Subsample Size Research Articles

The EDGE-CALIFA survey: The role of spiral arms and bars in driving central molecular gas concentrations

Shocks and torques produced by non-axisymmetric structures such as spiral arms and bars may transport gas to galaxy central regions. We test this hypothesis by studying the dependence of the concentration of CO luminosity (CCO) and molecular gas (Cmol) and the star formation rate (CSFR) in the central ∼2 kpc on the strength of non-axisymmetric disk structure using a sample of 57 disk galaxies selected from the EDGE-CALIFA survey. The Cmol is calculated using a CO-to-H2 conversion factor that decreases with higher metallicity and higher stellar surface density. We find that Cmol is systematically 0.22 dex lower than CCO. We confirm that high Cmol and strong non-axisymmetric disk structure are more common in barred galaxies than in unbarred galaxies. However, we find that spiral arms also increase Cmol. We show that there is a good correlation between Cmol and the strength of non-axisymmetric structure (which can be due to a bar, spiral arms, or both). This suggests that the stronger the bars and spirals, the more efficient the galaxy is at transporting cold gas to its center. Despite the small subsample size, the Cmol of the four Seyferts are not significantly reduced compared to inactive galaxies of similar disk structure, implying that the active galactic nucleus feedback in Seyferts may not notably affect the molecular gas distribution in the central ∼2 kpc. We find that CSFR tightly correlates with Cmol in both unbarred and barred galaxies. Likewise, elevated CSFR is found in galaxies with strong disk structure. Our results suggest that the disk structure, either spirals or bars, can transport gas to the central regions, with higher inflow rates corresponding to stronger structure, and consequently boost central star formation. Both spirals and bars play, therefore, an essential role in the secular evolution of disk galaxies.

Estimation of predictive performance in high-dimensional data settings using learning curves

In high-dimensional prediction settings, it remains challenging to reliably estimate the test performance. To address this challenge, a novel performance estimation framework is presented. This framework, called Learn2Evaluate, is based on learning curves by fitting a smooth monotone curve depicting test performance as a function of the sample size. Learn2Evaluate has several advantages compared to commonly applied performance estimation methodologies. Firstly, a learning curve offers a graphical overview of a learner. This overview assists in assessing the potential benefit of adding training samples and it provides a more complete comparison between learners than performance estimates at a fixed subsample size. Secondly, a learning curve facilitates in estimating the performance at the total sample size rather than a subsample size. Thirdly, Learn2Evaluate allows the computation of a theoretically justified and useful lower confidence bound. Furthermore, this bound may be tightened by performing a bias correction. The benefits of Learn2Evaluate are illustrated by a simulation study and applications to omics data.

Few Differences in Sexual Talk by Gender/Sex and Dyad Type: A Retrospective and Daily Diary Study with Couples.

Sexual talk is a type of verbal communication that occurs exclusively during sexual activity and that is specific to the sexual activity itself. Previous research has identified two types of sexual talk: individualistic (i.e., self-focused) and mutualistic (i.e., sharing/partner-focused), which have generally been linked to greater sexual and relationship well-being. Whether sexual talk use varies by gender/sex (i.e., men, women, gender/sex diverse individuals; GSD) or dyad type (i.e., same- vs. mixed-gender/sex) has not been examined. Given initial evidence that the types of sexual talk may contribute differently to sexual and relationship well-being, it is important to identify factors (e.g., gender/sex) that may be associated with the amount of sexual talk used. We examined differences by gender/sex and dyad type in the average sexual talk use among long-term couples (N = 229; 69 same-gender/sex) using retrospective cross-sectional dyadic data. We also examined these differences in the same sample (N = 217) using a 35-day dyadic daily diary study. Retrospectively, but not daily, women reported using more mutualistic talk than men, especially when partnered with a woman. There were no significant gender/sex or dyad type differences in use of individualistic talk retrospectively or daily. Exploratory analyses with the GSD couples suggested that there may be gender/sex and dyad type differences retrospectively and daily, for individualistic and not mutualistic talk; however, these analyses must be interpreted with caution due to the small subsample size of GSD couples.

Is cowpea a ‘women’s crop’ in Mali? Implications for value chain development

Cowpea, which is produced primarily in West Africa, is valued locally for its agronomic benefits in dryland farming, nutritional content, and contribution to the livelihoods of farming families. Many feel that more investment in cowpea research and development is needed for the crop to achieve its economic potential. Cowpea has long been labelled a ‘women’s crop’. We tested whether this is the case in Mali by exploring five indicators with a combination of primary and secondary databases, and interpreting our results in the context of the regional literature. We conclude that, in Mali, cowpea is better characterised as a ‘women’s enterprise’. Men are more likely than women to plant cowpea as either a primary or secondary crop and tend to plant larger areas, with cowpea intercropped. In drier agro-ecologies, women are more likely to grow cowpea than other crops. Although subsample sizes are very small, women cowpea growers appear to earn more on average from selling the harvest from their individual plots than do men. Women represent 99% of traders of processed cowpea products in the open-air markets we surveyed. In the production segment of the value chain, investments to facilitate women’s access to improved cowpea seed and local markets would support their commercial orientation. Investments in women’s trade in processed cowpea products, including the provision of credit, storage, and training in organisational capacity, would enhance their incomes and their livelihood

Subsampling bootstrap in network DEA

Data Envelopment Analysis (DEA), provides an empirical estimation of the production frontier, based on an observed sample of decision making units (DMUs). Except for the single input-single output case, the asymptotic distribution of the DEA estimator can only be approximated through bootstrapping approaches. Therefore, bootstrapping techniques have been widely applied in the DEA literature to make statistical inference for the cases when the production process has a single-stage structure. However, in many cases, the transformation of inputs into outputs has an inner structure that needs to be considered. This paper examines the applicability of the subsampling bootstrap procedure in the approximation of the asymptotic distribution of the DEA estimator when the production process has a network structure, and in the presence of undesirable factors. Evidence on the performance of subsampling bootstrap is obtained through Monte Carlo experiments for the case of two-stage series structures, where overall and stage efficiency estimates are calculated using the additive decomposition approach. Results indicate great sensitivity both to the sample and subsample size, as well as to the data generating process. Subsampling methodology is then applied to construct confidence interval estimates for the overall and stage efficiency scores of railways in 22 European countries, where the railway transport process is decomposed into two stages and the railway noise pollution problem is considered as an undesirable output.

Quantifying uncertainty of subsampling-based ensemble methods under a U-statistic framework

ABSTRACT This paper addresses the problem of variance estimation of predictions obtained from a subsampling-based ensemble estimator, such as subbagging and sub-random forest. We first recognize that a subsampling-based ensemble can be written as an infinite-order U-statistic of degree , where is the subsample size that may depend on the learning sample size n. As a result, one can study the uncertainty of predictions obtained from a subsampling-based ensemble under a U-statistic framework, such as approximating its asymptotic variance. However, existing methods used to estimate the asymptotic variance relies on some regularity conditions. In addition, they tend to yield variance estimations with large bias in finite sample scenarios. Motivated by the work of Wang and Lindsay (2014), we propose to construct an unbiased variance estimator for a subsampling-based ensemble. It is efficient to realize with the help of a partition-resampling scheme. We show by simulation studies that the proposed variance estimator yields better performance in terms of mean, standard deviation, and mean squared error compared to both the infinitesimal jackknife and internal variance estimation methods under either a simple linear regression model or a multivariate adaptive regression splines model. Furthermore, we present how to construct an asymptotic confidence interval for the expected prediction at a given test instance using the proposed variance estimator, and compare its coverage probability to that of competing methods. In the end, we demonstrate the practical applications of the methodology using two real data examples.

Subsampling sparse graphons under minimal assumptions

Summary We study the properties of two subsampling procedures for networks, vertex subsampling and $p$-subsampling, under the sparse graphon model. The consistency of network subsampling is demonstrated under the minimal assumptions of weak convergence of the corresponding network statistics and an expected subsample size growing to infinity more slowly than the number of vertices in the network. Furthermore, under appropriate sparsity conditions, we derive limiting distributions for the nonzero eigenvalues of an adjacency matrix under the sparse graphon model. Our weak convergence result implies the consistency of our subsampling procedures for eigenvalues under appropriate conditions.

Edgeworth expansions for network moments

Network method of moments (Ann. Statist. 39 (2011) 2280–2301) is an important tool for nonparametric network inference. However, there has been little investigation on accurate descriptions of the sampling distributions of network moment statistics. In this paper, we present the first higher-order accurate approximation to the sampling CDF of a studentized network moment by Edgeworth expansion. In sharp contrast to classical literature on noiseless U-statistics, we show that the Edgeworth expansion of a network moment statistic as a noisy U-statistic can achieve higher-order accuracy without nonlattice or smoothness assumptions but just requiring weak regularity conditions. Behind this result is our surprising discovery that the two typically-hated factors in network analysis, namely, sparsity and edgewise observational errors, jointly play a blessing role, contributing a crucial self-smoothing effect in the network moment statistic and making it analytically tractable. Our assumptions match the minimum requirements in related literature. For sparse networks, our theory shows that our empirical Edgeworth expansion and a simple normal approximation both achieve the same gradually depreciating Berry–Esseen-type bound as the network becomes sparser. This result also significantly refines the best previous theoretical result. For practitioners, our empirical Edgeworth expansion is highly accurate and computationally efficient. It is also easy to implement and convenient for parallel computing. We demonstrate the clear advantage of our method by several comprehensive simulation studies. As a byproduct, we also provide a finite-sample analysis of the network jackknife. We showcase three applications of our results in network inference. We prove, to our knowledge, the first theoretical guarantee of higher-order accuracy for some network bootstrap schemes, and moreover, the first theoretical guidance for selecting the subsample size for network subsampling. We also derive a one-sample test and the Cornish–Fisher confidence interval for a given moment with higher-order accurate controls of confidence level and type I error, respectively.

Rates of convergence for random forests via generalized U-statistics

Random forests are among the most popular off-the-shelf supervised learning algorithms. Despite their well-documented empirical success, however, until recently, few theoretical results were available to describe their performance and behavior. In this work we push beyond recent work on consistency and asymptotic normality by establishing rates of convergence for random forests and other supervised learning ensembles. We develop the notion of generalized U-statistics and show that within this framework, random forest predictions can remain asymptotically normal for larger subsample sizes and under weaker conditions than previously established. Moreover, we provide Berry-Esseen bounds in order to quantify the rate at which this convergence occurs, making explicit the roles of the subsample size and the number of trees in determining the distribution of random forest predictions. When these generalized estimators are reduced to their classical U-statistic form, the rates we establish are faster than any available in the existing literature.

Understanding Bias and Variance of Learning-to-Rank Algorithms: An Empirical Framework

ABSTRACT Learning-to-rank (LtR) algorithms are at the heart of modern day information retrieval systems. While a good number of LtR algorithms have been developed and scrutinized over the past decade, theoretical underpinnings of these algorithms are not thoroughly investigated so far. Amongst the theoretical aspects of a supervised learning algorithm, the bias-variance profiles are of utmost importance. In this article we aim to better understand the bias-variance profiles of rank-learning algorithms. Firstly, we formalize the bias and variance from a pointwise perspective where each query-document pair is treated as an individual training example. Secondly, we develop a framework to analyze the variability and systematic error of a rank-learner in terms of its ranking error, i.e., we analyze the bias-variance from a listwise perspective where a query and all of its associated documents are treated as a single training example. After developing the theoretical framework, we move on to test its applicability in practice. In particular, we choose a promising algorithm, namely random forest-based rank-learning algorithms for our investigation. We study the effect of varying an important parameter of the algorithm, namely the sub-sample size used to learn each tree, on bias and variance. Our hypothesis is that as the sub-sample size (per tree) increases, classical bias-variance tradeoff should be observed. On two large LtR benchmark datasets, experimental results show that our hypothesis holds true. We also explain the relative performance of two of the top performing LtR algorithms using their bias and variance profiles. To the best of our knowledge, this article presents the first thorough investigation into bias and variance analysis of rank-learning algorithms.

Impact of irrigation water type and sampling frequency on Microbial Water Quality Profiles required for compliance with U.S. Food Safety Modernization Act Produce Safety Rule standards

The U.S. Food Safety Modernization Act (FSMA) Produce Safety Rule (PSR) requires that farmers generate a Microbial Water Quality Profile (MWQP) from 20 samples per agricultural water source, taken over 2–4 years and five annual samples thereafter. Farmers must use the MWQP to ascertain a geometric mean (GM) of ≤126 CFU/100 mL and statistical threshold value (STV) of ≤410 CFU/100 mL of generic Escherichia coli. Farmers are responsible for collecting samples and paying for testing, incurring a financial and time burden. To determine if testing frequency can be reduced without compromising accuracy, water samples (n = 279) were collected from twelve sites in the U.S. Mid-Atlantic region from 2016 to 2018 comprising tidal brackish river, non-tidal fresh river, pond, vegetable processing, and reclaimed water. The GM and STV were calculated for all sites and water types using all samples, and for multiple sub-samples of <20 from each site and water type. A Monte Carlo simulation was used to determine the proportion of sub-sample sizes that yielded the same determination as the entire sample size of PSR standard compliance. Four sites, two pond and two reclaimed water sites, complied with PSR GM and STV requirements when using the entire sample set. When a water source's calculated GM and STV using the entire sample set hovered close to the PSR thresholds, sub-sample sizes approached the recommended 20 samples to reach a congruent compliance determination. However, 99% agreement was obtained with a sub-sample of five when the absolute difference between the GM and STV from total samples and the PSR thresholds was ≥2.6 and 4.5 log CFU/100 mL E. coli, respectively. These findings suggest that under certain conditions the MWQP may be generated with well below 20 samples, reducing the economic burden on farmers while still maintaining a representative MWQP.

Are sample size and sample preparation for mycotoxin quantitation in grain products getting trivialized?

Sampling and sample preparation (grinding and subsampling) are largest sources of variability that negate precision and accuracy of mycotoxin quantitation, particularly in grains. In general, halving sample or sub-sample (e.g., ground test portion) size doubles variance. Therefore, this paper reports on trends in sample and test portion masses used for the quantification of mycotoxin in maize between 1991 and 2020 by reviewing articles on mycotoxin quantitation in maize (grain and flour) published during this period. About 50% of the articles did not explicitly state the sample mass that was ground. Sample and test portion masses show a significant (p < 0.05) decline over the study period. In addition, over two-thirds of the articles did not specify the type of grinder and sieve sizes used in their analysis. Therefore, our findings suggest that standardized sampling plans with emphasis on laboratory sample size and sample preparation methods for maize are increasingly being overlooked during mycotoxin quantitation and increasing the uncertainty associated with estimating the true mycotoxin concentration in grain lots.

Copula methods for evaluating relative tail forecasting performance

PurposeThe authors apply their method to analyze which portfolios are capable of providing superior performance to those based on the Sharpe ratio (SR).Design/methodology/approachIn this paper the authors illustrate the use of conditional copulas for identifying differences in alternative portfolio performance strategies. The authors analyze which portfolios are capable of providing superior performance to those based on the SR.FindingsThe results show that under the Gaussian copula, both expected tail ratio (ETR) and skewness-kurtosis ratio portfolios exhibit remarkably low correlations respecting the SR portfolio. This means that these two portfolios are different respecting the SR one. The authors also find that copulas which focus on either the upper tail (Gumbel) or the lower tail (Clayton) render significant differences. In short, the copula analysis is useful to understand what kind of equity-screening strategy based on its corresponding performance measure (PM) performs better in relation to the SR portfolio.Practical implicationsCopula methods for evaluating relative tail forecasting performance provide an alternative tool when forecast differences are very small or found non statistically significant through standard tests.Originality/valueOur copula methods to evaluate models' performance differences are significant because when models' performance is rather similar, conclusions on statistical differences, can be defective as they may hinge on the subsample type or size used, leading to inefficient investment decisions. Our method based in copula is novel in this research topic.

Imputation to predict height to crown base for trees with predicted heights

Height-to-crown-base (HTCB) measurements are frequently used as inputs for growth and yield models. They are essential for reliable projections of stand structure over time required for sustainable forest management. Often, HTCB is measured alongside total height (HT) for only a subsample of trees and must be imputed for unobserved measurements in the sample. Typically, HTCB models require HT as a covariate, requiring a “double” imputation for missing HTCB measurements, leading to potentially substantial error propagation issues. We compared the effects of subsample size, imputation method, and use of imputed rather than measured HT on the accuracy of HTCB predictions. HT and HTCB were imputed using a nonlinear fixed-effects model (NFEM), a NFEM with a correction factor estimated using an ordinary least squares (OLS) regression on the subsampled measurements, and a nonlinear mixed-effects model (NMEM) with stand- and plot-level random parameters. Using cross-validated bias and root mean squared error (RMSE), our results indicated that NMEM obtained the smallest RMSE at subsample sizes greater than one, with RMSEs ranging between 2.19 and 2.51 m. However, while NMEM provided generally smaller RMSEs, biases ranging between −0.92 and −0.29 m existed at subsample sizes less than three trees per plot, and −2.86 m when no subsample was available. We observed negative bias when using imputed rather than measured HT for both the correction factor and mixed-effects model that is mitigated to a range of −0.51 m to −0.08 m with subsample sizes of at least two trees. Following this, we recommend using NMEMs for HTCB imputation with at least four trees per plot.

Using 26,000 diary entries to show ovulatory changes in sexual desire and behavior.

[Correction Notice: An Erratum for this article was reported online in Journal of Personality and Social Psychology on Jun 3 2019 (see record 2019-34417-001). In the original article the number of participants available for robustness checks should have been 1,054, not 1,043; this applies to the third sentence in the abstract, the first sentence of the second paragraph in the Participants section, the first sentence of the second paragraph in the Robustness Checks section, and the subsample size of women in Table 3. The correct number of naturally cycling usable data should have been 429, not 421. The correct number of diary days should have been 26,680, not 25,948. The correct percentage of diary days in the fourth sentence in the Exclusion Criteria section should have been 5%. Figure 1 should have included guessing hypotheses (n 40) and long diary interruptions (n 41) as further reasons for exclusion, and an error in the effsize R package led to the reporting of inflated effect sizes for the differences between hormonal contraceptive users and non-users in Table 1. Figure 1, Table 1, and Table 3 have been corrected. All versions of this article have been corrected.] Previous research reported ovulatory changes in women's appearance, mate preferences, extra- and in-pair sexual desire, and behavior, but has been criticized for small sample sizes, inappropriate designs, and undisclosed flexibility in analyses. In the present study, we sought to address these criticisms by preregistering our hypotheses and analysis plan and by collecting a large diary sample. We gathered more than 26,000 usable online self-reports in a diary format from 1,054 women, of which 429 were naturally cycling. We inferred the fertile period from menstrual onset reports. We used hormonal contraceptive users as a quasi-control group, as they experience menstruation, but not ovulation. We probed our results for robustness to different approaches (including different fertility estimates, different exclusion criteria, adjusting for potential confounds, moderation by methodological factors). We found robust evidence supporting previously reported ovulatory increases in extra-pair desire and behavior, in-pair desire, and self-perceived desirability, as well as no unexpected associations. Yet, we did not find predicted effects on partner mate retention behavior, clothing choices, or narcissism. Contrary to some of the earlier literature, partners' sexual attractiveness did not moderate the cycle shifts. Taken together, the replicability of the existing literature on ovulatory changes was mixed. We conclude with simulation-based recommendations for reading the past literature and for designing future large-scale preregistered within-subject studies to understand ovulatory cycle changes and the effects of hormonal contraception. Interindividual differences in the size of ovulatory changes emerge as an important area for further study. (PsycInfo Database Record (c) 2021 APA, all rights reserved).

Optimal architecture of a convolutional neural network to estimate structural responses for safety evaluation of the structures

A convolutional neural network (CNN) is a deep learning algorithm, which can be utilized in various engineering fields due to its superior prediction and classification performance. In recent years, CNN that is known to be outstanding to handle large volumes of data, it is has been in the spotlight to solve the problems of sensor defects and data loss, which may have resulted from the limitations of the current structural health monitoring (SHM) techniques. However, although the architecture of CNN should be constructed differently depending on the characteristics of each problem, there is no rational nor reasonable method for the construction of the architecture. In this regard, this study seeks to propose a technique for constructing an optimal architecture for the effective utilization of CNN in recovery and estimation of measured data dealt in the field of SHM. In the proposed technique, the number of kernels, the kernel size, and the subsampling size are set as the decision variables, among the variables that determine the CNN architecture. To prevent CNN from being trained with bias toward specific datasets, root mean square errors are calculated for each of the training datasets and validation datasets, and set as objective functions, respectively. Then these two objective functions are minimized at the same time. In this case, non-dominated sorting genetic algorithm-II, a multi-objective optimization algorithm, is introduced to minimize these two objective functions. The proposed technique is verified by a numerical study on beam-like structures and an experimental study on reinforced concrete structures. These two studies explore the optimal CNN architecture, which estimates the dynamic strain of the structure, and evaluates the performance of the explored architecture.

SubTSBR to tackle high noise and outliers for data-driven discovery of differential equations

Data-driven discovery of differential equations has been an emerging research topic. We propose a novel algorithm subsampling-based threshold sparse Bayesian regression (SubTSBR) to tackle high noise and outliers. The subsampling technique is used for improving the accuracy of the Bayesian learning algorithm. It has two parameters: subsampling size and the number of subsamples. When the subsampling size increases with fixed total sample size, the accuracy of our algorithm goes up and then down. When the number of subsamples increases, the accuracy of our algorithm keeps going up. We demonstrate how to use our algorithm step by step and compare our algorithm with threshold sparse Bayesian regression (TSBR) for the discovery of differential equations. We show that our algorithm produces better results. We also discuss the merits of discovering differential equations from data and demonstrate how to discover models with random initial and boundary condition as well as models with bifurcations. The numerical examples are: (1) predator-prey model with noise, (2) shallow water equations with outliers, (3) heat diffusion with random initial and boundary condition, and (4) fish-harvesting problem with bifurcations.

Most Likely Optimal Subsampled Markov Chain Monte Carlo

Markov Chain Monte Carlo (MCMC) requires to evaluate the full data likelihood at different parameter values iteratively and is often computationally infeasible for large data sets. This paper proposes to approximate the log-likelihood with subsamples taken according to nonuniform subsampling probabilities, and derives the most likely optimal (MLO) subsampling probabilities for better approximation. Compared with existing subsampled MCMC algorithm with equal subsampling probabilities, the MLO subsampled MCMC has a higher estimation efficiency with the same subsampling ratio. The authors also derive a formula using the asymptotic distribution of the subsampled log-likelihood to determine the required subsample size in each MCMC iteration for a given level of precision. This formula is used to develop an adaptive version of the MLO subsampled MCMC algorithm. Numerical experiments demonstrate that the proposed method outperforms the uniform subsampled MCMC.

Disparities in Electronically Monitored Antiretroviral Adherence and Differential Adherence Predictors in Latinx and Non-Latinx White Persons Living with HIV.

Antiretroviral therapy (ART) adherence is vital for optimal HIV treatment. However, there is limited ART adherence research on the US Latinx population, who are at increased risk for HIV infection and worse HIV health outcomes. This study examined electronically measured ART adherence (Medication Event Monitoring System) and its association with demographic, clinical, neurocognitive, and sociocultural variables in Latinx and non-Latinx white (NLW) persons living with HIV [PLWH (N = 128)]. Latinx participants demonstrated worse adherence than NLW participants (p = 0.04). Linear regressions revealed different predictors of adherence. Among Latinx participants, recent cocaine use, stress, and, unexpectedly, higher US acculturation predicted worse adherence (ps < 0.05). Among NLW participants, recent cocaine use predicted worse adherence (p < 0.05). Intergroup comparisons within the Latinx group were not conducted due to subsample size. Thus, ethnicity, sociocultural variables, and cocaine use are important considerations for ART adherence, and poor ART adherence may be one pathway explaining worse outcomes in Latinx PLWH. Culturally tailored adherence interventions incorporating substance use treatment, acculturation, and stress management are warranted to improve health outcomes.

Hangi Sektörlerde Getiri Öngörülebilirliği İçin Tarihsel Fiyatlar Kullanılabilir Otomatik Portmanteau Testi ile Borsa İstanbul Üzerine Ampirik Bir Çalışma

The aim of this study, to compare return predictability in other words, the weak form of market efficiency of different sectors in Turkey. For this purpose, analyses were carried out by automatic portmanteau test with a 2-year sub-sample size using the daily data between 19.04.2000-07.02.2020 of 19 primary sector indexes within Borsa Istanbul. As a result of the analyses, it was understood that Transportation, Insurance, Electricity, and Metal Products Machinery sectors have higher return predictability periods and therefore their weak form market efficiency is lower than other sectors. In addition, it was determined that the sectors that have the least predictable periods of return, in other words, that have more weak form efficiency than other sectors, are Food Beverage, Banks, Wholesale and Retail Trade, and Wood Paper Printing.