# Soft Sensor Transferability: A Survey
## 2 Soft Sensors  
In industrial plants, many variables are monitored through online sensors. There are cases though in which some of them cannot be measured online due to the lack of online measurement instruments or hostile environments. This requires frequent sensor maintenance and/or introduces high delays due to laboratory analysis. SSs can be a solution to these issues. They are inferential models that make use of the underlying relations between the accessible process variables to provide an estimation of the required hard-to-measure physical variables as schematized in Figure 1.  
Depending on the problem complexity an, SS can be realized by using static/dynamic, linear/nonlinear, time-invariant/variant models [2]. SSs allow for real-time plant control, measuring system back-up, what-if analysis, sensor validation and fault diagnosis as well [3; 16]. Figure 1: SS working principle. SSs, also named as virtual sensors, are adopted in different fields such as refineries [17; 18], chemical plants [19], cement kilns [20], power plants [21], pulp and paper mills [22], food processing [23], polymerization processes [24], and wastewater treatment systems [25]. SSs can be designed by using first principle models when a priori physical knowledge of the plant is given [3]. When such knowledge is not available, or the modelling process is overcomplicated, the design of SSs relies on data-driven black-box techniques [26]. Data-driven SS design is a highly time-consuming task that involves pattern recognition [27] and system identification [28] steps, as summarized in the following:  
* Data collection and filtering;
* Input variables selection;
* Model choice;
* Model identification;
* Model validation. Data are retrieved from industries' historical databases and must be selected to represent the whole dynamics of the system. Historical databases usually suffer from oversampling, outliers, missing data, offsets, seasonal effects and high-frequency noise. An accurate pre-processing is therefore needed for the successive step of input selection, in which highly informative inputs concerning the chosen output are selected among the many available inputs [29]. The model choice should be conducted taking into consideration different characteristics, i.e., linear/nonlinear, static/dynamic, time-variant/invariant. Linear models, in general, do not show good performances for industrial processes. Nonlinear models are therefore widely used. The following model classes for linear systems are usually considered:  
\[y(t)=G(z^{-1})u(t)+H(z^{-1})e(t), \tag{1}\]  
where \(G(\cdot)\) and \(H(\cdot)\) are transfer functions, \(z^{-1}\) is the time delay operator and \(e(t)\) a white noise signal. The identification procedure aims at determining a good estimate of \(G(\cdot)\) and \(H(\cdot)\), so the model can produce one-step-ahead predictions with a low variance error. The one-step-ahead predictor can be written in its regressor form as:  
\[\hat{y}(t|t-1,\theta)=\varphi(t)\theta, \tag{2}\]  
where \(\theta\) is the parameter vector and \(\varphi\) the regression vector that can contain past samples of system inputs and outputs and/or residuals. The model is then determined by identifying the parameters of the transfer functions. Models; structures are defined by imposing the structure of the transfer functions i.e., of the regression vector. The main parametric structure families are  
* FIR, characterized by the following regression vector: \[\varphi(t)=[u(t-d)\ldots u(t-d-m)]^{T},\] (3) with \(d\) and \(m\) being the delay of the samples;
* ARX, characterized by the following regression vector: \[\varphi(t)=[y(t-1),\ldots,y(t-l),u(t-d),\ldots,u(t-d),u(t-d-m)]^{T};\] (4) where \(l\) is the maximum delay needed for the output variables;
* ARMAX, characterized by the following regression vector: \[\varphi(t,\theta)= [y(t-1),\ldots,y(t-l),u(t-d),\ldots,u(t-d-m),u(t-d-m),\] \[,\varepsilon(t,\theta),\ldots,\varepsilon(t-k,\theta)]^{T}.\] where \(\varepsilon\) is the model residual and \(k\) is the associated maximum time delay.