# Input Selection Methods for Soft Sensor Design: A Survey
## 1 Introduction  
When dealing with industrial processes, many variables are monitored through online sensors. Some of these variables can be very hard to measure though or can be measured only sporadically due to high cost sensors or lack of the latter. In some cases, variables are measured with high delays because of slow hardware sensors or laboratory analysis, leading to the impossibility of real-time monitoring of the process. Inferential models can then be created to estimate these hard-to-measure variables on the basis of online measured ones. Such models are referred to as Soft Sensors (SSs), or Virtual Sensors [1, 2]. Their working principle is summarized in Figure 1.  
SSs were originally exploited in the science of chemometrics, a discipline that studies how to extract information from data sets of chemical systems by using multivariate statistics, mathematics and  
Figure 1: Basic working principle of a Soft Sensor (SS). computer science [3; 4]. Chemometrics solves prediction problems by learning models from data and exploiting machine learning, system identification, artificial intelligence and statistical learning theory [1; 2; 5]. SSs' industrial use ranges over a number of different types of processes, such as refineries [6; 7; 8; 9; 10; 11], chemical plants [12], cement kilns [13; 14], power plants [15; 16], pulp and paper mills [17; 18], food processing [2; 19], nuclear plants [20; 21], pollution monitoring [22], polymerization processes [23; 24; 25] or wastewater treatment systems [26; 27; 28; 29], just to mention a few. In Figure 2, a distillation column process from a refinery is shown through its control software: displayed online measurements are collected in a historical database for future use. SSs' real-time estimation capability allows developing tight control policies and since they represent a low-cost alternative to expensive hardware devices, SSs allow the realization of more comprehensive monitoring networks. The real-time estimation obtained by the SS can be used by a controller, while the corresponding delayed measurements allow the model to be improved, by avoiding an error propagation effect. Besides their use for plant control, they are used to approach a number of other different problems as well, such as measuring system back-up, what-if analysis, sensor validation and fault diagnosis [7; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40; 41]. The back-up of measuring instrumentation consists of substituting unavailable measuring equipment, to avoid degradation of plant performance and rises in costs. This can become necessary since measuring devices, and their data transmission systems, generally face harsh working environments that impose periodic maintenance procedures or lead to faults. Therefore, when a maintenance intervention is performed, measuring hardware needs to be turned off and suitably substituted throughout the entire process. What-if analysis consists of using the model to perform simulations of the system dynamics with respect to types of input that are of interest, for a given time span, with the aim of both achieving a deeper understanding of the system behavior or designing suitable control policies. SSs allow to reduce the need for measuring devices as well, improving system reliability and decreasing sensors acquisition and maintenance costs. They can eventually work in parallel with hardware sensors, giving this way useful information for fault detection tasks and they can be easily implemented on existing hardware and retuned when system parameters change. The main goals of fault detection are to perform early detection of faults providing as much information as possible about it, to provide a support system for scheduled maintenance interventions, and to provide a basis for the development of fault-tolerant systems. SSs can be built with different approaches.