The explanation of Logistic Regression as a Generalized Linear Model and use as a classifier is often confusing.

In this article, I try to explain this idea from first principles. This blog is part of my forthcoming book on the Mathematical foundations of Data Science. If you are interested in knowing more, please follow me on linkedin Ajit Jaokar

We take the following approach:

- We see first briefly how linear regression works
- We then explore the assumptions and limitations of linear regression.
- Following this, we then show how these limitations can be overcome by the Generalized Linear Model (GLM)
- Finally, we explore Logistic Regression as a GLM

Machine learning involves creating a model of a process. To create a model of a process, we need to identify patterns in data. Broadly, patterns in data can be of two types: The signal (data generating process) and the variation (error generation process). The simplest model to start off with is the Linear Regression model. Linear models have some advantages – for example, they are relatively simple to implement, and many phenomenon can be modelled using linear regression

Linear regression has the following requirements (assumptions for use)

- As per the name, Linear regression needs the relationship between the independent and dependent variables to be linear.
- the linear regression analysis requires all variables to be multivariate normal distribution.
- No multicollinearity in the data. Multicollinearity occurs when the independent variables are highly correlated with each other.
- linear regression analysis requires that there is little or no autocorrelation in the data. Autocorrelation is the correlation of a signal with a delayed copy of itself as a function of delay.
- Homoscedasticity (the residuals are equal across the regression line).

The requirement that the response variable is of normal distribution excludes many cases such as:

- Where the response variable is expected to be always positive and varying over a wide range or
- Where constant input changes lead to geometrically varying, rather than continually varying, output changes.

We can illustrate these using examples: Suppose we have a model which predicts that a 10 degree temperature decrease would lead to 1,000 fewer people visiting the beach. This model does not work over small and large beaches. (Here, we could consider a small beach as one where expected attendance is 50 people and a large beach as one where the expected attendance was 10,000.). For the small beach (50 people), the model implies that -950 people would attend the beach. This prediction is obviously not correct. This model would also not work if we had a situation where we had an output that was bounded on both sides – for example in the case of a yes/no choice. This is represented by a **Bernoulli variable** where the probabilities are bounded on both ends (they must be between 0 and 1). If our model predicted that a change in 10 degrees makes a person twice as likely to go to the beach, as temperatures increases by 10 degrees, this model does not work because probabilities cannot be doubled.

**Generalised linear models** (GLM) cater to these situations by allowing for response variables that have arbitrary distributions (other than only normal distributions), and by using an arbitrary function of the response variable (called **the link function**) to vary linearly with the predicted values (rather than assuming that the response itself must vary linearly with the predictor). Thus, in a generalised linear model (GLM), each outcome Y of the dependent variables is assumed to be generated from **the exponential family of distributions** (which includes distributions such as the **normal, binomial, Poisson and gamma distributions**, among others). GLM thus expands the scenarios in which linear regression can apply by expanding the possibilities of the outcome variable. GLM uses the **maximum likelihood estimation** of the model parameters for the exponential family and least squares for normal linear models. (Note the section is adapted from Wikipedia)

To understand how logistic regression can be seen as GLM, we can elaborate this approach as follows:

Logistic regression measures the relationship between the dependent variable and one or more independent variables(features) by estimating probabilities using the underlying **logit function.** In statistics, the logit function or the log-odds is the logarithm of the odds. Given a probability p, the corresponding odds are calculated as p / (1 – p). For The logit function is the logarithm of the odds: logit(x) = log(x / (1 – x)). The **Odds** describes the ratio of success to ratio of failure. The **Odds ratio** is the ratio of odds and is calculated as the ratio of odds for each group.

The inverse of the logit function is the **sigmoid function**. The formula for the sigmoid function is *σ(x) = 1/(1 + exp(-x))*. The sigmoid function maps probabilities to the range [0, 1] – and this makes logistic regression as a classifier. Thus, many models have data generating processes that can be linearized by considering the inverse

The logit and the sigmoid functions are useful in analysis because their gradients are simple to calculate. Many optimization and machine learning techniques make use of gradients ( for example in neural networks). The biggest drawback of the sigmoid function for many analytics practitioners is the so-called “vanishing gradient” problem.

This blog is part of my forthcoming book on the Mathematical foundations of Data Science. If you are interested in knowing more, please follow me on linkedin Ajit Jaokar

logit of logistic regression understanding the fundamentals

nathanbrixius -logit-and-sigmoid

quora.com – Why is logistic regression called regression if it does...

© 2019 Data Science Central ® Powered by

Badges | Report an Issue | Privacy Policy | Terms of Service

**Most Popular Content on DSC**

To not miss this type of content in the future, subscribe to our newsletter.

- Book: Statistics -- New Foundations, Toolbox, and Machine Learning Recipes
- Book: Classification and Regression In a Weekend - With Python
- Book: Applied Stochastic Processes
- Long-range Correlations in Time Series: Modeling, Testing, Case Study
- How to Automatically Determine the Number of Clusters in your Data
- New Machine Learning Cheat Sheet | Old one
- Confidence Intervals Without Pain - With Resampling
- Advanced Machine Learning with Basic Excel
- New Perspectives on Statistical Distributions and Deep Learning
- Fascinating New Results in the Theory of Randomness
- Fast Combinatorial Feature Selection

**Other popular resources**

- Comprehensive Repository of Data Science and ML Resources
- Statistical Concepts Explained in Simple English
- Machine Learning Concepts Explained in One Picture
- 100 Data Science Interview Questions and Answers
- Cheat Sheets | Curated Articles | Search | Jobs | Courses
- Post a Blog | Forum Questions | Books | Salaries | News

**Archives:** 2008-2014 |
2015-2016 |
2017-2019 |
Book 1 |
Book 2 |
More

**Most popular articles**

- Free Book and Resources for DSC Members
- New Perspectives on Statistical Distributions and Deep Learning
- Time series, Growth Modeling and Data Science Wizardy
- Statistical Concepts Explained in Simple English
- Machine Learning Concepts Explained in One Picture
- Comprehensive Repository of Data Science and ML Resources
- Advanced Machine Learning with Basic Excel
- Difference between ML, Data Science, AI, Deep Learning, and Statistics
- Selected Business Analytics, Data Science and ML articles
- How to Automatically Determine the Number of Clusters in your Data
- Fascinating New Results in the Theory of Randomness
- Hire a Data Scientist | Search DSC | Find a Job
- Post a Blog | Forum Questions

## You need to be a member of Data Science Central to add comments!

Join Data Science Central