Support vector regression (SVR) has been widely used to reduce the high computational cost of computer simulation. SVR assumes the input parameters have equal sample sizes, but unequal sample sizes are often encountered in engineering practices. To solve this issue, a new prediction approach based on SVR, namely as high-low level SVR approach (HL-SVR) is proposed for data modeling of input parameters of unequal sample sizes in this paper. The proposed approach consists of low-level SVR models for the input parameters of larger sample sizes and high-level SVR model for the input parameters of smaller sample sizes. For each training point of the input parameters of smaller sample sizes, one low-level SVR model is built based on its corresponding input parameters of larger sample sizes and their responses of interest. The high-level SVR model is built based on the obtained responses from the low-level SVR models and the input parameters of smaller sample sizes. A number of numerical examples are used to validate the performance of HL-SVR. The experimental results indicate that HL-SVR can produce more accurate prediction results than SVR. The proposed approach is applied to the stress analysis of dental implant, in which the structural parameters have massive samples but the material of implant can only be selected from Ti and its alloys. The obtained prediction results of the HL-SVR approach are much better than SVR. The proposed approach can be used for the design, optimization, and analysis of engineering systems with input parameters of unequal sample sizes.

中文翻译：

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具有不等样本大小的输入参数的数据建模的高低级支持向量回归预测方法 (HL-SVR)

支持向量回归 (SVR) 已被广泛用于降低计算机模拟的高计算成本。SVR 假设输入参数具有相等的样本量，但在工程实践中经常会遇到不相等的样本量。针对这一问题，本文提出了一种新的基于SVR的预测方法，即高-低级SVR方法（HL-SVR），用于对不等样本量的输入参数进行数据建模。所提出的方法包括用于较大样本量的输入参数的低级 SVR 模型和用于较小样本量的输入参数的高级 SVR 模型。对于较小样本量的输入参数的每个训练点，根据其对应的较大样本量的输入参数及其感兴趣的响应，构建一个低级SVR模型。高级 SVR 模型是基于从低级 SVR 模型获得的响应和较小样本量的输入参数构建的。许多数值示例用于验证 HL-SVR 的性能。实验结果表明HL-SVR比SVR能产生更准确的预测结果。该方法应用于牙种植体的应力分析，其中结构参数有大量样本，但种植体材料只能从钛及其合金中选择。HL-SVR方法得到的预测结果比SVR好很多。所提出的方法可用于具有不等样本量的输入参数的工程系统的设计、优化和分析。