Multivariate Gaussian/Normal Distribution (MVN)
- is a type of multivariate probability distribution
- is a type of Process
- not to be confused with NMM)
- is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions
- is often used to approximate any set of (possibly) correlated real-valued random variables each of which clusters around a mean value
MVN - Probability Density Function
A multivariate normal distribution is where each random variable is normally distributed and their joint distribution is also normal.
The multivariate normal distribution is defined by:
- mean vector 𝝁
- covariance matrix 𝚺
Visually, the multivariate normal distribution is:
- centered around the mean 𝝁
- its shape is defined by the covariance matrix 𝚺
Covariance Matrix Types
Non-Degenerate Case
The multivariate normal distribution is said to be “non-degenerate” when the symmetric covariance matrix is positive definite. In this case, the distribution has a density
where:
- 𝝁 is the 𝑘-dimensional mean vector
- 𝐱 is a real 𝑘-dimensional column vector
- 𝚺 is a 𝑘x𝑘 covariance matrix
- 𝚺-1 is the inverse matrix of 𝚺
- |𝚺| ≡ 𝑑𝑒𝑡 𝚺 is the determinant of 𝚺 (also known as the generalized variance)
The equation above reduces to that of the univariate normal distribution if:
- 𝝁 is a 1-dimensional vector
- 𝐱 is a 1-dimensional vector
- 𝚺 is a 1x1 matrix
Degenerate Case
TODO
MVN - Bivariate Unimodal Model (Example)
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