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Univariate continuous distributions are one of the fundamental components on which statistical modelling, ancient and modern, frequentist and Bayesian, multi-dimensional and complex, is based. In this article, I review and compare some of the main general techniques for providing families of typically unimodal distributions on ℝ with one or two, or possibly even three, shape parameters, controlling skewness and/or tailweight, in addition to their all-important location and scale parameters. One important and useful family is comprised of the 'skew-symmetric' distributions brought to prominence by Azzalini. As these are covered in considerable detail elsewhere in the literature, I focus more on their complements and competitors. Principal among these are distributions formed by transforming random variables, by what I call 'transformation of scale'—including two-piece distributions—and by probability integral transformation of nonuniform random variables. I also treat briefly the issues of multi-variate extension, of distributions on subsets of ℝ and of distributions on the circle. The review and comparison is not comprehensive, necessarily being selective and therefore somewhat personal.

Nitrogen-rich heterocyclic energetic compounds (NRHECs) and their salts have witnessed widespread synthesis in recent years. The substantial energy-density content within these compounds can lead to potentially dangerous explosive reactions when subjected to external stimuli such as electrical discharge. Therefore, developing a reliable model for predicting their electrostatic discharge sensitivity (ESD) becomes imperative. This study proposes a novel and straightforward model based on the presence of specific groups (–NH2 or -NH-, −N=N+−O− and –NNO2, -ONO2 or -NO2) under certain conditions to assess the ESD of NRHECs and their salts, employing interpretable structural parameters. Utilizing a comprehensive dataset comprising 54 ESD measurements of NRHECs and their salts, divided into 49/5 training/test sets, the model achieves promising results. The Root Mean Square Error (RMSE), Mean Absolute Error (MAE), and Maximum Error for the training set are reported as 0.16 J, 0.12 J, and 0.5 J, respectively. Notably, the ratios RMSE(training)/RMSE(test), MAE(training)/MAE(test), and Max Error(training)/Max Error(test) are all greater than 1.0, indicating the robust predictive capabilities of the model. The presented model demonstrates its efficacy in providing a reliable assessment of ESD for the targeted NRHECs and their salts, without the need for intricate computer codes or expert involvement. •A reliable model is derived for prediction of electrostatic discharge sensitivity (ESD).•It predicts ESD values of nitrogen-rich heterocyclic energetic compounds (NRHECs) and their salts.•It needs only interpretable structural parameters.•It assesses ESD values of newly proposed NRHECs.•Various statistical parameters are used to confirm the reliability of the new model.

Complex astrophysical systems often exhibit low-scatter relations between observable properties (e.g., luminosity, velocity dispersion, oscillation period). These scaling relations illuminate the underlying physics, and can provide observational tools for estimating masses and distances. Machine learning can provide a fast and systematic way to search for new scaling relations (or for simple extensions to existing relations) in abstract high-dimensional parameter spaces. We use a machine learning tool called symbolic regression (SR), which models patterns in a dataset in the form of analytic equations. We focus on the Sunyaev-Zeldovich flux–cluster mass relation (Y SZ – M), the scatter in which affects inference of cosmological parameters from cluster abundance data. Using SR on the data from the Illustris TNG hydrodynamical simulation, we find a new proxy for cluster mass which combines Y SZ and concentration of ionized gas ( c gas ) : M ∝ Y conc 3 / 5 ≡ Y SZ 3 / 5 ( 1 − c gas ) Y conc reduces the scatter in the predicted M by ∼ 20 – 30% for large clusters (M ≳ 1014 h −1 M ☉), as compared to using just Y SZ. We show that the dependence on c gas is linked to cores of clusters exhibiting larger scatter than their outskirts. Finally, we test Y conc on clusters from CAMELS simulations and show that Y conc is robust against variations in cosmology, subgrid physics, and cosmic variance. Our results and methodology can be useful for accurate multiwavelength cluster mass estimation from upcoming CMB and X-ray surveys like ACT, SO, eROSITA and CMB-S4.

Abstract Learning minimal interpretable models (e.g., decision trees, decision sets, and binary decision diagrams) is computationally challenging, yet increasingly important in high-stakes settings. We use decision trees as a canonical case study, but the proposed structural parameter is solver-agnostic. Recent parameterized-complexity results show fixed-parameter tractability when parameterized by model size s and a data-dependent conflict parameterδ δ , the maximum Hamming disagreement between oppositely labeled examples. We show thatδ δ is highly noise-sensitive: under small relevant support and independent irrelevant features,δ δ typically scales with ambient dimension, makingδ δ -based branching uninformative. We introduce a distribution-aware alternative, the effective conflict widthκ _(τ) κ τ , obtained by restricting conflicts to features whose relevance exceeds a threshold. We instantiate this idea as structure-guided branching (SGB), which branches on relevance-filtered conflict features and safely falls back to fullδ δ -branching. Using conflict-driven branching simulations to isolate search-tree effects, we find thatκ _(τ) κ τ can remain stable as dimension grows and yields substantial reductions in explored search nodes on synthetic data and multiple real datasets. These results suggest structural parameters can improve the noise robustness of exact interpretable learning and can serve as solver-agnostic pruning signals.

Neural networks (NNs) usually hinder any insight into the reasoning behind their predictions. We demonstrate how influence functions can unravel the black box of NN when trained to predict the phases of the one-dimensional extended spinless Fermi-Hubbard model at half-filling. Results provide strong evidence that the NN correctly learns an order parameter describing the quantum transition in this model. We demonstrate that influence functions allow to check that the network, trained to recognize known quantum phases, can predict new unknown ones within the data set. Moreover, we show they can guide physicists in understanding patterns responsible for the phase transition. This method requires no a priori knowledge on the order parameter, has no dependence on the NN's architecture or the underlying physical model, and is therefore applicable to a broad class of physical models or experimental data.

As an important indicator of the surface quality of workpieces, surface roughness has a great impact on production costs and the quality performance of the finished components. Effective surface roughness prediction can not only increase productivity but also reduce costs. However, the current methods for surface roughness prediction have some limitations. On the one hand, the prediction accuracy of classical experimental and statistical-based surface roughness prediction methods is low. On the other hand, the results of deep learning-based surface roughness prediction methods are uninterpretable due to their black-box learning mechanism. Therefore, this paper presents an ensemble learning with a differential evolution algorithm, applies it to the prediction of surface roughness of abrasive water jet polishing (AWJP), and conducts an interpretability analysis to identify key factors contributing to the prediction accuracy of surface roughness. First, we proposed automatically constructing features by an Evolution Forest algorithm to train the base regression models. The differential evolution algorithm with a simplified encoding mechanism was then used to search for the best weighted-ensemble to integrate the base regression models for obtaining highly accurate prediction results. Extensive experiments have been conducted on AWJP to validate the effectiveness of our proposed methods. The results show that the prediction accuracy of our proposed method is higher than the existing machine learning algorithms. In addition, this is the first of its time for the contributions of machining parameters (i.e., features) on surface roughness prediction by using interpretable analysis methods. The analysis results can provide a reference basis for subsequent experiments and studies.

Existing methods for filter pruning mostly rely on specific data-driven paradigms but lack the interpretability. Besides, these approaches usually assign layer-wise compression ratios automatically only under given FLOPs by neural architecture search algorithms or just manually, which are short of efficiency. In this paper, we propose a novel interpretable task-inspired adaptive filter pruning method for neural networks to solve the above problems. First, we treat filters as semantic detectors and develop the task-inspired importance criteria by evaluating correlations between input tasks and feature maps, and observing the information flow through filters between adjacent layers. Second, we refer to the human neurobiological mechanism for the better interpretability, where the retained first layer filters act as individual information receivers. Third, inspired by the phenomenon that each filter has a deterministic impact on FLOPs and network parameters, we provide an efficient adaptive compression ratio allocation strategy based on differentiable pruning approximation under multiple budget constraints, as well as considering the performance objective. The proposed method is validated with extensive experiments on the state-of-the-art neural networks, which significantly outperforms all the existing filter pruning methods and achieves the best trade-off between neural network compression and task performance. With ResNet-50 on ImageNet, our approach reduces 75.49% parameters and 70.90% FLOPs, only suffering from 2.31% performance degradation.

The cavity pressure profile representing the effective molding condition in a cavity is closely related to part quality. Analysis of the effect of the cavity pressure profile on quality requires prior knowledge and understanding of the injection-molding process and polymer materials. In this work, an analysis methodology to examine the effect of the cavity pressure profile on part quality is proposed. The methodology uses the interpretation of a neural network as a metamodel representing the relationship between the cavity pressure profile and the part weight as a quality index. The process state points (PSPs) extracted from the cavity pressure profile were used as the input features of the model. The overall impact of the features on the part weight and the contribution of them on a specific sample clarify the influence of the cavity pressure profile on the part weight. The effect of the process parameters on the part weight and the PSPs supported the validity of the methodology. The influential features and impacts analyzed using this methodology can be employed to set the target points and bounds of the monitoring window, and the contribution of each feature can be used to optimize the injection-molding process.

The conical pick is the most essential tool of excavation machinery such as roadheaders, continuous miners, and shearers for breaking rock in mining and civil engineering operations. For rock cuttability, however, the geometry of conical picks and mechanical parameters of rocks are the most important factors. This study aims to construct an optimized data-driven predictive model to establish a quantitative correlation between strength of rock, geometry of tool, and cutting action data with the mean cutting force (CF). For this purpose, 157 datasets of 47 different materials including rocks, ores, coals, and artificial rocks with uniaxial compressive strength (σc), tensile strength (σt), cone angle (θ), attack angle (γ), cutting depth (d), and mean CF (MCF) are accumulated from the literature. Then, extreme gradient boosting (XGBoost) model is constructed by fine-tuning hyperparameters using grid search, random search, genetic algorithm (GA), particle swarm optimization (PSO), and differential evolution (DE). Based on performance indices that are calculated for each model, i.e., coefficient of determination (R2), root mean square error (RMSE), and mean absolute percentage error (MAE) for the best performed model, i.e., DE-XGboost are R2=0.812, RMSE = 2256.90 N, and MAE = 1313.66 N for training stage and R2=0.875, RMSE = 2104.86 N, and MAE = 1140.42 N for testing stage, respectively. The findings also suggest that using a metaheuristic algorithm to fine-tune the hyperparameters of the XGBoost model can increase prediction accuracy. In the last step, three model interpretation methods viz., the permutation-based variable importance, H-statistic-based variable interaction, and accumulated local effects are applied to sensitivity analysis of the input parameters to predict MCF, providing key insights to model and researchers. The ALE plot demonstrated a complex non-linear relationship between predictors and the response variable. It is revealed that parameters d and θ have the highest and lowest impact on the MCF, respectively. Finally, the successful implementation of this approach provides a solid platform for future studies and can be an alternative to complicated conventional and theoretical methods. HighlightsA predictive model to relate the strength of rock, geometry of tool, and cutting action data to mean cutting force is established.Extreme gradient boosting method is optimized by meta‑heuristic algorithms.Differential evolution outperformed grid search, random search, genetic algorithm, and particle swarm optimization algorithms in tunning processApplying adaptive sampling with a minimum resampling number of five and fivefold CV with three repetitions to validate the results.Application of model interpretation methods to assess the influence, interaction, and relation of parameters with the mean cutting force response.

Temporal knowledge graphs (KGs) have recently attracted increasing attention. The temporal KG forecasting task, which plays a crucial role in such applications as event prediction, predicts future links based on historical facts. However, current studies pay scant attention to the following two aspects. First, the interpretability of current models is manifested in providing reasoning paths, which is an essential property of path-based models. However, the comparison of reasoning paths in these models is operated in a black-box fashion. Moreover, contemporary models utilize separate networks to evaluate paths at different hops. Although the network for each hop has the same architecture, each network achieves different parameters for better performance. Different parameters cause identical semantics to have different scores, so models cannot measure identical semantics at different hops equally. Inspired by the observation that reasoning based on multi-hop paths is akin to answering questions step by step, this paper designs an Interpretable Multi-Hop Reasoning (IMR) framework based on consistent basic models for temporal KG forecasting. IMR transforms reasoning based on path searching into stepwise question answering. In addition, IMR develops three indicators according to the characteristics of temporal KGs and reasoning paths: the question matching degree, answer completion level, and path confidence. IMR can uniformly integrate paths of different hops according to the same criteria; IMR can provide the reasoning paths similarly to other interpretable models and further explain the basis for path comparison. We instantiate the framework based on common embedding models such as TransE, RotatE, and ComplEx. While being more explainable, these instantiated models achieve state-of-the-art performance against previous models on four baseline datasets.