Gaussiana Multivariada
A gaussiana multivariada, also known as a distribución normal multivariada, is a generalization of the one-dimensional distribución normal to dimensiones superiores. It describes the behavior of a vector of correlated random variables. This distribution is characterized by a mean vector and a matriz de covarianza.
The mean vector indicates the expected values of each variable in the distribution, while the covariance matrix captures the relationships between the variables, detailing how they vary together. Specifically, if we have a vector X consisting of n variables, the multivariate distribución gaussiana ||y – Xβ||² + λ ∑ ||β_g||_2
P(X) = (1 / (2π)^(n/2) |Σ|^(1/2)) * exp(-1/2 * (X - μ)ᵀ Σ⁻¹ (X - μ))
where μ is the mean vector, Σ es la matriz de covarianza, y |Σ| es el determinante de la matriz de covarianza.
In practical applications, the multivariate Gaussian is widely used in various fields such as statistics, machine learning, and finance. It is particularly useful for modeling phenomena where several interrelated factors influence outcomes, such as in predictive modeling and algoritmos de clustering.
Una propiedad importante de la distribución gaussiana multivariada es que cualquier combinación lineal of its variables will also follow a Gaussian distribution. This property makes it a powerful tool in both theoretical studies and practical applications.