Observation Noise Research Articles

Online estimation of colored observation‐noise parameters within an ensemble Kalman filtering framework

AbstractThis work addresses the problem of data assimilation in large‐dimensional systems with colored observation noise of unknown statistics, a scenario that will become more common in the near future with the deployment of denser observational networks of high spatio‐temporal coverage. Here, we are interested in the ensemble Kalman filter (EnKF) framework, which has been derived around a white observation noise assumption. Recently, colored observation‐noise aware EnKFs in which the noise was modeled as a first‐order autoregressive (AR) model were introduced. This work generalizes the above‐mentioned filters to learn the statistics of the AR model further online. We follow the state augmentation approach first to estimate the state and the AR model transfer matrix simultaneously, then the variational Bayesian approach to estimate the AR model noise covariance parameters further. We accordingly derive two filtering EnKF‐like algorithms, which estimate those statistics together with the system state. We demonstrate the effectiveness of the proposed colored observation‐noise aware filtering schemes and compare their performance based on several numerical experiments conducted with the Lorenz‐96 model.

Accurate and Efficient Self-Localization of AGV Relying on Trusted Area Information in Dynamic Industrial Scene

Automatic Guided Vehicles (AGVs) are crucial for improving the operational efficiency of industrial scenes, however the self-localization of AGVs faces severe challenges in a dynamic environment due to a large number of observation noises. To address the above issue, this paper presents a trusted area information based self-localization method to achieve efficient and accurate localization in a highly dynamic scene, where the proportion of dynamic obstacles is close to 50%. First of all, the “coarse-noise filter-fine” self-localization framework is proposed to establish the foundation of accurate and efficient localization. Then, the morphology-based trusted area selection is adopted to efficaciously extract the trusted area, in which the probability of appearing dynamic objects is low. After that, based on dynamic angular resolution filtering considering the trusted area, a robust point cloud generation is designed to generate trusted area information. Trusted area information contains dense reliable point cloud in the statice area and spares observation noise in the dynamic area, thus improving the performance of observation models and scan matching. Finally, the self-developed AGV is adopted to verify the effectiveness of the proposed self-localization system in real and simulated dynamic scenes.

A Gaussian process cross-correlation approach to time-delay estimation in active galactic nuclei

Context. We present a probabilistic cross-correlation approach to estimate time delays in the context of reverberation mapping (RM) of active galactic nuclei (AGN). Aims. We reformulate the traditional interpolated cross-correlation method as a statistically principled model that delivers a posterior distribution for the delay. Methods. The method employs Gaussian processes as a model for observed AGN light curves. We describe the mathematical formalism and demonstrate the new approach using both simulated light curves and available RM observations. Results. The proposed method delivers a posterior distribution for the delay that accounts for observational noise and the non-uniform sampling of the light curves. This feature allows us to fully quantify the uncertainty on the delay and propagate it to subsequent calculations of dependant physical quantities, such as black hole masses. The method delivers out-of-sample predictions, which enables us to subject it to model selection, and can calculate the joint posterior delay for more than two light curves. Conclusions. Because of the numerous advantages of our reformulation and the simplicity of its application, we anticipate that our method will find favour not only in the specialised community of RM, but also in all fields where cross-correlation analysis is performed. We provide the algorithms and examples of their application as part of our Julia GPCC package.

Unsteady flow enhancement on an airfoil using sliding window weak-constraint four-dimensional variational data assimilation

This study establishes a continuous sliding window weak-constraint four-dimensional variational approach for reproducing a complete instantaneous flow from sparse spatiotemporal velocity observations. The initial condition, boundary condition, and model-form uncertainties are corrected simultaneously by a spatiotemporally varying additive forcing, coupled with the large eddy simulation (LES) framework, which reinforces subgrid-scale viscosity stresses and simplifies gradient computation. The additive force undergoes a Stokes–Helmholtz decomposition to ensure divergence-free projection and natural pressure determination. The model is theoretically derived to minimize discrepancies between the sparse velocity observations and the numerical predictions of the primary-adjoint system, enabling optimal contribution of the additive force. Synthetic data from a fine-grid LES of the vortical flow over an NACA0012 airfoil are used as observations. The algorithm is evaluated on a benchmark case, where observations are subsampled at 1/400 000 spatiotemporal resolution required for an LES. The sliding window strategy expands the dependence domain of the observations and mitigates the impact of primary-adjoint chaos, achieving over 90% pointwise correlation for filtered parameters and 80% spectral correlation for all of the resolved wavenumbers. Despite the lack of near-wall observations, streaks are accurately recovered due to the convective sensitivity of the observations from the outer flow. While the pressure fluctuation in the inflow region is not as well excited as in LES, recovery is augmented downstream. In both the inner and outer wall layers, the pressure distributions are obtained reasonably well by capturing the signatures of the vortical structure and their downstream convection. The robustness of the algorithm to observation noise is demonstrated. Finally, the impact of temporal resolution on estimation is evaluated, establishing a resolution threshold for successful reconstruction.

Harnessing Unresolved Lensed Quasars: The Mathematical Foundation of the Fluctuation Curve

Strong gravitational lensed quasars (QSOs) have emerged as powerful and novel cosmic probes as they can deliver crucial cosmological information, such as a measurement of the Hubble constant, independent of other probes. Although the upcoming LSST survey is expected to discover 103–104 lensed QSOs, a large fraction will remain unresolved due to seeing. The stochastic nature of the quasar intrinsic flux makes it challenging to identify lensed quasars and measure the time delays using unresolved light-curve data only. In this regard, Bag et al. introduced a data-driven technique based on the minimization of the fluctuation in the reconstructed image light curves. In this article, we delve deeper into the mathematical foundation of this approach. We show that the lensing signal in the fluctuation curve is dominated by the autocorrelation function (ACF) of the derivative of the joint light curve. This explains why the fluctuation curve enables the detection of the lensed QSOs only using the joint light curve, without making assumptions about QSO flux variability, nor requiring any additional information. We show that the ACF of the derivative of the joint light curve is more reliable than the ACF of the joint light curve itself because intrinsic quasar flux variability shows significant autocorrelation up to a few hundred days (as they follow a red power spectrum). In addition, we show that the minimization of fluctuation approach provides even better precision and recall as compared to the ACF of the derivative of the joint light curve when the data have significant observational noise.

Adaptive SLAM Methodology Based on Simulated Annealing Particle Swarm Optimization for AUV Navigation

Simultaneous localization and mapping (SLAM) is crucial and challenging for autonomous underwater vehicle (AUV) autonomous navigation in complex and uncertain ocean environments. However, inaccurate time-varying observation noise parameters may lead to filtering divergence and poor mapping accuracy. In addition, particles are easily trapped in local extrema during the resampling, which may lead to inaccurate state estimation. In this paper, we propose an innovative simulated annealing particle swarm optimization-adaptive unscented FastSLAM (SAPSO-AUFastSLAM) algorithm. To cope with the unknown observation noise, the maximum a posteriori probability estimation algorithm is introduced into SLAM to recursively correct the measurement noise. Firstly, the Sage–Husa (SH) based unscented particle filter (UPF) algorithm is proposed to estimate time-varying measurement noise adaptively in AUV path estimation for improving filtering accuracy. Secondly, the SH-based unscented Kalman filter (UKF) algorithm is proposed to enhance mapping accuracy in feature estimation. Thirdly, SAPSO-based resampling is proposed to optimize posterior particles. The random judgment mechanism is used to update feasible solutions iteratively, which makes particles disengage local extreme values and achieve optimal global effects. The effectiveness and accuracy of the proposed algorithm are evaluated through simulation and sea trial data. The average AUV navigation accuracy of the presented SAPSO-AUFastSLAM method is improved by 18.0% compared to FastSLAM, 6.5% compared to UFastSLAM, and 5.9% compared to PSO-UFastSLAM.

Absolute and relative POD of LEO satellites in formation flying: Undifferenced and uncombined approach

Absolute or relative precise orbit determination (POD) is an essential prerequisite for many low earth orbit (LEO) missions. The POD of LEO satellites typically relays on processing the onboard global navigation satellite system (GNSS) measurements. The absolute POD is usually based on an ionosphere-free (IF) combination, and currently, integer ambiguity resolution (IAR) can be achieved only when external GNSS satellite phase bias (SPB) products are used. The use of these products is not flexible in multi-frequency/multi-constellation scenarios and is difficult to achieve in real-time missions. For relative POD, the double-differenced (DD) with IAR is the most general method. However, the differencing process amplifies observation noise and loses the opportunity to impose dynamic constraints on some eliminated parameters. In this contribution, based on the use of undifferenced and uncombined (UDUC) observations, a new model for both absolute and relative POD is proposed. In this model, the ambiguities of common-view satellites are constructed into DD form, thus IAR can be achieved without any external SPB products. Working with the UDUC observations, multi-frequency scenarios can be easily applied, and residuals can be separated for each frequency. In addition, with precise GNSS satellite clock/orbit products, both the absolute and relative orbits can be derived, which supports absolute and relative LEO POD. Based on onboard GPS observations of T-A and T-B satellites in formation flying, the performance of the UDUC POD model with DD ambiguity was evaluated. With the UDUC algorithm and IAR, the proposed model presented a consistency of 2.8–3.8 cm in 3D with the reference orbits, and the orbit difference was reduced by 16.3% and 10.6% for T-A and T-B compared with the IF-based POD, respectively. In addition, the relative orbit of the two satellites derived from the proposed model showed a consistency of 1.1–1.5 mm, which proved the feasibility of the UDUC POD model with DD ambiguity for formation flying missions.

Sequential Fusion Filter for State Estimation of Nonlinear Multi-Sensor Systems with Cross-Correlated Noise and Packet Dropout Compensation

This paper is concerned with the problem of state estimation for nonlinear multi-sensor systems with cross-correlated noise and packet loss compensation. In this case, the cross-correlated noise is modeled by the synchronous correlation of the observation noise of each sensor, and the observation noise of each sensor is correlated with the process noise at the previous moment. Meanwhile, in the process of state estimation, since the measurement data may be transmitted in an unreliable network, data packet dropout will inevitably occur, leading to a reduction in estimation accuracy. To address this undesirable situation, this paper proposes a state estimation method for nonlinear multi-sensor systems with cross-correlated noise and packet dropout compensation based on a sequential fusion framework. Firstly, a prediction compensation mechanism and a strategy based on observation noise estimation are used to update the measurement data while avoiding the noise decorrelation step. Secondly, a design step for a sequential fusion state estimation filter is derived based on an innovation analysis method. Then, a numerical implementation of the sequential fusion state estimator is given based on the third-degree spherical-radial cubature rule. Finally, the univariate nonstationary growth model (UNGM) is combined with simulation to verify the effectiveness and feasibility of the proposed algorithm.

High-Precision State of Charge Estimation of Urban-Road-Condition Lithium-Ion Batteries Based on Optimized High-Order Term Compensation-Adaptive Extended Kalman Filtering

It is significant to improve the accuracy of estimating the state of charge (SOC) of lithium-ion batteries under different working conditions on urban roads. In this study, an improved second-order polarized equivalent circuit (SO-PEC) modeling method is proposed. Accuracy test using segmented parallel exponential fitting parameter identification method. Online parameter identification using recursive least squares with variable forgetting factors(VFFRLS). An optimized higher-order term compensation-adaptive extended Kalman filter (HTC-AEKF) is proposed in the process of estimating SOC. The algorithm incorporates a noise-adaptive algorithm that introduces noise covariance into the recursive process in real-time to reduce the impact of process noise and observation noise on the accuracy of SOC estimation. Multiple iterations are performed for some of the processes in the extended Kalman filter(EKF) to compensate for the accuracy impact of missing higher-order terms in the linearization process. Model validation results show over 98% accuracy. The results after comparing with the EKF algorithm show a 4.1% improvement in SOC estimation accuracy under Hybrid Pulse Power Characterization(HPPC) working conditions. 2.7% improvement in accuracy in Dynamic Stress Test(DST) working conditions. 2.1% improvement in Beijing Bus Dynamic Stress Test(BBDST) working conditions. The superiority of the algorithm is demonstrated.

Modulation Effect of the Annual Cycle on Interdecadal Warming Trends over the Tibetan Plateau during 1998–2020

Abstract The Tibetan Plateau is a sensitive area of global climate change, where few conventional observations exist. Satellite AMSU-A microwave temperature sounding observations of brightness temperature (TB) are located in the absorption band of oxygen, which is well mixed in the atmosphere, and have microwave frequencies varying from 50.3 to 57.6 GHz. Therefore, AMSU-A TB observations at different sounding channels reflect atmospheric temperatures at different altitudes. In this study, AMSU-A TB observations during 1998–2020 from five polar-orbiting environmental meteorological satellites (POESs) are employed to investigate the interdecadal warming/cooling trends over the Tibetan Plateau. A limb correction is first applied to all AMSU-A channels before using TB observations at all fields of view for examining geographic distributions and differences of global warming/cooling trends. It is found that interdecadal trends of upper-tropospheric warming and stratospheric cooling are stronger over the Qinghai Tibetan Plateau than its eastern plain areas. An interdecadal variation of the annual cycle over the Tibetan Plateau is an important factor for the enhanced tropospheric warming trend. We also applied a different approach of significance testing that is based on counting signs of local trends (sign test) and confirmed that the detected significant local trends were not a result of chance. In addition, high-frequency noise in TB observations with periods smaller than annual and semiannual oscillations do not affect the climate trends of TB very much, but significantly reduced the uncertainty of the TB trends over the Tibetan Plateau.

Evaluation of error characteristics of derived TEC with IRI-2016

The total electron content (TEC) derived from the Global Navigation Satellite System (GNSS) has been widely used to study the ionosphere. The evaluation of the derivation method is usually based on the fitting residuals, which are the difference between the estimate and observation. However, there are few studies on the statistical characteristics of fitting residuals. The characteristics of fitting residuals and estimation errors are studied with international reference ionosphere (IRI-2016) and GNSS observations. First, the systematic estimation errors caused by mathematical modeling and related assumptions are discussed with IRI-2016. The errors are 1 TECU and 0.5 TECU for slant TEC (STEC) and vertical TEC (vTEC), respectively. The fitting residuals of STEC observations (ΔSTECr) are studied by different noise models, including independent and identically distributed (IID) Gaussian noise, independent and non identical distributed (INID) Gaussian noise and INID Laplace noise. The statistical analysis shows that the distribution of ΔSTECr for the whole observations of all receivers is Gaussian when the noise is IID Gaussian, while it is Laplace for INID Gaussian or Laplace noise. The ΔSTECr reflects the statistical characteristic of noise. The estimation error of STEC (ΔSTEC), which is the difference between the estimate (STECe) and the model STEC (STEC) is discussed with IRI-2016. The distribution of ΔSTEC is neither Gaussian nor Laplace, and STECe is a biased estimate of STEC. The distribution of estimation error of vTEC (ΔvTEC) is neither Gaussian nor Laplace. Finally, the ΔSTECr is studied with GNSS observations. The distribution of ΔSTECr is more consistent with Laplace than Gaussian, indicating that the noise of the GNSS observations should not be modeled as IID Gaussian distribution.

Bridge weigh-in-motion considering dynamic response in observation noise with application to multiple driving conditions

In addition to causing a static response when traversing a bridge, moving vehicles also produce a dynamic response, which is thought to be one of the factors adversely affecting the accuracy of bridge weigh-in-motion approaches. This paper proposes a novel method for improving the accuracy of such approaches. Rather than modeling the dynamic response directly, the proposed model considers it as noise with autocorrelation. A Bayesian inference approach is considered as it allows a rational way of assimilating data into the model. Here, the dynamic response is considered in the form of a covariance matrix of observation noise, composed of the residuals of the measured and calculated responses, in a Bayesian framework. In a validation study using actual measured data, it was confirmed that the estimation of axle weights in the presence of multiple vehicles was significantly improved. From the result, it can be concluded that proper modeling of the dynamic response in observation data leads to more accurate weigh-in-motion estimates of axle weights and GVW, particularly when complicated vehicle travel is involved. A comparison of static and dynamic influence lines is also discussed using actual measured data. It is shown that the weights estimated by the dynamic influence line are generally more accurate than those by the static influence line when correlation of noise is not considered. The dynamic component in influence line can be interpreted as a regularization term. The proposed method, which uses static influence line and covariance matrix considering correlation of noise, is more accurate than the method with dynamic influence line and also, it does not cause any bias.

Manifold lifting: scaling Markov chain Monte Carlo to the vanishing noise regime

Abstract Standard Markov chain Monte Carlo methods struggle to explore distributions that concentrate in the neighbourhood of low-dimensional submanifolds. This pathology naturally occurs in Bayesian inference settings when there is a high signal-to-noise ratio in the observational data but the model is inherently over-parametrised or nonidentifiable. In this paper, we propose a strategy that transforms the original sampling problem into the task of exploring a distribution supported on a manifold embedded in a higher-dimensional space; in contrast to the original posterior this lifted distribution remains diffuse in the limit of vanishing observation noise. We employ a constrained Hamiltonian Monte Carlo method, which exploits the geometry of this lifted distribution, to perform efficient approximate inference. We demonstrate in numerical experiments that, contrarily to competing approaches, the sampling efficiency of our proposed methodology does not degenerate as the target distribution to be explored concentrates near low-dimensional submanifolds. Python code reproducing the results is available at https://doi.org/10.5281/zenodo.6551654.

Multi-Mode Imaging of Ambient Background Noise for Karst Detection in the Limestone Area Based on Frequency-Bessel Transform

In response to the challenges of karst geophysical exploration in an environment with strong external interference, this paper proposes a new method, namely the frequency-Bessel transform method, for extracting multi-order dispersion curves of surface waves from background noise to characterize karst. The observation noise data of the Wuhan karst development area are used as an example, where the dolomitic limestone and limestone mixed with dolomite of the Jialing River Formation of the middle lower Triassic are widely developed in the observation area. The frequency-Bessel transform method involves performing a Bessel integral transformation on the cross-correlation coefficient of background noise in the frequency domain. Firstly, by synthesizing theoretical noise data and comparing it with the spatial autocorrelation method—which is currently the main method for extracting the fundamental dispersion curve of surface waves—it is verified that the frequency-Bessel transform method can extract the higher-mode dispersion curve. Then, by taking the actual measured single-point noise data as an example, the effect of applying the frequency-Bessel transform to the actual noise data is tested, and the inversion of the fine structure of the strata by the addition of higher-mode dispersion, the use of the damped least squares inversion method, and the joint inversion of fundamental and higher-mode dispersion curves are analyzed. The higher-mode dispersion curve of Rayleigh surface wave extracted by the frequency-Bessel transform is much clearer, and the 2D shear wave velocity structure profile obtained from inversion explains the karst development area, karst strip area, and thickness of the Quaternary overburden. The inferred results match with the actual borehole data. Multi-mode imaging of background noise based on the frequency-Bessel method can be applied to depict karst in complex backgrounds, and has significant potentiality in the field of ambient seismic noise tomography, providing a new idea and method for karst detection in near-surface engineering.

Studying the multifrequency angular power spectrum of the cosmic dawn 21-cm signal

ABSTRACT The light-cone (LC) anisotropy arises due to cosmic evolution of the cosmic dawn (CD) 21-cm signal along the line-of-sight (LoS) axis of the observation volume. The LC effect makes the signal statistically non-ergodic along the LoS axis. The multifrequency angular power spectrum (MAPS) provides an unbiased alternative to the popular three-dimensional (3D) power spectrum as it does not assume statistical ergodicity along every direction in the signal volume. Unlike the 3D power spectrum which mixes the cosmic evolution of the 21-cm signal along the LoS k modes, MAPS keeps the evolution information disentangled. Here, we first study the impact of different underlying physical processes during CD on the behaviour of the 21-cm MAPS using simulations of various different scenarios and models. We also make error predictions in 21-cm MAPS measurements considering only the system noise and cosmic variance for mock observations of Hydrogen Epoch of Reionization Array (HERA), NenuFAR, and SKA-Low. We find that 100 h of HERA observations will be able to measure 21-cm MAPS at ≥3σ for ℓ ≲ 1000 with $0.1\, {\rm MHz}$ channel-width. The better sensitivity of SKA-Low allows reaching this sensitivity up to ℓ ≲ 3000. Note that due to the difference in the frequency coverage of the various experiments, the CD–epoch of reionization model considered for NenuFAR is different than those used for the HERA and SKA-Low predictions. Considering NenuFAR with the new model, measurements ≥2σ are possible only for ℓ ≲ 600 with $0.2\, {\rm MHz}$ channel-width and for a 10 times longer observation time of tobs = 1000 h. However, for the range 300 ≲ ℓ ≲ 600 and tobs = 1000 h more than 3σ measurements are still possible for NenuFAR when combining consecutive frequency channels within a 5 MHz band.

Dynamic Compensation of a Piezoelectric Accelerometer Obtained through a General Probabilistic Approach.

Dynamic compensation is the (partial) correction of the measurement signals for the effects due to bandwidth limitations of measurement systems and constitutes a research topic in dynamic measurement. The dynamic compensation of an accelerometer is here considered, as obtained by a method that directly comes from a general probabilistic model of the measurement process. Although the application of the method is simple, the analytical development of the corresponding compensation filter is quite complex and had been previously developed only for first-order systems, whilst here a second-order system is considered, thus moving from a scalar to a vector problem. The effectiveness of the method has been tested both through simulation and by a dedicated experiment. Both tests have shown the capability of the method of significantly improve the performance of the measurement system when dynamic effects are more prevalent than additive observation noise.

Enhancing the Accuracy of Water-Level Forecasting with a New Parameter-Inversion Model for Estimating Bed Roughness in Hydrodynamic Models

The accurate and efficient estimation of bed roughness using limited historical observational data is well-established. This paper presents a new parameter-inversion model for estimating bed roughness in hydrodynamic models that constrains the roughness distribution between river sections. The impact of various factors on the accuracy of inversed roughness was analyzed through a numerical experiment with the number of measurement stations, observed data amount, initial bed roughness, observational noise, and the weight of the regularization term. The results indicate that increasing the number of measurement stations and the amount of observed data significantly improves the robustness of the model, with an optimal parameter setting of 3 stations and 30 observed data. The initial roughness had little impact on the model, and the model showed good noise resistance capacity, with the error significantly reduced by controlling the smoothness level of inversed roughness using a small weight of the regularization term (i.e., 100). An experiment conducted on a real river using the calibrated model parameters shows a forecasted water level RMSE of 0.041 m, 31% less than that from the Federal Emergency Management Agency. The proposed model provides a new approach to estimating bed roughness parameters in hydrodynamic models and can help in improving the accuracy of water-level forecasting.

Earth as an Exoplanet. II. Earth’s Time-variable Thermal Emission and Its Atmospheric Seasonality of Bioindicators

We assess the dependence of Earth’s disk-integrated mid-infrared thermal emission spectrum on observation geometries and investigate which and how spectral features are impacted by seasonality on Earth. We compiled an exclusive data set containing 2690 disk-integrated thermal emission spectra for four different full-disk observing geometries (North and South Pole-centered and Africa and Pacific-centered equatorial views) over four consecutive years. The spectra were derived from 2378 spectral channels in the wavelength range from 3.75–15.4 μm (nominal resolution ≈1200) and were recorded by the Atmospheric Infrared Sounder on board the Aqua satellite. We learned that there is significant seasonal variability in Earth’s thermal emission spectrum, and the strength of spectral features of bioindicators, such as N2O, CH4, O3, and CO2 depends strongly on both season and viewing geometry. In addition, we found a strong spectral degeneracy with respect to the latter two indicating that multi-epoch measurements and time-dependent signals may be required in order to fully characterize planetary environments. Even for Earth and especially for equatorial views, the variations in flux and strength of absorption features in the disk-integrated data are small and typically ≤ 10%. Disentangling these variations from the noise in future exoplanet observations will be a challenge. However, irrespectively of when the planet will be measured (i.e., day or night or season) the results from mid-infrared observations will remain the same to the zeroth order, which is an advantage over reflected light observations.

Semi-supervised invertible neural operators for Bayesian inverse problems

Neural Operators offer a powerful, data-driven tool for solving parametric PDEs as they can represent maps between infinite-dimensional function spaces. In this work, we employ physics-informed Neural Operators in the context of high-dimensional, Bayesian inverse problems. Traditional solution strategies necessitate an enormous, and frequently infeasible, number of forward model solves, as well as the computation of parametric derivatives. In order to enable efficient solutions, we extend Deep Operator Networks (DeepONets) by employing a RealNVP architecture which yields an invertible and differentiable map between the parametric input and the branch-net output. This allows us to construct accurate approximations of the full posterior, irrespective of the number of observations and the magnitude of the observation noise, without any need for additional forward solves nor for cumbersome, iterative sampling procedures. We demonstrate the efficacy and accuracy of the proposed methodology in the context of inverse problems for three benchmarks: an anti-derivative equation, reaction-diffusion dynamics and flow through porous media.

The Efficiency of the Proposed Smoothing Method over the Classical Cubic Smoothing Spline Regression Model with Autocorrelated Residual

Spline smoothing is a technique used to filter out noise in time series observations when predicting nonparametric regression models. Its performance depends on the choice of the smoothing parameter. Most of the existing smoothing methods applied to time series data tend to over fit in the presence of autocorrelated errors. This study aims to determine the optimum performance value, goodness of fit and model overfitting properties of the proposed Smoothing Method (PSM), Generalized Maximum Likelihood (GML), Generalized Cross-Validation (GCV), and Unbiased Risk (UBR) smoothing parameter selection methods. A Monte Carlo experiment of 1,000 trials was carried out at three different sample sizes (20, 60, and 100) and three levels of autocorrelation (0.2, 05, and 0.8). The four smoothing methods' performances were estimated and compared using the Predictive Mean Squared Error (PMSE) criterion. The findings of the study revealed that: for a time series observation with autocorrelated errors, provides the best-fit smoothing method for the model, the PSM does not over-fit data at all the autocorrelation levels considered ( the optimum value of the PSM was at the weighted value of 0.04 when there is autocorrelation in the error term, PSM performed better than the GCV, GML, and UBR smoothing methods were considered at all-time series sizes (T = 20, 60 and 100). For the real-life data employed in the study, PSM proved to be the most efficient among the GCV, GML, PSM, and UBR smoothing methods compared. The study concluded that the PSM method provides the best fit as a smoothing method, works well at autocorrelation levels (ρ=0.2, 0.5, and 0.8), and does not over fit time-series observations. The study recommended that the proposed smoothing is appropriate for time series observations with autocorrelation in the error term and econometrics real-life data. This study can be applied to; non – parametric regression, non – parametric forecasting, spatial, survival, and econometrics observations.