Linear regression as likelihood maximization
Tags: machinelearning, beginners
Published: 2023-09-19 21:07:30

Linear regression is a tool for forecasting outcomes by analyzing the connections among variables. It's about crafting an optimal line to represent a dataset, aiming to minimize discrepancies between observed data values and model predictions. To gauge the model's effectiveness, we rely on benchmarks such as Mean Squared Error (MSE) and Mean Absolute Error (MAE). However, these metrics aren't isolated in their function. MSE and MAE can be understood in the context of likelihood maximization. This means that when we work with MSE or MAE, we are operating within a framework of maximizing the likelihood that our model's predictions mirror the actual data, and with the support of advanced algorithms, this accuracy can be further refined. 

The combination of linear regression and likelihood maximization is widely used across various industries, such as financial predictions, scientific research, or engineering analyses. Generally, this method provides a reliable way to understand data. Below, we'll discuss more about its significance.

## Mean Squared Error (MSE)
The Mean Squared Error, often abbreviated as MSE, is a fundamental metric in regression analysis, particularly in linear models. Represented mathematically, it's given by:

![Image description](https://dev-to-uploads.s3.amazonaws.com/uploads/articles/m8shktj1o9fxahqcy3js.png)

Here, yi, is the actual observed value, while ŷi is the value predicted by the model, and n is the number of observations. 

MSE isn't just an error metric in linear regression. When you minimize MSE, you're essentially maximizing the likelihood, assuming errors follow a normal distribution. So, under Gaussian noise, the best model is the one that minimizes squared differences between what we see and predict. A more detailed explanation showing the equivalence between minimizing MSE and maximizing the likelihood under the assumption of normally distributed errors can be found in this lecture by the UCSD professor Peter Gerstoft (http://noiselab.ucsd.edu/ECE285/lecture3.pdf). 

MSE can be particularly useful in a number of industries and situations:

1. Financial forecasting. In predicting stock prices or economic indicators, MSE provides a reliable measure to understand the accuracy of financial models. By minimizing the MSE, financial analysts can optimize models to generate more reliable predictions, which are critical in decision-making processes.

2. Healthcare. MSE finds its application in predicting patient outcomes, especially in regression models that estimate future health metrics such as blood pressure, glucose levels, or cholesterol. By having a low MSE, healthcare professionals gain confidence in the predictive capacity of their models, leading to more informed medical interventions.

3. Real estate. Estimating property values based on features like location, size, and amenities is common in real estate. A model with a low MSE indicates a robust estimation of property prices. This helps both buyers and sellers make better, more informed decisions.

4. Energy consumption. Predicting power usage for cities or large facilities is important for efficient energy management. Models optimized using MSE can reliably forecast energy demands. This is useful when designing strategies for sustainable energy consumption.

5. Marketing analytics. MSE can assist marketing specialists in optimizing advertising spend. By predicting consumer responses to various campaigns, and refining these predictions through minimizing MSE, businesses can allocate resources more effectively.

Tech tools and algorithms, especially Gradient Descent, are very useful in these scenarios. By prioritizing the minimization of MSE, tech professionals can optimize linear regression to achieve more accurate and consistent predictions.

## Mean Absolute Error (MAE)
Mean Absolute Error, usually referred to as MAE, is another important metric in regression analysis. It emphasizes the magnitude of errors without squaring them. The formula for MAE is expressed as:

![Image description](https://dev-to-uploads.s3.amazonaws.com/uploads/articles/c2o6shzzzqi47p12gyr4.png)

In this equation, yi represents the model’s prediction values, x denotes true values, and stands for the quantity of the data points.

When we explore the details of MAE, we see a clear link with likelihood maximization. Specifically, when errors are presumed to follow a Laplace distribution, minimizing MAE is directly tied to maximizing the likelihood of the observed data. In simpler terms, under this distribution assumption, a model that minimizes the MAE is one that most closely aligns with the data's inherent patterns. A more detailed explanation of how it works and when it’s best to use MAE, you can find here (https://gmd.copernicus.org/preprints/gmd-2022-64/gmd-2022-64-manuscript-version2.pdf). 

Again, it can be useful in several specific contexts:

1. Medical research. With monitoring patient recovery or the efficacy of new treatments, MAE offers a better understanding of discrepancies. It's especially useful in cases where extreme deviations from predicted outcomes can have serious consequences.

2. Retail and E-commerce. MAE is very useful in analyzing customer behavior and sales forecasts. This methric is less sensitive to extreme deviations, so retailers gain a more balanced view of their sales predictions.

3. Transport and logistics. Here, MAE can be used for predicting transit times or assessing transportation models. When timely delivery matters most, focusing on absolute error helps recognize and address any discrepancies.

4.. Agriculture. MAE aids in predicting crop yields based on variables like rainfall, temperature, and soil quality. Such predictions are important for farmers and stakeholders to make decisions on planting, harvesting, or selling.

5. Urban planning. For predictions about urban growth, traffic patterns, or infrastructure needs, MAE delivers a consistent evaluation of model accuracy. This aids city planners in designing better and more responsive urban environments.

In each of these scenarios, using optimization methods like Coordinate Descent with MAE ensures more accurate and useful insights. 

## Final thoughts 

When linear regression is paired with likelihood maximization, it provides tech professionals with enhanced prediction accuracy. This combination ensures that models effectively represent data patterns, which is invaluable in technology.

Choosing the right error metric, such as MSE or MAE, affects how models are evaluated and adapted, and the decision should be tailored to the specific data and objectives.

In short, using linear regression with likelihood maximization offers clarity in data analysis. Likelihood maximization can be viewed as an interpretation of metrics like MSE/MAE because it represents the probability of observing the given data under specific model parameters. When we minimize the MSE or MAE, we're essentially maximizing this likelihood, assuming certain error distributions. Therefore, selecting the right error metric helps us get more reliable outcomes and insights.

Here’s a few things to keep in mind while using these techniques: 

1. Understand your data. Before exploring linear regression and likelihood maximization, get a grasp on your data. Know its patterns, be aware of outliers, and recognize any biases. 

2. Choose the right metric. While both MSE and MAE are valuable, it’s best to select the one that aligns best with your specific goals and the nature of your data.

3. Validate regularly. Test and validate your model against new data. This helps you make sure its predictions remain accurate and relevant over time.

4. Stay updated with tools. The tech world is evolving. Use the latest algorithms and optimization techniques to maximize the efficiency of your models.

5. Iterate and refine. No model is perfect from the start. Review and refine based on results and feedback. Iterative improvement is key to long-term success.

Using these practices with linear regression and likelihood maximization can improve your analysis for clearer and more reliable outcomes.