Lower Estimation Error Research Articles

Hole-Free Symmetric Complementary Sparse Array Design for High-Precision DOA Estimation

Direction of arrival (DOA) estimation plays a critical role in remote sensing, where it aids in identifying and tracking multiple targets across complex environments, from atmospheric monitoring to resource mapping. Leveraging difference covariance array (DCA) for DOA estimation has become prevalent, particularly with sparse arrays capable of resolving more targets than the number of sensors. This paper proposes a new hole-free sparse array configuration for remote sensing applications to achieve improved DOA estimation performance using DCA. By symmetrically placing a minimum redundancy array (MRA) and its complementary MRA on both sides of a sparse uniform linear array (ULA), this configuration maximizes degrees of freedom (DOFs) and minimizes mutual coupling effects. Expressions for calculating sensor positions and optimal element allocation methods to maximize DOFs are derived. Simulation experiments in various scenarios have shown the advantages of the proposed array in DOA estimation, including a strong ability to estimate multi-targets, high angular resolution, low estimation error, and strong robustness to mutual coupling.

Enhanced fault distance estimation for robust protection in unbalanced active distribution networks

Active Distribution Networks (ADN) have evolved significantly by integrating renewable energy sources. While these advancements have ushered in a more sustainable and efficient power system, these have also introduced challenges for traditional protection strategies. Conventional methods, often reliant on overcurrent relays, can struggle to adapt to the complexities of active distribution networks, where fault scenarios are diverse and dynamic. This paper identifies the limitations of conventional protection strategies and explores the redundancy concept applied to protection systems. To improve protection system reliability, an approach that overcomes these limitations using the advantages of distance protection is proposed here. The distance-based protection relay, traditionally employed in transmission networks, offers a promising redundancy solution for the challenges imposed by ADN. Considering the previous exposure, this paper introduces an algorithm for fault resistance and distance estimation designed to optimise the operation of distance relays in active distribution networks. The approach achieves remarkable accuracy with consistently low estimation errors for fault resistance and distance, regardless of fault type or distance. The proposed distance-based protection method is poised to revolutionise fault analysis in ADN, ultimately leading to faster fault isolation and more reliable and improved grid resilience.

Deep learning-enhanced atomic norm minimization for DOA estimation of coherent and incoherent sources using coprime array

Abstract In the field of array signal processing, accurate direction of arrival (DOA) estimation is crucial for handling both coherent and incoherent mixed signal sources. This paper proposes a deep learning-enhanced atomic norm minimization (DLEANM) algorithm, designed to improve DOA estimation for coprime arrays. The algorithm first models the mixed signal scenarios realistically and employs interpolation to transform coprime arrays into virtual uniform linear arrays, which facilitates the formulation of the atomic norm minimization problem. Solving this problem reconstructs an augmented Hermitian Toeplitz covariance matrix. To further enhance the accuracy of DOA estimation, we developed a multi-scale convolutional neural network with an attention mechanism, which uses the covariance matrix as input to better capture signal variations across different frequencies and spatial distributions. By processing multi-scale inputs in parallel, the model improves adaptability and robustness in estimating mixed signal sources. Simulation results show that, under equivalent SNR and snapshot conditions, the DLEANM algorithm achieves lower estimation errors, more accurate multi-target DOA estimation, and relatively faster computation speed compared to conventional methods. Field experiments further confirm the effectiveness of the proposed algorithm. Therefore, after training, the algorithm is able to accurately identify the directions of unknown signals received by sensor arrays.

MVImgNet2.0: A Larger-scale Dataset of Multi-view Images

MVImgNet is a large-scale dataset that contains multi-view images of ~220k real-world objects in 238 classes. As a counterpart of ImageNet, it introduces 3D visual signals via multi-view shooting, making a soft bridge between 2D and 3D vision. This paper constructs the MVImgNet2.0 dataset that expands MVImgNet into a total of ~520k objects and 515 categories, which derives a 3D dataset with a larger scale that is more comparable to ones in the 2D domain. In addition to the expanded dataset scale and category range, MVImgNet2.0 is of a higher quality than MVImgNet owing to four new features: (i) most shoots capture 360° views of the objects, which can support the learning of object reconstruction with completeness; (ii) the segmentation manner is advanced to produce foreground object masks of higher accuracy; (iii) a more powerful structure-from-motion method is adopted to derive the camera pose for each frame of a lower estimation error; (iv) higher-quality dense point clouds are reconstructed via advanced methods for objects captured in 360 ° views, which can serve for downstream applications. Extensive experiments confirm the value of the proposed MVImgNet2.0 in boosting the performance of large 3D reconstruction models. MVImgNet2.0 will be public at luyues.github.io/mvimgnet2 , including multi-view images of all 520k objects, the reconstructed high-quality point clouds, and data annotation codes, hoping to inspire the broader vision community.

Performance Evaluation of QT-RR Adaptation Time Lag Estimation in Exercise Stress Testing.

Slower adaptation of the QT interval to sudden changes in heart rate has been identified as a risk marker of ventricular arrhythmia. The gradual changes observed in exercise stress testing facilitates the estimation of the QT-RR adaptation time lag. The time lag estimation is based on the delay between the observed QT intervals and the QT intervals derived from the observed RR intervals using a memoryless transformation. Assuming that the two types of QT interval are corrupted with either Gaussian or Laplacian noise, the respective maximum likelihood time lag estimators are derived. Estimation performance is evaluated using an ECG simulator which models change in RR and QT intervals with a known time lag, muscle noise level, respiratory rate, and more. The accuracy of T-wave end delineation and the influence of the learning window positioning for model parameter estimation are also investigated. Using simulated datasets, the results show that the proposed approach to estimation can be applied to any changes in heart rate trend as long as the frequency content of the trend is below a certain frequency. Moreover, using a proper position of the learning window for exercise so that data compensation reduces the effect of nonstationarity, a lower mean estimation error results for a wide range of time lags. Using a clinical dataset, the Laplacian-based estimator shows a better discrimination between patients grouped according to the risk of suffering from coronary artery disease. Using simulated ECGs, the performance evaluation of the proposed method shows that the estimated time lag agrees well with the true time lag.

Sea surface heat flux helps predicting thermocline in the South China Sea

In this study, a deep learning model called Four Dimensional Residual Network (4D-ResNet) was proposed, which can capture both temporal and spatial information. Temperatures at various depths were predicted for the next 40 days using the last month's sea surface variables, and a spatio-temporal prediction of the thermocline was achieved. In addition to the satellite-observed sea surface parameters: sea surface temperature (SST), sea level anomaly (SLA), and sea surface wind (SSW), net heat flux (Qnet) was also included in the model input. Qnet can alter the density of the upper water, resulting in convection or improved stratification stability. The results indicate that the additional input of Qnet improves the model's accuracy, especially at the depth of the thermocline, where the RMSE reduced by up to 13.7%. The 4D-ResNet model has much lower estimation error compared to other models and successfully captures the seasonal characteristics of the thermocline.

Accurate drift-invariant single-molecule force calibration using the Hadamard variance

Single-molecule force spectroscopy (SMFS) techniques play a pivotal role in unraveling the mechanics and conformational transitions of biological macromolecules under external forces. Among these techniques, multiplexed magnetic tweezers (MT) are particularly well suited to probe very small forces, ≤1 pN, critical for studying noncovalent interactions and regulatory conformational changes at the single-molecule level. However, to apply and measure such small forces, a reliable and accurate force-calibration procedure is crucial. Here, we introduce a new approach to calibrate MT based on thermal motion using the Hadamard variance (HV). To test our method, we perform bead-tether Brownian dynamics simulations that mimic our experimental system and compare the performance of the HV method against two established techniques: power spectral density (PSD) and Allan variance (AV) analyses. Our analysis includes an assessment of each method’s ability to mitigate common sources of additive noise, such as white and pink noise, as well as drift, which often complicate experimental data analysis. We find that the HV method exhibits overall similar or higher precision and accuracy, yielding lower force estimation errors across a wide range of signal/noise ratios (SNRs) and drift speeds compared with the PSD and AV methods. Notably, the HV method remains robust against drift, maintaining consistent uncertainty levels across the entire studied SNR and drift speed spectrum. We also explore the HV method using experimental MT data, where we find overall smaller force estimation errors compared with PSD and AV approaches. Overall, the HV method offers a robust method for achieving sub-pN resolution and precision in multiplexed MT measurements. Its potential extends to other SMFS techniques, presenting exciting opportunities for advancing our understanding of mechanosensitivity and force generation in biological systems. To make our methods widely accessible to the research community, we provide a well-documented Python implementation of the HV method as an extension to the Tweezepy package.

Extended Association Rule Mining and Its Application to Software Engineering Data Sets

Association rule mining is a highly effective approach to data analysis for datasets of varying sizes, accommodating diverse feature values. Nevertheless, deriving practical rules from datasets with numerical variables presents a challenge, as these variables must be discretized beforehand. Quantitative association rule mining addresses this issue, allowing the extraction of valuable rules. This paper introduces an extension to quantitative association rules, incorporating a two-variable function in their consequent part. The use of correlation functions, statistical test functions, and error functions is also introduced. We illustrate the utility of this extension through three case studies employing software engineering datasets. In case study 1, we successfully pinpointed the conditions that result in either a high or low correlation between effort and software size, offering valuable insights for software project managers. In case study 2, we effectively identified the conditions that lead to a high or low correlation between the number of bugs and source lines of code, aiding in the formulation of software test planning strategies. In case study 3, we applied our approach to the two-step software effort estimation process, uncovering the conditions most likely to yield low effort estimation errors.

Model-less multi-input analysis of pulmonary blood flow using deep learning convolution

The study investigates two categories of perfusion-based pulmonary blood flow analysis: model-based and model-less methods. The model-based approach yields plausible results, but requires strict parameter settings and presents challenges in handling. On the other hand, the model-less approach is simpler but limited to a single input analysis, necessitating an inverse problem to estimate the impulse response from input–output relationships. To overcome these limitations, this article proposes a model-less method that combines simplicity and accuracy, enabling multi-input system analysis and aiming for standardized analysis. They leverage deep learning convolution to directly estimate the impulse response, allowing for multi-input analysis. Comparative experiments demonstrate that the proposed method is easy to implement and exhibits a low estimation error within the measured signal-to-noise ratio (SNR) range, even though it is sensitive to noise. Furthermore, the proposed method is evaluated through waveform analysis, specifically Delay and Dispersion in Experiment 1, where it is compared with conventional methods. In Experiment 2, blood flow analysis is performed on a patient with a defect in the left pulmonary artery. The results indicate high convergence, independence from input waveforms, and effective analysis of cases with vascular stenosis. Moreover, the method enables multi-input system analysis, consistently yielding results consistent with medical findings, even for patients with left pulmonary artery defects.

Do people need guidance to estimate a food portion size? Evidence from an exploratory eye-tracker study

A promising strategy to encourage portion control involves clear guidance and portion information on product packaging alongside nutrition details. However, literature on the impact of this information on serving sizes is conflicting. To gather more objective evidence about how consumers interact with on-pack portion guidance, we conducted a laboratory-based eye-tracker study with 66 participants completing two portion-size estimation tasks. In the first, participants estimated serving sizes indirectly by gauging how many people a product pack could serve. The second task involved direct portion size selection using pictorial representations of the food portion on a virtual plate. Additionally, we explored how participants’ familiarity with products affected their portion estimations. Results: when portion guidance was presented, errors in portion estimation significantly decreased only in the indirect task, where 85% of participants showed an improved accuracy. This suggests that the way individuals are prompted to estimate portion sizes influences their approach and their reliance on the provided guidance. Higher error rates occurred when participants were more familiar with the products. For less familiar products, the lower portion estimation errors were associated with the longer fixations on the portion graphic in the indirect task. In addition, participants who quickly noticed the portion graphic provided more accurate portion size estimates. This emphasizes the practical importance of swiftly locating portion information on-pack to enhance estimation accuracy.

Estimation of land surface temperature (LST) using single-channel and multi-band methods in Sablan mountainous region

Studying Land Surface Temperature (LST) provides various scientific, environmental, and practical advantages. It aids climate monitoring, urban heat island studies, environmental assessments, agriculture, urban planning, infrastructure development, and remote sensing. Algorithm selection depends on research constraints and goals, considering different algorithm strengths and limitations. In this study, the LST around the Sabalan volcanic peak was estimated using Landsat 8 satellite images captured by the OLI and TIRS, employing the Single Channel JM&S and Split Window algorithms. The obtained surface temperatures from both algorithms were converted to air temperature. Subsequently, the pixel air temperatures were compared to the corresponding synoptic station air temperatures of the studied region, considering the satellite's passage time. This comparison was executed using regression analysis (R2 and correlation coefficients) and statistical indices such as Root Mean Square Error (RMSE) and Mean Absolute Error (MAE) to validate the data. The results indicated R2 values and correlation coefficients of 0.04 and 0.97, respectively, for the Single Channel algorithm, and 0.12 and 0.69 for the SW algorithm, signifying a positive relationship and close alignment between these datasets. Furthermore, the examination of the results revealed root mean square error and mean absolute error with minimum errors of 4.20, 3.88, and 8.14, 8.97 °C in the SW method. By assessing the difference between mean LST and the synoptic station air temperatures, the SW approach exhibited lower temperature discrepancies across all stations and lower estimation errors compared to the Single Channel method, thus displaying higher accuracy and closer conformity to the actual temperature.

Semi-blind sparse channel estimation using regularized expectation maximization

In Massive multiple-input multiple-output (MIMO) systems, channel estimation is crucial. The large size of the antennas causes a significant pilot and feedback overhead, making it challenging to estimate channels in massive MIMO systems. Besides, the studies have shown that mmWave channels are sparse due to the limited number of dominant propagation paths. Therefore, the motivation of this paper is to exploit the inherent sparsity of the massive mmWave MIMO channels to develop a semi-blind channel estimation with reduced number of pilots. To this goal, an expectation maximization (EM) based technique has been developed which leverages the sparsity of the underlying channel for its better estimation. An iterative approach is proposed to solve the modeled problem which simultaneously updates the channel coefficients and data symbols using available data and system structure at each iteration. The proposed method imposes sparsity with the aid of Smoothed L0 norm (SL0) in the M-step. The simulation results demonstrate the proposed method have quick convergence and lower channel estimation error compared to the existing methods. As a quantitative evaluation, the proposed method attains the normalized mean square error of 9×10−2 and 7×10−3 at SNR=5dB and SNR=15dB, respectively.

Rapid Mass Conversion for Environmental Microplastics of Diverse Shapes.

Rivers have been recognized as the primary conveyors of microplastics to the oceans, and seaward transport flux of riverine microplastics is an issue of global attention. However, there is a significant discrepancy in how microplastic concentration is expressed in field occurrence investigations (number concentration) and in mass flux (mass concentration). Of urgent need is to establish efficient conversion models to correlate these two important paradigms. Here, we first established an abundant environmental microplastic dataset and then employed a deep neural residual network (ResNet50) to successfully separate microplastics into fiber, fragment, and pellet shapes with 92.67% accuracy. We also used the circularity (C) parameter to represent the surface shape alteration of pellet-shaped microplastics, which always have a more uneven surface than other shapes. Furthermore, we added thickness information to two-dimensional images, which has been ignored by most prior research because labor-intensive processes were required. Eventually, a set of accurate models for microplastic mass conversion was developed, with absolute estimation errors of 7.1, 3.1, 0.2, and 0.9% for pellet (0.50 ≤ C < 0.75), pellet (0.75 ≤ C ≤ 1.00), fiber, and fragment microplastics, respectively; environmental samples have validated that this set is significantly faster (saves ∼2 h/100 MPs) and less biased (7-fold lower estimation errors) compared to previous empirical models.

Exploring the relationship between pavement mean texture depth and mean profile depth: from theoretical derivation to field results

Pavement mean texture depth (MTD) and mean profile depth (MPD) are common evaluation indicators for characterizing the pavement surface. Clarifying the relationship between the two contributes to coordinate various measurement methods and evaluation criterions. The calculation methods of MTD and MPD were first introduced to trace the similarity between the indicators. Subsequently, the tortuosity area of surface texture was observed using x-ray Computer Tomography (CT). To further determine the model parameter, over 3000 sets of field data in Research Institute of Highway Ministry of Transport (RIOH) track were measured. Meanwhile, the field results of another track and existing recommended model were also used to validate the model. The results show that the shift relationship between MTD and MPD is theoretically unified into a linear model with a slope of 1 and an intercept related to the surface tortuosity. These tortuous areas, the main difference source between the two indicators, would lead to a greater equivalent texture depth under the coarser gradation or lower the compactness of the asphalt mixture. Even for the different track roads, the determined model (MTD = MPD + 0.03) has a lower mean relative estimation error, indicating good applicability. This study provides theoretical explanation and empirical references for the automated laser-based texture detection.

Machine learning-based comparison of factors influencing estimated glomerular filtration rate in Chinese women with or without non-alcoholic fatty liver.

The prevalence of non-alcoholic fatty liver (NAFLD) has increased recently. Subjects with NAFLD are known to have higher chance for renal function impairment. Many past studies used traditional multiple linear regression (MLR) to identify risk factors for decreased estimated glomerular filtration rate (eGFR). However, medical research is increasingly relying on emerging machine learning (Mach-L) methods. The present study enrolled healthy women to identify factors affecting eGFR in subjects with and without NAFLD (NAFLD+, NAFLD-) and to rank their importance. To uses three different Mach-L methods to identify key impact factors for eGFR in healthy women with and without NAFLD. A total of 65535 healthy female study participants were enrolled from the Taiwan MJ cohort, accounting for 32 independent variables including demographic, biochemistry and lifestyle parameters (independent variables), while eGFR was used as the dependent variable. Aside from MLR, three Mach-L methods were applied, including stochastic gradient boosting, eXtreme gradient boosting and elastic net. Errors of estimation were used to define method accuracy, where smaller degree of error indicated better model performance. Income, albumin, eGFR, High density lipoprotein-Cholesterol, phosphorus, forced expiratory volume in one second (FEV1), and sleep time were all lower in the NAFLD+ group, while other factors were all significantly higher except for smoking area. Mach-L had lower estimation errors, thus outperforming MLR. In Model 1, age, uric acid (UA), FEV1, plasma calcium level (Ca), plasma albumin level (Alb) and T-bilirubin were the most important factors in the NAFLD+ group, as opposed to age, UA, FEV1, Alb, lactic dehydrogenase (LDH) and Ca for the NAFLD- group. Given the importance percentage was much higher than the 2nd important factor, we built Model 2 by removing age. The eGFR were lower in the NAFLD+ group compared to the NAFLD- group, with age being was the most important impact factor in both groups of healthy Chinese women, followed by LDH, UA, FEV1 and Alb. However, for the NAFLD- group, TSH and SBP were the 5th and 6th most important factors, as opposed to Ca and BF in the NAFLD+ group.

Plant Density and Health Evaluation in Green Stormwater Infrastructure Using Unmanned-Aerial-Vehicle-Based Imagery

Over the past few decades, there has been a notable surge in interest in green stormwater infrastructure (GSI). This trend is a result of the need to effectively address issues related to runoff, pollution, and the adverse effects of urbanization and impervious surfaces on waterways. Concurrently, umanned aerial vehicles (UAVs) have gained prominence across applications, including photogrammetry, military applications, precision farming, agricultural land, forestry, environmental surveillance, remote-sensing, and infrastructure maintenance. Despite the widespread use of GSI and UAV technologies, there remains a glaring gap in research focused on the evaluation and maintenance of the GSIs using UAV-based imagery. This study aimed to develop an integrated framework to evaluate plant density and health within GSIs using UAV-based imagery. This integrated framework incorporated the UAV (commonly known as a drone), WebOpenDroneMap (WebDOM), ArcMap, PyCharm, and the Canopeo application. The UAV-based images of GSI components, encompassing trees, grass, soil, and unhealthy trees, as well as entire GSIs (e.g., bioretention and green roofs) within the Morgan State University (MSU) campus were collected, processed, and analyzed using this integrated framework. Results indicated that the framework yielded highly accurate predictions of plant density with a high R2 value of 95.8% and lower estimation errors of between 3.9% and 9.7%. Plant density was observed to vary between 63.63% and 75.30% in the GSIs at the MSU campus, potentially attributable to the different types of GSI, varying facility ages, and inadequate maintenance. Normalized difference vegetation index (NDVI) maps and scales of two GSIs were also generated to evaluate plant health. The NDVI and plant density results can be used to suggest where new plants can be added and to provide proper maintenance to achieve proper functions within the GSIs. This study provides a framework for evaluating plant performance within the GSIs using the collected UAV-based imagery.

From laboratory to industrial use: Understanding process variation during enzymatic protein hydrolysis with dry film fourier-transform infrared spectroscopy

Industries are continuously looking for dedicated sensor solutions for evaluating the chemical composition of products that can provide valuable insights into process control, process optimization, and product quality. A rapid and robust analytical method, Fourier-Transform Infrared (FTIR) spectroscopy, holds significant potential in this regard as it provides detailed compositional values of food nutrients. The main aim of the present study was to use dry film FTIR spectroscopy as an analytical tool to characterize products from an industrial enzymatic protein hydrolysis process and link this to understand industrial process variations. For this purpose, 463 protein hydrolysate samples were obtained from an industrial enzymatic protein hydrolysis plant. In the same period, systematic variations in process parameters such as raw material composition, enzyme type, and water addition, were performed. All samples were analyzed using dry film FTIR spectroscopy. Two subsets containing 200 and 68 hydrolysate samples were chosen for measuring average molecular weight (AMW) and collagen content respectively, providing reference data for constructing calibration models based on partial least squares regression (PLSR). The percentage of low molecular weight constituents was derived from the molecular weight distribution of size exclusion chromatograms of protein hydrolysates and also used in the modeling. Calibration models for the prediction of AMW, low molecular weight constituents, and collagen content were obtained with a good fit and lower estimation errors. Principal component analysis (PCA) of protein hydrolysates' FTIR spectra effectively differentiated process variations (enzyme types and raw materials) without extensive reference analysis. One-factor analysis of variance (ANOVA) tests was used to observe the impact of process variation on product quality. FTIR proved to be a sensitive method for liquid protein analysis and process control with a significant potential in process optimization approaches. To the authors’ knowledge, this is the first time dry film FTIR spectroscopy has been used to evaluate an industrial bioprocess with designed process variations.

Full inference for the anisotropic fractional Brownian field

The anisotropic fractional Brownian field (AFBF) is a non-stationary Gaussian random field which has been used for the modeling of textured images. In this paper, we address the open issue of estimating the functional parameters of this field, namely the topothesy and Hurst functions. We propose an original method which fits the empirical semi-variogram of an image to the semi-variogram of a turning-band field that approximates the AFBF. Expressing the fitting criterion in terms of a separable non-linear least square criterion, we design a minimization algorithm inspired by the variable projection approach. This algorithm also includes a coarse-to-fine multigrid strategy based on approximations of functional parameters. Compared to existing methods, the new method enables to estimate both functional parameters on their whole definition domain. On simulated textures, we show that it has a low estimation error, even when the parameters are approximated with a high precision. We also apply the method to characterize mammograms and sample images with synthetic parenchymal patterns.

Dynamic Performance Evaluation of a Brushless AC Motor Drive Using Different Sensorless Schemes

The presented study concerns with evaluating the dynamic performance of an isotropic sinusoidal brushless motor drive while utilizing different sensorless schemes. Three estimation algorithms are considered: the first depends on extracting the speed and position via comparing two values of motor's voltage in two co-ordinate systems; the second extracts the speed and position signal via comparing two different values of motor's current defined in two co-ordinates; while the third depends on estimating the motor's flux and use it to get the speed and position. The vector control is adopted to manage the drive dynamics. The detailed mathematical derivations for all system components are presented to facilitate the performance analysis. The theoretical base of each sensorless scheme is also described in detail. The target of the provided comparative analysis is to outline the weakness and strength points of each adopted sensorless schemes while estimating the speed and rotor position for a wide operating speed range. The judgment is measured in terms of the speed and rotor position estimation errors and the dynamic response as well. The performance evaluation process is carried out using MATLAB/Simulink software in which all system parts are simulated using their mathematical models. The findings from the study state that when it comes to dynamic speed behaviour, the voltage-based sensorless technique dominates, while the current-based sensorless approach gives stability in speed estimate priority. Alternatively, the third adopted sensorless scheme offers an acceptable high-speed performance and respectable performance at lower speeds. Statistically, it is found that the voltage-based estimation technique gives respectively lower speed and position estimation errors with percentages of 35% and 10% lower than their values under the current-based estimation technique, and with percentages of 35% and 30% lower than their values under the third adopted scheme.

State of Charge Estimation for Lithium Battery Based on Fractional Order Square Root Cubature Kalman Filter and Adaptive Multi-innovation Unscented Kalman Filter

Accurate state of charge (SOC) estimation of batteries is of great significance for electric vehicles. A SOC estimation method based on a fractional order square root cubature Kalman filter (FOSRCKF) and an adaptive multi-innovation unscented Kalman filter (AMIUKF) is proposed. The battery is modelled using fractional order calculus theory and the model parameters are identified by adaptive genetic algorithm. The FOSRCKF estimates the battery SOC, while the AMIUKF online updates the internal resistance in the model, and there exchanges information between two filters. The experimental results under the Urban Dynamometer Driving Schedule (UDDS) and the US06 Highway Driving Schedule show that the proposed method has lower SOC estimation error and lower terminal prediction error compared with the traditional SRCKF method based on integer order models, which demonstrates the effectiveness, accuracy and robustness of the proposed method.