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Degrees-of-freedom penalized piecewise regression
Methodology · Submitted 27 Dec 2023
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim A constrained variant of the proposed method gives state-of-the-art results in the Turing benchmark for unsupervised changepoint detection.
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The abstract, sentence by sentence
Many popular piecewise regression models rely on minimizing a cost function on the model fit with a linear penalty on the number of segments. However, this penalty does not take into account varying complexities of the model functions on the segments potentially leading to overfitting when models with varying complexities, such as polynomials of different degrees, are used. In this work, we enhance on this approach by instead using a penalty on the sum of the degrees of freedom over all segments, called degrees-of-freedom penalized piecewise regression (DofPPR). We show that the solutions of the resulting minimization problem are unique for almost all input data in a least squares setting. We develop a fast algorithm which does not only compute a minimizer but also determines an optimal hyperparameter -- in the sense of rolling cross validation with the one standard error rule -- exactly. This eliminates manual hyperparameter selection. Our method supports optional user parameters for incorporating domain knowledge. We provide an open-source Python/Rust code for the piecewise polynomial least squares case which can be extended to further models. We demonstrate the practical utility through a simulation study and by applications to real data. A constrained variant of the proposed method gives state-of-the-art results in the Turing benchmark for unsupervised changepoint detection.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Many popular piecewise regression models rely on minimizing a cost function on the model fit with a linear penalty on the number of segments. | background 1.00 | 0.04 | 0.03 | asserted 1.00 |
| s1 | However, this penalty does not take into account varying complexities of the model functions on the segments potentially leading to overfitting when models with varying complexities, such as polynomials of different degrees, are used. | gap 0.88 | 0.10 | 0.04 | hedged possibility 0.50 |
| s2 | In this work, we enhance on this approach by instead using a penalty on the sum of the degrees of freedom over all segments, called degrees-of-freedom penalized piecewise regression (DofPPR). | contribution 0.77 | 0.87 | 0.22 | asserted 0.96 |
| s3 | We show that the solutions of the resulting minimization problem are unique for almost all input data in a least squares setting. | result 0.97 | 0.95 | 0.14 | asserted 1.00 |
| s4 | We develop a fast algorithm which does not only compute a minimizer but also determines an optimal hyperparameter -- in the sense of rolling cross validation with the one standard error rule -- exactly. | method 0.55 | 0.97 | 0.13 | asserted 1.00 |
| s5 | This eliminates manual hyperparameter selection. | interpretation 0.66 | 0.74 | 0.08 | asserted 1.00 |
| s6 | Our method supports optional user parameters for incorporating domain knowledge. | method 0.51 | 0.84 | 0.03 | asserted 0.98 |
| s7 | We provide an open-source Python/Rust code for the piecewise polynomial least squares case which can be extended to further models. | contribution 0.93 | 0.93 | 0.04 | asserted 1.00 |
| s8 | We demonstrate the practical utility through a simulation study and by applications to real data. | contribution 0.66 | 0.95 | 0.14 | asserted 1.00 |
| s9 | A constrained variant of the proposed method gives state-of-the-art results in the Turing benchmark for unsupervised changepoint detection. | result 0.99 | 0.89 | 0.21 | asserted 1.00 |
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The paper
Original abstract
Many popular piecewise regression models rely on minimizing a cost function on the model fit with a linear penalty on the number of segments. However, this penalty does not take into account varying complexities of the model functions on the segments potentially leading to overfitting when models with varying complexities, such as polynomials of different degrees, are used. In this work, we enhance on this approach by instead using a penalty on the sum of the degrees of freedom over all segments, called degrees-of-freedom penalized piecewise regression (DofPPR). We show that the solutions of the resulting minimization problem are unique for almost all input data in a least squares setting. We develop a fast algorithm which does not only compute a minimizer but also determines an optimal hyperparameter -- in the sense of rolling cross validation with the one standard error rule -- exactly. This eliminates manual hyperparameter selection. Our method supports optional user parameters for incorporating domain knowledge. We provide an open-source Python/Rust code for the piecewise polynomial least squares case which can be extended to further models. We demonstrate the practical utility through a simulation study and by applications to real data. A constrained variant of the proposed method gives state-of-the-art results in the Turing benchmark for unsupervised changepoint detection.
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What this paper does with the arXiv papers it cites
What it cites · 6 sentences without a confident label
Uses data or values from 1 paper
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We now evaluate the proposed method using the full TCPD benchmark [ref] .
An Evaluation of Change Point Detection Algorithms · Results on the TCPD benchmark for unsupervised changepoint detection · 0.86
Cites as background 3 papers
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Several papers consider piecewise regression with continuity contraints such as change in slope problems [ref] .
Change-in-Slope Optimal Partitioning Algorithm in a Finite-Size Parameter Space · Prior and related work · 0.96
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We refer to [ref] and the references therein for a more detailed overview and a comparison over selected changepoint detection methods.
An Evaluation of Change Point Detection Algorithms · Prior and related work · 0.99
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[ref] discuss pitfalls and solutions when using an interleaved cross validation procedure.
Cross-validation for change-point regression: pitfalls and solutions · Prior and related work · 0.92
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