## 3 Applications to Kinetic Models  
### A Recursive KF Algorithm Application  
We considered an object moving along a straight line. Its movements are captured by a video camera. The object is moving along a straight line, with random acceleration \(a_{t}\), and \(\Delta_{t}\) is the time interval. Its location is described by the \(x\)-coordinate. We also assumed that the acceleration has a Gaussian distribution with zero mean and constant variance (\(\sigma_{a}^{2}\)). The position of the object, denoted as \(y_{t}\), is also noisy (Gaussian with zero mean and constant variance). We denote the variance as \(\sigma_{y}^{2}\). The state vector of this motion is given by \(v_{t}=(x_{t},v_{t})\), where the speed is the first derivative of the distance, i.e., \(v_{t}=\dot{x}_{t}\). We also assumed that there was no process control and get:  
\[x_{t}=\left( \begin{array}{cc}1&\Delta_{t}\\ 0&1\end{array} \right)\left( \begin{array}{c}x_{t-1}\\ v_{t-1}\end{array} \right)+a_{t}\left( \begin{array}{c}\frac{\Delta_{t}^{2}} {2}\\ \Delta_{t}\end{array} \right)=F_{t}x_{t-1}+w_{t},\;Q_{t}=\mathrm{var}(w_{t}), \tag{13}\]  
where  
* \(F_{t}\) is a known state transition matrix applied to the \(t-1\) state \(x_{t-1}\),
* \(Q_{t}\) is the variance matrix, and
* \(w_{t}\) is a process noise vector, which has the joint distribution as a multivariate Gaussian with a variance matrix \(Q_{t}\) and \(\mu_{t,t}=E[w_{t,t}]=0\) for \(i=\overline{1,n}\). Now, let \(G_{t}=\left(\frac{\Delta_{t}^{2}}{2},\Delta_{t}\right)^{T}\). Then, we get:  
\[Q_{t}=E\Big{[}w_{t},w_{t}^{T}\Big{]}=G_{t}E\Big{[}a_{t}^{2}\Big{]};G_{t}^{T}= \sigma_{a}^{2}G_{t}G_{t}^{T}=\sigma_{a}^{2}\left[ \begin{array}{cc}\frac{ \Delta_{t}^{4}}{4}&\frac{\Delta_{t}^{3}}{2}\\ \frac{\Delta_{t}^{3}}{2}&\Delta_{t}{}^{2}\end{array} \right]. \tag{14}\]  
This means that \(F_{t}\), \(Q_{t}\), and \(G_{t}\) are independent of \(t\). Hence, we will call them just \(F\), \(Q_{t}\) and \(G\). To measure the object's position at time \(t\), we get:  
\[y_{t}=\left( \begin{array}{c}v_{t}\\ 0\end{array} \right)\left( \begin{array}{cc}1&0\\ 0&0\end{array} \right)x_{t}=v_{t}+H_{t}\cdot x_{t}, \tag{15}\]  
where  
* \(v_{t}\) is the observation noise and
* \(H_{t}\) is an observation matrix. Hence, the variance matrix is:  
\[R=E\Big{[}v_{t}v_{t}^{T}\Big{]}=\left( \begin{array}{cc}\sigma_{y}^{2}&0\\ 0&0\end{array} \right). \tag{16}\]  
We denote \(\hat{x}_{t|t}\) as the estimate of state \(x_{t}\) at time \(t\) and \(P_{t|t}\) as the variance matrix of the error \(x_{t}-\hat{x}_{t|t}\). Our goal was to minimize \(P_{t|t}\). The initial position of the object is \(\hat{x}_{0|0}=\left(0,0\right)^{T}\). If this position vector is known accurately, then the variance matrix is zero, i.e.,  
\[P_{0|0}=\left( \begin{array}{cc}0&0\\ 0&0\end{array} \right);\]  
otherwise, we will have that:  
\[P_{0|0}=\left( \begin{array}{cc}a&0\\ 0&a\end{array} \right),\]  
where \(a\) is real and suitably large, \(a>0\). When \(t=1\), we first predicted \(\hat{x}_{1|0}\) and computed its variance matrix \(P_{1|0}\) by using the prediction equations:  
\[\hat{x}_{t|t-1}=F_{t}\hat{x}_{t-1|t-1}+B_{t}u_{t}=F\hat{x}_{t-1|t-1}, \tag{17}\]  
\[P_{t|t-1}=F_{t}P_{t-1|t-1}F_{t}^{T}+Q_{t}=FP_{t-1|t-1}F_{t}^{T}+Q_{t}=FP_{t-1 |t-1}+Q, \tag{18}\]  
where \(B_{t}\) is the control matrix, which is applied to a control vector \(u_{t}\). We then calculated the auxiliary data \(S_{1}\) and \(z_{1}\) from the equations:  
\[\widetilde{z}_{t}=y_{t}-H_{t}\hat{x}_{t|t-1}=y_{t}-H\hat{x}_{t|t-1}, \tag{19}\]  
\[S_{t}=H_{t}P_{t|t-1}H_{t}^{T}+R_{t}=HP_{t|t-1}H^{T}+R, \tag{20}\]  
where \(R_{t}\) is the variance matrix of a multivariate Gaussian distribution vector \(v_{t}\).