## 4 Luminescence Quenching for Oxygen and Temperature Sensing  
To demonstrate its advantages, the MTL approach was applied to the simultaneous determination of the oxygen concentration and temperature of a medium. There are different optical methods used to determine oxygen concentration since this is of great relevance for numerous research and application fields, ranging from biomedical imaging, packaging, environmental monitoring, process control, and chemical industry, to mention only a few [26]. Among the optical methods, a well-known approach is based on luminescence quenching [27; 28; 29]. The measuring principle is based on the quenching of the luminescence of a specific molecule (luminophore) by oxygen molecules. Because of the collisions of the luminophore with oxygen, both the luminescence intensity and decay time are reduced. Sensors based on this principle rely on approximate empirical models to parametrize the dependence of the sensing quantity (e.g., luminescence intensity or intensity decay time) on influencing factors. The most relevant parameter, which can be a major source of error in sensors based on luminescence sensing, is the temperature of the luminophore, since both the luminescence and the quenching phenomena are strongly dependent on temperature [26]. The conventional approach consists in relating the change of the luminescence decay time from the oxygen concentration through a multi-parametric model, called Stern-Volmer equation [28]. The value of the device-specific constants is then determined through calibration. The decay time can be easily measured by modulating the intensity of the excitation. The emitted luminescence is also modulated but shows a phase shift \(\theta\) which depends on the decay time. Without going into the details of the analytical model, the measured quantity, the phase shift \(\theta\), is most frequently related to the oxygen concentration \([O_{2}]\) and temperature \(T\) through the approximate equation [30]  
\[\frac{\tan\theta(\omega,T,[O_{2}])}{\tan\theta_{0}(\omega,T)}=\left(\frac{f( \omega,T)}{1+K_{SV1}(\omega,T)\cdot[O_{2}]}+\frac{1-f(\omega,T)}{1+K_{SV2}( \omega,T)\cdot[O_{2}]}\right)^{-1} \tag{7}\]  
where \(\theta_{0}\) and \(\theta\), respectively, are the phase shifts in the absence and presence of oxygen, \(f\) and \(1-f\) indicate the fraction of the total emission of two components under unquenched conditions, \(K_{SV1}\) and \(K_{SV2}\) are associated (Stern-Volmer) constants for each component. Since the phenomena of luminescence and luminescence quenching are strongly influenced by the temperature, the parameters \(\theta_{0}\), \(K_{SV1}\), \(K_{SV2}\), and \(f\) need to be modeled through different temperature dependencies [30]. The value of the parametrisation quantities is determined through non-linear regression. \(\omega\) is the angular frequency of the modulation of the excitation light. Finally, Equation (7) must be inverted to obtain \([O_{2}]\) as a function of \(\theta\), T, and \(\omega\). To be able to have more information as input to our network, we will not use a single \(\omega\) frequency value, but 16. Let's define  
\[r(\omega,T,[O_{2}])\equiv\frac{\tan\theta(\omega,T,[O_{2}])}{\tan\theta(\omega,T,[O_{2}]=0)}. \tag{8}\]  
The goal of the network is to predict the oxygen concentration and temperature from an array of values of \(r(\omega,T,[O_{2}])\) evaluated at a discrete set of sixteen \(\omega_{i}\), with \(i=1,...16\), that have been used for the measurements. The \(j\)th measurement can be written as \(x^{[j]}=(r_{1}^{[j]},r_{2}^{[j]},...,r_{16}^{[j]})\) with \(r_{i}^{[j]}=r(\omega_{i},T^{[j]},[O_{2}]^{[j]})\) and \(i=1,...16\). Each measurement \(j\) corresponds to a specific tuple of the oxygen concentration and temperature \((T^{[j]},[O_{2}]^{[j]})\).