Mixture Regression Research Articles

GeM-LR: Discovering predictive biomarkers for small datasets in vaccine studies.

Despite significant progress in vaccine research, the level of protection provided by vaccination can vary significantly across individuals. As a result, understanding immunologic variation across individuals in response to vaccination is important for developing next-generation efficacious vaccines. Accurate outcome prediction and identification of predictive biomarkers would represent a significant step towards this goal. Moreover, in early phase vaccine clinical trials, small datasets are prevalent, raising the need and challenge of building a robust and explainable prediction model that can reveal heterogeneity in small datasets. We propose a new model named Generative Mixture of Logistic Regression (GeM-LR), which combines characteristics of both a generative and a discriminative model. In addition, we propose a set of model selection strategies to enhance the robustness and interpretability of the model. GeM-LR extends a linear classifier to a non-linear classifier without losing interpretability and empowers the notion of predictive clustering for characterizing data heterogeneity in connection with the outcome variable. We demonstrate the strengths and utility of GeM-LR by applying it to data from several studies. GeM-LR achieves better prediction results than other popular methods while providing interpretations at different levels.

Weighted fast-trimmed likelihood estimator for mixture regression models

The fast-trimmed likelihood estimate is a robust method to estimate the parameters of a mixture regression model. However, this method is vulnerable to the presence of bad leverage points, which are outliers in the direction of independent variables. To address this issue, we propose the weighted fast-trimmed likelihood estimate to mitigate the impact of leverage points. The proposed method applies the weights of the minimum covariance determinant to the rows suspected of containing leverage points. Notably, both real data and simulation studies were considered to determine the efficiency of the proposed method compared to the previous methods. The results reveal that the weighted fast-trimmed estimate method is more robust and reliable than the fast-trimmed likelihood estimate and the expectation–maximization (EM) methods, particularly in cases with small sample sizes.

Gaussian Mixture Regression Model with Sparsity for Clustering of Territory Risk in Auto Insurance

Abstract Insurance rating territory design and accurate estimation of territory risk relativities are fundamental aspects of auto insurance rate regulation. It is crucial to develop methodologies that can facilitate the effective design of rating territories and their risk relativities estimate, as they directly impact the rate filing and the decision support of the rate change review process. This article proposes a Gaussian Mixture Regression model clustering approach for territory design. The proposed method incorporates a linear regression model, taking spatial location as model covariates, which helps estimate the cluster mean more accurately. Also, to further enhance the estimation of territory risk relativities, we impose sparsity through sparse matrix decomposition of the membership coefficient matrix obtained from the Gaussian Mixture Regression model. By transitioning from the current hard clustering method to a soft approach, our methodology could improve the evaluation of territory risk for rate-making purposes. Moreover, using non-negative sparse matrix approximation ensures that the estimation of risk relativities for basic rating units remains smooth, effectively eliminating data noise from the territory risk relativity estimate. Overall, our novel methodology aims to significantly enhance the accuracy and reliability of risk analysis in auto insurance. Furthermore, the proposed method exhibits potential for extension to various other domains that involve spatial clustering of data, thereby broadening its applicability and expanding its usefulness beyond auto insurance rate regulation.

Learning Underwater Intervention Skills Based on Dynamic Movement Primitives

Improving the autonomy of underwater interventions by remotely operated vehicles (ROVs) can help mitigate the impact of communication delays on operational efficiency. Currently, underwater interventions for ROVs usually rely on real-time teleoperation or preprogramming by operators, which is not only time-consuming and increases the cognitive burden on operators but also requires extensive specialized programming. Instead, this paper uses the intuitive learning from demonstrations (LfD) approach that uses operator demonstrations as inputs and models the trajectory characteristics of the task through the dynamic movement primitive (DMP) approach for task reproduction as well as the generalization of knowledge to new environments. Unlike existing applications of DMP-based robot trajectory learning methods, we propose the underwater DMP (UDMP) method to address the problem that the complexity and stochasticity of underwater operational environments (e.g., current perturbations and floating operations) diminish the representativeness of the demonstrated trajectories. First, the Gaussian mixture model (GMM) and Gaussian mixture regression (GMR) are used for feature extraction of multiple demonstration trajectories to obtain typical trajectories as inputs to the DMP method. The UDMP method is more suitable for the LfD of underwater interventions than the method that directly learns the nonlinear terms of the DMP. In addition, we improve the commonly used homomorphic-based teleoperation mode to heteromorphic mode, which allows the operator to focus more on the end-operation task. Finally, the effectiveness of the developed method is verified by simulation experiments.

Investigating ingredient influence on reconstitution of powdered cocoa beverage via constrained mixture design

Reconstitution of a powdered cocoa beverage (PCB) produced by wet-mixing and oven-drying were studied. Sixteen PCB samples with different proportions of core ingredients, namely cocoa (5–25 %), skimmed milk (10–50 %), hydrolysed cereal extract (15–60 %), sucrose (0–30 %) and palm olein (0–15 %), were generated via a constrained mixture design. PCB samples had varying physical properties which included bulk porosity (57.3–74.8 %), particle surface hydrophilicity (contact angle = 60.0–94.5°) and glass transition temperature (51.2–86.3 °C). Consequently, PCB samples differed in reconstitution properties like wetting time (11–>300 s), reconstitution time (14–141 s) and soluble content (51.3–86.8 %). Subsequently, mixture regression models were developed to varying success (adjusted R2 = 70–97 %) for identifying the influence of each core ingredient on physical and reconstitution properties of PCB. PCB wettability was most improved by increasing the proportion of hydrolysed cereal extract, through forming powders with moderate particle surface hydrophilicities but higher glass transition temperatures and higher bulk porosities. Conversely, PCB wettability became exceptionally poor when it contained >35 % skimmed milk and <5 % palm olein. Additionally, a higher ratio of protein-rich skimmed milk against carbohydrate-rich hydrolysed cereal extract and sucrose led to solubility loss in PCB.

Risk Perception, Cooperation, and Emotional Distress During the COVID-19 Pandemic: Exploring Adaptive Risk Perception

During the COVID-19 pandemic, while risk perception may promote public cooperation with pandemic prevention, it may also increase emotional distress and thus endanger mental health. This study aimed to examine whether there is an adaptive risk perception pattern that fits both needs of pandemic control and mental health protection. Two waves of Chinese participants (N sample 1 = 1633, N sample 2 = 1899) completed the Scale of Pandemic Risk Perception, the Scale of Public Cooperation with Pandemic Prevention, the Epidemic Worry Scale, and the Positive and Negative Affective Schedule during Feb 3rd to 5th, and during Feb 18th to 20th, 2021 respectively. Four risk perception profiles were identified by using latent profile analysis based on pandemic risk perception. Regression mixture models found that individuals in the perceived-controllable-high-perceived-risk profile were the most cooperative and reported the least worries and negative affect. The perceived-uncontrollable-high-perceived-risk profile demonstrated high cooperation but serious worry and negative affect. Individuals in the ignoring-risk profile reported the least levels of cooperation and worry but the highest levels of negative affect. Finally, the perceived-moderate-perceived-risk profile reported moderate levels of both cooperation and emotional distress. These results were well repeated in two samples. Present findings point towards an adaptive risk perception pattern (the controllable-high-perceived-risk profile) which may optimize cooperation while also avoid serious emotional distress.

Mixtures of Probit Regression Models with Overlapping Clusters

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

Global Optimality of the EM Algorithm for Mixtures of Two-Component Linear Regressions

Global Optimality of the EM Algorithm for Mixtures of Two-Component Linear Regressions

First Results From a Calibrated Network of Low-Cost PM2.5 Monitors in Mombasa, Kenya Show Exceedance of Healthy Guidelines.

The paucity of fine particulate matter (PM2.5) measurements limits estimates of air pollution mortality in Sub-Saharan Africa. Well calibrated low-cost sensors can provide reliable data especially where reference monitors are unavailable. We evaluate the performance of Clarity Node-S PM monitors against a Tapered element oscillating microbalance (TEOM) 1400a and develop a calibration model in Mombasa, Kenya's second largest city. As-reported Clarity Node-S data from January 2023 through April 2023 was moderately correlated with the TEOM-1400a measurements (R 2=0.61) and exhibited a mean absolute error (MAE) of 7.03μgm-3. Employing three calibration models, namely, multiple linear regression (MLR), Gaussian mixture regression and random forest (RF) decreased the MAE to 4.28, 3.93, and 4.40μgm-3 respectively. The R 2 value improved to 0.63 for the MLR model but all other models registered a decrease (R 2=0.44 and 0.60 respectively). Applying the correction factor to a five-sensor network in Mombasa that was operated between July 2021 and July 2022 gave insights to the air quality in the city. The average daily concentrations of PM2.5 within the city ranged from 12 to 18μgm-3. The concentrations exceeded the WHO daily PM2.5 limits more than 50% of the time, in particular at the sites nearby frequent industrial activity. Higher averages were observed during the dry and cold seasons and during early morning and evening periods of high activity. These results represent some of the first air quality monitoring measurements in Mombasa and highlight the need for more study.

Fruit flexible collecting trajectory planning based on manual skill imitation for grape harvesting robot

The flexible collection is vital for the intelligent grape-picking robot. In view of the reliable operation method obtained by long-term training of human prior skill, a fruit flexible collecting trajectory planning method based on manual skill imitation for grape harvesting robot is proposed. The method involves capturing manual teaching trajectory data using a motion capture system, preprocessing the data, extracting features from multiple teaching trajectories, and forming a probability distribution through Gaussian mixture model – Gaussian mixture regression (GMM-GMR). And combined with the key point of manual operation trajectory, the general trajectory generated by GMR is segmented and further imitated by kernelized movement primitives (KMP) to obtain the reference trajectory, respectively. An optimization method for hyperparameter adaptation KMP (O-KMP) was proposed to meet the trajectory fitting effect of multiple key points. Mean square error (MSE) was used to evaluate the deviation of the trajectory from the reference trajectory. The partial optimal trajectory is selected and integrated into a single trajectory. Two experiments were conducted to investigate imitation: For the same starting and ending points task, the MSE of the trajectory generated by O-KMP decreased by 15.274% compared to the original fixed hyperparameter KMP. For different placement tasks, the MSE of the trajectory generated by the O-KMP decreased by 7.296% compared to the original KMP. Finally, the robot arm of the actuator visually presents different task states for the compliant placement of fresh grapes.

Change point detection via Gaussian mixture model

Change point detection aims to find abrupt changes in time series data. These changes denote substantial modifications to the process; these can be modeled as a change in the distribution (in location, scale, or trend). Traditional changepoint detection methods often rely on a cost function to assess if a change occurred in a series. Here, change point detection is investigated in a mixture-model-based clustering framework and a novel change point detection algorithm is developed using a finite mixture of regressions with concomitant variables. Through the introduction of a label correction mechanism, the unstructured clustering-based labels are treated as ordered and distinct segment labels. This approach can detect change points in both univariate and multivariate time series, and different kinds of change can be captured using a parsimonious family of models. Performance is illustrated on both simulated and real data. Journal of Statistical Research 2024, Vol. 58, No. 1, pp. 197-219.

Prenatal air pollution and children's autism traits score: Examination of joint associations with maternal intake of vitamin D, methyl donors, and polyunsaturated fatty acids using mixture methods.

Maternal nutrient intake may moderate associations between environmental exposures and children's neurodevelopmental outcomes, but few studies have assessed joint effects. We aimed to evaluate whether prenatal nutrient intake influences the association between air pollutants and autism-related trait scores. We included 126 participants from the EARLI (Early Autism Risk Longitudinal Investigation, 2009-2012) cohort, which followed US pregnant mothers who previously had a child with autism. Bayesian kernel machine regression and traditional regression models were used to examine joint associations of prenatal nutrient intake (vitamins D, B12, and B6; folate, choline, and betaine; and total omega 3 and 6 polyunsaturated fatty acids, reported via food frequency questionnaire), air pollutant exposure (particulate matter <2.5 μm [PM2.5], nitrogen dioxide [NO2], and ozone [O3], estimated at the address level), and children's autism-related traits (measured by the Social Responsiveness Scale [SRS] at 36 months). Most participants had nutrient intakes and air pollutant exposures that met US standards. Bayesian kernel machine regression mixture models and traditional regression models provided little evidence of individual or joint associations of nutrients and air pollutants with SRS scores or of an association between the overall mixture and SRS scores. In this cohort with a high familial likelihood of autism, we did not observe evidence of joint associations between air pollution exposures and nutrient intake with autism-related traits. Future work should examine the use of these methods in larger, more diverse samples, as our results may have been influenced by familial liability and/or relatively high nutrient intakes and low air pollutant exposures.

Exposure to metal mixtures and telomere length in Bangladeshi children.

Telomere length is associated with chronic diseases and in younger populations, may represent a biomarker of disease susceptibility. As growing evidence suggests that environmental factors, including metals, may impact telomere length, we investigated the association between 17 metals measured in toenail samples and leukocyte relative telomere length (RTL), among 472 five- to seven-year-old children enrolled in the Bangladesh Environmental Research in Children's Health (BiRCH) cohort. In single exposure linear regression models, a doubling of arsenic (As) and mercury (Hg) (μg/g) were associated with a -0.21 (95%CI: -0.032, -0.010; p=0.0005) and -0.017 (95%CI: -0.029, -0.004; p=0.006) difference in RTL, respectively. In Bayesian Kernel Machine Regression (BKMR) mixture models, the overall metal mixture was inversely associated with RTL (P-for-trend <0.001). Negative associations with RTL were observed with both log2-As and log2-Hg, while an inverted U-shaped association was observed for log2-zinc (Zn) with RTL. We found little evidence of interaction among metals. Sex-stratification identified stronger associations of the overall mixture and log2-As with RTL among females, compared to males. Our study suggests that As and Hg may independently influence RTL in mid-childhood. Further studies are needed to investigate potential long-term impacts of metal-associated telomere shortening in childhood on health outcomes in adult life.

Using Learning from Demonstration (LfD) to perform the complete apple harvesting task

Learning from Demonstration (LfD) can be used to make it easier for robots to learn to perform tasks such as apple harvesting. The required apple harvesting motions can be split into four steps: 1. Approaching, 2. Grasping, 3. Detaching, 4. Placing. So far, only parts of the apple harvesting task have been learned using LfD. In our work, we learned an abstracted version of the complete apple harvesting task using a cube. We used two methods and tested the sensitivity to model parameters and the number of demonstrations. We selected (1) the Gaussian Mixture Model (GMM) with Gaussian Mixture Regression (GMR) and (2) the Hidden Markov Model (HMM) with Linear Quadratic Regression (LQR). Both methods make use of Task Parameterization (TP), which allows models to combine multiple phases from the apple harvesting task. Combining multiple harvesting phases is important for effective LfD. We analysed the performance of these algorithms on four performance metrics, namely the grasp pose, the detachment motion, the place pose, and the success rate when using the real robot. A high grasp pose accuracy was achieved, specifically a position error of 0.004 and an orientation error of 0.004 for the GMM with GMR, trained with 100 demonstrations and 9 Gaussian components. The detachment motion was performed but with a reduced range. The model that performed the best was the GMM with GMR, trained with 60 demonstrations and 5 Gaussian components. A high place pose accuracy was achieved, specifically a position error of 0.003 and an orientation error of 0.003 for the GMM with GMR, trained with 100 demonstrations and 15 Gaussian components. The task was executed by a real robot successfully, resulting in a success rate of 67% for HMM with LQR, trained with 40 demonstrations, 5 states, and a scaling factor of 1.0. For the number of components, no clear relation was found. The same was true for the number of states in the HMM with LQR. For the scaling factor, a clear relation could be found. For this task, a value in the range of -1.0 to 1.0 should be used. For the number of demonstrations, there was an optimum of 40 demonstrations. Further improvements are needed to deal with challenges such as apple shift and detecting detachment of fruits.

Performance evaluation of nursing homes using finite mixtures of logistic models and M-quantile regression for binary data

Evaluating the performance of health care institutions is of paramount interest and it is often conducted using generalized linear mixed models. In this paper, we focus on the evaluation of Nursing Homes for elderly residents in a region of Italy and concentrate on binary outcomes (death and worsening). We propose to use a routinely assessed covariate such as the Resource Utilization Group to account for case-mix. We fit finite mixtures of logistic models to check the assumption of normality of the random effects in the generalized linear mixed model approach and to obtain a clustering of the Nursing Homes with respect to their performance. Since the distribution of the random effects is very skew, we propose to use scores based on robust M-Quantile regression for binary data and estimate their standard error using block-bootstrap. A sensitivity analysis is also conducted to evaluate the assumption of missing at random for non-observed data on discharged residents.

The Relationship Between Social Media Information Sharing Characteristics and Problem Behaviors Among Chinese College Students Under Recommendation Algorithms.

With the development of information technology and various social media, recommendation algorithms have increasingly more influence on users' social media usage. To date, there has been limited research focused on analyzing the impact of recommendation algorithms on social media use and their corresponding role in the development of problematic behaviors. The present study analyzes the impact of recommendation algorithms on college students' information sharing and internalizing, externalizing problem behaviors to address the aforementioned shortcomings. An online questionnaire survey was conducted among 34,752 college students in China. A latent profile analysis was conducted to explore the various behavioral patterns of Chinese college students' information sharing across the three social media platforms identified for this study. The Bolck-Croon-Hagenaars (BCH) method Regression Mixture Modeling was then used to analyze the differences in internalizing and externalizing problem behaviors among the different subgroups of Chinese college students. The level of information sharing by college students across different social media platforms could be divided into "WeChat Moments low-frequency information sharing", "middle-frequency comprehensive information sharing", "TikTok high-frequency information sharing", and "Sina Weibo high-frequency information sharing". Significant differences were observed regarding internalizing and externalizing problem behaviors among college students in different information-sharing subgroups. This study identified four subgroups with different information-sharing characteristics using latent profile analysis. Among them, college students who are in subgroup of social media information sharing influenced by recommendation algorithms exhibit higher frequency of information sharing and higher level of internalizing and externalizing problematic behaviors. These results expand our understanding of college students' social media usage and problem behaviors from a technological perspective. In future, the negative impacts of recommendation algorithms on college students can be reduced by improving their awareness of these algorithms and optimizing the algorithms themselves.

Who's most at risk? A person-centered approach to understanding the long-term relationship between early social media use and later depression across adolescence.

Person-centered analyses examined the relationship between social media use and depression over an 8-year period. The purpose was to examine the varying ways early social media use was associated with the development of depressive symptoms with a hypothesis that social media would not have a uniform association with depressive symptoms across adolescents. Participants included 488 adolescents (52% female), living in the United States, who were surveyed once a year for 8 years (beginning in 2010 when the average age for participants was 13.33 years old). Longitudinal mixture regression was used to identify classes of adolescents representing unique ways their early social media use was related to the development of depressive symptoms over an 8-year period. Five classes were found representing unique ways social media use was related to depression. Findings suggest social media use does not impact all adolescents in the same way. Social media use was related to increased depression for adolescents with greater parental hostility, peer bullying, anxiety, reactivity to stressors, and lower parental media monitoring. In other instances, social media use was related to less depression or was unrelated to depression. By identifying which adolescents may be most at risk from social media use, health providers, schools, and caregivers can tailor interventions to fit the needs of each adolescent.

Lower Prevalence of Asymptomatic Alzheimer's Disease Among Healthy African Americans.

Alzheimer's disease (AD) is believed to be more common in African Americans (AA), but biomarker studies in AA populations are limited. This report represents the largest study to date examining cerebrospinal fluid AD biomarkers in AA individuals. We analyzed 3,006 cerebrospinal fluid samples from controls, AD cases, and non-AD cases, including 495 (16.5%) self-identified black/AA and 2,456 (81.7%) white/European individuals using cutoffs derived from the Alzheimer's Disease Neuroimaging Initiative, and using a data-driven multivariate Gaussian mixture of regressions. Distinct effects of race were found in different groups. Total Tauand phospho181-Tau were lower among AA individuals in all groups (p < 0.0001), and Aβ42 was markedly lower in AA controls compared with white controls (p < 0.0001). Gaussian mixture of regressions modeling of cerebrospinal fluid distributions incorporating adjustments for covariates revealed coefficient estimates for AA race comparable with 2-decade change in age. Using Alzheimer's Disease Neuroimaging Initiative cutoffs, fewer AA controls were classified as biomarker-positive asymptomatic AD (8.0% vs 13.4%). After adjusting for covariates, our Gaussian mixture of regressions model reduced this difference, but continued to predict lower prevalence of asymptomatic AD among AA controls (9.3% vs 13.5%). Although the risk of dementia is higher, data-driven modeling indicates lower frequency of asymptomatic AD in AA controls, suggesting that dementia among AA populations may not be driven by higher rates of AD. ANN NEUROL 2024;96:463-475.

Culture, sex and social context influence brain-to-brain synchrony: an fNIRS hyperscanning study

BackgroundUnique interpersonal synchrony occurs during every social interaction, and is shaped by characteristics of participating individuals in these social contexts. Additionally, depending on context demands, interpersonal synchrony is also altered. The study therefore aims to investigate culture, sex, and social context effects simultaneously in a novel role-play paradigm. Additionally, the effect of personality traits on synchrony was investigated across cultures, and a further exploratory analysis on the effects of these variables on pre- and post-session empathy changes was conducted.Methods83 dyads were recruited in two waves from Singapore and Italy and took part in a within-subjects session where they interacted with each other as themselves (Naturalistic Conversation) and as others (Role-Play and Role Reversal). Big Five Inventory (administered pre-session) and Interpersonal Reactivity Index (administered pre- and post-session) were used as measures of personality and empathy respectively, while synchrony was measured using hyperscanning functional near-infrared spectroscopy in the prefrontal cortex. After data-preprocessing and preliminary analyses, a mixture of multiple linear regression and exploratory forward stepwise regression models were used to address the above study aims.ResultsResults revealed significant main and interaction effects of culture, sex and social context on brain-to-brain synchrony, particularly in the medial left cluster of the prefrontal cortex, and a unique contribution of extraversion and openness to experience to synchrony in the Italian cohort only. Finally, culture-driven differences in empathy changes were identified, where significant increases in empathy across sessions were generally only observed within the Singaporean cohort.ConclusionsMain findings indicate lowered brain-to-brain synchrony during role-playing activities that is moderated by the dyad’s sex make-up and culture, implying differential processing of social interactions that is also influenced by individuals’ background factors. Findings align with current literature that role-playing is a cognitively demanding activity requiring greater levels of self-regulation and suppression of self-related cognition as opposed to interpersonal co-regulation characterized by synchrony. However, the current pattern of results would be better supported by future studies investigating multimodal synchronies and corroboration.

A Data-Driven Model with Hysteresis Compensation for I2RIS Robot.

Retinal microsurgery is a high-precision surgery performed on a delicate tissue requiring the skill of highly trained surgeons. Given the restricted range of instrument motion in the confined intraocular space, snake-like robots may prove to be a promising technology to provide surgeons with greater flexibility, dexterity, and positioning accuracy during retinal procedures such as retinal vein cannulation and epiretinal membrane peeling. Kinematics modeling of these robots is an essential step toward accurate position control. Unlike conventional manipulators, modeling these robots does not follow a straightforward method due to their complex mechanical structure and actuation mechanisms. The hysteresis problem can especially impact the positioning accuracy significantly in wire-driven snake-like robots. In this paper, we propose a data-driven kinematics model using a probabilistic Gaussian mixture model (GMM) and Gaussian mixture regression (GMR) approach with a hysteresis compensation algorithm. Experimental results on the two-degree-of-freedom (DOF) integrated robotic intraocular snake (I2RIS) show that the proposed model with the hysteresis compensation can predict the snake tip bending angle for pitch and yaw with 0.45° and 0.39° root mean square error (RMSE), respectively. This results in overall 60% and 70% improvements of accuracy for yaw and pitch over the same model without the hysteresis compensation.