La Équation Normale is a mathematical formula used in statistics and apprentissage automatique, particularly in the context of régression linéaire. It provides a way to compute the parameters (coefficients) of a modèle linéaire that minimize the difference between the predicted and actual values of the target variable.
En régression linéaire, nous visons à trouver un relation linéaire between the input features (independent variables) and the output (dependent variable). The Normal Equation is derived from the principle of least squares, which minimizes the cost function defined as the sum of the squared differences between the observed values and the values predicted by the linear model.
L'équation normale s'exprime mathématiquement comme :
θ = (X^T * X)^{-1} * X^T * y
Où :
- θ représente le vecteur de paramètres que nous souhaitons estimer.
- X is the matrix of input features, where each row represents an observation and each column represents a feature.
- y est le vecteur des valeurs de sortie observées.
- X^T est la transposée de la matrice X.
- (X^T * X)^{-1} denotes the inverse of the product of X transposed and X.
One of the key advantages of using the Normal Equation is that it provides a direct analytical solution to the problem of parameter estimation, eliminating the need for iterative des techniques d'optimisation like gradient descent. However, it is important to note that the Normal Equation can be computationally expensive for large datasets, particularly when the number of features is high, due to the matrix inversion involved.
In summary, the Normal Equation is a foundational concept in statistics and machine learning, particularly useful for efficiently solving linear regression problems when the dataset est de taille gérable.