A real-time data processing method based on twice weighted improved Kalman filter algorithm

BeiDou Navigation Satellite System (BDS) is composed of satellites in geostationary Earth orbit (GEO), medium Earth orbit (MEO) and inclined geosynchronous orbit (IGSO). However, the orbit determination of geostationary Earth orbits and of geosynchronous orbits (GSO) with small inclination angle and small eccentricity is a challenging task that is addressed in this paper using Extended Kalman Filter (EKF). The satellite positions were predicted in Earth-centred inertial (ECI) reference frame when propagated through Keplerian model and perturbation force model for different values of right ascension of ascending node (RAAN). Root mean square (RMS) errors of 9.61 cm, 6.73 cm and 11.46 cm were observed in ECI X, Y and Z satellite position coordinates of GSO respectively, whereas, the RMS errors for GEO satellite were 8.89 cm, 7.92 cm, and 0.93 cm respectively in ECI X, Y and Z coordinates; for perturbation force model with maximum value of RAAN when compared with dynamic orbit determination model. Kolmogorov-Smirnov test for EKF reported a p-value > 0.05, indicating a good fit of perturbation force model for orbit propagation. Orbit determination using EKF with perturbation force model were compared with that using EKF with Kepler's model. Wilcoxon Rank Sum test was used to compare the residuals from EKF algorithm through Kepler's model and perturbation force model. EKF with Perturbation force model showed improvement in predicting the satellite positions as compared to Kepler's model. EKF with Perturbation force model was further applied to International GNSS Service (IGS) station data and kilometre level accuracy was achieved. RMS errors of 0.75 km, 2.53 km and 1.91 km were observed in ECI X, Y and Z satellite position coordinates of GSO, respectively, whereas, the RMS errors for GEO satellite were 3.89 km, 4.20 km and 6.66 km respectively in ECI X, Y and Z coordinates for perturbation force model.

Acoustic emission (AE) detection, as a non-electrical detection method, is very suitable for effective fault detection of power equipment with a strong electromagnetic field. However, the AE signal collected at industrial sites often contains a lot of interference noise, affecting the analysis and prediction of faults. In this study, a wavelet denoising method based on a new threshold function is proposed, to achieve a noise reduction in the low signal-to-noise ratio (SNR) signals. Simulation experiment results show that the proposed threshold function not only overcomes the shortcomings of the discontinuous hard threshold function, but also solves the constant deviation of the soft threshold function. What’s more, the proposed function achieves a good adaptability. When SNR = 10 dB: The SNR of the new threshold is 20.6622, and the RMSE is 0.0026. The SNR of the hard threshold is 20.2246 and the RMSE is 0.0027, compared with the traditional hard threshold method, the SNR of the new threshold is increased by 2.16% and the root mean square error (RMSE) is reduced by 3.7%; the SNR of the soft threshold is 15.5656, and the RMSE is 0.0047, compared with the traditional soft threshold method, the new threshold has a 32.74% increase in SNR and a 40.43% reduction in RMSE. When SNR = -10 dB: The SNR of the new threshold is 4.2602, and the RMSE is 0.0172. The SNR of the hard threshold is 3.8558 and the RMSE is 0.0182, compared with the traditional hard threshold method, the SNR of the new threshold is increased by 10.49% and the RMSE is reduced by 5.49%; the SNR of the soft threshold is 2.1625, and the RMSE is 0.0212, compared with the soft threshold method, SNR is improved by 97% and RMSE is reduced by 18.87%. Performance analyses have proved that the improved wavelet denoising method can obtain a good noise reduction effect. It is very helpful for AE signal analysis with the generally low SNR, which can improve the accuracy of failure identification in subsequent acts.

Tracking of wireless nodes such as robots in wireless sensor network (WSN) has been widely studied in literature. Most of these studies are based on the Kalman filter (KF) for linear models corrupted by Gaussian noise, the extended Kalman filter (EKF) for non linear models and the particles filter (PF) which does no require the model o be linear nor the noise be Gaussian. In his paper, we present a comparative study of mobile target node (TN) racking via the KF, EKF and PF based on the received power of the signal. A constant velocity model is considered for the motion of TN, depicting an indoor environment. The performance of the filters are compared in terms of the root mean square error (RMSE). Extensive simulations are performed to evaluate the performance of the discussed filters.

The state estimation problem for the nonlinear system is a most important issue for industrial control application. It is also preferable that the estimation technique should be accurate and easily implementable. In the present work, we compare the performance of two estimation algorithm for state estimation of the Twin Rotor MIMO system (TRMS). Here, the Kalman filter (KF)and extended Kalman filter (EKF) are used for realtime state estimation for the system. It is observed that in both simulation and experimental result, that the performance of EKF is better than the KF in terms of robustness and root mean square error (RMSE) in state estimation.

This paper compares two different recursive tracking techniques for precisely localizing a mobile vehicle in an indoor harsh industrial environment. An extended Kalman filter (EKF) and unscented Kalman filter (UKF), the corresponding algorithms and mathematical models are presented and analysed. Experimental range measurements generated from local positioning radar system are used to test the performance of these algorithms with respect to position and velocity root mean square errors. True and estimated trajectories of the mobile vehicle with associated means and error covariances are illustrated with the number of samples required in each case. Results obtained show that UKF outer performs EKF with respect to positioning accuracy and root mean square error. Both filters show comparable computational complexity with more robustness obtained by applying UKF for non linear estimation since there are no linearization errors as in the case of EKF.

The aim of the present research is to calculate the Relative Accuracy (R.A) and Root Mean Square Error (RMSE) for the location of Technical Institute, Mosul in Nineveh Governorate, Iraq. at the northern part of Mosul city by using surveying Instruments (GPS 1230 , Total Station 06 & Level NAK2), to calculate the coordinates and elevations of (16) Ground Control Points which covered the study area. The field data, remote sensing techniques and GIS software (Arc Map GIS10.3) are used to check the digital map of study area that helped the administrator to take the correct decision, determining positions of the future expansionist for buildings, roads, measuring lengths, areas and there coordinates, directions as well as the total costs estimation needed for any development from the final map. The gradient percentage between the highest and lowest points is calculated to be (9.954%). Relative accuracy for the based measured points (R.A) = 1:60000), and Root Mean Square error in Easting coordinates (RMSE) = ± 0.102m, in Northing coordinates (RMSN) = ± 0.096m, the Root Mean Square for the resultant (RMSR) = ± 0.099m, and Root Mean Square for Elevations ( RMSRL)= ± 0.005m. We Matched and correlated (10) measured points from study area with their graphic data positions on the aerial photo, determining the offsets for position and elevations as follows: (RMSE = ± 0.076m), (RMSN = ± 0.125m), (RMSR = ± 0.088m), and (Root Mean Square for Elevations = RMSRL= ± 0.045m). This research enable us after checking the results to produce new maps from Aerial photograph or Digital Image to use it for land uses and future extensions in whole study area.

Heart health becomes more important to monitor, which is analyzed using Electrocardiography (ECG) signal. ECG signal can be corrupted by different forms of noises, when it is acquired. These noises mislead the ECG analysis and reduce the chance of accurate identification of the ECG signal features. To remove these noises, various adaptive and non adaptive filter techniques are used. This paper proposes a methodology to remove power line interference noise, using various adaptive techniques, like least mean square (LMS) algorithm, normalized least mean square (NLMS) algorithm and Kalman least mean square (KLMS) algorithm. All three filters results are compared based on evaluation metrics like signal to noise ratio (SNR), Root mean square error and convergence. KLMS algorithm is analyzed, as it overcomes the drawback of LMS and NLMS algorithm. And the advantage of KLMS algorithm is that; it is step size independent, and has least root mean square error (RMSE) and converges quickly. The results have been concluded with the MIT-BIH arrhythmia data base and simulated using MATLAB software. The algorithms are implemented in TMS320C6713 DSP kit using Code composer studio v3. TMS320C6713 is use Very long instruction word (VLIW), which helps to compute numerically complex algorithms.

In order to reduce the influence of historical measurement data errors in the process of vehicle state estimation and improve the accuracy of the vehicle state estimation, a limited memory random weighted extended Kalman filter (LMRWEKF) algorithm is proposed. Firstly, a 3-DOF nonlinear vehicle dynamics model is established. Secondly, the limited memory extended Kalman filter is formed by fusing the limited memory filter and the extended Kalman filter. Then, according to the random weighting theory, the weighting coefficients that obey Dirichlet distribution are introduced to further improve the filtering estimation accuracy. Finally, a virtual test based on the ADAMS/CAR is used for the experimental verification. The results show that the error in the longitudinal velocity and the yaw rate is small, especially in the mean value of estimation error of side slip angle which is different in just 0.015 degrees between the virtual test and the simulation result. And also, the results compared with traditional methods indicate that the proposed LMRWEKF algorithm can solve the problem of vehicle state estimation with the performance of noise fluctuation suppression and higher estimation accuracy. The mean absolute error (MAE) and root mean square error (RMSE) are considered to verify the estimation accuracy of the proposed algorithm. And the comparison results indicate that the estimation accuracy of the LMRWEKF algorithm is significantly higher than those of the EKF and DEKF methods.

Accurate flood water level prediction are essential for reliable flood forecasting modelling. Although back propagation neural network (BPN) offer advantages for flood water level prediction, nonlinearity due to input parameters are the major issue to this modelling. A novel Extended Kalman Filter (EKF) optimization algorithm was employed in this study to overcome the nonlinearity problem and come out with an optimal ANN for the prediction of flood water level 3 hours in advance. The inputs used in the algorithm were current values of rainfall at the flood location and three upstream locations of river water levels. The BPN model was trained and tested successfully with Root Mean Square Error (RMSE) and loss function (V) close to zero.

Kalman filter based on a fractional discrete-time stochastic augmented CoVid-19 model

The concept of orthometric heights system determination plays a major key role in geodesy, and it has broad applications in various fields and activities. In geodesy, one significant quantity is the orthometric height, the height above or below the geoid along the gravity plumbline. Conventionally, the orthometric height is determined by gravimetry and levelling techniques. However, the aforementioned techniques has its own demerits. Thus, the error is accumulated with the increase of the propagation measurement line, it is difficult to convert two separated points which is located in two continents or islands separated by sea. These techniques are tedious, time consuming and expensive. In order to resolve this challenge, many researchers resort to various techniques and approaches of obtaining orthometric heights for an area using various mathematical models. It is in this quest that, this study seek to estimate orthometric heights of a mine by utilizing plausible alternative techniques based on artificial neural networks (ANN), multivariate adaptive regression splines (MARS), polynomial regression models and multiple linear regression (MLR). The working efficiency and performance of each model has been assessed based on statistical indicators of Mean (M), Mean Square Error (MSE), Root Mean Square Error (RMSE), Mean Bias Error (MBE), Mean Absolute Error (MAE), Standard Deviation (SD), Correlation coefficient (R), Correction of determination (R2), and Signal to Noise Ratio (SNR). The statistical findings reveal that all the models produce satisfactory results in estimating the orthometric heights in the mine. MARS and ANN models compare to the MLR and polynomial models achieved higher results in terms of accuracy with mean and standard deviation of -0.000001888 m, +2.24736 m, and +0.005835 m and 0.095063 m respectively. This study will create the opportunity for geospatial practitioners to recognize the significant of ANN, MARS, MLR, and Polynomial model in solving some of the problems in geoscientific community.

In this study, Adaptive Network-Based Fuzzy Inference System (ANFIS) is presented with sensor data fusion approach to estimate satellite attitude. The active sensors are sun and earth sensors. Satellite attitude dynamic, including attitude quaternion and angular velocities are estimated simultaneously utilizing the measured values by the sensors. The Extended Kalman Filter (EKF) is employed to verify and evaluate the efficiency of the presented method. Additionally, the neural networks with Radial Basis Function (RBF) and Multi-Layer Perceptron (MLP) are also designed to prove the superiority of the proposed ANFIS network among the smart methods of sensor data fusion for satellite attitude estimation. Root Mean Square Error (RMSE) as a numerical criterion and graphical analysis of residues are utilized to evaluate the simulation results. The simulations confirm that the obtained estimations from ANFIS network have more accuracy in modeling of nonlinear complex systems compared to EKF, MLP and RBF networks. In general, using intelligent data fusion, especially ANFIS, reduces attitude estimation error and time in comparison to the classical EKF method.

OPINION article Front. Physiol., 22 January 2014Sec. Integrative Physiology Volume 4 - 2013 | https://doi.org/10.3389/fphys.2013.00403

Single User, Transmission Control Protocol Downstream Throughput (TCPDST) probability models in an IEEE802.11b WLAN have been developed, validated and evaluated for performance. Measurement of single user TCPDST were taken using Tamosoft throughput test while that of signal to noise ratio (SNR) were taken using inSSIDer 2.1. The Tamosoft throughput tests were conducted using different quality of service (QoS) traffic. These QoS traffic (which were sent through an infrastructure based network) correspond to different wireless multimedia tags. Measurements were taken in free space, small offices and open corridor environments. By assuming a normal distribution, single user TCPDST Cumulative distribution function (CDF) probability models were developed for different signal categories namely: (i) all the SNR considered, (ii) strong signals only, (iii) grey signals only and (iv) weak signals only. The models were validated and their performances evaluated using root mean square (RMS) errors. RMS errors were computed by comparing model predicted values with validation data. The RMS errors for single user CDF all signals model was 0.1466%. RMS errors for strong signals models, grey signals model and weak signals model respectively were 0.1466%, 0.6756% and 0.1233% indicating acceptable performances. All signals, strong signals, grey signals and weak signals CDF probability models predicted probabilities of obtaining TCPDST values greater than 5Mbps as 74.79%, 90.55%, 13.00% and 4.77% respectively while probabilities of obtaining TCPDST values less than 2Mbps were predicted as 4.91%, 0.00%, 18.98% and 52.41% respectively. These probability models will provide additional useful information needed to design efficient distributed data networks.
 Keywords: Throughput, TCP, WLAN, probability models

Full text Figures and data Side by side Abstract Editor's evaluation Introduction Materials and methods Results Discussion Data availability References Decision letter Author response Article and author information Metrics Abstract Background: Oxygen uptake (VO2) is one of the most important measures of fitness and critical vital sign. Cardiopulmonary exercise testing (CPET) is a valuable method of assessing fitness in sport and clinical settings. There is a lack of large studies on athletic populations to predict VO2max using somatic or submaximal CPET variables. Thus, this study aimed to: (1) derive prediction models for maximal VO2 (VO2max) based on submaximal exercise variables at anaerobic threshold (AT) or respiratory compensation point (RCP) or only somatic and (2) internally validate provided equations. Methods: Four thousand four hundred twenty-four male endurance athletes (EA) underwent maximal symptom-limited CPET on a treadmill (n=3330) or cycle ergometer (n=1094). The cohort was randomly divided between: variables selection (nrunners = 1998; ncyclist = 656), model building (nrunners = 666; ncyclist = 219), and validation (nrunners = 666; ncyclist = 219). Random forest was used to select the most significant variables. Models were derived and internally validated with multiple linear regression. Results: Runners were 36.24±8.45 years; BMI = 23.94 ± 2.43 kg·m−2; VO2max=53.81±6.67 mL·min−1·kg−1. Cyclists were 37.33±9.13 years; BMI = 24.34 ± 2.63 kg·m−2; VO2max=51.74±7.99 mL·min−1·kg−1. VO2 at AT and RCP were the most contributing variables to exercise equations. Body mass and body fat had the highest impact on the somatic equation. Model performance for VO2max based on variables at AT was R2=0.81, at RCP was R2=0.91, at AT and RCP was R2=0.91 and for somatic-only was R2=0.43. Conclusions: Derived prediction models were highly accurate and fairly replicable. Formulae allow for precise estimation of VO2max based on submaximal exercise performance or somatic variables. Presented models are applicable for sport and clinical settling. They are a valuable supplementary method for fitness practitioners to adjust individualised training recommendations. Funding: No external funding was received for this work. Editor's evaluation The authors have established new formulas to predict maximum oxygen uptake for cyclists and runners based on submaximal exercise testing and anthropometric characteristics. This is an important study with a large and comprehensive dataset, which may be helpful for many exercise labs. The work is convincing, using appropriate and validated methodology in line with the current state-of-the-art, as shown by references to common exercise books. https://doi.org/10.7554/eLife.86291.sa0 Decision letter Reviews on Sciety eLife's review process Introduction The oxygen uptake (VO2) is considered an important metric in assessing cardiorespiratory fitness, health status, or endurance performance potential (Guazzi et al., 2012). With the application of standardised procedures and interpretation protocols, during graded exercise tests (GXT), the (maximal oxygen uptake) VO2max can be established (Bentley et al., 2007). GXT is the most widely used assessment to examine the dynamic relationship between exercise and integrated physiological systems (Albouaini et al., 2007; Bentley et al., 2007). The information from GXT during cardiopulmonary exercise testing (CPET) can be applied across the spectrum of sport performance, occupational safety screening, research, and clinical diagnostics (Guazzi et al., 2017). VO2 max is often used as a boundary between severe and extreme intensity domains and by definition requires maximal effort from the tested subject (Gaesser and Poole, 1996). However, it is not always recommended or possible to undertake a test to exhaustion (Guazzi et al., 2012). For the athletes, the proximity of competition or injury history can allow submaximal testing, but not testing to exhaustion (Sassi et al., 2006). Testing that requires maximal effort may be disruptive to the training process or interfere with race performance (Coutts et al., 2007; Lamberts et al., 2011). Due to practical constraints, tests to exhaustion or peak-power-output tests are often performed only two or three times a year (Coutts et al., 2007). However, VO2 values are widely used in sport science and the decision-making process (Mann et al., 2013). VO2 is widely considered one of the major endurance performance determinants (Joyner and Coyle, 2008). Using VO2max to guide the selection process, prescribing training intensity, assessing training adaptations, or predicting race times is a common practice in high-performance sports (Bassett and Howley, 2000; Bentley et al., 2007; Hawley and Noakes, 1992; Noakes et al., 1990). VO2max is also one of the critical vital signs coordinating the function of the cardiovascular, respiratory, and muscular systems, it is an indicator of overall body health status (Kaminsky et al., 2017). Quantifying VO2max provides additional input regarding clinical decision-making, risk stratification, evaluation of therapy, and physical activity guidelines (Guazzi et al., 2012). For patients undertaking a test to exhaustion is rarely needed or possible due to health restraints or cardiac risk (Guazzi et al., 2016). For many years researchers have studied indirect methods of estimating VO2max(Sartor et al., 2013). Protocols such as the Astrand-Ryhming Test, Six-Minute Walk Test, or YMCA Step Test have been established and validated (Astrand and Ryhming, 1954; Beutner et al., 2015; Carey, 2022; Jalili et al., 2018). Moreover, estimation of the VO2 and heart rate (HR) values below the ventilatory threshold can be based on cardiorespiratory kinetics assessment using randomised changes in the work rate known as a pseudo-random binary sequences testing (Hoffmann et al., 2022). However, with the development of technology, the accessibility of laboratory testing and mobile testing improved (Montoye et al., 2020; Pritchard et al., 2021). Therefore, new opportunities to develop more precise yet simple and accessible methods and models to assess VO2max occur (Jurov et al., 2023). This appears to be especially important considering the low prediction accuracy of most of the VO2max formulae that were validated in our previous study (Wiecha et al., 2023). Recently, we have been observing the development of prediction methods with the usage of machine learning (ML) and artificial intelligence (AI) (Ashfaq et al., 2022). Both ML and AI are used in sport science as forecasting and decision-making support tools (Abut and Akay, 2015; Bobowik and Wiszomirska, 2022; Chmait and Westerbeek, 2021; Hammes et al., 2022; Rossi et al., 2021). There is growing evidence that VO2max prediction based on ML models, especially support vector ML and artificial neural network models, exhibits more robust and accurate results compared to MLR only (Abut and Akay, 2015; Ashfaq et al., 2022). Therefore, in this research, with the support of ML, we look for algorithms and prediction patterns that allow us to use values obtained during submaximal CPET and somatic measurements to estimate maximal VO2max values in male runners and cyclists. We stipulate that prediction models allow for accurate calculation of VO2max based on somatic or submaximal CPET variables. Materials and methods We have applied the development and validation of the prediction TRIPOD guidelines to conduct the study (see Supplementary Material 1TRIPOD Checklist for Prediction Model Development and Validation) (Collins et al., 2015). The study is based on retrospective data analysis from the CPET registry collected from 2013 to 2021 at the medical clinic (Sportslab, Warsaw, Poland). All CPET have been performed at the individual request of participants, as a part of regular training monitoring or performance assessment. Ethical approval Request a detailed protocol The Institutional Review Board of the Bioethical Committee at the Medical University of Warsaw (AKBE/32/2021) has approved the study protocol. The regulations of the Declaration of Helsinki were met during all parts of the study. Each study participant delivered written consent to undergo CPET and participate in the study. Derivation cohort Request a detailed protocol We selected the cohort with the use of rigorous exclusion/inclusion criteria. Due to the insufficient number of women in our database and the number of potential variables in the regression models for adequate power, we had to limit ourselves to conduct analysis in the male population only (Martens and Logan, 2021). Out of 6439 healthy, adult male cyclists and long-distance runners that undergone CPET, 4423 met the criteria as further: (1) age ≥18 years, (2) declared regular cycling or running training for ≥3 months, (3) had no extreme outliers ≤ or ≥±3 standard deviations (SD) from mean for all of the testing variables (beyond ≥±3 SD in VO2max), (4) lack of any injury, medical condition, or addiction in medical history that may affect exercise capacity, (5) not taking any medications with a modifying effect on exercise capacity, (6) maximum exertion achieved during CPET. We defined the maximum exertion in CPET as the fulfilment of the minimum six of the following criteria: (1) respiratory exchange ratio (RER) ≥1.10, (2) present VO2 plateau (growth <100 mL·min–1 in VO2 despite increased running speed or cycling power), (3) respiratory frequency (fR) ≥45 breaths·min–1, (4) declared subjective exertion intensity during CPET ≥18 in the Borg scale (Borg, 1970), (5) blood lactate concentration [La-]b ≥8 mmol·L–1, (6) growth in speed/power ≥10% of respiratory compensation point (RCP) values after exceeding the RCP, (7) peak heart rate (HRpeak) ≥15 beats·min–1 below predicted maximal heart rate (HRmax) (Lach et al., 2021). Participants’ selection procedure has been shown in Figure 1. Figure 1 Download asset Open asset Flowchart of the preliminary inclusion and exclusion process. Abbreviations: EA, endurance athlete; CPET, cardiopulmonary exercise testing; SD, standard deviation; TE, treadmill; RER, respiratory exchange ratio; VO2, oxygen uptake (mL·min−1·kg−1); [La−]b, lactate concentration (mmol·L−1); fR, breathing frequency (breaths·min−1); RCP, respiratory compensation point; HRpeak, peak heart rate (beats·min−1); HRmax, maximal heart rate (bpm). At both stages of the selection, some participants met several (>1) exclusion criteria. Somatic measurements and CPET protocols Request a detailed protocol Body mass was measured with a body composition (BC) analyser (Tanita, MC 718, Japan) with the multifrequency of 5 kHz/50 kHz/250 kHz via the bioimpedance analysis and normal testing mode. The participants’ skin was cleaned with alcohol before placing the electrodes on the skin. Prior to the test, the participants received instructions to refrain from exercising for 2 hr, consume a light meal rich in carbohydrates 2–3 hr beforehand, and maintain hydration by drinking isotonic beverages. Additionally, they were advised to abstain from medications, caffeine, and cigarettes on the day of the test. Running CPET (TE) was performed on a mechanical treadmill (h/p/Cosmos Quasar, Germany). Cycling CPET (CE) was performed on Cyclus-2 (RBM elektronik-automation GmbH, Leipzig, Germany). Hans Rudolph V2 mask (Hans Rudolph, Inc, Shawnee, KS, USA), breath-by-breath method with Cosmed Quark CPET gas exchange analysing device (Cosmed Srl, Rome, Italy), and Quark PFT Suite to Omnia 1.6 software were utilised. The gas analyser device was regularly calibrated with the reference gas (16% O2; 5% CO2) in accordance with the manufacturer’s instructions (Airgas USA, LLC, Plumsteadville, PA, USA). From 2013 to 2021, three Cosmed Quark CPET units were used. HR was measured with the Cosmed torso belt (Cosmed srl, Rome, Italy). [La-]b was measured via enzymatic-amperometric electrochemical technique with Super GL2 analyser (Müller Gerätebau GmbH, Freital, Germany). The [La-]b analyser was regularly calibrated before each measurement series. The 40 m2 indoor, air-conditioned laboratory with 20–22°C temperature and 40–60% humidity, and 100 m ASL provided the same conditions for all BC and CPET. Each CPET began with a 5 min personalised warm-up (walk or easy jog with ‘conversational’ intensity for running, easy pedalling with ‘conversational’ intensity for cycling). Then after the preparation (about 5 min), the continuous progressive step test was conducted. Due to the population diversity (training status), the running test speed started from 7 to 12 km·hr–1 with a 1% treadmill incline. The choice of initial starting speed was determined by the interview and sports results achieved. For example, those running less than 60 min at a distance of 10 km started the test at 7 km/hr, while those running 10 km for less than 35 min started the test at an initial speed of 12 km/hr. The pace increased by 1 km·hr–1 every 2 min with no change in incline. The cycling test began at 60–150 W, depending on the athletes training status. The power increased by 20–30 W every 2 min. It was recommended to maintain a constant cadence of 80–90 (repetition·min–1) during the test. The tests were terminated due to exhaustion: volitional inability to continue the activity or/and VO2 and HR plateau with increasing load or/and observed disturbance of coordination in running or/and inability to maintain the set cadence. Due to the graded protocol used, the cycling power and running speed values have been calculated as a function of time to better reflect the actual level for the test moment being determined (Kuipers et al., 1985). Before the test, after every step, and 3 min after the termination of the effort technician took a 20 µL blood sample from a fingertip. Samples were collected during the test without interrupting the effort. The samples were taken from the initial puncture. The first blood drop was collected into the swab and the second blood drop was drawn for further analysis into the capillary. VO2max was recorded as the highest value (15 s intervals) before the termination of the test. HRmax was recorded as the highest value obtained at the end of the test, without averaging. The anaerobic threshold (AT) was established with the following criteria: (1) common start of VE/VO2 and VE/VCO2 curves, (2) end-tidal partial pressure of oxygen raised constantly with the end-tidal partial pressure of carbon dioxide (Beaver et al., 1986). The was established with the following criteria: (1) PetCO2 must decrease after reaching maximal amount, (2) the presence of fast nonlinear growth in VE (second deflection), (3) the VE/VCO2 ratio achieved minimum and started to rise, and (4) a nonlinear increase in VCO2 versus VO2 (lack of linearity) (Beaver et al., 1986). The [La-]b was estimated for AT and RCP in relation to power or speed (Wiecha et al., 2022). Data analysis Request a detailed protocol Our comprehensive ML approach enables the evaluation of each formula by preliminary variables precision (at the stage of selection), then accuracy (during the model’s building) and recall (in internal validation). Individual CPET results were saved into the Excel file (Microsoft Corporation, Redmond, WA, USA) and a custom-made script was used to generate the database in Excel (Python programming). Further, mean, SD, and 95% confidence intervals (CI) were calculated. The normality of the distribution of the data was examined using the Shapiro-Wilk test and intergroup differences were calculated using the Student’s t-test for independent variables. Three-step variable selection procedures based on random forests were applied using the R package VSURF in RStudio software (R Core Team, Vienna, Austria; version 3.6.4) (Genuer et al., 2016). For each level of measurement (AT, RCP) and their combination (AT+RCP), significant variables were identified separately. The first step was dedicated to eliminate irrelevant variables from the dataset. Second step aimed to select all variables related to the response for interpretation purposes. The third step refined the selection by eliminating redundancy in the set of variables selected by the second step, for prediction purposes (Genuer et al., 2017). Each time for variables selection, the anthropometric variables as in Tables 1–2 and the CPET parameters given in Tables 3–4 from a specific level of measurement (AT; RCP) and their combinations were visible. Table 1 Basic anthropometric characteristics for runners. Variable (unit)Derivation group n=1998Testing group n=666Validation group n=666MeanCISDMeanCISDMeanCISDAge (years)36.235.6–36.98.4535.935.5–36.38.0535.534.9–36.28.14Height (cm)180.0179.6–180.56.04179.4179.1–179.76.13179.7179.2–180.26.61BM (kg)77.777.0–78.49.3577.777.3–78.19.2977.977.1–78.610.1BMI (kg·m–2)23.923.8–24.12.4324.124.0–24.22.4124.123.9–24.32.56BF (%)15.415.1–15.74.5515.515.3–15.74.5215.415.1–15.84.55FM (kg)12.211.9–12.64.6812.312.1–12.54.6512.311.9–12.74.92FFM (kg)65.565.0–66.06.4365.465.1–65.76.3165.665.1–66.16.86 BM, body mass; BMI, body mass index; BF, body fat; FM, fat mass; FFM, fat-free mass; CI, 95% confidence interval; SD, standard deviation. Table 2 Basic anthropometric characteristics for cyclists. Variable (unit)Derivation group n=656Testing group n=219Validation group n=219MeanCISDMeanCISDMeanCISDAge (years)37.336.6–38.09.1337.135.9–38.49.5037.636.5–38.88.46Height (cm)179.9179.4–180.46.27180.1179.2–181.06.96180.2179.4–181.06.13BM (kg)78.878.1–79.69.8079.177.7–80.510.479.878.4–81.310.9BMI (kg·m–2)24.324.1–24.62.6324.424.0–24.72.8024.624.2–25.02.96BF (%)16.415.7–17.14.9916.115.7–16.54.8116.215.5–16.84.87FM (kg)13.312.6–14.15.6613.012.6–13.45.2713.312.5–14.05.85FFM (kg)65.864.9–66.66.2565.865.4–66.36.0666.665.7–67.46.58 BM, body mass; BMI, body mass index; BF, body fat; FM, fat mass; FFM, fat-free mass; CI, 95% confidence interval; SD, standard deviation. Table 3 Cardiopulmonary exercise testing (CPET) characteristics for runners. Variable (unit)Derivation group n=1998Testing group n=666Validation group n=666MeanCISDMeanCISDMeanCISDrVO2AT (mL·min–1·kg–1)38.438.1–38.85.0138.538.3–38.74.8838.137.7–38.55.16RERAT0.870.86–0.870.040.870.86–0.870.040.870.86–0.870.04HRAT (beats·min–1)151.5150.8–152.310.3151.0150.6–151.510.8152.0151.2–152.810.8VEAT (L·min–1)79.178.1–80.012.278.377.8–78.912.077.276.3–78.212.0SPEEDAT (km·h–1)11.010.9–11.11.4511.011.0–11.11.3610.910.8–11.01.42LAAT (mmol·L–1)2.082.02–2.140.631.801.76–1.830.622.352.27–2.420.72rVO2RCP (mL·min–1·kg–1)47.547.0–48.05.8847.747.4–48.06.1547.346.8–47.86.16RERRCP1.001.00–1.000.041.001.00–1.000.041.001.00–1.000.03HRRCP (beats·min–1)173.4172.7–174.19.21173.2172.8–173.69.30174.3173.5–175.09.50VERCP (L·min–1)114.7113.5–116.015.9113.9113.1–114.616.7112.7111.4–114.016.2SPEEDRCP (km·h–1)14.013.9–14.11.7714.114.0–14.11.7013.913.8–14.11.75LARCP (mmol·L–1)4.724.63–4.821.044.404.34–4.451.044.814.69–4.931.14rVO2max (mL·min–1·kg–1)53.853.3–54.36.6754.354.0–54.66.9553.853.3–54.37.09 CI, 95% confidence interval; SD, standard deviation; rVO2AT, oxygen uptake at anaerobic threshold relative to body mass; RERAT, respiratory exchange ratio at anaerobic threshold; HRAT, heart rate at anaerobic threshold; VEAT, pulmonary ventilation at anaerobic threshold; SPEEDAT, velocity at anaerobic threshold; LAAT, blood lactate concentration at anaerobic threshold; rVO2RCP, oxygen uptake at respiratory compensation point relative to body mass; RERRCP, respiratory exchange ratio at respiratory compensation point; HRRCP, heart rate at respiratory compensation point; VERCP, pulmonary ventilation at respiratory compensation point; SPEEDRCP, velocity at respiratory compensation point; LARCP, blood lactate concentration at respiratory compensation point; rVO2max, maximal oxygen uptake relative to body mass. Table 4 Cardiopulmonary exercise testing (CPET) characteristics for cyclists. Variable (unit)Derivation group n=656Testing group n=219Validation group n=219MeanCISDMeanCISDMeanCISDrVO2AT (mL·min–1·kg–1)33.032.5–33.45.8433.232.4–33.95.6833.732.9–34.55.89RERAT0.870.87–0.870.040.870.87–0.880.040.870.87–0.880.04HRAT CI, 95% confidence interval; SD, standard deviation; rVO2AT, oxygen uptake at anaerobic threshold relative to body mass; RERAT, respiratory exchange ratio at anaerobic threshold; HRAT, heart rate at anaerobic threshold; VEAT, pulmonary ventilation at anaerobic threshold; power at anaerobic threshold relative to body mass; LAAT, blood lactate concentration at anaerobic threshold; rVO2RCP, oxygen uptake at respiratory compensation point relative to body mass; RERRCP, respiratory exchange ratio at respiratory compensation point; HRRCP, heart rate at respiratory compensation point; VERCP, pulmonary ventilation at respiratory compensation point; LARCP, blood lactate concentration at respiratory compensation point; power at respiratory compensation point relative to body mass; rVO2max, maximal oxygen uptake relative to body mass. selection variables were in the further only selected parameters were into multiple linear regression The data for MLR model building were randomly into that is testing, validation and of the a only significant were in the Derived are by the of mean and mean analysis was used to the model’s precision and accuracy during validation and tests to the fulfilment of MLR test the of in MLR test assessment between and test of Each model was examined the and any have not been 2 package in RStudio (R Core Team, Vienna, Austria; version version for and software version were used in was considered as the Results Somatic measurements and CPET results data of the runners models for testing, and validation are in Table while cyclists are in Table The runners of and for testing, and validation the cyclists and differences between of runners and cyclists were in BMI and between testing in all between validation only in CPET results for runners models are in Table 3 and for cyclists in Table Runners in the cohort achieved relative to body mass VO2max of in testing group and in validation group cyclists mean was and for testing, and validation to body mass oxygen uptake at anaerobic threshold in runners for ± ± and ± of in testing, and validation it was ± ± and ± of rVO2max, relative to body mass oxygen uptake at respiratory compensation point in runners for ± ± and ± of for testing, and validation while in cyclists for ± ± and ± of rVO2max, There were no significant differences in values between testing, and validation the runners and cyclists between runners and cyclists results were all significant Prediction models based on AT and RCP Full of MLR prediction models for cyclists are in Table for runners in Table The models prediction performance is as with and for cyclists from for somatic parameters to for RCP equations. For runners from for to for AT and equations. for cyclists models was the for RCP and the highest for For from for AT and to for equation. observed for cyclists was the for RCP in the validation group and the highest for while in runners the for AT and and the highest for The performance of prediction is in Figure Figure 2 Download asset Open asset of prediction for Abbreviations: maximal oxygen anaerobic threshold; RCP, respiratory compensation point; All values are in performance for running while the performance for cycling equations. performance of the prediction model for for for AT and for somatic-only equation. Table 5 VO2max prediction for cyclists. linear regression group group = = = = based on anaerobic threshold; RCP, based on respiratory compensation point; based on somatic variables mean mean maximal oxygen uptake relative to body mass rVO2AT, oxygen uptake at anaerobic threshold relative to body mass power at anaerobic threshold relative to body mass rVO2RCP, oxygen uptake at respiratory compensation point relative to body mass VERCP, pulmonary ventilation at respiratory compensation point BF, body fat BM, body mass Table VO2max prediction for runners. linear regression group group = = = = based on anaerobic threshold; RCP, based on respiratory compensation point; based on somatic variables mean mean maximal oxygen uptake relative to body mass rVO2AT, oxygen uptake at anaerobic threshold relative to body mass SPEEDAT, velocity at anaerobic threshold FFM, fat mass VEAT, pulmonary ventilation at anaerobic threshold HRAT, heart rate at anaerobic threshold BF, body fat rVO2RCP, oxygen uptake at respiratory compensation point relative to body mass SPEEDRCP, velocity at respiratory compensation point Models validation of each model for cyclists is in Table while for runners in Table the performance of our prediction was to that observed in the