E. Notation and Symbols

Some mathematical notations are used throughout this book:

$\bf x$

a vector

$\bf A$

a matrix

x_i

component of the vector $\bf x$

a_ij

component of the matrix $\bf A$

$\bf a^{T}_{}$$\bf b$

scalar product of the vectors $\bf a$ and $\bf b$

$\bf a$$\bf b^{T}_{}$

matrix with components a_ib_j (direct product)

$\left\langle\vphantom{ x }\right.$x$\left.\vphantom{ x }\right\rangle$

expectation value of a random variable x

{$\bf x_{i}^{}$}

set of vectors with index i

p($\bf x$| j)

probability of $\bf x$ given the condition j (conditional probability)

The meaning of often used symbols:

t

time (discrete)

S_t

sensory state at time t

M_t

motor command at time t

IR

set of all real numbers

n

number of training patterns

d

dimension of training patterns

m

number of units in a mixture, or for kernel PCA, the number of points in a reduced set

q

number of principal components

$\bf c_{j}^{}$

code-book vector or the center of the unit j

$\bf C$

covariance matrix of a data distribution

$\bf W$

d×q matrix containing the principal components as columns

$\bf w$

a principal component

$\lambda^{l}_{}$

eigenvalue belonging to the principal component l

$\sigma^{2}_{}$

residual variance per dimension. $\sigma$ is also used as the width of a Gaussian function

$\bf K$

kernel matrix

In this book, the following abbreviations appear:

PCA

principal component analysis (or analyzer)

MLP

multi-layer perceptron

RNN

recurrent neural network

SOM

self-organizing map

PSOM

parametrized self-organizing map

NGPCA

neural gas extended to principal component analysis

MPPCA

mixture of probabilistic principal component analyzers

RRLSA

robust recursive least square algorithm