E. Notation and Symbols

Some mathematical notations are used throughout this book:

- a vector
- a matrix
*x*_{i}- component of the vector
*a*_{ij}- component of the matrix
- scalar product of the vectors and
- matrix with components
*a*_{i}*b*_{j}(direct product) -
*x* - expectation value of a random variable
*x* - {}
- set of vectors with index
*i* -
*p*(|*j*) - probability of 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
- code-book vector or the center of the unit
*j* - covariance matrix of a data distribution
*d*×*q*matrix containing the principal components as columns- a principal component
- eigenvalue belonging to the principal component
*l* - residual variance per dimension. is also used as the width of a Gaussian function
- 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

2005-03-22