text:
\section{Introduction}

In recent years, estimation problems involving filtering and smoothing based on dynamic state space models (SSMs) have received significant attentions. These problems are frequently encountered in many areas, such as target tracking, fault detection and diagnosis, parameter estimation, navigation, and many others. The celebrated Kalman filter (KF)~\cite{Kalman:1960} offers optimal estimation with the minimum mean square error (MMSE) for a linear SSM when both the process and measurement noises are Gaussian. Nevertheless, most applications in practice inherently have nonlinear SSMs, for which an optimal nonlinear filter or smoother is typically intractable. Several sub-optimal solutions were proposed for nonlinear SSMs, including the extended KF~\cite{anderson:1979}, unscented KF~\cite{Julier:2000}, cubature Kalman filter (CKF)~\cite{arasaratnam:2009}, and others. Some interesting relations among these solutions can be found  in~\cite{gustafsson:2012,wu:2006}. Meanwhile, for smoothing, the Rauch-Tung-Striebel (RTS) smoother was introduced in~\cite{RTS:1965} for linear Gaussian SSMs. A general framework of Gaussian optimal smoothing for nonlinear SSMs was proposed in~\cite{sarkka:2010}, following which several sub-optimal nonlinear RTS smoothers were developed, such as the extended RTS smoother, unscented RTS smoother, cubature Kalman smoother (CKS), and others.

Although the aforementioned filtering and smoothing methods in general perform well under the Gaussian assumption, they can potentially break down in the presence of heavy-tailed non-Gaussian noises, which may appear in either the process procedure or measurement procedure. This happens in, for instance, tracking a maneuvering target with outliers in observations~\cite{bilik:2010}. A primary reason behind the degradation is that the traditional KF/RTS and their sub-optimal extensions were derived from the MMSE criterion, and the resulting estimates are sensitive to heavy-tailed noises~\cite{schick:1994}.

Persistent efforts have been devoted to tackling heavy-tailed noises in order to obtain more robust state estimates. Multiple models based techniques and sequential Monte Carlo sampling methods can be used to handle non-Gaussian noises~\cite{kotecha:2003,arulampalam:2002}. Unfortunately, their heavy computational burden makes them considerably more difficult to implement in real time. Another common strategy to enhance the robustness is robust statistics (or influence functions). Masreliez and Martin~\cite{masreliez:1977} introduced the Huber cost function~\cite{Huber:2009} to a linear SSM to construct a robust KF by recasting the filtering problem as a linear regression. After that, this type of robust filters were extended to nonlinear SSMs via linearization~\cite{Gandhi:2010}. More recently, to reduce the approximation error arising from linearization, nonlinear regression based Huber's robust filters were proposed in~\cite{karlgaard:2015,Chang:2017}. Other approaches for robust estimation, e.g., the $H_{\infty}$ filter~\cite{hu2011second} and heavy-tailed distributions based filters and smoothers~\cite{Huang:2016,agamennoni:2012,huang:2016_2}, were also studied.

In addition, several recent studies explored using optimization to design robust smoothers since smoothing is in essence an optimization problem. Specifically, an $\ell_1$ robust Kalman smoother was presented in~\cite{aravkin2011ell} based on an interior point method, where the Laplace distribution was utilized to model the measurement noise. Quadratic programming was employed to design a student's t Kalman smoother in~\cite{aravkin2014robust}, where both the process and measurement noise were modeled by student's t distribution. Quadratic support functions were introduced in~\cite{aravkin2013sparse} to present generalized Kalman smoothing methods. Other optimization methods for robust Kalman smoothing were discussed in ~\cite{aravkin:2017generalized}. Although optimization based methods provide reasonable smoothing results, they still have some limitations. A major one is that the methods ignore covariance propagation, and hence some important statistic information about the state may get lost. Besides, these methods were primarily designed for smoothing, and it is difficult to extend them for filtering, which is more frequently encountered in real-time applications.

Recently a concept called correntropy in information theoretic learning and an associated maximum correntropy criterion (MCC)~\cite{Liu:2007} was utilized as a new cost function for robust filtering~\cite{liu:2017}. As shown in~\cite{Liu:2007}, MCC has a close relationship with M-estimators. In fact, correntropy is a robust recasting of the Welsch cost function~\cite{dennis:1978}. Compared with Huber's cost function, the Welsch function penalizes a large fitting errors~\cite{Liu:2007}. Furthermore, the Welsch M-estimator outperforms Huber's M-estimator in dealing with different types of outliers. Meanwhile the kernel size in the Welsch is less sensitive to select than the threshold in Huber's M-estimator, which can have a significant impact on the estimation performance~\cite{He:2014}.

In the present paper, we focus on developing a general robust framework, including both robust filtering and robust smoothing, to estimate the states of nonlinear SSMs where heavy-tailed noises may be present in both the process and measurement procedures. The proposed framework is based on the MCC and a half-quadratic (HQ) optimization method. Numerical results show that the proposed algorithms can provide reliable state estimation, and are robust to almost all levels of contamination. In comparison with existing robust algorithms, our methods yield more accurate results with a moderate increase of computation time.

\subsection{Related Work}

Correntropy and its associated MCC criterion was originally proposed in~\cite{Liu:2007}, where the relationship between MCC and M-estimation was established. The advantages of the MCC in dealing with non-Gaussian noises with heavy tails were demonstrated in~\cite{Liu:2007} via numerical simulations. Thereafter, the MCC criterion was employed to improve the robustness in several areas, such as feature selection~\cite{Yan:2011}, robust face recognition~\cite{he:2011maximum}, principal component analysis~\cite{he:2011_2}, and so on. During last few years, several researchers started to employ the MCC to deal with heavy-tailed noises in Kalman filtering. It was first utilized to improve the robustness of the KF in~\cite{Cinar:2011,Cinar:2012}, where gradient descent was used for state estimation. Those methods, however, ignored covariance propagation, which is an important part in the KF.

In~\cite{izanloo:2016}, a robust KF called MCC-KF was proposed, which integrated both the MCC and weighted least squares into the traditional KF framework. A square-root format of the MCC-KF was proposed in~\cite{kulikova:2017} to improve the numerical stability. MCC-KF and its square-root version only modified the weights of the measurements and thus had the same prediction step as the traditional KF which was derived under the Gaussian assumption. Consequently, the MCC-KF and its square-root version were sensitive to non-Gaussian process noise. To handle this issue, another robust KF called MCKF~\cite{chen:2017} was developed using an approach similar to that of the Huber based robust KF, i.e., recasting the filtering problem as a linear regression. In MCKF, a fixed-point algorithm was employed to iteratively provide robust estimation, and both the prediction step and measurement update were adapted. Henceforth, several variants of the MCKF were proposed for nonlinear systems by linearizing the nonlinear mappings, e.g., the Taylor series~\cite{liu2:2016}, unscented transformation~\cite{liu:2017}, and Gauss-Hermite quadrature rule~\cite{qin:2017}. Most recently, the MCC was employed to solve the robust estimation problem in the continuous-discrete system~\cite{kulikov:2018}. In addition, the MCC was also utilized to suppress outliers in measurements assuming Gaussian noise in the process~\cite{wang:2016,wang:2017}.

Although above studies have clearly demonstrated the potential of the MCC for robust Kalman filtering, there are some deficiencies with the state-of-the-art research in this area. Specifically, a general framework of using MCC for robust Kalman filtering and smoothing is missing.  Most existing works have addressed the problem under a specific scenario, but the resulting methods may degrade significantly under a different scenario. For example, the method in \cite{liu:2017} was developed under the assumption that the process and measurement are both contaminated, its performance may experience considerable degradation under a different scenario, e.g., when only the process or the measurement is contaminated. In addition, most recently developed robust Kalman filtering schemes using the MCC were based on linear regression~\cite{chen:2017,liu2:2016,liu:2017,qin:2017}, where a linearization procedure is inevitable, resulting in a loss of accuracy. Another limitation is that no existing work has explored MCC for robust Kalman smoothing.

\subsection{Main Contributions}

The main contributions of this paper are summarized as follows.
\begin{itemize}
  \item We propose a general framework that integrates MCC for robust Kalman filtering. We take an optimization perspective based approach, rather than linear regression as employed in most existing MCC based robust Kalman filtering solutions. This approach bypasses the linearization procedure, which not only makes our methods derivative-free but also reduces approximation errors caused by linearization.
  \item A distinctive feature of the proposed framework involves replacing the quadratic cost in the traditional KF by a sum of weighted instantaneous correntropy terms to improve robustness. The weight of each correntropy is automatically tuned to cope with different scenarios involving Gaussian or heavy-tailed non-Gaussian noises.
  \item The optimization problem underlying the proposed framework is solved by a half-quadratic method. This enables our robust Kalman filtering algorithm to be cast using the conventional KF iterative procedures, thus inheriting benefits associated with the latter.
  \item We also extend the above robust filtering framework for robust smoothing, resulting in a MCC based robust Kalman smoother. To the best of our knowledge, our approach is the first robust nonlinear Kalman smoother based on the MCC.
\end{itemize}

The paper is organized as follows. We first formulate the problem in Section~\ref{Set:prom}, and derive the robust nonlinear Kalman filter based on MCC in Section~\ref{Sec:fmcc}. In Section~\ref{smcc}, the robust nonlinear Kalman smoother is presented. We show numerical examples to illustrate the performance of the proposed robust filtering and smoothing algorithms in Section~\ref{simu}. Finally, we draw conclusions and summarize future related research in Section~\ref{con}.

\emph{Notations:} In this paper, normal font letters are scalars, boldface lowercase letters denote column vectors, and boldface uppercase letters mean matrices. The estimate of state $\bm x_t$ given the measurements from $t=1$ to $t=m$ is denoted by $\hat{\bm x}_{t|m}$ where $m=t$ means filtering, and $m=T$ means smoothing, with $T$ denoting the total number of observations. $\bm I_n$ denotes the $n$-dimensional identity matrix. $\mathcal{N}(\cdot,\cdot)$ denotes a Gaussian distribution.

\section{Problem Formulation}
\label{Set:prom}

In this paper, we address a sequential estimation problem in nonlinear systems involving non-Gaussian noises. More specifically, both the process noise and measurement noise are primarily Gaussian random processes; but may occasionally be contaminated by outliers with an unknown distribution. To model this, we consider the following nonlinear discrete-time state-space model (SSM)
\begin{align}
\bm x_t &= \bm f(\bm x_{t-1})+(1-\rho_1)\bm v_{t-1}+\rho_1\bm\varepsilon_1\label{SSM1}\\
\bm y_t &= \bm h(\bm x_t) +(1-\rho_2) \bm w_t+\rho_2\bm\varepsilon_2\label{SSM2}
\end{align}
where $t=\{1,\cdots,T\}$ is the time index; $\bm x_t\in\mathbb{R}^n$ is the state of interest to be estimated; $\bm y_t\in\mathbb{R}^m$ is the observation; $\bm f(\cdot)$ and $\bm h(\cdot)$ are some known nonlinear mappings, representing the state transition and measurement procedure, respectively; $\bm v_{t-1}$ and $\bm w_t$ are zero-mean Gaussian distribution vectors with $\bm Q_{t-1}$ and $\bm R_t$ as covariance, representing the nominal process noise and observation noise, respectively; $\bm\varepsilon_1$ and $\bm\varepsilon_2$ are some contaminating noises; $\rho_1$ and $\rho_2$ are the corresponding contamination ratios. The initial state ${\bm x}_{0}$ is assumed to follow a known Gaussian prior distribution $p(\bm x_0)=\mathcal{N}(\hat{{\bm x}}_{0|0},\bm P_{0|0})$. Furthermore, $\bm x_0$, $\bm v_{t-1}$, $\bm w_t$, $\bm\varepsilon_1$ and $\bm \varepsilon_2$ are assumed to be mutually independent.

Practically, the distributions of the contaminating noises and their corresponding contamination ratios are often unknown. If one chooses to ignore the contaminating noises and run a Gaussian approximation filter and smoother, the performance will likely to be unsatisfying or even diverge~\cite{Huber:2009}. Although ineffective in dealing with the contaminating noise, the Gaussian approximation filter and smoother are popular in those nonlinear sequential estimation when both process and measurement noises are Gaussian due to their computational efficiency and relative high accuracy. This motivates us to explore extended versions of the Gaussian approximation filter and smoother to deal with the contaminating noise in nonlinear systems.

The aim of this work is to develop a unified framework for robust nonlinear filter and smoother design with the information $p(\bm x_0)$, $p(\bm v_t)$, $p(\bm w_t)$ and the observations (namely, $\bm y_{1:t}$ for filtering and $\bm y_{1:T}$ for smoothing, respectively). To address non-Gaussian contaminating noise, the proposed framework employs a new maximum correntropy criterion (MCC)~\cite{Liu:2007} which integrated with the Gaussian approximation filter and smoother, resulting in a class of MCC based filter and smoother solutions. The developed robust filtering and smoothing techniques are expected to have a similar performance as the Gaussian approximation counterparts in the absence of outliers, meanwhile exhibiting less sensitivity in scenarios involving contaminating noise.

In the following, our robust framework is presented in conjunction with the CKF~\cite{arasaratnam:2009} and CKS~\cite{arasaratnam:2011}, which are popular for nonlinear system estimation. A brief summary of the CKF and CKS is included in Appendix \ref{A} to facilitate the presentation of our framework. It is straightforward to extend the framework for use with other nonlinear Gaussian approximation filters and smoothers.

\section{Robust Nonlinear Kalman filter with MCC}
\label{Sec:fmcc}
\subsection{MCC-Based Filtering}
\label{par_dis}

In the Bayesian filtering paradigm, the core problem is to calculate the posterior density through Bayes' rule based on the measurements up to $t$,
\begin{align}
p(\bm x_t|\bm y_{1:t})\propto{p(\bm y_t|\bm x_t)p(\bm x_t|\bm y_{1:t-1})}
\label{meaupt}
\end{align}
where $p(\bm y_t|\bm x_t)$ is the likelihood function which is defined by the measurement process and $p(\bm x_k|\bm y_{1:t-1})$ is the predictive density which is controlled by Chapman-Kolmogorov equation
\begin{align}
p(\bm x_t|\bm y_{1:t-1})=\int{p(\bm x_t|\bm x_{t-1})p(\bm x_{t-1}|\bm y_{1:t-1})d{\bm x_{t-1}}}
\label{timeupt}
\end{align}
where $p(\bm x_{t-1}|\bm y_{1:t-1})$ is the posterior density at time $t-1$ and $p(\bm x_t|\bm x_{t-1})$ is the transition density which is defined by the state transition process. The maximum a posteriori estimation (MAP) of $\bm x_t$ can be obtained through minimizing the negative log posterior density, i.e.,
\begin{align}
\hat{\bm x}_{t|t}=&\text{arg}\min_{\bm x_t}-\log p(\bm x_t|\bm y_{1:t})\notag\\
=&\text{arg}\min_{\bm x_t}\Big(-\log p(\bm x_t|\bm y_{1:t-1})-\log p(\bm y_t|\bm x_t)\Big)
\label{cost1}
\end{align}
Under the Gaussian approximation, i.e., both the process noise and measurement noise are assumed Gaussian and the predictive density is a Gaussian distribution, dropping terms that do not depend on $\bm x_t$, we can rewrite (\ref{cost1}) as~\cite{Chang:2017}
\begin{align}
\hat{\bm x}_{t|t}=\text{arg}\min_{\bm x_t}\Big(\frac{1}{2}\|\bm x_t-\hat{\bm x}_{t|t-1}\|^2_{{\bm P}_{t|t-1}^{-1}}
+\frac{1}{2}\|\bm y_t-h(\bm x_t)\|^2_{{\bm R}_{t}^{-1}}\Big)
\label{cost2}
\end{align}
where $\|\bm z\|_{\bm A}^2=\bm z^T\bm A\bm z$ and $\hat{\bm x}_{t|t-1}$ is the predicted state with the prediction error covariance $\bm P_{t|t-1}$. We define the normalized error as
\begin{align}
\bm \alpha_{t}&=\bm P_{t|t-1}^{-1/2}\left(\bm x_t-\hat{\bm x}_{t|t-1}\right)\label{def1} \\
\bm \beta_{t}&=\bm R_{t}^{-1/2}\left(\bm y_t-h(\bm x_t)\right)\label{def2}
\end{align}
Then (\ref{cost2}) can be further interpreted as
\begin{align}
\hat{\bm x}_{t|t}=&\text{arg}\min_{\bm x_t}\ \left(\frac{1}{2}\bm \alpha_{t}^T\bm \alpha_{t}+\frac{1}{2}\bm \beta_{t}^T\bm \beta_{t}\right)\notag\\
=&\text{arg}\min_{\bm x_t}\ \left(\frac{1}{2}\sum_{i=1}^n\alpha_{t,i}^2
+\frac{1}{2}\sum_{j=1}^m\beta_{t,j}^2\right)
\label{cost:2}
\end{align}
where $\alpha_{t,i}$ is the $i$-th component of $\bm \alpha_t$ and $\beta_{t,j}$ is the $j$-th component of $\bm \beta_t$.

Equation \eqref{cost:2} shows that the Gaussian approximation filtering approach is actually a minimum mean square error (MMSE) estimator. It is clear that the quadratic function (i.e., $\ell_2$ norm) in MMSE has an apparent effect of emphasizing the contribution of large errors, which makes the MMSE estimator sensitive to impulsive noise. To improve the robustness, the $\ell_2$-norm objective function should be replaced by another cost function which is insensitive to the impulsive noise.

We propose to integrate the concept of correntropy~\cite{Liu:2007}, which has been shown effective in dealing with outliers~\cite{mandanas:2017}, for robust Kalman filtering and smoothing, by utilizing the maximum correntropy criterion (MCC) to replace the MMSE criterion. Specifically, we apply the instantaneous correntropy to each component of the normalized error, resulting in the following cost function for robust nonlinear filtering:
\begin{align}
\hat{\bm x}_{t|t}=\text{arg}\max_{\bm x_t}\left(\sum_{i=1}^n a_{t,i}\kappa_{\sigma_{t,i}}(\alpha_{t,i})
+\sum_{j=1}^m b_{t,j}\kappa_{\eta_{t,j}}(\beta_{t,j})\right)
\label{cost_ro}
\end{align}
where $\kappa_\sigma(\cdot)$ is the kernel function which satisfies Mercer's Theorem~\cite{Vapnik:1995}; $a_{t,i}$ and $b_{t,j}$ are weighting coefficients to be further determined. The kernel function plays a central role in MCC and different kernel functions provide different estimation results.

Without loss of generality,  we apply the the Gaussian kernel function with bandwidth $\sigma$,
\begin{align*}
\kappa_{\sigma}(e)=\exp\left({-\frac{e^2}{2\sigma^2}}\right).
\end{align*}
The Gaussian kernel based MCC is closely related to the Welsch M-estimator, where outliers are given small weights in optimization~\cite{He:2014}. For completeness, in Appendix \ref{B}, we provide a short review of the correntropy, MCC, and related properities.

There are two aspects that should be noted in \eqref{cost_ro}. First, a weighting coefficient (namely $a_{t,i}$ and $b_{t,j}$) is introduced to each correntropy. These coefficients should be properly selected to maintain consistency to the Gaussian approximation filter when the SSM does not involve impulse noises (see Section \ref{sle_par} for details on selection for these parameters). Second, having distinct bandwidth for each correntropy offers additional flexibility of adaptiveness to different applications. One possible strategy is to use the same kernel bandwidth for each component of the predictive error and a different kernel bandwidth for the measurement error terms in (\ref{cost_ro}). This makes the proposed algorithms suitable for handling different process noise and measurement noise. For example, in a case where outliers only occur in the measurement procedure, we can select a large $\sigma_t$ so that the resulting MCC behaves like a quadratic criterion, and a smaller $\eta_t$ to defend the outliers.

\subsection{A Half-Quadratic Optimization Based Solution}

\label{sle_par}

A number of optimization techniques such as the steepest descent~\cite{Liu:2007} and the fixed point iteration approach~\cite{Chen:2015} can in principle be applied to solve MCC based problems. However, our correntropy criterion~\eqref{cost_ro} involves different kernel bandwidths. The problem can be more efficiently solved by a half-quadratic approach~\cite{Geman:1992}. Specifically, our proposed solution to \eqref{cost_ro} applies the half-quadratic approach in an iterative manner, through which auxiliary variables are introduced to transform the original problem into a quadratic problem. The resulting quadratic problem can be cast in the conventional Gaussian approximation filtering framework. This is a major advantage of our proposed solution. The half-quadratic approach is based on the concept of the convex conjugate function and, in particular, the following result.

\begin{proposition}
\label{propo}
There exists a convex function $\psi$: $[-1,0)$ $\to$ $ [-1,0)$, such that
\begin{align}
\kappa_{\sigma}(x)=\sup_{-1\le p<0}\left(\frac{p}{2\sigma^2}x^2
-\psi(p)\right)
\label{pro1}
\end{align}
and for a fixed $x$, the supremum is reached at $p=-\kappa_{\sigma}(x)$.

Proof: see Appendix~\ref{prop_proof}.
\end{proposition}

Using (\ref{pro1}), the robust filtering problem \eqref{cost_ro} can be reinterpreted in an augmented format as
\begin{align}
\hat{\bm x}_{t|t}=\text{arg}\max_{\bm p_t,\bm q_t,\bm x_t}\
\left\{\sum_{i=1}^n a_{t,i}\left(\frac{p_{t,i}}{2\sigma_{t,i}^2}\alpha_{t,i}^2
-\psi(p_{t,i})\right)
+\sum_{j=1}^m b_{t,j}\left(\frac{q_{t,j}}{2\eta_{t,j}^2}\beta_{t,j}^2
-\psi(q_{t,j})\right)\right\}
\label{cost_aug}
\end{align}
where $\bm p_t=\{p_{t,i}\}_{i=1}^n$ and $\bm q_t=\{q_{t,j}\}_{j=1}^m$ are the collections of the auxiliary variables. As shown in Section~\ref{proof}, maximizing \eqref{cost_ro} with respect to $\bm x_t$ is equivalent to maximizing the augmented cost function (\ref{cost_aug}) in the enlarged parameter domain, i.e., $\{\bm p_t,\bm q_t, \bm x_t\}$. The augmented optimization problem can be solved in an alternating maximization manner. Specifically, we first maximize (\ref{cost_aug}) with respect to $\bm p_t$ and $\bm q_t$ while keeping $\bm x_t$ fixed. Then $\bm x_t$ is updated with the newest $\bm p_t$ and $\bm q_t$. This process is repeated until a convergence criterion is met.

For the first step of the $k$-th iteration after $\bm x_t^{k-1}$ is given and $\{\bm \alpha^{k-1}_{t},\ \bm \beta^{k-1}_{t}\}$ are computed via~\eqref{def1}\eqref{def2}, by Proposition~\ref{propo}, $\bm p_t^k$ and $\bm q_t^k$ are updated as follows,
\begin{align}
p_{t,i}^k&=-
\exp\left({-\frac{(\alpha_{t,i}^{k-1})^2}{2\sigma_{t,i}^2}}\right)\label{p_up}\\
q_{t,j}^k&=-
\exp\left({-\frac{(\beta_{t,j}^{k-1})^2}{2\eta_{t,j}^2}}\right)\label{q_up}
\end{align}
At the beginning of the iterative procedure, i.e., $k=1$, we set $\bm \alpha^{0}_{t}=\bm \beta^{0}_{t}=0$ as the initial condition.

For the second step of the $k$-th iteration, we fix $\{\bm p_t^k,\bm q_t^k\}$ and drop the items that do not depend on $\bm x_t$, and then (\ref{cost_aug}) can be simplified as
\begin{align}
\hat{\bm x}_{t|t}^k=\text{arg}\max_{\bm x_t}\Big(
\sum_{i=1}^n\frac{a_{t,i}p^k_{t,i}}{2\sigma_{t,i}^2}\alpha_{t,i}^2
+\sum_{j=1}^m\frac{b_{t,j}q^k_{t,j}}{2\eta_{t,j}^2}\beta_{t,j}^2
\Big)
\label{cost_last}
\end{align}

Before optimizing the above cost function, it is necessary to discuss the choices of the constants $a_{t,i}$ and $b_{t,j}$. In order to maintain the consistency of the Gaussian approximation filter, $a_{t,i}$ and $b_{t,j}$ should be properly selected so that (\ref{cost_last}) is equivalent to (\ref{cost:2}) when both process and measurement noises in the SSM are Gaussian. Specifically, when both noises follow the Gaussian distribution, the following constraints must be in place
\begin{align}
\frac{a_{t,i}p^k_{t,i}}{\sigma_{t,i}^2}&=-1\\
\frac{b_{t,j}q^k_{t,j}}{\eta_{t,j}^2}&=-1
\end{align}
In such a case, all bandwidths should be chosen as large as possible so that the correntropy behaves like the $\ell_2$ norm, and it follows from Proposition \ref{propo} that we can approximate $\{\bm p^k_{t,i}, \bm q^k_{t,j}\}$ by constant $-1$. Then the coefficients are obtained as follows
\begin{align}
{a_{t,i}}&={\sigma_{t,i}^2},\label{k1}\\
{b_{t,j}}&={\eta_{t,j}^2}
\label{k2}
\end{align}
Substituting \eqref{k1} and \eqref{k2} into \eqref{cost_last} leads to the updated cost function:
\begin{align}
\hat{\bm x}_{t|t}^k=\text{arg}\max_{\bm x_t}\left(
\sum_{i=1}^n\frac{p^k_{t,i}}{2}\alpha_{t,i}^2
+\sum_{j=1}^m\frac{q^k_{t,j}}{2}\beta_{t,j}^2\right)
\label{cost_last1}
\end{align}
which can equivalently expressed in a matrix format as
\begin{align}
\hat{\bm x}_{t|t}^k=\text{arg}\min_{\bm x_t}\left(
\frac{1}{2}\bm \alpha_{t}^T\bm \Psi_t\bm \alpha_{t}+\frac{1}{2}\bm \beta_{t}^T\bm \Phi_t\bm \beta_{t}\right)
\label{cost_last2}
\end{align}
where $\bm \Psi_t\in\mathbb{R}^{n\times n}$ and $\bm \Phi_t\in\mathbb{R}^{m\times m}$ are diagonal matrices given by
\begin{align}
\bm \Psi_t&=\text{diag}(-p_{t,1}^k,\cdots,-p_{t,n}^k)\label{ww1}\\
\bm \Phi_t&=\text{diag}(-q_{t,1}^k,\cdots,-q_{t,m}^k)\label{ww2}
\end{align}
Furthermore, substituting \eqref{def1} and \eqref{def2} into \eqref{cost_last2}, we obtain
\begin{align}
\hat{\bm x}_{t|t}^k=\text{arg}\min_{\bm x_t}\Big(\frac{1}{2}\|\bm x_t-\hat{\bm x}_{t|t-1}\|^2_{\bar{\bm P}_{t|t-1}^{-1}}+\frac{1}{2}\|\bm y_t-h(\bm x_t)\|^2_{\bar{\bm R}_{t}^{-1}}\Big)
\label{cost_last3}
\end{align}
where
\begin{align}
&\bar{\bm P}_{t|t-1}={\bm P}_{t|t-1}^{1/2}\bm \Psi_t^{-1}{\bm P}_{t|t-1}^{T/2}\label{P_upt}\\
&\bar{\bm R}_{t}={\bm R}_{t}^{1/2}\bm \Phi_t^{-1}{\bm R}_{t}^{T/2}\label{R_upt}
\end{align}

We notice that, expect for a different weighting matrix, \eqref{cost_last3} is similar to (\ref{cost2}), which can be efficiently solved in the Kalman filtering framework. Hence the second step of the $k$-th iteration can be implemented in the Kalman filtering framework with the modified weighting matrices. Once the $\hat{\bm x}_{t|t}^k$ is obtained, we can update the weighting matrices and then solve the problem to obtain $\hat{\bm x}_{t|t}^{k+1}$. The iterative procedure is repeated until a convergence criterion is met. The resulting robust cubature Kalman filter (RCKF) is summarized in Algorithm 1.

\begin{algorithm}[!t]
  \caption{Robust Cubature Kalman Filter Based on MCC} \label{alg:lrkf}
  \begin{algorithmic}[0]
    \State \textbf{Input:} $\bm y_{1:T}$, $\hat{\bm x}_{0|0}$, $\bm P_{0|0}$, $\bm Q_{1:T}$, $\bm R_{1:T}$.
    \State \textbf{Output:} $\bm \hat{\bm x}_{t|t}$ and $\bm P_{t|t}$ for $t=1:T$.

  \For{$t = 1,\cdots,T$}
  \State 1. Run the prediction step of CKF to calculate $\hat{\bm x}_{t|t-1}$
  \State \ and $\bm P_{t|t-1}$;
  \State 2. Initialize $\bm \Psi_t^{0}=\bm I_n,\ \bm \Phi_t^{0}=\bm I_m,\ k=0;$
  \Repeat
    \State 1. Update $\bar{\bm {P}}_{t|t-1}$ via \eqref{P_upt} and $\bar{\bm R}_t$ via \eqref{R_upt};
    \State 2. Update the innovation covariance $\bm P_{yy}$ and the
    \State  \quad\ filtering gain $\bm K_{t}$ via \eqref{G_upt} and \eqref{I_upt};
    \State 3. Calculate the filtering state $\hat{\bm x}_{t|t}^{k}$ via \eqref{s_upt};
    \State 4. Update $\bm\Phi_t$ and $\bm\Psi_t$ via \eqref{def1}, \eqref{def2},\eqref{p_up}, \eqref{q_up}, \eqref{ww1} and
    \State \quad \eqref{ww2};
    \State 5. $k=k+1$
  \Until a convergence criterion is met
  \State $\bm \hat{\bm x}_{t|t} = \hat{\bm x}_{t|t}^{k-1};\ \bm P_{t|t}=\bm P_{t|t-1}-\bm K_t\bm P_{yy}\bm K_t^T$.
  \EndFor
  \end{algorithmic}
\end{algorithm}

A closer examination of the proposed algorithm reveals that it is similar to the conventional Gaussian approximation Kalman filter, with the same prediction update step and iterative measurement update step. As proven by simulation results, the iteration number is relatively small in general (see Section~\ref{simu} for more details). Hence the proposed MCC based robust nonlinear Kalman filter has only a slightly higher computational complexity compared with the corresponding Gaussian approximation filer.

\subsection{Convergence}

\label{proof}

In this section, we provide a convergence analysis for the MCC based robust nonlinear Kalman filter. For brevity, we denote the original objective function~\eqref{cost_ro} and the transformed cost function~\eqref{cost_aug} by $\mathcal{Q}(\bm x_t)$ and $\mathcal{J}(\bm x_t,\bm p_t,\bm q_t)$, respectively. We first show that $\mathcal{Q}(\bm x_t)$ and $\mathcal{J}(\bm x_t,\bm p_t,\bm q_t)$ have the same optimal solution. We assume that $\bm x_t^+$ and $\{\bm x_t^*,\bm p_t^*,\bm q_t^*\}$ are the optimal solutions to $\mathcal{Q}(\bm x_t)$ and $\mathcal{J}(\bm x_t,\bm p_t,\bm q_t)$, respectively; in addition, $\{\bm p_t^+,\bm q_t^+\}$ maximize $\mathcal{J}(\bm x_t,\bm p_t,\bm q_t)$ when $\bm x_t$ equals $\bm x_t^+$. It is clear that
\begin{align}
\mathcal{Q}(\bm x_t^+)&\ge\mathcal{Q}(\bm x_t^*)\label{pro_1}\\
\mathcal{J}(\bm x_t^*,\bm p_t^*,\bm q_t^*)&\ge\mathcal{J}(\bm x_t^+,\bm p_t^+,\bm q_t^+)
\label{pro_2}
\end{align}
According to Proposition 1, the following equalities hold
\begin{align}
\mathcal{Q}(\bm x_t^+)&=\mathcal{J}(\bm x_t^+,\bm p_t^+,\bm q_t^+)\label{pro_3}\\
\mathcal{Q}(\bm x_t^*)&=\mathcal{J}(\bm x_t^*,\bm p_t^*,\bm q_t^*)
\label{pro_4}
\end{align}
Combing~\eqref{pro_1}-\eqref{pro_4}, we obtain that $\mathcal{Q}(\bm x_t^+)=\mathcal{Q}(\bm x_t^*)$, which illustrates that the original optimization problem (\ref{cost_ro}) and the transformed one (\ref{cost_aug}) have the same optimal solution.

Next, we prove that the sequence $\{\bm x_{t|t}^k,\bm p_t^k,\bm q_t^k\}$ generated by Algorithm 1 converges, i.e.,
\begin{align}
\lim\limits_{k\rightarrow\infty}\left(\mathcal{J}(\bm x_{t|t}^{k+1},\bm p_t^{k+1},\bm q_t^{k+1})-\mathcal{J}(\bm x_{t|t}^k,\bm p_t^k,\bm q_t^k)\right)=0
\label{conn}
\end{align}
According to Algorithm 1, the following equations hold
\begin{align}
\mathcal{J}(\bm x_{t|t}^k,\bm p_t^{k+1}\bm q_t^{k+1})&\ge\mathcal{J}(\bm x_{t|t}^k,\bm p_t^k,\bm q_t^k)\label{11}\\
\mathcal{J}(\bm x_{t|t}^{k+1},\bm p_t^{k+1}\bm q_t^{k+1})&\ge\mathcal{J}(\bm x_{t|t}^k,\bm p_t^{k+1}\bm q_t^{k+1})\label{22}
\end{align}
Combining (\ref{11}) and (\ref{22}), we conclude that the sequence $\mathcal{J}(\bm x_{t|t}^k,\bm p_t^k,\bm q_t^k)$ produced by Algorithm 1 is non-decreasing. Furthermore, the objective function (\ref{cost_aug}) is bounded above. According to the monotone convergence theorem~\cite{James:2006}, we can draw the conclusion that (\ref{conn}) holds.

To sum up, our proposed robust filtering algorithm yields the same solution of the original problem through the half-quadratic technique and  its iteratively generated sequence converges. The proof of convergence is now complete.

\section{\!Robust Nonlinear Kalman Smoother with MCC}
\label{smcc}
In this section, we derive the MCC based robust Kalman smoother along with the similar clue in Section~\ref{Sec:fmcc}. In the Bayesian smoothing paradigm, the posterior density of the states $p(\bm x_{0:T}|\bm y_{1:T})$ is given by
\begin{align}
p(\bm x_{0:T}|\bm y_{1:T})\propto p(\bm x_0)\prod_{t=1}^Tp(\bm x_{t}|\bm x_{t-1})\prod_{t=1}^Tp(\bm y_t|\bm x_t)
\end{align}
The MAP estimate of the smoothed states is obtained by minimizing the negative log posterior density $p(\bm x_{0:T}|\bm y_{1:T})$, i.e.,
\begin{align}
\hat{\bm x}_{0:T|T}=\arg\min_{{\bm x}_{0:T}}\Big(-\log\left(p(\bm x_0)\right)-\sum_{t=1}^T\log\left(p(\bm x_{t}|\bm x_{t-1})\right)-\sum_{t=1}^T\log\left(p(\bm y_t|\bm x_t)\right)\Big)
\end{align}
Under the Gaussian assumption and dropping the terms that do not relate on ${\bm x}_{0:T}$, $\hat{\bm x}_{0:T|T}$ can be obtained by
\begin{align}
\hat{\bm x}_{0:T|T}=\arg\min_{{\bm x}_{0:T}}\Big(
\frac{1}{2}\sum_{t=1}^T\|\bm x_t-f(\bm x_{t-1})\|^2_{{\bm Q}_{t-1}^{-1}}+\frac{1}{2}\sum_{t=1}^T\|\bm y_t-h(\bm x_t)\|^2_{{\bm R}_{t}^{-1}}+\frac{1}{2}\|\bm x_0-\hat{\bm x}_{0|0}\|_{{\bm P_{0|0}^{-1}}}^2\Big)
\label{cost:sm}
\end{align}
Similarly, (\ref{cost:sm}) can be written in terms of fitting errors as
\begin{align}
\hat{\bm x}_{0:T|T}&=\arg\min_{{\bm x}_{0:T}}\left(
\frac{1}{2}\sum_{t=0}^T\bm\alpha_t^T\bm\alpha_t
+\frac{1}{2}\sum_{t=1}^T\bm\beta_t^T\bm\beta_t\right)\notag\\
&=\text{arg}\min_{{\bm x}_{0:T}}\left(\frac{1}{2}\sum_{t=0}^T\sum_{i=1}^n\alpha_{t,i}^2
+\frac{1}{2}\sum_{t=1}^T\sum_{j=1}^m\beta_{t,j}^2\right)
\label{cost:sm_e}
\end{align}
where $\bm \beta_t$ has a same definition in~\eqref{def2}, $\bm \alpha_t$ is redefined as
\begin{align}
\bm \alpha_t=\left\{
\begin{array}{ll}
\bm P_{0|0}^{-1/2}\left(\bm x_0-\hat{\bm x}_{0|0}\right),&t=0\\
\bm Q_{t-1}^{-1/2}\left(\bm x_t-h(\bm x_{t-1})\right),&t\ne 0
\end{array}\right.
\label{def0}
\end{align}

Akin to the robust nonlinear Kalman filter, we replace the quadratic loss with the weighted instantaneous correntropy for each component of the normalized error, resulting in the following objective function for the nonlinear robust smoother
\begin{align}
\hat{\bm x}_{0:T|T}=\text{arg}\max_{{\bm x}_{0:T}}\Big( \sum_{t=0}^T\sum_{i=1}^n &a_{t,i}\kappa_{\sigma_{t,i}}(\alpha_{t,i})+\sum_{t=1}^T\sum_{j=1}^m b_{t,j}\kappa_{\eta_{t,j}}(\beta_{t,j})\Big)
\label{cost:sr_cor}
\end{align}
where $\{a_{t,i},b_{t,j}\}$ are parameters that should be selected to maintain the consistency of the Gaussian approximation smoother when the SSM does not involve the impulsive noises, similarly to the filtering case discussed in Section~\ref{par_dis}. According to Proposition 1, \eqref{cost:sr_cor} can be transformed as
\begin{align}
\!\!\hat{\bm x}_{0:T|T}=\text{arg}\max_{{\bm x},{\bm p},{\bm q}}&\Bigg( \sum_{t=0}^T\sum_{i=1}^n a_{t,i}\Big(\frac{p_{t,i}}{2\sigma_{t,i}^2}\alpha_{t,i}^2-\psi(p_{t,i})\Big)+\sum_{t=1}^T\sum_{j=1}^m b_{t,i}\Big(\frac{q_{t,i}}{2\eta_{t,i}^2}\beta_{t,i}^2-\psi(q_{t,i})\Big)\Bigg)
\label{cost:sr_cor1}
\end{align}
where $\bm x=\{\bm x_t\}_{t=0}^T$, $\bm p=\{\bm p_t\}_{t=0}^T$ and $\bm q=\{\bm q_t\}_{t=1}^T$ are the collections of all variables to be optimized.

Again, the augmented objective function can be solved in an iterative manner via the half-quadratic method. First of all, fixing the states and optimizing~\eqref{cost:sr_cor1} with respect to $\bm p$ and $\bm q$ according to Proposition 1, we obtain the same updating equations for $\{p_{t,i},\ q_{t,j}\}$ as shown in~\eqref{p_up}\eqref{q_up}. Next, keeping $\bm p$ and $\bm q$ fixed and dropping the items that do not relate to the state, one can update the smoothed state $\hat{\bm x}_{0:T|T}$ though
\begin{align}
\hat{\bm x}_{0:T|T}=\text{arg}\max_{{\bm x_{0:T}}}& \Big(\sum_{t=0}^T\sum_{i=1}^n \frac{p_{t,i}a_{t,i}}{2\sigma_{t,i}^2}\alpha_{t,i}^2+\sum_{t=1}^T\sum_{j=1}^m \frac{q_{t,i}b_{t,i}}{2\eta_{t,i}^2}\beta_{t,i}^2\Big)
\label{cost:sr_cor2}
\end{align}
Similarly, to maintain the consistency of the Gaussian approximation smoother when the SSM dose not involve any impulsive noise, the coefficients should be selected by~\eqref{k1} and \eqref{k2}. Substituting \eqref{k1} and \eqref{k2} into (\ref{cost:sr_cor2}), the objective function is updated as
\begin{align}
\hat{\bm x}_{0:T|T}=\text{arg}\max_{{\bm x_{0:T}}}\Big( \sum_{t=0}^T\sum_{i=1}^n&\frac{1}{2} p_{t,i}\alpha_{t,i}^2+\sum_{t=1}^T\sum_{j=1}^m \frac{1}{2}q_{t,i}\beta_{t,i}^2\Big)
\label{cost:sr_cor3}
\end{align}
Writing in matrix format, we get
\begin{align}
\hat{\bm x}_{0:T|T}=\text{arg}\min_{\bm x_{0:T}}\left(
\frac{1}{2}\sum_{t=0}^T\bm \alpha_{t}^T\bm \Psi_t\bm \alpha_{t}+\frac{1}{2}\sum_{t=1}^T\bm \beta_{t}^T\bm \Phi_t\bm \beta_{t}
\right)
\label{obj_last2}
\end{align}
where $\bm \Psi_t\in\mathbb{R}^{n\times n}$ and $\bm \Phi_t\in\mathbb{R}^{m\times m}$ are diagonal matrixes, given by \eqref{ww1} and \eqref{ww2}, respectively. Furthermore, substituting the expression of $\bm \alpha_{t}$ and $\bm \beta_{t}$, namely \eqref{def0} and \eqref{def2}, into (\ref{obj_last2}), we obtain
\begin{align}
\hat{\bm x}_{0:T|T}=\text{arg}\min_{\bm x_{0:T}}\Big(\frac{1}{2}\sum_{t=1}^T\|\bm x_t-f({\bm x}_{t-1})\|^2_{\bar{\bm Q}_{t-1}^{-1}}+\frac{1}{2}\sum_{t=1}^T\|\bm y_t-h(\bm x_t)\|^2_{\bar{\bm R}_{t}^{-1}}+\frac{1}{2}\|\bm x_0-\hat{\bm x}_{0|0}\|_{\bar{\bm P}_{0|0}^{-1}}^2\Big)
\label{obj_last3}
\end{align}
where
\begin{align}
&\bar{\bm Q}_{t-1}={\bm Q}_{t-1}^{1/2}\bm \Psi_t^{-1}{\bm Q}_{t-1}^{T/2}\label{w1}\\
&\bar{\bm R}_{t}={\bm R}_{t}^{1/2}\bm \Phi_t^{-1}{\bm R}_{t}^{T/2}\\
&\bar{\bm P}_{0|0}={\bm P}_{0|0}^{1/2}\bm \Psi_0^{-1}{\bm P}_{0|0}^{T/2}\label{w2}
\end{align}

The objective function (\ref{obj_last3}) can be solved by CKS, which provides updates of the state $\hat{\bm x}_{0:T|T}$. The iterative procedure is repeated till some convergence criterion is met. The detailed robust cubature Kalman smoother (RCKS) is summarized in Algorithm 2.

\begin{algorithm}[thbp]
  \caption{MCC Based Robust Cubature Kalman Smoother} \label{alg:lrkf}
  \begin{algorithmic}[0]
    \State \textbf{Input:} $\bm y_{1:T}$, $\hat{\bm x}_0$, $\bm P_0$, $\bm Q_{1:T}$, $\bm R_{1:T}$.
    \State \textbf{Output:} $\bm \hat{\bm x}_{t|T}$ and $\bm P_{t|T}$ for $t=1:T$.
    \State \textbf{Initialized:} $k=0$, $\bm \Psi_t=\bm I_n$, $\bm \Phi_t=\bm I_m$.

  \Repeat
  \State 1. Update $\bar{\bm Q}_{t-1},\bar{\bm R}_{t}\ \text{and }\bar{\bm P}_0$ via \eqref{w1}-\eqref{w2};
  \State 2. Update $\hat{\bm x}_{0:T|T}^k\ \text{and }{\bm P}_{0:T|T}^k$ via CKS;
  \State 3. Update $\bm\Phi_t$ and $\bm\Psi_t$ via \eqref{def0}, \eqref{def2}, \eqref{p_up}, \eqref{q_up}, \eqref{ww1} and
  \State  \quad \eqref{ww2};
  \State 4. $k=k+1$;
  \Until a convergence criterion is met;
  \State $\bm \hat{\bm x}_{t|T}=\bm \hat{\bm x}_{t|T}^{(k-1)}$ and $\bm P_{t|T}=\bm P_{t|T}^{(k-1)}$;
  \end{algorithmic}
\end{algorithm}

It can be seen that the proposed robust nonlinear Kalman smoother actually contains two steps in each iteration, i.e., updating the noise covariance and running the conventional Gaussian approximation smoother. One can follow similar steps used in Section~\ref{proof} and show that the proposed nonlinear robust Kalman smoother converges to a local minimum of the cost function.

\begin{comment}
\begin{table*}[t]
\centering
\caption{Failure ratio of different estimation methods}
\subfloat[CKF]{
\begin{tabular}{|c|c|c|c|c|} \hline
      \backslashbox{$p_2$}{$p_1$} &0&0.1&0.2&0.3\\
      \hline
      0& 0&0&0&0 \\
      \hline
      0.1&0.015&0.011&0.018&0.02 \\
      \hline
      0.2&0.071&0.072&0.087&0.085\\
      \hline
      0.3&0.171&0.195&0.166&0.181\\
      \hline
\end{tabular}}\quad
\subfloat[CKS]{
\begin{tabular}{|c|c|c|c|c|} \hline
      \backslashbox{$p_2$}{$p_1$} &0&0.1&0.2&0.3\\
      \hline
      0& 0&0&0&0 \\
      \hline
      0.1&0.006&0.001&0.001&0.006 \\
      \hline
      0.2&0.035&0.034&0.037&0.040\\
      \hline
      0.3&0.107&0.128&0.113&0.113\\
      \hline
\end{tabular}}\\
\subfloat[RCKF]{
\begin{tabular}{|c|c|c|c|c|} \hline
      \backslashbox{$p_2$}{$p_1$} &0&0.1&0.2&0.3\\
      \hline
      0& 0&0&0&0 \\
      \hline
      0.1&0.016&0.011&0.015&0.008 \\
      \hline
      0.2&0.065&0.069&0.080&0.060\\
      \hline
      0.3&0.189&0.200&0.194&0.192\\
      \hline
\end{tabular}}\quad
\subfloat[RCKS]{
\begin{tabular}{|c|c|c|c|c|} \hline
      \backslashbox{$p_2$}{$p_1$} &0&0.1&0.2&0.3\\
      \hline
      0& 0&0&0&0 \\
      \hline
      0.1&0&0.002&0&0 \\
      \hline
      0.2&0.010&0.006&0.008&0.005\\
      \hline
      0.3&0.037&0.041&0.036&0.041\\
      \hline
\end{tabular}}
\end{table*}

\begin{figure*}[]
\centering
\includegraphics[width = 1.5\columnwidth]{trmse_kernel_size.pdf}\label{sfig:on_slt_sp_t}
\caption{Two randomly selected trails. The left two panels show the estimate of $x_1$ (top) and $x_2$ (bottom) when $p_1=0$ and $p_2=0.2$ while the right two panels show the estimate of $x_1$ (top) and $x_2$ (bottom) when $p_1=0.2$ and $p_2=0.2$.}
\label{fig:online_slight}
\vspace{-2mm}
\end{figure*}
\end{comment}

\begin{comment}
\section{Several Implantation details}
\label{remarks}

This section presents some details of the implementations of the proposed RCKF and RCKS.

\begin{itemize}
\item kernel size;
\item SVD based update ;
\item terminal criterion ;
\item how to calculate the mean of the innovation;
\end{itemize}
\end{comment}

\section{Numerical Simulations}
\label{simu}
Numerical examples are presented in this section to \mbox{illustrate} the performance of the proposed RCKF and RCKS in the presence of heavy-tailed noises. We also compare the state estimates of the proposed algorithms with those of the conventional CKF~\cite{arasaratnam:2009}, conventional CKS~\cite{arasaratnam:2011}, linear regression and MCC based robust Kalman filter (LRKF)~\cite{liu:2017}, nonlinear regression based Huber Kalman filter (HRKF)~\cite{karlgaard:2015} and variational Bayesian based student's cubature Kalman smoother (TCKS)~\cite{huang:2016_2}. The LRKF is originally proposed in the unscented Kalman filter (UKF) framework and here for consistency with the other compared algorithms, we set parameters $(\alpha,\beta,\kappa)=(1,0,0)$ in LRKF, which makes it functionally equivalent to a CKF. Other design parameters in the aforementioned algorithms are set as recommended in their literatures. In the simulation, we take the following convergence criterion, $\|\hat{\bm x}_{t|t}^{k}-\hat{\bm x}_{t|t}^{k-1}\|/\|\hat{\bm x}_{t|t}^{k-1}\|\le 10^{-6}$ in filtering and $\text{max}(\|\hat{\bm x}_{t|T}^{k}-\hat{\bm x}_{t|T}^{k-1}\|/\|\hat{\bm x}_{t|T}^{k-1}\|)\le 10^{-6}$ for $t=1,\cdots,T$ in smoothing.

\subsection{Van der Pol Oscillator}

The Van der Pol oscillator (VPO) model~\cite{Kandepu:2008} is usually used as a benchmark for testing the performance of nonlinear filters and smoothers. The continuous-time nonlinear dynamics of the VPO is governed by the following differential equations
\begin{align}
\dot {x}_1&=x_2\\
\dot {x}_2 &= \mu\left(1-x_1^2\right)x_2-x_1
\end{align}
where $\mu$ is a scalar parameter indicating the nonlinearity. Discretization of the VPO yields
\begin{align}
\!\!\bm x_{t+1} =
\left(
\begin{array}{>{\displaystyle}l}
\!\!x_{1,t}+\int_{t}^{t+\delta}x_2dt\!\!\!\!\\
\!\!x_{2,t}+\int_{t}^{t+\delta}\left(\mu\left(1-x_1^2\right)x_2-x_1\right)dt\!\!\!\!
\end{array}
\right)+\bm\omega_t
\label{VPO_sys}
\end{align}
where $\bm x_t=[x_{1,t},x_{2,t}]^T$ is the state, $\bm \omega_t$ is the process noise to model the discretization error, and $\delta$ is the sampling interval. The integral terms in \eqref{VPO_sys} are numerically calculated by the fourth-order Runge-Kutta scheme with a time step of the sampling interval. Furthermore, we assume that the state is measured by the following noise disturbed nonlinear procedure
\begin{align}
y_t = (x_{1,t}-1)^2+1+v_{t}
\end{align}
Both noises are generated from the mixed-Gaussian model~\cite{Huang:2016}:
\begin{align}
\bm\omega_{t-1}&\thicksim(1-p_1)\mathcal{N}(0,\bm Q_{t-1})+p_1\mathcal{N}(0,\phi_1\bm Q_{t-1})\\
v_{t}&\thicksim(1-p_2)\mathcal{N}(0,R_t)+p_2\mathcal{N}(0,\phi_2R_t)
\end{align}
where $p_1$ and $p_2$ are the contaminating parameters, $\phi_1$ and $\phi_2$ are the scaling factors indicating the strength of the contaminating noises, and $\bm Q_{t-1}$ and $R_t$ are the covariance matrices of the nominal process noise and measurement noise, respectively.

In the simulation, we select $\mu=1$ and total samples $T=120$ with the sampling interval $\delta=0.1$s. The initial state $\hat{\bm x}_{0|0}$ follows $\mathcal{N}(\bm x_0,0.01\bm I_2)$ with $\bm x_0=[0,-0.5]^T$. The covariance of the noises are set as $\bm Q_{t-1}=0.01\bm I_2$ and $R_t=1$, respectively. The covariance of the contaminating part in each noise is controlled by $\phi_1=10$ and $\phi_2=50$. Specifically, three scenarios are considered: 1) all noises are Gaussian, i.e., $p_1=p_2=0$; 2) the process noise is a Gaussian noise while the measurement noise is a heavy-tailed noise, i.e., $p_1=0$ and $p_2=0.2$; 3) both process and measurement noises are heavy-tailed noises, i.e., $p_1=p_2=0.2$.

For fair comparison, we implement $L=1000$ Monte Carol runs in each scenario. We use the implementation time (IT) and time-averaged root mean square error (TRMSE) as performance metrics. The TRMSE is defined as
\begin{align}
\text{TRMSE} = \frac{1}{T}\sum_{t=1}^T\sqrt{\frac{1}{L}\sum_{i=1}^L\left(x_t^i-\hat{x}_t^i\right)^2}
\label{trmse}
\end{align}
where $x_{t}^i$ and $\hat{x}_{t}^i$, respectively, denote the true and estimated state component at time $t$ in the $i$-th Monte Carlo run.

\begin{figure}[!h]
  \centering
  \includegraphics[width = 1.0\columnwidth]{trmse_kernel_size.pdf}
  \caption{The TRMSEs of different methods for $x_1$ (top) and $x_2$ (bottom).}
  \label{fig_trmse1}

\end{figure}

\begin{figure}[!h]
  \centering
  \includegraphics[width = 1.0\columnwidth]{trmse_comp.pdf}\label{fig_trmse2}
  \caption{The TRMSEs of the different methods for $x_1$ (top) and $x_2$ (bottom).}

  \label{fig:2}
\end{figure}

The kernel size is the only design parameter in the proposed algorithms and it has a significant impact on the performance of the RCKF and RCKS. Therefore, we studied the performance of the proposed algorithms with three different kernel sizes, i.e., $\sigma_t=\eta_t=2$; $\sigma_t=20,\ \eta_t=2$ and $\sigma_t=\eta_t=20$, and the results are labeled as RCKF1/RCKS1, RCKF2/RCKS2 and RCKF3/RCKS3, respectively.  The conventional CKF and CKS are also implemented as a performance benchmark.

Fig.~\ref{fig_trmse1} shows the TRMSEs while the IT is collected in Table~\ref{Table:1}. It can be seen, as expected, the smoothers have higher accuracies at the cost of more computational time than their filtering counterparts due to the use of backward procedure. The IT of the robust estimators with different kernel sizes in three scenarios are similar, which implies the IT of the proposed algorithms is insensitive to the kernel size.

\begin{table}[!h]
\centering
\caption{The implementation time of different methods}
\label{Table:1}
\begin{tabular}[width = 0.85\columnwidth]{@{\extracolsep{\fill}}c|ccc|c|ccc}
\hline
\multirow{2}{*}{Method} & \multicolumn{3}{c|}{IT/s} & \multirow{2}{*}{Method} & \multicolumn{3}{c}{IT/s} \\
\cline{2-4}\cline{6-8}
                        & S1     & S2    & S3    &                         & S1     & S2    & S3    \\
                        \hline
CKF                     &37.1& 36.8  & 36.6   & RCKF1                   & 142.6     & 140.9     &130.8  \\
CKS                     & 44.5      & 43.8     & 42.1     & RCKS1                   & 643.0      & 590.7     & 533.3     \\
RCKF2                   & 136.7      & 133.0     & 122.4     & RCKF3                   & 127.8      & 130.8     & 122.0     \\
RCKS2                   & 648.5      & 588.6     & 532.4     & RCKS3                   & 642.8      & 589.8     & 531.9\\
\hline
\end{tabular}
\footnotetext{a ha}
\end{table}

In the scenario 1, the robust algorithms are similar to the conventional ones, which suggest that the robust algorithms are consistent with the conventional ones. When heavy-tailed noises are involved, taking their characteristics into account in general pays off in terms of accuracy, while the computational time increase because of their iterative procedure. It is observed that CKF3/CKS3, which employ a larger kernel size, perform the worst among the robust algorithms. CKF1/CKS1 and CKF2/CKS2 are slightly different in their perfermance, which is caused by their different kernel sizes.

\begin{table}[!h]
\centering
\caption{The implementation time of different methods}
\label{Table:2}
\begin{tabular}[width = 0.85\columnwidth]{@{\extracolsep{\fill}}c|ccc|c|ccc}
\hline
\multirow{2}{*}{Method} & \multicolumn{3}{c|}{IT/s} & \multirow{2}{*}{Method} & \multicolumn{3}{c}{IT/s} \\
\cline{2-4}\cline{6-8}
                        & S1     & S2    & S3    &                         & S1     & S2    & S3    \\
                        \hline
CKF                     & 37.1& 36.8  & 36.6 &    CKS                   & 44.5      & 43.8     & 42.1 \\
RCKF                     & 136.7      & 133.0     & 132.4     & RCKS                   & 648.5      & 588.6     & 532.4    \\
LRKF                   & 95.1      & 98.3     & 90.4     &  TCKS                   & 1097.2      & 964.4     & 941.1     \\
HRKF                   & 66.1      & 68.1     & 63.9     &              &       &   &\\
\hline
\end{tabular}
\label{T2}
\footnotetext{a ha}
\end{table}

\begin{figure*}[!t]
\centering
\subfloat[]{
\includegraphics[width=0.5\textwidth]{vp1fp2_first.pdf}}
\subfloat[]{
\includegraphics[width=0.5\textwidth]{vp1fp2_second.pdf}}
\\
\subfloat[]{
\includegraphics[width=0.5\textwidth]{fp1_vp2_1.pdf}}
\subfloat[]{
\includegraphics[width=0.5\textwidth]{fp1_vp2_2.pdf}}
\caption{TRMSE of different estimators via the contamination ratio. Top: varying $p_1$ with fixed $p_2=0.2$, (a) TRMSE of the first component; (b) TRMSE of the second component. Bottom: varying $p_2$ with fixed $p_1=0.2$, (c) TRMSE of the first component; (d) TRMSE of the second component.}
\label{f3}
\end{figure*}

Next we present comparison results for different algorithms. For fairness, we set all kernel sizes to $2$ in LRKF, RCKF and RCKS. Fig.~\ref{fig:2} shows the TRMSEs of the different algorithms in the three scenarios and the IT of the different algorithms are given in Table~\ref{T2}. It is clear that all robust algorithms have a similar performance as that of the conventional estimators in the scenario when all noises are Gaussian. However, in the presence of the heavy-tailed noises, the robust estimators have a significant improvement over the conventional algorithms, which does not account for the presence of outliers. It is observed that the proposed RCKF outperforms HRKF. This is mainly because the MCC is instinctively more suitable for heavy-tailed noise than Huber's function. It is also seen that RCKF has a slight gain than LRKF. This is because, while both algorithms are derived from MCC, LRKF is based on a linear regression which introduces the linearzation error. Although RCKF is better than LRKF and HRKF in accuracy, the former has a higher IT than the latter, by $30\%$ and $95\%$, respectively. On the other hand, all three smoothers have smaller TRMSEs than their filtering counterparts. The proposed RCKS has a similar performance to that of TCKS and both outperform the traditional CKS. TCKS, however, needs about $40\%$ more IT than the proposed RCKS.

The TRMSE versus the increase of the contamination ratio is also studied, as shown in Fig.~\ref{f3}. The TRMSEs of the two traditional algorithms rise considerably, while the inverse is true for all robust algorithms except HRKF. The slightly increasing TRMSEs of the robust algorithms suggests that these algorithms are relatively insensitive to the level of the heavy-tailed noise. The TRMSEs of HRKF have a slight increase when the contamination ratio is relatively small and then the increase is more notable when the contamination ratio becomes larger, confirming that Huber's function based robust estimators performs well only at relativly low levels of contamination.

\subsection{Agile Target Tracking}

Here we consider the problem of tracking an agile target which is observed by an active radar located at the original point. The outliers may occur in both the process and measurement procedure due to the rapid motion of the target or disturbances experienced by the radar. The SSM is formulated as~\cite{li2003survey, bar2004estimation, arasaratnam2007discrete,huang:2016_2}
\begin{align}
\bm x_t &= \left[
\begin{array}{cc}
\bm I_2&\delta\bm I_2\\\bm 0 & \bm I_2
\end{array}
\right]\bm x_{t-1}+\bm\omega_{t-1} \\
\bm y_t &= \left[
\begin{array}{c}
\sqrt{a_t^2+b_t^2}\\ \text{atan2}(b_t,a_t)
\end{array}\right]
+\bm v_t
\end{align}
where $\bm x_t =[a_t,b_t,\dot{a}_t,\dot{b}_t]^T$ is the state; $a_t$, $b_t$, $\dot{a}_t$ and $\dot{b}_t$ denote the positions and corresponding velocities in Cartesian coordinates; $\delta=0.5$s is the sampling interval and $\text{atan2}$ is the four-quadrant inverse tangent function. In the simulation, the initial state $\hat{\bm x}_{0|0}$ is randomly chosen from $\mathcal{N}(\bm x_0,\bm P_0)$ with the true initial state $\bm x_0=[-10000;10000;30;-40]^T$ and corresponding error covariance $\bm P_0=100\bm I_4$. Outlier-corrupted process and measurement noises are generated from the following Gaussian mixture model:
\begin{align}
\bm\omega_{t-1}&\sim 0.8\mathcal{N}(0,\bm Q_{t-1})+0.2\mathcal{N}(0,10\bm Q_{t-1})\\
\bm v_t&\sim 0.8\mathcal{N}(0,\bm R_t)+0.2\mathcal{N}(0,50\bm R_t)
\end{align}
where the nominal covariance matrices for the process and measurement noise, i.e. $\bm Q$ and $\bm R$, are given as
\begin{align}
\!\!\!
\bm Q_{t-1}=\left[
\begin{array}{cc}
\frac{\delta^3}{3}\bm{I}_2&\!\frac{\delta^2}{2}\bm{I}_2\\
\frac{\delta^2}{2}\bm{I}_2&\!\delta\bm{I}_2
\end{array}\right],\
\bm R_t=\left[
\begin{array}{cc}
100\text{m}^2&\!\!0\\
0&\!\!16\text{mrad}^2
\end{array}\right]
\end{align}
$L=1000$ independent Monte-Carlo runs are implemented and in each run $T=200$ noisy measurements are collected. The root mean square error (RMSE) of position and velocity are performed as performance metrics, which are defined as
\begin{align}
\text{RMSE}_{\text{pos}}(t)&=\sqrt{\frac{1}{L}\sum_{i=1}^L\left(( a_{t}^i-\hat{ a}_{t}^i)^2+( b_{t}^i-\hat{ b}_{t}^i)^2\right)}
\end{align}
where $a_{t}^i$ and $\hat{ a}_{t}^i$ have similar meanings in~\eqref{trmse}. Akin to position, the RMSE of velocity can also be formulated.

\begin{comment}
\begin{figure}[!h]
  \centering
  \includegraphics[width = \columnwidth]{tt_trmse.pdf}\label{fig_trmse2}
  \caption{The TRMSEs of the different methods for position (top) and velocity (bottom).}
  \vspace{-2mm}
\end{figure}
\end{comment}

\begin{comment}
\begin{figure*}[]
\centering
\subfloat[]{
\includegraphics[width=0.45\textwidth]{rmse_tt_0002_pos.pdf}}
\subfloat[]{
\includegraphics[width=0.45\textwidth]{rmse_tt_0002_vol.pdf}}
\\
\subfloat[]{
\includegraphics[width=0.45\textwidth]{rmse_tt_0202_pos.pdf}}
\subfloat[]{
\includegraphics[width=0.45\textwidth]{rmse_tt_0202_vol.pdf}}
\caption{The RMSE of different estimators in two scenarios. Top: $p_1=0,\ p_2=0.2$, (a) RMSE of position; (b) RMSE of velocity. Bottom: $p_1=0.2,\ p_2=0.2$, (c) RMSE of position; (d) RMSE of velocity.}
\end{figure*}
\end{comment}
\begin{figure}[]
\centering
\subfloat[]{
\includegraphics[width=0.45\textwidth]{rmse_tt_0202_pos.pdf}}\\
\subfloat[]{
\includegraphics[width=0.45\textwidth]{rmse_tt_0202_vol.pdf}}
\caption{The RMSE of different estimators: (a) RMSE of position; (b) RMSE of velocity.}
\label{rmse}
\end{figure}

Fig.~\ref{rmse} shows the RMSEs of the position and velocity for the compared algorithms. It can be seen that the robust filters have a significant improvement except HRKF. Although both RCKF and LRKF are based on MCC, the proposed RCKF performs better than the LRKF, mainly because LRKF is based a linear regression, which introduces the linearzation error. The superiority of both RCKS and TCKS are also illustrated in Fig.~\ref{rmse} where RCKS has a similar performance as the TCKS.

\begin{comment}

\section{Numerical Simulation}
\label{simu}

\subsection{Van der Pol oscillator}

In the simulation, both heavy-tailed noise are realized by the mix-Gaussian model~\cite{Roth:2013}:
\begin{align}
\bm\omega_{t-1}&\thicksim(1-p_1)\mathcal{N}(0,0.01I)+p_1\mathcal{N}(0,\phi_1I)\\
v_{t}&\thicksim(1-p_2)\mathcal{N}(0,0.5)+p_2\mathcal{N}(0,\phi_2)
\end{align}

In the simulation, we select $\mu=2$ and total samples $T=164$ with the sample time $\delta=0.1$s. The initial state $\bm x_0$ is randomly selected from $\mathcal{N}(\hat{\bm x}_0,0.1I)$ with $\hat{\bm x}_0=[0,-0.5]^T$. The covariance of contaminated part in each noise is selected as 100 times greater than their counterpart, i.e., $\phi_1=1$ and $\phi_2=50$. $p_1$ and $p_2$ are chosen from $\{0,0.1,0.2,0.3\}$. We simulated 1000 realizations of estimation in each possible combination of $p_1$ and $p_2$. As for the bandwidths, we set all bandwidths as 5.

During the simulation, we found that the estimate in each trail may not always succeed to track the true state. In each trail, if the mean of the square error is larger than three times' maximum square of the true state, we said the corresponding method is failure. Furthermore, we define that failure ratio to illustrate the tracking performance of each method. The failure ratio is obtained though total trails dividing the number of failure. The failure ratio for different approaches in each possible combination of contamination ratio is showed in Table. 1. In general, it is clear that for all methods, the failure ratio increase dramatically along with the increase of $p_2$ while a slight increment with the growth of $p_1$. The smoother types have a smaller failure ratio than their filter counterparts since the smoother based approaches utilize more information than filtering methods. RCKF has a litter larger failure ratio than CKF while the inverse is truth for RCKS and CKS.

Fig. 1 provides an illustration of two randomly selected realizations. It can be seen that RCKS provided the best estimate, followed by CKS. As for first component $x_1$, RCKF can provide the better estimate than CKF due to the infomation lying in the measurements. For the second component $x_2$, however, RCKF provide the worse estimate than CKF in several time instant. This is because RCKF is heavily dependent on the predicated state. This also can be identified by RMSE showed in Fig. 2. In those two combinations of $p_1$ and $p_2$, the robust versions always offer the better results that their normal counterparts when only considering $x_1$. As for $x_2$, the robust versions also provide better estimate except a few instants.

\begin{figure}
\centering
\subfloat{
\quad\quad\includegraphics[width = 0.8\columnwidth]{head_of_rmse.pdf}}\\
\subfloat{
\includegraphics[width = 0.95\columnwidth]{rmse_1.pdf}\label{sfig:on_slt_sp_t}}\\
\subfloat{
\includegraphics[width = 0.95\columnwidth]{rmse_2.pdf}\label{sfig:on_slt_sp_t}}
\caption{The RMSE of $x_1$ and $x_2$ in the case $p_1=0,\ p_2=0.2$ (top) and $p_1=0.2,\ p_2=0.2$ (bottom).}
\vspace{-2mm}
\end{figure}

The TRMSE is summarized in Table. 2. Again, it has been proved that robust approaches have the better performance that the corresponding normal approaches and the smoothing based methods have the superiority than filtering based. For all methods, the TRMSE has a small addition with the increase of $p_1$ while the TRMSE witness a dramatic increase with $p_2$. Especially, when $p_1=p_2=0$, i.e., the SSM does not contain any impulsive noise, the proposed RCKF/RCKS have a similar performance with CKF/CKS.

\begin{table*}
\centering
\caption{TRMSE of different estimation methods}
\subfloat[First component $x_1$]{
\begin{tabular}{|c|c|c|c|c|c|}\hline
$p_1$&$p_2$&CKF&CKS&RCKF&RCKS\\ \hline
\multirow{4}{*}{0}&0&0.2901&0.1982&0.2901&0.1987\\ \cline{2-6}
&0.1&0.6434&0.4840&0.4019&0.2430 \\ \cline{2-6}
&0.2&0.7934&0.6165&0.5094&0.2985 \\ \cline{2-6}
&0.3&0.8048&0.6377&05472&0.3211 \\ \hline
\multirow{4}{*}{0.1}&0&0.2928&0.1953&0.2928&0.1963\\ \cline{2-6}
&0.1&0.6356&0.4834&0.4063&0.2450 \\ \cline{2-6}
&0.2&0.7747&0.5954&0.4998&0.2922 \\ \cline{2-6}
&0.3&0.7882&0.6227&0.5475&0.3225 \\ \hline
\multirow{4}{*}{0.2}&0&0.2916&0.1957&0.2915&0.1968\\ \cline{2-6}
&0.1&0.6393&0.4928&0.3979&0.2424 \\ \cline{2-6}
&0.2&0.7779&0.6052&0.5016&0.2935 \\ \cline{2-6}
&0.3&0.8110&0.6319&0.5646&0.3252 \\ \hline
\multirow{4}{*}{0.3}&0&0.2919& 0.1965& 0.2918& 0.1975\\ \cline{2-6}
&0.1&0.6437& 0.4905& 0.4062& 0.2459 \\ \cline{2-6}
&0.2&0.7795& 0.6022& 0.4970& 0.2908 \\ \cline{2-6}
&0.3&0.8039& 0.6415& 0.5424& 0.3250 \\ \hline
\end{tabular}}\quad
\subfloat[Second component $x_2$]{
\begin{tabular}{|c|c|c|c|c|c|}\hline
$p_1$&$p_2$&CKF&CKS&RCKF&RCKS\\ \hline
\multirow{4}{*}{0}&0&0.4673&0.3625&0.4670&0.3642\\ \cline{2-6}
&0.1&1.1016&0.9004&0.7202&0.4873 \\ \cline{2-6}
&0.2&1.3546&1.1420&1.0018&0.7084 \\ \cline{2-6}
&0.3&1.4494&1.2210&1.0232&0.8495 \\ \hline
\multirow{4}{*}{0.1}&0&0.4599&0.3529&0.4593&0.3547\\ \cline{2-6}
&0.1&1.0526&0.8867&0.7213&0.5114 \\ \cline{2-6}
&0.2&1.3118&1.1390&0.9354&0.6778 \\ \cline{2-6}
&0.3&1.3815&1.1958&1.0886&0.8639 \\ \hline
\multirow{4}{*}{0.2}&0&0.4629&0.3584&0.4621&0.3598\\ \cline{2-6}
&0.1&1.1125&0.9316&0.6883&0.4736 \\ \cline{2-6}
&0.2&1.3561&1.1774&0.9803&0.7494 \\ \cline{2-6}
&0.3&1.3611&1.1921&1.1627&0.8788 \\ \hline
\multirow{4}{*}{0.3}&0&0.4580&0.3542&0.4579&0.3559\\ \cline{2-6}
&0.1&1.0381&0.9281&0.6983&0.4814 \\ \cline{2-6}
&0.2&1.2603&1.1071&0.9406&0.6869 \\ \cline{2-6}
&0.3&1.3798&1.2152&1.0313&0.9149 \\ \hline
\end{tabular}}
\end{table*}

Table. 3 illustrates the average iteration number of RCKF and RCKS. It seems that the iteration number is insensitive to $p_1$, i.e., the average iteration number stay almost in the same level for different $p_1$. It also can be found the positive correlation between average iteration number and $p_2$. In general, RCKS need more iteration number compared with RCKF.
\begin{table}[h]
\centering
\caption{Iterative number of RCKF and RCKS}
\subfloat[RCKF]{
\begin{tabular}{|c|c|c|c|c|} \hline
      \diagbox{$p_2$}{$p_1$} &0&0.1&0.2&0.3\\
      \hline
      0&3.357&3.38&3.38&3.375 \\
      \hline
      0.1&3.5385&3.5443&3.544&3.546 \\
      \hline
      0.2&3.723&3.7189&3.72&3.714\\
      \hline
      0.3&3.883&3.8916&3.891&3.8773\\
      \hline
\end{tabular}}\quad
\subfloat[RCKS]{
\begin{tabular}{|c|c|c|c|c|} \hline
      \diagbox{$p_2$}{$p_1$} &0&0.1&0.2&0.3\\
      \hline
      0&3.412&3.364&3.341&3.35 \\
      \hline
      0.1&5.75&5.79&5.852&5.732 \\
      \hline
      0.2&7.8&7.811&8.14&7.76\\
      \hline
      0.3&10.406&11.19&10.89&10.885\\
      \hline
\end{tabular}}
\end{table}

\subsection{Re-entry Ballistic Target Tracking}
In recent years, the problem of re-entry ballistic target tracking has attracted more and more attention because of its wide applications in both military and manned spaceflight. Generally, the re-entry vehicles can be divided into two types, i.e., cooperative target and non-cooperative target. The main difference is that we know the control input of cooperative targets while not the case in non-cooperative ones.

The dynamics model of a reentry ballistic vehicle is given as:
\begin{align}
\bm x_{t+1}=\bm \Phi(\bm x_{t})+\bm G\left[\begin{array}{c}u_{x,t}\\u_{y,t}-g\end{array}\right]+\bm \omega_{t},
\label{rbtt}
\end{align}
where $\bm x_{t}=[r_t^x,v_t^x,r_t^y,v_t^y]^T$ is the defined target state at time step $t$, in which $r_t^x$ and $r_t^y$ represent positions, $v_t^x$ and $v_t^y$ denotes velocities in $X,Y$ axis, respectively; $g$ is the gravity acceleration, $u_{x,t}$ and $u_{y,t}$ are the acceleration input in in $X,Y$ axis at $t$ instant, respectively. The process noise $\omega_{t-1}$ is modeled as zero mean white Gaussian noise with covariance $\bm Q_t$, given by
\begin{align*}
\bm Q_t=q\left[\begin{array}{cc}\Theta&0\\0&\Theta\end{array}\right]
\quad\text{with}\quad\Theta=
\left[\begin{array}{cc}\delta^3/3&\delta^2/2\\\delta^2/2&\delta\end{array}\right]
\end{align*}
where $q$ denotes the noise intensity and $\delta$ represents the sample time. The nonlinear part in \eqref{rbtt} is given as
\begin{align}
\bm \Phi(\bm x_{t})=\bm F\bm x_{t}+\bm Gf(\bm x_{t})
\end{align}
with
\begin{align*}
\bm F=\left[\begin{array}{cccc}1&\delta&0&0\\0&1&0&0\\0&0&1&\delta
\\0&0&0&1\end{array}\right],\quad
\bm G=\left[\begin{array}{cc}\delta^2/2&0\\ \delta&0\\0&\delta^2/2\\0&\delta\end{array}\right],
\end{align*}
and $f(\bm x_{t})$ means the aerodynamic drag
\begin{align}
f(\bm x_{t})=-\frac{g\rho(r_{t}^y)\sqrt{(v_{t}^x)^2+(v_{t}^y)^2}\left[\begin{array}{c}
v_{t}^x\\v_{t}^y
\end{array}\right]
}
{2\beta}
\end{align}
where $\beta$ is the ballistic coefficient and $\rho(\cdot)$ is a density function as a function of the $r_{t}^y$.

The measurement of ballistic reentry target is provided a radar located at original point, containing the ranges and bearings. The measurement equation is written as
\begin{align}
\bm y_t=
\left[\begin{array}{c}
\sqrt{(r_{t}^x)^2+(r_{t}^y)^2}\\
\tan^{-1}\left(\frac{r_{t}^y}{r_{t}^x}\right)
\end{array}
\right]
+\bm v_t
\end{align}
where the measurement noise is a heavy-tailed noise which is modeled by a mixed-Gaussian density, i.e.,
\begin{align*}
\bm v_t=0.8\mathcal{N}(0,\ \bm R_t)+0.2\mathcal{N}(0,\ 50\bm R_t).
\end{align*}

Here we consider two scenarios of re-entry target tracking, i.e., cooperative target and non-cooperative target. Both scenarios have the same initial setups except the input vector. The simulation initiations are assumed to be
$\bm x_0=[232000,\ 2290\cos(190^\circ),\ 88000,\ 2290\sin(190^\circ)]^T$, $\bm P_0=\text{diag}\{1000^2,\ 20^2,\ 1000^2,\ 20^2\}$, $\delta=1$, $q=10$, $\beta=40000$, $\bm R_t=\text{diag}\{100^2,\ 0.05^2\}$.

For the cooperative target, we set the input vector as zero all the time. However for non-cooperative target, we assume that the target maneuver for $t=38,39,40$ with $u_{x,t}=u_{y,t}=2g$. The tracker do not know this information, hence the tracker still apply the system model with zero-input vector. This will cause the state outlier.

The Fig.~\ref{fig:rbtt1} depicts the RMSE of the velocity estimates and the position estimates in the cooperative scenario. It is clear that the robust algorithms perform better than their conventional counterparts.

\begin{figure}[h]
\centering
\includegraphics[width = 1\columnwidth]{RMSE_tbm_1.pdf}\label{sfig:on_slt_sp_t}
\vspace{-1mm}
\caption{The RMSE of velocity and position.}
\label{fig:rbtt1}
\vspace{-2mm}
\end{figure}

The RMSE of the velocity estimates and the position estimates in the non-cooperative scenario is shown in Fig.~\ref{fig:rbtt2}. Again, the robust versions have a priority than the conventional ones. When the state outliers happen at $38,39$ and $40$ instant. All algorithms have the fluctuations. It is clear that the robust algorithms have a much smaller fluctuations than the conventional ones.
\begin{figure}[h]
\centering
\includegraphics[width = 1\columnwidth]{RMSE_tbm_2.pdf}\label{sfig:on_slt_sp_t}
\vspace{-1mm}
\caption{The RMSE of velocity and position.}
\label{fig:rbtt2}
\vspace{-2mm}
\end{figure}
\end{comment}

\section{conclusion}
\label{con}

A new unified framework for robust Kalman filtering and smoothing based on the MCC has been proposed in this work. To ease the understanding of the proposed approach, we integrate the conventional cubature Kalman filter and smoother into our framework. The proposed robust Kalman filter and smoother is derived under the MCC, instead of the well-known MMSE, to cope with heavy-tailed noises in both the process and observation procedures. The intractable objective function based on MCC is then solved by the half-quadratic optimization method in an iterative manner. At each iteration the problem is reduced to a quadratic optimization problem, which can be efficiently solved by the conventional nonlinear Kalman filter and smoother. Numerical simulations demonstrate the superiority of the proposed robust filter and smoother when dealing with outliers in both states and measurements. The simulation results show that the proposed algorithms outperform state-of-art methods at comparable computational burden. One interesting future research direction is to investigate how to choose the kernel bandwidth of the MCC.

redpajama_set_name: RedPajamaArXiv


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The Minister of Information and Communication Technology, Mr. Stanley Simataa, urged Namibian to refrain from making derogatory or insulting statements about the President, Cabinet members and other senior Government officials on social media platforms.
In a statement the minister says insults circulating on social media platforms is not only contrary to the letter and spirit of the Namibian Constitution but also goes against our cultural values and norms as human beings and Africans.
The Ministry of Information and Communication Technology (MICT) has noted with great concern a deluge of derogatory and insulting messages (audio, video and text) directed at the Head of State, Cabinet members and the entire Government currently circulating on social media platforms.
Such derogatory and insulting language directed at the Head of State for that matter, Cabinet members and the entire Government is not only contrary to the letter and spirit of the Constitution but also goes against our cultural values and norms as human beings and as Africans.
In a democratic state, of which Namibia is one, it is a norm for citizens to express views or criticize their government or leaders on a myriad of issues. However, established protocols dictate that such views / criticism where merited, be advanced in a civilized and constructive manner, using language which conforms with the decorum befitting the stature of those being criticized.
As government, we are fully conscious of the fact that the recent wave of insults and derogatory language, this coupled with the heightened concert of agitation of our masses, are all orchestrated to tempt government to intervene in a heavy handed manner a temptation that we will continue to resist.
That notwithstanding, however, government will not shy away from using the provisions of the law, to hold the perpetrators of such morally and culturally moribund acts, to account. There must be consequences for such morally repugnant acts!
We implore all Namibians to consciously and jealously work towards embracing the tenets of a truly mature democratic society where public discourse is exercised within the framework of established norms and values!
As a nation, we must reclaim our seemly lost values!


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redpajama_set_name: RedPajamaCommonCrawl


Applicants who are interested in applying are encouraged to download the Externships application form. To download the application form*, click on the links below.
The Dental School Externship application form is in Adobe Acrobat and Microsoft Word format.


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Here's How Long It Takes to Reach a City From Every Place on Earth
A new map measures accessibility to the world's cities.
by Sarah Laskow January 10, 2018
Here's How Long It Takes to Reach a City From Every Place on Earth
Red indicates places an hour or more away from a city; purple indicates place a day or more away from a city. The Malaria Atlas Project, University of Oxford
The last time that researchers created a map showing the accessibility of cities worldwide, in 2008, Google Maps was three years old. The data used to make the map was from 2000, when printing out directions from Mapquest still felt like an exciting new convenience.
That map showed that there were only a few places left in the world where it would take days of travel to reach a city of 50,000 people or more. Most of the world's cultivated land was within a two-hour journey of a city. As the geographer Alastair Bonnett wrote in his book New Views, the map showed that "the old distinction between rural and urban society is no longer useful."
But a new global map of access to cities, published today in Nature, shows that in some parts of the world, there still are disparities between rural and urban life, defined by how quickly a person can reach a city and all the amenities located there.
The new map, created by a team led by the University of Oxford's Big Data Institute, shows the travel time needed to reach a city from anywhere in the world in 2015. They found that 80.7 percent of the world's human population—5.88 billion people—live within an hour of a city.
To come to that conclusion, the researchers used data from Open Street Maps and Google, which they write, "capture transportation networks with unprecedented detail and precision." At a resolution of one square kilometer, they analyze the availability and condition of roads, railways, water, and other elements that might speed or slow a person's progress towards an urban center. The result is the most detailed map ever created that measures the possibilities of human movement from place to place.
Compare high-income countries to low-income countries, though, and a division appears. In richer countries (in North America and Europe, predominantly) 90.7 percent of people live within an hour of a city. In poorer countries (in sub-Saharan Africa, most notably), 50.9 percent of people live within an hour of the city.
Not everyone wants to live close to a city, of course, but ease of access to urban areas connects people to the resources concentrated there—jobs, hospitals, schools, banks and other financial institutions. Traveling into a city is still the best way to take advantage of these key societal services.
Even with the advances in infrastructure and tools to document it, though, there are still places in the world that are distant from the bustle of city life. The purple spots on the map represent places that are still far away from a city—from a day to more than a week's journey. These are challenging places in which few people live, but even today there are a few spots left that are isolated from the great agglomerations of humanity that have spread across the globe.
Wishlist: Two Hours of Prince-Adjacent Funk From the Twin Cities
"Purple Snow" is full of boundary-pushing musicians from Minnesota.
citiesmapstransportationnews
This Map Shows Every Adorably Named Salt Truck in Scotland
From Luke Snowalker to Sir Salter Scott.
Natasha Frost January 17, 2018
Here Are the Real Boundaries of American Metropolises, Decided by an Algorithm
How is the U.S. actually split geographically?
The Worst Bus Stop in North America Just Got a Makeover
Noor Al-Samarrai January 25, 2019
The 'Folded Map' Project Bridges the Distance Between Chicago's Neighborhoods
Tonika Lewis Johnson, a local artist and activist, connects individuals from the North and South Sides through her art.
Hannah Steinkopf-Frank January 23, 2019
One District in Paris Is Creating a Map of Rat Sightings
The rodent map marks reports of animals both dead and alive.


Directory listings of Campgrounds and RV sites for businesses in South Dakota.
Cabins, full RV service, swimming pool, hot tub and wooded hiking trails, all just a 15 minutes' drive from Mt. Rushmore.
Elk Creek Resort offers a variety of room & camping accommodations to meet your every need. Amenities Elk Creek Resort Lodge & Big Rack Bar, Hiking Trails, Petrified Forest & Museum, Gallery of Stone Gift Shop, Pine Tree Gift Shop, Outdoor Wedding Chapel, Outdoor Pool,Hot Tub, Game Room,Bath House, Laundry Facilities.
Nestled in the Southern Black Hills. 76 sites including 10 Kamping Kabins. Near Wind Cave National Park, Custer State Park, Mammoth site, Evans Plunge and Wild Horse Sanctuary.
A summer home for snow birds. Facilities for the largest campers and motorhomes. 50 ft pull throughs with 50 amp breakers. Large lawn areas, security gate, close to interstate 90, 10 miles to Rapid City, full medical facilities available, and regional airport.
Campgrounds and camping directory, including the Black Hills of South Dakota. Complete guide of places to camp while vacationing at Mount Rushmore and Crazy Horse.
Directory of campgrounds in the Sturgis and Black Hills area.


Piazza Castello is the main square and historic center of Turin. It is located important city buildings, including the Royal Palace and the Senate. Piazza Castello merge four of the main roads of the center: Garibaldi Street (pedestrian), Via Po, Via Roma and Via Pietro Micca.
Castle Square was the center of Turin also in the Savoy period and even earlier. The square's name is derived from the ancient castle can still be seen as part of the Senate, which stands on the ruins of the ancient Roman gate, which was then transformed into a castle.
the first is, in front Via Garibaldi, the sculpture dedicated to the Messenger Army Sardo, opened in April 1859 by Vincenzo Vela.
On the side towards Via Roma, there is a monument to the Knight of Italy by Pietro Canonica in 1923.
Finally, on the side facing Via Po, it is the work of Eugenio Baroni dedicated to Emanuele Filiberto di Savoia-Aosta.


text:
/*
 *
 *
 *
 */
package com.yhy.core.dto;

import lombok.Data;

/**
 * 带选中状态的DTO.
 *
 * @author YHY
 * @version 2014-10-20
 * @----------------------------------------------------------------------------------------
 * @updated 修改描述.
 * @updated by YHY
 * @updated at 2014-10-20
 */
@Data
public class Checked {

	private String id;
	private String name;
	private String genre;
	private String pId;
	private Boolean checked;
	private String memo;
	private Boolean open = true;

	public Checked() {
		super();
	}

	public Checked(String id, String name) {
		super();
		this.id = id;
		this.name = name;
	}

	public Checked(String id, String name, Boolean checked) {
		super();
		this.id = id;
		this.name = name;
		this.checked = checked;
	}

	public Checked(String id, String name, Boolean checked, String memo) {
		super();
		this.id = id;
		this.name = name;
		this.checked = checked;
		this.memo = memo;
	}

	public Checked(String id, String name, String genre) {
		super();
		this.id = id;
		this.name = name;
		this.genre = genre;
	}

	public Checked(String id, String name, String pId, Boolean checked) {
		super();
		this.id = id;
		this.name = name;
		this.pId = pId;
		this.checked = checked;
	}

	public Checked(String id, String name, String pId, Boolean checked, String memo) {
		super();
		this.id = id;
		this.name = name;
		this.pId = pId;
		this.checked = checked;
		this.memo = memo;
	}

	public Checked(String id, String name, String genre, String pId, Boolean checked) {
		super();
		this.id = id;
		this.name = name;
		this.genre = genre;
		this.pId = pId;
		this.checked = checked;
	}

	public Checked(String id, String name, String genre, String pId, Boolean checked, String memo) {
		super();
		this.id = id;
		this.name = name;
		this.genre = genre;
		this.pId = pId;
		this.checked = checked;
		this.memo = memo;
	}

	public Checked(String id, String name, String pId, Boolean checked, Boolean open) {
		super();
		this.id = id;
		this.name = name;
		this.pId = pId;
		this.checked = checked;
		this.open = open;
	}

	public Checked(String id, String name, String genre, String pId, Boolean checked, Boolean open) {
		super();
		this.id = id;
		this.name = name;
		this.genre = genre;
		this.pId = pId;
		this.checked = checked;
		this.open = open;
	}

	public String getpId() {
		return pId;
	}

	public void setpId(String pId) {
		this.pId = pId;
	}

}

redpajama_set_name: RedPajamaGithub


text:
Q: A weirdly difficult question on probability of flipping a coin. The Question:
Assume the coin is perfectly fair, and that it cannot land on its side. Now let's imagine I throw the coin T times, it lands on the same side, X times in a row in T throws, what is the probability (P) this happens N times?
The question with numbers
Let's say I throw a coin 200 (T) times, what is the probability (P) it lands exactly 7 (X) times in a row on the same face, twice (N)?
To Note

*

*The coin has to land exactly X times in a row, if it lands less or more it does not count. This means if I do 7 heads in a row at the start, the flip after must be tails, if I do it in the middle the flip before and after must be tails and if I do it at the end the flip before has to be tails.

*If I land seven heads in a row and seven tails in a row then N is 2.

*I found this question out myself, I have no idea how hard it is or if someone has already done it, I study math in high school but I have not been able to find a solution to my own question. If you find a solution can you show the steps as well as the reasoning behind it.

*Sorry if it is difficult to understand or there are mistakes in my English but it is not my first language.

redpajama_set_name: RedPajamaStackExchange


During October 2016 I took part in "Inktober", producing one drawing in ink per day, and I used the "31 Witches" prompts to inspire each days drawing. I really enjoyed designing each day's witch, making up backstories and developing contect for each one. I liked them so much I decided to put them together into an art book! I hope you enjoy reading it as much as I enjoyed making it.
Bi The Way - Physical Book!
Bi The Way - Pun Sticker Set!
I love this so much. It's so creative and lovely to look at. I'd love to have some of these on my wall but i'd hate to ruin the book. Fab artwork and I can even relate to a few of them myself!


text:
<!DOCTYPE html>

    <meta charset="utf-8">
    <title>iframe</title>


We had two Americans on the podium in Chicago this year including the first American-born winner in the men's race in 35 years. Steve and Chris break it all down with this week's Chicago Marathon recap episode. Enjoy!


A 19-year-old California student who went missing while celebrating New Year's Eve at a Lake Tahoe music festival was found dead in a snowdrift Friday.
Police do not suspect foul play in the death of Alyssa Byrne, of Petaluma, California, who was last seen leaving the SnowGlobe Music Festival at Lake Tahoe Community College on the night of December 31.
Five days later, the 19-year-old's body was found in a mound of snow by a utility worker. Investigators believe Byrne was walking along the Pioneer Trail from the concert site to the Horizon Casino Resort where she was staying when she strayed from the road and fell into a snowdrift.
The El Dorado County coroner will perform an autopsy to determine the exact cause of death. Preliminary examination of the body revealed no outward signs of trauma, the San Francisco Chronicle reported.
The victim told friends that she planned to take a shuttle back to her hotel about three miles away from the event venue, but she apparently decided to walk it instead since there were long lines of people waiting for transport at the music festival.
While it was not snowing on the night of Byrne's disappearance, it was extremely cold, with temperatures only reaching three degrees above freezing. Byrne was wearing a sweatshirt, yoga pants and snow boots, but no coat.
Police, however, do not believe the posts are connected to the young woman's death.
Byrne, whom relatives and friends described as a happy person who was nice to everyone, worked at a steakhouse in Petaluma while studying to become a paramedic and firefighter at Santa Rosa Junior College.
Following her disappearance, the student's parents and friends organized a searching party, canvassing the area of Lake Tahoe and handing out 'missing' flyers in hopes of finding Byrne.
On December 31, the young woman had made the trip from Petaluma to Lake Tahoe along with three friends. Sometime around 11pm, Byrne's companions lost track of her.
Petaluma360.com reported that about half an hour later, Byrne called her friends to tell them that she was taking a shuttle back to the hotel, where they were sharing a room with two other acquaintances, according to Jay Donnellan, who was in Byrne's party that night.
Following a frantic three-day search involving local police, the FBI and loved ones, a utility worker came across the woman's body at around 8.30am Friday.
Police said she was not discovered earlier likely because Byrne fell into a pile of snow about 10 feet off the road.
Byrne graduated from Casa Grande High School in 2011. Her mother, Kim Byrne, told Patch.com the New Year's Eve trip to the music festival was the first time her daughter had traveled on her own.
Tragically, it ended up being her last.


text:
›Digital Film Production›
Digital Film Production Certificate
Romus Reece
rreece@contracosta.edu
Digital filmmakers create the videos that entertain and inform us all. They write the material, plan the production process, record the audio and video, then edit it all into something that viewers can understand and appreciate.
Some work at desks writing scripts. Others work in studios or on location, building sets, lighting scenes, running cameras or recording audio. Still others work in editing bays or other pre- or post-production facilities. All of that work goes into images we see every day.
Films, television programs, informational videos, newscasts, sporting events and many other kinds of productions need qualified, talented people to make them. Even smaller events like weddings are often documented by professional videographers. But it takes directors, cinematographers, camera operators, writers, storyboard artists, editors and countless others to bring major projects to life.
Most of these people work in entertainment hubs, like New York and Los Angeles. But thanks to the rise of streaming, more video content than ever is being produced in other places, as well. Today, it isn't uncommon for major businesses to hire video production teams to meet their day-to-day needs in-house.
If you want to know more information about entry-level positions in digital film production, like how much Camera Operators and Editors earn and how many jobs are available in the area, check out our online Career Coach.
You can bring your dreams to the screen right here at Contra Costa College.
In CCC's digital film production program, you'll plan, format and write your own treatments, scripts and screenplays. You'll learn how to operate film production equipment by actually using it, hands-on. You'll edit films with Adobe Premiere. You'll create motion graphics, 3-D CG models and other visual effects with post-production programs like After Effects, Speed Grade, Encore and Media Encoder. You'll even direct and produce a short narrative film, managing it through every step of planning and production.
Whether you're looking for the confidence and experience to shoot your own videos or planning to transfer to a four-year school to study film, we've got you.
So why not check out the program, see if we can help you pay for college or even get the ball rolling and apply now?
LIVE IN SAN PABLO? GET 50% OF THIS PROGRAM'S COSTS COVERED
Digital Film Grad Hector Ortiz talks about his grant-funded DACA film
find out what's happening around campus
Drama Professor Takes One-Man Show to Dublin
CCC Journalism Alums on Pulitzer Prizewinning Team


And what can the Citymaster 600 do for you?
The pro in the compact class.
It is your aim to carry out reliable work and to satisfy outdoor cleaning needs or your customer's needs in the long run! What you are looking for is a professional, powerful and compact sweeper, right? The Citymaster 600 meets your demands exactly – with best-possible operating comfort and innovative quickchange system for a range of attachments designed to suit your individual requirements.
Hako's years of experience in the field of sweeping machines are all concentrated in this one machine – use it to your advantage!


Conveyor belts -- Specification for rubber- or plastics-covered conveyor belts of textile construction for general use 90.93: 53.040.20; ISO 15147:1999 [Withdrawn] Light conveyor belts -- Tolerances on widths and lengths of cut light conveyor belts 95.99: 53.040.20; ISO 15147:2012.
Alibaba.com offers 6,290 rubber conveyor belt price products. About 50% of these are conveyors, 25% are rubber belts, and 5% are transmission belts. A wide variety of rubber conveyor belt price options are available to you, such as free samples, paid samples.
Heavy Conveyor Belt Price Rubber WHOLESALE . ... Exact specifications yield maximum service in the most demanding applications. Long, ... NET prices on heavy belt (pages 6-20) are for any quantity from warehouse inventories, prepared any way you instruct.
Ashland Conveyor supplies several types of conveyor belts. PVC black smooth top is a general purpose belt. SBR rough top has better product "grabbing" and is suitable for incline.
PVC-PU -Conveyor Belts As leading importer & distributor of various conveyor belts, timing belts, transmission belt, pu conveyor belts, pvc conveyor belt, silicon coated conveyor belts, teflon coated conveyor belt etc, we have created a broad understanding of the industry to be able to offer our customers advanced solutions to their belting needs.
wire rope conveyor belt is a rubber conveyor belt with steel skeleton,constitute by core glue, rope, cover and side glue. It's advantage is tensile strength, impact resistance, long life, small elongation, good trough ability, flex resistance, good benefits.
The world is "Transported" with Japan-made standard belt. Bridgestone is the global leader in the tire industry and has also been supplying conveyor belts to the world for 60 years.
€WEAR RESISTANT RUBBER CONVEYOR BELTS ... Cover combinations and cover qualities according to specification. Open or endless lengths. Max. length unit 400 m/roll or according to agreement, however, allowance must be made for practical handling possibilities (weight and roll diameter).
Sparks Belting product support team provides the right technical support and service you need to make belting recommendations, meet price points, and meet on-time delivery. Our MRO national account will document savings by helping you standardize conveyor belt types, make applications recommendations, and minimize transaction costs and paperwork.


text:
In recent weeks, it seems like America has arrived at a crucial crossroads in its long and painful history of gun violence. In the wake of several high-profile mass shootings in Atlanta, in Boulder, Colo., and most recently, in Indianapolis, Ind., there is a growing sense of outrage and demand for solutions. And in Washington, for the first time in several years, political leadership in Congress and the White House is aligned in seeking to create new laws to rein in the worst aspects of American gun violence. So there is a glimmer of hope that Washington might make some progress on an issue that has been gridlocked for many years.
Already, President Biden has announced executive orders and other policy changes at the Justice Department and elsewhere throughout the federal government, including rules to stop the proliferation of homemade "ghost guns" and new guidelines on braces that can turn certain pistols into a short-barreled rifle. And the Biden Administration will deploy new resources for violence intervention through five federal agencies.
But major efforts to stem gun violence will still require new legislation and it remains challenging to get new gun laws passed, especially through a Senate that is evenly split down party lines. But at least the Senate is controlled by Democrats and may therefore at least consider bringing gun legislation forward, where previously, in a Republican-controlled Senate, such proposals were not even allowed to be brought forward for consideration.
With this political background in mind, it's useful to remind ourselves of the many ways that media funders and media makers have devoted tremendous creative and journalistic resources to illuminate this troubled issue. Each of these efforts—reporting initiatives, documentary films, research projects and more—have contributed to our collective understanding of the impact of gun violence on our country. And over the years, we have highlighted many of them.
Going back nearly a decade, Media Impact Funders presented several screenings and discussions concerning The Interrupters, which portrayed the brave work of community activists in Chicago who created an innovative effort to interrupt gun violence. In a similar vein, we organized screenings and discussions concerning Newtown, the deeply personal and heartbreaking story of the lives lost in the 2012 shooting at Sandy Hook Elementary School, in Newtown, Conn.
We also reported on the extensive efforts of some of the leading funders who have devoted significant resources to study and advocate around the issue of gun violence, like the Joyce Foundation, which has distinguished itself as one of the nation's leading funders in this area. Another major funding initiative on gun violence was the Kendeda Fund's special project, Guns & America, a collaboration between 10 public radio stations to create a national reporting network to cover myriad reporting angles on this complex story. And more recently, The Trace, a nonprofit news organization focused on investigating gun violence in America, has created Up the Block, a new resource hub for Philadelphians affected by gun violence, with support from the Emerson Collective.
These are but a few of the media projects supported by philanthropy, illuminating the grim territory of gun violence in America. As we confront the major milestones and contemplate the possibility of long-sought reforms, it's important to recognize the many contributions that media makers and media funders have made to light the way.
Audio/Radio
Before joining Media Impact Funders in 2011 as executive director, Vince was program director for Nonprofit Sector Support at the Surdna Foundation, a family foundation based in New York City. Prior to joining Surdna, Stehle worked for 10 years as a reporter for the Chronicle of Philanthropy, where he covered a broad range of issues about the nonprofit sector. Stehle has served as chairperson of Philanthropy New York and on the governing boards of VolunteerMatch and the Nonprofit Technology Network (NTEN). Currently, he serves on the board of directors of the Center for Effective Philanthropy.


With so many things to do in Atouguia da Baleia, it's important to choose a good Hotel.
Beach Apartment is a self-catering accommodation located in Atouguia da Baleia. Accommodation will provide you with a TV, a terrace and a seating area. There is a full kitchen with a dishwasher and a microwave.