## 2 Time Series Modeling
### ARMA Model in Structural Identification
\[\mathbf{x}(k+1)=e^{\mathbf{A}\Delta_{t}}\mathbf{x}(k)+\int_{k\Delta_{t}}^{(k+1 )\Delta_{t}}e^{\mathbf{A}[(k+1)\Delta_{t}-\tau]}\mathbf{B}w(\tau)d\tau. \tag{5}\]  
Define \(s=(k+1)\Delta_{t}-\tau\) and \(ds=-d\tau\), and the second term in Equation (5) is rewritten as follows,  
\[\mathbf{z}(k)=\int_{0}^{\Delta_{t}}e^{\mathbf{A}s}\mathbf{B}w((k+1)\Delta_{t} -s)ds. \tag{6}\]  
Define the discrete version of the state matrix \(\Phi_{2n\times 2n}=e^{\mathbf{A}\Delta_{t}}\), and its characteristic polynomial \(P(\lambda)\) with the coefficients \(\alpha_{i}\) (\(\alpha_{0}=1\)) is expressed as follows,  
\[P(\lambda)=det(\Phi-\lambda\mathbf{I})=\lambda^{2n}+\alpha_{1}\lambda^{2n-1}+ \cdots+\alpha_{2n-1}\lambda+\alpha_{2n}=0. \tag{7}\]  
For \(i=0,1,\ldots,2n\), the outputs \(y(k+i)\), from the second equation of Equations (4)-(6), are expressed as follows,  
\[y(k) = \mathbf{D}\mathbf{x}(k)+v(k),\] \[y(k+1) = \mathbf{D}\mathbf{x}(k+1)+v(k+1)=\mathbf{D}(\Phi\mathbf{x}(k)+ \mathbf{z}(k))+v(k+1),\] \[\cdots = \cdots,\] \[y(k+2n) = \mathbf{D}(\Phi^{2n}\mathbf{x}(k)+\Phi^{2n-1}\mathbf{z}(k)+\Phi^ {2n-2}\mathbf{z}(k+1)+\cdots+ \tag{8}\] \[\Phi\mathbf{z}(k+2n-2)+\mathbf{z}(k+2n-1))+v(k+2n).\]  
Multiply \(y(k+i)\) by \(\alpha_{2n-i}\) and, adding all terms from \(i=0\) to \(2n\), it follows that,  
\[y(k+2n)+\alpha_{1}y(k+2n-1)+\cdots+\alpha_{2n-1}y(k+1)+\alpha_{2n}y(k) =\] \[\mathbf{D}P(e^{\mathbf{A}\Delta_{t}})\mathbf{x}(k)+\vec{\beta}_{ 1}\mathbf{z}(k+2n-1)+\vec{\beta}_{2}\mathbf{z}(k+2n-2)+\cdots+\vec{\beta}_{2 n}\mathbf{z}(k)+\sum_{i=0}^{2n}\alpha_{i}v(k+2n-i), \tag{9}\]  
where \(\vec{\beta}_{i}\) is a \(1\times n\) vector for a single output \(y\), i.e., \(\vec{\beta}_{1}=\mathbf{D}\), \(\vec{\beta}_{2}=\mathbf{D}\Phi+\alpha_{1}\mathbf{D}\), \(\dots\), \(\vec{\beta}_{2n}=\mathbf{D}\Phi^{2n-1}+\alpha_{1}\mathbf{D}\Phi^{2n-2}+\cdots+ \alpha_{2n-2}\mathbf{D}\Phi+\alpha_{2n-1}\mathbf{D}\). Analogous to Equation (7), \(P(e^{\mathbf{A}\Delta_{t}})=P(\Phi)=\Phi^{2n}+\alpha_{1}\Phi^{2n-1}+\cdots+ \alpha_{2n}=0\) according to the Cayley-Hamilton theorem [24]. The dot-product between a deterministic vector \(\vec{\beta}_{i}\) and a discrete Gaussian white noise vector process \(\mathbf{z}(k+i)\) generates a sequence of random scalar quantities. Based on [23], these dot-product terms can be represented by a weighted sum of a discrete Gaussian white noise \(e(t)\) with adapted scalar coefficients \(\beta_{j}\) according to certain dependency constraints. This step is referred to as the \(\beta\)-trick in the sequel. In addition, the weighted sum of white noise \(\sum_{i=0}^{2n}\alpha_{i}v(k+2n-i)\) can be replaced by \(e(k+2n)\). After replacing \(k+2n\) by \(t\), Equation (9) can be expressed as follows,  
\[\mathbf{z}(k)=\int_{0}^{\Delta_{t}}e^{\mathbf{A}s}\mathbf{B}w((k+1)\Delta_{t} -s)ds. \tag{6}\]0  
Re-denote \(-\alpha_{i}\) as \(\alpha_{i}\), and then Equation (10) can be rewritten as follows,  
\[\mathbf{z}(k)=\int_{0}^{\Delta_{t}}e^{\mathbf{A}s}\mathbf{B}w((k+1)\Delta_{t} -s)ds. \tag{6}\]1  
Based on several assumptions and constraints [23] used in the \(\beta\)-trick and \(y(t)\) being a stationary TS, Equation (11) shares the same form with an ARMA(\(2n,2n\)) model. In other words, \(y(t)\) can be estimated by \(\widehat{y}(t)\) using the previous \(2n\) time steps, \(y(t-1),\ldots,y(t-2n)\), a random Gaussian noise sequence, \(\epsilon(t)\), and a constant \(c\), as follows,  
\[\mathbf{z}(k)=\int_{0}^{\Delta_{t}}e^{\mathbf{A}s}\mathbf{B}w((k+1)\Delta_{t} -s)ds. \tag{6}\]2  
The above derivation links the arbitrary floor response \(y(t)\) from structural dynamics to an ARMA model. According to [25], the eigen values \(\lambda_{i}\) obtained from characteristic polynomial of discrete time state matrix \(\Phi\), Equation (7), can be used to identify the modal frequency and damping ratio of the structure. In other words, coefficients \(\alpha_{i}\) of the ARMA(\(2n,2n\)) model are related to the structural dynamic properties, which can directly contribute to the SHM [3; 9; 23; 25].