dw_dfddm: Partial Derivative of 5-parameter DDM PDF with respect to w...
In fddm: Fast Implementation of the Diffusion Decision Model
dw_dfddm R Documentation
Partial Derivative of 5-parameter DDM PDF with respect to w (initial bias)
Description
Partial Derivative of the density function for the 5-parameter variant of
the Ratcliff diffusion decision model (DDM) with respect to w, the initial
bias. This variant contains the following parameters:
v (drift rate), a (threshold separation),
t0 (non-decision time/response time constant), w
(relative starting point), sv (inter-trial variability of drift),
and sigma (diffusion coefficient of underlying Wiener process).
Usage
dw_dfddm(
rt,
response,
v,
a,
t0,
w = 0.5,
sv = 0,
sigma = 1,
sl_thresh = 1,
err_tol = 1e-06
)
Arguments
rt
A vector of response times (in seconds). If a response time is
non-positve, then its density will evaluate to 0 if log = FALSE and
-∞ if log = TRUE.
response
Binary response(s) that correspond(s) to either the "lower"
or "upper" threshold. This model parameter can either be a singular value
or a vector. The value(s) in 'response' can be of the following data
types:
integers or doubles (1
→ "lower", 2
→ "upper");
factors (the first level gets mapped to "lower", and the second
level gets mapped to "upper"; any additional levels are ignored).
strings (only the first character is checked, "L"
→ "lower" or "U"
→ "upper", case insensitive);
logicals (FALSE → "lower",
TRUE → "upper");
v
Drift rate. Average slope of the information accumulation process.
The drift gives information about the speed and direction of the
accumulation of information. Large (absolute) values of drift indicate a
good performance. If received information supports the response linked to
the upper threshold, then the sign will be positive; similarly a negative
value indicates that the received information supports the response linked
to the lower threshold. Allowed range: v is a real number. Typical
range: -5 < v < 5.
a
Threshold separation. Amount of information that is considered for
a decision. Large values indicate a conservative decisional style. Allowed
range: 0 < a. Typical range: 0.5 < a < 2.
t0
Non-decision time or response time constant (in seconds). Lower
bound for the duration of all non-decisional processes (encoding and
response execution). If this value is greater than rt, then the
resulting density is returned as if rt \le 0. Allowed range:
0 \le t0. Typical range: 0.1 < t0 < 0.5.
w
Relative starting point. Indicator of an a priori bias in decision
making. When the relative starting point w deviates from 0.5,
the amount of information necessary for a decision differs between
response alternatives. Allowed range: 0 < w < 1.
Default value is 0.5 (i.e., no bias).
sv
Inter-trial-variability of drift rate. Standard deviation of a
normal distribution with mean v describing the distribution of
actual drift rates from specific trials. Values different from 0 can
predict slow errors. Allowed range: 0 \le sv. Typical range:
0 < sv < 2. Default value is 0, which indicates
no drift in the function call. See Details for more information.
sigma
Diffusion coefficient of the underlying Wiener process. Allowed
range: 0 < sigma. Default value is 1. This parameter
simply scales the parameters v, a, and sv as follows.
See Details for more information.
-
a
→ a / sigma
-
v → v /
sigma
-
sv →
sv / sigma
.
sl_thresh
Threshold for deciding when to use the "small-time" variant
or the "large-time" variant in one half of the calculation of the partial
derivative of the density function. This acts in exactly the same way as
the parameter max_terms_large in the dfddm() function: it is
the maximum number of terms to use for the "large-time" variant before
switching to using the "small-time" variant. The other half of the
calculation leverages a different type of comparison between the
"large-time" and "small-time" variants that requires no input from the
user. Allowed values are any real number; however any non-positive number
means that the "small-time" variant will always be used. Similarly, any
very large positive number (e.g., +Inf) means that the "large-time"
variant will always be used. Default value is 1.
err_tol
Allowed error tolerance of the overall calculation. Since the
partial derivative of the density function contains the sum of two
infinite sums, each approximation of these two infinite sums will have an
individual error tolerance of err_tol / 2; thus the total overall
error of the calculation will be at most err_tol. If the provided
error tolerance is less than 1e-300, it is set to 1e-300.
Default is 1e-6.
Details
All of the model inputs and parameters (rt, response,
v, a, t0, w, sv, sigma) can be
input as a single value or as a vector of values. If input as a vector of
values, then the standard R input wrapping will occur.
sv - Both the "small-time" and "large-time" variants of the density
function have two further variants: one with a constant drift rate v
(i.e., sv = 0), and one with a variable drift rate v
(i.e., sv > 0). The details of the differences between these
two density functions can be found in our associated paper. To use the
density function with a constant drift rate, leave the parameter sv
to its default value of sv = 0, as this will indicate no drift to the
function. To use the density function with a variable drift rate, set the
parameter sv to some non-negative value, i.e., sv > 0.
sigma - The default value of this parameter is 1 because it
only scales the parameters v, a, and sv, as shown
above. However, other formulations of the DDM may set sigma = 0.1
(see Ratcliff (1978), the fourth reference), so care must be taken when
comparing the results of different formulations.
Value
A vector containing the partial derivatives of the DDM PDF with
precision err_tol whose length matches that of the longest input
parameter (usually rt).
References
Navarro, D. J., & Fuss, I. G. (2009). Fast and accurate
calculations for first-passage times in Wiener diffusion models. Journal
of Mathematical Psychology, 53(4), 222-230.
Gondan, M., Blurton, S. P., & Kesselmeier, M. (2014). Even faster and even
more accurate first-passage time densities and distributions for the
Wiener diffusion model. Journal of Mathematical Psychology, 60, 20-22.
Blurton, S. P., Kesselmeier, M., & Gondan, M. (2017). The first-passage
time distribution for the diffusion model with variable drift. Journal of
Mathematical Psychology, 76, 7-12.
Hartmann, R., Klauer, K. C. (2021). Partial derivatives for the
first-passage time distribution in Wiener diffusion models. Journal of
Mathematical Psychology, 103, 102550.
Ratcliff, R. (1978). A theory of memory retrieval. Psychological review,
85(2), 59.
Examples
# derivative with respect to rt (response time)
dt_dfddm(rt = 1.2, response = "lower", a = 1, v = -1, t0 = 0.3, w = 0.5,
sv = 0.4, err_tol = 1e-6)
# derivative with respect to t0 (non-decision time)
dt0_dfddm(rt = 1.2, response = "lower", a = 1, v = -1, t0 = 0.3, w = 0.5,
sv = 0.4, err_tol = 1e-6)
# derivative with respect to a (threshold separation)
da_dfddm(rt = 1.2, response = "lower", a = 1, v = -1, t0 = 0.3, w = 0.5,
sv = 0.4, err_tol = 1e-6)
# derivative with respect to v (drift rate)
dv_dfddm(rt = 1.2, response = "lower", a = 1, v = -1, t0 = 0.3, w = 0.5,
sv = 0.4, err_tol = 1e-6)
# derivative with respect to w (relative starting point)
dw_dfddm(rt = 1.2, response = "lower", a = 1, v = -1, t0 = 0.3, w = 0.5,
sv = 0.4, err_tol = 1e-6)
# derivative with respect to sv (inter-trial variability in the drift rate)
dsv_dfddm(rt = 1.2, response = "lower", a = 1, v = -1, t0 = 0.3, w = 0.5,
sv = 0.4, err_tol = 1e-6)
#-------- Log Versions --------#
# # derivative with respect to rt (response time)
# dt_log_dfddm(rt = 1.2, response = "lower", a = 1, v = -1, t0 = 0.3, w = 0.5,
# sv = 0.4, err_tol = 1e-6)
# # derivative with respect to t0 (non-decision time)
# dt0_log_dfddm(rt = 1.2, response = "lower", a = 1, v = -1, t0 = 0.3, w = 0.5,
# sv = 0.4, err_tol = 1e-6)
# # derivative with respect to a (threshold separation)
# da_log_dfddm(rt = 1.2, response = "lower", a = 1, v = -1, t0 = 0.3, w = 0.5,
# sv = 0.4, err_tol = 1e-6)
# # derivative with respect to v (drift rate)
# dv_log_dfddm(rt = 1.2, response = "lower", a = 1, v = -1, t0 = 0.3, w = 0.5,
# sv = 0.4, err_tol = 1e-6)
# # derivative with respect to w (relative starting point)
# dw_log_dfddm(rt = 1.2, response = "lower", a = 1, v = -1, t0 = 0.3, w = 0.5,
# sv = 0.4, err_tol = 1e-6)
# # derivative with respect to sv (inter-trial variability in the drift rate)
# dsv_log_dfddm(rt = 1.2, response = "lower", a = 1, v = -1, t0 = 0.3, w = 0.5,
# sv = 0.4, err_tol = 1e-6)
fddm documentation built on June 26, 2024, 5:10 p.m.
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| dw_dfddm | R Documentation |
Partial Derivative of 5-parameter DDM PDF with respect to w (initial bias)
Description
Partial Derivative of the density function for the 5-parameter variant of
the Ratcliff diffusion decision model (DDM) with respect to w, the initial
bias. This variant contains the following parameters:
v (drift rate), a (threshold separation),
t0 (non-decision time/response time constant), w
(relative starting point), sv (inter-trial variability of drift),
and sigma (diffusion coefficient of underlying Wiener process).
Usage
dw_dfddm(
rt,
response,
v,
a,
t0,
w = 0.5,
sv = 0,
sigma = 1,
sl_thresh = 1,
err_tol = 1e-06
)
Arguments
rt |
A vector of response times (in seconds). If a response time is
non-positve, then its density will evaluate to |
response |
Binary response(s) that correspond(s) to either the "lower" or "upper" threshold. This model parameter can either be a singular value or a vector. The value(s) in 'response' can be of the following data types:
|
v |
Drift rate. Average slope of the information accumulation process.
The drift gives information about the speed and direction of the
accumulation of information. Large (absolute) values of drift indicate a
good performance. If received information supports the response linked to
the upper threshold, then the sign will be positive; similarly a negative
value indicates that the received information supports the response linked
to the lower threshold. Allowed range: |
a |
Threshold separation. Amount of information that is considered for
a decision. Large values indicate a conservative decisional style. Allowed
range: |
t0 |
Non-decision time or response time constant (in seconds). Lower
bound for the duration of all non-decisional processes (encoding and
response execution). If this value is greater than |
w |
Relative starting point. Indicator of an a priori bias in decision
making. When the relative starting point |
sv |
Inter-trial-variability of drift rate. Standard deviation of a
normal distribution with mean |
sigma |
Diffusion coefficient of the underlying Wiener process. Allowed
range:
. |
sl_thresh |
Threshold for deciding when to use the "small-time" variant
or the "large-time" variant in one half of the calculation of the partial
derivative of the density function. This acts in exactly the same way as
the parameter |
err_tol |
Allowed error tolerance of the overall calculation. Since the
partial derivative of the density function contains the sum of two
infinite sums, each approximation of these two infinite sums will have an
individual error tolerance of |
Details
All of the model inputs and parameters (rt, response,
v, a, t0, w, sv, sigma) can be
input as a single value or as a vector of values. If input as a vector of
values, then the standard R input wrapping will occur.
sv - Both the "small-time" and "large-time" variants of the density
function have two further variants: one with a constant drift rate v
(i.e., sv = 0), and one with a variable drift rate v
(i.e., sv > 0). The details of the differences between these
two density functions can be found in our associated paper. To use the
density function with a constant drift rate, leave the parameter sv
to its default value of sv = 0, as this will indicate no drift to the
function. To use the density function with a variable drift rate, set the
parameter sv to some non-negative value, i.e., sv > 0.
sigma - The default value of this parameter is 1 because it
only scales the parameters v, a, and sv, as shown
above. However, other formulations of the DDM may set sigma = 0.1
(see Ratcliff (1978), the fourth reference), so care must be taken when
comparing the results of different formulations.
Value
A vector containing the partial derivatives of the DDM PDF with
precision err_tol whose length matches that of the longest input
parameter (usually rt).
References
Navarro, D. J., & Fuss, I. G. (2009). Fast and accurate calculations for first-passage times in Wiener diffusion models. Journal of Mathematical Psychology, 53(4), 222-230.
Gondan, M., Blurton, S. P., & Kesselmeier, M. (2014). Even faster and even more accurate first-passage time densities and distributions for the Wiener diffusion model. Journal of Mathematical Psychology, 60, 20-22.
Blurton, S. P., Kesselmeier, M., & Gondan, M. (2017). The first-passage time distribution for the diffusion model with variable drift. Journal of Mathematical Psychology, 76, 7-12.
Hartmann, R., Klauer, K. C. (2021). Partial derivatives for the first-passage time distribution in Wiener diffusion models. Journal of Mathematical Psychology, 103, 102550.
Ratcliff, R. (1978). A theory of memory retrieval. Psychological review, 85(2), 59.
Examples
# derivative with respect to rt (response time)
dt_dfddm(rt = 1.2, response = "lower", a = 1, v = -1, t0 = 0.3, w = 0.5,
sv = 0.4, err_tol = 1e-6)
# derivative with respect to t0 (non-decision time)
dt0_dfddm(rt = 1.2, response = "lower", a = 1, v = -1, t0 = 0.3, w = 0.5,
sv = 0.4, err_tol = 1e-6)
# derivative with respect to a (threshold separation)
da_dfddm(rt = 1.2, response = "lower", a = 1, v = -1, t0 = 0.3, w = 0.5,
sv = 0.4, err_tol = 1e-6)
# derivative with respect to v (drift rate)
dv_dfddm(rt = 1.2, response = "lower", a = 1, v = -1, t0 = 0.3, w = 0.5,
sv = 0.4, err_tol = 1e-6)
# derivative with respect to w (relative starting point)
dw_dfddm(rt = 1.2, response = "lower", a = 1, v = -1, t0 = 0.3, w = 0.5,
sv = 0.4, err_tol = 1e-6)
# derivative with respect to sv (inter-trial variability in the drift rate)
dsv_dfddm(rt = 1.2, response = "lower", a = 1, v = -1, t0 = 0.3, w = 0.5,
sv = 0.4, err_tol = 1e-6)
#-------- Log Versions --------#
# # derivative with respect to rt (response time)
# dt_log_dfddm(rt = 1.2, response = "lower", a = 1, v = -1, t0 = 0.3, w = 0.5,
# sv = 0.4, err_tol = 1e-6)
# # derivative with respect to t0 (non-decision time)
# dt0_log_dfddm(rt = 1.2, response = "lower", a = 1, v = -1, t0 = 0.3, w = 0.5,
# sv = 0.4, err_tol = 1e-6)
# # derivative with respect to a (threshold separation)
# da_log_dfddm(rt = 1.2, response = "lower", a = 1, v = -1, t0 = 0.3, w = 0.5,
# sv = 0.4, err_tol = 1e-6)
# # derivative with respect to v (drift rate)
# dv_log_dfddm(rt = 1.2, response = "lower", a = 1, v = -1, t0 = 0.3, w = 0.5,
# sv = 0.4, err_tol = 1e-6)
# # derivative with respect to w (relative starting point)
# dw_log_dfddm(rt = 1.2, response = "lower", a = 1, v = -1, t0 = 0.3, w = 0.5,
# sv = 0.4, err_tol = 1e-6)
# # derivative with respect to sv (inter-trial variability in the drift rate)
# dsv_log_dfddm(rt = 1.2, response = "lower", a = 1, v = -1, t0 = 0.3, w = 0.5,
# sv = 0.4, err_tol = 1e-6)
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