Celestial navigation is required to improve the long-term accuracy preservation capability of near space vehicles. However, it takes a long time for traditional celestial navigation methods to identify the star map, which limits the improvement of the dynamic response ability. Meanwhile, the aero-optical effects caused by the near space environment can lead to the colorization of measurement noise, which affects the accuracy of the integrated navigation filter. In this paper, an INS/CNS deeply integrated navigation method, which includes a deeply integrated model and a second-order state augmented H-infinity filter, is proposed to solve these problems. The INS/CNS deeply integrated navigation model optimizes the attitude based on the gray image error function, which can estimate the attitude without star identification. The second-order state augmented H-infinity filter uses the state augmentation algorithm to whiten the measurement noise caused by the aero-optical effect, which can effectively improve the estimation accuracy of the H-infinity filter in the near space environment. Simulation results show that the proposed INS/CNS deeply integrated navigation method can reduce the computational cost by 50%, while the attitude accuracy is kept within 10” (3  Near space refers to the airspace 20 to 100 km above the surface . TherefoNear space vehicles fly so fast that attitude errors can greatly affect their position accuracy ,7. As thFurthermore, the complex flow structures around the optical window of the star sensor produce target image offsets , which is an important factor affecting the navigation accuracy of the near space vehicle ,22. The At present, the Kalman filter is the most widely used technology in integrated navigation systems . HoweverBased on the above analyses, there are two problems that need to be solved: First, due to the strong mobility and high speed of near space vehicles, attitude data delay has a great impact on the navigation accuracy ,5,20. ThTo solve the above problems, an INS/CNS deeply integrated navigation model is proposed in this paper. In loosely and tightly integrated mode, IMU and star sensor work independently, but in deeply integrated mode, IMU will be involved in the calculation process of star sensor. IMU data is used to predict the navigation star and ensure that the algorithm can converge. In the deeply integrated navigation system, the star sensor only outputs a gray image (star identification is unnecessary). The attitude is obtained by optimizing the gray image error function, which is constructed with the help of IMU. Therefore, compared with the loosely and tightly integrated modes, the integration will be deeper in deeply integrated mode. Meanwhile, a state augmentation algorithm is introduced to whiten the measurement noise, which can effectively reduce the uncertainty of the measurement noise model. Then, a second-order state augmented H-infinity filter is proposed by using prior information of the colored noise, which can effectively improve the estimation accuracy of the near space vehicle navigation system.This paper is organized as follows: In The main idea of INS/CNS integrated navigation is to use the error between the measurement information of the star sensor and the prediction information of the INS to estimate the misalignment angle. The difference is the navigation information processing level; for example, a loosely integrated model deals with navigation information at the attitude angle level , while aThe gray image error function is the gray error between the measurement star image of the star sensor and the predicted star image by the inertial navigation system.As shown in  satisfy :(1)p1=[uDue to the INS misalignment angle, i represents the number of pixels in the image, and According to the gray scale invariant, image transformation does not change the gray value of the same star point . When ϕ^According to Equation (4), the principle of the deeply integrated model is to minimize the mean square error between two images by adjusting A and one obvious trough B in A and B will come closer and closer by iteration, and finally become the same as the At the beginning, star identification is needed to modify INS to make sure that According to the gray scale invariant,Equation (3) can be further written asAs shown in It is difficult to calculate Next, we calculate the three partial derivatives. Supposing that It is difficult to calculate atisfies :(11)∂ρ∂ϕAccording to the Baker–Campbell–Hausdorff formula ,(12){κ=The Jacobian matrix,N pixels to be optimized, each When there are The damped Newton method is used to update Following which, There is a strong interaction between the aircraft and the surrounding airflow when the near space vehicle re-enters the atmosphere. This effect is called the aero-optical effect, which causes colorization of the attitude noise . This seX-axis and Y-axis, respectively, caused by aero-optical effects. The pixel offset effect is shown in When light travels through a rapidly varying flow field, the imaging position on the focal plane may be biased. The image distortion due to aero-optical effects can be described approximately as follows:It has been pointed out, in , that thHowever, sinusoidal mathematical models are difficult to directly apply in the navigation systems of near space vehicles. This is because, in the actual flight environment, it is difficult to obtain the real-time phases Therefore, the discrete recursive model of pixel offset can be obtained as follows:The H-infinity filter can obtain the optimal estimation of state variables under the condition of noise with unknown statistics ,34. AccoSuppose the filter model isThe first- and second-order terms of the noise sequence The state equation can be expressed as follows:SupposeThen, the second-order state augmented model (after whitening) can be written as follows:The noise characteristics of Although Equation (31) can be abbreviated as follows:According to Equation (30), The covariance matrix of system noise and measurement noise is as follows:Obviously, if The variance matrix of If Suppose the estimator of the H-infinity Filter, after expanding its dimension, isA second-order state augmented H-infinity filter can be obtained by substituting Equation (38) and Equation (40) into the standard H-infinity formula, thus satisfyingThe optimization object of the deeply integrated model is the global gray error of all navigation stars, where the optimization result is the optimal solution (in the least-squares sense). Theoretically, the mismatching of individual stars does not affect the final result; this was verified by simulation.The Smithsonian Astrophysical Observatory (SAO) catalog was used for simulation validation, where precession and nutation were compensated for by the IAU1980 model, aberration only considered first-order corrections, and polar shift correction was provided by the International Earth Rotation Service (IERS). As shown in The attitude calculation results obtained under the three conditions are shown in From To explain this phenomenon, suppose Equation (37) shows that the constant term does not affect the final optimization result, such that optimizing all stars is equivalent to only optimizing the correctly matched stars, which means that unmatched stars do not affect the optimization result.Star identification is not necessary in the deeply integrated mode and the navigation system can still work when the number of navigation stars is less than three. Therefore, the simulation should cover three conditions: The number of navigation satellites is more than three, equal to three, and less than three. The star map of the star sensor in the simulation is shown in The damped Newton method was used to optimize the misalignment angle iteratively. Taking the time As shown in As shown in The accuracies and calculation times of loosely, tightly, and deeply integrated modes are presented in The simulation results show that the attitude accuracies of the loosely, tightly, and deeply integrated navigation modes were of the same magnitude when the number of navigation stars was sufficient, the non-optical axis attitude accuracy was 10” (3 Each star occupies about 9 pixels in the star image, and the deeply integrated mode needs 4 iterations. On average, each star needs 36 iterations; the tightly and loosely integrated mode require star identification, there are 196 navigation stars in a sub catalogue on average, and the argument search space is The attitude accuracy of the star sensor on the optical axis was lower than the remaining two axes, so celestial navigation systems of near space vehicles should be configured with double star sensors. The star sensor configuration schemes are shown in An accuracy comparison of the single- and double-star sensor configurations is shown in As shown in The vehicle attitude profile, star sensor and gyro specification, initial attitude estimation error are shown in The second-order state augmented H-infinity filter and standard H-infinity filter were compared in a near space environment. As can be seen from At the beginning of filtering, the SOSA filter exhibit a large error at around 3 s, this is because the initial value of The core of the second-order state augmented H-infinity filter is the colored noise model, where the core parameter of the colored noise model is the digital angular frequency, In this paper, an INS/CNS deeply integrated navigation method was presented for near space vehicles. This method does not need star identification and can significantly reduce the required computational cost. Meanwhile, the proposed second-order state augmented H-infinity filter can weaken the influence of aero-optical effects on the measurement noise and effectively improve the filtering accuracy in near space environments. The simulation results show that the attitude accuracy of the INS/CNS deeply integrated navigation method is kept within 10” (3


PLOS ONE Editors to re-evaluate the article, together with a member of our Editorial Board.After this article  was publThis study aimed to (1) evaluate whether the rhythmic sensory stimulation (RSS) treatments with different parameters would yield different effects on participants; and (2) evaluate the efficacy of RSS in reducing the severity of fibromyalgia symptoms. The reported experiments involved two RSS treatment groups, as described in the Methods, both of which received RSS treatment albeit with different specifications. The study did not include a true sham control treatment. The results of pre- versus post-treatment analyses reported in the article suggested that RSS stimulation may impact fibromyalgia symptoms. However, given the lack of a sham control group, one cannot distinguish effects of either treatment from placebo effects as would be needed to address the second study aim.PLOS ONE’s publication criteria, and so the PLOS ONE Editors retract this article. We regret that this issue in the study design was not fully addressed during the pre-publication peer review process.The lack of a placebo group was raised during pre-publication peer review and was acknowledged as a limitation in the article’s Discussion. However, after further consideration and post-publication input from a member of our Editorial Board, we concluded that the claims made in the article are not adequately supported by the results given the lack of the control group and the implications of this issue for the interpretation of the data. Therefore, the article does not meet TBJ and LRB did not agree with the retraction. DP, LP, and AG either did not respond directly or could not be reached.At the time of retraction, the article  was repu


We study the minimally nonlinear irreversible heat engines in which the time-reversal symmetry for the systems may be broken. The expressions for the power and the efficiency are derived, in which the effects of the nonlinear terms due to dissipations are included. We show that, as within the linear responses, the minimally nonlinear irreversible heat engines can enable attainment of Carnot efficiency at positive power. We also find that the Curzon-Ahlborn limit imposed on the efficiency at maximum power can be overcome if the time-reversal symmetry is broken. Heat engines as energy converters provide a good platform for studying the nature of thermodynamics, in addition to its relation with utilization of energy resources. Exploring the efficient heat engines at large power is therefore an issue of significance in thermodynamics. The second law of thermodynamics tells us that the efficiency of a heat engine working between two heat reservoirs of constant temperatures rn model ,25,26,27rn model ,35,36,37In the seminal paper  the bounIn the present paper, we investigate the questions of whether the maximum efficiency can approach the Carnot limit at positive power and whether the Curzon–Ahlborn limit for the efficiency at maximum power can be exceeded in the nonlinear response regime. We propose a minimally nonlinear irreversible heat engine , in whicThe heat engine model under consideration, which may be cyclic or steady state and where broken time-reversal symmetry may be induced, for instance, by interaction with an external magnetic field elations ,51 with In the heat engine, the heat flux P can be expressed as F is an external force and x is its corresponding thermodynamically conjugate variable. As the entropy production rate can be expressed in terms of the thermodynamic fluxes Without loss of generality, the power output Equation  we haveLet Equation  becomes With consideration of Equations , 4) and and4) anAs the efficiency x, the relation 11) that We now turn to the maximum power output Equation  and settSubstituting Equation (21) into Equation , we findx. When To see how the time-broken asymmetry induced by the external field influences the performance on the heat engine, in We emphasize that, for the time-reversal symmetry  and right (r) electronic reservoirs. Specifically, at the time-reversal symmetry with x can affect the efficiency and power, for some values of x with tuations  in the ntuations ) in contilibrium  enhance ilibrium ,49,50. Wonsessee .x satisfies a certain condition. We pointed out that a two-terminal thermoelectric device with broken time-reversal symmetry can be mapped onto the engine model discussed here. Our analytical results provide a theoretical framework for understanding of minimally nonlinear heat engines, but should also be helpful for studying the heat devices in which higher nonlinear terms due to dissipations are involved.For systems with broken time-reversal symmetry, we have investigated the performance of minimally nonlinear irreversible heat engines (based on these systems). For these nonlinear irreversible heat engines, the maximum efficiency can tend to be the Carnot limit at nonzero power and efficiency at maximum power can go beyond the Curzon–Ahlborn limit when the asymmetric parameter


Multiple site damage (MSD) cracks are small fatigue cracks that may accumulate at the sides of highly loaded holes in aging aircraft structures. The presence of MSD cracks can drastically reduce the residual strength of fuselage panels. In this paper, artificial neural networks (ANN) modeling is used for predicting the residual strength of aluminum panels with MSD cracks. Experimental data that include 147 unique configurations of aluminum panels with MSD cracks are used. The experimental dataset includes three different aluminum alloys , four different test panel configurations , many different panel widths and thicknesses, and the sizes of the lead and MSD cracks. The results presented in this paper demonstrate that a single ANN model can predict the residual strength for all materials and configurations with high accuracy. Specifically, the overall mean absolute error for the ANN model predictions is 3.82%. Furthermore, the ANN model residual strength predictions are compared to those obtained using the most accurate semi-analytical and computational approaches from the literature. The ANN model predictions are found to be at the same accuracy level of these approaches, and they even outperform the other approaches for many configurations. The concern about multiple site damage (MSD) typically arises for aging passenger and transport aircraft, especially since many of these aircraft are being used beyond their original design life. MSD cracks are small fatigue cracks that may accumulate at the sides of highly loaded holes (rivet holes in particular) in the aircraft’s fuselage or internal structure. These MSD cracks usually appear after an extended period of time due to the large number of loading cycles. Aircraft manufacturers design the fuselage of their airplanes to be able to carry the design load with the presence of a relatively large crack (in the fuselage) spanning several adjacent rivet holes. However, the presence of MSD cracks can significantly reduce the structure’s ability to carry loads ,2,3. TheMany researchers have investigated the ability of several methodologies to estimate the residual strength of panels with MSD, and the accuracy of these methodologies varied substantially. The methodologies ranged all the way from simple engineering models to sophisticated, robust computational techniques. The vast majority of these methodologies relied on analytical or semi-analytical  models ,15,16,17The artificial neural networks (ANN) is one of the advanced computational intelligence methods, which is inspired by the way the human nervous system works. The ANN resembles the work principle of the human brain by acquiring knowledge through a learning process and storing this knowledge through interneuron connections of different synaptic weight . Based oSince its introduction in the 1960s, ANNs continued to provide a powerful framework for modeling nonlinear systems, and they were used in a wide variety of engineering applications, including automatic control , solar eP) and the root mean square percentage error (RMSEP). The ANN optimization covered three of the most commonly used learning algorithms, 12 different activation functions and up to 30 nodes in the hidden layer. Based on this optimization, the Bayesian Regularization (BR) learning algorithm, the Elliot symmetric sigmoid activation function, and seven hidden nodes are used in the ANN model. Our results show that the ANN is able to predict the residual strength for all the different materials and panel configurations with a mean absolute error of about 3.82%. The results also show that the ANN predictions are generally accurate for all the different materials and panel configurations. The obtained ANN predictions are also compared with the residual strength predictions of the best available fracture mechanics semi-analytical and computational models. The comparison shows that the ANN results are of comparable accuracy and even give more accurate results in many cases.Previous experimental investigations done by Smith et al. ,13,14 leC). The fracture toughness is a material property, but for thin sheets, it is also slightly dependent on the thickness, grain orientation, and crack length. Another theory that can be used for predicting the residual strength is the Net Section Yielding (NSY), and it is more applicable to ductile materials. However, experiments have shown that neither of the LEFM nor NSY theories is able to accurately predict the residual strength of panels with MSD for neither ductile nor brittle materials [The residual strength of panels having a lead crack along with adjacent MSD cracks, as illustrated in aterials .LU), is found as:An analytical model that is especially formulated for the prediction of residual strength of panels with MSD was introduced by Swift . This moy: The yield strength of the material.σL: Length of the ligament between the lead crack and MSD crack.a: Lead crack half-length.The SIF geometric correction factors  are usually readily available in the literature in the form of equations or charts, and in case they are not available for some configurations, they can be determined using FEA .As mentioned previously, as the linkup occurs, the entire panel will fail ; therefore, the linkup stress (Equation (1)) is equal to the residual strength of the panel. Experimental investigations have shown that the linkup model is not accurate for many crack configurations ,14,15,19The CTOA criterion is one of the more recent sophisticated Elastic–Plastic Fracture Mechanics (EPFM)-based techniques that can be used for predicting residual strength . The assIn addition to the typical fracture mechanics approaches mentioned earlier, some attempts were also made to use data-driven methods for predicting the residual strength of panels with MSD. Wu et al.  used a lThe review conducted by Nasiri et al.  concludeApplications of ANN in fracture mechanics are mostly concerned with crack propagation, fatigue life, and failure mode prediction. Nechval et al.  adopted ANN applications associated with fracture mechanics in aircraft structures are relatively scarce. Pidaparti and Palakal  presenteThe experimental data used in this investigation were obtained from several sources in the literature ,13,14,19As a matter of fact, testing panels with MSD cracks is not a quick and easy experimental task, and it is usually a time-and-cost-intensive process in both preparation and testing. For instance, the panels need to be relatively wide to avoid the dominance of the NSY type of failure and in order to resemble the actual case of aircraft structures realistically. For this reason, it is not possible to find large experimental datasets for panels with MSD in the literature. The most extensive testing program for the residual strength of panels with MSD was carried out by Smith et al. ,13,14,15The experimental data used in this investigation include a wide variety of panel configurations, materials, material conditions, widths, thicknesses, and lead and MSD cracks geometries. The experimental data presented here are grouped according to the panel configuration into three general groups: unstiffened panels, stiffened panels (it includes two different configurations), and lap-joint panels. The data for each of the three groups are presented in each of the succeeding subsections.The unstiffened panel is the simplest panel configuration that can be tested for the effect of MSD cracks, and therefore, it is the most commonly found configuration in the literature. The test panels’ configuration is generally similar to that shown in stf) which are also made of aluminum and fixed at the front and back sides of the panel using bolts, as seen in the figure. For the first stiffened panel configuration , three different sets of stiffeners were used in the experiments, where each set has a different cross-sectional area, as can be seen in the table.The experimental data for the stiffened panels included two general configurations that are of interest in the aircraft industry, which are a lead crack centered between two stiffeners and a lead crack centered under a broken stiffener. The experimental data for the lap-joint panels includes 27 unique configurations, all of the same material (Al 2024-T3 clad), grain orientation, width, and thickness . Figure The main objective of this study is to develop a single ANN model that can accurately predict the residual strength of panels with MSD for any panel configuration and geometry, material type and condition, and lead and MSD cracks geometry. Several parameters such as geometry, material properties, and panel configuration identifiers are used as inputs to the ANN. Only one hidden layer is used in the ANN structure, since the number of input/output nodes is relatively small. In addition, the use of ANN with one hidden layer  generally gives better performance than multi-hidden-layer ANN (deep network) when dealing with small datasets, such as the case being considered here . The leaP value, it is calculated by averaging 40 MAEP values of the 40 different random combinations used in the simulation trials.To avoid any bias in the results that might come from choosing particular data points for testing the network, the individual data points that are used for training, validation, and testing are chosen randomly. Furthermore, to get more reliable results and robustly evaluate the ANN residual strength prediction accuracy, a total of 40 random combinations of data points are used for training, validation, and testing. This approach of choosing several random combinations is usually referred to as cross-validation, and it is commonly used in data-driven analyses . In othestf), it will have a value for the stiffened panels only, while its magnitude will be zero for the unstiffened and lap-joint panels. As a matter of fact, the geometric input parameters being used here are carefully chosen based on the researchers’ experience with this type of fracture mechanics problems and in order to account for the experimental data, which are obtained from different sources. For the material input parameters, two material properties that have significance in fracture mechanics problems are used; these are the yield strength and fracture toughness. In addition to these two material properties, a material identification number is used to designate each of the three different materials being included here. Identification numbers 1, 2, and 3 are assigned to the 2024-T3, 2524-T3, and 7075-T6 aluminum alloys, respectively. A preliminary sensitivity analysis is performed to evaluate the significance of each of the ANN inputs and to see how it affects the residual strength. The sensitivity analysis is done by calculating the Pearson linear correlation coefficient between each of the inputs and the residual strength [A schematic illustration of the structure of the implemented ANN model is shown in strength . AlthougP and RMSEP. Although the MAEP is more meaningful for reflecting the level of the error, the two metrics are used to get a better understanding of the performance, since the RMSEP reflects the closeness of the errors to the mean value. The MAEP and RMSEP are calculated here as:The validation datasets are used for optimizing the ANN’s configuration to give the best performance , where the performance is evaluated using two metrics: MAEP value for the 40 different random datasets combinations. The configuration of the ANN model is optimized in terms of the following:The optimum ANN configuration is determined by inspecting the average MAE(1)The adopted back-propagation learning algorithm ((2)The hidden nodes activation function ((3)The number of nodes . These 147 data points represent different material types and conditions, panel configuration and geometry, cracks geometry, etc. Nine independent inputs are used for the ANN model where these inputs cover the different panel and cracks geometry, the different materials, and the different panel configurations. It is needless to say that the proper selection of the input parameters and how the different configurations are handled is essential to obtain good results. Overall, the ANN model has demonstrated good efficiency in predicting the residual strength for panels with MSD, as will be shown in the following sections.P by about 0.2%. Accordingly, this partially randomized training dataset’s selection approach is adapted in here. It should be mentioned here that the MAEP improvement that results from using the partially randomized approach seems to be small (only 0.2%) because the MAEP values are averaged for 40 different random combinations (as mentioned previously). However, for some particular data points, the error in the ANN residual strength predictions can be noticeable if the ANN is extrapolating out of the range used for training. It should also be indicated that this approach, which is being followed here to avoid the extrapolation for the ANN predictions, is consistent with the findings reported by Mortazavi and Ince [As mentioned previously, a total of 97 data points  are used for ANN training, where these include 34 unstiffened 2024-T3 panels, 14 unstiffened 2524-T3 panels, 8 unstiffened 7075-T6 panels, 24 stiffened panels, and 17 lap-joint panels. These data points are chosen randomly, and 40 different random combinations of the 97 training data points are used in this investigation. This randomized selection procedure is meant to ensure the credibility of the obtained ANN results. However, by inspecting the individual data points within the 40 randomly selected combinations, it is noticed that occasionally, some of the data points that represent the upper or lower limit values of some of the important inputs (such as the crack length or ligament length) are not included in the training dataset. Therefore, in order to avoid any extrapolation in the ANN residual strength predictions (for the testing dataset), some fixed manually selected data points are always included in the training datasets. For each group of data points , the fixand Ince  about thP value (averaged over the 40 different random combinations). For identifying the best of the three learning algorithms , an ANN is developed using the training dataset based on each algorithm, and the validation dataset is used to determine the optimum configuration for each. P, RMSEP, and the coefficient of determination (R2). From the table, it can also be seen that the BR learning algorithm gives the best performance using the Elliot symmetric sigmoid (elliotsig) activation function along with 30 hidden nodes . To further clarify the effect of the number of hidden nodes on the ANN performance, the evolution of the MAEP versus the considered numbers of hidden nodes for each learning algorithm (using the optimum activation function) is shown in P value using the validation datasets. The figure shows that for the BR algorithm, the optimum number of hidden nodes is 30, which corresponds to the lowest MAEP value. By carefully inspecting the curve, it can be seen that the MAEP value dropped rabidly at the beginning (at seven hidden nodes), and it remained relatively steady afterwards. Based on that, it can be seen that taking the number of hidden nodes to be seven would probably be good enough, and it will not make much difference in the residual strength prediction accuracy. As a matter of fact, some researchers use some rules of thumb for choosing the number of hidden nodes to be used in ANN models. One of the most commonly used rules of thump suggests that the number of hidden nodes should be somewhere in between the number of input nodes and the number of output nodes  [P value (3.38% for 30 nodes vs. 3.43% for seven nodes). To further investigate that, the residual strength prediction performance of the seven hidden nodes and the 30 hidden nodes ANNs is also compared using the testing datasets . The compression based on the testing datasets (averages for the 40 random dataset combinations) shows that the seven hidden nodes gives slightly more accurate residual strength predictions where the MAEP values using the seven and 30 hidden nodes are 3.82% and 3.92%, respectively. Consequently, the number of hidden nodes is taken to be seven, especially that such a small number of hidden nodes makes the ANN less complex and reduces the computational effort. Therefore, in summary, the ANN model that is being adopted in this study uses the “BR” learning algorithm, the “elliotsig” activation function, and seven nodes in the hidden layer. All the residual strength predictions presented in the succeeding sections are obtained using this ANN configuration.As mentioned previously, the ANN optimization is done using three different learning algorithms, 12 different activation functions, and up to 30 hidden nodes. The optimum ANN configuration is determined based on the MAEur case) . Based oP = 2.35%, and the testing dataset that gives the worst performance has an MAEP = 6.35%, while the average MAEP for the 40 testing datasets is 3.82%. This in fact shows the importance of using the cross-validation technique that makes the results more credible, since it eliminates the variations associated with the selection of the data points used in the testing dataset. It should be stressed here that the overall performance metrics reported in this paper are calculated by averaging over the 40 testing dataset combinations. In order to evaluate the performance of the ANN in predicting the residual strength of panels with MSD, the selected ANN configuration is applied on the testing “unseen” datasets, which include 40 different combinations, each consisting of 25 randomly selected data points. Of course, there are some differences in the calculated performance metrics for each of the 40 testing datasets. For instance, the testing dataset that gives the best performance has an MAE2) values for the training, validation, and testing datasets are also shown in the figure. It should be noticed here that the coefficient of determination for the training dataset is clearly higher than those for the validation and training datasets. This is in fact quite expected, since the training data are already "seen" by the ANN (since they are used for training); therefore, the ANN can predict the residual strength for the training dataset more accurately that the "unseen" validation and testing datasets. The overall coefficient of determination for all the 147 data points shown in In order to better visualize the overall accuracy for the ANN residual strength predictions, the correlation between the experimental results and ANN predictions for all the 147 data points is shown in P and RMSEP values for the three different materials and for the three general panel configurations, as well as the totals for all materials and panel configurations. From the table, it is evident that the ANN predictions are generally accurate for all the deferent materials. However, it also can be seen that there are some relatively small differences in the residual strength predictions error levels between the different materials and panel configurations. The largest error is observed for the Al 7075-T6 material (MAEP = 8.2%). The relatively high error associated with this material is rather expected, since the number of data points used for training the ANN is smaller than that for other materials (only eight data points are used for training). On the other hand, the best ANN performance is observed for the lap-joint panels where the MAEP is 1.81%. This error value is lower than that observed for the unstiffened panels of the same Al 2024-T3 material (MAEP = 4.61%), even though the number for panels used for ANN training for the unstiffened panels is larger than that for the lap-joint panels. In fact, this difference can be attributed to the fact that the unstiffened panels data have more variation where they include different thicknesses, material conditions, and grain orientations; while on the other hand, the lap-joint panels are all of the same thickness and grain orientation. Therefore, it is quite normal that the ANN is able to give better predictions for the lap-joint panels, since there is no variation in their material properties. It is probably worth mentioning here that using the actual values of the material properties  might slightly improve the ANN prediction accuracy. However, as mentioned previously, adopting the standard handbook values of the material properties is much more convenient for engineering use. By comparing the values of the two performance metrics given in the table, it can be seen that the RMSEP values are consistently higher than the MAEP values, but the difference is generally not very high. This difference between the RMSEP and the MAEP values comes from the variability of the error values for the individual predictions, and the fact that the difference is not very high indicates that the variability is not that significant. Finally, the table also shows the overall error for all the testing datasets where the MAEP is equal to 3.82%. For the same group of testing datasets, the average error for the residual strength predictions obtained using the semi-analytical models (Equation (2) to Equation (4)) is 4.1%. Comparing the overall error values of the ANN and semi-analytical models predictions shows that both have the same level of accuracy. However, obtaining residual strength predictions using the ANN is easier, and the fact that a single ANN model is used for all materials and panel configurations makes it even more convenient.Finally, P) is 3.82%, which puts it at the same accuracy level (and even better for many configurations) of the best available semi-analytical and computational approaches. The main conclusions of this study can be summarized in the following points:The presence of MSD cracks and their effect on residual strength is a serious concern for aging aircraft fleets. Several methodologies ranging from simple engineering models to sophisticated computational approaches can be used for estimating the residual strength of panels with MSD cracks. Data-driven techniques, and in particular ANN, have been successfully applied and are increasingly being used in a wide variety of engineering applications. In this paper, ANN modeling is used for predicting the residual strength of aluminum panels with MSD. The experimental data used for developing the ANN model  include 147 unique data points that cover several material types and conditions, panels and cracks geometry, and test panel configurations. The ANN model has nine independent inputs that deal with the geometry, materials, and panel configurations. The results presented in this paper demonstrate that the ANN model can predict the residual strength for all materials and configurations with high accuracy. The overall ANN model average prediction error  and a designation number are used as inputs to account for the three different materials being used here. The material properties also "indirectly" account for sheet thickness, material condition, and grain orientation, and this eliminates the need for some inputs. In addition, using the standard handbook values of the material properties  makes the ANN modeling approach being used here simpler and more convenient for engineering use.Optimizing the ANN model is essential for obtaining high-accuracy predictions. The ANN optimization carried out here showed that there could be quite a significant difference in the prediction accuracy when using different learning algorithms, hidden node activation functions, and numbers of hidden nodes. In this investigation, the best ANN performance was obtained using the Bayesian Regularization learning algorithm with the Elliot symmetric sigmoid activation function and seven hidden nodes.In order to avoid bias and get more reliable results, it is essential to implement a randomized selection procedure for the data points used in the training, validation, and testing datasets; also, the randomized selection needs to be repeated several times . Additionally, it is beneficial to include some fixed, manually selected data points that cover the upper and lower limit values of the different inputs within the training group. This will avoid the extrapolation in the ANN predictions and thus improve the accuracy.P = 8.2%) since only eight data points are used for training. In fact, eight data points is a very small number, and it is definitely not sufficient for training an ANN; however, the error level is still reasonable, since this material is not used alone for developing an ANN model but rather among a larger group of data points that share many common geometric and configuration inputs, although the materials are different. On the other hand, the best accuracy (MAEP = 1.81%) is observed for the lap-joint panels, since they all share the same sheet thickness, material condition, and grain orientation.There are some differences  in the average residual strength prediction error values between the different materials and panel configurations. These differences in the error values can be attributed to two factors: (i) the different number of data points available for training the ANN for the different materials and configurations and (ii) the amount of variation in the different inputs parameters within the different materials and configurations. For instance, the highest error is observed for the 7075-T6 material (MAE


Derma roller, a device rolled onto the skin to form micropores, is extensively used for cosmetic purposes. The pores thus created are utilized to either result in the induction of collagen production, leading to glowing and wrinkle-free skin or for permeating the applied formulations to the site of action within the skin. Recent studies have shown the benefits of using derma rollers for transdermal delivery of drugs. In the nascent stage, this approach paves a way to successfully breach the stratum corneum and aid in the movement of medications directed towards the dermis and the hair follicles. The review essentially summarizes the evidence of the use of derma rollers in cosmetic setup, their designing, and the preclinical and clinical reports of efficacy, safety, and concerns when translated for pharmaceutical purposes and transdermal drug delivery. Microneedling, a term familiar to most dermatologists today, has continuously evolved since its inception in the early twentieth century. The technique of microneedling is built upon non-pathogenic puncturing of the skin with micro-sized needles, thereby stimulating the underlying cells to increase the production of growth factors and the vital dermal ingredient, collagen ; the dermis (approximate thickness of 0.7 to 4 mm), and the hypodermis. Several layers of dead cells line the outer surface of the epidermis exposed to the external air and environment, collectively called the ‘stratum corneum’ which is regarded as the primary barrier of the body . Although the success of the transdermal application of derma roller applications is minimal, numerous examples merit the need for research in the area of systemic drug delivery. Ahmed and colleagues pointed out that pretreatment of healthy skin with derma rollers can increase the penetration of the drug-loaded liposomal formulations more than two folds compared to the passive delivery  introduced pulse stimulus . The first therapy was carried out by another dermatologist.in which derma roller was rolled over melasma areas vertically, horizontally, and diagonally. Treatment was repeated twice a month with an interval of 1 mo. As the first treatment ended, each patient was taught to use derma roller C8. This device does not require any anesthetic application, and also it doesn’t cause any bleeding. They were informed to apply the depigmenting serum on the right hemifacial areas of melasma. Patients were asked to do this every day for 2 mo. Results were analyzed after the completion of the two-session of study. As per the detailed evaluation, it was found that the treated areas with a combination of derma roller and depigmenting serum showed better results compared to the use of only depigmenting serum. The combination therapy had a baseline MASI mean score of 19.1. It decreased to 14.4 on the first treatment and further decreased to 9.2 on the second treatment. The application of the depigmenting agent alone had a baseline score of 20.4, which dropped to 17.4 on the first treatment and also reduced to 13.3 on the second treatment.The clinical study conducted by Fabbrocini et al.  justifieMost of the cosmetic procedures available today pose a particular risk to the patient seeking treatment. Along similar lines, microneedling or therapy with derma rollers are not entirely free of side-effects and contraindications Sharad . These uAlthough the dermaroller and microneedle patchwork on the same principle, dermaroller can be advantageous over the microneedle patch. Dissolving microneedles patch being water-soluble, it is essential to keep them away from moisture. Storage becomes a problem in humid areas for dissolving microneedles, whereas storage conditions won’t affect derma rollers as their needles are stainless steel. In conditions where cream and microneedle patch has to be applied simultaneously, attachment of microneedle patch becomes difficult because of the left out the residue of cream (2) of the FD&C Act, also known as the De Novo classification system. This procedure provides a path to class I or class II classification for low-moderate risk devices for which general and unique controls or general controls provide a guarantee of efficacy and safety but for which there is no legally marketed predicate product. If a De Novo application for a microneedling device is approved, the particular product and system type would be listed as Class I or Class II and maybe subsequently marketed and act as a predicate for future products.To facilitate a review, the De Novo request should provide all the information necessary to demonstrate a guarantee of the safety and efficacy of the system. The De Novo application will define the benefits and risks of the system and the general controls or general and special controls required to ensure safety, such as mitigation of any identified health risks and effectiveness. Data should be presented, showing that general controls or general and special controls endorse class I or class II classification. Based on the information currently available, the F.D.A. considers that risks associated with microneedling devices include infection, nerve and blood vessel disruption, user-to-user disease transmission, scar formation, hyperpigmentation, skin inflammation, allergic reactions, and skin irritation. Depending on the technical characteristics and intended uses of the specific microneedling tool, additional risks can be established Guidance .Derma rollers have moved a long way from the traditional cosmetic applications to more complex transdermal drug delivery. Current data indicate that epidermal preservation is beneficial in aesthetic situations as the derma roller facilitates the development of collagen and elastin. Extensive studies are underway to examine responses to skin repair at a cellular level. In particular, its effectiveness, safety, and accessibility make the derma roller a successful therapeutic option. With many exciting studies published over the past few years, the use of derma roller technologies to facilitate the efficient delivery of pharmaceuticals is rapidly advancing. Derma roller results in disruption of the stratum corneum cells, which weakened the main barrier of the skin, causing an increase in the absorption of applied compounds. It is necessary to remember that most derma roller comparative studies are case reports, case series, or minor randomized controlled studies. Before microneedling is accepted for extensive use in patient care, there is much work to be done, but novel models address these issues. It is expected that, with the advancement in dermal and transdermal technology, derma roller will be approved clinically in the near future.


In such a scenario, the user herself must try to either use the full count that the model counter returns to compute the count up to isomorphism, or ensure that the input formula to the model counter adequately captures the symmetry breaking predicates so it can directly report the count she desires.Propositional model counting is a classic problem that has recently witnessed many technical advances and novel applications. While the basic model counting problem requires computing the number of all solutions to the given formula, in some important application scenarios, the desired count is not of CNF-level and domain-level symmetry breaking predicates in the context of the state-of-the-art in model counting, specifically the leading approximate model counter ApproxMC and the recently introduced exact model counter ProjMC. As benchmarks, we use a range of problems, including structurally complex specifications of software systems and constraint satisfaction problems. The results show that while it is sometimes feasible to compute the model counts up to isomorphism using the full counts that are computed by the model counters, doing so suffers from poor scalability. The addition of symmetry breaking predicates substantially assists model counters. Domain-specific predicates are particularly useful, and in many cases can provide full symmetry breaking to enable highly efficient model counting up to isomorphism. We hope our study motivates new research on designing model counters that directly account for symmetries to facilitate further applications of model counting.We study the use of


This work introduces a new population-based stochastic search technique, named multi-variant differential evolution (MVDE) algorithm for solving fifteen well-known real world problems from UCI repository and compared to four popular optimization methods. The MVDE proposes a new self-adaptive scaling factor based on cosine and logistic distributions as an almost factor-free optimization technique. For more updated chances, this factor is binary-mapped by incorporating an adaptive crossover operator. During the evolution, both greedy and less-greedy variants are managed by adjusting and incorporating the binary scaling factor and elite identification mechanism into a new multi-mutation crossover process through a number of sequentially evolutionary phases. Feature selection decreases the number of features by eliminating irrelevant or misleading, noisy and redundant data which can accelerate the process of classification. In this paper, a new feature selection algorithm based on the MVDE method and artificial neural network is presented which enabled MVDE to get a combination features’ set, accelerate the accuracy of the classification, and optimize both the structure and weights of Artificial Neural Network (ANN) simultaneously. The experimental results show the encouraging behavior of the proposed algorithm in terms of the classification accuracies and optimal number of feature selection. The most challenging matter in ANN models is the selection of the appropriate weights, number of layers, and number of nodes in each layer. The complexity of the network is affected by the number of layers and nodes, so the difficulty for the training process will be increased. Therefore, selecting the suitable ANN model is required which should not be very small network that has a limited potential to be able to characterize the real state nor a large network which doing complex training process and may provide noise in the training data and thus cannot represent superior capability6. A dataset usually includes a large number of features in classification problems, so irrelevant and redundant data are not applicable for classification and they may reduce the performance of the classification due to the large search space. FS strategies can be divided generally into three categories: filter, wrapper, and hybrid techniques. Filter technique dependently operates on data itself using appointed methods such as Principal Component Analysis (PCA) which is a common method. On the other hand, wrapper techniques are beneficial in finding feature subsets that satisfy a predetermined classifier. Consequently, they are broadly examined for the accuracy of the class. Besides that wrapper approaches are expensive and can be collapsed with a very large number of features due to utilizing learning algorithms in evaluating feature subset every time. However, hybrid techniques try to gather merits of the filter and wrapper techniques by manipulating their correlative strengths8. FS plays very important role in many areas such as pattern classification, multimedia information retrieval, data mining, machine learning applications and so on, which can influence the classification accuracy rate and reducing the time required for training. Classifying any given input feature vector into pre-defined set of classes of patterns needs assigning this vector to one of a set of classes10.Feature selection (FS) supplies a way to reduce the number of features from a large number of available features to capture better classification performance than using all features by removing or reducing irrelevant and redundant features13.Meta-heuristics have been very dependable for solving diverse optimization problems in the last two decades and overcoming the challenging problem of searching optimal subset from all the original set. A new FS approaches were generated based on evolutionary optimization techniques since they can lead to a faster way to find optimal solutions. Moreover, by considering an effective fitness function a high dimensional data can be managed by limited number training samples. Whereas the complete search generates all possible solutions for the problem, meta-heuristics present outstanding performance compared to other conventional search techniques20. In21, Menghour and Meslati introduced a hybrid feature selection algorithm based on Ant Colony Optimization (ACO) and Particle Swarm Optimization (PSO) algorithms. The algorithm was designed to solve six well-known datasets. Artificial Bee Colony optimization (ABC) and DE technique was proposed in22 as a new combinable method for feature selection of classification tasks for solving fifteen public datasets. In23, a new hybrid ACO-ABC algorithm was introduced to validate thirteen datasets’ problems, in which ants decide the best ant and best feature subset by exploiting the bees and adjusting as their sources of food. In addition, a feature selection approach was proposed based on two different implementation of multi-objective ABC algorithm combined with non-dominated sorting procedure and genetic operators for examining twelve benchmark datasets24. In25, a new method called PSO-DFS using bare-bone particle swarm optimization (BBPSO) for discretization and feature selection in a single stage was proposed for solving ten high-dimensional datasets. In26, a comprehensive study to investigate the use of Genetic programming (GP) for feature construction and selection on high-dimensional classification problems was presented and tested on seven high-dimensional gene expression problems. Chen et al. proposed two novel Bacterial Foraging Optimization algorithms (BFO), which named Adaptive Chemotaxis Bacterial Foraging Optimization algorithm (ACBFO) and Improved Swarming and Elimination-Dispersal Bacterial Foraging Optimization algorithm (ISEDBFO) to create the mapping relationship between the bacterium and the feature subset and to evaluate the importance of features. This method dealt with feature selection problems and tested ten public datasets of UCI27. Majdi et al. proposed a Grasshopper Optimization Algorithm (GOA) as a search strategy to design a wrapper-based feature selection method in the form of four different strategies to moderate the immature convergence and stagnation drawbacks of the conventional GOA. These approaches were benchmarked on twenty-two public UCI datasets28.In recent years, several meta-heuristics have been utilized by many researchers in the field of optimization to search feature subset space for selecting optimal feature set. The strategy of meta-heuristic may determine a satisfactory solution in a reasonable time in spite of it doesn’t assure finding the best solution in every run. These algorithms showed superior performance in solving many practical problems which can be original, modified, or hybrid algorithms29, the antlion multi-objective wrapper-based feature selection method (CALO) was proposed by using different chaotic Maps and tested on eighteen datasets to balance between exploration and exploitation in the search space. The proposed method achieved better performance by converging to the optimal solution than PSO and GA methods and it was more effective than the original ALO method.In30.Li et al. introduced a new multi-objective ranking binary artificial bee colony method for the gene selection on eight microarray datasets. They first used the Fisher Markov Selector method to assort and choose the features that will be used as inputs to the binary ranking artificial bee colony. After that, the binary ranking artificial bee colony selected the genes subset. This method achieved the best performance compared to other methods with different classifiers. The results show also outperforms of that method in selecting smaller number of selected features31. Thus, all classification/ feature selection problems cannot be solved by only one of the heuristic feature selection methods and there is always a possibility to improve the current methods to solve better the current new classification/feature selection problems. This is our motivation for attempting to propose another optimization algorithm for feature selection on classification.In spite of the advantages of the above mentioned heuristic algorithms for feature selection on classification problems, someone may inquire if we need another new heuristic algorithms. The theory of No-Free-Lunch (NFL) illustrated that all the optimization problems cannot be solved by one optimizer32. The CSO algorithm has two modes: seeking mode and tracing mode. In the beginning of the iteration, the number of cats is specified and cats broadcast arbitrarily in M-dimensions space. Then, applying cats to solve the problem, in which every cat has position, velocity for each dimension, fitness value, and a flag to determine if the cat in seeking or tracing mode. The one of the cats with the last solution will have the finest position. At the end of the iterations, the best solution will be kept33.Cat Swarm Optimization algorithm was proposed in 2007 by Chu and Tsai34.This algorithm comprises of two main stages; encircling prey and spiral updating position in the first stage (exploitation stage). In the second stage, a random searching for a prey is carried out (exploration stage). In the beginning, whales are allocated by arbitrary solutions and the minimum or maximum value of the objective function will be assumed as the best optimal value relying on the problem is solved. Then, every search agent of the objective function is calculated. Every search agent modifies its position relying on the best solution or on a random choice search agent for every iteration.Whale Optimization Algorithm was presented by Mirjalili and Lewis in 201635.In SCA, the algorithm started by arbitrary solutions’ set. The objective function calculated recurrently this arbitrary set and rules’ set which is the core of this method was used to improve it. It consists of two phases: in the first phase (exploration phase), the arbitrary solutions in the solutions’ set were combined suddenly by the optimization method to find the encouraging search space areas. Random solutions were changed gradually in the exploitation phase. In addition, arbitrary differences were greatly fewer than those in the first phase.Sine Cosine Algorithm was proposed in 2015 by Mirjalili36. It is one of the most popular evolutionary algorithms to solve the global optimization problems. Global optimization is necessary in fields such as engineering, statistics and finance. It is stochastic and population-based optimization algorithm. It is developed to optimize real valued functions and real parameter. A population of candidate solutions for the optimization problem to be solved is randomly initialized. By applying crossover and mutation, new individuals are created for each generation of the evolution process. Recombination of the target individual with mutant individual to create the trial individual incorporates successful solutions from the previous generation. The target individual is compared with the trial individual and the one with the lowest function value is admitted to the next generation. Mutation, recombination and selection continue until some stopping criterion is reached38.Differential Evolution (DE) algorithm was introduced by Storn and Price in 1996Xr1, g to disorganize it. A mutant vector is generated by the following formula:r1, r2 and r3 are indexes selected randomly over , N is the number of individuals in the population, g is the current generation, and F is a constant mutation factor from .The mutation is performed by computing the vector differences between other two individuals in the same population which are selected randomly. Generating the mutant individual . The crossover factor i = 1, 2. . . N; N is the population size, j = 1, 2. . . D; D is the dimension of a single vector. The trial vector g + 1 using the fitness function formula:Finally, the trial vector Recently, DE has arisen as an encouraging approach in several real world challenges. Effectiveness, robustness, capability to deal with complex large-dimensional optimization problems, and needing few control parameters are some merits of DE algorithm over other meta-heuristic algorithms. Furthermore, because of the fitness of offspring is competed and compared one-to-one with the fitness of corresponding parent, DE has sufficiently fast convergence characteristics. Although this approach raises the possibility of trapping in local optimal  and leads to premature convergence, it may be efficient to find an optimal solution rapidly. The control parameters of DE are needed a fine-tuning while remain fixed over the process of optimization is considering another demerit of DE.39.To overcome the drawbacks of DE algorithm, it has been integrated with other optimization algorithms in a hybridized form to improve the performance of DE. On contrast, it is an effective method on a large range of classic optimization problems. DE is one of the most popular heuristic algorithms to solve single-objective optimization problems and it has been extended to solve multi-objective optimization problemsA novel optimization algorithm is proposed in this work to optimize both the weights and the structure of ANN simultaneously by presenting a new solution representation. Two main phases are composed of the proposed method: arrangement optimization and weights update. However, the proposed MVDE algorithm with multi variant mutation and adaptive scaling factor has been developed to choose the optimal number of features used for classification which has an impact on the accuracy of the classification.However, the proposed MVDE is presented to overcome the drawbacks of DE which mentioned above. On this issue, different five mutation approaches incorporated with two high random scaling factors based on cosine and logistic distributions will be used to maintain the population diversity during the optimization process which leads to preventing premature convergence. Moreover, the requirements for tuning of the control parameters can be reduced by the proposed adaptive crossover and adaptive selection.33, WOA34, SCA35, and PSO21 for evaluating the proposed algorithm performance.In addition, the experimental results are compared to the results in the literature and to another four optimization algorithms; DE, CSOThe rest of this paper is organized as follows: The methodology of the proposed approach is outlined in "36 is a promising tool that can be regarded to deal with complicated high-dimensional problems with enhanced search qualities. DE algorithm is selected here to use as a search engine because it has meaningful merits over other meta-heuristic methods in terms of robustness, effectiveness, and fast convergence characteristics in searching high-scale problems. On contrast, the premature convergence to local optima and fixed control parameters are the problems of DE39. Additional improvements are essential before using this method in practical problems to gain better performance. In this concern, a new algorithm named Multi-Variant Differential Evolution (MVDE) has been proposed as an almost parameter-free optimization method. The proposed approach was designed mainly to enhance the global search ability of the original DE, i.e., reducing the probability of trapping in local optima and preventing the premature convergence. The proposed MVDE uses five different mutation strategies integrated with two high random scaling factors (based on cosine and logistic distributions) to maintain the population diversity during the optimization process, thereby prevent premature convergence. Additionally, the proposed adaptive crossover integrated with adaptive selection to reduce the needs for tuning of the control parameters. The overall steps for proposed MVDE algorithm are shown in Fig. Differential evolution (DE) algorithmThe proposed MVDE is designed as population-based meta-heuristic algorithm. It begins to solve d-dimensional optimization problem with initial solutions  and adaptive parent selection. In the beginning, it is essential to explain the mutation-crossover process by describing its associated operators.40, a self-adaptive scaling factor of DE was introduced based on the elitist scheme of the learning rate to keep track of the fittest vector . In spite of the widespread use of elitist scheme in GAs for attempting to help for fast convergence, it might be suffering from the problem of premature convergence. In41, biological genetic strategy has been proposed to inspire another adaptive scaling factor by changing (increasing or decreasing) F exponentially between pre-specified initial and maximum scaling factors besides to two of adjustable elements. However, extra parameters have been involved by this adaptive scaling factor which need to be chosen or adjusted.Scaling factor In this work, to enhance the adaptive and dynamic performance of MVDE, a new adaptive scaling factor is proposed. The values of random variables which are generated from two different probability distributions: cosine and logistic distributions are used to update the scale factor of each individual for each generation.The two parametric distributions with different thicknesses tails which can be identified completely are cosine and logistic distributions. Depending on the development of optimization operation, the probability of selecting each random generator is proposed.In the first generations, F has a value with a high probability to be produced according to the cosine distribution. On the other hand, at the end of optimization, the probability of utilizing the logistic distribution is increased. Hence, both of these distributions are used simultaneously. Certainly, small and large values of independent variables can be generated from each random distribution . Therefore, the control of the exploration  and the exploitation  are attempted to be supported by this proposed scheme for controlling the exploration– exploitation balance.41, the proposed two types of distributions that are used by the proposed MVDE have been introduced.As shown in Fig. 43:The cosine or half-cosine distribution is able to transact with cases where the central orientation is not obvious enough. In spite of that, it presents a low central orientation with a heavy tail in which samples appear likely nearby the limits of distribution compared to others such as normal distribution, as presented in Fig. 45:On the other hand, the logistic density curve is symmetrical, bell-shaped, and has a larger amplitude in the central area of distribution and lighter tails on the limits’ sides compared to the curve of cosine, as shown in Fig. This stage involves increasing in exploitation strength whereas preserving some exploration for the problems which offer a large search space at the end of optimization. In the meantime, the location and scale parameters can identify the logistic distribution completely. In consequence, in this work, the scale parameter is set to 0.1 for achieving lighter tail with more exploitation. According to the values drawn from Gaussian distribution in the range , the location parameter is shifted from zero to preserve the population diversity and offer adequate exploration capability till the end of the optimization process. The variance of the proposed logistic distribution is about 0.033 which is lesser than the cosine distribution and calculated as b^2 π^2/3.30. Hence, the crossover probability constant is chosen for linearly increasing across the track of iterations from nearby zero to 0.5, as follows:jth element of the ith individual and is produced from the uniform distribution in the range . The above mentioned continuous scaling factor is mapped using the (0–1) matrix A as follows:The use of low crossover rate or high scale factor produces large disturbances which are beneficial for population diversity but decrease the speed of convergence. The convergence will be rapid, but also premature if the contrary happens, so that, these operators can be valuable in case of integratingObserve that if Mutations schemes of the MVDE method need a suitable number of chosen individuals to attend as parents in order to lead the search operation. So, the qualified parents of size Depending on the strategy size, an acceptable number of parents is recognized among the qualified parents. The MVDE algorithm construction retains the ability to integrate and handle five mutation schemes.The time horizon of the optimization process is divided evenly into five successive iterative sub-processes (phases/periods) after initialization. According to choose one of the five kinds of mutation schemes, the population in each phase is modified. During the first iteration, the solutions are updated utilizing the proposed less greedy evolution variants such as (DE/rand/k) to cover the search area and to recognize the encouraging areas by benefiting from their explorative capability. The less greedy variants discover the most encouraging area which is used to lead the search at the end of optimization operation to share more information among the used schemes through the greedy strategies such as (DE/best/k). To focus only on the optimal solution guiding to the desired rapid convergence, the low disturbances of these variants are essential.According to previous suggestions, the chosen five mutation schemes incorporated with the updated scaling factor  are applied through mutation-crossover operation, as follows below.Actually, the differences between population individuals are the basis of the disturbance utilized in all variants of the evolution. The “DE/rand/2” variant is the famous least greedy mutation scheme which has a top priority for exploration, and is applied to the first 20% of iterations, as follows:Two binary scaling factors are produced independently in this variant, therefore each element of the group member The less greedy and explorative “DE/rand/1” variant is implemented in the next group of iterations as follows:As a result of the existence of a single binary scaling factor, this scheme comprises only two updated chances.The most appropriate current solution is integrated into the “DE/current-to-best/1” strategy for leading the search to the global optimal with more aesthetic rapid convergence (exploitative conduct). Differences between random solutions are utilized to balance such conduct for updating the robustness as explorative conduct. So, at the middle of the search area, this variant is applied to attain a good balance between exploration and exploitation as follows:The population in the next sub-domain handled due to the greedy and exploitative “DE/best/2” variant where the four kinds of updated chances are available as follows:At last, the positions of the population are updated by implementing the highly greedy and exploitative “DE/best/1” variant during the last sub-iterations as follows:46.According to the description and difficulty of the problem, the above mentioned mutation schemes can be redescribed or reduced, respectively. At the end of each generation, the mutated individuals compared to the previous population, and the best solutions permitted for surviving to the next generation. We are the original source and the owners of this new algorithm. The original source code is available at 47.In this work, the optimization of the weights and structure of ANN is regarded by applying MVDE algorithm where each solution in the population of the MVDE holds both weight and structure solution. In this strategy, a specific fitness function which is dependent on the weights and structure of ANN is the base to measure inputs, different weights, number of hidden-layers, and number of nodes in each hidden-layer49.Structure optimization: finding a compact topology with the minimum number of hidden nodes is the goal of the structure optimization stage. This can be achieved by choosing the important nodes from the network while neglecting the reminder.Weight update: selecting the most important hidden nodes from the initial network is proposed to produce a compact structure of the network during the structure optimization.The training phase consists of two main stages: structure optimization and weight update. The network structure is optimized during the training process by selecting important hidden nodes that minimize the output error. The weights of the network are updated by maximizing the diversity of the output from hidden nodes. The weight update stage is proposed to enhance the performance of the structure optimizationThe selected hidden node is then divided into two new hidden nodes which have the same number of weight connections as their parents. The new weight connections are calculated as follows:The search process can be accelerated by identifying the suitable number of hidden nodes in ANN architecture.47.In this work, two one-dimensional vectors are considered for solution representation. One vector describes the structure solution which holds binary values of 0 and 1 while another vector contains weights and biases with real numbers in range . The first vector has three sections; two sections are for the number of hidden-layers and the number of nodes in each of these layers which occupy three cells each. These three cells also keep the binary values of the number of hidden-layers and the number of nodes in each of these layersith feature in the complete feature set is held in the subset of a selected feature if the value of ith in this section is equal to 1 while the subset of selected feature does not contain this feature if the cell presents 0 value.The feature selection section is added to the structure solution representation for classification since the values of the input nodes, and the number of input nodes is essential which its dimension is equal to the number of features in the dataset. The 48.For weights and biases, the second vector contains weights and biases in real values in which the number of them is calculated based on the structure solutionThe use of an effective fitness function is necessary to select a solution that minimizes the objective function and to evaluate the quality of it in successive iterations. In this paper, the fitness function is used to minimize the classification error, the number of selected features, and the size of ANN with good generalization capability which is calculated by the average of the error ε, the ration of the selected features, and the ration of the number of weights and biases (connections). The classification error which presents the percentage of misclassified training samples is calculated as follows:Classification problems are implemented using MATLAB R2017a software on Windows 10 and executed on a PC with an Intel Core i7-5600U processor of 2.6 GHZ 8.0 GB. The maximum number of iterations is set to 100 and the population size is set to 40.52. Datasets with different issues of instances and attributes are chosen for validating MVDE. Four heuristic algorithms: DE, CSO, WOA, SCA, and PSO algorithms are used to compare and valuate with the MVDE. Table Fifteen classification datasets are evaluated using the MVDE method. The datasets are from different sourcesThe performance of the proposed algorithm and compared methods for finding solutions are measured using eight metric indices. The metric indices are the average of classification accuracy, best, mean, worst, standard deviation (Std), the number of selected features, complexity of the network , and time cost (Seconds) of solutions for denoting the sturdiness and firmness of all implemented methods. The chance to find optimal solutions is increased when candidate solutions broadcast over a wide area of search space. Moreover, the average ranking test is used to judge the performance of MVDE and other methods. The best results for all algorithms using different metric indices are presented in bold.In Fig. Table Table We can conclude that the MVDE is an outstanding method with various benchmarked problems which attains the best results of different metric indices to avoid trapping in local minima.The average of selected features are outlined in Table The consuming time is introduced in Table In the case of the complexity of the network, Table In a conclusion, the judgment on the different optimizers are presented in Table Table 30. Eleven datasets that are common among this study and the compared studies in the recent researches. The results due to the comparison among the MVDE and the methods in the literature review indicate that the MVDE outperforms the other four methods on iris, Australian credit, German credit, hillvalley, and waveform datasets. In contrast, a slight bad performance of classification on ionosphere, vehicle, wine, zoo, WBC, and heart datasets. For the heart benchmark, MVDE attains well performance of classification over one method, but worse than the other one. On the other side, the MVDE chooses a fewer number of features compared to all other algorithms for all benchmarks. However, it selects a slight more selected features’ number on ionosphere dataset than only one method. In addition, the computational time that is achieved by MVDE algorithm is typically shorter for all datasets except for WBC dataset since the number of selected features is smaller. Additionally, the results validate that the MVDE algorithm’s performance is competitive to the state-of-the art algorithms. MVDE benefits from the advantages of DE algorithm in addition to the modifications applied to it for overcoming DE drawbacks. MVDE is a parameter free optimization method with a high performance and applicable to deal with complex high-dimensional problems. The design of MVDE algorithm with multi variant mutation and adaptive scaling factor help it to select the optimal number of features used for classification which has an impact on the accuracy of the classification.To authenticate the performance of the MVDE technique, the comparison among the MVDE and the lately four published algorithms by other researchers is presented in Table 12This work presents a novel optimization algorithm (MVDE), which has multi variant mutation with adaptive scaling factor is developed by integrating adaptive crossover rate with mutation factors and adaptive selection of parent to achieve better performance. The performance of the MVDE algorithm is verified using fifteen real-world problems to ensure its stability, quality, and simplicity. In this paper, the performance and the complexity of the ANN for training process are optimized simultaneously. Different weights, biases, number of nodes of the hidden layers, and selecting inputs during the search process have the opportunity to be checked by providing this method. So, an effective model of ANN with low complexity and classification error has the chance to be found. The goal challenge to balance between exploration (diversification) and exploitation (intensification) is achieved utilizing the MVDE and a varied population is preserved through iterations.The evaluation is implemented using a set of evaluation criteria to evaluate different aspects of the MVDE algorithm. In addition, DE, CSO, WOA, and SCA methods are applied to solve the problems and to compare the results to MVDE method. The results of the methods in the literature review and the MVDE algorithm are compared to evaluate the performances.The investigative results infer that the MVDE optimization method is a useful and an appropriate technique to classify data and can move to a particular group of benchmarks. The results also investigate the ability of the MVDE algorithm for dodging the local minima better than the compared DE, CSO, WOA, and SCA methods. The performance of the selected features is promising and better for the features selected by the MVDE. Furthermore, the superiority of the MVDE performance is obviously detected for training ANNs in terms of evaluation metrics.


This paper documents daily compound indicators on physical mobility and sales activity in Switzerland during the Corona crisis. We report several insights from these indicators: The Swiss population substantially reduced its activities already before the shops closed and before the authorities introduced containment policies in mid-March 2020. Activity started to gradually recover from the beginning of April onwards, again substantially before the first phase of the shutdown easing started at the end of April. Low physical mobility during the second half of March and during April likely contributed to the quick fall in new COVID-19 infections since mid-March. The sharp drop in economic activity in consumer-related services during March and April and the gradual recovery in these sectors since May correlate strongly with the reduction and subsequent gradual resurgence of mobility. In addition, while activity within Switzerland was back to normal levels by late June, activity of Swiss residents outside of Switzerland was still below normal. The Corona pandemic caused a sharp drop in Swiss economic activity in March and April 2020To overcome these challenges, shortly after the start of the Corona crisis, we started collecting a diverse set of daily and weekly data series. We eliminated seasonal patterns, dealt with missing data and ragged edges, extracted common cyclical components via principal component analysis and applied different scaling methods. From this procedure, we obtained daily compound indicators on physical mobility, on sales activity, on economic activity inside Switzerland and on international travel activity of Swiss residents. Also, we constructed an encompassing activity indicator. Since March, we publish the indicators in the Our focus on mobility and spending activity of the Swiss population has two reasons. First, social distancing is a key factor to contain the spread of the virus. Physical mobility is a proxy for social distancing, since the latter is largely achieved by a reduction in the former. Thus, by providing an indicator on the degree of physical (im)mobility, we intended to help monitoring the degree of social distancing exerted by the Swiss population during the crisis. Second, there was an increasing tendency of the Swiss people to reduce their activities well before the strict imposition of physical distancing rules. This response to the spread of the virus was certain to induce a steep fall in economic activity in consumer-related services. We wanted to help tracking this development by providing a compound indicator on spending activity in Switzerland. Indeed, our compound indicators have been cited by Swiss sources in order to report on mobility, sales activity and economic activity in general during the Corona crisisThis paper documents the construction procedure of the daily compound indicators. In addition, the paper provides several insights on mobility and sales activity in Switzerland during the Corona crisis. These insights are summarized in the conclusion of the paper.The paper is structured as follows. Section The series underlying the indicators cover aspects of physical mobility and/or economic activity related to purchases and sales of goods and services. The majority of the series cover Switzerland as a whole, whereas some series are limited to the Zurich region. We take these latter series as the best available proxies for overall Switzerland. The data come from various sources and are assembled and provided by the Statistical Office of the Canton of Zurich as part of its Corona crisis monitoring initiative (Gesellschaftsmonitoring COVID19 ). FurtheMost time series used in this paper are observed at daily frequency and start on January 7, 2020. Some series come at weekly frequency and start in the second week of January 2020. We temporarily disaggregate these series to daily frequency by linear interpolation, assuming the actual observation to take place in the middle of the week. The inclusion of the weekly variables provides additional information on the broad trajectory of the principal component. This benefit outweighs the potential drawback of less precision at the highest frequency. The tradeoff between information and precision is further investigated in Appendix Most of the daily series have strong weekly seasonality patterns. We adjust for the weekly seasonality using the seasonal-trend decomposition (STL) based on locally estimated scatterplot smoothing (LOESS) . Some seIn order to construct the compound indicators from the underlying series, we compute the first principal component using a singular value decomposition of the series. All series are centred at zero and are scaled to have unit variance before the principal component analysis (PCA) is applied. To check for robustness, we also experimented with various forms of factor analysis e.g.,  and founFor the purpose of standardization, we index the average of the period from January 7, 2020 to February 28, 2020 to 100 for each resulting indicator. Underlying this standardization is the presumption that the levels of the series have been fluctuating in a normal range during the aforementioned period, which was prior to the outbreak of the crisis in Switzerland. We further standardize the indicators by setting the individual minimum value of each indicator to zero. We call this standardization the “minimum-maximum standardization”. The standardization procedure acknowledges the fact that, due to the shortness of the indicators, one should be careful in making statements on how the indicators compare to each other in terms of the size of their decrease during the crisis. Nevertheless, the standardization allows for statements on the relative position of the indicators with respect to their pre-Corona level (= 100) and their minimum level (= 0) during the peak of the crisismobility indicator includes series related to physical mobility, such as the frequency of persons passing at important train stations, the frequency of private transport vehicles (cars etc.) counted at important measuring stations and the daily average geographical distance covered by a representative sample of the Swiss population. Table 1 in Appendix sales activity indicator encompasses series that are related to the purchase and sales of goods and services such as the volume and number of credit and debit card transactions or Google searches for shopping goods. The full variable list is given in Table 2 in Appendix indicator on activity inside Switzerland comprises all available series that reflect economic activity inside Switzerland, whereas the indicator on international travel activity of Swiss residents includes series such as Swiss credit card transactions abroad and the number of flight arrivals/departures at Zurich Airport. Tables 1 and 2 list all variables included in the aforementionend indicators. Finally, the encompassing actvity indicator comprises all available high-frequency series and, thereby, attempts to capture the daily activity of the Swiss population as broadly as possible given the available data.We compile five compound indicators. First, the The first recorded Swiss COVID-19 case occurred in the canton of Ticino and was confirmed on February 25 by the Swiss Federal Office of Public Health (FOPH). On February 28, the Swiss government declared the situation in Switzerland a “special situation” according to Swiss pandemic law and banned events with more than 1000 participants. On March 1, the FOPH launched the prevention campaign “How we protect ourselves” including public posters, leaflets, a telephone hotline and a website. The campaign included recommendations on personal hygiene and social distancing measures.The virus quickly started to spread throughout Switzerland with 29 recorded cases by March 1, 125 recorded cases by March 5 and 542 recorded cases by March 10. The first recorded Swiss COVID-19 fatality occurred on March 5.On March 16, the Swiss government declared an “extraordinary situation” according to Swiss pandemic law. All shops (except food stores and pharmacies), markets, restaurants, bars, entertainment and recreation facilities had to be closed (“lockdown”). Private and public events were prohibited. Cross-border mobility for non-resident foreigners was heavily restricted. Further, companies started to introduce home office as a temporary working standard. On March 20, the government banned gatherings of more than five people. In its prevention campaign, the FOPH now appealed to everyone to generally stay at home. The slogan “Stay at home” was disseminated by virtually all Swiss media following the recommendations of the FOPH and the Swiss Media Association.The reopening of the economy and social life took place in several stages: From April 27 onwards, selected shops, e.g., hairdressers, do-it-yourself stores and garden centres, were allowed to reopen, provided that they introduced certain social distancing and health measures (“Easing Phase 1”). Shops, restaurants, markets, museums and libraries could reopen on May 11, again provided the implementation of certain social distancing and health measures. Also, compulsory schools reopened (“Easing Phase 2”). From June 6 onwards, private and public events up to 300 persons were again permitted, and secondary schools opened. Further, leisure and tourism businesses, such as mountain railways, camping sites, casinos, amusement parks, zoos, botanical gardens and swimming pools, were allowed to reopen (“Easing Phase 3”). On June 15, Switzerland completely opened its borders to all EU/EFTA states. Cross-country shopping tourism, which is important for border regions, was also permitted again.Figure From the end of March onwards, activity gradually started to recover. When the government introduced the first stage of relaxations on April 27, activity was already back to around one fourth between its minimum level and its pre-Corona level. One explanation is that the Swiss population increased its activity as a reaction to the strong decrease in the daily COVID-19 infections (see below). An additional factor might be that the Swiss government had announced on April 16 a timetable for relaxing the containment measures. Activity started to increase quickly after this announcement.The pace of normalization increased further since early May. The temporary flattening of the indicator at the end of May and the beginning of June may be attributed to poor weather. Since then, normalization has continued with the activity being back to pre-Corona levels in the last week of June.Figure Figure Figure Shortly after the start of the Corona crisis in Switzerland, we began to publish daily compound indicators on physical mobility and spending activity of the Swiss population. Daily updates were made publically available via the This paper presented compound indicators on physical mobility, sales activity, activity inside Switzerland and international travel activity of Swiss residents, as well as an encompassing activity indicator. The indicators have been constructed from a set of time series including the frequency of persons passing at important train stations, the frequency of private transport vehicles (cars etc.) counted at important measuring stations, flight departures and arrivals, the volume and number of credit and debit card transactions and Google searches for shopping goods, amongst others.We gained several insights from the daily indicators. To begin with, the Swiss population had substantially reduced its activities already before the shop closings, the general conversion to home office and the appeal to generally stay at home in mid-March 2020. This behaviour may be a response to the perceived infection risk. Next, from the beginning of April onwards, activity gradually recovered. When the Swiss government introduced the first phase of relaxations on April 27, activity was already back to one fourth of its pre-Corona level relative to its minimum level. Also, a comparison of physical mobility and COVID-19 infection numbers suggests that social distancing behaviour (proxied by immobility) of the Swiss population during the second half of March and during April correlates strongly with the quick fall in new infection cases since mid-March. Further, physical mobility and spending activity moved very closely together during the Corona crisis. The joint pattern suggests that the decisions of the Swiss people to reduce their mobility has contributed to the steep fall during March and April in value added of the consumer-related service industries. The pattern further suggests that the gradual recovery in these industries since May is related to the gradual resurgence of mobility. Finally, while the activity inside Switzerland was back to normal levels by the end of June, the activity of Swiss residents travelling abroad was still below normal.Physical distancing has proven to be effective in suppressing the Corona pandemic until June 2020. In order to contain a second virus wave, precautionary measures will stay in place for the foreseeable future. While it is too early to determine the level of social distancing that keeps the pandemic in check, it seems certain that this level lies above pre-Corona levels. The Swiss population decreased its mobility already well before the lockdown, and it has increased its activity steadily instead of abruptly during the easing steps. This behaviour may be interpreted as a sign for the individual responsibility of the Swiss people. As a consequence, timely information from the authorities on new infections seems to be important to contain new virus waves. Timely and effective dissemination of new insights on how individuals should adapt their personal behaviour in order to reduce the infection risk seems to be another key ingredient to overcome the crisis.The following tables provide an overview of the data used in this paper, along with meta information such as frequency, coverage and source. The tables furthermore show which series contribute to which compound indicator.Figure Figure Scaling using mean-variance standardization hinges on the assumption that an indicator’s standard deviation of the period Jan.-Feb. 2020 is representative for its standard deviation during “normal times.” As far as this assumption is fulfilled for each indicator, they can be readily compared to each other in terms of the size of their decrease during the crisis.Figure