Interpretation of High-Dimensional Linear Regression: Effects of Nullspace and Regularization Demonstrated on Battery Data

• URL: http://arxiv.org/abs/2309.00564v2
• Date: Wed, 6 Sep 2023 17:35:10 GMT
• Title: Interpretation of High-Dimensional Linear Regression: Effects of Nullspace and Regularization Demonstrated on Battery Data
• Authors: Joachim Schaeffer, Eric Lenz, William C. Chueh, Martin Z. Bazant, Rolf Findeisen, Richard D. Braatz
• Abstract summary: This article considers discrete measured data of underlying smooth latent processes, as is often obtained from chemical or biological systems. The nullspace and its interplay with regularization shapes regression coefficients. We show that regularization and z-scoring are design choices that, if chosen corresponding to prior physical knowledge, lead to interpretable regression results.
• Score: 0.019064981263344844
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: High-dimensional linear regression is important in many scientific fields. This article considers discrete measured data of underlying smooth latent processes, as is often obtained from chemical or biological systems. Interpretation in high dimensions is challenging because the nullspace and its interplay with regularization shapes regression coefficients. The data's nullspace contains all coefficients that satisfy $\mathbf{Xw}=\mathbf{0}$, thus allowing very different coefficients to yield identical predictions. We developed an optimization formulation to compare regression coefficients and coefficients obtained by physical engineering knowledge to understand which part of the coefficient differences are close to the nullspace. This nullspace method is tested on a synthetic example and lithium-ion battery data. The case studies show that regularization and z-scoring are design choices that, if chosen corresponding to prior physical knowledge, lead to interpretable regression results. Otherwise, the combination of the nullspace and regularization hinders interpretability and can make it impossible to obtain regression coefficients close to the true coefficients when there is a true underlying linear model. Furthermore, we demonstrate that regression methods that do not produce coefficients orthogonal to the nullspace, such as fused lasso, can improve interpretability. In conclusion, the insights gained from the nullspace perspective help to make informed design choices for building regression models on high-dimensional data and reasoning about potential underlying linear models, which are important for system optimization and improving scientific understanding.

Related papers

• Branch and Bound to Assess Stability of Regression Coefficients in Uncertain Models [0.6990493129893112]
  We introduce our algorithm, along with supporting mathematical results, an example application, and a link to our computer code. It helps researchers summarize high-dimensional data and assess the stability of regression coefficients in uncertain models.
  arXiv  Detail & Related papers  (2024-08-19T01:37:14Z)
• Deep Generative Symbolic Regression [83.04219479605801]
  Symbolic regression aims to discover concise closed-form mathematical equations from data. Existing methods, ranging from search to reinforcement learning, fail to scale with the number of input variables. We propose an instantiation of our framework, Deep Generative Symbolic Regression.
  arXiv  Detail & Related papers  (2023-12-30T17:05:31Z)
• Understanding Augmentation-based Self-Supervised Representation Learning via RKHS Approximation and Regression [53.15502562048627]
  Recent work has built the connection between self-supervised learning and the approximation of the top eigenspace of a graph Laplacian operator. This work delves into a statistical analysis of augmentation-based pretraining.
  arXiv  Detail & Related papers  (2023-06-01T15:18:55Z)
• Kernel-based off-policy estimation without overlap: Instance optimality beyond semiparametric efficiency [53.90687548731265]
  We study optimal procedures for estimating a linear functional based on observational data. For any convex and symmetric function class $mathcalF$, we derive a non-asymptotic local minimax bound on the mean-squared error.
  arXiv  Detail & Related papers  (2023-01-16T02:57:37Z)
• Experimental Design for Linear Functionals in Reproducing Kernel Hilbert Spaces [102.08678737900541]
  We provide algorithms for constructing bias-aware designs for linear functionals. We derive non-asymptotic confidence sets for fixed and adaptive designs under sub-Gaussian noise.
  arXiv  Detail & Related papers  (2022-05-26T20:56:25Z)
• Bias-variance decomposition of overparameterized regression with random linear features [0.0]
  "Over parameterized models" avoid overfitting even when the number of fit parameters is large enough to perfectly fit the training data. We show how each transition arises due to small nonzero eigenvalues in the Hessian matrix. We compare and contrast the phase diagram of the random linear features model to the random nonlinear features model and ordinary regression.
  arXiv  Detail & Related papers  (2022-03-10T16:09:21Z)
• Statistical Inference for High-Dimensional Linear Regression with Blockwise Missing Data [13.48481978963297]
  Blockwise missing data occurs when we integrate multisource or multimodality data where different sources or modalities contain complementary information. We propose a computationally efficient estimator for the regression coefficient vector based on carefully constructed unbiased estimating equations. Numerical studies and application analysis of the Alzheimer's Disease Neuroimaging Initiative data show that the proposed method performs better and benefits more from unsupervised samples than existing methods.
  arXiv  Detail & Related papers  (2021-06-07T05:12:42Z)
• A Hypergradient Approach to Robust Regression without Correspondence [85.49775273716503]
  We consider a variant of regression problem, where the correspondence between input and output data is not available. Most existing methods are only applicable when the sample size is small. We propose a new computational framework -- ROBOT -- for the shuffled regression problem.
  arXiv  Detail & Related papers  (2020-11-30T21:47:38Z)
• A Nonconvex Framework for Structured Dynamic Covariance Recovery [24.471814126358556]
  We propose a flexible yet interpretable model for high-dimensional data with time-varying second order statistics. Motivated by the literature, we quantify factorization and smooth temporal data. We show that our approach outperforms existing baselines.
  arXiv  Detail & Related papers  (2020-11-11T07:09:44Z)
• Non-parametric Models for Non-negative Functions [48.7576911714538]
  We provide the first model for non-negative functions from the same good linear models. We prove that it admits a representer theorem and provide an efficient dual formulation for convex problems.
  arXiv  Detail & Related papers  (2020-07-08T07:17:28Z)
• Linear Tensor Projection Revealing Nonlinearity [0.294944680995069]
  Dimensionality reduction is an effective method for learning high-dimensional data. We propose a method that searches for a subspace that maximizes the prediction accuracy while retaining as much of the original data information as possible.
  arXiv  Detail & Related papers  (2020-07-08T06:10:39Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.