## 3 Extended Kalman Filter with Maximum Correntropy  
A strategy for non-parametrically adapting systems based on entropy and divergence is information theoretic learning. In information theory, entropy measures the uncertainty associated with random variables. Since [PERSON] initially proposed entropy in 1948 [23], several definitions, including those by [PERSON] and [PERSON], have been developed for various purposes. For example, correntropy has been widely employed in machine learning and adaptive filtering to ascertain the local similarity of random variables. The MCC-based algorithms are further resistant to non-Gaussian noise, mainly when impulse noise is present. Figure 2: The block diagram of the IMM filter algorithm (one cycle with \(r\) models). Information theoretic and kernel approaches define the correntropy between two random variables provided in a kernel space as a generalized local similarity measure. MCC has approved many applications. For instance, MCC has benefited from signal processing, pattern recognition, and machine learning. This section introduces MCC and extended Kalman filters to deal with non-white Gaussian noise, notably the outliers. The correntropy of two scalar random variables measures the second and higher orders of statistical data in the joint probability density function. The metric similarity between two variables is measured by correntropy. The correntropy of two scalar random variables, X and Y, is defined as  
\[V_{\sigma}(X,Y)=E[k_{\sigma}(X,Y)]=\iint k_{\sigma}(x,y)f_{XY}(x,y)dxdy \tag{16}\]  
where \(k_{\sigma}(x,y)=G_{\sigma}(e)\) is a positive definite kernel function that satisfies the Mercer theory, \(f_{XY}(x,y)\) is the joint probability density function, and \(E[\cdot]\) represents the expectation operator. The provided Gaussian kernel is one of the kernel functions that are most frequently utilized. \[k_{\sigma}(x,y)=G_{\sigma}(e)=\frac{1}{\sqrt{2\pi}\sigma}\exp\biggl{(}-\frac{ e^{2}}{2\sigma^{2}}\biggr{)} \tag{17}\]  
When the kernel bandwidth is determined by \(e=x-y\), \(\sigma>0\), the joint distribution \(F(X,Y)\) is usually inaccessible. This is so because we might only have access to a certain amount of data in most real-world circumstances. The correntropy can be calculated using a sample mean square of the data to deal with this problem. Since there is often a dearth of data and a limited amount of information, the joint distribution is typically unknown in most real-world settings. When determining the correntropy from \(N\) samples, the sample mean estimator can be employed in the following circumstances:  
\[\hat{V}_{\sigma}(X,Y)=\frac{1}{N}\sum_{i=1}^{N}k_{\sigma}(x_{i},y_{i}) \tag{18}\]  
which, in terms of Gaussian kernel, can be written as  
\[\hat{V}_{\sigma}(X,Y)=\frac{1}{N}\sum_{i=1}^{N}G_{\sigma}(x_{i}-y_{i})=\frac{ 1}{N}\sum_{i=1}^{N}G_{\sigma}(e_{i}) \tag{19}\]  
Taking the Taylor series expansion of the Gaussian kernel, we have  
\[V_{\sigma}(X,Y)=\frac{1}{\sqrt{2\pi}\sigma}\sum_{n=0}^{\infty}\frac{\left(-1 \right)^{n}}{2^{n}\sigma^{2n}n!}E\Bigl{[}(X-Y)^{2n}\Bigr{]} \tag{20}\]  
The weighted average of all the even-order error variable moments is correntropy, as seen in \((X-Y)\). The kernel bandwidth is a factor that determines how heavily the second and higher-order moments are weighted. The second-order moment will predominate in the correntropy in the presence of a large kernel bandwidth \(\sigma\) (in comparison to the dynamic range of the data). The MCC serves as the basis for deriving the nonlinear version of the linear Kalman filter in detail from the linear version in [15]. An alternative objective function might be created, and its robustness increased using the weighted least squares (WLS) approach.