Multidimensional Models of Perception and Cognition

By F. Gregory Ashby | Go to book overview

multivariate normal pdf for the momentary psychological magnitudes xi. Ennis and Mullen ( 1992) discuss one such case leading to a unidimensional normal pdf.

In other models discussed in this chapter, no attention was given to stimulus parameters or to processes that might have led to the percepts. In this sense, the probabilistic models already discussed are much more restrictive than the models in this section. These models, however, require a great deal of work before they can be put to use in modeling experimental results.


SOME COMPUTING NOTES

Integral expressions such as Equations 6, 9, 13, and 16 were evaluated numerically on Gould 32/97 and Trace Multiflow computers. An adaptive routine by Genz and Malik ( 1980) was found to be useful for multiple integration. A check for gross errors in the numerical computations was achieved by conducting large- scale (100,000 trials per estimate) Monte Carlo evaluations.

In some cases, such as the evaluation of Equation 13, significant saving in computer time can be achieved by using Cholesky Factorization to avoid the need to compute fi(z). Taking the standard multivariate normal pdf, one can select values on each dimension (for example, the median of equal-probability intervals) and convert them to values in probabilistically equivalent intervals from the multivariate normal pdf of interest. This is achieved by computing Z = AY + μ, where A is a lower triangular matrix such that LL′ = Σ,Y is a vector with a standard multivariate normal pdf, and μ is the mean of the desired pdf. Since the interval bounding each Z is probabilistically equivalent to each corresponding interval bounding Y, it is only necessary to compute the probability contents of these intervals once. Numerical integration of f(Z)g(Z) then becomes a dot product operation with a constant vector of probability weights [a vector computed from f(Z) at particular values of Z] for all values of Σ and μ. These values can be computed once, stored, and reused as needed. It is still necessary, of course, to compute g(Z) for all values of Z. In some cases, this approach to numerical integration can lead to significant improvements in computing speed compared with adaptive numerical integration of the entire function. See Chapter 1 for a more thorough discussion of this technique.


CONCLUDING REMARKS

Changes occur continuously in the physical and chemical properties of stimuli in the world. Neither do our biological transduction and information processing systems remain static from moment to moment. Although universal principles governing acquisition of information, judgment, and behavior may exist, they may not be revealed by using deterministic or static models of these processes.

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