# MIMO LS-SVR-Based Multi-Point Vibration Response Prediction in the Frequency Domain
###### Abstract
LS-SVR also solves the primal-dual problem, and it can replace the quadratic programming problem in SVR by solving a system of linear equations (caused by linear constraints in the optimization objective), and be applied to the regression task in high-dimensional input spaces. Many multiple regression analysis prediction models are essentially optimization problems. They seek the optimal solution for a certain goal using traditional optimization methods, such as gradient descent methods and Newton iterative methods, and fall easily into the local optimal solution [18; 19]. To find the global optimal solution for these multiple regression analysis prediction models, the application of intelligent optimization algorithms is also a major trend at present. For example, the use of the particle swarm optimization algorithm to determine the optimal solution of LS-SVR to replace the SVR using traditional gradient descent methods for response prediction is described in [20; 21]. Moreover, some response prediction methods have been developed in special fields, such as aeroelasticity and rotorcraft [22; 23; 24]. At present, most research is focused on vibration response prediction under the condition of known multiple loads excitation [25; 26; 27; 28; 29; 30]. However, multiple loads excitation is often very difficult to measure and unknown in reality. For the study of response prediction under an unknown load, [PERSON] [31] solves the response prediction under the single load unknown condition. In the multi-point vibration response prediction problem under the condition of unknown uncorrelated multi-source loads, researchers [32; 33; 34; 35; 36; 37] have used the transmissibility matrix to build a relationship between the unmeasured multi-point vibration response and measured multi-point vibration response, and predict the unmeasured vibration responses using the transmissibility matrix. However, the transmissibility matrix or transfer function of the LTI structure is often very difficult to obtain. When the transmissibility matrix or transfer function of the LTI structure is unknown, [PERSON] [38] used the measured vibration response based multiple linear regression model to predict the unmeasured vibration response in the frequency domain without the information of loads. Although this method can achieve good prediction results, the prediction accuracy is often affected by ill-posed problems at resonance frequencies. Multi-input multi-output least squares SVR (MIMO LS-SVR) methods can effectively alleviate the above problems [39]. Therefore, based on the above research, using the measured multi-point vibration response to predict the unmeasured multi-point vibration response in the frequency domain using MIMO LS-SVR methods is proposed. The main innovations of this study are as follows:  
1. In the case of unknown uncorrelated multiple loads excitation and transfer functions, the relationship between measured multi-point vibration response and unmeasured multi-point vibration response is derived. Then, the response prediction model based on data is established by using response history data. 2. Using the measured vibration response to predict the unmeasured vibration response in the frequency domain using MIMO LS-SVR methods is proposed. Its theoretical correctness, solvability, and applicability are discussed by comparing it with the transmissibility matrix, transfer function with known uncorrelated multi-source loads methods, multiple linear regression methods, MIMO neural network, and MIMO LS-SVR with known loads methods.