Complex mechanical parts such as a blisk of aero-engines are commonly used in aerospace industry. These parts are complex in shape and their rough machining are conducted in 4-axis machine tools with end mills. The end mills are fully engaged into the workpiece material to be removed. Because of the complex cutter motion in 4-axis milling, the bottom edges of the end mills are involved in cutting with high possibility, resulting in an undesirable increase of cutting forces, tool deflection, and quick tool wear. To address this technical challenge, an analytical method is proposed to identify and evaluate the bottom edge cutting in 4-axis milling in this work. The motion of the cutter’s tool tip with respect to the workpiece is analyzed and the equations are formulated based on a basic interpolation algorithm. An approach to identifying and evaluating the bottom edge cutting is proposed. The increment of the cutting forces caused by the bottom edge cutting is taken into consideration to precisely evaluate the overall cutting forces. A feed rate optimization model is then established to control the cutting forces. The simulation and the experiment of rough milling of a blisk verify that the bottom edge cutting can be identified and the cutting force can be controlled by optimizing the feed rates without losing much machining efficiency. Complex mechanical parts such as blisks of aero-engines are commonly used in aeronautic and astronautic industries. These parts are complex in shape and their rough machining is conducted in 4-axis Computer Numerically Controlled (CNC) machine tools usually with end mills. The end mills are fully engaged into the workpiece material to be removed see . BecauseThe bottom edges of cutters engage in cutting in many machining processes such as orbital drilling and plunge milling. Kong et al.  pointed However, the bottom edge cutting in rough milling with multi-axis machine tools has been more and more in the focus in the recent years. Zhu et al.  argued tDuring the rough milling of complex parts, cutting parameters need to be carefully determined to achieve the objectives of rough machining. The cutting parameters include feed rate, spindle speed, width and depth of cut, etc., and the objectives could be different such as minimum machining time, maximum material removal rate, and maximum uniformity of the remaining volume at the end of roughing . The cutUnfortunately, the bottom edge cutting in 4-axis rough machining has not been successfully addressed in these existing research works. To address this technical challenge, an analytical method is proposed to identify and evaluate the bottom edge cutting in 4-axis milling in this paper. In In aerospace industry, the rough milling of complex parts is conducted in multi-axis machine tools, and the CNC controllers of the machine tools use different CNC interpolation algorithms. In order to evaluate the bottom edge cutting with high fidelity, the instantaneous cutter positions and orientations (or cutter locations) in machining should be accurately computed by using the actual machine kinematics and the interpolation algorithm of the CNC controller. Unfortunately, most conventional methods approximately calculate cutter locations without taking the interpolation algorithm into consideration, by which the large deviations from the actual locations are inevitable. This work adopts a 4-axis horizonal machine tool  and a basic 4-axis CNC interpolation algorithm of constant-acceleration interpolation algorithm as example. The methodology of this work can be applied to 4-axis machines with different CNC controllers. The main steps of formulating bottom edge cutting are (1) according to two cutter locations in steps of an NC program, several instantaneous cutter locations in the machine coordinate system are sampled and calculated by using the constant-acceleration interpolation algorithm, (2) the instantaneous cutter locations are converted into the workpiece coordinate system by using the machine kinematics, and (3) the feed rate of the tool tip is formulated in the workpiece coordinate system, and it is adopted to evaluate the bottom edge cutting. The technical details are given as follows.To cut the parts on 4-axis machine tool, discrete cutter locations of tool paths are preplanned using the CAM software, in which cutter positions  and orientations (B-coordinate) are calculated, and these Cutter Location Data (CLData) are translated to G-code feeding to the CNC controller. The feed rate for each cutter location is preplanned as well. In milling, the CNC controller interpolates many instantaneous cutter locations according to consecutive cutter locations in the NC program, and the cutter is controlled to move from one location to the next. Since the interpolation algorithm is not disclosed, it is reasonable to adopt a basic CNC interpolation algorithm, which is a constant-acceleration interpolation algorithm, in this work as an example. Although the adopted algorithm is different from those in other CNC controllers, this method is generic by using the actual algorithm.The constant-acceleration interpolation algorithm is briefly described here. Suppose a step including two cutter locations By using the machine kinematics, these cutter locations are converted into the workpiece coordinate system.B of the machine tool’s table is zero; (b) B when the rotation angle B is not zero. In this case, angle B represents the tool orientation. At last, the tool coordinate system The machine kinematics is established to convert the instantaneous cutter locations from the machine coordinate system into the workpiece coordinate system. Three coordinate systems are defined see . The oriBased on these coordinate systems, the machine kinematics is established. The transformation matrix The equivalent transformation matrix For a flat end mill, its periphery cutting edges always cut the workpiece material at a preplanned cutting speed see . The botIn the workpiece coordinate system, the instantaneous cutter locations, which include the tool tip position Due to the complex motion in 4-axis machining, the feed directions of the tool tip at the instantaneous cutter locations are different. The feed direction At some moments, the feed direction tter see , in whicFrom Equation (8), a positive feed rate To evaluate the bottom edge cutting, a number of instantaneous feed rates of the tool tip on a step are sampled. The maximum of them is called the maximum instantaneous bottom edge feed rate on the step, and it is denoted as fiM see . The botThe process of milling is unstable if the cutting forces are larger than the normal, resulting in chatters, quick tool wear or cutting-edge chipping. In the worst case, the cutter breaks. In the end of the last section, we have theoretically demonstrated that the bottom edge may be engaged in cutting and it is represented by the maximum instantaneous bottom edge federate computed with Equation (8). Because the bottom edge cutting increases the cutting forces, the effect of bottom edge cutting is taking into consideration to control the cutting forces, achieving stable cutting. In this research study, an optimization model is constructed to control the resultant cutting forces. The objective of the optimization is to minimize the cutting time because the purpose of rough milling is to remove the large amount of material of billets as quickly as possible. To identify the optimization variables, a few key factors of the rough milling process are analyzed. The tool paths are workpiece geometry-dependent and the spindle speed is determined by the workpiece material and the cutters; these two factors are not modified in machining. Therefore, the feed rates are selected as optimization variables. Each cutter location in a tool path can have its own feed rate. For a step, if the maximum instantaneous bottom edge feed rate Constraint 1The cutting forces must be controlled within an acceptable range. Here, in this work, a practical and simple control method is proposed. Based on the well-established mechanistic model of cutting force, milling forces are proportional to the area of the swept cross-section of the cutting edges. For the periphery cutting edges, the area research , CB is fAfter optimization, the feed rate To eliminate the additional effect of the bottom edge cutting on the resultant cutting force, the cutting force caused only by the periphery cutting edges Constraint 2To ensure that the cutter’s acceleration does not exceed the dynamics of the machine tools, the second constraint isTo demonstrate its validity, this approach is applied to a tool path of rough machining a blisk in a 4-axis machine tool. Twenty channels of the blisk need to be machined see a. Each cAccording to the preplanned tool path and its feed rate, the maximum instantaneous bottom edge feed rates are computed by using the evaluation method proposed in The optimization model is established as described in A further experiment is conducted to verify the reduction in cutting force. A tool path is programed using UG NX to cut the channel. In the experiment, a dynameter of Kistler 9367C is employed to measure the cutting forces before and after the optimization see .By evaluating the bottom edge cutting, the feed rates all over the tool path are marked as to-be-optimized. The preplanned and the optimized feed rates are plotted in As can be seen in This paper proposes an analytical approach to identify and evaluate the bottom edge cutting in the rough milling of complex parts. By using the CNC interpolation algorithm, the motion of the cutter’s tool tip with respect to the workpiece material is formulated. As the benefit of doing so, the motion of the bottom edges of the cutters are represented precisely in accordance with the specific CNC controllers. The mechanism of the bottom edge cutting is analyzed. The motion vector of the tool tip is projected onto the opposite of the cutter’s axis to calculate the signed feed rate of the tool tip engaging into the workpiece material. A number of the feed rates are sampled within the step of tool paths and the maximum of these feed rates is used to evaluate the bottom edge cutting. The sign of this feed rate is employed to identify the bottom edge cutting. Then, the cutting forces caused by bottom edges are estimated by computing the area of the cross-section swept by the bottom edges multiped by the cutting force coefficient. An optimization model is established to achieve the high roughness efficiency by optimizing the feed rates, as well as constraining the combination of bottom edge cutting forces and the periphery edge cutting forces. The results of simulation and experiment show that the bottom edge cutting is identified and the cutting force is controlled by optimizing the feed rates without losing much efficiency in rough milling blisks of aero-engines. We believe that this approach can be directly implemented in rough milling impellers of aero-engines, and other complex parts in industry.


With the advancement in next-generation communication technologies, the so-called Tactile Internet is getting more attention due to its smart applications, such as haptic-enabled teleoperation systems. The stringent requirements such as delay, jitter, and packet loss of these delay-sensitive and loss-intolerant applications make it more challenging to ensure the Quality of Service (QoS) and Quality of Experience (QoE). In this regard, different haptic codec and control schemes were proposed for QoS and QoE provisioning in the Tactile Internet. However, they maximize the QoE while degrading the system’s stability under varying delays and high packet rates. In this paper, we present a reinforcement learning-based Intelligent Tactile Edge (ITE) framework to ensure both transparency and stability of teleoperation systems with high packet rates and variable time delay communication networks. The proposed ITE first estimates the network challenges, including communication delay, jitter, and packet loss, and then utilizes a Q-learning algorithm to select the optimal haptic codec scheme to reduce network load. The proposed framework aims to explore the optimal relationship between QoS and QoE parameters and make the tradeoff between stability and transparency during teleoperations. The simulation result indicates that the proposed strategy chooses the optimal scheme under different network impairments corresponding to the congestion level in the communication network while improving the QoS and maximizing the QoE. The end-to-end performance of throughput (1.5 Mbps) and average RTT (70 ms) during haptic communication is achieved with a learning rate and discounted factor value of 0.5 and 0.8, respectively. The results indicate that the communication system can successfully achieve the QoS and QoE requirements by employing the proposed ITE framework. The dramatic growth in communication technologies, specifically Fifth-Generation 5G), has obtained enormous interest due to its evolving generic services like massive machine-type communication, Ultrareliable and Low Latency Communication (URLLC), and enhanced mobile broadband ,2,3. ApaG, has ob latency . The six latency ,7. The w latency  explored latency  presente latency  explored latency  revived In order to achieve 1 ms latency, Software-Defined Networking (SDN) and Mobile Edge Computing (MEC) are utilized in the core network of 5G and beyond networks . SDN proVarious standard organizations, for instance, the International Telecommunication Union, 3GPP, and the Institute of Electrical and Electronics Engineering (IEEE) have been operating to empower the existing and design new architecture to incorporate haptic traffic over a network. The Tactile Internet IEEE P1819.1 standard group has been introduced to describe the basic definition of the Tactile Internet and design for standard communication architecture along with technical functions . In addiThe scope of this paper is to ensure the Quality of Services (QoS) and Quality of Experience (QoE) requirements for teleoperation systems with haptic feedback. The reason behind considering the teleoperation systems only is that most haptic-driven applications utilized teleoperations to interact with the object at a remote location. The basic architecture, along with key components of the haptic-enabled teleoperation system, is illustrated in s (1 ms) ,17. MoreThese stringent requirements expose different challenges to guaranteeing the QoS and QoE provisioning for the emerging services of Tactile Internet. In the literature, various solutions have been proposed based on haptic codecs and control approaches to tackle the above-mentioned teleoperation challenges . The recWe discuss the Tactile Internet challenges in the context of high pack rate/packet size and extensively review different haptic codec solutions and summarized their key contributions.An Intelligent Tactile Edge (ITE) framework is proposed that incorporates the ECN mechanism, Machine Learning (ML), and particularly, RL policies for optimal haptic codecs selection based on network congestion level employing the Markov decision process (MDP).In the proposed framework, an LSTM model is utilized to predict the congestion level in the network by exploiting ECN information in the TCP header of the data packet. To select the best haptic codec scheme corresponding to the network congestion, an RL-based Q-learning algorithm is employed.We conduct different experiments to demonstrate that the proposed system selects the best haptic data compression and reduction scheme for haptic-enabled bilateral teleoperations to ensure QoS and QoE requirements.To this end, there is a strong need for a strategy to achieve the tradeoff between stability and transparency of the system and meet the QoS and QoE requirements under variable time delay and high packet rate. Therefore, in this paper, an RL-based dynamic switching strategy has been proposed where the haptic codecs govern the allocated environment’s (operator/teleoperator) dynamics and network resources. The proposed strategy estimates the network resources in real-time and utilizes one of the famous RL algorithms, Q-learning, to select the optimal haptic codec schemes. The proposed switching strategy aims to gain both stability and transparency while meeting the stringent QoS and QoE requirements of tactile Internet services like teleoperations. The proposed strategy collects the Explicit Congestion Notification (ECN) marking packets during communication and inputs to the Long Short-Term Memory (LSTM) model to estimate the congestion levels in the network and select the best codec scheme to reduce the congestion and provide the desired QoS and QoE. The core network design of the proposed system comprises SDN, and MEC technology to realize reduce network latency and support Tactile Internet applications. As we discussed, the SDN paradigm reduces the network complexity and provides dynamic sharing of network resources. Moreover, it provides flexibility to the network, where the operators can define virtual slices corresponding to use cases with a focus to ensure QoS. The major contributions of this paper are summarized below as:The remainder of this paper is structured as follows. In the literature, various schemes have been suggested to address the communication challenges, such as the high packet rate of the haptic data and the time-varying delay of the network. The haptic data sampling rate challenge imposes a strong demand to packetize and transmit haptic information with a data rate 1 kHz or higher to provide stability and transparency to the teleoperation systems .In , an apprThe work in  offers aSimilarly, the study in  presentsIn this paper, we proposed an ITE framework that dynamically selects the haptic codec scheme corresponding to the level of congestion in the network while attaining the desired QoE performance under variable network resources and required QoS.The proposed strategy aims to predict the level of congestion in the network, compare them with the QoS and QoE requirements of the teleoperation system, and finally provide the optimal QoS and QoE solution to ensure the stringent requirements. To achieve this goal, the proposed RL-based ITE framework deals with the main two challenges: (1) estimate the network congestion and comparison with domain-specific requirements in real-time, and (2) select the optimal haptic codec to decrease the data rate without degradation in QoE performance and improve the system stability and transparency. This section presents a detailed discussion of the proposed ITE to deal with the aforementioned challenges. At first, we will discuss the ECN mechanism and how ECN-marked packets are used to predict the levels of congestion in communication networks. Secondly, we briefly discuss the Q-learning algorithm to select the optimal codecs based on the level of congestion.N refers to the number of samples and L is the number of hidden layers. Around 30 neurons are selected for each hidden layer in the LSTM model. The model is trained on 6000 samples at 100 ms intervals for ten minutes period. The proposed ITE deploys the trained model at the edge, which is also known as the tactile support engine in the IEEE P1918.1 Tactile Internet reference architecture. At each iteration, the model predicts the congestion level in the network and inputs the Q-learning algorithm to select the appropriate action to provide the required QoS/QoE. The work in [ work in  motivatet, and move it into the next state Agent: An agent is an entity, which takes out learning tasks in the system. In the proposed framework the ITE that recommends the haptic codec at the application layer acts as an agent.State: A state of the understudy system provides the observation of the environment that can be examined by the agent. In our proposed framework, we define the discrete level of network congestion as the states Action: An action reveals how an intelligent agent responds to the environment based on the observed state. In ITE, the action A is the set of the haptic codecs schemes, which reduce and compress the packet size or packet rate of haptic traffic during communication to ensure QoS and QoE provisioning. The agent performs an action from the action space of databases of haptic codecs approaches and is denoted as n refers to the total available haptic codecs as listed in Reward: The reward to the system is a scalar value. At each time, when an agent executes an action, the environment returns a reward as a response to the agent. The reward characterizes the behavior (good/bad) of the environment to action and the agent adjusts its policies based on the reward. In this system, our aim is to increase the network performance and provide QoS/QoE by minimizing the congestion that leads to delay and packet loss. The ITE uses the power function of the connection as a reward function and defined it as follows:In this paper, we formulate the dynamic section of the QoS, and QoE provision approaches as the Markov decision process (MDP), where the ITE algorithm selects optimal policy as an agent and the communication network behaves as an environment. The intelligent agent communicate with the environment in sequential decision-making process to observe the current state In the MDP, the successor state In the literature, Q-learning and its variant algorithms have shown robust results to solve MDP problems. It is an off-policy approach that can be employed for any solution in the MDP framework. On the other hand, Q-learning uses, parametric operations to approximate the Q-function and defined it as follows:https://github.com/zubair1811/IntelligentTactileEdge2022, accessed on 15 September 2022) for the reproducibility of our work and the ease of fellow researchers.In this section, we discuss and highlight the efficacy of the proposed ITE framework with haptic-enabled bilateral teleoperation system scenarios using different simulation parametric settings. Default parameter values and settings for experiments and the list of haptic codecs are presented in Our simulations environment is created in Python on a machine with an Intel Core i7 processor, 16GB memory, NVIDIA GeForce GT 1030 graphics card, and 64bit Linux (Ubuntu 18.04) operating system. The LSTM algorithm is programmed in the TensorFlow framework, and the Q-learning algorithm is programmed in Python. A Mininet emulator is utilized to design the communication network. Users may simulate actual network topologies with this tool. It also has built-in support for SDN architecture. For the experimentation, we utilized the publicly available 3 Degrees of Freedom (3DoF) teleoperation system haptic dataset . A humanIn haptic communication, QoS parameters are characterized by E2E delay, reliability, and synchronization . The lat work in  presentsTo investigate the effect of network congestion with no external host and with multiple external hosts during haptic communication among the master and the slave, a simple haptic-enabled teleoperation system is developed and simulated for the number of external hosts = 1 to 20. The latency investigation with direct haptic communication without the external host, with the number of hosts = 10, 15, 20 is illustrated in c refers to the control signals, i is the sample that is transmitted, and T indicates the time instantly. In simple, it can be written the interarrival time of the packet i at the time instant From This section presents the study of the Temporal Difference (TD) error made by the proposed system in the learning process over time to explore the different parametric settings to get faster convergence of the system. The main goal is to attain a better convergence with a smaller number of training epochs to reduce the computation overhead. The TD is defined as follows:To study the effect of the learning rate The E2E performance of RTT during haptic communication with the external host are shown in In , we propIn this work, we proposed an Intelligent Tactile Edge framework to predict the congestion level at the network layer by utilizing the ECN mechanism and adjusting the packet size and packet rate of the haptic transmission at the application layer by employing different haptic codec schemes with a focus ensure QoS and QoE. The proposed ITE utilizes the LSTM model to estimate the network congestion and then the Q-learning algorithm applies to take the action and select optimal haptic codecs to solve the communication network issues . We employ the power function of the connection as a reward function to optimize the Q-learning algorithm and find the pattern. The proposed ITE was investigated on the Tactile Internet standard 3DoF teleoperation traffic dataset. The simulation results suggest that the ITE is able to ensure the QoS and QoE requirements of the haptic-enabled bilateral teleoperation application. The proposed framework was simulated on different parametric settings to tune the algorithm and show the algorithm convergence to select the optimal parametric configuration. In the recent future, we plan to implement the proposed system in real-work application with a haptic device and extended it with advanced machine learning techniques such as federated learning-aware approaches at the ITE to support emerging technologies.


Pythagorean fuzzy sets (PFSs) proved to be powerful for handling uncertainty and vagueness in multi-criteria group decision-making (MCGDM). To make a compromise decision, comparing PFSs is essential. Several approaches were introduced for comparison, e.g., distance measures and similarity measures. Nevertheless, extant measures have several defects that can produce counter-intuitive results, since they treat any increase or decrease in the membership degree the same as the non-membership degree; although each parameter has a different implication. This study introduces the differential measure (DFM) as a new approach for comparing PFSs. The main purpose of the DFM is to eliminate the unfair arguments resulting from the equal treatment of the contradicting parameters of a PFS. It is a preference relation between two PFSs by virtue of position in the attribute space and according to the closeness of their membership and non-membership degrees. Two PFSs are classified as identical, equivalent, superior, or inferior to one another giving the degree of superiority or inferiority. The basic properties of the proposed DFM are given. A novel method for multiple criteria group decision-making is proposed based on the introduced DFM. A new technique for computing the weights of the experts is developed. The proposed method is applied to solve two applications, the evaluation of solid-state drives and the selection of the best photovoltaic cell. The results are compared with the results of some extant methods to illustrate the applicability and validity of the method. A sensitivity analysis is conducted to examine its stability and practicality. Globalization, internationalization, and cross-border activities can have a great impact on the decision-making processes of managers responsible for making these complex interrelated decisions . ManagerMultiple criteria decision‐making (MCDM) is a branch of operations research and an advanced mathematical tool that explicitly evaluates the feasible alternatives over conflicting multiple criteria in decision‐making to identify the optimal solution . As it iThe fuzzy theory was first introduced by Zadeh  to imitaIntuitionistic fuzzy sets (IFSs) are one of the most prominent extensions. IFSs were initially introduced by Atanassov  to generThe suitability of PFSs for expressing ambiguous information has attained many researchers’ attention. Remarkable contributions in the family of multi-criteria decision-making (MCDM) and multi-criteria group decision-making (MCGDM) methods have been made utilizing PF information . Zhang aIn the Pythagorean fuzzy environment, measuring the difference between PFSs plays an important role in inference problems . TherefoSimilarity measures were introduced as an important and useful tool to find the degree of similarity between two objects. Several functions were proposed to express the degree of similarity between sets and were applied in different applications, e.g., physical anthropology, automatic classification, ecology, psychology, information retrieval, medical diagnosis, and pattern recognition . The simSimilarity and distance measures are two faces of a coin. Most similarity measures are based on distance measures. Comparing two PFSs depends to a large extent on both measures. Although many studies proposed several distance measures, still extant distance measures have some flaws . First, Up to now, the extant distance approaches that have been proposed in the literature compute precise distance measures for PFSs. According to Habib et al. , how canThe main defect in the extant measures can be attributed to the treatment of the parameters of a PFS equally, although each parameter has a different influence, i.e., an increase in the membership degree has the same influence as an increase in the non-membership degree, although the former is an advantage and the latter is a disadvantage. Hence, it is important to find a discrimination method that preserves the identity of each parameter of a PFS to avoid the deficiencies of both distance and similarity measures that result in counter-intuitive results.Since the two parameters of a PFS, the membership rdinates . In thisIn this article, the concept of differential measure (DFM) is introduced as a new approach for comparing PFSs. It is a preference relation between two PFSs by virtue of position in the attribute space and according to the closeness of their membership and non-membership degrees. The extant distance measures and similarity measures have several drawbacks due to the equal treatment of the membership (support) and non-membership degree (opposition), although each direction has a different implication. The DFM utilizes signed distance to account for the impact of each component of a Pythagorean fuzzy assessment. An increment in the support direction is considered a positive step, while an increment in the opposition direction is considered a negative step. Since similarity measures indicate how much two PFSs are similar and do not reveal which one is better and, to what extent, the DFM can better indicate the discrimination degree of PFSs when making a decision. Two Pythagorean fuzzy sets (PFSs) are classified as identical, equivalent, superior, or inferior to one another, giving the degree of superiority or inferiority. A new method for MCGDM is proposed based on the introduced DFM. Furthermore, a new technique for computing the weights of the experts is developed. Two practical problems, namely, the evaluation of solid-state drives and the selection of the best photovoltaic cell, are solved in the Pythagorean fuzzy environment using the proposed method. Then, the applicability and feasibility of the model developed in this article are demonstrated by comparison with some existing techniques. Additionally, a sensitivity analysis is conducted to illustrate its stability and practicality. Consequently, the contribution of this study is triple manifold.The concept of DFM is introduced as a new approach for comparing PFSs. It is a preference relation between two PFSs according to their position in the attribute space. A DFM preserves the identity of the parameters of PFSs to avoid the drawbacks of the extant distance and similarity measures in discrimination.A new MCGDM method is introduced based on the proposed DFM.A new technique for computing the experts’ weights in MCGDM problems is proposed. This technique can also yield a solution to the MCGDM problem.Since PFSs are more flexible than IFSs to deal with absurdness and uncertainty and to cope with human evaluation information, it is necessary to pay more attention to group decision‐making in this context. ThereforeThe article is organized as follows. In the section “Researchers classified the methods of comparing PFSs to information measures and comparison methods. Information measures encompass several approaches, such as distance measures, similarity measures, entropy, inclusion measures, divergence measure, and correlation coefficients , 54. ComThe information measures have played a crucial role in the development of PFSs theory and its applications . Peng etThe similarity measures between fuzzy sets attracted researchers’ attention due to their wide applications in various fields, such as machine learning, decision-making, image processing, and pattern recognition . The simZhang  developeWei and Wei  proposedWang et al.  presenteThe Pythagorean fuzzy entropy (PFE) is used to measure fuzziness and uncertainty occurring in a PFS. High entropy indicates more information in the process; a high entropy value represents low uncertainty. The PFE is mainly used in Pythagorean fuzzy MCDM to determine the objective weights of the criteria which are essential in the evaluation of alternatives. Peng et al.  proposedGarg  introducThe Pythagorean fuzzy inclusion measure also called the Pythagorean fuzzy subset hood measure indicates the degree to which one PFS is contained in another PFS. Peng et al.  construcXiao and Ding  considerFour comparison methods have been presented to compare PFSs : score fZhang and Xu  defined Zhang  proposedYager  introducIt has been observed from the above studies that almost all the proposed methods for comparing PFSs produce crisp values, regardless of the information structure of a PFS that contains two contradicting pairs. As a result, several drawbacks might arise in some cases which will influence the ranking values of the alternatives whenever PFSs are used to express the values of the criteria. The main aim of the study is to propose a new measure that can better discriminate PFSs and avoid generating counter-intuitive results while preserving their information structure to reduce information loss and retain more original information. This, in turn, will increase the validity and reliability of the obtained results.. A PFS \The degree of hesitation, i.e., the degree of indeterminacy, is represented by(i)(ii)(iii)(iv). For anyY) and the Pythagorean fuzzy weighted geometric operator (PFWGY) are defined by(i)(ii). For a s. The sco. The acc. The comHussian and Yang  and Xiao, 63. The, 63 The Recently, Song et al.  pointed function , alternadistance  or the ndistance , we get and Ding  and Huanand Ding .Sure, When comparing the alternatives Consider the alternatives function  and the function , we get\The parameters of a PFS ectively . Hence, function  which ac another . This brSince the Pythagorean fuzzy human assessment encompasses two conflicting parameters, it is more convenient to express the differential measure by a PFS The differential measure between two PFSs A differential measure between two Pythagorean fuzzy sets (PFSs) is a preference relation between two PFSs based on the closeness of their membership and non-membership degrees according to their position in the attribute space. The differential measure between two PFSs If If If If Two PFSs For the PFSs(i)if and then(ii)if and thenThe proof follows from the definition of the score function .If B~2 from , hence \If Then, If B~2 from , hence \If Then, For the PFSsAnd if In this section, some examples are presented to illustrate the application of differential measures.Let Then, Let Then, It is clear that the degree of superiority of Let Then, in this case, Let Here, by Zhang 16\documeSuppose distance \documentAlthough Calculating the degrees of similarityBoth points have the same degree of similarity to the positive ideal solution. Then, both have the same effect when making a decision. Although the accuracy function  of \docue choice . By intuLet Let Hence, This subsection aims to illustrate how DFM works versus how some extant information measures work. Hence, the proposed DFM is compared with some extant information measures to demonstrate the difference in the results and the reasonability of the DFM.The DFM between two PFSs Three distance measures are employed in comparison, the normalized Hamming distance , the norgiven in .The most commonly used similarity measures are distance-based similarity measures, and vector-based similarity measures, for instance, Cosine, Dice, and Jaccard similarity measures. Distance-based similarity measures utilize different distance measures, i.e., distance and similarity measure is a dual concept. A general form of distance-based similarity measure between two PFSs can be defined ast}fx=e-x . Since tAdditionally, the previously mentioned three vector-based similarity measures are employed in comparison. The cosine similarity measure is given by 21\documeThe Dice similarity measure is defined by 22\documeThe Jaccard similarity measure is defined by 23\documeA correlation measure is employed in the comparison as well. Garg’s correlation measure defined by 24\documeIt can be seen from the previous formulas that some information measures use the two independent degrees Two PFSs A~ using , 18), , , a, a\documA~ using . The PFSIt is clear by intuition that mentclass2pt{minim, consequently the same degree of similarity to , but in opposite directions, From Table  Hamming  and the  Hamming . The com the DFM  directlydistance  and the distance  but moreRegarding the computational complexity of the proposed DFM relative to the extant information measures, the number of operations required by the methods used in the comparison is calculated. For distance measures, the number of operations ranges from distance  with thr measure  with twocument}$$n$$\end{dBased on the aforementioned DFM, an algorithm for MCGDM problems can be summarized as follows:Step 1: Identify the alternatives Step 2: (i) Form the decision matrices and determine the weights of the criteriaith alternative for the jth criterion as evaluated by the pth expert, and jth criteria.(ii) Normalize the decision matrices by replacing the rating of the cost criteria with its conjugate .Step 3: Compute the aggregated decision matrix using  for each alternativeStep 6: Compute the score function of the CDFM and rank.Using the score function , the criThe CDFM represents the total degree of superiority of the ideal ratings to the ratings of an alternative for the evaluation criteria. The smaller the degree of superiority of the ideal rating, the better the alternative is. Hence, the alternatives are arranged in ascending order and the alternative with the smallest value is the best.Moving from a single expert to multiple experts became essential with the increasing complexity of the socio-economic environment as it is hard for a single expert to handle all the relevant aspects of a complex problem . In grouThe experts’ weights are ordinarily determined subjectively. They are set by a supervisor or mutual evaluations of each other. This way of determining the experts’ weights has low credibility. Hence, objective methods that utilize quantitative techniques are employed to get the experts’ weight from a more objective perspective .Zhang  proposedThis study proposes a simple method to determine the unknown experts’ weights based on their consensus on the ranking of the alternatives. The ranking of each expert is obtained using the weighted sum method (WSM). Then, the correlation between the rankings of each two experts is calculated using Spearman’s correlation coefficient. The total correlation of an expert is calculated by adding his/her correlation with other experts. Finally, the total correlation of each expert is divided by the sum of the experts’ total correlation.First, the judgments of each expert for an alternative for the evaluation criteria are aggregated using the Pythagorean fuzzy weighted averaging operator .Step 1: Form the decision vector of each expert.pth expert for the ith alternative are aggregated using  are widely used as secondary memory in modern computing systems instead of magnetic hard disk drives (HDDs). The performance of HDDs became stagnant due to the limitations in the seek time of actuator arms and the rotational speed of magnetic platters. On the other hand, SSDs do not have complex mechanical parts. This lowers both latency and failure rates than HDDs. Moreover, SSDs offer superior bandwidth, lower power consumption, higher random I/O performance, higher shock resistance, and improved system reliability compared to HDDs .An enterprise wants to choose a type from five available types of solid-state drives (SSDs) denoted by entclass1pt{minimaUsing the decision matrix, the differential measure between the ideal ratings and the ratings of each alternative for the criteria is calculated using . Then, tThe differential measures of each alternative are aggregated to get its collective differential measure. Then, using the score function , the CDFThe result of the proposed method is compared with the results of other decision- making methods namely, the TOPSIS method proposed by Zhang and Xu , the TODIt can be seen from Table The current energy crisis in the world due to the scarcity of natural fuels in the earth’s crust has made great attention to the photovoltaic (PV) energy conversion technology . PhotovoA photovoltaic cell is a device that converts light energy into electricity by a process called the photovoltaic effect which is a known physical and chemical phenomenon. Several types of PV cells are categorized by generation. The first generation is the traditional type made of monocrystalline silicon or polycrystalline silicon and is most commonly used; they dominate the current solar energy market (over 90%). Attempts were made to develop new materials with low-cost fabrication technology due to the high-cost manufacturing of crystalline silicon. The second generation is the thin-film cells that were fabricated by cheaper technologies. The most commonly used materials are amorphous silicon (a-Si), nanocrystalline silicon (nc-Si), cadmium telluride (CdTe), and copper indium gallium selenide (CIGS). Yet, their performance is not higher than the first-generation cells. The third generation was designed to achieve both high efficiency and low cost. They include a variety of thin-film technologies, some of them generate electricity using organic materials, and others use inorganic substances, e.g., dye-sensitized cells (DSSCs), quantum dot-sensitized cells (QDSSCs), organic solar cells, and hybrid perovskite cells .After the selection of the correct location to install a photovoltaic solar plant, it is required to select the kind of cell, from the various available PV cells, that optimizes the installation, e.g., increases production or efficiency, minimizes costs, and grants high maturity and reliability .It is required to select the best photovoltaic cell from the following alternatives: crystalline silicon nt}PFWAY  togetherIR using . The PytThen, the differential measure between the ideal rating and the rating of an alternative for the evaluation criteria is calculated using , and theThe differential measures of each alternative are aggregated to get its collective differential measure. Then, the score function  is compuIt is clear from Table Zhang  introducFrom Table In the previous PV cell selection problem, the experts’ weights can be determined using the proposed technique. The steps are illustrated as follows.Step 1: Form the decision vector of each expert When the PV cell selection problem is resolved using the weights obtained by the proposed technique, the results obtained are given in Table It is worth noting that this technique can yield the best alternative while determining the experts’ weights. Having obtained the ranking of the alternatives for each expert, if an alternative is in the first position by all experts, then it is the best choice, and no need to proceed unless a complete ranking list is required. In the previous example, the alternative Procedure 1 The best alternative does not change when a non-optimal alternative is replaced by another worse alternative, keeping the relative importance of each decision criterion unchanged.Procedure 2 When the decision problem is decomposed into smaller subproblems, the rankings of alternatives should follow the transitivity property.Procedure 3 When the rankings of these subproblems are combined, the new overall ranking of the alternatives should be identical to the original overall ranking of the undecomposed problem.This section aims to explore the reliability and stability of the proposed MCGDM method. Wang and Triantaphyllou  came up The previously two solved problems are examined using these procedures as follows.In the SSD evaluation problem, to apply Procedure 1, one of the non-optimal alternatives is replaced by a worse alternative. Since mentclasspt{minimaFrom Table To apply Procedure 2 and Procedure 3, three subproblems are solved. Each sub-problem consists of three alternatives, i.e., two alternatives are omitted each time from the original problem. The results obtained are given in Table From Table Regarding the PV cells’ problem, the alternative From Table To apply Procedure 2 and procedure 3, six subproblems are solved. First, three subproblems are solved each having two alternatives. Then, another three subproblems are solved each consisting of three alternatives. The results obtained are given in Table The transitivity property is demonstrated in Table From the previous illustrations, the proposed method successfully fulfilled the evaluation procedures. Therefore, it can be concluded that it is consistent and stable.This article proposed the concept of the DFM for the discrimination of PFSs. It is a preference relation between two PFSs by virtue of position in the attribute space and according to the closeness of their membership and non-membership degrees. The main objective of the DFM is to overcome the deficiencies that might arise in the existing measures. The drawbacks in the extant distance measures and similarity measures result from handling the membership (support) and non-membership degree (opposition) equally, although each direction has a different implication. A DFM treats the changes in the parameters of a Pythagorean fuzzy assessment differently. An increment in the support direction is considered a positive step, while an increment in the opposition direction is considered a negative step. Two PFSs are classified as identical, equivalent, superior, or inferior to one another giving the degree of superiority or inferiority. An MCGDM method is proposed based on the introduced DFM and a new technique for computing the weights of the experts is developed.Two practical problems were solved to illustrate the method. The results of the proposed method are compared with the results of some extant MCDM methods. The main aim of the comparison is to verify and demonstrate the applicability of the method. None of the MCDM methods can be considered better than the others; there is no worst or best technique. Different results can be obtained through distinct methods, even if the processed data and information are the same. We can find a method that is most relevant in a given particular situation . The proThe first example is the evaluation of solid-state drives. The result of the proposed method is compared with the results of other decision-making methods, namely, the TOPSIS method proposed by Zhang and Xu , the TODThe second example is the selection of the best photovoltaic cell. The result of the proposed method is compared with the approach of Zhang  and the A sensitivity analysis was conducted to examine the performance of the proposed MCGDM method. Various MCDM methods have different stabilities. For example, the stability of the MULTIMOORA method is good; for the TOPSIS method, it is poor; for the VIKOR and the ELECTRE methods, it is medium . The proFinally, a new technique for determining the weights of the experts is developed based on Spearman’s correlation coefficient and applied to the photovoltaic cell problem. The obtained experts’ weights are slightly different from the weights obtained by Zhang . Yet, thPractically, the proposed DFM can be applied in different applications, e.g., image processing, pattern recognition, machine learning, information retrieval, medical diagnosis, and decision-making. In the decision-making process, the proposed framework can eliminate the biases that may arise due to assigning subjective weights to the decision-makers that can lead to unreliable results in many cases . The fraPFSs are designed to address only two components of human perception independently, namely, preference and non-preference represented by the membership and non-membership degrees. The third component of human perception, hesitation, is handled as a dependent component. Actually, hesitation can be handled as an independent component with the previous two degrees. This is not covered by the frame of PFSs.The proposed differential measure lacks symmetry, i.e., The proposed framework has the following limitations:Hence, further investigations are required to propose a symmetric differential measure in future work. The concept will also be developed in the spherical fuzzy environment which encompasses three independent degrees, membership, non-membership, and hesitation.


As the typical application of computational intelligence in software engineering, cross-project defect prediction (CPDP) uses labeled data from other projects (source projects) for building models to predict the defects in the current projects (target projects), helping testers quickly locate the defective modules. But class imbalance and different data distribution among projects make CPDP a challenging topic. To address the above two problems, we propose a two-phase feature importance amplification (TFIA) CPDP model in this paper which can solve these two problems from domain adaptation phase and classification phase. In the domain adaptation phase, the differences in data distribution among projects are reduced by filtering both source and target projects, and the correlation-based feature selection with greedy best-first search amplifies the importance of features with strong feature-class correlation. In the classification phase, Random Forest works as the classifier to further amplify the importance of highly correlated features and establish a model which is sensitive to highly correlated features. We conducted both ablation experiments and comparison experiments on the widely used AEEEM database. Experimental results show that TFIA can yield significant improvement on CPDP. And the performance of TFIA CPDP model in all experiments is stable and efficient, which lays a solid foundation for its further application in practical engineering. Software defects may cause unexpected disasters in the application scenarios, threatening the security of software and even people's lives . During Software defect prediction (SDP) is a method that can assist testers in quickly identifying potentially defective modules  so as toAlthough CPDP focuses on the problem of lack of historical data, there are still two major problems to be solved. Due to a variety of factors such as project functionality and developers' habits, data distribution varies greatly from project to project. Differences in data distribution between projects are evident even when the same metrics are used to evaluate projects , which mTo address the above problems, in this paper we propose a two-phase feature importance amplification (TFIA) CPDP model. Specifically, TFIA divides CPDP into two phases. In the domain adaptation phase, TFIA reduces differences in data distribution among source and target projects. After adding a certain proportion of samples from target projects to the source projects, the correlation-based feature selection method with a greedy best-first search strategy is proposed to amplify the importance of features with high feature-class correlation. The subset made up of the final selected items is used to filter the source and target projects for the purpose of removing redundant features as well as interfering items and reducing the dimensionality of the data. In the classification phase, resampling is conducted on the source project's data to make the numbers of clean and defective samples balanced. And an integrated learning method, Random Forest, is adopted to build the prediction model. The reason for adopting Random Forest is that integrated learning methods have shown their advantages in solving the class imbalanced problem . In our We propose a two-phase feature importance amplification CPDP model (TFIA). TFIA reduces the effect of data distribution and class imbalance problems in domain adaptation phase and classification phase.We conducted experiments on the AEEEM database to verify the validity of the method. In addition, we performed a number of ablation experiments to carefully analyse and dissect the detailed components of our method. And we also conducted comparison experiments to compare TFIA with other methods.In summary, our contributions lie in twofold.CPDP usually includes two phases, called domain adaptation and classification. Domain adaptation is mainly addressed by data processing methods, such as data migration and feature selection. Classification is mainly handled by machine learning classifiers . The resk samples with the smallest Euclidean distance as training samples. Finally, the Naive Bayes method was used to predict the processed dataset. Burak filter improves the performance of CPDP model by filtering the samples of the source project datasets based on the target project datasets. Peters et al. . AUC for random prediction is 0.5. The advantage of AUC is that AUC is insensitive to decision thresholds such as precision and recall. The higher the AUC, the better the prediction.Matthews correlation coefficient (MCC) is used in machine learning as a measure of binary (2-category) quality of classification, which is introduced from biochemistry by Brian W. Matthews in 1975 .MCC takes into account true and false, positives and negatives, and is generally regarded as a balanced measure that can be used even if the classes are of very different sizes . MCC is The formula for the MCC is shown byIn this section, we present the experimental results and give the answers to questions from To answer RQ1, we designed two groups of experiments on the AEEEM database, where we formed a total of 20 source-target project pairs from the five projects, including EQ, JDT, ML, LC, and PDE. The first group of experiments performed no filtering on these 20 source-target project pairs. And the second group of experiments were filtered by CFS based on greedy best-first search strategy proposed in this paper. Both sets of experiments used Random Forest as the classifier. Both groups of experiments randomly added 40% of the target project data to the source project dataset and source project datasets were dealt with through resample.From To answer RQ2, 20 source-target projects from the AEEEM database were trained and validated using Naive Bayes (NB), Logistic Regression (LR), Decision Tree (DT), Support Vector Classification (SVC), and Random Forest (RF) as classifiers. Respectively, all the datasets were processed by the filtering method proposed in this paper. Every set of experiments randomly added 40% sample from the target project datasets to the source project datasets and the source project datasets were dealt with through resample.To analyse the effect of the proportion of samples from target projects datasets added to the source projects datasets on the prediction results, we added different proportions of labeled target item samples to the source projects datasets in steps of 10% from 0% and trained with TFIA. Every source project dataset had been dealt with through resample.Then we smoothed the values to plot the graphs as To answer RQ4, we used TCA, Burak filter, Peters filter, and ALTRA on 20 source-target project pairs in the AEEEM database for prediction and compared with our proposed method, respectively. To ensure the consistency of the experimental conditions, the source items in the dataset were randomly added to 40% samples of the target project datasets, and the source project datasets were dealt with through resample.As can be seen from the data in In this paper, the threats to validity are mainly divided into internal and external validity.The internal validity comes from the setting of classifier parameters. In this paper, we set the parameters defaulted by the references as well as the tool of WEKA, which may lead to some differences in the classification results. To mitigate this difference, we use the same default parameters provided by WEKA for the classification phase of our proposed method.The external validity mainly comes from the dataset used. We use five project datasets from the publicly available AEEEM database and combine 20 source-target project pairs for our experiments. If other data sources are selected, especially those from real-world engineering datasets, different experimental results may be obtained. The attempt on more scenarios will be included in our future research plan.To tackle the problems of class imbalance and different data distribution among projects in cross-project defect prediction (CcrossPDP) which is the typical application of pattern recognition, we propose a model based on two-phase feature importance amplification (TFIA) in this paper. In the domain adaptation phase, TFIA reduces the differences in data distribution between the source and target projects by adding a certain percentage of samples from the target project to the source project. Meanwhile, correlation-based feature selection (CFS) with a greedy best-first search strategy for feature selection is utilized to obtain a feature subset with a weak feature-feature correlation and a strong feature-class correlation. The source and target project datasets filtered by this feature subset are input to the following classification phase. As the tree-based classifiers are sensitive to features with high feature-class correlation, Random Forest, the integrated method based on Decision Tree for pattern recognition, functions in the classification phase. Multiple decision trees can amplify the importance of features and help each other to improve the performance of the whole prediction model.We conducted four experiments to validate TFIA on 20 project pairs from the widely adopted AEEEM dataset. The first experiment demonstrates that the domain adaptation approach we designed is effective and has a significant improvement on the overall model performance. The second experiment demonstrates that Random Forest outperforms other classifiers on the dataset processed by our domain adaptation approach. The third experiment analyses the influence of the proportion of samples added from the target projects on the model. The experiment proves that adding 40% of the target item data makes great improvement on the model performance considering the practical application scenarios. In the fourth experiment, we compare TFIA with four other classical research methods, and the experimental result proves that TFIA provides better and more stable performance.In this paper, our method has been proved to have an improvement effect on CPDP, but there are still some parts that need to be further studied. We will follow two aspects in our future research. The first is to validate the effectiveness of TFIA on other datasets and make it more scalable and robust since our experiments use the subprojects from the same root project. Secondly, since the parameters used in this paper are defaulted, which have a large impact on the prediction model, we will seek to find the effect of adjusting parameters on the model performance. We will also look for the most suitable parameters and the methods that can automatically adjust the parameters according to the real application scenarios.


Accurate tide level prediction is crucial to human activities in coastal areas. Many practical applications show that compared with traditional harmonic analysis, long short-term memory (LSTM), gated recurrent units (GRUs) and other neural networks, along with ensemble learning models, such as light gradient boosting machine (LightGBM) and eXtreme gradient boosting (XGBoost), can achieve extremely high prediction accuracy in relatively stationary time series. Therefore, this paper proposes a variable weight combination model based on LightGBM and CNN-BiGRU with relevant research. It uses the variable weight combination method to weight and synthesize the prediction results of the two base models so that the combination model has a stronger ability to capture time series features and fits the data well. The experimental results show that in contrast to the base model LightGBM, the RMSE value and MAE value of the combination model are reduced by 43.2% and 44.7%, respectively; in contrast to the base model CNN-BiGRU, the RMSE value and MAE value of the combination model are reduced by 35.3% and 39.1%, respectively. This means that the variable weight combination model can greatly improve the accuracy of tide level prediction. In addition, we use tidal data from different geographical environments to further verify the good universality of the model. This study provides a new idea and method for tide prediction. Tide refers to seawater moving periodically under the influence of celestial bodies’ gravitation (moon and sun). The change in the water level caused by tides has a profound impact on engineering construction, the ecological environment and people's development and utilization of marine resources in coastal areas. For a long time, tide forecasting not only provided safety guarantees for port operations, shipping traffic and coastal protection but also provided important information for offshore aquaculture and the prevention of meteorological disasters. Currently, while promoting the use of clean energy such as tidal energy, we find that accurate tidal level prediction is not only directly related to the power generation efficiency of tidal power stations but also the key to formulating a reasonable dispatching scheme for tidal power stations. Therefore, the accurate prediction of the tide level is of great significance to both human activities and the construction and development of the ocean in coastal areas.1 put forward the equilibrium tide theory, Doodson4 used the least squares method to process tide observation data and determine the harmonic analysis constant, but this method is only applicable to the equilibrium tide. To overcome the shortcomings of the harmonic analysis method and improve the prediction accuracy, scholars have been studying and improving it for a hundred years. Kukulka et al.6 introduced nonstationary external force forcing (river runoff and open sea tide) into the basic function of harmonic analysis and proposed a nonstationary fluvial tide model to make the harmonic analysis applicable to nonequilibrium tides such as estuarine tides. Jin et al.7 proposed the nonstationary tidal harmonic analysis method EHA , which turned the amplitude and epoch of partial tide (both are constants) in harmonic analysis into a function varying with time, used the independent point scheme11 and cubic spline interpolation method to optimize the solution of each component tide’s amplitude and delay angle, and generalized the harmonic analysis to the analysis of internal tidal velocity. With the development of computer science and technology, Matte12, Pan13 and others developed NS_TIDE, S_TIDE and other harmonic analysis toolkits based on the above models. Now, the S_Tide toolkit can be applied to the analysis and prediction of all types of tides in theory. Although the harmonic analysis method can effectively predict tides after continuous development, there will still be large errors in the prediction of tides if we only use the harmonic analysis method due to the influence of environmental factors such as air pressure, wind force and terrain in reality14.To carry out tide prediction, many domestic and foreign scholars have established models to describe and study tide observation data. Harmonic analysis is a classical tidal data analysis method. Based on the assumption that the amplitude and lag angle of each component tide are constant, we can calculate the harmonic constants and then predict the component tide. After Darwin15 first used a backpropagation (BP) neural network to predict full-day and half-day tides and then extended the research to multiple sites16. Zhang et al.17 used an adaptive particle swarm optimization algorithm to improve a BP neural network for tide prediction. Later, Zhang et al.18 proposed the gray GMDH model combined with harmonic analysis to predict astronomical tides and nonastronomical tides and achieved good results. Zhu et al.19 used bidirectional long short-term memory (BiLSTM) networks to predict the tide level of the Isabel port. The simulation results show that the prediction result of the BiLSTM network is better than that of the unidirectional long short-term memory (LSTM) networks. Yang et al.20 established an LSTM model and used nearly 21 years of data to predict the tide level of 17 ports in Taiwan. By comparing various models, it is found that the LSTM model has higher stability and stronger prediction ability. Huang et al.21 noted that only using recurrent neural network models22 such as LSTM cannot effectively mine the local features and potential relationships of tide level sequences, so they use a one-dimensional convolutional neural network (CNN) to extract the local features in tide level sequences, which improves the prediction accuracy of the model.With the continuous development of machine-learning theory, analysis and prediction models such as neural networks are emerging, which not only provide a new method for the simulation, prediction and control of complex systems but also bring a new method for tide prediction research. Tsai et al.23 (LightGBM) and eXtreme gradient boosting24 (XGBoost), are also widely used in time series prediction27, which can achieve good results in predicting relatively stationary time series. Moreover, some researchers have also tried to improve the accuracy of time series prediction by combining several single machine-learning models. Zhang et al.28 constructed a prediction model based on the combination of wavelet noise reduction and LSTM, which improved the accuracy of coal mine gas concentration prediction. Han et al.29 constructed a gas concentration residual correction model based on a Markov model and a gray neural network, and the combination model had better results than the single model. However, most combination models put the first prediction results into another model for secondary prediction or simply add the prediction results of the two models to obtain the average value, which does not improve the prediction accuracy of the model.At the same time, we note that in addition to neural networks, models based on ensemble learning, such as light gradient boosting machine21. Through many experiments, we found that BiGRU enjoys a better prediction ability than BiLSTM, so we use the BiGRU network rather than the BiLSTM network for subsequent prediction tasks. The LightGBM-CNN-BiGRU (combination model) will carry out parallel prediction through the CNN-BiGRU network and LightGBM model and then generate the most accurate tide level prediction through the variable weight combination method.To further improve the accuracy of tide prediction and overcome the shortcomings of the above research, we propose a variable weight combination model based on LightGBM and CNN-BiGRU, which will be simply written as LightGBM-CNN-BiGRU (combination model). The combination model is integrated with the LightGBM and CNN-BiGRU models to predict the tide level in a short time. Aiming at the disadvantages that the traditional LSTM network generally has a slow training speed and is weak in extracting sequence features from tidal observations, we add a local feature preextraction module  before the recurrent neural network to preextract the local features of the tide level sequenceThe experimental results show that compared with the base models  and another variable weight combination model based on LightGBM and BiGRU (LightGBM-BiGRU (combination model)), the model we propose (LightGBM-CNN-BiGRU (combination model)) effectively improves the accuracy of tide level prediction and thus can provide more reliable data guarantees for marine construction and development. Our experiment also shows that the combination model can reduce the absolute error to approximately 0.03 m with tide observations of only one quarter, and its architecture is not complicated, so it deserves to be used in practice.To carry out effective tide prediction, most coastal countries will set up tide measurement stations in their main ports to collect enough tidal observations. The data we used in this study come from the real-time tide monitoring network INTGN , which is operated by the Irish Marine Institute. INTGN consists of 20 tide measuring stations. We select the observation data of Howth Harbor station in Dublin, the capital of Ireland, to build our tide prediction model. In addition, we also selected data from four other sites along the Irish coastline to analyze the generality and generalization ability of our model. The location distribution of the five stations is shown in Fig. After simple statistics, only a small number of tidal observations are missing. As shown in Fig. Given a piece of time series, construct a curve passing through those data points and use the curve to predict values of the curve at other time points. We want to interpolate a curve through the three time points shown below. Since we have three time points, the simplest curve that passes through all three of those points will be a quadratic polynomial. Our job is to compute that polynomial and use it to predict the value of y at other time points between The first method to compute the polynomial starts with the assumption that the polynomial takes the formSubstituting the three time points into this curve gives a set of three equations in three unknowns.These equations can be rewritten as a matrix–vector system.By computing the inverse of the matrix, we can then solve this system for the coefficients of the polynomial.We use neural networks when constructing the combination model, and the normalized data can reduce the computational complexity and accelerate the convergence of the network. Therefore, it is necessary to normalize the data. We normalize the tidal observation sequences by applying the Min–Max normalization. The normalization formula is:Let Time series forecasting, in terms of a supervised learning problem, can be formulated as follows:Given a set The predictors in Eq.  consist On the basis of the problem formulation in “entclass1pt{minima23, it is necessary to introduce XGBoost24, which is also based on the gradient boosting decision tree (GBDT) algorithm30. XGBoost integrates multiple classification and regression trees (CART) to compensate for the lack of prediction accuracy of a single CART. It is an improved boosting algorithm based on GBDT, which is popular due to its high processing speed, high regression accuracy and ability to process large-scale data31. However, XGBoost uses a presorted algorithm to find data segmentation points, which takes up considerable memory in the calculation and seriously affects cache optimization.Before explaining LightGBMLightGBM is improved based on XGBoost. It uses a histogram algorithm to find the best data segmentation point, which occupies less memory and has a lower complexity of data segmentation. The flow of the histogram algorithm to find the optimal segmentation point is shown in Fig. Moreover, LightGBM abandons the levelwise decision tree growth strategy used by most GBDT tools and uses the leafwise algorithm with depth limitations. This leaf-by-leaf growth strategy can reduce more errors and obtain better accuracy. Decision trees in boosting algorithms may grow too deep while training, leading to model overfitting. Therefore, LightGBM adds a maximum depth limit to the leafwise growth strategy to prevent this from happening and maintains its high computational efficiency. To summarize, LightGBM can be better and faster used in industrial practice and is also very suitable as the base model in our tide level prediction task. The layer-by-layer growth strategy and leaf-by-leaf growth strategy are shown in Fig. 33. In recent years, CNNs have also been widely used in time series processing. For example, Bai et al.34 proposed a temporal convolutional network (TCN) based on a convolutional neural network and residual connections, which is not worse than recurrent neural networks such as LSTM in some time series analysis tasks. At present, a convolutional neural network is generally composed of convolution layers, pooling layers and a fully connected layer. Its network structure is shown in Fig. A convolutional neural network (CNN) is a deep feedforward neural network with the characteristics of local connection and weight sharing. It was first used in the field of computer vision and achieved great success35 have achieved good prediction results. However, in our subsequent experiments, the bidirectional gated recurrent unit achieved higher prediction accuracy than BiLSTM, so we used the BiGRU network for subsequent prediction tasks.In previous attempts at tide level prediction by scholars, bidirectional long short-term memory networks36 adds a gating mechanism to control information updating in a recurrent neural network. Different from the mechanism in LSTM, GRU consists of only two gates called the update gate The GRU networkThe recurrent unit structure of the GRU network is shown in Fig. Each unit of GRU is calculated as follows:In the above formula, 37 is a combination of two GRUs whose information propagating directions are reversed, and it has independent parameters in each, which makes it able to fit both forward and backward data at first and then join up the results from two directions. BiGRU can capture sequence patterns that may be ignored by unidirectional GRU. The structure of BiGRU is shown in Fig. A bidirectional gated recurrent unit networkTaking the BiGRU’s forward hidden state vector at time Because CNN has significant advantages in extracting useful features from a picture or a sequence and BiGRU is good at processing time series, we combine CNN and BiGRU to build the CNN-BiGRU model. The model can be mainly divided into an input layer, a convolution layer, a BiGRU network layer, a dropout layer, a fully connected layer and an output layer. The CNN layer and BiGRU layer are the core structures of the model. The function of the dropout layer is to avoid model overfitting. The CNN layer consists of two one-dimensional convolution (Conv1D) layers and a one-dimensional maximum pooling (MaxPooling1D) layer. The input of BiGRU is the output sequence of the CNN layer, and the BiGRU network is set as a one-hidden-layer structure. The structure of the CNN-BiGRU combination model is shown in Fig. When we analyze and predict relatively stationary tide level time series, LightGBM can perform well. However, due to environmental factors such as air pressure, wind force and terrain in reality, most tide level observation sequences are sometimes not relatively stationary, which requires that our tide level prediction model be reasonably able to "extrapolate" based on the sample observations, that is, be capable of generating values that are not in the sample. LightGBM is a tree-based model, which leads to our prediction results being between the maximum and minimum values of sequences. Therefore, LightGBM will not be able to accurately predict the situation or tidal change trend that did not appear in previous observations. However, the CNN-BiGRU model, which is a kind of neural network, has no such problem in theory and will be able to find the trend information that may be hidden in the tide level series. Therefore, we consider providing an appropriate weight for a single base model to build a combination model to improve the accuracy of the tide level prediction task.To improve the prediction accuracy of the combination model, a simple and effective idea is to determine the base models’ weights in the combination model according to the error between the prediction value and the real value. This method is also called the residual weight method, and its calculation formulas for determining the weights are:Our LightGBM-CNN-BiGRU (combination model) is based on the improved residual weight method. We call it the variable weight combination model. We use the weights calculated by formula  and formAfter obtaining a series of weights through formula  and formComparing Because the LightGBM-CNN-BiGRU (combination model) is a variable weight combination of the prediction results from two base models, the performance of the combination model can be directly improved by separately optimizing the super parameters of the two base models. We mainly use the grid search algorithm and K-fold cross validation method to optimize the parameters. The grid search algorithm is a method to improve the performance of a certain model by iterating over a given set of parameters. With the help of the K-fold cross validation method, we can calculate the performance score of the LightGBM model on the training set and easily optimize its superparameters. The final parameters of the LightGBM model are set to num_leaves = 26, learning_rate = 0.05, and n_estimators = 46.For the CNN-BiGRU network, we mainly improve the prediction accuracy of the model by adjusting the size and number of hidden layers in the BiGRU structure and prevent the model from overfitting by changing the dropout ratio and tracking the validation loss of the network while training.Data preprocessing: The quality of the data directly determines the upper limit of the prediction and generalization ability of a certain machine learning model. Standard, clean and continuous data are conducive to model training. The data used in this study are from the Irish National Tide Gauge Network, and all of them are subject to quality control. We filled in a small number of missing values and normalized the data to speed up the model training.Construction and optimization of base models: We divide the dataset into a training set, a validation set and a test set according to the proportion of 7:1:2 and train the LightGBM model and CNN-BiGRU model with data on the training set. We optimize the parameters and monitor whether the model has been overfitted by tracking the validation loss of the network while training. Finally, we put the data into two base models for training and then obtain the prediction results of a single base model.Construction of the variable weight combination model. Based on the prediction results of two single base models obtained in step (2), we calculate the weight of each base model according to the principle of the improved variable weight combination method and then obtain the prediction results of the variable weight combination model.Model evaluation and analysis: According to the indexes of the model evaluation, the variable weight combination model is compared with other basic models to analyze its prediction performance after being improved.The workflow of our tide level prediction model is shown in Fig. To evaluate the performance of the LightGBM-CNN-BiGRU (combination model) we proposed on the task of short-term tidal level prediction, we took the tidal observations of the Howth Harbor site in the first quarter of 2017 as the sample, used a sliding data window (We use the root mean square error (RMSE) and mean absolute error (MAE) as the evaluation criteria of model prediction performance. The calculation formula is:21. In many industrial practices, the prediction performances of LSTM and GRU are considered to have no significant difference. Based on the existing data, we designed a simple comparative experiment to show that the BiGRU unit can achieve higher prediction accuracy than BiLSTM in the tide level prediction task.When building tidal prediction models, previous scholars mostly used LSTM or BiLSTM as the constituent unit of recurrent neural networksAfter many experiments, we found that the network structure of single-layer BiLSTM or single-layer BiGRU is sufficient for tide level prediction. Therefore, we control the number of training epochs of the two models to be the same and the size of their hidden layer to be similar and then carry out the prediction on the tidal observations from five ports. The prediction accuracy of the two kinds of networks is shown in Fig. To verify the good prediction ability of our variable weight combination model based on LightGBM and CNN-BiGRU, we use the observations of the Howth Harbor site to construct the tide prediction model we proposed and select the simple model BiGRU, the base model LightGBM, CNN-BiGRU, and another variable weight combination model called LightGBM-BiGRU (combination model) as our comparison models. The comparison of absolute prediction errors between different models is shown in Fig. Table Due to environmental factors such as air pressure, wind force and topography, the tidal observations obtained in different areas should have a degree of pattern differences. To further determine whether the LightGBM-CNN-BiGRU (combination model) has good universality and generalization ability for tide level prediction, under the same simulation conditions, we selected the data of the other four sites on the Irish coastline for prediction experiments. The subsequent prediction results are shown in Table It is not difficult to find that the RMSE and MAE indexes of the variable weight combination model (LightGBM—CNN-BiGRU (combination model)) are better than those of the single base model. After calculation, compared with the single base model, the RMSE of the variable weight combination model is reduced by at least 16.2%, and the MAE is reduced by at least 16.7%. To conclude, the prediction results of the variable weight combination model based on LightGBM and CNN-BiGRU are more accurate, and the universality and generalization ability of the model have been verified.Accurate tidal prediction is of great significance for human activities in coastal areas. The traditional harmonic analysis method for tide level prediction needs to take the local hydrological, meteorological and geographical conditions into consideration and depends on a large amount of tide observation data. Even still, there is a large prediction error when using it. Machine learning models, represented by LSTM, XGBoost and LightGBM, with their wide applicability and strong fitting ability, have been proven by many studies to be capable of predicting tide levels accurately. Based on previous research, this paper proposes a variable weight combination model based on LightGBM and CNN-BiGRU. It combines the predictions of LightGBM and CNN-BiGRU models with variable weights to further improve the accuracy of tide prediction and to compensate for the lack of "extrapolation" of LightGBM to a certain extent.We use the data of five stations from the real-time tide monitoring network INTGN operated by the Irish Marine Institute for all model construction and analysis and decide to predict the Howth Harbor’s tide level in the next time step by using the observed values within 1 h, with the purpose of verifying the good short-term tide level prediction ability of our combination model. As shown in Fig. In addition, we can conclude from the example analysis that using GRU as the constituent unit can achieve better results in the tidal prediction task when the structure of the recurrent neural network is almost the same. While using the BiGRU network for tide level prediction, adding a CNN module to the network at first for information extraction can effectively improve the accuracy of model prediction. Moreover, the variable weight combination method has good universality to some extent. Both LightGBM-BiGRU (combination model) and LightGBM-CNN-BiGRU (combination model) predict more accurately than their respective base models.Indeed, our research still has many areas that can be improved. Due to the limited data collection, we only carried out prediction research on tide observations along the coast of Ireland. If we can collect tidal observation data from other places of the world in the future, we will be able to further verify the good prediction performance and the strong generalization ability of the variable weight combination model based on LightGBM and CNN-BiGRU. We will also try the multistep prediction ability of the combination model to realize tide prediction in a longer time. In short, our research shows that the LightGBM-CNN-BiGRU (combination model) we proposed can achieve higher prediction accuracy than traditional tide prediction models such as BiLSTM, and the variable weight combination method can enable the combination model to achieve better prediction performance than its single base models. Our experiment also shows that the combination model can reduce the absolute error to approximately 0.03 m with only one quarter of the tide observations, and the construction of the model is not complicated, so it deserves to be used in practice.