Given the birds, where is the flock? Visual estimation of the location of collections of points

How does the visual system assign locations to collections of points? The perceived location of a flock of birds is influenced by the locations of individual birds but is not synonymous with them. If the collection is a statistical sample from a particular distribution, the location estimation task is analogous to parametric estimation in statistics. The statistician knows the sample and also has partial knowledge of the distribution from which the sample was drawn. We examine how observers carry out parametric estimation tasks that are also visual organization tasks.

In our experiment the collections were one-dimensional statistical samples plotted along a horizontal line. They were drawn from one of three location families – the Gaussian, the Laplacian, and the Uniform. Observers saw only the sample and had to estimate the location of the generating probability density function. We compared human performance to unbiased minimum variance estimators (UMVUEs) for each family. We tested whether observers used the same or different estimators for all three distributional families and whether their estimators were UMVUEs.

Observers used different estimators for different distributional families. The estimators chosen for samples from the Gaussian and Uniform distributions were significantly different from the corresponding UMVUEs while that for the Laplacian was not. We developed a model of human performance based on clusters in samples. Clusters convey information about the distributions from which they are drawn. We propose a Two-Stage Visual Cluster Model (VCM) of location estimation. The observer first partitions the sample into non-overlapping clusters assigning a location to each cluster and then computes a weighted average of the cluster locations with weights reflecting how well each cluster fit the entire sample. The VCM captured human performance across all three distributions. These findings suggest that the visual system is sensitive to the structural information underlying the locations of individual points and bases their location estimates on Gestalt principles.