Author: Niek Stevenson & Steven Miletić

# Trends

#### Niek Stevenson & Steven Miletić

## 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).
    ##   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
    ##   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, 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 = "delta2lr"
    )

Delta rules are useful for modelling learning processes. The standard delta rule updates each trial based on prediction error. The dual‑kernel 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.

## 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"
    )

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)

For sequential kernels (learning rules), when a covariate value is `NA` on trial t, the last-known updated covariate value is carried over (in delta rules, this may be the `q0` value if no update has taken place yet). For non-sequential kernels (`lin`, `exp`, `pow`, `poly`), `NA`-values contribute 0.

In custom kernels, the user needs to handle `NA` values (e.g., it is good practice to check for `ISNAN()` on each trial) themselves. EMC2 cannot know whether the last-known value should be carried forward, or whether the covariate should be set to 0 or some other value.

### Trial‑wise evaluation and conditional covariates

- Delta‑rule kernels (`delta`, `delta2lr`) 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`, `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
    )

Note that, when loading an emc object from disk, the pointers to the custom kernels need to be re-added before the emc object can be used for sampling or posterior data generation, which can be done as follows:

    load('./emc_samples_with_custom_trend.RData')
    emc <- fix_custom_kernel_pointers(emc, trend_custom)

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`)
- `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.