Classical Clustering Research Articles

Investigating 10 Yr of Volcanoacoustic Activity at Tungurahua Volcano, Ecuador, Aided by Machine Learning

Abstract Here, we present the results of applying diverse data processing and machine learning tools to investigate a very large dataset obtained from single station infrasonic recordings from the last 10 yr of the most recent period of explosive activity at Tungurahua volcano, Ecuador. To increase the quality and quantity of information extracted from the large data set and enhance pattern recognition, we combined traditional techniques with more recent ones. We divided the investigation into sequential steps: detection, discrimination, cleaning, and clustering. For the detection step, we tested the classical short-term average/long-term average algorithm and an algorithm specific for explosions detection called “Volcanic INfrasound Explosions Detector Algorithm (VINEDA)” and detected 118,516 events. To clean the detected signals from potential false positives, we used supervised classification that reduced the events to 75,483, and a catalog cleaning procedure using shallow learners including support vector machines, random forests, and a single layer neural network, trained using data from a manual catalog, to a final number of 36,359 events. This led to a sixfold increase in detected explosions compared to the manual catalog. Then, we applied hierarchical clustering to a well-studied time window of activity using two independent difference metrics: dynamic time warping and waveform cross correlation and showed the insights and drawbacks from this approach. We showed that the different techniques were able to reveal repeating and striving events between selected different eruptive phases and associated them to possible changes in eruptive dynamics. Finally, to analyze the whole dataset at once we used a convolutional autoencoder network and obtained similar results to the classical clustering in a fraction of the time. We identified different families of explosions that appeared, sometimes intermittently, and revealed various potentially competing eruptive processes during the whole time period.

Scalable Clustering: Large Scale Unsupervised Learning of Gaussian Mixture Models with Outliers

Clustering is a widely used technique with a long and rich history in a variety of areas. However, most existing algorithms do not scale well to large datasets, or are missing theoretical guarantees of convergence. This paper introduces a provably robust clustering algorithm based on loss minimization that performs well on Gaussian mixture models with outliers. It provides theoretical guarantees that the algorithm obtains high accuracy with high probability under certain assumptions. Moreover, it can also be used as an initialization strategy for k-means clustering. Experiments on real-world large-scale datasets demonstrate the effectiveness of the algorithm when clustering a large number of clusters, and a k-means algorithm initialized by the algorithm outperforms many of the classic clustering methods in both speed and accuracy, while scaling well to large datasets such as ImageNet.

Boost clustering with Gaussian Boson Sampling: a quantum-classical hybrid approach

Gaussian Boson Sampling (GBS) is a recently developed paradigm of quantum computing consisting of sending a Gaussian state through a linear interferometer and then counting the number of photons in each output mode. When the system encodes a symmetric matrix, GBS can be viewed as a tool to sample subgraphs: the most sampled are those with a large number of perfect matchings, and thus are the densest ones. This property has been the foundation of the novel clustering approach we propose in this work, called GBS-based clustering, which relies solely on GBS, without the need of classical algorithms. The GBS-based clustering has been tested on several datasets and benchmarked with two well-known classical clustering algorithms. Results obtained by using a GBS simulator show that on average, our approach outperforms the two classical algorithms in two out of the three chosen metrics, proposing itself as a viable quantum-classical hybrid clustering option.

Clustering of Electric Power Systems into Reliability Zones in Adequacy Assessment. Part 2

The article presents an analysis of clustering methods in relation to the problem of the formation of energy calculation models (ECM) of electric power systems (EPS). Classical clustering methods are considered, such as: the k-means method, the shortest path method, as well as methods for detecting communities in graphs according to given features, which are the most suitable for solving the problem. Based on the analysis, it was found that for graph clustering, the most adequate result can be obtained by applying the Leiden method. The experimental part of the article presents the results of applying the Leiden method for the formation of the ECM of the unified energy system of Siberia.

A novel unsupervised adaptive density-based clustering filter for remaining useful life prediction of bearings

Abstract Constructing the health indicator (HI) and predicting the remaining useful life (RUL) are essential steps in bearing health management. Some prediction methods depend on prior information about HIs, especially when these indicators are generated by deep learning models. However, acquiring such prior information can be challenging in practical applications. This paper introduces a novel unsupervised adaptive density-based clustering filter (UADCF) for RUL prediction of bearings, which operates without the need for prior knowledge. Firstly, a post-hoc interpretation HI model (PIHIM) is proposed to characterize the deep learning constructed HIs from the perspective of what the deep learning has done. Then, leveraging the classical density-based clustering algorithm, we introduce the UADCF for unsupervised estimation of model parameters, which can dynamically adjust density parameters based on the current conditions. Finally, we develop a prediction framework combining PIHIM and UADCF, enabling unsupervised RUL prediction of bearings. The experimental studies validate the effectiveness of the proposed method.

Framework for Regional to Global Extension of Optical Water Types for Remote Sensing of Optically Complex Transitional Water Bodies

Water quality indicator algorithms often separate marine and freshwater systems, introducing artificial boundaries and artifacts in the freshwater to ocean continuum. Building upon the Ocean Colour- (OC) and Lakes Climate Change Initiative (CCI) projects, we propose an improved tool to assess the interactions across river–sea transition zones. Fuzzy clustering methods are used to generate optical water types (OWT) representing spectrally distinct water reflectance classes, occurring within a given region and period (here 2016–2021), which are then utilized to assign membership values to every OWT class for each pixel and seamlessly blend optimal in-water algorithms across the region. This allows a more flexible representation of water provinces across transition zones than classic hard clustering techniques. Improvements deal with expanded sensor spectral band-sets, such as Sentinel-3 OLCI, and increased spatial resolution with Sentinel-2 MSI high-resolution data. Regional clustering was found to be necessary to capture site-specific characteristics, and a method was developed to compare and merge regional cluster sets into a pan-regional representative OWT set. Fuzzy clustering OWT timeseries data allow unique insights into optical regime changes within a lagoon, estuary, or delta system, and can be used as a basis to improve WQ algorithm performance.

Analysis of Arctic Sea Ice Concentration Anomalies Using Spatiotemporal Clustering

The dynamic changes of sea ice exhibit spatial clustering, and this clustering has characteristics extending from its origin, through its development, and to its dissipation. Current research on sea ice change primarily focuses on spatiotemporal variation trends and remote correlation analysis, and lacks an analysis of spatiotemporal evolution characteristics. This study utilized monthly sea ice concentration (SIC) data from the National Snow and Ice Data Center (NSIDC) for the period from 1979 to 2022, utilizing classical spatiotemporal clustering algorithms to analyze the clustering patterns and evolutionary characteristics of SIC anomalies in key Arctic regions. The results revealed that the central-western region of the Barents Sea was a critical area where SIC anomaly evolutionary behaviors were concentrated and persisted for longer durations. The relationship between the intensity and duration of SIC anomaly events was nonlinear. A positive correlation was observed for shorter durations, while a negative correlation was noted for longer durations. Anomalies predominantly occurred in December, with complex evolution happening in April and May of the following year, and concluded in July. Evolutionary state transitions mainly occurred in the Barents Sea. These transitions included shifts from the origin state in the northwestern margin to the dissipation state in the central-north Barents Sea, from the origin state in the central-north to the dissipation state in the central-south, and from the origin state in the northeastern to the dissipation state in the central-south Barents Sea and southeastern Kara Sea. Various evolutionary states were observed in the same area on the southwest edge of the Barents Sea. These findings provide insights into the evolutionary mechanism of sea ice anomalies.

Nanoscale single-vesicle analysis: High-throughput approaches through AI-enhanced super-resolution image analysis

The analysis of membrane vesicles at the nanoscale level is crucial for advancing the understanding of intercellular communication and its implications for health and disease. Despite their significance, the nanoscale analysis of vesicles at the single particle level faces challenges owing to their small size and the complexity of biological fluids. This new vesicle analysis tool leverages the single-molecule sensitivity of super-resolution microscopy (SRM) and the high-throughput analysis capability of deep-learning algorithms. By comparing classical clustering methods (k-means, DBSCAN, and SR-Tesseler) with deep-learning-based approaches (YOLO, DETR, Deformable DETR, and Faster R–CNN) for the analysis of super-resolution fluorescence images of exosomes, we identified the deep-learning algorithm, Deformable DETR, as the most effective. It showed superior accuracy and a reduced processing time for detecting individual vesicles from SRM images. Our findings demonstrate that image-based deep-learning-enhanced methods from SRM images significantly outperform traditional coordinate-based clustering techniques in identifying individual vesicles and resolving the challenges related to misidentification and computational demands. Moreover, the application of the combined Deformable DETR and ConvNeXt-S algorithms to differently labeled exosomes revealed its capability to differentiate between them, indicating its potential to dissect the heterogeneity of vesicle populations. This breakthrough in vesicle analysis suggests a paradigm shift towards the integration of AI into super-resolution imaging, which is promising for unlocking new frontiers in vesicle biology, disease diagnostics, and the development of vesicle-based therapeutics.

A Novel Hierarchical High-Dimensional Unsupervised Active Learning Method

This paper processes a novel hierarchical high-dimensional clustering algorithm based on the Active Learning Method (ALM), which is a fuzzy-learning algorithm. The hierarchical part of the algorithm is composed of two phases: divisible and agglomerative. The divisible phase, a zooming-in-process, searches for sub-clusters in already-found clusters hierarchically. At each level of the hierarchy, the clusters are found by an ensemble clustering method based on the density of data. This part of the algorithm blurs each data point as multiple one-dimensional fuzzy membership functions called ink-drop patterns; then, it accumulates the ink-drop patterns of all data points on every dimension separately. Next, it performs one-dimensional density partitioning to produce an ensemble of clustering solutions; after that, combining the results is done based on a novel consensus method with the aid of prime numbers. An agglomerative phase is a bottom-up approach that merges clusters based on a novel distance metric, named K2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${K}^{2}$$\\end{document}-nearest neighbor. The algorithm is named as the Hierarchical High-Dimensional Unsupervised Active Learning Method (HiDUALM) and is explained in more detail throughout this paper. Although the classical clustering methods are not suitable for high-dimensional data clustering, the proposed method solves the problems related to speed and memory using ensemble learning, while due to its hierarchy and the use of different distance criteria, different levels of the cluster provide the clause. Experiments on synthetic and real-world datasets are presented to show the effectiveness of the proposed-clustering algorithm.

Grouping Districts / Cities in Central Sulawesi Province Based on Poverty Indicators Using the Fuzzy Geographically Weighted Clustering -Artificial Bee Colony Method

Introduction: Poverty is the main problem that is the focus of attention of the government in Indonesia. In general, poverty is a person's inability to meet basic basic needs in every aspect of life. Cluster analysis is a solution to map this problem. Method: Fuzzy Geographically Weighted Clustering-Artificial Bee Colony (FGWC-ABC) is one clustering method that is an integration of classical fuzzy clustering methods and geodemographic elements. Artificial Bee Colony is a metaheuristic algorithm that is used as a global optimization to increase cluster accuracy. Artificial Bee Colony can efficiently and effectively solve various function optimization problems in various cases. Result and Discussion: The research results obtained 3 optimum clusters with each cluster characteristic relatively different based on poverty indicators. Cluster 1 with low poverty, cluster 2 with high poverty, and cluster 3 with moderate poverty. Conclusion: By using the IFV validity index, 3 optimum clusters were obtained with different characteristics of each cluster based on its indicators. Cluster 1 consists of three regencies/cities with low poverty status, cluster 2 consists of seven regencies/cities with high poverty status, and cluster 3 consists of six regencies/cities with moderate poverty status.

Evolution-aware Constraint Derivation Approach for Software Remodularization

Existing software clustering techniques tend to ignore prior knowledge from domain experts, leading to results (suggested big-bang remodularization actions) that can not acceptable to developers. Incorporating domain experts knowledge or constraints during clustering ensures the obtained modularization aligns with developers’ perspectives, enhancing software quality. However, manual review by knowledgeable domain experts for constraint generation is time-consuming and labor-intensive. In this paper, we propose an evolution-aware constraint derivation approach, Escort, which automatically derives clustering constraints based on the evolutionary history from the analyzed software. Specifically, Escort can serve as an alternative approach to derive implicit and explicit constraints in situations where domain experts are absent. In the subsequent constrained clustering process, Escort can be considered as a framework to help supplement and enhance various unconstrained clustering techniques to improve their accuracy and reliability. We evaluate Escort based on both quantitative and qualitative analysis. In quantitative validation, Escort, using generated clustering constraints, outperforms seven classic unconstrained clustering techniques. Qualitatively, a survey with developers from five IT companies indicates that 89% agree with Escort’s clustering constraints. We also evaluate the utility of refactoring suggestions from our constrained clustering approach, with 54% acknowledged by project developers, either implemented or planned for future releases.

An adaptive kernel dictionary-based low-rank representation method for subspace clustering

Low-rank representation (LRR) is a classic subspace clustering (SC) algorithm, and many LRR-based methods have been proposed. Generally, LRR-based methods use denoized data as dictionaries for data reconstruction purpose. However, the dictionaries used in LRR-based algorithms are fixed, leading to poor clustering performance. In addition, most of these methods assume that the input data are linearly correlated. However, in practice, data are mostly nonlinearly correlated. To address these problems, we propose a novel adaptive kernel dictionary-based LRR (AKDLRR) method for SC. Specifically, to explore nonlinear information, the given data are mapped to the Hilbert space via the kernel technique. The dictionary in AKDLRR is not fixed; it adaptively learns from the data in the kernel space, making AKDLRR robust to noise and yielding good clustering performance. To solve the AKDLRR model, an efficient procedure including an alternative optimization strategy is proposed. In addition, a theoretical analysis of the convergence performance of AKDLRR is presented, which reveals that AKDLRR can converge in at most three iterations under certain conditions. The experimental results show that AKDLRR can achieve the best clustering performance and has excellent speed in comparison with other algorithms.

A New Approach In Metaheuristic Clustering: Coot Clustering

As a result of technological advancements, the increase in vast amounts of data in today's world has made artificial intelligence and data mining significantly crucial. In this context, the clustering process, which aims to explore hidden patterns and meaningful relationships within complex datasets by grouping similar features to conduct more effective analyses, holds vital importance. As an alternative to classical clustering methods that face challenges such as large volumes of data and computational complexities, a metaheuristic clustering method utilizing Coot Optimization (COOT), a swarm intelligence-based algorithm, has been proposed. COOT, inspired by the hunting stages of eagles and recently introduced into the literature, is a metaheuristic method. Through the proposed COOT metaheuristic clustering method, the aim is to contribute to the literature by leveraging COOT's robust exploration and exploitation processes, utilizing its dynamic and flexible structure. Comprehensive experimental clustering studies were conducted to evaluate the consistency and effectiveness of the COOT-based algorithm using randomly generated synthetic data and the widely used Iris dataset in the literature. The same datasets underwent analysis using the traditional clustering algorithm K-Means, renowned for its simplicity and computational speed, for comparative purposes. The performance of the algorithms was assessed using cluster validity measures such as Silhouette Global, Davies-Bouldin, Krznowski-Lai, and Calinski-Harabasz indices, along with the Total Squared Error (SSE) objective function. Experimental results indicate that the proposed algorithm performs clustering at a competitive level with K-Means and shows potential, especially in multidimensional datasets and real-world problems. Despite not being previously used for clustering purposes, the impressive performance of COOT in some tests compared to the K-Means algorithm showcases its success and potential to pioneer different studies aimed at expanding its usage in the clustering domain.

Clustering time series by extremal dependence

AbstractThe goal of this paper is to characterize the temporal dependence structure on the extremes of time series and use such dependency to group them. In particular, three similarity measures to capture extremal dependence are proposed, being their performance assessed in different scenarios. This will involve the use of classical time series clustering algorithms, as well as rigorous evaluation of their performance in both simulated scenarios and real-world time series data sets. The focus will be on comparing the performance of these similarity measures with different clustering methods, and illustrate the efficacy of extremal dependence-based clustering in meteorological data. To achieve this, we will consider a dataset consisting of daily maximum temperatures recorded at 500 stations across Europe.

Estimation curve of multivariate adaptive biresponse fuzzy clustering means regression splines approach to stunting and wasting cases in Southeast Sulawesi

This article offers a new method of clustering data. This method is constructed by combining the multivariate adaptive regression spline biresponse continuous model (MARSBC) with the fuzzy clustering means (FCM) approach, called the multivariate adaptive biresponse fuzzy clustering means regression splines (MABFCMRS) model. This method uses patterns obtained from the MARSBC model to separate data into specific groups. Observing unobserved heterogeneity that has not been obtained from previous models. Unlike the classic fuzzy clustering methods that use euclid distances to determine the weight of the object, this method uses the total square of the massed residual distance generated by the MARSBC model. Theoretical studies were conducted to obtain predictions for the MABFCMRS model parameters. Furthermore, this method was applied to stunting and wasting cases in southeastern Sulawesi province. The results of the research show that in the case of stunting modeling and wasting in southeast Sulawesi province, the best clusters were obtained based on the criteria of partition coefficient (PC) and modification of PC (MPC). This research is able to show that the clustering process using the MABFCMRS model has been able to improve generalized cross-validation (GCV) values and determination coefficients.•This paper presents a new, effective method for clustering data based on unobserved heterogeneity.•This is applicable to sizable samples with 3 to 20 predictors.

Classification of synoptic circulation patterns with a two-stage clustering algorithm using the modified structural similarity index metric (SSIM)

Abstract. We develop a new classification method for synoptic circulation patterns with the aim to extend the evaluation routine for climate simulations. This classification is applicable to any region of the globe of any size given the reference data. Its unique novelty is the use of the modified structural similarity index metric (SSIM) instead of traditional distance metrics for cluster building. This classification method combines two classical clustering algorithms used iteratively, hierarchical agglomerative clustering (HAC) and k-medoids, with only one pre-set parameter – the threshold on the similarity between two synoptic patterns expressed as the structural similarity index measure (SSIM). This threshold is set by the user to imitate the human perception of the similarity between two images (similar structure, luminance, and contrast), whereby the number of final classes is defined automatically. We apply the SSIM-based classification method to the geopotential height at the pressure level of 500 hPa from the ERA-Interim reanalysis data for 1979–2018 and demonstrate that the built classes are (1) consistent with the changes in the input parameter, (2) well-separated, (3) spatially stable, (4) temporally stable, and (5) physically meaningful. We demonstrate an exemplary application of the synoptic circulation classes obtained with the new classification method for evaluating Coupled Model Intercomparison Project Phase 6 (CMIP6) historical climate simulations and an alternative reanalysis (for comparison purposes): output fields of CMIP6 simulations (and of the alternative reanalysis) are assigned to the classes and the Jensen–Shannon distance is computed for the match in frequency, transition, and duration probabilities of these classes. We propose using this distance metric to supplement a set of commonly used metrics for model evaluation.

Multi cascaded transformer network and hybrid heuristic-aided optimal bi-clustering mechanism for patent retrieval system using query expansion

To enhance the reliability of the document retrieval system, the most efficient techniques such as Query Expansion (QE) are utilized. It has offered more adequate queries for the user when assimilated over original or initial queries by adding up one or more expansion keywords. Moreover, these techniques are more effective to enhance the performance of document retrieval and return the unnecessary information. In recent times, searching the suitable documents in the huge datasets is tiresome work. Generally, the automatic QE is used to address the refining query. A typical technique for QE has included the extracted close expression and the related documents clustering by utilizing the clusters. However, classical clustering poses some issues to QE. Hence, a novel optimized bi-clustering mechanism is proposed in this paper for patent retrieval by QE. The ultimate aim of this implemented model is to retrieve the patent information by expanding the request query. Initially, the patent-related data is collected from standard data sources in terms of abstract and text. It is then given to the text pre-processing stage. Consequently, the pre-processed text or word is converted into vector formation by using the Multi-cascade Transformer Network (MTN). Finally, the retrieval process is done by proposing the Optimal Bi-Clustering (OptBi-C) process, in which the parameters are optimally determined by a hybrid algorithm of Reptile Search Algorithm (RSA) and Lion Algorithm (LA) termed as Iteration-based Reptile Search and Lion Algorithm (IRSLA). Thus, the performance of the model is examined with certain metrics and compared with traditional techniques. The precision of the implemented patent retrieval system using the QE model is maximized by 8.82% of DHOA-OptBi-C, 7.35% of HHO-OptBi-C, 10.29% of RSA-OptBi-C, and 7.35% of LA-OptBi-C respectively when the number of retrieved data is 10. Moreover, the recall of the designed patent retrieval system using the QE model is enhanced by 21.83% of KNN, 24.13% of CNN, 19.54% of FUZZY, and 11.49% of Bi-clustering respectively when the number of retrieved data is 6. Thus, the findings demonstrate that the system improves the retrieval performance.

A computational topology-based spatiotemporal analysis technique for honeybee aggregation

A primary challenge in understanding collective behavior is characterizing the spatiotemporal dynamics of the group. We employ topological data analysis to explore the structure of honeybee aggregations that form during trophallaxis, which is the direct exchange of food among nestmates. From the positions of individual bees, we build topological summaries called CROCKER matrices to track the morphology of the group as a function of scale and time. Each column of a CROCKER matrix records the number of topological features, such as the number of components or holes, that exist in the data for a range of analysis scales, at a given point in time. To detect important changes in the morphology of the group from this information, we first apply dimensionality reduction techniques to these matrices and then use classic clustering and change-point detection algorithms on the resulting scalar data. A test of this methodology on synthetic data from an agent-based model of honeybees and their trophallaxis behavior shows two distinct phases: a dispersed phase that occurs before food is introduced, followed by a food-exchange phase during which aggregations form. We then move to laboratory data, successfully detecting the same two phases across multiple experiments. Interestingly, our method reveals an additional phase change towards the end of the experiments, suggesting the possibility of another dispersed phase that follows the food-exchange phase.

ZSGEEHCP: Zonal Stable Gateway Based Energy Efficient Heterogeneous Clustering Protocol For WSN- Based IoT.

In future most of the devices on the earth connected to the internet for making world of intelligent networks. The internet of things (IoT) and Industrial IoT play a very significant role for implementing such kind of smart networks. So, in that case, the transmission of packets of data sent to the base station from sensor nodes with minimum energy consumption in a homogeneous or heterogeneous environment to hike the stability of the network appears as a big challenge for IoT-based networks. But, such kind of problem can be overcome by using a proper clustering algorithm. In this work, different kinds of classical clustering algorithms are studied precisely. Most of the traditional clustering algorithm protocols presume that all the nodes have the same amount of energy, they are unable to fully exploit the presence of node heterogeneity and the same problem will be faced with IoT-based networks. In this specific piece of work, we recommend a gateway based heterogeneous energy-based clustering algorithm which is crucial for many wireless sensor networks along with IoT-based applications. In this concept geographical area is divided into three zone, Advanced, Intermediate and normal zone. Here, advance nodes send data through gateway to the base station while intermediate node use clustering algorithm for transferring data and normal nodes communicate data to the base station directly. ZSGHCP is driven by the mounded election protocol probabilities of each node elected as a cluster head depending on the rest of the energy. By simulation, we show that ZSGHCP invariably extends the stability time as compared to current clustering procedures. Finally, we investigate the sensitivity of our ZSGHCP protocol to network heterogeneity characteristics that capture energy imbalance, throughput, and no of alive nodes. We construct that ZSGHCP payout extends the durability region for excessive values of additional energy brought by extra higher-powered nodes.

Tailored clustering method to identify quasi-regional sites

Classical clustering (CC) normally divides a database into a predetermined number of fixed clusters before assigning each target site into one of these clusters to build a quasi-local model. Because the range of target sites is much larger than the range of clusters, it is possible that the identified cluster contains database sites that are weakly correlated to some assigned target sites, especially when these target sites are located near the cluster boundaries. The existence of weakly correlated sites in the cluster may make the quasi-local model less effective in predicting soil properties at the target site. This paper proposes a Tailored Clustering Enabled Regionalization (TCER) framework that can minimize the inference uncertainty at a target site. It mainly consists of two parts: (1) a site similarity measure to quantify the similarity between database sites and the target site and (2) a novel tailored clustering (TC) approach to identify the optimal cluster (called quasi-regional cluster) of database sites with the highest site similarity measures. In comparison to the traditional TC, the novel TC does not require users to specify a site similarity threshold that divides database sites into those within the cluster and those outside the cluster. Finally, a hierarchical Bayesian model is applied to the quasi-regional cluster (rather than the entire database) to infer geotechnical/geological properties at the target site. The capability of TCER is verified using synthetic and real examples. It is shown that the proposed TCER is computationally efficient and can reduce inference uncertainty.