The empirical likelihood is a powerful nonparametric tool, that emulates its parametric counterpart -- the parametric likelihood -- preserving many of its large-sample properties. This article tackles the problem of assessing the discriminatory power of three-class diagnostic tests from an empirical likelihood perspective. In particular, we concentrate on interval estimation in a three-class ROC analysis, where a variety of inferential tasks could be of interest. We present novel theoretical results and tailored techniques studied to efficiently solve some of such tasks. Extensive simulation experiments are provided in a supporting role, with our novel proposals compared to existing competitors, when possible. It emerges that our new proposals are extremely flexible, being able to compete with contestants and being the most suited to accommodating flexible distributions for target populations. We illustrate the application of the novel proposals with a real data example. The article ends with a discussion and a presentation of some directions for future research.
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In uncertainty quantification, variance-based global sensitivity analysis quantitatively determines the effect of each input random variable on the output by partitioning the total output variance into contributions from each input. However, computing conditional expectations can be prohibitively costly when working with expensive-to-evaluate models. Surrogate models can accelerate this, yet their accuracy depends on the quality and quantity of training data, which is expensive to generate (experimentally or computationally) for complex engineering systems. Thus, methods that work with limited data are desirable. We propose a diffeomorphic modulation under observable response preserving homotopy (D-MORPH) regression to train a polynomial dimensional decomposition surrogate of the output that minimizes the number of training data. The new method first computes a sparse Lasso solution and uses it to define the cost function. A subsequent D-MORPH regression minimizes the difference between the D-MORPH and Lasso solution. The resulting D-MORPH surrogate is more robust to input variations and more accurate with limited training data. We illustrate the accuracy and computational efficiency of the new surrogate for global sensitivity analysis using mathematical functions and an expensive-to-simulate model of char combustion. The new method is highly efficient, requiring only 15% of the training data compared to conventional regression.
The widespread use of maximum Jeffreys'-prior penalized likelihood in binomial-response generalized linear models, and in logistic regression, in particular, are supported by the results of Kosmidis and Firth (2021, Biometrika), who show that the resulting estimates are also always finite-valued, even in cases where the maximum likelihood estimates are not, which is a practical issue regardless of the size of the data set. In logistic regression, the implied adjusted score equations are formally bias-reducing in asymptotic frameworks with a fixed number of parameters and appear to deliver a substantial reduction in the persistent bias of the maximum likelihood estimator in high-dimensional settings where the number of parameters grows asymptotically linearly and slower than the number of observations. In this work, we develop and present two new variants of iteratively reweighted least squares for estimating generalized linear models with adjusted score equations for mean bias reduction and maximization of the likelihood penalized by a positive power of the Jeffreys-prior penalty, which eliminate the requirement of storing $O(n)$ quantities in memory, and can operate with data sets that exceed computer memory or even hard drive capacity. We achieve that through incremental QR decompositions, which enable IWLS iterations to have access only to data chunks of predetermined size. We assess the procedures through a real-data application with millions of observations, and in high-dimensional logistic regression, where a large-scale simulation experiment produces concrete evidence for the existence of a simple adjustment to the maximum Jeffreys'-penalized likelihood estimates that delivers high accuracy in terms of signal recovery even in cases where estimates from ML and other recently-proposed corrective methods do not exist.
Mixtures of factor analysers (MFA) models represent a popular tool for finding structure in data, particularly high-dimensional data. While in most applications the number of clusters, and especially the number of latent factors within clusters, is mostly fixed in advance, in the recent literature models with automatic inference on both the number of clusters and latent factors have been introduced. The automatic inference is usually done by assigning a nonparametric prior and allowing the number of clusters and factors to potentially go to infinity. The MCMC estimation is performed via an adaptive algorithm, in which the parameters associated with the redundant factors are discarded as the chain moves. While this approach has clear advantages, it also bears some significant drawbacks. Running a separate factor-analytical model for each cluster involves matrices of changing dimensions, which can make the model and programming somewhat cumbersome. In addition, discarding the parameters associated with the redundant factors could lead to a bias in estimating cluster covariance matrices. At last, identification remains problematic for infinite factor models. The current work contributes to the MFA literature by providing for the automatic inference on the number of clusters and the number of cluster-specific factors while keeping both cluster and factor dimensions finite. This allows us to avoid many of the aforementioned drawbacks of the infinite models. For the automatic inference on the cluster structure, we employ the dynamic mixture of finite mixtures (MFM) model. Automatic inference on cluster-specific factors is performed by assigning an exchangeable shrinkage process (ESP) prior to the columns of the factor loading matrices. The performance of the model is demonstrated on several benchmark data sets as well as real data applications.
Analysis of high-dimensional data, where the number of covariates is larger than the sample size, is a topic of current interest. In such settings, an important goal is to estimate the signal level $\tau^2$ and noise level $\sigma^2$, i.e., to quantify how much variation in the response variable can be explained by the covariates, versus how much of the variation is left unexplained. This thesis considers the estimation of these quantities in a semi-supervised setting, where for many observations only the vector of covariates $X$ is given with no responses $Y$. Our main research question is: how can one use the unlabeled data to better estimate $\tau^2$ and $\sigma^2$? We consider two frameworks: a linear regression model and a linear projection model in which linearity is not assumed. In the first framework, while linear regression is used, no sparsity assumptions on the coefficients are made. In the second framework, the linearity assumption is also relaxed and we aim to estimate the signal and noise levels defined by the linear projection. We first propose a naive estimator which is unbiased and consistent, under some assumptions, in both frameworks. We then show how the naive estimator can be improved by using zero-estimators, where a zero-estimator is a statistic arising from the unlabeled data, whose expected value is zero. In the first framework, we calculate the optimal zero-estimator improvement and discuss ways to approximate the optimal improvement. In the second framework, such optimality does no longer hold and we suggest two zero-estimators that improve the naive estimator although not necessarily optimally. Furthermore, we show that our approach reduces the variance for general initial estimators and we present an algorithm that potentially improves any initial estimator. Lastly, we consider four datasets and study the performance of our suggested methods.
Self-supervised learning (SSL) has proven effective in solving various problems by generating internal supervisory signals. Unsupervised anomaly detection, which faces the high cost of obtaining true labels, is an area that can greatly benefit from SSL. However, recent literature suggests that tuning the hyperparameters (HP) of data augmentation functions is crucial to the success of SSL-based anomaly detection (SSAD), yet a systematic method for doing so remains unknown. In this work, we propose DSV (Discordance and Separability Validation), an unsupervised validation loss to select high-performing detection models with effective augmentation HPs. DSV captures the alignment between an augmentation function and the anomaly-generating mechanism with surrogate losses, which approximate the discordance and separability of test data, respectively. As a result, the evaluation via DSV leads to selecting an effective SSAD model exhibiting better alignment, which results in high detection accuracy. We theoretically derive the degree of approximation conducted by the surrogate losses and empirically show that DSV outperforms a wide range of baselines on 21 real-world tasks.
We compare two different linear dimensionality reduction strategies for the multigroup classification problem: the trace ratio method and Fisher's discriminant analysis. Recently, trace ratio optimization has gained in popularity due to its computational efficiency, as well as the occasionally better classification results. However, a statistical understanding is still incomplete. We study and compare the properties of the two methods. Then, we propose a robust version of the trace ratio method, to handle the presence of outliers in the data. We reinterpret an asymptotic perturbation bound for the solution to the trace ratio, in a contamination setting. Finally, we compare the performance of the trace ratio method and Fisher's discriminant analysis on both synthetic and real datasets, using classical and robust estimators.
Aphids are one of the main threats to crops, rural families, and global food security. Chemical pest control is a necessary component of crop production for maximizing yields, however, it is unnecessary to apply the chemical approaches to the entire fields in consideration of the environmental pollution and the cost. Thus, accurately localizing the aphid and estimating the infestation level is crucial to the precise local application of pesticides. Aphid detection is very challenging as each individual aphid is really small and all aphids are crowded together as clusters. In this paper, we propose to estimate the infection level by detecting aphid clusters. We have taken millions of images in the sorghum fields, manually selected 5,447 images that contain aphids, and annotated each aphid cluster in the image. To use these images for machine learning models, we crop the images into patches and created a labeled dataset with over 151,000 image patches. Then, we implement and compare the performance of four state-of-the-art object detection models.
A fundamental goal of scientific research is to learn about causal relationships. However, despite its critical role in the life and social sciences, causality has not had the same importance in Natural Language Processing (NLP), which has traditionally placed more emphasis on predictive tasks. This distinction is beginning to fade, with an emerging area of interdisciplinary research at the convergence of causal inference and language processing. Still, research on causality in NLP remains scattered across domains without unified definitions, benchmark datasets and clear articulations of the remaining challenges. In this survey, we consolidate research across academic areas and situate it in the broader NLP landscape. We introduce the statistical challenge of estimating causal effects, encompassing settings where text is used as an outcome, treatment, or as a means to address confounding. In addition, we explore potential uses of causal inference to improve the performance, robustness, fairness, and interpretability of NLP models. We thus provide a unified overview of causal inference for the computational linguistics community.
Graph Neural Networks (GNNs) have been shown to be effective models for different predictive tasks on graph-structured data. Recent work on their expressive power has focused on isomorphism tasks and countable feature spaces. We extend this theoretical framework to include continuous features - which occur regularly in real-world input domains and within the hidden layers of GNNs - and we demonstrate the requirement for multiple aggregation functions in this context. Accordingly, we propose Principal Neighbourhood Aggregation (PNA), a novel architecture combining multiple aggregators with degree-scalers (which generalize the sum aggregator). Finally, we compare the capacity of different models to capture and exploit the graph structure via a novel benchmark containing multiple tasks taken from classical graph theory, alongside existing benchmarks from real-world domains, all of which demonstrate the strength of our model. With this work, we hope to steer some of the GNN research towards new aggregation methods which we believe are essential in the search for powerful and robust models.
With the rapid increase of large-scale, real-world datasets, it becomes critical to address the problem of long-tailed data distribution (i.e., a few classes account for most of the data, while most classes are under-represented). Existing solutions typically adopt class re-balancing strategies such as re-sampling and re-weighting based on the number of observations for each class. In this work, we argue that as the number of samples increases, the additional benefit of a newly added data point will diminish. We introduce a novel theoretical framework to measure data overlap by associating with each sample a small neighboring region rather than a single point. The effective number of samples is defined as the volume of samples and can be calculated by a simple formula $(1-\beta^{n})/(1-\beta)$, where $n$ is the number of samples and $\beta \in [0,1)$ is a hyperparameter. We design a re-weighting scheme that uses the effective number of samples for each class to re-balance the loss, thereby yielding a class-balanced loss. Comprehensive experiments are conducted on artificially induced long-tailed CIFAR datasets and large-scale datasets including ImageNet and iNaturalist. Our results show that when trained with the proposed class-balanced loss, the network is able to achieve significant performance gains on long-tailed datasets.
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