Best linear prediction: from covariance to linear model with additive noise and vice-versa (theory)

Antonio Sala, UPV

Difficulty: *** ,       Relevance: ,      Duration: 09:20

*Enlace a Spanish version

Summary:

This video discusses how two related statistical computations are related:

First, the video discusses how to obtain a linear model with additive independent noise from a given covariance matrix of a pair of random variables $(a,b)$. Indeed, using the best-linear prediction formula, the prediction can be expressed asserting that, if we believe linearity, the conditional distribution of b given a may be written as $b=𝜃a+𝜖$, being $𝜃={\mathrm{\Sigma }}_{ba}{\mathrm{\Sigma }}_{aa}^{-1}$ and $𝜖$ having variance ${\mathrm{\Sigma }}_{bb}-{\mathrm{\Sigma }}_{ba}{\mathrm{\Sigma }}_{aa}^{-1}{\mathrm{\Sigma }}_{ba}^T$. This does not mean that the actual underlying physics is linear but, well, it explains the observed variance.

Second, the inverse process is illustrated, i.e., how to obtain the associated covariance matrix from a linear model with additive noise $b=𝜃a+𝜖$.

Numerical examples are left for videos [vcinv1bEN] and [vcinv2EN].

*Link to my [ whole collection] of videos in English. Link to larger [ Colección completa] in Spanish.