Nonstationary Error Research Articles

A long-term, high-accuracy and seamless 1km soil moisture dataset over the Qinghai-Tibet Plateau during 2001–2020 based on a two-step downscaling method

ABSTRACTLong-term, high-resolution soil moisture (SM) is a vital variable for understanding the water-energy cycle and the impacts of climate change on the Qinghai-Tibet Plateau (QTP). However, most existing satellite SM data are only available at coarse scale (~25 km) and suffer a lot from data gaps due to satellite orbit coverage and snow cover, especially on the QTP. Although substantial efforts have been devoted to downscale SM utilizing multiple soil moisture indices (SMIs) or diverse machine learning (ML) methods, the potentials of different SMIs and ML approaches in SM downscaling on the complex plateau remain unclear, and there is still a necessity to obtain an accurate, long-term, high-resolution and seamless SM data over the QTP. To address this issue, this study generated the long-term, high-accuracy and seamless soil moisture dataset (LHS-SM) over the QTP during 2001–2020 using a two-step downscaling method (first downscaling then merging). Firstly, the daily SM data from the Climate Change Initiative program of the European Space Agency (ESA CCI) was downscaled to 1 km utilizing five ML approaches. Then, a dynamic data merging method that considers spatiotemporal nonstationary error was applied to derive the final LHS-SM data. The performance of fifteen SMIs was also assessed and the optimal indexes for downscaling were identified. Results indicated that the shortwave infrared band-based indices had better performance than the near infrared band-based and energy-based indices. The generated LHS-SM data exhibited satisfying accuracy (mean R = 0.52, ubRMSE = 0.047 m3/m3) and certain improvement to the ESA CCI SM data both at station and network scales. Compared with existing 1 km SM datasets, the LHS-SM data also showed the best performance (mean R = 0.62, ubRMSE = 0.047 m3/m3), while existing datasets either failed to fully characterize the spatial details or had some data gaps and unreasonable distributions. Strong spatial heterogeneity was observed in the SM dynamics during 2001–2020 with the southwest and northeast showing a “dry gets wetter” scheme and the southeast presenting a “wet gets drier” trend. Overall, the LHS-SM dataset gained its added values by compensating the drawbacks of existing 1 km SM products over the QTP and was much valuable for many regional applications.

Nonstationary Clutter Compensation for Airborne Surveillance Radar Systems With Crab Angle

In this letter, the clutter range dependence issue caused by crab angle is investigated in an airborne multi-channel radar system. A new method is proposed to accomplish the crab angle estimation and the non-stationary error compensation. Firstly, from the center-biased and spectrum-broadened characteristics of the clutter space-time spectrum in the case of non-neglected crab angle, a 1-D cost function based on space-time spectrum broadening degree is constructed to achieve the accurate crab angle estimation. Then, the clutter non-stationary errors can be effectively compensated in the post-Doppler domain, significantly improving the subsequent space time adaptive processing (STAP) performance. Simulation results are presented to validate the feasibility and effectiveness of the proposed method.

Visual extensions and anomaly detection in the pNEUMA experiment with a swarm of drones

The usage of Unmanned Aerial Systems (UAS) in traffic monitoring has advantages such as broader vision, flexibility, privacy, and cost-efficiency compared to other traffic monitoring sensors like loop detectors or fixed surveillance cameras. UAS has been instrumental in the recent pNEUMA experiment, where a swarm of drones collected a large-scale urban traffic dataset containing vehicle trajectories. These trajectory data are subject to challenges such as noise due to visual restrictions, perspective distortions, and human-induced errors. While the pNEUMA dataset is missing its imagery part, we present an extended version of it named pNEUMA Vision, which incorporates imagery data and annotations of vehicles in the form of image coordinates along with newly added vehicle trajectory features like azimuth. Moreover, we demonstrate that visually restricted trajectories are also highly prone to become anomalies, and vice-versa, through novel anomaly detection proposed in this literature. Specifically, we distinguish between stationary and non-stationary errors and argue that the latter account for the largest part of the noise. Furthermore, we analyze the new visual dataset with two different computer vision methods for estimating the number of vehicles on the roads from the input images. Particularly, we show that using a density map to count vehicles from drone images achieves comparable results to conventional vehicle counting methods via detection. Results show that the time–space diagrams plotted from density map predictions could identify the congested urban roads’ queues better than a widely used vehicle detection method.

A triple collocation-based 2D soil moisture merging methodology considering spatial and temporal non-stationary errors

Random error in remotely sensed (RS) and modeled soil moisture (SM) products is typically assumed to be statistically stationary for the purpose of SM merging applications. In reality, such error is often non-stationary, which may undermine applications based on a stationary assumption. Here, we introduce a dual time-space (2D) SM merging approach that considers a class of inferred error that is non-stationary and undetectable in time (or space) but stationary and detectable in space (or time). Such 2D merging is realized in a least-squares framework where spatial and temporal error variances for each product are estimated via a triple collocation (TC) analysis. As a test case, a 2D-merged SM product is obtained by combining three independent SM products – including two RS SM products and one modeled SM product. Results show that the 2D merging method can effectively handle non-stationary errors and, as a result, produces a superior merged SM product than classical 1D merging methods. The spatial and temporal correlations with in-situ observations are 0.62 and 0.63 [−] on average for 2D merging methods, and 0.58 and 0.59 [−] on average for classical 1D merging methods. By providing a more effective way to detect and remove non-stationary errors during the process of merging multi-source SM products, this approach will improve future global multi-source merged SM products.

Accounting for theory errors with empirical Bayesian noise models in nonlinear centroid moment tensor estimation

SUMMARYCentroid moment tensor (CMT) parameters can be estimated from seismic waveforms. Since these data indirectly observe the deformation process, CMTs are inferred as solutions to inverse problems which are generally underdetermined and require significant assumptions, including assumptions about data noise. Broadly speaking, we consider noise to include both theory and measurement errors, where theory errors are due to assumptions in the inverse problem and measurement errors are caused by the measurement process. While data errors are routinely included in parameter estimation for full CMTs, less attention has been paid to theory errors related to velocity-model uncertainties and how these affect the resulting moment-tensor (MT) uncertainties. Therefore, rigorous uncertainty quantification for CMTs may require theory-error estimation which becomes a problem of specifying noise models. Various noise models have been proposed, and these rely on several assumptions. All approaches quantify theory errors by estimating the covariance matrix of data residuals. However, this estimation can be based on explicit modelling, empirical estimation and/or ignore or include covariances. We quantitatively compare several approaches by presenting parameter and uncertainty estimates in nonlinear full CMT estimation for several simulated data sets and regional field data of the Ml 4.4, 2015 June 13 Fox Creek, Canada, event. While our main focus is at regional distances, the tested approaches are general and implemented for arbitrary source model choice. These include known or unknown centroid locations, full MTs, deviatoric MTs and double-couple MTs. We demonstrate that velocity-model uncertainties can profoundly affect parameter estimation and that their inclusion leads to more realistic parameter uncertainty quantification. However, not all approaches perform equally well. Including theory errors by estimating non-stationary (non-Toeplitz) error covariance matrices via iterative schemes during Monte Carlo sampling performs best and is computationally most efficient. In general, including velocity-model uncertainties is most important in cases where velocity structure is poorly known.

Changepoint in Error-Prone Relations

Linear relations, containing measurement errors in input and output data, are considered. Parameters of these so-called errors-in-variables models can change at some unknown moment. The aim is to test whether such an unknown change has occurred or not. For instance, detecting a change in trend for a randomly spaced time series is a special case of the investigated framework. The designed changepoint tests are shown to be consistent and involve neither nuisance parameters nor tuning constants, which makes the testing procedures effortlessly applicable. A changepoint estimator is also introduced and its consistency is proved. A boundary issue is avoided, meaning that the changepoint can be detected when being close to the extremities of the observation regime. As a theoretical basis for the developed methods, a weak invariance principle for the smallest singular value of the data matrix is provided, assuming weakly dependent and non-stationary errors. The results are presented in a simulation study, which demonstrates computational efficiency of the techniques. The completely data-driven tests are illustrated through problems coming from calibration and insurance; however, the methodology can be applied to other areas such as clinical measurements, dietary assessment, computational psychometrics, or environmental toxicology as manifested in the paper.

A distribution free test for changes in the trend function of locally stationary processes

In the common time series model Xi,n=μ(i∕n)+εi,n with non-stationary errors we consider the problem of detecting a significant deviation of the mean function μ from a benchmark g(μ) (such as the initial value μ(0) or the average trend ∫01μ(t)dt). The problem is motivated by a more realistic modelling of change point analysis, where one is interested in identifying relevant deviations in a smoothly varying sequence of means (μ(i∕n))i=1,…,n and cannot assume that the sequence is piecewise constant. A test for this type of hypotheses is developed using an appropriate estimator for the integrated squared deviation of the mean function and the threshold. By a new concept of self-normalization adapted to non-stationary processes an asymptotically pivotal test for the hypothesis of a relevant deviation is constructed. The results are illustrated by means of a simulation study and a data example.

Semiparametric estimation of generalized transformation panel data models with nonstationary error

Summary Early studies of the generalized transformation panel data model resorted to the identical marginal distribution of the error term over time. This stationarity condition is restrictive for many applications, especially as the number of time periods increases. This paper considers nonstationary censored generalized transformation panel data models where the idiosyncratic error has an unknown nonseparable form and admits a flexible relationship between the observable and the unobservable. We propose an estimation method, and establish the consistency and asymptotic normality for the proposed estimator. Simulation results illustrate the good performance of our estimator in a finite sample. We apply the proposed method to bilateral trade issues of the U.S.A. and foreign countries.

Uncertainties in GPS-based operational orbit determination: A case study of the Sentinel-1 and Sentinel-2 satellites

ABSTRACTThe European Space Operations Centre currently operates five Copernicus Sentinel satellites in the framework of Europe’s Copernicus Earth observation programme. The routine operations rely on a daily orbit determination, carried out on-ground, consisting in a least-squares fit of a dynamical model to GPS navigation solutions generated on-board. The purpose of this paper is the estimation of realistic uncertainties on this daily determined state vector. By comparison with the orbit derived by Precise Orbit Determination, we estimate the 1-sigma errors at approximately 0.5m and 0.5mm/s. Non-stationary errors in the navigation solution preclude their characterisation with a constant covariance matrix. Error whitening is achieved by decreasing the signal-to-noise ratio in the errors through the use of underestimated weights on the data. The approach keeps the errors on the derived state vector unchanged and allows the covariance on the state vector to become realistic.

China’s energy consumption prediction considering error correction based on decompose–ensemble method

Energy consumption forecasting plays a vital role in rational energy and economic planning formulation for a country or an area around the world. A novel forecasting method based on error correction and decompose-ensemble strategy combined with linear regression model (LR) and triple exponential smoothing model (TESM) is proposed in this study. Firstly, original energy consumption is decomposed into the main trend subseries and non-stationary errors subseries by use of LR model. And the mainly trend subseries is forecasted by LR. Then, the non-stationary subseries (forecasting errors of LR) is decomposed into several intrinsic mode functions (high-frequency subseries, middle-frequency subseries and low-frequency subseries) and residual subseries by empirical mode decomposition (EMD). With respect to their different dynamic changing features and influenced factors, each intrinsic mode functions subseries and residual subseries are forecasted, respectively by TESM. Finally, the prediction of energy consumption is obtained by summing the trend subseries prediction results and these errors subseries prediction results. Forecasting results prove that error correction is a useful strategy for improve the forecasting performance. Most of all, the origin complex errors correction forecasting problem has been resolved into some simple forecasting problem based on decompose-ensemble strategy, which have better forecasting performance, compared with individual models (LR, auto regression model (AR) and TESM), traditional error correction method, combination models (which is developed by using of LR, AR and TESM based on equal weight method, entropy weight method and optimal weight method). The proposed novel method can provide accurate and reliable forecasting results, which is a feasible forecasting approach for China’s annual energy consumption. By use of the proposed forecasting method considering error correction based on EMD decompose-ensemble strategy combined with LR and TESM, China’s energy consumption in 2021 will increase to 484,555.30 ten thousand standard tons coal equivalent (tce), and the average annual growth of the coming 5 years is 2.135%. Since China’s energy consumption is still on its growing process, China should pay more attention to change its economic development model from energy-intensive economy to low-carbon economy.

Analysis of GRACE range-rate residuals with focus on KBR instrument system noise

We investigate the post-fit range-rate residuals after the gravity field parameter estimation from the inter-satellite ranging data of the gravity recovery and climate experiment (grace) satellite mission. Of particular interest is the high-frequency spectrum (f> 20 mhz) which is dominated by the microwave ranging system noise. Such analysis is carried out to understand the yet unsolved discrepancy between the predicted baseline errors and the observed ones. The analysis consists of two parts. First, we present the effects in the signal-to-noise ratio (snrs) of the k-band ranging system. The snrs are also affected by the moon intrusions into the star cameras’ field of view and magnetic torquer rod currents in addition to the effects presented by Harvey et al. (2016). Second, we analyze the range-rate residuals to study the effects of the kbr system noise. The range-rate residuals are dominated by the non-stationary errors in the high-frequency observations. These high-frequency errors in the range-rate residuals are found to be dependent on the temperature and effects of sun intrusion into the star cameras’ field of view reflected in the snrs of the k-band phase observations.

BOOTSTRAP-ASSISTED UNIT ROOT TESTING WITH PIECEWISE LOCALLY STATIONARY ERRORS

In unit root testing, a piecewise locally stationary process is adopted to accommodate nonstationary errors that can have both smooth and abrupt changes in second- or higher-order properties. Under this framework, the limiting null distributions of the conventional unit root test statistics are derived and shown to contain a number of unknown parameters. To circumvent the difficulty of direct consistent estimation, we propose to use the dependent wild bootstrap to approximate the nonpivotal limiting null distributions and provide a rigorous theoretical justification for bootstrap consistency. The proposed method is compared through finite sample simulations with the recolored wild bootstrap procedure, which was developed for errors that follow a heteroscedastic linear process. Furthermore, a combination of autoregressive sieve recoloring with the dependent wild bootstrap is shown to perform well. The validity of the dependent wild bootstrap in a nonstationary setting is demonstrated for the first time, showing the possibility of extensions to other inference problems associated with locally stationary processes.

Consistent estimator of nonparametric structural spurious regression model for high frequency data

We propose a new nonparametric estimator for continuous-time regression models with nonstationary error terms. While other conventional nonparametric estimators such as the Nadaraya–Watson and local linear estimators are not consistent, our estimator achieves consistency and asymptotic normality.

PIAPF for manoeuvring target tracking in the presence of multiplicative noise

In this study, a prediction-interval-based adaptive particle filter (PIAPF) is developed to track a manoeuvring target in the presence of multiplicative measurement noise. In PIAPF, input augmentation technique is utilised to estimate the state variables of target and unknown inputs (manoeuvres) simultaneously. To cope with unknown sudden changes of system state variables (caused by manoeuvres), the covariance matrix of the importance density function is adaptively adjusted based on the prediction interval of the output estimation. In addition, a theorem is developed which confirms that the output estimation error is upper bounded by a given probability. The likelihood function of the non-stationary state-dependent error of sensors, which is modelled as multiplicative noise, is then obtained for weight calculation of PIAPF. The proposed PIAPF is then used to track a manoeuvring target in a wireless sensor network with distance-measuring sensor nodes. Simulation results demonstrate the effectiveness of the proposed PIAPF in terms of tracking accuracy and computational load.

Cointegration as an explanation for the Meese–Rogoff puzzle

ABSTRACTSome economists suggest that the failure of exchange-rate models to outperform the random walk in exchange rate forecasting out of sample can be attributed to failure to take into account cointegration when it is present. We attempt to find out if cointegration matters for forecasting accuracy by examining the relation between the stationarity and size of the forecasting error. Results based on three macroeconomic models of exchange rates do not provide strong support for the proposition that cointegration matters for forecasting accuracy. The simulation results show that while stationary errors tend to be smaller than non-stationary errors, this is not a universal rule. Irrespective of the presence or absence of cointegration, none of the three models can outperform the random walk in out-of-sample forecasting, which means that cointegration cannot solve the Meese–Rogoff puzzle.

Testing for jumps in the presence of smooth changes in trends of nonstationary time series

Nonparametric smoothing methods have been widely used in trend analysis. However, the inference procedure usually requires the crucial assumption that the underlying trend function is smooth. This paper considers the situation where the trend function has potential jumps in addition to smooth changes. In order to determine the existence of jumps, we propose a nonparametric test that can survive under dependent and nonstationary errors, where existing tests assuming independence or stationarity can fail. When the existence of jumps is affirmative, we further consider the problem of estimating the number, location and size of jumps. The results are illustrated via both Monte Carlo simulations and a real data example.

Stochastic Characterization of a MEMs based Inertial Navigation Sensor using Interval Methods

The aim here remains to introduce effectiveness of interval methods in analyzing dynamic uncertainties for marine navigational sensors. The present work has been carried out with an integrated sensor suite consisting of a low cost MEMs inertial sensor, GPS receiver of moderate accuracy, Doppler velocity profiler and a magnetic fluxgate compass. Error bounds for all the sensors have been translated into guaranteed intervals. GPS based position intervals are fed into a forward- backward propagation method in order to estimate interval valued inertial data. Dynamic noise margins are finally computed from comparisons between the estimated and measured inertial quantities It has been found that the intervals as estimated by proposed approach are supersets of 95% confidence levels of dynamic errors of accelerations. This indicates a significant drift of dynamic error in accelerations which may not be clearly defined using stationary error bounds. On the other side bounds of non-stationary error for rate gyroscope are found to be in consistence with the intervals as predicted using stationary noise coefficients. The guaranteed intervals estimated by the proposed forward backward contractor, are close to 95% confidence levels of stationary errors computed over the sampling period. Index Terms—Stochastic, dynamic, error, MEMs, inertial, interval, methods, INS, GPS.

Improved Chirp Scaling Algorithm Based on Fractional Fourier Transform and Motion Compensation

Fractional fourier transform (FrFT) is a new tool of time-frequency analysis, and a good tool to compress chirp signal and to process non-stationary motion error. Aiming at the problem of the conventional SAR imaging algorithm based on FFT and two-step motion compensation, that of the effect of non-stationary error elimination is not obvious and the improvement of resolution is restricted, an improved Chirp Scaling imaging algorithm based on FrFT and motion compensation is put forward in this paper, which is expected to eliminate the influence of range motion error, so as to ob- tain high quality SAR images. Simulation results and imaging results of real SAR data show that the proposed algorithm can eliminate the influence of range motion error effectively. (the real SAR data provided by Institute of Electronics, Chi- nese Academy of Sciences).

An empirical examination of currency ETFs tracking error

In this paper, I study on intradaily basis six currency ETFs' tracking error volatility around the Great Recession - 12/30/2002 to 04/05/2013. I study the Australian Dollar ETF (FXA), the British Pound ETF (FXB), the Canadian Dollar ETF (FXC), the Euro ETF (FXE), the Swiss Franc ETF (FXF) and the Japanese Yen ETF (FXY). I find that the FXA, FXB and FXE tracking errors are non-stationary, whereas FXC, FXF and FXY tracking errors are stationary. The FXC, FXF and FXY ETFs tracking errors which are stationary and do not exhibit clustering of volatility are best described by a simple AR(1) model specification. The FXA, FXB and FXE ETFs tracking errors which are non-stationary and exhibit ARCH effects are best described by a AR(1)-GARCH(1,1) model. An arbitrageur can use these models to forecast these six currency ETFs tracking errors and identify a signal for an arbitrage opportunity.

Motion Compensation Technique Based On Fractional Fourier Transform

Fractional Fourier transform(FrFT) is a kind of generalized Fourier transform, which processes signals in the unified time-frequency domain and the linear frequency modulation signal can be well focused after FrFT. Motion error is an important factor affecting the SAR resolution, pointing to the problem that the effect of error elimination is not obvious in processing non-stationary motion error using the traditional FFT based motion compensation combined SAR imaging algorithm, FrFT based two-step motion compensation combined wavenumber domain algorithm and sub-aperture wide beam motion compensation algorithm are put forward in this paper, which are expected to eliminate the influence of motion error more effectively, so as to obtain high quality SAR images. The simulation results and the imaging results of real SAR data show that the proposed algorithms indeed eliminate the influence of motion error effectively. (the real SAR data provided by Institute of Electronics, Chinese Academy of Sciences). DOI : http://dx.doi.org/10.11591/telkomnika.v12i6.5478 Full Text: PDF