With advancements in sensor technology, a heterogeneous set of data, containing samples of scalar, waveform signal, image, or even structured point cloud are becoming increasingly popular. Developing a statistical model, representing the behavior of the underlying system based upon such a heterogeneous set of data can be used in monitoring, control, and optimization of the system. Unfortunately, available methods only focus on the scalar and curve data and do not provide a general framework that can integrate different sources of data to construct a model. This paper poses the problem of estimating a process output, measured by a scalar, curve, an image, or a point cloud by a set of heterogeneous process variables such as scalar process setting, sensor readings, and images. We introduce a general multiple tensor on tensor regression (MTOT) approach in which each set of input data (predictor) as well as the output measurements are represented by tensors. We formulate a linear regression model between the input and output tensors and estimate the parameters by minimizing a least square loss function. In order to avoid overfitting and to reduce the number of parameters to be estimated, we decompose the model parameters using several bases, spanning the input and output spaces. Next, we learn both the bases and their spanning coefficients when minimizing the loss function using an alternating least square (ALS) algorithm. We show that such a minimization has a closed-form solution in each iteration and can be computed very efficiently. Through several simulation and case studies, we evaluate the performance of the proposed method. The results reveal the advantage of the proposed method over some benchmarks in the literature in terms of the mean square prediction error.

中文翻译：

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多张量对张量回归：一种用异构数据源建模过程的方法

随着传感器技术的进步，包含标量、波形信号、图像甚至结构化点云样本的异构数据集正变得越来越流行。开发统计模型，表示基于此类异构数据集的底层系统的行为，可用于系统的监视、控制和优化。遗憾的是，现有的方法只关注标量和曲线数据，并没有提供可以整合不同数据源来构建模型的通用框架。本文提出了通过一组异质过程变量（例如标量过程设置、传感器读数和图像）估计由标量、曲线、图像或点云测量的过程输出的问题。我们介绍了一种通用的多张量张量回归 (MTOT) 方法，其中每组输入数据（预测器）以及输出测量值都由张量表示。我们在输入和输出张量之间制定了一个线性回归模型，并通过最小化最小二乘损失函数来估计参数。为了避免过度拟合并减少要估计的参数数量，我们使用多个基分解模型参数，跨越输入和输出空间。接下来，我们在使用交替最小二乘 (ALS) 算法最小化损失函数时同时学习基数和它们的跨度系数。我们表明这种最小化在每次迭代中都有一个封闭形式的解决方案，并且可以非常有效地计算。通过多次模拟和案例研究，我们评估了所提出方法的性能。结果揭示了所提出的方法在均方预测误差方面优于文献中的一些基准。