CSSD

May 11, 2026 · View on GitHub

Cubic smoothing splines for signals with a-priori unknown discontinuities — joint estimation of the spline and the discontinuity set via dynamic programming, with K-fold cross-validation for the regularisation parameters.

This is a reference implementation for the algorithms described in the paper

M. Storath, A. Weinmann, "Smoothing splines for discontinuous signals", Journal of Computational and Graphical Statistics, 2023, [Preprint]

Overview of main functionalities

cssd.m computes a cubic smoothing spline with discontinuities (CSSD) for data (x,y). It is a solution of the following model of a smoothing spline $f$ with a-priori unknown discontinuities $J$

$\min_{f, J} p \sum_{i=1}^N \left(\frac{y_i - f(x_i)}{\delta_i}\right)^2 + (1-p) \int_{[x_1, x_N] \setminus J} (f''(t))^2 dt + \gamma |J|.$

where

$y_i = g(x_i) + \epsilon_i$ are samples of piecewise smooth function $g$ at data sites $x_1, \ldots, x_N$ , and an estimate $\delta_i$ of the standard deviation of the errors $\epsilon_i$
the minimum is taken over all possible sets of discontinuities between two data sites $J \subset [x_1, x_N]\setminus \{x_1, \ldots, x_N\}$ and all functions $f$ that are twice continuously differentiable away from the discontinuities.
The model parameter $p \in (0, 1)$ controls the relative weight of the smoothness term (second term) and the data fidelity term.
The last term is a penalty for the number of discontinuities $|J|$ weighted by a parameter $\gamma > 0.$

cssd_cv.m automatically determines values for the model parameters $p$ and $\gamma$ based on K-fold cross validation.

Quickstart

Python (Rust core)

pip install cssd

import numpy as np
from cssd import cssd, cssd_cv

x = np.linspace(0, 1, 100)
y = np.sin(4 * np.pi * x) - np.sign(x - 0.3) - np.sign(0.72 - x)

out = cssd(x, y, p=0.999, gamma=8.0)
out.discont      # detected jump locations
out.pp(x)        # evaluate the piecewise spline

cv = cssd_cv(x, y, cv_type="random", cv_arg=5)
cv.p, cv.gamma, cv.fit.discont

The Python package wraps a Rust extension built with PyO3 and maturin; see crates/cssd-core for the algorithm crate and crates/cssd-py for the bindings.