---
title: "Mathematical Description of Methods in `dfddm`"
author: "Kendal Foster and Henrik Singmann"
date: '`r format(Sys.Date(), "%B %d, %Y")`'
bibliography: references.bib
output:
  rmarkdown::html_vignette:
    css: stile.css
    toc: false
vignette: >
  %\VignetteIndexEntry{Mathematical Description of Methods in `dfddm`}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  error = TRUE,
  comment = "#>"
)
```

<script type="text/x-mathjax-config">
MathJax.Hub.Config({
  TeX: { equationNumbers: { autoNumber: "AMS" } }
});
</script>

<div id="TOC">
<ul>
  <li><a href="#background">Mathematical Background</a></li>
  <li><a href="#dens">The Density Function Approximations</a>
  <ul>
    <li><a href="#dens-lt">Large-Time</a></li>
    <li><a href="#dens-st">Small-Time</a></li>
    <li><a href="#dens-ct">Combining Large-Time and Small-Time</a></li>
  </ul></li>
  <li><a href="#references">References</a></li>
</ul>
</div>






Function `dfddm()` evaluates the density function (or probability density function, PDF) for the Ratcliff diffusion decision model (DDM) using different methods for approximating the full PDF, which contains an infinite sum. An empirical validation of the implemented methods is provided in the [Validity Vignette](validity.html). Timing benchmarks for the present methods and comparison with existing methods are provided in the [Benchmark Vignette](benchmark.html). Examples of using `dfddm()` for parameter estimation are provided in the [Example Vignette](example.html).

Our implementation of the DDM has the following parameters: $a \in (0, \infty)$ (threshold separation), $v \in (-\infty, \infty)$ (drift rate), $t_0 \in [0, \infty)$ (non-decision time/response time constant), $w \in (0, 1)$ (relative starting point), $sv \in (0, \infty)$ (inter-trial-variability of drift), and $\sigma \in (0, \infty)$ (diffusion coefficient of the underlying Wiener Process). Please note that for this vignette, we will refer to the inter-trial variability of drift as $\eta$ instead of $sv$ to make the notation in the equations less confusing.
<br><br>





# Mathematical Background {#background}
<hr class="sec1">

There are several different methods for approximating the PDF of the DDM, and there are three optional control parameters in `dfddm()` that can be used to indicate which method should be used in the function call: `switch_mech`, `n_terms_small`, and `summation_small`. For each method we describe, we include the parameter settings for the function call to `dfddm()` so that it uses the desired method. As these parameters are optional, leaving them blank results in the [default implementation](#default-implementation) that is indicated later in this vignette. For general purpose use, we recommend ignoring these optional parameters so that the default settings are used as this will be the fastest and most stable algorithm. Note that precedence for the optional parameters is first given to checking if the [default implementation](#default-implementation) is selected. If not, precedence is then given to the `switch_mech` parameter value; for example, `switch_mech = "large"` will ignore the `summation_small` input value.

Note that there are actually two probability density functions of the DDM: the PDF with respect to the upper boundary, and the PDF with respect to the lower boundary. Following the precedent set by the literature, we use the general terminology "PDF" or "density function" to mean the probability density function with respect to the lower boundary. Should the probability density function with respect to the upper boundary be required, it may be calculated using the simple transformation $f_\text{upper}(t ~|~ v, \eta, a, w) = f_\text{lower}(t ~|~ -v, \eta, a, 1-w)$.

Since the DDM is widely used in parameter estimation usually involving numerical optimization, significant effort has been put into making the evaluation of its density as fast as possible. However, the density function for the DDM is notorious for containing an unavoidable infinite sum; hence, the literature has produced a few different methods of approximating the density by truncating the infinite sum. This vignette details the various methods used in the literature to approximate the infinite sum in the density function for the DDM.

The author of the seminal book where the density function originates, @feller1968introduction explains the derivation from first principles. In this derivation there is a step that requires taking a limit, and @feller1968introduction provides two different &ndash; but equivalent &ndash; limiting processes that yield two different &ndash; but equal &ndash; forms of the density function. Each of these forms contains an infinite sum, and they are known individually as the large-time approximation and the small-time approximation because the former is on average faster when calculating the density for large response times and the latter is on average faster when calculating the density for small response times. The improved speed is due to the ease at which the infinite sums can be adequately approximated with a finite truncation. In other words, the efficiency of each approximation is dictated by the number of terms required in the truncated sum to achieve the pre-specified precision; fewer required terms translates to fewer computations and thus generally faster computation time.

When the drift rate is held constant (i.e. $\eta = 0$), the density function for the DDM is often written in a factorized form [@navarro2009fast]:
\begin{equation}
	f(t ~|~ v, a, w) = \frac{1}{a^2} \exp \left( -vaw -\frac{v^2 t}{2} \right) f_i \left( \frac{t}{a^2} ~\Big\vert~ 0, 1, w \right), \nonumber
\end{equation}
where $f_i(\frac{t}{a^2} | 0, 1, w)$ determines whether the large-time or small-time model will be used:
\begin{equation}
\begin{aligned}
	f_{i=\ell} (t ~|~ 0, 1, w) &= \pi \sum_{j = 1}^{\infty} j \exp \left( -\frac{j^2 \pi^2 t}{2a^2} \right) \sin \left( j w \pi \right),\\
	f_{i=s} (t ~|~ 0, 1, w) &= \frac{1}{\sqrt{2 \pi t^3}} \sum_{j = -\infty}^{\infty} (w + 2j) \exp \left( -\frac{(w + 2j)^2}{2t} \right).
\end{aligned}
\nonumber
\end{equation}

In an effort to simplify the terms inside the infinite summations as much as possible, we instead rewrite the constant drift rate density function as two separate functions without the factorization:
\begin{align}
	f_\ell(t | v, a, w) &= \frac{\pi}{a^2} e^{ \left( -vaw-\frac{v^2 t}{2} \right)}
										     \sum_{j = 1}^{\infty} j \sin \left( j w \pi \right) \exp{ \left( -\frac{j^2 \pi^2 t}{2a^2} \right)}, \label{eq:con-l} \\
  f_s(t | v, a, w) &= \frac{a}{\sqrt{2 \pi t^3}} e^{ \left( -vaw-\frac{v^2 t}{2} \right)}
										  \sum_{j = -\infty}^{\infty} (w + 2j) \exp{ \left( -\frac{a^2}{2t} \left( w + 2j \right)^2 \right)}. \label{eq:con-s}
\end{align}

In addition to having large-time and small-time variants, there exist two mathematically equivalent formulations for the infinite summation in the small-time density functions. The details and proof of equivalence of these two formulations are provided in the paper accompanying `fddm`, but we will continue to use the traditional formulation for the remainder of this vignette.

Now allowing the drift rate to vary across trials (i.e. $\eta > 0$), we should have two density functions. However, as only the small-time variable drift rate density function has been available in the literature [@blurton2017first], we provide the derivation of the large-time variable drift rate density function in the `fddm` paper. The large-time and small-time variable drift rate density function are:
\begin{align}
f_\ell(t | v, \eta, a, w) &= \frac{\pi}{a^2 \sqrt{1 + \eta^2 t}}
                               \exp{ \left( \frac{\eta^2 a^2 w^2 -2vaw -v^2 t}{2 (1 + \eta^2 t)} \right)}
															 \sum_{j = 1}^{\infty} j \sin \left( j w \pi \right) \exp{ \left( -\frac{j^2 \pi^2 t}{2a^2} \right)}, \label{eq:var-l}\\
f_s(t | v, \eta, a, w) &= \frac{a}{\sqrt{2 \pi t^3 \left( 1 + \eta^2 t \right)}}
											 			\exp{ \left( \frac{\eta^2 a^2 w^2 -2vaw -v^2 t}{2 (1 + \eta^2 t)} \right)}
											 		 	\sum_{j = -\infty}^{\infty} (w + 2j) \exp{ \left( -\frac{a^2}{2t} \left( w + 2j \right)^2 \right)}. \label{eq:var-s}
\end{align}

Immediately of note is that the infinite summation for each time scale is the same regardless of the inclusion of variability in the drift rate. It then follows that there exists a term $M$ such that the density function for the constant drift rate multiplied by $M$ yields the density function for the variable drift rate. That is, $M \cdot f(t | v, a, w) = f(t | v, a, w, \eta^2)$ from the above equations; this value $M$ works for converting both the large-time and small-time constant drift rate densities to variable drift rate densities. Although we do not use this term, it may be useful in adapting current algorithms to easily outputting the density with variable drift rate. Note that there are some issues with simply scaling the constant drift rate density, so please see the [Validity Vignette](#validity.html) for more information about the potential problems with this conversion. The multiplicative term $M$ is given below:
\begin{equation}
  M = \frac{1}{\sqrt{1 + \eta^2 t}} \exp \left( vaw + \frac{v^2 t}{2} + \frac{\eta^2 a^2 w^2 -2vaw -v^2 t}{2 (1 + \eta^2 t)} \right). \nonumber
\end{equation}





# The Density Function Approximations {#dens}
<hr class="sec1">

The main issue of these families of density functions is that they all contain an infinite sum that must be approximated. Since there is no closed form analytical solution to this infinite sum, we instead calculate only a partial sum by truncating the sequence of terms after a certain point. We cannot actually calculate the true value of the density function, but we can mathematically prove that we can get arbitrarily close to the true value; the proof of this fact is provided in the paper accompanying the `fddm` package. The nature of this truncation has been the topic of many papers in the literature, but the underlying idea supporting all of the methods is the same: the user specifies an allowable error tolerance, and the algorithm calculates terms of the infinite sum so that the truncated result is within the allowed error tolerance of the true value.

The methods in the literature pre-calculate the number of terms required for the infinite sum to converge within the allowed error tolerance, and this number of terms is referred to as $k_\ell$ and $k_s$ for the large-time and small-time infinite sums, respectively. @navarro2009fast include a method for calculating $k_c$, the number of required terms for the infinite sum when combining the density functions of the two time scales. In addition to these existing methods, we add a novel method that does not perform this pre-calculation, and we also provide two new combinations of the large-time and small-time density functions. Note that in each method that pre-calculates the number of terms, the response time $t$ is scaled inversely by $a^2$, that is $t' := \tfrac{t}{a^2}$. Also note that for the rest of this vignette, the ceiling function will be denoted by $\lceil \cdot \rceil$.




<div class="indent2">

## Large-Time {#dens-lt}

The large-time density functions, Equations <span class="math inline eqref">\(\eqref{eq:con-l}\)</span> and <span class="math inline eqref">\(\eqref{eq:var-l}\)</span>, have an infinite sum that runs for all of the positive integers. For a given error tolerance $\epsilon$, @navarro2009fast provide an expression for $k_\ell$, the number of terms required for the large-time infinite sum to be within $\epsilon$ of the true value of the density function. Thus the infinite sum becomes finite:
\begin{equation} \label{eq:kl}
  \sum_{j = 1}^{k_\ell^\text{Nav}} j \sin \left( j w \pi \right) \exp{ \left( -\frac{j^2 \pi^2 t'}{2} \right)}. \nonumber
\end{equation}

It remains to find the value of $k_\ell^\text{Nav}$ that ensures the truncated sum is $\epsilon$-close to the true value. @navarro2009fast provide a derivation in their paper that finds an upper bound for the tail of the sum, the sum of all terms greater than $k_\ell^\text{Nav}$ (i.e., the error). Then they back-calculate the number of terms required to force this upper bound on the error to be less than $\epsilon$, since then the actual error must also be less than $\epsilon$. The resulting number of terms is:
\begin{equation} \label{eq:kl-Nav}
	k_\ell^{\text{Nav}} \left( t', \epsilon \right) = \left\lceil \max \left\{ \sqrt{\frac{-2 \log(\pi t' \epsilon)}{\pi^2 t'}}, \frac{1}{\pi \sqrt{t'}} \right\} \right\rceil.
\tag{L.1}
\end{equation}

<p class="method">

This method is often viewed as the most inefficient of the available options in the literature; however, this method proves to be extremely efficient in particular areas of the parameter space (typically for large $t'$). To implement this method in `dfddm()` (and completely excluding the small-time methods), the user must set the parameter `switch_mech = "large"` in the function call; in this case, the other parameters are ignored.
</p>




## Small-Time {#dens-st}

The small-time approximations, Equations <span class="math inline eqref">\(\eqref{eq:con-s}\)</span> and <span class="math inline eqref">\(\eqref{eq:var-s}\)</span>, also contain an infinite sum, but this sum runs over all of the integers &ndash; from negative infinity to positive infinity. Given this infinite nature in both directions, it is impossible to rigorously define the number of terms required to achieve the $\epsilon$-accuracy because we don't know where to start counting the terms. To solve this issue, we rearrange the terms in the sum into the sequence $\left\{ b_0, b_{-1}, b_1, \dots, b_{-j}, b_j, b_{-(j+1)}, b_{j+1}, \dots \right\}$; this allows us not only to count the terms in a sensible manner but also to define $k_s$ as the index of the sequence where the truncation should occur. Then we can write the truncated version of the sum:
\begin{equation} \label{eq:ks}
  \sum_{j = -\left\lfloor\frac{k_s}{2}\right\rfloor}^{\left\lfloor\frac{k_s}{2}\right\rfloor} (w + 2j) \exp{ \left( -\frac{a^2}{2t} \left( w + 2j \right)^2 \right)}. \nonumber
\end{equation}

To choose the small-time methods when using `dfddm()` (and completely excluding the large-time method), set the optional parameter `switch_mech = "small"` in the function call. You can also set the optional control parameter `summation_small = "2017"` or `summation_small = "2014"`, but it is recommended to ignore this parameter so it retains its default value of "2017" that evaluates slightly faster than its counterpart. This parameter controls the style of summation used in the small-time approximation, and more details on the differences between these two styles can be found in the paper accompanying `fddm`. The final control parameter, `n_terms_small`, will be discussed in the following three subsections.



<div class="indent3">

### Navarro & Fuss {#dens-st-nav}

Similarly to their large-time solution, @navarro2009fast provide an expression for $k_s$ given an error tolerance $\epsilon$. They follow a similar idea to their large-time derivation by bounding the error of the truncation with integrals then back-calculating the number of terms required to keep the error less than $\epsilon$. The resulting approximation yields:
\begin{equation} \label{eq:ks-Nav}
  k_s^{\text{Nav}} \left( t', \epsilon \right) = \left\lceil \max \left\{ 2 + \sqrt{-2t' \log(2 \epsilon \sqrt{2 \pi t'})}, 1 + \sqrt{t'} \right\} \right\rceil.
\tag{S.1}
\end{equation}

<p class="method">

To use this method, set `switch_mech = "small"` and `n_terms_small = "Navarro"` in the function call. The parameter `summation_small` should be ignored so that it retains its default value to obtain the best performance.
</p>



### Gondan, Blurton, Kesselmeier {#dens-st-gon}

After @navarro2009fast published their paper, @gondan2014even introduced another method for calculating the required number of terms in the truncated small-time summation. It is important to note, however, that @gondan2014even provided the number of required _pairs_ of terms in the $S_{14}$ summation style, and not the number of required _individual_ terms. As we want the number of _individual_ terms, we adapt their formula and define it given a desired precision $\epsilon$:
\begin{equation} \label{eq:ks-Gon}
\begin{aligned}
  k_s^{\text{Gon}} \left( t', w, \epsilon \right) &= \left\lceil \max \left\{ \tfrac{1}{2} \left( \sqrt{2t'} - w \right), \tfrac{1}{2} \left( \sqrt{-t' (u_\epsilon - \sqrt{-2 u_\epsilon -2})} - w \right) \right\} \right\rceil,\\
  u_\epsilon &= \min \left\{ -1, \log(2 \pi t'^2 \epsilon^2) \right\}.
\end{aligned}
\tag{S.2}
\end{equation}

<p class="method">

To use this method, set `switch_mech = "small"` and `n_terms_small = "Gondan"` in the function call. The parameter `summation_small` should be ignored so that it retains its default value to obtain the best performance.
</p>


### Stop When Small Enough (SWSE) {#dens-st-swse}

If we consider the terms of the infinite sum as the sequence defined above, the series alternates in sign $(+, -, +, \dots)$; moreover, the series eventually decreases monotonically (in absolute value) due to the exponential term. Combining and exploiting these two mathematical properties has been the cornerstone of the previous approximations, but we will instead truncate the sum using a method suggested by @gondan2014even. This method does not pre-calculate the number of terms required to achieve the given error tolerance. Instead, the general idea of this method is to take full advantage of the alternating and decreasing nature of the terms in the infinite sum by applying a handy theorem (commonly known as the alternating series test) to place an upper bound on the truncation error after including so many terms. It has been proven that this upper bound is in fact the absolute value of the next term in the sequence, thus we can truncate the infinite sum once one of its terms (in absolute value) is less than the desired error tolerance, $\epsilon$. Hence we do not consider the number of terms in the sum, rather just that the terms in the summation will eventually be small enough. The validity of this method is proven in the paper that accompanies the `fddm` package.

<p class="method">

To use this method, set `swich_mech = "small"` and `n_terms_small = "SWSE"` in the function call. The parameter `summation_small` should be ignored so that it retains its default value to obtain the best performance.
</p>
</div>



## Combining Large-Time and Small-Time {#dens-ct}

A sensible next approach to approximating the density of the DDM is to use some combination of the large-time and small-time density functions. As their names suggest, each density function approximation has a section of the parameter space where it outperforms the other one. Essentially, these methods involve calculating the number of terms required for both the large-time and small-time density functions, then using whichever approximation requires fewer terms. The goal is to use each approximation where it is efficient and avoid the areas of the parameter space where the approximations perform poorly. The main control parameter used to indicate this preference is `switch_mech`.

Fundamentally, each combined time scale method uses either the large-time or small-time approximation for each calculation of the density function. As there is only one option for the large-time approximation, there are no optional control parameters to set for this part of the combined time scale approximation. In contrast, there are multiple options for the small-time part of the combined time scale approximation. However, the effect of the control parameter `summation_small` is consistent throughout the small-time methods; we recommend leaving it to its default value of "2017" for the best performance. The remaining subsections of this vignette detail how to set the rest of the control parameters in the function call to use a particular set of methods for calculating the combined time scale density function.



<div class="indent3">

### Navarro Small & Navarro Large {#dens-ct-nav}

@navarro2009fast initially suggested an algorithm wherein both $k_s^\text{Nav}$ and $k_l^\text{Nav}$ were calculated, and the preferred method was whichever one had the smaller value of $k$. However, one issue with their original algorithm arises when the user inputs a vector of response times. They calculate the maximum value of $k_s^\text{Nav}$ for _all_ of the input response times and then compare that to the maximum value of $k_l^\text{Nav}$ for _all_ of the input response times. They end up with one $k_c^\text{Nav}$ for _all_ of the input response times, and this one value can be larger than necessary for most of the input response times due to the maximization of $k_c^\text{Nav}$ over _all_ of the response times. This over-accuracy is not necessarily a bad thing, but it does mean that it is possible to get different densities for the same input response time depending on the other response times that have been input. To correct for this, we simply do not maximize $k_s^\text{Nav}$ or $k_l^\text{Nav}$ over all of the input response times and instead calculate one value of $k_c^\text{Nav}$ for _each_ input response time. For more details, see the "Implementation" section in the paper accompanying `fddm`.
\begin{equation} \label{eq:kc-Nav}
  k_c^\text{Nav} \left( t', w, \epsilon \right) = \min \left\{ k_s^{\text{Nav}}, k_l^{\text{Nav}} \right\}.
\tag{B.1}
\end{equation}

<p class="method">

To use this method, set `switch_mech = "terms"` and `n_terms_small = "Navarro"` in the function call. The parameter `summation_small` should be ignored so that it retains its default value to obtain the best performance.
</p>


### Gondan Small & Navarro Large {#dens-ct-gon}

This combination of methods has not been explored in the literature before, but it works very similarly to the Navarro-Navarro combination above. The only difference is that we use the @gondan2014even approximation for the small-time instead of the one provided by @navarro2009fast. Since $k_s^\text{Gon} \leq k_s^\text{Nav}$, this method should never be less efficient than the previous combined time scale method.
\begin{equation} \label{eq:kc-Gon}
  k_c^{\text{Gon}} \left( t', w, \epsilon \right) = \min \left\{ k_s^{\text{Gon}}, k_l^{\text{Nav}} \right\}.
\tag{B.2}
\end{equation}

<p class="method">

To use this method, set `switch_mech = "terms"` and `n_terms_small = "Gondan"` in the function call. The parameter `summation_small` should be ignored so that it retains its default value to obtain the best performance.
</p>


### Stop When Small Enough (SWSE) Small & Navarro Large {#dens-ct-swse}

The SWSE approximation to the small-time density function differs from the Navarro or Gondan approximations in that it does not pre-calculate $k_s$, the number of terms in the infinite sum that are required to achieve the desired precision; instead, the infinite sum is truncated when the individual terms of the sum become "small enough." This method of truncating the infinite sum poses a problem of how to incorporate this method with the Navarro large-time approximation that relies on pre-calculating $k_\ell^{\text{Nav}}$. We will introduce two heuristics for determining when to use the small-time approximation or the large-time approximation. Both heuristics will use a fourth parameter called `switch_thresh`, but each heuristic will use the new parameter in a slightly different way. The validity of these two methods is proven in the paper that accompanies the `fddm` package. Note that in this paper, the parameter `max_terms_large` is labelled $\delta$.

<div class="indent3">

##### First Heuristic: Large-Time Efficiency

The first heuristic compares `switch_thresh` to $k_\ell^{\text{Nav}}$; in this case, `switch_thresh` is treated as the proxy for the required number of terms for approximating the truncated small-time infinite sum. If $k_\ell^{\text{Nav}}$ $\le$ `switch_thresh`, then the Navarro large-time approximation is used. On the other hand, if $k_\ell^{\text{Nav}}$ $>$ `switch_thresh`, then the SWSE small-time approximation is used. This method essentially checks the efficiency of the large-time approximation relative to `switch_thresh`. The user can alter the behavior of this method by setting the optional parameter `switch_thresh` to any non-negative integer; the default value for this parameter when using this method is $1$.

<p class="method">

To use this method, set `switch_mech = "terms_large"`. The user may wish to specify a particular threshold (measured in the required number of terms for the truncated sum) for switching between the small-time and large-time approximations by setting the parameter `switch_thresh` (i.e., `switch_thresh = 1`). The parameter `summation_small` should be ignored so that it retains its default value to obtain the best performance.
</p>

<a id="default-implementation"></a>

##### Second Heuristic: Effective Response Time

The second heuristic avoids all comparison with $k_\ell^{\text{Nav}}$ as it simply considers the effective response time, $t' := \tfrac{t}{a^2}$, to be the indicator of a small or large response time. $t'$ is compared to a new parameter `switch_thresh` so that if $t'$ $>$ `switch_thresh`, then the Navarro large-time approximation is used; otherwise, the SWSE small-time approximation is used. The user can alter the behavior of this method by setting the optional parameter `switch_thresh` to any non-negative real number; the default value for this parameter when using this method is $0.8$.

<p class="method">

Since this is the default implementation, all four of the optional parameters (`switch_mech`, `switch_thresh`, `n_terms_small`, and `summation_small`) can be ignored. The user may wish to specify a particular threshold (measured in seconds) for switching between the small-time and large-time approximations by setting the parameter `switch_thresh` (i.e., `switch_thresh = 0.8`). We recommend ignoring the parameter `summation_small` so that it retains its default value and achieves optimal performance. Note that the parameter `n_terms_small` is ignored because the only option is "SWSE".
</p>
</div>
</div>

</div>





</div>
# {.unlisted .unnumbered}
#### R Session Info {.unlisted .unnumbered}
```{r session-info, collapse=TRUE}
sessionInfo()
```



# References
