Forschungsmethoden und Mathematische Psychologie

Prof. Dr. Jürgen Heller


Research Interests

My primary goal is to develop psychological theories, and to solve substantial psychological problems, both in basic research as well as in practical applications (e.g. technology-enhanced learning). In many cases meeting this objective cannot be achieved by simply employing well-known methods of psychological research, but requires to develop new approaches.

As my research interests are spread across a spectrum of quite different psychological areas I am using and developing a whole variety of methods: Principles of representational measurement theory are applied to a formal treatment of binocular distance perception. Classical and modern methods of psychophysics, including functional equations techniques, are used to characterize invariants in lightness and brightness perception, and to determine their impact on the form of the psychophysical function. A new experimental method for a psychological investigation of individual verbal concepts was developed within a set-theoretic framework. Sophisticated methods for parameter estimation and model testing are developed for probabilistic knowledge structures.

Lightness and Brightness Perception

In a classical study, M.J. Plateau (1872) gave a pair of painted disks - one white, one black - to each of eight artists and instructed them to return to their respective studios and paint a gray disk midway between the two. As Plateau reported, the resulting gray disks were virtually identical for all artists, in spite of the variation in the illumination conditions.

Looking for the impact of Plateau's classical result on the current research in lightness and brightness perception, one is confronted with the following situation:

On the one hand, the midgray operation is not used as an experimental paradigm. Nearly all empirical studies confine themselves to collecting cross-context matches. On the other hand, J.-Cl. Falmagne (1985) shows that the illumination invariance observed by Plateau puts severe constraints on the possible form of the psychophysical function, which turns out to be either a logarithmic function, or a power function. Falmagne's arguments, however, rely on assumptions on the cross-context matching that may be valid in complex scenes, but do not hold in the settings that are used in experimental research (e.g. simple center-surround stimulus configurations).

Current projects:

Probabilistic Knowledge Structures

In a seminal paper Doignon & Falmagne (1985) suggested a set-theoretical framework for designing efficient procedures for knowledge assessment. Up to now the so-called theory of knowledge spaces was developed further and applied successfully over more than two decades. A state of the art report is provided by Doignon & Falmagne (1999). One of the main problems in the field is to devise appropriate methods for empirically validating knowledge structures. Off-the-shelf techniques of parameter estimation and statistical testing cannot serve this purpose, and at the present only more or less ad-hoc procedures are available.

Current projects:

Individual Semantic Structures

It is common practice in the psychological investigation of verbal concepts to capture their underlying semantic features by analysing proximity data, which arise either directly or indirectly from judgments of similarity in meaning. The analysis often consists in applying algorithmic procedures, like multidimensional scaling or clustering methods. Proceeding in this way, however, causes severe problems to any interpretation of the results. Since the assumptions of the used methods are most often not tested in the applications, it is not clear how the obtained results are related to the data. Moreover, the data usually have to be aggregated over subjects, so it is not possible to consider individual semantic structures.

By adapting basic ideas from knowledge space theory and Formal Concept Analysis an extensional, set-theoretic representation of individual semantic structures can be established. The notion of a semantic space is introduced, which is formally an intersection-stable family of subsets of the set of stimuli under study. The representation is based on a new experimental paradigm in which the subject has to decide whether pairs of stimuli have some features in common that are not shared by a third stimulus. This paradigm formally resembles the procedure for querying an expert, which has been devised in the context of knowledge space theory. Sufficient conditions for the representation have been presented, and its uniqueness has been characterized (cf. Heller, 2000).

Current projects:

Binocular Space Perception

Binocular vision gives rise to a perceptual space that is endowed with a rich geometrical structure, which according to R.K. Luneburg (1947) is a non-Euclidean Riemannian geometry of constant Gaussian curvature. This hypothesis, together with certain psychophysical assumptions, provides a qualitative explanation of classical empirical phenomena, but is less successful in quantitatively predicting experimental data. At least some of these shortcomings can be attributed to the systematic empirical failure of psychophysical assumptions that are independent of the presumed internal structure of visual space.

Heller (1997) generalized Luneburg's psychophysical mapping by introducing visual polar coordinates within a conjoint measurement approach. The conjoint structure is formulated for the physical horizontal plane at eye level, with its components being constituted by the monocular directions with respect to both eyes. The underlying empirical assumptions characterize basic properties of the visual system, like the independence of the eyes, and the functional relationship between corresponding and symmetrical retinal points. The proposed generalized theory is able to explain some of the empirical observations that have been considered to contradict Luneburg's conception of a psychophysical theory of binocular space perception.

In collaboration with János Aczél, Che Tat Ng (University of Waterloo, Canada), and Zóltan Boros (University of Debrecen, Hungary), an axiomatization of Luneburg's psychophysical mapping, as well as that of a straightforward generalization, was developed by considering the so-called iseikonic transformations as automorphisms of the conjoint structure (Aczél, Boros, Heller & Ng, 1999).

Experimental investigations were conducted in a worldwide unique laboratory for binocular space perception. To control for monocular cues, observers were presented with point-like light sources in complete darkness. The setup was completely computer-cotrolled and, by presenting stimuli in real space, avoided some artifacts that occur with stereoscopic displays.

Current projects: