Discrete Time Series Research Articles

Discrete Time Series Forecasting of Hive Weight, In-Hive Temperature, and Hive Entrance Traffic in Non-Invasive Monitoring of Managed Honey Bee Colonies: Part I.

From June to October, 2022, we recorded the weight, the internal temperature, and the hive entrance video traffic of ten managed honey bee (Apis mellifera) colonies at a research apiary of the Carl Hayden Bee Research Center in Tucson, AZ, USA. The weight and temperature were recorded every five minutes around the clock. The 30 s videos were recorded every five minutes daily from 7:00 to 20:55. We curated the collected data into a dataset of 758,703 records (280,760-weight; 322,570-temperature; 155,373-video). A principal objective of Part I of our investigation was to use the curated dataset to investigate the discrete univariate time series forecasting of hive weight, in-hive temperature, and hive entrance traffic with shallow artificial, convolutional, and long short-term memory networks and to compare their predictive performance with traditional autoregressive integrated moving average models. We trained and tested all models with a 70/30 train/test split. We varied the intake and the predicted horizon of each model from 6 to 24 hourly means. Each artificial, convolutional, and long short-term memory network was trained for 500 epochs. We evaluated 24,840 trained models on the test data with the mean squared error. The autoregressive integrated moving average models performed on par with their machine learning counterparts, and all model types were able to predict falling, rising, and unchanging trends over all predicted horizons. We made the curated dataset public for replication.

Data-driven reduced order surrogate modeling for coronary in-stent restenosis

Background:The intricate process of coronary in-stent restenosis (ISR) involves the interplay between different mediators, including platelet-derived growth factor, transforming growth factor-β, extracellular matrix, smooth muscle cells, endothelial cells, and drug elution from the stent. Modeling such complex multiphysics phenomena demands extensive computational resources and time. Methods:This paper proposes a novel non-intrusive data-driven reduced order modeling approach for the underlying multiphysics time-dependent parametrized problem. In the offline phase, a 3D convolutional autoencoder, comprising an encoder and decoder, is trained to achieve dimensionality reduction. The encoder condenses the full-order solution into a lower-dimensional latent space, while the decoder facilitates the reconstruction of the full solution from the latent space. To deal with the 5D input datasets (3D geometry + time series + multiple output channels), two ingredients are explored. The first approach incorporates time as an additional parameter and applies 3D convolution on individual time steps, encoding a distinct latent variable for each parameter instance within each time step. The second approach reshapes the 3D geometry into a 2D plane along a less interactive axis and stacks all time steps in the third direction for each parameter instance. This rearrangement generates a larger and complete dataset for one parameter instance, resulting in a singular latent variable across the entire discrete time-series. In both approaches, the multiple outputs are considered automatically in the convolutions. Moreover, Gaussian process regression is applied to establish correlations between the latent variable and the input parameter. Results:The constitutive model reveals a significant acceleration in neointimal growth between 30−60 days post percutaneous coronary intervention (PCI). The surrogate models applying both approaches exhibit high accuracy in pointwise error, with the first approach showcasing smaller errors across the entire evaluation period for all outputs. The parameter study on drug dosage against ISR rates provides noteworthy insights of neointimal growth, where the nonlinear dependence of ISR rates on the peak drug flux exhibits intriguing periodic patterns. Applying the trained model, the rate of ISR is effectively evaluated, and the optimal parameter range for drug dosage is identified. Conclusion:The demonstrated non-intrusive reduced order surrogate model proves to be a powerful tool for predicting ISR outcomes. Moreover, the proposed method lays the foundation for real-time simulations and optimization of PCI parameters.

Heteroscedasticity Detection in the existence of Outliers in Discrete-Time Series

Heteroscedasticity Detection in the existence of Outliers in Discrete-Time Series

Posterior Representations for Bayesian Context Trees: Sampling, Estimation and Convergence

We revisit the Bayesian Context Trees (BCT) modelling framework for discrete time series, which was recently found to be very effective in numerous tasks including model selection, estimation and prediction. A novel representation of the induced posterior distribution on model space is derived in terms of a simple branching process, and several consequences of this are explored in theory and in practice. First, it is shown that the branching process representation leads to a simple variable-dimensional Monte Carlo sampler for the joint posterior distribution on models and parameters, which can efficiently produce independent samples. This sampler is found to be more efficient than earlier MCMC samplers for the same tasks. Then, the branching process representation is used to establish the asymptotic consistency of the BCT posterior, including the derivation of an almost-sure convergence rate. Finally, an extensive study is carried out on the performance of the induced Bayesian entropy estimator. Its utility is illustrated through both simulation experiments and real-world applications, where it is found to outperform several state-of-the-art methods.

Subgroup detection in linear growth curve models with generalized linear mixed model (GLMM) trees.

Growth curve models are popular tools for studying the development of a response variable within subjects over time. Heterogeneity between subjects is common in such models, and researchers are typically interested in explaining or predicting this heterogeneity. We show how generalized linear mixed-effects model (GLMM) trees can be used to identify subgroups with different trajectories in linear growth curve models. Originally developed for clustered cross-sectional data, GLMM trees are extended here to longitudinal data. The resulting extended GLMM trees are directly applicable to growth curve models as an important special case. In simulated and real-world data, we assess performance of the extensions and compare against other partitioning methods for growth curve models. Extended GLMM trees perform more accurately than the original algorithm and LongCART, and similarly accurate compared to structural equation model (SEM) trees. In addition, GLMM trees allow for modeling both discrete and continuous time series, are less sensitive to (mis-)specification of the random-effects structure and are much faster to compute.

A method for calculating fall risk parameters from discrete stride time series regardless of sensor placement

BackgroundTo complement traditional clinical fall risk assessments, research is oriented towards adding real-life gait-related fall risk parameters (FRP) using inertial sensors fixed to a specific body position. While fixing the sensor position can facilitate data processing, it can reduce user compliance. A newly proposed step detection method, Smartstep, has been proven to be robust against sensor position and real-life challenges. Moreover, FRP based on step variability calculated from stride times (Standard deviation (SD), Coefficient of Variance (Cov), fractal exponent, and sample entropy of stride duration) proved to be useful to prospectively predict the fall risk. Research questionsTo evaluate whether Smartstep is convenient for calculating FRP from different sensor placements. Methods29 elderly performed a 6-minute walking test with IMU placed on the waist and the wrist. FRP were computed from step-time estimated from Smartstep and compared to those obtained from foot-mounted inertial sensors: precision and recall of the step detection, Root mean square error (RMSE) and Intraclass Correlation Coefficient (ICC) of stride durations, and limits of agreement of FRP. ResultsThe step detection precision and recall were respectively 99.5% and 95.9% for the waist position, and 99.4% and 95.7% for the wrist position. The ICC and RMSE of stride duration were 0.91 and 54 ms respectively for both the waist and the hand position. The limits of agreement of Cov, SD, fractal exponent, and sample entropy of stride duration are respectively 2.15%, 25 ms, 0.3, 0.5 for the waist and 1.6%, 16 ms, 0.23, 0.4 for the hand. SignificanceRobust against the elderly's gait and different body locations, especially the wrist, this method can open doors toward ambulatory measurements of steps, and calculation of different discrete stride-related falling risk indicators.

Identifying Effective Connectivity between Stochastic Neurons with Variable-Length Memory Using a Transfer Entropy Rate Estimator.

Information theory explains how systems encode and transmit information. This article examines the neuronal system, which processes information via neurons that react to stimuli and transmit electrical signals. Specifically, we focus on transfer entropy to measure the flow of information between sequences and explore its use in determining effective neuronal connectivity. We analyze the causal relationships between two discrete time series, X:=Xt:t∈Z and Y:=Yt:t∈Z, which take values in binary alphabets. When the bivariate process (X,Y) is a jointly stationary ergodic variable-length Markov chain with memory no larger than k, we demonstrate that the null hypothesis of the test-no causal influence-requires a zero transfer entropy rate. The plug-in estimator for this function is identified with the test statistic of the log-likelihood ratios. Since under the null hypothesis, this estimator follows an asymptotic chi-squared distribution, it facilitates the calculation of p-values when applied to empirical data. The efficacy of the hypothesis test is illustrated with data simulated from a neuronal network model, characterized by stochastic neurons with variable-length memory. The test results identify biologically relevant information, validating the underlying theory and highlighting the applicability of the method in understanding effective connectivity between neurons.

Evaluation and Selection of Multi-Spectral Indices to Classify Vegetation Using Multivariate Functional Principal Component Analysis

The identification, classification and mapping of different plant communities and habitats is of fundamental importance for defining biodiversity monitoring and conservation strategies. Today, the availability of high temporal, spatial and spectral data from remote sensing platforms provides dense time series over different spectral bands. In the case of supervised mapping, time series based on classical vegetation indices (e.g., NDVI, GNDVI, …) are usually input characteristics, but the selection of the best index or set of indices (which guarantees the best performance) is still based on human experience and is also influenced by the study area. In this work, several different time series, based on Sentinel-2 images, were created exploring new combinations of bands that extend the classic basic formulas as the normalized difference index. Multivariate Functional Principal Component Analysis (MFPCA) was used to contemporarily decompose the multiple time series. The principal multivariate seasonal spectral variations identified (MFPCA scores) were classified by using a Random Forest (RF) model. The MFPCA and RF classifications were nested into a forward selection strategy to identify the proper and minimum set of indices’ (dense) time series that produced the most accurate supervised classification of plant communities and habitat. The results we obtained can be summarized as follows: (i) the selection of the best set of time series is specific to the study area and the habitats involved; (ii) well-known and widely used indices such as the NDVI are not selected as the indices with the best performance; instead, time series based on original indices (in terms of formula or combination of bands) or underused indices (such as those derivable with the visible bands) are selected; (iii) MFPCA efficiently reduces the dimensionality of the data (multiple dense time series) providing ecologically interpretable results representing an important tool for habitat modelling outperforming conventional approaches that consider only discrete time series.

Approximated Characteristics of Bivariate Discrete Time Series with Missing Data

Approximated Characteristics of Bivariate Discrete Time Series with Missing Data

Sorting ECGs by lag irreversibility

In this work we introduce the lag irreversibility function as a method to assess time-irreversibility in discrete time series. It quantifies the degree of time-asymmetry for the joint probability function of the state variable under study and the state variable lagged in time. We test its performance in a time-irreversible Markov chain model for which theoretical results are known. Moreover, we use our approach to analyze electrocardiographic recordings of four groups of subjects: healthy young individuals, healthy elderly individuals, and persons with two different disease conditions, namely, congestive heart failure and atrial fibrillation. We find that by studying jointly the variability of the amplitudes of the different waves in the electrocardiographic signals, one can obtain an efficient method to discriminate between the groups already mentioned. Finally, we test the accuracy of our method using the ROC analysis.

Advancing diabetes prediction with a progressive self-transfer learning framework for discrete time series data

Although diabetes mellitus is a complex and pervasive disease, most studies to date have focused on individual features, rather than considering the complexities of multivariate, multi-instance, and time-series data. In this study, we developed a novel diabetes prediction model that incorporates these complex data types. We applied advanced techniques of data imputation (bidirectional recurrent imputation for time series; BRITS) and feature selection (the least absolute shrinkage and selection operator; LASSO). Additionally, we utilized self-supervised algorithms and transfer learning to address the common issues with medical datasets, such as irregular data collection and sparsity. We also proposed a novel approach for discrete time-series data preprocessing, utilizing both shifting and rolling time windows and modifying time resolution. Our study evaluated the performance of a progressive self-transfer network for predicting diabetes, which demonstrated a significant improvement in metrics compared to non-progressive and single self-transfer prediction tasks, particularly in AUC, recall, and F1 score. These findings suggest that the proposed approach can mitigate accumulated errors and reflect temporal information, making it an effective tool for accurate diagnosis and disease management. In summary, our study highlights the importance of considering the complexities of multivariate, multi-instance, and time-series data in diabetes prediction.

Properties and interpretation of the Skellam model—A discrete‐time contest competition population model

AbstractThe Skellam model describes discrete‐time population dynamics of a single species assuming uniform (i.e., random or Poissonian) individual distribution and intraspecific contest competition. Apart from studies on individual‐based models derived from first principles it has been rarely applied in ecological research although in specific situations it may be more appropriate than, for instance, the frequently used Ricker model, which is derived assuming scramble competition among the individuals. In this article, we offer an insight into the first principles underlying the Skellam model and provide an alternative parameterization of the model in terms of two commonly used parameters: intrinsic rate of population increase and carrying capacity. We also provide guidelines and software for fitting the Skellam model to discrete population time series data. In light of these findings, the Skellam model may be a useful alternative for a range of purposes where it has been earlier overlooked.

Mitigation of assembly constraints for floating offshore wind turbines using discrete event simulation

There is a large and increasing pipeline of floating offshore wind projects with total global floating offshore wind capacity projected to grow year on year by, on average, between 59 and 104 % in the 2020s. This will lead to competition for infrastructure resources, in particular, port facilities for the construction and marshalling of the floating foundations and turbines. It is likely that multiple ports will need to be combined to provide the necessary fabrication capacity for a floating offshore wind farm of commercial scale. To enable an efficient and coordinated utilisation of multiple fabrication ports, it is crucial to understand the likely duration of different assembly and construction activities at different locations. However, at present this task is difficult to perform using top-down estimation models, as commercial-scale floating offshore wind farms comprising many tens of units have not been built to date. In this work we present a methodology, based on discrete event simulation (DES) and time series analysis, to produce an explicit simulation-based estimate of assembly activity durations, which are sensitive to setting specific factors. Three example case studies are outlined to demonstrate the ability to capture the variation in activity duration due to resource availability, and the season and location of activity. The methodology will be of use to project planners as it can be used at an early stage in the project life-cycle to appraise and adopt different construction strategies.

Reconstruction of Missing Well-Logs Using Facies-Informed Discrete Wavelet Transform and Time Series Regression

Summary Geophysical logging is widely used in lithofacies identification, reservoir parameter prediction, and geological modeling. However, it is common to have well-log sections with low-quality and/or missing segments. Repeating the well-log measurements is not only expensive but might also be impossible depending on the condition of the borehole walls. In these situations, reliable and accurate well-log prediction is, therefore, necessary in different stages of the geomodeling workflow. In this study, we propose a time series regression model to predict missing well-log data, incorporating facies information as an additional geological input and using discrete wavelet transform (DWT) to denoise the input data set. The main contributions of this work are threefold: (i) We jointly use facies information with well logs as the input data set; (ii) we use DWT to denoise the input data and consequently improve the signal-to-noise ratio of the input data; and (iii) we regard the depth domain as the time domain and use a time series regression algorithm for log reconstruction modeling. We show a real application example in two distinct scenarios. In the first, we predict missing well-log intervals. In the second, we predict complete well logs. The experimental results show the ability of the proposed prediction model to recover missing well-log data with high accuracy levels.

Dynamics near the edge-of-chaos in a fiber laser model

We consider the dynamics in a fiber laser model for pump modulation frequencies much smaller than that of the relaxation oscillations of the model. We find that for low enough subharmonic pump frequencies and those near the Edge-of-Chaos, long-term correlations arise in the discrete time series of the laser intensity. Moreover, within this low frequency range we demonstrate that both frequency locking and, imperfect phase synchronization occur. These are determined through the notion of mean-frequency. We conjecture that these dynamical effects may be experimentally observed in suitable time series.

Learnable Path in Neural Controlled Differential Equations

Neural controlled differential equations (NCDEs), which are continuous analogues to recurrent neural networks (RNNs), are a specialized model in (irregular) time-series processing. In comparison with similar models, e.g., neural ordinary differential equations (NODEs), the key distinctive characteristics of NCDEs are i) the adoption of the continuous path created by an interpolation algorithm from each raw discrete time-series sample and ii) the adoption of the Riemann--Stieltjes integral. It is the continuous path which makes NCDEs be analogues to continuous RNNs. However, NCDEs use existing interpolation algorithms to create the path, which is unclear whether they can create an optimal path. To this end, we present a method to generate another latent path (rather than relying on existing interpolation algorithms), which is identical to learning an appropriate interpolation method. We design an encoder-decoder module based on NCDEs and NODEs, and a special training method for it. Our method shows the best performance in both time-series classification and forecasting.

Synthesizing forecasts to inform decision‐making and advance ecological theory

Synthesizing forecasts to inform decision‐making and advance ecological theory

Epidemic change-point detection in general integer-valued time series

ABSTRACT In this paper, we consider the structural change in a class of discrete valued time series, where the true conditional distribution of the observations is assumed to be unknown. The conditional mean of the process depends on a parameter θ ∗ which may change over time. We provide sufficient conditions for the consistency and the asymptotic normality of the Poisson quasi-maximum likelihood estimator (QMLE) of the model. We consider an epidemic change-point detection and propose a test statistic based on the QMLE of the parameter. Under the null hypothesis of a constant parameter (no change), the test statistic converges to a distribution obtained from increments of a Browninan bridge. The test statistic diverges to infinity under the epidemic alternative, which establishes that the proposed procedure is consistent in power. The effectiveness of the proposed procedure is illustrated by simulated and real data examples.

Stationary Density Estimation of Itô Diffusions Using Deep Learning

In this paper, we consider the density estimation problem associated with the stationary measure of ergodic Itô diffusions from a discrete-time series that approximate the solutions of the stochastic differential equations. To take advantage of the characterization of density function through the stationary solution of a parabolic-type Fokker–Planck PDE, we proceed as follows: First, we employ deep neural networks to approximate the drift and diffusion terms of the SDE by solving appropriate supervised learning tasks. Subsequently, we solve a steady-state Fokker–Planck equation associated with the estimated drift and diffusion coefficients with a neural-network–based least squares method. We establish the convergence of the proposed scheme under appropriate mathematical assumptions, accounting for the generalization errors induced by regressing the drift and diffusion coefficients and the PDE solvers. This theoretical study relies on a recent perturbation theory of Markov chain result that shows a linear dependence of the density estimation to the error in estimating the drift term and generalization error results of nonparametric regression and PDE regression solution obtained with neural-network models. We demonstrate the effectiveness of this method by numerical simulations of a two-dimensional Student t-distribution and a 20-dimensional Langevin dynamics.

Generative Adversarial Networks in Time Series: A Systematic Literature Review

Generative adversarial network (GAN) studies have grown exponentially in the past few years. Their impact has been seen mainly in the computer vision field with realistic image and video manipulation, especially generation, making significant advancements. Although these computer vision advances have garnered much attention, GAN applications have diversified across disciplines such as time series and sequence generation. As a relatively new niche for GANs, fieldwork is ongoing to develop high-quality, diverse, and private time series data. In this article, we review GAN variants designed for time series related applications. We propose a classification of discrete-variant GANs and continuous-variant GANs, in which GANs deal with discrete time series and continuous time series data. Here we showcase the latest and most popular literature in this field—their architectures, results, and applications. We also provide a list of the most popular evaluation metrics and their suitability across applications. Also presented is a discussion of privacy measures for these GANs and further protections and directions for dealing with sensitive data. We aim to frame clearly and concisely the latest and state-of-the-art research in this area and their applications to real-world technologies.