Parameter estimation of hyper-spherical diffusion models with a time-dependent threshold: An integral equation method

Abstract

Over the past several decades, decision-making research has been dominated by the study of binary choice tasks, with key models assuming that people remain equally cautious regardless of how long they have spent on the choice problem. Recent research has begun to place a greater focus on studying tasks with a continuous-response scale, as well as models that allow for decreases in caution over decision time; however, these research topics have remained separate from one another. For instance, proposed models of continuous-response scales have assumed constant caution over time, and studies investigating whether caution decreases over time have focused on binary choice tasks. One reason for this separation is the lack of methodology for estimating the parameters of the decision models with time-dependent parameters for continuous responses. This paper aims to provide a stable and efficient parameter estimation technique for hyper-spherical diffusion models with a time-dependent threshold. Here, we propose an integral equation method for estimating the first-passage time distribution of hyper-spherical diffusion models. We assessed the robustness of our method through parameter recovery studies for constant and time-dependent threshold models, with our results demonstrating efficient and precise estimates for the parameters in both situations.

Keywords: Collapsing thresholds; Continuous option space; Decision making; Parameter estimation; Sequential sampling models.

Fig. 1

The schematic plots of a one-dimensional (left panel; diffusion decision model), a two-dimensional (middle panel; circular diffusion model), and a three-dimensional (right panel; spherical diffusion model) diffusion model. The figure is adapted from Smith and Corbett (2019)

Fig. 2

A schematic view of two-dimensional HSDM (circular diffusion model) with a linear collapsing threshold. The radius of the circular decision threshold shrinks linearly over time

Fig. 3

The marginal response time distribution for a circular diffusion model with drift rate $\mathit{\mu }={[6,6]}^{\prime }$, diffusion coefficient $\sigma =1$, and decision threshold $b=5$. Upper-Left panel: The marginal distributions for short response times based on Bessel series estimation with 500 terms. Upper-tight panel: The absolute value of the first 50 terms of the Bessel series (6) for different time points. Each dot shows the absolute value of a single term. Lower panel: The marginal response time distribution of 10,000 simulated trajectories against the Bessel series and the integral equation estimations. The red dashed line represents the Bessel series estimation with 500 terms for the marginal distribution of response time longer than 0.1, and the blue dashed line shows the integral equation estimation

Fig. 4

The parameter recovery plots for three HSDMs with a constant threshold. In each panel, the x-axis corresponds to the actual parameter value (true parameter), and the y-axis corresponds to the estimated parameter. The first row shows the parameter recovery of a two-dimensional HSDM (i.e., circular diffusion model). The second row presents the parameter recovery of the three-dimensional HSDM (i.e., spherical diffusion model), and the last row exhibits the parameter recovery of the four-dimensional HSDM

Fig. 5

The quality of parameter estimation for constant threshold models based on the integral equation method (with $\mathrm{\Delta }t=0.05$) as a function of the number of trials. The first row shows the root mean square error (RMSE) for different numbers of trials (i.e., 50, 100, 150, 200, 250), the second row shows the correlation value ($\rho$), and the third row shows the R-squared value ($R^2$). Each column corresponds to one parameter, and different colors represent different models

Fig. 6

The convergence plot for parameter estimation based on the integral equation method for different $\mathrm{\Delta }t$ values (i.e., 0.2, 0.15, 0.1, and 0.05). The first row shows the root mean square error (RMSE) as a function of $\mathrm{\Delta }t$, the second row shows the correlation value ($\rho$), and the third row shows the R-squared value ($R^2$). Each column corresponds to one parameter, and different colors represent different models

Fig. 7

Illustration of the effect of $\mathrm{\Delta }t$ on the precision of parameter recovery for the fixed threshold circular diffusion model. In each panel, the x-axis corresponds to the true generating parameter, and the y-axis corresponds to the estimated parameter

Fig. 8

The marginal distribution of response time for 10,000 simulated trajectories with $\mathit{\mu }={[1,1]}^{\prime }$, and linear collapsing threshold in two-dimensional HSDM (i.e., linear collapsing circular diffusion model). The left panel shows the response time distribution with $b_0=3$ and different slopes. The right panel shows the response time distribution for $\lambda =1.5$ and different starting thresholds. The solid gray lines represent the kernel density estimation for the simulated data, while the blue dashed lines denote the predictions obtained using the integral equation method

Fig. 9

The parameter recovery plots for three HSDMs with the linear collapsing threshold. In each panel, the x-axis corresponds to the actual parameter value (true parameter), and the y-axis corresponds to the estimated parameter. The first row shows the parameter recovery of a two-dimensional HSDM (i.e., circular diffusion model). The second row presents the parameter recovery of the three-dimensional HSDM (i.e., spherical diffusion model), and the last row exhibits the parameter recovery of the four-dimensional HSDM. All parameters were being estimated simultaneously

Fig. 10

The quality of parameter estimation for linear collapsing threshold models based on the integral equation method (with $\mathrm{\Delta }t=0.02$) as a function of the number of trials. The first row shows the root mean square error (RMSE) as a function of $\mathrm{\Delta }t$, the second row shows the correlation value ($\rho$), and the third row shows the R-squared value ($R^2$). Each column corresponds to one parameter, and different colors represent different models

Fig. 11

The quality of parameter estimation for linear collapsing threshold models based on 500 trials using the integral equation method as a function of $\mathrm{\Delta }t$. The first row shows the root mean square error (RMSE) as a function of $\mathrm{\Delta }t$, the second row shows the correlation value ($\rho$), and the third row shows the R-squared value ($R^2$). Each column corresponds to one parameter, and different colors represent different models

Fig. 12

The parameter recovery plots based on the Bessel series method for three HSDMs with a constant threshold. In each panel, the x-axis corresponds to the actual parameter value (true parameter), and the y-axis corresponds to the estimated parameter. The first row shows the parameter recovery of a two-dimensional HSDM (i.e., circular diffusion model). The second row presents the parameter recovery of the three-dimensional HSDM (i.e., spherical diffusion model), and the last row exhibits the parameter recovery of the four-dimensional HSDM

Fig. 13

Illustration of the effect of $\mathrm{\Delta }t$ on the precision of parameter recovery for the fixed threshold spherical diffusion model. In each panel, the x-axis corresponds to the true generating parameter, and the y-axis corresponds to the estimated parameter

Fig. 14

The illustration of the effect of $\mathrm{\Delta }t$ on the precision of parameter recovery for the fixed threshold hyper-spherical diffusion model. In each panel, the x-axis corresponds to the true generating parameter, and the y-axis corresponds to the estimated parameter

Fig. 15

The parameter recovery plots for three HSDMs with constant boundary in the presence of across-trial variability in drift rate. In each panel, the x-axis corresponds to the actual parameter value (true parameter), and the y-axis corresponds to the estimated parameter. The first row shows the parameter recovery of a two-dimensional HSDM (i.e., circular diffusion model). The second row presents the parameter recovery of the three-dimensional HSDM (i.e., spherical diffusion model), and the last row exhibits the parameter recovery of the four-dimensional HSDM

Fig. 16

The marginal distribution of response time for $10000$ simulated trajectories with $\mathit{\mu }=(1,1)$, and hyperbolic collapsing threshold in two-dimensional HSDM (i.e., hyperbolic collapsing circular diffusion model). The left panel shows the response time distribution with $b_0=4$ and different decay rates. The right panel shows the response time distribution for $\lambda =2$ and different starting thresholds. The solid gray lines represent the kernel density estimation for the simulated data, while the blue dashed lines denote the predictions obtained using the integral equation method

Fig. 17

The marginal distribution of response time for $10000$ simulated trajectories with $\mathit{\mu }=(1,1)$, and exponential collapsing threshold in two-dimensional HSDM (i.e., exponential collapsing circular diffusion model). The left panel shows the response time distribution with $b_0=3$ and different decay rates. The right panel shows the response time distribution for $\lambda =1$ and different starting thresholds. The solid gray lines represent the kernel density estimation for the simulated data, while the blue dashed lines denote the predictions obtained using the integral equation method