We investigate a novel non-parametric regression-based clustering algorithm for longitudinal data analysis. Combining natural cubic splines with Gaussian mixture models (GMM), the algorithm can produce smooth cluster means that describe the underlying data well. However, there are some shortcomings in the algorithm: high computational complexity in the parameter estimation procedure and a numerically unstable variance estimator. Therefore, to further increase the usability of the method, we incorporated approaches to reduce its computational complexity, we developed a new, more stable variance estimator, and we developed a new smoothing parameter estimation procedure. We show that the developed algorithm, SMIXS, performs better than GMM on a synthetic dataset in terms of clustering and regression performance. We demonstrate the impact of the computational speed-ups, which we formally prove in the new framework. Finally, we perform a case study by using SMIXS to cluster vertical atmospheric measurements to determine different weather regimes.
相關內容
The method of instrumental variables provides a fundamental and practical tool for causal inference in many empirical studies where unmeasured confounding between the treatments and the outcome is present. Modern data such as the genetical genomics data from these studies are often high-dimensional. The high-dimensional linear instrumental-variables regression has been considered in the literature due to its simplicity albeit a true nonlinear relationship may exist. We propose a more data-driven approach by considering the nonparametric additive models between the instruments and the treatments while keeping a linear model between the treatments and the outcome so that the coefficients therein can directly bear causal interpretation. We provide a two-stage framework for estimation and inference under this more general setup. The group lasso regularization is first employed to select optimal instruments from the high-dimensional additive models, and the outcome variable is then regressed on the fitted values from the additive models to identify and estimate important treatment effects. We provide non-asymptotic analysis of the estimation error of the proposed estimator. A debiasing procedure is further employed to yield valid inference. Extensive numerical experiments show that our method can rival or outperform existing approaches in the literature. We finally analyze the mouse obesity data and discuss new findings from our method.
Heterogeneity is a dominant factor in the behaviour of many biological processes. Despite this, it is common for mathematical and statistical analyses to ignore biological heterogeneity as a source of variability in experimental data. Therefore, methods for exploring the identifiability of models that explicitly incorporate heterogeneity through variability in model parameters are relatively underdeveloped. We develop a new likelihood-based framework, based on moment matching, for inference and identifiability analysis of differential equation models that capture biological heterogeneity through parameters that vary according to probability distributions. As our novel method is based on an approximate likelihood function, it is highly flexible; we demonstrate identifiability analysis using both a frequentist approach based on profile likelihood, and a Bayesian approach based on Markov-chain Monte Carlo. Through three case studies, we demonstrate our method by providing a didactic guide to inference and identifiability analysis of hyperparameters that relate to the statistical moments of model parameters from independent observed data. Our approach has a computational cost comparable to analysis of models that neglect heterogeneity, a significant improvement over many existing alternatives. We demonstrate how analysis of random parameter models can aid better understanding of the sources of heterogeneity from biological data.
Numerical methods for random parametric PDEs can greatly benefit from adaptive refinement schemes, in particular when functional approximations are computed as in stochastic Galerkin and stochastic collocations methods. This work is concerned with a non-intrusive generalization of the adaptive Galerkin FEM with residual based error estimation. It combines the non-intrusive character of a randomized least-squares method with the a posteriori error analysis of stochastic Galerkin methods. The proposed approach uses the Variational Monte Carlo method to obtain a quasi-optimal low-rank approximation of the Galerkin projection in a highly efficient hierarchical tensor format. We derive an adaptive refinement algorithm which is steered by a reliable error estimator. Opposite to stochastic Galerkin methods, the approach is easily applicable to a wide range of problems, enabling a fully automated adjustment of all discretization parameters. Benchmark examples with affine and (unbounded) lognormal coefficient fields illustrate the performance of the non-intrusive adaptive algorithm, showing best-in-class performance.
We develop an R package SPQR that implements the semi-parametric quantile regression (SPQR) method in Xu and Reich (2021). The method begins by fitting a flexible density regression model using monotonic splines whose weights are modeled as data-dependent functions using artificial neural networks. Subsequently, estimates of conditional density and quantile process can all be obtained. Unlike many approaches to quantile regression that assume a linear model, SPQR allows for virtually any relationship between the covariates and the response distribution including non-linear effects and different effects on different quantile levels. To increase the interpretability and transparency of SPQR, model-agnostic statistics developed by Apley and Zhu (2020) are used to estimate and visualize the covariate effects and their relative importance on the quantile function. In this article, we detail how this framework is implemented in SPQR and illustrate how this package should be used in practice through simulated and real data examples.
We present an alternating least squares type numerical optimization scheme to estimate conditionally-independent mixture models in $\mathbb{R}^n$, with minimal additional distributional assumptions. Following the method of moments, we tackle a coupled system of low-rank tensor decomposition problems. The steep costs associated with high-dimensional tensors are avoided, through the development of specialized tensor-free operations. Numerical experiments illustrate the performance of the algorithm and its applicability to various models and applications. In many cases the results exhibit improved reliability over the expectation-maximization algorithm, with similar time and storage costs. We also provide some supporting theory, establishing identifiability and local linear convergence.
Bayesian nonparametric mixture models are common for modeling complex data. While these models are well-suited for density estimation, their application for clustering has some limitations. Miller and Harrison (2014) proved posterior inconsistency in the number of clusters when the true number of clusters is finite for Dirichlet process and Pitman--Yor process mixture models. In this work, we extend this result to additional Bayesian nonparametric priors such as Gibbs-type processes and finite-dimensional representations of them. The latter include the Dirichlet multinomial process and the recently proposed Pitman--Yor and normalized generalized gamma multinomial processes. We show that mixture models based on these processes are also inconsistent in the number of clusters and discuss possible solutions. Notably, we show that a post-processing algorithm introduced by Guha et al. (2021) for the Dirichlet process extends to more general models and provides a consistent method to estimate the number of components.
Seeking legal advice is often expensive. Recent advancement in machine learning for solving complex problems can be leveraged to help make legal services more accessible to the public. However, real-life applications encounter significant challenges. State-of-the-art language models are growing increasingly large, making parameter-efficient learning increasingly important. Unfortunately, parameter-efficient methods perform poorly with small amounts of data, which are common in the legal domain (where data labelling costs are high). To address these challenges, we propose parameter-efficient legal domain adaptation, which uses vast unsupervised legal data from public legal forums to perform legal pre-training. This method exceeds or matches the fewshot performance of existing models such as LEGAL-BERT on various legal tasks while tuning only approximately 0.1% of model parameters. Additionally, we show that our method can achieve calibration comparable to existing methods across several tasks. To the best of our knowledge, this work is among the first to explore parameter-efficient methods of tuning language models toward the legal domain.
A biomechanical model often requires parameter estimation and selection in a known but complicated nonlinear function. Motivated by observing that data from a head-neck position tracking system, one of biomechanical models, show multiplicative time dependent errors, we develop a modified penalized weighted least squares estimator. The proposed method can be also applied to a model with non-zero mean time dependent additive errors. Asymptotic properties of the proposed estimator are investigated under mild conditions on a weight matrix and the error process. A simulation study demonstrates that the proposed estimation works well in both parameter estimation and selection with time dependent error. The analysis and comparison with an existing method for head-neck position tracking data show better performance of the proposed method in terms of the variance accounted for (VAF).
As soon as abstract mathematical computations were adapted to computation on digital computers, the problem of efficient representation, manipulation, and communication of the numerical values in those computations arose. Strongly related to the problem of numerical representation is the problem of quantization: in what manner should a set of continuous real-valued numbers be distributed over a fixed discrete set of numbers to minimize the number of bits required and also to maximize the accuracy of the attendant computations? This perennial problem of quantization is particularly relevant whenever memory and/or computational resources are severely restricted, and it has come to the forefront in recent years due to the remarkable performance of Neural Network models in computer vision, natural language processing, and related areas. Moving from floating-point representations to low-precision fixed integer values represented in four bits or less holds the potential to reduce the memory footprint and latency by a factor of 16x; and, in fact, reductions of 4x to 8x are often realized in practice in these applications. Thus, it is not surprising that quantization has emerged recently as an important and very active sub-area of research in the efficient implementation of computations associated with Neural Networks. In this article, we survey approaches to the problem of quantizing the numerical values in deep Neural Network computations, covering the advantages/disadvantages of current methods. With this survey and its organization, we hope to have presented a useful snapshot of the current research in quantization for Neural Networks and to have given an intelligent organization to ease the evaluation of future research in this area.
Clustering is one of the most fundamental and wide-spread techniques in exploratory data analysis. Yet, the basic approach to clustering has not really changed: a practitioner hand-picks a task-specific clustering loss to optimize and fit the given data to reveal the underlying cluster structure. Some types of losses---such as k-means, or its non-linear version: kernelized k-means (centroid based), and DBSCAN (density based)---are popular choices due to their good empirical performance on a range of applications. Although every so often the clustering output using these standard losses fails to reveal the underlying structure, and the practitioner has to custom-design their own variation. In this work we take an intrinsically different approach to clustering: rather than fitting a dataset to a specific clustering loss, we train a recurrent model that learns how to cluster. The model uses as training pairs examples of datasets (as input) and its corresponding cluster identities (as output). By providing multiple types of training datasets as inputs, our model has the ability to generalize well on unseen datasets (new clustering tasks). Our experiments reveal that by training on simple synthetically generated datasets or on existing real datasets, we can achieve better clustering performance on unseen real-world datasets when compared with standard benchmark clustering techniques. Our meta clustering model works well even for small datasets where the usual deep learning models tend to perform worse.
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191