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  "body": "\n## Definition\n\nLet $p \\ge 2$.\nThe *independent component analysis model* for a random vector $X$ in $\\R^p$ is\n$$X = A S,$$\nwhere:\n\n1. $A$ is an unknown [[invertible|0EB9N0S33C]] $p \\times p$ matrix, the *mixing matrix*.\n2. $S = (S_1, \\ldots, S_p)$ is an unobserved random vector whose components are [[mutually independent|0GQ938GDE0]] and non-constant.\n3. At most one of the $S_i$ has a [[normal distribution|05AQE0QR0D]].\n\n*Independent component analysis* estimates, from observations of $X$, an unmixing matrix $W$ such that $W X$ has independent components.\nThen $W X$ is the estimate of the sources $S$.\n\n## Remarks\n\nUnder clauses 1 to 3, $A$ is determined up to a permutation of its columns and a nonzero scaling of each column.\nSo the sources are recovered up to order and scale.\nExample: let $S_1, S_2$ be independent and uniform on $[-1, 1]$, and $A = \\begin{pmatrix} 1 \u0026 1 \\\\ 0 \u0026 1 \\end{pmatrix}$.\nThen $X = (S_1 + S_2, S_2)$ has dependent components, and $W = A^{-1}$ gives back $W X = S$.\nNon-example: if $S_1$ and $S_2$ are independent standard normal, clause 3 fails.\nFor every rotation matrix $Q$, the vector $Q S$ is again standard normal with independent components, so $A$ and $A Q^T$ give the same law of $X$ and cannot be told apart.\nAlgorithms usually whiten $X$ first and then maximise a measure of non-Gaussianity, such as kurtosis or negentropy.\n",
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