# 6. Extensions and advanced usage

This vignette covers the extensibility of `rLifting` beyond the built-in wavelets and standard denoising pipelines: defining custom wavelets via predict and update steps, validating them with the `diagnose_wavelet` suite, and assembling low-level pipelines with custom logic between decomposition and reconstruction. Thresholding theory and the empirical decision guide live in `vignette("v02-thresholding-and-tuning")`; this vignette uses the threshold primitives but does not re-derive them.

For the deep dives that this vignette only summarises, see `vignette("v02-thresholding-and-tuning")` (empirical thresholding decision guide), `vignette("v03-causal-stream")` (causal-mode mechanics), `vignette("v04-boundary-modes")` (per-mode empirical impact), and `vignette("v05-irregular-grids")` (irregular-grid handling).

    library(rLifting)

    if (!requireNamespace("ggplot2", quietly = TRUE)) {
      knitr::opts_chunk$set(eval = FALSE)
      message("'ggplot2' is required to render plots. Vignette code will not run.")
    } else {
      library(ggplot2)
    }

## 1. Built-in wavelet families

`rLifting` ships with six built-in lifting schemes:

Name | Alias | Steps | Vanishing moments | Properties
--- | --- | --- | --- | ---
`haar` |  | 2 | 1 | Orthogonal; step-like signals; fastest
`db2` |  | 3 | 2 | Orthogonal; maximises vanishing moments
`cdf53` | `bior2.2` | 2 | 2 | Biorthogonal; linear interpolation predict
`cdf97` | `bior4.4` | 4 | 4 | Biorthogonal; JPEG 2000; smoothest output
`dd4` | `interp4` | 2 | 4 | Interpolating; cubic Lagrange predict
`lazy` |  | 0 | 0 | Identity polyphase split only

Each is constructed via `lifting_scheme()`:

    haar = lifting_scheme("haar")
    cdf53 = lifting_scheme("cdf53")
    cdf97 = lifting_scheme("cdf97")
    dd4 = lifting_scheme("dd4")
    print(cdf97)
    #> Lifting Scheme: cdf97
    #> Steps: 4
    #> Norm:  Approx=1.1496, Detail=0.8699

The CDF 9/7 wavelet has four lifting steps (two predict, two update), making it significantly smoother than Haar but with wider filter support. The trade-off between smoothness and locality is fundamental in wavelet analysis.

Irregular-grid support: `cdf53` and `dd4` have interpolating predict steps (`sum(coeffs) ≈ 1`) and activate Lagrange interpolation at the actual physical positions when `t` is supplied. `haar` works correctly on irregular grids without any position correction: its nearest-neighbour predict (`degree = 0`) captures the local difference `d[i] = x[2i+1] - x[2i]` regardless of spacing, so the result is identical with or without `t` (position-aware ≡ position-ignoring). `db2` and `cdf97` have non-interpolating predict steps and fall back to fixed coefficients on irregular grids (with a warning). See `vignette("v05-irregular-grids")` for details.

For empirical guidance on which wavelet to choose given a signal and operation mode, see `vignette("v01-introduction")` Section 3 (regular-grid bench) and `vignette("v07-benchmarks")` (full picture).

## 2. Defining custom wavelets

One of the strengths of the Lifting Scheme is its modularity. Unlike convolution-based implementations, lifting allows you to construct wavelets by composing simple predict and update steps. Each step is defined by:

- `type`: `"predict"` or `"update"`
- `coeffs`: a numeric vector of filter coefficients
- `start_idx`: integer offset for the filter window (or use `position = "center"`, `"left"`, or `"right"`)
- `degree`: inferred automatically. If `sum(coeffs) ≈ 1`, the step is treated as interpolating and Lagrange correction is applied on irregular grids.

### Example: recreating CDF 5/3 manually

The CDF 5/3 wavelet uses linear interpolation for prediction and averaging for update:

\[d[i] \mathbin{-}= 0.5 \cdot s[i] + 0.5 \cdot s[i+1]\]

\[s[i] \mathbin{+}= 0.25 \cdot d[i-1] + 0.25 \cdot d[i]\]

    p_step = lift_step(
      type = "predict",
      coeffs = c(0.5, 0.5),
      start_idx = 0
    )

    u_step = lift_step(
      type = "update",
      coeffs = c(0.25, 0.25),
      start_idx = -1
    )

    my_cdf53 = custom_wavelet(
      name = "MyCDF53",
      steps = list(p_step, u_step),
      norm = c(sqrt(2), 1/sqrt(2))
    )

    print(my_cdf53)
    #> Lifting Scheme: MyCDF53
    #> Steps: 2
    #> Norm:  Approx=1.4142, Detail=0.7071

The `degree` field is inferred automatically: since `sum(c(0.5, 0.5)) = 1`, the predict step is recognised as a linear interpolator (`degree = 1`). This means `my_cdf53` will automatically use position-aware Lagrange interpolation when `t` is supplied, with the same behaviour as the built-in `cdf53`.

### Using the custom wavelet

Once defined, a custom wavelet works in every function (`lwt`, `ilwt`, `denoise_signal_offline`, `denoise_signal_causal`, and `new_wavelet_stream`):

    x = 1:16
    # local_linear extension preserves the linear trend at the boundary,
    # so all detail coefficients vanish for a degree-1 input (2 VM).
    fwd = lwt(
      x, my_cdf53,
      levels = floor(log2(length(x))),
      extension = "local_linear"
    )
    #> Warning in lwt(x, my_cdf53, levels = floor(log2(length(x))), extension =
    #> "local_linear"): Residual signal at level 4 has only 1.0 samples.

    # CDF 5/3 has 2 VM: detail coefficients are zero for a linear signal
    print(round(fwd$coeffs$d1, 10))
    #> [1] 0 0 0 0 0 0 0 0

    # Perfect reconstruction
    rec = ilwt(fwd)
    all.equal(as.numeric(x), rec)
    #> [1] TRUE

## 3. Diagnosing a wavelet

`rLifting` includes a diagnostic suite that verifies the mathematical properties of any lifting scheme. Pass a `config` list specifying the expected properties:

    config = list(
      is_ortho = FALSE, # CDF 5/3 is biorthogonal, not orthogonal
      vm_degrees = c(0, 1), # degree 0 = constant, 1 = ramp
      max_taps = 5 # expected filter support width
    )

    diag = diagnose_wavelet(
      my_cdf53,
      config = config,
      plot = FALSE # suppress basis plot
    )
    #>
    #> === DIAGNOSIS: MYCDF53 ===
    #> [PASS] Perfect Reconstruction (Stress Test) | Max Error (Global): 1.42e-14
    #> [PASS] Orthogonality (Energy Conservation) | Ratio: 1.077141 (Non-Orthogonal) [Exp: FALSE]
    #> [PASS] Vanishing Moments (Degree 0)        | Residual Energy: 0.00e+00 (Tol: 1.00e-09)
    #> [PASS] Vanishing Moments (Degree 1)        | Residual Energy: 2.08e-28 (Tol: 1.00e-09)
    #> [PASS] Compact Support (FIR)               | Active Taps: 5 (Max Expected: 5)
    #> [INFO] Shift Sensitivity                   | Energy Variation (Shift 1): 50.00%

The five sub-tests are also exported individually, so they can be invoked in unit tests or in custom workflows that only need a subset:

Function | What it tests
--- | ---
`validate_perfect_reconstruction` | \(\|\text{ilwt}(\text{lwt}(x)) - x\|_\infty < 10^{-9}\)
`validate_vanishing_moments` | Detail coefficients vanish for polynomials of given degree
`validate_orthogonality` | Energy preservation (Parseval’s theorem)
`validate_compact_support` | Impulse response has finite support
`validate_shift_sensitivity` | Quantifies shift-variance (informational)

Common failure modes:

- A failed perfect reconstruction usually indicates a sign error in a predict or update coefficient, an incorrect `start_idx` that shifts the filter window asymmetrically, or a wrong normalization factor.
- A failed vanishing-moment test for degree \(d\) means the predict step does not reproduce polynomials of that degree, typically because the coefficients do not sum to the value required or the filter window is asymmetric in a way that breaks polynomial reproduction.
- A failed orthogonality test is expected for biorthogonal wavelets (CDF families, DD4); the test passes only when `is_ortho = TRUE` is supplied and the energy ratio is close to 1.
- A failed compact-support test typically reflects a large `start_idx` offset that pushes the effective filter support wider than `max_taps + 2`.

Typical configs by built-in wavelet:

    diagnose_wavelet("haar", list(is_ortho = TRUE,  vm_degrees = 0,    max_taps = 2))
    diagnose_wavelet("db2", list(is_ortho = TRUE,  vm_degrees = 0:1,  max_taps = 4))
    diagnose_wavelet("cdf53", list(is_ortho = FALSE, vm_degrees = 0:1,  max_taps = 5))
    diagnose_wavelet("dd4", list(is_ortho = FALSE, vm_degrees = 0:3,  max_taps = 11))
    diagnose_wavelet("cdf97", list(is_ortho = FALSE, vm_degrees = 0:3,  max_taps = 12))

## 4. Low-level transform pipeline

For fine-grained control over the denoising flow, the four primitives at the heart of the high-level functions (`denoise_signal_offline`, `denoise_signal_causal`, `new_wavelet_stream`) can be invoked directly:

- `lwt(x, scheme, levels, ...)` — forward lifting wavelet transform, returns an `lwt` object holding the detail coefficients `d1, d2, …, dn` and the approximation `an`.
- `compute_adaptive_threshold(lwt_obj, alpha, beta)` — MAD-based noise estimate plus the recursive per-level threshold \(\lambda_k = \lambda_{k-1} (k - 1)/(k + \alpha - 1)\) (corresponds to `threshold_method = "universal"`).
- `threshold(x, lambda, method, a)` — applies one of the four shrinkage rules (`"hard"`, `"soft"`, `"semisoft"`, `"scad"`) to a vector of coefficients.
- `ilwt(lwt_obj)` — inverse transform that reuses the `extension`, `ll_k`, and `t` carried by the `lwt` object.

The math behind these primitives lives in `vignette("v02-thresholding-and-tuning")`: §3 covers the universal vs SURE threshold rules, §4 the α/β recursion, §5 the `tune_alpha_beta()` tuner, §6 the four shrinkage functions with bench-grounded recommendations. This vignette focuses on composing the primitives in non-standard ways.

The example below decomposes a noisy two-frequency signal at five levels, applies a different shrinkage rule per level (semisoft on the finest details, hard on a middle level, no threshold on the coarsest), and reconstructs:

    set.seed(42)
    n = 512
    t_v = seq(0, 1, length.out = n)
    pure = sin(2 * pi * 5 * t_v) + 0.5 * sin(2 * pi * 20 * t_v)
    noisy = pure + rnorm(n, sd = 0.4)

    sch = lifting_scheme("cdf97")

    # 1. Forward transform
    decomp = lwt(noisy, sch, levels = 5)

    # 2. Compute adaptive thresholds
    lambdas = compute_adaptive_threshold(decomp, alpha = 0.3, beta = 1.2)

    # 3. Custom strategy: semisoft on fine levels, hard on d4; leave d5 untouched
    for (lev in 1:3) {
      dname = paste0("d", lev)
      decomp$coeffs[[dname]] = threshold(
        decomp$coeffs[[dname]], lambdas[[dname]], method = "semisoft"
      )
    }
    decomp$coeffs$d4 = threshold(decomp$coeffs$d4, lambdas$d4, method = "hard")
    # d5 untouched to preserve coarse structure

    # 4. Inverse transform
    reconstructed = ilwt(decomp)

    df_pipe = data.frame(
      t = rep(t_v, 3),
      value = c(noisy, pure, reconstructed),
      Signal = rep(c("Noisy", "Truth", "Filtered"), each = n)
    )

    ggplot(df_pipe, aes(x = t, y = value, colour = Signal)) +
      geom_line(alpha = 0.7, linewidth = 0.4) +
      labs(
        title = "Custom denoising pipeline (CDF 9/7, 5 levels)",
        x = "Time", y = "Amplitude"
      ) +
      scale_colour_manual(
        values = c(
          "Noisy" = "grey70", "Truth" = "black",
          "Filtered" = "#D55E00"
        )
      ) +
      theme_minimal()

Figure 1: Custom denoising pipeline on a synthetic two-frequency signal. The noisy input (grey) is decomposed with CDF 9/7 at 5 levels; details d1–d3 are denoised with semisoft, d4 with hard, and d5 is left untouched to preserve coarse structure. The reconstructed signal (orange) tracks the underlying truth (black) closely.

Note on argument naming: all four shrinkages (`"hard"`, `"soft"`, `"semisoft"`, `"scad"`) are supported by both the primitive `threshold()` and the top-level denoising functions. Only the argument name differs: `threshold(x, lambda, method = ...)` (the primitive’s original name), versus `denoise_signal_offline(..., shrinkage = ...)` and the other top-level functions (where `method` is still accepted as a deprecated alias with a warning). For SCAD, pass the shape parameter as `a` (default 3.7) to either form.

## 5. Boundary extension modes

Every transform function accepts an `extension` parameter controlling how the signal is extrapolated when a filter reaches the edge of available data. Five modes are available:

Mode | Behaviour | Best for
--- | --- | ---
`"symmetric"` | Mirror reflection at the boundary (default) | Smooth signals; general use
`"periodic"` | Wraps signal (assumes periodicity) | Genuinely periodic signals
`"zero"` | Pads with zeros beyond the edge | Signals with compact support
`"local_linear"` | OLS line through `ll_k` boundary samples (default 4) | Trending signals; drifting baselines
`"one_sided"` | Drops out-of-bounds taps; renormalises filter by sum | Causal mode with `haar`; no assumption beyond edge

For the empirical impact in offline mode (boundary choice changes MSE by at most a few percent) and the wavelet-conditional picture in causal/stream (`one_sided` and `zero` dominate with `haar`; `symmetric` is the safe default with longer filters), see `vignette("v04-boundary-modes")`. The two short subsections below illustrate the two non-default modes that have user-facing parameters or constraints.

### local_linear: k-point extrapolation

`"local_linear"` fits an ordinary least-squares line through the `ll_k` nearest boundary samples and extrapolates. The default `ll_k = 4` is more robust than 2-point slope estimation when boundary samples are noisy.

\[\hat{x}[i] = \hat{a} + \hat{b} \cdot i, \quad \hat{b} = \frac{k \sum j \cdot x[j] - \sum j \cdot \sum x[j]}{k \sum j^2 - (\sum j)^2}\]

    # Compare 2-point vs 4-point extrapolation on a noisy boundary
    res_k2 = denoise_signal_offline(
      noisy, sch,
      extension = "local_linear", ll_k = 2L
    )

    res_k4 = denoise_signal_offline(
      noisy, sch,
      extension = "local_linear", ll_k = 4L
    )

If `ll_k` exceeds the signal length, it is clamped to `n` with a warning.

### one_sided: filter renormalisation

`"one_sided"` does not extend the signal at all. Instead, it drops out-of-bounds filter taps and renormalises the remaining coefficients by their sum. The filter changes shape near boundaries rather than relying on virtual signal values, which is the only mode with this behaviour.

    set.seed(42)
    n_bnd = 256
    t_bnd = seq(0, 1, length.out = n_bnd)
    trend = 3 * t_bnd
    sig_b = trend + 0.3 * rnorm(n_bnd) + 0.5 * sin(2 * pi * 8 * t_bnd)
    sch_b = lifting_scheme("cdf97")

    res_sym = denoise_signal_offline(
      sig_b, sch_b, levels = 4,
      extension = "symmetric"
    )

    res_ll = denoise_signal_offline(
      sig_b, sch_b, levels = 4,
      extension = "local_linear"
    )
    res_os = denoise_signal_offline(
      sig_b, sch_b, levels = 4,
      extension = "one_sided"
    )

    df_bnd = data.frame(
      t = rep(t_bnd, 4),
      value = c(sig_b, res_sym, res_ll, res_os),
      Signal = rep(
        c("Noisy", "symmetric", "local_linear", "one_sided"),
        each = n_bnd
      )
    )

    ggplot(df_bnd, aes(x = t, y = value, colour = Signal, linewidth = Signal)) +
      geom_line(alpha = 0.85) +
      scale_linewidth_manual(
        values = c(
          "Noisy" = 0.3, "symmetric" = 0.7, "local_linear" = 0.9, "one_sided" = 0.9
        )
      ) +
      scale_colour_manual(values = c(
        "Noisy" = "grey75",
        "symmetric" = "#0072B2",
        "local_linear" = "#009E73",
        "one_sided" = "#CC79A7"
      )) +
      coord_cartesian(xlim = c(0, 0.15)) +
      labs(
        title = "Left-boundary zoom: symmetric vs local_linear vs one_sided",
        subtitle = "Linearly trending signal, CDF 9/7, 4 levels",
        x = "Time", y = "Amplitude"
      ) +
      theme_minimal()

Figure 2: Left-boundary zoom of an offline denoising on a trending signal (3·t + noise + sinusoid) using CDF 9/7 at 4 levels. symmetric (blue) and local_linear (green) reflect or extrapolate the boundary; one_sided (pink) uses only existing samples. On a smooth trending signal in offline mode all three are visually similar; differences become important in causal mode.

`one_sided` is incompatible with irregular grids: when `extension = "one_sided"` and `t` is supplied, `one_sided` takes priority in C++ and Lagrange interpolation is not applied. A warning is raised at the R level.

### Causal stream: symmetric vs one_sided

In causal and stream modes, both edges of the sliding window are virtual, and the boundary choice affects every filtered output rather than just the edge samples. For the recommended causal default (`haar`), `one_sided` makes no assumption about samples beyond the window and consistently outperforms `symmetric` on the regular-grid benchmark; the full per-wavelet comparison lives in `vignette("v04-boundary-modes")` Section 5.

    set.seed(1)
    n_c = 512
    t_c = seq(0, 2, length.out = n_c)
    x_c = 2 * t_c + 0.4 * rnorm(n_c) + sin(2 * pi * 4 * t_c)

    sch_c = lifting_scheme("haar")
    proc_sym = new_wavelet_stream(
      sch_c, window_size = 63, levels = 3,
      extension = "symmetric"
    )

    proc_os = new_wavelet_stream(
      sch_c, window_size = 63, levels = 3,
      extension = "one_sided"
    )

    out_sym = out_os = numeric(n_c)
    for (i in seq_len(n_c)) {
      out_sym[i] = proc_sym(x_c[i])
      out_os[i]  = proc_os(x_c[i])
    }

    df_c = data.frame(
      t = rep(t_c, 3),
      value = c(x_c, out_sym, out_os),
      Signal = rep(c("Noisy", "symmetric", "one_sided"), each = n_c)
    )

    ggplot(df_c, aes(x = t, y = value, colour = Signal, linewidth = Signal)) +
      geom_line(alpha = 0.85) +
      scale_linewidth_manual(
        values = c("Noisy" = 0.3, "symmetric" = 0.8, "one_sided" = 0.8)
      ) +
      scale_colour_manual(
        values = c(
          "Noisy" = "grey75",
          "symmetric" = "#0072B2",
          "one_sided" = "#CC79A7"
        )
      ) +
      labs(
        title = "Causal stream: symmetric vs one_sided (Haar, window = 63)",
        subtitle = "Trending signal, differences emerge at window boundaries",
        x = "Time", y = "Amplitude"
      ) +
      theme_minimal()

Figure 3: Causal-stream denoising of a trending signal with sinusoidal component (Haar, window = 63). symmetric (blue) reflects the most recent sample inward at each step; one_sided (pink) renormalises the filter against the available window. Both follow the trend; the differences emerge in regions where the filter window contains a transition.

For the full mechanics of causal and stream modes (warm-up, `window_size`, `update_freq`, latency), see `vignette("v03-causal-stream")`.

## 6. Irregular grids and custom wavelets

Custom wavelets natively support irregular grids when predict steps have `sum(coeffs) ≈ 1`. The `degree` field is inferred automatically in `lift_step`. The example below defines a 4-point cubic interpolating custom wavelet (structurally similar to the built-in `dd4`) and applies it to a smooth signal sampled on a non-uniform grid:

    # 4-point cubic interpolating predict (sum = 1 -> degree = 3)
    p_cubic = lift_step(
      "predict",
      coeffs = c(-1/16, 9/16, 9/16, -1/16),
      start_idx = -1
    )

    u_cubic = lift_step(
      "update",
      coeffs = c(-1/32, 9/32, 9/32, -1/32),
      start_idx = -1
    )

    my_dd4 = custom_wavelet(
      "MyDD4", list(p_cubic, u_cubic), c(sqrt(2), 1/sqrt(2))
    )

    # Confirm the degree was inferred correctly (3 means cubic Lagrange)
    my_dd4$steps[[1]]$degree
    #> [1] 3

    set.seed(20260523)
    n_irr = 256

    # Irregular grid: cumulative spacing with high variance
    t_irr = cumsum(c(0, abs(rnorm(n_irr - 1, mean = 1, sd = 0.6))))

    # Smooth underlying signal
    pure_irr = sin(t_irr / 8) + 0.3 * cos(t_irr / 3)
    noisy_irr = pure_irr + rnorm(n_irr, sd = 0.3)

    # Denoise using the custom wavelet on the irregular grid
    clean_irr = denoise_signal_offline(
      noisy_irr, my_dd4,
      levels = 4, t = t_irr,
      shrinkage = "semisoft", alpha = 0.3, beta = 1.2
    )

    df_irr = data.frame(
      t = rep(t_irr, 3),
      value = c(noisy_irr, pure_irr, clean_irr),
      Signal = factor(
        rep(c("Noisy", "Truth", "Denoised (MyDD4)"), each = n_irr),
        levels = c("Noisy", "Truth", "Denoised (MyDD4)")
      )
    )

    ggplot(df_irr, aes(x = t, y = value, colour = Signal)) +
      geom_point(
        data = subset(df_irr, Signal == "Noisy"),
        size = 0.6, alpha = 0.5
      ) +
      geom_line(
        data = subset(df_irr, Signal == "Truth"),
        linewidth = 0.5, linetype = "dashed"
      ) +
      geom_line(
        data = subset(df_irr, Signal == "Denoised (MyDD4)"),
        linewidth = 0.8
      ) +
      scale_colour_manual(values = c(
        "Noisy" = "grey60",
        "Truth" = "black",
        "Denoised (MyDD4)" = "#D55E00"
      )) +
      theme_minimal() +
      labs(
        title = "Custom wavelet on an irregular grid",
        subtitle = sprintf(
          "n = %d samples; spacing CV = %.0f%%",
          n_irr, 100 * sd(diff(t_irr)) / mean(diff(t_irr))
        ),
        x = "Physical time t", y = "Amplitude"
      )

Figure 4: Custom 4-point cubic wavelet applied to a smooth signal on an irregular grid. The noisy input (grey dots) is sampled at non-uniform time positions drawn from a log-normal-ish process; the underlying signal (black dashed) is recovered by `denoise_signal_offline` with the custom wavelet and `t = t_irr` (orange). The predict step uses Lagrange interpolation evaluated at each sample’s physical position rather than fixed coefficients.

The Lagrange path is activated automatically because the predict step’s `degree` was inferred as 3 (cubic) from `sum(c(-1/16, 9/16, 9/16, -1/16)) = 1`. No additional configuration is needed in the user code; the same `denoise_signal_offline` call would work with the built-in `dd4` or `cdf53` on the same irregular grid. Note that `haar` (`degree = 0`) needs no position correction: its nearest-neighbour predict captures `d[i] = x[2i+1] - x[2i]` regardless of spacing, so the output is identical with or without `t` — this is correct behaviour, not a limitation.

Non-interpolating predict steps (`sum ≠ 1`) fall back to fixed coefficients when `t` is supplied. This is the correct behaviour for orthogonal wavelets like `db2`, whose predict steps are designed for regular spacing; the package issues a warning so the user knows the irregular path was not activated.

The full discussion of irregular-grid mechanics, wavelet eligibility, and per-mode behaviour lives in `vignette("v05-irregular-grids")`, with empirical recommendations grounded in `data(benchmark_rlifting_irregular)`.

## 7. Visualizing wavelet basis functions

`plot(scheme)` renders the scaling function (\(\phi\)) and wavelet function (\(\psi\)) by cascade iteration of the lifting steps:

    plot(lifting_scheme("cdf97"))

Figure 5: Scaling and wavelet basis functions of CDF 9/7 obtained by cascade iteration of the lifting steps. Useful for verifying that a custom wavelet produces reasonable basis functions and for diagnosing coefficient specification errors.

`visualize_wavelet_basis(scheme, plot = TRUE)` is the standalone function that `plot.lifting_scheme` dispatches to; call it directly when finer control over the cascade depth or plot output is needed.

## Summary

`rLifting` is designed to be both high-performance and extensible:

- Six built-in wavelets cover common use cases. `cdf53` and `dd4` adapt to irregular grids; `cdf97` is the smoothest; `haar` is the fastest and the recommended default for causal/stream mode.
- Custom wavelets via `lift_step()` plus `custom_wavelet()` work in every function; `degree` is inferred automatically for irregular-grid support.
- Diagnostic suite validates perfect reconstruction, vanishing moments, orthogonality, compact support, and shift sensitivity, both via the wrapper `diagnose_wavelet()` and via the five standalone `validate_*` functions.
- Low-level API (`lwt`, `ilwt`, `compute_adaptive_threshold`, `threshold`) for fully custom pipelines that interleave bespoke logic between decomposition and reconstruction. Threshold mathematics and the empirical decision guide live in `vignette("v02-thresholding-and-tuning")`.
- Five boundary modes with mode-specific trade-offs and wavelet-conditional empirical impact (`vignette("v04-boundary-modes")`); `one_sided` is especially suited for causal/stream mode with short filters (`haar`) at regular grids; `local_linear`’s `ll_k` is the only numeric tunable.
- Irregular-grid support via the `t` vector activates Lagrange interpolation for `cdf53` (linear) and `dd4` (cubic). `haar` works correctly on irregular grids without position correction — its nearest-neighbour predict is spacing-agnostic by design, so the output is identical with or without `t`. `db2` and `cdf97` fall back to fixed coefficients with a warning.

## References

Antoniadis, A., & Fan, J. (2001). Regularization of wavelet approximations. Journal of the American Statistical Association, 96(455), 939–967.

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