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The problem of extracting as much information as possible from a sequence of observations of a stationary stochastic process $X_0,X_1,...X_n$ has been considered by many authors from different points of view. It has long been known through the work of D. Bailey that no universal estimator for $\textbf{P}(X_{n+1}|X_0,X_1,...X_n)$ can be found which converges to the true estimator almost surely. Despite this result, for restricted classes of processes, or for sequences of estimators along stopping times, universal estimators can be found. We present here a survey of some of the recent work that has been done along these lines.

This thesis examines the topic of chaotic time series. An overview of chaos, dynamical systems, and traditional approaches to time series analysis is provided, followed by an examination of state space reconstruction. State space reconstruction is a nonlinear, deterministic approach whose goal is to use the immediate past behavior of the time series to reconstruct the current state of the system. The choice of delay parameter and embedding dimension are crucial to this reconstruction. Once the state space has been properly reconstructed, one can address the issue of whether apparently random data has come from a low- dimensional, chaotic (deterministic) source or from a random process. Specific techniques for making this determination include attractor reconstruction, estimation of fractal dimension and Lyapunov exponents, and short-term prediction. If the time series data appears to be from a low-dimensional chaotic source, then one can predict the continuation of the data in the short term. This is the inverse problem of dynamical systems. In this thesis, the technique of local fitting is used to accomplish the prediction. Finally, the issue of noisy data is treated, with the purpose of highlighting where further research may be beneficial

Careful blood glucose monitoring and consistent insulin administration are necessary for managing diabetes. People with demanding schedules or little access to medical personnel may find this difficult. Fortunately, without having to visit a doctor every day, daily insulin dosage may now be customized to a person’s unique needs using technology and customised algorithms based on their food intake, exercise routines, and blood glucose levels. This information can be entered into a diabetes management app or device, where an algorithm will determine the proper insulin dosage and offer real-time feedback to assist maintain ideal blood glucose levels. A patient's dietary preferences, degree of physical activity, and blood sugar are taken into account for determining the proper bolus and basal insulin dosages in this study. According to the tracked body data, a patient’s appropriate insulin dosage is predicted using artificial neural network (ANN)-based models. Based on patient activity, food intake, exercise, and past insulin administration, insulin projections are created. To forecast an individual’s basal and bolus insulin requirements, long short-term memory (LSTM) and random forest regression models are employed. Accuracy of both models are tested and random forest regression shows better accuracy which is used in the prediction system. 

A computer program is described and presented for calculating finite memory predictors and prediction variances for autoregressive moving average time series models. The Cholesky decomposition algorithm is used, and a number of simplifying results are described and implemented in the program. (Author)

A computer program is described and presented for calculating finite memory predictors and prediction variances for autoregressive moving average time series models. The Cholesky decomposition algorithm is used, and a number of simplifying results are described and implemented in the program. (Author)

Thesis advisors, Christopher Frenzen, Philip Beaver

Thesis advisors, Christopher Frenzen, Philip Beaver

This article is from Healthcare Informatics Research , volume 16 . Abstract Objectives: To develop and evaluate time series models to predict the daily number of patients visiting the Emergency Department (ED) of a Korean hospital. Methods: Data were collected from the hospital information system database. In order to develop a forecasting model, we used, 2 years of data from January 2007 to December 2008 data for the following 3 consecutive months were processed for validation. To establish a Forecasting Model, calendar and weather variables were utilized. Three forecasting models were established: 1) average; 2) univariate seasonal auto-regressive integrated moving average (SARIMA); and 3) multivariate SARIMA. To evaluate goodness-of-fit, residual analysis, Akaike information criterion and Bayesian information criterion were compared. The forecast accuracy for each model was evaluated via mean absolute percentage error (MAPE). Results: The multivariate SARIMA model was the most appropriate for forecasting the daily number of patients visiting the ED. Because it's MAPE was 7.4%, this was the smallest among the models, and for this reason was selected as the final model. Conclusions: This study applied explanatory variables to a multivariate SARIMA model. The multivariate SARIMA model exhibits relativelyhigh reliability and forecasting accuracy. The weather variables play a part in predicting daily ED patient volume.

Maintaining inventory level to avoid high inventory costs is an issue for Cardiac Product Distribution Companies (CPDCs) because of the shortage of their products which affect their sale and causes loss of the customer. This research aims to provide a method for predicting the upcoming demand of the Balloon and Stents by using time series analysis (Auto Regression Integrated Moving Average) and Deep learning (Long-Short Term Memory). To conduct this research, data was collected from Pakistan’s leading cardiac product distributors to determine the method's performance. The findings were compared using Mean absolute error (MAE) and Root Mean Square Error (RMSE). Resulst conclude that the ARIMA algorithm successfully forecasts cardiac products sale.

This article is from BMC Bioinformatics , volume 15 . Abstract Background: Time series models can play an important role in disease prediction. Incidence data can be used to predict the future occurrence of disease events. Developments in modeling approaches provide an opportunity to compare different time series models for predictive power. Results: We applied ARIMA and Random Forest time series models to incidence data of outbreaks of highly pathogenic avian influenza (H5N1) in Egypt, available through the online EMPRES-I system. We found that the Random Forest model outperformed the ARIMA model in predictive ability. Furthermore, we found that the Random Forest model is effective for predicting outbreaks of H5N1 in Egypt. Conclusions: Random Forest time series modeling provides enhanced predictive ability over existing time series models for the prediction of infectious disease outbreaks. This result, along with those showing the concordance between bird and human outbreaks (Rabinowitz et al. 2012), provides a new approach to predicting these dangerous outbreaks in bird populations based on existing, freely available data. Our analysis uncovers the time-series structure of outbreak severity for highly pathogenic avain influenza (H5N1) in Egypt.

In the prediction of oscillating time series, the interest is in the turning points of successive oscillations rather than the samples themselves. For this purpose a scheme has been proposed; the state space reconstruction is limited to the turning points and the local (nearest neighbor) model is modified in order to predict the turning point magnitudes and times. This approach is extended here using a local dynamic regression model on both turning point magnitudes and times. Simulations on oscillating nonlinear systems show that the proposed approach gives better predictions of turning points than the standard local model applied to all the samples of the oscillating time series.

Prediction of events is fundamental to both human and artificial agents. The main problem with previous prediction techniques is that they cannot predict events that have never been experienced before. This dissertation addresses the problem of predicting such novelty by developing algorithms and computational models inspired from recent cognitive science theories conceptual blending theory and event segmentation theory. We were able to show that prediction accuracy for event or state prediction can be significantly improved using these methods. The main contribution of this dissertation is a new class of prediction techniques inspired by conceptual blending that improves prediction accuracy overall and has the ability to predict even events that have never been experienced before. We also show that event segmentation theory, when integrated with these techniques, results in greater computational efficiency. We implemented the new prediction techniques, and more traditional alternatives such as Markov and Bayesian techniques, and compared their prediction accuracy quantitatively for three domains a role-playing game, intrusion-system alerts, and event prediction of maritime paths in a discrete-event simulator. Other contributions include two new unification algorithms that improve over a nave one, and an exploration of ways to maintain a minimum-size knowledge base without affecting prediction accuracy.

The Deterministic Versus Stochastic algorithm developed by Martin Casdagli is modified to produce two new, methodologies, each of which selectively uses embedding space nearest neighbors. Neighbors which are considered prediction relevant are retained for local linear prediction, while those which are considered likely to represent noise are ignored. For many time series, it is shown possible to improve on local linear prediction with both of the new algorithms. Furthermore, the theory of embedology is applied to determine a length of test sequence sufficient for accurate classification of moving objects. Sequentially recorded feature vectors of a moving object form a training trajectory in feature space. Each of the sequences of feature vector components is a time series, and under certain conditions, each of these time series has approximately the same fractal dimension. The embedding theorem is applied to this fractal dimension to establish a number of observations sufficient to determine the feature space trajectory of the object. It is argued that this number is a reasonable test sequence length for use in object classification. Experiments with data corresponding to five military vehicles (observed following a projected Lorenz trajectory on a viewing sphere) show that this number is indeed adequate. Time series prediction, Embedology, Motion analysis

The Deterministic Versus Stochastic algorithm developed by Martin Casdagli is modified to produce two new, methodologies, each of which selectively uses embedding space nearest neighbors. Neighbors which are considered prediction relevant are retained for local linear prediction, while those which are considered likely to represent noise are ignored. For many time series, it is shown possible to improve on local linear prediction with both of the new algorithms. Furthermore, the theory of embedology is applied to determine a length of test sequence sufficient for accurate classification of moving objects. Sequentially recorded feature vectors of a moving object form a training trajectory in feature space. Each of the sequences of feature vector components is a time series, and under certain conditions, each of these time series has approximately the same fractal dimension. The embedding theorem is applied to this fractal dimension to establish a number of observations sufficient to determine the feature space trajectory of the object. It is argued that this number is a reasonable test sequence length for use in object classification. Experiments with data corresponding to five military vehicles (observed following a projected Lorenz trajectory on a viewing sphere) show that this number is indeed adequate. Time series prediction, Embedology, Motion analysis

In this research the technology of complex Markov chains is applied to predict financial time series. The main distinction of complex or high-order Markov Chains and simple first-order ones is the existing of aftereffect or memory. The technology proposes prediction with the hierarchy of time discretization intervals and splicing procedure for the prediction results at the different frequency levels to the single prediction output time series. The hierarchy of time discretizations gives a possibility to use fractal properties of the given time series to make prediction on the different frequencies of the series. The prediction results for world's stock market indices is presented.

Multi-task learning employs shared representation of knowledge for learning multiple instances from the same or related problems. Time series prediction consists of several instances that are defined by the way they are broken down into fixed windows known as embedding dimension. Finding the optimal values for embedding dimension is a computationally intensive task. Therefore, we introduce a new category of problem called dynamic time series prediction that requires a trained model to give prediction when presented with different values of the embedding dimension. This can be seen a new class of time series prediction where dynamic prediction is needed. In this paper, we propose a co-evolutionary multi-task learning method that provides a synergy between multi-task learning and coevolution. This enables neural networks to retain modularity during training for building blocks of knowledge for different instances of the problem. The effectiveness of the proposed method is demonstrated using one-step-ahead chaotic time series problems. The results show that the proposed method can effectively be used for different instances of the related time series problems while providing improved generalisation performance.

Developed intelligent technologies are become play a promising role in providing better decision-making and improving the medical services provided to the patients. A risk prediction task for short-term is big challenge task; however, it is a great importance for recommendation systems in health care field to provide patients with accurate and reliable recommendations. In this work, clustering method and least square support vector machine are used for prediction a short-term disease risk prediction. The clustering similar method is based on euclidean distance which used to identify the similar sliding windows. The proposed model is trained by using the slide windows samples. Finally, the appropriate recommendations are generated for heart diseases patients who need to take a medical test or not for following day using least square support vector machine. A real dataset which collected from heart diseases patient is used for evaluation. The proposed method yields a good results related by the recommendations accuracy generated to chronicle heart patients and reduce the risk of incorrect recommendations. 

This study investigates deep learning models for financial data prediction and examines whether the architecture of a deep learning model and time-series data properties affect prediction accuracy. Comparing the performance of convolutional neural network (CNN), long short-term memory (LSTM), Stacked-LSTM, CNN-LSTM, and convolutional LSTM (ConvLSTM) when used as a prediction approach to a collection of financial time-series data is the main methodology of this study. In this instance, only those deep learning architectures that can predict multivariate time-series data sets in parallel are considered. This research uses the daily movements of 4 (four) Asian stock market indices from 1 January 2020 to 31 December 2020. Using data from the early phase of the spread of the Covid-19 pandemic that has created worldwide economic turmoil is intended to validate the performance of the analyzed deep learning models. Experiment results and analytical findings indicate that there is no superior deep learning model that consistently makes the most accurate predictions for all states' financial data. In addition, a single deep learning model tends to provide more accurate predictions for more stable time-series data, but the hybrid model is preferred for more chaotic time-series data.

We establish rates of convergences in time series forecasting using the statistical learning approach based on oracle inequalities. A series of papers extends the oracle inequalities obtained for iid observations to time series under weak dependence conditions. Given a family of predictors and $n$ observations, oracle inequalities state that a predictor forecasts the series as well as the best predictor in the family up to a remainder term $\Delta_n$. Using the PAC-Bayesian approach, we establish under weak dependence conditions oracle inequalities with optimal rates of convergence. We extend previous results for the absolute loss function to any Lipschitz loss function with rates $\Delta_n\sim\sqrt{c(\Theta)/ n}$ where $c(\Theta)$ measures the complexity of the model. We apply the method for quantile loss functions to forecast the french GDP. Under additional conditions on the loss functions (satisfied by the quadratic loss function) and on the time series, we refine the rates of convergence to $\Delta_n \sim c(\Theta)/n$. We achieve for the first time these fast rates for uniformly mixing processes. These rates are known to be optimal in the iid case and for individual sequences. In particular, we generalize the results of Dalalyan and Tsybakov on sparse regression estimation to the case of autoregression.

A computer program is described and presented for calculating finite memory predictors and prediction variances for autoregressive moving average time series models. The Cholesky decomposition algorithm is used, and a number of simplifying results are described and implemented in the program. (Author)

An algorithm is presented for conveniently calculating h step ahead minimum mean square linear predictors and prediction variances given a finite number of observations from a covariance stationary time series Y. It is shown that elements of the modified Cholesky decomposition of the covariance matrix of observations play the role in finite memory prediction that the coefficients in the infinite order moving average representation of Y play in infinite memory prediction. The algorithm is applied to autoregressive-moving average time series where further simplifications are shown to occur. A numerical example illustrating the basic points of the general algorithm is presented. (Author)

Greater accuracy is required in predicting hurricane landfall in order to insure timely evacuation. A significant result of this research is the classification of past storms by time series stationarity category which relates to direction of movement. Also, a psi-weight representation of the forecast is used to develop a bivariate Normal confidence ellipse for the threshold autoregressive model. It is shown that the landfall of North Atlantic hurricane and tropical storms can be accurately predicted by modeling the storm track as a bivariate (latitude and longitude) fifth-order autoregressive process. A threshold approach is used to allow model parameters to change as the storm moves to a new region of the ocean. For test cases, operational average 72 hours prediction error is at least three standard deviations below the average error of the official forecasts issued by the National Hurricane Center.

Greater accuracy is required in predicting hurricane landfall in order to insure timely evacuation. A significant result of this research is the classification of past storms by time series stationarity category which relates to direction of movement. Also, a psi-weight representation of the forecast is used to develop a bivariate Normal confidence ellipse for the threshold autoregressive model. It is shown that the landfall of North Atlantic hurricane and tropical storms can be accurately predicted by modeling the storm track as a bivariate (latitude and longitude) fifth-order autoregressive process. A threshold approach is used to allow model parameters to change as the storm moves to a new region of the ocean. For test cases, operational average 72 hours prediction error is at least three standard deviations below the average error of the official forecasts issued by the National Hurricane Center.

We report on a technique based on multi-agent games which has potential use in the prediction of future movements of financial time-series. A third-party game is trained on a black-box time-series, and is then run into the future to extract next-step and multi-step predictions. In addition to the possibility of identifying profit opportunities, the technique may prove useful in the development of improved risk management strategies.

We present two approaches for next step linear prediction of long memory time series. The first is based on the truncation of the Wiener-Kolmogorov predictor by restricting the observations to the last $k$ terms, which are the only available values in practice. Part of the mean squared prediction error comes from the truncation, and another part comes from the parametric estimation of the parameters of the predictor. By contrast, the second approach is non-parametric. An AR($k$) model is fitted to the long memory time series and we study the error made with this misspecified model.

We tackle the issue of the blind prediction of a Gaussian time series. For this, we construct a projection operator build by plugging an empirical covariance estimation into a Schur complement decomposition of the projector. This operator is then used to compute the predictor. Rates of convergence of the estimates are given.

Forecasting a time series from multivariate predictors constitutes a challenging problem, especially using model-free approaches. Most techniques, such as nearest-neighbor prediction, quickly suffer from the curse of dimensionality and overfitting for more than a few predictors which has limited their application mostly to the univariate case. Therefore, selection strategies are needed that harness the available information as efficiently as possible. Since often the right combination of predictors matters, ideally all subsets of possible predictors should be tested for their predictive power, but the exponentially growing number of combinations makes such an approach computationally prohibitive. Here a prediction scheme that overcomes this strong limitation is introduced utilizing a causal pre-selection step which drastically reduces the number of possible predictors to the most predictive set of causal drivers making a globally optimal search scheme tractable. The information-theoretic optimality is derived and practical selection criteria are discussed. As demonstrated for multivariate nonlinear stochastic delay processes, the optimal scheme can even be less computationally expensive than commonly used sub-optimal schemes like forward selection. The method suggests a general framework to apply the optimal model-free approach to select variables and subsequently fit a model to further improve a prediction or learn statistical dependencies. The performance of this framework is illustrated on a climatological index of El Ni\~no Southern Oscillation.

We present a simple randomized procedure for the prediction of a binary sequence. The algorithm uses ideas from recent developments of the theory of the prediction of individual sequences. We show that if the sequence is a realization of a stationary and ergodic random process then the average number of mistakes converges, almost surely, to that of the optimum, given by the Bayes predictor. The desirable finite-sample properties of the predictor are illustrated by its performance for Markov processes. In such cases the predictor exhibits near optimal behavior even without knowing the order of the Markov process. Prediction with side information is also considered.

We present a simple randomized procedure for the prediction of a binary sequence. The algorithm uses ideas from recent developments of the theory of the prediction of individual sequences. We show that if the sequence is a realization of a stationary and ergodic random process then the average number of mistakes converges, almost surely, to that of the optimum, given by the Bayes predictor. The desirable finite-sample properties of the predictor are illustrated by its performance for Markov processes. In such cases the predictor exhibits near optimal behavior even without knowing the order of the Markov process. Prediction with side information is also considered.

In this paper we apply a new approach of the string theory to the real financial market. It is direct extension and application of the work [1] into prediction of prices. The models are constructed with an idea of prediction models based on the string invariants (PMBSI). The performance of PMBSI is compared to support vector machines (SVM) and artificial neural networks (ANN) on an artificial and a financial time series. Brief overview of the results and analysis is given. The first model is based on the correlation function as invariant and the second one is an application based on the deviations from the closed string/pattern form (PMBCS). We found the difference between these two approaches. The first model cannot predict the behavior of the forex market with good efficiency in comparison with the second one which is, in addition, able to make relevant profit per year.

We address the problem of nonparametric estimation of characteristics for stationary and ergodic time series. We consider finite-alphabet time series and real-valued ones and the following four problems: i) estimation of the (limiting) probability (or estimation of the density for real-valued time series), ii) on-line prediction, iii) regression and iv) classification (or so-called problems with side information). We show that so-called archivers (or data compressors) can be used as a tool for solving these problems. In particular, firstly, it is proven that any so-called universal code (or universal data compressor) can be used as a basis for constructing asymptotically optimal methods for the above problems. (By definition, a universal code can "compress" any sequence generated by a stationary and ergodic source asymptotically till the Shannon entropy of the source.) And, secondly, we show experimentally that estimates, which are based on practically used methods of data compression, have a reasonable precision.

In this paper we address the problem of predicting a time series using the ARMA (autoregressive moving average) model, under minimal assumptions on the noise terms. Using regret minimization techniques, we develop effective online learning algorithms for the prediction problem, without assuming that the noise terms are Gaussian, identically distributed or even independent. Furthermore, we show that our algorithm's performances asymptotically approaches the performance of the best ARMA model in hindsight.

The predictive capability of a modification of Rissanen's accumulated prediction error (APE) criterion, APE$_{\delta_n}$, is investigated in infinite-order autoregressive (AR($\infty$)) models. Instead of accumulating squares of sequential prediction errors from the beginning, APE$_{\delta_n}$ is obtained by summing these squared errors from stage $n\delta_n$, where $n$ is the sample size and $1/n\leq \delta_n\leq 1-(1/n)$ may depend on $n$. Under certain regularity conditions, an asymptotic expression is derived for the mean-squared prediction error (MSPE) of an AR predictor with order determined by APE$_{\delta_n}$. This expression shows that the prediction performance of APE$_{\delta_n}$ can vary dramatically depending on the choice of $\delta_n$. Another interesting finding is that when $\delta_n$ approaches 1 at a certain rate, APE$_{\delta_n}$ can achieve asymptotic efficiency in most practical situations. An asymptotic equivalence between APE$_{\delta_n}$ and an information criterion with a suitable penalty term is also established from the MSPE point of view. This offers new perspectives for understanding the information and prediction-based model selection criteria. Finally, we provide the first asymptotic efficiency result for the case when the underlying AR($\infty$) model is allowed to degenerate to a finite autoregression.

Let $\{X_n\}$ be a stationary and ergodic time series taking values from a finite or countably infinite set ${\cal X}$. Assume that the distribution of the process is otherwise unknown. We propose a sequence of stopping times $\lambda_n$ along which we will be able to estimate the conditional probability $P(X_{\lambda_n+1}=x|X_0,...,X_{\lambda_n})$ from data segment $(X_0,...,X_{\lambda_n})$ in a pointwise consistent way for a restricted class of stationary and ergodic finite or countably infinite alphabet time series which includes among others all stationary and ergodic finitarily Markovian processes. If the stationary and ergodic process turns out to be finitarily Markovian (among others, all stationary and ergodic Markov chains are included in this class) then $ \lim_{n\to \infty} {n\over \lambda_n}>0$ almost surely. If the stationary and ergodic process turns out to possess finite entropy rate then $\lambda_n$ is upperbounded by a polynomial, eventually almost surely.

Let $\{X_n\}$ be a stationary and ergodic time series taking values from a finite or countably infinite set ${\cal X}$. Assume that the distribution of the process is otherwise unknown. We propose a sequence of stopping times $\lambda_n$ along which we will be able to estimate the conditional probability $P(X_{\lambda_n+1}=x|X_0,...,X_{\lambda_n})$ from data segment $(X_0,...,X_{\lambda_n})$ in a pointwise consistent way for a restricted class of stationary and ergodic finite or countably infinite alphabet time series which includes among others all stationary and ergodic finitarily Markovian processes. If the stationary and ergodic process turns out to be finitarily Markovian (among others, all stationary and ergodic Markov chains are included in this class) then $ \lim_{n\to \infty} {n\over \lambda_n}>0$ almost surely. If the stationary and ergodic process turns out to possess finite entropy rate then $\lambda_n$ is upperbounded by a polynomial, eventually almost surely.

A novel method for predicting time series is described and demonstrated. This method inputs time series data points and outputs multiple "spaghetti" functions from which predictions can be made. Spaghetti prediction has desirable properties that are not realized by classic autoregression, moving average, spline, Gaussian process, and other methods. It is particularly appropriate for short time series because it allows asymmetric prediction distributions and produces prediction functions which are robust in that they use multiple independent models.

We here present a simple and effective model to predict the popularity of web content. Our solution, which is the winner of two of the three tasks of the ECML/PKDD 2014 Predictive Analytics Challenge, aims at predicting user engagement metrics, such as number of visits and social network engagement, that a web page will achieve 48 hours after its upload, using only information available in the first hour after upload. Our model is based on two steps. We first use time series clustering techniques to extract common temporal trends of content popularity. Next, we use linear regression models, exploiting as predictors both content features (e.g., numbers of visits and mentions on online social networks) and metrics that capture the distance between the popularity time series to the trends extracted in the first step. We discuss why this model is effective and show its gains over state of the art alternatives.

In real-time monitoring of hospital patients, high-quality inference of patients' health status using all information available from clinical covariates and lab tests are essential to enable successful medical interventions and improve patient outcomes. In this work, we develop and explore a Bayesian nonparametric model based on Gaussian process (GP) regression for hospital patient monitoring. Our method, MedGP, incorporates 24 clinical and lab covariates and supports a rich reference data set from which the relationships between these observed covariates may be inferred and exploited for high-quality inference of patient state over time. To do this, we develop a highly structured sparse GP kernel to enable tractable computation over tens of thousands of time points while estimating correlations among clinical covariates, patients, and periodicity in high-dimensional time series measurements of physiological signals. We apply MedGP to data from hundreds of thousands of patients treated at the Hospital of the University of Pennsylvania. MedGP has a number of benefits over current methods, including (i) not requiring an alignment of the time series data, (ii) quantifying confidence intervals in the predictions, (iii) exploiting a vast and rich database of patients, and (iv) providing interpretable relationships among clinical covariates. We evaluate and compare results from MedGP on the task of online state prediction for three different patient subgroups.

We consider $\alpha$-mixing observations and deal with the estimation of the conditional mode of a scalar response variable $Y$ given a random variable $X$ taking values in a semi-metric space. We provide a convergence rate in $L^p$ norm of the estimator. A useful and typical application to functional times series prediction is given.

This paper is the second in a series of two, and describes the current state of the art in modelling and prediction of chaotic time series. Sampled data from deterministic non-linear systems may look stochastic when analysed with linear methods. However, the deterministic structure may be uncovered and non-linear models constructed that allow improved prediction. We give the background for such methods from a geometrical point of view, and briefly describe the following types of methods: global polynomials, local polynomials, multi layer perceptrons and semi-local methods including radial basis functions. Some illustrative examples from known chaotic systems are presented, emphasising the increase in prediction error with time. We compare some of the algorithms with respect to prediction accuracy and storage requirements, and list applications of these methods to real data from widely different areas.

Motivated by a broad range of potential applications, we address the quantile prediction problem of real-valued time series. We present a sequential quantile forecasting model based on the combination of a set of elementary nearest neighbor-type predictors called "experts" and show its consistency under a minimum of conditions. Our approach builds on the methodology developed in recent years for prediction of individual sequences and exploits the quantile structure as a minimizer of the so-called pinball loss function. We perform an in-depth analysis of real-world data sets and show that this nonparametric strategy generally outperforms standard quantile prediction methods

In this paper we show that different physiological states and pathological conditions may be characterized in terms of predictability of time series signals from the underlying biological system. In particular we consider systolic arterial pressure time series from healthy subjects and Chronic Heart Failure patients, undergoing paced respiration. We model time series by the regularized least squares approach and quantify predictability by the leave-one-out error. We find that the entrainment mechanism connected to paced breath, that renders the arterial blood pressure signal more regular, thus more predictable, is less effective in patients, and this effect correlates with the seriousness of the heart failure. The leave-one-out error separates controls from patients and, when all orders of nonlinearity are taken into account, alive patients from patients for which cardiac death occurred.

For oscillating time series, the prediction is often focused on the turning points. In order to predict the turning point magnitudes and times it is proposed to form the state space reconstruction only from the turning points and modify the local (nearest neighbor) model accordingly. The model on turning points gives optimal prediction at a lower dimensional state space than the optimal local model applied directly on the oscillating time series and is thus computationally more efficient. Monte Carlo simulations on different oscillating nonlinear systems showed that it gives better predictions of turning points and this is confirmed also for the time series of annual sunspots and total stress in a plastic deformation experiment.

In the subject of railway operation, predicting railway passenger volume has always been a hot topic. Accurately forecasting railway passenger volume is the foundation for railway transportation companies to optimize transit efficiency and revenue. The goal of this research is to use a combination of the fuzzy time series approach based on the rate of change algorithm and the Holt double exponential smoothing method to forecast the number of train passengers. In contrast to prior investigations, we focus primarily on determining the next time period in this research. The fuzzy time series is employed as the forecasting basis, the rate of change is used to build the set of universes, and the Holt's double exponential smoothing method is utilized to forecast the following period in this case study. The number of railway passengers predicted for January 2020 is 38199, with a tiny average forecasting error rate of 0.89 percent and a mean square error of 131325. It can also help rail firms identify future passenger needs, which can be used to decide whether to expand train cars or run new trains, as well as how to distribute tickets.

Shang and Hyndman (2016) proposed grouped functional time series forecasting approach as a combination of individual forecasts using generalized least squares regression. We modify their methodology using generalized exponential smoothing technique for the most disaggregated series in order to obtain more robust predictor. We show some properties of our proposals using simulations and real data related to electricity demand prediction.

Results of analysis of long time variations of solar activities based on monthly Wolf number series (1749-2011 years) are represented. Shown the presence of stable oscillations in autocorrelation function for the Wolf numbers series with a period near 42.5 years. The statistical model of variations of autocorrelation function based on assumption of nonstationarity process are constructed. The question of using the model to predict the solar activity is discussed.

In the nonlinear prediction of scalar time series, the common practice is to reconstruct the state space using time-delay embedding and apply a local model on neighborhoods of the reconstructed space. The method of false nearest neighbors is often used to estimate the embedding dimension. For prediction purposes, the optimal embedding dimension can also be estimated by some prediction error minimization criterion. We investigate the proper state space reconstruction for multivariate time series and modify the two abovementioned criteria to search for optimal embedding in the set of the variables and their delays. We pinpoint the problems that can arise in each case and compare the state space reconstructions (suggested by each of the two methods) on the predictive ability of the local model that uses each of them. Results obtained from Monte Carlo simulations on known chaotic maps revealed the non-uniqueness of optimum reconstruction in the multivariate case and showed that prediction criteria perform better when the task is prediction.

In the nonlinear prediction of scalar time series, the common practice is to reconstruct the state space using time-delay embedding and apply a local model on neighborhoods of the reconstructed space. The method of false nearest neighbors is often used to estimate the embedding dimension. For prediction purposes, the optimal embedding dimension can also be estimated by some prediction error minimization criterion. We investigate the proper state space reconstruction for multivariate time series and modify the two abovementioned criteria to search for optimal embedding in the set of the variables and their delays. We pinpoint the problems that can arise in each case and compare the state space reconstructions (suggested by each of the two methods) on the predictive ability of the local model that uses each of them. Results obtained from Monte Carlo simulations on known chaotic maps revealed the non-uniqueness of optimum reconstruction in the multivariate case and showed that prediction criteria perform better when the task is prediction.

We study the occurrence of events, subject to threshold, in a representative SOC sandpile model and in high-resolution rainfall data. The predictability in both systems is analyzed by means of a decision variable sensitive to event clustering, and the quality of the predictions is evaluated by the receiver operating characteristics (ROC) method. In the case of the SOC sandpile model, the scaling of quiet-time distributions with increasing threshold leads to increased predictability of extreme events. A scaling theory allows us to understand all the details of the prediction procedure and to extrapolate the shape of the ROC curves for the most extreme events. For rainfall data, the quiet-time distributions do not scale for high thresholds, which means that the corresponding ROC curves cannot be straightforwardly related to those for lower thresholds.

A computer program is described and presented for calculating finite memory predictors and prediction variances for autoregressive moving average time series models. The Cholesky decomposition algorithm is used, and a number of simplifying results are described and implemented in the program. (Author)