# Getting Started with smoothbp

## Overview

`smoothbp` fits hierarchical piecewise regression models with multiple change-points, each smoothed by a logistic sigmoid. The general mean function for \(K\) change-points is:

\[ \mu_i = b0_i + b1_i(\tau_i - \omega_{1i}) + \sum_{k=1}^{K} \delta_{ki}(\tau_i - \omega_{ki})\, \text{logistic}\!\left[(\tau_i - \omega_{ki})\rho_{ki}\right] \]

where

Symbol | Role
--- | ---
\(\tau\) | time / covariate
\(\omega_k\) | location of the \(k\)-th change-point
\(\rho_k\) | transition sharpness for change-point \(k\) (\(\rho \to \infty\) → hard kink)
\(b0\) | level at the first change-point
\(b1\) | pre-change slope (anchored at \(\omega_1\))
\(\delta_k\) | slope change attributable to change-point \(k\)

When \(K = 0\) the model reduces to a standard hierarchical linear regression. When \(K = 1\) it matches the classic two-phase piecewise model.

Each parameter has its own linear predictor specified via a formula. `b0` may also include a random intercept `(1 | group)` for clustered / repeated-measures data.

Posterior inference uses a Metropolis-within-Gibbs sampler written in Rust:

- \(b0\), \(b1\), \(\delta_k\), random intercepts — exact Gibbs (conjugate normal).
- \(\omega_k\), \(\rho_k\) — Hamiltonian Monte Carlo with adaptive mass matrix during warmup.
- \(\sigma\), \(\sigma_u\) — exact Gibbs (inverse-gamma conjugate).
- \(\gamma_k\) (spike-and-slab) — Gibbs with Bernoulli likelihood.

## Real-World Applications

While `smoothbp` is a general-purpose statistical tool, it is uniquely powerful for modeling structural changes across various scientific domains:

- Pharmacokinetics / Medicine: Modeling the delayed onset of a drug’s effect and identifying the precise time threshold where clearance rates exponentially shift.
- Ecology / Climate Science: Identifying “tipping points” where species population growth trajectories abruptly change in response to accumulated temperature variations.
- Epidemiology: Tracking structural shifts in infection rates (\(R_0\)) to pinpoint exactly when a public health intervention successfully flattened the curve.
- Finance: Discovering the exact timing of market regime shifts and measuring how individual assets respond differently to global economic shocks.

## Simulating data

`simulate_smoothbp()` generates synthetic data from the model. It is useful for testing and for understanding how the parameters relate to the observable trajectory shape.

    library(smoothbp)

    dat <- simulate_smoothbp(
      n_subj    = 20,
      n_obs     = 8,
      b0        = 5.0,   # level at change-point
      b1        = -0.3,  # pre-change slope
      delta     =  1.2,  # slope change (delta_1)
      omega     =  3.0,  # change-point location
      rho       =  4.0,  # transition sharpness
      sigma     =  0.4,  # residual SD
      sigma_u   =  0.5,  # between-subject SD
      tau_range = c(0, 6),
      seed      = 42
    )

    head(dat)
    true_params(dat)

## Fitting the model

### Zero change-points (linear fallback)

When `deltas`, `omega`, and `rho` are empty lists, `smoothbp` fits a standard hierarchical linear regression:

    fit0 <- smoothbp(
      formula = y ~ tau,
      b0      = ~ 1 + (1 | subject),
      b1      = ~ 1,
      deltas  = list(),   # no change-points
      omega   = list(),
      rho     = list(),
      data    = dat,
      chains  = 2L, iter = 1000L, warmup = 500L, seed = 42L, .verbose = FALSE
    )
    summary(fit0)

### One change-point

    fit1 <- smoothbp(
      formula = y ~ tau,
      b0      = ~ 1 + (1 | subject),
      b1      = ~ 1,
      deltas  = list(~ 1),          # one slope change
      omega   = list(~ 1),          # one change-point location
      rho     = list(~ 1),          # one sharpness
      data    = dat,
      priors  = smoothbp_priors(
        omega = list(prior_normal(3, 2, lb = 0))
      ),
      chains  = 4L, iter = 2000L, warmup = 1000L, seed = 42L
    )
    summary(fit1)

### Multiple change-points (The List-of-Formulas API)

For models with multiple structural breaks, `smoothbp` uses a “List-of-Formulas” API. You pass lists of equal length to `deltas`, `omega`, and `rho`. Each element in the list corresponds to the sub-model for that specific breakpoint.

    # Fit a model with 3 candidate breakpoints
    fit3 <- smoothbp(
      formula = y ~ tau,
      b0      = ~ 1 + (1 | subject),
      b1      = ~ 1,
      # Each segment gets its own formula (here, just an intercept)
      deltas  = list(~ 1, ~ 1, ~ 1),
      omega   = list(~ 1, ~ 1, ~ 1),
      rho     = list(~ 1, ~ 1, ~ 1),
      data    = dat,
      priors  = smoothbp_priors(
        # Use the space_omega_priors helper to initialize the search
        omega = space_omega_priors(K = 3, tau_min = 0, tau_max = 6)
      )
    )

### Known Change-Points with `fixed()`

If you are performing an intervention analysis (e.g., a Regression Discontinuity Design or a Stepped-Wedge trial), you may already know exactly where the change points should be.

By using the `fixed()` helper, you tell the model not to estimate the location or sharpness, but only the magnitude of the change (\(\delta\)). This is much faster and allows for direct hypothesis testing of the intervention effect.

    # Test for a hard kink at exactly tau = 3.0
    fit_fixed <- smoothbp_ss(
      formula = y ~ tau,
      omega   = list(fixed(3.0)),   # Location is known
      rho     = list(fixed(100)),   # Sharpness is fixed (hard kink)
      data    = dat
    )

    # The PIP tells us the probability that the intervention had an effect
    pip(fit_fixed)

> Note: `fixed()` can also accept a vector of the same length as your data, allowing for observation-specific change points (e.g., different intervention times per cluster).

> Tip: If you are unsure how many change-points are present, use `smoothbp_ss()` with spike-and-slab regularization — see `vignette("spike-and-slab")`.

### Progress and parallel chains

    fit1 <- smoothbp(
      formula = y ~ tau,
      b0      = ~ 1 + (1 | subject),
      data    = dat,
      chains  = 4L, iter = 2000L, warmup = 1000L, seed = 42L,
      cores   = 4L    # run all 4 chains concurrently
    )

To make parallel the default for a project, add this to `.Rprofile`:

    options(smoothbp.cores = parallel::detectCores())

## Posterior summary

`print()` displays results in distinct, labelled sections; `summary()` returns a clean data frame. By default, `effects = c("fixed", "ran_pars")` is used, matching the conventions of other mixed-modeling packages like `brms`.

You can filter which parameters are returned or printed by passing a vector to the `effects` argument. The parameters are classified into three strict categories: 1. `"fixed"`: Population-level structural parameters (\(b0, b1, \delta, \omega, \rho, \sigma\)) 2. `"ran_pars"`: Group-level variance parameters (e.g., `sigma_u`) 3. `"ran_vals"`: The actual group-level deviations (e.g., the individual `u[j]` random intercepts)

    print(fit1)                                        # fixed + ran_pars (default)
    print(fit1, effects = "fixed")                     # population-level only
    summary(fit1, effects = "all")                     # returns everything, including u[j]

Parameter names follow the convention:

Name | Meaning
--- | ---
`b0_(Intercept)` | Intercept (level at first change-point)
`b1_(Intercept)` | Pre-change slope
`delta1_(Intercept)` | Slope change at change-point 1
`omega1_(Intercept)` | Change-point 1 location
`rho1_(Intercept)` | Change-point 1 sharpness
`sigma` | Residual SD
`sigma_u` | Between-group SD (random intercept)

## Diagnostics

### Trace plots with mixing indicators

    plot(fit1)        # alias for trace_plot(fit1)
    trace_plot(fit1, type = "both")   # trace + density

Parameters with \(\hat{R} > 1.05\) or bulk-ESS \(< 100\) are flagged with a ⚠ symbol and a red-tinted background. Thresholds are adjustable:

    trace_plot(fit1, rhat_thresh = 1.01, ess_thresh = 400)

### Posterior predictive check

    pp_check(fit1)

## Prior specification

`smoothbp_priors()` accepts either a single prior applied to all coefficients of a component, or a named list. For multi-breakpoint models, `omega` and `rho` accept a list of priors (one per breakpoint):

    smoothbp_priors(
      b0    = prior_normal(0, 10),
      b1    = prior_normal(0, 5),
      omega = list(
        prior_normal(2, 1),   # change-point 1
        prior_normal(5, 1)    # change-point 2
      ),
      rho   = list(
        prior_normal(4, 2, lb = 0),
        prior_normal(4, 2, lb = 0)
      ),
      sigma = prior_invgamma(shape = 2, scale = 1)
    )

The `lb` and `ub` arguments impose hard bounds. `rho` should typically have `lb = 0` (sharpness must be positive). For `omega` and `deltas`, it is often better to avoid hard bounds unless they are physically necessary for your domain.

> Performance Tip: `smoothbp` is significantly faster when `b1`, `deltas`, and `omega` are unconstrained. Without bounds, the sampler can use exact conjugate Gibbs updates for `b1` and `deltas`, and standard HMC for `omega`. Truncated priors require additional checks and rejection sampling steps.

If you do use bounds for `omega`, ensure they cover the actual range of your time variable (e.g. if your time starts at -10, do not use `lb = 0`).

## Breakpoint selection with spike-and-slab

When the number of change-points is unknown, fit a model with \(K\) potential breakpoints using `smoothbp_ss()`. Each slope change \(\delta_k\) has a binary inclusion indicator \(\gamma_k\); the posterior mean of \(\gamma_k\) is the posterior inclusion probability (PIP) for that breakpoint.

    fit_ss <- smoothbp_ss(
      formula = y ~ tau,
      b0      = ~ 1 + (1 | subject),
      b1      = ~ 1,
      deltas  = list(~ 1, ~ 1, ~ 1),   # 3 candidate breakpoints
      omega   = list(~ 1, ~ 1, ~ 1),
      rho     = list(~ 1, ~ 1, ~ 1),
      data    = dat,
      spike   = prior_spike_slab(pi = 0.1, learn_pi = TRUE),
      chains  = 4L, iter = 2000L, warmup = 1000L, seed = 42L
    )

    # Posterior inclusion probabilities per breakpoint
    pip(fit_ss)
    #> gamma_delta1_(Intercept) gamma_delta2_(Intercept) gamma_delta3_(Intercept)
    #>                    0.987                    0.063                    0.041

A PIP near 1 indicates that breakpoint is supported by the data; a low PIP indicates the slope change is not needed.

See `vignette("spike-and-slab")` for a full tutorial.

## Hypothesis tests and evidence ratios

`hypothesis()` evaluates directional hypotheses against the posterior draws.

    # Is the slope change at breakpoint 1 positive?
    smoothbp::hypothesis(fit1, "delta1_(Intercept) > 0")

    # Complex linear hypothesis: is the final slope (b1 + delta1) negative?
    smoothbp::hypothesis(fit1, "b1_(Intercept) + delta1_(Intercept) < 0")

    # Does the change-point fall before time 4?
    smoothbp::hypothesis(fit1, "omega1_(Intercept) < 4")

Stars | ER | \(P(H)\)
--- | --- | ---
`***` | > 99 | > 0.99
`**` | > 19 | > 0.95
`*` | > 3 | > 0.75

## Model comparison

### Leave-one-out cross-validation (LOO-IC)

    # Compare 0-BP (linear) vs 1-BP vs 3-BP models
    loo0 <- loo(fit0)
    loo1 <- loo(fit1)
    loo3 <- loo(fit3)

    loo::loo_compare(loo0, loo1, loo3)

### Bayes factors via bridge sampling

    library(bridgesampling)
    bs0 <- bridge_sampler(fit0)
    bs1 <- bridge_sampler(fit1)
    bayes_factor(bs1, bs0)

## Extracting results

### Fitted values

    # Posterior mean + 95% CI at training observations
    fitted(fit1)

    # Full posterior draws matrix (n_draws × n_obs)
    draws_mat <- fitted(fit1, summary = FALSE)

### Working with `posterior` draws

The `$draws` component of a `smoothbp` fit is a standard `draws_array` object from the `posterior` package. You can use any of the `posterior` tools to manipulate and summarize the draws.

    library(posterior)

    # Convert to a data frame for use with ggplot2
    draws_df <- as_draws_df(fit1$draws)

    # Extract a specific parameter
    b1_draws <- draws_df$`b1_(Intercept)`

    # Compute custom summaries
    summarise_draws(fit1$draws, "mean", "sd", ~quantile(.x, c(0.1, 0.9)))

### Marginal predictions

    # Population-level predictions at new time points
    newdata_marginal <- data.frame(tau = seq(0, 6, by = 0.1))
    fitted(fit1, newdata = newdata_marginal)

### Conditional predictions

    # Subject-specific predictions
    fitted(fit1, newdata = data.frame(tau = seq(0, 6, by = 0.1), subject = "1"))

## Manuscript-ready reporting

`model_methods()` and `model_results()` generate structured, self-contained text reports from a fitted model. Both functions are designed to be readily adapted for use in a manuscript or report, or passed to a large language model (LLM): the output includes the full algebraic specification with every symbol defined and every parameter value or prior stated explicitly, so no package documentation is needed to interpret the report.

The return value of each function is the complete report as an invisible character string, so you can capture it for programmatic use:

    methods_txt <- model_methods(fit1)
    results_txt <- model_results(fit1)

`model_methods()` also emits a brief convergence note: if any Rhat values exceed 1.05, effective sample sizes fall below 100, or divergent transitions are present, a warning is printed with suggested diagnostics so that model-specification issues can be caught before results are reported.

### Statistical Analysis section — `model_methods()`

`model_methods()` describes how the analysis was done. It produces:

- a narrative paragraph readily adapted for use in the Methods section,
- the formula syntax and structural equation with all symbols defined,
- the prior distribution for every parameter,
- the MCMC algorithm details (sampler type, chains, iterations, seed), and
- a convergence note: silent when all diagnostics pass, prominent when Rhat, ESS, or divergent transitions indicate a potential problem.

    model_methods(fit1)

### Results section — `model_results()`

`model_results()` describes what was found. It produces:

- a narrative paragraph readily adapted for use in the Results section,
- the fitted model equation with posterior mean coefficients substituted to 3 d.p. (both the algebraic form and the fully numerical version),
- the asymptotic pre- and post-breakpoint slopes derived from \(b1\) and \(\delta_k\),
- per-breakpoint summaries (location, sharpness, slope change, all with 95% credible intervals),
- the full posterior estimates table, and
- convergence diagnostics.

    model_results(fit1)