# A Novel Reconstruction Method for Irregularly Sampled Observation Sequences for Digital Twin
## 2 Problem Formulation  
In this section, we first give formal descriptions of multivariate regular and irregular observation sequences. **Definition 1**. : _Consider an observation dataset \(\mathcal{D}=\{(\mathbf{y}_{n},t_{n})\}_{n=0}^{N}\), where \(\mathbf{y}_{n}\in\mathbb{R}^{d}\) is the observation vector at sampling time \(t_{n}\), and \(N+1\) is the total number of samples. Let NaN denote the missing data in the observation vector. And when an observation vector has missing data, its dimension satisfies \(\dim(\mathbf{y}_{n})<d\). Then, if dataset \(\mathcal{D}\) meets the following two conditions:__For any two consecutive observation vectors_ \(\mathbf{y}_{n}\) _and_ \(\mathbf{y}_{n+1}\) _in_ \(\mathcal{D}\)_, the sampling interval_ \(\Delta t_{n}=t_{n+1}-t_{n}\) _is a constant_ \(C\)_, i.e.,_ \(t_{n+1}-t_{n}\equiv C\) _for_ \(\forall n\in[0,N]\)_._
2 _No observation vector_ \(\mathbf{y}_{n}\) _in_ \(\mathcal{D}\) _contains missing data NaN, i.e.,_ \(\dim(\mathbf{y}_{n})=d\) _for_ \(\forall n\in[0,N]\)_._  
_Then,_ \(\mathcal{D}\) _forms a regular sampled time series. Conversely, if_ \(\mathcal{D}\) _does not satisfy both conditions, it forms an irregular sampled time series._  
Condition (1) in Definition 1 describes the regularity of the sampling sequence distribution on the time dimension, that is, whether a fixed sampling interval is followed. Condition (2) describes whether the sampling time of every variable in the multivariate observation sequence is aligned. Only when \(\mathcal{D}\) satisfies both conditions, which means that the sampling times of all the variables are aligned and there is an unified and fixed sampling interval, it can be regarded as a regular multivariate observation sequence. If either of the conditions is not met, it is considered as an irregular multivariate observation sequence. Overall, a regular multivariate sampling sequence needs to meet both conditions: a fixed sampling interval and aligned sampling times. Due to various uncertainties in communication networks and non-synchronous sampling of sensors, twin systems usually receive non-aligned, irregular sampling sequences. Consider the nonlinear discrete-time system corrupted by measurement noise as follows:  
\[\left\{ \begin{array}{l}\mathbf{x}_{n+1}=\mathbf{f}(\mathbf{x}_{n},\mathbf{u}_{n})\\ \mathbf{y}_{n+1}=\mathbf{h}(\mathbf{x}_{n+1},\mathbf{u}_{n})+\mathbf{v}_{n}\end{array} \right. \tag{1}\]  
where \(\mathbf{x}_{n}\in\mathbb{R}^{s}\), \(\mathbf{u}_{n}\in\mathbb{R}^{c}\), and \(\mathbf{y}_{n}\in\mathbb{R}^{d}\) are the state vector, control vector, and observation vector of the system at sampling time step \(t_{n}\), respectively. \(\mathbf{f}\colon\mathbb{R}^{s}\times\mathbb{R}^{c}\to\mathbb{R}^{s}\) and \(\mathbf{h}:\mathbb{R}^{s}\times\mathbb{R}^{c}\to\mathbb{R}^{d}\) are the nonlinear state transition function and measurement function, respectively. For system (1), we make the following assumptions:  
**Assumption 1**. : _The process noise of the system is ignored, and it is considered that system (1) only contains measurement noise \(\mathbf{v}_{n}\), which follows a Gaussian distribution with mean \(\mathbf{0}\) and covariance \(\mathbf{R}\)._  
**Assumption 2**. : _Due to the uncertainties of network communication and non-synchronous sampling of sensors, the time series formed by the observations \(\mathbf{y}_{n}\) of system (1) is no longer regular.