Logistic regression is a widely used **statistical modeling technique** in **various fields**, including **machine learning**, social sciences, and healthcare. It is often considered a **fundamental tool for binary classification problems**.

Despite its popularity, there exists an intriguing question surrounding logistic regression — **the Linearity Paradox**. At first glance, logistic regression **appears to be a linear model **due to its **name** and **mathematical formulation**. However, when delving deeper, we **uncover the non-linearity** **hidden** within its structure.

This intricate **interplay** between linearity and nonlinearity has sparked debates within the data science community, questioning the **true nature** of logistic regression, **whether it can still be classified (pun intended) as a linear model**. In this article I will be referencing **binary classification** for establishing an easy understanding.

To understand the foundation of logistic regression, we start with **linear regression**, a well-established technique for predicting **continuous numerical values**. Linear regression models the relationship between input features and the target variable using a linear equation.

In logistic regression, we adopt a similar concept, **but rather than predicting continuous values**, we focus on estimating the **probability of a binary outcome**.

At first glance, logistic regression’s mathematical formulation **resembles that of linear regression**, which outputs a continuous response variable.

This similarity leads to the **assumption that logistic regression is a linear model**. However, this assumption is kind of flawed.

The linearity in logistic regression **pertains to the relationship between the predictors and the log-odds (logarithm of the odds) of the response variable, rather than the relationship between the predictors and the response variable itself.**

The linearity in logistic regression **pertains to the relationship between the predictors and the log-odds (logarithm of the odds) of the response variable, rather than the relationship between the predictors and the response variable itself.**

The transformation performed by the **logistic function** introduces a non-linear element, complicating the interpretation and analysis of logistic regression models.

## Introducing the Sigmoid Function:

The **sigmoid **function, also known as the** logistic function**, is a key element that transforms the linear combination of input features into a **bounded probability between 0 and 1**. It takes the form of:

**P(y=1|x) = 1 / (1 + e^(-z))**

where **P(y=1|x) or S(x) or φ(x)** represents the **probability of the target variabl**e being 1 given the** input features (x)**, and **z **denotes the **linear combination** of the input features with **associated weights**.

The sigmoid curve has an S-shaped form,** introducing nonlinearity to the model and allowing for capturing complex relationships between variables**.

## The Nonlinear Transformation:

To better understand the non-linearity in logistic regression, let’s examine a simple **example**.

Suppose we have a **binary response variable**, such as **“success” or “failure,”** and a single predictor variable, X. In a linear regression model, we would directly predict the response variable using a linear relationship:

**Y = β₀ + β₁X**

However, in logistic regression, the** log-odds of success** are modeled as a linear combination of predictors: **log(odds) = β₀ + β₁X**

This linear relationship is then transformed using the logistic function, converting the **log-odds into a probability:**

**P(Y=1) = 1 / (1 + exp(-(β₀ + β₁X)))**

The **non-linearity arises from the sigmoid transformation**, which maps the linear relationship onto the probability scale.

It’s arguable that the sigmoid function **inherently introduces nonlinearity** into the logistic regression model, **challenging** its classification as a linear model.

**This nonlinearity enables logistic regression to capture intricate patterns and interactions between variables that linear models cannot handle effectively.**

## Linear Decision Boundaries:

Despite the presence of nonlinearity in logistic regression, **it is important to note that the decision boundaries separating the classes are still linear.**

The linearity arises from the fact that the decision boundary is determined by a **threshold value of 0.5 on the sigmoid curve**. This means that logistic regression **draws linear decision boundaries in the input feature space,** dividing it into regions corresponding to different classes.

## Model Complexity and Flexibility:

While the sigmoid transformation introduces nonlinearity, logistic regression is still considered a **relatively simple and interpretable model** compared to more complex nonlinear models, such as neural networks.

The linearity debate in logistic regression **revolves around the extent to which the model captures complex relationships **between variables and whether it provides sufficient flexibility for accurate predictions.

## The Role of Feature Engineering:

To address the **limitations of linearity**, feature engineering plays a crucial role in logistic regression.

By **creating nonlinear features through transformations, interactions, or polynomial expansions, the model can capture more complex relationships beyond the linear terms**.

Feature engineering allows logistic regression to leverage the power of nonlinearities indirectly, further enhancing its capability to handle intricate patterns in the data.

The essence of linearity surrounding logistic regression highlights the **intricate relationship between linearity and nonlinearity** in this popular classification algorithm.

The Linearity Paradox **stems from the misconception that logistic regression follows the same linear structure as its counterpart**.

In reality, **logistic regression combines linear relationships between the predictors and the log-odds with non-linearity induced by the logistic function.**