Linear Regression

Linear regression is a statistical model which estimates the linear relationship between a scalar response and one or more explanatory variables (also known as dependent and independent variables).

The case of one explanatory variable is called simple linear regression; for more than one, the process is called multiple linear regression.This term is distinct from multivariate linear regression, where multiple correlated dependent variables are predicted, rather than a single scalar variable. If the explanatory variables are measured with error then errors-in-variables models are required, also known as measurement error models.

Formulation

Given a dataset

of n statistical units, a linear regression model assumes that the relationship between the dependent variable y and the vector of regressors x is linear.

This relationship is modeled through a disturbance term or error variable ε — an unobserved random variable that adds "noise" to the linear relationship between the dependent variable and regressors. Thus the model takes the form

where T denotes the transpose, so that xiTβ is the inner product between vectors xi and β.

Often these n equations are stacked together and written in matrix notation as

where

Notation and terminology

• is a vector of observed values of the variable called the regressand, endogenous variable, response variable, target variable, measured variable, criterion variable, or dependent variable. This variable is also sometimes known as the predicted variable, but this should not be confused with predicted values, which are denoted . The decision as to which variable in a data set is modeled as the dependent variable and which are modeled as the independent variables may be based on a presumption that the value of one of the variables is caused by, or directly influenced by the other variables. Alternatively, there may be an operational reason to model one of the variables in terms of the others, in which case there need be no presumption of causality.
• may be seen as a matrix of row-vectors or of n-dimensional column-vectors , which are known as regressors, exogenous variables, explanatory variables, covariates, input variables, predictor variables, or independent variables (not to be confused with the concept of independent random variables). The matrix is sometimes called the design matrix.

  • Usually a constant is included as one of the regressors. In particular, for . The corresponding element of β is called the intercept. Many statistical inference procedures for linear models require an intercept to be present, so it is often included even if theoretical considerations suggest that its value should be zero.
  • Sometimes one of the regressors can be a non-linear function of another regressor or of the data values, as in polynomial regression and segmented regression. The model remains linear as long as it is linear in the parameter vector β.

  • The values xij may be viewed as either observed values of random variables Xj or as fixed values chosen prior to observing the dependent variable. Both interpretations may be appropriate in different cases, and they generally lead to the same estimation procedures; however different approaches to asymptotic analysis are used in these two situations.

• is a -dimensional parameter vector, where is the intercept term (if one is included in the model—otherwise is p-dimensional). Its elements are known as effects or regression coefficients (although the latter term is sometimes reserved for the estimated effects). In simple linear regression, p=1, and the coefficient is known as regression slope. Statistical estimation and inference in linear regression focuses on β. The elements of this parameter vector are interpreted as the partial derivatives of the dependent variable with respect to the various independent variables.
• is a vector of values . This part of the model is called the error term, disturbance term, or sometimes noise (in contrast with the "signal" provided by the rest of the model). This variable captures all other factors which influence the dependent variable y other than the regressors x. The relationship between the error term and the regressors, for example their correlation, is a crucial consideration in formulating a linear regression model, as it will determine the appropriate estimation method.

Fitting a linear model to a given data set usually requires estimating the regression coefficients such that the error term is minimized. For example, it is common to use the sum of squared errors as a measure of for minimization.

Applications:

Linear regression is widely used in various fields for prediction, forecasting, and understanding the relationships between variables. Some common applications include:

• Predicting house prices based on features such as size, number of bedrooms, and location.

• Forecasting sales based on advertising spending, economic indicators, etc.

• Analyzing the impact of independent variables on a dependent variable in scientific research.

Advantages and Disadvantages:

• Advantages:

  • Simple and easy to understand.

  • Provides interpretable coefficients for each independent variable.

  • Can be applied to both numerical and categorical independent variables.

• Disadvantages:

  • Assumes a linear relationship between variables, which may not always be the case.

  • Sensitive to outliers and multicollinearity.

  • Limited to linear relationships and may not capture complex patterns in the data.

Implementation:

In Python, linear regression can be implemented using libraries such as scikit-learn or StatsModels. Here's a basic example using scikit-learn:

from sklearn.linear_model import LinearRegression

# Create a linear regression model
model = LinearRegression()

# Fit the model to the training data
model.fit(X_train, y_train)

# Make predictions on the test data
predictions = model.predict(X_test)