Introduction to Multivariate Regression Analysis

Contributed by: Pooja Korwar

Introduction to Multivariate Regression

In today’s world, data is everywhere. Data itself is just facts and figures, and this needs to be explored to get meaningful information. Hence, data analysis is important. Data analysis is the process of applying statistical analysis and logical techniques to describe and visualize, reduce, revise, summarize, and assess data into useful information that provides a better context for the data. Check out the Statistical Analysis course to learn the statistical methods involved in data analysis.

Data analysis plays a significant role in finding meaningful information which will help businesses make better decisions basis the output.

Along with Data analysis, Data science also comes into the picture. Data science is a field combining many methods of scientific methodology, processes, algorithms, and tools to extract information from, particularly huge datasets for insights on structured and unstructured data. A different range of terms related to data mining, cleaning, analyzing, and interpreting data are often used interchangeably in data science.

Let us look at one of the important models of data science.

Regression analysis

Regression analysis is one of the most sought out methods used in data analysis. It follows a supervised machine-learning algorithm. Regression analysis is an important statistical method that allows us to examine the relationship between two or more variables in the dataset.

Regression analysis is a way of mathematically differentiating variables that have an impact. It answers the questions: the important variables? Which can be ignored? How do they interact with each other? And most important is how certain we are about these variables.

We have a dependent variable — the main factor that we are trying to understand or predict. And then we have independent variables — the factors we believe have an impact on the dependent variable.

Simple linear regression is a regression model that estimates the relationship between a dependent variable and an independent variable using a straight line.

On the other hand, Multiple linear regression estimates the relationship between two or more independent variables and one dependent variable. The difference between these two models is the number of independent variables.

Sometimes the above-mentioned regression models will not work. Here’s why.

As known, regression analysis is mainly used in understanding the relationship between a dependent and independent variable. In the real world, there are an ample number of situations where many independent variables get influenced by other variables for that we have to look for other options rather than a single regression model that can only work with one independent variable.

With these setbacks in hand, we would want a better model that will fill up the disadvantages of Simple and Multiple Linear Regression and that model is Multivariate Regression. If you are a beginner in the field and wish to learn more such concepts to start your career in Machine Learning, you can head over to Great Learning Academy and take up the Basics of machine learning , Linear Regression. The course will cover all the basic concepts required for you to kick-start your machine learning journey.

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What is Multivariate Regression?

Multivariate Regression is a supervised machine learning algorithm involving multiple data variables for analysis. Multivariate regression is an extension of multiple regression with one dependent variable and multiple independent variables. Based on the number of independent variables, we try to predict the output.

Multivariate regression tries to find out a formula that can explain how factors in variables respond simultaneously to changes in others.

There are numerous areas where multivariate regression can be used. Let’s look at some examples to understand multivariate regression better.

1. Praneeta wants to estimate the price of a house. She will collect details such as the location of the house, number of bedrooms, size in square feet, amenities available, or not. Basis these details price of the house can be predicted and how each variables are interrelated.
2. An agriculture scientist wants to predict the total crop yield expected for the summer. He collected details of the expected amount of rainfall, fertilizers to be used, and soil conditions. By building a Multivariate regression model scientists can predict his crop yield. With the crop yield, the scientist also tries to understand the relationship among the variables.
3. If an organization wants to know how much it has to pay to a new hire, they will take into account many details such as education level, number of experience, job location, has niche skill or not. Basis this information salary of an employee can be predicted, how these variables help in estimating the salary.
4. Economists can use Multivariate regression to predict the GDP growth of a state or a country based on parameters like total amount spent by consumers, import expenditure, total gains from exports, total savings, etc.
5. A company wants to predict the electricity bill of an apartment, the details needed here are the number of flats, the number of appliances in usage, the number of people at home, etc. With the help of these variables, the electricity bill can be predicted.

The above example uses Multivariate regression, where we have many independent variables and a single dependent variable.

Mathematical equation

The simple regression linear model represents a straight line meaning y is a function of x. When we have an extra dimension (z), the straight line becomes a plane.

Here, the plane is the function that expresses y as a function of x and z. The linear regression equation can now be expressed as:

y = m1.x + m2.z+ c

y is the dependent variable, that is, the variable that needs to be predicted.
x is the first independent variable. It is the first input.

m1 is the slope of x1. It lets us know the angle of the line (x).
z is the second independent variable. It is the second input.
m2 is the slope of z. It helps us to know the angle of the line (z).
c is the intercept. A constant that finds the value of y when x and z are 0.

The equation for a model with two input variables can be written as:

y = β0 + β1.x1 + β2.x2

What if there are three variables as inputs? Human visualizations can be only three dimensions. In the machine learning world, there can be n number of dimensions. The equation for a model with three input variables can be written as:

y = β0 + β1.x1 + β2.x2 + β3.x3

Below is the generalized equation for the multivariate regression model-

y = β0 + β1.x1 + β2.x2 +….. + βn.xn

Where n represents the number of independent variables, β0~ βn represents the coefficients, and x1~xn is the independent variable.

The multivariate model helps us in understanding and comparing coefficients across the output. Here, the small cost function makes Multivariate linear regression a better model.

Also Read: 100+ Machine Learning Interview Questions

What is Cost Function?

The cost function is a function that allows a cost to samples when the model differs from observed data. This equation is the sum of the square of the difference between the predicted value and the actual value divided by twice the length of the dataset. A smaller mean squared error implies better performance. Here, the cost is the sum of squared errors.

Steps of Multivariate Regression analysis

Steps involved for Multivariate regression analysis are feature selection and feature engineering, normalizing the features, selecting the loss function and hypothesis, setting hypothesis parameters, minimizing the loss function, testing the hypothesis, and generating the regression model.

• Feature selection-
The selection of features is an important step in multivariate regression. Feature selection also known as variable selection. It becomes important for us to pick significant variables for better model building.
• Normalizing Features-
We need to scale the features as it maintains general distribution and ratios in data. This will lead to an efficient analysis. The value of each feature can also be changed.
• Select Loss function and Hypothesis-
The loss function predicts whenever there is an error. Meaning, when the hypothesis prediction deviates from actual values. Here, the hypothesis is the predicted value from the feature/variable.
• Set Hypothesis Parameters-
The hypothesis parameter needs to be set in such a way that it reduces the loss function and predicts well.
• Minimize the Loss Function-
The loss function needs to be minimized by using a loss minimization algorithm on the dataset, which will help in adjusting hypothesis parameters. After the loss is minimized, it can be used for further action. Gradient descent is one of the algorithms commonly used for loss minimization.
• Test the hypothesis function-
The hypothesis function needs to be checked on as well, as it is predicting values. Once this is done, it has to be tested on test data.