Random Forest Weighted Local Fréchet Regression with Random Objects

• URL: http://arxiv.org/abs/2202.04912v4
• Date: Sat, 16 Mar 2024 10:39:18 GMT
• Title: Random Forest Weighted Local Fréchet Regression with Random Objects
• Authors: Rui Qiu, Zhou Yu, Ruoqing Zhu,
• Abstract summary: We propose a novel random forest weighted local Fr'echet regression paradigm. Our first method uses these weights as the local average to solve the conditional Fr'echet mean. Second method performs local linear Fr'echet regression, both significantly improving existing Fr'echet regression methods.
• Score: 18.128663071848923
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Statistical analysis is increasingly confronted with complex data from metric spaces. Petersen and M\"uller (2019) established a general paradigm of Fr\'echet regression with complex metric space valued responses and Euclidean predictors. However, the local approach therein involves nonparametric kernel smoothing and suffers from the curse of dimensionality. To address this issue, we in this paper propose a novel random forest weighted local Fr\'echet regression paradigm. The main mechanism of our approach relies on a locally adaptive kernel generated by random forests. Our first method uses these weights as the local average to solve the conditional Fr\'echet mean, while the second method performs local linear Fr\'echet regression, both significantly improving existing Fr\'echet regression methods. Based on the theory of infinite order U-processes and infinite order $M_{m_n}$-estimator, we establish the consistency, rate of convergence, and asymptotic normality for our local constant estimator, which covers the current large sample theory of random forests with Euclidean responses as a special case. Numerical studies show the superiority of our methods with several commonly encountered types of responses such as distribution functions, symmetric positive-definite matrices, and sphere data. The practical merits of our proposals are also demonstrated through the application to New York taxi data and human mortality data.

Related papers

• Deep Fréchet Regression [4.915744683251151]
  We propose a flexible regression model capable of handling high-dimensional predictors without imposing parametric assumptions. The proposed model outperforms existing methods for non-Euclidean responses.
  arXiv  Detail & Related papers  (2024-07-31T07:54:14Z)
• Multivariate root-n-consistent smoothing parameter free matching estimators and estimators of inverse density weighted expectations [51.000851088730684]
  We develop novel modifications of nearest-neighbor and matching estimators which converge at the parametric $sqrt n $-rate. We stress that our estimators do not involve nonparametric function estimators and in particular do not rely on sample-size dependent parameters smoothing.
  arXiv  Detail & Related papers  (2024-07-11T13:28:34Z)
• von Mises Quasi-Processes for Bayesian Circular Regression [57.88921637944379]
  We explore a family of expressive and interpretable distributions over circle-valued random functions. The resulting probability model has connections with continuous spin models in statistical physics. For posterior inference, we introduce a new Stratonovich-like augmentation that lends itself to fast Markov Chain Monte Carlo sampling.
  arXiv  Detail & Related papers  (2024-06-19T01:57:21Z)
• Relaxed Quantile Regression: Prediction Intervals for Asymmetric Noise [51.87307904567702]
  Quantile regression is a leading approach for obtaining such intervals via the empirical estimation of quantiles in the distribution of outputs. We propose Relaxed Quantile Regression (RQR), a direct alternative to quantile regression based interval construction that removes this arbitrary constraint. We demonstrate that this added flexibility results in intervals with an improvement in desirable qualities.
  arXiv  Detail & Related papers  (2024-06-05T13:36:38Z)
• Conformal inference for regression on Riemannian Manifolds [49.7719149179179]
  We investigate prediction sets for regression scenarios when the response variable, denoted by $Y$, resides in a manifold, and the covariable, denoted by X, lies in Euclidean space. We prove the almost sure convergence of the empirical version of these regions on the manifold to their population counterparts.
  arXiv  Detail & Related papers  (2023-10-12T10:56:25Z)
• Medoid splits for efficient random forests in metric spaces [0.0]
  This paper revisits an adaptation of the random forest for Fr'echet regression, addressing the challenge of regression in metric spaces. We introduce a new splitting rule that circumvents the computationally expensive operation of Fr'echet means by substituting with a medoid-based approach.
  arXiv  Detail & Related papers  (2023-06-29T15:32:11Z)
• Instance-Dependent Generalization Bounds via Optimal Transport [51.71650746285469]
  Existing generalization bounds fail to explain crucial factors that drive the generalization of modern neural networks. We derive instance-dependent generalization bounds that depend on the local Lipschitz regularity of the learned prediction function in the data space. We empirically analyze our generalization bounds for neural networks, showing that the bound values are meaningful and capture the effect of popular regularization methods during training.
  arXiv  Detail & Related papers  (2022-11-02T16:39:42Z)
• Communication-Efficient Distributed Quantile Regression with Optimal Statistical Guarantees [2.064612766965483]
  We address the problem of how to achieve optimal inference in distributed quantile regression without stringent scaling conditions. The difficulties are resolved through a double-smoothing approach that is applied to the local (at each data source) and global objective functions. Despite the reliance on a delicate combination of local and global smoothing parameters, the quantile regression model is fully parametric.
  arXiv  Detail & Related papers  (2021-10-25T17:09:59Z)
• Dimension Reduction and Data Visualization for Fr\'echet Regression [8.713190936209156]
  Fr'echet regression model provides a promising framework for regression analysis with metric spacevalued responses. We introduce a flexible sufficient dimension reduction (SDR) method for Fr'echet regression to achieve two purposes.
  arXiv  Detail & Related papers  (2021-10-01T15:01:32Z)
• Generalized Gumbel-Softmax Gradient Estimator for Various Discrete Random Variables [16.643346012854156]
  Esting the gradients of nodes is one of the crucial research questions in the deep generative modeling community. This paper proposes a general version of the Gumbel-Softmax estimator with continuous relaxation.
  arXiv  Detail & Related papers  (2020-03-04T01:13:15Z)
• Censored Quantile Regression Forest [81.9098291337097]
  We develop a new estimating equation that adapts to censoring and leads to quantile score whenever the data do not exhibit censoring. The proposed procedure named it censored quantile regression forest, allows us to estimate quantiles of time-to-event without any parametric modeling assumption.
  arXiv  Detail & Related papers  (2020-01-08T23:20:23Z)

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.