Understanding Multivariate Regression in Machine Learning: Techniques and Implementation

Regression analysis is used to understand how independent variables influence a dependent variable, such as estimating property prices based on the area of a house. In simple regression, a single independent variable is used to predict the dependent variable. 

However, in real-world scenarios, multiple factors often influence outcomes, which is where multivariate regression comes in.

Multivariate regression in machine learning can predict healthcare treatment outcomes by analyzing patient demographics and clinical data.

What is Multivariate Regression? Applications and Benefits

Multivariate regression is a statistical technique used to model the relationship between multiple independent variables (predictors) and a single dependent variable (outcome). Multivariate regression models relationships between multiple predictors and a single outcome. 

The ability of multivariate regression in predictive analysis, risk analysis, and optimization of processes makes it suitable to be used in industries like healthcare and manufacturing.

Let’s explore in detail the reasons for performing multivariate regression.

Why Perform Multivariate Regression Analysis?

Multivariate regression analysis can uncover the relationship between multiple independent variables (predictors) and a single dependent variable (outcome), making it a powerful tool for understanding complex data. 

It enhances the accuracy of predictions, aids in driving outcomes, and supports better decision-making across industries like finance, healthcare, and marketing. This ability to analyze and predict based on multiple factors ensures more informed, data-driven choices in real-world scenarios.

Here’s why multivariate regression in machine learning is used.

• Accurate Predictions

Multivariate regression makes precise predictions by analyzing multiple factors simultaneously.

Example: A retail store predicts monthly sales based on factors like advertising spend, seasonality, and customer traffic.

• Understand Relationships

It helps identify the relationships between multiple predictors and a dependent variable, making it easier to understand complex interactions.

Example: In healthcare, doctors use multivariate regression to understand how factors like age and blood pressure affect the risk of developing heart disease.

• Control Confounding Variables

By accounting for multiple predictors, it reduces the influence of confounding variables that might distort the relationship between the main variables of interest.

Example: Ensuring results are solely influenced by the drug" could expand on statistical controls (e.g., adjusting for patient demographics).

• Improved Decision Making

It provides valuable insights into which factors are most influential, helping businesses make more informed, data-driven decisions.

Example: A marketing team uses multivariate regression to evaluate how factors like product pricing and social media engagement influence customer purchase decisions. 

• Model Complex Scenarios

When outcomes are influenced by more than one factor, multivariate regression can model these complex scenarios more effectively.

Example: A car manufacturer uses multivariate regression to predict vehicle fuel efficiency based on multiple factors like engine size, weight, tire type, and aerodynamics. 

• Assess the Impact of Multiple Factors

It helps in evaluating the individual and combined impact of several predictors on the outcome variable.

Example: In real estate, a company uses multivariate regression to assess how location, square footage, and property age collectively influence home prices. 

Now that you know why multivariate regression in machine learning is used in industries, let’s understand an important component of this concept, which is the cost function.

What is the Cost Function of Multivariate Regression?

The cost function in multivariate regression measures how well the prediction of the model matches the actual values (observed data). By minimizing this error during model training, you can ultimately improve the model’s accuracy. 

Mean Squared Error (MSE) is one of the most common cost functions for regression tasks. By penalizing large errors more significantly than smaller ones, it encourages the model to make precise predictions.

By using techniques like parameter tuning, you can reduce MSE, thereby improving the model’s accuracy over iterations.

Here’s how the Mean Squared Error (MSE) is calculated.

Where, 

yi is actual value (true value)

^yi ​is the predicted value

n is the number of data points

To implement the Mean Squared Error (MSE) cost function, first split the dataset into training and testing sets and then train a linear regression model on the training data.

After making predictions on the test data, it calculates the difference between the actual values (y_test) and predicted values (y_pred).

Here’s a code snippet for implementing Multivariate Regression with MSE Using Scikit-Learn: 

# Import necessary 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 mean_squared_error

# Sample data (assuming this data is already cleaned)
data = {'Feature1': [1, 2, 3, 4, 5],
        'Feature2': [2, 3, 4, 5, 6],
        'Target': [3, 4, 5, 6, 7]}
df = pd.DataFrame(data)
# Define features (independent variables) and target (dependent variable)
X = df[['Feature1', 'Feature2']]  # Independent variables
y = df['Target']  # Dependent variable
# Split the data into training and testing sets
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)
# Create a linear regression model
model = LinearRegression()
# Fit the model on training data
model.fit(X_train, y_train)

# Make predictions on the test data
y_pred = model.predict(X_test)

# Calculate the Mean Squared Error (MSE)
mse = mean_squared_error(y_test, y_pred)

print("Mean Squared Error (MSE):", mse)

Output:

For a given data set:

data = {'Feature1': [1, 2, 3, 4, 5],
        'Feature2': [2, 3, 4, 5, 6],
        'Target': [3, 4, 5, 6, 7]}

Result: The following is an idealized output due to simple linear relationships. The result may vary for complex relationships. The smaller MSE value indicates that the model's predictions are closer to the actual values, demonstrating better performance.

Mean Squared Error (MSE): 3.6e-30

With the cost function understood, let’s explore the steps to implement multivariate regression.

Essential Steps to Implement Multivariate Regression in Machine Learning

Implementing multivariate regression in machine learning involves selecting relevant features, normalizing them for consistency, and defining a hypothesis to model relationships between inputs and outputs. 

Here are the steps involved in implementing multivariate regression.

Selection of Features

Feature selection is the process of choosing the most relevant variables that contribute to predicting the outcome. Through feature selection, you can avoid redundant features that can degrade the model’s performance.

Example: In predicting house prices using multivariate regression, the features might include factors like the square foot of the house, number of bedrooms, color of house and proximity to public transport.

Features like the color of the house are not relevant and can be removed during the process.

Feature Normalizing

Features in your dataset may have different units or scales (e.g., age and marital status), which can affect the regression analysis. Use techniques like min-max scaling or standardization to scale the features so that they are all on a similar scale.

Example: Consider a model to predict employee salaries based on features such as years of experience, education level, and location. Here, the features may have different scales.

Without normalization, features like years of experience (0-30) might dominate the model since they are on a much larger scale.

Selecting Loss Function and Hypothesis

The loss function measures how well the model’s predictions align with the actual outcomes. A common loss function is Mean Squared Error (MSE), which penalizes larger errors more heavily. 

The hypothesis is the model or equation that explains the relationship between the independent variables (features) and the dependent variable (outcome).

Example: If you’re building a forecast sales model for a retail store based on features like holiday season, advertising budget, and customer foot traffic. The hypothesis (model) could be represented as:

The loss function in this case would be MSE, and the goal is to minimize the MSE by adjusting the model parameters 

Fixing Hypothesis Parameter

The parameters (0,1 etc) in the hypothesis are initially set randomly. The model must learn to adjust these parameters based on the training data in order to minimize the error (loss function). This process is usually done through optimization techniques like Gradient Descent.

Example: In predicting car prices based on features like mileage, age, and car brand, the initial hypothesis will be:

Initially, the parameters 

will be set to random values. These parameters will be adjusted during training to find the best fit.

Also Read: Comprehensive Guide to Hypothesis in Machine Learning: Key Concepts, Testing and Best Practices

Reducing the Loss Function and Analyzing the Hypothesis Function

The loss function is reduced using optimization techniques like Gradient Descent. After training, you have to analyze the model to see if it makes sense logically and aligns with expectations. 

Gradient Descent iteratively minimizes the loss function by updating parameters in the direction of the steepest descent.

Example: After training a model to predict hospital readmissions based on features like age, health history, treatment received, and insurance status, the parameters of the hypothesis will be fine-tuned to minimize the MSE. 

After training, you need to check whether the parameter for age is negatively correlated with readmission risk (older patients might have a higher risk).

If the obtained results align with medical knowledge and the loss function has been minimized, the model is considered successful.

Also Read: How to Perform Multiple Regression Analysis?

The above steps will guide you in successfully implementing multivariate regression in machine learning. Now, let's explore different ways of using multivariate regression models.

Different Ways to Use Multivariate Regression: Linear & Logistic Approaches

Multivariate regression can be applied in two primary forms: Multivariate Linear Regression and Multivariate Logistic Regression. The linear regression handles continuous outcomes, and logistic regression focuses on categorical outcomes.

Here’s a detailed look at multivariate linear regression in machine learning, followed by logistic regression.

Multivariate Linear Regression in Machine Learning

A multivariate linear regression approach is used when the relationship between a dependent variable (target) and multiple independent variables (features) is assumed to be linear.

The objective is to model the target variable as a weighted sum of the input features, allowing for prediction based on these relationships. It is widely used for continuous outcome variables.

Multivariate linear regression in machine learning is calculated using the following formula.

Here,

• y is the predicted dependent variable (e.g., house price, salary).
• $x_1,x_2,x_3,.......,x_n$ are the independent variables (features)
• ${\beta }_0$ is the intercept.
• ${\beta }_1,{\beta }_2,....,{\beta }_3$ are the coefficients for each feature.
• $ϵ$ is the error term

Example: Imagine you want to predict the price of a house based on various features such as square footage, number of bedrooms, and age of the house. These features are all independent variables that likely influence the price of the house.

Using the formula, you get:

The model will learn the best values after training. It may give the value like:

Price = 50,000 + 200 (Square Footage) + 10,000 (Number of Bedrooms) − 2,000 (Age of House)

Here, 

• For each additional square foot, the price increases by USD 200.
• For each additional bedroom, the price increases by USD 10,000.
• Each year of age decreases the price by USD 2,000.

By giving values for square footage, number of bedrooms and age of house, you can predict the price of the house.

The function of linear regression is to predict a continuous dependent variable based on multiple independent variables. 

Here’s when you can use multivariate linear regression in machine learning.

• Continuous Dependent Variable: The dependent variable (target) must be continuous, meaning it can take on any value within a range.

Example: If the goal is to predict the price of a house based on features such as square footage, number of bedrooms, and location, multivariate linear regression is suitable since the price is continuous.

• Independence of Observations: The observations have to be independent of one another. It assumes that each data point is independent of the others.

Example: When predicting sales revenue based on various product features, the sales of each product should be independent of others to avoid biased results.

• Normal Distribution of Errors: The errors (residuals) should be normally distributed. This is important for making reliable confidence intervals and significance tests for the coefficients.

Example: If you are predicting production costs based on various factors such as machine time, labor hours, and material costs, the residuals should follow a normal distribution to ensure valid predictions and inference.

• Linear Relationship Between Predictors and Dependent Variable: There should be a linear relationship between the independent variables and the dependent variable. You can represent the dependent variable as a weighted sum of the independent variables plus a constant (intercept).

Example: In predicting salary based on experience and education level, the relationship should be approximately linear (e.g., salary increases by a fixed amount with each additional year of experience).

Also Read: Linear Regression Model: What is & How it Works?

Now that you’ve seen how to use multivariate linear regression to handle outcomes, let’s explore the logistic regression approach. 

Multivariate Logistic Regression in Machine Learning

Multivariate logistic regression is used when the dependent variable is binary or categorical. In this approach, the output is transformed into probabilities, which range between 0 and 1, using the logistic function (sigmoid). 

It is usually used to solve problems like predicting whether a customer will buy a product (yes/no) or whether a patient will develop a disease (yes/no).

The multivariate logistic regression is calculated using the formula:

Here, 

• P (y = 1)  represents the probability of the event (e.g., a customer churning).
• $x_1,x_2,x_3,.......,x_n$ are the independent variables (features)
• ${\beta }_0$ is the intercept
• ${\beta }_1,{\beta }_2,....,{\beta }_3$ are the coefficients for each feature
• e is the base of the natural logarithm

Example: Build a model for a telecom company to predict whether a customer will churn (leave) or stay based on features like monthly usage, number of support tickets, and contract type. 

After training, the model might output a result where:

By inputting values for monthly usage, support tickets, and contract type, you can calculate churn.

• If P (Churn = 1) > 0.5, classify as churn
• If P(Churn = 1) ≤ 0.5P, classify as not churn

Logistic regression is most suitable for classification problems, such as probability prediction or when data points are independent of each other.

Here’s when you can use logistic regression in machine learning.

• Binary Dependent Variable: If the dependent variable (target) is binary, meaning it has two possible outcomes.

Example: If the goal is to predict whether a customer will churn (yes/no) or whether an email is spam (spam/not spam), logistic regression is a good choice.

• Independence of Observations: The data points are independent of each other. Logistic regression assumes that each observation is independent from others.

Example: In customer churn prediction, each customer's decision to leave the service should be independent of other customers' decisions.

• Large Sample Size: Performs better with larger datasets. A small sample size may cause overfitting or inaccurate estimates.

Example: When predicting a rare event, like fraud detection in financial transactions, having a large dataset with numerous examples of both fraud and non-fraud cases will ensure accurate model training.

• Prediction of Probabilities: The model needs to output probabilities that indicate the likelihood of the occurrence of a certain event or category.

Example: In a marketing campaign, you may want to predict that a customer will respond to an offer rather than just a binary classification of whether they will respond or not.

Also Read: Logistic Regression for Machine Learning: A Complete Guide

Now that you’ve seen how to use multivariate logistic regression to handle outcomes, let’s explore the benefits and issues associated with multivariate regression in machine learning.

Advantages and Disadvantages of Multivariate Regression in Machine Learning

While multivariate regression has advantages like handling multiple predictors and large datasets, it has issues like overfitting and sensitivity to outliers.

Here are the advantages of multivariate regression in machine learning.

• Ability to Handle Multiple Predictors

It captures complex interactions between predictors and outcomes. This makes it suitable in conditions where multiple factors influence the outcome.

Example: Predicting housing prices based on multiple features such as square footage, number of bedrooms, neighborhood quality, etc.

• Interpretability

The model provides interpretable coefficients that show how each independent variable affects the dependent variable. This is useful in understanding feature importance.

Example: In predicting a company's sales revenue based on advertising spend, multivariate regression can assess the impact of each advertising channel (TV, digital, print).

• Linear Relationships:

Multivariate regression is effective in providing reliable predictions when the independent variables and the dependent variable have an approximately linear relationship.

Example: Predicting salary based on years of experience and education level can often be modeled well using linear regression.

• Scalability

It can be applied to large datasets with a high number of variables as long as the data does not violate assumptions like multicollinearity.

Example: Predicting customer lifetime value using numerous customer characteristics (e.g., age, income, spending habits, etc.) can scale to large datasets.

Multivariate regression fails when it comes to handling non-linear relationships and multicollinearity, making them prone to errors.

Here are the disadvantages of multivariate regression in machine learning.

• Multicollinearity

If the independent variables have a high correlation with each other, the model becomes unstable and cannot predict outcomes accurately.

Example: In predicting sales revenue based on advertising spend, if TV and radio advertising spending are highly correlated, the model may not correctly assess the impact of each channel.

• Overfitting

With too many predictors and a small dataset, the model might memorize the training data rather than learn general patterns.

Example: Predicting company performance using only a few months of data can lead to overfitting, as the model may capture noise rather than meaningful patterns.

Also Read: What is Overfitting & Underfitting In Machine Learning ? [Everything You Need to Learn]

• Sensitivity to Outliers

Multivariate regression can be highly sensitive to outliers, as they can influence the estimated coefficients and model predictions.

Example: Predicting employee performance based on age and tenure may be skewed if a few outliers exist, such as highly exceptional or underperforming individuals.

Also Read: Outlier Analysis in Data Mining: Techniques, Detection Methods, and Best Practices

• Non-linearity Limitations

If the relationship between predictors and the dependent variable is nonlinear, the model may not perform well.

Example: Predicting a stock's price based on factors like market sentiment and company performance might involve non-linear relationships that multivariate regression cannot model properly.

Also Read: 6 Types of Regression Models in Machine Learning: Insights, Benefits, and Applications in 2025

The advantages and limitations of multivariate regression can impact your decision to use it in machine learning. Let’s explore how to deepen your understanding of this technique to make the right choice for your model.

How upGrad Can Help You Advance Your Career?

The function of multivariate regression is to solve real-world problems, such as predicting sales revenue, by considering multiple factors simultaneously.

Professionals such as data scientists and business analysts require expertise in multivariate regression to solve complex problems. You need specialized learning to stay relevant in this field.

upGrad’s courses in machine learning will equip you with the expertise to apply multivariate regression and other techniques to solve industry-specific challenges.

Here are the related courses offered by upGrad:.

• Unsupervised Learning: Clustering 
• Linear Regression - Step-by-Step Guide
• Logistic Regression for Beginners
• Linear Algebra for Analysis

Do you need help deciding which courses can help you in machine learning? Contact upGrad for personalized counseling and valuable insights. For more details, you can also visit your nearest upGrad offline center. 

Similar Read:

• What is Multinomial Logistic Regression? Definition & Examples
• Assumptions of Linear Regression
• What is Regression: Regression Analysis Explained

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