit | Temperature | humidity | windspeed | cnt | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | spring | 0 | january | 0 | Monday | 1 | cloudy | 14.110847 | 80.5833 | 10.749882 | 985 |
| 1 | spring | 0 | january | 0 | Tuesday | 1 | cloudy | 14.902598 | 69.6087 | 16.652113 | 801 |
| 2 | spring | 0 | january | 0 | Wednesday | 1 | clear | 8.050924 | 43.7273 | 16.636703 | 1349 |
| 3 | spring | 0 | january | 0 | Thursday | 1 | clear | 8.200000 | 59.0435 | 10.739832 | 1562 |
| 4 | spring | 0 | january | 0 | Friday | 1 | clear | 9.305237 | 43.6957 | 12.522300 | 1600 |
In [51]:
import matplotlib.pyplot as plt
import seaborn as sns
Visulization on numerical columns¶
In [53]:
#visulaizing the numerical variable
sns.pairplot(data=data,x_vars=['Temperature','humidity','windspeed'],y_vars='cnt',kind='scatter',height=5,aspect=1);
Seeing the scatter plot Found some outliers so will remove outliers by taking reference of scatter plot as I can also see outliers are less in number we can also skip the step for removing an outliers but I have removed the outlier just to get the clean data
Handlig Outliers for numerical columns¶
In [56]:
data = data.drop(index = data[(data['Temperature'] > 15) & (data['Temperature'] < 20) & (data['cnt'] < 100)].index)
data = data.drop(index = data[(data['Temperature'] > 25) & (data['Temperature'] < 30) & (data['cnt'] < 2000)].index)
data = data.drop(index = data[(data['humidity'] < 20)].index)
data = data.drop(index = data[(data['windspeed'] > 30)].index)
As we can see the humidity and windspeed having outliers no much in values but we can still remove them
In [58]:
data.shape
Out[58]:
(725, 11)
Visulization on categorical columns¶
For categorical column i have used boxplot for the outlier analysis
In [65]:
cat_col = data[['holiday','workingday','season','weekday','weathersit','month']]
In [67]:
def plotting(data, column):
fig = plt.figure(figsize=(8,6))
ax1 = plt.subplot(2,2,1)
sns.boxplot(data = data, palette='Set1',x = column, y = 'cnt', ax = ax1)
plt.title('Plotting data for the column: '+ column)
plt.xticks(rotation=90)
plt.tight_layout() # Or equivalently, "plt.tight_layout()"
plt.show()
In [69]:
for i in cat_col:
plotting(data, i)
Working Day and Holiday :
- Rental counts are generally lower on holidays compared to working days.
- Holidays show greater fluctuations in rental demand.
Season
- Fall season has the highest demand.
Weekday
- There is no trend found in weekdays
Weathersit
- When Weathersit is clear bike demand is high
Month
- May, June , July and August are highly month in bike demand
Handling Outliers for categorical variable¶
In [73]:
# Dropping outliers in Categorical Variables
data = data.drop(index = data[(data['season'] == 'spring') & (data['cnt'] > 7000)].index)
In [75]:
data.shape
Out[75]:
(724, 11)
In [77]:
data.head()
Out[77]:
| season | year | month | holiday | weekday | workingday | weathersit | Temperature | humidity | windspeed | cnt | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | spring | 0 | january | 0 | Monday | 1 | cloudy | 14.110847 | 80.5833 | 10.749882 | 985 |
| 1 | spring | 0 | january | 0 | Tuesday | 1 | cloudy | 14.902598 | 69.6087 | 16.652113 | 801 |
| 2 | spring | 0 | january | 0 | Wednesday | 1 | clear | 8.050924 | 43.7273 | 16.636703 | 1349 |
| 3 | spring | 0 | january | 0 | Thursday | 1 | clear | 8.200000 | 59.0435 | 10.739832 | 1562 |
| 4 | spring | 0 | january | 0 | Friday | 1 | clear | 9.305237 | 43.6957 | 12.522300 | 1600 |
In [79]:
sns.pairplot(data,vars=["Temperature","humidity","windspeed","cnt"])
Out[79]:
<seaborn.axisgrid.PairGrid at 0x200b3821ac0>
- As I can see
Temperatureas some linear relationahipp with target variablecnt - Lets further analyse using heatmap for numerical variables
In [82]:
plt.figure(figsize = (6, 5))
sns.heatmap(data.corr(numeric_only = True), annot = True, cmap = 'OrRd')
plt.show()
- Here we can see
Temperatureis higly corelated withcntfollowed byYear
Data Preparation¶
- Encoding: - Categorical Variables to dummy variables
Creating Dummy Variables¶
In [87]:
## creating dummy variable
season_dummies = pd.get_dummies(data['season'] ,dtype= int ,drop_first=True)
month_dummies = pd.get_dummies(data['month'],dtype= int,drop_first=True)
weekday_dummies = pd.get_dummies(data['weekday'],dtype= int,drop_first= True)
weathersit_dummies = pd.get_dummies(data['weathersit'],dtype= int,drop_first= True)
In [89]:
# Concat dummy dataframe with original one
data = pd.concat([data, season_dummies,month_dummies,weekday_dummies,weathersit_dummies], axis =1)
data.head()
Out[89]:
| season | year | month | holiday | weekday | workingday | weathersit | Temperature | humidity | windspeed | ... | october | september | Monday | Saturday | Sunday | Thursday | Tuesday | Wednesday | cloudy | light_rain | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | spring | 0 | january | 0 | Monday | 1 | cloudy | 14.110847 | 80.5833 | 10.749882 | ... | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | spring | 0 | january | 0 | Tuesday | 1 | cloudy | 14.902598 | 69.6087 | 16.652113 | ... | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 |
| 2 | spring | 0 | january | 0 | Wednesday | 1 | clear | 8.050924 | 43.7273 | 16.636703 | ... | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 3 | spring | 0 | january | 0 | Thursday | 1 | clear | 8.200000 | 59.0435 | 10.739832 | ... | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 4 | spring | 0 | january | 0 | Friday | 1 | clear | 9.305237 | 43.6957 | 12.522300 | ... | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
5 rows × 33 columns
In [91]:
# Deleting original variables
data = data.drop(columns=['season','month','weekday','weathersit'] )
In [93]:
data.columns
Out[93]:
Index(['year', 'holiday', 'workingday', 'Temperature', 'humidity', 'windspeed',
'cnt', 'spring', 'summer', 'winter', 'august', 'december', 'february',
'january', 'july', 'june', 'march', 'may', 'november', 'october',
'september', 'Monday', 'Saturday', 'Sunday', 'Thursday', 'Tuesday',
'Wednesday', 'cloudy', 'light_rain'],
dtype='object')
In [95]:
data.shape
Out[95]:
(724, 29)
Splitting data into train and test¶
In [98]:
from sklearn.model_selection import train_test_split
In [99]:
data_train,data_test = train_test_split(data,train_size=0.7,random_state=100)
In [102]:
print(data_train.shape)
print(data_test.shape)
(506, 29) (218, 29)
In [104]:
pd.set_option('display.max_columns',None)
In [106]:
data.head()
Out[106]:
| year | holiday | workingday | Temperature | humidity | windspeed | cnt | spring | summer | winter | august | december | february | january | july | june | march | may | november | october | september | Monday | Saturday | Sunday | Thursday | Tuesday | Wednesday | cloudy | light_rain | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 0 | 0 | 1 | 14.110847 | 80.5833 | 10.749882 | 985 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 1 | 14.902598 | 69.6087 | 16.652113 | 801 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 |
| 2 | 0 | 0 | 1 | 8.050924 | 43.7273 | 16.636703 | 1349 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 3 | 0 | 0 | 1 | 8.200000 | 59.0435 | 10.739832 | 1562 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 1 | 9.305237 | 43.6957 | 12.522300 | 1600 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
In [108]:
data.shape
Out[108]:
(724, 29)
Rescaling variables using MinMax Scaler¶
In [111]:
from sklearn.preprocessing import MinMaxScaler
In [113]:
scaler = MinMaxScaler()
num_vars = ['Temperature','humidity','windspeed','cnt']
data_train[num_vars] = scaler.fit_transform(data_train[num_vars] )
data_train.head()
Out[113]:
| year | holiday | workingday | Temperature | humidity | windspeed | cnt | spring | summer | winter | august | december | february | january | july | june | march | may | november | october | september | Monday | Saturday | Sunday | Thursday | Tuesday | Wednesday | cloudy | light_rain | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 422 | 1 | 0 | 1 | 0.383206 | 0.281444 | 0.615256 | 0.469757 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 728 | 1 | 0 | 1 | 0.245101 | 0.270255 | 0.822447 | 0.164795 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 614 | 1 | 0 | 0 | 0.802708 | 0.647560 | 0.373836 | 0.853918 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 113 | 0 | 0 | 1 | 0.651106 | 0.758823 | 0.425255 | 0.453942 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 |
| 579 | 1 | 0 | 0 | 0.880586 | 0.507702 | 0.484409 | 0.814198 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 0 |
In [115]:
data_train.describe()
Out[115]:
| year | holiday | workingday | Temperature | humidity | windspeed | cnt | spring | summer | winter | august | december | february | january | july | june | march | may | november | october | september | Monday | Saturday | Sunday | Thursday | Tuesday | Wednesday | cloudy | light_rain | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| count | 506.000000 | 506.000000 | 506.000000 | 506.000000 | 506.000000 | 506.000000 | 506.000000 | 506.000000 | 506.000000 | 506.000000 | 506.000000 | 506.000000 | 506.000000 | 506.000000 | 506.000000 | 506.000000 | 506.000000 | 506.000000 | 506.000000 | 506.000000 | 506.000000 | 506.000000 | 506.000000 | 506.000000 | 506.000000 | 506.000000 | 506.000000 | 506.000000 | 506.000000 |
| mean | 0.507905 | 0.029644 | 0.683794 | 0.536409 | 0.490968 | 0.423239 | 0.492988 | 0.243083 | 0.245059 | 0.256917 | 0.084980 | 0.092885 | 0.069170 | 0.088933 | 0.084980 | 0.069170 | 0.086957 | 0.088933 | 0.088933 | 0.084980 | 0.081028 | 0.136364 | 0.142292 | 0.148221 | 0.138340 | 0.158103 | 0.146245 | 0.369565 | 0.019763 |
| std | 0.500432 | 0.169772 | 0.465454 | 0.227992 | 0.210552 | 0.190072 | 0.238332 | 0.429369 | 0.430548 | 0.437366 | 0.279128 | 0.290559 | 0.253994 | 0.284928 | 0.279128 | 0.253994 | 0.282050 | 0.284928 | 0.284928 | 0.279128 | 0.273148 | 0.343514 | 0.349696 | 0.355671 | 0.345598 | 0.365198 | 0.353701 | 0.483165 | 0.139322 |
| min | 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 | 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 | 0.000000 | 0.000000 | 0.000000 |
| 25% | 0.000000 | 0.000000 | 0.000000 | 0.333364 | 0.323713 | 0.281934 | 0.322226 | 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 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |
| 50% | 1.000000 | 0.000000 | 1.000000 | 0.532212 | 0.490297 | 0.408108 | 0.496499 | 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 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |
| 75% | 1.000000 | 0.000000 | 1.000000 | 0.736253 | 0.646471 | 0.532182 | 0.677472 | 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 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 1.000000 | 0.000000 |
| max | 1.000000 | 1.000000 | 1.000000 | 1.000000 | 1.000000 | 1.000000 | 1.000000 | 1.000000 | 1.000000 | 1.000000 | 1.000000 | 1.000000 | 1.000000 | 1.000000 | 1.000000 | 1.000000 | 1.000000 | 1.000000 | 1.000000 | 1.000000 | 1.000000 | 1.000000 | 1.000000 | 1.000000 | 1.000000 | 1.000000 | 1.000000 | 1.000000 | 1.000000 |
In [117]:
data.shape
Out[117]:
(724, 29)
In [119]:
# Let's check the correlation coefficients to see which variables are highly correlated
plt.figure(figsize = (30,15))
sns.heatmap(data_train.corr(),annot=True, cmap = "YlGnBu")
plt.show()
Training The Model¶
In [122]:
data.shape
Out[122]:
(724, 29)
In [124]:
y_train = data_train.pop('cnt')
X_train = data_train
In [126]:
X_train.head()
Out[126]:
| year | holiday | workingday | Temperature | humidity | windspeed | spring | summer | winter | august | december | february | january | july | june | march | may | november | october | september | Monday | Saturday | Sunday | Thursday | Tuesday | Wednesday | cloudy | light_rain | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 422 | 1 | 0 | 1 | 0.383206 | 0.281444 | 0.615256 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 728 | 1 | 0 | 1 | 0.245101 | 0.270255 | 0.822447 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 614 | 1 | 0 | 0 | 0.802708 | 0.647560 | 0.373836 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 113 | 0 | 0 | 1 | 0.651106 | 0.758823 | 0.425255 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 |
| 579 | 1 | 0 | 0 | 0.880586 | 0.507702 | 0.484409 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 0 |
In [128]:
y_train.head()
Out[128]:
422 0.469757 728 0.164795 614 0.853918 113 0.453942 579 0.814198 Name: cnt, dtype: float64
Model 1¶
Lets try all all the vaibles in linear regression model using statistical analysis
In [132]:
import statsmodels.api as sm
In [133]:
X_train_lm = sm.add_constant(X_train)
In [134]:
m1 = sm.OLS(y_train, X_train_lm).fit()
In [135]:
m1.params
Out[135]:
const 0.377624 year 0.242509 holiday -0.164266 workingday -0.104056 Temperature 0.467470 humidity -0.147343 windspeed -0.119137 spring -0.090295 summer 0.017066 winter 0.106043 august -0.002092 december -0.068920 february -0.035135 january -0.061048 july -0.037722 june -0.004828 march 0.004660 may 0.035879 november -0.075464 october 0.007204 september 0.067729 Monday -0.010415 Saturday -0.094819 Sunday -0.094949 Thursday 0.008495 Tuesday -0.028485 Wednesday -0.002381 cloudy -0.048702 light_rain -0.182129 dtype: float64
In [140]:
print(m1.summary())
OLS Regression Results
==============================================================================
Dep. Variable: cnt R-squared: 0.859
Model: OLS Adj. R-squared: 0.851
Method: Least Squares F-statistic: 103.9
Date: Tue, 11 Mar 2025 Prob (F-statistic): 2.65e-183
Time: 11:11:02 Log-Likelihood: 504.01
No. Observations: 506 AIC: -950.0
Df Residuals: 477 BIC: -827.4
Df Model: 28
Covariance Type: nonrobust
===============================================================================
coef std err t P>|t| [0.025 0.975]
-------------------------------------------------------------------------------
const 0.3776 0.086 4.388 0.000 0.209 0.547
year 0.2425 0.008 28.564 0.000 0.226 0.259
holiday -0.1643 0.066 -2.470 0.014 -0.295 -0.034
workingday -0.1041 0.072 -1.450 0.148 -0.245 0.037
Temperature 0.4675 0.048 9.684 0.000 0.373 0.562
humidity -0.1473 0.030 -4.888 0.000 -0.207 -0.088
windspeed -0.1191 0.024 -4.977 0.000 -0.166 -0.072
spring -0.0903 0.031 -2.899 0.004 -0.152 -0.029
summer 0.0171 0.028 0.610 0.542 -0.038 0.072
winter 0.1060 0.028 3.820 0.000 0.052 0.161
august -0.0021 0.036 -0.059 0.953 -0.072 0.068
december -0.0689 0.035 -1.993 0.047 -0.137 -0.001
february -0.0351 0.035 -1.012 0.312 -0.103 0.033
january -0.0610 0.035 -1.741 0.082 -0.130 0.008
july -0.0377 0.036 -1.037 0.300 -0.109 0.034
june -0.0048 0.026 -0.184 0.854 -0.056 0.047
march 0.0047 0.026 0.179 0.858 -0.047 0.056
may 0.0359 0.022 1.664 0.097 -0.007 0.078
november -0.0755 0.037 -2.021 0.044 -0.149 -0.002
october 0.0072 0.037 0.196 0.845 -0.065 0.079
september 0.0677 0.033 2.027 0.043 0.002 0.133
Monday -0.0104 0.016 -0.648 0.517 -0.042 0.021
Saturday -0.0948 0.072 -1.309 0.191 -0.237 0.048
Sunday -0.0949 0.072 -1.311 0.191 -0.237 0.047
Thursday 0.0085 0.016 0.535 0.593 -0.023 0.040
Tuesday -0.0285 0.016 -1.832 0.068 -0.059 0.002
Wednesday -0.0024 0.016 -0.150 0.881 -0.034 0.029
cloudy -0.0487 0.012 -4.182 0.000 -0.072 -0.026
light_rain -0.1821 0.035 -5.252 0.000 -0.250 -0.114
==============================================================================
Omnibus: 109.188 Durbin-Watson: 1.939
Prob(Omnibus): 0.000 Jarque-Bera (JB): 296.697
Skew: -1.050 Prob(JB): 3.74e-65
Kurtosis: 6.108 Cond. No. 68.0
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
Checking Multicolinearity Using VIF¶
In [143]:
from statsmodels.stats.outliers_influence import variance_inflation_factor
In [145]:
vif = pd.DataFrame()
X = X_train
vif['Features'] = X.columns
vif['VIF'] = [variance_inflation_factor(X.values,i) for i in range (X.shape[1])]
vif['VIF'] = round(vif['VIF'],2)
vif = vif.sort_values(by = 'VIF', ascending= False)
vif
Out[145]:
| Features | VIF | |
|---|---|---|
| 2 | workingday | 55.75 |
| 3 | Temperature | 42.31 |
| 4 | humidity | 15.48 |
| 6 | spring | 13.28 |
| 22 | Sunday | 12.05 |
| 21 | Saturday | 11.84 |
| 8 | winter | 11.57 |
| 7 | summer | 10.05 |
| 5 | windspeed | 7.24 |
| 17 | november | 6.79 |
| 18 | october | 6.40 |
| 13 | july | 6.35 |
| 9 | august | 5.98 |
| 10 | december | 5.93 |
| 12 | january | 5.88 |
| 19 | september | 4.95 |
| 11 | february | 4.57 |
| 15 | march | 3.30 |
| 26 | cloudy | 2.99 |
| 1 | holiday | 2.95 |
| 14 | june | 2.79 |
| 16 | may | 2.45 |
| 24 | Tuesday | 2.26 |
| 0 | year | 2.18 |
| 25 | Wednesday | 2.17 |
| 20 | Monday | 2.08 |
| 23 | Thursday | 2.07 |
| 27 | light_rain | 1.42 |
The VIF value should be less than 5 it can be seen there very high vif value and high p value lets take top 20 features with RFE
Recursive Feature Elimination¶
In [149]:
from sklearn.feature_selection import RFE
from sklearn.linear_model import LinearRegression
In [150]:
lm = LinearRegression()
lm.fit(X_train,y_train)
rfe = RFE(lm, n_features_to_select=20, step = 1)
rfe = rfe.fit(X_train, y_train)
In [151]:
rfe_ranking = pd.DataFrame(list(zip(X_train.columns, rfe.support_,rfe.ranking_)), columns=['features','Support',"Rank"])
rfe_ranking.sort_values(by='Rank', ascending = True)
Out[151]:
| features | Support | Rank | |
|---|---|---|---|
| 0 | year | True | 1 |
| 24 | Tuesday | True | 1 |
| 22 | Sunday | True | 1 |
| 21 | Saturday | True | 1 |
| 19 | september | True | 1 |
| 17 | november | True | 1 |
| 16 | may | True | 1 |
| 26 | cloudy | True | 1 |
| 12 | january | True | 1 |
| 11 | february | True | 1 |
| 13 | july | True | 1 |
| 4 | humidity | True | 1 |
| 1 | holiday | True | 1 |
| 2 | workingday | True | 1 |
| 3 | Temperature | True | 1 |
| 10 | december | True | 1 |
| 5 | windspeed | True | 1 |
| 27 | light_rain | True | 1 |
| 8 | winter | True | 1 |
| 6 | spring | True | 1 |
| 7 | summer | False | 2 |
| 23 | Thursday | False | 3 |
| 20 | Monday | False | 4 |
| 18 | october | False | 5 |
| 15 | march | False | 6 |
| 14 | june | False | 7 |
| 25 | Wednesday | False | 8 |
| 9 | august | False | 9 |
In [155]:
col = X_train.columns[rfe.support_]
col
Out[155]:
Index(['year', 'holiday', 'workingday', 'Temperature', 'humidity', 'windspeed',
'spring', 'winter', 'december', 'february', 'january', 'july', 'may',
'november', 'september', 'Saturday', 'Sunday', 'Tuesday', 'cloudy',
'light_rain'],
dtype='object')
In [157]:
X_train.columns[~rfe.support_]
Out[157]:
Index(['summer', 'august', 'june', 'march', 'october', 'Monday', 'Thursday',
'Wednesday'],
dtype='object')
In [159]:
X_train_rfe = X_train[col]
In [161]:
X_train_rfe.head()
Out[161]:
| year | holiday | workingday | Temperature | humidity | windspeed | spring | winter | december | february | january | july | may | november | september | Saturday | Sunday | Tuesday | cloudy | light_rain | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 422 | 1 | 0 | 1 | 0.383206 | 0.281444 | 0.615256 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 728 | 1 | 0 | 1 | 0.245101 | 0.270255 | 0.822447 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 614 | 1 | 0 | 0 | 0.802708 | 0.647560 | 0.373836 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 113 | 0 | 0 | 1 | 0.651106 | 0.758823 | 0.425255 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 |
| 579 | 1 | 0 | 0 | 0.880586 | 0.507702 | 0.484409 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 |
Model 2 : Using RFE¶
In [164]:
X_train_lm = sm.add_constant(X_train_rfe)
In [166]:
lm = sm.OLS(y_train, X_train_rfe).fit()
In [168]:
print(lm.summary())
OLS Regression Results
=======================================================================================
Dep. Variable: cnt R-squared (uncentered): 0.972
Model: OLS Adj. R-squared (uncentered): 0.971
Method: Least Squares F-statistic: 836.4
Date: Tue, 11 Mar 2025 Prob (F-statistic): 0.00
Time: 11:11:07 Log-Likelihood: 489.37
No. Observations: 506 AIC: -938.7
Df Residuals: 486 BIC: -854.2
Df Model: 20
Covariance Type: nonrobust
===============================================================================
coef std err t P>|t| [0.025 0.975]
-------------------------------------------------------------------------------
year 0.2472 0.009 28.910 0.000 0.230 0.264
holiday 0.1302 0.034 3.853 0.000 0.064 0.197
workingday 0.2350 0.027 8.863 0.000 0.183 0.287
Temperature 0.5014 0.034 14.833 0.000 0.435 0.568
humidity -0.1380 0.030 -4.606 0.000 -0.197 -0.079
windspeed -0.0944 0.024 -3.977 0.000 -0.141 -0.048
spring -0.0814 0.019 -4.295 0.000 -0.119 -0.044
winter 0.1145 0.015 7.733 0.000 0.085 0.144
december -0.0694 0.018 -3.811 0.000 -0.105 -0.034
february -0.0317 0.023 -1.391 0.165 -0.076 0.013
january -0.0560 0.022 -2.536 0.012 -0.099 -0.013
july -0.0437 0.018 -2.427 0.016 -0.079 -0.008
may 0.0517 0.016 3.168 0.002 0.020 0.084
november -0.0812 0.019 -4.180 0.000 -0.119 -0.043
september 0.0619 0.017 3.709 0.000 0.029 0.095
Saturday 0.2423 0.028 8.751 0.000 0.188 0.297
Sunday 0.2416 0.028 8.764 0.000 0.187 0.296
Tuesday -0.0254 0.012 -2.114 0.035 -0.049 -0.002
cloudy -0.0457 0.012 -3.892 0.000 -0.069 -0.023
light_rain -0.1798 0.035 -5.160 0.000 -0.248 -0.111
==============================================================================
Omnibus: 113.859 Durbin-Watson: 1.973
Prob(Omnibus): 0.000 Jarque-Bera (JB): 325.332
Skew: -1.076 Prob(JB): 2.26e-71
Kurtosis: 6.287 Cond. No. 20.0
==============================================================================
Notes:
[1] R² is computed without centering (uncentered) since the model does not contain a constant.
[2] Standard Errors assume that the covariance matrix of the errors is correctly specified.
We can see all p value is less then 0.05 lets see the vif
Checking VIF
In [172]:
vif = pd.DataFrame()
X = X_train_rfe
vif['Features'] = X.columns
vif['VIF'] = [variance_inflation_factor(X.values,i) for i in range (X.shape[1])]
vif['VIF'] = round(vif['VIF'],2)
vif = vif.sort_values(by = 'VIF', ascending= False)
vif
Out[172]:
| Features | VIF | |
|---|---|---|
| 2 | workingday | 27.61 |
| 3 | Temperature | 22.29 |
| 4 | humidity | 14.72 |
| 5 | windspeed | 6.96 |
| 16 | Sunday | 6.47 |
| 15 | Saturday | 6.27 |
| 6 | spring | 5.01 |
| 7 | winter | 3.24 |
| 18 | cloudy | 2.93 |
| 10 | january | 2.49 |
| 0 | year | 2.13 |
| 9 | february | 2.06 |
| 1 | holiday | 1.94 |
| 13 | november | 1.93 |
| 8 | december | 1.77 |
| 11 | july | 1.58 |
| 19 | light_rain | 1.38 |
| 12 | may | 1.36 |
| 17 | Tuesday | 1.31 |
| 14 | september | 1.30 |
Dropping Feature workingday based on high vif variable
In [175]:
X_train_rfe = X_train_rfe.drop(['workingday'], axis = 1)
Model 3¶
In [178]:
X_train_lm = sm.add_constant(X_train_rfe)
m3 = sm.OLS(y_train, X_train_lm).fit()
print(m3.summary())
OLS Regression Results
==============================================================================
Dep. Variable: cnt R-squared: 0.858
Model: OLS Adj. R-squared: 0.852
Method: Least Squares F-statistic: 154.2
Date: Tue, 11 Mar 2025 Prob (F-statistic): 7.94e-192
Time: 11:11:09 Log-Likelihood: 501.56
No. Observations: 506 AIC: -963.1
Df Residuals: 486 BIC: -878.6
Df Model: 19
Covariance Type: nonrobust
===============================================================================
coef std err t P>|t| [0.025 0.975]
-------------------------------------------------------------------------------
const 0.2914 0.028 10.316 0.000 0.236 0.347
year 0.2439 0.008 29.131 0.000 0.227 0.260
holiday -0.0726 0.024 -2.980 0.003 -0.121 -0.025
Temperature 0.4482 0.035 12.907 0.000 0.380 0.516
humidity -0.1489 0.029 -5.087 0.000 -0.206 -0.091
windspeed -0.1156 0.024 -4.917 0.000 -0.162 -0.069
spring -0.0978 0.019 -5.202 0.000 -0.135 -0.061
winter 0.1031 0.015 7.045 0.000 0.074 0.132
december -0.0784 0.018 -4.388 0.000 -0.113 -0.043
february -0.0418 0.022 -1.872 0.062 -0.086 0.002
january -0.0696 0.022 -3.201 0.001 -0.112 -0.027
july -0.0410 0.018 -2.333 0.020 -0.076 -0.006
may 0.0465 0.016 2.913 0.004 0.015 0.078
november -0.0865 0.019 -4.557 0.000 -0.124 -0.049
september 0.0618 0.016 3.796 0.000 0.030 0.094
Saturday 0.0087 0.012 0.716 0.474 -0.015 0.033
Sunday 0.0080 0.012 0.671 0.502 -0.015 0.032
Tuesday -0.0271 0.012 -2.315 0.021 -0.050 -0.004
cloudy -0.0472 0.011 -4.121 0.000 -0.070 -0.025
light_rain -0.1791 0.034 -5.268 0.000 -0.246 -0.112
==============================================================================
Omnibus: 109.357 Durbin-Watson: 1.938
Prob(Omnibus): 0.000 Jarque-Bera (JB): 302.518
Skew: -1.045 Prob(JB): 2.04e-66
Kurtosis: 6.159 Cond. No. 17.4
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
In [180]:
vif = pd.DataFrame()
X = X_train_rfe
vif['Features'] = X.columns
vif['VIF'] = [variance_inflation_factor(X.values,i) for i in range (X.shape[1])]
vif['VIF'] = round(vif['VIF'],2)
vif = vif.sort_values(by = 'VIF', ascending= False)
vif
Out[180]:
| Features | VIF | |
|---|---|---|
| 3 | humidity | 13.42 |
| 2 | Temperature | 12.43 |
| 4 | windspeed | 5.33 |
| 5 | spring | 4.45 |
| 17 | cloudy | 2.92 |
| 6 | winter | 2.90 |
| 9 | january | 2.35 |
| 0 | year | 2.12 |
| 8 | february | 2.00 |
| 12 | november | 1.86 |
| 7 | december | 1.71 |
| 10 | july | 1.56 |
| 18 | light_rain | 1.37 |
| 11 | may | 1.36 |
| 16 | Tuesday | 1.30 |
| 13 | september | 1.29 |
| 14 | Saturday | 1.26 |
| 15 | Sunday | 1.26 |
| 1 | holiday | 1.06 |
Dropping Feature humidity based on high vif variable
In [183]:
X_train_rfe = X_train_rfe.drop(['humidity'], axis = 1)
Model 4¶
In [186]:
X_train_lm = sm.add_constant(X_train_rfe)
m4 = sm.OLS(y_train, X_train_lm).fit()
print(m4.summary())
OLS Regression Results
==============================================================================
Dep. Variable: cnt R-squared: 0.850
Model: OLS Adj. R-squared: 0.845
Method: Least Squares F-statistic: 153.5
Date: Tue, 11 Mar 2025 Prob (F-statistic): 1.75e-187
Time: 11:11:11 Log-Likelihood: 488.44
No. Observations: 506 AIC: -938.9
Df Residuals: 487 BIC: -858.6
Df Model: 18
Covariance Type: nonrobust
===============================================================================
coef std err t P>|t| [0.025 0.975]
-------------------------------------------------------------------------------
const 0.2487 0.028 8.994 0.000 0.194 0.303
year 0.2508 0.008 29.622 0.000 0.234 0.267
holiday -0.0707 0.025 -2.827 0.005 -0.120 -0.022
Temperature 0.4016 0.034 11.695 0.000 0.334 0.469
windspeed -0.0841 0.023 -3.618 0.000 -0.130 -0.038
spring -0.1065 0.019 -5.546 0.000 -0.144 -0.069
winter 0.0935 0.015 6.281 0.000 0.064 0.123
december -0.0885 0.018 -4.865 0.000 -0.124 -0.053
february -0.0425 0.023 -1.854 0.064 -0.087 0.003
january -0.0720 0.022 -3.228 0.001 -0.116 -0.028
july -0.0305 0.018 -1.704 0.089 -0.066 0.005
may 0.0349 0.016 2.154 0.032 0.003 0.067
november -0.0918 0.019 -4.725 0.000 -0.130 -0.054
september 0.0531 0.017 3.198 0.001 0.020 0.086
Saturday 0.0120 0.012 0.960 0.337 -0.013 0.037
Sunday 0.0110 0.012 0.896 0.371 -0.013 0.035
Tuesday -0.0306 0.012 -2.552 0.011 -0.054 -0.007
cloudy -0.0851 0.009 -9.528 0.000 -0.103 -0.068
light_rain -0.2573 0.031 -8.276 0.000 -0.318 -0.196
==============================================================================
Omnibus: 105.198 Durbin-Watson: 1.892
Prob(Omnibus): 0.000 Jarque-Bera (JB): 292.588
Skew: -1.004 Prob(JB): 2.92e-64
Kurtosis: 6.138 Cond. No. 16.3
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
In [188]:
vif = pd.DataFrame()
X = X_train_rfe
vif['Features'] = X.columns
vif['VIF'] = [variance_inflation_factor(X.values,i) for i in range (X.shape[1])]
vif['VIF'] = round(vif['VIF'],2)
vif = vif.sort_values(by = 'VIF', ascending= False)
vif
Out[188]:
| Features | VIF | |
|---|---|---|
| 2 | Temperature | 6.37 |
| 3 | windspeed | 5.23 |
| 4 | spring | 4.22 |
| 5 | winter | 2.70 |
| 8 | january | 2.32 |
| 0 | year | 2.08 |
| 7 | february | 2.00 |
| 11 | november | 1.83 |
| 6 | december | 1.65 |
| 16 | cloudy | 1.62 |
| 9 | july | 1.52 |
| 10 | may | 1.32 |
| 15 | Tuesday | 1.29 |
| 12 | september | 1.28 |
| 13 | Saturday | 1.26 |
| 14 | Sunday | 1.26 |
| 17 | light_rain | 1.09 |
| 1 | holiday | 1.06 |
Dropping Feature windspeed based on high vif variable
In [191]:
X_train_rfe = X_train_rfe.drop(['windspeed'], axis = 1)
Model 5¶
In [194]:
X_train_lm = sm.add_constant(X_train_rfe)
m5 = sm.OLS(y_train, X_train_lm).fit()
print(m5.summary())
OLS Regression Results
==============================================================================
Dep. Variable: cnt R-squared: 0.846
Model: OLS Adj. R-squared: 0.841
Method: Least Squares F-statistic: 157.9
Date: Tue, 11 Mar 2025 Prob (F-statistic): 8.41e-186
Time: 11:11:13 Log-Likelihood: 481.73
No. Observations: 506 AIC: -927.5
Df Residuals: 488 BIC: -851.4
Df Model: 17
Covariance Type: nonrobust
===============================================================================
coef std err t P>|t| [0.025 0.975]
-------------------------------------------------------------------------------
const 0.2021 0.025 8.159 0.000 0.153 0.251
year 0.2502 0.009 29.191 0.000 0.233 0.267
holiday -0.0712 0.025 -2.814 0.005 -0.121 -0.021
Temperature 0.4156 0.035 12.030 0.000 0.348 0.483
spring -0.1097 0.019 -5.652 0.000 -0.148 -0.072
winter 0.1021 0.015 6.864 0.000 0.073 0.131
december -0.0850 0.018 -4.618 0.000 -0.121 -0.049
february -0.0367 0.023 -1.588 0.113 -0.082 0.009
january -0.0647 0.022 -2.877 0.004 -0.109 -0.021
july -0.0262 0.018 -1.448 0.148 -0.062 0.009
may 0.0368 0.016 2.247 0.025 0.005 0.069
november -0.0933 0.020 -4.748 0.000 -0.132 -0.055
september 0.0592 0.017 3.543 0.000 0.026 0.092
Saturday 0.0122 0.013 0.968 0.333 -0.013 0.037
Sunday 0.0110 0.012 0.888 0.375 -0.013 0.035
Tuesday -0.0321 0.012 -2.641 0.009 -0.056 -0.008
cloudy -0.0835 0.009 -9.246 0.000 -0.101 -0.066
light_rain -0.2701 0.031 -8.639 0.000 -0.331 -0.209
==============================================================================
Omnibus: 101.379 Durbin-Watson: 1.943
Prob(Omnibus): 0.000 Jarque-Bera (JB): 282.326
Skew: -0.968 Prob(JB): 4.94e-62
Kurtosis: 6.106 Cond. No. 15.1
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
In [196]:
vif = pd.DataFrame()
X = X_train_rfe
vif['Features'] = X.columns
vif['VIF'] = [variance_inflation_factor(X.values,i) for i in range (X.shape[1])]
vif['VIF'] = round(vif['VIF'],2)
vif = vif.sort_values(by = 'VIF', ascending= False)
vif
Out[196]:
| Features | VIF | |
|---|---|---|
| 2 | Temperature | 4.01 |
| 3 | spring | 3.83 |
| 4 | winter | 2.69 |
| 7 | january | 2.31 |
| 0 | year | 2.07 |
| 6 | february | 1.99 |
| 10 | november | 1.79 |
| 5 | december | 1.64 |
| 15 | cloudy | 1.62 |
| 8 | july | 1.48 |
| 9 | may | 1.31 |
| 14 | Tuesday | 1.28 |
| 11 | september | 1.27 |
| 12 | Saturday | 1.26 |
| 13 | Sunday | 1.26 |
| 1 | holiday | 1.06 |
| 16 | light_rain | 1.06 |
Dropping Feature Sunday based on high p Value
In [199]:
X_train_rfe = X_train_rfe.drop(['Sunday'], axis = 1)
Model 6¶
In [202]:
X_train_lm = sm.add_constant(X_train_rfe)
m6 = sm.OLS(y_train, X_train_lm).fit()
print(m6.summary())
OLS Regression Results
==============================================================================
Dep. Variable: cnt R-squared: 0.846
Model: OLS Adj. R-squared: 0.841
Method: Least Squares F-statistic: 167.8
Date: Tue, 11 Mar 2025 Prob (F-statistic): 9.45e-187
Time: 11:11:14 Log-Likelihood: 481.32
No. Observations: 506 AIC: -928.6
Df Residuals: 489 BIC: -856.8
Df Model: 16
Covariance Type: nonrobust
===============================================================================
coef std err t P>|t| [0.025 0.975]
-------------------------------------------------------------------------------
const 0.2047 0.025 8.329 0.000 0.156 0.253
year 0.2503 0.009 29.225 0.000 0.234 0.267
holiday -0.0721 0.025 -2.853 0.005 -0.122 -0.022
Temperature 0.4150 0.035 12.017 0.000 0.347 0.483
spring -0.1097 0.019 -5.653 0.000 -0.148 -0.072
winter 0.1020 0.015 6.863 0.000 0.073 0.131
december -0.0851 0.018 -4.627 0.000 -0.121 -0.049
february -0.0374 0.023 -1.617 0.107 -0.083 0.008
january -0.0650 0.022 -2.894 0.004 -0.109 -0.021
july -0.0260 0.018 -1.437 0.151 -0.061 0.010
may 0.0368 0.016 2.249 0.025 0.005 0.069
november -0.0938 0.020 -4.772 0.000 -0.132 -0.055
september 0.0594 0.017 3.555 0.000 0.027 0.092
Saturday 0.0099 0.012 0.799 0.425 -0.014 0.034
Tuesday -0.0344 0.012 -2.906 0.004 -0.058 -0.011
cloudy -0.0833 0.009 -9.226 0.000 -0.101 -0.066
light_rain -0.2711 0.031 -8.679 0.000 -0.332 -0.210
==============================================================================
Omnibus: 98.326 Durbin-Watson: 1.942
Prob(Omnibus): 0.000 Jarque-Bera (JB): 270.326
Skew: -0.944 Prob(JB): 1.99e-59
Kurtosis: 6.043 Cond. No. 15.0
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
In [204]:
vif = pd.DataFrame()
X = X_train_rfe
vif['Features'] = X.columns
vif['VIF'] = [variance_inflation_factor(X.values,i) for i in range (X.shape[1])]
vif['VIF'] = round(vif['VIF'],2)
vif = vif.sort_values(by = 'VIF', ascending= False)
vif
Out[204]:
| Features | VIF | |
|---|---|---|
| 2 | Temperature | 3.82 |
| 3 | spring | 3.82 |
| 4 | winter | 2.69 |
| 7 | january | 2.31 |
| 0 | year | 2.07 |
| 6 | february | 1.99 |
| 10 | november | 1.79 |
| 5 | december | 1.64 |
| 14 | cloudy | 1.61 |
| 8 | july | 1.48 |
| 9 | may | 1.31 |
| 11 | september | 1.27 |
| 13 | Tuesday | 1.23 |
| 12 | Saturday | 1.21 |
| 1 | holiday | 1.06 |
| 15 | light_rain | 1.06 |
Dropping Feature Saturday based on high p Value
In [207]:
X_train_rfe = X_train_rfe.drop(['Saturday'], axis = 1)
Model 7¶
In [210]:
X_train_lm = sm.add_constant(X_train_rfe)
m7 = sm.OLS(y_train, X_train_lm).fit()
print(m7.summary())
OLS Regression Results
==============================================================================
Dep. Variable: cnt R-squared: 0.846
Model: OLS Adj. R-squared: 0.841
Method: Least Squares F-statistic: 179.0
Date: Tue, 11 Mar 2025 Prob (F-statistic): 9.53e-188
Time: 11:11:16 Log-Likelihood: 480.99
No. Observations: 506 AIC: -930.0
Df Residuals: 490 BIC: -862.4
Df Model: 15
Covariance Type: nonrobust
===============================================================================
coef std err t P>|t| [0.025 0.975]
-------------------------------------------------------------------------------
const 0.2065 0.024 8.437 0.000 0.158 0.255
year 0.2503 0.009 29.237 0.000 0.234 0.267
holiday -0.0727 0.025 -2.879 0.004 -0.122 -0.023
Temperature 0.4150 0.035 12.023 0.000 0.347 0.483
spring -0.1100 0.019 -5.669 0.000 -0.148 -0.072
winter 0.1019 0.015 6.855 0.000 0.073 0.131
december -0.0850 0.018 -4.624 0.000 -0.121 -0.049
february -0.0373 0.023 -1.615 0.107 -0.083 0.008
january -0.0657 0.022 -2.927 0.004 -0.110 -0.022
july -0.0258 0.018 -1.430 0.153 -0.061 0.010
may 0.0365 0.016 2.230 0.026 0.004 0.069
november -0.0936 0.020 -4.766 0.000 -0.132 -0.055
september 0.0588 0.017 3.524 0.000 0.026 0.092
Tuesday -0.0361 0.012 -3.093 0.002 -0.059 -0.013
cloudy -0.0830 0.009 -9.206 0.000 -0.101 -0.065
light_rain -0.2717 0.031 -8.705 0.000 -0.333 -0.210
==============================================================================
Omnibus: 99.001 Durbin-Watson: 1.939
Prob(Omnibus): 0.000 Jarque-Bera (JB): 274.529
Skew: -0.947 Prob(JB): 2.44e-60
Kurtosis: 6.071 Cond. No. 14.9
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
In [212]:
vif = pd.DataFrame()
X = X_train_rfe
vif['Features'] = X.columns
vif['VIF'] = [variance_inflation_factor(X.values,i) for i in range (X.shape[1])]
vif['VIF'] = round(vif['VIF'],2)
vif = vif.sort_values(by = 'VIF', ascending= False)
vif
Out[212]:
| Features | VIF | |
|---|---|---|
| 3 | spring | 3.81 |
| 2 | Temperature | 3.66 |
| 4 | winter | 2.68 |
| 7 | january | 2.31 |
| 0 | year | 2.07 |
| 6 | february | 1.99 |
| 10 | november | 1.79 |
| 5 | december | 1.63 |
| 13 | cloudy | 1.61 |
| 8 | july | 1.48 |
| 9 | may | 1.31 |
| 11 | september | 1.26 |
| 12 | Tuesday | 1.19 |
| 1 | holiday | 1.06 |
| 14 | light_rain | 1.06 |
Dropping Feature july based on high p Value
In [215]:
X_train_rfe = X_train_rfe.drop(['july'], axis = 1)
Model 8¶
In [218]:
X_train_lm = sm.add_constant(X_train_rfe)
m8 = sm.OLS(y_train, X_train_lm).fit()
print(m8.summary())
OLS Regression Results
==============================================================================
Dep. Variable: cnt R-squared: 0.845
Model: OLS Adj. R-squared: 0.841
Method: Least Squares F-statistic: 191.3
Date: Tue, 11 Mar 2025 Prob (F-statistic): 1.87e-188
Time: 11:11:17 Log-Likelihood: 479.93
No. Observations: 506 AIC: -929.9
Df Residuals: 491 BIC: -866.5
Df Model: 14
Covariance Type: nonrobust
===============================================================================
coef std err t P>|t| [0.025 0.975]
-------------------------------------------------------------------------------
const 0.2140 0.024 8.945 0.000 0.167 0.261
year 0.2508 0.009 29.272 0.000 0.234 0.268
holiday -0.0739 0.025 -2.924 0.004 -0.123 -0.024
Temperature 0.3957 0.032 12.444 0.000 0.333 0.458
spring -0.1096 0.019 -5.645 0.000 -0.148 -0.071
winter 0.1039 0.015 7.011 0.000 0.075 0.133
december -0.0873 0.018 -4.765 0.000 -0.123 -0.051
february -0.0396 0.023 -1.718 0.086 -0.085 0.006
january -0.0693 0.022 -3.106 0.002 -0.113 -0.025
may 0.0418 0.016 2.621 0.009 0.010 0.073
november -0.0955 0.020 -4.868 0.000 -0.134 -0.057
september 0.0643 0.016 3.956 0.000 0.032 0.096
Tuesday -0.0371 0.012 -3.190 0.002 -0.060 -0.014
cloudy -0.0829 0.009 -9.183 0.000 -0.101 -0.065
light_rain -0.2740 0.031 -8.782 0.000 -0.335 -0.213
==============================================================================
Omnibus: 101.339 Durbin-Watson: 1.935
Prob(Omnibus): 0.000 Jarque-Bera (JB): 282.861
Skew: -0.967 Prob(JB): 3.78e-62
Kurtosis: 6.111 Cond. No. 14.0
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
In [220]:
vif = pd.DataFrame()
X = X_train_rfe
vif['Features'] = X.columns
vif['VIF'] = [variance_inflation_factor(X.values,i) for i in range (X.shape[1])]
vif['VIF'] = round(vif['VIF'],2)
vif = vif.sort_values(by = 'VIF', ascending= False)
vif
Out[220]:
| Features | VIF | |
|---|---|---|
| 3 | spring | 3.73 |
| 2 | Temperature | 2.76 |
| 4 | winter | 2.58 |
| 7 | january | 2.30 |
| 0 | year | 2.06 |
| 6 | february | 1.99 |
| 9 | november | 1.79 |
| 5 | december | 1.63 |
| 12 | cloudy | 1.60 |
| 8 | may | 1.22 |
| 10 | september | 1.20 |
| 11 | Tuesday | 1.19 |
| 1 | holiday | 1.06 |
| 13 | light_rain | 1.06 |
Dropping Feature february based on high P value
In [223]:
X_train_rfe = X_train_rfe.drop(['february'], axis = 1)
Now as you can see, the VIFs and p-values both are within an acceptable range. So we go ahead and make our predictions this model only.
Model 9¶
In [227]:
X_train_lm = sm.add_constant(X_train_rfe)
m9 = sm.OLS(y_train, X_train_lm).fit()
print(m9.summary())
OLS Regression Results
==============================================================================
Dep. Variable: cnt R-squared: 0.844
Model: OLS Adj. R-squared: 0.840
Method: Least Squares F-statistic: 204.9
Date: Tue, 11 Mar 2025 Prob (F-statistic): 5.54e-189
Time: 11:11:19 Log-Likelihood: 478.42
No. Observations: 506 AIC: -928.8
Df Residuals: 492 BIC: -869.7
Df Model: 13
Covariance Type: nonrobust
===============================================================================
coef std err t P>|t| [0.025 0.975]
-------------------------------------------------------------------------------
const 0.2065 0.024 8.761 0.000 0.160 0.253
year 0.2505 0.009 29.188 0.000 0.234 0.267
holiday -0.0761 0.025 -3.009 0.003 -0.126 -0.026
Temperature 0.4060 0.031 12.980 0.000 0.345 0.468
spring -0.1255 0.017 -7.336 0.000 -0.159 -0.092
winter 0.1023 0.015 6.904 0.000 0.073 0.131
december -0.0772 0.017 -4.440 0.000 -0.111 -0.043
january -0.0483 0.019 -2.582 0.010 -0.085 -0.012
may 0.0423 0.016 2.644 0.008 0.011 0.074
november -0.0905 0.019 -4.655 0.000 -0.129 -0.052
september 0.0648 0.016 3.974 0.000 0.033 0.097
Tuesday -0.0369 0.012 -3.162 0.002 -0.060 -0.014
cloudy -0.0822 0.009 -9.097 0.000 -0.100 -0.064
light_rain -0.2726 0.031 -8.723 0.000 -0.334 -0.211
==============================================================================
Omnibus: 96.153 Durbin-Watson: 1.934
Prob(Omnibus): 0.000 Jarque-Bera (JB): 259.618
Skew: -0.930 Prob(JB): 4.21e-57
Kurtosis: 5.976 Cond. No. 13.7
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
In [229]:
vif = pd.DataFrame()
X = X_train_rfe
vif['Features'] = X.columns
vif['VIF'] = [variance_inflation_factor(X.values,i) for i in range (X.shape[1])]
vif['VIF'] = round(vif['VIF'],2)
vif = vif.sort_values(by = 'VIF', ascending= False)
vif
Out[229]:
| Features | VIF | |
|---|---|---|
| 2 | Temperature | 2.75 |
| 4 | winter | 2.52 |
| 0 | year | 2.06 |
| 3 | spring | 2.00 |
| 8 | november | 1.77 |
| 6 | january | 1.66 |
| 11 | cloudy | 1.60 |
| 5 | december | 1.49 |
| 7 | may | 1.22 |
| 9 | september | 1.20 |
| 10 | Tuesday | 1.19 |
| 12 | light_rain | 1.06 |
| 1 | holiday | 1.05 |
Dropping Feature january based on high P value
In [232]:
X_train_rfe = X_train_rfe.drop(['january'], axis = 1)
Final Model¶
In [235]:
X_train_lm = sm.add_constant(X_train_rfe)
Final_model= sm.OLS(y_train, X_train_lm).fit()
print(Final_model.summary())
OLS Regression Results
==============================================================================
Dep. Variable: cnt R-squared: 0.842
Model: OLS Adj. R-squared: 0.838
Method: Least Squares F-statistic: 219.0
Date: Tue, 11 Mar 2025 Prob (F-statistic): 9.90e-189
Time: 11:11:22 Log-Likelihood: 475.01
No. Observations: 506 AIC: -924.0
Df Residuals: 493 BIC: -869.1
Df Model: 12
Covariance Type: nonrobust
===============================================================================
coef std err t P>|t| [0.025 0.975]
-------------------------------------------------------------------------------
const 0.1944 0.023 8.367 0.000 0.149 0.240
year 0.2499 0.009 28.966 0.000 0.233 0.267
holiday -0.0794 0.025 -3.129 0.002 -0.129 -0.030
Temperature 0.4242 0.031 13.834 0.000 0.364 0.484
spring -0.1367 0.017 -8.218 0.000 -0.169 -0.104
winter 0.1026 0.015 6.888 0.000 0.073 0.132
december -0.0667 0.017 -3.925 0.000 -0.100 -0.033
may 0.0433 0.016 2.691 0.007 0.012 0.075
november -0.0849 0.019 -4.370 0.000 -0.123 -0.047
september 0.0648 0.016 3.956 0.000 0.033 0.097
Tuesday -0.0374 0.012 -3.191 0.002 -0.060 -0.014
cloudy -0.0831 0.009 -9.147 0.000 -0.101 -0.065
light_rain -0.2716 0.031 -8.642 0.000 -0.333 -0.210
==============================================================================
Omnibus: 90.162 Durbin-Watson: 1.928
Prob(Omnibus): 0.000 Jarque-Bera (JB): 229.215
Skew: -0.894 Prob(JB): 1.68e-50
Kurtosis: 5.770 Cond. No. 13.4
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
In [237]:
vif = pd.DataFrame()
X = X_train_rfe
vif['Features'] = X.columns
vif['VIF'] = [variance_inflation_factor(X.values,i) for i in range (X.shape[1])]
vif['VIF'] = round(vif['VIF'],2)
vif = vif.sort_values(by = 'VIF', ascending= False)
vif
Out[237]:
| Features | VIF | |
|---|---|---|
| 2 | Temperature | 2.72 |
| 4 | winter | 2.49 |
| 0 | year | 2.06 |
| 7 | november | 1.76 |
| 10 | cloudy | 1.59 |
| 5 | december | 1.44 |
| 3 | spring | 1.36 |
| 6 | may | 1.22 |
| 8 | september | 1.20 |
| 9 | Tuesday | 1.19 |
| 11 | light_rain | 1.06 |
| 1 | holiday | 1.05 |
In [239]:
data.head()
Out[239]:
| year | holiday | workingday | Temperature | humidity | windspeed | cnt | spring | summer | winter | august | december | february | january | july | june | march | may | november | october | september | Monday | Saturday | Sunday | Thursday | Tuesday | Wednesday | cloudy | light_rain | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 0 | 0 | 1 | 14.110847 | 80.5833 | 10.749882 | 985 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 1 | 14.902598 | 69.6087 | 16.652113 | 801 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 |
| 2 | 0 | 0 | 1 | 8.050924 | 43.7273 | 16.636703 | 1349 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 3 | 0 | 0 | 1 | 8.200000 | 59.0435 | 10.739832 | 1562 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 1 | 9.305237 | 43.6957 | 12.522300 | 1600 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
Residual Analysis of Train Data¶
In [242]:
X_train_lm = sm.add_constant(X_train_rfe)
In [244]:
y_train_pred = Final_model.predict(X_train_lm)
In [246]:
fig = plt.figure()
sns.distplot((y_train - y_train_pred), bins = 20)
fig.suptitle('Error terms', fontsize = 20)
plt.xlabel('Errors', fontsize= 18)
Out[246]:
Text(0.5, 0, 'Errors')
Predictions on test Data¶
In [249]:
num_vars = ['Temperature','humidity','windspeed','cnt']
data_test[num_vars] = scaler.transform(data_test[num_vars])
In [251]:
data_test.describe()
Out[251]:
| year | holiday | workingday | Temperature | humidity | windspeed | cnt | spring | summer | winter | august | december | february | january | july | june | march | may | november | october | september | Monday | Saturday | Sunday | Thursday | Tuesday | Wednesday | cloudy | light_rain | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| count | 218.000000 | 218.000000 | 218.000000 | 218.000000 | 218.000000 | 218.000000 | 218.000000 | 218.000000 | 218.000000 | 218.000000 | 218.000000 | 218.000000 | 218.000000 | 218.000000 | 218.000000 | 218.000000 | 218.000000 | 218.000000 | 218.000000 | 218.000000 | 218.000000 | 218.000000 | 218.000000 | 218.00000 | 218.000000 | 218.000000 | 218.000000 | 218.000000 | 218.000000 |
| mean | 0.486239 | 0.027523 | 0.711009 | 0.560655 | 0.474024 | 0.408393 | 0.498156 | 0.247706 | 0.275229 | 0.215596 | 0.082569 | 0.068807 | 0.091743 | 0.077982 | 0.087156 | 0.114679 | 0.073394 | 0.077982 | 0.068807 | 0.082569 | 0.082569 | 0.155963 | 0.137615 | 0.12844 | 0.155963 | 0.110092 | 0.137615 | 0.261468 | 0.036697 |
| std | 0.500961 | 0.163978 | 0.454337 | 0.230041 | 0.199583 | 0.193546 | 0.215383 | 0.432674 | 0.447658 | 0.412182 | 0.275863 | 0.253709 | 0.289327 | 0.268760 | 0.282713 | 0.319367 | 0.261383 | 0.268760 | 0.253709 | 0.275863 | 0.275863 | 0.363656 | 0.345288 | 0.33535 | 0.363656 | 0.313724 | 0.345288 | 0.440446 | 0.188450 |
| min | 0.000000 | 0.000000 | 0.000000 | 0.046591 | -0.071617 | 0.049875 | 0.009055 | 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 | 0.000000 | 0.000000 | 0.000000 | 0.00000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |
| 25% | 0.000000 | 0.000000 | 0.000000 | 0.366546 | 0.336799 | 0.277259 | 0.353042 | 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 | 0.000000 | 0.000000 | 0.000000 | 0.00000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |
| 50% | 0.000000 | 0.000000 | 1.000000 | 0.576343 | 0.476623 | 0.373049 | 0.501751 | 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 | 0.000000 | 0.000000 | 0.000000 | 0.00000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |
| 75% | 1.000000 | 0.000000 | 1.000000 | 0.771557 | 0.616670 | 0.507796 | 0.637239 | 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 | 0.000000 | 0.00000 | 0.000000 | 0.000000 | 0.000000 | 1.000000 | 0.000000 |
| max | 1.000000 | 1.000000 | 1.000000 | 0.984424 | 0.985082 | 1.049896 | 0.933961 | 1.000000 | 1.000000 | 1.000000 | 1.000000 | 1.000000 | 1.000000 | 1.000000 | 1.000000 | 1.000000 | 1.000000 | 1.000000 | 1.000000 | 1.000000 | 1.000000 | 1.000000 | 1.000000 | 1.00000 | 1.000000 | 1.000000 | 1.000000 | 1.000000 | 1.000000 |
In [253]:
y_test = data_test.pop('cnt')
X_test = data_test
In [255]:
X_test_new = sm.add_constant(X_test)
In [257]:
X_train_rfe.head()
Out[257]:
| year | holiday | Temperature | spring | winter | december | may | november | september | Tuesday | cloudy | light_rain | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 422 | 1 | 0 | 0.383206 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 728 | 1 | 0 | 0.245101 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 614 | 1 | 0 | 0.802708 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 113 | 0 | 0 | 0.651106 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 |
| 579 | 1 | 0 | 0.880586 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
In [259]:
# DataFrames of Selected Features
col = ['year', 'holiday', 'Temperature', 'spring', 'winter','december','may',"november",'september',
'Tuesday', 'cloudy','light_rain']
In [261]:
X_test_new = X_test[col]
In [263]:
X_test_new = sm.add_constant(X_test_new)
In [265]:
y_test_pred = Final_model.predict(X_test_new)
In [267]:
fig = plt.figure()
plt.scatter(y_test, y_test_pred)
fig.suptitle('y_test vs y_test_pred', fontsize = 20) # Plot heading
plt.xlabel('y_test', fontsize = 18) # X-label
plt.ylabel('y_pred', fontsize = 16) # y - label
Out[267]:
Text(0, 0.5, 'y_pred')
In [269]:
# plotting a Regression plot
plt.figure()
sns.regplot(x=y_test, y=y_test_pred, ci=68, fit_reg=True,scatter_kws={"color": "blue"}, line_kws={"color": "red"})
plt.title('y_test vs y_test_pred', fontsize=15)
plt.xlabel('y_test', fontsize=15)
plt.ylabel('y_pred', fontsize=15)
plt.show()
In [271]:
from sklearn.metrics import mean_squared_error,r2_score
In [273]:
# Calculate mean squared error of the test set
mse_test = mean_squared_error(y_test, y_test_pred)
# Calculate RMSE
rmse_test = np.sqrt(mse_test)
print(rmse_test)
0.09449604122254024
In [275]:
# R squared of training and test data
rsquared_train = r2_score(y_train, y_train_pred)
rsquared_test = r2_score(y_test, y_test_pred)
#print the squared
print('R-squared for train data:',rsquared_train)
print('R-squared for test data:',rsquared_test)
R-squared for train data: 0.8420110687421533 R-squared for test data: 0.8066246590011492
In [277]:
# Coefficients of the final model
round(Final_model.params, 4)
Out[277]:
const 0.1944 year 0.2499 holiday -0.0794 Temperature 0.4242 spring -0.1367 winter 0.1026 december -0.0667 may 0.0433 november -0.0849 september 0.0648 Tuesday -0.0374 cloudy -0.0831 light_rain -0.2716 dtype: float64
Conclusion¶
Regression Equation¶
The equation of the best fit line is given by:
cnt = 0.1944 + 0.2499 x year - 0.0794 x holiday + 0.4242 x Temperature - 0.1367 x spring +
0.1026 x winter - 0.0667 x december + 0.0433 x may - 0.0849 x november + 0.0648
x september - 0.0374 x Tuesday - 0.0831 x cloudy - 0.2716 x light_rain
- Analysis is been done both recursive feature elimination and Mannual elimination
- 20 features has been selected using recursive feature elimination algorithium
- The bike demand is influenced by the feature year, holiday , temperature , spring, winter, light_rain
- In final Model we have seen 84% of variability in training data and 80% of variability in testing data
**************¶
In [ ]: