Introduction to Multivariate Regression in Machine Learning: Complete Guide

It’s no secret that today’s technology is data-driven. Data may only be a compilation of figures but it can be processed meaningfully to extract productivity and resourcefulness for businesses to remain competitive and sustainable in the long term. As it happens, data analysis is the answer to deriving accurate estimations from raw information.

Data Analysis is a technique that involves statistical and logical ideas to scrutinize, process, and transform data into a usable form. The solutions that are drawn by data analysis are used in businesses to make vital decisions. Data science along with data analysis is used to predict future outcomes with high accuracy. It is a process of employing scientific techniques, and algorithms to procure viable information from a pool of data.

A common problem faced by data professionals is the manner in which to determine if a statistical relationship exists between a response variable (denoted by Y) and explanatory variables (denoted by Xi).

The answer to this concern is regression analysis. Let’s understand this in further detail.

What is Regression Analysis?

Regression analysis is one of the popular methods in data analysis that follows a controlled or supervised machine learning algorithm. It is an effective technique to identify and establish a relationship among variables in data. 

Regression analysis involves sorting out viable variables using mathematical strategies to draw highly accurate conclusions about those sorted variables. 

What is Multivariate Regression?

Multivariate is a controlled or supervised Machine Learning algorithm that analyses multiple data variables. It is a continuation of multiple regression that involves one dependent variable and many independent variables. The output is predicted based on the number of independent variables. 

Multivariate regression figures out a formula that explains the simultaneous response of the factors present in variables to the changes in others. They are used to study the data in various fields. For instance, in real estate multivariate regression is used to predict the price of a house based on several factors like its location, number of rooms, and the available amenities.  

Cost Function in Multivariate Regression

The cost function allocates a cost to samples when the outcome of a model deviates from the observed data. The equation of cost function is the total of the square of the difference between the predicted value and the actual value divided by two times the length of the dataset.

Here’s an example:

Source 

How to use Multivariate Regression Analysis?

The processes involved in multivariate regression analysis include the selection of features, engineering the features, feature normalization, selection loss functions, hypothesis analysis, and creating a regression model.

1. Selection of features: It is the most important step in multivariate regression. Also known as variable selection, this process involves selecting viable variables to build efficient models. 
2. Feature Normalizing: This involves feature scaling to maintain streamlined distribution and data ratios. This helps in better data analysis. The value of all the features can be changed according to the requirement.
3. Selecting Loss function and hypothesis: The loss function is used for predicting errors. The loss function comes into play when the hypothesis prediction changes from the actual figures. Here, the hypothesis represents the value predicted from the feature or variable.
4. Fixing hypothesis parameter: The parameter of the hypothesis is fixed or set in such a way that it minimizes the loss function and enhances better prediction.
5. Reducing the loss function: The loss function is minimized by generating an algorithm specifically for loss minimization on the dataset which in turn facilitates the alteration of hypothesis parameters. Gradient descent is the most commonly used algorithm for loss minimization. The algorithm can also be used for other actions once the loss minimization is complete.
6. Analyzing the hypothesis function: The function of the hypothesis needs to be analyzed as it is crucial for predicting the values. After the function is analyzed, it is then tested on test data. 

Let us now look at the two ways multivariate regression can be used.

1. Multivariate Linear Regression

Multivariate linear regression resembles simple linear regression except that in multivariate linear regression, multiple independent variables contribute to the dependent variables and so multiple coefficients are used in the computation. 

• It is used to derive a mathematical relationship amongst multiple random variables. It explains how many multiple independent variables are associated with one dependent variable. 
• The details of the multiple independent variables are used to make an accurate prediction of the influence they have on the outcome variable.
• Multivariate linear regression model generates a relationship in a linear form (a form of a straight line) with the best approximation of each data point.
• The equation of the Multivariate linear regression model is:

yi​=β0​+β1​xi1​+β2​xi2​+…+βp​xip​+

where for i=n observations:

Source 

When can linear regression be used?

The linear regression model can be used only when there are two continuous variables of which one is dependent and the other one is independent. 

The independent variable is used as a parameter to determine the value or outcome of the dependent variable.

2. Multivariate Logistic Regression

Logistic regression is an algorithm used to predict a binary outcome based on multiple independent variables. A binary outcome has two possibilities, either the scenario happens( represented by 1) or it doesn’t happen ( denoted by 0).

Logistic regression is used while working on binary data, the data where the outcome (or the dependent variable) is dichotomous.

Where can logistic regression be used?

Logistic regression is primarily used to deal with classification issues. For instance, to ascertain if an email is spam or not and if a particular transaction is malicious or not. In data analysis, it is used to make calculated decisions to minimize loss and increase profits.

Multivariate logistic regression is used when there is one dependent variable and multiple outcomes. It differs from logistic regression by having more than two possible outcomes.

X1 to Xp are distinct independent variables. 

b0 to bp are the regression coefficients

The multiple logistic regression model can also be written in a different form. In the form below, the outcome is the expected log of the odds that the outcome is present,

The multiple logistic regression model can also be written in a different form. In the form below, the outcome is the expected log of the odds that the outcome is present.

The right side of the above equation resembles the linear regression equation but the method of finding out the regression coefficients differs.

Assumptions in the Multivariate Regression Model

• The dependent and the independent variables have a linear relationship.
• The independent variables do not have a strong correlation among themselves.
• The observations of yi​ are chosen randomly and individually from the population.

Assumptions in Multivariate Logistic Regression Model

• The dependent variable is nominal or ordinal. The nominal variables have two or more categories without any meaningful organization. Ordinal variables can also have two or more categories, but they have a structure and can be ranked.
• There can be single or multiple independent variables that can be ordinal, continuous, or nominal. Continuous variables are those that can have infinite values within a specific range.
• The dependent variables are mutually exclusive and exhaustive.
• The independent variables do not have a strong correlation among themselves.

Advantages of Multivariate Regression

1. Multivariate regression helps us to study the relationships among multiple variables in the dataset.
2. The correlation between dependent and independent variables helps in predicting the outcome.
3. It is one of the most convenient and popular algorithms used in machine learning.

Disadvantages of Multivariate Regression

• The complexity of multivariate techniques requires complex mathematical calculations. 
• It is not easy to interpret the output of the multivariate regression model since there are inconsistencies in the loss and error outputs.
• Multivariate regression models cannot be applied to smaller datasets; they are designed for producing accurate outputs when it comes to larger datasets.

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