View source for Joint Approximation Diagonalization of Eigen-matrices

{{Short description|Independent component analysis algorithm}}
'''Joint Approximation Diagonalization of Eigen-matrices (JADE)''' is an algorithm for [[independent component analysis]] that separates observed mixed signals into [[Latent variable|latent]] source signals by exploiting [[Moment (mathematics)#Kurtosis|fourth order moments]].<ref name = "main">{{cite journal|last1=Cardoso|first1=Jean-François|last2=Souloumiac|first2=Antoine|title=Blind beamforming for non-Gaussian signals|journal=IEE Proceedings F - Radar and Signal Processing |date=1993|volume=140|issue=6|pages=362–370|doi=10.1049/ip-f-2.1993.0054|citeseerx=10.1.1.8.5684}}</ref> The fourth order moments are a measure of non-Gaussianity, which is used as a proxy for [[Independent component analysis#Defining component independence|defining independence between the source signals]]. The motivation for this measure is that Gaussian distributions possess zero excess [[kurtosis]], and with non-Gaussianity being a canonical assumption of ICA, JADE seeks an orthogonal rotation of the observed mixed vectors to estimate source vectors which possess high values of excess kurtosis.

== Algorithm ==
Let <math>\mathbf{X} = (x_{ij}) \in \mathbb{R}^{m \times n}</math> denote an observed data matrix whose <math>n</math> columns correspond to observations of <math>m</math>-variate mixed vectors. It is assumed that <math>\mathbf{X}</math> is ''prewhitened'', that is, its rows have a sample mean equaling zero and a sample covariance is the <math>m \times m</math> dimensional identity matrix, that is,
{{Center|<math> \frac{1}{n}\sum_{j=1}^n x_{ij} = 0 \quad \text{and} \quad \frac{1}{n}\mathbf{X}{\mathbf X}^{\prime}  = \mathbf{I}_m </math>. }}
Applying JADE to <math>\mathbf{X}</math> entails
# computing ''fourth-order cumulants'' of <math>\mathbf{X}</math> and then
# optimizing a ''contrast function'' to obtain a <math>m \times m</math> rotation matrix <math>O</math>
to estimate the source components given by the rows of the <math>m \times n</math> dimensional matrix <math>\mathbf{Z} := \mathbf{O}^{-1} \mathbf{X}</math>.<ref name=JADE-main>{{cite journal|last1=Cardoso|first1=Jean-François|title=High-order contrasts for independent component analysis|journal=Neural Computation|date=Jan 1999|volume=11|issue=1|pages=157–192|doi=10.1162/089976699300016863|citeseerx=10.1.1.308.8611}}</ref>

==References==

{{Reflist}}

[[Category:Computational statistics]]


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