Assumption Of Local Stationarity Research Articles

Time-varying coefficient DAR model and stability measures for stablecoin prices: An application to Tether

This paper examines the dynamics of Tether, the stablecoin with the largest market capitalization. We show that the distributional and dynamic properties of Tether/USD rates have been evolving between 2017 - 2021. We use local analysis methods to detect local trends and time-varying volatility and persistence. To accommodate these dynamic patterns, we introduce the time-varying parameter Double Autoregressive tvDAR(1) process under the assumption of local stationarity of Tether/USD rates. We also propose a set of simple plug-in measures of stability based on the Lyapunov exponent for assessing and comparing the stability of Tether and other stablecoins. We show the consistency of stability measure estimators and study their distributional properties. The tvDAR model is estimated on Tether prices and compared with other stablecoins. We find that it provides good fit to data with time-varying persistence and volatility and produces reliable out-of-sample forecasts at short horizons. In the application, the proposed stability measures provide reliable indicators of local explosive patterns for stablecoin monitoring.

Local Likelihood Estimation of Complex Tail Dependence Structures, Applied to U.S. Precipitation Extremes

To disentangle the complex nonstationary dependence structure of precipitation extremes over the entire contiguous United States (U.S.), we propose a flexible local approach based on factor copula models. Our subasymptotic spatial modeling framework yields nontrivial tail dependence structures, with a weakening dependence strength as events become more extreme; a feature commonly observed with precipitation data but not accounted for in classical asymptotic extreme-value models. To estimate the local extremal behavior, we fit the proposed model in small regional neighborhoods to high threshold exceedances, under the assumption of local stationarity, which allows us to gain in flexibility. By adopting a local censored likelihood approach, we make inference on a fine spatial grid, and we perform local estimation by taking advantage of distributed computing resources and the embarrassingly parallel nature of this estimation procedure. The local model is efficiently fitted at all grid points, and uncertainty is measured using a block bootstrap procedure. We carry out an extensive simulation study to show that our approach can adequately capture complex, nonstationary dependencies, in addition, our study of U.S. winter precipitation data reveals interesting differences in local tail structures over space, which has important implications on regional risk assessment of extreme precipitation events. Supplementary materials for this article, including a standardized description of the materials available for reproducing the work, are available as an online supplement.

Modeling and emulation of nonstationary Gaussian fields

Geophysical and other natural processes often exhibit nonstationary covariances and this feature is important for statistical models that attempt to emulate the physical process. A convolution-based model is used to represent nonstationary Gaussian processes that allows for variation in the correlation range and the variance of the process across space. We apply this model in two steps: windowed estimates of the covariance function under the assumption of local stationarity and encoded local estimates into a single spatial process model that allows for efficient simulation. We show that nonstationary covariance functions based on the Matérn family can be reproduced by the LatticeKrig (LK) model, a flexible, multi-resolution representation of Gaussian processes. Stationary models based on the Matérn covariance are fit in local windows and these estimates are assembled into a single, global LK model. The LK model is efficient for simulating nonstationary fields even at 105 locations. This work is motivated by the interest in emulating spatial fields derived from numerical model simulations such as Earth system models. We successfully apply these ideas to emulate fields that describe the uncertainty in the pattern scaling of mean summer (JJA) surface temperature from a series of climate model experiments. The spatial covariance structure developed in this paper is not limited to emulation, and could also be used for spatial prediction and conditional simulation for observational data and leverages embarrassingly parallel strategies for computational efficiency.

The dimensionless dissipation rate and the Kolmogorov (1941) hypothesis of local stationarity in freely decaying isotropic turbulence

An expression for the dimensionless dissipation rate was derived from the Kármán-Howarth equation by an asymptotic expansion of the second- and third-order structure functions in powers of the inverse Reynolds number. The implications of the time-derivative term for the assumption of local stationarity (or local equilibrium) which underpins the derivation of the Kolmogorov “4/5” law for the third-order structure function were studied. It was concluded that neglect of the time-derivative cannot be justified by reason of restriction to certain scales (the inertial range) nor to large Reynolds numbers. In principle, therefore, the hypothesis cannot be correct, although it may be a good approximation. It follows, at least in principle, that the quantitative aspects of the hypothesis of local stationarity could be tested by a comparison of the asymptotic dimensionless dissipation rate for free decay with that for the stationary case. But in practice this is complicated by the absence of an agreed evolution time te for making the measurements during the decay. However, we can assess the quantitative error involved in using the hypothesis by comparing the exact asymptotic value of the dimensionless dissipation in free decay calculated on the assumption of local stationarity to the experimentally determined value (e.g., by means of direct numerical simulation), as this relationship holds for all measuring times. Should the assumption of local stationarity lead to significant error, then the “4/5” law needs to be corrected. Despite this, scale invariance in wavenumber space appears to hold in the formal limit of infinite Reynolds numbers, which implies that the “−5/3” energy spectrum does not require correction in this limit.

Interferogram denoising using an iteratively refined nonlocal InSAR filter

ABSTRACTInterferogram denoising is important for height reconstruction and deformation measurement. For high-resolution interferograms over heterogeneous areas, local biases and resolution losses may appear due to the violation of the local stationarity assumption. To address this problem, an iteratively refined nonlocal filter is proposed, whose estimation is refined iteratively by jointly using the amplitude, interferometric phase, coherence, and the pre-estimated patches. For nonlocal interferometric synthetic aperture radar (InSAR) techniques, the similarity of two pixels is computed via their surrounding patches, followed by the maximum likelihood weighted averaging of similar pixels. In this letter, the outliers in the search window is identified before the weight calculation. If the normalized probability density function (NPDF) of the central pixel and other surrounding pixels in the search window is less than the preset threshold, the pixel will be assigned with the minimum weight in the search window. Moreover, the denoising weight is calculated not only depending on the probabilistic patch-based (PPB) similarity, but also the coherence of the pixels in the search window. Both simulated and real data experiments are used to validate the effectiveness of the proposed method.

Predictive Geological Mapping Using Closed-Form Non-stationary Covariance Functions with Locally Varying Anisotropy: Case Study at El Teniente Mine (Chile)

This paper is concerned with the problem of predicting the surface elevation of the Braden breccia pipe at the El Teniente mine in Chile. This mine is one of the world’s largest and most complex porphyry-copper ore systems. As the pipe surface constitutes the limit of the deposit and the mining operation, predicting it accurately is important. The problem is tackled by applying a geostatistical approach based on closed-form non-stationary covariance functions with locally varying anisotropy. This approach relies on the mild assumption of local stationarity and involves a kernel-based experimental local variogram a weighted local least-squares method for the inference of local covariance parameters and a kernel smoothing technique for knitting the local covariance parameters together for kriging purpose. According to the results, this non-stationary geostatistical method outperforms the traditional stationary geostatistical method in terms of prediction and prediction uncertainty accuracies.

Noise Filtering of High-Resolution Interferograms Over Vegetation and Urban Areas With a Refined Nonlocal Filter

Interferometric synthetic aperture radar (SAR) filtering is a crucial processing step in topography reconstruction and deformation measurement. The high-resolution interferograms over vegetation or urban areas are heterogeneous, which will violate the local stationarity assumption and make it difficult to obtain a large number of independent and identically distributed samples for interferometric noise suppression. To overcome this problem, a refined nonlocal filter is proposed in this letter. Given coherence, topography phase, and expected SAR intensity of the pixel to be filtered, the similarity distance between the central pixel and other pixels in the searching window is measured by the normalized probability density function of the interferogram. Then, the outliers whose normalized probability density is smaller than the given threshold are removed from the filtering process. Finally, both simulated and real high-resolution interferograms are used to evaluate the effectiveness of the proposed method.

Modeling and Control of Nonstationary Noise Characteristics in Filtered-Backprojection and Penalized Likelihood Image Reconstruction.

Nonstationarity of CT noise presents a major challenge to the assessment of image quality. This work presents models for imaging performance in both filtered backprojection (FBP) and penalized likelihood (PL) reconstruction that describe not only the dependence on the imaging chain but also the dependence on the object as well as the nonstationary characteristics of the signal and noise. The work furthermore demonstrates the ability to impart control over the imaging process by adjusting reconstruction parameters to exploit nonstationarity in a manner advantageous to a particular imaging task. A cascaded systems analysis model was used to model the local noise-power spectrum (NPS) and modulation transfer function (MTF) for FBP reconstruction, with locality achieved by separate calculation of fluence and system gain for each view as a function of detector location. The covariance and impulse response function for PL reconstruction (quadratic penalty) were computed using the implicit function theorem and Taylor expansion. Detectability index was calculated under the assumption of local stationarity to show the variation in task-dependent image quality throughout the image for simple and complex, heterogeneous objects. Control of noise magnitude and correlation was achieved by applying a spatially varying roughness penalty in PL reconstruction in a manner that improved overall detectability. The models provide a foundation for task-based imaging performance assessment in FBP and PL image reconstruction. For both FBP and PL, noise is anisotropic and varies in a manner dependent on the path length of each view traversing the object. The anisotropy in turn affects task performance, where detectability is enhanced or diminished depending on the frequency content of the task relative to that of the NPS. Spatial variation of the roughness penalty can be exploited to control noise magnitude and correlation (and hence detectability). Nonstationarity of image noise is a significant effect that can be modeled in both FBP and PL image reconstruction. Prevalent spatial-frequency-dependent metrics of spatial resolution and noise can be analyzed under assumptions of local stationarity, providing a means to analyze imaging performance as a function of location throughout the image. Knowledgeable selection of a spatially-varying roughness penalty in PL can potentially improve local noise and spatial resolution in a manner tuned to a particular imaging task.

Multiscale Modeling for Polymeric Flow: Particle-Fluid Bridging Scale Methods

Multiscale simulation methods have been developed based on the local stress sampling strategy and applied to three flow problems with different difficulty levels: (a) general flow problems of simple fluids, (b) parallel (one-dimensional) flow problems of polymeric liquids, and (c) general (two- or three-dimensional) flow problems of polymeric liquids. In our multiscale methods, the local stress of each fluid element is calculated directly by performing microscopic or mesoscopic simulations according to the local flow quantities instead of using any constitutive relations. For simple fluids (a), such as the Lenard-Jones liquid, a multiscale method combining molecular dynamics (MD) and computational fluid dynamics (CFD) simulations is developed rather straightforwardly based on the local stationarity assumption without memories of the flow history. For polymeric liquids in parallel flows (b), the multiscale method is extended to take into account the memory effects that arise in hydrodynamic stress due to the slow relaxation of polymer-chain conformations. The memory of polymer dynamics on each fluid element is thus resolved by performing MD simulations in which cells are fixed at the mesh nodes of the CFD simulations. The complicated viscoelastic flow behaviors of a polymeric liquid confined between oscillating plates are simulated using the multiscale method. For general (two- or three-dimensional) flow problems of polymeric liquids (c), it is necessary to trace the history of microscopic information such as polymer-chain conformation, which carries the memories of past flow history, along the streamline of each fluid element. A Lagrangian-based CFD is thus implemented to correctly advect the polymerchain conformation consistently with the flow. On each fluid element, coarse-grained polymer simulations are carried out to consider the dynamics of entangled polymer chains that show extremely slow relaxation compared to microscopic time scales. This method is successfully applied to simulate a flow around a cylindrical obstacle.

Hypothesis of local stationarity in the problem of stochastic prognosis by kriging method

One of the most widely-used stochastic prognosis methods for estimating the reserves of mineral resources, namely the method of kriging, is considered. A modification of this method based on the assumption of local stationarity of the spatial variable is proposed. The modified kriging method, in combination with a morphological algorithm for detecting the subdomains of local stationarity is shown to be in good agreement with the a priori ideas on data structure. This makes it possible to decrease computational efforts for the forecasting procedure as compared to the standard kriging technique.

Analytic-Wavelet-Ridge-Based Detection of Dynamic Eccentricity in Brushless Direct Current (BLDC) Motors Functioning Under Dynamic Operating Conditions

A new method using the analytic wavelet transform of the stator-current signal is proposed for detecting dynamic eccentricity in brushless direct current (BLDC) motors operating under rapidly varying speed and load conditions. As wavelets are inherently suited for nonstationary signal analysis, this method does not require the use of any windows, nor is it dependent on any assumption of local stationarity as in the case of the short-time Fourier transform. The proposed technique uses analytic wavelets, which are smooth wavelets that possess both magnitude and phase information. This makes them particularly suitable for motor-fault diagnostics. Experimental results are provided to show that the proposed method works over a wide speed range of motor operation and provides an effective and robust way of detecting rotor faults such as dynamic eccentricity in BLDC motors

Statistical analysis of financial time series under the assumption of local stationarity

The aim of this paper is to apply a nonparametric methodology developed by Donoho et al(2003 IEEE Trans. Signal Processing 53614–27) for estimating an autocovariance sequence to the statistical analysis of the return of securities and discuss the advantages offered by this approach over other existing methods such as fixed-window-length segmentation procedures. Theoretical properties of adaptivity of this estimation method have been proved for a specific class of time series, namely the class of locally stationary processes, with an autocovariance structure which varies slowly over time in most cases but might exhibit abrupt changes of regime. This method is based on an algorithm that selects empirically from the data the tiling of the time–frequency plane which exposes best in the least-squares sense the underlying second-order time-varying structure of the time series, and so may properly describe the time-inhomogeneous variations of speculative prices. The applications we consider here mainly concern the analysis of structural changes occurring in stock market returns, VaR estimation and the comparison between the variation structure of stock index returns in developed markets and in developing markets.

Statistical analysis of financial time series under the assumption of local stationarity

The aim of this paper is to apply a nonparametric methodology developed by Donoho et al(2003 IEEE Trans. Signal Processing 53614–27) for estimating an autocovariance sequence to the statistical analysis of the return of securities and discuss the advantages offered by this approach over other existing methods such as fixed-window-length segmentation procedures. Theoretical properties of adaptivity of this estimation method have been proved for a specific class of time series, namely the class of locally stationary processes, with an autocovariance structure which varies slowly over time in most cases but might exhibit abrupt changes of regime. This method is based on an algorithm that selects empirically from the data the tiling of the time–frequency plane which exposes best in the least-squares sense the underlying second-order time-varying structure of the time series, and so may properly describe the time-inhomogeneous variations of speculative prices. The applications we consider here mainly concern the analysis of structural changes occurring in stock market returns, VaR estimation and the comparison between the variation structure of stock index returns in developed markets and in developing markets.

A time-frequency approach for newborn seizure detection.

Techniques previously designed for seizure detection in newborns using the electroencephalogram (EEG) have been relatively inefficient due to their assumption of local stationarity of the EEG. To overcome the problem raised by the nonstationarity of the EEG signal, current methods are extended to a time-frequency approach. This allows the analysis and characterization of the different newborn EEG patterns that are intended to be the first step toward an automatic time-frequency seizure detection and classification. An in-depth analysis of both the autocorrelation and spectrum seizure detection techniques identified the detection criteria that can be extended to the time-frequency domain. The selected method uses a high-resolution reduced interference time-frequency distribution referred to as the B-distribution (BD). Here, the authors present the various patterns of observed time-frequency seizure signals and relate them to current knowledge of seizures. In particular, initial results indicate that a quasilinear instantaneous frequency (IF) can be used as a critical feature of the EEG seizure characteristics.

Using phase-state modelling for inferring forecasting uncertainty in nonlinear stochastic decision schemes

The paper introduces the use of phase-state modelling as a means of estimating expected benefits or losses when dealing with decision processes under uncertainty of future events. For this reason the phase-space approach to time series, which generally aims at forecasting the expected value of a future event, is here also used to assess the forecasting uncertainty. Under the assumption of local stationarity the ensemble of generated future trajectories can be used to estimate a probability density that represents the a priori uncertainty of forecasts conditional on the latest measurements. This a priori density can then be used directly in the optimisation schemes if no additional information is available, or after deriving an a posteriori distribution in the Bayesian sense, by combining it with forecasts from deterministic models, here taken as noise-corrupted ‘pseudo-measurements’ of future events. Examples of application are given in the case of the Lake Como real-time management system as well as in the case of rainfall ensemble forecasts on the River Reno.

Detection of phase- or frequency-modulated signals in reverberation noise

This paper is devoted to the detection of phase- or frequency-modulated signals corrupted by the reverberation noise they have created. When the reverberation is modeled as a time-varying stochastic filtering of the emitted signal, the optimal detector consists of prewhitening the observation before applying a matched filter. This theory is very difficult to implement because of the nonstationarity of the reverberation noise. In this paper, a suboptimal detector proposed by Kay and Salisbury in 1990 is generalized to phase or frequency modulated signals. The reverberation is assumed locally stationary; the signal is cut into blocks and the whitening is performed as follows: An autoregressive (AR) modeling of one block allows the whitening of the following block. After whitening, an appropriate matched filter is applied. This scheme is explained in detail. The local stationarity assumption of the reverberation noise is discussed in light of the Itakura spectral distance. Performance of the proposed approach is evaluated theoretically when all parameters are known, and experimentally using three different real-data sets.

Estimation of time‐varying autoregressive model parameters by multilayered neural networks: Identification of local stationary sections from noise variance

This paper proposes a method of estimating time-varying autoregressive (AR) model parameters by means of multilayered neural networks (NNs). First, an estimation method for time-varying AR parameters is discussed under the assumption of local stationarity. Then it is shown that the section length T satisfying local stationarity is very important in maintaining the balance of estimation precision for both noise variance and time-varying AR parameters. Next, on the basis of the nonlinear Yule-Walker equation, the noise variance is estimated by NNs. The section length T satisfying local stationarity is determined by estimated noise variance, and for the specified section length T, time-varying AR parameters are estimated by the Yule-Walker method. Finally, from numerical simulation results it is shown that the proposed method is very effective for the estimation of time-varying AR model parameters. © 1997 Scripta Technica, Inc. Electron Comm Jpn Pt 3, 80(7): 20–27, 1997