A mathematical formalization of the replaced elements model.

In this work, I develop a mathematical formalization of the Replaced Elements Model (Wagner. Quarterly Journal of Experimental Psychology: Section B, 56(1), 7, 2003), within a general framework proposed by Ghirlanda (Journal of Mathematical Psychology, 64/65, 8-16, 2015, Journal of Mathematical Psychology, 85, 55-61, 2018), which provides a new way to apply and study the model. The result derived here has the novelty of explicitly stating how the model computes associative values without requiring either the application of complex algorithms or the use of special software. As a way of showing how to use this formalization, I apply it to the study of varied learning phenomena and several models, by either analytic means or simulations. In the process, I reproduce conclusions drawn previously for the Replaced Elements Model by other methods (Glautier. Behavior Research Methods, 39(4), 993-1000, 2007; Schultheis et al. Behavior Research Methods, 40, 435-441, 2008; Wagner. Experimental Psychology: Section B, 56(1), 7, 2003). As an interesting byproduct, I provide a general algorithm which may be applied to simulate the predictions of the replaced elements model, Rescorla-Wagner's model (Rescorla & Wagner. Classical conditioning, Current research and theory, 2, 64-69, 1972), and Pearce's configural model (Pearce. Psychological Review, 94(1), 61, 1994) among others. Concrete instances of the algorithm, coded in Python, are provided in the Appendix.