# `pp_second_derivative` — Second derivative

_Group_: **Preprocessing** · _Binding_: `n4m.sklearn.SecondDerivative` · _C ABI_: `n4m_pp_second_derivative_*`

## Description

Two passes of ``np.gradient`` (shape-preserving).

### Parameters

| Name | Type | Default |
|------|------|---------|
| `delta` | `float` | `1.0` |
| `edge_order` | `int` | `2` |

## Explanations

### Bibliographic source

Standard finite-difference / gap derivative; see Savitzky & Golay (1964) and Norris & Williams (1984).

### Mathematical principle

Approximates $\mathrm{d}^2\mathbf{x}/\mathrm{d}\lambda^2$. The second derivative removes both constant and linear baselines and resolves overlapping peaks into sharp negative lobes at the original band positions, at the cost of amplifying high-frequency noise (hence it is usually paired with smoothing).

### Implementation

C ABI `n4m_pp_second_derivative_*` in libn4m (create / apply / destroy lifecycle), wrapped by `n4m.sklearn.SecondDerivative`. The same numerical kernel backs every language binding.

### Usage

```python
from n4m.sklearn import SecondDerivative
op = SecondDerivative()
X_transformed = op.fit_transform(X)
```

### Benchmarks

Adaptive wall-clock per cell measured against [`full_matrix.csv`](../benchmarks/overview.md). Only backends that implement this method are listed; libraries without the method are omitted.

**Verdict**  ·  ✓ ref / ≈ ref / ~ shape mark a reference-gate pass at strict / relaxed / qualitative tolerance  ·  ✓ bind = pls4all binding agrees with the C++ baseline  ·  ✗ divergent  ·  ⚠ error  ·  — not run. The fastest backend per column is marked 🏆.

**Reference gate**: strict — numeric equivalence (`rmse_rel_tol ≤ 1e-12`).

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<div class="parity-table-wrap">
<table class="docutils parity-grouped">
<thead><tr><th scope="col">Backend</th><th scope="col">Parity</th><th class="size-col" scope="col">50×250 (ms)</th><th class="size-col" scope="col">250×50 (ms)</th></tr></thead>
<tbody class="lang-band lang-cpp"><tr class="lang-band-row" data-lang="cpp"><th colspan="4" scope="rowgroup"><span class="lang-band-dot"></span>C++ native · libn4m</th></tr>
<tr class="bk-row"><td class="bk-name"><code>pls4all.cpp.blas+omp</code></td><td class="parity parity-ref-strict">✓ ref</td><td class="ms">—</td><td class="ms">—</td></tr>
</tbody>
<tbody class="lang-band lang-python"><tr class="lang-band-row" data-lang="python"><th colspan="4" scope="rowgroup"><span class="lang-band-dot"></span>Python · external</th></tr>
<tr class="bk-row"><td class="bk-name"><code>ref.python_numpy</code></td><td class="parity parity-ref-source">source</td><td class="ms">—</td><td class="ms">—</td></tr>
</tbody>
</table>
</div>

:::

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_See also_: [methods index](index.md) · [interactive dashboard](../landing/dashboard.md)