Trends

Introduction

Trends in EMC2 allow you to model how parameters change as a function of covariates or other parameters. They provide a flexible way to capture systematic changes in model parameters across conditions or over time. This vignette shows:

• How to specify trends with kernels and bases
• How to attach trends to models via design() and run via make_emc() and fit()
• How to control when a trend is applied (premap, pretransform, posttransform)
• How to share trend parameters
• How to use other parameters as trend inputs (par_input)
• How to register and use a custom C++ trend kernel

Trend Composition

A trend is composed of a kernel and a base. The kernel maps inputs to a per‑trial value k; the base then maps k into the parameter scale. Use trend_help() to list kernels and bases. The timing of when a trend is applied is controlled by phase (see Phases below).

trend_help()

## Available kernels:
##   custom: Custom C++ kernel: provided via register_trend().
##   lin_decr: Decreasing linear kernel: k = -c
##   lin_incr: Increasing linear kernel: k = c
##   exp_decr: Decreasing exponential kernel: k = exp(-d_ed * c)
##   exp_incr: Increasing exponential kernel: k = 1 - exp(-d_ei * c)
##   pow_decr: Decreasing power kernel: k = (1 + c)^(-d_pd)
##   pow_incr: Increasing power kernel: k = 1 - (1 + c)^(-d_pi)
##   poly2: Quadratic polynomial: k = d1 * c + d2 * c^2
##   poly3: Cubic polynomial: k = d1 * c + d2 * c^2 + d3 * c^3
##   poly4: Quartic polynomial: k = d1 * c + d2 * c^2 + d3 * c^3 + d4 * c^4
##   delta: Standard delta rule kernel: k = q[i].
##          Updates q[i] = q[i-1] + alpha * (c[i-1] - q[i-1]).
##          Parameters: q0 (initial value), alpha (learning rate).
##   delta2kernel: Dual kernel delta rule: k = q[i].
##           Combines fast and slow learning rates
##           and switches between them based on dSwitch.
##           Parameters: q0 (initial value), alphaFast (fast learning rate),
##           propSlow (alphaSlow = propSlow * alphaFast), dSwitch (switch threshold).
##   delta2lr: Dual learning rate delta rule: k = q[i].
##           Like the standard delta rule, but with separate
##           learning rates for positive and negative prediction errors.
##           Parameters: q0 (initial value), alphaPos (learning rate for positive PEs),
##           alphaNeg (learning rate for negative PEs).
## 
## Available base types:
##   lin: Linear base: parameter + w * k
##   exp_lin: Exponential linear base: exp(parameter) + exp(w) * k
##   centered: Centered mapping: parameter + w*(k - 0.5)
##   add: Additive base: parameter + k
##   identity: Identity base: k
## 
## Phase options:
##   premap: Trend is applied before parameter mapping. This means the trend parameters
##           are mapped first, then used to transform cognitive model parameters before 
##           their mapping.
##   pretransform: Trend is applied after parameter mapping but before transformations.
##                 Cognitive model parameters are mapped first, then trend is applied, 
##                 followed by transformations.
##   posttransform: Trend is applied after both mapping and transformations.
##                  Cognitive model parameters are mapped and transformed first, 
##                  then trend is applied.

Quick Start: From trend to model

Below is a minimal pipeline using the data in samples_LNR (three subjects from Forstmann 2008) to demonstrate where a trend fits in. We create a trend, attach it in the design, and show how it plugs into make_emc() and fit() (calls not evaluated here).

# Example trend: log-mean increases linearly with trial
trend_quick <- make_trend(
  par_names = "m",
  cov_names = "trial_nr",
  kernels   = "lin_incr",
  bases     = "lin",
  phase     = "pretransform"
)

data <- get_data(samples_LNR)
# This does not take subject id into account
data$trial_nr <- 1:nrow(data)
data$covariate1 <- rnorm(nrow(data))
data$covariate2 <- rnorm(nrow(data))


# Build a design with the trend
design_trend <- design(
  data     = data,
  trend    = trend_quick,
  matchfun = function(d) d$S == d$lR,
  formula  = list(m ~ lM, s ~ 1, t0 ~ 1),
  contrasts = list(lM = matrix(c(-1/2, 1/2), ncol = 1, dimnames = list(NULL, "d"))),
  model    = LNR
)

## Intercept formula added for trend_pars: m.w

## 
##  Sampled Parameters: 
## [1] "m"     "m_lMd" "s"     "t0"    "m.w"  
## 
##  Design Matrices: 
## $m
##     lM m m_lMd
##   TRUE 1   0.5
##  FALSE 1  -0.5
## 
## $s
##  s
##  1
## 
## $t0
##  t0
##   1
## 
## $m.w
##  m.w
##    1

# How you would run (not executed here)
# emc <- make_emc(data, design_trend, type = "single")
# fit <- fit(emc)

Defining a Trend

A trend is created using the make_trend() function with the following main arguments:

• par_names: Which cognitive model parameters to apply the trend to
• cov_names: Which covariates to use for each trend
• kernels: Which kernel function to use for each trend
• bases: Optional base functions for each trend

Let’s take the following example:

trend_lin_decr <- make_trend(
  par_names = "v",
  cov_names = "trial_nr",
  kernels = "lin_incr",
  bases = "lin"
)

This trend applies a linear increasing kernel to v using trial as input. This captures a hypothesis that the drift rate (v) increases over trials. To see how the trend is formulated use trend_help().

trend_help(kernel = "lin_incr")

## Description: 
## Increasing linear kernel: k = c 
##  
## Available bases, first is the default: 
## lin, exp_lin, centered 
## 

The function tells us how the kernel maps the covariate to the kernel k. In this case the kernel k is just the covariate c. The kernel is then used in the base function. The trend_help() function also shows us the available base functions for this kernel. In this case we will use the lin base function. To see how this base function is formulated we can use the trend_help() function again.

trend_help(base = "lin")

## Description: 
## Linear base: parameter + w * k 
##  
## Default transformations: 
## list(w = "identity")
## 

Together these specify how the trend affects the model parameter. With base lin, the mapping is v <- v + B0 * k, where B0 is the base parameter and k comes from the kernel (here k = c, the covariate). Although this separation may seem verbose for simple cases, it makes more complex specifications clear and composable.

EMC2 automatically creates parameter names for the trend, which can be accessed using the get_trend_pnames() function.

get_trend_pnames(trend_lin_decr)

## [1] "v.w"

These parameter names are now included in the parameter types of the model.

Available Kernel Types

EMC2 provides several kernel functions for modeling how cognitive model parameters change as a function of covariates.

Linear Trends

# Linear decreasing trend
trend_lin_decr <- make_trend(
  par_names = "v",
  cov_names = "trial_nr",
  kernels = "lin_decr"
)

# Linear increasing trend
trend_lin_incr <- make_trend(
  par_names = "v",
  cov_names = "trial",
  kernels = "lin_incr"
)

Linear trends model straight‑line relationships between parameters and covariates. They are often a good starting point. You could encode linear effects via design() covariates, but bundling them into a trend enables shared control via bases and phases and keeps specifications local to parameters.

Exponential Trends

# Exponential decreasing trend
trend_exp_decr <- make_trend(
  par_names = "v",
  cov_names = "trial_nr",
  kernels = "exp_decr"
)

# Exponential increasing trend
trend_exp_incr <- make_trend(
  par_names = "v",
  cov_names = "trial_nr",
  kernels = "exp_incr"
)

Exponential trends are useful for rapid initial changes that level off. exp_decr captures decay; exp_incr captures learning.

Power Trends

# Power decreasing trend
trend_pow_decr <- make_trend(
  par_names = "v",
  cov_names = "trial_nr",
  kernels = "pow_decr"
)

# Power increasing trend
trend_pow_incr <- make_trend(
  par_names = "v",
  cov_names = "trial_nr",
  kernels = "pow_incr"
)

Power trends provide another way to model non‑linear relationships, often used for practice effects.

Polynomial Trends

# Quadratic trend
trend_poly2 <- make_trend(
  par_names = "v",
  cov_names = "trial_nr",
  kernels = "poly2"
)

# Cubic trend
trend_poly3 <- make_trend(
  par_names = "v",
  cov_names = "trial_nr",
  kernels = "poly3"
)

# Quartic trend
trend_poly4 <- make_trend(
  par_names = "v",
  cov_names = "trial_nr",
  kernels = "poly4"
)

Polynomial trends allow for complex relationships, including multiple turning points. Use higher orders cautiously to avoid overfitting.

Learning Rule Trends

# Standard delta learning rule
trend_delta <- make_trend(
  par_names = "v",
  cov_names = "trial_nr",
  kernels = "delta"
)

# Dual learning rate delta rule
trend_delta2 <- make_trend(
  par_names = "v",
  cov_names = "trial_nr",
  kernels = "delta2kernel"
)

Delta rules are useful for learning processes. The standard delta rule updates based on prediction error. The dual‑rate rule uses fast and slow learning rates, switching between them based on discrepancies.

Each kernel has supported bases. You can leave bases to defaults or set them explicitly. Changing bases can reflect different interpretations of how k enters the parameter:

trend_exp_incr <- make_trend(
  par_names = "v",
  cov_names = "trial_nr",
  kernels = "exp_incr",
  bases =  "exp_lin"
)

For example with base lin: v <- v + B0 * k. With base exp_lin: v <- exp(v) + exp(B0) * k. The latter can be useful when the parameter operates on a multiplicative or positively constrained scale.

Multiple Trends

Some models have multiple parameters that are affected by the same covariate. In this case, we can specify multiple trends for the same covariate. Note that with the kernels argument, we can specify different kernels for each parameter.

# Applying different trends to multiple parameters
trend_multi <- make_trend(
  par_names = c("v", "t0"),
  cov_names = c("trial_nr"),
  kernels = c("exp_incr", "poly2")
)

Alternatively, different parameters may be affected by different covariates:

# Specifying different covariates for each trend
trend_multi <- make_trend(
  par_names = c("v", "t0"),
  cov_names = c("trial_nr", "covariate1"),
  kernels = c("exp_incr", "poly2")
)

Lastly, a single parameter may be affected by multiple covariates:

# Specifying multiple covariates for a trend
trend_multi <- make_trend(
  par_names = c("v", "t0"),
  cov_names = list(c("trial", "covariate1"), "covariate1"),
  kernels = c("exp_incr", "poly2")
)

Sharing Parameters

Sometimes multiple trend entries should use the same parameter(s). For example, share a base parameter across different target parameters:

# Sharing parameters between trends
trend_shared <- make_trend(
  par_names = c("v", "a"),
  cov_names = "trial_nr",
  kernels = c("exp_incr", "exp_incr"),
  shared = list(intercept = list("v.B0", "a.B0"))
)

This shares B0 between v and a trends. Instead of two B0 parameters, there is one named intercept. Use get_trend_pnames() to inspect the generated names:

get_trend_pnames(trend_shared)

## [1] "v.w"       "v.d_ei"    "a.w"       "a.d_ei"    "intercept"

We can also specify multiple shared parameters, for example shared = list(intercept = list("v.B0", "a.B0"), slope = list("v.d_ei", "a.d_ei")).

Phases

Control when a trend is applied via the phase argument:

• premap: before parameter mapping (i.e., modifies raw parameter vectors)
• pretransform: after mapping but before transforms
• posttransform: after transforms

# Pre-mapping trend
trend_premap <- make_trend(
  par_names = "v",
  cov_names = "trial_nr",
  kernels   = "exp_incr",
  phase     = "premap"
)

# Pre-transform trend
trend_pretrans <- make_trend(
  par_names = "v",
  cov_names = "trial_nr",
  kernels   = "exp_incr",
  phase     = "pretransform"
)

# Post-transform trend
trend_posttrans <- make_trend(
  par_names = "v",
  cov_names = "trial_nr",
  kernels   = "exp_incr",
  phase     = "posttransform"
)

Phases change how trends interact with mapping and transformations. For example, a premap trend can feed transformed outputs into parameter mapping, while posttransform trends act on the final, transformed parameter scale.

Parameter Inputs (par_input)

Trends can also depend on other parameters via par_input. For example, use t0 as an input to a trend on m:

trend_par_input <- make_trend(
  par_names = "m",
  cov_names = NULL,
  kernels   = "lin_incr",
  par_input = list("t0"),
  phase     = "pretransform"
)

This example mirrors the tests: the input matrix provided to the kernel will contain the covariate columns (none here) followed by the par_input columns (here t0).

Apply a trend only at a factor level (at)

Use at to apply a trend only for rows at a specific factor level (e.g., first level of lR) and expand its contribution to the other levels within the same subject.

trend_at <- make_trend(
  par_names = c("m"),
  cov_names = list("covariate1"),
  kernels   = c("exp_incr"),
  phase     = "pretransform",
  at        = "lR"   # apply only at first level of lR, then expand within subject
)

This is useful when a covariate (or its effect) should only be applied to a specific level of a factor (most commonly lR), so that if the design is replicated across levels of the factor, the trend is only applied to the first level of the factor.

Multiple contributions to the same parameter

You can place multiple trend entries on the same target parameter. Their kernel outputs will be summed (after any optional per‑column map weights) before the base is applied.

trend_multi_same_par <- make_trend(
  par_names = c("m", "m"),
  cov_names = list("covariate1", c("covariate2")),  # second entry could also use par_input
  kernels   = c("exp_incr", "delta"),
  phase     = "pretransform",
  at        = "lR"
)

In the internal input matrix, covariate columns come first, followed by any par_input columns. The per‑column kernel contributions are summed row‑wise.

Phase per trend entry

Instead of one phase for all entries, you can provide a vector matching par_names:

trend_phases <- make_trend(
  par_names = c("m", "s", "t0"),
  cov_names = list("covariate1", "covariate1", "covariate2"),
  kernels   = c("lin_incr", "exp_decr", "pow_incr"),
  phase     = c("premap", "pretransform", "posttransform")
)

Non‑premap targets must be actual model parameter names (checked in the code), since these act after mapping.

Sharing kernel parameters (not only base weights)

Sharing can also target kernel parameters, not just the base weight. For example, share d1 across two parameters:

trend_shared_kernel <- make_trend(
  par_names = c("m", "s"),
  cov_names = list("covariate1", "covariate2"),
  kernels   = c("poly3", "poly4"),
  shared    = list(shrd = list("m.d1", "s.d1"))
)

After sharing, get_trend_pnames() will include the shared name once (here shrd) and remove duplicates.

Missing input values (NA handling)

Trend kernels ignore rows where any input used by the kernel is NA. For those rows, the kernel contributes zero. This applies to both built‑in and custom kernels. In effect:

• Built‑in kernels evaluate only on “good” rows and expand results back; NA rows contribute 0.
• Custom kernels receive only the subset of non‑NA rows; any NA in their outputs is coerced to 0 before being expanded back.

This behavior makes trends robust when covariates are intermittently missing (as exercised in tests that set some covariate entries to NA).

Trial‑wise evaluation and conditional covariates

• Delta‑rule kernels (delta, delta2) are sequential by construction; they update across trials within a subject.
• Using behavioral covariates (e.g., rt, response R, or outputs of functions in the design) can force a trial‑wise evaluation path so that data generation is not conditional on the observed data, but rather on the simulated outcomes.

# Example delta trend, capturing trial-wise dynamics
trend_delta <- make_trend(
  par_names = "m",
  cov_names = "trial_nr",
  kernels   = "delta",
  phase     = "pretransform"
)

design_delta <- design(
  factors    = list(subjects = 1, S = 1:2),
  Rlevels    = 1:2,
  covariates = "trial_nr",
  matchfun   = function(d) d$S == d$lR,
  trend      = trend_delta,
  formula    = list(m ~ lM, s ~ 1, t0 ~ 1),
  contrasts  = list(lM = matrix(c(-1/2, 1/2), ncol = 1, dimnames = list(NULL, "d"))),
  model      = LNR
)

## Intercept formula added for trend_pars: m.w, m.q0, m.alpha

## 
##  Sampled Parameters: 
## [1] "m"       "m_lMd"   "s"       "t0"      "m.w"     "m.q0"    "m.alpha"
## 
##  Design Matrices: 
## $m
##     lM m m_lMd
##   TRUE 1   0.5
##  FALSE 1  -0.5
## 
## $s
##  s
##  1
## 
## $t0
##  t0
##   1
## 
## $m.w
##  m.w
##    1
## 
## $m.q0
##  m.q0
##     1
## 
## $m.alpha
##  m.alpha
##        1

# Retrieve trial-wise parameters alongside generated data
# (not executed here)
# res <- make_data(p_vector, design_delta, n_trials = 10,
#                  conditional_on_data = FALSE,
#                  return_trialwise_parameters = TRUE)
# str(attr(res, "trialwise_parameters"))

Column names in the returned trial‑wise matrix are formed as parameter_inputName, where inputName comes from cov_names followed by any par_input names.

Vary trend parameters via the design formula

Trend parameters (base and kernel parameters) can vary by experimental factors using the formula argument of design(). Use get_trend_pnames(trend) to get the full names to place on the left-hand side of a formula. Names follow:

• Base parameter: <targetParam>.w (for bases like lin, exp_lin, centered)
• Kernel parameters: <targetParam>.<default_par_name> (e.g., d1, d2, … for poly*; q0, alpha for delta)

Premap example: vary a kernel parameter (d1) by lR

trend_premap <- make_trend(
  par_names = c("m", "lMd"),
  cov_names = list("covariate1", "covariate2"),
  kernels   = c("exp_incr", "poly2"),
  phase     = "premap"
)

design_premap <- design(
  data     = data,                   
  trend    = trend_premap,
  formula  = list(m ~ 1, s ~ 1, t0 ~ 1, lMd.d1 ~ lR),
  model    = LNR
)

## Intercept formula added for trend_pars: m.w, m.d_ei, lMd.d2

## 
##  Sampled Parameters: 
## [1] "m"              "s"              "t0"             "lMd.d1"        
## [5] "lMd.d1_lRright" "m.w"            "m.d_ei"         "lMd.d2"        
## 
##  Design Matrices: 
## $m
##  m
##  1
## 
## $s
##  s
##  1
## 
## $t0
##  t0
##   1
## 
## $lMd.d1
##     lR lMd.d1 lMd.d1_lRright
##   left      1              0
##  right      1              1
## 
## $m.w
##  m.w
##    1
## 
## $m.d_ei
##  m.d_ei
##       1
## 
## $lMd.d2
##  lMd.d2
##       1

# mapped_pars(design_premap)  # inspect mapped parameter structure

Pretransform example: vary the base weight (w) for s by lR

trend_pretrans <- make_trend(
  par_names = c("m", "s"),
  cov_names = list("covariate1", "covariate2"),
  kernels   = c("delta", "exp_decr"),
  phase     = "pretransform"
)

design_pretrans <- design(
  data     = data,
  trend    = trend_pretrans,
  formula  = list(m ~ 1, s ~ 1, t0 ~ 1, s.w ~ lR),
  model    = LNR
)

## Intercept formula added for trend_pars: m.w, m.q0, m.alpha, s.d_ed

## 
##  Sampled Parameters: 
## [1] "m"           "s"           "t0"          "s.w"         "s.w_lRright"
## [6] "m.w"         "m.q0"        "m.alpha"     "s.d_ed"     
## 
##  Design Matrices: 
## $m
##  m
##  1
## 
## $s
##  s
##  1
## 
## $t0
##  t0
##   1
## 
## $s.w
##     lR s.w s.w_lRright
##   left   1           0
##  right   1           1
## 
## $m.w
##  m.w
##    1
## 
## $m.q0
##  m.q0
##     1
## 
## $m.alpha
##  m.alpha
##        1
## 
## $s.d_ed
##  s.d_ed
##       1

# mapped_pars(design_pretrans)

Posttransform example: again vary s.w by lR

trend_posttrans <- make_trend(
  par_names = c("m", "s"),
  cov_names = list("covariate1", "covariate2"),
  kernels   = c("pow_decr", "pow_incr"),
  phase     = "posttransform"
)

design_posttrans <- design(
  data     = data,
  trend    = trend_posttrans,
  formula  = list(m ~ 1, s ~ 1, t0 ~ 1, s.w ~ lR),
  model    = LNR
)

## Intercept formula added for trend_pars: m.w, m.d_pd, s.d_pi

## 
##  Sampled Parameters: 
## [1] "m"           "s"           "t0"          "s.w"         "s.w_lRright"
## [6] "m.w"         "m.d_pd"      "s.d_pi"     
## 
##  Design Matrices: 
## $m
##  m
##  1
## 
## $s
##  s
##  1
## 
## $t0
##  t0
##   1
## 
## $s.w
##     lR s.w s.w_lRright
##   left   1           0
##  right   1           1
## 
## $m.w
##  m.w
##    1
## 
## $m.d_pd
##  m.d_pd
##       1
## 
## $s.d_pi
##  s.d_pi
##       1

# mapped_pars(design_posttrans)

Notes: - If a trend parameter is not listed in formula, an intercept-only formula is added automatically so it can still be estimated. - For non-premap phases, the target parameter names must exist in the cognitive model (e.g., m, s, t0 for LNR).

Custom C++ Trends

You can register your own C++ kernel function. The function takes per‑trial kernel parameters trend_pars (base params excluded) and an input matrix, and returns a vector of outputs. To keep this vignette fast and portable, we show the code and how to register it but do not actually compile or run it here.

library(EMC2)

# Write a custom kernel to a separate file
tf <- tempfile(fileext = ".cpp")
writeLines(c(
  "// [[Rcpp::depends(EMC2)]]",
  "#include <Rcpp.h>",
  "#include \"EMC2/userfun.hpp\"",
  "",
  "// Example: two params (a, b) and two inputs (covariate1, t0)",
  "Rcpp::NumericVector custom_kernel(Rcpp::NumericMatrix trend_pars, Rcpp::NumericMatrix input) {",
  "  int n = input.nrow();",
  "  Rcpp::NumericVector out(n, 0.0);",
  "  for (int i = 0; i < n; ++i) {",
  "    double a = (trend_pars.ncol() > 0) ? trend_pars(i, 0) : 0.0;",
  "    double b = (trend_pars.ncol() > 1) ? trend_pars(i, 1) : 0.0;",
  "    double in1 = input(i, 0);  // covariate1",
  "    double in2 = input(i, 1);  // t0",
  "    if ((i % 2) == 0) out[i] = (Rcpp::NumericVector::is_na(in1) ? 0.0 : in1) + a;",
  "    else             out[i] = (Rcpp::NumericVector::is_na(in2) ? 0.0 : in2) * b;",
  "  }",
  "  return out;",
  "}",
  "",
  "// Export pointer maker for registration",
  "// [[Rcpp::export]]",
  "SEXP EMC2_make_custom_kernel_ptr();",
  "EMC2_MAKE_PTR(custom_kernel)"
), tf)

# Register with parameter names, transforms, and a default base
ct <- register_trend(
  trend_parameters = c("a", "b"),
  file = tf,
  transforms = c(a = "identity", b = "pnorm"),
  base = "add"
)

# Use in a trend (note par_input to add t0 as an input column)
trend_custom <- make_trend(
  par_names  = "m",
  cov_names  = "covariate1",
  kernels    = "custom",
  par_input  = list("t0"),
  phase      = "pretransform",
  bases      = NULL,         # uses ct$base (here: add)
  custom_trend = ct
)

design_custom_trend <- design(
  data = data,
  trend = trend_custom,
  formula = list(m ~ 1, s ~ 1, t0 ~ 1, m.a ~ lR),
  model = LNR
)

Interpretation: the base controls how the custom kernel’s output enters the parameter (e.g., add means additive). The transforms declared in register_trend() are attached to the model so custom kernel parameters can be estimated under appropriate constraints.

Tips for Choosing Trends

1. Start simple: Begin with linear trends before moving to more complex options
2. Match theory: Choose trends that align with theoretical expectations
3. Consider interpretability: Simpler trends are often easier to interpret
4. Watch for overfitting: More complex trends (like high-order polynomials) may overfit
5. Use shared parameters when you expect related processes
6. Choose phase based on how the trend should interact with mapping and transformations

Interpreting Bases and Transforms

• lin: additive on the parameter scale (θ <- θ + B0 * k)
• exp_lin: additive on an exponentialized parameter (θ <- exp(θ) + exp(B0) * k)
• centered: centers k at 0.5 before scaling (θ <- θ + B0*(k - 0.5))
• add: parameter is θ <- θ + k (no base parameter)
• identity: parameter is θ <- k (no base parameter)

Trend parameters can have default transforms (from trend_help()), and custom kernels can declare their own transforms via register_trend(). These transforms feed into the model’s transform pipeline, ensuring parameters respect intended bounds.