Neural Network in R, Neural Network is just like a human nervous system, which is made up of interconnected neurons, in other words, a neural network is made up of interconnected information processing units. The neural network draws from the parallel processing of information, which is the strength of this method. A neural network helps us to extract meaningful information and detect hidden patterns from complex data sets. A neural network is considered one of the most powerful techniques in the data science world. This method is developed to solve problems that are easy for humans and difficult for machines.

Learning exists in the context of data, yet notions of confidence typically focus on model predictions, not label quality. Confident learning (CL) is an alternative approach which focuses instead on label quality by characterizing and identifying label errors in datasets, based on the principles of pruning noisy data, counting with probabilistic thresholds to estimate noise, and ranking examples to train with confidence. Whereas numerous studies have developed these principles independently, here, we combine them, building on the assumption of a class-conditional noise process to directly estimate the joint distribution between noisy (given) labels and uncorrupted (unknown) labels. This results in a generalized CL which is provably consistent and experimentally performant. We present sufficient conditions where CL exactly finds label errors, and show CL performance exceeding seven recent competitive approaches for learning with noisy labels on the CIFAR dataset. Uniquely, the CL framework is not coupled to a specific data modality or model (e.g., we use CL to find several label errors in the presumed error-free MNIST dataset and improve sentiment classification on text data in Amazon Reviews). We also employ CL on ImageNet to quantify ontological class overlap (e.g., estimating 645 missile images are mislabeled as their parent class projectile), and moderately increase model accuracy (e.g., for ResNet) by cleaning data prior to training. These results are replicable using the open-source cleanlab release.

We consider the problem of forecasting the daily number of hospitalized COVID-19 patients at a single hospital site, in order to help administrators with logistics and planning. We develop several candidate hierarchical Bayesian models which directly capture the count nature of data via a generalized Poisson likelihood, model time-series dependencies via autoregressive and Gaussian process latent processes, and can share statistical strength across related sites. We demonstrate our approach on public datasets for 8 hospitals in Massachusetts, U.S.A. and 10 hospitals in the United Kingdom. Further prospective evaluation compares our approach favorably to baselines currently used by stakeholders at 3 related hospitals to forecast 2-week-ahead demand by rescaling state-level forecasts. The COVID-19 pandemic has created unprecedented demand for limited hospital resources across the globe. Emergency resource allocation decisions made by hospital administrators (such as planning additional personnel or provisioning beds and equipment) are crucial for achieving successful patient outcomes and avoiding overwhelmed capacity. However, at present hospitals often lack the ability to forecast what will be needed at their site in coming weeks. This may be especially true in under-resourced hospitals, due to constraints on funding, staff time and expertise, and other issues.

Probability theory is far from being the most general mathematical theory of uncertainty. A number of arguments point at its inability to describe second-order ('Knightian') uncertainty. In response, a wide array of theories of uncertainty have been proposed, many of them generalisations of classical probability. As we show here, such frameworks can be organised into clusters sharing a common rationale, exhibit complex links, and are characterised by different levels of generality. Our goal is a critical appraisal of the current landscape in uncertainty theory.

Uncertainty quantification provides quantitative measures on the reliability of candidate solutions of ill-posed inverse problems. Due to their sequential nature, Monte Carlo sampling methods require large numbers of sampling steps for accurate Bayesian inference and are often computationally infeasible for large-scale inverse problems, such as seismic imaging. Our main contribution is a data-driven variational inference approach where we train a normalizing flow (NF), a type of invertible neural net, capable of cheaply sampling the posterior distribution given previously unseen seismic data from neighboring surveys. To arrive at this result, we train the NF on pairs of low- and high-fidelity migrated images. In our numerical example, we obtain high-fidelity images from the Parihaka dataset and low-fidelity images are derived from these images through the process of demigration, followed by adding noise and migration. During inference, given shot records from a new neighboring seismic survey, we first compute the reverse-time migration image. Next, by feeding this low-fidelity migrated image to the NF we gain access to samples from the posterior distribution virtually for free. We use these samples to compute a high-fidelity image including a first assessment of the image's reliability. To our knowledge, this is the first attempt to train a conditional network on what we know from neighboring images to improve the current image and assess its reliability.

This paper presents a Gaussian process (GP) model for estimating piecewise continuous regression functions. In scientific and engineering applications of regression analysis, the underlying regression functions are piecewise continuous in that data follow different continuous regression models for different regions of the data with possible discontinuities between the regions. However, many conventional GP regression approaches are not designed for piecewise regression analysis. We propose a new GP modeling approach for estimating an unknown piecewise continuous regression function. The new GP model seeks for a local GP estimate of an unknown regression function at each test location, using local data neighboring to the test location. To accommodate the possibilities of the local data from different regions, the local data is partitioned into two sides by a local linear boundary, and only the local data belonging to the same side as the test location is used for the regression estimate. This local split works very well when the input regions are bounded by smooth boundaries, so the local linear approximation of the smooth boundaries works well. We estimate the local linear boundary jointly with the other hyperparameters of the GP model, using the maximum likelihood approach. Its computation time is as low as the local GP's time. The superior numerical performance of the proposed approach over the conventional GP modeling approaches is shown using various simulated piecewise regression functions.

In critical situations involving discrimination, gender inequality, economic damage, and even the possibility of casualties, machine learning models must be able to provide clear interpretations for their decisions. Otherwise, their obscure decision-making processes can lead to socioethical issues as they interfere with people's lives. In the aforementioned sectors, random forest algorithms strive, thus their ability to explain themselves is an obvious requirement. In this paper, we present LionForests, which relies on a preliminary work of ours. LionForests is a random forest-specific interpretation technique, which provides rules as explanations. It is applicable from binary classification tasks to multi-class classification and regression tasks, and it is supported by a stable theoretical background. Experimentation, including sensitivity analysis and comparison with state-of-the-art techniques, is also performed to demonstrate the efficacy of our contribution. Finally, we highlight a unique property of LionForests, called conclusiveness, that provides interpretation validity and distinguishes it from previous techniques.

Application of deep neural networks to medical imaging tasks has in some sense become commonplace. Still, a "thorn in the side" of the deep learning movement is the argument that deep networks are somehow prone to overfitting and are thus unable to generalize well when datasets are small. The claim is not baseless and likely stems from the observation that PAC bounds on generalization error are usually so large for deep networks that they are vacuous (i.e., logically meaningless). Contrary to this, recent advances using the PAC-Bayesian framework have instead shown non-vacuous bounds on generalization error for large (stochastic) networks and standard datasets (e.g., MNIST and CIFAR-10). We apply these techniques to a much smaller medical imagining dataset (the ISIC 2018 challenge set). Further, we consider generalization of deep networks on segmentation tasks which has not commonly been done using the PAC-Bayesian framework. Importantly, we observe that the resultant bounds are also non-vacuous despite the sharp reduction in sample size. In total, our results demonstrate the applicability of PAC-Bayesian bounds for deep stochastic networks in the medical imaging domain.

B\'ezier simplex fitting algorithms have been recently proposed to approximate the Pareto set/front of multi-objective continuous optimization problems. These new methods have shown to be successful at approximating various shapes of Pareto sets/fronts when sample points exactly lie on the Pareto set/front. However, if the sample points scatter away from the Pareto set/front, those methods often likely suffer from over-fitting. To overcome this issue, in this paper, we extend the B\'ezier simplex model to a probabilistic one and propose a new learning algorithm of it, which falls into the framework of approximate Bayesian computation (ABC) based on the Wasserstein distance. We also study the convergence property of the Wasserstein ABC algorithm. An extensive experimental evaluation on publicly available problem instances shows that the new algorithm converges on a finite sample. Moreover, it outperforms the deterministic fitting methods on noisy instances.

Due to the scarcity and often high acquisition cost of labelled data, machine learning methods that make effective use of large quantities of unlabelled data are being increasingly used. One such method is semi-supervised learning (SSL) where, in addition to labelled data, possibly large numbers of unlabelled observations are available at the time of the construction of the classification rule (classifier) to be used. Not surprisingly, semisupervised learning approaches have been gaining much attention in both the application oriented and the theoretical machine learning communities. However, theoretical analysis of SSL has so far been scarce. But last year, Ahfock and McLachlan (2020) provided an asymptotic basis on how to increase in certain situations the accuracy of the commonly used linear discriminant function formed from a partially classified sample as in SSL (Ahfock and McLachlan, 2020). The increase in accuracy can be of sufficient magnitude for this SSL-based classifier to have smaller error rate than that if it were formed from a completely classified sample.