Error Model Research Articles

Deep evidential learning for radiotherapy dose prediction

Background:As we navigate towards integrating deep learning methods in the real clinic, a safety concern lies in whether and how the model can express its own uncertainty when making predictions. In this work, we present a novel application of an uncertainty-quantification framework called Deep Evidential Learning in the domain of radiotherapy dose prediction. Method:Using medical images of the Open Knowledge-Based Planning Challenge dataset, we found that this model can be effectively harnessed to yield uncertainty estimates that inherited correlations with prediction errors upon completion of network training. This was achieved only after reformulating the original loss function for a stable implementation. Results:We found that (i)epistemic uncertainty was highly correlated with prediction errors, with various association indices comparable or stronger than those for Monte-Carlo Dropout and Deep Ensemble methods, (ii)the median error varied with uncertainty threshold much more linearly for epistemic uncertainty in Deep Evidential Learning relative to these other two conventional frameworks, indicative of a more uniformly calibrated sensitivity to model errors, (iii)relative to epistemic uncertainty, aleatoric uncertainty demonstrated a more significant shift in its distribution in response to Gaussian noise added to CT intensity, compatible with its interpretation as reflecting data noise. Conclusion:Collectively, our results suggest that Deep Evidential Learning is a promising approach that can endow deep-learning models in radiotherapy dose prediction with statistical robustness. We have also demonstrated how this framework leads to uncertainty heatmaps that correlate strongly with model errors, and how it can be used to equip the predicted Dose-Volume-Histograms with confidence intervals.

Robust Driving Control Design for Precise Positional Motions of Permanent Magnet Synchronous Motor Driven Rotary Machines with Position-Dependent Periodic Disturbances

Position-dependent periodic disturbances often limit the accuracy and smoothness of the positional motion of permanent magnet synchronous motor (PMSM)-driven rotary machines. Because the period of these disturbances varies with the motion velocity of the rotary machine, spatial domain control methods such as spatial iterative learning control (SILC) and spatial repetitive control (SRC) have been proposed and applied to improve rotary machine motion control designs. However, problems with learning period convergence and rotary machine dynamics significantly affect transient motion, further constraining the overall motion performance. To address these challenges, this study developed a robust driving control (RDC) that integrates a robust control design with position-dependent periodic disturbance feedforward compensation, rotary machine dynamics compensation, and proportional–proportional integral feedback control. A position-dependent periodic disturbance model was developed using multiple position–sinusoidal signals and identified using a spatial fast Fourier transform. RDC compensates for disturbances and dynamics and considers the effects of model parameter uncertainty and modeling error on the stability of the control system. Several motion control experiments were conducted on a PMSM test bench to compare the RDC, SILC, and SRC. The experimental results demonstrated that although both SILC and SRC can effectively suppress position-dependent periodic disturbances, SILC provides slower position error convergence owing to the learning process, and SILC and SRC result in significant position errors because of the influence of the PMSM-driven rotary machine dynamics. RDC not only suppresses position-dependent periodic disturbances, but also significantly reduces position errors with a reduction rate of 90%. Therefore, the RDC developed in this study effectively suppressed position-dependent periodic disturbances and significantly improved both the transient-state and steady-state position-tracking performances of the PMSM-driven rotary machine.

Identification of heart rate dynamics during treadmill and cycle ergometer exercise: the role of model zeros and dead time.

The response of heart rate to changes in exercise intensity is comprised of several dynamic modes with differing magnitudes and temporal characteristics. Investigations of empirical identification of dynamic models of heart rate showed that second-order models gave substantially and significantly better model fidelity compared to the first order case. In the present work, we aimed to reanalyse data from previous studies to more closely consider the effect of including a zero and a pure delay in the model. This is a retrospective analysis of 22 treadmill (TM) and 54 cycle ergometer (CE) data sets from a total of 38 healthy participants. A linear, time-invariant plant model structure with up to two poles, a zero and a dead time is considered. Empirical estimation of the free parameters was performed using least-squares optimisation. The primary outcome measure is model fit, which is a normalised root-mean-square model error. A model comprising parallel connection of two first-order transfer functions, one with a dead time and one without, was found to give the highest fit (56.7 % for TM, 54.3 % for CE), whereby the non-delayed component appeared to merely capture initial transients in the data and the part with dead time likely represented the true dynamic response of heart rate to the excitation. In comparison, a simple first-order model without dead time gave substantially lower fit than the parallel model (50.2 % for TM, 47.9 % for CE). This preliminary analysis points to a linear first-order system with dead time as being an appropriate model for heart rate response to exercise using treadmill and cycle ergometer modalities. In order to avoid biased estimates, it is vitally important that, prior to parameter estimation and validation, careful attention is paid to data preprocessing in order to eliminate transients and trends.

A New Framework for Understanding Systematic Errors in Cluster Lens Modeling. III. Deflection from Large-Scale Structure

Interpreting and reconstructing distant sources that are gravitationally lensed by galaxy clusters requires accurate and precise lens models. While high-quality data sets have reduced statistical errors in such models, systematic errors remain important. We examine systematic lensing effects caused by density fluctuations due to large-scale structure along the line of sight. We use a multiplane ray-tracing algorithm with the IllustrisTNG 100-3 cosmological simulation of matter distribution and compute the statistical distributions of shear, convergence, and higher-order deflections using two Hubble Frontier Field clusters as examples (A2744 and MACS J0416.1−2403). The cosmic shear distribution is Gaussian in each component, while the cosmic convergence distribution is skewed such that 1 + κ is consistent with a log-normal distribution; the standard deviations for these quantities are at the level of a few to 10%, depending on the redshift of the source. The deflection from higher-order terms beyond convergence and shear has significant scatter: the rms deflection is ∼15″, considerably larger than the image position residuals for current lens models. These results indicate that line-of-sight deflection effects due to large-scale structure can significantly impact lens models and should not be neglected. We present results in forms that can be incorporated into future cluster lens models.

A baseline decomposition ultra-short baseline localization algorithm for arbitrary array structures.

With the rapid development of the marine economy, hydroacoustic positioning technology plays an increasingly important role in marine engineering. The ultra-short baseline (USBL) hydroacoustic positioning system has the advantages of small size, simple operation, and flexible use, and has been widely used. Aiming at the existing USBL acoustic positioning algorithm with low positioning accuracy and complex calculation, a baseline decomposition localization algorithm with arbitrary array structure is proposed. The algorithm is based on the theory of coordinate system transformation, establishes positioning observation equations for each baseline in the base array, and adopts the least squares method to obtain positioning results by selecting different combinations of baselines. The systematic errors of different positioning models themselves are simulated, and then the effects of the three parameter errors, namely, time delay, element coordinates, and sound speed, on the positioning results are analyzed, respectively. Finally, the simulation results and sea trial data show that, compared with the existing algorithms, this algorithm not only simplifies the complicated computation process, but also improves the positioning accuracy and robustness, and has a better application effect.

Prediction of building cooling load: an innovative design for GA-SVM and IG-SVM system for the estimation of hourly precision and computational fluctuation range

ABSTRACT Anticipating the loads that buildings will bear is a crucial strategy in the pursuit of energy efficiency and the reduction of emissions. In this paper, we introduce an interactive and comprehensive joint prediction model, specifically crafted to elevate the precision of forecasts related to building loads. The resultant Genetic Algorithm-Support Vector Machine (GA-SVM) prediction model is put into action to provide hourly predictions of cooling loads for buildings. In scenarios where the prediction of building cooling load (BCL) fluctuations becomes imperative, especially in the face of extreme weather conditions, the information granulation (IG) method is employed. The findings of this validation process unveil significant improvements. The coefficient of variation of root mean square error (CV-RMSE) and mean absolute percentage error (MAPE) for the GA-SVM model are reduced by 58.85% and 68.04%, respectively, when compared to the conventional SVM model. Furthermore, in a comparative analysis against three widely utilized prediction models, the SVM model demonstrates a reduction in CV-RMSE and MAPE ranging from 2.04% to 68.04%. The application of the joint prediction model showcases impressive results, with R 2 values ranging from 97.27% to 99.68%, MAPE ranging from 2.59% to 2.84%, and CV-RMSE ranging from only 0.0249 to 0.0319.

Load-bearing characteristics of surface conductor in deepwater gas hydrate formations

The surface conductor is the first structural casing in deepwater natural gas hydrate (NGH) development, bearing the top load while suspending the casings of various layers. NGH decomposition leads to formation settlement, changing the mechanical properties of the formation and reducing the bearing capacity of the surface conductor, threatening the safety and stability of the wellhead. Understanding the bearing characteristics of the surface conductor in the hydrate formation can guide the safe drilling operation in the field. By introducing the negative skin friction theory of pile foundations and based on conventional bearing capacity models, a method for calculating the bearing capacity of surface conductors in NGH formations was developed. Using an NGH drilling simulation apparatus, the accuracy of the bearing capacity theoretical model was verified, empirical coefficients under different conditions were obtained, and the influence of soil parameters, hydrate saturation, and decomposition temperature on the bearing capacity of surface conductors was quantified. The results indicated that compared to clay, sandy soils have higher porosity and significantly weakened strength after the decomposition of NGH; when the hydrate saturation in the formation is 20%, the reduction in bearing capacity of the surface conductor in sand exceeds 30%, and in clay soils, it decreases by 25% after complete decomposition of NGH; as the hydrate saturation increases, the reduction in the bearing capacity of surface conductors after decomposition becomes more significant. Verified through Experimentation, the error of the hydrate-bearing strata-bearing capacity model is around 10%. For short-term test production operations of NGH in water, the design depth for surface conductors is around 100 meters. These research results can provide a scientific theoretical basis for the design of conductor depth below the mud， and reduce operational risks.

Long Duration Forecasting and its Performance Capability for Seasonal Variation Modelling of Residual Chlorine Concentrations: A Comparative Evaluation of two Small-scale Water Distribution Systems in Japan

Chlorine dosing control is a critical task for health security, complying with drinking water quality standards while achieving customer satisfaction. In Japan, this became more difficult due to decreasing number of technical personnel as a result of declining population and huge retirement of veterans. As experienced-based dosing control still exists and is considered inefficient due to risks of inaccurate dosing and need of experienced manpower, streamlining of operations is needed. This study aims to address these concerns through the development and evaluation of a deep learning model for residual chlorine concentrations capable of forecasting long durations. In this paper, the model is investigated in seasonal timeframe analyzing trends and variations. Two water distributions systems of varying network—simple and complex, are also compared to further analyze model performance and versatility. The model utilizes the past 24 hours of flow rate, chlorine level at treatment plant, water temperature, and residual chlorine at private homes as input to predict the 12 hours ahead of residual chlorine at private homes evaluated at hourly training lengths of 0.5, 1, 1.5, 2 years. Results revealed that the model achieved high accuracy in predicting hourly residual chlorine with general increasing model error from winter, spring, autumn, and summer due to the progressing instabilities in chlorine concentrations from low to high temperatures. Moreover, smaller system tends to be more unstable incurring lower model performance relative to larger system. In terms of optimal training length, ≥1Y training models are found to have lesser chance of data drift occurrence prospectively reducing model retraining frequency in the future.

Statistical Response of ENSO Complexity to Initial Condition and Model Parameter Perturbations

Abstract Studying the response of a climate system to perturbations has practical significance. Standard methods in computing the trajectory-wise deviation caused by perturbations may suffer from the chaotic nature that makes the model error dominate the true response after a short lead time. Statistical response, which computes the return described by the statistics, provides a systematic way of reaching robust outcomes with an appropriate quantification of the uncertainty and extreme events. In this paper, information theory is applied to compute the statistical response and find the most sensitive perturbation direction of different El Niño–Southern Oscillation (ENSO) events to initial value and model parameter perturbations. Depending on the initial phase and the time horizon, different state variables contribute to the most sensitive perturbation direction. While initial perturbations in sea surface temperature (SST) and thermocline depth usually lead to the most significant response of SST at short and long ranges, respectively, initial adjustment of the zonal advection can be crucial to trigger strong statistical responses at medium range around 5–7 months, especially at the transient phases between El Niño and La Niña. It is also shown that the response in the variance triggered by external random forcing perturbations, such as the wind bursts, often dominates the mean response, making the resulting most sensitive direction very different from the trajectory-wise methods. Finally, despite the strong non-Gaussian climatology distributions, using Gaussian approximations in the information theory is efficient and accurate for computing the statistical response, allowing the method to be applied to sophisticated operational systems. Significance Statement The purpose of this work is to better understand how El Niño–Southern Oscillation (ENSO) responds to changes in its initial state and internal dynamics or external forcings. A statistical quantification of this response allows for the comprehension of the triggering conditions and the effect of climate change on the occurrence frequency and strength of each type of ENSO event. Such a study also allows to detect the most sensitive perturbation directions, which has practical significance in guiding anthropogenic activities. The approach used to study the response in this work is through the framework of information theory, which allows for an unbiased and robust assessment of the statistical response that is not affected by the turbulent dynamics of the system.

A spatial one-sided error model to identify where unarrested criminals live

A spatial one-sided error model to identify where unarrested criminals live

Modelling motion energy in psychotherapy: A dynamical systems approach.

In this study we introduce an innovative mathematical and statistical framework for the analysis of motion energy dynamics in psychotherapy sessions. Our method combines motion energy dynamics with coupled linear ordinary differential equations and a measurement error model, contributing new clinical parameters to enhance psychotherapy research. Our approach transforms raw motion energy data into an interpretable account of therapist-patient interactions, providing novel insights into the dynamics of these interactions. A key aspect of our framework is the development of a new measure of synchrony between the motion energies of therapists and patients, which holds significant clinical and theoretical value in psychotherapy. The practical applicability and effectiveness of our modelling and estimation framework are demonstrated through the analysis of real session data. This work advances the quantitative analysis of motion dynamics in psychotherapy, offering important implications for future research and therapeutic practice.

Triple junction benchmark for multiphase-field models combining capillary and bulk driving forces

Abstract A benchmark problem is formulated which is well suited for the validation of mesoscopic phase-field models for grain-boundary migration in polycrystals. First, an analytical steady-state solution of the sharp moving boundary problem is derived for a symmetric lamellar structure, which is valid for arbitrary bulk driving forces and triple junction angles. Characteristic quantities are identified to reduce the parameter space which in turn allows a systematic comparison of simulations and analytical results. Various multiphase-field (MPF) formulations are compared which approximate the sharp interface problem in terms of a diffuse regularization. An interfacial thickness convergence study reveals that the model error is largely dependent on the ratio of bulk to interfacial stabilizing force as well as the underlying model formulation. An additional grid convergence study highlights the efficiency of a more advanced discretization scheme. The results can be used to guide the selection of appropriate models and to estimate the interface thickness and spatial resolution required to achieve a given accuracy target. The post-processing framework consists of a fully automated determination of well-defined metrics from the phase field simulation data, eliminating human bias and facilitating reproducibility. The corresponding code is made openly available to assist the materials science and engineering community in validating MPF, multi-order parameter and similar model developments. We believe that this work provides a reliable benchmark procedure to better understand the potentials and limitations of current multiphase-field models as well as alternative approaches.

Assessing the potential accuracy of a small-sized goniometer with extended dynamic range based on nuclear magnetic resonance

This paper evaluates potential accuracy characteristics of a small-sized goniometer based on nuclear magnetic resonance with an extended dynamic range. This required constructing a model of goniometer errors, estimating its accuracy based on this model, and formulating practical recommendations for the design of such a device based on the accuracy assessment. To evaluate the accuracy of a nuclear goniometer, a theoretical model was built that makes it possible to determine the optimal operating parameters of the cell gas mixture, the ranges of their permissible changes, the sensitivity of the goniometer, and the dependence of its characteristics on external and internal factors. In particular, the dependence of output signal of the device on the parameters of gas mixture and optical pumping has been determined. For a goniometer with a cell volume of 8 cm3, the optimum temperature is 130 °C, and the optimum intensity of the pumping radiation is 5 mW. The dependence of output signal on the measured angle of rotation was also established, as well as the noise and error dependence of the device on the permissible values of its parameters. Based on the model built, parameters of a goniometer with a cell volume of 8 cm3 were determined; the maximum angular sensitivity of such a goniometer with complete suppression of technical noise is δφsen=1.0 arcsec. The greatest contribution to the angle measurement error is from the instability of pumping power Ip (ΔIp/Ip=0.05) – 85 %, magnetic field B0 (ΔB0/B0=10-8) – 13 %, temperature T (ΔT/T=0,1) – 2 %. The goniometer under consideration corresponds to the medium accuracy class, δφtot≥10 arcsec. It could be used in optical manufacturing for operational control, calibration, and certification of optical products. To improve the angular accuracy of the goniometer, it is necessary to increase the stability of the laser pumping intensity

Data Quality and Data Governance: Investigating the Impact on Data Science Outcomes

Abstract: The purpose of this study is to evaluate the effects that high data quality and effective data governance have on the outcomes of data science, with a particular emphasis on the vital roles that these factors play in improving decision-making within businesses. The rising use of data science projects by enterprises for the purpose of digital transformation presents major hurdles, particularly with regard to the trustworthiness of data and compliance with appropriate legal frameworks. A high data quality improves model accuracy, decision-making efficiency, and error reduction, while good governance boosts compliance with rules, data security, and accessibility, according to the findings of the study, which were derived from an analysis of data collected from professionals working in data science and related sectors. These findings demonstrate the synergy that exists between data quality and governance, underscoring the fact that businesses may obtain more dependable and trustworthy outcomes by prioritizing these components, which in turn supports digital transformation programs that are effective and responsible.

Исследование влияния отклонений параметров моделей синхронных генераторов на результаты расчета динамической устойчивости электроэнергетической системы

One of the important tasks solved in planning the electric power system (EPS) operation is the assessment of its transient stability. Use of incorrect parameters for EPS element models, in particular synchronous generators (SG), contributes to the occurrence of errors when determining the operating conditions of power systems, and, therefore, can lead to a decrease in the reliability of EPS operation. This article is devoted to the study of the impact of errors in SG model parameters on the transient stability assessment results. The analysis presented in this article allows us to substantiate the relevance of the SG parameter estimation task using a “passive experiment”, such as synchronized phasor measurements. This research is based on the theory of electrical circuits, mathematical description of electrical machines and transients in EPS, as well as statistical analysis. Computational experiments have been carried out in the software package MATLAB using the simulation modeling environment “Simulink”. During the research, two different multi-machine EPS models have been used, including the well-known “IEEE 9-bus” system. To quantitatively assess the influence of SG parameter uncertainties on transient stability analysis, the values of the maximum permissible fault clearance time are determined, as well as the values of the maximum power output of generating equipment in prefault conditions. The authors have carried out a comprehensive analysis of the influence of deviations in SG model parameters on the maximum fault clearance time, as well as on the maximum power output of generating equipment in the prefault conditions for a given fault duration. During the study, two approaches have been considered; in each case, the EPS transient stability assessment has been performed almost automatically, and quantitative indicators have been computed for the influence of SG model parameter deviations on the outcome of EPS transient stability analysis. Analysis of the computational experiment results has demonstrated a significant influence of the SG model errors on the errors in calculating certain transient stability-related parameters. For the considered scenarios, the spread of errors in calculating the maximum fault clearance time reached 100 % relative to a base-case (errorless) result, and the error in calculating the maximum power output of generating equipment in prefault conditions for a given fault duration may reach 20 % relative to the errorless result.

Bias Analysis and Correction in Weighted-L1 Estimators for the First-Order Bifurcating Autoregressive Model

This study examines the bias in weighted least absolute deviation (WL1) estimation within the context of stationary first-order bifurcating autoregressive (BAR(1)) models, which are frequently employed to analyze binary tree-like data, including applications in cell lineage studies. Initial findings indicate that WL1 estimators can demonstrate substantial and problematic biases, especially when small to moderate sample sizes. The autoregressive parameter and the correlation between model errors influence the volume and direction of the bias. To address this issue, we propose two bootstrap-based bias-corrected estimators for the WL1 estimator. We conduct extensive simulations to assess the performance of these bias-corrected estimators. Our empirical findings demonstrate that these estimators effectively reduce the bias inherent in WL1 estimators, with their performance being particularly pronounced at the extremes of the autoregressive parameter range.

Evaluation of forecasted wind speed at turbine hub height and wind ramps by five NWP models with observations from 262 wind farms over China

AbstractAccurate wind speed forecasts are essential for optimizing the efficiency of wind energy operations. Currently, there is limited research on nationwide assessment of predictive performance in multiple numerical weather prediction (NWP) models for wind speed at turbine hub height over China, especially concerning wind ramp events. Utilizing observed measurements from 262 wind farms, this study evaluated the performance of five NWP models in forecasting the mean state and spatiotemporal variations of wind speed as well as wind ramps. The results indicated that the European Center for Medium‐Range Weather Forecast Integrated Forecasting System (ECMWF–IFS) performed the best in forecasting climatological wind speed with a temporal correlation coefficient (TCC) of 0.74 and root mean square error (RMSE) of 2.34 m s−1. Although not widely utilized in China, the model from Meteo‐France (MF–ARPEGE) showed promising potential for wind energy forecasting with a TCC of 0.72 and RMSE of 2.45 m s−1. In terms of temporal variations of wind speed, all the models could reasonably predict the seasonal variations of wind speed, whereas only three NWP models were able to capture the characteristics of the observed diurnal variation. An error decomposition analysis further revealed that the primary source of predicted error for wind speed was the sequence error component (SEQU), indicating the model errors were mainly attributed from the temporal inconsistency between forecasts and observations. Furthermore, the occurrences of wind ramps were generally underestimated by NWP models, while this shortcoming can be partly overcome by improving the time resolution of NWP models.

Assessment of digital elevation models accuracy for local geoid modeling

One of the critical factors influencing the accuracy of a local geoid model is the quality of the digital elevation model (DEM). A properly selected high-resolution DEM can significantly mitigate errors in geoid modeling, gravity anomaly processing, and topography and downward continuation correction. Purpose. Evaluating the accuracy of five global DEMs obtained from open sources to identify the most suitable model for creating a local geoid. Methodology. The vertical accuracy of the DEMs was assessed by comparing the heights between the DEM and control points across different types of terrain. The reference values are based on 344 ground benchmarks, where GNSS observations were performed with subsequent adjustment of coordinates and heights. The accuracy analysis involved calculating statistical indicators of the height differences between the GNSS data and the DEM data. Findings. The standard deviation assessment showed favorable values for the COPERNICUS and ALOS DEMs, followed by SRTM, ASTER, and ETOPO. In the mean absolute error calculations for mountainous areas, the ALOS model performed best, followed by COPERNICUS, SRTM, ASTER, and ETOPO. For other types of terrain, COPERNICUS demonstrated the best results in mean absolute error. Originality. This study distinguishes itself through the incorporation of advanced high-resolution DEMs, such as GLO30, providing a modern and thorough evaluation of DEM accuracy specifically for Kazakhstan. What is new is a detailed analysis of the impact of terrain features (plain, hilly, mountainous) on modeling accuracy. This approach advances beyond previous assessments, delivering new and significant insights into the performance of contemporary DEMs. Practice value. The practical value of the results obtained consists in issuing recommendations regarding the possibility of using the studied DEM for the regions of Kazakhstan which differ among themselves in terms of landscape characteristics. The findings indicate that COPERNICUS and ALOS DEMs are highly suitable for precise geoid modeling in southern Kazakhstan. These models can significantly improve the accuracy of local geoid models, benefiting applications in geospatial science and engineering.

Confidence ellipses for simple regression parameters with Quasi-associated random errors

The work presented in this paper introduces a novel methodology for constructing confidence regions for linear regression parameters using ellipses, particularly in cases where the random errors are quasi-associated. Linear regression is widely employed in statistical analysis to predict a dependent variable using one or more independent variables. Traditionally, it is assumed that random errors in regression models follow a normal distribution. However, this assumption may not hold in many practical applications where errors exhibit dependencies. Thus, this study focuses on quasi-associated random variables, which include both positively and negatively associated variables.We derive exponential inequalities that allow the construction of confidence ellipses for least squares estimates of model parameters. This approach is applied to a Keynesian consumption function, which models the relationship between consumption and national income. Using data on monthly per capita income in Africa over 12 years, we demonstrate the effectiveness of our method in estimating regression parameters when the random errors are quasi-associated. The findings show that this technique can improve the accuracy of statistical models used in fields such as economics and engineering, especially in cases where the traditional assumptions of regression analysis are not met.

Dynamic Analysis of Helical Gears Considering Friction

The use of high contact ratio (HCR) gears is an effective method to reduce single tooth loads, as well as vibration and noise, making the application of helical gears increasingly widespread. Although the presence of helix angles can cause significant relative sliding, the presence of more teeth in contact makes HCR gears more sensitive to surface quality. The existing literature mainly investigates geometric optimization, load distribution, or efficiency calculation of helical gears. The research on the influence of rough surfaces on the dynamic performance of helical gears is very limited. To this end, a multi degree of freedom mathematical model of static transmission error was established based on the 2-DOF system. The relative sliding friction was studied using different friction coefficients, and the influence of different friction coefficients on the gear system was analyzed in detail. The numerical results indicate that surface roughness has a relatively small impact on the dynamic transmission error of the system, but has a significant impact on the motion and friction in the direction of departure.