What is the difference between linear and logistic regression?

Overview of linear and logistic regression

Linear and logistic regression models are widely used in machine learning. Both these models fall into the category of supervised learning.

The linear regression model uses a linear combination of the inputs to predict the output. After successful training, the outcome is always a continuous value in the range of $(-\infty,\infty)$.

On the other hand, the logistic regression model is a probabilistic model that uses log oddsOdds provide the measure of the likelihood of a particular outcome. The mathematical logs of this are defined as log odds. of independent variables to make data predictions. This model’s output is usually a discrete categorical label.

Linear regression

We mainly use linear regression for regression problems. For example, we can use this model to estimate the housing price in a particular locality. In this case, the data contains independent and dependent variables that are linearly related.

The formula of linear regression

The following is the mathematical formulation of the model:

Here, $\alpha$ is the weight obtained after training, and $b$ represents the bias term. These together constitute the parameters of the model. $\hat{y}$ is the prediction of the model, and $x$ is the input vector in a high dimensional subspace.

The graphical representation of the output after training is the best-fit line predicted using the training data’s trend. The visualization below depicts the model after successful training:

We optimize the model’s weights using the mean squared error loss function. Therefore, the weights that minimize the loss function are always optimal. Moreover, to apply the linear regression model, the independent variables must satisfy the property of collinearityCollinearity refers to the correlation between independent variables..

Logistic regression

Logistic regression, on the other hand, is a model used for classification problems. For example, predicting rain (“yes” or “no”) is a binary classification problem we can solve using a logistic regression model.

The training data in the case of logistic regression can support variable relationships between the independent and dependent variables. This model usually outputs a value in the range of $(0,1)$ with the help of the sigmoid (σ) function.

The formula of logistic regression

The following is the formulation of the model:

Here, $\hat{y}$ is the output vector with the probabilities of all the classes. $x$ is the input vector that is mapped to the output after the linear combination of inputs, that is, $\theta^Tx$ . Here, $\theta^{T}$ refers to the transpose of weights, which are tuned during training.

The sigmoid function is as follows:

If the output of the sigmoid is greater than the threshold of 0.5, we classify the data point as Class 1. Otherwise, it belongs to class 2.

As we can see above, an S-shaped curve (the sigmoid) fits the data and separates it into different linearly separable classes. The model uses the concept of maximum likelihood estimation to find optimal weights and accuracy. Also, the data to which we apply the model must not be collinear in the case of independent variables.

Key Differences between linear and logistic regression

The following table lists the main differences between linear and logistic regression:

Applications of linear regression

There are several real-world applications of linear regression. We can use it to:

1. Forecast stocks or prices: Linear regression models can predict trends in stocks using stock price data. You can get a hands-on understanding of it by predicting car prices using different regression models in the R programming language.

2. Analyze market effectiveness: After training on the past data of large-scale businesses, a linear regression model can predict their market effectiveness. We can use it to get a general idea of their position in the competitive market.

3. Study the operational efficiency of machines: We can use this model to study the efficiency of machines.

Applications of logistic regression

The real-world applications of logistic regression are quite different from those of linear regression. We can use logistic regression to:

• Detect pollution levels using images: We can use this model to classify or detect pollution levels after training on a dataset containing relevant images.

• Analyze text: We can use this model in various natural language processing tasks to determine the tone of text after training it on appropriate sentiments.

• Calculate credit scores: We can use this model to reduce the number of features that exhibit high correlation, and calculate an individual’s credit score.

Conclusion

In summary, linear and logistic regression are powerful machine learning tools, each serving distinct purposes based on the nature of the problem at hand. While linear regression excels at predicting continuous outcomes, logistic regression is ideal for classification tasks. Understanding the differences between these models can significantly enhance your ability to choose the right approach for your data analysis needs.