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PCA: Principal Component Analysis

ICA: Independent Component Analysis

pdf: probability density function

**Variables and constants:**

*i*: General-purpose index, also: imaginary unit

*m*: Dimension of the observed data

*n*: Dimension of the transformed component vector

*t*: Time or iteration index

*x* and *y*: General-purpose scalar random variables

*y*_{i}: Output of the *i*-th neuron in a neural network

:
A scalar constant

:
Learning rate constant or sequence

All the vectors are printed in boldface lowercase letters, and are column
vectors:

:
Observed data, an *m*-dimensional random vector

Also: the input vector of a neural network

:
*n*-dimensional random vector of transformed components *s*_{i}

:
*m*-dimensional random noise vector

:
*m*-dimensional constant vector

:
Weight vectors of a neural network indexed by *i*

:
*m*-dimensional general-purpose random vector

Also: the output vector of a neural network

All the matrices are printed in boldface capital letters:

:
The constant
mixing matrix in the ICA model

:
A transformed
mixing matrix

:
Covariance matrix of ,

:
The weight matrix of an artificial neural network, with rows

Also: A general transformation matrix

**Functions:**

:
Mathematical expectation

*f*(.): A probability density function

*f*_{i}(.): Marginal probability density functions

:
The characteristic function of a random variable

*g*(.): A scalar non-linear function

*H*(.): Differential entropy

*I*(.): Mutual information

*J*(.): Negentropy

:
Kullback-Leibler divergence

*J*_{G}(.): Generalized contrast function

:
A general transformation from *R*^{m} to *R*^{n}

*h*(*t*): A FIR filter

:
The *i*-th order cumulant of a scalar random variable

:
Kurtosis, or fourth-order cumulant

:
Cumulant (cross-cumulant) of several random variables

**Other notation:**

:
Change in parameter

:
Proportional to (proportionality constant may change with *t*)

*f*': First derivative of function *f*

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*Aapo Hyvarinen*

*1999-04-23*