The Projected Spherical Diffusion Model: An Evidence Accumulation Theory for Estimation

People often encounter estimation problems in everyday life. Recently, evidence accumulation models (EAMs), originally proposed for discrete choice problems, have been extended to account for continuous estimation. The models offer a significant advantage over existing estimation theories by providing a mechanistic account of the processes underlying estimation and generating predictions for response and response-time distributions. Despite these advances, existing EAMs for estimation are largely restricted to circular response formats. As a result, they are difficult to apply to estimation tasks in which responses are constrained to a bounded range. Moreover, many current models lack mathematical tractability, complicating model estimation and limiting their practical applicability. To address these limitations, we introduce a diffusion model that combines a radial process with a Wiener process, yielding a positively valued process constrained to a bounded range. Geometrically, the model can be interpreted as a three-dimensional diffusion process projected onto a two-dimensional semicircular subspace. We derive analytical expressions for the joint distribution of response time and choice, making the model computationally tractable. We evaluate the model using two newly conducted sampling-based estimation experiments and three existing datasets involving numerosity and line-length estimations. The modeling results demonstrate that the model accurately captures important behavioral patterns, including a right-skewed response-time distribution, a skewed distribution of estimates, systematic changes in the shape of the response distribution, and the interaction between response time and response precision. Overall, the projected diffusion model provides a theoretically principled and mathematically tractable framework for explaining human estimation across a wide range of domains.