Time Warping Distance Research Articles

Building electricity load forecasting based on spatiotemporal correlation and electricity consumption behavior information

Accurate prediction of building electricity load is essential for grid management and building optimization operations. This paper proposes a novel approach based on spatiotemporal correlations and electricity consumption behavior information. The K-Medoids algorithm and the Derivative Dynamic Time Warping (DDTW) distance are employed to explore the correlation between electricity consumption behaviors among different partitions and floors. Different partitions and floors are clustered and grouped, followed by modifying the adjacency matrix with electricity consumption behaviors. The hybrid model and K-Medoids-LSTM model are proposed separately for clusterable nodes and non-clustered nodes. For clusterable nodes, spatial-temporal features are extracted, trained, and predicted with the hybrid model based on graph neural networks (GNNs) and LSTM models. A K-Medoids-LSTM model based on the K-Medoids algorithm is proposed to predict the electricity load of the non-clustered nodes. To explore the model's practicality, we predicted the building electrical load under different dataset sizes. The model achieves an R2 above 0.89, and the MAE, MSE, and RMSE of the GCN-LSTM and GAT-LSTM models all remain below 0.1, indicating strong predictive capabilities. The results demonstrate that, without relying on other external features, the proposed method can accurately predict the building electricity load for different partitions and floors simultaneously.

A data fusion method for maritime traffic surveillance: The fusion of AIS data and VHF speech information

The more comprehensive information supply for Vessel Traffic Service (VTS), the greater capacity for maritime traffic surveillance. To supply ship intention information contained in very high frequency (VHF) speech for VTS, we propose an automatic identification system (AIS) data and VHF speech information fusion method. In the first part of this method, the feature information of VHF speech is extracted by Bert-BiLSTM-CRF-based entity extraction method, likes location, ship name, and ship intention information, etc. In the second part, we design a spatio-temporal range to filter AIS data, the temporal range is determined by receive time of VHF speech, and the spatial range is determined by the location information extracted from VHF speech. In the third part, by measuring the ship name similarity between AIS data and VHF speech, the barriers to fusion are successfully overcome. The similarity is calculated by combining with Chinese pinyin character similarity and Arabic number similarity and Chinese pinyin characters are encoded by pronunciation characters, calculated by the dynamic time warping (DTW) distance. Through experiments, the method can effectively provide comprehensive information for maritime traffic surveillance, and even when crews misreport ship name within a small deviation, our method still possesses a great fusion capability.

PaFESD: Patterns augmented by Features Epileptic Seizure Detection.

The detection of epileptic seizures can have a significant impact on the patients' quality of life and on their caregivers. In this paper we propose a method for detecting such seizures from electroencephalogram (EEG) data named Patterns augmented by Features Epileptic Seizure Detection (PaFESD). The main novelty of our proposal consists in a detection model that combines EEG signal features with pattern matching. After cleaning the signal and removing artifacts (as eye blinking or muscle movement noise), time-domain and frequency-domain features are extracted to filter out non-seizure regions of the EEG. Jointly, pattern matching based on Dynamic Time Warping (DTW) distance is also leveraged to identify the most discriminative patterns of the seizures, even under scarce training data. The proposed model is evaluated on all patients in the CHB-MIT database, and the results show that it is able to detect seizures with an average F1 score of 98.9%. Furthermore, our approach achieves a F1 score of 100% (no false alarms or missed true seizures) for 20 of the patients (out of 24). Additionally, we automatically detect the most seizure/non-seizure discriminative EEG channel so that a wearable with only two electrodes would suffice to warn patients of seizures.

Enhancing Financial Time Series Prediction with Quantum-Enhanced Synthetic Data Generation: A Case Study on the S&P 500 Using a Quantum Wasserstein Generative Adversarial Network Approach with a Gradient Penalty

This study introduces a novel Quantum Wasserstein Generative Adversarial Network approach with a Gradient Penalty (QWGAN-GP) model that leverages a quantum generator alongside a classical discriminator to synthetically generate time series data. This approach aims to accurately replicate the statistical properties of the S&P 500 index. The synthetic data generated by this model were compared to the original series using various metrics, including Wasserstein distance, Dynamic Time Warping (DTW) distance, and entropy measures, among others. The outcomes demonstrate the model’s robustness, with the generated data exhibiting a high degree of fidelity to the statistical characteristics of the original data. Additionally, this study explores the applicability of the synthetic time series in enhancing prediction models. An LSTM (Long-Short Term Memory)-based model was developed to evaluate the impact of incorporating synthetic data on forecasting accuracy, particularly focusing on general trends and extreme market events. The findings reveal that models trained on a mix of synthetic and real data significantly outperform those trained solely on historical data, improving predictive performance.

ANALYSIS OF RAINFALL IN INDONESIA USING A TIME SERIES-BASED CLUSTERING APPROACH

Indonesia has a tropical climate and has two seasons: dry and rainy. Prolonged drought can cause drought disasters, and rain can cause floods and landslides. According to information from the Meteorology, Climatology, and Geophysics Agency (BMKG), natural disasters such as floods and landslides due to heavy rains have been a severe problem in Indonesia for the past five years. Different regional characteristics can affect the intensity of rain that falls in every province in Indonesia. It can be grouped to determine which provinces have similar characteristics to natural disasters due to rainfall. Later, it can provide information to the government and the public so that they are more aware of natural disasters. So, it is necessary to research and classify provinces in Indonesia for rainfall with cluster analysis. The data used is secondary rainfall data taken from the official BMKG website. Cluster analysis of rainfall in 34 provinces in Indonesia used hierarchical and non-hierarchical methods in this study. The approach that is used in this research limits our clustering of the data. Further research with a machine learning approach is recommended. For the clustering method, the agglomerative hierarchical method includes single, average, and complete linkage. The non-hierarchical method includes k-medoids and fuzzy c-means. The cluster analysis results show that the dynamic time warping (DTW) distance measurement method with the average linkage method has the most optimal cluster results with a silhouette coefficient value of 0.813.

Analyzing the Impact of the Pandemic on Indonesia’s Economic Growth Using Dynamic Time Warping

Asia’s GDP experienced the most drastic decline during the COVID-19 compared to other economic crises. This study collected data on economic indicators for each province/city to observe economic growth in Indonesia, such as Gross Regional Domestic Product (GRDP), unemployment rate, and economic growth. The clustering method on time series data found several provinces/cities with similar economic growth patterns to observe the pandemic's impact on their economies. Knowing the pattern of economic growth will help the regulation holder support provinces with the right policy. For this purpose, we utilized the Dynamic Time Warping (DTW) distance with the k-medoids procedure. The DTW is an algorithm for measuring the similarity between two temporal sequences. The clustering of the three economic indicators had three clusters with the most optimal validation index. Each cluster had almost the same pattern since the trend tended to increase from before the pandemic and then decrease during the pandemic. The decrease in GRDP was less significant than the minimal data on GRDP that happened before the pandemic. Most provinces had negative economic growth during the pandemic, which skyrocketed even for the first quarter of 2023, almost the same as before the pandemic.

Analyzing the Impacts of Land Use and Network Features on Passenger Flow Distribution at Urban Rail Stations from a Classification Perspective

This study employed big data analytics to investigate the impacts of land use and network features on passenger flow distribution at urban rail stations. The aim was to provide decision support for differentiated operational management strategies for various types of rail stations, thereby achieving refined operation and the sustainable development of urban rail systems. First, this study compared clustering results using different similarity measurement functions within the K-means algorithm framework, selecting the optimal similarity measurement function to construct clustering models. Second, factors influencing passenger flow distribution were selected from land use and network features, forming a feature set that when combined with clustering model results, served as input for the XGBoost model to analyze the relationship between various features and the station passenger flow distribution. The case study showed that (1) the clustering results using a dynamic time-warping distance as the similarity measurement function was optimal; (2) the results of the XGBoost model highlighted commercial services and closeness centrality as the most important factors that affected rail station passenger flow distribution; (3) urban rail stations in Nanjing could be categorized into four types: “strong traffic attraction stations”, “balanced traffic attraction stations”, “suburban strong traffic occurrence stations”, and “distant suburban strong traffic occurrence stations”. Differentiated operational and management strategies were developed for these station types. This paper offers a novel approach for enhancing the operational management of urban rail transit, which not only boosts operational efficiency but also aligns with the goals of sustainable development by promoting resource-efficient transportation solutions.

Multi-View Gait Analysis by Temporal Geometric Features of Human Body Parts.

A gait is a walking pattern that can help identify a person. Recently, gait analysis employed a vision-based pose estimation for further feature extraction. This research aims to identify a person by analyzing their walking pattern. Moreover, the authors intend to expand gait analysis for other tasks, e.g., the analysis of clinical, psychological, and emotional tasks. The vision-based human pose estimation method is used in this study to extract the joint angles and rank correlation between them. We deploy the multi-view gait databases for the experiment, i.e., CASIA-B and OUMVLP-Pose. The features are separated into three parts, i.e., whole, upper, and lower body features, to study the effect of the human body part features on an analysis of the gait. For person identity matching, a minimum Dynamic Time Warping (DTW) distance is determined. Additionally, we apply a majority voting algorithm to integrate the separated matching results from multiple cameras to enhance accuracy, and it improved up to approximately 30% compared to matching without majority voting.

Transformer based day-ahead cooling load forecasting of hub airport air-conditioning systems with thermal energy storage

The air conditioning system constitutes more than half of the total energy demand in hub airport buildings. To enhance the energy efficiency and to enable intelligent energy management, it is vital to build an accurate cooling load prediction model. However, the current models face challenges in dealing with dispersed load patterns and lack interpretability when black box approaches are adopted. To tackle these challenges, we propose a novel k-means-Temporal Fusion Transformer (TFT) based hybrid load prediction model. Specifically, the daily load patterns are grouped using an improved k-means clustering method that considers both input feature weights and dynamic time warping (DTW) distances. Additionally, the statistical features of the clustering output are inputted into the TFT. By further incorporating context information, the integration of data between different schema categories is achieved, thus reducing errors that may occur during the transition process. As a result, the prediction performance and interpretability are significantly improved. The Chongqing Jiangbei Airport T3A terminal is used as a case study, and experiments are conducted using cooling data from the No.1 energy station, as well as the airport traffic data and the meteorological station data. Results are compared with other mainstream models, confirming that the proposed day-ahead load forecasting model achieves improvements in several performance indicators, including MAE, MAPE, CV-RMSE, and R2, which are 384 kW, 3%, 5%, and 0.058 respectively.

Pengelompokan Provinsi di Indonesia Menggunakan Time Series Clustering pada Sektor Ekspor Nonmigas

Indonesia's export activities are dominated by non-oil and gas exports consisting of four sectors, namely the processing industry, agriculture, mining, and others. The government must pay attention to non-oil and gas exports for each province because exports can play an essential role in a country's economic growth. This study was conducted to cluster provinces in Indonesia using time series clustering in the non-oil and gas export sector based on data patterns concerning Dynamic Time Warping (DTW) distance. The sectors used in this study are the manufacturing industry sector and the agricultural sector in 34 Indonesian provinces in the period 2017 - 2021. Time series clustering analysis uses the average linkage method with DTW distance and the selection of the optimum number of clusters using the silhouette coefficient method. The results of the analysis in the processing industry sector resulted in 3 optimum clusters, namely cluster 1 consisting of 1 province that has high processing industry exports, cluster 2 consisting of 8 provinces that have medium processing industry exports, and cluster 3 consisting of 25 provinces that have low processing industry exports. As for the agricultural sector, it produces 2 optimum clusters, namely cluster 1 consisting of 5 provinces that have high agricultural industry exports, and cluster 2 consisting of 29 provinces that have low agricultural industry exports. The clustering results in the processing industry sector and the agricultural sectors have a silhouette coefficient value of 0.778 and 0.798, so it is said to have a strong cluster structure.

Discriminating Internal Multiparameter Difference From Shape Dissimilarity Between Voltage Curves of Lithium-Ion Batteries

Discriminating internal multi-parameter difference between cells is a basis of battery screening, diagnosis, consistency assessment and early warning. However, the existing methods based on voltage features can only identify one or two parameter difference sources and usually work under specific conditions. To solve this problem, a modified multidimensional scaling (MDS) framework that incorporates the concept of shape dissimilarity is newly proposed here. First, the shape dissimilarities between voltage curves are presented by derivative dynamic time warping (DDTW) distances. Then, by utilizing MDS, the dissimilarities are converted into the relative locations between points in two-dimensional space. Finally, based on the complementarity of DDTW-MDS and traditional MDS, the parameter difference sources are determined visually on a spatial map. The effectiveness of proposed method is validated on 484 cells with different combinations of three internal parameters including capacity, state of charge and internal resistance, and various working conditions including constant charging, constant discharging and dynamic condition.

Network analysis of the foreign exchange market using minimum spanning trees constructed from the dynamic time warping distance measure

This paper evaluates the changes that occurred in the topological structure of the foreign exchange market due to the COVID-19 pandemic and after the Russian invasion of Ukraine. A network of 17 currencies was analysed from 1.01.2019 to 31.07.2022 in four sub-periods: before the pandemic, in the year of the pandemic outbreak, in the further course of the pandemic, and the year the war started. Dynamic time warping (DTW) distances between pairs of time series for individual currencies were calculated, and based on these distances, minimum spanning trees (MST) were constructed whose topological characteristics were then analysed. The results show that the topological structure of the foreign exchange market varied in the sub-periods studied. The financial crises caused by the COVID-19 pandemic and the Russian invasion of Ukraine affected the FX network in different ways. However, the two strongest monetary clusters (i.e. clusters centred on the USD and EUR, respectively) have still emerged. Comparing the MSTs for the crisis sub-periods, it can be seen that the USD took a dominant position during the pandemic outbreak, while the EUR played a more significant role during the war. This study contributes to the literature by characterising the topological structure of the FX market in specific sub-periods – during the COVID-19 pandemic and after the Russian invasion of Ukraine. Thus, it represents a particularly relevant study of the impact of crises on the global financial market.

Variational Autoencoders for Biomedical Signal Morphology Clustering and Noise Detection.

Accurate estimation of physiological biomarkers using raw waveform data from non-invasive wearable devices requires extensive data preprocessing. An automatic noise detection method in time-series data would offer significant utility for various domains. As data labeling is onerous, having a minimally supervised abnormality detection method for input data, as well as an estimation of the severity of the signal corruptness, is essential. We propose a model-free, time-series biomedical waveform noise detection framework using a Variational Autoencoder coupled with Gaussian Mixture Models, which can detect a range of waveform abnormalities without annotation, providing a confidence metric for each segment. Our technique operates on biomedical signals that exhibit periodicity of heart activities. This framework can be applied to any machine learning or deep learning model as an initial signal validator component. Moreover, the confidence score generated by the proposed framework can be incorporated into different models' optimization to construct confidence-aware modeling. We conduct experiments using dynamic time warping (DTW) distance of segments to validated cardiac cycle morphology. The result confirms that our approach removes noisy cardiac cycles and the remaining signals, classified as clean, exhibit a 59.92% reduction in the standard deviation of DTW distances. Using a dataset of bio-impedance data of 97885 cardiac cycles, we further demonstrate a significant improvement in the downstream task of cuffless blood pressure estimation, with an average reduction of 2.67 mmHg root mean square error (RMSE) of Diastolic Blood pressure and 2.13 mmHg RMSE of systolic blood pressure, with increases of average Pearson correlation of 0.28 and 0.08, with a statistically significant improvement of signal-to-noise ratio respectively in the presence of different synthetic noise sources. This enables burden-free validation of wearable sensor data for downstream biomedical applications.

An anomaly detection method for identifying locations with abnormal behavior of temperature in school buildings

Time series data collected using wireless sensors, such as temperature and humidity, can provide insight into a building’s heating, ventilation, and air conditioning (HVAC) system. Anomalies of these sensor measurements can be used to identify locations of a building that are poorly designed or maintained. Resolving the anomalies present in these locations can improve the thermal comfort of occupants, as well as improve air quality and energy efficiency levels in that space. In this study, we developed a scoring method to identify sensors that shows collective anomalies due to environmental issues. This leads to identifying problematic locations within commercial and institutional buildings. The Dynamic Time Warping (DTW) based anomaly detection method was applied to identify collective anomalies. Then, a score for each sensor was obtained by taking the weighted sum of the number of anomalies, vertical distance to an anomaly point, and dynamic time-warping distance. The weights were optimized using a well-defined simulation study and applying the grid search algorithm. Finally, using a synthetic data set and the results of a case study we could evaluate the performance of our developed scoring method. In conclusion, this newly developed scoring method successfully detects collective anomalies even with data collected over one week, compared to the machine learning models which need more data to train themselves.

Transmission Tower category Identification from Airborne LiDAR Point Clouds based on Shape Curve

Using airborne LiDAR point cloud data to reconstruct the three-dimensional (3D) models of the transmission towers is crucial for ensuring power transmission safety. To enhance the models’ accuracy, knowing the tower categories is essential. However, there are few studies on the identification of tower categories at present. The existing studies have various shortcomings. Aiming at the problem of tower category identification in the application of airborne LiDAR point clouds in a smart grid, a transmission tower category identification method based on the shape curve is proposed. Firstly, a tower curve database is established based on relevant standards. Then, we employ three methods, namely dynamic time warping (DTW) distance, Hausdorff distance, and DTW-Hausdorff distance, to calculate the similarity indexes between the tower point cloud shape curves and those in the database. These indexes are used to determine the tower category. Finally, this method is tested using point clouds from transmission corridors. The experimental results show that the tower identification accuracy of DTW-Hausdorff distance is 94.0%. The tower category identification accuracy can reach 88.0%. The F-score used as the overall evaluation index of the tower identification effect is 90.9%. When employed as a similarity index, the tower identification effect is the best.

The Application of Numerical Measure Variations in K-Means Clustering for Grouping Data

The K-Means Clustering algorithm is commonly used by researchers in grouping data. The main problem in this study was that it has yet to be discovered how optimal the grouping with variations in distance calculations is in K-Means Clustering. The purpose of this research was to compare distance calculation methods with K-Means such as Euclidean Distance, Canberra Distance, Chebychev Distance, Cosine Similarity, Dynamic TimeWarping Distance, Jaccard Similarity, and Manhattan Distance to find out how optimal the distance calculation is in the K-Means method. The best distancecalculation was determined from the smallest Davies Bouldin Index value. This research aimed to find optimal clusters using the K-Means Clustering algorithm with seven distance calculations based on types of numerical measures. This research method compared distance calculation methods in the K-Means algorithm, such as Euclidean Distance, Canberra Distance, Chebychev Distance, Cosine Smilirity, Dynamic Time Warping Distance, Jaccard Smilirity and Manhattan Distance to find out how optimal the distance calculation is in the K-Means method. Determining the best distance calculation can be seen from the smallest Davies Bouldin Index value. The data used in this study was on cosmetic sales at Devi Cosmetics, consisting of cosmetics sales from January to April 2022 with 56 product items. The result of this study was a comparison of numerical measures in the K-Means Clustering algorithm. The optimal cluster was calculating the Euclidean distance with a total of 9 clusters with a DBI value of 0.224. In comparison, the best average DBI value was the calculation of the Euclidean Distance with an average DBI value of 0.265.

Monitoring maize lodging severity based on multi-temporal Sentinel-1 images using Time-weighted Dynamic time Warping

Lodging is one of the most important stresses in maize growth, which affects the yield, quality and mechanical harvesting ability of maize. Monitoring the lodging range and severity of large-scale maize by satellite remote sensing technology can support agricultural insurance claims, planting structure adjustment and field management. The purpose of this study is to investigate the application of Synthetic Aperture Radar (SAR) remote sensing images in monitoring the severity of maize lodging at the county scale. A time series feature dataset was obtained from multi-temporal Sentinel-1 SAR images before and after lodging. The sensitive features of maize lodging severity were screened by Jeffreys-Matusita (J-M) distance. The study compared the performance of three different methods for monitoring maize lodging severity, including Time-weighted Dynamic Time Warping (TwDTW), random forest (RF) and minimum distance (MinD). The results showed that all time phases of SAR images selected in the study had an important contribution to the differentiation of maize lodging severities. Among all features, the VH (Vertical Send and Horizontal Receive) time series features composed of five time-phase had better discrimination for different maize lodging severities. The accuracy of extracting maize lodging severities based on TwDTW was 76% and the Kappa coefficient was 0.70, which was higher than RF and MinD. Thus, by utilizing the multi-time phase features of Sentinel-1, we can effectively monitor the grades of maize lodging at the county scale.

Three-Stage Sampling Algorithm for Highly Imbalanced Multi-Classification Time Series Datasets

To alleviate the data imbalance problem caused by subjective and objective factors, scholars have developed different data-preprocessing algorithms, among which undersampling algorithms are widely used because of their fast and efficient performance. However, when the number of samples of some categories in a multi-classification dataset is too small to be processed via sampling or the number of minority class samples is only one or two, the traditional undersampling algorithms will be less effective. In this study, we select nine multi-classification time series datasets with extremely few samples as research objects, fully consider the characteristics of time series data, and use a three-stage algorithm to alleviate the data imbalance problem. In stage one, random oversampling with disturbance items is used to increase the number of sample points; in stage two, on the basis of the latter operation, SMOTE (synthetic minority oversampling technique) oversampling is employed; in stage three, the dynamic time-warping distance is used to calculate the distance between sample points, identify the sample points of Tomek links at the boundary, and clean up the boundary noise. This study proposes a new sampling algorithm. In the nine multi-classification time series datasets with extremely few samples, the new sampling algorithm is compared with four classic undersampling algorithms, namely, ENN (edited nearest neighbours), NCR (neighborhood cleaning rule), OSS (one-side selection), and RENN (repeated edited nearest neighbors), based on the macro accuracy, recall rate, and F1-score evaluation indicators. The results are as follows: of the nine datasets selected, for the dataset with the most categories and the fewest minority class samples, FiftyWords, the accuracy of the new sampling algorithm was 0.7156, far beyond that of ENN, RENN, OSS, and NCR; its recall rate was also better than that of the four undersampling algorithms used for comparison, corresponding to 0.7261; and its F1-score was 200.71%, 188.74%, 155.29%, and 85.61% better than that of ENN, RENN, OSS, and NCR, respectively. For the other eight datasets, this new sampling algorithm also showed good indicator scores. The new algorithm proposed in this study can effectively alleviate the data imbalance problem of multi-classification time series datasets with many categories and few minority class samples and, at the same time, clean up the boundary noise data between classes.

New pilot protection method based on the current fault component waveform similarity

AbstractIn order to improve the ability of the modular multi‐level converter to quickly clear and isolate faults in the multi‐terminal flexible DC grid, this paper analyzes the characteristics of current faults in the multi‐terminal flexible direct current grid, introduces the comprehensive distance of current fault waveform, and proposes a line protection method based on the similarity of current fault waveform. This protection method firstly uses Euclid Dynamic Time Warping (DTW) distance and entropy weight method to process electrical quantity signals, calculates the comprehensive distance of current fault waveform of two converter stations, and builds the protection criterion based on the similarity of current fault waveform. Then, the fault identification is realized by using the calculated comprehensive distance, and the fault pole identification is realized by using the ratio of comprehensive distance between positive and negative poles. Finally, PSCAD was used to build a simulation model to output the fault data under different fault conditions, and MATLAB was used to process the fault data to verify the protection algorithm. The simulation results show that the proposed method has good resistance to high resistance, and its rapidity and adaptability are further improved. Compared with other similarity methods, this method combines Euclidean DTW distance and entropy method to process the signal, realizes the fault pole identification, and can still operate correctly under the interference of 500 Ω transition resistance and 20 dB noise ratio. The research results will provide a theoretical basis for the safe and reliable operation of flexible DC power grid.

Comparing Levenshtein distance and dynamic time warping in predicting listeners’ judgments of accent distance

Listeners attend to variation in segmental and prosodic cues when judging accent strength. The relative contributions of these cues to perceptions of accentedness in English remains open for investigation, although objective accent distance measures (such as Levenshtein distance) appear to be reliable tools for predicting perceptual distance. Levenshtein distance, however, only accounts for phonemic information in the signal. The purpose of the current study was to examine the relative contributions of phonemic (Levenshtein) and holistic acoustic (dynamic time warping) distances from the local accent to listeners’ accent rankings for nine non-local native and nonnative accents. Listeners (n = 52) ranked talkers on perceived distance from the local accent (Midland American English) using a ladder task for three sentence-length stimuli. Phonemic and holistic acoustic distances between Midland American English and the other accents were quantified using both weighted and unweighted Levenshtein distance measures, and dynamic time warping (DTW). Results reveal that all three metrics contribute to perceived accent distance, with the weighted Levenshtein slightly outperforming the other measures. Moreover, the relative contribution of phonemic and holistic acoustic cues was driven by the speaker's accent. Both nonnative and non-local native accents were included in this study, and the benefits of considering both of these accent groups in studying phonemic and acoustic cues used by listeners is discussed.