Nonparametric Time Series Research Articles

Tutorial on Empirical Mode Decomposition: Basis Decomposition and Frequency Adaptive Graduation in Non-Stationary Time Series

This tutorial explores the class of non-parametric time series basis decomposition methods particularly suited for non-stationary time series known as Empirical Mode Decomposition (EMD). A detailed review of the state of the art statistical approaches that combine finite basis signal decomposition methods with smoothing graduation methods to characterise a non-stationary time series, where the representation is locally adapted to the frequency content of the signal, is addressed. Furthermore, the proposed EMD method also accommodates a variety of statistical properties including robustness to outliers and noise of various forms, regular or irregular sampling intervals such as with event-driven observations, and a variety of forms of non-stationarity. In outlining a statistical perspective of the EMD method, it will be contrasted with other existing basis decomposition methods such as functional Independent Component Analysis, Short-Time Fourier Transform, and Morlet Wavelet Analysis methods. Furthermore, the basis representations considered will be connected to versions of local frequency graduation smoothing methods, demonstrating that these can be adapted to a local frequency adaptive framework within the EMD context. This will provide new practical insights into the interface between time series basis decomposition and graduation smoothed representations.

Singular spectrum analysis as a tool for early detection of centrifugal compressor flow instability

Centrifugal compressor machinery is subject to a potentially damaging phenomenon called surge at low mass flow rates. This effect may be preceded by a phenomena known as inlet recirculation — a flow reversal upstream of the impeller. A methodology to isolate inlet recirculation as a characteristic feature for monitoring of centrifugal compressor instability is presented in this study. The methodology is based on a nonparametric time series analysis technique called as singular spectrum analysis (SSA). SSA decomposes a signal into a number of Reconstructive Components (RCs), from which data trends and oscillatory components may be extracted. The frequency spectra of each RC and their relative contributions to the reconstruction of the original signal were examined and comparisons were made with spectral maps in existing literature. Individual and independent RCs were chosen to construct a compressor’s instability monitoring system. Additionally, the performance of SSA was determined by the Window length parameter. The effect of modification of this parameter was also studied, and the various viable choices of component for the basis of inlet recirculation diagnosis were considered. The methodology was implemented in pressure dynamical signals measured in an experimental centrifugal compressor rig. High frequency pressure measurements were taken at a number of flow conditions and locations within the compressor. The results demonstrated the potential of a methodology based on SSA to identify and extract oscillatory components with information about the local effect of inlet recirculation and eventually successfully monitor centrifugal compressor’s instability.

Nonparametric time series summary statistics for high-frequency accelerometry data from individuals with advanced dementia

Accelerometry data has been widely used to measure activity and the circadian rhythm of individuals across the health sciences, in particular with people with advanced dementia. Modern accelerometers can record continuous observations on a single individual for several days at a sampling frequency of the order of one hertz. Such rich and lengthy data sets provide new opportunities for statistical insight, but also pose challenges in selecting from a wide range of possible summary statistics, and how the calculation of such statistics should be optimally tuned and implemented. In this paper, we build on existing approaches, as well as propose new summary statistics, and detail how these should be implemented with high frequency accelerometry data. We test and validate our methods on an observed data set from 26 recordings from individuals with advanced dementia and 14 recordings from individuals without dementia. We study four metrics: Interdaily stability (IS), intradaily variability (IV), the scaling exponent from detrended fluctuation analysis (DFA), and a novel nonparametric estimator which we call the proportion of variance (PoV), which calculates the strength of the circadian rhythm using spectral density estimation. We perform a detailed analysis indicating how the time series should be optimally subsampled to calculate IV, and recommend a subsampling rate of approximately 5 minutes for the dataset that has been studied. In addition, we propose the use of the DFA scaling exponent separately for daytime and nighttime, to further separate effects between individuals. We compare the relationships between all these methods and show that they effectively capture different features of the time series.

A weighted sieve estimator for nonparametric time series models with nonstationary variables

We study a class of nonparametric regression models that includes deterministic time trends and both stationary and nonstationary stochastic processes (whose shocks are allowed to be mutually correlated). We propose a unified approach to estimation based on the weighted sieve method to tackle the issue of unbounded support of the covariates. This approach improves on the existing technology in terms of some key regularity conditions such as moment conditions and the α-mixing coefficients for the stationary process. We establish self-normalized central limit theorems for the sieve estimator and other related quantities. Monte Carlo simulation confirms the theoretical results. We use our methodology to study the effect of CO2 and solar irradiance on global sea level rise.

Censored Nonparametric Time-Series Analysis with Autoregressive Error Models

This paper focuses on nonparametric regression modeling of time-series observations with data irregularities, such as censoring due to a cutoff value. In general, researchers do not prefer to put up with censored cases in time-series analyses because their results are generally biased. In this paper, we present an imputation algorithm for handling auto-correlated censored data based on a class of autoregressive nonparametric time-series model. The algorithm provides an estimation of the parameters by imputing the censored values with the values from a truncated normal distribution, and it enables unobservable values of the response variable. In this sense, the censored time-series observations are analyzed by nonparametric smoothing techniques instead of the usual parametric methods to reduce modelling bias. Typically, the smoothing methods are updated for estimating the censored time-series observations. We use Monte Carlo simulations based on right-censored data to compare the performances and accuracy of the estimates from the smoothing methods. Finally, the smoothing methods are illustrated using a meteorological time- series and unemployment datasets, where the observations are subject to the detection limit of the recording tool.

Illuminating Subsurface Microbial Water Quality Patterns Using Adenosine Triphosphate and Dynamic Time Warping Approaches

AbstractAquifer microbial water quality evaluations are often performed by collecting groundwater samples from monitoring wells. While samples collected from continuously pumped sources are seldom disputed as representative of the aquifer, natural biofilm present in the vicinity of well screens may introduce unwanted microbial artefacts in monitoring wells that are only periodically sampled. The need for well water purging to obtain samples void of these artefacts has been widely recognized. However, purging methods are not standardized; many approaches presume that physico‐chemical water quality stability achieved through the removal of 3 to 5 well volumes is indicative of the stability of target analytes. Using a data set collected from a shallow unconfined aquifer in Southern Ontario, Canada, the need for using dedicated approaches that account for the time‐dependent nature of microbial water quality changes was demonstrated. Specifically, the utility of adenosine triphosphate (ATP) as a rapid, field‐ready biochemical indicator of microbial water quality stability was investigated. This work shows that ATP concentrations reflect time‐limited (bio)colloid transport processes that are consistent with other microbial water quality parameters monitored, but different from commonly measured physical and chemical water quality indicators of well purging adequacy. ATP concentrations occasionally fluctuated even after 3 or 4 h of purging, indicating that microbial artefacts attributable to biofilms in the vicinity of the well screen can still persist. The recurrence of characteristic ATP patterns in each well was systematically examined through the novel application of dynamic time warping (DTW), a nonparametric time series analysis approach. These patterns are believed to be linked with seasonal hydrogeological conditions, which warrant consideration in the design and interpretation of subsurface microbial water quality investigations.

A nonparametric analysis of energy environmental Kuznets Curve in Chinese Provinces

Energy resources are an important material foundation for the survival and development of human society, and the relationship between energy and economy is interactive and complementary. This paper analyzes the energy consumption–economic growth nexus in Chinese provinces using novel and recent nonparametric time-series as well as panel data empirical approaches. The dataset covers 30 provinces over the period of 1980-2018. The empirical analysis indicates the presence of a nonlinear functional form and smooth structural changes in most of the provinces. The nonparametric empirical analysis validates the presence of a nonlinear unit root problem in energy consumption and economic growth, and nonlinear cointegration between the variables. Additionally, the nonparametric panel cointegration test reports evidence of convergence in energy consumption and economic growth patterns across the provinces. The nonparametric regression analysis finds economic growth to have a positive effect, on average, on energy consumption in all provinces, except for Beijing. Further, the energy environmental Kuznets curve exists between economic growth and energy consumption in 20 out of 30 Chinese provinces. The Granger causality analysis reveals the presence of a mixed causal relationship between economic growth and energy consumption. The empirical findings have important implications for Chinese authorities in planning for improving energy efficiency, decoupling between economic growth and energy consumption, and reducing the environmental footprint of provinces.

Forecasting of jabodetabek train passengers using singular spectrum analysis and holt-winters methods

Train is one of the convenient transportation, it also reduce traffic jam in Jabodetabek. In order to maintain the convenience of passengers using train services, it is necessary to predict the number of train passengers as a consideration in the planning of Jabodetabek train services. In this study the Singular Spectrum Analysis (SSA) and Holt-Winters methods are used to predict the number of Jabodetabek train passengers. SSA is a powerful technique for nonparametric time series analysis and forecasting, which decomposes the original time series into the sum of a small number of independent and interpretable components such as a slowly varying trend, oscillatory components and noise. Holt-Winters method are suitable methods for data that contains trends and seasonality. The Holt-Winters method is based on three smoothing equations, one for level, one for trend, and one for seasonality. SSA method in forecasting the number of Jabodetabek train passengers give lower Mean Absolute Percentage Error (MAPE) value than Holt-Winters method. This means SSA method is better than Holt-Winters method in forecasting the number of Jabodetabek train passengers.

High-frequency factor models and regressions

We consider a nonparametric time series regression model. Our framework allows precise estimation of betas without the usual assumption of betas being piecewise constant. This property makes our framework particularly suitable to study individual stocks. We provide an inference framework for all components of the model, including idiosyncratic volatility and idiosyncratic jumps. Our empirical analysis investigates the largest dataset in the high-frequency literature. First, we use all traded stocks from NYSE, AMEX, and NASDAQ stock markets for 1996–2017 to construct the five Fama–French factors and the momentum factor at the 5-minute frequency. Second, we document the key empirical properties across all the stocks and the new factors, and apply the nonparametric time series regression model with the new high-frequency Fama–French factors. We find that this factor model is effective in explaining the systematic component of the risk of individual stocks. In addition, we provide evidence that idiosyncratic jumps are related to idiosyncratic events such as earnings disappointments.

Estimating change points in nonparametric time series regression models

In this paper we consider a regression model that allows for time series covariates as well as heteroscedasticity with a regression function that is modelled nonparametrically. We assume that the regression function changes at some unknown time lfloor ns_0rfloor , s_0in (0,1), and our aim is to estimate the (rescaled) change point s_0. The considered estimator is based on a Kolmogorov-Smirnov functional of the marked empirical process of residuals. We show consistency of the estimator and prove a rate of convergence of O_P(n^{-1}) which in this case is clearly optimal as there are only n points in the sequence. Additionally we investigate the case of lagged dependent covariates, that is, autoregression models with a change in the nonparametric (auto-) regression function and give a consistency result. The method of proof also allows for different kinds of functionals such that Cramér-von Mises type estimators can be considered similarly. The approach extends existing literature by allowing nonparametric models, time series data as well as heteroscedasticity. Finite sample simulations indicate the good performance of our estimator in regression as well as autoregression models and a real data example shows its applicability in practise.

Nearest neighbor time series bootstrap for generating influent water quality scenarios

Understanding influent water quality variability is essential for the long-term planning of potable water systems. To quantify variability and generate realistic influent scenarios, we propose a nonparametric time series approach based on k-nearest neighbor (k-NN) bootstrap resampling. The k-NN approach resamples historical data conditioned on a “feature vector” at a given time to generate values at subsequent times. We modified this algorithm by adding random perturbations to the resampled values to generate realistic extremes unobserved in the historical record. k-NN is widely used in stochastic hydrology and hydroclimatology; however, it is adapted here for the multivariate, data-limited context of water treatment. To examine the performance of the algorithm, we applied it to an eleven-year, monthly water quality dataset of alkalinity, temperature, total organic carbon, and pH from the Cache la Poudre River in Colorado. We found that the k-NN simulations captured the relevant distributional statistics of the historical record, which suggests that the algorithm produces realistic and varied scenarios. When used in conjunction with modeling and optimization, these scenarios have the potential to improve the sustainability, resilience, and efficiency of potable water systems.

A Monte Carlo subsampling method for estimating the distribution of signal-to-noise ratio statistics in nonparametric time series regression models

Signal-to-noise ratio (SNR) statistics play a central role in many applications. A common situation where SNR is studied is when a continuous time signal is sampled at a fixed frequency with some noise in the background. While estimation methods exist, little is known about its distribution when the noise is not weakly stationary. In this paper we develop a nonparametric method to estimate the distribution of an SNR statistic when the noise belongs to a fairly general class of stochastic processes that encompasses both short and long-range dependence, as well as nonlinearities. The method is based on a combination of smoothing and subsampling techniques. Computations are only operated at the subsample level, and this allows to manage the typical enormous sample size produced by modern data acquisition technologies. We derive asymptotic guarantees for the proposed method, and we show the finite sample performance based on numerical experiments. Finally, we propose an application to electroencephalography (EEG) data.

Precipitation amounts and variability in a cool climate wine region, southern Quebec, Canada

ABSTRACTRecent climatic changes to warmer conditions, in both winter and summer, are creating more amenable conditions for growing vitis vinifera grapes in southern Quebec, an area considered to be a cool climate wine region. Gradual increases in growing season length, growing and dormant season temperatures, both maximum and minimum, are creating a general amelioration in wine grape growing conditions here. Although these recent increases in temperatures, as has been documented in southern Quebec, are well established, results from precipitation investigations are lacking. Both the intensity and magnitude of precipitation events require analysis. The present study uses non-parametric time series to analyze changes in growing season and September total precipitation, and changes in precipitation intensity during the September ripening period. This mid-latitude climatic zone has historically seen relatively high precipitation amounts. The results from this study indicate that in the southern Quebec wine grape growing regions little change has occurred in both the total amount of precipitation received and the intensity of that precipitation over the 35-year study period. Wet growing season and ripening period conditions continue to present a challenge for growers in this region.

Recent land surface temperature patterns in Antarctica using satellite and reanalysis data

Antarctica is among the most important regions when it comes to observing the effects of climate change. One important part of the variability of the land surface temperature (LST) observed on this continent is related to the sea surface temperature of the Tropical Pacific Ocean, which is associated with El Niño-Southern Oscillation (ENSO). This large-scale phenomenon is called tele-connection. In this study, we investigate the recent trends of LST in Antarctica over the last few decades and its relationship with one of the most intense El Niño events in 2015–2016. LST anomalies derived from satellite data (MODIS) and skin temperature from re-analysis (ERA-5 and ERA-Interim) were used, in addition to the Oceanic Niño Index (ONI) and Southern Annular Mode (SAM) time series. A non-parametric time series analysis was carried out to estimate LST trends over different zones of Antarctica. The results show a warming trend obtained from ERA-5 of about 0.9 K/decade in the interior of Antarctica, while MODIS showed the lowest temperature trends of −1 K/decade in East Antarctica. The analysis between LST and the SAM index shows a negative relationship in the extreme periods of ENSO 2015–2016, showing colder autumns and warmer springs than in previous years. These results aide in understanding the effects of extreme phases of ENSO on Antarctica, which are of great importance given the projection of more intense phases over the next century as foreseen in future scenarios of climate change.

A Self-Calibrated Non-Parametric Time Series Analysis Approach for Assessing Insect Defoliation of Broad-Leaved Deciduous Nothofagus pumilio Forests

Folivorous insects cause some of the most ecologically and economically important disturbances in forests worldwide. For this reason, several approaches have been developed to exploit the temporal richness of available satellite time series data to detect and quantify insect forest defoliation. Current approaches rely on parametric functions to describe the natural annual phenological cycle of the forest, from which anomalies are calculated and used to assess defoliation. Quantification of the natural variability of the annual phenological baseline is limited in parametric approaches, which is critical to evaluating whether an observed anomaly is “true” defoliation or only part of the natural forest variability. We present here a fully self-calibrated, non-parametric approach to reconstruct the annual phenological baseline along with its confidence intervals using the historical frequency of a vegetation index (VI) density, accounting for the natural forest phenological variability. This baseline is used to calculate per pixel (1) a VI anomaly per date and (2) an anomaly probability flag indicating its probability of being a “true” anomaly. Our method can be self-calibrated when applied to deciduous forests, where the winter VI values are used as the leafless reference to calculate the VI loss (%). We tested our approach with dense time series from the MODIS enhanced vegetation index (EVI) to detect and map a massive outbreak of the native Ormiscodes amphimone caterpillars which occurred in 2015–2016 in Chilean Patagonia. By applying the anomaly probability band, we filtered out all pixels with a probability <0.9 of being “true” defoliation. Our method enabled a robust spatiotemporal assessment of the O. amphimone outbreak, showing severe defoliation (60–80% and >80%) over an area of 15,387 ha of Nothofagus pumilio forests in only 40 days (322 ha/day in average) with a total of 17,850 ha by the end of the summer. Our approach is useful for the further study of the apparent increasing frequency of insect outbreaks due to warming trends in Patagonian forests; its generality means it can be applied in deciduous broad-leaved forests elsewhere.

High-Frequency Factor Models and Regressions

We consider a nonparametric time series regression model. Our framework allows precise estimation of betas without the usual assumption of betas being piecewise constant. This property makes our framework particularly suitable to study individual stocks. We provide an inference framework for all components of the model, including idiosyncratic volatility and idiosyncratic jumps. Our empirical analysis investigates the largest data set in the high-frequency literature. First, we use all traded stocks from NYSE, AMEX, and NASDAQ stock markets for 1996-2017 to construct the five Fama-French factors and the momentum factor at the 5-minute frequency. Second, we document the key empirical properties across all the stocks and the new factors, and apply the nonparametric time series regression model with the new high-frequency Fama-French factors. We find that this factor model is effective in explaining the systematic component of the risk of individual stocks. In addition, we provide evidence that idiosyncratic jumps are related to idiosyncratic events such as earnings disappointments.

A Nonparametric Bayesian Framework for Short-Term Wind Power Probabilistic Forecast

To improve the energy system resilience and economic efficiency, the wind power as a renewable energy starts to be deeply integrated into smart power grids. However, the wind power forecast uncertainty brings operational challenges. In order to provide a reliable guidance on operational decisions, in this paper, we propose a short-term wind power probabilistic forecast. Specifically, to model the rich dynamic behaviors of underlying physical wind power stochastic process occurring in various meteorological conditions, we first introduce an infinite Markov switching autoregressive model. This nonparametric time series model can capture the important properties in the real-world data to improve the prediction accuracy. Then, given finite historical data, the posterior distribution of flexible forecast model can correctly quantify the model estimation uncertainty. Built on it, we develop the posterior predictive distribution to rigorously quantify the overall forecasting uncertainty accounting for both inherent stochastic uncertainty and model estimation error. Therefore, the proposed approach can provide accurate and reliable short-term wind power probabilistic forecast, which can be used to support smart power grids real-time risk management.

A copula approach for dependence modeling in multivariate nonparametric time series

This paper is concerned with modeling the dependence structure of two (or more) time series in the presence of a (possibly multivariate) covariate which may include past values of the time series. We assume that the covariate influences only the conditional mean and the conditional variance of each of the time series but the distribution of the standardized innovations is not influenced by the covariate and is stable in time. The joint distribution of the time series is then determined by the conditional means, the conditional variances and the marginal distributions of the innovations, which we estimate nonparametrically, and the copula of the innovations, which represents the dependency structure. We consider a nonparametric and a semiparametric estimator based on the estimated residuals. We show that under suitable assumptions, these copula estimators are asymptotically equivalent to estimators that would be based on the unobserved innovations. The theoretical results are illustrated by simulations and a real data example.

The ENBIS‐17 Quality and Reliability Engineering International Special Issue

The ENBIS‐17 Quality and Reliability Engineering International Special Issue

The Decomposition Issue of a Time Series in the Forecasting Process

Abstract Decomposition of time series is the estimate and extraction of deterministic part of the series - trend, cyclical and seasonal fluctuations in the hope that the rest of the data, that is, theoretically, a random variable will be stationary random process. During the process of predicting the time series elements affects significantly on the determination of the future values, which are characterized by a low forecast error. Therefore, the purpose of this article is to identify the elements of the time series decomposition and to determine the extent to which they affect the forecasting process. Problems that often appear when you run the forecast and methods of building models and forecasts based on time series will be presented. Observations will be described on the basis of nonparametric time series modeling.