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Machine Learning Basics for a Newbie
6 Steps of Machine learning Lifecycle Introduction to Predictive Modeling
Introduction to Exploratory Data Analysis & Data Insights Descriptive Statistics Inferential Statistics How to Understand Population Distributions?
Reading Data Files into Python Different Variable Datatypes
Probability for Data Science Basic Concepts of Probability Axioms of Probability Conditional Probability
Central Tendencies for Continuous Variables Spread of Data KDE plots for Continuous Variable Overview of Distribution for Continuous variables Normal Distribution Skewed Distribution Skeweness and Kurtosis Distribution for Continuous Variable
Central Tendencies for Categorical Variables Understanding Discrete Distributions Performing EDA on Categorical Variables
Dealing with Missing Values Understanding Outliers Identifying Outliers in Data Outlier Detection in Python Outliers Detection Using IQR, Z-score, LOF and DBSCAN
Sample and Population Central Limit Theorem Confidence Interval and Margin of Error
Bivariate Analysis Introduction
Covariance Pearson Correlation Spearman's Correlation & Kendall's Tau Correlation versus Causation Tabular and Graphical methods for Bivariate Analysis Performing Bivariate Analysis on Continuous-Continuous Variables
Tabular and Graphical methods for Continuous-Categorical Variables Introduction to Hypothesis Testing P-value Two sample Z-test T-test T-test vs Z-test Performing Bivariate Analysis on Continuous-Catagorical variables
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Multivariate Analysis A Comprehensive Guide to Data Exploration The Data Science behind IPL
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Dealing with Missing Values Replacing Missing Values Imputing Missing Values in Data Working with Categorical Variables Working with Outliers Preprocessing Data for Model Building
Understanding Cost Function Understanding Gradient Descent Math Behind Gradient Descent Assumptions of Linear Regression Implement Linear Regression from Scratch Train Linear Regression in Python Implementing Linear Regression in R Diagnosing Residual Plots in Linear Regression Models Generalized Linear Models Introduction to Logistic Regression Odds Ratio Implementing Logistic Regression from Scratch Introduction to Scikit-learn in Python Train Logistic Regression in python Multiclass using Logistic Regression How to use Multinomial and Ordinal Logistic Regression in R ? Challenges with Linear Regression Introduction to Regularisation Implementing Regularisation Ridge Regression Lasso Regression
Introduction to K Nearest Neighbours Determining the Right Value of K in KNN Implement KNN from Scratch Implement KNN in Python
Bias Variance Tradeoff Introduction to Overfitting and Underfitting Visualizing Overfitting and Underfitting Selecting the Right Model What is Validation? Hold-Out Validation Understanding K Fold Cross Validation
Introduction to Feature Selection Feature Selection Algorithms Missing Value Ratio Low Variance Filter High Correlation Filter Backward Feature Elimination Forward Feature Selection Implement Feature Selection in Python Implement Feature Selection in R
Introduction to Decision Tree Purity in Decision Tree Terminologies Related to Decision Tree How to Select Best Split Point in Decision Tree? Chi-Squares Information Gain Reduction in Variance Optimizing Performance of Decision Tree Train Decision Tree using Scikit Learn Pruning of Decision Trees
Introduction to Feature Engineering Feature Transformation Feature Scaling Feature Engineering Frequency Encoding Automated Feature Engineering: Feature Tools
Introduction to Naive Bayes Conditional Probability and Bayes Theorem Introduction to Bayesian Adjustment Rating: The Incredible Concept Behind Online Ratings! Working of Naive Bayes Math behind Naive Bayes Types of Naive Bayes Implementation of Naive Bayes
Understanding how to solve Multiclass and Multilabled Classification Problem Evaluation Metrics: Multi Class Classification
Introduction to Ensemble Techniques Basic Ensemble Techniques Implementing Basic Ensemble Techniques Finding Optimal Weights of Ensemble Learner using Neural Network Why Ensemble Models Work well?
Introduction to Stacking Implementing Stacking Variants of Stacking Implementing Variants of Stacking Introduction to Blending Bootstrap Sampling Introduction to Random Sampling Hyper-parameters of Random Forest Implementing Random Forest Out-of-Bag (OOB) Score in the Random Forest IPL Team Win Prediction Project Using Machine Learning Introduction to Boosting Gradient Boosting Algorithm Math behind GBM Implementing GBM in python Regularized Greedy Forests Extreme Gradient Boosting Implementing XGBM in python Tuning Hyperparameters of XGBoost in Python Implement XGBM in R/H2O Adaptive Boosting Implementing Adaptive Boosing LightGBM Implementing LightGBM in Python Catboost Implementing Catboost in Python
Different Hyperparameter Tuning methods Implementing Different Hyperparameter Tuning methods GridsearchCV RandomizedsearchCV Bayesian Optimization for Hyperparameter Tuning Hyperopt
Understanding SVM Algorithm SVM Kernels In-depth Intuition and Practical Implementation SVM Kernel Tricks Kernels and Hyperparameters in SVM Implementing SVM from Scratch in Python and R
Introduction to Principal Component Analysis Steps to Perform Principal Compound Analysis Computation of Covariance Matrix Finding Eigenvectors and Eigenvalues Implementing PCA in python Visualizing PCA A Brief Introduction to Linear Discriminant Analysis Introduction to Factor Analysis
Introduction to Clustering Applications of Clustering Evaluation Metrics for Clustering Understanding K-Means Implementation of K-Means in Python Implementation of K-Means in R Choosing Right Value for K Profiling Market Segments using K-Means Clustering Hierarchical Clustering Implementation of Hierarchial Clustering DBSCAN Defining Similarity between clusters Build Better and Accurate Clusters with Gaussian Mixture Models
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8 Ways to Improve Accuracy of Machine Learning Models
Introduction to Dask Working with CuML
Introduction to Machine Learning Interpretability Framework and Interpretable Models model Agnostic Methods for Interpretability Implementing Interpretable Model Understanding SHAP Out-of-Core ML Introduction to Interpretable Machine Learning Models Model Agnostic Methods for Interpretability Game Theory & Shapley Values
Introduction to Machine Learning Interpretability Framework and Interpretable Models model Agnostic Methods for Interpretability Implementing Interpretable Model Understanding SHAP Out-of-Core ML Introduction to Interpretable Machine Learning Models Model Agnostic Methods for Interpretability Game Theory & Shapley Values
Introduction to AutoML Implementation of MLBox Introduction to PyCaret TPOT Auto-Sklearn EvalML
Pickle and Joblib Introduction to Model Deployment
Deploying Machine Learning Model using Streamlit Deploying ML Models in Docker Deploy Using Streamlit Deploy on Heroku Deploy Using Netlify Introduction to Amazon Sagemaker Setting up Amazon SageMaker Using SageMaker Endpoint to Generate Inference Deploy on Microsoft Azure Cloud Introduction to Flask for Model Deploying ML model using Flask
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Multiple Linear Regression using Python

Amrutha
Amrutha Last Updated : 21 Oct, 2024
6 min read

This article, we will be dealing with multi-linear regression, and we will take a dataset that contains information about 50 startups. Features include R&D Spend, Administration, Marketing Spend, State, and finally, Profit. Here we have to build the machine learning model to predict the profit of the startups.

Let’s get started.

Multiple Linear Regression is a machine learning algorithm where we provide multiple independent variables for a single dependent variable. However, linear regression only requires one independent variable as input.

In this article, you will explore the multiple linear regression formula, understand a multiple linear regression example, learn how to implement it using multiple linear regression in Python, and discover its significance in machine learning. By the end, you’ll have a solid grasp of this essential statistical technique and its applications in data analysis.

This article was published as a part of the Data Science Blogathon.

Table of contents

Working with Dataset

Let’s start by importing some libraries.

import numpy as np
import pandas as pd
from sklearn.model_selection import train_test_split
from sklearn.linear_model import LinearRegression
from sklearn.metrics import r2_score
import matplotlib.pyplot as plt
import seaborn as sns
import warnings
warnings.filterwarnings("ignore")

Import train_test_split to split the dataset into training and testing datasets. And Linear Regression is the model on which we have to work. Import this model from scikit learn library. r2_score is to find the accuracy of the model. Matplotlib and seaborn are used for visualizations. Finally, import warnings and set it to ignore so that it will ignore all the warnings that we will come throughout.

Here is the link for the dataset. Download it and import it by passing the path of the dataset file into read_csv().

Let us view our data frame.

Python Code:

import numpy as np
import pandas as pd
from sklearn.model_selection import train_test_split
from sklearn.linear_model import LinearRegression
from sklearn.metrics import r2_score
# import matplotlib.pyplot as plt
# import seaborn as sns
import warnings
warnings.filterwarnings("ignore")

#import dataset
startup_df=pd.read_csv(r'50_Startups.csv')

print(startup_df.head())

View the shape of the data frame.

shape=startup_df.shape
print("Dataset contains {} rows and {} columns".format(shape[0],shape[1]))

The dataset contains 50 rows and 5 columns.

View all the columns in the data frame.

startup_df.columns
Multiple Linear Regression

Data frame contains R&D Spend, Administration, Marketing Spend, State, and Profit.

View the statistical description of the dataset which includes the total count of each column, mean of all values, standard deviation, minimum, maximum values, and 25th, 50th, 75th per cent values of the dataset.

#Statistical Details of the dataset
startup_df.describe()

Define X and Y

This is like extracting dependent and independent variables.

We have to define x and y for the model. x and y are input and output features of the dataset. So taking x features as input values that are independent, our model will predict the outcome which is y that are dependent.

x=startup_df.iloc[:,:4]
y=startup_df.iloc[:,4]

Perform One-Hot Encoding

We use one-hot encoding when there are categorical values in our dataset. Here for us, there is a state column that is categorical, so we have to use one-hot encoding to convert them.

So, import One-HotEncoder from scikit learn library.

from sklearn.preprocessing import OneHotEncoder
ohe=OneHotEncoder(sparse=False)
x=ohe.fit_transform(startup_df[['State']])

View x.

x

array([[0., 0., 1.],
[1., 0., 0.],
[0., 1., 0.],
[0., 0., 1.],
[0., 1., 0.],
[0., 0., 1.],
[1., 0., 0.],
[0., 1., 0.],
[0., 0., 1.],
[1., 0., 0.],
[0., 1., 0.],
[1., 0., 0.],
[0., 1., 0.],
[1., 0., 0.],
[0., 1., 0.],
[0., 0., 1.],
[1., 0., 0.],
[0., 0., 1.],
[0., 1., 0.],
[0., 0., 1.],
[1., 0., 0.],
[0., 0., 1.],
[0., 1., 0.],
[0., 1., 0.],
[0., 0., 1.],
[1., 0., 0.],
[0., 1., 0.],
[0., 0., 1.],
[0., 1., 0.],
[0., 0., 1.],
[0., 1., 0.],
[0., 0., 1.],
[1., 0., 0.],
[0., 1., 0.],
[1., 0., 0.],
[0., 0., 1.],
[0., 1., 0.],
[1., 0., 0.],
[0., 0., 1.],
[1., 0., 0.],
[1., 0., 0.],
[0., 1., 0.],
[1., 0., 0.],
[0., 0., 1.],
[1., 0., 0.],
[0., 0., 1.],
[0., 1., 0.],
[1., 0., 0.],
[0., 0., 1.],
[1., 0., 0.]])

It will give an array like this. Let us see what are those three categories.

ohe.categories_
[array([‘California’, ‘Florida’, ‘New York’], dtype=object)]

Here [0., 0., 1.] indicates NewYork, [0., 1., 0.] indicates Florida  and [1., 0., 0.] indicates California.

Change Columns using Column Transformer

For this import make_column_transformer from scikit learn library and pass the column that we want to transfer.

from sklearn.compose import make_column_transformer
col_trans=make_column_transformer(
    (OneHotEncoder(handle_unknown='ignore'),['State']),
    remainder='passthrough')
x=col_trans.fit_transform(x)

Now view x.

It will look like this.

Split the Dataset into Train Set and Test Set

Now, split your dataset into two parts in which 80% of the dataset will go to the training set, and 20% of the dataset will go to the testing set. Actually, you can divide it as per your wish by setting the value into test_size.

x_train,x_test,y_train,y_test=train_test_split(x,y,test_size=0.2,random_state=0)

View the shapes of splitter data.

#shapes of splitted data
print("X_train:",x_train.shape)
print("X_test:",x_test.shape)
print("Y_train:",y_train.shape)
print("Y_test:",y_test.shape)

X_train: (40, 6)
X_test: (10, 6)
Y_train: (40,)
Y_test: (10,)

Train the Model

To train the model, we have to import the Linear Regression model, which we have already created at the beginning. Use the fit method, and pass the training sets into it to train the model.

linreg=LinearRegression()
linreg.fit(x_train,y_train)

Predict the Test Results

Use the predict method to predict the results, then pass the independent variables into it and view the results. It will give the array with all the values in it.

y_pred=linreg.predict(x_test)
y_pred

Evaluate the Model

We have different metrics to find the accuracy score of the model, and here we use r2_score to evaluate our model and find its accuracy.

Accuracy=r2_score(y_test,y_pred)*100
print(" Accuracy of the model is %.2f" %Accuracy)

The accuracy of the model is 93.47.

Plot the Results

We will plot the scatter plot between actual values and predicted values. Use xlabel to label the x-axis and use ylabel to label the y-axis.

plt.scatter(y_test,y_pred);
plt.xlabel('Actual');
plt.ylabel('Predicted');
Ploting graph , X- axis and Y - axis

Regression plot of our model.

A regression plot is useful to understand the linear relationship between two parameters. It creates a regression line in-between those parameters and then plots a scatter plot of those data points.

sns.regplot(x=y_test,y=y_pred,ci=None,color ='red');
regression plot and scatter plots

Predicted Values

Let us create a new data frame that contains actual values, predicted values, and differences between them so that we will understand how near the model predicts its actual value.

pred_df=pd.DataFrame({'Actual Value':y_test,'Predicted Value':y_pred,'Difference':y_test-y_pred})

View the data frame.

pred_df
Predicted Values

Here we can see the difference between Actual values and predicted values which are not very high. When values are in the range of lakhs, then the difference in thousands is not much.
We have already seen that the accuracy of this model is about 93 percent.

Conclusion

In this article, we have created a new Linear Regression model, and we learned how to perform One-Hot Encoding and where to perform it. We used a column transformer and then trained the model, predicted the results, evaluated the model using r2_score metrics, and plotted the results.

Hope you guys found it useful.

Q1. What is the formula slope of multiple linear regression?

A. The formula for the slope coefficients (β) in multiple linear regression is:
β = (X’X)^(-1) X’Y
where X is the design matrix (containing the independent variables), Y is the vector of the dependent variable, and “^(-1)” denotes the inverse of a matrix.

Q2. What is the equation for multiple regression?

A. The equation for multiple regression is:
Y = β0 + β1X1 + β2X2 + … + βkXk + ε
where Y is the dependent variable, X1, X2, …, Xk are the independent variables, β0 is the intercept, β1, β2, …, βk are the coefficients of the independent variables, and ε is the error term.

Q3.What are the 5 assumptions of multiple linear regression?

5 Assumptions of Multiple Linear Regression:
Linearity: Relationship between variables is linear.
Independence: Observations are independent.
Homoscedasticity: Constant variance of residuals.
Normality: Residuals follow normal distribution.
No Multicollinearity: Independent variables are not highly correlated.

Q4.How to detect multicollinearity?

Correlation Matrix: High correlations between independent variables.
VIF: Values greater than 5 indicate multicollinearity.
Eigenvalues/Condition Indices: Small eigenvalues or high condition indices suggest problems.
Tolerance: Values close to 0 indicate multicollinearity.
Auxiliary Regression: High R-squared values for regressions of independent variables against each other.

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Connect with me on LinkedIn: https://www.linkedin.com/in/amrutha-k-6335231a6vl/

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Amrutha
Amrutha

This is Amrutha, I am pursuing B.Tech in the Computer science Department. I am interested in developing ML Models with python and Data Analysis. And also I have an interest in Web Development. I hope my articles in Analytics Vidhya help you to learn better. Thank you!!

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