Introduction to Regression

Regression Analysis: The Know-How

Jump to a specific topic

Regression analysis is like tinder (a dating app) for data – you try to find meaningful relationships that will generalize to upcoming data. Just like only swiping right is not a great way to find relationships, implementing regression using pre-built packages is not helpful for an in-depth understanding of it.

Some people might think it’s better to hop on to the practical aspects of regression analysis and ditch the theory part. But to build something powerful you have to understand the underlying concepts behind it.

Before getting deep into regression analysis, if you are just getting started with machine learning – I would strongly recommend going through Introduction to Machine Learning and Supervised Learning. This will help you get the basics right and not scratch your head amid reading this blog 😜

What is Regression Analysis?

In simple terms, It is the process of finding the relationship between a dependent variable(s) and an independent variable(s).

βœ… Dependent Variable – Target value or outcome variable
βœ… Independent Variable – Predictors or Feature columns

Let’s understand this with an example most of you could relate with…!

Problem Statement – Consider you’re brewing a cup of coffee. And we want to apply some regression analysis to your cup of coffee.

So here, the dependent variable is the sweetness of the coffee, which is the end result. And independent variables could be coffee beans and sugar.

Some patterns which you will agree with, might be…

  • The sweetness of the coffee decreases⬇ as I increase the number of coffee beans.
  • The sweetness of the coffee increases⬆ as more sugar is added to the coffee.

In brief, the taste of the coffee would actually differ with the amount of coffee beans and sugar added to the milk.

There it is! The relationship of coffee beans and sugar to the sweetness of the coffee is the relationship we wanted to model.

In some ideal world, the equations might look something like, sweetness of the coffee = amount of sugar added / amount of coffee beans added.

πŸ™†β€β™‚οΈ Free Advice alert! Ditch coffee and eat an apple πŸ˜‚- Source

We will discuss some practical use cases later in the blog.

πŸ’‘ The earliest form of regression was the method of least squares, published by Legendre in 1805.

Types of Regression

3 factors distinguish different types of regression algorithms:

  1. The number of independent variables (features)
  2. The shape of the regression line.
  3. The type of the dependent variable (target label)
Types of Regression

Although there are numerous regression algorithms out there, you can even build your own algorithms considering the above 3 factors. The two most commonly used and easy to implement algorithms include:

Linear Regression

This is one of the simplest and well-studied forms of regression. With its origin to statistics, linear regression has been used by data scientists to model relationships within data and get promising results.

Linear Regression builds a hypothesis or model which reflects a linear relationship between feature variables and target variables. In other terms, it assumes that the relationship between variables is linear and tries to map it using a straight line.

The best fit straight line in linear regression is represented as:

    \[Y=mx+c+e\]

Here m is the slope of the line, c is the intercept and e is the error term.

Linear Regression Explained

We can further divide linear regression into two categories, depending on the number of independent variables (or feature columns):

  • Simple LR(when independent variables = 1)
  • Multiple LR (when independent variables > 1)

To come up with the best fit straight line, we need to minimize the error between the predicted line and actual output. We can use different error metrics like Least Square (commonly used), Ordinary Least Squares, Weighted Least Square, etc to minimize the error.

If you want to read more about Linear Regression and python implementation(from scratch) of the same, read the blog – Linear Regression.

Logistic Regression

This form of regression is commonly used when we want to predict the probabilities of an event which is binary. In data terms, if we need to predict the target column with values like 0/1, True/False, Yes/No we use logistic regression.

Logistic regression measures the relationship between a dependent and independent variable(s) using a logistic function which represents a sigmoid function as shown in the image. It takes an input and outputs a value between zero and one.

It generates a hypothesis that is used to validate a relationship rather than state the relationship. A logistic regression model will build such a hypothesis which can then be used with a threshold to build a classifier. For example,

    \[if \begin{Bmatrix} probability >= 0.8; y=1\\probability < 0.8; y=0\end{Bmatrix}\]

Logistic Regression Explained

If you want to read more about Logistic Regression, its types and implementing it in python from scratch, read the blog – Logistic Regression.

Other Regression Algorithms

Although you may not use the algorithms listed below very often, but they are equally important. Because we can use them for certain types of data and target variables. And being a data scientist you should have a simple understanding of these as well:

  • Polynomial
  • Stepwise
  • Moving
  • Lasso
  • Ridge
  • Elastic-Net

Discussing each regression algorithm in detail is out of the scope of this blog. In any case you can read more about other algorithms here – Top 10 regression algorithms to know about.

Mathematics behind Regression Analysis

It is always good to know the mathematics behind the derivations as it clears out the basic idea of the execution of the algorithms.

You can skip this section if you are having a hard time understanding the maths. It’s not absolutely important but I have included this as it is always good to know.

A regression analysis involves the following parameters:

  • {\Theta} – Unknown parameter which needs to be calculated
  • X – Rows of independent variables or the feature columns from the data
  • Y – Rows of the dependent variable or the target column from the data
  • e – When we do computer regression analysis, we only compute an approximation of the real relationship between variables. This is the error term that denotes the difference between our approximation and the real relationship( theoretically). Please note that I am not talking about the difference between our model’s prediction and actual values. Refer to this video for a better understanding.

For every i_{th} row in the data, most of the regression algorithms approximate Y_i as a function of X_i and \Theta_i with a random statistical noise or error term e_i. We represent it as,

    \[Y_i = f(X_i, \Theta_i) + e_i\]

Function estimation

Here, the f(X_i, \Theta_i) term, is the hypothesis. The better the hypothesis, the better the predictions of the model. To select the appropriate hypothesis, a data scientist needs to evaluate the data and understand which function would work the best.

\Theta Parameter Estimation

Once the function is estimated, different cost calculation parameters are used to estimate the parameters \Theta_i. It is sometimes denoted by \hat{\Theta_i} to distinguish it from the original parameters used to produce the data. There are several approaches like least square which help in approximating the value of the parameters.

Prediction

The value of the parameters estimated (\hat{\Theta_i}) is then used to predict and assess the performance of the model. Output is denoted by, \hat{Y_i}.

The flow of the regression analysis looks something like,

Regression workflow
I have tried my best to explain this portion in a simple way as possible. Hope you have a clear understanding of the mathematical concepts or at least have got the intuition on the concepts. As mentioned earlier it’s okay to take your time with mathematics.
0
On a scale of 0(Very Easy) to 5(Very Hard), how hard was the mathematics part for you?x

Applications of Regression analysis

In machine learning, we use regression analysis normally for the following:

Prediction and Forecasting

Regression models are used to learn the mapping of the target variable with the features from data and trends, which is then used to predict on new data points.

Example:

  • Predicting real estate prices
  • Predicting death caused by bike accidents

You can also use it to forecast from trends and historical data like forecasting weather or sales of a predict.

Cause and Effect Relationship

Additionally, you can also use it to learn the cause and effect relationships. Like, the sweetness of the coffee will increase as the amount of sugar is increased. Although it is important to understand that cause and effect relationship might not always justify the prediction. Simply speaking, It is not necessary that a relationship within the dataset might also hold true for new data points.

Filtering important features

Regression analysis allows us to determine which factors matter and which can be ignored. It also tells us the magnitude of the impact of important factors.

Problems with Regression

Depending on the type of data, sometimes regression algorithms may fall prey to the below conditions:

  • Multi-Collinearity: Sometimes the independent variables in the data may be highly correlated which causes problems in approximating the parameters and selecting the important features.

πŸ’‘ Two variables A and B are collinear, if they have a linear relationship between them.

  • Outliers: Outliers are the data points that are too high or low values when compared to other data points. They can cause issues as they do not represent the distribution of original data and should be removed.
  • In-sufficient Data: If the data points available are lower than the independent variables then regression analysis results in infinite solutions and modeling is not possible.

Regression vs Classification

Both regression and classification are types of supervised learning algorithms. The only difference between regression and classification is the type of the target label. For a target label with,

  • Continuous value – We use Regression
  • Discrete Value – We use Classification

Continuous values can take on all real values between an interval. Eg. 11.3, 11.6, 20.3 etc. Discrete values belonging to a set are distinct and separate. Eg: 10, 20, 11 etc..

Quiz

It is equally important to test your understanding of the concepts and quizzes are a fun way to do it. Attempt the below quiz on regression algorithms to get your score.

Regression Quiz

Next Steps

I hope you have a clear understanding of regression analysis, its types, and its applications. Going ahead, I would recommend you to read about the top 10 regression algorithms and start putting this knowledge into actual projects.

You can also refer to Introduction to Classification Algorithms if you would want to learn more about classification.

Was this blog useful to you? Tell me your views and doubts(if any) in the comments section below.

Regression Analysis: The Know-How
Share on facebook
Share on twitter
Share on linkedin
Share on whatsapp
Hardik Munjal

Hardik Munjal

I am a technology enthusiast with around 2 years of experience in Software Development. I love to share my learning from my experience or something I am exploring myself. I also consult college grads with their doubts to help them in their professional and personal life.
If my content adds value to you, do consider supporting me! 😊

Hardik Munjal

Hardik Munjal

I am a technology enthusiast with around 2 years of experience in Software Development. I love to share my learning from my experience or something I am exploring myself. I also consult college grads with their doubts to help them in their professional and personal life.
If my content adds value to you, do consider supporting me! 😊

Subscribe
Notify of
guest
0 Comments
Inline Feedbacks
View all comments
Share on facebook
Share on twitter
Share on linkedin
Share on whatsapp

Join the Club!

Get all the latest blog updates directly in your mailbox and stay ahead of the curve.

Yes, I'm IN!

Join and Stay ahead of the curve.