[Paper Review] Another Approximation of the First-Passage Time Densities for the Ratcliff Diffusion Decision Model

Kendal B. Foster, Henrik Singmann|arXiv (Cornell University)|Apr 5, 2021

Transportation Planning and Optimization5 references4 citations

TL;DR

This paper introduces a novel, faster approximation method for the first-passage time density in the Ratcliff diffusion decision model (DDM) by eliminating the need to precompute the number of terms in the infinite sum. Implemented in C++ via the R package fddm, the new method—combined with the large-time approximation—achieves superior speed and stability, outperforming existing methods in both benchmark tests and real-data fitting scenarios.

ABSTRACT

We present a novel method for approximating the probability density function (PDF) of the first-passage times in the Ratcliff diffusion decision model (DDM). We implemented this approximation method in $ exttt{C++}$ using the $ exttt{R}$ package $ exttt{Rcpp}$ to utilize the faster $ exttt{C++}$ language while maintaining the $ exttt{R}$ language interface. In addition to our novel approximation method, we also compiled all known approximation methods for the DDM density function (with fixed and variable drift rate), including previously unused combinations of techniques found in the relevant literature. We ported these approximation methods to $ exttt{C++}$ and optimized them to run in this new language. Given an acceptable error tolerance in the value of the PDF approximation, we benchmarked all of these approximation methods to compare their speed against each other and also against commonly used $ exttt{R}$ functions from the literature. The results of these tests show that our novel approximation method is not only orders of magnitude faster than the current standards, but it is also faster than all of the other approximation methods available even after translation and optimization to the faster $ exttt{C++}$ language. All of these approximation methods are bundled in the $ exttt{fddm}$ package for the $ exttt{R}$ statistical computing language; this package is available via CRAN, and the source code is available on GitHub.

Motivation & Objective

• To develop a faster and more stable approximation method for the probability density function (PDF) of first-passage times in the Ratcliff diffusion decision model (DDM).
• To address the computational inefficiency of existing methods, particularly the high cost of precomputing terms in infinite sums for small- and large-time response times.
• To improve numerical stability and performance in optimization settings, such as fitting the DDM to empirical data.
• To implement and benchmark all known DDM density approximation methods—fixed and variable drift rate—in a unified, optimized C++-based R package.
• To provide a robust, high-performance software solution for researchers using the DDM in cognitive modeling and response time analysis.

Proposed method

• Proposes a novel 'SWSE' (small-time) approximation method that avoids precomputing the number of terms in the infinite sum, reducing computational overhead.
• Combines the new SWSE small-time method with the established large-time approximation from [11] to create a hybrid 'combined-time' approach that adapts to response time range.
• Implements all approximation methods—including previously unused combinations—in C++ using Rcpp to leverage C++'s speed while retaining an R interface.
• Optimizes all implementations for performance, including efficient summation styles (S14 and S17), with S17 preferred for its balance of speed and term count.
• Uses a heuristic to dynamically select between small-time and large-time approximations based on effective response time, enhancing both speed and accuracy.
• Benchmarks all methods under identical conditions, including fixed response times and full optimization routines, to assess real-world performance.

Experimental results

Research questions

• RQ1 Can a new approximation method for the DDM density function be developed that eliminates the need for precomputing the number of terms in the infinite sum?
• RQ2 How does the performance of the new SWSE small-time approximation compare to existing methods in terms of speed and numerical accuracy?
• RQ3 Does combining the SWSE small-time method with the large-time approximation from [11] yield a more robust and faster overall algorithm across diverse parameter spaces?
• RQ4 To what extent does the overhead of R function calls affect the relative performance of different DDM approximation methods in optimization settings?
• RQ5 Is the new combined approximation method suitable for reliable and efficient fitting of the DDM to real empirical data?

Key findings

• The novel SWSE small-time approximation method is significantly faster than existing methods because it avoids the computationally expensive precomputation of sum terms.
• The combined SWSE + large-time approximation method is the fastest and most consistent implementation across all tested parameter ranges and response time values.
• Even after translation and optimization to C++, the new method outperforms all previously available approximation techniques in speed and stability.
• The combined method reduces convergence issues in optimization routines by avoiding the numerical instability of pure small-time methods at large response times.
• Benchmark results show that while the performance gap narrows in full optimization settings due to R’s function call overhead, the new method remains the fastest and most reliable choice.
• The fddm R package, which includes the new method, is recommended as the default for DDM fitting due to its superior speed, accuracy, and robustness.