Error Propagation Analysis Research Articles

On the cluster of the families of hybrid polynomial kernels in kernel density estimation

This study introduces a novel cluster of hybrid polynomial kernel families, designed to achieve significantly lower asymptotic mean integrated squared error compared to traditional kernels. These hybrid kernels are developed by heuristically combining classical polynomial kernels using probability axioms. An in-depth analysis of error propagation within these kernels is conducted, utilizing both simulation experiments and real-life datasets, including the Life Span of Batteries and COVID-19 datasets. The findings consistently demonstrate that the proposed hybrid kernels outperform their classical counterparts in various density estimation tasks across different distribution types and sample sizes. This research highlights the potential of hybrid polynomial kernels to enhance accuracy in density estimation, advocating for their adoption in statistical modelling and analysis.

CRISP-DM-Based Data-Driven Approach for Building Energy Prediction Utilizing Indoor and Environmental Factors

The significant energy consumption associated with the built environment demands comprehensive energy prediction modelling. Leveraging their ability to capture intricate patterns without extensive domain knowledge, supervised data-driven approaches present a marked advantage in adaptability over traditional physical-based building energy models. This study employs various machine learning models to predict energy consumption for an office building in Berkeley, California. To enhance the accuracy of these predictions, different feature selection techniques, including principal component analysis (PCA), decision tree regression (DTR), and Pearson correlation analysis, were adopted to identify key attributes of energy consumption and address collinearity. The analyses yielded nine influential attributes: heating, ventilation, and air conditioning (HVAC) system operating parameters, indoor and outdoor environmental parameters, and occupancy. To overcome missing occupancy data in the datasets, we investigated the possibility of occupancy-based Wi-Fi prediction using different machine learning algorithms. The results of the occupancy prediction modelling indicate that Wi-Fi can be used with acceptable accuracy in predicting occupancy count, which can be leveraged to analyze occupant comfort and enhance the accuracy of building energy models. Six machine learning models were tested for energy prediction using two different datasets: one before and one after occupancy prediction. Using a 10-fold cross-validation with an 8:2 training-to-testing ratio, the Random Forest algorithm emerged superior, exhibiting the highest R2 value of 0.92 and the lowest RMSE of 3.78 when occupancy data were included. Additionally, an error propagation analysis was conducted to assess the impact of the occupancy-based Wi-Fi prediction model’s error on the energy prediction model. The results indicated that Wi-Fi-based occupancy prediction can improve the data inputs for building energy models, leading to more accurate energy consumption predictions. The findings underscore the potential of integrating the developed energy prediction models with fault detection systems, model predictive controllers, and energy load shape analysis, ultimately enhancing energy management practices.

Development of Detailed and Reduced Chemical Kinetic Models for Ammonia Gas Turbines

Abstract The power generation sector has been recently moving towards decarbonization and there is an increased interest in replacing conventional fossil fuels with fuels that produce reduced/zero carbon emissions. One such fuel is ammonia (NH3). However, ammonia is hard to ignite, has a low flame speed, and produces a significantly large amount of nitrogen oxide (NOx) emissions. Hence, using 100% ammonia as fuel in gas turbines requires significant modifications and the development of novel combustors. Blending hydrogen with ammonia, however, helps in having better control over the combustion properties. Before utilizing hydrogen-blended ammonia in an actual gas turbine combustor, thorough simulation studies are required to evaluate its performance, possible hazards, and emissions. The literature lacks well-validated chemical kinetic models for the combustion of hydrogen-blended ammonia for undiluted mixtures at gas turbine-relevant conditions (~20 bar). Hence, in this work, we develop a detailed chemical kinetic model for hydrogen blended ammonia combustion and validate it with a wide range of experimental data for both dilute and undiluted mixtures relevant to gas turbine operating conditions. The detailed chemical kinetic mechanism was reduced to a smaller version (32 species mechanism) without significant loss in accuracy using the direct relation graph with error propagation (DRGEP) and full species sensitivity analysis. The resultant mechanism can predict a wide range of experimental results with the least cumulative error and will be a valuable tool in CFD simulations that will enable the development of gas turbines for zero-carbon power generation.

On the Representativity of Electrode Microstructure Parameters and Their Electrochemical Response for Lithium Ion Batteries

Lithium-ion battery electrochemical models require an accurate description of the electrodes microstructures to be predictive that can be achieved through nanoscale imaging. Such observations are however limited by their field of view (FOV), as they provide only a subset of the whole electrode volume that does not necessarily represent the whole electrode microstructure heterogeneity, and therefore can bias the microstructure analysis. A microstructure scale electrochemical model was used to investigate lithium plating onset, material non-uniform utilization, and in-plane heterogeneities for an NMC-graphite full cell [1]. To evaluate the representativeness, and thus relevance, of these model predictions, a coupled representativity analysis has been performed on the microstructure parameters and, in a novel way, on the full cell electrochemical response [2].Electrode microstructure parameters representativeness has been first quantified using the representative volume element (RVE) methodology [3]. The RVE major flaw is that ultimately it can only conclude if a FOV contains representative subvolumes of the FOV, but not if the FOV itself is representative of the electrode volume. Analysis can conclude negatively (“FOV is not representative”), but not positively (“FOV is representative”). One major contribution of this work was to quantify the convergence of the RVE size with the FOV, to actually investigate the FOV representativeness and thus partly remedy this intrinsic limitation. The analysis determined that performing a standard RVE calculation, without exploring its FOV convergence, is likely to strongly underestimate the actual RVE size (cf. Fig. 1). The new RVE methodology has been automated in the NREL open-source Microstructure Analysis Toolbox (MATBOX) and is available to the battery community* [4].Representativeness of microstructure parameters is however only an intermediate step, as the end-results of an electrochemical model are performances predictions. Indeed, what is the practical consequence of a given deviation for a microstructure parameter? The microstructure parameter deviation propagations to the 3D microstructure scale electrochemical response have been then quantified for different charge rates. This defines a threshold for the microstructure parameters FOV for a desired maximum deviation of the electrochemical response. Such deviation propagation analysis is analogous to error propagation analysis and is necessary to determine the relevance of microstructure scale model predictions for macroscale predictions. Electrochemical model shows cell representative section areas are increasing with C-rate, due to higher in-plane heterogeneities, indicating larger FOVs are required specifically for fast charge modeling. Therefore, we introduced the novel concept of electrochemical RVE (eRVE) that is a function of the operating conditions (thus defined as a dynamic RVE), with an increasing dependence with the C-rate. Representativity analysis of the investigated cell determined a FOV of 144.4 ×154.4 µm2 is large enough to establish a convergence on the representative section areas for low to intermediate C-rate (≤2.5C), but not large enough to conclude for higher rates.This work aims to emphasize the importance of representativity analysis for LIB electrode microstructures, as it is required to estimate the error, and thus the relevance, of microstructure parameters intended to be used in macroscale models. The methodology and results can help researchers to select the relevant imaging and associated FOV required to provide accurate enough microstructure parameters. *New features will be released after article [2] publication. [1] F. L. E. Usseglio-Viretta, A. M. Colclasure, J. Allen, D. P. Finegan, P. Graf, K. Smith, Microstructure Scale Lithium-ion Battery Modeling, Part I: on lithium plating prediction and heterogeneity, in preparation.[2] F. L. E. Usseglio-Viretta, A. M. Colclasure, J. Allen, P. Graf, K. Smith, Microstructure Scale Lithium-ion Battery Modeling, Part II: on the representativity of microstructure parameters and electrochemical response, in preparation.[3] T. Kanit, S. Forest, I. Galliet, V. Mounoury, D. Jeulin, Determination of the size of the representative volume element for random composites: statistical and numerical approach, Int. J. of Solids and Structures, 40 (2003) 3647–3679.[4] F. L. E. Usseglio-Viretta, P. Patel, E. Bernhardt, A. Mistry, P. P. Mukherjee, J. Allen, S. J. Cooper, J. Laurencin, K. Smith, MATBOX: An Open-source Microstructure Analysis Toolbox for microstructure generation, segmentation, characterization, visualization, correlation, and meshing, SoftwareX, 17 (2022) 100915. Figure 1

Spiking neural networks in the Alexiewicz topology: A new perspective on analysis and error bounds

In order to ease the analysis of error propagation in neuromorphic computing and to get a better understanding of spiking neural networks (SNN), we address the problem of mathematical analysis of SNNs as endomorphisms that map spike trains to spike trains. A central question is the adequate structure for a space of spike trains and its implication for the design of error measurements of SNNs including time delay, threshold deviations, and the design of the reinitialization mode of the leaky-integrate-and-fire (LIF) neuron model. First, we identify the underlying topology by analyzing the closure of all sub-threshold signals of a LIF model. For zero leakage this approach yields the Alexiewicz topology, which we adopt to LIF neurons with arbitrary positive leakage. As a result, LIF can be understood as spike train quantization in the corresponding norm. This way we obtain various error bounds and inequalities such as a quasi-isometry relation between incoming and outgoing spike trains. Another result is a Lipschitz-style global upper bound for the error propagation and a related resonance-type phenomenon.

Error propagation and model update analysis in three-dimensional CSEM inversion

SUMMARY This study examines error propagation from data to model space during 3-D inversion of controlled-source electromagnetic (CSEM) data using a Gauss–Newton-based algorithm. An expression for model parameter correction is obtained using higher order generalized singular value decomposition for various regularization strategies. Inverse modelling is performed for different types of noise employing distinct regularization schemes to investigate the impact of error. Data corrupted with random noise suggest that the random noise mainly propagates when regularization parameters are small, owing to the high-frequency nature of random noise. Furthermore, the random noise predominantly causes artefacts in the shallower part of the inverted model. However, it has little impact on the estimation of major anomalies because the anomaly primarily depends on the smoothly varying parts of data. These observations are valid for both isotropic and anisotropic inversions. Resistive geological anomalies, like vertical dyke or vertical fractures, may pose a significant challenge for isotropic inversion in terms of convergence and data fit, even if the subsurface is isotropic. On the other hand, anisotropic inversion performs remarkably well in such cases, showing faster convergence and better data fit than isotropic inversion. Anisotropic inversion is indispensable in the case of an anisotropic host medium, as isotropic inversion produces significant artefacts and poorer data fit. Numerical experiments suggest that, in general, anisotropic inversion produces relatively better data fit and faster convergence, even in the case of isotropic subsurface. However, due to the varying degree of sensitivity of CSEM data on thin resistive bodies, caution is required in interpreting an anisotropy obtained using anisotropic inversion. An investigation of field data also supports the observations obtained using synthetic experiments.

An improved manifold-projection trajectory based method for chemical kinetic mechanism reduction

An improved manifold-projection trajectory based dimension reduction algorithm (MTDR) to automatically generate skeletal chemical kinetic mechanisms was proposed and validated in this study. This algorithm is implemented by constructing an approximate manifold-based trajectory of the combustion system using Euclidean distances between two adjacent points and cosine similarities between two adjacent vectors in the manifold space, and then evaluating the importance of species by calculating the error of manifold-based trajectories before and after the projection of the manifold. In this method, the nature of the combustion system is well reflected since the combustion process information can be captured by the location of the state point in the manifold space. The calculation during the reduction process is simplified by using the manifold-based trajectory instead of the complex differential equations. Compared to our previous work, this study improves the prediction accuracy of the reduced mechanisms through a more comprehensive error evaluation in the mechanism reduction process. The detailed mechanisms of fuels including iso-cetane, iso-octane and gasoline surrogate mixtures of toluene, iso-octane and n-heptane were reduced to portray the prowess of the algorithm. It’s demonstrated that the number of species of the fuels is reduced from 492, 254 and 1389 to 95, 87 and 427 within an acceptable error, respectively. Meanwhile, the case of iso-octane mechanism reduction was chosen to compare the performance of MTDR with traditional methods including path flux analysis (PFA) and directed relation graph with error propagation and sensitivity analysis (DRGEPSA), and it is shown that the MDTR can generate more compact mechanism under a broad range of error limits.

Ephemeris Tracking and Error Propagation Analysis of LEO Satellites With Application to Opportunistic Navigation

Ephemeris Tracking and Error Propagation Analysis of LEO Satellites With Application to Opportunistic Navigation

Dynamic error modeling and analysis of articulated arm coordinate measuring machine with integrated joint module

The self-driven articulated arm coordinate measuring machine (AACMM) is a new non-orthogonal flexible coordinate measuring equipment providing automatic positioning and measurement with integrated joint modules introduced to its rotary joints. The dynamic measuring accuracy is the crucial indicator of the self-driven AACMM performance. However, the part manufacturing assembly error and structural dynamic deformation error will seriously reduce the dynamic measuring accuracy. Therefore, a modeling compensation approach based on error propagation and accumulation mechanism analysis of the above two aspects is necessarily required to preserve the automatic measuring accuracy of the self-driven AACMM. In this paper, the authors propose an innovative modeling method for evaluating and compensating for the dynamic measuring errors of the self-driven AACMM. The ideal measuring model is constructed assisted with on the classical Denavit–Hartenberg parameter methodology and the complex structure of the self-driven AACMM. The source and its propagation mechanism of the dynamic error from the self-driven AACMM were analyzed systematically to decouple the superposed error impact on measuring precision. The dynamic measurement error compensation mechanism model is presented considering the influence of instantaneous spatial attitude on the dynamic structural parameters of self-driven AACMM. Experimental results indicate that the single-point repeatability measurement error fluctuation of the self-driven AACMM is 0.046 mm. The radius measuring error of the gauge ball reduces from −0.26 mm to −0.19 mm after the error model compensation. Additionally, the measuring accuracy is demonstrated to be effectively improved by the dynamic error compensation mechanism model.

A Visual Comparison of Silent Error Propagation.

High-performance computing (HPC) systems play a critical role in facilitating scientific discoveries. Their scale and complexity (e.g., the number of computational units and software stack) continue to grow as new systems are expected to process increasingly more data and reduce computing time. However, with more processing elements, the probability that these systems will experience a random bit-flip error that corrupts a program's output also increases, which is often recognized as silent data corruption. Analyzing the resiliency of HPC applications in extreme-scale computing to silent data corruption is crucial but difficult. An HPC application often contains a large number of computation units that need to be tested, and error propagation caused by error corruption is complex and difficult to interpret. To accommodate this challenge, we propose an interactive visualization system that helps HPC researchers understand the resiliency of HPC applications and compare their error propagation. Our system models an application's error propagation to study a program's resiliency by constructing and visualizing its fault tolerance boundary. Coordinating with multiple interactive designs, our system enables domain experts to efficiently explore the complicated spatial and temporal correlation between error propagations. At the end, the system integrated a nonmonotonic error propagation analysis with an adjustable graph propagation visualization to help domain experts examine the details of error propagation and answer such questions as why an error is mitigated or amplified by program execution.

Analyzing the effect of shot noise in indirect Time-of-Flight cameras

Continuous wave indirect Time-of-Flight cameras obtain depth images by emitting a modulated continuous light wave and measuring the delay of the received signal. In this paper we generalize the estimation of the effect of the shot noise when obtaining the phase delay with an arbitrary number of points in the Discrete Fourier Transform, extending and generalizing the analysis done in previous works for the case of four points. For that particular case, we compare our analysis with the state of art. Moreover, we extend the error model using a second order approximation in the error propagation analysis, which provides more accurate estimations according to the Montecarlo simulation experiments. The analysis, based on both analytical and numerical methods, shows that the phase error is, in general, related to the exposure time and weakly to the number of points in the Discrete Fourier Transform. It also depends on the background illumination level, on the amplitude of the received signal, and, when using a three point DFT, on the distance to the objects.

ERROR ASSESSMENT AND PROPAGATION IN DIGITAL TERRAIN MODELING: A REVIEW

Abstract. Digital Terrain Modeling (DTM) is widely used in various applications such as terrain analysis, hydrological modeling, and urban planning. However, the accuracy of DTM heavily relies on the quality of input data and the algorithms used to process the data. Errors in DTM can have significant impacts on downstream applications, such as flood forecasting and land-use planning. Error assessment and propagation in DTM is the process of identifying and quantifying errors in the DTM and evaluating how these errors propagate through downstream applications. This paper provides an overview of error assessment and propagation in DTM, including sources of errors, methods of error assessment, and techniques for error propagation analysis. The paper also discusses the importance of error assessment and propagation in ensuring the accuracy and reliability of DTM and downstream applications.

Analysis of error propagation: from raw light-field data to depth estimation.

In micro-lens-array-based light-field imaging, the micro-lens centers serve as the origins of local micro-lens coordinate systems. Each micro-lens receives angular/depth information coded according to its center location. Therefore, the errors in positioning the micro-lens centers will lead to errors in depth estimation. This paper proposes a method that resolves error propagation from raw light-field data to depth estimation based on analyzing large amounts of simulated images with various aperture sizes, noise levels, and object distance values. The simulation employs backward ray tracing and Monte Carlo sampling to improve computational efficiency. The errors are counted and accumulated stepwise from center positioning and generation of sub-aperture images to depth estimation. The disparity errors calculated during depth estimation are shown to be more apparent either with more significant center positioning errors or with a greater defocusing distance. An experiment using an industrial light-field camera is conducted, confirming that disparity errors at considerable object distances can be reduced significantly when the micro-lens centers are positioned with higher accuracy.

Quantification of Statistical Error in the Estimate of Strain Power Spectral Density Transmissibility for Operational Strain Modal Analysis

The use of strain modes in structural health monitoring has been constantly increasing because of their superior sensitivity to local structural anomalies. This study aims to investigate the applicability and robustness of power spectral density transmissibility (PSDT) in operational strain modal analysis (OSMA). By noting that OSMA in the frequency domain is vulnerable to the error of spectral estimates, uncertainty quantification stemming from strain spectral estimates and the error propagation analysis in OSMA are conducted from an analytical perspective. The main contributions include the following: (i) the mean and variance of strain PSDT estimates are asymptotically derived based on statistical moment theory and the statistics of PSD estimate error, (ii) the coefficients of variation (c.o.v.) of the strain PSDT estimate and strain spectral estimates are compared with each other through asymptotic analysis to elaborate the robustness of strain PSDT, and (iii) the variability of the strain mode shape is quantified based on the asymptotic formula of strain PSDT estimates tending to local minima of asymptotic zero variance at the resonances. The accuracy and efficiency of the quantification and propagation analysis are validated through numerical and experimental test data accompanied by various parametric studies.

Reactive molecular dynamics simulation and chemical kinetic modeling of ammonia/methane co-combustion

Under the vision of “low carbonization”, ammonia (NH3), as a carbon-free hydrogen-rich fuel, blended with reactive additives, is an effective means to achieve low carbon emissions and reduce the consumption of traditional fossil fuels. In this work, the combustion behaviors of NH3 and CH4, the distribution of main products and the formation and reduction characteristics of NO at different temperatures, CH4 mixing ratios, and O2 equivalence ratios are explored by means of reactive molecular dynamics simulation to reveal the microscopic mechanism of NH3/CH4 co-combustion. The calculated results show that CH4 takes the lead in producing H radicals which react with O2 to generate OH radicals to promote the oxidation of NH3, and it is found that the co-combustion process of NH3/CH4 in the radical chain transfer stage is similar to the respective pyrolysis and oxidation reactions of CH4 and NH3. Accordingly, an optimized kinetic mechanism of NH3/CH4 co-combustion is constructed based on the mechanisms proposed by Glarborg et al. and Okafor et al. by incorporating the NH3 pyrolysis sub-mechanisms related to NH2, HNO, and N2H2 and the skeletal mechanism of CH4 oxidation. The optimized mechanisms of NH3/CH4 co-combustion are found to improve the prediction of the ignition delay time (IDT) and laminar burning velocity (LBV) of NH3/CH4 mixture under wide ranges of conditions compared to the original mechanisms of Glarborg et al. and Okafor et al. To decrease the computational cost of three-dimensional combustion numerical simulations, the skeletal mechanism models are established by simplification of the optimized mechanisms of NH3/CH4 co-combustion using direct relation graph method with error propagation and sensitivity analysis, which can provide nearly the same estimates of IDTs and LBVs as the optimized mechanisms mentioned above.

Monitoring ice flow velocity of Petermann glacier combined with Sentinel-1 and −2 imagery

Synthetic Aperture Radar (SAR) images are commonly used to monitor glacier flow velocity at Greenland Ice Sheet (GrIS). However, offset-tracking with SAR imagery in summer usually show poor quality because the rapid ice surface freezing-melting cycles contaminating the surface backscattering characteristic, while optical images are less sensitive to this phenomenon. In this study, we combine Sentinel-1 and -2 images to create the glacier velocity time series for the Petermann glacier, located in the northern GrIS. Firstly, the offset-tracking technique is employed to acquire the initial deformation fields with SAR and optical sensors separately, each SAR and/or optical acquisition is tracked with its closest next three acquisitions. Next, after removing the bad matchings, the least squares method based on connected components is employed to calculate the time series of glacier velocity for Sentinel-1 and −2, separately. Finally, these two kinds of derived time series are integrated with a weighted least squares method, where weights are evaluated according to the estimated RMSEs in the last step. Error propagation analysis suggests RMSEs of the single pair of Sentinel-1 and −2 images offset-tracking are ∼ 0.22 m and ∼ 2.5 m for Petermann glaciers. Standard deviation of the difference between Sentinel-1 and Sentinel-2 measured velocity are ∼ 0.25 m/day. Compared with 6-day velocity fields product, NSIDC (National Snow and Ice Data Center) −0766, which is only derived with Sentinel-1observations, our results show good agreement and less defects in summer. The differences are ∼ 0.20 m/day in non-melting seasons and ∼ 0.34 m/day in summer. Longitudinal velocity differences growing in 2019 and 2020 at ∼ 20 Km up to the terminus are consistency with the crevasse expansion, indicating another calving event is approaching. This research finds that the fusion of Sentinel-1 and −2 offset-tracking results improves the completeness of the ice movement time series for polar glaciers.

The Uncertainty of SNO Cross-Calibration for Satellite Infrared Channels

The on-orbit radiometric calibration is a fundamental task in quantitative remote sensing applications. A widely used calibration method is the cross-calibration based on Simultaneous Nadir Observation (SNO), which involves using high-precision reference instruments to calibrate lower-precision onboard instruments. However, despite efforts to match the observation time, spatial location, field geometry, and instrument spectra, errors can still be introduced during the matching processes and linear regression analysis. This paper focuses on the error generated by sample matching and the error fitting method generated by the sample fitting method. An error propagation analysis is performed to develop a generic model for assessing the uncertainty of the SNO cross-calibration method itself in meteorological satellite infrared channels. The model is validated using the payload parameters of the Hyperspectral Infrared Atmospheric Sounder (HIRAS) and the Medium Resolution Spectral Imager (MERSI) instruments aboard the FengYun-3D (FY-3D). Simulation experiments are performed considering typical bright temperatures, different background fields, and varying matching threshold conditions. The results demonstrate the effectiveness of the proposed model in capturing the error propagation chain in the SNO cross-calibration process. The model provides valuable insight into error analysis in the SNO cross-calibration method and can assist in determining the optimal sample matching threshold for achieving radiometric calibration accuracy.

Investigation of the signal uncertainty in laser-induced breakdown spectroscopy based on error propagation considering self-absorption

The relatively high signal uncertainty of laser-induced breakdown spectroscopy (LIBS) is a key obstacle to its analytical performance and large-scale application. Despite the extensive research efforts to reduce signal uncertainty, few fundamental studies have been conducted because of the complexity of the microscopic mechanisms involved in plasma evolution. Recently, error propagation analysis has provided an inspiring perspective by linking signal uncertainty with plasma property fluctuations, while further investigation is urgently needed due to the limitations of the employed plasma diagnostic techniques. In this study, a spectrum fitting model was proposed for optically thick and homogeneous plasmas to provide more reliable plasma properties for error propagation analysis for the first time. The model was applied to laser-induced copper plasmas using a laser wavelength of 1064 nm and a pulse duration of 25 ns. The uncertainty of an atomic copper line was analyzed at different delay times. Our findings revealed that the fluctuation of columnar total number density was the most significant factor under the investigated conditions, and that the correlation term of temperature and columnar total number density made a considerable negative contribution. Furthermore, self-absorption suppressed the magnitudes of major contribution terms for the signal uncertainty of LIBS.

Correction to “Error propagation and uncertainty analysis: Application to fault tree analysis”

Correction to “Error propagation and uncertainty analysis: Application to fault tree analysis”

Simultaneous Measurement of Strain and Temperature Distributions Using Optical Fibers with Different GeO2 and B2O3 Doping.

Compensating for the effects of temperature is a crucial issue in structural health monitoring when using optical fiber sensors. This study focused on the change in sensitivity due to differences in GeO2 and B2O3 doping and then verified the accuracy when measuring the strain and temperature distributions simultaneously. Four types of optical fiber sensors were utilized to measure the strain and temperature in four-point bending tests, and the best combination of the sensors resulted in strain and temperature errors of 28.4 μϵ and 1.52 °C, respectively. Based on the results obtained from the four-point bending tests, we discussed the error factors via an error propagation analysis. The results of the error propagation analysis agreed well with the experimental results, thus indicating the effectiveness of the analysis as a method for verifying accuracy and error factors.