Random Change Point Research Articles

Improving piecewise linear snow density models through hierarchical spatial and orthogonal functional smoothing

AbstractSnow density estimates as a function of depth are used for understanding climate processes, evaluating water accumulation trends in polar regions, and estimating glacier mass balances. The common and interpretable physically derived differential equation models for snow density are piecewise linear as a function of depth (on a transformed scale); thus, they can fail to capture important data features. Moreover, the differential equation parameters show strong spatial autocorrelation. To address these issues, we allow the parameters of the physical model, including random change points over depth, to vary spatially. We also develop a framework for functionally smoothing the physically motivated model. To preserve inference on the interpretable physical model, we project the smoothing function into the physical model's spatially varying null space. The proposed spatially and functionally smoothed snow density model better fits the data while preserving inference on physical parameters. Using this model, we find significant spatial variation in the parameters that govern snow densification.

Detecting pattern transitions in psychological time series - A validation study on the Pattern Transition Detection Algorithm (PTDA).

With the increasing use of real-time monitoring procedures in clinical practice, psychological time series become available to researchers and practitioners. An important interest concerns the identification of pattern transitions which are characteristic features of psychotherapeutic change. Change Point Analysis (CPA) is an established method to identify the point where the mean and/or variance of a time series change, but changes of other and more complex features cannot be detected by this method. In this study, an extension of the CPA, the Pattern Transition Detection Algorithm (PTDA), is optimized and validated for psychological time series with complex pattern transitions. The algorithm uses the convergent information of the CPA and other methods like Recurrence Plots, Time Frequency Distributions, and Dynamic Complexity. These second level approaches capture different aspects of the primary time series. The data set for testing the PTDA (300 time series) is created by an instantaneous control parameter shift of a simulation model of psychotherapeutic change during the simulation runs. By comparing the dispersion of random change points with the real change points, the PTDA determines if the transition point is significant. The PTDA reduces the rate of false negative and false positive results of the CPA below 5% and generalizes its application to different types of pattern transitions. RQA quantifiers also can be used for the identification of nonstationary transitions in time series which was illustrated by using Determinism and Entropy. The PTDA can be easily used with Matlab and is freely available at Matlab File Exchange (https://www.mathworks.com/matlabcentral/fileexchange/80380-pattern-transition-detection-algorithm-ptda).

Stochastic Version of EM Algorithm for Nonlinear Random Change-Point Models.

Random effect change-point models are commonly used to infer individual-specific time of event that induces trend change of longitudinal data. Linear models are often employed before and after the change point. However, in applications such as HIV studies, a mechanistic nonlinear model can be derived for the process based on the underlying data-generation mechanisms and such nonlinear model may provide better ``predictions". In this article, we propose a random change-point model in which we model the longitudinal data by segmented nonlinear mixed effect models. Inference wise, we propose a maximum likelihood solution where we use the Stochastic Expectation-Maximization (StEM) algorithm coupled with independent multivariate rejection sampling through Gibbs's sampler. We evaluate the method with simulations to gain insights.

Least square and Empirical Bayes Approaches for Estimating Random Change Points

Here we develop methods for applications where random change points are known to be present a priori and the interest lies in their estimation and investigating risk factors that influence them. A simple least square method estimating each individual’s change point based on one’s own observations is first proposed. An easy-to-compute empirical Bayes type shrinkage is then proposed to pool information from separately estimated change points. A method to improve the empirical Bayes estimates is developed. Simulations are conducted to compare least-square estimates and Bayes shrinkage estimates. The proposed methods are applied to the Berkeley Growth Study data to estimate the transition age of the puberty height growth.

Least square and Empirical Bayes Approaches for Estimating Random Change Points

Here we develop methods for applications where random change points are known to be present a priori and the interest lies in their esti- mation and investigating risk factors that influence them. A simple least- square method estimating each individual's change point based on one's own observations is first proposed. An easy-to-compute empirical Bayes type shrinkage is then proposed to pool information from separately esti- mated change points. A method to improve the empirical Bayes estimates is developed. Simulations are conducted to compare least-square estimates and Bayes shrinkage estimates. The proposed methods are applied to the Berkeley Growth Study data to estimate the transition age of the puberty height growth.

Multivariate piecewise joint models with random change-points for skewed-longitudinal and survival data

Methodological development and application of joint models for longitudinal and time-to-event data have mostly coupled a single longitudinal outcome-based linear mixed-effects model with normal distribution and Cox proportional hazards model. In practice, however, (i) profile of subject's longitudinal response may follow a `broken-stick nonlinear' (piecewise) trajectory. Such multiple phases are an important indicator to help quantify treatment effect, disease diagnosis and clinical decision-making. (ii) Normality in longitudinal models is a routine assumption, but it may be unrealistically obscuring important features of subject variations. (iii) Data collected are often featured by multivariate longitudinal outcomes which are significantly correlated, ignoring their correlation may lead to biased estimation. (iv) It is of importance to investigate how multivariate longitudinal outcomes are associated with event time of interest. In the article, driven by a motivating example, we propose Bayesian multivariate piecewise joint models with a skewed distribution and random change-points for longitudinal measures with an attempt to cope with correlated multivariate longitudinal data, adjust departures from normality, mediate accuracy from longitudinal trajectories with random change-point and tailor linkage in specifying a time-to-event process. A real example is analyzed to demonstrate methodology and simulation studies are conducted to evaluate performance of the proposed models and method.

Acceleration of health deficit accumulation in late-life: evidence of terminal decline in frailty index three years before death in the US Health and Retirement Study

BackgroundLittle is known about within-person frailty index (FI) changes during the last years of life. In this study, we assess whether there is a phase of accelerated health deficit accumulation (terminal health decline) in late-life. Material and methodsA total of 23,393 observations from up to the last 21 years of life of 5713 deceased participants of the AHEAD cohort in the Health and Retirement Study were assessed. A FI with 32 health deficits was calculated for up to 10 successive biannual, self- and proxy-reported assessments (1995–2014), and FI changes according to time-to-death were analyzed with a piecewise linear mixed model with random change points. ResultsThe average normal (preterminal) health deficit accumulation rate was 0.01 per year, which increased to 0.05 per year at approximately 3 years before death. Terminal decline began earlier in women and was steeper among men. The accelerated (terminal) rate of health deficit accumulation began at a FI-value of 0.29 in the total sample, 0.27 for men, and 0.30 for women. ConclusionWe found evidence for an observable terminal health decline in the FI following declining physiological reserves and failing repair mechanisms. Our results suggest a conceptually meaningful cut-off value for the continuous FI around 0.30.

To what degree is late life cognitive decline driven by age-related neuropathologies?

The ageing brain is vulnerable to a wide array of neuropathologies. Prior work estimated that the three most studied of these, Alzheimer's disease, infarcts, and Lewy bodies, account for ∼40% of the variation in late life cognitive decline. However, that estimate did not incorporate many other diseases that are now recognized as potent drivers of cognitive decline [e.g. limbic predominant age-related TDP-43 encephalopathy (LATE-NC), hippocampal sclerosis, other cerebrovascular conditions]. We examined the degree to which person-specific cognitive decline in old age is driven by a wide array of neuropathologies. Deceased participants (n = 1164) from two longitudinal clinical-pathological studies, the Rush Memory and Aging Project and Religious Orders Study, completed up to 24 annual evaluations including 17 cognitive performance tests and underwent brain autopsy. Neuropathological examinations provided 11 pathological indices, including markers of Alzheimer's disease, non- Alzheimer's disease neurodegenerative diseases (i.e. LATE-NC, hippocampal sclerosis, Lewy bodies), and cerebrovascular conditions (i.e. macroscopic infarcts, microinfarcts, cerebral amyloid angiopathy, atherosclerosis, and arteriolosclerosis). Mixed effects models examined the linear relation of pathological indices with global cognitive decline, and random change point models examined the relation of the pathological indices with the onset of terminal decline and rates of preterminal and terminal decline. Cognition declined an average of about 0.10 unit per year (estimate = -0.101, SE = 0.003, P < 0.001) with considerable heterogeneity in rates of decline (variance estimate for the person-specific slope of decline was 0.0094, P < 0.001). When considered separately, 10 of 11 pathological indices were associated with faster decline and accounted for between 2% and 34% of the variation in decline, respectively. When considered simultaneously, the 11 pathological indices together accounted for 43% of the variation in decline; Alzheimer's disease-related indices accounted for 30-36% of the variation, non-Alzheimer's disease neurodegenerative indices 4-10%, and cerebrovascular indices 3-8%. Finally, the 11 pathological indices combined accounted for less than a third of the variation in the onset of terminal decline (28%) and rates of preterminal (32%) and terminal decline (19%). Although age-related neuropathologies account for a large proportion of the variation in late life cognitive decline, considerable variation remains unexplained even after considering a wide array of neuropathologies. These findings highlight the complexity of cognitive ageing and have important implications for the ongoing effort to develop effective therapeutics and identify novel treatment targets.

An Empirical Comparison of Segmented and Stochastic Linear Mixed Effects Models to Estimate Rapid Disease Progression in Longitudinal Biomarker Studies

Longitudinal studies of rapid disease progression often rely on noisy biomarkers; the underlying longitudinal process naturally varies between subjects and within an individual subject over time; the process can have substantial memory in the form of within-subject correlation. Cystic fibrosis lung disease progression is measured by changes in a lung function marker (FEV1), such as a prolonged drop in lung function, clinically termed rapid decline. Choosing a longitudinal model that estimates rapid decline can be challenging, requiring covariate specifications to assess drug effect while balancing choices of covariance functions. Two classes of longitudinal models have recently been proposed: segmented and stochastic linear mixed effects (LMEs) models. With segmented LMEs, random changepoints are used to estimate the timing and degree of rapid decline, treating these points as structural breaks in the underlying longitudinal process. In contrast, stochastic LMEs, such as random walks, are locally linear but use continuously changing slopes, viewing bouts of rapid decline as localized, sharp changes. We compare commonly used variants of these approaches through an application using the Cystic Fibrosis Foundation Patient Registry. Changepoint modeling had the worst fit and predictive accuracy but certain covariance forms in stochastic LMEs produced problematic variance estimates.

Remaining useful life prediction for multi-phase deteriorating process based on Wiener process

Owing to the environmental stress and internal materials, the degradation signals show multiple phases characteristics, which have frequently been encountered in practice. In this paper, a multi-phase degradation model with jumps based on Wiener process is formulated to describe the multi-phase degradation pattern. The modified information criterion is adopted to determine the change-point number, and a simple yet effective algorithm is proposed for obtaining the change-point locations, which are critical for remaining useful life prediction. In the proposed model, to take into account the unit heterogeneity, all model parameters are assumed to be random variables. A Bayesian approach is used for integrating historical data and real-time data, which involves two stages, the off-line stage and on-line stage. Meanwhile, by treating the drift parameter and the diffusion parameter of each phase as latent parameters, the corresponding hyper-parameters are estimated based on the expectation maximization (EM) algorithm. The model parameters are updated under Bayesian rule at the on-line stage. Then, considering the multiple random change points and the corresponding jumps, the expressions of remaining useful life are derived under the concept of first passage time. Finally, a numerical simulation and a practical case study are provided for demonstrating the effectiveness.

Bayesian Change-Point Joint Models for Multivariate Longitudinal and Time-to-Event Data

Joint modeling of longitudinal and survival data is an active area of statistical research that has received much attention recently. Although it is a common practice to analyze complex longitudinal data using nonlinear mixed-effects (NLME) or nonparametric mixed-effects (NPME) models in literature, the following issues may standout: (i) In practice, the profile of each subject’s longitudinal response may follow a “broken-stick” like trajectory, indicating multiple phases of increase and/or decline in response trajectory. Such multiple phases (with random change-points) may be an important indicator to help quantify treatment effect and clinical diagnosis of a disease. To estimate random change-points, NLME or NPME models become a challenge. (ii) Many studies are often to collect data of multivariate longitudinal variables which may be significantly correlated, ignoring their correlation may lead to biased estimate and reduce efficiency. Moreover, the time-to-event may be dependent of the multivariate longitudinal measures and it is of importance to explore their association. (iii) Missing observations in the longitudinal responses are often encountered. The missing data are likely to be informative (nonignorable) and ignoring this phenomenon may result in inaccurate statistical inference. In this article, under a Bayesian framework we consider a piecewise joint model for multivariate longitudinal and time-to-event data to accurately estimate change rates of longitudinal trajectory patterns and timing of change-point which may be critical indicators to quantify the effect of longitudinal profile on the risk of an event. The proposed models and method are applied to analyze a longitudinal dataset arising from a diabetes study. Simulation studies are conducted to assess the performance of the proposed models under various scenarios.

Random changepoint segmented regression with smooth transition.

We consider random changepoint segmented regression models to analyse data from a study conducted to verify whether treatment with stem cells may delay the onset of a symptom of amyotrophic lateral sclerosis in genetically modified mice. The proposed models capture the biological aspects of the data, accommodating a smooth transition between the periods with and without symptoms. An additional changepoint is considered to avoid negative predicted responses. Given the nonlinear nature of the model, we propose an algorithm to estimate the fixed parameters and to predict the random effects by fitting linear mixed models iteratively via standard software. We compare the variances obtained in the final step with bootstrapped and robust ones.

A Bayesian hierarchical change point model with parameter constraints.

Alzheimer's disease is the leading cause of dementia among adults aged 65 or above. Alzheimer's disease is characterized by a change point signaling a sudden and prolonged acceleration in cognitive decline. The timing of this change point is of clinical interest because it can be used to establish optimal treatment regimens and schedules. Here, we present a Bayesian hierarchical change point model with a parameter constraint to characterize the rate and timing of cognitive decline among Alzheimer's disease patients. We allow each patient to have a unique random intercept, random slope before the change point, random change point time, and random slope after the change point. The difference in slope before and after a change point is constrained to be nonpositive, and its parameter space is partitioned into a null region (representing normal aging) and a rejection region (representing accelerated decline). Using the change point time, the estimated slope difference, and the threshold of the null region, we are able to (1) distinguish normal aging patients from those with accelerated cognitive decline, (2) characterize the rate and timing for patients experiencing cognitive decline, and (3) predict personalized risk of progression to dementia due to Alzheimer's disease. We apply the approach to data from the Religious Orders Study, a national cohort study of aging Catholic nuns, priests, and lay brothers.

A curvilinear bivariate random changepoint model to assess temporal order of markers.

In biomedical research, various longitudinal markers measuring different quantities are often collected over time. For example, repeated measures of psychometric scores are very informative about the degradation process toward dementia. These trajectories are generally nonlinear with an acceleration of the decline a few years before the diagnosis and a large heterogeneity between psychometric tests depending on the underlying cognitive function to be evaluated and the metrological properties of the test. Comparing the times of acceleration of the decline before diagnosis between cognitive tests is useful to better understand the natural history of the disease. Our objective is to propose a bivariate random changepoint model that allows for the comparison of the mean time of change between two markers. A frequentist approach is proposed that gives validated statistical tests to assess the temporal order of the changepoints. Using a spline transformation function, the model is designed to handle non-Gaussian data, that are common for cognitive scores which frequently exhibit a strong ceiling effect. The procedure is assessed through a simulation study and applied to a French cohort of elderly to identify the order of the decline of several cognitive scores. The whole methodology has been implemented in a R package freely available.

Modeling repeated labor curves in consecutive pregnancies: Individualized prediction of labor progression from previous pregnancy data.

The measurement of cervical dilation of a pregnant woman is used to monitor the progression of labor until 10 cm when pushing begins. There is anecdotal evidence that labor tracks across repeated pregnancies; moreover, no statistical methodology has been developed to address this important issue, which can help obstetricians make more informed clinical decisions about an individual woman's progression. Motivated by the NICHD Consecutive Pregnancies Study (CPS), we propose new methodology for analyzing labor curves across consecutive pregnancies. Our focus is both on studying the correlation between repeated labor curves on the same woman and on using the cervical dilation data from prior pregnancies to predict subsequent labor curves. We propose a hierarchical random effects model with a random change point that characterizes repeated labor curves within and between women to address these issues. We employ Bayesian methodology for parameter estimation and prediction. Model diagnostics to examine the appropriateness of the hierarchical random effects structure for characterizing the dependence structure across consecutive pregnancies are also proposed. The methodology was used in analyzing the CPS data and in developing a predictor for labor progression that can be used in clinical practice.

Distinct Growth Phases in Early Life Associated With the Risk of Type 1 Diabetes: The TEDDY Study.

This study investigates two-phase growth patterns in early life and their association with development of islet autoimmunity (IA) and type 1 diabetes (T1D). The Environmental Determinants of Diabetes in the Young (TEDDY) study followed 7,522 genetically high-risk children in Sweden, Finland, Germany, and the U.S. from birth for a median of 9.0 years (interquartile range 5.7-10.6) with available growth data. Of these, 761 (10.1%) children developed IA and 290 (3.9%) children were diagnosed with T1D. Bayesian two-phase piecewise linear mixed models with a random change point were used to estimate children's individual growth trajectories. Cox proportional hazards models were used to assess the effects of associated growth parameters on the risks of IA and progression to T1D. A higher rate of weight gain in infancy was associated with increased IA risk (hazard ratio [HR] 1.09 [95% CI 1.02, 1.17] per 1 kg/year). A height growth pattern with a lower rate in infancy (HR 0.79 [95% CI 0.70, 0.90] per 1 cm/year), higher rate in early childhood (HR 1.48 [95% CI 1.22, 1.79] per 1 cm/year), and younger age at the phase transition (HR 0.76 [95% CI 0.58, 0.99] per 1 month) was associated with increased risk of progression from IA to T1D. A higher rate of weight gain in early childhood was associated with increased risk of progression from IA to T1D (HR 2.57 [95% CI 1.34, 4.91] per 1 kg/year) in children with first-appearing GAD autoantibody only. Growth patterns in early life better clarify how specific growth phases are associated with the development of T1D.

Mandelbrot's Stochastic Time Series Models

I survey and illustrate the main time series models that Mandelbrot introduced into time series analysis in the 1960s and 1970s. I focus particularly on the members of the additive fractional stable family including Lévy flights and fractional Brownian motion (fBm), noting some of the less well‐known aspects of this family, such as the cases when the self‐similarity exponent H and the Hurst exponent J differ. I briefly discuss the role of multiplicative models in modeling the physics of cascades. I then recount the still little‐known story of Mandelbrot's work on fractional renewal models in the late 1960s, explaining how these differ from their more familiar fBm counterpart and form a “missing link” between fBm and the problem of random change points. I conclude by highlighting the frontier problem of damped fractional models.

Reliability modeling for a two-phase degradation system with a change point based on a Wiener process

A two-phase degradation system with a change point is commonly seen in many real-world degradation systems due to influence of internal mechanisms and external environments, such as lithium-ion batteries, light-emitting diodes (LEDs), plasma display panels (PDPs), and vacuum fluorescent displays (VFDs). Random jumps exist at the change points of the degradation processes. Motivated by this phenomenon, this paper attempts to model such kinds of systems based on the Wiener process. Specifically, In Model 1, a constant change point model is first considered. The analytical results of the system reliability, lifetime distribution, and the remaining useful lifetime (RUL) are obtained. Then, based on Model 1, a random change point model, i.e., Model 2, is proposed to explain the change point phenomenon which is influenced by the accumulative effect of a shock process. Two phase-change patterns (PCPs) are considered in Model 2. For PCP I, the change point is triggered when the number of shocks reaches a predetermined threshold. For PCP II, the change point is triggered when the accumulative shock damage exceeds the corresponding threshold. The system reliability indexes are studied by using analytical and simulation methods. Finally, several numerical examples are given to illustrate the approaches and results.

A hypothesis testing procedure for random changepoint mixed models.

In biomedical research, random changepoint mixed models are used to take into account an individual breakpoint in a biomarker trajectory. This may be observed in the cognitive decline measured by psychometric tests in the prediagnosis phase of Alzheimer's disease. The existence, intensity and duration of this accelerated decline can depend on individual characteristics. The main objective of our work is to propose inferential methods to assess the existence of this phase of accelerated decline, ie, the existence of a random changepoint. To do so, we use a mixed model with two linear phases and test the nullity of the parameter measuring the difference of slopes between the two phases. Because we face the issue of nuisance parameters being unidentifiable under the null hypothesis, the supremum of the classic score test statistic on these parameters is used. The asymptotic distribution of the supremum under the null is approached with a perturbation method based on the multiplier bootstrap. The performance of our testing procedure is assessed via simulations and the test is applied to the French cohort PAQUID of elderly subjects to study the shape of the prediagnosis decline according to educational level. The test is significant for both educational levels and the estimated trajectories confirmed that educational level is a good marker for cognitive reserve.

83-OR: Distinct Growth Phases in Early Life Associated with Risk of Type 1 Diabetes Prodrome: The TEDDY Study

This study investigates two-phase growth patterns in early life and their associations with development of islet autoimmunity (IA) and type 1 diabetes (T1D). There were 7521 genetically high-risk children from Sweden, Finland, U.S. and Germany followed from birth for a median of 9.0 (IQR 5.7-10.6) years with available growth data. Of these, 761 children developed IA and 290 children progressed to T1D. Bayesian two-phase piecewise linear mixed models with a random change point were used to evaluate distinct growth phases in early life (Figure 1). Cox proportional hazard models were used to assess growth phase effects on risk of IA and progression to T1D. A higher rate of weight gain (kg/year) in the first phase was associated with an increased IA risk (HR =1.10, 95% CI 1.02, 1.18). A height growth pattern with a slower growth rate (cm/year) in the first phase, a higher growth rate in the second phase and younger age (months) at the phase transition was associated with an increased risk of progression from IA to T1D (HR=0.80, 95% CI 0.70, 0.91; HR=1.47, 95% CI 1.21, 1.78; HR=0.69, 95% CI 0.52, 0.93, respectively). A higher rate of weight gain in the second phase was associated with an increased risk of progression from IA to T1D (HR=2.54, 95% CI 1.36, 4.74) in children with first appearing GADA-only. Considering growth by phases in early life better clarified how specific growth phases contribute to the risk of IA and T1D. Disclosure X. Liu: None. K. Vehik: None. Y. Huang: None. H. Elding Larsson: None. J. Toppari: None. A. Ziegler: None. J. She: None. M. Rewers: None. W. Hagopian: Research Support; Self; Novo Nordisk A/S. B. Akolkar: None. J. Krischer: None. Funding National Institute of Diabetes and Digestive and Kidney Diseases; National Institute of Allergy and Infectious Diseases; Eunice Kennedy Shriver National Institute of Child Health and Human Development; National Institute of Environmental Health Sciences; Centers for Disease Control and Prevention; JDRF; University of Florida (UL1TR000064); University of Colorado (UL1TR001082)