CSSD - Cubic smoothing splines for discontinuous signals

This is a reference implementation in Matlab 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

1. 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 $\gamma > 0.$

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

Quickstart

1. Execute "install_cssd.m" which adds the folder and all subfolders to the Matlab path.
2. Execute any m-file from the demos folder

Examples

Synthetic data

A synthetic signal is sampled at $N = 100$ random data sites $x_i$ and corrupted by zero mean Gaussian noise with standard deviation $0.1.$ The results of the discussed model are shown for $p=0.999$ and different parameters of $\gamma,$ where $\gamma=\infty$ corresponds to classical smoothing splines. The thick lines represent the results of the shown sample realization. The shaded areas depict the $2.5 \%$ to $97.5 \%$ (pointwise) quantiles of $1000$ realizations. The histograms under the plots show the frequency of the detected discontinuity locations over all realizations.

Stock data

The dots represent the logarithm of the closing prices of the Meta stock from May 18, 2012, to May 19, 2022. The curve represents the CSSD with parameters determined by K-fold CV ($p = 0.4702$, $\gamma = 0.0069$). The dashed vertical lines indicate the discontinuities of the CSSD, and the ticks correspond to the date before the discontinuity.

Geyser data

Fitting a CSSD to the Old Faithful data (circles): If the parameter is selected based on K-fold CV we obtain a result without discontinuities which coincides with a classical smoothing spline (solid curve). Keeping the selected $p$-parameter and lowering the $\gamma$ parameter sufficiently gives a two-phase regression curve (dashed curves) with a breakpoint near $x = 3$ (dashed vertical line), and the two curve segments are nearly linear. Both of the above parameter sets yield better CV-scores than a linear model (dotted line).

Reference

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

See also

• Higher order Mumford-Shah models (discrete splines with discontinuities)
• Degrees of freedom penalized regression
• Pottslab
• L1TV regularization