## 3 Methods
### Autoregressive Asymptotic Method  
The asymptotic method consists of the development of the amplitude and time delay between a reference temperature signal and a measured or generated one. The main assumption is that this pseudothermal contrast is sensitive only to the intensity variations \(A\) (due to the presence of the thermal resistance) and a delay \(\tau(s)\) proportional to both the depth of the thermal resistance and its thickness. If both parameters are estimated only the amplitude will be used to quantitatively estimated the thermal resistance.  
\[T_{m}(t)=AT_{ref}(t)+\tau\frac{dT_{ref}(t)}{dt} \tag{3}\]From the asymptotic Equation (3), and to avoid the influence of noisy data, an autoregressive inverse algorithm [27] is applied for a long time, as described by the Equation (4). The inverse processing is based on the Moore-Penrose inversion method [28; 29]. The advantage of using this autoregressive method is the ability to estimate the parameters as a function of time to create a strong link with the maximum sensitivity corresponding to the early \"separation\" between the reference signal and the signal to process. With the above remarks under consideration, the inverse of the system can be written as follows  
\[\left\{\begin{aligned} \int_{0}^{t_{1}}T_{m}(t)dt\\ \vdots\\ \int_{0}^{t_{l}}T_{m}(t)dt\\ \vdots\\ \int_{0}^{t_{N}}T_{m}(t)dt\\ \end{aligned}\right\}=\left[\begin{aligned} \int_{0}^{t_{1}}T_{ref}(t)dt& T_{ref}(0,t_{1})\\ \vdots&\vdots\\ \int_{0}^{t_{1}}T_{ref}(t)dt& T_{ref}(0,t_{i})\\ \vdots&\vdots\\ \int_{0}^{t_{N}}T_{ref}(t)dt& T_{ref}(0,t_{N})\\ \end{aligned}\right]\left[\begin{aligned}  A_{1}&\cdots& A_{i}& \cdots& A_{N}\\ \tau_{1}&\cdots&\tau_{i}&\cdots&\tau_{N}\\ \end{aligned}\right]\Leftrightarrow Y=X\beta \tag{4}\]  
where \(A(-)\) is the amplitude of the asymptotic development, \(\tau(s)\) is the time delay, \(t(s)\) is the time, and the indices \({}_{m}\) and \({}_{ref}\) correspond to the measured temperature and the reference one. From Equation (4), the pseudoinverse \(X\) is obtained by using the singular value decomposition (SVD) methods [20; 21; 22; 23]. The principle of SVD is to decompose a matrix I of size m by n into a product of three matrices, where \(U\) is an orthogonal matrix of size m by m, \(V\) is an orthogonal matrix of size n by n, and \(S\) is a \"diagonal\" matrix of size m by n. The diagonal elements of S are called singular values. They are positive and ranked from largest to smallest. If only the first p singular values are different from zero (\(p<\) n), the SVD can then be further reduced. Thus, the pseudoinversion of Equation (4) gives:  
\[\beta=\sum_{k=1}^{p}\frac{U_{k}^{T}Y}{s_{k}} \tag{5}\]  
Figure 4: Loglog dimensionless temperature response of the front face to a pulse of 1 ms duration and under adiabatic boundary conditions as a function of the thermal resistance depth and thickness of the reference sample composed of polycarbonate and aluminum.  
From the reference sample and the temperature response calculated and plotted in Figure 4, the autoregressive method is applied to all the data, with the \(Rth=0\) plot as the reference, corresponding to an assembly without any thermal resistance. The estimated parameters \(A(-)\) is reported in Figure 5 as a function of the generated thermal resistance \(Rth\). The time required to calculate all the time steps of the autoregressive method is approximately 6 s with a 2.3 GHz quad-core i5 MacBook Pro with 16 GB of RAM. The data can be fitted by using polynomial regression for the thermal resistance (Figure 5b). These results can be qualified by using a calibrating curve to link the nonquantitative estimation given by the asymptotic development method to the absolute value of he thermal resistance.