Introduction

Linear regression is one of the most well-known and widely used statistical modeling techniques. Its aim is to describe and interpret the relationship between variables using mathematical models. A mathematical model, in general, is the mathematical description of a phenomenon based on observations or experimental data. The process of creating such a model is called modeling, and it is particularly important in the field of statistics, as it allows the extraction of useful conclusions and the prediction of future values. The techniques of linear regression are divided into two main categories: simple and multiple linear regression.

Linear Regression Models

Linear regression models have been extensively studied and are considered easy to understand. They are used in a wide range of applications, as many problems can be described through linear relationships. At their core, these models focus on estimating the parameters β, which represent the coefficients of the model. The independent and dependent variables are known from the dataset, while the coefficients are estimated using statistical methods.

Least Squares Method

The most common method for estimating the parameters is the least squares method. It is based on the idea of minimizing the differences between the actual values and the values predicted by the model. Specifically, the cost criterion is the sum of the squares of these differences. In this way, the coefficients that provide the best possible fit of the model to the data are obtained.

Gradient Descent Method

An alternative method for estimating the coefficients is the gradient descent method. Unlike the least squares method, which gives closed-form solutions for the coefficients, gradient descent works iteratively. It starts from an initial guess and gradually improves the values of the coefficients by reducing the cost function at each step. This is an optimization process that is often used when the dataset is large or when closed-form solutions are difficult to compute.

Evaluation and Interpretation of Models

The evaluation of a linear regression model is crucial, as it ensures that the results are meaningful and reliable. The simple estimation of coefficients is not sufficient; it is also necessary to examine whether the relationship between the variables is significant. Various indicators are used, such as the coefficient of determination (R²), which shows the percentage of variance explained by the model. Equally important is the interpretation of the coefficients, so that meaningful conclusions can be drawn about the relationship of the variables.

Simple Linear Regression

Simple regression examines the relationship between one independent and one dependent variable. Its mathematical form is Yi = β0 + β1Xi + εi, where β0 is the intercept, β1 is the coefficient that indicates the slope of the line, and εi is the error term. Simple regression is used both for interpretation and for predicting future values, aiming to find the line that best fits the data.

Multiple Linear Regression

Multiple regression extends the simple case by taking into account more than one independent variable. Its general model is Y = β0 + β1X1 + β2X2 + … + βpXp + ε. This technique is particularly useful when the studied phenomenon is influenced by many factors. It allows the creation of more complex and accurate models, which can be used for prediction or for understanding intricate relationships.

Conclusions

In summary, linear regression is a powerful tool of statistical analysis and modeling. In simple linear regression, the goal is to identify the best possible regression line, while in multiple regression the focus is on finding the best possible plane that describes the relationships among multiple variables. Through estimation methods such as least squares or gradient descent, the model is fitted to the data and becomes capable of providing both interpretations and predictions. Proper evaluation and interpretation are essential in order to determine whether the conclusions drawn truly have scientific and practical value.