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Supervised Learning : Implementing Gradient Descent Algorithm in Estimating Regression Coefficients

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John Pauline Pineda

February 5, 2024

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• 1. Table of Contents
  • 1.1 Data Background
  • 1.2 Data Description
  • 1.3 Data Quality Assessment
  • 1.4 Data Preprocessing
    • 1.4.1 Data Cleaning
    • 1.4.2 Missing Data Imputation
    • 1.4.3 Outlier Treatment
    • 1.4.4 Collinearity
    • 1.4.5 Shape Transformation
    • 1.4.6 Centering and Scaling
    • 1.4.7 Data Encoding
    • 1.4.8 Preprocessed Data Description
  • 1.5 Data Exploration
    • 1.5.1 Exploratory Data Analysis
    • 1.5.2 Hypothesis Testing
  • 1.6 Linear Regression Model Coefficient Estimation
    • 1.6.1 Premodelling Data Description
    • 1.6.2 Normal Equations
    • 1.6.3 Gradient Descent Algorithm with Very High Learning Rate and Low Epoch Count
    • 1.6.4 Gradient Descent Algorithm with Very High Learning Rate and High Epoch Count
    • 1.6.5 Gradient Descent Algorithm with High Learning Rate and Low Epoch Count
    • 1.6.6 Gradient Descent Algorithm with High Learning Rate and High Epoch Count
    • 1.6.7 Gradient Descent Algorithm with Low Learning Rate and Low Epoch Count
    • 1.6.8 Gradient Descent Algorithm with Low Learning Rate and High Epoch Count
  • 1.7 Consolidated Findings
• 2. Summary
• 3. References

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1. Table of Contents

This project manually implements the Gradient Descent algorithm using various helpful packages in Python, and evaluates a range of values for the learning rate and epoch count parameters to optimally estimate the coefficients of a linear regression model. The cost function optimization profiles of the different candidate parameter settings were compared, with the resulting estimated coefficients assessed against those obtained using normal equations which served as the reference baseline values. All results were consolidated in a Summary presented at the end of the document.

Linear Regression explores the linear relationship between a scalar response and one or more covariates by having the conditional mean of the dependent variable be an affine function of the independent variables. The relationship is modeled through a disturbance term which represents an unobserved random variable that adds noise. The algorithm is typically formulated from the data using the least squares method which seeks to estimate the coefficients by minimizing the squared residual function. The linear equation assigns one scale factor represented by a coefficient to each covariate and an additional coefficient called the intercept or the bias coefficient which gives the line an additional degree of freedom allowing to move up and down a two-dimensional plot.

Regression Coefficients represent the changes in the independent variable which explain the variation of the dependent variable in the model. The methods applied in this study attempt to estimate the unknown model coefficients by optimizing a loss function - that which measures the quality of the estimated parameters based on how well the model-induced scores agree with the ground truth labels in the data set.

Normal Equations are a system of equations whose solution is the Ordinary Least Squares (OLS) estimator of the regression coefficients and which are derived from the first-order condition of the least squares minimization problem. These equations are obtained by setting equal to zero the partial derivatives of the sum of squared errors (least squares). This approach is a closed-form solution and a one-step algorithm used to analytically find the coefficients that minimize the loss function.

Gradient Descent minimizes the loss function parameterized by the model’s coefficients based on the direction and learning rate factors which determine the partial derivative calculations of future iterations, allowing the algorithm to gradually arrive at the local or global minimum considered the point of convergence. This particular implementation used Batch Gradient Descent which computes the gradient of the loss function with respect to the parameters for the entire data set.

1.1. Data Background

Datasets used for the analysis were separately gathered and consolidated from various sources including:

1. Cancer Rates from World Population Review
2. Social Protection and Labor Indicator from World Bank
3. Education Indicator from World Bank
4. Economy and Growth Indicator from World Bank
5. Environment Indicator from World Bank
6. Climate Change Indicator from World Bank
7. Agricultural and Rural Development Indicator from World Bank
8. Social Development Indicator from World Bank
9. Health Indicator from World Bank
10. Science and Technology Indicator from World Bank
11. Urban Development Indicator from World Bank
12. Human Development Indices from Human Development Reports
13. Environmental Performance Indices from Yale Center for Environmental Law and Policy

This study hypothesized that various global development indicators and indices influence cancer rates across countries.

The target variable for the study is:

• CANRAT - Age-standardized cancer rates, per 100K population (2022)

The predictor variables for the study are:

• GDPPER - GDP per person employed, current US Dollars (2020)
• URBPOP - Urban population, % of total population (2020)
• PATRES - Patent applications by residents, total count (2020)
• RNDGDP - Research and development expenditure, % of GDP (2020)
• POPGRO - Population growth, annual % (2020)
• LIFEXP - Life expectancy at birth, total in years (2020)
• TUBINC - Incidence of tuberculosis, per 100K population (2020)
• DTHCMD - Cause of death by communicable diseases and maternal, prenatal and nutrition conditions, % of total (2019)
• AGRLND - Agricultural land, % of land area (2020)
• GHGEMI - Total greenhouse gas emissions, kt of CO2 equivalent (2020)
• RELOUT - Renewable electricity output, % of total electricity output (2015)
• METEMI - Methane emissions, kt of CO2 equivalent (2020)
• FORARE - Forest area, % of land area (2020)
• CO2EMI - CO2 emissions, metric tons per capita (2020)
• PM2EXP - PM2.5 air pollution, population exposed to levels exceeding WHO guideline value, % of total (2017)
• POPDEN - Population density, people per sq. km of land area (2020)
• GDPCAP - GDP per capita, current US Dollars (2020)
• ENRTER - Tertiary school enrollment, % gross (2020)
• HDICAT - Human development index, ordered category (2020)
• EPISCO - Environment performance index , score (2022)

1.2. Data Description

1. The dataset is comprised of:
   • 177 rows (observations)
   • 22 columns (variables)
     • 1/22 metadata (object)
       • COUNTRY
     • 1/22 target (numeric)
       • CANRAT
     • 19/22 predictor (numeric)
       • GDPPER
       • URBPOP
       • PATRES
       • RNDGDP
       • POPGRO
       • LIFEXP
       • TUBINC
       • DTHCMD
       • AGRLND
       • GHGEMI
       • RELOUT
       • METEMI
       • FORARE
       • CO2EMI
       • PM2EXP
       • POPDEN
       • GDPCAP
       • ENRTER
       • EPISCO
     • 1/22 predictor (categorical)
       • HDICAT

##################################
# Loading Python Libraries
##################################
import numpy as np
import pandas as pd
import seaborn as sns
import matplotlib.pyplot as plt
import itertools
import os
%matplotlib inline

from operator import add,mul,truediv
from sklearn.experimental import enable_iterative_imputer
from sklearn.impute import IterativeImputer
from sklearn.linear_model import LinearRegression
from sklearn.preprocessing import PowerTransformer, StandardScaler
from scipy import stats

from sklearn.linear_model import RidgeCV, LassoCV, ElasticNetCV
from sklearn.metrics import r2_score,mean_squared_error,mean_absolute_error
from sklearn.model_selection import train_test_split, LeaveOneOut
from sklearn.preprocessing import PolynomialFeatures 
from sklearn.pipeline import Pipeline

##################################
# Defining file paths
##################################
DATASETS_ORIGINAL_PATH = r"datasets\original"

##################################
# Loading the dataset
# from the DATASETS_ORIGINAL_PATH
##################################
cancer_rate = pd.read_csv(os.path.join("..", DATASETS_ORIGINAL_PATH, "NumericCancerRates.csv"))

##################################
# Performing a general exploration of the dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate.shape)

Dataset Dimensions: 

(177, 22)

##################################
# Listing the column names and data types
##################################
print('Column Names and Data Types:')
display(cancer_rate.dtypes)

Column Names and Data Types:

COUNTRY     object
CANRAT     float64
GDPPER     float64
URBPOP     float64
PATRES     float64
RNDGDP     float64
POPGRO     float64
LIFEXP     float64
TUBINC     float64
DTHCMD     float64
AGRLND     float64
GHGEMI     float64
RELOUT     float64
METEMI     float64
FORARE     float64
CO2EMI     float64
PM2EXP     float64
POPDEN     float64
ENRTER     float64
GDPCAP     float64
HDICAT      object
EPISCO     float64
dtype: object

##################################
# Taking a snapshot of the dataset
##################################
cancer_rate.head()

	COUNTRY	CANRAT	GDPPER	URBPOP	PATRES	RNDGDP	POPGRO	LIFEXP	TUBINC	DTHCMD	...	RELOUT	METEMI	FORARE	CO2EMI	PM2EXP	POPDEN	ENRTER	GDPCAP	HDICAT	EPISCO
0	Australia	452.4	98380.63601	86.241	2368.0	NaN	1.235701	83.200000	7.2	4.941054	...	13.637841	131484.763200	17.421315	14.772658	24.893584	3.335312	110.139221	51722.06900	VH	60.1
1	New Zealand	422.9	77541.76438	86.699	348.0	NaN	2.204789	82.256098	7.2	4.354730	...	80.081439	32241.937000	37.570126	6.160799	NaN	19.331586	75.734833	41760.59478	VH	56.7
2	Ireland	372.8	198405.87500	63.653	75.0	1.23244	1.029111	82.556098	5.3	5.684596	...	27.965408	15252.824630	11.351720	6.768228	0.274092	72.367281	74.680313	85420.19086	VH	57.4
3	United States	362.2	130941.63690	82.664	269586.0	3.42287	0.964348	76.980488	2.3	5.302060	...	13.228593	748241.402900	33.866926	13.032828	3.343170	36.240985	87.567657	63528.63430	VH	51.1
4	Denmark	351.1	113300.60110	88.116	1261.0	2.96873	0.291641	81.602439	4.1	6.826140	...	65.505925	7778.773921	15.711000	4.691237	56.914456	145.785100	82.664330	60915.42440	VH	77.9

5 rows × 22 columns

##################################
# Setting the levels of the categorical variables
##################################
cancer_rate['HDICAT'] = cancer_rate['HDICAT'].astype('category')
cancer_rate['HDICAT'] = cancer_rate['HDICAT'].cat.set_categories(['L', 'M', 'H', 'VH'], ordered=True)

##################################
# Performing a general exploration of the numeric variables
##################################
print('Numeric Variable Summary:')
display(cancer_rate.describe(include='number').transpose())

Numeric Variable Summary:

	count	mean	std	min	25%	50%	75%	max
CANRAT	177.0	183.829379	7.974340e+01	78.400000	118.100000	155.300000	240.400000	4.524000e+02
GDPPER	165.0	45284.424283	3.941794e+04	1718.804896	13545.254510	34024.900890	66778.416050	2.346469e+05
URBPOP	174.0	59.788121	2.280640e+01	13.345000	42.432750	61.701500	79.186500	1.000000e+02
PATRES	108.0	20607.388889	1.340683e+05	1.000000	35.250000	244.500000	1297.750000	1.344817e+06
RNDGDP	74.0	1.197474	1.189956e+00	0.039770	0.256372	0.873660	1.608842	5.354510e+00
POPGRO	174.0	1.127028	1.197718e+00	-2.079337	0.236900	1.179959	2.031154	3.727101e+00
LIFEXP	174.0	71.746113	7.606209e+00	52.777000	65.907500	72.464610	77.523500	8.456000e+01
TUBINC	174.0	105.005862	1.367229e+02	0.770000	12.000000	44.500000	147.750000	5.920000e+02
DTHCMD	170.0	21.260521	1.927333e+01	1.283611	6.078009	12.456279	36.980457	6.520789e+01
AGRLND	174.0	38.793456	2.171551e+01	0.512821	20.130276	40.386649	54.013754	8.084112e+01
GHGEMI	170.0	259582.709895	1.118550e+06	179.725150	12527.487367	41009.275980	116482.578575	1.294287e+07
RELOUT	153.0	39.760036	3.191492e+01	0.000296	10.582691	32.381668	63.011450	1.000000e+02
METEMI	170.0	47876.133575	1.346611e+05	11.596147	3662.884908	11118.976025	32368.909040	1.186285e+06
FORARE	173.0	32.218177	2.312001e+01	0.008078	11.604388	31.509048	49.071780	9.741212e+01
CO2EMI	170.0	3.751097	4.606479e+00	0.032585	0.631924	2.298368	4.823496	3.172684e+01
PM2EXP	167.0	91.940595	2.206003e+01	0.274092	99.627134	100.000000	100.000000	1.000000e+02
POPDEN	174.0	200.886765	6.453834e+02	2.115134	27.454539	77.983133	153.993650	7.918951e+03
ENRTER	116.0	49.994997	2.970619e+01	2.432581	22.107195	53.392460	71.057467	1.433107e+02
GDPCAP	170.0	13992.095610	1.957954e+04	216.827417	1870.503029	5348.192875	17421.116227	1.173705e+05
EPISCO	165.0	42.946667	1.249086e+01	18.900000	33.000000	40.900000	50.500000	7.790000e+01

##################################
# Performing a general exploration of the object variable
##################################
print('Object Variable Summary:')
display(cancer_rate.describe(include='object').transpose())

Object Variable Summary:

	count	unique	top	freq
COUNTRY	177	177	Australia	1

##################################
# Performing a general exploration of the categorical variable
##################################
print('Categorical Variable Summary:')
display(cancer_rate.describe(include='category').transpose())

Categorical Variable Summary:

	count	unique	top	freq
HDICAT	167	4	VH	59

1.3. Data Quality Assessment

Data quality findings based on assessment are as follows:

1. No duplicated rows observed.
2. Missing data noted for 20 variables with Null.Count>0 and Fill.Rate<1.0.
   • RNDGDP: Null.Count = 103, Fill.Rate = 0.418
   • PATRES: Null.Count = 69, Fill.Rate = 0.610
   • ENRTER: Null.Count = 61, Fill.Rate = 0.655
   • RELOUT: Null.Count = 24, Fill.Rate = 0.864
   • GDPPER: Null.Count = 12, Fill.Rate = 0.932
   • EPISCO: Null.Count = 12, Fill.Rate = 0.932
   • HDICAT: Null.Count = 10, Fill.Rate = 0.943
   • PM2EXP: Null.Count = 10, Fill.Rate = 0.943
   • DTHCMD: Null.Count = 7, Fill.Rate = 0.960
   • METEMI: Null.Count = 7, Fill.Rate = 0.960
   • CO2EMI: Null.Count = 7, Fill.Rate = 0.960
   • GDPCAP: Null.Count = 7, Fill.Rate = 0.960
   • GHGEMI: Null.Count = 7, Fill.Rate = 0.960
   • FORARE: Null.Count = 4, Fill.Rate = 0.977
   • TUBINC: Null.Count = 3, Fill.Rate = 0.983
   • AGRLND: Null.Count = 3, Fill.Rate = 0.983
   • POPGRO: Null.Count = 3, Fill.Rate = 0.983
   • POPDEN: Null.Count = 3, Fill.Rate = 0.983
   • URBPOP: Null.Count = 3, Fill.Rate = 0.983
   • LIFEXP: Null.Count = 3, Fill.Rate = 0.983
3. 120 observations noted with at least 1 missing data. From this number, 14 observations reported high Missing.Rate>0.2.
   • COUNTRY=Guadeloupe: Missing.Rate= 0.909
   • COUNTRY=Martinique: Missing.Rate= 0.909
   • COUNTRY=French Guiana: Missing.Rate= 0.909
   • COUNTRY=New Caledonia: Missing.Rate= 0.500
   • COUNTRY=French Polynesia: Missing.Rate= 0.500
   • COUNTRY=Guam: Missing.Rate= 0.500
   • COUNTRY=Puerto Rico: Missing.Rate= 0.409
   • COUNTRY=North Korea: Missing.Rate= 0.227
   • COUNTRY=Somalia: Missing.Rate= 0.227
   • COUNTRY=South Sudan: Missing.Rate= 0.227
   • COUNTRY=Venezuela: Missing.Rate= 0.227
   • COUNTRY=Libya: Missing.Rate= 0.227
   • COUNTRY=Eritrea: Missing.Rate= 0.227
   • COUNTRY=Yemen: Missing.Rate= 0.227
4. Low variance observed for 1 variable with First.Second.Mode.Ratio>5.
   • PM2EXP: First.Second.Mode.Ratio = 53.000
5. No low variance observed for any variable with Unique.Count.Ratio>10.
6. High skewness observed for 5 variables with Skewness>3 or Skewness<(-3).
   • POPDEN: Skewness = +10.267
   • GHGEMI: Skewness = +9.496
   • PATRES: Skewness = +9.284
   • METEMI: Skewness = +5.801
   • PM2EXP: Skewness = -3.141

##################################
# Counting the number of duplicated rows
##################################
cancer_rate.duplicated().sum()

np.int64(0)

##################################
# Gathering the data types for each column
##################################
data_type_list = list(cancer_rate.dtypes)

##################################
# Gathering the variable names for each column
##################################
variable_name_list = list(cancer_rate.columns)

##################################
# Gathering the number of observations for each column
##################################
row_count_list = list([len(cancer_rate)] * len(cancer_rate.columns))

##################################
# Gathering the number of missing data for each column
##################################
null_count_list = list(cancer_rate.isna().sum(axis=0))

##################################
# Gathering the number of non-missing data for each column
##################################
non_null_count_list = list(cancer_rate.count())

##################################
# Gathering the missing data percentage for each column
##################################
fill_rate_list = map(truediv, non_null_count_list, row_count_list)

##################################
# Formulating the summary
# for all columns
##################################
all_column_quality_summary = pd.DataFrame(zip(variable_name_list,
                                              data_type_list,
                                              row_count_list,
                                              non_null_count_list,
                                              null_count_list,
                                              fill_rate_list), 
                                        columns=['Column.Name',
                                                 'Column.Type',
                                                 'Row.Count',
                                                 'Non.Null.Count',
                                                 'Null.Count',                                                 
                                                 'Fill.Rate'])
display(all_column_quality_summary)

	Column.Name	Column.Type	Row.Count	Non.Null.Count	Null.Count	Fill.Rate
0	COUNTRY	object	177	177	0	1.000000
1	CANRAT	float64	177	177	0	1.000000
2	GDPPER	float64	177	165	12	0.932203
3	URBPOP	float64	177	174	3	0.983051
4	PATRES	float64	177	108	69	0.610169
5	RNDGDP	float64	177	74	103	0.418079
6	POPGRO	float64	177	174	3	0.983051
7	LIFEXP	float64	177	174	3	0.983051
8	TUBINC	float64	177	174	3	0.983051
9	DTHCMD	float64	177	170	7	0.960452
10	AGRLND	float64	177	174	3	0.983051
11	GHGEMI	float64	177	170	7	0.960452
12	RELOUT	float64	177	153	24	0.864407
13	METEMI	float64	177	170	7	0.960452
14	FORARE	float64	177	173	4	0.977401
15	CO2EMI	float64	177	170	7	0.960452
16	PM2EXP	float64	177	167	10	0.943503
17	POPDEN	float64	177	174	3	0.983051
18	ENRTER	float64	177	116	61	0.655367
19	GDPCAP	float64	177	170	7	0.960452
20	HDICAT	category	177	167	10	0.943503
21	EPISCO	float64	177	165	12	0.932203

##################################
# Counting the number of columns
# with Fill.Rate < 1.00
##################################
len(all_column_quality_summary[(all_column_quality_summary['Fill.Rate']<1)])

20

##################################
# Identifying the columns
# with Fill.Rate < 1.00
##################################
display(all_column_quality_summary[(all_column_quality_summary['Fill.Rate']<1)].sort_values(by=['Fill.Rate'], ascending=True))

	Column.Name	Column.Type	Row.Count	Non.Null.Count	Null.Count	Fill.Rate
5	RNDGDP	float64	177	74	103	0.418079
4	PATRES	float64	177	108	69	0.610169
18	ENRTER	float64	177	116	61	0.655367
12	RELOUT	float64	177	153	24	0.864407
21	EPISCO	float64	177	165	12	0.932203
2	GDPPER	float64	177	165	12	0.932203
16	PM2EXP	float64	177	167	10	0.943503
20	HDICAT	category	177	167	10	0.943503
15	CO2EMI	float64	177	170	7	0.960452
13	METEMI	float64	177	170	7	0.960452
11	GHGEMI	float64	177	170	7	0.960452
9	DTHCMD	float64	177	170	7	0.960452
19	GDPCAP	float64	177	170	7	0.960452
14	FORARE	float64	177	173	4	0.977401
6	POPGRO	float64	177	174	3	0.983051
3	URBPOP	float64	177	174	3	0.983051
17	POPDEN	float64	177	174	3	0.983051
10	AGRLND	float64	177	174	3	0.983051
7	LIFEXP	float64	177	174	3	0.983051
8	TUBINC	float64	177	174	3	0.983051

##################################
# Identifying the rows
# with Fill.Rate < 0.90
##################################
column_low_fill_rate = all_column_quality_summary[(all_column_quality_summary['Fill.Rate']<0.90)]

##################################
# Gathering the metadata labels for each observation
##################################
row_metadata_list = cancer_rate["COUNTRY"].values.tolist()

##################################
# Gathering the number of columns for each observation
##################################
column_count_list = list([len(cancer_rate.columns)] * len(cancer_rate))

##################################
# Gathering the number of missing data for each row
##################################
null_row_list = list(cancer_rate.isna().sum(axis=1))

##################################
# Gathering the missing data percentage for each column
##################################
missing_rate_list = map(truediv, null_row_list, column_count_list)

##################################
# Identifying the rows
# with missing data
##################################
all_row_quality_summary = pd.DataFrame(zip(row_metadata_list,
                                           column_count_list,
                                           null_row_list,
                                           missing_rate_list), 
                                        columns=['Row.Name',
                                                 'Column.Count',
                                                 'Null.Count',                                                 
                                                 'Missing.Rate'])
display(all_row_quality_summary)

	Row.Name	Column.Count	Null.Count	Missing.Rate
0	Australia	22	1	0.045455
1	New Zealand	22	2	0.090909
2	Ireland	22	0	0.000000
3	United States	22	0	0.000000
4	Denmark	22	0	0.000000
...	...	...	...	...
172	Congo Republic	22	3	0.136364
173	Bhutan	22	2	0.090909
174	Nepal	22	2	0.090909
175	Gambia	22	4	0.181818
176	Niger	22	2	0.090909

177 rows × 4 columns

##################################
# Counting the number of rows
# with Missing.Rate > 0.00
##################################
len(all_row_quality_summary[(all_row_quality_summary['Missing.Rate']>0.00)])

120

##################################
# Counting the number of rows
# with Missing.Rate > 0.20
##################################
len(all_row_quality_summary[(all_row_quality_summary['Missing.Rate']>0.20)])

14

##################################
# Identifying the rows
# with Missing.Rate > 0.20
##################################
row_high_missing_rate = all_row_quality_summary[(all_row_quality_summary['Missing.Rate']>0.20)]

##################################
# Identifying the rows
# with Missing.Rate > 0.20
##################################
display(all_row_quality_summary[(all_row_quality_summary['Missing.Rate']>0.20)].sort_values(by=['Missing.Rate'], ascending=False))

	Row.Name	Column.Count	Null.Count	Missing.Rate
35	Guadeloupe	22	20	0.909091
39	Martinique	22	20	0.909091
56	French Guiana	22	20	0.909091
13	New Caledonia	22	11	0.500000
44	French Polynesia	22	11	0.500000
75	Guam	22	11	0.500000
53	Puerto Rico	22	9	0.409091
85	North Korea	22	6	0.272727
168	South Sudan	22	6	0.272727
132	Somalia	22	6	0.272727
117	Libya	22	5	0.227273
73	Venezuela	22	5	0.227273
161	Eritrea	22	5	0.227273
164	Yemen	22	5	0.227273

##################################
# Formulating the dataset
# with numeric columns only
##################################
cancer_rate_numeric = cancer_rate.select_dtypes(include='number')

##################################
# Gathering the variable names for each numeric column
##################################
numeric_variable_name_list = cancer_rate_numeric.columns

##################################
# Gathering the minimum value for each numeric column
##################################
numeric_minimum_list = cancer_rate_numeric.min()

##################################
# Gathering the mean value for each numeric column
##################################
numeric_mean_list = cancer_rate_numeric.mean()

##################################
# Gathering the median value for each numeric column
##################################
numeric_median_list = cancer_rate_numeric.median()

##################################
# Gathering the maximum value for each numeric column
##################################
numeric_maximum_list = cancer_rate_numeric.max()

##################################
# Gathering the first mode values for each numeric column
##################################
numeric_first_mode_list = [cancer_rate[x].value_counts(dropna=True).index.tolist()[0] for x in cancer_rate_numeric]

##################################
# Gathering the second mode values for each numeric column
##################################
numeric_second_mode_list = [cancer_rate[x].value_counts(dropna=True).index.tolist()[1] for x in cancer_rate_numeric]

##################################
# Gathering the count of first mode values for each numeric column
##################################
numeric_first_mode_count_list = [cancer_rate_numeric[x].isin([cancer_rate[x].value_counts(dropna=True).index.tolist()[0]]).sum() for x in cancer_rate_numeric]

##################################
# Gathering the count of second mode values for each numeric column
##################################
numeric_second_mode_count_list = [cancer_rate_numeric[x].isin([cancer_rate[x].value_counts(dropna=True).index.tolist()[1]]).sum() for x in cancer_rate_numeric]

##################################
# Gathering the first mode to second mode ratio for each numeric column
##################################
numeric_first_second_mode_ratio_list = map(truediv, numeric_first_mode_count_list, numeric_second_mode_count_list)

##################################
# Gathering the count of unique values for each numeric column
##################################
numeric_unique_count_list = cancer_rate_numeric.nunique(dropna=True)

##################################
# Gathering the number of observations for each numeric column
##################################
numeric_row_count_list = list([len(cancer_rate_numeric)] * len(cancer_rate_numeric.columns))

##################################
# Gathering the unique to count ratio for each numeric column
##################################
numeric_unique_count_ratio_list = map(truediv, numeric_unique_count_list, numeric_row_count_list)

##################################
# Gathering the skewness value for each numeric column
##################################
numeric_skewness_list = cancer_rate_numeric.skew()

##################################
# Gathering the kurtosis value for each numeric column
##################################
numeric_kurtosis_list = cancer_rate_numeric.kurtosis()

numeric_column_quality_summary = pd.DataFrame(zip(numeric_variable_name_list,
                                                numeric_minimum_list,
                                                numeric_mean_list,
                                                numeric_median_list,
                                                numeric_maximum_list,
                                                numeric_first_mode_list,
                                                numeric_second_mode_list,
                                                numeric_first_mode_count_list,
                                                numeric_second_mode_count_list,
                                                numeric_first_second_mode_ratio_list,
                                                numeric_unique_count_list,
                                                numeric_row_count_list,
                                                numeric_unique_count_ratio_list,
                                                numeric_skewness_list,
                                                numeric_kurtosis_list), 
                                        columns=['Numeric.Column.Name',
                                                 'Minimum',
                                                 'Mean',
                                                 'Median',
                                                 'Maximum',
                                                 'First.Mode',
                                                 'Second.Mode',
                                                 'First.Mode.Count',
                                                 'Second.Mode.Count',
                                                 'First.Second.Mode.Ratio',
                                                 'Unique.Count',
                                                 'Row.Count',
                                                 'Unique.Count.Ratio',
                                                 'Skewness',
                                                 'Kurtosis'])
display(numeric_column_quality_summary)

	Numeric.Column.Name	Minimum	Mean	Median	Maximum	First.Mode	Second.Mode	First.Mode.Count	Second.Mode.Count	First.Second.Mode.Ratio	Unique.Count	Row.Count	Unique.Count.Ratio	Skewness	Kurtosis
0	CANRAT	78.400000	183.829379	155.300000	4.524000e+02	135.300000	130.600000	3	2	1.500000	167	177	0.943503	0.881825	0.063467
1	GDPPER	1718.804896	45284.424283	34024.900890	2.346469e+05	98380.636010	77541.764380	1	1	1.000000	165	177	0.932203	1.517574	3.471992
2	URBPOP	13.345000	59.788121	61.701500	1.000000e+02	100.000000	86.699000	2	1	2.000000	173	177	0.977401	-0.210702	-0.962847
3	PATRES	1.000000	20607.388889	244.500000	1.344817e+06	6.000000	2.000000	4	3	1.333333	97	177	0.548023	9.284436	91.187178
4	RNDGDP	0.039770	1.197474	0.873660	5.354510e+00	1.232440	3.422870	1	1	1.000000	74	177	0.418079	1.396742	1.695957
5	POPGRO	-2.079337	1.127028	1.179959	3.727101e+00	1.235701	2.204789	1	1	1.000000	174	177	0.983051	-0.195161	-0.423580
6	LIFEXP	52.777000	71.746113	72.464610	8.456000e+01	83.200000	82.256098	1	1	1.000000	174	177	0.983051	-0.357965	-0.649601
7	TUBINC	0.770000	105.005862	44.500000	5.920000e+02	12.000000	4.100000	4	3	1.333333	131	177	0.740113	1.746333	2.429368
8	DTHCMD	1.283611	21.260521	12.456279	6.520789e+01	4.941054	4.354730	1	1	1.000000	170	177	0.960452	0.900509	-0.691541
9	AGRLND	0.512821	38.793456	40.386649	8.084112e+01	46.252480	38.562911	1	1	1.000000	174	177	0.983051	0.074000	-0.926249
10	GHGEMI	179.725150	259582.709895	41009.275980	1.294287e+07	571903.119900	80158.025830	1	1	1.000000	170	177	0.960452	9.496120	101.637308
11	RELOUT	0.000296	39.760036	32.381668	1.000000e+02	100.000000	80.081439	3	1	3.000000	151	177	0.853107	0.501088	-0.981774
12	METEMI	11.596147	47876.133575	11118.976025	1.186285e+06	131484.763200	32241.937000	1	1	1.000000	170	177	0.960452	5.801014	38.661386
13	FORARE	0.008078	32.218177	31.509048	9.741212e+01	17.421315	37.570126	1	1	1.000000	173	177	0.977401	0.519277	-0.322589
14	CO2EMI	0.032585	3.751097	2.298368	3.172684e+01	14.772658	6.160799	1	1	1.000000	170	177	0.960452	2.721552	10.311574
15	PM2EXP	0.274092	91.940595	100.000000	1.000000e+02	100.000000	100.000000	106	2	53.000000	61	177	0.344633	-3.141557	9.032386
16	POPDEN	2.115134	200.886765	77.983133	7.918951e+03	3.335312	19.331586	1	1	1.000000	174	177	0.983051	10.267750	119.995256
17	ENRTER	2.432581	49.994997	53.392460	1.433107e+02	110.139221	75.734833	1	1	1.000000	116	177	0.655367	0.275863	-0.392895
18	GDPCAP	216.827417	13992.095610	5348.192875	1.173705e+05	51722.069000	41760.594780	1	1	1.000000	170	177	0.960452	2.258568	5.938690
19	EPISCO	18.900000	42.946667	40.900000	7.790000e+01	29.600000	43.600000	3	3	1.000000	137	177	0.774011	0.641799	0.035208

##################################
# Counting the number of numeric columns
# with First.Second.Mode.Ratio > 5.00
##################################
len(numeric_column_quality_summary[(numeric_column_quality_summary['First.Second.Mode.Ratio']>5)])

1

##################################
# Identifying the numeric columns
# with First.Second.Mode.Ratio > 5.00
##################################
display(numeric_column_quality_summary[(numeric_column_quality_summary['First.Second.Mode.Ratio']>5)].sort_values(by=['First.Second.Mode.Ratio'], ascending=False))

	Numeric.Column.Name	Minimum	Mean	Median	Maximum	First.Mode	Second.Mode	First.Mode.Count	Second.Mode.Count	First.Second.Mode.Ratio	Unique.Count	Row.Count	Unique.Count.Ratio	Skewness	Kurtosis
15	PM2EXP	0.274092	91.940595	100.0	100.0	100.0	100.0	106	2	53.0	61	177	0.344633	-3.141557	9.032386

##################################
# Counting the number of numeric columns
# with Unique.Count.Ratio > 10.00
##################################
len(numeric_column_quality_summary[(numeric_column_quality_summary['Unique.Count.Ratio']>10)])

0

##################################
# Counting the number of numeric columns
# with Skewness > 3.00 or Skewness < -3.00
##################################
len(numeric_column_quality_summary[(numeric_column_quality_summary['Skewness']>3) | (numeric_column_quality_summary['Skewness']<(-3))])

5

##################################
# Identifying the numeric columns
# with Skewness > 3.00 or Skewness < -3.00
##################################
display(numeric_column_quality_summary[(numeric_column_quality_summary['Skewness']>3) | (numeric_column_quality_summary['Skewness']<(-3))].sort_values(by=['Skewness'], ascending=False))

	Numeric.Column.Name	Minimum	Mean	Median	Maximum	First.Mode	Second.Mode	First.Mode.Count	Second.Mode.Count	First.Second.Mode.Ratio	Unique.Count	Row.Count	Unique.Count.Ratio	Skewness	Kurtosis
16	POPDEN	2.115134	200.886765	77.983133	7.918951e+03	3.335312	19.331586	1	1	1.000000	174	177	0.983051	10.267750	119.995256
10	GHGEMI	179.725150	259582.709895	41009.275980	1.294287e+07	571903.119900	80158.025830	1	1	1.000000	170	177	0.960452	9.496120	101.637308
3	PATRES	1.000000	20607.388889	244.500000	1.344817e+06	6.000000	2.000000	4	3	1.333333	97	177	0.548023	9.284436	91.187178
12	METEMI	11.596147	47876.133575	11118.976025	1.186285e+06	131484.763200	32241.937000	1	1	1.000000	170	177	0.960452	5.801014	38.661386
15	PM2EXP	0.274092	91.940595	100.000000	1.000000e+02	100.000000	100.000000	106	2	53.000000	61	177	0.344633	-3.141557	9.032386

##################################
# Formulating the dataset
# with object column only
##################################
cancer_rate_object = cancer_rate.select_dtypes(include='object')

##################################
# Gathering the variable names for the object column
##################################
object_variable_name_list = cancer_rate_object.columns

##################################
# Gathering the first mode values for the object column
##################################
object_first_mode_list = [cancer_rate[x].value_counts().index.tolist()[0] for x in cancer_rate_object]

##################################
# Gathering the second mode values for each object column
##################################
object_second_mode_list = [cancer_rate[x].value_counts().index.tolist()[1] for x in cancer_rate_object]

##################################
# Gathering the count of first mode values for each object column
##################################
object_first_mode_count_list = [cancer_rate_object[x].isin([cancer_rate[x].value_counts(dropna=True).index.tolist()[0]]).sum() for x in cancer_rate_object]

##################################
# Gathering the count of second mode values for each object column
##################################
object_second_mode_count_list = [cancer_rate_object[x].isin([cancer_rate[x].value_counts(dropna=True).index.tolist()[1]]).sum() for x in cancer_rate_object]

##################################
# Gathering the first mode to second mode ratio for each object column
##################################
object_first_second_mode_ratio_list = map(truediv, object_first_mode_count_list, object_second_mode_count_list)

##################################
# Gathering the count of unique values for each object column
##################################
object_unique_count_list = cancer_rate_object.nunique(dropna=True)

##################################
# Gathering the number of observations for each object column
##################################
object_row_count_list = list([len(cancer_rate_object)] * len(cancer_rate_object.columns))

##################################
# Gathering the unique to count ratio for each object column
##################################
object_unique_count_ratio_list = map(truediv, object_unique_count_list, object_row_count_list)

object_column_quality_summary = pd.DataFrame(zip(object_variable_name_list,
                                                 object_first_mode_list,
                                                 object_second_mode_list,
                                                 object_first_mode_count_list,
                                                 object_second_mode_count_list,
                                                 object_first_second_mode_ratio_list,
                                                 object_unique_count_list,
                                                 object_row_count_list,
                                                 object_unique_count_ratio_list), 
                                        columns=['Object.Column.Name',
                                                 'First.Mode',
                                                 'Second.Mode',
                                                 'First.Mode.Count',
                                                 'Second.Mode.Count',
                                                 'First.Second.Mode.Ratio',
                                                 'Unique.Count',
                                                 'Row.Count',
                                                 'Unique.Count.Ratio'])
display(object_column_quality_summary)

	Object.Column.Name	First.Mode	Second.Mode	First.Mode.Count	Second.Mode.Count	First.Second.Mode.Ratio	Unique.Count	Row.Count	Unique.Count.Ratio
0	COUNTRY	Australia	New Zealand	1	1	1.0	177	177	1.0

##################################
# Counting the number of object columns
# with First.Second.Mode.Ratio > 5.00
##################################
len(object_column_quality_summary[(object_column_quality_summary['First.Second.Mode.Ratio']>5)])

0

##################################
# Counting the number of object columns
# with Unique.Count.Ratio > 10.00
##################################
len(object_column_quality_summary[(object_column_quality_summary['Unique.Count.Ratio']>10)])

0

##################################
# Formulating the dataset
# with categorical columns only
##################################
cancer_rate_categorical = cancer_rate.select_dtypes(include='category')

##################################
# Gathering the variable names for the categorical column
##################################
categorical_variable_name_list = cancer_rate_categorical.columns

##################################
# Gathering the first mode values for each categorical column
##################################
categorical_first_mode_list = [cancer_rate[x].value_counts().index.tolist()[0] for x in cancer_rate_categorical]

##################################
# Gathering the second mode values for each categorical column
##################################
categorical_second_mode_list = [cancer_rate[x].value_counts().index.tolist()[1] for x in cancer_rate_categorical]

##################################
# Gathering the count of first mode values for each categorical column
##################################
categorical_first_mode_count_list = [cancer_rate_categorical[x].isin([cancer_rate[x].value_counts(dropna=True).index.tolist()[0]]).sum() for x in cancer_rate_categorical]

##################################
# Gathering the count of second mode values for each categorical column
##################################
categorical_second_mode_count_list = [cancer_rate_categorical[x].isin([cancer_rate[x].value_counts(dropna=True).index.tolist()[1]]).sum() for x in cancer_rate_categorical]

##################################
# Gathering the first mode to second mode ratio for each categorical column
##################################
categorical_first_second_mode_ratio_list = map(truediv, categorical_first_mode_count_list, categorical_second_mode_count_list)

##################################
# Gathering the count of unique values for each categorical column
##################################
categorical_unique_count_list = cancer_rate_categorical.nunique(dropna=True)

##################################
# Gathering the number of observations for each categorical column
##################################
categorical_row_count_list = list([len(cancer_rate_categorical)] * len(cancer_rate_categorical.columns))

##################################
# Gathering the unique to count ratio for each categorical column
##################################
categorical_unique_count_ratio_list = map(truediv, categorical_unique_count_list, categorical_row_count_list)

categorical_column_quality_summary = pd.DataFrame(zip(categorical_variable_name_list,
                                                    categorical_first_mode_list,
                                                    categorical_second_mode_list,
                                                    categorical_first_mode_count_list,
                                                    categorical_second_mode_count_list,
                                                    categorical_first_second_mode_ratio_list,
                                                    categorical_unique_count_list,
                                                    categorical_row_count_list,
                                                    categorical_unique_count_ratio_list), 
                                        columns=['Categorical.Column.Name',
                                                 'First.Mode',
                                                 'Second.Mode',
                                                 'First.Mode.Count',
                                                 'Second.Mode.Count',
                                                 'First.Second.Mode.Ratio',
                                                 'Unique.Count',
                                                 'Row.Count',
                                                 'Unique.Count.Ratio'])
display(categorical_column_quality_summary)

	Categorical.Column.Name	First.Mode	Second.Mode	First.Mode.Count	Second.Mode.Count	First.Second.Mode.Ratio	Unique.Count	Row.Count	Unique.Count.Ratio
0	HDICAT	VH	H	59	39	1.512821	4	177	0.022599

##################################
# Counting the number of categorical columns
# with First.Second.Mode.Ratio > 5.00
##################################
len(categorical_column_quality_summary[(categorical_column_quality_summary['First.Second.Mode.Ratio']>5)])

0

##################################
# Counting the number of categorical columns
# with Unique.Count.Ratio > 10.00
##################################
len(categorical_column_quality_summary[(categorical_column_quality_summary['Unique.Count.Ratio']>10)])

0

1.4. Data Preprocessing

1.4.1 Data Cleaning

1. Subsets of rows and columns with high rates of missing data were removed from the dataset:
   • 4 variables with Fill.Rate<0.9 were excluded for subsequent analysis.
     • RNDGDP: Null.Count = 103, Fill.Rate = 0.418
     • PATRES: Null.Count = 69, Fill.Rate = 0.610
     • ENRTER: Null.Count = 61, Fill.Rate = 0.655
     • RELOUT: Null.Count = 24, Fill.Rate = 0.864
   • 14 rows with Missing.Rate>0.2 were exluded for subsequent analysis.
     • COUNTRY=Guadeloupe: Missing.Rate= 0.909
     • COUNTRY=Martinique: Missing.Rate= 0.909
     • COUNTRY=French Guiana: Missing.Rate= 0.909
     • COUNTRY=New Caledonia: Missing.Rate= 0.500
     • COUNTRY=French Polynesia: Missing.Rate= 0.500
     • COUNTRY=Guam: Missing.Rate= 0.500
     • COUNTRY=Puerto Rico: Missing.Rate= 0.409
     • COUNTRY=North Korea: Missing.Rate= 0.227
     • COUNTRY=Somalia: Missing.Rate= 0.227
     • COUNTRY=South Sudan: Missing.Rate= 0.227
     • COUNTRY=Venezuela: Missing.Rate= 0.227
     • COUNTRY=Libya: Missing.Rate= 0.227
     • COUNTRY=Eritrea: Missing.Rate= 0.227
     • COUNTRY=Yemen: Missing.Rate= 0.227
2. No variables were removed due to zero or near-zero variance.
3. The cleaned dataset is comprised of:
   • 163 rows (observations)
   • 18 columns (variables)
     • 1/18 metadata (object)
       • COUNTRY
     • 1/18 target (numeric)
       • CANRAT
     • 15/18 predictor (numeric)
       • GDPPER
       • URBPOP
       • POPGRO
       • LIFEXP
       • TUBINC
       • DTHCMD
       • AGRLND
       • GHGEMI
       • METEMI
       • FORARE
       • CO2EMI
       • PM2EXP
       • POPDEN
       • GDPCAP
       • EPISCO
     • 1/18 predictor (categorical)
       • HDICAT

##################################
# Performing a general exploration of the original dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate.shape)

Dataset Dimensions: 

(177, 22)

##################################
# Filtering out the rows with
# with Missing.Rate > 0.20
##################################
cancer_rate_filtered_row = cancer_rate.drop(cancer_rate[cancer_rate.COUNTRY.isin(row_high_missing_rate['Row.Name'].values.tolist())].index)

##################################
# Performing a general exploration of the filtered dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate_filtered_row.shape)

Dataset Dimensions: 

(163, 22)

##################################
# Filtering out the columns with
# with Fill.Rate < 0.90
##################################
cancer_rate_filtered_row_column = cancer_rate_filtered_row.drop(column_low_fill_rate['Column.Name'].values.tolist(), axis=1)

##################################
# Formulating a new dataset object
# for the cleaned data
##################################
cancer_rate_cleaned = cancer_rate_filtered_row_column

##################################
# Performing a general exploration of the filtered dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate_cleaned.shape)

Dataset Dimensions: 

(163, 18)

1.4.2 Missing Data Imputation

Iterative Imputer is based on the Multivariate Imputation by Chained Equations (MICE) algorithm - an imputation method based on fully conditional specification, where each incomplete variable is imputed by a separate model. As a sequential regression imputation technique, the algorithm imputes an incomplete column (target column) by generating plausible synthetic values given other columns in the data. Each incomplete column must act as a target column, and has its own specific set of predictors. For predictors that are incomplete themselves, the most recently generated imputations are used to complete the predictors prior to prior to imputation of the target columns.

Linear Regression explores the linear relationship between a scalar response and one or more covariates by having the conditional mean of the dependent variable be an affine function of the independent variables. The relationship is modeled through a disturbance term which represents an unobserved random variable that adds noise. The algorithm is typically formulated from the data using the least squares method which seeks to estimate the coefficients by minimizing the squared residual function. The linear equation assigns one scale factor represented by a coefficient to each covariate and an additional coefficient called the intercept or the bias coefficient which gives the line an additional degree of freedom allowing to move up and down a two-dimensional plot.

1. Missing data for numeric variables were imputed using the iterative imputer algorithm with a linear regression estimator.
   • GDPPER: Null.Count = 1
   • FORARE: Null.Count = 1
   • PM2EXP: Null.Count = 5
2. Missing data for categorical variables were imputed using the most frequent value.
   • HDICAP: Null.Count = 1

##################################
# Formulating the summary
# for all cleaned columns
##################################
cleaned_column_quality_summary = pd.DataFrame(zip(list(cancer_rate_cleaned.columns),
                                                  list(cancer_rate_cleaned.dtypes),
                                                  list([len(cancer_rate_cleaned)] * len(cancer_rate_cleaned.columns)),
                                                  list(cancer_rate_cleaned.count()),
                                                  list(cancer_rate_cleaned.isna().sum(axis=0))), 
                                        columns=['Column.Name',
                                                 'Column.Type',
                                                 'Row.Count',
                                                 'Non.Null.Count',
                                                 'Null.Count'])
display(cleaned_column_quality_summary)

	Column.Name	Column.Type	Row.Count	Non.Null.Count	Null.Count
0	COUNTRY	object	163	163	0
1	CANRAT	float64	163	163	0
2	GDPPER	float64	163	162	1
3	URBPOP	float64	163	163	0
4	POPGRO	float64	163	163	0
5	LIFEXP	float64	163	163	0
6	TUBINC	float64	163	163	0
7	DTHCMD	float64	163	163	0
8	AGRLND	float64	163	163	0
9	GHGEMI	float64	163	163	0
10	METEMI	float64	163	163	0
11	FORARE	float64	163	162	1
12	CO2EMI	float64	163	163	0
13	PM2EXP	float64	163	158	5
14	POPDEN	float64	163	163	0
15	GDPCAP	float64	163	163	0
16	HDICAT	category	163	162	1
17	EPISCO	float64	163	163	0

##################################
# Formulating the cleaned dataset
# with categorical columns only
##################################
cancer_rate_cleaned_categorical = cancer_rate_cleaned.select_dtypes(include='object')

##################################
# Formulating the cleaned dataset
# with numeric columns only
##################################
cancer_rate_cleaned_numeric = cancer_rate_cleaned.select_dtypes(include='number')

##################################
# Taking a snapshot of the cleaned dataset
##################################
cancer_rate_cleaned_numeric.head()

	CANRAT	GDPPER	URBPOP	POPGRO	LIFEXP	TUBINC	DTHCMD	AGRLND	GHGEMI	METEMI	FORARE	CO2EMI	PM2EXP	POPDEN	GDPCAP	EPISCO
0	452.4	98380.63601	86.241	1.235701	83.200000	7.2	4.941054	46.252480	5.719031e+05	131484.763200	17.421315	14.772658	24.893584	3.335312	51722.06900	60.1
1	422.9	77541.76438	86.699	2.204789	82.256098	7.2	4.354730	38.562911	8.015803e+04	32241.937000	37.570126	6.160799	NaN	19.331586	41760.59478	56.7
2	372.8	198405.87500	63.653	1.029111	82.556098	5.3	5.684596	65.495718	5.949773e+04	15252.824630	11.351720	6.768228	0.274092	72.367281	85420.19086	57.4
3	362.2	130941.63690	82.664	0.964348	76.980488	2.3	5.302060	44.363367	5.505181e+06	748241.402900	33.866926	13.032828	3.343170	36.240985	63528.63430	51.1
4	351.1	113300.60110	88.116	0.291641	81.602439	4.1	6.826140	65.499675	4.113555e+04	7778.773921	15.711000	4.691237	56.914456	145.785100	60915.42440	77.9

##################################
# Defining the estimator to be used
# at each step of the round-robin imputation
##################################
lr = LinearRegression()

##################################
# Defining the parameter of the
# iterative imputer which will estimate 
# the columns with missing values
# as a function of the other columns
# in a round-robin fashion
##################################
iterative_imputer = IterativeImputer(
    estimator = lr,
    max_iter = 10,
    tol = 1e-10,
    imputation_order = 'ascending',
    random_state=88888888
)

##################################
# Implementing the iterative imputer 
##################################
cancer_rate_imputed_numeric_array = iterative_imputer.fit_transform(cancer_rate_cleaned_numeric)

##################################
# Transforming the imputed data
# from an array to a dataframe
##################################
cancer_rate_imputed_numeric = pd.DataFrame(cancer_rate_imputed_numeric_array, 
                                           columns = cancer_rate_cleaned_numeric.columns)

##################################
# Taking a snapshot of the imputed dataset
##################################
cancer_rate_imputed_numeric.head()

	CANRAT	GDPPER	URBPOP	POPGRO	LIFEXP	TUBINC	DTHCMD	AGRLND	GHGEMI	METEMI	FORARE	CO2EMI	PM2EXP	POPDEN	GDPCAP	EPISCO
0	452.4	98380.63601	86.241	1.235701	83.200000	7.2	4.941054	46.252480	5.719031e+05	131484.763200	17.421315	14.772658	24.893584	3.335312	51722.06900	60.1
1	422.9	77541.76438	86.699	2.204789	82.256098	7.2	4.354730	38.562911	8.015803e+04	32241.937000	37.570126	6.160799	59.475540	19.331586	41760.59478	56.7
2	372.8	198405.87500	63.653	1.029111	82.556098	5.3	5.684596	65.495718	5.949773e+04	15252.824630	11.351720	6.768228	0.274092	72.367281	85420.19086	57.4
3	362.2	130941.63690	82.664	0.964348	76.980488	2.3	5.302060	44.363367	5.505181e+06	748241.402900	33.866926	13.032828	3.343170	36.240985	63528.63430	51.1
4	351.1	113300.60110	88.116	0.291641	81.602439	4.1	6.826140	65.499675	4.113555e+04	7778.773921	15.711000	4.691237	56.914456	145.785100	60915.42440	77.9

##################################
# Formulating the cleaned dataset
# with categorical columns only
##################################
cancer_rate_cleaned_categorical = cancer_rate_cleaned.select_dtypes(include='category')

##################################
# Imputing the missing data
# for categorical columns with
# the most frequent category
##################################
cancer_rate_cleaned_categorical['HDICAT'] = cancer_rate_cleaned_categorical['HDICAT'].fillna(cancer_rate_cleaned_categorical['HDICAT'].mode()[0])
cancer_rate_imputed_categorical = cancer_rate_cleaned_categorical.reset_index(drop=True)

##################################
# Formulating the imputed dataset
##################################
cancer_rate_imputed = pd.concat([cancer_rate_imputed_numeric,cancer_rate_imputed_categorical], axis=1, join='inner')  

##################################
# Gathering the data types for each column
##################################
data_type_list = list(cancer_rate_imputed.dtypes)

##################################
# Gathering the variable names for each column
##################################
variable_name_list = list(cancer_rate_imputed.columns)

##################################
# Gathering the number of observations for each column
##################################
row_count_list = list([len(cancer_rate_imputed)] * len(cancer_rate_imputed.columns))

##################################
# Gathering the number of missing data for each column
##################################
null_count_list = list(cancer_rate_imputed.isna().sum(axis=0))

##################################
# Gathering the number of non-missing data for each column
##################################
non_null_count_list = list(cancer_rate_imputed.count())

##################################
# Gathering the missing data percentage for each column
##################################
fill_rate_list = map(truediv, non_null_count_list, row_count_list)

##################################
# Formulating the summary
# for all imputed columns
##################################
imputed_column_quality_summary = pd.DataFrame(zip(variable_name_list,
                                                  data_type_list,
                                                  row_count_list,
                                                  non_null_count_list,
                                                  null_count_list,
                                                  fill_rate_list), 
                                        columns=['Column.Name',
                                                 'Column.Type',
                                                 'Row.Count',
                                                 'Non.Null.Count',
                                                 'Null.Count',                                                 
                                                 'Fill.Rate'])
display(imputed_column_quality_summary)

	Column.Name	Column.Type	Row.Count	Non.Null.Count	Null.Count	Fill.Rate
0	CANRAT	float64	163	163	0	1.0
1	GDPPER	float64	163	163	0	1.0
2	URBPOP	float64	163	163	0	1.0
3	POPGRO	float64	163	163	0	1.0
4	LIFEXP	float64	163	163	0	1.0
5	TUBINC	float64	163	163	0	1.0
6	DTHCMD	float64	163	163	0	1.0
7	AGRLND	float64	163	163	0	1.0
8	GHGEMI	float64	163	163	0	1.0
9	METEMI	float64	163	163	0	1.0
10	FORARE	float64	163	163	0	1.0
11	CO2EMI	float64	163	163	0	1.0
12	PM2EXP	float64	163	163	0	1.0
13	POPDEN	float64	163	163	0	1.0
14	GDPCAP	float64	163	163	0	1.0
15	EPISCO	float64	163	163	0	1.0
16	HDICAT	category	163	163	0	1.0

1.4.3 Outlier Detection

1. High number of outliers observed for 5 numeric variables with Outlier.Ratio>0.10 and marginal to high Skewness.
   • PM2EXP: Outlier.Count = 37, Outlier.Ratio = 0.226, Skewness=-3.061
   • GHGEMI: Outlier.Count = 27, Outlier.Ratio = 0.165, Skewness=+9.299
   • GDPCAP: Outlier.Count = 22, Outlier.Ratio = 0.134, Skewness=+2.311
   • POPDEN: Outlier.Count = 20, Outlier.Ratio = 0.122, Skewness=+9.972
   • METEMI: Outlier.Count = 20, Outlier.Ratio = 0.122, Skewness=+5.688
2. Minimal number of outliers observed for 5 numeric variables with Outlier.Ratio<0.10 and normal Skewness.
   • TUBINC: Outlier.Count = 12, Outlier.Ratio = 0.073, Skewness=+1.747
   • CO2EMI: Outlier.Count = 11, Outlier.Ratio = 0.067, Skewness=+2.693
   • GDPPER: Outlier.Count = 3, Outlier.Ratio = 0.018, Skewness=+1.554
   • EPISCO: Outlier.Count = 3, Outlier.Ratio = 0.018, Skewness=+0.635
   • CANRAT: Outlier.Count = 2, Outlier.Ratio = 0.012, Skewness=+0.910

##################################
# Formulating the imputed dataset
# with numeric columns only
##################################
cancer_rate_imputed_numeric = cancer_rate_imputed.select_dtypes(include='number')

##################################
# Gathering the variable names for each numeric column
##################################
numeric_variable_name_list = list(cancer_rate_imputed_numeric.columns)

##################################
# Gathering the skewness value for each numeric column
##################################
numeric_skewness_list = cancer_rate_imputed_numeric.skew()

##################################
# Computing the interquartile range
# for all columns
##################################
cancer_rate_imputed_numeric_q1 = cancer_rate_imputed_numeric.quantile(0.25)
cancer_rate_imputed_numeric_q3 = cancer_rate_imputed_numeric.quantile(0.75)
cancer_rate_imputed_numeric_iqr = cancer_rate_imputed_numeric_q3 - cancer_rate_imputed_numeric_q1

##################################
# Gathering the outlier count for each numeric column
# based on the interquartile range criterion
##################################
numeric_outlier_count_list = ((cancer_rate_imputed_numeric < (cancer_rate_imputed_numeric_q1 - 1.5 * cancer_rate_imputed_numeric_iqr)) | (cancer_rate_imputed_numeric > (cancer_rate_imputed_numeric_q3 + 1.5 * cancer_rate_imputed_numeric_iqr))).sum()

##################################
# Gathering the number of observations for each column
##################################
numeric_row_count_list = list([len(cancer_rate_imputed_numeric)] * len(cancer_rate_imputed_numeric.columns))

##################################
# Gathering the unique to count ratio for each categorical column
##################################
numeric_outlier_ratio_list = map(truediv, numeric_outlier_count_list, numeric_row_count_list)

##################################
# Formulating the outlier summary
# for all numeric columns
##################################
numeric_column_outlier_summary = pd.DataFrame(zip(numeric_variable_name_list,
                                                  numeric_skewness_list,
                                                  numeric_outlier_count_list,
                                                  numeric_row_count_list,
                                                  numeric_outlier_ratio_list), 
                                        columns=['Numeric.Column.Name',
                                                 'Skewness',
                                                 'Outlier.Count',
                                                 'Row.Count',
                                                 'Outlier.Ratio'])
display(numeric_column_outlier_summary)

	Numeric.Column.Name	Skewness	Outlier.Count	Row.Count	Outlier.Ratio
0	CANRAT	0.910128	2	163	0.012270
1	GDPPER	1.554434	3	163	0.018405
2	URBPOP	-0.212327	0	163	0.000000
3	POPGRO	-0.181666	0	163	0.000000
4	LIFEXP	-0.329704	0	163	0.000000
5	TUBINC	1.747962	12	163	0.073620
6	DTHCMD	0.930709	0	163	0.000000
7	AGRLND	0.035315	0	163	0.000000
8	GHGEMI	9.299960	27	163	0.165644
9	METEMI	5.688689	20	163	0.122699
10	FORARE	0.556183	0	163	0.000000
11	CO2EMI	2.693585	11	163	0.067485
12	PM2EXP	-3.061617	37	163	0.226994
13	POPDEN	9.972806	20	163	0.122699
14	GDPCAP	2.311079	22	163	0.134969
15	EPISCO	0.635994	3	163	0.018405

##################################
# Formulating the individual boxplots
# for all numeric columns
##################################
for column in cancer_rate_imputed_numeric:
        plt.figure(figsize=(17,1))
        sns.boxplot(data=cancer_rate_imputed_numeric, x=column)

1.4.4 Collinearity

Pearson’s Correlation Coefficient is a parametric measure of the linear correlation for a pair of features by calculating the ratio between their covariance and the product of their standard deviations. The presence of high absolute correlation values indicate the univariate association between the numeric predictors and the numeric response.

1. Majority of the numeric variables reported moderate to high correlation which were statistically significant.
2. Among pairwise combinations of numeric variables, high Pearson.Correlation.Coefficient values were noted for:
   • GDPPER and GDPCAP: Pearson.Correlation.Coefficient = +0.921
   • GHGEMI and METEMI: Pearson.Correlation.Coefficient = +0.905
3. Among the highly correlated pairs, variables with the lowest correlation against the target variable were removed.
   • GDPPER: Pearson.Correlation.Coefficient = +0.690
   • METEMI: Pearson.Correlation.Coefficient = +0.062
4. The cleaned dataset is comprised of:
   • 163 rows (observations)
   • 16 columns (variables)
     • 1/16 metadata (object)
       • COUNTRY
     • 1/16 target (numeric)
       • CANRAT
     • 13/16 predictor (numeric)
       • URBPOP
       • POPGRO
       • LIFEXP
       • TUBINC
       • DTHCMD
       • AGRLND
       • GHGEMI
       • FORARE
       • CO2EMI
       • PM2EXP
       • POPDEN
       • GDPCAP
       • EPISCO
     • 1/16 predictor (categorical)
       • HDICAT

##################################
# Formulating a function 
# to plot the correlation matrix
# for all pairwise combinations
# of numeric columns
##################################
def plot_correlation_matrix(corr, mask=None):
    f, ax = plt.subplots(figsize=(11, 9))
    sns.heatmap(corr, 
                ax=ax,
                mask=mask,
                annot=True, 
                vmin=-1, 
                vmax=1, 
                center=0,
                cmap='coolwarm', 
                linewidths=1, 
                linecolor='gray', 
                cbar_kws={'orientation': 'horizontal'})  

##################################
# Computing the correlation coefficients
# and correlation p-values
# among pairs of numeric columns
##################################
cancer_rate_imputed_numeric_correlation_pairs = {}
cancer_rate_imputed_numeric_columns = cancer_rate_imputed_numeric.columns.tolist()
for numeric_column_a, numeric_column_b in itertools.combinations(cancer_rate_imputed_numeric_columns, 2):
    cancer_rate_imputed_numeric_correlation_pairs[numeric_column_a + '_' + numeric_column_b] = stats.pearsonr(
        cancer_rate_imputed_numeric.loc[:, numeric_column_a], 
        cancer_rate_imputed_numeric.loc[:, numeric_column_b])

##################################
# Formulating the pairwise correlation summary
# for all numeric columns
##################################
cancer_rate_imputed_numeric_summary = cancer_rate_imputed_numeric.from_dict(cancer_rate_imputed_numeric_correlation_pairs, orient='index')
cancer_rate_imputed_numeric_summary.columns = ['Pearson.Correlation.Coefficient', 'Correlation.PValue']
display(cancer_rate_imputed_numeric_summary.sort_values(by=['Pearson.Correlation.Coefficient'], ascending=False).head(20))

	Pearson.Correlation.Coefficient	Correlation.PValue
GDPPER_GDPCAP	0.921009	8.173822e-68
GHGEMI_METEMI	0.905121	1.087643e-61
POPGRO_DTHCMD	0.759470	7.124695e-32
GDPPER_LIFEXP	0.755792	2.052275e-31
CANRAT_EPISCO	0.712599	1.445594e-26
CANRAT_GDPCAP	0.696991	4.991271e-25
GDPCAP_EPISCO	0.696707	5.312642e-25
CANRAT_LIFEXP	0.692318	1.379448e-24
CANRAT_GDPPER	0.686787	4.483016e-24
LIFEXP_GDPCAP	0.683834	8.321371e-24
GDPPER_EPISCO	0.680814	1.554608e-23
GDPPER_URBPOP	0.666399	2.778872e-22
GDPPER_CO2EMI	0.654956	2.451320e-21
TUBINC_DTHCMD	0.643615	1.936081e-20
URBPOP_LIFEXP	0.623997	5.669778e-19
LIFEXP_EPISCO	0.620271	1.048393e-18
URBPOP_GDPCAP	0.559181	8.624533e-15
CO2EMI_GDPCAP	0.550221	2.782997e-14
URBPOP_CO2EMI	0.550046	2.846393e-14
LIFEXP_CO2EMI	0.531305	2.951829e-13

##################################
# Plotting the correlation matrix
# for all pairwise combinations
# of numeric columns
##################################
cancer_rate_imputed_numeric_correlation = cancer_rate_imputed_numeric.corr()
mask = np.triu(cancer_rate_imputed_numeric_correlation)
plot_correlation_matrix(cancer_rate_imputed_numeric_correlation,mask)
plt.show()

##################################
# Formulating a function 
# to plot the correlation matrix
# for all pairwise combinations
# of numeric columns
# with significant p-values only
##################################
def correlation_significance(df=None):
    p_matrix = np.zeros(shape=(df.shape[1],df.shape[1]))
    for col in df.columns:
        for col2 in df.drop(col,axis=1).columns:
            _ , p = stats.pearsonr(df[col],df[col2])
            p_matrix[df.columns.to_list().index(col),df.columns.to_list().index(col2)] = p
    return p_matrix

##################################
# Plotting the correlation matrix
# for all pairwise combinations
# of numeric columns
# with significant p-values only
##################################
cancer_rate_imputed_numeric_correlation_p_values = correlation_significance(cancer_rate_imputed_numeric)                     
mask = np.invert(np.tril(cancer_rate_imputed_numeric_correlation_p_values<0.05)) 
plot_correlation_matrix(cancer_rate_imputed_numeric_correlation,mask)  

##################################
# Filtering out one among the 
# highly correlated variable pairs with
# lesser Pearson.Correlation.Coefficient
# when compared to the target variable
##################################
cancer_rate_imputed_numeric.drop(['GDPPER','METEMI'], inplace=True, axis=1)

##################################
# Performing a general exploration of the filtered dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate_imputed_numeric.shape)

Dataset Dimensions: 

(163, 14)

1.4.5 Shape Transformation

Yeo-Johnson Transformation applies a new family of distributions that can be used without restrictions, extending many of the good properties of the Box-Cox power family. Similar to the Box-Cox transformation, the method also estimates the optimal value of lambda but has the ability to transform both positive and negative values by inflating low variance data and deflating high variance data to create a more uniform data set. While there are no restrictions in terms of the applicable values, the interpretability of the transformed values is more diminished as compared to the other methods.

1. A Yeo-Johnson transformation was applied to all numeric variables to improve distributional shape.
2. Most variables achieved symmetrical distributions with minimal outliers after transformation.
3. One variable which remained skewed even after applying shape transformation was removed.
   • PM2EXP
4. The transformed dataset is comprised of:
   • 163 rows (observations)
   • 15 columns (variables)
     • 1/15 metadata (object)
       • COUNTRY
     • 1/15 target (numeric)
       • CANRAT
     • 12/15 predictor (numeric)
       • URBPOP
       • POPGRO
       • LIFEXP
       • TUBINC
       • DTHCMD
       • AGRLND
       • GHGEMI
       • FORARE
       • CO2EMI
       • POPDEN
       • GDPCAP
       • EPISCO
     • 1/15 predictor (categorical)
       • HDICAT

##################################
# Conducting a Yeo-Johnson Transformation
# to address the distributional
# shape of the variables
##################################
yeo_johnson_transformer = PowerTransformer(method='yeo-johnson',
                                          standardize=False)
cancer_rate_imputed_numeric_array = yeo_johnson_transformer.fit_transform(cancer_rate_imputed_numeric)

##################################
# Formulating a new dataset object
# for the transformed data
##################################
cancer_rate_transformed_numeric = pd.DataFrame(cancer_rate_imputed_numeric_array,
                                               columns=cancer_rate_imputed_numeric.columns)

##################################
# Formulating the individual boxplots
# for all transformed numeric columns
##################################
for column in cancer_rate_transformed_numeric:
        plt.figure(figsize=(17,1))
        sns.boxplot(data=cancer_rate_transformed_numeric, x=column)

##################################
# Filtering out the column
# which remained skewed even
# after applying shape transformation
##################################
cancer_rate_transformed_numeric.drop(['PM2EXP'], inplace=True, axis=1)

##################################
# Performing a general exploration of the filtered dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate_transformed_numeric.shape)

Dataset Dimensions: 

(163, 13)

1.4.6 Centering and Scaling

1. All numeric variables were transformed using the standardization method to achieve a comparable scale between values.
2. The scaled dataset is comprised of:
   • 163 rows (observations)
   • 15 columns (variables)
     • 1/15 metadata (object)
       • COUNTRY
     • 1/15 target (numeric)
       • CANRAT
     • 12/15 predictor (numeric)
       • URBPOP
       • POPGRO
       • LIFEXP
       • TUBINC
       • DTHCMD
       • AGRLND
       • GHGEMI
       • FORARE
       • CO2EMI
       • POPDEN
       • GDPCAP
       • EPISCO
     • 1/15 predictor (categorical)
       • HDICAT

##################################
# Conducting standardization
# to transform the values of the 
# variables into comparable scale
##################################
standardization_scaler = StandardScaler()
cancer_rate_transformed_numeric_array = standardization_scaler.fit_transform(cancer_rate_transformed_numeric)

##################################
# Formulating a new dataset object
# for the scaled data
##################################
cancer_rate_scaled_numeric = pd.DataFrame(cancer_rate_transformed_numeric_array,
                                          columns=cancer_rate_transformed_numeric.columns)

##################################
# Formulating the individual boxplots
# for all transformed numeric columns
##################################
for column in cancer_rate_scaled_numeric:
        plt.figure(figsize=(17,1))
        sns.boxplot(data=cancer_rate_scaled_numeric, x=column)

1.4.7 Data Encoding

1. One-hot encoding was applied to the HDICAP_VH variable resulting to 4 additional columns in the dataset:
   • HDICAP_L
   • HDICAP_M
   • HDICAP_H
   • HDICAP_VH

##################################
# Formulating the categorical column
# for encoding transformation
##################################
cancer_rate_categorical_encoded = pd.DataFrame(cancer_rate_cleaned_categorical.loc[:, 'HDICAT'].to_list(),
                                               columns=['HDICAT'])

##################################
# Applying a one-hot encoding transformation
# for the categorical column
##################################
cancer_rate_categorical_encoded = pd.get_dummies(cancer_rate_categorical_encoded, columns=['HDICAT'])

1.4.8 Preprocessed Data Description

1. The preprocessed dataset is comprised of:
   • 163 rows (observations)
   • 18 columns (variables)
     • 1/18 metadata (object)
       • COUNTRY
     • 1/18 target (numeric)
       • CANRAT
     • 12/18 predictor (numeric)
       • URBPOP
       • POPGRO
       • LIFEXP
       • TUBINC
       • DTHCMD
       • AGRLND
       • GHGEMI
       • FORARE
       • CO2EMI
       • POPDEN
       • GDPCAP
       • EPISCO
     • 4/18 predictor (categorical)
       • HDICAT_L
       • HDICAT_M
       • HDICAT_H
       • HDICAT_VH

##################################
# Consolidating both numeric columns
# and encoded categorical columns
##################################
cancer_rate_preprocessed = pd.concat([cancer_rate_scaled_numeric,cancer_rate_categorical_encoded], axis=1, join='inner')  

##################################
# Performing a general exploration of the consolidated dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate_preprocessed.shape)

Dataset Dimensions: 

(163, 17)

1.5. Data Exploration

1.5.1 Exploratory Data Analysis

1. Bivariate analysis identified individual predictors with generally linear relationship to the target variable based on visual inspection.
2. Increasing values for the following predictors correspond to higher CANRAT measurements:
   • URBPOP
   • LIFEXP
   • CO2EMI
   • GDPCAP
   • EPISCO
   • HDICAP_VH
3. Decreasing values for the following predictors correspond to higher CANRAT measurements:
   • POPGRO
   • TUBINC
   • DTHCMD
   • HDICAP_L
   • HDICAP_M
4. Values for the following predictors did not affect CANRAT measurements:
   • AGRLND
   • GHGEMI
   • FORARE
   • POPDEN
   • HDICAP_H

##################################
# Segregating the target
# and predictor variable lists
##################################
cancer_rate_preprocessed_target = ['CANRAT']
cancer_rate_preprocessed_predictors = cancer_rate_preprocessed.drop('CANRAT', axis=1).columns

##################################
# Segregating the target
# and predictor variable names
##################################
y_variable = 'CANRAT'
x_variables = cancer_rate_preprocessed_predictors

##################################
# Defining the number of 
# rows and columns for the subplots
##################################
num_rows = 8
num_cols = 2

##################################
# Formulating the subplot structure
##################################
fig, axes = plt.subplots(num_rows, num_cols, figsize=(15, 40))

##################################
# Flattening the multi-row and
# multi-column axes
##################################
axes = axes.ravel()

##################################
# Formulating the individual scatterplots
# for all scaled numeric columns
##################################
for i, x_variable in enumerate(x_variables):
    ax = axes[i]
    ax.scatter(cancer_rate_preprocessed[x_variable],cancer_rate_preprocessed[y_variable])
    ax.set_title(f'{y_variable} Versus {x_variable}')
    ax.set_xlabel(x_variable)
    ax.set_ylabel(y_variable)

##################################
# Adjusting the subplot layout
##################################
plt.tight_layout()

##################################
# Presenting the subplots
##################################
plt.show()

1.5.2 Hypothesis Testing

1. The relationship between the numeric predictors to the CANRAT target variable was statistically evaluated using the following hypotheses:
   • Null: Pearson correlation coefficient is equal to zero
   • Alternative: Pearson correlation coefficient is not equal to zero
2. There is sufficient evidence to conclude of a statistically significant linear relationship between the CANRAT target variable and 10 of the 12 numeric predictors given their high Pearson correlation coefficient values with reported low p-values less than the significance level of 0.05.
   • GDPCAP: Pearson.Correlation.Coefficient=+0.735, Correlation.PValue=0.000
   • LIFEXP: Pearson.Correlation.Coefficient=+0.702, Correlation.PValue=0.000
   • DTHCMD: Pearson.Correlation.Coefficient=-0.687, Correlation.PValue=0.000
   • EPISCO: Pearson.Correlation.Coefficient=+0.648, Correlation.PValue=0.000
   • TUBINC: Pearson.Correlation.Coefficient=+0.628, Correlation.PValue=0.000
   • CO2EMI: Pearson.Correlation.Coefficient=+0.585, Correlation.PValue=0.000
   • POPGRO: Pearson.Correlation.Coefficient=-0.498, Correlation.PValue=0.000
   • URBPOP: Pearson.Correlation.Coefficient=+0.479, Correlation.PValue=0.000
   • GHGEMI: Pearson.Correlation.Coefficient=+0.232, Correlation.PValue=0.002
   • FORARE: Pearson.Correlation.Coefficient=+0.165, Correlation.PValue=0.035
3. The relationship between the categorical predictors to the CANRAT target variable was statistically evaluated using the following hypotheses:
   • Null: Difference in the means between groups 0 and 1 is equal to zero
   • Alternative: Difference in the means between groups 0 and 1 is not equal to zero
4. There is sufficient evidence to conclude of a statistically significant difference between the means of CANRAT measurements obtained from groups 0 and 1 in 3 of the 4 categorical predictors given their high t-test statistic values with reported low p-values less than the significance level of 0.05.
   • HDICAT_VH: T.Test.Statistic=-10.605, T.Test.PValue=0.000
   • HDICAT_L: T.Test.Statistic=+6.559, T.Test.PValue=0.000
   • HDICAT_M: T.Test.Statistic=+5.104, T.Test.PValue=0.000

##################################
# Computing the correlation coefficients
# and correlation p-values
# between the target variable
# and numeric predictor columns
##################################
cancer_rate_preprocessed_numeric_correlation_target = {}
cancer_rate_preprocessed_numeric = cancer_rate_preprocessed.drop(['HDICAT_L','HDICAT_M','HDICAT_H','HDICAT_VH'], axis=1)
cancer_rate_preprocessed_numeric_columns = cancer_rate_preprocessed_numeric.columns.tolist()
for numeric_column in cancer_rate_preprocessed_numeric_columns:
    cancer_rate_preprocessed_numeric_correlation_target['CANRAT_' + numeric_column] = stats.pearsonr(
        cancer_rate_preprocessed_numeric.loc[:, 'CANRAT'], 
        cancer_rate_preprocessed_numeric.loc[:, numeric_column])

##################################
# Formulating the pairwise correlation summary
# between the target variable
# and numeric predictor columns
##################################
cancer_rate_preprocessed_numeric_summary = cancer_rate_preprocessed_numeric.from_dict(cancer_rate_preprocessed_numeric_correlation_target, orient='index')
cancer_rate_preprocessed_numeric_summary.columns = ['Pearson.Correlation.Coefficient', 'Correlation.PValue']
display(cancer_rate_preprocessed_numeric_summary.sort_values(by=['Correlation.PValue'], ascending=True).head(13))

	Pearson.Correlation.Coefficient	Correlation.PValue
CANRAT_CANRAT	1.000000	0.000000e+00
CANRAT_GDPCAP	0.735131	5.617239e-29
CANRAT_LIFEXP	0.702430	1.491302e-25
CANRAT_DTHCMD	-0.687136	4.164564e-24
CANRAT_EPISCO	0.648431	8.136735e-21
CANRAT_TUBINC	-0.628877	2.503346e-19
CANRAT_CO2EMI	0.585452	2.251585e-16
CANRAT_POPGRO	-0.498457	1.278437e-11
CANRAT_URBPOP	0.479386	9.543704e-11
CANRAT_GHGEMI	0.232488	2.822914e-03
CANRAT_FORARE	0.165265	3.500992e-02
CANRAT_AGRLND	-0.024520	7.560347e-01
CANRAT_POPDEN	0.001902	9.807807e-01

##################################
# Computing the t-test 
# statistic and p-values
# between the target variable
# and categorical predictor columns
##################################
cancer_rate_preprocessed_categorical_ttest_target = {}
cancer_rate_preprocessed_categorical = cancer_rate_preprocessed[['CANRAT','HDICAT_L','HDICAT_M','HDICAT_H','HDICAT_VH']]
cancer_rate_preprocessed_categorical_columns = ['HDICAT_L','HDICAT_M','HDICAT_H','HDICAT_VH']
for categorical_column in cancer_rate_preprocessed_categorical_columns:
    group_0 = cancer_rate_preprocessed_categorical[cancer_rate_preprocessed_categorical.loc[:,categorical_column]==0]
    group_1 = cancer_rate_preprocessed_categorical[cancer_rate_preprocessed_categorical.loc[:,categorical_column]==1]
    cancer_rate_preprocessed_categorical_ttest_target['CANRAT_' + categorical_column] = stats.ttest_ind(
        group_0['CANRAT'], 
        group_1['CANRAT'], 
        equal_var=True)

##################################
# Formulating the pairwise ttest summary
# between the target variable
# and categorical predictor columns
##################################
cancer_rate_preprocessed_categorical_summary = cancer_rate_preprocessed_categorical.from_dict(cancer_rate_preprocessed_categorical_ttest_target, orient='index')
cancer_rate_preprocessed_categorical_summary.columns = ['T.Test.Statistic', 'T.Test.PValue']
display(cancer_rate_preprocessed_categorical_summary.sort_values(by=['T.Test.PValue'], ascending=True).head(4))

	T.Test.Statistic	T.Test.PValue
CANRAT_HDICAT_VH	-10.605706	2.909971e-20
CANRAT_HDICAT_L	6.559780	7.003957e-10
CANRAT_HDICAT_M	5.104986	9.237518e-07
CANRAT_HDICAT_H	-0.635957	5.257075e-01

1.6. Linear Regression Model Coefficient Estimation

1.6.1 Premodelling Data Description

1. Among the predictor variables determined to have a statistically significant linear relationship between the CANRAT target variable, only 2 were retained with absolute Pearson correlation coefficient values greater than 0.70.
   • GDPCAP: Pearson.Correlation.Coefficient=+0.735, Correlation.PValue=0.000
   • LIFEXP: Pearson.Correlation.Coefficient=+0.702, Correlation.PValue=0.000

##################################
# Consolidating relevant numeric columns
# and encoded categorical columns
# after hypothesis testing
##################################
cancer_rate_premodelling = cancer_rate_preprocessed.drop(['DTHCMD','EPISCO','TUBINC','CO2EMI','AGRLND','POPDEN','GHGEMI','FORARE','POPGRO','URBPOP','HDICAT_VH','HDICAT_H','HDICAT_M','HDICAT_L'], axis=1)

##################################
# Performing a general exploration of the filtered dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate_premodelling.shape)

Dataset Dimensions: 

(163, 3)

##################################
# Listing the column names and data types
##################################
print('Column Names and Data Types:')
display(cancer_rate_premodelling.dtypes)

Column Names and Data Types:

CANRAT    float64
LIFEXP    float64
GDPCAP    float64
dtype: object

##################################
# Taking a snapshot of the dataset
##################################
cancer_rate_premodelling.head()

	CANRAT	LIFEXP	GDPCAP
0	2.076468	1.643195	1.549766
1	1.962991	1.487969	1.407752
2	1.742760	1.537044	1.879374
3	1.690866	0.664178	1.685426
4	1.634224	1.381877	1.657777

##################################
# Gathering the pairplot for all variables
##################################
sns.pairplot(cancer_rate_premodelling, kind='reg')
plt.show()

##################################
# Separating the target 
# and predictor columns
##################################
X = cancer_rate_premodelling.drop('CANRAT', axis = 1)
y = cancer_rate_premodelling.CANRAT

##################################
# Formulating the train and test data
# using a 70-30 ratio
##################################
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size = 0.3, random_state= 88888888)

##################################
# Performing a general exploration of the train dataset
##################################
print('Dataset Dimensions: ')
display(X_train.shape)

Dataset Dimensions: 

(114, 2)

1.6.2 Normal Equations

Normal Equations are a system of equations whose solution is the Ordinary Least Squares (OLS) estimator of the regression coefficients and which are derived from the first-order condition of the least squares minimization problem. These equations are obtained by setting equal to zero the partial derivatives of the sum of squared errors (least squares). This approach is a closed-form solution and a one-step algorithm used to analytically find the coefficients that minimize the loss function.

1. Applying normal equations, the estimated linear regression coefficients for the given data are as follows:
   • INTERCEPT = -0.00061
   • LIFEXP = +0.23342
   • GDPCAP = +0.50808
2. The linear regression model from the sklearn.linear_model Python library API was implemented which generated the same regression coefficient estimates.
3. These estimated coefficients will be the baseline values from which all gradient descent algorithm-derived coefficients will be compared with.

##################################
# Defining the components
# for matrix algebra computations
# using Normal Equations
##################################
num_observations = X_train.shape[0]
constant_array = np.ones(num_observations)
x_train_matrix = np.array([constant_array,X_train.LIFEXP,X_train.GDPCAP]).T

##################################
# Consolidating the regression coefficients
# obtained using the Normal Equations
##################################
linear_regression_coefficients = pd.DataFrame(["INTERCEPT","LIFEXP","GDPCAP"])
linear_regression_estimates = pd.DataFrame(np.linalg.inv(np.dot(x_train_matrix.T,x_train_matrix)).dot(x_train_matrix.T).dot(y_train))
linear_regression_normal_equations = pd.concat([linear_regression_coefficients, linear_regression_estimates], axis=1)
linear_regression_normal_equations.columns = ['Coefficient', 'Estimate']
linear_regression_normal_equations.reset_index(inplace=True, drop=True)
display(linear_regression_normal_equations)

	Coefficient	Estimate
0	INTERCEPT	-0.000606
1	LIFEXP	0.233418
2	GDPCAP	0.508083

##################################
# Defining the linear regression model
# using the Scikit-Learn package
##################################
linear_regression = LinearRegression()

##################################
# Fitting a linear regression model
##################################
linear_regression.fit(X_train, y_train)

LinearRegression()

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##################################
# Consolidating the regression coefficients
# obtained using the Scikit-Learn package
##################################
linear_regression_intercept = pd.DataFrame(zip(["INTERCEPT"], [linear_regression.intercept_]))
linear_regression_predictors = pd.DataFrame(zip(X_train.columns, linear_regression.coef_))
linear_regression_scikitlearn_estimates = pd.concat([linear_regression_intercept, linear_regression_predictors], axis=0)
linear_regression_scikitlearn_estimates.reset_index(inplace=True, drop=True)
linear_regression_method = pd.DataFrame(["Normal_Equations"]*3)
linear_regression_scikitlearn_computations = pd.concat([linear_regression_scikitlearn_estimates,linear_regression_method], axis=1)
linear_regression_scikitlearn_computations.columns = ['Coefficient', 'Estimate','Method']
linear_regression_scikitlearn_computations.reset_index(inplace=True, drop=True)
display(linear_regression_scikitlearn_computations)

	Coefficient	Estimate	Method
0	INTERCEPT	-0.000606	Normal_Equations
1	LIFEXP	0.233418	Normal_Equations
2	GDPCAP	0.508083	Normal_Equations

##################################
# Setting the regression coefficients
# determined using normal equations
# as the target coefficient estimates
##################################
intercept = -0.000606
theta_1 = 0.233418
theta_2 = 0.508083

1.6.3 Gradient Descent Algorithm with Very High Learning Rate and Low Epoch Count

Gradient descent minimizes the loss function parameterized by the model’s coefficients based on the direction and learning rate factors which determine the partial derivative calculations of future iterations, allowing the algorithm to gradually arrive at the local or global minimum considered the point of convergence. This particular implementation used Batch Gradient Descent which computes the gradient of the loss function with respect to the parameters for the entire data set. A very high learning rate (also referred to as step size or the alpha) and low epoch count were applied resulting in larger steps with lesser risks of overshooting the minimum due to a lower number of iterations.

1. The gradient descent algorithm was implemented with parameter settings described as follows:
   • Learning Rate = 1.00 (Very High)
   • Epochs = 30 (Low)
2. The final squared loss estimate determined as 53.54705 at the 30th epoch was not optimally low as compared to those obtained using the other parameter settings.
3. Applying the gradient descent algorithm with a very high learning rate and low epoch count, the estimated linear regression coefficients for the given data are as follows:
   • INTERCEPT = +0.00801 (Baseline = -0.00061)
   • LIFEXP = +0.28176 (Baseline = +0.23342)
   • GDPCAP = +0.54996 (Baseline = +0.50808)
4. The estimated coefficients using the gradient descent algorithm with a very high learning rate and low epoch count were not fully optimized and comparable with the baseline coefficients using normal equations.

##################################
# Formulating a function
# for computing the regression coefficients
# using gradient descent
##################################
def gradient_descent(learning_rate, num_iterations, theta_initial):
    # Setting the initialization values
    # Initializing the coefficient estimates
    theta = theta_initial
    # Initializing the gradient descent trajectory path
    theta_path = np.zeros((num_iterations+1,3))
    theta_path[0,:]= theta_initial
    # Initializing the loss function values
    loss_vector = np.zeros(num_iterations)
    ## Implementing the main Gradient Descent loop givne a fixed number of iterations
    for i in range(num_iterations):
        # Generating predictions
        y_predicted = np.dot(theta.T,x_train_matrix.T)
        # Computing for the loss function values
        loss_vector[i] = np.sum((y_train-y_predicted)**2)
        # Summing up the gradients across all observations and divide by the number of observations
        gradient_vector = (y_train-y_predicted).dot(x_train_matrix)/num_observations  
        # Updating the gradients
        gradient_vector = gradient_vector
        # Updating the coefficient estimates
        theta = theta + learning_rate*gradient_vector
        # Updating the trajectory of the gradient descent process
        theta_path[i+1,:]=theta
    return theta_path, loss_vector

##################################
# Formulating a function
# for plotting the gradient descent trajectory
# with respect to the initialized
# and target coefficient estimates
##################################
target_coefficient = [intercept, theta_1, theta_2]
target_coefficient_name = ["INTERCEPT","LIFEXP","GDPCAP"]
def plot_ij(theta_path, i, j, ax):
    ax.plot(target_coefficient[i], target_coefficient[j],
            marker='p', markersize=15, label='Target Coefficient', 
            color='#FF0000')
    ax.plot(theta_path[:, i],theta_path[:, j],
            color='k', linestyle='--', marker='^', 
            markersize=5, markevery=1)
    ax.plot(theta_path[0, i], theta_path[0, j], marker='d', 
            markersize=10, label='Gradient Descent Path Start', color='#00FF0088')
    ax.plot(theta_path[-1, i], theta_path[-1, j], marker='o', 
            markersize=10, label='Gradient Descent Path End', color='#0000FF88')
    ax.set(
        xlabel=target_coefficient_name[i],
        ylabel=target_coefficient_name[j])
    ax.axis('equal')
    ax.grid(True)
    ax.legend(loc='upper left')
    
##################################
# Consolidating all 
# gradient descent trajectory plots
# by pairwise combinations
# of regression coefficients
##################################
def plot_all(theta_path, loss_vector, learning_rate, num_iterations, theta_initial, gdtype='Gradient Descent'):
    fig = plt.figure(figsize=(12, 12))
    title = '{gdtype}: Learning Rate = {lr} | Epoch Count = {iters}'
    title = title.format(gdtype=gdtype, lr=learning_rate, 
                         iters=num_iterations, initial=theta_initial)
    fig.suptitle(title, fontsize=15)
    ax = fig.add_subplot(2, 2, 1)
    plot_ij(theta_path, 0, 1, ax)
    ax = fig.add_subplot(2, 2, 2)
    plot_ij(theta_path, 0, 2, ax)
    ax = fig.add_subplot(2, 2, 3)
    plot_ij(theta_path, 1, 2, ax)
    ax = fig.add_subplot(2, 2, 4)
    ax.plot(loss_vector)
    ax.set_ylim([0, 5000])
    ax.set_xlim([0, 300])
    ax.set(xlabel='Iterations', ylabel='Squared Loss')
    ax.grid(True)

##################################
# Setting the initial parameters
##################################
theta_initial = np.array([3,3,3])
##################################
# Using a very high learning rate
##################################
learning_rate = 1.00
##################################
# Using a low epoch count
##################################
num_iterations = 30

##################################
# Implementing the gradient descent process
# for determining the regression coefficients
##################################
theta_path, loss_vector = gradient_descent(learning_rate, num_iterations, theta_initial)

##################################
# Consolidating the gradient descent
# trajectory plots in a 3D parameter space
# for each pair of regression coefficients
##################################
plot_all(theta_path, loss_vector, learning_rate, num_iterations, theta_initial)

##################################
# Determining the final estimated loss
##################################
loss_vector_vhlearningrate_lepochcount = loss_vector[-1]
loss_vector_vhlearningrate_lepochcount

np.float64(53.54705756819414)

##################################
# Consolidating the regression coefficients
# obtained using the Gradient Descent process
# with very high learning rate
# and low epoch count
##################################
linear_regression_coefficients = pd.DataFrame(["INTERCEPT","LIFEXP","GDPCAP"])
linear_regression_vhlearningrate_lepochcount = pd.DataFrame(theta_path[-1])
linear_regression_vhlearningrate_lepochcount_method = pd.DataFrame(["VeryHighLearningRate_LowEpochCount"]*3)
linear_regression_gradientdescent_vhlearningrate_lepochcount = pd.concat([linear_regression_coefficients, 
                                                                          linear_regression_vhlearningrate_lepochcount,
                                                                          linear_regression_vhlearningrate_lepochcount_method], axis=1)
linear_regression_gradientdescent_vhlearningrate_lepochcount.columns = ['Coefficient', 'Estimate', 'Method']
linear_regression_gradientdescent_vhlearningrate_lepochcount.reset_index(inplace=True, drop=True)
display(linear_regression_gradientdescent_vhlearningrate_lepochcount)

	Coefficient	Estimate	Method
0	INTERCEPT	0.008008	VeryHighLearningRate_LowEpochCount
1	LIFEXP	0.281761	VeryHighLearningRate_LowEpochCount
2	GDPCAP	0.549961	VeryHighLearningRate_LowEpochCount

1.6.4 Gradient Descent Algorithm with Very High Learning Rate and High Epoch Count

Gradient descent minimizes the loss function parameterized by the model’s coefficients based on the direction and learning rate factors which determine the partial derivative calculations of future iterations, allowing the algorithm to gradually arrive at the local or global minimum considered the point of convergence. This particular implementation used Batch Gradient Descent which computes the gradient of the loss function with respect to the parameters for the entire data set. A very high learning rate (also referred to as step size or the alpha) and high epoch count were applied resulting in larger steps with more risks of overshooting the minimum due to a higher number of iterations.

1. The gradient descent algorithm was implemented with parameter settings described as follows:
   • Learning Rate = 1.00 (Very High)
   • Epochs = 300 (High)
2. The final squared loss estimate determined as 52.38026 at the 300th epoch was optimally low as compared to those obtained using the other parameter settings.
3. Applying the gradient descent algorithm with a very high learning rate and high epoch count, the estimated linear regression coefficients for the given data are as follows:
   • INTERCEPT = -0.00061 (Baseline = -0.00061)
   • LIFEXP = +0.23342 (Baseline = +0.23342)
   • GDPCAP = +0.50808 (Baseline = +0.50808)
4. The estimated coefficients using the gradient descent algorithm with a very high learning rate and high epoch count were fully optimized and comparable with the baseline coefficients using normal equations.

##################################
# Setting the initial parameters
##################################
theta_initial = np.array([3,3,3])
##################################
# Using a very high learning rate
##################################
learning_rate = 1.00
##################################
# Using a high epoch count
##################################
num_iterations = 300

##################################
# Implementing the gradient descent process
# for determining the regression coefficients
##################################
theta_path, loss_vector = gradient_descent(learning_rate, num_iterations, theta_initial)

##################################
# Consolidating the gradient descent
# trajectory plots in a 3D parameter space
# for each pair of regression coefficients
##################################
plot_all(theta_path, loss_vector, learning_rate, num_iterations, theta_initial)

##################################
# Determining the final estimated loss
##################################
loss_vector_vhlearningrate_hepochcount = loss_vector[-1]
loss_vector_vhlearningrate_hepochcount

np.float64(52.38026912784727)

##################################
# Consolidating the regression coefficients
# obtained using the Gradient Descent process
# with very high learning rate
# and high epoch count
##################################
linear_regression_coefficients = pd.DataFrame(["INTERCEPT","LIFEXP","GDPCAP"])
linear_regression_vhlearningrate_hepochcount = pd.DataFrame(theta_path[-1])
linear_regression_vhlearningrate_hepochcount_method = pd.DataFrame(["VeryHighLearningRate_HighEpochCount"]*3)
linear_regression_gradientdescent_vhlearningrate_hepochcount = pd.concat([linear_regression_coefficients, 
                                                                          linear_regression_vhlearningrate_hepochcount,
                                                                          linear_regression_vhlearningrate_hepochcount_method], axis=1)
linear_regression_gradientdescent_vhlearningrate_hepochcount.columns = ['Coefficient', 'Estimate', 'Method']
linear_regression_gradientdescent_vhlearningrate_hepochcount.reset_index(inplace=True, drop=True)
display(linear_regression_gradientdescent_vhlearningrate_hepochcount)

	Coefficient	Estimate	Method
0	INTERCEPT	-0.000606	VeryHighLearningRate_HighEpochCount
1	LIFEXP	0.233418	VeryHighLearningRate_HighEpochCount
2	GDPCAP	0.508083	VeryHighLearningRate_HighEpochCount

1.6.5 Gradient Descent Algorithm with High Learning Rate and Low Epoch Count

Gradient descent minimizes the loss function parameterized by the model’s coefficients based on the direction and learning rate factors which determine the partial derivative calculations of future iterations, allowing the algorithm to gradually arrive at the local or global minimum considered the point of convergence. This particular implementation used Batch Gradient Descent which computes the gradient of the loss function with respect to the parameters for the entire data set. A sufficiently high learning rate (also referred to as step size or the alpha) and low epoch count were applied resulting in average steps with more risks of not reaching the minimum due to a lower number of iterations.

1. The gradient descent algorithm was implemented with parameter settings described as follows:
   • Learning Rate = 0.10 (High)
   • Epochs = 30 (Low)
2. The final squared loss estimate determined as 54.31567 at the 30th epoch was not optimally low as compared to those obtained using the other parameter settings.
3. Applying the gradient descent algorithm with a high learning rate and low epoch count, the estimated linear regression coefficients for the given data are as follows:
   • INTERCEPT = +0.11066 (Baseline = -0.00061)
   • LIFEXP = +0.31784 (Baseline = +0.23342)
   • GDPCAP = +0.41305 (Baseline = +0.50808)
4. The estimated coefficients using the gradient descent algorithm with a high learning rate and low epoch count were not fully optimized and comparable with the baseline coefficients using normal equations.

##################################
# Setting the initial parameters
##################################
theta_initial = np.array([3,3,3])
##################################
# Using a high learning rate
##################################
learning_rate = 0.1
##################################
# Using a low epoch count
##################################
num_iterations = 30

##################################
# Implementing the gradient descent process
# for determining the regression coefficients
##################################
theta_path, loss_vector = gradient_descent(learning_rate, num_iterations, theta_initial)

##################################
# Consolidating the gradient descent
# trajectory plots in a 3D parameter space
# for each pair of regression coefficients
##################################
plot_all(theta_path, loss_vector, learning_rate, num_iterations, theta_initial)

##################################
# Determining the final estimated loss
##################################
loss_vector_hlearningrate_lepochcount = loss_vector[-1]
loss_vector_hlearningrate_lepochcount

np.float64(54.31566869139367)

##################################
# Consolidating the regression coefficients
# obtained using the Gradient Descent process
# with high learning rate
# and low epoch count
##################################
linear_regression_coefficients = pd.DataFrame(["INTERCEPT","LIFEXP","GDPCAP"])
linear_regression_hlearningrate_lepochcount = pd.DataFrame(theta_path[-1])
linear_regression_hlearningrate_lepochcount_method = pd.DataFrame(["HighLearningRate_LowEpochCount"]*3)
linear_regression_gradientdescent_hlearningrate_lepochcount = pd.concat([linear_regression_coefficients, 
                                                                         linear_regression_hlearningrate_lepochcount,
                                                                         linear_regression_hlearningrate_lepochcount_method], axis=1)
linear_regression_gradientdescent_hlearningrate_lepochcount.columns = ['Coefficient', 'Estimate', 'Method']
linear_regression_gradientdescent_hlearningrate_lepochcount.reset_index(inplace=True, drop=True)
display(linear_regression_gradientdescent_hlearningrate_lepochcount)

	Coefficient	Estimate	Method
0	INTERCEPT	0.110662	HighLearningRate_LowEpochCount
1	LIFEXP	0.317843	HighLearningRate_LowEpochCount
2	GDPCAP	0.413052	HighLearningRate_LowEpochCount

1.6.6 Gradient Descent Algorithm with High Learning Rate and High Epoch Count

Gradient descent minimizes the loss function parameterized by the model’s coefficients based on the direction and learning rate factors which determine the partial derivative calculations of future iterations, allowing the algorithm to gradually arrive at the local or global minimum considered the point of convergence. This particular implementation used Batch Gradient Descent which computes the gradient of the loss function with respect to the parameters for the entire data set. A sufficiently high learning rate (also referred to as step size or the alpha) and low epoch count were applied resulting in average steps with lesser risks of not reaching the minimum as compensated by the higher number of iterations.

1. The gradient descent algorithm was implemented with parameter settings described as follows:
   • Learning Rate = 0.10 (High)
   • Epochs = 300 (High)
2. The final squared loss estimate determined as 52.38063 at the 300th epoch was not optimally low as compared to those obtained using the other parameter settings.
3. Applying the gradient descent algorithm with a high learning rate and high epoch count, the estimated linear regression coefficients for the given data are as follows:
   • INTERCEPT = -0.00058 (Baseline = -0.00061)
   • LIFEXP = +0.23703 (Baseline = +0.23342)
   • GDPCAP = +0.50442 (Baseline = +0.50808)
4. The estimated coefficients using the gradient descent algorithm with a high learning rate and high epoch count, while not fully optimized, were sufficiently comparable with the baseline coefficients using normal equations.

##################################
# Setting the initial parameters
##################################
theta_initial = np.array([3,3,3])
##################################
# Using a high learning rate
##################################
learning_rate = 0.1
##################################
# Using a high epoch count
##################################
num_iterations = 300

##################################
# Implementing the gradient descent process
# for determining the regression coefficients
##################################
theta_path, loss_vector = gradient_descent(learning_rate, num_iterations, theta_initial)

##################################
# Consolidating the gradient descent
# trajectory plots in a 3D parameter space
# for each pair of regression coefficients
##################################
plot_all(theta_path, loss_vector, learning_rate, num_iterations, theta_initial)

##################################
# Determining the final estimated loss
##################################
loss_vector_hlearningrate_hepochcount = loss_vector[-1]
loss_vector_hlearningrate_hepochcount

np.float64(52.380634424224674)

##################################
# Consolidating the regression coefficients
# obtained using the Gradient Descent process
# with high learning rate
# and high epoch count
##################################
linear_regression_coefficients = pd.DataFrame(["INTERCEPT","LIFEXP","GDPCAP"])
linear_regression_hlearningrate_hepochcount = pd.DataFrame(theta_path[-1])
linear_regression_hlearningrate_hepochcount_method = pd.DataFrame(["HighLearningRate_HighEpochCount"]*3)
linear_regression_gradientdescent_hlearningrate_hepochcount = pd.concat([linear_regression_coefficients, 
                                                                         linear_regression_hlearningrate_hepochcount,
                                                                         linear_regression_hlearningrate_hepochcount_method], axis=1)
linear_regression_gradientdescent_hlearningrate_hepochcount.columns = ['Coefficient', 'Estimate', 'Method']
linear_regression_gradientdescent_hlearningrate_hepochcount.reset_index(inplace=True, drop=True)
display(linear_regression_gradientdescent_hlearningrate_hepochcount)

	Coefficient	Estimate	Method
0	INTERCEPT	-0.000585	HighLearningRate_HighEpochCount
1	LIFEXP	0.237033	HighLearningRate_HighEpochCount
2	GDPCAP	0.504420	HighLearningRate_HighEpochCount

1.6.7 Gradient Descent Algorithm with Low Learning Rate and Low Epoch Count

Gradient descent minimizes the loss function parameterized by the model’s coefficients based on the direction and learning rate factors which determine the partial derivative calculations of future iterations, allowing the algorithm to gradually arrive at the local or global minimum considered the point of convergence. This particular implementation used Batch Gradient Descent which computes the gradient of the loss function with respect to the parameters for the entire data set. A low learning rate (also referred to as step size or the alpha) and low epoch count were applied resulting in smaller steps with higher risks of not reaching the minimum due to the smaller number of iterations.

1. The gradient descent algorithm was implemented with parameter settings described as follows:
   • Learning Rate = 0.01 (Low)
   • Epochs = 30 (Low)
2. The final squared loss estimate determined as 1631.70371 at the 30th epoch was not optimally low as compared to those obtained using the other parameter settings.
3. Applying the gradient descent algorithm with a low learning rate and low epoch count, the estimated linear regression coefficients for the given data are as follows:
   • INTERCEPT = +2.13385 (Baseline = -0.00061)
   • LIFEXP = +1.81162 (Baseline = +0.23342)
   • GDPCAP = +1.83111 (Baseline = +0.50808)
4. The estimated coefficients using the gradient descent algorithm with a low learning rate and low epoch count were not fully optimized and comparable with the baseline coefficients using normal equations.

##################################
# Setting the initial parameters
##################################
theta_initial = np.array([3,3,3])
##################################
# Using a low learning rate
##################################
learning_rate = 0.01
##################################
# Using a low epoch count
##################################
num_iterations = 30

##################################
# Implementing the gradient descent process
# for determining the regression coefficients
##################################
theta_path, loss_vector = gradient_descent(learning_rate, num_iterations, theta_initial)

##################################
# Consolidating the gradient descent
# trajectory plots in a 3D parameter space
# for each pair of regression coefficients
##################################
plot_all(theta_path, loss_vector, learning_rate, num_iterations, theta_initial)

##################################
# Determining the final estimated loss
##################################
loss_vector_llearningrate_lepochcount = loss_vector[-1]
loss_vector_llearningrate_lepochcount

np.float64(1631.7037156372803)

##################################
# Consolidating the regression coefficients
# obtained using the Gradient Descent process
# with low learning rate
# and low epoch count
##################################
linear_regression_coefficients = pd.DataFrame(["INTERCEPT","LIFEXP","GDPCAP"])
linear_regression_llearningrate_lepochcount = pd.DataFrame(theta_path[-1])
linear_regression_llearningrate_lepochcount_method = pd.DataFrame(["LowLearningRate_LowEpochCount"]*3)
linear_regression_gradientdescent_llearningrate_lepochcount = pd.concat([linear_regression_coefficients, 
                                                                         linear_regression_llearningrate_lepochcount,
                                                                         linear_regression_llearningrate_lepochcount_method], axis=1)
linear_regression_gradientdescent_llearningrate_lepochcount.columns = ['Coefficient', 'Estimate', 'Method']
linear_regression_gradientdescent_llearningrate_lepochcount.reset_index(inplace=True, drop=True)
display(linear_regression_gradientdescent_llearningrate_lepochcount)

	Coefficient	Estimate	Method
0	INTERCEPT	2.133856	LowLearningRate_LowEpochCount
1	LIFEXP	1.811625	LowLearningRate_LowEpochCount
2	GDPCAP	1.831115	LowLearningRate_LowEpochCount

1.6.8 Gradient Descent Algorithm with Low Learning Rate and High Epoch Count

Gradient descent minimizes the loss function parameterized by the model’s coefficients based on the direction and learning rate factors which determine the partial derivative calculations of future iterations, allowing the algorithm to gradually arrive at the local or global minimum considered the point of convergence. This particular implementation used Batch Gradient Descent which computes the gradient of the loss function with respect to the parameters for the entire data set. A low learning rate (also referred to as step size or the alpha) and low epoch count were applied resulting in smaller steps with lesser risks of not reaching the minimum as compensated by the smaller number of iterations.

1. The gradient descent algorithm was implemented with parameter settings described as follows:
   • Learning Rate = 0.01 (Low)
   • Epochs = 300 (High)
2. The final squared loss estimate determined as 54.50204 at the 300th epoch was not optimally low as compared to those obtained using the other parameter settings.
3. Applying the gradient descent algorithm with a low learning rate and high epoch count, the estimated linear regression coefficients for the given data are as follows:
   • INTERCEPT = +0.12806 (Baseline = -0.00061)
   • LIFEXP = +0.32058 (Baseline = +0.23342)
   • GDPCAP = +0.41551 (Baseline = +0.50808)
4. The estimated coefficients using the gradient descent algorithm with a low learning rate and high epoch count were not fully optimized and comparable with the baseline coefficients using normal equations.

##################################
# Setting the initial parameters
##################################
theta_initial = np.array([3,3,3])
##################################
# Using a low learning rate
##################################
learning_rate = 0.01
##################################
# Using a high epoch count
##################################
num_iterations = 300

##################################
# Implementing the gradient descent process
# for determining the regression coefficients
##################################
theta_path, loss_vector = gradient_descent(learning_rate, num_iterations, theta_initial)

##################################
# Consolidating the gradient descent
# trajectory plots in a 3D parameter space
# for each pair of regression coefficients
##################################
plot_all(theta_path, loss_vector, learning_rate, num_iterations, theta_initial)

##################################
# Determining the final estimated loss
##################################
loss_vector_llearningrate_hepochcount = loss_vector[-1]
loss_vector_llearningrate_hepochcount

np.float64(54.50204042343666)

##################################
# Consolidating the regression coefficients
# obtained using the Gradient Descent process
# with low learning rate
# and high epoch count
##################################
linear_regression_coefficients = pd.DataFrame(["INTERCEPT","LIFEXP","GDPCAP"])
linear_regression_llearningrate_hepochcount = pd.DataFrame(theta_path[-1])
linear_regression_llearningrate_hepochcount_method = pd.DataFrame(["LowLearningRate_HighEpochCount"]*3)
linear_regression_gradientdescent_llearningrate_hepochcount = pd.concat([linear_regression_coefficients, 
                                                                         linear_regression_llearningrate_hepochcount,
                                                                         linear_regression_llearningrate_hepochcount_method], axis=1)
linear_regression_gradientdescent_llearningrate_hepochcount.columns = ['Coefficient', 'Estimate', 'Method']
linear_regression_gradientdescent_llearningrate_hepochcount.reset_index(inplace=True, drop=True)
display(linear_regression_gradientdescent_llearningrate_hepochcount)

	Coefficient	Estimate	Method
0	INTERCEPT	0.128061	LowLearningRate_HighEpochCount
1	LIFEXP	0.320584	LowLearningRate_HighEpochCount
2	GDPCAP	0.415517	LowLearningRate_HighEpochCount

1.7. Consolidated Findings

1. The gradient descent parameter setting which estimated sufficiently comparable coefficients with the baseline values are as follows:
   • VHLR_HEC = Very High Learning Rate (1.0) and High Epoch Count (300)
   • HLR_HEC = High Learning Rate (0.1) and High Epoch Count (300)
2. The choice of Learning Rate and Epoch Count in the implementation of the gradient descent algorithm are critical to achieving fully optimized coefficients while maintaining minimal squared loss estimates.

##################################
# Consolidating the regression coefficients
# obtained using all estimation methods
##################################
linear_regression_methods = pd.concat([linear_regression_scikitlearn_computations, 
                                       linear_regression_gradientdescent_vhlearningrate_lepochcount,
                                       linear_regression_gradientdescent_vhlearningrate_hepochcount,
                                       linear_regression_gradientdescent_hlearningrate_lepochcount,
                                       linear_regression_gradientdescent_hlearningrate_hepochcount,
                                       linear_regression_gradientdescent_llearningrate_lepochcount,
                                       linear_regression_gradientdescent_llearningrate_hepochcount], axis=0)
linear_regression_methods.reset_index(inplace=True, drop=True)
display(linear_regression_methods)

	Coefficient	Estimate	Method
0	INTERCEPT	-0.000606	Normal_Equations
1	LIFEXP	0.233418	Normal_Equations
2	GDPCAP	0.508083	Normal_Equations
3	INTERCEPT	0.008008	VeryHighLearningRate_LowEpochCount
4	LIFEXP	0.281761	VeryHighLearningRate_LowEpochCount
5	GDPCAP	0.549961	VeryHighLearningRate_LowEpochCount
6	INTERCEPT	-0.000606	VeryHighLearningRate_HighEpochCount
7	LIFEXP	0.233418	VeryHighLearningRate_HighEpochCount
8	GDPCAP	0.508083	VeryHighLearningRate_HighEpochCount
9	INTERCEPT	0.110662	HighLearningRate_LowEpochCount
10	LIFEXP	0.317843	HighLearningRate_LowEpochCount
11	GDPCAP	0.413052	HighLearningRate_LowEpochCount
12	INTERCEPT	-0.000585	HighLearningRate_HighEpochCount
13	LIFEXP	0.237033	HighLearningRate_HighEpochCount
14	GDPCAP	0.504420	HighLearningRate_HighEpochCount
15	INTERCEPT	2.133856	LowLearningRate_LowEpochCount
16	LIFEXP	1.811625	LowLearningRate_LowEpochCount
17	GDPCAP	1.831115	LowLearningRate_LowEpochCount
18	INTERCEPT	0.128061	LowLearningRate_HighEpochCount
19	LIFEXP	0.320584	LowLearningRate_HighEpochCount
20	GDPCAP	0.415517	LowLearningRate_HighEpochCount

consolidated_ne = linear_regression_methods[linear_regression_methods['Method']=='Normal_Equations'].loc[:,"Estimate"]
consolidated_vhlr_lec = linear_regression_methods[linear_regression_methods['Method']=='VeryHighLearningRate_LowEpochCount'].loc[:,"Estimate"]
consolidated_vhlr_hec = linear_regression_methods[linear_regression_methods['Method']=='VeryHighLearningRate_HighEpochCount'].loc[:,"Estimate"]
consolidated_hlr_lec = linear_regression_methods[linear_regression_methods['Method']=='HighLearningRate_LowEpochCount'].loc[:,"Estimate"]
consolidated_hlr_hec = linear_regression_methods[linear_regression_methods['Method']=='HighLearningRate_HighEpochCount'].loc[:,"Estimate"]
consolidated_llr_lec = linear_regression_methods[linear_regression_methods['Method']=='LowLearningRate_LowEpochCount'].loc[:,"Estimate"]
consolidated_llr_hec = linear_regression_methods[linear_regression_methods['Method']=='LowLearningRate_HighEpochCount'].loc[:,"Estimate"]
linear_regression_methods_plot = pd.DataFrame({'NE': consolidated_ne.values,
                                               'VHLR_LEC': consolidated_vhlr_lec.values,
                                               'VHLR_HEC': consolidated_vhlr_hec.values,
                                               'HLR_LEC': consolidated_hlr_lec.values,
                                               'HLR_HEC': consolidated_hlr_hec.values,
                                               'LLR_LEC': consolidated_llr_lec.values,
                                               'LLR_HEC': consolidated_llr_hec.values},
                                              linear_regression_methods['Coefficient'].unique())
linear_regression_methods_plot

	NE	VHLR_LEC	VHLR_HEC	HLR_LEC	HLR_HEC	LLR_LEC	LLR_HEC
INTERCEPT	-0.000606	0.008008	-0.000606	0.110662	-0.000585	2.133856	0.128061
LIFEXP	0.233418	0.281761	0.233418	0.317843	0.237033	1.811625	0.320584
GDPCAP	0.508083	0.549961	0.508083	0.413052	0.504420	1.831115	0.415517

linear_regression_coefficent_estimation_methods_plot = linear_regression_methods_plot.plot.barh(figsize=(10, 6),width=0.90)
linear_regression_coefficent_estimation_methods_plot.set_xlim(-0.5,3)
linear_regression_coefficent_estimation_methods_plot.set_title("Linear Regression Coefficient Comparison by Estimation Method")
linear_regression_coefficent_estimation_methods_plot.set_xlabel("Linear Regression Coefficient Estimates")
linear_regression_coefficent_estimation_methods_plot.set_ylabel("Linear Regression Coefficients")
linear_regression_coefficent_estimation_methods_plot.grid(False)
linear_regression_coefficent_estimation_methods_plot.legend(loc='center left', bbox_to_anchor=(1.0, 0.5))
for container in linear_regression_coefficent_estimation_methods_plot.containers:
    linear_regression_coefficent_estimation_methods_plot.bar_label(container, fmt='%.5f', padding=5, color='black')

2. Summary

3. References

• [Book] Data Preparation for Machine Learning: Data Cleaning, Feature Selection, and Data Transforms in Python by Jason Brownlee
• [Book] Feature Engineering and Selection: A Practical Approach for Predictive Models by Max Kuhn and Kjell Johnson
• [Book] Feature Engineering for Machine Learning by Alice Zheng and Amanda Casari
• [Book] Applied Predictive Modeling by Max Kuhn and Kjell Johnson
• [Book] Data Mining: Practical Machine Learning Tools and Techniques by Ian Witten, Eibe Frank, Mark Hall and Christopher Pal
• [Book] Data Cleaning by Ihab Ilyas and Xu Chu
• [Book] Data Wrangling with Python by Jacqueline Kazil and Katharine Jarmul
• [Book] Regression Modeling Strategies by Frank Harrell
• [Python Library API] NumPy by NumPy Team
• [Python Library API] pandas by Pandas Team
• [Python Library API] seaborn by Seaborn Team
• [Python Library API] matplotlib.pyplot by MatPlotLib Team
• [Python Library API] itertools by Python Team
• [Python Library API] operator by Python Team
• [Python Library API] sklearn.experimental by Scikit-Learn Team
• [Python Library API] sklearn.impute by Scikit-Learn Team
• [Python Library API] sklearn.linear_model by Scikit-Learn Team
• [Python Library API] sklearn.preprocessing by Scikit-Learn Team
• [Python Library API] sklearn.metrics by Scikit-Learn Team
• [Python Library API] sklearn.model_selection by Scikit-Learn Team
• [Python Library API] sklearn.pipeline by Scikit-Learn Team
• [Python Library API] scipy by SciPy Team
• [Article] Exploratory Data Analysis in Python — A Step-by-Step Process by Andrea D'Agostino (Towards Data Science)
• [Article] Exploratory Data Analysis with Python by Douglas Rocha (Medium)
• [Article] 4 Ways to Automate Exploratory Data Analysis (EDA) in Python by Abdishakur Hassan (BuiltIn)
• [Article] 10 Things To Do When Conducting Your Exploratory Data Analysis (EDA) by Alifia Harmadi (Medium)
• [Article] How to Handle Missing Data with Python by Jason Brownlee (Machine Learning Mastery)
• [Article] Statistical Imputation for Missing Values in Machine Learning by Jason Brownlee (Machine Learning Mastery)
• [Article] Imputing Missing Data with Simple and Advanced Techniques by Idil Ismiguzel (Towards Data Science)
• [Article] Missing Data Imputation Approaches | How to handle missing values in Python by Selva Prabhakaran (Machine Learning +)
• [Article] Master The Skills Of Missing Data Imputation Techniques In Python(2022) And Be Successful by Mrinal Walia (Analytics Vidhya)
• [Article] How to Preprocess Data in Python by Afroz Chakure (BuiltIn)
• [Article] Easy Guide To Data Preprocessing In Python by Ahmad Anis (KDNuggets)
• [Article] Data Preprocessing in Python by Tarun Gupta (Towards Data Science)
• [Article] Data Preprocessing using Python by Suneet Jain (Medium)
• [Article] Data Preprocessing in Python by Abonia Sojasingarayar (Medium)
• [Article] Data Preprocessing in Python by Afroz Chakure (Medium)
• [Article] Detecting and Treating Outliers | Treating the Odd One Out! by Harika Bonthu (Analytics Vidhya)
• [Article] Outlier Treatment with Python by Sangita Yemulwar (Analytics Vidhya)
• [Article] A Guide to Outlier Detection in Python by Sadrach Pierre (BuiltIn)
• [Article] How To Find Outliers in Data Using Python (and How To Handle Them) by Eric Kleppen (Career Foundry)
• [Article] Statistics in Python — Collinearity and Multicollinearity by Wei-Meng Lee (Towards Data Science)
• [Article] Understanding Multicollinearity and How to Detect it in Python by Terence Shin (Towards Data Science)
• [Article] A Python Library to Remove Collinearity by Gianluca Malato (Your Data Teacher)
• [Article] 8 Best Data Transformation in Pandas by Tirendaz AI (Medium)
• [Article] Data Transformation Techniques with Python: Elevate Your Data Game! by Siddharth Verma (Medium)
• [Article] Data Scaling with Python by Benjamin Obi Tayo (KDNuggets)
• [Article] How to Use StandardScaler and MinMaxScaler Transforms in Python by Jason Brownlee (Machine Learning Mastery)
• [Article] Feature Engineering: Scaling, Normalization, and Standardization by Aniruddha Bhandari (Analytics Vidhya)
• [Article] How to Normalize Data Using scikit-learn in Python by Jayant Verma (Digital Ocean)
• [Article] What are Categorical Data Encoding Methods | Binary Encoding by Shipra Saxena (Analytics Vidhya)
• [Article] Guide to Encoding Categorical Values in Python by Chris Moffitt (Practical Business Python)
• [Article] Categorical Data Encoding Techniques in Python: A Complete Guide by Soumen Atta (Medium)
• [Article] Categorical Feature Encoding Techniques by Tara Boyle (Medium)
• [Article] Ordinal and One-Hot Encodings for Categorical Data by Jason Brownlee (Machine Learning Mastery)
• [Article] Hypothesis Testing with Python: Step by Step Hands-On Tutorial with Practical Examples by Ece Işık Polat (Towards Data Science)
• [Article] 17 Statistical Hypothesis Tests in Python (Cheat Sheet) by Jason Brownlee (Machine Learning Mastery)
• [Article] A Step-by-Step Guide to Hypothesis Testing in Python using Scipy by Gabriel Rennó (Medium)
• [Article] Gradient Descent and Stochastic Gradient Descent in R by Jason Anastasopoulos
• [Article] Linear Regression Tutorial Using Gradient Descent for Machine Learning by Jason Brownlee
• [Article] An Overview of Gradient Descent Optimization Algorithms by Sebastian Ruder
• [Article] What is Gradient Descent? by IBM Team
• [Article] Gradient Descent in Machine Learning: A Basic Introduction by Niklas Donges
• [Article] Gradient Descent for Linear Regression Explained, Step by Step by Boris Giba
• [Article] Gradient Descent Explained Simply with Examples by Ajitesh Kumar
• [Article] Implementing the Gradient Descent Algorithm in R by Richter Walsh
• [Article] Regression via Gradient Descent in R by Matt Bogard
• [Article] Stochastic Gradient Descent by Jonathon Price, Alfred Wong, Tiancheng Yuan, Joshua Mathew and Taiwo Olorunniwo
• [Article] Difference Between Batch Gradient Descent and Stochastic Gradient Descent by Geeks For Geeks Team
• [Article] Stochastic Gradient Descent Vs Gradient Descent: A Head-To-Head Comparison by SDS Club Team
• [Article] Differences Between Gradient, Stochastic and Mini Batch Gradient Descent by Baeldung
• [Article] Difference Between Backpropagation and Stochastic Gradient Descent by Jason Brownlee
• [Article] ML | Normal Equation in Linear Regression by Geeks For Geeks Team
• [Article] Derivation of the Normal Equation for Linear Regression by Eli Bendersky
• [Article] Normal Equation by ML Wiki Team
• [Article] Normal Equations by Marco Taboga
• [Article] Fitting a Model via Closed-form Equations versus Gradient Descent Versus Stochastic Gradient Descent Versus Mini-Batch Learning. What is the Difference? by Sebastian Raschka
• [Article] Gradient Descent Versus Normal Equation For Regression Problems by Pushkara Sharma
• [Publication] New Methods for the Determination of Comet Orbits by Adrien-Marie Legendre
• [Publication] General Method for the Resolution of a System of Simultaneous Equations by Augustine Cauchy
• [Publication] Data Quality for Machine Learning Tasks by Nitin Gupta, Shashank Mujumdar, Hima Patel, Satoshi Masuda, Naveen Panwar, Sambaran Bandyopadhyay, Sameep Mehta, Shanmukha Guttula, Shazia Afzal, Ruhi Sharma Mittal and Vitobha Munigala (KDD ’21: Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery & Data Mining)
• [Publication] Overview and Importance of Data Quality for Machine Learning Tasks by Abhinav Jain, Hima Patel, Lokesh Nagalapatti, Nitin Gupta, Sameep Mehta, Shanmukha Guttula, Shashank Mujumdar, Shazia Afzal, Ruhi Sharma Mittal and Vitobha Munigala (KDD ’20: Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining)
• [Publication] Multiple Imputation of Discrete and Continuous Data by Fully Conditional Specification by Stef van Buuren (Statistical Methods in Medical Research)
• [Publication] Mathematical Contributions to the Theory of Evolution: Regression, Heredity and Panmixia by Karl Pearson (Royal Society)
• [Publication] A New Family of Power Transformations to Improve Normality or Symmetry by In-Kwon Yeo and Richard Johnson (Biometrika)
• [Course] IBM Data Analyst Professional Certificate by IBM Team (Coursera)
• [Course] IBM Data Science Professional Certificate by IBM Team (Coursera)
• [Course] IBM Machine Learning Professional Certificate by IBM Team (Coursera)

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