Abstract
Models comprising, in sequence, linear analysers,
, non-linear transducer functions, and the Minkowski decision rule, are widely used to fit detection and discrimination data, especially when necessary to take into account the effect of probability summation. However, the analysers’ characteristics cannot be derived from detection/discrimination data because of an unavoidable trade-off between these characteristics and the decision rule. Here we show how to overcome this problem, i.e. how to identify the analysers despite the probability summation between them. The observer's performance is assumed to be quantitatively defined in terms of an equi-detection (discrimination) surface. Each analyser Ψi is expressed as a weighted sum of linear (coordinate) functionals φj:Ψ1,…,Ψn, so that an identification of the analysers ϕi is then reduced to evaluating the weight matrix A={aij}. It is proved that A can be uniquely recovered from a quadratic approximation of the equi-detection (discrimination) surface at the neighbourhood of at least two points. More specifically, the following equation holds true: ATD(AT)#=H1H2#, where A# is the generalised inverse of the matrix A, D is an unknown diagonal matrix, H1 and H2 are the matrices of the quadratic forms determining the quadratic surfaces approximating the equi-detection (discrimination) surface at two different points. Thus, the matrix H1H2# known from experiment is a similarity transform of the diagonal matrix, the rows of A being the eigenvectors of H1H2#. Hence, any eigensystem routine can be used to derive A from H1H2#.
, non-linear transducer functions, and the Minkowski decision rule, are widely used to fit detection and discrimination data, especially when necessary to take into account the effect of probability summation. However, the analysers’ characteristics cannot be derived from detection/discrimination data because of an unavoidable trade-off between these characteristics and the decision rule. Here we show how to overcome this problem, i.e. how to identify the analysers despite the probability summation between them. The observer's performance is assumed to be quantitatively defined in terms of an equi-detection (discrimination) surface. Each analyser Ψi is expressed as a weighted sum of linear (coordinate) functionals φj:Ψ1,…,Ψn, so that an identification of the analysers ϕi is then reduced to evaluating the weight matrix A={aij}. It is proved that A can be uniquely recovered from a quadratic approximation of the equi-detection (discrimination) surface at the neighbourhood of at least two points. More specifically, the following equation holds true: ATD(AT)#=H1H2#, where A# is the generalised inverse of the matrix A, D is an unknown diagonal matrix, H1 and H2 are the matrices of the quadratic forms determining the quadratic surfaces approximating the equi-detection (discrimination) surface at two different points. Thus, the matrix H1H2# known from experiment is a similarity transform of the diagonal matrix, the rows of A being the eigenvectors of H1H2#. Hence, any eigensystem routine can be used to derive A from H1H2#.
Original language | English |
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Pages (from-to) | 495-506 |
Number of pages | 12 |
Journal | Journal of Mathematical Psychology |
Volume | 47 |
Issue number | 5-6 |
DOIs | |
Publication status | Published - Oct 2003 |
Externally published | Yes |
Keywords
- Analysers
- Channels
- Detection
- Discrimination
- Identification
- Linear system
- Masking
- Vision
ASJC Scopus subject areas
- General Psychology
- Applied Mathematics