Squares Of Residuals Research Articles

Simulation case study of precise orbit determination for IGSO satellites using onboard Ka/GNSS observations

Abstract Inter-satellite links (ISLs) can provide a more accurate spatial reference for inclined geosynchronous orbit satellite (IGSO) satellites than onboard global navigation satellite system (GNSS) side-lobe signals. In this study, precise orbit determination (POD) quality is analyzed based on onboard Ka/GNSS observations using a simulated IGSO satellite orbit. Considering the similar orbit altitude, we first select three IGSO satellites from BeiDou Navigation Satellite System (BDS-3) as target satellites, which have real Ka-band measurements available, along with other BDS-3 satellites and ground anchor stations, enabling us to evaluate POD performance. For the POD results using real Ka-band measurements, the root mean square of the residuals can reach 10.9 cm, and the orbit consistency in the 3D direction is better than 0.35 m. Additionally, simulations are conducted for onboard GNSS, ISL, and ground anchor-satellite link (GSL), considering the different constraints on the Ka-band resources. When only GNSS measurements are used, the 3D orbit accuracy estimated by the simulated theoretical orbit is approximately 1.5 m, and when only ISL measurements are used, the 3D orbit accuracy can reach approximately 0.35 m. However, when additional ISL and GSL measurements are included, the POD accuracy can exceed 0.28 m. These results demonstrate that incorporating Ka-band measurements can effectively enhance the POD accuracy for high-orbit satellites.

Machine learning approach to investigate pregnancy and childbirth risk factors of sleep problems in early adolescence: Evidence from two cohort studies

BackgroundThis study aimed to predict early adolescent sleep problems using pregnancy and childbirth risk factors through machine learning algorithms, and to evaluate model performance internally and externally. MethodsData from the China Jintan Child Cohort study (CJCC; n=848) for model development and the US Healthy Brain and Behavior Study (HBBS; n=454) for external validation were employed. Maternal pregnancy histories, obstetric data, and adolescent sleep problems were collected. Several machine learning techniques were employed, including least absolute shrinkage and selection operator, logistic regression, random forest, naïve bayes, extreme gradient boosting, decision tree, and neural network. The area under the receiver operating characteristic curve, sensitivity, specificity, accuracy, and root mean square of residuals were used to evaluate model performance. ResultsKey predictors for CJCC adolescents’ sleep problems include gestational age, birthweight, duration of delivery, and maternal happiness during pregnancy. In HBBS adolescents, the duration of postnatal depressive emotions was the primary perinatal predictor. The prediction models developed in the CJCC had good-to-excellent internal validation performance but poor performance in predicting the sleep problems in HBBS adolescents. ConclusionThe identification of specific perinatal risk factors associated with adolescent sleep problems can inform targeted interventions during and after pregnancy to mitigate these risks. Health providers should consider integrating these predictive factors into routine pre- and postnatal assessments to identify at-risk populations. The variability in model performance across different cohorts highlights the need for context-specific models and the cautious application of predictive analytics across diverse populations. Future research should focus on refining predictive models to account for such variations, potentially through the incorporation of additional socio-cultural factors and genetic markers. This study emphasizes the importance of personalized and culturally sensitive approaches in the prediction and management of adolescent sleep problems, leveraging advanced computational methods to enhance maternal and child health outcomes.

Treating Tropospheric Phase Delay in Large-scale Sentinel-1 Stacks to Analyze Land Subsidence

Variations in the tropospheric phase delay pose a primary challenge to achieving precise displacement measurements in Interferometric Synthetic Aperture Radar (InSAR) analysis. This study presents a cluster-based empirical tropospheric phase correction approach to analyze land subsidence rates from large-scale Sentinel‑1 data stacks. Our method identifies the optimum number of clusters in individual interferograms for K‑means clustering, and segments extensive interferograms into areas with consistent tropospheric phase delay behaviors. It then performs tropospheric phase correction based on empirical topography-phase correlation, addressing stratified and broad-scale tropospheric phase delays. Applied to a six-year data stack along a 1000-km track in Iran, we demonstrate that this approach enhances interferogram quality by reducing the standard deviation by 50% and lowering the semivariance of the interferograms to 20 cm2 at distances up to 800 km in 97% of the interferograms. Additionally, the corrected time series of deformation shows a 40% reduction in the root mean square of residuals at the most severely deformed points. By analyzing the corrected interferograms, we show that our method improves the efficiency of country-scale InSAR surveys to detect and quantify present-day land subsidence in Iran, which is essential for groundwater management and sustainable water resource planning.

Exploring the role of spending on malaria incidence in Uganda using the auto-regressive distributed lag approach.

Malaria has remained a persistent global health problem. Despite multiple government and donor initiatives to eradicate malaria and its detrimental effects on Uganda's health outcomes, the incidence of malaria is worrying as it appears higher than the average of 219 cases per 1000 for sub-Saharan Africa for the period 2017-2018. This study investigated the effect of public and private healthcare spending on the incidence of malaria in Uganda. Employing time series data spanning over 20years from the first quarter of 2000 to the last quarter of 2019, the study builds a model based on the Grossman framework for analysing demand for health. The estimation technique used was the ARDL approach that takes into account reverse causality and incidental relationships. Prior to the adoption of the technique, a bounds test was performed to determine whether the variables contained in the model have a long-term relationship. Several diagnostic tests for serial correlation, functional normality, and heteroskedastic specification error were carried out to verify the ARDL model's goodness of fit. Additionally, the cumulative sum of recursive (CUSUM) and cumulative sum of squares of recursive residuals (CUSUMSQ) were used to test model stability. The results indicate that in the long run, an increase in public spending of one percent significantly reduces malaria incidence by 0.196 at the 10 percent level of significance. On the other hand, there is no significant evidence of private health expenditure's effect on malaria incidence. However, in the short run, public spending reduces malaria incidence by a smaller magnitude of 0.158 percent relative to the long-run. Still, private expenditure is found to exhibit no significant effect. Additional findings point to the importance of GDP per capita and urban population growth in reducing malaria incidence, whereas female unemployment, income inequality, as well as female-headed household. In the short run, however, the female-headed households and urban population growth are found to significantly reduce malaria incidence while an improvement in regulatory quality decreases malaria incidence by 0.129 percent. There is need for further government interventions to reduce malaria incidence in the country via budget allocation, as well as the strengthening of programmes to raise household income to support private health spending, in addition to the development of strategies to promote well-planned and organized urban centres.

Parallel observations process of Tianwen-1 orbit determination

Abstract To meet various requirements of the future deep space missions of China, State Key Laboratory of Astronautic Dynamics constructs a new orbit determinatin software with parallel observations process. By using 32 threads, the computational efficiency per iteration (including spacecraft’s integration) could be promoted to 10 times as that of single-threaded orbit determination. Suppose the number of observations is p p , the number of estimated parameters (including spacecraft’s state) is q q , the amount of computation of one observation is x x , the amount of computation of one Givens transformation is y y and the best number of threads is proved to be [ p ⋅ ( x + y ) / ( q ⋅ y ) ] 1 / 2 {\left[p\cdot \left(x+y)\text{/}\left(q\cdot y)]}^{1\text{/}2} for one-step threads combination. The root mean square of the postfit residuals of China’s deep space monitoring network (CDSMN) and China’s Very Long Baseline Interferometry (VLBI) network (CVN) observations in the Earth-Mars transfer phase and the Mars-orbiting phase are almost the same: about 0.3 m for Ranging, about 0.3 mm/s for Doppler, about 3 cm for VLBI delay and about 0.5 mm/s for VLBI delay rate. It is also found that all the four types of observations of CDSMN and CVN are needed in orbit determination for deep space maneuver and braking at periareion calculation. In the Mars-orbiting phase, the position accuracy after orbit determination under CDSMN-only tracking mode can reach about 1 km.

Robust 3D coordinate transformation based on genetic algorithm intelligent weighting

In order to improve the solution quality of 3D coordinate transformation parameters, a robust optimization model is obtained by integrating the 3D coordinate transformation model within the overall framework of the genetic algorithm in this paper. The minimum of the weighted sum of squares of the point residuals of the common points is taken as the objective function for optimization, and the parallel search is started from the initial population, and the updating process of the solution set is used to generate new and better individuals, and the adaptive weight combination corresponding to the optimum of the fitness function is obtained at the end of the generation iteration. The generated simulated data and measured data were analyzed using the Hefei Light Source tunnel control network as an application scenario. The results show that the method in this paper can automatically reduce the influence of low-quality data on the model, and compared with the two robust conversion models of IGG III iterative weighting, the method in this paper is not affected by the empirical parameters in determining the weight combinations and the weighted sum of squares of the residuals of the solved points are smaller, so it can effectively improve the quality of the solved conversion parameters.

GPS displacement dataset for the study of elastic surface mass variations

Abstract. Quantification of uncertainty in surface mass change signals derived from Global Positioning System (GPS) measurements poses challenges, especially when dealing with large datasets with continental or global coverage. We present a new GPS station displacement dataset that reflects surface mass load signals and their uncertainties. We assess the structure and quantify the uncertainty of vertical land displacement derived from 3045 GPS stations distributed across the continental US. Monthly means of daily positions are available for 15 years. We list the required corrections to isolate surface mass signals in GPS estimates and screen the data using GRACE(-FO) as external validation. Evaluation of GPS time series is a critical step, which identifies (a) corrections that were missed, (b) sites that contain non-elastic signals (e.g., close to aquifers), and (c) sites affected by background modeling errors (e.g., errors in the glacial isostatic model). Finally, we quantify uncertainty of GPS vertical displacement estimates through stochastic modeling and quantification of spatially correlated errors. Our aim is to assign weights to GPS estimates of vertical displacements, which will be used in a joint solution with GRACE(-FO). We prescribe white, colored, and spatially correlated noise. To quantify spatially correlated noise, we build on the common mode imaging approach by adding a geophysical constraint (i.e., surface hydrology) to derive an error estimate for the surface mass signal. We study the uncertainty of the GPS displacement time series and find an average noise level between 2 and 3 mm when white noise, flicker noise, and the root mean square (rms) of residuals about a seasonality and trend fit are used to describe uncertainty. Prescribing random walk noise increases the error level such that half of the stations have noise > 4 mm, which is systematic with the noise level derived through modeling of spatially correlated noise. The new dataset is available at https://doi.org/10.5281/zenodo.8184285 (Peidou et al., 2023) and is suitable for use in a future joint solution with GRACE(-FO)-like observations.

Study of Linear Model Fitting and Distribution Analysis on Random Numbers Generated in Fortran

In this study, we employed Fortran 90 programming to generate pseudorandom numbers ranging from 0 to 1 at various sample sizes (n = 1000, 500, 250, 125, 63, 32, 16, 8). Subsequently, linear fitting models were applied to the generated data. Our findings indicated that the higher sample size of n=1000 yielded the least asymptotic standard error for both slope ‘a’ and intercept ‘b’ parameters in the linear equation. We observed a decrease in the sum of squares of residuals as the sample size (n) decreased, indicating that the linear model also provided a better fit to the data with smaller sample sizes. The consistent nature of the kernel density estimate plots suggests that as the sample size increases, the estimation becomes more precise and less affected by random noise or sampling variability, further enhancing the reliability of the estimated PDF.

Exploring the influence of gender diversity and women's empowerment on family entrepreneurship performance: the moderating impact of firm characteristic

PurposeThis paper aims to examine how gender diversity and women’s empowerment influence the performance of family entrepreneurships and explores the role of firm characteristics as a moderating factor.Design/methodology/approachThe study used a structured questionnaire as the survey tool to collect data from 91 women managers working in family entrepreneurships, which originated from entrepreneurial initiatives, located in various Lebanese regions. The validity of the construct was assumed using the fitness of extracted index, incremental fit-index, non-normal fit-index, root mean square of residuals and standard root mean square residual. Composite reliability, Cronbach's alpha and value confirmatory factor analysis were used to measure the internal consistency. Data were analyzed using the structural equation modeling method.FindingsThis study reveals that gender equality, education level and family support significantly affect women's empowerment while an insignificant association was found between empowerment and earning social status and achieving financial independence. This paper also showed a significant interaction between women’s empowerment and the performance of family entrepreneurships. Additionally, the results showed that women holding managerial positions in family entrepreneurships is positively associated with firm performance. Finally, it was concluded that the location of the family firm moderates the relationship between gender diversity and firm performance.Originality/valueThis research contributes to theory and practice regarding the role of women in family entrepreneurships and sheds light on gender differences influencing family entrepreneurships and women empowerment issues.

Development and reliability testing of a risk factor and risk outcome assessment scale for nurses in “internet + nursing services” for the elderly

BackgroundChina is experiencing an aging population, leading to a significant demand for “Internet + nursing services” tailored for elderly individuals. However, there are many risk problems in the process of nurse service, which hinder the development of the service, and a scale is needed to assess the risk problems faced by nurses in “Internet + nursing services” for the elderly.ObjectiveThe purpose of this study is to develop an assessment scale for risk factors and outcomes related to nurses’ involvement in the “Internet + Nursing Service” for the elderly and to assess its reliability and validity.MethodsBased on literature analysis, focus group, the Delphi method, and a presurvey, we designed an initial scale. The initial scale comprised two sections: risk factors and risk outcomes for nurses. In January and February of 2023, nurses engaged in “Internet + nursing services” for the elderly in Shanxi Province were chosen through a convenience sampling technique for a questionnaire survey. Subsequently, item analysis and exploratory factor analysis were employed to refine and develop a test version of the scale further. A follow-up questionnaire survey was carried out in March and April 2023 using a similar approach. The reliability and validity of the scale were assessed through confirmatory factor analysis, culminating in the formation of the final scale.ResultsThe initial survey yielded 244 valid responses. The cumulative variance contributions of the two segments from the exploratory factor analysis were 84.584% and 90.089%, respectively. A subsequent survey garnered 220 valid responses. The confirmatory factor analysis results indicated: χ2/df = 2.086, comparative fit index (CFI) = 0.918, normative fit index (NLI) = 0.855, root mean square of residuals (RMR) = 0.045, and root mean square of error of approximation (RMSEA) = 0.070. These results demonstrate good structural, convergent, and discriminant validity. The content validity index at the item level (I-CVI) ranged between 0.875 and 1.000, while the content validity index at the scale level (S-CVI/Ave) was 0.941. Cronbach’s alpha coefficient for the entire scale stood at 0.970. Moreover, the scale exhibited a split-half reliability of 0.876 and a retest reliability of 0.980 (p < 0.01).ConclusionThe risk factors and risk outcomes associated with nurses involved in “Internet + nursing services” for elderly individuals, as developed in this study, demonstrate strong reliability and validity. They are well suited to the Chinese national context.

Characteristics of Inter-System Bias between BDS-2 and BDS-3 and Its Impact on BDS Orbit and Clock Solutions

The inter-system-like bias between the regional (BDS-2) and global (BDS-3) constellation of the BeiDou Navigation Satellite System (BDS) has been identified on common B1I pseudo-range observations. In this study, its characteristics are investigated with tracking data from the International GNSS Service (IGS) and International GNSS Monitoring and Assessment System (iGMAS) network. Firstly, the satellite-specific inter-system-like bias is calculated and the dependency on satellite is observed. Clearly noticeable discrepancies on BDS-2 and BDS-3 can be identified. Hence, the constellation-specific inter-system-like bias is estimated. Biases for all receivers are quite stable, with standard derivation (STDev) less than 0.2 m in average. The bias shows clear dependence on the receiver, while the firmware and antenna have limited but not negligible impacts, particularly for Trimble NetR9 and Alloy receivers. The Trimble NetR9 with TRM59800.00 antenna shows noticeable discrepancy up to about 1.5 m with different antenna, and the bias of the Trimble Alloy 5.37 jumps about 2.4 m with respect to later firmware. In addition, clear annual variations are observed for stations ABPO and MIZU with Septentrio POLARX5 5.3.2 and ASTERX4 4.4.2 receivers, respectively. Furthermore, the impacts of the biases on the BDS orbit and clock solutions are analyzed. Once BDS-2 and BDS-3 are treated as two independent systems, the root mean square (RMS) of code and carrier phase residuals can be reduced by around 9.3 cm and 0.23 mm, respectively, while the three-dimensional orbit consistency is improved by 6.8%, mainly in the tracking direction. Satellite laser ranging (SLR) shows marginal impacts on IGSO and MEO satellites. However, the SLR residual of C01 shifts −13.2 cm, resulting in a smaller RMS value. In addition, the RMS of linear clock fitting is reduced from 0.050 ns to 0.038 ns for BDS-3 MEO satellites in average.

Particle Swarm Optimization-based co-state initialization for low-thrust minimum-fuel trajectory optimization

In this paper, Particle Swarm Optimization with energy-to-fuel continuation is proposed for initializing the co-state variables for low-thrust minimum-fuel trajectory optimization problems in the circular restricted three-body problem. Particle Swarm Optimization performs a search of the solution space by minimizing the weighted sum of squares of the two-point boundary-value problem final boundary condition residuals for the minimum-energy problem. Next, an energy-to-fuel homotopy is employed to transition the minimum-energy trajectory to a minimum-fuel trajectory, starting from the generated guess. The proposed methodology is applied to two low-thrust transfer problems in the Earth–Moon system: a transfer from a geostationary transfer orbit to an L1 halo orbit, and a transfer from an L2 halo orbit to an L1 halo orbit. The resulting minimum-fuel trajectories are validated with the literature. It is demonstrated that the methodology can successfully generate guesses for the initial co-state variables which converge to a solution for both scenarios. A strategically chosen particle swarm size is shown to improve the efficiency of the methodology. The proposed approach is of simple implementation, can be easily extended to other trajectory optimization problems, facilitates the discovery of multiple candidate trajectories, and does not require a user-provided guess, all of which are advantageous features for the preliminary phase of mission design.

High-Precision Orbit Determination of the Small TJU-1 Satellite Using GPS, GLONASS, and BDS

Tianjin University-1 (TJU-1) is a small low-Earth-orbiting (LEO) meteorological satellite equipped with a multi-GNSS receiver that provides the dual-frequency pseudorange and carrier-phase measurements from GPS, GLONASS, and BDS, including BDS-2 and BDS-3. In this contribution, the quality of GNSS data during the period of 2022/31~2022/37 is assessed, and the precise orbit determination (POD) based on the onboard GNSS data is conducted. Assessing the pseudorange measurement accuracy, it reveals that the GPS and BDS-3 are comparable, followed by BDS-2, and GLONASS the worst. The POD experiments demonstrate that, the orbit precision of all single-GNSS solutions is almost at an equivalent level of 2.2~3.4 cm, in terms of the overlapped orbit consistency, and the GPS-based solution is slightly worse than the BDS-based one and the GLONASS-based one is worse than other two. According to the orbit differences between different solutions, the orbit accuracy of single-GNSS solutions is inferred to be at the level of about 4.7~6.2 cm. With the multi-GNSS fusion, the precision and inferred accuracy of orbits could be improved significantly to 1.7~2.1 cm and about 1.7~3.7 cm, respectively. Furthermore, the root mean square (RMS) of posteriori carrier-phase residuals for all the IGSO/MEO satellites is at the level of 7~15 mm in different single-GNSS and multi-GNSS fusion solutions. These performances have fully demonstrated that, the centimeter-level POD is easily achievable for small LEO satellites by using the multi-GNSS receiver.

Evaluation of Station Timing Errors from Demeaned Arrival Time Delays

ABSTRACT We present an analytical solution for the evaluation of timing errors at seismological stations. The method makes use of differential P- and S-wave arrival time measurements demeaned over a network that recorded a set of densely located seismic events. In this configuration, one can assume coincident P and S ray paths between sources and receivers, and cancel out dependencies associated with absolute event origin times, event locations, and P- or S-wave velocities in the problem. Relative timing errors can be obtained by linear inversion, using only a limited amount of input data: differential P- and S-wave arrival times, and a local VP/VS ratio. By setting at least one reference station in the network, supposed to be devoid of any timing error, one can retrieve reliable timing errors for other stations. We validate the approach against synthetic and real data. We also analyze the sensitivity of results on errors in the input data. Although picking uncertainties do affect the variability of estimates, we also identified a significant bias when an incorrect VP/VS ratio is used. However, this bias can be reduced if one uses the optimal VP/VS value that minimizes the root mean square of travel-time residuals. Application of the method to a collection of manually picked arrival times for the 2002–2003 Tricastin, France, earthquake, swarm allowed us to identify nonstationary timing errors from tenth to tens of seconds during the monitoring campaign.

A three-dimensional Oral health-related quality of life framework for temporomandibular joint disorders: A structural analysis of the Oral Health Impact Profile-14

Objectivesthe objective is to analyze the structure and relationship between the components of the Oral health-related quality of life using the Oral Health Impact Profile-14 (OHIP-14) for patients with temporomandibular joint disorders (TMJD). Methodstwo studies were examinated by means of factor analysis, multidimensional scaling and their spatial display in a cross-sectional design (n = 153). Resultsthree-dimensional solutions with an adequate fit were found: residual distribution, root mean square of residuals (RMSR) = 0.04, Goodness of Fit Index (GFI) = 0.96, and Adjusted Goodness of Fit Index (AGFI) = 0.93; and Stress I = 0.023, respectively. The overall ordinal alpha coefficient was 0.92 (CI=0.90 to 0.94). The ordinal alpha coefficients for functional impact, psychosocial impact and psychological impact factors were 0.87 (CI=0.84 to 0.90), 0.88 (CI=0.84 to 0.91) and 0.78 (CI= 0.72 to 0.83), respectively. Item 9 was the most central, followed by 7 and 11. Items 1 and 2 had minor associations. There was no evidence of differences between centers: edge weight differences (M = 0.226, p = 0.94), global strength invariance test (S = 1.88, p = 0.1), centrality invariance test (p >0.05), edge invariance test (p >0.05). Conclusionsthe links between the dimensions and the items seem to involve psychological components. Oral health-related quality of life emerged as a three-dimensional structure (Functional Impact, Psychosocial Impact and Psychological Impact) of functional and psychosocial elements in which physical, psychological, and social disability were essential, while theoretical functional limitation was least important. Clinical significancethe three-dimensional OHRQoL system for temporomandibular disorders (TMD) is a worthwhile alternative to interpret psychological and psychosocial aspects.

Modeling of the land surface temperature as a function of the soil-adjusted vegetation index

Land surface temperature is a physical-environmental variable that is the target of studies in climatology and heat island phenomena resulting from the urbanization model of the 21st century. It is known that each object has a different thermal capacity, which results in higher or lower temperatures. The modeling of temperature as a function of objects on the land surface can allow understanding between these variables. It can corroborate temperature forecasts with the alteration of objects in an area. The objects on the Earth's surface can be computed with geoprocessing techniques that aim to detail Land Use and Occupation. This paper evaluates the Linear, Exponential, and Sinusoidal models to determine which of these models is more expressive for the study of the land surface temperature as a function of land surface objects. For this purpose, images from the Landsat 8 satellite were used to calculate the Earth's surface temperature and determine the scene's objects. To determine the scene's objects, the Soil Adjusted Vegetation Index (SAVI) was used, which is one of the techniques for obtaining Land Use and Occupation. For model analysis, Akaike's Information Criterion (AIC), Bayesian Information Criterion (BIC), and Sum of Square of Residuals (SSR). According to the AIC, BIC, and SSR criteria, the sinusoidal model presented better performance when compared to the other models. However, there are large variations in SSR between classes, especially for the pasture class, which makes the models not highly explanatory.

Development and validation of convective‐diffusion models to analyse the transient behaviour of a packed bed: Factors influencing aqueous phase adsorption of a bisbiguanide

Chlorhexidine gluconate [CHG] in combination with cetrimide [CET], widely used in various pharmaceutical compositions, is potentially hazardous to aquatic ecosystems. Continuous removal of these compounds using commercial [GAC] and functionalized activated carbons [FACs] in a packed bed column, is reported. Transient forms of convective diffusion models are developed with depletion terms being represented by the first and second-order kinetics. Performance of the packed bed is analysed with varying bed height, flow rate, initial concentrations, and estimated Damköhler numbers. On the basis of minimum value of sum of the squares of residuals [SSQ], it is evident that the second-order model fits better for the removal of CHG and CET by FACHF and FACNH3, while the first-order model fits better for GAC. Damköhler number increases with a decrease in flow rate for both the compounds. Ratio of Damköhler numbers does not change with flow rate. Irrespective of flow rates, for GAC: Da2(CHG):Da2(CET) ≈ 0.0526; for FACHF: Da2(CHG):Da2(CET)=0.30; for FACNH3: Da2(CHG):Da2(CET) = 0.0076. Changes in the breakthrough volume (nBT,max), causing the analytes to migrate from the front of the adsorbent bed into the back, are in the order: for CHG: nBT,max(FACHF) >nBT,max (FACNH3) >nBT,max (GAC); for CET: nBT,max (FAC- NH3) >nBT,max (FACHF) >nBT,max (GAC).

Validation of Machine Learning Prediction Models

We address the estimation of the Integrated Squared Error (ISE) of a predictor $\eta (x)$ of an unknown function f learned using data acquired on a given design ${\mathbf{X}_{n}}$. We consider ISE estimators that are weighted averages of the residuals of the predictor $\eta (x)$ on a set of selected points ${\mathbf{Z}_{m}}$. We show that, under a stochastic model for f, minimisation of the mean squared error of these ISE estimators is equivalent to minimisation of a Maximum Mean Discrepancy (MMD) for a non-stationary kernel that is adapted to the geometry of ${\mathbf{X}_{n}}$. Sequential Bayesian quadrature then yields sequences of nested validation designs that minimise, at each step of the construction, the relevant MMD. The optimal ISE estimate can be written in terms of the integral of a linear reconstruction, for the assumed model, of the square of the interpolator residuals over the domain of f. We present an extensive set of numerical experiments which demonstrate the good performance and robustness of the proposed solution. Moreover, we show that the validation designs obtained are space-filling continuations of ${\mathbf{X}_{n}}$, and that correct weighting of the observed interpolator residuals is more important than the precise configuration ${\mathbf{Z}_{m}}$ of the points at which they are observed.

A New Optimal Subset Selection Method of Partial Ambiguity Resolution for Precise Point Positioning

Rapid and accurate ambiguity resolution is the core of high-precision precise point positioning (PPP) data processing. However, the ambiguity parameters in PPP observation models are easily affected by atmospheric residual and gross errors, which lead to the probability of successfully fixing decreases and computational burden increases in full ambiguity resolution. Therefore, an increasing number of partial ambiguity resolution (PAR) strategies have been proposed. The selection of the optimal subset of PAR is crucial in this method. The traditional optimal subset selection method of PAR commonly leads to a single judgment criterion and weakened geometric configuration strength because the satellites with low elevation angles are often easily eliminated during the optimal subset selection. In this paper, a multi-factor constrained optimal subset selection method for PAR was proposed, which incorporates the ambiguity variance, the ambiguity dilution of precision (ADOP), satellite position dilution of precision (PDOP) and ratio test values. In order to verify the feasibility of the proposed optimal subset selection method, PAR tests under two schemes were performed for GPS/Galileo based on the static observation data of 15 Multi-GNSS Experiment (MGEX) tracking stations. The results show that, compared with the ambiguity variance sorting method, the proposed subset selection method can further improve the accuracy of the coordinate solution and the strength of geometric figure positioning. The average root mean square of the coordinate residuals is found to decrease by about 12.90%, 6.83% and 9.39% in the eastern, northern and vertical directions, respectively. The increase in the fixed epoch rate ranged from 0.87% to 33.33%, with an average of about 8.71%.

Inter-Satellite Single-Difference Ionospheric Delay Interpolation Model for PPP-RTK and Its Positioning Performance Verification

In PPP-RTK, obtaining accurate atmospheric delay information for the user through interpolation is one of the keys to achieving high-precision real-time positioning. The ionospheric delay that is extracted by a reference network based on uncalibrated phase delay (UPD) products is often difficult to separate from errors such as receiver code hardware delay and UPD reference error. Inter-satellite single-difference (SD) ionospheric delay information is typically provided to the user. This paper proposes an interpolation model that uses the atmospheric delay coefficient to represent the SD ionospheric delay, based on the mean position of the ionospheric pierce point (IPP) of each satellite pair and the center position of the network, which is called the differenced surface model (DSM). We chose four scenarios to compare the interpolation accuracy of the proposed model with the inverse distance-based linear interpolation method (DIM) and USM based on the difference between the longitude and latitude of the reference and ionospheric pierce point (IPP) of every satellite (here, we call it USM for short). The four scenarios involve a medium-scale reference network with an average distance to the reference station of 41 km, a large-scale reference network with an average distance to the reference station of 98 km, and out-of-network users, and a network with a common minimum of three reference stations. The results show that the root mean square (RMS) of the SD residuals of ionospheric delay for DSM were 1.4, 3.2, 2.2, and 1.4 cm, respectively, for the four scenarios that were considered, which are slightly better delay values than those that were achieved using DIM and USM. For the scenario with three reference stations, the interpolation accuracies of DIM and DSM were no different from those for four reference stations, indicating that the server can still try to provide ionospheric correction service under the condition of fewer reference stations. In contrast, USM could not provide service because it lacked the sufficient number of reference stations. DSM was used as the ionospheric delay interpolation model to analyze GPS and Galileo dual-system PPP-RTK positioning performance. In addition, the atmospheric parameter constraint method of users was used in PPP-RTK in reference networks of different scales. For the 41-km and 98-km reference networks, the time to first fix (TTFF) were 14.5 s and 33.1 s, respectively, and the mean RMS values for the east (E), north (N), and up (U) directions were 0.80, 0.93, and 2.72 cm, respectively, and 1.0, 1.1, and 4.0 cm, respectively, for a period of 5 min after convergence. The fixing rate and positioning accuracy of DSM during the 5-min period were better than those of DIM when the same empirical model was used to determine the mean square error of atmospheric delay.