Multi - Multiple Linear Regression

Multivariate multiple linear regression has the potential to be a very powerful tool in many fields of work and research. This method is used when we have a problem consisting of two or more predictor variables and two or more response variables.

In the context of predicting multiple dependent variables from multiple independent variables, the model can be conceptualized as an extension of the linear regression framework to accommodate multiple outcomes. This scenario is sometimes referred to as multivariate multiple linear regression, though more commonly, it's just part of multivariate linear regression.

The equation for such a model, when predicting more than one dependent variable, can be represented in a matrix form rather than a single equation form. If we have m independent variables and p dependent variables, the model can be expressed as:

                                                               X = Yβ + E
  • Y = is an n × p matrix of p dependent variables (with n observations each).
  • X = is an n × (m +1) matrix of m independent variables (with n observations each), including a column of ones for the intercept.
  • β is an (m +1) x p matrix of coefficients, including the intercepts and the slopes of the m
  •  independent variables for each of the p dependent variables.
  • E is an n×p matrix of errors or residuals, representing the differences between the observed and predicted values of the dependent variables.

Each column of Y represents a different dependent variable, and each column of β represents the coefficients (including intercept) corresponding to one of the dependent variables. This formulation allows for the simultaneous prediction of multiple dependent variables based on the same set of independent variables, capturing the relationships between all variables in a single model framework.

This approach is powerful for analyzing complex datasets where multiple outcomes may be influenced by a common set of predictors and can reveal insights into the interdependencies among the outcome variables.

NOTE : Multivariate - Multiple linear regression consider the data you have is multi-independent variables(feature) and multi - dependent variables (target).