Benchmark Test Functions Research Articles

Improved Honey Badger Algorithm and Its Application to K-Means Clustering

As big data continues to evolve, cluster analysis still has a place. Among them, the K-means algorithm is the most widely used method in the field of clustering, which can cause unstable clustering results due to the random selection of the initial clustering center of mass. In this paper, an improved honey badger optimization algorithm is proposed: (1) The population is initialized using sin chaos to make the population uniformly distributed. (2) The density factor is improved to enhance the optimization accuracy of the population. (3) A nonlinear inertia weight factor is introduced to prevent honey badger individuals from relying on the behavior of past individuals during position updating. (4) To improve the diversity of solutions, random opposition learning is performed on the optimal individuals. The improved algorithm outperforms the comparison algorithm in terms of performance through experiments on 23 benchmark test functions. Finally, in this paper, the improved algorithm is applied to K-means clustering and experiments are conducted on three data sets from the UCI data set. The results show that the improved honey badger optimized K-means algorithm improves the clustering effect over the traditional K-means algorithm.

Improved dynamic programming method for solving multi-objective and multi-stage decision-making problems

Multi-objective and multi-stage decision-making problems require balancing multiple objectives at each stage and making optimal decision in multi-dimensional control variables, where the commonly used intelligent optimization algorithms suffer from low solving efficiency. To this end, this paper proposes an efficient algorithm named non-dominated sorting dynamic programming (NSDP), which incorporates non-dominated sorting into the traditional dynamic programming method. To improve the solving efficiency and solution diversity, two fast non-dominated sorting methods and a dynamic-crowding-distance based elitism strategy are integrated into the NSDP algorithm. The proposed algorithm has been verified on 12 benchmark test functions and a multi-objective travelling salesman problem. The results demonstrate that the NSDP algorithm achieves better outcome in multiple performance metrics and higher solving efficiency, compared with non-dominated sorting genetic algorithm II and multi-objective particle swarm optimization.

Improved Sparrow Search Algorithm Based on Multistrategy Collaborative Optimization Performance and Path Planning Applications

To address the problems of limited population diversity and a tendency to converge prematurely to local optima in the original sparrow search algorithm (SSA), an improved sparrow search algorithm (ISSA) based on multi-strategy collaborative optimization is proposed. ISSA employs three strategies to enhance performance: introducing one-dimensional composite chaotic mapping SPM to generate the initial sparrow population, thus enriching population diversity; introducing the dung beetle dancing search behavior strategy to strengthen the algorithm’s ability to jump out of local optima; integrating the adaptive t-variation improvement strategy to balance global exploration and local exploitation capabilities. Through experiments with 23 benchmark test functions and comparison with algorithms such as PSO, GWO, WOA, and SSA, the advantages of ISSA in convergence speed and optimization accuracy are verified. In the application of robot path planning, compared with SSA, ISSA exhibits shorter path lengths, fewer turnings, and higher planning efficiency in both single-target point and multi-target point path planning. Especially in multi-target point path planning, as the obstacle rate increases, ISSA can more effectively find the shortest path. Its traversal order is different from that of SSA, making the planned path smoother and with fewer intersections. The results show that ISSA has significant superiority in both algorithm performance and path planning applications.

Arithmetic optimization algorithm with three-dimensional chaotic mapping in spherical coordinate system for combined economic emission dispatch problem

The objective of the Combined Economic Emissions Dispatch (CEED) problem is to reduce pollutant emissions by lowering the total cost of generating electricity while complying with all other constraints. The multiple objective functions of the CEED problem may be converted into a single objective by using a price penalty factor. Then it was solved with the Arithmetic Optimization Algorithm (AOA) with three-dimensional chaotic mapping in spherical coordinate system. Firstly, five three-dimensional chaotic mappings in Cartesian coordinate system are embedded in the Mathematical Optimization Accelerator (MOA) and the Mathematical Optimization Probability (MOP) of the original algorithm. Secondly, 15 three-dimensional chaotic mappings in spherical coordinate system are constructed from the values of 5 three-dimensional chaotic mappings in Cartesian coordinate system through the mathematical expressions of mode length, polar angle and azimuth angle in spherical coordinate system. Then through a large number of simulation experiments, five best three-dimensional chaotic mappings in the spherical coordinate system are selected to be embedded into the two parameters (MOA and MOP) of the original algorithm to better balance the algorithm's global and local searching ability. The superiority of the improved algorithm is verified by employing 12 benchmark test functions in CEC2022. Eventually, the improved optimal algorithm (IAOA-r) to solve the power system CEED problem is tested on six generating units with four different loads (150 MW, 175 MW, 200 MW and 225 MW). The results of the improved algorithm are compared and analyzed with the results of the Reptile Search Algorithm (RSA), Prairie Dog Optimization (PDO), Bat Algorithm (BAT), Whale Optimization Algorithm (WOA), Harris Hawk Optimization (HHO), Rat Swarm Optimization (RSO). The results demonstrate that the improved algorithm is capable of obtaining optimal fuel costs and smaller pollution emissions in every test case.

Optimal Allocation of Urban Water Resources Based on Multi-Objective Nutcracker Optimization Algorithm

The imbalance between water supply and demand (WSD) has been growing noticeable as a result of the economy’s fast expansion which can be effectively alleviated using optimal allocation of water resources. An urban water resources allocation (WRA) model based on the innovative Multi-Objective Nutcracker Optimization Algorithm (MONOA) is proposed in this study. Taking into account economic, social and ecological benefits, a comprehensive multi-objective optimization (MOO) model is established. By introducing the opposite learning strategy, non-dominated sorting approach and crowding distance mechanism to a recently reported intelligent optimization algorithm called the Nutcracker Optimization Algorithm (NOA), the novel nature-inspired metaheuristic algorithm MONOA is proposed to solve the multi-objective optimization model. The MONOA is evaluated on ten benchmark test functions, and it exhibits superior distribution and convergence by comparing with some highly cited algorithms. The proposed model is applied to Handan, China, in order to obtain a reasonable water allocation scheme in the planning year. The simulation results reveal that the economic benefit is in the range CNY [1.36, 1.44] × 1011, water shortage is in the range [0.66, 0.98] × 108 m3 and COD emission is in the range [3.70, 3.91] × 104 t in all the obtained Pareto solutions. The water resources management departments might create customized water allocation plans by balancing different goals and taking preferences into account. Moreover, the proposed method is a general approach that can be applied to many other cities. Hence, it is of great significance to the sustainable development and utilization of urban water resources.

Mixed Multi-Strategy Improved Aquila Optimizer and Its Application in Path Planning

With the growing prevalence of drone technology across various sectors, efficient and safe path planning has emerged as a critical research priority. Traditional Aquila Optimizers, while effective, face limitations such as uneven population initialization, a tendency to get trapped in local optima, and slow convergence rates. This study presents a multi-strategy fusion of the improved Aquila Optimizer, aiming to enhance its performance by integrating diverse optimization techniques, particularly in the context of path planning. Key enhancements include the integration of Bernoulli chaotic mapping to improve initial population diversity, a spiral stepping strategy to boost search precision and diversity, and a “stealing” mechanism from the Dung Beetle Optimization algorithm to enhance global search capabilities and convergence. Additionally, a nonlinear balance factor is employed to dynamically manage the exploration–exploitation trade-off, thereby increasing the optimization of speed and accuracy. The effectiveness of the mixed multi-strategy improved Aquila Optimizer is validated through simulations on benchmark test functions, CEC2017 complex functions, and path planning scenarios. Comparative analysis with seven other optimization algorithms reveals that the proposed method significantly improves both convergence speed and optimization accuracy. These findings highlight the potential of mixed multi-strategy improved Aquila Optimizer in advancing drone path planning performance, offering enhanced safety and efficiency.

A hybrid metaheuristic algorithm for antimicrobial peptide toxicity prediction

The development of new algorithms can aid researchers and professionals in resolving problems that were once unsolvable or discovering superior solutions to problems that were already settled. By recognizing the importance of continuous research on creating novel algorithms, this paper introduced a hybrid metaheuristic algorithm-h-PSOGNDO, which is a combination of Particle Swarm Optimization (PSO) and Generalized Normal Distribution Optimization (GNDO). The proposed algorithm utilizes the Particle Swarm Optimization’s strategy for exploitation and the Generalized Normal Distribution Optimization’s global search strategy for exploration. Through this combination, h-PSOGNDO is believed to be an effective algorithm that can promote the advantages of its parents’ algorithms. Different assessment methods are used to assess the proposed novel algorithm. First, the h-PSOGNDO is set to conduct experiments on two sets of mathematical functions, including twenty-eight IEEE CEC2017 and ten IEEE CEC2019 benchmark test functions, respectively. Then, the h-PSOGNDO algorithm is applied to a case study on the prediction of antimicrobial peptides’ toxicity to evaluate its performance on real-life problems. The statistical findings collected from both the test function sets and the case study show that the h-PSOGNDO algorithm works effectively, proving its astonishing ability to yield highly competitive outcomes for complex problems.

Multi-strategy cooperative enhancement dung beetle optimizer and its application in obstacle avoidance navigation

Path planning is a crucial component of unmanned driving systems designed to address the challenge of autonomous navigation. This article introduces a novel approach called the multi-strategy cooperative enhanced dung beetle optimization algorithm (RCDBO) and applies it to dispose of path planning issues. Its primary aim is to mitigate the shortcomings of the fundamental dung beetle optimization algorithm (DBO), namely, its tendency to prematurely converge to local optima and its limited global planning capability. Initially, population initialization leverages Bernoulli-based chaotic mapping to enhance diversity and ensure uniform randomness, thereby improving both the quality of initial solutions and optimization efficiency. Subsequently, a random walk strategy is employed to perturb the rolling behavior of the dung beetle population during the initial stage, thus mitigating potential algorithmic local stagnation. Finally, adopting a vertical and horizontal crossover strategy introduces perturbations to the current best value of the dung beetle population, thereby enhancing the DBO method’s global optimization capability during the later stages of evolution. The RCDBO method was evaluated against several well-established swarm intelligence algorithms using twelve benchmark test functions, the CEC2021 test suite, the Wilcoxon rank-sum test, and the Friedman test, to rigorously assess its feasibility. Additionally, the effectiveness of each strategy within the RCDBO framework was validated, and the algorithm’s capability to achieve an optimal balance between global exploration and local exploitation was systematically analyzed through the CEC2021 test suite. Furthermore, in path planning simulation experiments targeting maps of varying sizes and complexities, the RCDBO algorithm, compared to the basic DBO algorithm, results in shorter planned path lengths. In simpler map environments, it exhibits fewer turning points and smoother paths, indicating that the RCDBO algorithm also possesses certain advantages in addressing path-planning challenges.

Competition of tribes and cooperation of members algorithm: An evolutionary computation approach for model free optimization

Metaheuristic algorithms are a series of intelligent algorithms that are inspired by natural phenomenon and achieve the optimal searching via utilizing behaviors from nature. The rising complexity and dimensions of practical engineering problem in reality, a significant number of metaheuristic algorithms are promoted and applied in various of fields. Inspired by ancient tribes competition and members cooperative behavior, this paper proposes the Competition of Tribes and Cooperation of Members Algorithm (CTCM). Subsequent experiments are conducted on 23 benchmark test functions and exhaustively compared with other state-of-the-art algorithms, including particle swarm optimization (PSO), grey wolf optimizer (GWO), sparrow search algorithm (SSA), egret swarm optimization (ESOA), beetle antennae search (BAS) and whale optimization (WOA). The standard deviation and average, as well as statistic test are utilized to compare the performance of each algorithms, which reveal that CTCM outperforms in most kinds of problem. And from the result of Wilcoxon and Friedman rank test, the CTCM achieve the first place in all categories of problems, which indicate that CTCM possesses strong global optimization search capability and stability, and has faster convergence speed. The superiority of CTCM is then proofed on practical engineering optimization problems, in which CTCM achieve all the optimal solution for each engineering problem.

Improved Osprey Optimization Algorithm with Multi-Strategy Fusion.

The osprey optimization algorithm (OOA) is an effective metaheuristic algorithm. Although the OOA has the characteristics of strong optimality-seeking ability and fast convergence speed, it also has the disadvantages of imbalance between global exploration and local exploitation ability, easily falling into local optima in the later stage, and reduced population diversity and convergence speed. Therefore, this paper proposes an improved osprey optimization algorithm (IOOA) with multi-strategy fusion. First, Fuch chaotic mapping is used to initialize the ospreys' population and increase the population diversity. Then, an adaptive weighting factor is introduced in the exploration phase of the algorithm to help the algorithm improve the convergence accuracy. The Cauchy variation strategy is integrated in the algorithm's exploitation stage to enhance the diversity of the ospreys' population and avoid falling into local optima. Finally, a Warner mechanism for the sparrow search algorithm is introduced to coordinate the algorithm's local optimization and global search capabilities. The IOOA with various optimization algorithms is tested in a simulation for 10 benchmark test functions and 15 CEC2017 test functions, and non-parametric tests are performed on the IOOA. Experimental results show that the IOOA achieves improved accuracy and stability. The application of the IOOA to the three-bar truss engineering design problem further verifies its superiority in dealing with practical optimization problems.

Improved black-winged kite algorithm based on chaotic mapping and adversarial learning

Abstract Aiming at the shortcomings of the black-winged kite algorithm, such as poor uniformity of initial population distribution, subjective search and single direction, this paper proposes an improved black-winged kite algorithm based on chaotic mapping and adversarial learning. It then progresses to adopting Tent chaotic mapping to achieve a uniform initial solution distribution, introducing Beta random distribution and optimizing the nonlinear factor in the attack phase to make the search trend more in line with the demand, and introducing an adversarial learning mechanism to extend the search direction. Twelve benchmark test functions are tested and compared with five other algorithms, and the result shows that the algorithm significantly outperforms the other algorithms in terms of searching ability and optimization-finding accuracy.

Application of spiral enhanced whale optimization algorithm in solving optimization problems

The Whale Optimization Algorithm (WOA) is regarded as a classic metaheuristic algorithm, yet it suffers from limited population diversity, imbalance between exploitation and exploration, and low solution accuracy. In this paper, we propose the Spiral-Enhanced Whale Optimization Algorithm (SEWOA), which incorporates a nonlinear time-varying self-adaptive perturbation strategy and an Archimedean spiral structure into the original WOA. The Archimedean spiral structure enhances the diversity of the solution space, aiding the algorithm in escaping local optima. The nonlinear time-varying self-adaptive optimization dynamic perturbation strategy improves the algorithm’s local search capability and enhances solution accuracy. The effectiveness of the proposed algorithm is validated from multiple perspectives using CEC2014 test functions, CEC2017 test functions, and 23 benchmark test functions. The experimental results demonstrate that the enhanced Whale Optimization Algorithm significantly improves population diversity, balances global and local search, and enhances solution accuracy. Additionally, SEWOA exhibits excellent performance in solving three engineering design problems, showcasing its value and wide range of potential applications.

Gorilla algorithm based on double random perturbation and its engineering application

Aiming at the problems of artificial gorilla troop optimization algorithm, such as easy to fall into local optimum, slow convergence speed and low optimization accuracy, an artificial gorilla troop optimization algorithm based on double random perturbation strategy is proposed. Firstly, the Halton sequence is introduced to initialize the population to increase the diversity of the population; secondly, a multi-dimensional random number strategy is used in the algorithm optimization stage and an adaptive position search mechanism is proposed in the exploration stage to improve the convergence speed of the algorithm; thirdly, a double random perturbation strategy is proposed to solve the group effect of gorillas and enhance the ability of the algorithm to jump out of the local optimum; finally, a dimension-by-dimensional update strategy is adopted to update individual positions, which improves the convergence accuracy of the algorithm. Through the comparison of the optimization results of 10 benchmark test functions and the Wilcoxon rank sum test, it can be seen that the improved algorithm has greatly improved the optimization accuracy and convergence speed. In addition, through the experimental comparative analysis of an engineering optimization problem, the superiority of the proposed algorithm in dealing with real engineering problems is further verified.

A novel improvement of particle swarm optimization using an improved velocity update function based on local best murmuration particle

Improvement of particle swarm optimization (PSO) is relevant to solving the inherent local optima and premature convergence problem of the PSO. In this paper, a novel improvement of the particle swarm optimization is provided to curb the problem of the classical PSO. The proposed improvement modifies the updating velocity function of the PSO, and it uses a local best murmuration particle which is found using the k-means clustering technique. In this contribution, each particle moves towards the global best position by not only using the personal best and global best, but particles are modelled to move in murmuration towards the global best using the personal best, global best and a local best particle known as the local best murmuration particle. The improved model was tested against the traditional PSO and two other variants of the PSO and genetic algorithm (GA) using 18 benchmark test functions. The proposed improvement demonstrated superior exploration abilities by achieving the best optimum values in 15 out of 18 functions, particularly in the multimodal functions, where it achieved the best optimum value in all 6 cases. It also achieved the best worst-case values in 12 out of 18 functions, especially in the variable-dimension functions, where other algorithms showed significant escalation, indicating the proposed improvement’s reliability and robustness. In terms of convergence, the proposed improvement exhibited the best convergence rate in all 18 functions. These findings highlight the impressive ability of the proposed improvement to converge swiftly without compromising accuracy.

A novel elite guidance-based social learning particle swarm optimization algorithm

To improve the premature convergence and poor global search capability of the classical particle swarm algorithm(PSO), this paper proposes a novel elite guidance-based social learning particle swarm optimization (ESLPSO) algorithm. In the ESLPSO algorithm, a hierarchical topological search method is proposed. The method divides particles into optimal elite particles and other civilian particles according to their fitness performance, revolutionizing the update sample of the traditional population iterative search and enhancing the guidance of the whole population evolution information. In addition, an elite particle-guided social learning strategy is designed to better utilize the multidimensional information on population evolution by increasing the uncertainty of state superposition. On this basis, the extremum perturbation migration mechanism motivates the particles to experience new search paths and regions, increase population diversity and balance the population's exploration and exploitation in the search process. Moreover, the Cubic chaos initialization is employed to endow the initial particle population with a wide coverage in the search space. Finally, 12 benchmark test function sets covering unimodal, multimodal and rotated-multimodal functions are used to validate the performance of the proposed algorithm. The results on comparing the ESLPSO algorithm with other eight improved PSO algorithms show that the ESLPSO algorithm has excellent search performances in solving different types of functions, having efficient robustness and excellent solutions.

Somersault Foraging and Elite Opposition-Based Learning Dung Beetle Optimization Algorithm

To tackle the shortcomings of the Dung Beetle Optimization (DBO) Algorithm, which include slow convergence speed, an imbalance between exploration and exploitation, and susceptibility to local optima, a Somersault Foraging and Elite Opposition-Based Learning Dung Beetle Optimization (SFEDBO) Algorithm is proposed. This algorithm utilizes an elite opposition-based learning strategy as the method for generating the initial population, resulting in a more diverse initial population. To address the imbalance between exploration and exploitation in the algorithm, an adaptive strategy is employed to dynamically adjust the number of dung beetles and eggs with each iteration of the population. Inspired by the Manta Ray Foraging Optimization (MRFO) algorithm, we utilize its somersault foraging strategy to perturb the position of the optimal individual, thereby enhancing the algorithm’s ability to escape from local optima. To verify the effectiveness of the proposed improvements, the SFEDBO algorithm is utilized to optimize 23 benchmark test functions. The results show that the SFEDBO algorithm achieves better solution accuracy and stability, outperforming the DBO algorithm in terms of optimization results on the test functions. Finally, the SFEDBO algorithm was applied to the practical application problems of pressure vessel design, tension/extension spring design, and 3D unmanned aerial vehicle (UAV) path planning, and better optimization results were obtained. The research shows that the SFEDBO algorithm proposed in this paper is applicable to actual optimization problems and has better performance.

A Reinforced Whale Optimization Algorithm for Solving Mathematical Optimization Problems.

The whale optimization algorithm has several advantages, such as simple operation, few control parameters, and a strong ability to jump out of the local optimum, and has been used to solve various practical optimization problems. In order to improve its convergence speed and solution quality, a reinforced whale optimization algorithm (RWOA) was designed. Firstly, an opposition-based learning strategy is used to generate other optima based on the best optimal solution found during the algorithm's iteration, which can increase the diversity of the optimal solution and accelerate the convergence speed. Secondly, a dynamic adaptive coefficient is introduced in the two stages of prey and bubble net, which can balance exploration and exploitation. Finally, a kind of individual information-reinforced mechanism is utilized during the encircling prey stage to improve the solution quality. The performance of the RWOA is validated using 23 benchmark test functions, 29 CEC-2017 test functions, and 12 CEC-2022 test functions. Experiment results demonstrate that the RWOA exhibits better convergence accuracy and algorithm stability than the WOA on 20 benchmark test functions, 21 CEC-2017 test functions, and 8 CEC-2022 test functions, separately. Wilcoxon's rank sum test shows that there are significant statistical differences between the RWOA and other algorithms.

A Multi-Strategy Enhanced Hybrid Ant–Whale Algorithm and Its Applications in Machine Learning

Based on the principles of biomimicry, evolutionary algorithms (EAs) have been widely applied across diverse domains to tackle practical challenges. However, the inherent limitations of these algorithms call for further refinement to strike a delicate balance between global exploration and local exploitation. Thus, this paper introduces a novel multi-strategy enhanced hybrid algorithm called MHWACO, which integrates a Whale Optimization Algorithm (WOA) and Ant Colony Optimization (ACO). Initially, MHWACO employs Gaussian perturbation optimization for individual initialization. Subsequently, individuals selectively undertake either localized exploration based on the refined WOA or global prospecting anchored in the Golden Sine Algorithm (Golden-SA), determined by transition probabilities. Inspired by the collaborative behavior of ant colonies, a Flight Ant (FA) strategy is proposed to guide unoptimized individuals toward potential global optimal solutions. Finally, the Gaussian scatter search (GSS) strategy is activated during low population activity, striking a balance between global exploration and local exploitation capabilities. Moreover, the efficacy of Support Vector Regression (SVR) and random forest (RF) as regression models heavily depends on parameter selection. In response, we have devised the MHWACO-SVM and MHWACO-RF models to refine the selection of parameters, applying them to various real-world problems such as stock prediction, housing estimation, disease forecasting, fire prediction, and air quality monitoring. Experimental comparisons against 9 newly proposed intelligent optimization algorithms and 9 enhanced algorithms across 34 benchmark test functions and the CEC2022 benchmark suite, highlight the notable superiority and efficacy of MSWOA in addressing global optimization problems. Finally, the proposed MHWACO-SVM and MHWACO-RF models outperform other regression models across key metrics such as the Mean Bias Error (MBE), Coefficient of Determination (R2), Mean Absolute Error (MAE), Explained Variance Score (EVS), and Median Absolute Error (MEAE).

Levy Sooty Tern Optimization Algorithm Builds DNA Storage Coding Sets for Random Access.

DNA molecules, as a storage medium, possess unique advantages. Not only does DNA storage exhibit significantly higher storage density compared to electromagnetic storage media, but it also features low energy consumption and extremely long storage times. However, the integration of DNA storage into daily life remains distant due to challenges such as low storage density, high latency, and inevitable errors during the storage process. Therefore, this paper proposes constructing a DNA storage coding set based on the Levy Sooty Tern Optimization Algorithm (LSTOA) to achieve an efficient random-access DNA storage system. Firstly, addressing the slow iteration speed and susceptibility to local optima of the Sooty Tern Optimization Algorithm (STOA), this paper introduces Levy flight operations and propose the LSTOA. Secondly, utilizing the LSTOA, this paper constructs a DNA storage encoding set to facilitate random access while meeting combinatorial constraints. To demonstrate the coding performance of the LSTOA, this paper consists of analyses on 13 benchmark test functions, showcasing its superior performance. Furthermore, under the same combinatorial constraints, the LSTOA constructs larger DNA storage coding sets, effectively reducing the read-write latency and error rate of DNA storage.

A Method for Optimizing Terminal Sliding Mode Controller Parameters Based on a Multi-Strategy Improved Crayfish Algorithm

This paper proposes a parameter optimization method for a terminal sliding mode controller (TSMC) based on a multi-strategy improved crayfish algorithm (JLSCOA) to enhance the performance of ship dynamic positioning systems. The TSMC is designed for the “Xinhongzhuan” vessel of Dalian Maritime University. JLSCOA integrates subtractive averaging, Levy Flight, and sparrow search strategies to overcome the limitations of traditional crayfish algorithms. Compared to COA, WOA, and SSA algorithms, JLSCOA demonstrates superior optimization accuracy, convergence performance, and stability across 12 benchmark test functions. It achieves the optimal value in 83% of cases, outperforms the average in 83% of cases, and exhibits stronger robustness in 75% of cases. Simulations show that applying JLSCOA to TSMC parameter optimization significantly outperforms traditional non-optimized controllers, reducing the average time for three degrees of freedom position changes by over 300 s and nearly eliminating control force and velocity oscillations.