Discrete Time Series Research Articles

Quantifying the Association Between Discrete Event Time Series with Applications to Digital Forensics

SummaryWe consider the problem of quantifying the degree of association between pairs of discrete event time series, with potential applications in forensic and cybersecurity settings. We focus in particular on the case where two associated event series exhibit temporal clustering such that the occurrence of one type of event at a particular time increases the likelihood that an event of the other type will also occur nearby in time. We pursue a non-parametric approach to the problem and investigate various score functions to quantify association, including characteristics of marked point processes and summary statistics of interevent times. Two techniques are proposed for assessing the significance of the measured degree of association: a population-based approach to calculating score-based likelihood ratios when a sample from a relevant population is available, and a resampling approach to computing coincidental match probabilities when only a single pair of event series is available. The methods are applied to simulated data and to two real world data sets consisting of logs of computer activity and achieve accurate results across all data sets.

Study on the Satellite Telemetry Data Classification Based on Self-Learning

Since great redundancy of telemetry data of spacecraft, telemetry data compression is a good solution for the limited bandwidth and contact wireless links. It is important to obtain accurate data characteristic firstly. State-of-the-art machine learning methods work well on data mining and pattern recognition under conditions of the given test data set, which could be used as the available tools for post-event data processing and analysis, such as trend forecasting and outlier detection, but they have not provided the proper solution from the source on-board. In this paper, four base classes of the telemetry data are suggested and studied through the time series feature and information entropy analysis, then a new on-board lightweight self-learning algorithm named Classification Probability calculation - Window Step optimization (CP-WS) is proposed to obtain the class features and make the decision of each single parameter from the continuous discrete telemetry time series. Simulation results show that, our algorithm correctly classifies the simulation and real mission data into the appropriate base class with advantages of high classification accuracy as 100% and adaptive computational complexity from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O(L^{2})$ </tex-math></inline-formula> to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O(L)$ </tex-math></inline-formula> , which could be used in satellite on-board data compression for space-to-ground transmission, especially for the deep space explorers to save important status with less on-board storage space.

ПРАКТИКА ПРОГНОЗУВАННЯ ПОКАЗНИКІВ ДІЯЛЬНОСТІ ПІДПРИЄМСТВА ЩОДО ЗАБЕЗПЕЧЕННЯ КОНКУРЕНТОСПРОМОЖНОСТІ

Prospective forecast of enterprise competitiveness on the basis of extrapolation of enterprise performance indicators with necessary accuracy is investigated and analyzed. Analytical dependencies on the performance indicators of the enterprise and the relevant factors of influence for the enterprise of PJSC “NORTHERN MINING AND PROCESSING PLANT” were constructed. Particular attention is paid to the comparative analysis of the results of the forecast of factors of influence on the competitiveness of the mining enterprise. The main components of the impact on competitiveness are analyzed: expenses on wages (thousand UAH), working capital (thousand UAH), fixed assets (thousand UAH), net income (thousand UAH). A feature of extrapolation in the narrow sense is that it can rely on a rather small amount of data, which is often related to the properties of the processes under study. In the case of industrial enterprises, the data volumes are usually small, which is primarily explained by the presentation of information in the form of discrete time series with a time interval of one year. As a result, the use of extrapolation methods and conclusions of mathematical statistics is not justified and leads to significant errors. In such calculations, it is advisable to rely on point forecasts, since the methods of mathematical statistics are used in the calculation of interval forecasts. It is therefore considered appropriate to extrapolate in such cases by carefully examining the features of the data. Moreover, in analyzing the results of forecasting it is necessary to rely on the opinion of decision makers, that is, experts. The processes of transformation of Ukrainian society require comprehensive forecasting of the results of the enterprise in order to ensure adequate competitiveness. Competitiveness of the enterprise's potential involves exploring the factors of uncertainty and taking into account various risks allows to profit from new opportunities. Extrapolation allows to transfer the regularities of formation of the basic indicators on the activity of the enterprise beyond the current time interval and to provide the scientific basis for the managers to make the appropriate effective decisions. The results obtained lead to the following conclusions: harmonization of regression analysis methods and quantitative forecasting methods is the basis for ensuring the competitiveness of the enterprise.

Obiective ale analizei trendurilor seriilor de timp discrete (Objectives of the Analysis of Trends in Discrete Time Series)

Romanian Abstract: Aceasta lucrare abordeaza câteva dintre principalele obiective ale analizei trendurilor seriilor de timp discrete. Un aspect major al acestei analize consta in identificarea unui model matematic ce descrie tendinţa persistenta pe termen lung a evoluţiei variabilei studiate. Modelul poate evidenţia câteva caracteristici importante ale tendinţei pe termen lung a seriei de timp. Poate fi, de asemenea, folositor in previziunea viitoarelor valori ale seriei de timp. Un alt aspect important al analizei unui trend consta in detectarea schimbarilor semnificative care au survenit (sau care ar putea surveni in viitor) in tendinţa pe termen lung a unei variabile. English Abstract: This paper approaches some main objectives of the analysis of trends in discrete time series. A major aspect of this analysis is to identify a mathematical model that describes the persistent long-term movement of the studied variable. The model could reveal some important characteristics of the long-term time series pattern. It may also be useful for predicting the time series future values. Another important aspect of a trend analysis is to detect the significant changes that occurred (or that could occur in the future) in the long-term pattern of a variable.

Bayesian forecasting of multivariate time series: scalability, structure uncertainty and decisions

I overview recent research advances in Bayesian state-space modeling of multivariate time series. A main focus is on the decouple/recouple concept that enables application of state-space models to increasingly large-scale data, applying to continuous or discrete time series outcomes. The scope includes large-scale dynamic graphical models for forecasting and multivariate volatility analysis in areas such as economics and finance, multi-scale approaches for forecasting discrete/count time series in areas such as commercial sales and demand forecasting, and dynamic network flow models for areas including internet traffic monitoring. In applications, explicit forecasting, monitoring and decision goals are paramount and should factor into model assessment and comparison, a perspective that is highlighted.

The partial visibility curve of the Feigenbaum cascade to chaos

A family of classical mathematical problems considers the visibility properties of geometric figures in the plane, e.g. curves or polygons. In particular, the {\it domination problem} tries to find the minimum number of points that are able to dominate the whole set, the so called, {\it domination number}. Alternatively, other problems try to determine the subsets of points with a given cardinality, that maximize the basin of domination, the {\it partial dominating set}. Since a discrete time series can be viewed as an ordered set of points in the plane, the dominating number and the partial dominating set can be used to obtain additional information about the visibility properties of the series; in particular, the total visibility number and the partial visibility set. In this paper, we apply these two concepts to study times series that are generated from the logistic map. More specifically, we focus this work on the description of the Feigenbaum cascade to the onset of chaos. We show that the whole cascade has the same total visibility number, $v_T=1/4$. However, a different distribution of the partial visibility sets and the corresponding partial visibility curves can be obtained inside both periodic and chaotic regimes. We prove that the partial visibility curve at the Feigenbaum accumulation point $r_{\infty} \approx 3.5699$ is the limit curve of the partial visibility curves ($n+1$-polygonals) that correspond to the periods $T=2^{n}$ for $n=1,2,\ldots$. We analytically calculate the length of these $n+1$-polygonals and, as a limit, we obtain the length of the partial visibility curve at the onset of chaos, $L_{\infty} = L(r_{\infty}) \approx 1.0414387863$. Finally, we compare these results with those obtained from the period 3-cascade, and with the partial visibility curve of the chaotic series at the crossing point $r_c \approx 3.679$.

Development of a combined method for predicting discrete time series with non-stability for forecasting military goods demand

The object of research is a model of the production system of military goods with non-stationary processes. In the study of the time series of the characteristics of the production system, various competing models, as a rule, are obtained under production conditions with stochastic data on the output of products due to bottleneck problems. So, the choice of the best model that describes the production system becomes difficult and critical, because some models that most closely correspond to the observed data may not foresee future values in accordance with the complexity of the model. This study seeks to demonstrate the procedure for selecting a model in a random data system using adjusted weights. This paper presents a method for combining two sets of forecasts. The obtained measurements serve as input with an autocorrelation function and a partial autocorrelation function to obtain the order of predictive models. The model parameters are evaluated and used for forecasting and compared with the original and converted data to obtain the sum of squared errors in (SSE). Models are evaluated for adequacy and subsequently tested against Akaike and Schwarz criteria. Two separate sets of forecasts of time series data are combined to form a combined set of forecasts. It should be noted that when each set of forecasts contains some independent information, combined forecasts can provide an improvement. The proposed method for combining forecasts allows to change weights, can lead to better forecasts. The main conclusion is that a set of forecasts can lead to a lower standard error than any of the initial forecasts. Past errors of each of the initial forecasts are used to determine the weight for joining two original forecasts in the formation of combined forecasts. However, the effectiveness of the forecast may change over time.

Fuel savings model after aero-engine washing based on convolutional neural network prediction

In order to accurately calculate the fuel savings after aero-engine washing, therefore the engine cleaning process flow can be arranged more scientifically and reasonably. In this paper, a derivation method of fuel savings model after engine washing based on delta fuel flow (DFF) degradation baseline and convolutional neural network prediction technology is proposed. Firstly, the fuel savings model of cruise phase is established based on gas path parameters related to the engine fuel consumption, and the QAR data’s trend on entire flight is mined, and the regularity of the gas path parameters in the cruise phase is extended to the entire flight, thereby the fuel savings model after engine washing of whole Flight is derived. In the model derivation process, the unwashed engine’s DFF degradation baseline after the wash point is predicted. According to the characteristics of DFF discrete time series, a discrete input process neural network prediction model based on discrete data convolution operation is established. The convolution operation of discrete data is used to replace the integral operation of continuous function, which effectively avoids the loss of fitting precision caused by the smooth processing of discrete time series data before the input of the process neural network model, and improves the prediction accuracy of discrete time series data. A denoising method combining Singular Value Decomposition (SVD) and Empirical Mode Decomposition (EMD) is used to denoise the DFF before and after engine washing, and the trend analysis method is used to solve the intersection of the unwashed state engine’s DFF degradation baseline and washed state engine’s DFF degradation baseline, thereby the washing effect cycle numbers is predicted, and in which the total fuel savings model of whole fleet after engine washing is derived by accumulating every flight fuel savings. The post-washed engine fuel saving calculation of an airline is used as a verification experiment. The results show that post-washed engine fuel saving prediction model presented in the paper is the closest to the actual fuel savings model.

Recurrence quantification analysis of complex-fractionated electrograms differentiates active and passive sites during atrial fibrillation.

To differentiate electrograms representing sites of active atrial fibrillation (AF) drivers from passive ones. Ablation of complex-fractionated atrial electrograms (CFAEs) is controversial due to difficulty in distinguishing CFAEs representing sites of active AF drivers from passive mechanisms. We hypothesized that active CFAE sites exhibit repetitive wavefront directionality, thereby inscribing an electrogram conformation (Egm-C) that is more recurrent compared with passive CFAE sites; and that can be differentiated from passive CFAEs using nonlinear recurrence quantification analysis (RQA). We developed multiple computer models of active CFAE mechanisms (ie, rotors) and passive CFAE mechanisms (ie,wavebreak, slow conduction, and double potentials). CFAE signals were converted into discrete time-series representing Egm-C. The RQA algorithm was used to compare signals derived from active CFAE sites to those from passive CFAEs sites. The RQA algorithm was then applied to human CFAE signals collected during AF ablation (n = 17 patients). RQA was performed in silico on simulated bipolar CFAEs within active (n = 45) and passive (n = 60) areas. Recurrence of Egm-C was significantly higher in active compared with passive CFAE sites (31.8% ± 19.6% vs 0.3% ± 0.5%, respectively, P < .0001) despite no difference in mean cycle length (CL). Similarly, for human AF (n = 39 signals), Egm-C recurrence was higher in active vs passive CFAE areas despite similar CLs (%recurrence 13.6% ± 15.5% vs 0.1% ± 0.3%, P < .002; mean CL 102.5 ± 14.3 vs 106.6 ± 14.4, P = NS). Active CFAEs critical to AF maintenance exhibit higher Egm-C recurrence and can be differentiated from passive bystander CFAE sites using RQA.

Approximations and forecasting quasi-stationary processes with sudden runs

The object of research is heteroskedastic processes in the formation and evaluation of the offset policy of states in international markets, in particular, in the conclusion and execution of offset contracts. The process of concluding and fulfilling the conditions of an offset contract is weakly unsteady, because at its conclusion there can be a wide variety of sudden events, force majeure circumstances that can’t be described and predicted in detail, with acceptable accuracy, in full. It is shown that the term «quasi-stationary process», which has some approximation to the stationary process, is more suitable for describing such processes. The most promising approach to constructing mathematical models of such processes is the use of combined fractal autoregression models and an integrated moving average. During the study, methods of the theory of non-stationary random processes and the theory of runs of random processes were used, it is shown that the combination of a quasi-stationary process with a sequence of random runs is quite satisfactorily modeled by the so-called fractal or self-similar process. As a universal mathematical model of self-similar processes with slowly decreasing dependencies, the model of fractal integrated autoregression and the moving average FARIMA are used. However, this model does not take into account the effect of runs of random processes on the coefficients of the numerator and denominator of the finely rational function, which approximates the process of autoregression and the moving average. Therefore, in the study, the relationship in the parameters of the runs and the coefficients of the approximating function. Mathematical models of discrete time series are considered, which are characterized by quasi-stationary and the presence of sudden runs. It is shown that for approximation of such series, models such as autoregression and moving average are quite suitable, modified to the classes of autoregression models and integrated moving average. And for self-similar (fractal) processes – modified to the classes of models of autoregression and fractal integrated moving average. The relationship between the shear length when calculating autocorrelation coefficients and the total sample length can serve as an acceptable indicator of the correctness of the solution.

Accelerating pattern-based time series classification: a linear time and space string mining approach

Subsequences-based time series classification algorithms provide interpretable and generally more accurate classification models compared to the nearest neighbor approach, albeit at a considerably higher computational cost. A number of discretized time series-based algorithms have been proposed to reduce the computational complexity of these algorithms; however, the asymptotic time complexity of the proposed algorithms is also cubic or higher-order polynomial. We present a remarkably fast and resource-efficient time series classification approach which employs a linear time and space string mining algorithm for extracting frequent patterns from discretized time series data. Compared to other subsequence or pattern-based classification algorithms, the proposed approach only requires a few parameters, which can be chosen arbitrarily and do not require any fine-tuning for different datasets. The time series data are discretized using symbolic aggregate approximation, and frequent patterns are extracted using a string mining algorithm. An independence test is used to select the most discriminative frequent patterns, which are subsequently used to create a transformed version of the time series data. Finally, a classification model can be trained using any off-the-shelf algorithm. Extensive empirical evaluations demonstrate the competitive classification accuracy of our approach compared to other state-of-the-art approaches. The experiments also show that our approach is at least one to two orders of magnitude faster than the existing pattern-based methods due to the extremely fast frequent pattern extraction, which is the most computationally intensive process in pattern-based time series classification approaches.

Copula directional dependence of discrete time series marginals

To understand the dynamic relationship of discrete time series processes, we adopted copula directional dependence via beta regression model applied with generalized autoregressive conditional heteroscedasticity (INGARCH) marginals. To validate the proposed method, we completed simulations of two INGARCH processes from asymmetric bivariate copula function with members such as Gaussian and Plackett copula functions. The simulations show that the proposed method is consistent for deriving directional dependent measurements regardless of the choice of the symmetric members. The proposed method is applied to the bivariate discrete time series data of the monthly counts of sandstorms and dust haze phenomena in Saudi Arabia.

A new Neuro-Fuzzy Inference System with Dynamic Neurons (NFIS-DN) for system identification and time series forecasting

A new Neuro-Fuzzy Inference System with Dynamic Neurons or NFIS-DN is presented here for discrete time dynamic system identification and time series forecasting problems. The proposed dynamic system based neuron, referred to as Dynamic Neuron (DN) is realized by a discrete-time nonlinear state-space model. The DN is designed such way, that the output considers only the effect of finite past instances, enabling the system with finite memory. The NFIS-DN model has five layers, and DNs are employed only in the layers handling crisp values. The antecedent and the consequent parameters of NFIS-DN are updated using a self-regulated backpropagation through time learning algorithm. The performance evaluation of NFIS-DN has been carried-out using benchmark problems in the areas of nonlinear system identification and time series forecasting. The results are compared with the state-of-the-art method on the neural fuzzy networks. The obtained results clearly suggest that the NFIS-DN performs significantly better while using a smaller or similar number of fuzzy rules. Finally the practical application of the NFIS-DN has been demonstrated using two real-world problems.

Machine Learning-Based Pre-Impact Fall Detection Model to Discriminate Various Types of Fall.

Pre-impact fall detection can send alarm service faster to reduce long-lie conditions and decrease the risk of hospitalization. Detecting various types of fall to determine the impact site or direction prior to impact is important because it increases the chance of decreasing the incidence or severity of fall-related injuries. In this study, a robust pre-impact fall detection model was developed to classify various activities and falls as multiclass and its performance was compared with the performance of previous developed models. Twelve healthy subjects participated in this study. All subjects were asked to place an inertial measuring unit module by fixing on a belt near the left iliac crest to collect accelerometer data for each activity. Our novel proposed model consists of feature calculation and infinite latent feature selection (ILFS) algorithm, auto labeling of activities, and application of machine learning classifiers for discrete and continuous time series data. Nine machine-learning classifiers were applied to detect falls prior to impact and derive final detection results by sorting the classifier. Our model showed the highest classification accuracy. Results for the proposed model that could classify as multiclass showed significantly higher average classification accuracy of 99.57 ± 0.01% for discrete data-based classifiers and 99.84 ± 0.02% for continuous time series-based classifiers than previous models (p < 0.01). In the future, multiclass pre-impact fall detection models can be applied to fall protector devices by detecting various activities for sending alerts or immediate feedback reactions to prevent falls.

Clustering electricity consumers using high‐dimensional regression mixture models

AbstractA massive amount of data about individual electrical consumptions are now provided with new metering technologies and smart grids. These new data are especially useful for load profiling and load modeling at different scales of the electrical network. A new methodology based on mixture of high‐dimensional regression models is used to perform clustering of individual customers. It leads to uncovering clusters corresponding to different regression models. Temporal information is incorporated in order to prepare the next step, the fit of a forecasting model in each cluster. Only the electrical signal is involved, slicing the electrical signal into consecutive curves to consider it as a discrete time series of curves. Interpretation of the models is given on a real smart meter dataset of Irish customers.

Regression Derivatives and Their Application to the Study of Geomagnetic Jerks

New mathematical constructions are developed for the regression smoothing of discrete time series defined on an irregular grid. The new method is used to study secular trends for the period 1991–2015 and relies on observations of three components of the magnetic field at five geomagnetic observatories of the INTERMAGNET network. The detected changes in the trend coincided in time with the jerk properties calculated by conventional methods, which indicates the applicability of the method of regression derivatives to geophysical problems.

What Differentiates Poor and Good Outcome Psychotherapy? A Statistical-Mechanics-Inspired Approach to Psychotherapy Research

Statistical mechanics investigates how emergent properties of macroscopic systems (such as temperature and pressure) relate to microscopic state fluctuations. The underlying idea is that global statistical descriptors of order and variability can monitor the relevant dynamics of the whole system at hand. Here we test the possibility of extending such an approach to psychotherapy research investigating the possibility of predicting the outcome of psychotherapy on the sole basis of coarse-grained empirical macro-parameters. Four good-outcome and four poor-outcome brief psychotherapies were recorded, and their transcripts coded in terms of standard psychological categories (abstract, positive emotional and negative emotional language pertaining to patient and therapist). Each patient-therapist interaction is considered as a discrete multivariate time series made of subsequent word-blocks of 150-word length, defined in terms of the above categories. “Static analyses” (Principal Component Analysis) highlighted a substantial difference between good-outcome and poor-outcome cases in terms of mutual correlations among those descriptors. In the former, the patient’s use of abstract language correlated with therapist’s emotional negative language, while in the latter it co-varied with therapist’s emotional positive language, thus showing the different judgment of the therapists regarding the same variable (abstract language) in poor and good outcome cases. On the other hand, the “dynamic analyses”, based on five coarse-grained descriptors related to variability, the degree of order and complexity of the series, demonstrated a relevant case-specific effect, pointing to the possibility of deriving a consistent picture of any single psychotherapeutic process. Overall, the results showed that the systemic approach to psychotherapy (an old tenet of psychology) is mature enough to shift from a metaphorical to a fully quantitative status.

A Concise Temporal Data Representation Model for Prediction in Biomedical Wearable Devices

Predictive analytics and event forecasting using deep learning techniques require processing long-term historical data which is infeasible in low-power wearable devices. Such devices are constrained in memory and computational power, and are pushed to their limits by resource hungry deep neural networks. Current techniques either ignore historical data, or convert temporal sequences to pattern sequences, eliminating valuable properties such as time and/or recency. The proposed model maps arbitrary-length multivariate discrete time series to a concise sequence, called mapped interval sequence (MIS). MIS retains original data properties such as time, recency, and scale, without being susceptible to missing values. Life model for time series (LMts) mapping, is capable of mapping billions of data elements with sampling rate of several kHz or higher into a sequence of 32 elements or fewer. Furthermore, a new loss function called as tolerance error is introduced to improve long-term forecasting events using LMts. In a smart health Internet of Things environment, LMts enables real-time health predictions depending on both recent and historical data. In addition, the LMts model can predict the approximate time of events, with granularity of seconds and up to years. Experimental results show that, compared to previous studies in fall prediction, LMts achieves the same 100% accuracy with a single long short-term memory layer, while covering $16{\boldsymbol \times }$ longer time period and using $80{\boldsymbol \times }$ less weight parameters. LMts is also used to forecast human fall up to 14 s in advance even with 50% missing values.

Nonlinear least-squares spline fitting with variable knots

In this paper, we present a nonlinear least-squares fitting algorithm using B-splines with free knots. Since its performance strongly depends on the initial estimation of the free parameters (i.e. the knots), we also propose a fast and efficient knot-prediction algorithm that utilizes numerical properties of first-order B-splines. Using ℓp(p=1,2,∞) norm solutions, we also provide three different strategies for properly selecting the free knots. Our initial predictions are then iteratively refined by means of a gradient-based variable projection optimization. Our method is general in nature and can be used to estimate the optimal number of knots in cases in which no a-priori information is available.To evaluate the performance of our method, we approximated a one-dimensional discrete time series and conducted an extensive comparative study using both synthetic and real-world data. We chose the problem of electrocardiogram (ECG) signal compression as a real-world case study. Our experiments on the well-known PhysioNet MIT-BIH Arrhythmia database show that the proposed method outperforms other knot-prediction techniques in terms of accuracy while requiring much lower computational complexity.

Big 5 ASEAN capital markets forecasting using WEMA method

ASEAN through ASEAN Economics Community (AEC) 2020 treaty has proposed financial integration via capital markets integration in order to aim comprehensive ASEAN economic integration. Therefore, the need to have a proper prediction of ASEAN capital market becomes a major issue. In this study, we took big 5 ASEAN capital markets, i.e. Straits Times Index (STI), Kuala Lumpur Stock Exchange (KLSE), Stock Exchange of Thailand (SET), Jakarta Stock Exchange (JKSE), and Philippine Stock Exchange (PSE) to be forecasted using WEMA method. Weighted Exponential Moving Average (WEMA) is a new hybrid moving average method which combines the weighting factor calculation in Weighted Moving Average (WMA) with the procedure of Exponential Moving Average (EMA). WEMA has successfully been implemented and used to forecaste discrete time series data, but never being used to forecast ASEAN capital markets. In this study, we took further action by implementing the WEMA method with brute force approach for scaling factor tuning on big 5 ASEAN capital markets. From the experimental results, we found that WEMA has successfully forecasted all those exchanges. By looking at the forecast error measurement, it gives the best performance on PSE and worst performance on SET dataset among all datasets being considered in this study.