The efficiency of detecting seabird behaviour from movement patterns: the effect of sampling frequency on inferring movement metrics in Procellariiformes.

BACKGROUND: Recent technological advances have resulted in low-cost GPS loggers that are small enough to be used on a range of seabirds, producing accurate location estimates (± 5 m) at sampling intervals as low as 1 s. However, tradeoffs between battery life and sampling frequency result in studies using GPS loggers on flying seabirds yielding locational data at a wide range of sampling intervals. Metrics derived from these data are known to be scale-sensitive, but quantification of these errors is rarely available. Very frequent sampling, coupled with limited movement, can result in measurement error, overestimating movement, but a much more pervasive problem results from sampling at long intervals, which grossly underestimates path lengths. METHODS: We use fine-scale (1 Hz) GPS data from a range of albatrosses and petrels to study the effect of sampling interval on metrics derived from the data. The GPS paths were sub-sampled at increasing intervals to show the effect on path length (i.e. ground speed), turning angles, total distance travelled, as well as inferred behavioural states. RESULTS: We show that distances (and per implication ground speeds) are overestimated (4% on average, but up to 20%) at the shortest sampling intervals (1-5 s) and underestimated at longer intervals. The latter bias is greater for more sinuous flights (underestimated by on average 40% when sampling > 1-min intervals) as opposed to straight flight (11%). Although sample sizes were modest, the effect of the bias seemingly varied with species, where species with more sinuous flight modes had larger bias. Sampling intervals also played a large role when inferring behavioural states from path length and turning angles. CONCLUSIONS: Location estimates from low-cost GPS loggers are appropriate to study the large-scale movements of seabirds when using coarse sampling intervals, but actual flight distances are underestimated. When inferring behavioural states from path lengths and turning angles, moderate sampling intervals (10-30 min) may provide more stable models, but the accuracy of the inferred behavioural states will depend on the time period associated with specific behaviours. Sampling rates have to be considered when comparing behaviours derived using varying sampling intervals and the use of bias-informed analyses are encouraged.

Insights into Blue Whale (Balaenoptera musculus L.) Population Movements in the Galapagos Archipelago and Southeast Pacific.

The Galapagos Marine Reserve is vital for cetaceans, serving as both a stopover and residency site. However, blue whales, occasionally sighted here, exhibit poorly understood migratory behavior within the Galapagos and the broader Eastern Tropical Pacific. This study, the first to satellite tag blue whales in the Galapagos (16 tagged between 2021 and 2023), explored their behavior in relation to environmental variables like chlorophyll-a concentration, sea surface temperature (SST), and productivity. Key findings show a strong correlation between foraging behavior, high chlorophyll-a levels, productivity, and lower SSTs, indicating a preference for food-rich areas. Additionally, there is a notable association with geomorphic features like ridges, which potentially enhance food abundance. Most tagged whales stayed near the Galapagos archipelago, with higher concentrations observed around Isabela Island, which is increasingly frequented by tourist vessels, posing heightened ship strike risks. Some whales ventured into Ecuador's exclusive economic zone, while one migrated southward to Peru. The strong 2023 El Niño-Southern Oscillation event led to SST and primary production changes, likely impacting whale resource availability. Our study provides crucial insights into blue whale habitat utilization, informing adaptive management strategies to mitigate ship strike risks and address altered migration routes due to climate-driven environmental shifts.

Multi-Cluster Approaches to Radio Sensor Array Channel Modeling and Simulation.

In this paper, we explore the physical propagation environment of radio waves by describing it in terms of distant scattering clusters. Each cluster consists of numerous scattering objects that may exhibit certain statistical properties. By utilizing geometry-based methods, we can study the channel second-order statistics (CSOS), where each distant scattering cluster corresponds to a CSOS, contributes a portion to the Doppler spectrum, and is associated with a state-space multiple-input and multiple-output (MIMO) radio channel model. Consequently, the physical propagation environment of radio waves can be modeled by summing multiple state-space MIMO radio channel models. This approach offers three key advantages: simplicity, the ability to construct the entire Doppler power spectrum from multiple uncorrelated distant scattering clusters, and the capability to obtain the channels contributed by these clusters by summing the individual channels. This methodology enables the reconstruction of the radio wave propagation environment in a simulated manner and is crucial for developing massive MIMO channel models.

Scale-dependent population drivers inform avian management in a declining saline lake ecosystem.

Shrinking saline lakes provide irreplaceable habitat for waterbird species globally. Disentangling the effects of wetland habitat loss from other drivers of waterbird population dynamics is critical for protecting these species in the face of unprecedented changes to saline lake ecosystems, ideally through decision-making frameworks that identify effective management options and their potential outcomes. Here, we develop a framework to assess the effects of hypothesized population drivers and identify potential future outcomes of plausible management scenarios on a saline lake-reliant waterbird species. We use 36 years of monitoring data to quantify the effects of environmental conditions on the population size of a regionally important breeding colony of American white pelicans (Pelecanus erythrorhynchos) at Great Salt Lake, Utah, US, then forecast colony abundance under various management scenarios. We found that low lake levels, which allow terrestrial predators access to the colony, are probable drivers of recent colony declines. Without local management efforts, we predicted colony abundance could likely decline approximately 37.3% by 2040, although recent colony observations suggest population declines may be more extreme than predicted. Results from our population projection scenarios suggested that proactive approaches to preventing predator colony access and reversing saline lake declines are crucial for the persistence of the Great Salt Lake pelican colony. Increasing wetland habitat and preventing predator access to the colony together provided the most effective protection, increasing abundance 145.4% above projections where no management actions are taken, according to our population projection scenarios. Given the importance of water levels to the persistence of island-nesting colonial species, proactive approaches to reversing saline lake declines could likely benefit pelicans as well as other avian species reliant on these unique ecosystems.

Individualized survival predictions using state space model with longitudinal and survival data.

Monitoring disease progression often involves tracking biomarker measurements over time. Joint models (JMs) for longitudinal and survival data provide a framework to explore the relationship between time-varying biomarkers and patients' event outcomes, offering the potential for personalized survival predictions. In this article, we introduce the linear state space dynamic survival model for handling longitudinal and survival data. This model enhances the traditional linear Gaussian state space model by including survival data. It differs from the conventional JMs by offering an alternative interpretation via differential or difference equations, eliminating the need for creating a design matrix. To showcase the model's effectiveness, we conduct a simulation case study, emphasizing its performance under conditions of limited observed measurements. We also apply the proposed model to a dataset of pulmonary arterial hypertension patients, demonstrating its potential for enhanced survival predictions when compared with conventional risk scores.

Learning capabilities to resolve tilt-translation ambiguity in goldfish.

Introduction: Spatial orientation refers to the perception of relative location and self-motion in space. The accurate formation of spatial orientation is essential for animals to survive and interact safely with their environment. The formation of spatial orientation involves the integration of sensory inputs from the vestibular, visual, and proprioceptive systems. Vestibular organs function as specialized head motion sensors, providing information regarding angular velocity and linear acceleration via the semicircular canals and otoliths, respectively. However, because forces arising from the linear acceleration (translation) and inclination relative to the gravitational axis (tilt) are equivalent, they are indistinguishable by accelerometers, including otoliths. This is commonly referred to as the tilt - translation ambiguity, which can occasionally lead to the misinterpretation of translation as a tilt. The major theoretical frameworks addressing this issue have proposed that the interpretation of tilt versus translation may be contingent on an animal's previous experiences of motion. However, empirical confirmation of this hypothesis is lacking. Methods: In this study, we conducted a behavioral experiment using goldfish to investigate how an animal's motion experience influences its interpretation of tilt vs. translation. We examined a reflexive eye movement called the vestibulo-ocular reflex (VOR), which compensatory-rotates the eyes in response to head motion and is known to reflect an animal's three-dimensional head motion estimate. Results: We demonstrated that the VORs of naïve goldfish do not differentiate between translation and tilt at 0.5 Hz. However, following prolonged visual-translation training, which provided appropriate visual stimulation in conjunction with translational head motion, the VORs were capable of distinguishing between the two types of head motion within 3 h. These results were replicated using the Kalman filter model of spatial orientation, which incorporated the variable variance of process noise corresponding to the accumulated motion experience. Discussion: Based on these experimental and computational findings, we discuss the neural mechanism underlying the resolution of tilt-translation ambiguity within a context analogous to, yet distinct from, previous cross-axis VOR adaptations.

Monitoring multistage healthcare processes using state space models and a machine learning based framework.

Monitoring healthcare processes, such as surgical outcomes, with a keen focus on detecting changes and unnatural conditions at an early stage is crucial for healthcare professionals and administrators. In line with this goal, control charts, which are the most popular tool in the field of Statistical Process Monitoring, are widely employed to monitor therapeutic processes. Healthcare processes are often characterized by a multistage structure in which several components, states or stages form the final products or outcomes. In such complex scenarios, Multistage Process Monitoring (MPM) techniques become invaluable for monitoring distinct states of the process over time. However, the healthcare sector has seen limited studies employing MPM. This study aims to fill this gap by developing an MPM control chart tailored for healthcare data to promote early detection, confirmation, and patient safety. As it is important to detect unnatural conditions in healthcare processes at an early stage, the statistical control charts are combined with machine learning techniques (i.e., we deal with Intelligent Control Charting, ICC) to enhance detection ability. Through Monte Carlo simulations, our method demonstrates better performance compared to its statistical counterparts. To underline the practical application of the proposed ICC framework, real data from a two-stage thyroid cancer surgery is utilized. This real-world case serves as a compelling illustration of the effectiveness of the developed MPM control chart in a healthcare setting.

Automated Seizure Detection Based on State-Space Model Identification.

In this study, we developed a machine learning model for automated seizure detection using system identification techniques on EEG recordings. System identification builds mathematical models from a time series signal and uses a small number of parameters to represent the entirety of time domain signal epochs. Such parameters were used as features for the classifiers in our study. We analyzed 69 seizure and 55 non-seizure recordings and an additional 10 continuous recordings from Thomas Jefferson University Hospital, alongside a larger dataset from the CHB-MIT database. By dividing EEGs into epochs (1 s, 2 s, 5 s, and 10 s) and employing fifth-order state-space dynamic systems for feature extraction, we tested various classifiers, with the decision tree and 1 s epochs achieving the highest performance: 96.0% accuracy, 92.7% sensitivity, and 97.6% specificity based on the Jefferson dataset. Moreover, as the epoch length increased, the accuracy dropped to 94.9%, with a decrease in sensitivity to 91.5% and specificity to 96.7%. Accuracy for the CHB-MIT dataset was 94.1%, with 87.6% sensitivity and 97.5% specificity. The subject-specific cases showed improved results, with an average of 98.3% accuracy, 97.4% sensitivity, and 98.4% specificity. The average false detection rate per hour was 0.5 ± 0.28 in the 10 continuous EEG recordings. This study suggests that using a system identification technique, specifically, state-space modeling, combined with machine learning classifiers, such as decision trees, is an effective and efficient approach to automated seizure detection.

Decreasing body size is associated with reduced calving probability in critically endangered North Atlantic right whales.

Body size is key to many life-history processes, including reproduction. Across species, climate change and other stressors have caused reductions in the body size to which animals can grow, called asymptotic size, with consequences for demography. A reduction in mean asymptotic length was documented for critically endangered North Atlantic right whales, in parallel with declines in health and vital rates resulting from human activities and environmental changes. Here, we tested whether smaller body size was associated with lower reproductive output, using a state-space model for individual health, survival and reproduction that quantifies the mechanistic links between these processes. Body size (as represented by the cube of length) was strongly associated with a female's calving probability at each reproductive opportunity. This relationship explained 62% of the variation in calving among reproductive females, along with their decreasing health (20%). The effects of decreasing mean body size on reproductive performance are another concerning indication of the worsening prospects for this species and many others affected by environmental change, requiring a focus of conservation and management interventions on improving conditions that affect reproduction as well as reducing mortality.

Joint iterative state and parameter estimation for bilinear systems with autoregressive noises via the data filtering.

This paper proposes a novel iterative algorithm for the joint state and parameter estimation of bilinear state-space systems disturbed by colored noise. Estimating the states and parameters of such systems is challenging due to their nonlinearity and greater number of parameters compared to linear systems. Our method is to modify the Kalman filtering appropriately to estimate the unknown states of bilinear systems. Once the unknown states are estimated, we develop the Kalman filtering-based multi-innovation gradient-based iterative (KF-MIGI) algorithm for parameter estimation. To further improve estimation accuracy and cope with colored noises, we introduce a data filtering-based KF-MIGI algorithm that uses an adaptive filter to filter input-output data. Additionally, we compare the gradient-based iterative algorithm and the stochastic gradient algorithm. The effectiveness of the proposed algorithm is demonstrated through numerical examples.

Estimation of spatial distribution of soil moisture on steep hillslopes by state-space approach (SSA).

Soil moisture is a critical variable that quantifies precipitation, floods, droughts, irrigation, and other factors with regard to decision-making and risk evaluation. An accurate prediction of soil moisture dynamics is important for soil and environmental management. However, the complex topographic condition and land use in hilly and mountainous areas make it a challenge to monitor and predict soil moisture dynamics in these areas. In this study, the determinants of soil moisture variability were determined by structural equation modeling, and then an attempt was made to estimate the spatial distribution of soil moisture content on steep hillslope using the state-space method. Herein, soil moisture at different depths (0-10, 10-20, and 20-30 cm) was monitored by portable time-domain reflectometer (TDR) along this hillslope (100 m × 180 m). It showed that the spatial variability of soil moisture decreased with increasing soil wetness, primarily in the topsoil (0-10 cm). Soil moisture was correlated with elevation (r = 0.28, 0.50, and 0.28), capillary porosity (r = 0.06, 0.37, and 0.28), soil texture (r for Clay: 0.20, 0.24, and 0.16; r for Sand: -0.25, -0.18, and -0.28), organic carbon (r = -0.31, -0.08, and 0.10) and land use (r = -0.01, 0.28, and 0.24) under different conditions (dry, moderate, and wet). Among these determinants, elevation made direct contributions to soil moisture variation, especially under moderate conditions, while land use made its impacts by altering soil texture. It is encouraging that the state-space approach yielded precise and cost-effective predictions of soil moisture dynamics along this steep hillslope since it gives the minimum root-mean-square error (RMSE) and Akaike information criterion (AIC). Moreover, soil organic carbon (AIC = -4.497, RMSE = 0.104, R2 = 0.899), rock fragment contents (AIC = -4.366, RMSE = 0.111, R2 = 0.878), and elevation (AIC = -3.693, RMSE = 0.156, R2 = 0.629) effectively anticipated the spatial distribution of soil moisture under dry, moderate, and wet conditions, respectively. This study confirms the efficacy of the state-space approach as a valuable tool for soil moisture prediction in areas characterized by complex and spatially heterogeneous conditions.

Evaluating swine disease occurrence on farms using the state-space model based on meat inspection data: a time-series analysis.

BACKGROUND: Data on abnormal health conditions in animals obtained from slaughter inspection are important for identifying problems in fattening management. However, methods to objectively evaluate diseases on farms using inspection data has not yet been well established. It is important to assess fattening management on farms using data obtained from slaughter inspection. In this study, we developed the state-space model to evaluate swine morbidity using slaughter inspection data. RESULTS: The most appropriate model for each disease was constructed using the state-space model. Data on 11 diseases in slaughterhouses over the past 4 years were used to build the model. The model was validated using data from 14 farms. The local-level model (the simplest model) was the best model for all diseases. We found that the analysis of slaughter data using the state-space model could construct a model with greater accuracy and flexibility than the ARIMA model. In this study, no seasonality or trend model was selected for any disease. It is thought that models with seasonality were not selected because diseases in swine shipped to slaughterhouses were the result of illness at some point during the 6-month fattening period between birth and shipment. CONCLUSION: Evaluation of previous diseases helps with the objective understanding of problems in fattening management. We believe that clarifying how farms manage fattening of their pigs will lead to improved farm profits. In that respect, it is important to use slaughterhouse data for fattening evaluation, and it is extremely useful to use mathematical models for slaughterhouse data. However, in this research, the model was constructed on the assumption of normality and linearity. In the future, we believe that we can build a more accurate model by considering models that assume non-normality and non-linearity.

Modeling ocean distributions and abundances of natural- and hatchery-origin Chinook salmon stocks with integrated genetic and tagging data.

Background: Considerable resources are spent to track fish movement in marine environments, often with the intent of estimating behavior, distribution, and abundance. Resulting data from these monitoring efforts, including tagging studies and genetic sampling, often can be siloed. For Pacific salmon in the Northeast Pacific Ocean, predominant data sources for fish monitoring are coded wire tags (CWTs) and genetic stock identification (GSI). Despite their complementary strengths and weaknesses in coverage and information content, the two data streams rarely have been integrated to inform Pacific salmon biology and management. Joint, or integrated, models can combine and contextualize multiple data sources in a single statistical framework to produce more robust estimates of fish populations. Methods: We introduce and fit a comprehensive joint model that integrates data from CWT recoveries and GSI sampling to inform the marine life history of Chinook salmon stocks at spatial and temporal scales relevant to ongoing fisheries management efforts. In a departure from similar models based primarily on CWT recoveries, modeled stocks in the new framework encompass both hatchery- and natural-origin fish. We specifically model the spatial distribution and marine abundance of four distinct stocks with spawning locations in California and southern Oregon, one of which is listed under the U.S. Endangered Species Act. Results: Using the joint model, we generated the most comprehensive estimates of marine distribution to date for all modeled Chinook salmon stocks, including historically data poor and low abundance stocks. Estimated marine distributions from the joint model were broadly similar to estimates from a simpler, CWT-only model but did suggest some differences in distribution in select seasons. Model output also included novel stock-, year-, and season-specific estimates of marine abundance. We observed and partially addressed several challenges in model convergence with the use of supplemental data sources and model constraints; similar difficulties are not unexpected with integrated modeling. We identify several options for improved data collection that could address issues in convergence and increase confidence in model estimates of abundance. We expect these model advances and results provide management-relevant biological insights, with the potential to inform future mixed-stock fisheries management efforts, as well as a foundation for more expansive and comprehensive analyses to follow.

Different Methods to Estimate the Phase of Neural Rhythms Agree But Only During Times of Low Uncertainty.

Rhythms are a common feature of brain activity. Across different types of rhythms, the phase has been proposed to have functional consequences, thus requiring its accurate specification from noisy data. Phase is conventionally specified using techniques that presume a frequency band-limited rhythm. However, in practice, observed brain rhythms are typically nonsinusoidal and amplitude modulated. How these features impact methods to estimate phase remains unclear. To address this, we consider three phase estimation methods, each with different underlying assumptions about the rhythm. We apply these methods to rhythms simulated with different generative mechanisms and demonstrate inconsistency in phase estimates across the different methods. We propose two improvements to the practice of phase estimation: (1) estimating confidence in the phase estimate, and (2) examining the consistency of phase estimates between two (or more) methods.

Dynamic prediction modeling of postoperative mortality among patients undergoing surgical aortic valve replacement in a statewide cohort over a 12-year period.

Objective: Clinical prediction models for surgical aortic valve replacement mortality, are valuable decision tools but are often limited in their ability to account for changes in medical practice, patient selection, and the risk of outcomes over time. Recent research has identified methods to update models as new data accrue, but their effect on model performance has not been rigorously tested. Methods: The study population included 44,546 adults who underwent an isolated surgical aortic valve replacement from January 1, 1999, to December 31, 2018, statewide in Pennsylvania. After chronologically splitting the data into training and validation sets, we compared calibration, discrimination, and accuracy measures amongst a nonupdating model to 2 methods of model updating: calibration regression and the novel dynamic logistic state space model. Results: The risk of mortality decreased significantly during the validation period (P < .01) and the nonupdating model demonstrated poor calibration and reduced accuracy over time. Both updating models maintained better calibration (Hosmer-Lemeshow χ2 statistic) than the nonupdating model: nonupdating (156.5), calibration regression (4.9), and dynamic logistic state space model (8.0). Overall accuracy (Brier score) was consistently better across both updating models: dynamic logistic state space model (0.0252), calibration regression (0.0253), and nonupdating (0.0256). Discrimination improved with the dynamic logistic state space model (area under the curve, 0.696) compared with the nonupdating model (area under the curve, 0.685) and calibration regression method (area under the curve, 0.687). Conclusions: Dynamic model updating can improve model accuracy, discrimination, and calibration. The decision as to which method to use may depend on which measure is most important in each clinical context. Because competing therapies have emerged for valve replacement models, updating may guide clinical decision making.

State Space Modeling of Event Count Time Series.

This paper proposes a class of algorithms for analyzing event count time series, based on state space modeling and Kalman filtering. While the dynamics of the state space model is kept Gaussian and linear, a nonlinear observation function is chosen. In order to estimate the states, an iterated extended Kalman filter is employed. Positive definiteness of covariance matrices is preserved by a square-root filtering approach, based on singular value decomposition. Non-negativity of the count data is ensured, either by an exponential observation function, or by a newly introduced "affinely distorted hyperbolic" observation function. The resulting algorithm is applied to time series of the daily number of seizures of drug-resistant epilepsy patients. This number may depend on dosages of simultaneously administered anti-epileptic drugs, their superposition effects, delay effects, and unknown factors, making the objective analysis of seizure counts time series arduous. For the purpose of validation, a simulation study is performed. The results of the time series analysis by state space modeling, using the dosages of the anti-epileptic drugs as external control inputs, provide a decision on the effect of the drugs in a particular patient, with respect to reducing or increasing the number of seizures.

State Space Mixture Modeling: Finding People with Similar Patterns of Change.

Increasingly, behavioral scientists encounter data where several individuals were measured on multiple variables over numerous occasions. Many current methods combine these data, assuming all individuals are randomly equivalent. An extreme alternative assumes no one is randomly equivalent. We propose state space mixture modeling as one possible compromise. State space mixture modeling assumes that unknown groups of people exist who share the same parameters of a state space model, and simultaneously estimates both the state space parameters and group membership. The goal is to find people that are undergoing similar change processes over time. The present work demonstrates state space mixture modeling on a simulated data set, and summarizes the results from a large simulation study. The illustration shows how the analysis is conducted, whereas the simulation provides evidence of its general validity and applicability. In the simulation study, sample size had the greatest influence on parameter estimation and the dimension of the change process had the greatest impact on correctly grouping people together, likely due to the distinctiveness of their patterns of change. State space mixture modeling offers one of the best-performing methods for simultaneously drawing conclusions about individual change processes while also analyzing multiple people.

Study of Polyvinyl Alcohol Hydrogels Applying Physical-Mechanical Methods and Dynamic Models of Photoacoustic Signals.

This study aims to analyze the physical-mechanical properties and dynamic models of tissue-simulating hydrogels, specifically the photoacoustic (PA) response signals, by varying the concentrations of polyvinyl alcohol (PVA) and molecular weight (MW). A state-space model (SSM) is proposed to study the PVA hydrogels to retrieve the PA-related signal's damping ratio and natural frequency. Nine box-shaped PVA hydrogels containing saline solution were used, with five concentrations of PVA (7, 9, 12, 15, 20%) for MW1 and four for MW2. The results indicated that the concentration of PVA and MW played an important role in the PA wave's amplitude, arrival time, and speed of sound over the hydrogels. The SSM parameters showed that increasing PVA and MW concentrations improved the hydrogels' ability to absorb and transfer energy under the PA effect. These parameters were also found to be correlated with density and modulus of elasticity. Additionally, the concentrations of PVA and MW affected the absorption and optical scattering coefficients. The physical-mechanical properties, including porosity, density, and modulus of elasticity, improved as the concentration of PVA and MW increased. The ultimate goal of this study is to develop hydrogels as phantoms that can be used for tissue simulation and imaging.

Modified Active Disturbance Rejection Predictive Control: A fixed-order state-space formulation for SISO systems.

This paper presents a novel control strategy that provides active disturbance rejection predictive control on constrained systems with no nominal identified model. The proposed loop relaxes the modelling requirement to a fixed discrete-time state-space realisation of a first-order plus integrator plant despite the nature of the controlled process. A third-order discrete Extended State Observer (ESO) estimates the model mismatch and assumed plant states. Moreover, the constraints handling is tackled by incorporating the compensation term related to the total perturbation in the definition of the optimisation problem constraints. The proposal merges in a new way state-space Model Predictive Control (MPC) and Active Disturbance Rejection Control (ADRC) into an architecture suitable for the servo-regulatory operation of linear and non-linear systems, as shown through validation examples.

Verification of the Mode Decomposition Technique for Closely Distributed Modal Systems in the State Space Domain.

This study aims to propose and validate the state space mode decomposition technique for precise mode separation of non-classical damping systems and closely distributed modal systems. To assess the reliability and applicability of this technique, a 40-story building with a tuned mass damper is investigated, and acceleration responses measured by the building's health monitoring system are used for the verification of the technique. The mode separation results reveal that the separated modal power spectrum becomes distorted at neighboring natural frequency ranges when the performance index only considers the concentration of power spectral energy at the target natural frequency. However, by introducing an augmented performance index that includes a constraint condition to account for distortion, more accurate mode decomposition can be achieved.