Octocat This notebook is part of a GitHub repository: https://github.com/pessini/moby-bikes
MIT Licensed
Author: Leandro Pessini

In [22]:
import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns
import numpy as np
from patsy import dmatrices

# Preprocessing
from sklearn.model_selection import train_test_split
from sklearn.preprocessing import StandardScaler, MinMaxScaler, OneHotEncoder, OrdinalEncoder
from sklearn.compose import ColumnTransformer
from sklearn.pipeline import Pipeline
from sklearn.preprocessing import PolynomialFeatures

# Model and Evaluation
from sklearn.linear_model import LinearRegression
from sklearn import linear_model
from sklearn import metrics

# statsmodel
import statsmodels.api as sm
import statsmodels.tsa.api as smt
import statsmodels.formula.api as smf
from statsmodels.stats.outliers_influence import variance_inflation_factor
from statsmodels.tools.tools import maybe_unwrap_results
from statsmodels.graphics.gofplots import ProbPlot
from typing import Type

# spicy
from scipy.stats import skew
from scipy.stats import boxcox
from scipy.stats import yeojohnson

import warnings
warnings.filterwarnings('ignore')
In [23]:
%reload_ext watermark
%watermark -a "Leandro Pessini" -n -u -v -iv -w
Author: Leandro Pessini

Last updated: Fri Sep 02 2022

Python implementation: CPython
Python version       : 3.9.13
IPython version      : 8.4.0

matplotlib : 3.5.2
pandas     : 1.4.3
sys        : 3.9.13 | packaged by conda-forge | (main, May 27 2022, 17:00:52) 
[Clang 13.0.1 ]
seaborn    : 0.11.2
statsmodels: 0.13.2
sklearn    : 1.1.1
numpy      : 1.23.0

Watermark: 2.3.1

Linear Regression

In [24]:
hourly_rentals = pd.read_csv('../data/processed/df_train.csv')
hourly_rentals.info()
<class 'pandas.core.frame.DataFrame'>
RangeIndex: 8760 entries, 0 to 8759
Data columns (total 23 columns):
 #   Column              Non-Null Count  Dtype  
---  ------              --------------  -----  
 0   rain                8760 non-null   float64
 1   temp                8760 non-null   float64
 2   rhum                8760 non-null   int64  
 3   wdsp                8760 non-null   int64  
 4   date                8760 non-null   object 
 5   hour                8760 non-null   int64  
 6   day                 8760 non-null   int64  
 7   month               8760 non-null   int64  
 8   year                8760 non-null   int64  
 9   count               8760 non-null   int64  
 10  holiday             8760 non-null   bool   
 11  dayofweek_n         8760 non-null   int64  
 12  dayofweek           8760 non-null   object 
 13  working_day         8760 non-null   bool   
 14  season              8760 non-null   object 
 15  peak                8760 non-null   bool   
 16  timesofday          8760 non-null   object 
 17  rainfall_intensity  8760 non-null   object 
 18  wind_bft            8760 non-null   int64  
 19  wind_speed_group    8760 non-null   object 
 20  temp_r              8760 non-null   int64  
 21  temp_bin            8760 non-null   float64
 22  rhum_bin            8760 non-null   float64
dtypes: bool(3), float64(4), int64(10), object(6)
memory usage: 1.4+ MB

OLS Assumptions

In [25]:
expr = \
     """ count ~ C(rainfall_intensity) + C(temp_bin) + C(rhum_bin) + C(wind_speed_group) + holiday + working_day + peak + C(timesofday)"""
y_train, X_train = dmatrices(expr, hourly_rentals, return_type = 'dataframe')
# nb_training_results = sm.GLM(y_train, X_train, family = sm.families.NegativeBinomial(alpha = 1)).fit(maxiter=5000, method='nm', cov_type='HC3')
X_with_constant = sm.add_constant(X_train)
model = sm.GLM(y_train, X_with_constant).fit(cov_type='HC3')
print(model.summary())
                 Generalized Linear Model Regression Results                  
==============================================================================
Dep. Variable:                  count   No. Observations:                 8760
Model:                            GLM   Df Residuals:                     8732
Model Family:                Gaussian   Df Model:                           27
Link Function:               identity   Scale:                          7.9395
Method:                          IRLS   Log-Likelihood:                -21491.
Date:                Fri, 02 Sep 2022   Deviance:                       69328.
Time:                        01:32:09   Pearson chi2:                 6.93e+04
No. Iterations:                     3   Pseudo R-squ. (CS):             0.4727
Covariance Type:                  HC3                                         
====================================================================================================================
                                                       coef    std err          z      P>|z|      [0.025      0.975]
--------------------------------------------------------------------------------------------------------------------
Intercept                                            6.7880      0.328     20.670      0.000       6.144       7.432
C(rainfall_intensity)[T.heavy rain]                 -0.2438      0.629     -0.388      0.698      -1.477       0.989
C(rainfall_intensity)[T.light rain]                 -0.0951      0.270     -0.352      0.725      -0.625       0.435
C(rainfall_intensity)[T.moderate rain]              -0.3793      0.166     -2.279      0.023      -0.705      -0.053
C(rainfall_intensity)[T.no rain]                     0.6175      0.117      5.264      0.000       0.388       0.847
C(temp_bin)[T.1.0]                                   0.1200      0.163      0.735      0.463      -0.200       0.440
C(temp_bin)[T.2.0]                                   0.3257      0.159      2.042      0.041       0.013       0.638
C(temp_bin)[T.3.0]                                   0.2212      0.154      1.434      0.151      -0.081       0.524
C(temp_bin)[T.4.0]                                   0.6052      0.157      3.847      0.000       0.297       0.914
C(temp_bin)[T.5.0]                                   0.8816      0.161      5.473      0.000       0.566       1.197
C(temp_bin)[T.6.0]                                   0.8592      0.160      5.383      0.000       0.546       1.172
C(temp_bin)[T.7.0]                                   1.5683      0.178      8.832      0.000       1.220       1.916
C(temp_bin)[T.8.0]                                   1.6133      0.211      7.652      0.000       1.200       2.027
C(temp_bin)[T.9.0]                                   1.6853      0.349      4.828      0.000       1.001       2.369
C(rhum_bin)[T.1.0]                                  -0.6582      0.286     -2.305      0.021      -1.218      -0.099
C(rhum_bin)[T.2.0]                                  -2.0287      0.265     -7.643      0.000      -2.549      -1.508
C(rhum_bin)[T.3.0]                                  -2.7681      0.263    -10.512      0.000      -3.284      -2.252
C(rhum_bin)[T.4.0]                                  -3.1979      0.265    -12.049      0.000      -3.718      -2.678
C(wind_speed_group)[T.Calm / Light Breeze]           0.0719      0.073      0.985      0.325      -0.071       0.215
C(wind_speed_group)[T.Gale / Storm]                 -0.6148      0.211     -2.920      0.003      -1.027      -0.202
C(wind_speed_group)[T.Moderate Breeze]              -0.1601      0.082     -1.961      0.050      -0.320   -6.34e-05
C(wind_speed_group)[T.Strong Breeze / Near Gale]    -1.1253      0.215     -5.229      0.000      -1.547      -0.704
holiday[T.True]                                     -0.5032      0.239     -2.105      0.035      -0.972      -0.035
working_day[T.True]                                 -0.2713      0.075     -3.604      0.000      -0.419      -0.124
peak[T.True]                                         0.3588      0.084      4.273      0.000       0.194       0.523
C(timesofday)[T.Evening]                            -1.3826      0.103    -13.447      0.000      -1.584      -1.181
C(timesofday)[T.Morning]                            -0.6480      0.116     -5.593      0.000      -0.875      -0.421
C(timesofday)[T.Night]                              -3.5116      0.098    -35.728      0.000      -3.704      -3.319
====================================================================================================================
In [26]:
model.resid_working
Out[26]:
0      -0.496577
1      -0.496577
2       0.503423
3      -0.496577
4      -0.496577
          ...   
8755   -1.534166
8756   -2.103445
8757   -0.745645
8758   -2.625601
8759    1.503423
Length: 8760, dtype: float64
In [27]:
def linearity_test(model, y):
    '''
    Function for visually inspecting the assumption of linearity in a linear regression model.
    It plots observed vs. predicted values and residuals vs. predicted values.
    
    Args:
    * model - fitted OLS model from statsmodels
    * y - observed values
    '''
    fitted_vals = model.predict()
    resids = model.resid_response

    fig, ax = plt.subplots(1,2, figsize=(12,6))
    
    sns.regplot(x=fitted_vals, y=y, lowess=True, ax=ax[0], line_kws={'color': 'red'})
    ax[0].set_title('Observed vs. Predicted Values', fontsize=16)
    ax[0].set(xlabel='Predicted', ylabel='Observed')

    sns.regplot(x=fitted_vals, y=resids, lowess=True, ax=ax[1], line_kws={'color': 'red'})
    ax[1].set_title('Residuals vs. Predicted Values', fontsize=16)
    ax[1].set(xlabel='Predicted', ylabel='Residuals')
    
linearity_test(model, y_train)   
In [28]:
#create instance of influence
influence = model.get_influence()
#obtain standardized residuals
standardized_residuals = influence.resid_studentized
In [29]:
p = sns.distplot(model.resid_response,kde=True)
p = plt.title('Normality of error terms/residuals')
In [30]:
model.resid_response.mean()
Out[30]:
2.3253241037409307e-14

Assumption 1 - Linearity

This assumes that there is a linear relationship between the predictors and the response variable.

In [31]:
#create instance of influence
influence = model.get_influence()
#obtain standardized residuals
standardized_residuals = influence.resid_studentized
#display standardized residuals
print(standardized_residuals)
[-0.17669343 -0.17669343  0.17912934 ... -0.26498742 -0.93460983
  0.5349521 ]

Distributions

In [32]:
fig = plt.figure(figsize=(16,12))
gs = fig.add_gridspec(2, 2)
ax0 = fig.add_subplot(gs[0, 0])
ax1 = fig.add_subplot(gs[0, 1])
ax2 = fig.add_subplot(gs[1, 0])
ax3 = fig.add_subplot(gs[1, 1])

sns.histplot(hourly_rentals['temp'],ax=ax0, stat='density', kde=True, 
             label= 'Skew :{0}'.format(np.round(skew(hourly_rentals['temp']),4)))
sns.histplot(hourly_rentals['rhum'],ax=ax1, stat='density', kde=True, 
             label= 'Skew :{0}'.format(np.round(skew(hourly_rentals['rhum']),4)))
sns.histplot(hourly_rentals['wdsp'],ax=ax2, stat='density', kde=True, 
             label= 'Skew :{0}'.format(np.round(skew(hourly_rentals['wdsp']),4)))
sns.histplot(hourly_rentals['count'],ax=ax3, stat='density', kde=True, 
             label= 'Skew :{0}'.format(np.round(skew(hourly_rentals['count']),4)))

ax0.set(xlabel='Temperature',title="Distribution - Temperature")
ax1.set(xlabel='Relative Humidity',title="Distribution - Relative Humidity")
ax2.set(xlabel='Wind Speed',title="Distribution - Wind Speed")
ax3.set(xlabel='Count', title="Distribution - Rentals Count")
ax0.legend(), ax1.legend(), ax2.legend(), ax3.legend()
plt.show()
In [33]:
poly = PolynomialFeatures(degree = 3)
X_poly = poly.fit_transform(X_train)
X_poly_constant = sm.add_constant(X_poly)
lin2 = sm.OLS(y_train, X_poly_constant).fit()
In [36]:
lin2.summary()
Out[36]:
OLS Regression Results
Dep. Variable: count R-squared: 0.514
Model: OLS Adj. R-squared: 0.449
Method: Least Squares F-statistic: 7.863
Date: Fri, 02 Sep 2022 Prob (F-statistic): 0.00
Time: 01:33:50 Log-Likelihood: -20501.
No. Observations: 8760 AIC: 4.308e+04
Df Residuals: 7720 BIC: 5.044e+04
Df Model: 1039
Covariance Type: nonrobust
coef std err t P>|t| [0.025 0.975]
const -5.521e+10 2.55e+11 -0.216 0.829 -5.56e+11 4.45e+11
x1 1.448e+11 5.48e+11 0.264 0.792 -9.29e+11 1.22e+12
x2 6.87e+10 1.61e+11 0.427 0.669 -2.47e+11 3.84e+11
x3 -1.043e+11 2.43e+11 -0.429 0.668 -5.81e+11 3.72e+11
x4 -5.361e+10 1.15e+11 -0.465 0.642 -2.79e+11 1.72e+11
x5 -2.579e+11 7.08e+11 -0.364 0.716 -1.65e+12 1.13e+12
x6 -6.82e+09 7.75e+10 -0.088 0.930 -1.59e+11 1.45e+11
x7 -4.744e+10 9.72e+10 -0.488 0.626 -2.38e+11 1.43e+11
x8 -4.593e+10 1.07e+11 -0.429 0.668 -2.56e+11 1.64e+11
x9 -4.458e+10 1.46e+11 -0.304 0.761 -3.32e+11 2.43e+11
x10 -3.415e+10 1.14e+11 -0.300 0.764 -2.57e+11 1.89e+11
x11 4.47e+10 1.26e+11 0.355 0.722 -2.02e+11 2.91e+11
x12 -2.099e+10 4.53e+10 -0.464 0.643 -1.1e+11 6.78e+10
x13 1.556e+10 7.7e+10 0.202 0.840 -1.35e+11 1.67e+11
x14 4.231e+09 1.25e+10 0.339 0.735 -2.02e+10 2.87e+10
x15 5.645e+09 7.4e+10 0.076 0.939 -1.39e+11 1.51e+11
x16 -3.214e+10 1.16e+11 -0.278 0.781 -2.59e+11 1.95e+11
x17 -8.489e+10 2.58e+11 -0.329 0.742 -5.91e+11 4.21e+11
x18 -1.221e+11 3.49e+11 -0.349 0.727 -8.07e+11 5.63e+11
x19 -1.455e+11 3.64e+11 -0.400 0.689 -8.58e+11 5.67e+11
x20 4.839e+09 1.28e+10 0.378 0.705 -2.02e+10 2.99e+10
x21 -7.887e+10 2.75e+11 -0.286 0.775 -6.19e+11 4.61e+11
x22 -4.097e+10 8.45e+10 -0.485 0.628 -2.07e+11 1.25e+11
x23 -2.586e+10 5.41e+10 -0.478 0.633 -1.32e+11 8.03e+10
x24 -2.997e+11 7.5e+11 -0.400 0.690 -1.77e+12 1.17e+12
x25 -9.989e+10 2.42e+11 -0.413 0.680 -5.74e+11 3.74e+11
x26 -6.211e+10 1.68e+11 -0.371 0.711 -3.91e+11 2.66e+11
x27 -6.271e+10 1.69e+11 -0.371 0.711 -3.94e+11 2.69e+11
x28 -9.752e+10 2.82e+11 -0.346 0.729 -6.5e+11 4.55e+11
x29 -8.215e+11 1.81e+12 -0.454 0.650 -4.37e+12 2.73e+12
x30 -2.908e+09 8.32e+09 -0.349 0.727 -1.92e+10 1.34e+10
x31 2.165e+09 1.78e+10 0.122 0.903 -3.27e+10 3.7e+10
x32 4.607e+09 9.96e+09 0.462 0.644 -1.49e+10 2.41e+10
x33 7.905e+10 1.85e+11 0.428 0.668 -2.83e+11 4.41e+11
x34 1.523e+10 3.17e+10 0.480 0.631 -4.7e+10 7.75e+10
x35 2.778e+10 5.71e+10 0.487 0.626 -8.41e+10 1.4e+11
x36 2.993e+10 6.28e+10 0.476 0.634 -9.32e+10 1.53e+11
x37 2.468e+10 5.68e+10 0.435 0.664 -8.67e+10 1.36e+11
x38 3.093e+10 6.68e+10 0.463 0.643 -1e+11 1.62e+11
x39 1.937e+10 4.37e+10 0.444 0.657 -6.62e+10 1.05e+11
x40 1.827e+10 3.81e+10 0.479 0.632 -5.65e+10 9.3e+10
x41 1.704e+10 3.9e+10 0.436 0.663 -5.95e+10 9.36e+10
x42 7.84e+09 1.69e+10 0.464 0.643 -2.53e+10 4.09e+10
x43 4.305e+09 1.66e+10 0.259 0.796 -2.83e+10 3.69e+10
x44 2.65e+10 5.72e+10 0.463 0.643 -8.57e+10 1.39e+11
x45 3.912e+10 8.75e+10 0.447 0.655 -1.32e+11 2.11e+11
x46 4.857e+10 1.1e+11 0.443 0.658 -1.66e+11 2.63e+11
x47 2.62e+10 6.51e+10 0.403 0.687 -1.01e+11 1.54e+11
x48 9.721e+08 2.1e+09 0.462 0.644 -3.15e+09 5.1e+09
x49 1.826e+10 5.54e+10 0.330 0.742 -9.04e+10 1.27e+11
x50 4.583e+09 9.51e+09 0.482 0.630 -1.41e+10 2.32e+10
x51 -1.129e+09 2.36e+09 -0.478 0.632 -5.76e+09 3.5e+09
x52 7.207e+10 1.67e+11 0.430 0.667 -2.56e+11 4e+11
x53 2.242e+10 4.76e+10 0.471 0.638 -7.09e+10 1.16e+11
x54 1.501e+10 3.92e+10 0.383 0.702 -6.19e+10 9.19e+10
x55 1.79e+10 4.33e+10 0.414 0.679 -6.69e+10 1.03e+11
x56 2.533e+10 5.68e+10 0.446 0.656 -8.6e+10 1.37e+11
x57 -2.418e+09 8.86e+09 -0.273 0.785 -1.98e+10 1.49e+10
x58 1.435e+08 1.36e+09 0.106 0.916 -2.52e+09 2.81e+09
x59 3.003e+09 6.22e+09 0.483 0.629 -9.19e+09 1.52e+10
x60 -3.54e+09 7.34e+09 -0.482 0.630 -1.79e+10 1.08e+10
x61 -2.521e+09 5.71e+09 -0.441 0.659 -1.37e+10 8.68e+09
x62 -2.046e+09 4.19e+09 -0.488 0.626 -1.03e+10 6.17e+09
x63 -6.599e+09 1.36e+10 -0.486 0.627 -3.32e+10 2e+10
x64 -3.24e+09 6.83e+09 -0.475 0.635 -1.66e+10 1.01e+10
x65 6.557e+08 1.77e+09 0.371 0.710 -2.81e+09 4.12e+09
x66 9.565e+08 3.33e+09 0.288 0.774 -5.56e+09 7.47e+09
x67 -3.862e+09 8.15e+09 -0.474 0.635 -1.98e+10 1.21e+10
x68 -3.702e+09 7.93e+09 -0.467 0.640 -1.92e+10 1.18e+10
x69 2.64e+08 1.97e+09 0.134 0.893 -3.59e+09 4.12e+09
x70 1.689e+09 3.51e+09 0.481 0.630 -5.19e+09 8.57e+09
x71 -5.038e+07 6.22e+08 -0.081 0.935 -1.27e+09 1.17e+09
x72 -3.038e+09 6.23e+09 -0.488 0.626 -1.52e+10 9.16e+09
x73 -7.353e+09 1.6e+10 -0.459 0.646 -3.88e+10 2.41e+10
x74 -2.754e+09 5.89e+09 -0.468 0.640 -1.43e+10 8.79e+09
x75 -1.639e+09 3.36e+09 -0.487 0.626 -8.23e+09 4.95e+09
x76 -3.868e+09 7.99e+09 -0.484 0.628 -1.95e+10 1.18e+10
x77 3.093e+08 2.28e+09 0.136 0.892 -4.16e+09 4.78e+09
x78 1.721e+09 3.64e+09 0.473 0.636 -5.41e+09 8.85e+09
x79 7.878e+07 9.49e+08 0.083 0.934 -1.78e+09 1.94e+09
x80 -8.864e+08 1.9e+09 -0.466 0.641 -4.61e+09 2.84e+09
x81 1.97e+08 1.47e+09 0.134 0.894 -2.69e+09 3.09e+09
x82 -2.988e+09 6.31e+09 -0.473 0.636 -1.54e+10 9.39e+09
x83 -1.874e+09 4.14e+09 -0.453 0.651 -9.98e+09 6.23e+09
x84 1.679e+10 3.85e+10 0.436 0.663 -5.86e+10 9.22e+10
x85 -6.105e+08 1.92e+09 -0.318 0.751 -4.38e+09 3.16e+09
x86 -1.459e+08 1.37e+09 -0.107 0.915 -2.83e+09 2.54e+09
x87 4.811e+09 1.02e+10 0.473 0.636 -1.51e+10 2.47e+10
x88 4.978e+09 1.03e+10 0.485 0.627 -1.51e+10 2.51e+10
x89 8.063e+08 2.28e+09 0.353 0.724 -3.67e+09 5.28e+09
x90 4.875e+09 1.03e+10 0.474 0.636 -1.53e+10 2.5e+10
x91 1.278e+09 2.76e+09 0.462 0.644 -4.14e+09 6.7e+09
x92 5.508e+09 1.18e+10 0.466 0.641 -1.77e+10 2.87e+10
x93 7.201e+08 1.7e+09 0.423 0.673 -2.62e+09 4.06e+09
x94 5.282e+08 1.36e+09 0.388 0.698 -2.14e+09 3.2e+09
x95 -1.092e+09 2.37e+09 -0.460 0.645 -5.74e+09 3.56e+09
x96 3.537e+09 7.57e+09 0.467 0.640 -1.13e+10 1.84e+10
x97 3.372e+09 7.08e+09 0.477 0.634 -1.05e+10 1.72e+10
x98 4.529e+09 1.06e+10 0.426 0.670 -1.63e+10 2.54e+10
x99 4.751e+09 1.11e+10 0.429 0.668 -1.7e+10 2.65e+10
x100 -4.742e+09 9.72e+09 -0.488 0.626 -2.38e+10 1.43e+10
x101 -1.288e+09 3.09e+09 -0.417 0.677 -7.34e+09 4.77e+09
x102 1.596e+09 3.3e+09 0.484 0.629 -4.87e+09 8.06e+09
x103 3.88e+09 7.97e+09 0.487 0.626 -1.17e+10 1.95e+10
x104 1.821e+09 3.77e+09 0.483 0.629 -5.56e+09 9.21e+09
x105 -1.014e+09 2.44e+09 -0.416 0.677 -5.8e+09 3.77e+09
x106 -3.22e+08 8.53e+08 -0.377 0.706 -1.99e+09 1.35e+09
x107 -2.801e+08 1.88e+09 -0.149 0.882 -3.97e+09 3.41e+09
x108 -1.843e+09 4.22e+09 -0.437 0.662 -1.01e+10 6.43e+09
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Omnibus: 1252.566 Durbin-Watson: 1.597
Prob(Omnibus): 0.000 Jarque-Bera (JB): 2616.853
Skew: 0.868 Prob(JB): 0.00
Kurtosis: 5.038 Cond. No. 6.26e+16


Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The smallest eigenvalue is 6.37e-29. This might indicate that there are
strong multicollinearity problems or that the design matrix is singular.
In [34]:
transformed_target, lam = yeojohnson(y_train)
fig,ax = plt.subplots(1,2,figsize=(13, 5))
sns.distplot(y_train, label= 'Original Skew :{0}'.format(np.round(skew(y_train),4)), color='r', ax=ax[0], axlabel='ORGINAL')
sns.distplot(transformed_target, label= 'Transformed Skew:{0}'.format(np.round(skew(transformed_target),4)), color='g', ax=ax[1], axlabel='BOX-COX TRANSFORMED')
ax[0].set(title='Distribution of Target Variable')
ax[1].set(title='After Transformation')
fig.legend()
plt.show()
In [35]:
corrMatt = hourly_rentals[['temp','wdsp','rhum', 'rain','count']].corr()
mask = np.array(corrMatt)
mask[np.tril_indices_from(mask)] = False
cmap = sns.diverging_palette(180, 20, as_cmap=True)

fig, ax = plt.subplots(figsize=(6, 6))
sns.heatmap(corrMatt, mask=mask,vmax=.3, annot=True, ax=ax, cmap=cmap)
plt.show()

GitHub Mark GitHub repository
Author: Leandro Pessini