Parametric Model for Coaxial Cavity Filter with Combined KCCA and MLSSVR

Aiming at the problems of poor data efectiveness, low modeling accuracy, and weak generalization in the tuning process of microwave cavity flters, a parametric model for coaxial cavity flter using kernel canonical correlation analysis (KCCA) and multioutput least squares support vector regression (MLSSVR) is proposed in this study. First, the low-dimensional tuning data is mapped to the high-dimensional feature space by kernel canonical correlation analysis, and the nonlinear feature vectors are fused by the kernel function; second, the multioutput least squares support vector regression algorithm is used for parametric modeling to solve the problems of low accuracy and poor prediction performance; third, the support vector of the parameter model is optimized by the diferential evolution whale algorithm (DWA) to improve the convergence and generalization ability of the model in actual tuning. Finally, the tuning experiments of two cavity flters with diferent topologies are carried out. Te experimental results show that the proposed method has an obvious improvement in generalization performance and prediction accuracy compared with the traditional methods.


Introduction
Te manufacture of coaxial cavity flters usually includes two steps: design and tuning. Te design is mainly based on the electromagnetic simulation software, which produces approximate theoretical errors. In addition, the diference between processing tolerance and metal coating makes it difcult to achieve the physical size of the microwave cavity flter after production, and its output response cannot match the theoretical results. Terefore, tuning has become an indispensable link. To facilitate the tuning after processing, the tuning screws are usually installed on the real object to replace the resonant and coupling rods in the cavity. By constantly changing the direction and amplitude of the screws, the waveform of the output response can meet the requirements of the performance indicators.
In view of the complex dynamic characteristics such as nonlinearity and strong coupling in the tuning process of coaxial cavity flters, how to efectively use a large number of of-line and online data generated in the tuning process to establish a relationship model that can accurately refect the tuning law is the focus of many scholars at home and abroad. At present, the relevant modeling methods mainly include feedback neural networks, fuzzy logic, and support vector regression algorithm. Te research on neural network methods is concentrated on the basis of electromagnetic simulation software, such as the model between microwave element size and parameters [1], the tuning model between tuning element and return loss [2], and the model between refection characteristic phase and center frequency [3]. However, the parameters in these models fuctuate greatly and are easily afected by the structure of microwave components, so it is difcult to extract the return loss and refection characteristic phases. To reduce the amount of training data and shorten the modeling time, a highprecision microwave flter parameter prediction model using the adaptive learning neural network is proposed in [4]. It is applied to the optimization design of microwave flters. Although the neural network has strong nonlinear mapping ability, it has strict requirements for data quality. For the cavity flter with insufcient feld tuning data, the modeling accuracy has great limitations.
Faced with a complex tuning process, Miraftab frst introduced engineers' experience in fuzzy rules into the tuning process of cavity flters [5]. Tis method uses experience and data fusion to establish the relationship model between coupling and tuning elements and is applied in loworder cavity flters. Te drawback of this method is that it is difcult to apply to multiple cross-coupled cavity flters, and the process of designing fuzzy rules is extremely complex. Diferent from fuzzy algorithms, support vector regression modeling is a novel small-sample learning method with a solid theoretical foundation [6]. It has better generalization ability than the neural network when the number of samples is small. On the basis of previous research, multicore support vector regression based on parameter self-adjustment is used to model the tuning process of cavity flters [7]. However, faced with multi-input and multioutput data from cavity flters, the above method needs to solve the complex quadratic programming problem and should consider the coupling relationship between output variables. To improve the solving efciency of multioutput systems, some studies using multicore machine learning have been proposed [8][9][10][11][12], which provides an important theoretical reference for the development of this study. Nonetheless, in industrial applications, the large-scale data collection and hyperparameter optimization time used for machine learning modeling often exceeds the tuning time. From a practical operational perspective, it is difcult to achieve.
To avoid these problems, parametric models based on vector feld and machine learning were proposed in [13,14], which provides a new idea for the modeling of the tuning process. However, with the increase in the order of the coaxial cavity flter, the input-output and action space dimensions of reinforcement learning increase sharply, resulting in an exponential growth of the value function network size and an increase in the training cost. In [15,16], a parametric modeling method based on hybrid neural networks and deep learning has been proposed successively, but the data acquisition process of cavity flter in industrial applications is cumbersome, and the time of large-scale data acquisition will even exceed the tuning time. Terefore, from a practical point of view, these methods are difcult to implement. On the other hand, although SVR has obvious advantages in parametric modeling, the parameter identifcation of the algorithm itself relies too much on expert experience and laboratory experiments. Terefore, intelligent biomimetic algorithms such as the genetic algorithm (GA), particle swarm optimization algorithm (PSO), and genetic whale algorithm (GWA) are generally used to calculate parameters [17][18][19]. However, these algorithms also have the problem of easily falling into local optima and complex parameter calculations.
Given the aforementioned research analysis, the parametric model of coaxial cavity flters will encounter three challenges. (1) Te difculty of feature data fusion under diferent tuning modes and the impact of data validity on model accuracy. (2) Low modeling accuracy and weak generalization ability of parameterized models for multiinput multioutput tuning processes based on multivariate, strongly coupled, and nonlinear relational data. (3) Dealing with the computational efciency and convergence speed of optimization algorithms in the solving process. Te goal of this article is to provide a parametric model for cavity flter using kernel canonical correlation analysis and multioutput least squares support vector regression.
Te contribution and novelty of this work includes three aspects: (i) the feature vectors under diferent modes of initial and fne tuning are fused by KCCA, which solves the problems of poor data validity and large input vector dimensionality of a single feature; (ii) the output coupling and complex computation process of quadratic programming in parameter modeling are avoided by combining the least squares method with the improved support vector regression; (iii) the diferential evolution whale algorithm is used for model parameter identifcation, which improves the optimization speed and global search ability of the model. Te rest of this study is organized as follows: Section 2 describes the theoretical synthesis of the cavity flter. Section 3 presents the principal component analysis of the raw data. Sections 4 and 5 mainly introduce the establishment of tuning model and parameter optimization. Section 6 shows the simulation results and analysis. Te last section is the conclusion.

Description of the Problem.
Te working process of this paper mainly includes data collection and kernel canonical correlation analysis, electromechanical characteristic modeling, and model parameter identifcation. First, collect the input and output data pairs (d, S) generated during the tuning process, where d denotes the screw tuning height and S denotes the corresponding output response; secondly, the coupling matrix is extracted from the scattering parameter (S-parameters) as mentioned in [20], and the characteristic parameters under diferent modes are fused through the kernel canonical correlation analysis technology. Build a data set for electromechanical characteristic modeling; fnally, the electromechanical relationship model of the cavity flter based on multioutput support vector regression is established according to the collected data sets of input and output relationships. On this basis, the diferential evolution whale algorithm is used to realize the adaptive identifcation of model parameters. Te fow chart of the parametric model for the cavity flter is shown in Figure 1.

Mechanism and Characteristics of Cavity Filter.
Te produced coaxial cavity flter is difcult to achieve in physical size, and its output response cannot be consistent with the theoretical results. Generally, the cavity flter is equipped with tuning screws, and the output S-parameters can meet the performance requirements by changing the length of the tuning screws. Te S-parameters have the following nonlinear relationship with the coupling matrix [21]: where R 1 and R 2 are the input and output port coupling, respectively. R denotes the diagonal matrix, I denotes the identity matrix. Te coupling matrix (M) extracted from the S-parameter contains diferent feature information, such as diagonal elements, adjacent coupling elements, and crosscoupling elements, which are interrelated in diferent tuning stages. To reduce the redundant information in the tuning model, this study uses the KCCA multifeature fusion method to fnd the key information from the multidimensional features and convert the original extracted feature vector into a new low-dimensional vector.

Kernel Canonical Correlation Analysis
Te tuning data of the cavity flter has serious nonlinear characteristics, especially for the cavity flter with initial tuning or multiple cross coupling. In addition, there is a strong coupling between the input variables, and it is difcult to describe the internal law of tuning with a single model. Terefore, data preprocessing and correlation analysis play a crucial role in modeling accuracy and generalization ability. Te KCCA is a nonlinear data analysis algorithm, which transforms the nonlinear relationship of input space into the linear relationship of feature space through the kernel function and carries out correlation analysis in the new space.
. , x n ), and Y � (y 1 , y 2 , . . . , y n ). Te random variables in the observation space are transformed into the kernel function of the following high-dimensional space by implicit mapping [22].
where R and H denote the observation space and highdimensional feature space, respectively. φ(x) and φ(y) denote the implicit mappings of x and y from the observation space to high-dimensional feature space through kernel functions, respectively.
, . . . , ϕ(y n )). Te essence of the kernel function is equivalent to the mapping of input data from low-dimensional space to high-dimensional space, and its mathematical expression is as follows: where c and d denote the constant vectors of the highdimensional space to be solved. * denotes a multiplication sign. Using equation (3), the problem of solving highdimensional space constant vectors c and d is transformed into the problem of solving low-dimensional space constant vectors φ x and ϕ y [23].
where X * and Y * denote the transformed characteristic components. T and 〈·〉 denote the transpose and inner product, respectively. Te core problem of KCCA is to solve the corresponding equations (3) and (4) when the correlation coefcient is the largest. Te mathematical expression of the correlation coefcient is as follows [24]: where Var(·) and Cov(·) denote the variance matrix and covariance matrix, respectively. Te variance and covariance of X * and Y * are calculated as follows: where K(x, y) � (φ x , φ y ), the following correlation coefcients can be obtained by substituting equations (5) into (6).
Te problem of solving the correlation coefcient is transformed into a constrained optimization problem, and its objective function and constraint conditions are as follows:  International Journal of Antennas and Propagation Te Lagrange multiplication is used to solve the above constrained extreme value problem, and the corresponding Lagrange equation is where ρ 1 and ρ 2 denote the Lagrange multiplier. Calculate the partial derivative of L with respect to φ x and ϕ y and make it zero. Te following equations are obtained by deriving φ x and ϕ y , respectively.
where ρ 1 � ρ 2 � λ, the solution of KCCA problem is equivalent to solving the eigenvector problem corresponding to the following generalized eigenequation: Tϕ y y , and φ x and ϕ y denote the eigenvectors to be solved. By solving the above equation, φ x , ϕ y , and the correlation coefcient ρ can be calculated.

Parametric Modeling
Te traditional multioutput regression problem is usually solved by transforming the multidimensional output into one dimensional output, which often ignores the correlation between variables, to reduce the accuracy of the model. To solve this problem and fully consider the correlation of output components, this study improves the loss function of the original algorithm and proposes a parametric modeling strategy combining KCCA and MLSSVR.

Support Vector Regression.
Te essence of support vector regression is to map the input space to the high-dimensional space through nonlinear mapping and perform linear regression through the estimation function in the highdimensional space. In this study, select the training set (x i , y i ) ∈ R p × R q of n group sample data, in which the input variable x i � (x i1 , x i2 , · · · x ip ) T and the output variable y i � (y i1 , y i2 , · · · y iq ) T . Tis research improves the insensitive loss function in the traditional single-output function regression algorithm, using the loss function on the hypersphere instead of the loss function on the hypercube. Te improved loss function is as follows [25]: and ε denotes the width of the neutral zone, which is the tolerance for errors, φ denotes a nonlinear mapping function, b � [b 1 b 2 · · · b n ], w � [w 1 w 2 · · · w n ], i denotes the number of input samples, and q denotes the dimension of the output variable. Te improved function takes the ftting error of each component into account, which can not only achieve the goal of overall optimization, but also weaken the noise.

Multioutput Least Squares Support Vector Regression.
Diferent from support vector machines, LS-SVR usually rewrites the primary loss function into a quadratic loss function and changes the inequality constraints into equality constraints, which avoids the complex calculation of solving quadratic programming problems and improves the efciency of optimization. However, the traditional LS-SVR algorithm is only applicable to the regression modeling of single-output systems. When faced with multioutput nonlinear systems, the single-output systems are usually simply combined, thus ignoring the potential correlation information between output variables. In view of this, this study improved the objective function and constraint equation and used the absolute error to describe the overall error of the sample, thus establishing a multioutput regression model. Te parametric model structure is shown in Figure 2, where u, v, and w denote the resonant cavity self-coupling, adjacent cavity coupling, and cross coupling variables, respectively. g, s, and t denote the number of corresponding variables. Te model has a simple structure and is easy to calculate. Te new eigenvector x i in the parametric model is mapped to the high-dimensional space φ(x i ), and then the linear modeling is realized in the new feature space [26].
where K(x, x i ) � 〈φ(x), φ(x i )〉 denotes the nonlinear mapping. In this study, the least squares method is introduced to change the inequality constraint in traditional support vector regression into an equality constraint, and the square term of error is used as the experience loss of the training set. Finally, the quadratic programming problem is transformed into solving linear equations, which improves the calculation speed and convergence accuracy of the algorithm. Te values of w and b are obtained through the following optimization and constraint functions [27]: where x i � [x i1 x i2 · · · x ip ] T , and y i � [y i1 y i2 · · · y iq ] T denote the input and output variables.
denotes the input mapping function, c 0 denotes the penalty coefcient of the overall error of the sample, C denotes the j-th dimension output error of the i-th sample, e ir denotes the j-dimensional output error of the i-th sample, and E i denotes the overall ftting error. Te objective function and constraints are transformed into the following Lagrange function: where α � [α 1 α 2 · · · α q ] and β � [β 1 β 2 · · · β q ] denote Lagrange factor. Use equation (3) to calculate the partial derivatives of w r , b r , e ir , and α ir , respectively, and obtain the following expression with KKT condition: By eliminating w r and e ir . Te solution of the optimization problem can be transformed into the solution of the following linear equations: where I denotes the unit matrix, k(x, x r ) � φ T (x)φ(x r ).
When the output error of the r-th dimension is greater than or equal to zero, ℏ r � 1. Te values of the variables b r and w r are obtained by solving the above linear equations, and the regression function of the output of the r-dimension is as follows: Te Gaussian radial basis function has good ability to handle complex nonlinear relationships between sample inputs and output. It requires fewer parameters to be determined and has high computational efciency. Terefore, the kernel function used in this study is the Gaussian radial basis function: where σ denotes the kernel width parameter, which refects the size of the training data sample space range. When its value is large, the space range is small. When the number of support vectors is large and the regression parameters have been calculated, the kernel method may sufer from the curse of dimensionality. To efectively utilize all sample information, a sparsization method based on singularity criteria is used to continuously update the sample subset. Tis subset efectively reduces the redundant information of input samples while maximizing the coverage of input sample information, thus signifcantly reducing the solution size. Te main steps include (1) initialize an empty sample set D 0 , and when obtaining the frst sample vector x 1 , make D 1 � x 1 ; (2) when the number of samples is greater than 1, calculate the minimum distance c � min x v ∈D i ‖x i − x ] ‖ between the new sample x i and the current sample set; (3) when the minimum distance is less than a preset threshold δ 1 , (x i , y i ) is not added to D i , otherwise the prediction error e i is calculated; and (4) when |e i | is greater than the preset threshold δ 2 , add (x i , y i ) to D i and update D i to D i+1 , otherwise proceed to (2). Te thresholds δ 1 and δ 2 are used to control the accuracy and scale of the solution. Increasing δ 1 and δ 2 is benefcial for reducing the size of sample set D, but it will lead to a decrease in the performance of the beamformer. Tis article takes δ 1 as 0.1 of the kernel radius and δ 2 as the root mean squares value of the average steady-state mean square error, which can achieve high computational accuracy on a smaller scale. Te steps to establish the parametric model of the cavity flter using MLSSVR algorithm are as follows: Step 1: Set the initial value of C, σ, and c 0 . Based on the collected sample dataset, the MLSSVR algorithm is used to establish the regression function of the multiinput and output tuning system for the cavity flter.
Step 2: Substitute the input and output matrices and relevant parameters of training samples into the subfunction. Use the established initial regression function to detect the test sample data and calculate the test error e ir ;

Input layer
Output layer Hidden layer

International Journal of Antennas and Propagation
Step 3: For any r-th dimension output, the value of ℏ r is determined according to e ir ; Step 4: According to the calculated α ir and b r , the regression function of the cavity flter multi-input and output tuning system is established. Te work fow chart of the improved whale algorithm is shown in Figure 3.

Parameters Identification
Te forecast accuracy of MLSSVR is mainly afected by the kernel function parameter σ, the penalty factor C and c 0 , where the value of the penalty factor corresponds to the empirical risk generated by the training samples. If the penalty factor is too small, the punishment for exceeding the insensitive band in the sample is too small, and the training error is too large. On the contrary, if the penalty factor is too large, the error penalty will be too large, while the model structure restriction will be reduced, resulting in the model being too complex; the kernel function parameters can refect the distribution of sample data in high-dimensional feature space information.

Diferential Evolution Whale
Algorithm. DWA is a heuristic-intelligent algorithm that imitates the behavior of whale predation. Compared with the traditional optimization algorithm, it has the advantages of less control parameters, a simple implementation, and high fexibility. DWA imitates the whale's predation method, which mainly includes three stages hunting predation, bubble net predation, and free search. Te position of each individual represents the potential solution, and the global optimal solution is obtained by constantly updating the position of the whale.
Shrink Surround: Since there is no prior knowledge of the global optimal solution of the search space before solving the optimization problem, a population of individuals is randomly selected as the target for predatory activities, and other whale individuals in the population are surrounded by the optimal individuals. Te position update equation is as follows [28]: where x i (k + 1) denotes the position of the i-th whale at iteration k. F → (k) and J → (k) denote the coefcient vector, and F → (k) � 2δ * rand(·) − δ, J → (k) � 2rand(·), δ denotes the convergence factor of linear descent. rand(·) denotes the random number in [0, 1]. In the process of bubble net predation, the position update between whale and prey is expressed by the following logarithmic spiral equation [29]: where d i ′ denotes the distance between the search individual and the current optimal solution, ϑ denotes the spiral shape parameter, l denotes a random number uniformly distributed in [−1, 1], and x * (k) denotes the current best position vector. Te algorithm can choose the predation behavior by setting the probability c of the predation mechanism. Te whale's position update equation is as follows [30]: where |c| < 1, each whale is in the phase of gradually surrounding the current optimal solution. To ensure that all whales can search completely in the solution space, the distance between whales in the whale algorithm is used to update the position to achieve the purpose of random search. Terefore, when |c| ≥ 1, the search experience will swim toward the random whale.
where x r (k) denotes the position of the current random individual. d ″ denotes the distance between the current search individual and the optimal solution. To increase the diversity of the population, the diferential evolution is introduced in the process of population position updating, which avoids the premature phenomenon brought by the local optimal solution. Te location updating formula is as follows: where x * and x r (k) denote the optimal individual position and the random individual, respectively, k denotes iterations and τ is a random number in [0 1]. Tis study combines the powerful exploration capabilities of diferential evolution algorithm with the whale algorithm to enhance the development space of the whale algorithm. First, the whale algorithm is used to preprocess the position of the individual population and generate an initial population. Ten, the diferential evolution algorithm is used to cross-select the generated population to achieve population update iteration. Te algorithm process is shown in Table 1.
To prevent over-and underftting of the model, the insensitivity coefcient is given empirically in this study. Select the following sample mean square deviation as the evaluation function of DWA: where f(x i ) and y i denote the measured and expected values of the model, respectively. Te optimization goal is to make f(C, σ, c 0 ) reach the minimum value. n denotes the population number. Te ftness function is 1 /1 + f. Parameter settings of the DWA algorithm are as follows: the evaluation number Iter � 45, the population size N � 30, diferential variability factor F � 0.75, and crossover rate CR � 0.9. Compare DWA with GA (population size N 1 � 30, maximum generation g max � 45, replication 6 International Journal of Antennas and Propagation probability P r � 0.65, crossover probability P c � 0.85, mutation probability P m � 0.05), PSO (particle number N 2 � 0.85, maximum particle velocity V max � 0.85, acceleration constant c 1 � c 2 � 2), and GWA (P r � 0.7, P c � 0.8, P m � 0.01), the parameter optimization curve is shown in Figure 4. Te optimal parameters for MLSSVR prediction model using the DWA as follows: C � 15.24.9, σ � 0.092, and c 0 � 9.74. Te optimization results show that the improved whale optimization algorithm has a higher convergence accuracy and better stability than the other optimization algorithms and basic whale optimization algorithms.

Simulation Results and Analysis
To evaluate the accuracy of the parametric model of the coaxial cavity flter, the standard deviation (STD), the maximum absolute error (MAE), and the correlation coefcient (R) of three diferent methods are used for analysis and comparison. Te calculation equation is as follows: where f(x ir ) and y ir denote the real and predicted values of the measured data, respectively. n and q denote the number of samples and output variables, respectively. Substitute the Randomly generated population, let k=1 Output optimal solution C,σ,c 0 Update the location of the whale population by equation (20) 10 Update the location of the whale population by equation (21) 12. end 13. else if 0.5 ≤ c < 1; 14.
Update the location of the whale population by equation (23)  tuning process relationship data into the parameter prediction model, and compare the kernel clustering back propagation neural network (KC-BPNN) and multioutput fuzzy support vector regression (MFSVR) with the KCCA-MLSSVR method proposed in this paper. Te prediction results are shown in Figures 5 and 6. It can be seen from this fgure that the STD and MAE of the proposed method prediction results are lower than those of the other two models; in addition, the absolute value of R by the proposed model is close to 1, which is higher than the other two methods. Te above results indicate that the proposed method can not only efectively reduce the impact of highdimensional data on the model's prediction accuracy but also improve the model's adaptability and generalization ability. Te S-parameter simulation curve of the ninth-order cavity flter with three diferent methods is shown in Figure 7.
Te purpose of cavity flter tuning is to make the S-parameters of the output response meet the key performance indexes (KPI) required before production. To further verify the feasibility of the method, a ten-order crosscoupled cavity flter with complicated topology is taken as the experimental object for simulation experiments. Kpis are as follows: center frequency f 0 � 2.38GHz, passband width bw � 0.10GHz, the maximum return loss RL � −34dB, maximum insertion loss IL � −0.35dB. It can be seen from Figure 8 that the return loss, insertion loss, and out-of-band suppression of the proposed method can meet the KPI.
For more quantitative comparison, the insertion loss (IL), return loss (RL), and left and right out-of-band suppression (RS or LS) of S-parameters calculated by different methods are shown in Tables 2 and 3. From the data given in the table, the proposed method in this study is

Conclusion
Te tuning process of the coaxial cavity flter has great nonlinearity, and the output characteristics vary greatly in diferent tuning stages. To solve this problem, a parametric model method for coaxial cavity flter fusing KCCA and MLSSVR is proposed in this study. First, the redundant eigenvectors are digested by KCCA, which not only reduces the input vector dimension of the prediction model but also accelerates the training speed of the system. Second, a parametric model using MLSSVR has higher prediction accuracy than traditional modeling methods. Finally, the model parameters are optimized by DWA, which improves the adaptive ability and convergence speed of the model. Te experimental results show that the proposed method in this study has higher prediction accuracy and generalization performance in the process of cavity flter model parameter ftting.

Data Availability
Te data used to support the fndings of this study are available from the corresponding author upon request.

Conflicts of Interest
Te authors declare that they have no conficts of interest.    International Journal of Antennas and Propagation 9