Clustering Procedure Research Articles

Changes in economic deprivation and parental self-efficacy: Unemployment, poverty, and the mediating effect of psychological distress

Objective: The article investigates the influence of three aspects of economic deprivation on parental self-efficacy, and whether, or to what extent, these effects are mediated by psychological distress. Background: Scholarship shows that psychological resources such as parental self-efficacy can attenuate the negative repercussions of poverty for families. The article builds on the family stress model that higlights the role of economic deprivation for family functioning by Conger and colleagues. It focuses on the consequences of transitioning into unemployment, objective poverty, and subjective poverty for parental self-efficacy, and the potential mediating effect of psychological distress. Method: Using data from the German Family Panel (pairfam) Waves 4 to 12 (2011–2020), Baron and Kenny mediation analyses are conducted based on fixed effects regressions with standard errors based on clustered bootstrap procedure. Results: The results show that transitioning into subjective poverty significantly decreases parental self-efficacy, and that only a marginal part of this effect is mediated by psychological distress. A transition into objective poverty, by contrast, has no effect on parental self-efficacy. Likewise, unemployment has no direct effect on parental self-efficacy. However, transitioning into short-term unemployment reduces psychological distress and thereby buffers parent’s self-efficacy. This does by contrast not apply for long-term unemployment. Conclusion: The findings demonstrate the importance of differentiating between specific aspects of economic deprivation when investigating its consequences for parental self-efficacy and considering psychological distress as a potential mediator. Moreover, the findings provide longitudinal evidence for the consequences of a transition into economic deprivation for parental self-efficacy and point to the presence of gender effects with regard to unemployment.

Artificial Algae Algorithm for Clustering of Benchmark Datasets

The best solution found for a problem under specific circumstances is called optimization. Algorithms for optimization can make the best use of the information at their disposal. Numerous optimization algorithms have been created thus far by researchers, and most of these algorithms are based on the characteristics of naturally occurring biological organisms. Optimization algorithms have proven to be highly effective in numerous fields, including finance, engineering, and medical. Apart from these applications, they have also been employed in data mining techniques including clustering and classification. In many different domains, the clustering method is widely applied. Finding the optimum cluster centers is the most crucial step in the clustering process. In this study, the Artificial Algae Algorithm (AAA) is used to perform the clustering procedure by using 12 datasets that were taken from the UCI Machine Learning Repository. For every dataset, the squared distance values between the cluster centers and the data were computed in order to assess the effectiveness of AAA. The study evaluated AAA's performance against that of the ALO, DEA, MFO, PSO, TSA, and WOA algorithms. The clustering performance of AAA on benchmark datasets was measured to be better than the performance of other algorithms.

Using Existing Clinical Data to Measure Older Adult Inpatients' Frailty at Admission and Discharge: Hospital Patient Register Study.

Frailty is a widespread geriatric syndrome among older adults, including hospitalized older inpatients. Some countries use electronic frailty measurement tools to identify frailty at the primary care level, but this method has rarely been investigated during hospitalization in acute care hospitals. An electronic frailty measurement instrument based on population-based hospital electronic health records could effectively detect frailty, frailty-related problems, and complications as well be a clinical alert. Identifying frailty among older adults using existing patient health data would greatly aid the management and support of frailty identification and could provide a valuable public health instrument without additional costs. We aim to explore a data-driven frailty measurement instrument for older adult inpatients using data routinely collected at hospital admission and discharge. A retrospective electronic patient register study included inpatients aged ≥65 years admitted to and discharged from a public hospital between 2015 and 2017. A dataset of 53,690 hospitalizations was used to customize this data-driven frailty measurement instrument inspired by the Edmonton Frailty Scale developed by Rolfson et al. A 2-step hierarchical cluster procedure was applied to compute e-Frail-CH (Switzerland) scores at hospital admission and discharge. Prevalence, central tendency, comparative, and validation statistics were computed. Mean patient age at admission was 78.4 (SD 7.9) years, with more women admitted (28,018/53,690, 52.18%) than men (25,672/53,690, 47.81%). Our 2-step hierarchical clustering approach computed 46,743 inputs of hospital admissions and 47,361 for discharges. Clustering solutions scored from 0.5 to 0.8 on a scale from 0 to 1. Patients considered frail comprised 42.02% (n=19,643) of admissions and 48.23% (n=22,845) of discharges. Within e-Frail-CH's 0-12 range, a score ≥6 indicated frailty. We found a statistically significant mean e-Frail-CH score change between hospital admission (5.3, SD 2.6) and discharge (5.75, SD 2.7; P<.001). Sensitivity and specificity cut point values were 0.82 and 0.88, respectively. The area under the receiver operating characteristic curve was 0.85. Comparing the e-Frail-CH instrument to the existing Functional Independence Measure (FIM) instrument, FIM scores indicating severe dependence equated to e-Frail-CH scores of ≥9, with a sensitivity and specificity of 0.97 and 0.88, respectively. The area under the receiver operating characteristic curve was 0.92. There was a strong negative association between e-Frail-CH scores at hospital discharge and FIM scores (rs=-0.844; P<.001). An electronic frailty measurement instrument was constructed and validated using patient data routinely collected during hospitalization, especially at admission and discharge. The mean e-Frail-CH score was higher at discharge than at admission. The routine calculation of e-Frail-CH scores during hospitalization could provide very useful clinical alerts on the health trajectories of older adults and help select interventions for preventing or mitigating frailty.

Modulation of Brain Activities in Healthy Individuals by Acupuncture at Quchi (LI11).

This research investigated the modulation of acupuncture at Quchi (LI11) on the brain activities in healthy individuals. Sub-bands power and EEG microstate analysis were carried out at pre-acupuncture, acupuncture, needle retaining and post-acupuncture periods in both the acupuncture group (n = 16) and control group (n = 18). Four microstate classes (A-D) were derived from the clustering procedure. Regression analysis was conducted, together with a two-way repeated measures ANOVA, which was then followed by Bonferroni correction. In the acupuncture group, we found the beta power during the acupuncture periods was significantly reduced. The channel-by-channel analysis revealed that acupuncture at LI11 mainly altered the power of delta, theta, and alpha waves in specific brain regions. The delta power increased predominantly in parietal, occipital, and central lobes, while theta and alpha power decreased predominantly in temporal, frontal, parietal, and occipital lobes. During the acupuncture period, participants in the acupuncture group showed a significant increase in both duration and contribution of microstate A, as well as the bidirectional transition probabilities A and B/D. Microstate analysis showed that acupuncture at LI11 significantly enhances the activity of microstate A and potentially strengthens the functional connectivity between the auditory network and either the visual network or the dorsal attention network. These correlational results indicate that acupuncture at LI11 mainly affects activities of the frontal, temporal, parietal, and occipital lobes. These findings highlight the potential of microstate as neuroimaging evidence and a specific index for elucidating the neuromodulatory effects of acupuncture at LI11.

Social Inequalities and Geographical Distribution in Caries Treatment Needs among Schoolchildren Living in Buenos Aires City: A Cross-Sectional Study.

Caries experience in Argentine children is quite high and no national data are present. This cross-sectional observational survey aimed to assess the oral health status of schoolchildren in Buenos Aires (CABA) using CTNI and to evaluate the ecological association between health inequalities and caries prevalence. Data were collected, via a clustered procedure, during the 2016-2018 school years. Children were examined at school to determine dental status according to the CTNI performed under standardized conditions, using light, dental mirrors, WHO probes and magnification (2.5×). Secondary sources were used to acquire socio-demographic and epidemiological data. A hierarchical cluster analysis was carried out with the municipalities of the CABA as the dependent variable and the socio-demographic factors. The study population consisted of 62,904 schoolchildren, 43,571 of whom were enrolled into the study. Total caries prevalence was 67.78% while severe caries prevalence was 28.29%, which was statistically associated with the year of the survey (p < 0.01). Four territorial conglomerates from a socio-epidemiological standpoint outline homogeneous areas of social risk; cluster 1 had the highest social risk. The prevalence of severe caries (CTNI 7-14) was highest in cluster 1 (44.7-53.2% of the population). The social inequalities between municipalities of the CABA were statistically associated with high inequalities in oral health needs.

Sleep Patterns, Social Media Usage, and Dietary Habits during COVID-19 Lockdown in Mexico: A Cross-Sectional Study.

During the COVID-19 lockdown, a high prevalence of disruption in lifestyle habits was reported, especially concerning sleep habits. The present study aimed to explore the relationship between lifestyles and the COVID-19 lockdown on subjects living in Mexico. A cross-sectional survey was conducted. An online questionnaire was administered to survey participants. A clustering procedure was performed to identify groups of survey respondents with similar characteristics. There were 155 survey respondents. Most of them were women (83%) of young age. The clustering identified two distinct groups of subjects, i.e., Cluster 1 and 2. The subjects in Cluster 2 were younger, more likely to use social media, and spent much more time watching TV/surfing the internet than subjects in Cluster 1. Furthermore, the prevalence of problems with falling and staying asleep during the lockdown was higher in Cluster 2 than in Cluster 1 (14% in Cluster 2 vs. 4% in Cluster 1) even though nobody from Cluster 2 had sleep problems before the lockdown. The study showed that younger respondents were those more affected by lockdown measures.

Dynamics-Adaptive Continual Reinforcement Learning via Progressive Contextualization.

A key challenge of continual reinforcement learning (CRL) in dynamic environments is to promptly adapt the reinforcement learning (RL) agent's behavior as the environment changes over its lifetime while minimizing the catastrophic forgetting of the learned information. To address this challenge, in this article, we propose DaCoRL, that is, dynamics-adaptive continual RL. DaCoRL learns a context-conditioned policy using progressive contextualization, which incrementally clusters a stream of stationary tasks in the dynamic environment into a series of contexts and opts for an expandable multihead neural network to approximate the policy. Specifically, we define a set of tasks with similar dynamics as an environmental context and formalize context inference as a procedure of online Bayesian infinite Gaussian mixture clustering on environment features, resorting to online Bayesian inference to infer the posterior distribution over contexts. Under the assumption of a Chinese restaurant process (CRP) prior, this technique can accurately classify the current task as a previously seen context or instantiate a new context as needed without relying on any external indicator to signal environmental changes in advance. Furthermore, we employ an expandable multihead neural network whose output layer is synchronously expanded with the newly instantiated context and a knowledge distillation regularization term for retaining the performance on learned tasks. As a general framework that can be coupled with various deep RL algorithms, DaCoRL features consistent superiority over existing methods in terms of stability, overall performance, and generalization ability, as verified by extensive experiments on several robot navigation and MuJoCo locomotion tasks.

Двоетапний метод розв’язання задачі комівояжера на основі генетичного алгоритму

One of the main tasks in logistics is to find the most efficient route in the traveling salesman problem on a given transportation network, allowing for the servicing of the maximum number of customers while considering certain criteria. In the typical traveling salesman problem, the objective function is most often the route length or duration. However, such a formulation does not account for the subjectivity in evaluating the travel durations at different stages, which may be influenced by various objective and subjective factors. The problem of the traveling salesman with fuzzily defined travel durations between network nodes is considered. The approach to solving the problem is based on a genetic algorithm with enhanced mutation procedures and population diversity generation. Trapezoidal fuzzy numbers are used to formalize fuzzy values, presented in the generalized case based on the application of Gaussian distribution and corresponding characteristics. A two-stage scheme for solving the optimization problem of the traveling salesman on a given transportation network is proposed, combining the genetic algorithm and Ward's clustering method. In the first stage, a clustering procedure is conducted using Ward's method. In the second stage, based on the obtained set of clusters and the identified optimal inter-cluster distances, route duration optimization is performed within the clusters. To test the efficiency of the developed method, two types of models were generated: one with a limited number of available paths and one with a fully connected topology. Computational experiments were conducted to apply the developed method to solving the traveling salesman problem with travel durations between cities represented as fuzzy numbers. A comparison of numerical calculation results was performed. The obtained results demonstrated the effectiveness of preliminary clustering when using the genetic algorithm to find the best local optima under the same input parameters.

Learning to Score: A Coding System for Constructed Response Items via Interactive Clustering

Constructed response items that require the student to give more detailed and elaborate responses are widely applied in large-scale assessments. However, the hand-craft scoring with a rubric for massive responses is labor-intensive and impractical due to rater subjectivity and answer variability. The automatic response coding method, such as the automatic scoring of short answers, has become a critical component of the learning and assessment system. In this paper, we propose an interactive coding system called ASSIST to efficiently score student responses with expert knowledge and then generate an automatic score classifier. First, the ungraded responses are clustered to generate specific codes, representative responses, and indicator words. The constraint set based on feedback from experts is taken as training data in metric learning to compensate for machine bias. Meanwhile, the classifier from responses to code is trained according to the clustering results. Second, the experts review each coded cluster with the representative responses and indicator words to score a rating. The coded cluster and score pairs will be validated to ensure inter-rater reliability. Finally, the classifier is available for scoring a new response with out-of-distribution detection, which is based on the similarity between response representation and class proxy, i.e., the weight of class in the last linear layer of the classifier. The originality of the system developed stems from the interactive response clustering procedure, which involves expert feedback and an adaptive automatic classifier that can identify new response classes. The proposed system is evaluated on our real-world assessment dataset. The results of the experiments demonstrate the effectiveness of the proposed system in saving human effort and improving scoring performance. The average improvements in clustering quality and scoring accuracy are 14.48% and 18.94%, respectively. Additionally, we reported the inter-rater reliability, out-of-distribution rate, and cluster statistics, before and after interaction.

Clustering and Interpretation of time-series trajectories of chronic pain using evidential c-means

The most well-known unsupervised classification algorithms allow for the identification of hard or probabilistic partitions. However, when working with complex datasets such as those found in the healthcare domain, it is important to note that these partitions may have limitations, particularly when considering the uncertainty associated with medical data. Indeed, these algorithms do not account for modeling outliers, imprecise, or uncertain observations. In this study, we focused on analyzing time series of barometers attributes related to pain intensity in individuals with chronic pain. Our goal was to identify different typologies of care trajectories, highlighting distinct trajectories of chronic pain through clustering techniques. We propose a soft clustering approach based on feature extraction and selection from sequential data, aiming to improve interpretability and enhance the performance of the clustering procedure. Time series feature extraction and selection effectively handles complex and noisy data, improving interpretability and clustering performance. Evidential soft clustering models data uncertainty and imprecision, providing a better representation of nuances and variations in complex, raw data. The first step involves extracting features from the available time series and selecting the most important attributes. For this purpose, we use a time series feature extractor (Tsfresh) and unsupervised feature selection methods, including unsupervised Random Forest, Laplacian Score, and unsupervised Spectral Feature Selection. The second step involves using the evidential c-means (ECM) clustering algorithm on the extracted attributes. The ECM method, based on belief functions, allows for generating a credal partition that has the ability to model various forms of uncertainty. This partition can then be transformed into a hard partition to study individuals based on this uncertainty criterion. The results reveal the existence of two clusters of chronic pain related to discomfort and well-being, exhibiting excellent separability and compactness. Additionally, an uncertain cluster emerges grouping patients with intermediate characteristics. The interpretability of the determined partitions through descriptive analysis, statistical tests, and repeated measures multinomial regression on clinical and demographic data allowed us to determine the profile of patients in the identified trajectories.

Quantum-inspired clustering with light

This article introduces a novel approach to perform the simulation of a single qubit quantum-inspired algorithm using laser beams. Leveraging the polarization states of photonic qubits, and inspired by variational quantum eigensolvers, we develop a variational quantum-inspired algorithm implementing a clustering procedure following the approach proposed by some of us in SciRep 13, 13284 (2023). A key aspect of our research involves the utilization of non-orthogonal states within the photonic domain, harnessing the potential of polarization schemes to reproduce unitary circuits. By mapping these non-orthogonal states into polarization states, we achieve an efficient and versatile quantum information processing unit which serves as a clustering device for a diverse set of datasets.

Recovering nested structures in networks: an evaluation of hierarchical clustering techniques

Abstract In this article, we present various algorithms to partition the nodes of a network into groups that show the property of nestedness. Since perfect nestedness is a rare phenomenon, we consider the task from a data mining perspective, and we search for groups having high-level of nestedness. We utilize both agglomerative and divisive hierarchical clustering procedures and compare them on several benchmark and real-life networks. Furthermore, we propose different metrics derived from the results of our algorithms. We show that average-linkage and complete-linkage clustering can recover the largest fully nested clusters, and that the cluster size-weighted mean nestedness was a more stable metric for measuring clustering performance. Our proposed algorithms allow us to create multiple resolution views of nestedness-based clustering of networks, extending the field of graph-based data mining.

Mixtures of Probit Regression Models with Overlapping Clusters

Studies with binary outcomes on a heterogeneous population are quite common. Typically, the heterogeneity is modelled through varying effect coefficients within some binary regression setting combined with a clustering procedure. Most of the existing methods assign statistical units to distinct and non-overlapping clusters. However, there are scenarios where units exhibit a more complex organization and the clusters can be thought as partially overlapping. In this case, the standard approach does not work. In this paper, we define a mixture of regression models that allows overlapping clusters. This approach involves an overlap function that maps the regression coefficients, either at the unit or response level, of the parent clusters into the coefficients of the multiple allocation clusters. In order to deal with this intrinsic heterogeneity, regression analyses have to be stratified for different groups of observations or clusters. We present a computationally efficient Monte Carlo Markov Chain (MCMC) scheme for the case of a mixture of probit regressions. A simulation study shows the overall performance of the method. We conclude with two illustrative examples of modelling voting behavior, involving United States (US) Supreme Court justices over a number of topics and members of the United Kingdom (UK) parliament over divisions related to Brexit. These applications provide insights on the usefulness of the method in real applications. The method described can be extended to the case of a generic mixture of multivariate generalized linear models under overlapping clusters.

Investigating the Relationship Between the Dynamics of the Structure of Intermediate Consumption and Output of Industries: Cross-Country Analysis

This paper examines one of the main prerequisites of the inter-industry balance methodology on the proportional dependence of the output of industries and their intermediate consumption. The statistical base of the study is the world input-output tables collected for the period from 2000 to 2014. According to the results of the work, it was revealed that the prerequisite of a proportional change in the vector of intermediate consumption with a change in output is not consistent. However, significant positive relationships were found between intermediate consumption and the output of industries. On the basis of the identified relationships, we conducted a clustering procedure on industries, where, in particular, clusters with strong, medium, and weak relationships between intermediate consumption and output are distinguished. Thus, the paper proposes to expand the input-output methodology by introducing proportionality coefficients for changes in the intermediate consumption of an industry with a change in its output. Proportionality coefficients are proposed to be obtained on the basis of regression equations built on statistical data from the input-output tables. It should be especially noted that in addition to obtaining point forecasts of intermediate consumption for each industry when its output changes, it is also possible to obtain probability distributions for the gross output of each industry, their value added, wages, taxes, and profits through Monte Carlo simulations based on the constructed regression equations, which will facilitate a scenario analysis of the development of the considered economic system. The results of this work can be useful to executive authorities in planning large state economic projects as well as in assessing the impact of external shocks on the countryʼs economy.

Optimisation of multiple clustering based undersampling using artificial bee colony: Application to improved detection of obfuscated patterns without adversarial training

Attack detection is one of the main features required in modern defence systems. Despite the ongoing research, it remains challenging for a typical mechanism like network-based intrusion detection system (NIDS) to catch up with evolving adversarial attacks. They specifically aim to confuse a machine-learning based predictor. Without the knowledge of adversarial patterns, the best approach is generalising signatures learned from a dataset of legitimate connections and known intrusions. This work focuses on analysing non-payload traffics so that the resulting techniques can be exploited to a range of network-based applications. It investigates a novel means to deal with the problem of imbalanced classes. An optimised undersampling method is introduced to select a subset of majority-class representatives initially created through an ensemble clustering procedure. A weighted combination of criteria representing distributions within and between classes is proposed as the objective function for a global optimisation using the artificial bee colony (ABC). This approach usually outperforms its baselines and other state-of-the-art undersampling models, with ABC being more effective using the global best strategy than a random selection of solutions or an iterative greedy search. The paper also details the parameter analysis offering a heuristic guide for potential taking up of the proposed techniques.

Distance-based Clustering of Functional Data with Derivative Principal Component Analysis

Functional data analysis (FDA) is an important modern paradigm for handling infinite-dimensional data. An important task in FDA is clustering, which identifies subgroups based on the shapes of measured curves. Considering that derivatives can provide additional useful information about the shapes of functionals, we propose a novel L 2 distance between two random functions by incorporating the functions and their derivative information to determine the dissimilarity of curves under a unified scheme for dense observations. The Karhunen–Loève expansion is used to approximate the curves and their derivatives. Cluster membership prediction for each curve intends to minimize the new distances between the observed and predicted curves through subspace projection among all possible clusters. We provide consistent estimators for the curves, curve derivatives, and the proposed distance. Identifiability issues of the clustering procedure are also discussed. The utility of the proposed method is illustrated via simulation studies and applications to two real datasets. The proposed method can considerably improve cluster performance compared with existing functional clustering methods. Supplementary materials for the article are available online.

ATHENA-FL: Avoiding Statistical Heterogeneity with One-versus-All in Federated Learning

Federated learning (FL) is a distributed approach to train machine learning models without disclosing private data from participating clients to a central server. Nevertheless, FL training struggles to converge when clients have distinct data distributions, which leads to an increased training time and model prediction error. We propose ATHENA-FL, a federated learning system that considers clients with heterogeneous data distributions to generate accurate models in fewer training epochs than state-of-the-art approaches. ATHENA-FL reduces communication costs, providing an additional positive aspect for resource-constrained scenarios. ATHENA-FL mitigates data heterogeneity by introducing a preliminary step before training that clusters clients with similar data distribution. To handle that, we use the weights of a locally trained neural network used as a probe. The proposed system also uses the one-versus-all model to train one binary detector for each class in the cluster. Thus, clients can compose complex models combining multiple detectors. These detectors are shared with all participants through the system's database. We evaluate the clustering procedure using different layers from the neural network and verify that the last layer is sufficient to cluster the clients efficiently. The experiments show that using the last layer as input for the clustering algorithm transmits 99.68% fewer bytes to generate clusters compared to using all the neural network weights. Finally, our results show that ATHENA-FL correctly identifies samples, achieving up to 10.9% higher accuracy than traditional training. Furthermore, ATHENA-FL achieves lower training communication costs compared with MobileNet architecture, reducing the number of transmitted bytes between 25% and 97% across evaluated scenarios.

Impact of Non-Residential Users on the Energy Performance of Renewable Energy Communities Considering Clusterization of Consumptions

Renewable energy communities foster the users’ engagement in the energy transition, paving the way to the integration of distributed renewable energy sources. So far, the scientific literature has focused on residential users in energy communities, thus overlooking the opportunities for industrial and commercial members. This paper seeks to bridge this gap by extending the analysis to the role of non-residential users. The proposed methodology develops an effective clustering approach targeted to actual non-residential consumption profiles. It is based on the k-means algorithm and statistical characterization based on relevant probability density function curves. The employed clusterization procedure allows for effectively reducing a sample of 49 real industrial load profiles up to 11 typical consumption curves, whilst capturing all the relevant characteristics of the initial population. Furthermore, a peer-to-peer sharing strategy is developed accounting for distributed and shared storage. Three scenarios are considered to validate the model with different shares of non-residential users, and the results are then evaluated by means of shared energy, self-consumption, and self-sufficiency indices. Moreover, the results show that the integration of a large non-residential prosumer in a REC may increase the self-sufficiency of residential members by 8.2%, self-consumption by 4.4%, and the overall shared energy by 37.3%. Therefore, the residential community consistently benefits from the presence of non-residential users, with larger users inducing more pronounced effects.

The Geography of Food Insecurity. A Taxonomical Analysis

The problem of food insecurity is worsening in a multi-crisis world simultaneously affected by socio-economic, health, governance and environmental problems. This article characterizes the geography of food insecurity based on six influential theories that explain the emergence of the most dramatic manifestation of food insecurity: famines. We build a taxonomy of 98 developing countries using a clustering procedure and identify four groups of countries with different vulnerabilities. The multidimensional analysis depicts a complex map of the diversity of human vulnerabilities that trigger hunger and famines across the world.

Data Mining Utilizing Various Leveled Clustering Procedures on the Position of Workers in a Data Innovation Firm

The reason of this paper is to clarify dynamic clustering, the divisive and agglomerative dynamic clustering techniques. It fundamentally centers on the concept of the divisive different leveled shapes as well known as the top-down approach by creating a workflow appear, dendrograms, clustered data table which accumulated the clusters based the chosen property, and appear the isolated between each cluster with the assistance of an data mining device called Python. The DIANA dynamic approach utilized data tests of the list of laborers in a Data Advancement firm to induce clusters from the position column inside the data test table. In this work, we in addition executed genuine infers by creating barchart that shows up the ages of the chosen agent sets plotted against the positions which are the Engineers, Assistants, Workers and Troughs.