text:
\section{Introduction}
	A SARX system is a special type of hybrid system composed of multiple subsystems/modes each with different parameters. At each time step only one subsystem is dominating and the dominant subsystem may switch over time. Given system inputs and outputs at each time step, our goal is to identify the switching sequence (discrete states) as well as to estimate the parameters of the subsystems every time we receive new data. This is a problem involving both clustering and estimation.

	In additional to applications in adaptive control, SARX system identification has been applied to video and texture segmentation \cite{vidal_recursive_2008,ozay_set_2015, ozay_sparsification_2012}. Due to the autoregressive nature of SARX model, it can also be applied to earthquake record analysis \cite{kozin1988autoregressive}, brain electrical activity mapping \cite{ogawa1993application}, meteorological objects identification \cite{bezruck2010application}, and financial time series analysis \cite{cochrane2005time}.

	\subsection{Prior Work}

	There have been many studies on the switched system identification problem in the offline/batch setting. A type of algebraic method was proposed in \cite{vidal_algebraic_2003}, which uses Veronese embedding to decouple the task of estimating the system parameters and switching sequence, and an exact solution is provided when the process and data are noise-free. Furthermore, the case when system orders are not necessarily equal or known is discussed in \cite{ma_identification_2005}. For systems with noise and measurements corrupted by outliers, \cite{ozay_set_2015} extends the algebraic method  by converting it to a rank minimization problem that is relaxed to a semi-definite program. Methods utilizing sparsity are proposed in \cite{bako_identification_2011, ozay_sparsification_2012}.

	As opposed to the offline/batch setting, where we have access to all the data at once, there are many problems in which the data appears in a streaming (online) fashion. That is, at each time step, we receive data with which we need to identify current dominant subsystem as well as give the latest estimate of the system parameters. Note that naively employing an offline algorithm by using all the data in the past at each step would be computationally intractable. The majority of online algorithms use a two-step approach that alternates between determining the switching sequence and updating the parameter estimates. The work in \cite{vidal_recursive_2008} is one of the first to study online identification of switched systems using an extension of the offline algebraic method \cite{vidal_algebraic_2003}. In the algorithms proposed in \cite{bako_recursive_2011}, \cite{goudjil2016convergence}, candidate estimates are built for each of the subsystems first. Then, every time a new data point arrives, the discrete state is determined by assigning the data to one of the candidates according to some criterion, and then the estimate of chosen candidate is updated with the new data. The algorithm in \cite{bako_recursive_2011} first identifies the discrete states based on prior or posterior residual error, and then updates the estimate using recursive least squares. The algorithm in \cite{goudjil2016convergence} identifies the discrete states by minimizing prior residual error similarly and then update the estimates with a modified Outer Bounding Ellipsoid (OBE) algorithm.

	\subsection{Contributions and Outline}

	We observe that in two-step algorithms, choosing a candidate based on minimum residual error can be sensitive to candidate initializations, since when a new subsystem dominates, it might ``take-over" a partially convergent candidate estimate if there is no candidate yet closer to its true parameters.

	The main contribution of our paper is a more robust two-step algorithm that can effectively overcome this issue. We initialize candidate estimates for each of the subsystems. Every time we receive new data, we determine the discrete state by assigning this data to one of candidates based on a robust criterion that incorporates both residual error and an upper bound of estimation error. After we assign the data to a candidate, we update the selected candidate using a variant of the randomized Kaczmarz algorithm proposed in \cite{strohmer2009randomized} or normalized least mean squares (NLMS) \cite{simon_s._haykin_[simon_2003}. We provide  partial and local convergence results for our algorithm. In our  partial convergence analysis we assume that we can always make correct assignments, i.e. identify the discrete state correctly, thus the parameter estimation updated for the candidates can be treated as if we are using data from a single subsystem. In local convergence analysis, we assume all candidate estimates have ``good enough" initializations, and show that with some probability no misassignment will ever be made and prove the convergence of parameter estimates. Our numerical simulations verify the convergence result and show obvious improvements of our algorithm over state of the art.

	The paper is organized as follows: in Section \ref{sctn_ProblemFormulation}, we present the problem formulation of online SARX system identification; Section \ref{sctn_Drawback} briefly discusses the drawbacks of existing algorithms; Section \ref{sctn_OurAlgorithm} introduces our algorithm; Section \ref{sctn_OurAlgorithm} gives the theoretical analyses of our algorithm; some discussions and extensions are provided in Section \ref{sctn_Extensions}; simulation evaluation are given in Section \ref{sctn_Experiments}.

	\section{Problem Formulation} \label{sctn_ProblemFormulation}

	\subsection{SARX System}

	A SARX system is defined by the following expression:
	\begin{equation} \label{eqn_ARXSystem}
	y_t = \sum\nolimits_{j=1}^{n_a}a_j(\sysidx_t) y_{t-j} + \sum\nolimits_{k=1}^{n_c} c_k(\sysidx_t) u_{t-k} + n_t
	\end{equation}
	where $u_t \in \R $ and $y_t \in \R$ are the input and output of the system, and $n_t \in \R$ is an additive noise term. The discrete state $\sysidx_t \in \{1, \dots, m\} \equiv [m]$ indexes the dominant/active subsystem at time $t$, and $\{\sysidx_t \}_t$ denotes the switching sequence. Coefficients $\{a_j(\sysidx_t)\}_{j=1}^{n_a}$ and $\{c_j(\sysidx_t)\}_{j=1}^{n_c}$ are the parameters of subsystem $\sysidx_t$.  Let $\bvltn{\phi}_{y,t} = [y_{t-1}, \dots, y_{t-n_a}]^\T, \bvltn{\phi}_{u,t} = [u_{t-1}, \dots, u_{t-n_c}]^\T, \bvltn{\phi}_t = [\bvltn{\phi}_{y,t}^\T, \bvltn{\phi}_{u,t}^\T]^\T$, and furthermore, let $\splitatcommas{\bv{w}_{\sysidx_t} = [a_1(\sysidx_t), \dots, a_{n_a}(\sysidx_t), c_1(\sysidx_t), \dots, c_{n_c}(\sysidx_t)]^\T}$. With this notation, the SARX system dynamics \eqref{eqn_ARXSystem} can be written in vector form:
	\begin{equation} \label{eqn_VecARXSystem}
	y_t = \bv{w}_{\sysidx_t}^\T \bvltn{\phi}_t + n_t \;.
	\end{equation}
	Let $n = n_a + n_c$ be the system order, which can also be viewed as the ambient dimension of our problem.

	\subsection{Assumptions}

	In this work, we make the following assumptions, where Assumption \ref{asmp_systemSpecs} and the noise upper bound in Assumption \ref{asmp_noise} are needed for the algorithm to work. (Case where noise is unbounded is discussed in Section \ref{subsctn_unboundedNoise}.) Assumption \ref{asmp_noise} to Assumption \ref{asmp_NoAmbiguous} are mainly for analysis purposes.
	\begin{assum} \label{asmp_systemSpecs}
		The model orders $n_a$, $n_c$ on RHS of \eqref{eqn_ARXSystem}, and the number of subsystems, $m$, are known.
	\end{assum}

	\begin{assum} \label{asmp_noise}
		The noise $n_t$ is random with $\expctn[n_t]=0$ and $\expctn[n_t^2]=\sigma_n^2$. $ |n_t| \leq n_{\max}$ and $n_{\max}$ is known. $n_t$ is independent of input $u_t$.
	\end{assum}

	\begin{assum} \label{asmp_data}
		For all $t$, $\norm{\bvltn{\phi}_t} \leq \phi_{\max}$. We also assume a lower bound on the SNR: for all $t, \frac{\norm{\bvltn{\phi}_t}}{ | n_t|} \geq S_{\min} $.
	\end{assum}

	\begin{assum} \label{asmp_singularvalue}
		There exists $\singval_{\max} \geq \singval_{\min}> 0$ such that $\forall \ S \subset \mathbb{N}^+$ with cardinality $N_R$ (defined in Section \ref{sctn_OurAlgorithm}),
		\begin{equation} \label{eqn_untitle52}
		\singval_{\min}^2 I_n \preceq \sum\nolimits_{t\in S} \bvltn{\phi}_t \bvltn{\phi}_t^\T \preceq \singval_{\max}^2 I_n \;.
		\end{equation}
	\end{assum}

	\begin{assum} \label{asmp_NoAmbiguous}
		If subsystem $i$ generates data pair $\{\bvltn{\phi}_t, y_t \}$, i.e. $y_t = \bv{w}_i^\T \ \bvltn{\phi}_t + n_t$, then $\forall j \neq i$, $|\bv{w}_j^\T \bvltn{\phi}_t - \bv{w}_i^\T \bvltn{\phi}_t | \geq \psi$.
	\end{assum}

	Assumption \ref{asmp_singularvalue} is similar to persistent excitation conditions in the literature, and it plays a critical role in the convergence rate. Assumption \ref{asmp_NoAmbiguous} guarantees there is no ambiguous data, since if data pair $\curlybrackets{\bvltn{\phi}_t, y_t}$ satisfies both $y_t = \bv{w}_1^\T \bvltn{\phi}_t$ and $y_t = \bv{w}_2^\T \bvltn{\phi}_t$, then even with the true parameters $\bv{w}_1$ and $\bv{w}_2$, we cannot tell which system generates $y_t$.

    \subsection{Goal}

	The goal of online system identification is as follows. After we collect the data pair $\curlybrackets{\bvltn{\phi}_t, y_t}$ at each time step, we want to identify discrete state $\sysidx_t$ and estimate parameters of the subsystem that generates $y_t$.

	\section{Drawbacks of Existing Algorithms} \label{sctn_Drawback}
	Existing algorithms e.g. \cite{bako_recursive_2011}, \cite{goudjil2016convergence}, commonly have a two-step structure after candidate estimate for each of the subsystem is initialized: (1) every time new data is available, it is assigned to the candidate with minimum prior/posterior residual error; (2) the parameter estimate of the chosen candidate is updated with this data. We will show that using only residual error as the criterion to assign data can be unreliable.
	\begin{figure}[h!]
		\begin{center}
			\includegraphics[width=10cm]{DownsideofPreviousAlgs.pdf}
			\caption{\small Demonstration of potential drawback of existing algorithms.}
			\label{fig_DownsideofPreviousAlgs}
		\end{center}
	\end{figure}

	Fig. \ref{fig_DownsideofPreviousAlgs} shows a toy example of what could go wrong with the above mentioned algorithms. There are 3 subsystems, and red circles illustrate their true parameter vectors in the ambient space; the 3 candidate estimates are initialized at the three dark blue points in the left box. Assume from $t=1 \text{ to } 10$, subsystem 1 is dominant (left box), and from $t=11 \text{ to } 20$, subsystem 2 is dominant (right box). Considering the positions and true and estimated parameters, it's \emph{likely} that from $t=1 \text{ to } 10$ data generated by subsystem 1 will be assigned to candidate 1 since it's the closest candidate.
    When $t=10$, candidate 1 is an improved estimate of system 1 parameters, given by the light blue point in the left box. At time $t=11$, subsystem 2 becomes dominant. Considering the current positions of all candidates, candidate 1 is still closest to subsystem 2, so it's \emph{likely} that data generated by subsystem 2 will also be assigned to candidate 1, and we could expect candidate 1 will start to drift from subsystem 1 parameter values toward subsystem 2, given by the trajectory in the right box. In this sense, all previous efforts used to let candidate 1 learn subsystem 1 will be wasted.

	In our algorithm, the basic idea to solve this drawback is to take the accuracy of the candidate estimates into account and be more cautious when assigning data to candidates with higher accuracy. The details will be discussed in Section \ref{subsctn_makeAssignment}.

	\section{Our Algorithm} \label{sctn_OurAlgorithm}
	\begin{algorithm}[h!]
		Initialize $N_R, N_C (N_R \geq n, N_C \geq N_R^2), \alpha, \beta, \nu$ \label{algline_untitled8}\\
		\For{$i = 1, \dots, m$}{
			$ \wht{\bv{w}}_{i,0} = \bv{0}_{n {\times} 1}, c_i = 0, $
			$\bvltn{\Phi}^R_{i,t} = \bv{0}_{n {\times} N_R}, \, \bv{y}^R_{i,t} = \bv{0}_{N_R {\times} 1},$ \\
			$\bvltn{\Phi}^C_{i,t} = \bv{0}_{n {\times} N_C}, \, \wht{\bv{W}}^C_{i,t} = \bv{0}_{n {\times} N_C}, \, \bv{h}^C_{i,t} = \bv{0}_{N_C {\times} 1}, \, \epsilon_{i,0}^u = \infty$ \\
		} \label{algline_untitled9}
		\For{$t = 1, 2, \dots$}{
			Receive $\curlybrackets{\bvltn{\phi}_t, y_t}$. \label{algline_untitled10}\\

			Compute normzlized residual errors and potential new estimates for all candidates: \\
			\Indp $r_i = |y_t - \wht{\bv{w}}_{i,t-1}^\T \bvltn{\phi}_t| \cdot \norm{\bvltn{\phi}_t}^{-1} \quad \forall i \in [m]$ \label{algline_untitled1}\\

			$\wtd{\bv{w}}_{i,t} {=} \wht{\bv{w}}_{i,t-1} {-} \norm{\bvltn{\phi}_t}^{-2} \bvltn{\phi}_t \parentheses{\wht{\bv{w}}_{i,t-1}^\T  \bvltn{\phi}_t {-} y_t }
			\quad \forall i \in [m]$ \label{algline_untitled2} \\ \Indm

			Choose a candidate to assign data: \\
			\Indp $\wht{\sysidx}_t = \arg \min_i \ r_i \cdot \max\parenthesesbig{1, \alpha  \frac{ \norm{\wtd{\bv{w}}_{i,t} - \wht{\bv{w}}_{i,t-1}}}{2 (\epsilon_{i,t-1}^u + \nu )}}^\beta$ \label{algline_makeAssignment}\\ \Indm

			Update counter and window varaibles: \\
			\Indp
			$c_{\wht{\sysidx}_t} = c_{\wht{\sysidx}_t} + 1$ \label{algline_untitled11}\\

			$\bvltn{\Phi}^R_{\hat{\sysidx}_t, t} {=} [\bvltn{\Phi}^R_{\hat{\sysidx}_t, {t-1}}[{:}, 2{:}\text{end}], \bvltn{\phi}_t], \,
				\bv{y}^R_{\hat{\sysidx}_t, t} {=} [\bv{y}^R_{\hat{\sysidx}_t, {t-1}}[2{:}\text{end}]; y_t]$ \label{algline_untitled12}\\	\Indm

			Update estimate of chosen candidate: \\
			\Indp
			\SetInd{2.05em}{-0.6em}
			\uIf{$c_{\wht{\sysidx}_t} < N_R$}{
				$\bvltn{\phi}_t^* = \bvltn{\phi}_t, y_t^* = y_t, \eta_t^* = \norm{\bvltn{\bvltn{\phi}}_t}^{-2}$ \label{algline_untitled4}\\
			}
			\Else{
				Sample $l_t {\in} [N_R]$  $\text{w.p.}\norm{\bvltn{\Phi}^R_{{\wht{\sysidx}_t}, t}[:, l_t]}^2 / \norm{\bvltn{\Phi}^R_{{\wht{\sysidx}_t}, t}}_F^2$ \label{algline_PickColumn}\\
				$\bvltn{\phi}_t^* {=} \bvltn{\Phi}^R_{{\wht{\sysidx}_t}, t}[:, l_t], \,  y_t^* {=} \bv{y}^R_{{\wht{\sysidx}_t}, t}[l_t], \, \eta_t^* {=} \norm{\bvltn{\phi}_t^{*}}^{-2}$ \label{algline_untitled5}\\
			}
			$\wht{\bv{w}}_{{\wht{\sysidx}_t},t} = \wht{\bv{w}}_{{\wht{\sysidx}_t},t-1} - \eta_t^* \bvltn{\phi}_t^* \parentheses{\wht{\bv{w}}_{{\wht{\sysidx}_t},t-1}^\T  \bvltn{\phi}_t^* - y_t^* }$ \label{algline_updateEstimate}\\
			\Indm

			Update error upper bound and window variables: \\
			\Indp
			$\bvltn{\Phi}^C_{\wht{\sysidx}_t,t}, \wht{\bv{W}}^C_{\wht{\sysidx}_t,t}, \bv{h}^C_{\wht{\sysidx}_t,t}, \epsilon_{\wht{\sysidx}_t, t}^u {=} \textbf{\small {UpdateUpperBound}}$ \label{algline_computeUpperbound} \\

			$\forall i\neq {\wht{\sysidx}_t}, \curlybrackets{\wht{\bv{w}}_{i,t}, \bvltn{\Phi}^R_{i,t}, \bv{y}^R_{i,t}, \bvltn{\Phi}^C_{i,t}, \wht{\bv{W}}^C_{i,t}, \bv{h}^C_{i,t}, \epsilon_{i,t}^u} = \curlybrackets{\wht{\bv{w}}_{i,\tau}, \bvltn{\Phi}^R_{i,\tau}, \bv{y}^R_{i,\tau}, \bvltn{\Phi}^C_{i,\tau}, \wht{\bv{W}}^C_{i,\tau}, \bv{h}^C_{i,\tau}, \epsilon_{i,\tau}^u}_{\tau=t-1}$ \label{algline_untitled3}\\
			\Indm
		}
		\caption{Our Main Algorithm} \label{alg_main}
	\end{algorithm}

	In this paper we propose Algorithm \ref{alg_main} for online identification of SARX models. This is also a two-step algorithm, but with an improved data assignment to consider not only the residual but also system estimation accuracy. This section gives an overview of the algorithm steps.

    Lines \ref{algline_untitled8} to \ref{algline_untitled9} show initialization. $\wht{\bv{w}}_{i,0}$ is the initial estimate for candidate $i$, and $c_i$ is number of assignments to candidate $i$. $\splitatcommas{\bvltn{\Phi}^R_{i,t} {\in} \R^{n \mathsf{x}\mskip1mu} 	% The product x sign used in dimension, e.g. R^{m \x n N_R}, \bv{y}^R_{i,t}{\in} \R^{N_R}, \bvltn{\Phi}^C_{i,t}{\in} \R^{n \mathsf{x}\mskip1mu} 	% The product x sign used in dimension, e.g. R^{m \x n N_C}, \wht{\bv{W}}^C_{i,t} {\in} \R^{n \mathsf{x}\mskip1mu} 	% The product x sign used in dimension, e.g. R^{m \x n N_C}, \bv{h}^C_{i,t} {\in} \R^{N_C}, \epsilon_{i,t}^u }$ are the corresponding window variables for candidate $i$ at time $t$, which will be explained in details later in this section. $N_R $ and $N_C$ are the number of columns of $\bvltn{\Phi}^R_{i,t}$ and $\bvltn{\Phi}^C_{i,t}$ respectively, which are also the window lengths for the randomized Kaczmarz algorithm and error upper bound estimation respectively.

	At each time step, via Lines \ref{algline_untitled10} to \ref{algline_untitled11}, we assign the data to one of the candidates using a new criterion to determine the discrete state. Then, we update the chosen candidate estimate using an idea similar to the randomized Kaczmarz algorithm in \cite{strohmer2009randomized} in Lines \ref{algline_untitled12} to Line \ref{algline_untitled3}.

	\subsection{Making Assignment/Identifying the Discrete State} \label{subsctn_makeAssignment}

	With data pair $\curlybrackets{\bvltn{\phi}_t, y_t}$, we compute the normalized residual error $r_i$ for each candidate in Line \ref{algline_untitled1}, where $\wht{\bv{w}}_{i, t-1}$ is the estimate of candidate $i$ at time $t-1$. We then compute the potential new estimate $\wtd{\bv{w}}_{i,t}$ for each candidate if we were to use $\curlybrackets{\bvltn{\phi}_t, y_t}$ to update $\wht{\bv{w}}_{i, t-1}$.

	The assignment criterion is given in Line \ref{algline_makeAssignment}. The criterion has two components: the first term is the normalized residual error $r_i$ and the second term measures whether $\wtd{\bv{w}}_{i,t}$ has a larger estimation error than $\wht{\bv{w}}_{i, t-1}$. The variables $\alpha$, $\beta$ and $\nu$ are tuning parameters. Variable $\epsilon_{i,t-1}^u$ is an estimate of upper bound on the magnitude of candidate $i$'s estimation error $\bvltn{\epsilon}_{i,t-1}\equiv\bv{w} - \wht{\bv{w}}_{i,t-1}$ with respect to \emph{some} true system parameter $\bv{w}$. The main difference between our algorithm and previous two-step algorithms mentioned in Section \ref{sctn_Drawback} is the incorporation of the second term, which makes assignment more robust.

	\begin{figure}[h!]
		\begin{center}
			\includegraphics[width=10cm]{IdeaofOurAlg.pdf}
			\caption{\small Idea of our algorithm}
			\label{fig_IdeaofOurAlg}
		\end{center}
	\end{figure}

	The idea behind the criterion is straightforward: letting $\alpha{=}1, \beta{=}1, \nu {=} 0$, and replacing $\epsilon_{i,t-1}^u$ with $\norm{\bvltn{\epsilon}_{i,t-1}}$, the second part of the criterion becomes $\max\parenthesesbig{1, \frac{\norm{\wtd{\bv{w}}_{i,t} - \wht{\bv{w}}_{i,t-1}}}{2 \norm{\bvltn{\epsilon}_{i,t-1}}}}$. The numerator $\norm{\wtd{\bv{w}}_{i,t} - \wht{\bv{w}}_{i,t-1}}$ is the magnitude of variation if we update the estimate $\wht{\bv{w}}_{i, t-1}$ of candidate $i$ with $\curlybrackets{\bvltn{\phi}_t, y_t}$. We could see that if the estimation error of $\wtd{\bv{w}}_{i,t}$ doesn't increase compared with the error of $\wht{\bv{w}}_{i,t-1}$, then we must have $\norm{\wtd{\bv{w}}_{i,t} - \wht{\bv{w}}_{i,t-1}} \leq 2 \norm{\bvltn{\epsilon}_{i,t-1}}$. And if $\norm{\wtd{\bv{w}}_{i,t} - \wht{\bv{w}}_{i,t-1}} > 2 \norm{\bvltn{\epsilon}_{i,t-1}}$, then the estimation error must get larger. Therefore, $\max\parenthesesbig{1, \frac{\norm{\wtd{\bv{w}}_{i,t} - \wht{\bv{w}}_{i,t-1}}}{2 \norm{\bvltn{\epsilon}_{i,t-1}}}}$ works against candidates whose error would increase if we update using $\curlybrackets{\bvltn{\phi}_t, y_t}$. The max operator shows that as long as $\norm{\wtd{\bv{w}}_{i,t} - \wht{\bv{w}}_{i,t-1}} \leq 2 \norm{\bvltn{\epsilon}_{i,t-1}}$, we don't penalize any further. Since we don't know the true estimation error $\norm{\bvltn{\epsilon}_{i,t-1}}$, we replace it with its estimated upper bound $\epsilon_{i,t-1}^u$ (we will show under some conditions, this is a valid upper bound in Theorem \ref{thrm_validUpperbound}) computed in Algorithm \ref{alg_computeUpperbound}.

	This idea is illustrated with Fig. \ref{fig_IdeaofOurAlg}. Consider the same experimental setup as the toy example in Section \ref{sctn_Drawback}. At time $t=11$, candidate 1 will be within the ball region (denoted with the red dotted circle) estimated by upper bound $\epsilon^u$. Once the potential update magnitude of candidate 1 exceeds the diameter $2\epsilon^u$ of this region, which implies its estimation accuracy will become worse if we update candidate 1 with this data, so candidate 1 should be penalized when making the assignment.

	We note that $\norm{\wtd{\bv{w}}_{i,t} - \wht{\bv{w}}_{i,t-1}} {\leq} 2 \norm{\bvltn{\epsilon}_{i,t-1}}$ is only a necessary but not sufficient condition to ensure non-increasing estimation error. Even though we are relying on a necessary condition, our algorithm empirically achieves significantly improved performance over previous algorithms.

	\subsection{Candidate Estimate Updates}

	After the assignment is made, from Line \ref{algline_untitled12} to Line \ref{algline_updateEstimate} we update the estimate of the candidate $\wht{\sysidx}_t$ to which the data has been assigned. Our updating approach is based on the randomized Kaczmarz method with a sliding window of data. We define window variables $\bvltn{\Phi}^R_{i,t}, \bv{y}^R_{i,t}$ to store previous $N_R$ data $\curlybrackets{\bvltn{\phi}, y}$ assigned to candidate $i$. If we have collected $N_R$ data, i.e. $c_{\wht{\sysidx}_t} \geq N_R$, we update the estimate with randomly picked historical data $\curlybrackets{\bvltn{\phi}^*_t, y^*_t}$; otherwise we simply update using current data $\curlybrackets{\bvltn{\phi}^*_t, y^*_t}=\curlybrackets{\bvltn{\phi}_t, y_t}$. The idea behind the update rule is that we project the current estimate $\wht{\bv{w}}_{{\wht{\sysidx}_t},t-1}$ onto the solution space of $\curlybrackets{\bvltn{\phi}^*_t, y^*_t}$ such that $y^*_t = \wht{\bv{w}}_{{\wht{\sysidx}_t},t}^\T \bvltn{\phi}^*_t$.

	\subsection{Computation of Error Upper Bound}

	\begin{algorithm}[h!]
		Update window variables for chosen candidate: \\	\Indp
		$\bvltn{\Phi}^C_{\wht{\sysidx}_t,t} = [\bvltn{\Phi}^C_{\wht{\sysidx}_t,t-1}[:\, , \, 2:\text{end}] \, , \, \bvltn{\phi}_t^*]$ \\
		$\wht{\bv{W}}^C_{\wht{\sysidx}_t,t} = [\wht{\bv{W}}^C_{\wht{\sysidx}_t,t-1}[:\, , \, 2:\text{end}] \, , \, \wht{\bv{w}}_{{\wht{\sysidx}_t},t}]$\\
		$\bv{h}^C_{\wht{\sysidx}_t,t} = [\bv{h}^C_{\wht{\sysidx}_t,t-1}[2:\text{end}]\, ; \, \eta_t^*]$ \\ \Indm

		Update the error upper bound for chosen candidate: \\	\Indp
		\SetInd{2.05em}{-0.6em}
		\uIf{$c_{\wht{\sysidx}_t} < N_C$}
		{
			$\epsilon_{\wht{\sysidx}_t, t}^u = \epsilon_{\wht{\sysidx}_t, t-1}^u$
		}
		\Else
		{
			$\Delta \wht{\bv{W}} = \wht{\bv{w}}_{{\wht{\sysidx}_t},t} \onevec_{N_C \times 1}^\T - \wht{\bv{W}}^C_{\wht{\sysidx}_t,t}$ \label{algline_untitled6}\\
			$\Delta \wht{\bv{w}} = \wht{\bv{w}}_{{\wht{\sysidx}_t},t}  - \wht{\bv{w}}_{{\wht{\sysidx}_t},t-N_C}$\\

			$\bv{H} = \diag (\bv{h}^C_{\wht{\sysidx}_t,t})$\\

			$\bv{A} = \parentheses{\bvltn{\Phi}^C_{\wht{\sysidx}_t,t} \bv{H} {\bvltn{\Phi}^C_{\wht{\sysidx}_t,t}}^\T}^{-1} \bvltn{\Phi}^C_{\wht{\sysidx}_t,t} \bv{H}$ \\

			$\bv{b} = \parentheses{\bvltn{\Phi}^C_{\wht{\sysidx}_t,t} \bv{H} {\bvltn{\Phi}^C_{\wht{\sysidx}_t,t}}^\T}^{-1}  \squarebracketsbig{\Delta \wht{\bv{w}} - \bvltn{\Phi}^C_{\wht{\sysidx}_t,t} \bv{H} \square \parenthesesbig{\bvltn{\Phi}^C_{\wht{\sysidx}_t,t}, \Delta \wht{\bv{W}}}}$ \label{algline_untitled7} \\

			Let $V = \{[\pm n_{\max}, \pm n_{\max}, \dots,  \pm n_{\max}]_{N_C}^\T \}$ \\
			$\epsilon_{\wht{\sysidx}_t, t}^u = \max_{\bv{n}\in V} \norm{A \bv{n} - \bv{b}}$ \label{algline_untitled13}

		}
		\Indm
		\nonl \textbf{Remark:} $\square \parentheses{A,B} $$ \equiv [ a_1^\T b_1, a_2^\T b_2, \dots, a_n^\T b_n ]^\T$, where $a_i, b_i$ are the $i$th columns of matrices $A, B$
		\caption{UpdateUpperBound} \label{alg_computeUpperbound}
	\end{algorithm}

	We update the error upper bound estimate $\epsilon_{\wht{\sysidx}_t, t}^u$ and related window variables $\bvltn{\Phi}^C_{\wht{\sysidx}_t,t}, \wht{\bv{W}}^C_{\wht{\sysidx}_t,t}, \bv{h}^C_{\wht{\sysidx}_t,t}$ of the chosen candidate in Line \ref{algline_computeUpperbound}. The details of this update are given in Algorithm \ref{alg_computeUpperbound}. If the window is not full, i.e. $c_{\wht{\sysidx}_t} {<} N_C$, we simply follow the previous upper bound estimate, i.e. $\infty$; otherwise, we update according to a slightly complicated rule whose justification is given in Theorem \ref{thrm_validUpperbound}.

	\section{Theoretical Results} \label{sctn_Theory}
	Our main theorems are Theorem \ref{thrm_PartialConstant} and Theorem \ref{thrm_LocalConvergence}, which show the partial and local convergence guarantees for the algorithm respectively. Note that the proofs for all the lemmas, theorems, and corollaries are provided at the appendices.

	To ease the exposition, we introduce some notation and concepts that are frequently used later:
	\begin{itemize}
		\setlength{\itemsep}{0pt}

        \item In Algorithm \ref{alg_main}, we sample a column index $l_t$ from the matrix $\bvltn{\Phi}^R_{i,t}$ in Line \ref{algline_PickColumn} of Algorithm \ref{alg_main}. Since $\bvltn{\Phi}^R_{i,t}$ is a matrix with columns being data vectors collected at different time, we essentially sampled a time index. Let $r_t(l_t)$ denote the true time corresponding to the collecting time of data of column $l_t$.

		\item Let $r(i,t)$ denote number of times subsystem $i$ is dominant up to time $t$.

		\item Setup(A): Assume hybrid SARX system only involves 1 subsystem, namely, subsystem $i$ with parameter $\bv{w}_i$. Then $\wht{\bv{w}}_{i,t}$ is the only candidate estimate. We let $\bvltn{\epsilon}_{i,t} = \bv{w}_i - \wht{\bv{w}}_{i,t}$ denote the estimation error.

	\end{itemize}

	\subsection{Preliminary Results} \label{subsctn_PreliminaryResult}

	In Section \ref{subsctn_PreliminaryResult}, we first present several lemmas that serve as the building blocks for later theorems.
	\begin{lem} \label{lemma_boundsonNorm}
		$\forall i$, after $c_i \geq N_R$, since $\bvltn{\Phi}^R_{i,t} {\bvltn{\Phi}^R_{i,t}}^\T = \sum \bvltn{\phi} \bvltn{\phi}^\T $,  and following Assumption \ref{asmp_singularvalue}, we know the singular values of $\bvltn{\Phi}^R_{i,t}$ is upper and lower bounded by $\singval_{\max}$ and $\singval_{\min}$ selectively. Following this, the following results hold trivially
		\begin{enumerate}[(i)]
			\setlength{\itemsep}{0pt}
			\item Let $F_{\max} = \sqrt{n} \singval_{\max}, F_{\min} = \sqrt{n} \singval_{\min}$, then we have
			\begin{equation}
			F_{\min} \leq \norm{\bvltn{\Phi}^R_{i,t}}_F \leq F_{\max}
			\end{equation}

			\item Let $\kappa(\bvltn{\Phi}^R_{i,t}) {=} \norm{\bvltn{\Phi}^R_{i,t}}_F \norm{{\bvltn{\Phi}^R_{i,t}}^{-1}}_2, \kappa_{\max} {=} \sqrt{\parentheses{{(n{-}1) \singval_{\max}^2 {+} \singval_{\min}^2 }}/\singval_{\min}^2 }$, and $\kappa_{\min} {=} \sqrt{n}$, where $-1$ denotes the right inverse, then we have
			\begin{equation}
			\kappa_{\min} \leq \kappa(\bvltn{\Phi}^R_{i,t}) \leq \kappa_{\max}
			\end{equation}

			\item Let $\xi(\bvltn{\Phi}^R_{i,t}) {=} \norm{\bvltn{\Phi}^R_{i,t}}_F / \norm{\bvltn{\Phi}^R_{i,t}}_2 $, $\xi_{\max} = \sqrt{n}$, and $\xi_{\min} = \sqrt{\parentheses{\singval_{\max}^2 + (n-1)\singval_{\min}^2 }/{\singval_{\max}^2}}$, then we have
			\begin{equation}
			\xi_{\min} \leq \xi(\bvltn{\Phi}^R_{i,t}) \leq \xi_{\max}
			\end{equation}
		\end{enumerate}
	\end{lem}

	\begin{lem} \label{lemma_NoiseData}
		Following Assumption \ref{asmp_noise}, we have
		\begin{enumerate}[(i)]
			\setlength{\itemsep}{0pt}
			\item $\expctn[n_{r_t(l_t)}]  = 0, \expctn[n_{r_t(l_t)}^2]  = \sigma_n^2$ \label{lemmaresult_1}
			\item $n_{r_t(l_t)}$ and $\bvltn{\phi}_{r_t(l_t)}$ are uncorrelated \label{lemmaresult_2}
			\Item
			\begin{equation} \label{eqn_untitle101}
			\frac{N_R}{F_{\max}^2} \sigma_n^2
			\leq
			\expctn \squarebracketsbig{\frac{n_{r_t(l_t)}^2}{\norm{\bvltn{\phi}_{r_t(l_t)}}^2}}
			\leq
			\frac{N_R}{F_{\min}^2} \sigma_n^2
			\end{equation} \label{lemmaresult_3}
		\end{enumerate}
	\end{lem}

	The following Lemma \ref{lemma_boundonExpectation} is extension of result in \cite{strohmer2009randomized}.
	\begin{lem} \label{lemma_boundonExpectation}
		For any random vector $\bv{z} \in \R^n$, we have
		\begin{equation} \label{eqn_untitle102}
		\kappa_{\max}^{-2} \expctn [\norm{\bv{z}}^2 ]
		\leq
		\expctn \squarebracketsbig{\parenthesesbig{\frac{\bvltn{\phi}_{r_t(l_t)}^\T \bv{z} }{\norm{\bvltn{\phi}_{r_t(l_t)}}}  }^2}
		\leq
		\xi_{\min}^{-2} \expctn [ \norm{\bv{z}}^2 ]
		\end{equation}
	\end{lem}

	\subsection{Valid Upper Bound} \label{subsctn_ValidUpBd}

	The following theorem gives justification for error upper bound $\epsilon_{\wht{\sysidx}_t, t}^u$ computed from Line \ref{algline_untitled6} to Line \ref{algline_untitled13} in Algorithm \ref{alg_computeUpperbound}. However, this is a restrictive result, as it requires that the candidate $\wht{\sysidx}_t$ is updated with data from the same subsystem for last $N_C$ steps, i.e. the elements in $\bvltn{\Phi}^C_{\wht{\sysidx}_t,t}, \wht{\bv{W}}^C_{\wht{\sysidx}_t,t}, \bv{h}^C_{\wht{\sysidx}_t,t}$ are collected from the same subsystem.

	\begin{thm} \label{thrm_validUpperbound}
	Assume at some time $t$, $\wht{\sysidx}_t = \sysidx_t = i$, $c_i \geq N_C$, and $\bvltn{\Phi}^C_{\wht{\sysidx}_t,t}, \wht{\bv{W}}^C_{\wht{\sysidx}_t,t}, \bv{h}^C_{\wht{\sysidx}_t,t}$ are constructed from data entirely from subsystem $i$, then $\epsilon_{i,t}^u$ is a valid upper bound for $\bvltn{\epsilon}_{i,t}$, i.e. $\epsilon_{i,t}^u \geq \norm{\bvltn{\epsilon}_{i,t}} = \norm{\bv{w}_i - \wht{\bv{w}}_{i,t}}$.
	\end{thm}

	\subsection{Partial Convergence Results} \label{subsctn_PartialConvrg}

	In this section, we list the results regarding partial convergence where we assume there is no misassignment, i.e. data generated from the same subsystem can all be assigned to one particular candidate. In this sense, to analyze the convergence properties, it suffices to consider the case where there is only one subsystem in the hybrid model, and one corresponding candidate, i.e. Setup(A).

	Lemma \ref{lemma_InitPhasePartial} provides the convergence analysis at the beginning phase of the algorithm, when $t<N_R$ and Algorithm \ref{alg_main} executes Line \ref{algline_untitled4}. Then Lemma \ref{lemma_2ndConvergenceConstant} provides the convergence analysis for the second phase of the algorithm, when $t\geq N_R$ and Algorithm \ref{alg_main} executes Line \ref{algline_PickColumn} and \ref{algline_untitled5}. Finally, we have the partial convergence result Theorem \ref{thrm_PartialConstant} simply by combining Lemma \ref{lemma_InitPhasePartial} and Lemma \ref{lemma_2ndConvergenceConstant}.

	\begin{lem} \label{lemma_InitPhasePartial}
		With Setup(A), we have
		\begin{equation} \label{eqn_InitPhasePartial}
		\frac{\sigma_n^2}{\phi_{\max}^2}
		\leq
		\expctn \squarebracketsbig{\norm{\bvltn{\epsilon}_{i, N_R-1}}^2}
		\leq
		\norm{\bvltn{\epsilon}_{i,0}}^2 + \frac{N_R-1}{S_{\min}^2}
		\end{equation}
	\end{lem}

	\begin{lem} \label{lemma_2ndConvergenceConstant}
		With Setup(A), for $t\geq N_R$ we have
		\begin{equation} \label{eqn_2ndConvergenceConstant_Upper}
		\expctn \squarebracketsbig{\norm{\bvltn{\epsilon}_{i,t}}^2}
		\leq
		\parenthesesbig{1-\kappa_{\max}^{-2}}^{t-N_R+1} \expctn \squarebracketsbig{\norm{\bvltn{\epsilon}_{i, N_R-1}}^2}
		+ N_R \frac{ \kappa_{\max}^2}{F_{\min}^2}  \squarebracketsbig{1-\parenthesesbig{1-\kappa_{\max}^{-2}}^{t-N_R+1}} \sigma_n^2
		\end{equation}
		\begin{equation} \label{eqn_2ndConvergenceConstant_Lower}
		\expctn \squarebracketsbig{\norm{\bvltn{\epsilon}_{i,t}}^2}
		\geq
		\parenthesesbig{1-\xi_{\min}^{-2}}^{t-N_R+1} \expctn \squarebracketsbig{\norm{\bvltn{\epsilon}_{i, N_R-1}}^2}
		+ N_R \frac{ \xi_{\min}^2}{F_{\max}^2}  \squarebracketsbig{1-\parenthesesbig{1-\xi_{\min}^{-2}}^{t-N_R+1}} \sigma_n^2
		\end{equation}
	\end{lem}

	\begin{thm}[Partial Convergence] \label{thrm_PartialConstant}
		WLOG, assume for any $i$, data generated by subsystem $i$ will all be assigned to candidate $i$. Let $\bvltn{\epsilon}_{i,t} = \bv{w}_i - \wht{\bv{w}}_{i,t}$ denote the estimation error of candidate $i$ at time $t$. Then $\forall i,t$ such that $r(i,t) \geq N_R$, we have
		\begin{equation} \label{eqn_untitle41}
		\expctn \squarebracketsbig{\norm{\bvltn{\epsilon}_{i,t}}^2}
		\leq
		\parenthesesbig{1 {-}\kappa_{\max}^{-2}}^{r(i,t)-N_R+1} \parenthesesbig{\norm{\bvltn{\epsilon}_{i,0}}^2 {+} \frac{(N_R-1) }{S_{\min}^2}} \\
		+ N_R \frac{ \kappa_{\max}^2}{F_{\min}^2}  \squarebracketsbig{1-\parenthesesbig{1-\kappa_{\max}^{-2}}^{r(i,t)-N_R+1}} \sigma_n^2
		\end{equation}
		\begin{equation} \label{eqn_untitle42}
		\expctn \squarebracketsbig{\norm{\bvltn{\epsilon}_{i,t}}^2}
		\geq
		\parenthesesbig{1-\xi_{\min}^{-2}}^{r(i,t)-N_R+1}  \frac{\sigma_n^2}{\phi_{\max}^2} \\
		+ N_R \frac{ \xi_{\min}^2}{F_{\max}^2}  \squarebracketsbig{1-\parenthesesbig{1-\xi_{\min}^{-2}}^{r(i,t)-N_R+1}} \sigma_n^2
		\end{equation}
		If as $t \rightarrow \infty$, we have $r(i,t) \rightarrow \infty$, i.e. subsystem $i$ can dominate infinitely often, then as $t\rightarrow\infty$, we shall have
		\begin{equation} \label{eqn_untitle43}
		N_R \frac{\xi_{\min}^2}{F_{\max}^2}   \sigma_n^2
		\leq
		\expctn \squarebracketsbig{\norm{\bvltn{\epsilon}_{i,t}}^2}
		\leq
		N_R \frac{\kappa_{\max}^2}{F_{\min}^2}   \sigma_n^2
		\end{equation}
	\end{thm}

	\subsection{Local Convergence Results} \label{subsctn_LocalConvrg}

	In this section, we present results regarding local convergence. Lemma \ref{lemma_noMisassignment} shows that when all candidates have accurate enough estimates, then the next assignment will be correct. Lemma \ref{lemma_ProbBoundforSingleSys}, which is derived from Lemma \ref{lemma_Supermartingale}, gives a lower bound on the probability that the estimates will stay accurate enough during the algorithm assuming assignments are correct. By Lemma \ref{lemma_ProbBoundforSingleSys} and Lemma \ref{lemma_noMisassignment}, we could obtain the local convergence result Theorem \ref{thrm_LocalConvergence}.

	For Lemma \ref{lemma_Supermartingale} to hold, we need a technical assumption given below to guarantee that estimation errors form a supermartingale, which allows us to use supermartingale maxima inequality to get the probability bound in Lemma \ref{lemma_Supermartingale}. Note that Assumption \ref{asmp_SNRUpperBound} is not mandatory for the algorithm to work, instead, it's solely for analysis purposes. Also note that if there is no noise, even though Assumption \ref{asmp_SNRUpperBound} fails, the estimation errors still form a supermartingale, which enables us to proceed with the analysis. Local convergence for the noiseless case is provided in Corollary \ref{crlry_LocalConvergenceWONoise}.

	\begin{assum}\label{asmp_SNRUpperBound}
		Assume there is an upper bound on the SNR: $\forall t, \frac{\norm{\bvltn{\phi}_t}}{ | n_t|} \leq S_{\max}$, which satisfies $S_{\max} \leq \kappa_{\max} S_{\min}$.
	\end{assum}

	\begin{lem} \label{lemma_Supermartingale}
		Assume Assumption \ref{asmp_SNRUpperBound} holds. With Setup(A), for $\forall t\geq N_R$ and some $\epsilon'>0$, we have
		\begin{equation} \label{eqn_supermartingaleMaximaIneq}
		P \parenthesesbig{\bigcap_{\tau = N_R}^t \curlybracketsbig{\norm{\bvltn{\epsilon}_{i, \tau}}^2 \leq \epsilon'^2} } \geq 1 - \frac{\expctn \squarebracketsbig{\norm{\bvltn{\epsilon}_{i, N_R-1}}^2}}{\epsilon'^2}
		\end{equation}
	\end{lem}

	\begin{lem} \label{lemma_ProbBoundforSingleSys}
		Assume Assumption \ref{asmp_SNRUpperBound} holds. With Setup(A), for some $\epsilon'>0$, assume $\norm{\bvltn{\epsilon}_{i,0}} \leq \epsilon_0$ such that $\sqrt{N_R \parenthesesbig{\epsilon_0^2 + \frac{N_R}{S_{\min}^2}}} \leq \epsilon'$, then for $\forall t$ we have
		\begin{equation} \label{eqn_ProbBoundforSingleSys}
		P \parenthesesbig{\bigcap_{\tau = 1}^t \curlybracketsbig{\norm{\bvltn{\epsilon}_{i, \tau}}^2 \leq \epsilon'^2} } {\geq} 1 - 2 \sqrt{ \frac{N_R}{\epsilon'^2} \parenthesesbig{\epsilon_0^2 + \frac{N_R}{S_{\min}^2}} }
		\end{equation}
	\end{lem}

	\begin{lem} \label{lemma_noMisassignment}
		Let $\epsilon' = \frac{1}{2 \phi_{\max}} \parenthesesbig{\psi - \frac{n_{\max}}{\nu S_{\min}} - 3n_{\max}}$, $\alpha = 2$, and $\beta = 1$. Assume at time $t$, candidates are one-to-one $\epsilon'$-close to subsystems. WLOG, we could assume $\forall i, \norm{\bvltn{\epsilon}_{i, t-1}}\equiv\norm{\bv{w}_i - \wht{\bv{w}}_{i, t-1}} \leq \epsilon'$. Furthermore, we assume that all assignments prior to time $t$ are made correctly, i.e. $\forall s<t, \wht{\sysidx}_s = \sysidx_s$. Then at time $t$, we will also assign data correctly, i.e. $\wht{\sysidx}_t = \sysidx_t$.
	\end{lem}

	\begin{thm}[Local Convergence]  \label{thrm_LocalConvergence}
		Assume Assumption \ref{asmp_SNRUpperBound} holds. Let $\epsilon' = \frac{1}{2 \phi_{\max}} \parenthesesbig{\psi - \frac{n_{\max}}{\nu S_{\min}} - 3n_{\max}}$, $\alpha = 2$, and $\beta = 1$. Let $\bvltn{\epsilon}_{i,t} = \bv{w}_i - \wht{\bv{w}}_{i,t}$ denote the estimation error of candidate $i$ at time $t$. WLOG, assume $\forall i, \norm{\bvltn{\epsilon}_{i,0}} \leq \epsilon_0$ such that $\sqrt{N_R \parenthesesbig{\epsilon_0^2 + \frac{N_R}{S_{\min}^2}}} \leq \epsilon'$. Then $\forall i,t$ such that $r(i,t) \geq N_R$, with probability at least $1 - 2 m \sqrt{ \frac{N_R}{\epsilon'^2} \parenthesesbig{\epsilon_0^2 + \frac{N_R}{S_{\min}^2}} }$, we have the following results: (i). We can correctly identify the switching sequence, i.e. $\forall t, \wht{\sysidx}_t = \sysidx_t$. In another way, $\forall i,t $, $\curlybrackets{\bvltn{\phi}_t, y_t}$ from subsystem $i$ will be assigned to candidate $i$. (ii). Results for \eqref{eqn_untitle41}, \eqref{eqn_untitle42} and \eqref{eqn_untitle43} will hold.
	\end{thm}

	\begin{cor}[Local Convergence Without Noise]  \label{crlry_LocalConvergenceWONoise} Let $n_t  = 0$, i.e. there is no noise. Let $\epsilon' {=} \frac{\psi}{2 \phi_{\max}}$, $\alpha {=} 2$, and $\beta {=} 1$. Let $\bvltn{\epsilon}_{i,t} {=} \bv{w}_i - \wht{\bv{w}}_{i,t}$ denote the estimation error of candidate $i$ at time $t$, and assume $\forall i, \norm{\bvltn{\epsilon}_{i,0}} \leq \epsilon_0$ such that $\sqrt{N_R \epsilon_0^2} \leq \epsilon'$.

	Then $\forall i,t$ such that $r(i,t) \geq N_R$, with probability at least $1 - 2 m \sqrt{ \frac{N_R}{\epsilon'^2} \epsilon_0^2 }$, we have the following results: (i). We can correctly identify the switching sequence, i.e. $\forall t, \wht{\sysidx}_t = \sysidx_t$. In another way, $\forall t, \forall i$, $\curlybrackets{\bvltn{\phi}_t, y_t}$ from subsystem $i$ will be assigned to candidate $i$. (ii). we have the following convergence results: $\forall i,t$ such that $r(i,t) \geq N_R$
	\begin{equation} \label{eqn_untitle44}
	\expctn \squarebracketsbig{\norm{\bvltn{\epsilon}_{i,t}}^2}
	\leq
	\parenthesesbig{1-\kappa_{\max}^{-2}}^{r(i,t)-N_R+1} \norm{\bvltn{\epsilon}_{i,0}}^2
	\end{equation}
	If as $t {\rightarrow} \infty$, we have $r(i,t) {\rightarrow} \infty$, i.e. subsystem $i$ can dominate infinitely often, then as $t{\rightarrow}\infty$, we have $\expctn \squarebracketsbig{\norm{\bvltn{\epsilon}_{i,t}}^2} {=} 0$.
	\end{cor}

	\section{Discussions and Extensions} \label{sctn_Extensions}

	\subsection{Poles, Condition Number, and Convergence Rate}\label{subsctn_PolesvsConvry}

	Systems with poles close to the unit circle are not preferable as they are close to be unstable. In algorithm convergence analysis, Hessian matrix or objective function with large condition number is usually not preferable as the convergence rate tends to get small. In this section, we will show how these two facts meet consistently in our algorithm. That is, as the system poles getting closer to the unit circle, the condition number of Hessian matrix will get larger, and the convergence rate of upper bound in \eqref{eqn_untitle41} will get smaller.

	To study the convergence rate, it suffices to study a single subsystem without any switching. We drop the subsystem subscript, and let $\bvltn{\epsilon}_t = \bv{w} - \wht{\bv{w}}_t$ denote the estimation error. Since the goal of this section is to provide insight into the relations between poles, condition number, and convergence rate, so several steps involve approximation. And when study how poles affect the condition number, we only consider a toy system with order 3, since it is challenging to find nice analytical expressions for systems with higher order.

	First we consider how condition number influences convergence the rate of upper bound in \eqref{eqn_untitle41}.

	\subsubsection{Condition Number vs. Convergence Rate} \label{subsubsctn_CondvsConvrg}
	The expression for single ARX system is given by $y_t = \sum_{j=1}^{n_a} a_j y_{t-j} + \sum_{k=1}^{n_c} c_k u_{t-k} + n_t = \bv{w}^\T \bvltn{\phi}_t + n_t$ following our notations in Section \ref{sctn_ProblemFormulation}. In Assumption \ref{asmp_singularvalue}, we have $\singval_{\min}^2 I_n \preceq \sum\nolimits_{t\in S} \bvltn{\phi}_t \bvltn{\phi}_t^\T \preceq \singval_{\max}^2 I_n$. We will see this equation is related to the correlation matrix $\bv{R}\equiv \expctn[\bvltn{\phi}_t \bvltn{\phi}_t^\T]$, if it exists.

	In \cite{simon_s._haykin_[simon_2003}, we could know for the ARX system given above, if (i) poles of system are within the unit circle, and (ii) noise is white Gaussian and input is wide-sense stationary, then there exists $\bv{R}$ such that $\lim_{t\rightarrow \infty} \expctn[\bvltn{\phi}_t \bvltn{\phi}_t^\T] = \bv{R}$.

	When $N_R$ is large, according to law of large numbers, equation \eqref{eqn_untitle52} and the result above, we have $\splitatcommas{ \lim_{\min(S)\rightarrow \infty} \sum\nolimits_{t\in S} \bvltn{\phi}_t \bvltn{\phi}_t^\T \approx N_R \bv{R} }$. We let $\lambda_{\max}, \lambda_{\min}$ denote the maximum and minimum eigenvalue of $\bv{R}$. Now dropping the ``$\lim$'' and replace ``$\approx$'' with ``$=$'', we could get $N_R \lambda_{\min} I_n \preceq \sum\nolimits_{t\in S} \bvltn{\phi}_t \bvltn{\phi}_t^\T \preceq N_R \lambda_{\max} I_n$. So $N_R \lambda_{\min}$ and $N_R \lambda_{\max}$ are equivalent to $\singval_{\min}^2$ and $\singval_{\max}^2$ defined in Assumption \ref{asmp_singularvalue}. Then according to Lemma \ref{lemma_boundsonNorm}, we could have $\kappa_{\max} = \sqrt{(n{-}1) \lambda_{\max} / \lambda_{\min} {+} 1 }$ and $\xi_{\min} = \sqrt{(n{-}1) \lambda_{\min} / \lambda_{\max} {+} 1 }$. So, for the asymptotic convergence upper bounds in \eqref{eqn_2ndConvergenceConstant_Upper}, \eqref{eqn_untitle41}, and \eqref{eqn_untitle44} which all involve $\kappa_{\max}$, when the condition number of $\bv{R}$, $\lambda_{\max} / \lambda_{\min}$, increases, $\kappa_{\max}$ will increase, and the convergence rate in upper bounds will decrease.

	\subsubsection{Poles vs. Condition Number} \label{subsubsctn_PolevsCond}
	We consider a toy example of system with order 3: $y_t = a_1 y_{t-1} + a_2 y_{t-2} + c_1 u_{t-1} + n_t = \bv{w}^\T \bvltn{\phi}_t + n_t$ where $\bv{w} \equiv [a_1, a_2, c_1]$ and $\bvltn{\phi}_t \equiv [y_{t-1}, y_{t-2}, u_{t-1}]$. We assume all poles are within the unit circle, $u_t \sim \N(0, \sigma_u^2)$, $n_t \sim \N(0, \sigma_n^2) $, $u_t \perp n_t$, $u_t \perp u_s$, $n_t \perp n_s, \forall t, s$, and $\sigma_u \gg \sigma_n$. Following \cite{simon_s._haykin_[simon_2003}, we have
	\begin{equation}
	\bv{R} = \expctn \squarebracketsbig{\bvltn{\phi}_t \bvltn{\phi}_t^\T}=
	\begin{bmatrix}
	r(0) & r(1) & 0\\
	r(1) & r(0) & 0\\
	0 &  0& \sigma_u^2
	\end{bmatrix}
	\end{equation}
	where $r(0), r(1)$ can be computed by solving
	\begin{equation}
	\begin{bmatrix}
	r(0)\\
	r(1)\\
	r(2)
	\end{bmatrix}
	=
	\begin{bmatrix}
	1 & -a_1 & -a_2\\
	-a_1 & 1{-}a_2 & 0\\
	-a_2 & -a_1 & 1
	\end{bmatrix}^{-1}
	\begin{bmatrix}
	\sigma_n^2 {+} c_1 \sigma_u^2\\
	0\\
	0
	\end{bmatrix}
	\end{equation}
	So, we have
	\begin{equation} \label{eqn_R}
	\bv{R} = \expctn \squarebracketsbig{\bvltn{\phi}_t \bvltn{\phi}_t^\T}=
	\begin{bmatrix}
	(a_2-1)c & (-a_1)c & 0\\
	(-a_1)c & (a_2-1)c & 0\\
	0 &  0& \sigma_u^2
	\end{bmatrix}
	\end{equation}
	where $c {=} \frac{\sigma_n^2 {+} c_1 \sigma_u^2}{(a_2{+}1)(a_1{+}a_2{-}1)(a_1{-}a_2{+}1)}$. We will drop $\sigma_n^2$ in the following computation as $\sigma_u \gg \sigma_n$. The eigenvalues of $\bv{R}$ are given by
	\begin{align}
	\lambda_1 &= -\frac{c_1 \sigma_u^2}{(a_2+1)(a_1+a_2-1)} \\
	\lambda_2 &= \frac{c_1 \sigma_u^2}{(a_2+1)(a_1-a_2+1)} \\
	\lambda_3 &= \sigma_u^2
	\end{align}
	Note that the poles $p_1,p_2$ satisfy $p_1 + p_2 = a_1$ and $p_1 p_2 = -a_2$, so we have
	\begin{align}
	\lambda_1 &= \frac{c_1 \sigma_u^2}{(1-p_1 p_2)(1-p_1)(1-p_2)} \\
	\lambda_2 &= \frac{c_1 \sigma_u^2}{(1-p_1 p_2)(1+p_1)(1+p_2)} \\
	\lambda_3 &= \sigma_u^2
	\end{align}
	Let $c_1 \geq 1$, since $p_1, p_2 < 1$, we can see the condition number of $\bv{R}$ will have the following lower bound
	\begin{equation}
	\frac{\lambda_{\max}}{\lambda_{\min}} \geq \frac{\lambda_1}{\lambda_3} = \frac{1}{(1-p_1 p_2)(1-p_1)(1-p_2)}
	\end{equation}
	It's easy to see as poles get closer to the unit circle, this lower bound will get larger and the condition number is likely to increase as well.

	\subsubsection{Poles vs. Convergence Rate}
	Finally, by combining the two results we just showed, we could see as the system poles getting closer to unit circle, the convergence rate of upper bound in \eqref{eqn_untitle41} will decrease.

	There are two comments regarding this conclusion. (i) Even though this result only involves the rate of \emph{upper bound}, empirical results show the true convergent rate follow accordingly; (ii) Our algorithm favors stable system which is a little counterintuitive as unstable system tends to have higher SNR.

	\subsection{Unbounded Noise and Monte Carlo Method} \label{subsctn_unboundedNoise}

	Note that we compute the error upper bound $\epsilon_{\wht{\sysidx}_t, t}^u$ in Line \ref{algline_untitled13} of Algorithm \ref{alg_computeUpperbound} by finding the maximum $\norm{A \bv{n} - \bv{b}}$ from cube vertices $V$ defined by the noise magnitude upper bound $n_{\max}$. However, if $n_{\max}$ is unknown or the noise itself is unbounded, e.g. Gaussian, Algorithm \ref{alg_computeUpperbound} is not applicable to evaluate $\epsilon_{\wht{\sysidx}_t, t}^u$. In this case, if we could have samples of noise instead, an alternative approach is to use Monte Carlo method to evaluate $\epsilon_{\wht{\sysidx}_t, t}^u$. Specifically, if we have $N_t$ samples of noise vector $\bv{n}_t$ (defined in the proof for Theorem \ref{thrm_validUpperbound}) given by $\curlybrackets{\bv{n}_t^{(i)}}_{i=1}^{N_t}$, we could let $\epsilon_{\wht{\sysidx}_t, t}^u = \max \norm{A \bv{n}_t^{(i)} - \bv{b}}$. Due to the Monte Carlo nature, this is not necessarily a valid upper bound. In another way, the result in Theorem \ref{thrm_validUpperbound} doesn't hold, i.e. $\epsilon_{\wht{\sysidx}_t, t}^u < \norm{\bvltn{\epsilon}_{\wht{\sysidx}_t, t}}$. Practically, algorithm still has satisfactory performance when using this Monte Carlo method, but theoretically, this may not guarantee local convergence since local convergence result Theorem \ref{thrm_LocalConvergence} implicitly Theorem \ref{thrm_validUpperbound}, i.e. $\epsilon_{\wht{\sysidx}_t, t}^u \geq \norm{\bvltn{\epsilon}_{\wht{\sysidx}_t, t}}$ , to hold for every time step.

	If we prefer the theoretical guarantees to practical implementation, by subtly designing the number of Monte Carlo samples $N_t$ at time $t$, there could be some probability guarantee to ensure $\epsilon_{\wht{\sysidx}_t, t}^u$ is a valid upper bound at every time step.

	For ease of illustration, we assume there is only one subsystem, then we could drop the subsystem index subscript, and replace $\epsilon_{\wht{\sysidx}_t, t}^u$ with $\epsilon_{t}^u$, $\bvltn{\epsilon}_{\wht{\sysidx}_t, t}$ with $\bvltn{\epsilon}_{t}$. And we assume the Monte Carlo method starts at time $1$. Then we have the following theorem:

	\begin{thm} \label{thrm_IncreaseMonteCarlo}
		If we use Monte Carlo method above to compute $\epsilon_{t}^u$, for some $\zeta_1, \zeta_2 \in (0,1)$, let $N_t \geq \frac{\zeta_2 t}{2\zeta_1^{2t}}$,  then
		\begin{equation} \label{eqn_untitle48}
			P\parenthesesbig{P\parenthesesbig{\bigcap_{t=1}^\infty \curlybracketsbig{\norm{\bvltn{\epsilon}_{t}} {\leq} \epsilon_{t}^u}  } \geq 1{-}\frac{\zeta_1}{1-\zeta_1}}
			\geq 1- \frac{\exp(-\zeta_2)}{1- \exp(-\zeta_2)}
		\end{equation}
		In another word, the probability that every $\epsilon_{t}^u$ is a valid upper bound is large with a large probability.
	\end{thm}

	The proof for this theorem is again in the appendices. We can immediately see from Theorem \ref{thrm_IncreaseMonteCarlo} that in order to make the probabilities large, the number of Monte Carlo samples need to increase exponentially with respect to time, which makes implementation intractable when time is long.

	\subsection{Extension to MIMO Case}

	So far we have been a considering SISO system in \eqref{eqn_ARXSystem}, where all the $y_t$ and $u_t$ are scalars. However, our algorithm can be applied to MIMO systems with some transformation of the system equation, and we will provide a potential direction in this section.

	Let $\bv{y}_t \in \R^{n_y}, \bv{u}_t \in \R^{n_u}$, then the MIMO SARX system is given by
	\begin{equation}
		\bv{y}_t = \sum_{j=1}^{n_a} \bv{A}_j(\sysidx_t) \bv{y}_{t-j} + \sum_{k=1}^{n_c} \bv{C}_k(\sysidx_t) \bv{u}_{t-k} + \bv{n}_t
	\end{equation}
	where $\curlybrackets{\bv{A}_j(\sysidx_t)}_{j=1}^{n_a}$, $\curlybrackets{\bv{C}_j(\sysidx_t)}_{j=1}^{n_c}$ are the parameters of subsystem $\sysidx_t$. Let $\splitatcommas{ \bv{W}_{\sysidx_t} = [\bv{A}_1(\sysidx_t), \dots, \bv{A}_{n_a}(\sysidx_t), \bv{C}_1(\sysidx_t), \dots \bv{C}_{n_c}(\sysidx_t) ]^\T }$, $\splitatcommas{ \bvltn{\phi}_t = [\bv{y}_{t-1}^\T, \dots,\bv{y}_{t-n_a}^\T, \bv{u}_{t-1}^\T, \dots, \bv{u}_{t-n_c}^\T]^\T }$. Let $\bv{w}_{\sysidx_t, i}$ denote the $i$th column of $\bv{W}_{\sysidx_t}$, and let $y_{t,i}, n_{t,i}$ denote the $i$th element in $\bv{y}_t$ and $\bv{n}_t$. Then the MIMO system can be broken into a set of equations: $\forall i \in [n_y]$,
	\begin{equation}
		y_{t,i} = \bv{w}_{\sysidx_t, i}^\T \bvltn{\phi}_t + n_{t,i}
	\end{equation}
		which has the same form as \eqref{eqn_ARXSystem}. So we could modify our algorithm to estimate each $\bv{w}_{\sysidx_t, i}$ in a parallel way, then combine them to estimate $\bv{W}_{\sysidx_t}$.

	\subsection{Multiple $N_C$'s and Forgetting Factor}

	\subsubsection{Multiple $N_C$'s}
	Note that in Algorithm \ref{alg_computeUpperbound}, we have window variables $\bvltn{\Phi}^C_{i,t} \in \R^{n \mathsf{x}\mskip1mu} 	% The product x sign used in dimension, e.g. R^{m \x n N_C}, \wht{\bv{W}}^C_{i,t} \in \R^{n \mathsf{x}\mskip1mu} 	% The product x sign used in dimension, e.g. R^{m \x n N_C}, \bv{h}^C_{i,t} \in \R^{N_C}$ for some window length $N_C$ to compute the error upper bound $\epsilon_{\wht{\sysidx}_t, t}^u$. Theorem \ref{thrm_validUpperbound} says  when all data stored in the window are from the same subsystem, then $\epsilon_{\wht{\sysidx}_t, t}^u$ will be a valid error upper bound with respect this subsystem. However, if there enters some \emph{outlier} data (data generated by subsystem that is different from the subsystem that generates the majority of data in the window variables), $\epsilon_{\wht{\sysidx}_t, t}^u$ computed using window variables might be an invalid upper bound. If the window length is too large, since the window is sliding, the effect of outlier will stay a longer time, but the correction effect of the majority of the inlier data might reduce the effect of outlier. On the contrary, if the window length is too small, the effect of outlier will quickly vanish, but the correction effect from the inlier data will reduce as well and we may have even worse $\epsilon_{\wht{\sysidx}_t, t}^u$ during the stay of outlier.

	Practically, we could use multiple $N_C$'s and corresponding window variables. Each set of window variables compute $\epsilon_{\wht{\sysidx}_t, t}^u$ separately, and we pick the maximum of them as the final decision. In this way, the disadvantages of large and small window lengths might cancel out each other thus making $\epsilon_{\wht{\sysidx}_t, t}^u$ more robust to misassignment.

	\subsubsection{Forgetting Factor}
	One interesting fact about our algorithm is, in Line \ref{algline_PickColumn} of Algorithm \ref{alg_main}, instead of using the latest data, we pick randomly from previous data to update the estimates. This idea is initially proposed in \cite{strohmer2009randomized}. The reason we incorporate this randomization into the algorithm is to acquire the asymptotic convergence result via Assumption \ref{asmp_singularvalue}, Lemma \ref{lemma_boundsonNorm}, Lemma \ref{lemma_NoiseData}, and Lemma \ref{lemma_boundonExpectation}.

	If no randomization scheme is utilized, the algorithm on a single subsystem is equivalent to the Kaczmarz algorithm or the normalized least mean squares (NLMS) algorithm in \cite{simon_s._haykin_[simon_2003}. This type of algorithm, however, does not have satisfactory convergence results yet. One linear convergence result provided in \cite{simon_s._haykin_[simon_2003} is valid only when the step size in estimate update is very small, which makes it little practical use. The difficulty to derive nice convergence results is that nearby data $\bvltn{\phi}_{t}$ could be highly correlated, which can be seen from the definition, and updating the estimate with data in chronological order aggravates the situation. The randomized scheme picks data for update randomly and independently, which brings independence into the algorithm and makes analysis tractable.

	Empirically, if we don't incorporate the random selection in Line \ref{algline_PickColumn}, and always use latest data as Line \ref{algline_untitled4}, the performance can sometimes be slightly better. This is potentially because when sampling previous data vectors, it's likely that we sample one data multiple times possibly due to its large norm, and, generally speaking, previously used data may not provide as much information as some new data.

	One potential way to balance between establishing theoretical results and exploiting new data is to incorporate a forgetting factor $\gamma$. Specifically, in Line \ref{algline_PickColumn}, we sample data according to the following distribution
	\begin{equation}
		P(l_t=i)
		\begin{cases}
		& \gamma \norm{\bvltn{\Phi}^R_{{\wht{\sysidx}_t}, t}[: \, , \, i]}^2 \frac{1}{F} \text{ if } i=N_R \\
		& (1-\gamma) \norm{\bvltn{\Phi}^R_{{\wht{\sysidx}_t}, t}[: \, , \, i]}^2 \frac{1}{F} \text{ if }\text{ if } i<N_R
		\end{cases}
	\end{equation}
	where $\gamma>0.5$, and $F = \gamma \norm{\bvltn{\Phi}^R_{{\wht{\sysidx}_t}, t}[: \, , \, N_R]}^2 + (1-\gamma) \sum_{i=1}^{N_R-1} \norm{\bvltn{\Phi}^R_{{\wht{\sysidx}_t}, t}[: \, , \, i]}^2$ is the normalization factor. With this distribution, we can see the probability of choosing the latest data ($l_t = N_R$) is larger compared with the distribution in Algorithm \ref{alg_main}. As $\gamma$ gets closer to $1$, we are more likely to sample the latest data.

	As for the convergence result, it suffices to only consider how the building block lemmas will change with this new distribution, and the main theorems will follow these lemmas. In the building block lemmas, only the expectations in Lemma \ref{lemma_NoiseData} (\ref{lemmaresult_3}), and Lemma \ref{lemma_boundonExpectation} involve the data sampling process. With the new sampling distribution, it's not difficult to see \eqref{eqn_untitle101} and \eqref{eqn_untitle102} will become

	\begin{equation}
	\tilde{\gamma}^{-1}\frac{N_R}{F_{\max}^2} \sigma_n^2
	\leq
	\expctn \squarebracketsbig{\frac{n_{r_t(l_t)}^2}{\norm{\bvltn{\phi}_{r_t(l_t)}}^2}}
	\leq
	\tilde{\gamma} \frac{N_R}{F_{\min}^2} \sigma_n^2
	\end{equation}

	\begin{equation}
	\tilde{\gamma}^{-1} \kappa_{\max}^{-2} \expctn [\norm{\bv{z}}^2 ]
	{\leq }
	\expctn \squarebracketsbig{\parenthesesbig{\frac{\bvltn{\phi}_{r_t(l_t)}^\T \bv{z} }{\norm{\bvltn{\phi}_{r_t(l_t)}}}  }^2}
	{\leq}
	\tilde{\gamma} \xi_{\min}^{-2} \expctn [ \norm{\bv{z}}^2 ]
	\end{equation}
	where $\tilde{\gamma} = \frac{\gamma}{1-\gamma}>1$. The rest of the lemmas, theorems, corollaries follow from these new results.

	\section{Numerical Results} \label{sctn_Experiments}

	In this section, we use simulation examples to evaluate the theoretical results as well as the performance of our algorithm.

	\subsection{Evaluation of Asymptotic Convergence Bounds}

	Since it is not convenient to visualize the convergence bounds for SARX system with multiple subsystems, and \eqref{eqn_2ndConvergenceConstant_Upper} and \eqref{eqn_2ndConvergenceConstant_Lower} give tighter performance than \eqref{eqn_untitle42} and \eqref{eqn_untitle43}, we will evaluate the lower and upper asymptotic convergence bounds in \eqref{eqn_2ndConvergenceConstant_Upper} and \eqref{eqn_2ndConvergenceConstant_Lower} on single ARX system by comparing the bounds with the actual convergence behavior.

	Consider a specific system
	\begin{equation}
	y_t = 0.7 y_{t-1} - 0.12 y_{t-2} + u_{t-1} + n_t
	\end{equation}
	where $n_t \sim \N(0, \sigma_n^2)$, $\sigma_n=10^{-4}$, $u_t \sim \N(0, 1)$. According to Section \ref{subsctn_PolesvsConvry}, the correlation matrix is given by
	\begin{equation}
	R=
	\begin{bmatrix}
	1.67 & 1.04 & 0\\
	1.04 & 1.67 & 0\\
	0 & 0 & 1
	\end{bmatrix}
	\end{equation}
	, and its minimum and maximum eigenvalues are $\lambda_{\min}=0.63$ and $\lambda_{\max}=2.71$.

	Since it's difficult to have exact knowledge of $\kappa_{\max}$ and $\xi_{\min}$, we will use the approximate values defined in Section \ref{subsctn_PolesvsConvry}, i.e. $\kappa_{\max} = \sqrt{(n{-}1) \lambda_{\max} / \lambda_{\min} {+} 1 }$ and $\xi_{\min} = \sqrt{(n{-}1) \lambda_{\min} / \lambda_{\max} {+} 1 }$. And similarly, we could have $F_{\max} = \sqrt{n N_R \lambda_{\max}}$ and $F_{\min}= \sqrt{n N_R \lambda_{\min}}$.

	We set $N_R = 10$ and simulation time horizon $T=1000$. To evaluate the expectation $\expctn \squarebracketsbig{\norm{\bvltn{\epsilon}_{i, t}}^2}$ in \eqref{eqn_2ndConvergenceConstant_Upper} and \eqref{eqn_2ndConvergenceConstant_Lower}, we run the algorithm $50$ times with different realizations of input $u_t$, noise $n_t$, and random data selection in Line \ref{algline_PickColumn} of Algorithm \ref{alg_main}, and take the average of estimation errors as the expectation.

	\begin{figure}[h!]
		\begin{center}
			\includegraphics[width=7.5cm]{BoundTest}
			\caption{\small Evaluation of convergence bounds}
			\label{fig_BoundsTest}
		\end{center}
	\end{figure}

	The simulation results are given in Fig. \ref{fig_BoundsTest}. We can see the estimation error $\expctn \squarebracketsbig{\norm{\bvltn{\epsilon}_{i, t}}^2}$ can be successfully bounded by the upper and lower bound in \eqref{eqn_2ndConvergenceConstant_Upper} and \eqref{eqn_2ndConvergenceConstant_Lower}.

	\subsection{Robust Behavior of our Algorithm}

	\subsubsection{Single Realization Experiment}
	First we evaluate our algorithm and compare it with the OBE algorithm in \cite{goudjil2016convergence} using SARX system given below
	\begin{itemize}
		\item Subsystem 1: $y_t = 0.2 y_{t-1} + 0.24 y_{t-2} + 2 u_{t-1} + n_t$
		\item Subsystem 2: $y_t = 0.7 y_{t-1} - 0.12 y_{t-2} + 1 u_{t-1} + n_t$
		\item Subsystem 3: $y_t = -1.4 y_{t-1} -0.53 y_{t-2} + 1 u_{t-1} + n_t$
		\item Subsystem 4: $y_t = 1.7 y_{t-1} -0.72 y_{t-2} + 0.5 u_{t-1} + n_t$
	\end{itemize}
	where $u_t {\sim} \N(0,1)$. $n_t$ follows $\N(0,\sigma_n^2)$ truncated to region $[-3\sigma_n, 3\sigma_n ]$ where $\sigma_n {=} 10^{-4}$, so noise is bounded with $n_{\max} {=} 3\sigma_n$.

	To fully evaluate the performance, we consider 3 different switching patterns of subsystems: (i) \emph{Slow Switching} (SS): subsystem 1 dominates from 1 to 500, subsystem 2 dominates from 501 to 1000, subsystem 3 dominates from 1001 to 1500, and subsystem 4 dominates from 1501 to 2000. (ii) \emph{Minimum Dwell Time} (MD): each subsystem dominates 30 time steps, and then the time it takes to switch to a new subsystem is a random variable following the geometric distribution with parameter $1/16$. When the subsystem switches, all subsystems are equally likely to be switched to, and after the switching, this process restarts again. (iii) \emph{Fast Switching} (FS): at every time step, every subsystem dominates with equal probabilities.

	In our algorithm, we set $N_R {=} 3, N_C{=}20, \alpha{=}4, \beta{=}3, \nu {=} 10^{-4}$,  and simulation time horizon $T=2000$. The candidates are initialized with standard multivariate Gaussian distribution. After the algorithms completes all $T$ time steps, we first relabel the candidates with a bijective mapping $h(\cdot): [m]\rightarrow [m]$ such that $\sum_{i \in [m]} \norm{\bv{w}_i - \wht{\bv{w}}_{h(i),T}}$ is minimized.
    \emph{In the following, the candidates are referring to the relabeled candidates.}

	We compute all the estimation errors, i.e. $\bvltn{\epsilon}_{i,t} = \bv{w}_i - \wht{\bv{w}}_{i, t}, \forall i, t$, which measure the distance between candidate $i$ and subsystem $i$ during the algorithm.

	Fig. \ref{fig_ConvrgComp} depicts the simulation results. The dots in the plots represent each $\norm{\bvltn{\epsilon}_{\wht{\sysidx}_t,t}}, \forall t$, which means there is only one dot plotted for one time step. Different colors correspond to different candidates and corresponding true subsystems. For example, if there exists a blue dot at time $t=1400$, this means we assign data generated at time $t=1400$ to candidate 3, and current error between candidate 3 and subsystem 3 is given by the y-axis value of the dot.

	\begin{figure}[h!]
		\centering
		\captionsetup[subfloat]{captionskip=0pt, farskip=0pt}
		\subfloat[]{\includegraphics[width=2in]{SS_Exp2_OBE_bound_delta_4.pdf} \label{1}} \
		\subfloat[]{\includegraphics[width=2in]{MD_Exp1_OBE_bound_delta_4.pdf} \label{3}} \
		\subfloat[]{\includegraphics[width=2in]{FS_Exp1_OBE_bound_delta_4.pdf} \label{5}} \\
		\subfloat[]{\includegraphics[width=2in]{SS_Exp2_kRandNLMSwConvrgFinFin.pdf} \label{2}} \
		\subfloat[]{\includegraphics[width=2in]{MD_Exp1_kRandNLMSwConvrgFinFin.pdf} \label{4}} \
		\subfloat[]{\includegraphics[width=2in]{FS_Exp1_kRandNLMSwConvrgFinFin.pdf} \label{6}}
		\caption{\small Estimation errors of OBE algorithm and our algorithm: (a) SS-OBE; (b) MD-OBE; (c) FS-OBE; (d) SS-Ours; (e) MD-OBE; (f) FS-Ours}
		\label{fig_ConvrgComp}
	\end{figure}

	From the plots, we see that our algorithm converges more quickly than the OBE algorithm. Since none of the colors have a sharp increase in error, we could claim the phenomenon described in Section \ref{sctn_Drawback} is effectively avoided in these realizations. For the OBE algorithm, the performance is obviously worse: in the FS case, none of the candidates even converge, and the algorithm even stops halfway due to numerical instability. In the plots for SS-OBE, we can see the undesired phenomenon described in Section \ref{sctn_Drawback}: candidate 2 has converged to the vicinity of subsystem 2 from $t=501$ to $t=1000$, but after time $t\geq 1500$, the error goes large again. This is because we are assigning data generated by subsystem 4 to candidate 2, making candidate 2 move towards subsystem 4. From these plots, we see that our algorithm outperforms OBE algorithm for all switching patterns.

	\subsubsection{Multiple Realizations Experiment}
	Since a single realization cannot comprehensively evaluates the performance, we further compare our algorithm with the OBE algorithm using multiple realizations. Specifically, we consider 9 experiment setups, given by all the combinations of switching patterns \{SS, MD, FS\} and noise level $\sigma_n \in \curlybrackets{10^{-1}, 10^{-2},  10^{-3}}$, and for each of the experiment setup, we run $M{=}100$ realizations. Each subsystem parameters are generated randomly in each realization: we first sample 2 real poles on $[-1, 1]$ uniformly, and then compute the parameters from the sampled poles. The rest of the setups, e.g. number and orders of subsystems, algorithm parameters, etc., follow the previous single realization experiment.

	For realization $i$, we define the two metrics: $\text{FE}(i) = \frac{1}{m} \sum_{j=1}^{m} \norm{\bvltn{\epsilon}_{j, T}}$ and $\text{CER}(i) = \frac{1}{T} \sum_{t=1}^T \indicator \{\sysidx_t \neq \wht{\sysidx}_{t} \}$. FE measures the final estimation error and CER is the classification error rate. Table \ref{tbl_table} lists the average FE and CER values over all 100 realizations. We could see, our algorithm exhibits better performance in each setup.
	\begin{table}[]
		\centering
		\caption{\small Results for Multiple Realizations Experiment} \label{tbl_table}
		\renewcommand{\arraystretch}{1.3}
		\begin{tabular}{|l|l|l|l|l|}
			\hline
			& Ours & OBE & Ours & OBE \\ \hline
			& FE & FE & CER & CER\\ \hline
			SS, $10^{-1}$ 	& $8.4{\mathsf{x}\mskip1mu} 	% The product x sign used in dimension, e.g. R^{m \x n}10^{-1}$ & $8.7{\mathsf{x}\mskip1mu} 	% The product x sign used in dimension, e.g. R^{m \x n}10^{-1}$ 	& $56.3\%$ 	& $59.1\%$ \\ \hline
			SS, $10^{-2}$	& $2.8{\mathsf{x}\mskip1mu} 	% The product x sign used in dimension, e.g. R^{m \x n}10^{-2}$ & $8.2{\mathsf{x}\mskip1mu} 	% The product x sign used in dimension, e.g. R^{m \x n}10^{-1}$ 	& $22.1\%$ 	& $55.5\%$ \\ \hline
			SS, $10^{-3}$	& $9.0{\mathsf{x}\mskip1mu} 	% The product x sign used in dimension, e.g. R^{m \x n}10^{-2}$ & $8.2{\mathsf{x}\mskip1mu} 	% The product x sign used in dimension, e.g. R^{m \x n}10^{-1}$ 	& $8.35\%$  & $56.4\%$ \\ \hline
			MD, $10^{-1}$	& $4.3{\mathsf{x}\mskip1mu} 	% The product x sign used in dimension, e.g. R^{m \x n}10^{-1}$ & $5.2{\mathsf{x}\mskip1mu} 	% The product x sign used in dimension, e.g. R^{m \x n}10^{-1}$ 	& $47.5\%$ 	& $50.3\%$ \\ \hline
			MD, $10^{-2}$	& $4.0{\mathsf{x}\mskip1mu} 	% The product x sign used in dimension, e.g. R^{m \x n}10^{-2}$ & $2.8{\mathsf{x}\mskip1mu} 	% The product x sign used in dimension, e.g. R^{m \x n}10^{-1}$ 	& $11.3\%$	& $31.3\%$ \\ \hline
			MD, $10^{-3}$	& $9.4{\mathsf{x}\mskip1mu} 	% The product x sign used in dimension, e.g. R^{m \x n}10^{-3}$ & $2.4{\mathsf{x}\mskip1mu} 	% The product x sign used in dimension, e.g. R^{m \x n}10^{-1}$ 	& $4.91\%$ 	& $28.8\%$ \\ \hline
			FS, $10^{-1}$	& $2.6{\mathsf{x}\mskip1mu} 	% The product x sign used in dimension, e.g. R^{m \x n}10^{-1}$ & $6.8{\mathsf{x}\mskip1mu} 	% The product x sign used in dimension, e.g. R^{m \x n}10^{-1}$ 	& $39.3\%$ 	& $53.9\%$ \\ \hline
			FS, $10^{-2}$	& $6.0{\mathsf{x}\mskip1mu} 	% The product x sign used in dimension, e.g. R^{m \x n}10^{-2}$ & $1.5{\mathsf{x}\mskip1mu} 	% The product x sign used in dimension, e.g. R^{m \x n}10^{-1}$ 	& $11.7\%$ 	& $22.1\%$ \\ \hline
			FS, $10^{-3}$	& $5.8{\mathsf{x}\mskip1mu} 	% The product x sign used in dimension, e.g. R^{m \x n}10^{-2}$ & $1.8{\mathsf{x}\mskip1mu} 	% The product x sign used in dimension, e.g. R^{m \x n}10^{-1}$ 	& $8.93\%$ 	& $18.9\%$ \\ \hline
		\end{tabular}
		\label{my-label}
	\end{table}

	\section{Conclusions}
	In this paper, we introduced a robust algorithm to solve online switched system identification problem. Our algorithm follows the conventional two-step framework, but the modified assignment criterion leads to a more robust assignment process. After we assign the data to some candidate, we update the candidate estimate based on the idea of randomized Kaczmarz algorithm. We showed partial and local convergence results. The partial convergence result is: assuming there is no misassignment, then the estimation error converges geometrically to some quantity related to noise variance in the expectation square sense. The local convergence result is: assuming all candidates have good enough initialization, with some probability, it can be guaranteed that no misassignment will be made, and the estimation error will converge geometrically as in the partial result. Numerical results verify the asymptotic convergence bounds we developed, and shows the efficiency of our proposed algorithm in comparison with the existing OBE algorithm.

    For future work, there are several aspects that we would focus on.
    \begin{itemize}
    	\item As for theories, we would seek to relax Assumption \ref{asmp_SNRUpperBound}, and analyze the local convergence in a more general setting. Also, we could relax even further to analyze the global convergence without good initialization requirement.
    	\item We plan to apply our algorithm to advanced and real world examples to further evaluate its applicability.
    	\item Our current algorithm finds the upper bound of estimation error by searching all the cube vertices $V$ defined in Algorithm \ref{alg_computeUpperbound}, which leads to heavy computation burden when $N_C$ is large. In the future, we would seek a way to estimate the error more efficiently without sacrificing theoretical guarantees.
    	\item Since so far we don't consider the case in which we have control over the system input, another interesting extension would be designing certain input, possibly closed-loop or open loop but with certain distribution, given which the system parameters can be learned faster.
    \end{itemize}

	\section*{Acknowledgement}
		The authors thank Yan Shuo Tan for suggesting the use of super-martingale theory, which proved to be crucial in the local convergence analysis.

	\bibliographystyle{abbrv}

redpajama_set_name: RedPajamaArXiv


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Mr. Atherton is also knowledgeable about a wide variety of other issues that affect his business clients, including employment law, taxation, securities and intellectual property. No matter what his clients needs are, however, Mr. Atherton is a team player who recommends and works well with the myriad of other experts (i.e. accountants, insurance companies, engineers and legal specialists) that businesses often need to accomplish their goals.


How Can Epidural Steroid Injections For Back Pain Help Me?
Home/Inside Pain Blog/How Can Epidural Steroid Injections For Back Pain Help Me?
Epidural steroid injections for back pain are one of the most useful and research-backed methods for helping people find pain relief. If you suffer from back pain, read on to learn more about how this interventional pain management option could help you.
Pain is a relatable experience for everyone and back pain is the number one culprit around the world. Most people will encounter back pain at least once throughout their lives. It is the leading cause of missed days at work as well as disabilities in the United States. 50% of people in the U.S. report having back pain symptoms each year and over $50 billion is spent annually just diagnosing and identifying it.
It is no wonder that lots of time and energy has been dedicated to researching this topic and finding new and creative ways to relieve this kind of pain. Medications are one of the top treatment options for back pain, but not all medications are delivered orally. The epidural steroid injection is the application of medications, via an image-guided needle, directly to the painful area. Epidural steroid injections for back pain are useful for both the management and diagnosis of back pain.
Watch the video to learn more about the basics of back pain science.
Epidural injections have been around in some form for over 100 years, but today's epidural steroid injections are much safer. The earliest epidural injections were performed in 1885 by neurologist James Leonard Corning. At that time, the injections contained cocaine. Sixteen years later, these injections were used specifically for pain relief and were being performed by at least two other physicians. Steroids were first described as being included in these injections in 1953.
Today the main component of an epidural steroid injection is the steroid, since steroids are believed to reduce painful inflammation. Cocaine is no longer used at all during epidural steroid injections obviously, but other, safer pain-relieving analgesic medications (such as lidocaine) are sometimes included. The inclusion of analgesics like lidocaine may allow for immediate pain relief, before the steroids begin to reduce painful inflammation.
Inflammation leads to painful stimuli in the body. Not only can the inflammation itself be painful, but in some conditions, such as sciatica, the inflammation can compress and irritate nearby nerves. Therefore, by reducing inflammation, epidural steroid injections can significantly reduce pain.
Epidural steroid injections are minimally-invasive injections that are administered into the epidural fluid surrounding the spinal cord. The pain doctor will guide a very thin needle into this space, injecting a steroid such as triamcinolone acetonide, dexamethasone, or methylprednisolone as an anti-inflammatory as well as a local anesthetic such as lidocaine or the longer-lasting bupivicaine to numb the area.
Additionally, epidural steroid injections may provide pain relief in another way. It's believed that the injection of fluid, whether or not it affects inflammation, can promote blood flow through the vessels of the epidural space. This may inhibit pain signals, cleanse damaged nerve cells, and wash away inflammatory molecules.
Epidural steroid injections can also be utilized in the diagnosis of different back pain conditions, too. For instance, if applying an epidural steroid injection to a specific area provides relief, it can confirm the location of the pain's source. If applying an epidural steroid injection to a specific area fails to provide relief, it suggests that the source of the pain is located elsewhere.
These injections may also provide a viable potential pain treatment for individuals who are unable to take oral pain medications. Certain health conditions or drug interactions can make oral pain medications a poor choice for pain management, so epidural steroid injections may be preferred.
To learn more about how epidural steroid injections work, and to watch a live procedure take place, watch the video below.
Epidural steroid injections can provide relief from several different back pain conditions. Steroid injections can provide a great deal of relief and assist in identifying the source of back pain. This treatment is nowhere near as invasive as surgery, but is a treatment option that can have many positive effects. It has been known to greatly increase a patient's quality of life and improve many chronic back pain conditions. The typical treatment is an outpatient procedure that usually takes no longer than 15 minutes. Even better, the reduction of pain is almost immediate.
Sciatica, facet joint arthritis, and lumbar disc herniation are all associated with increased inflammation and can be well-treated with epidural steroid injections. Steroids help treat inflammation that is already present while preventing further inflammation. When the body generates extra white blood cells as an immune response to infection, viruses, or bacteria, inflammation can also result. Epidural steroid injections for back pain help dampen that response so that healing can begin.
Conditions where the vertebrae are compressing nerves can also be helped by epidural steroid injections. Degenerative disc disease and spinal stenosis both cause back pain, the former by nerve impingement and the latter due to a narrowing of the spinal column that compresses the spinal cord. The injections do not treat the condition but may help with pain and inflammation caused by this constant pressure.
Interlaminar epidural steroid injections are perhaps the most common, largely because they are considered more specific in their aim. During this type of injection, the needle is placed between two vertebrae. The medications are then delivered directly to the nerves on both sides of the spine. However, interlaminar epidural steroid injections are associated with a higher risk of dural puncture, or puncture of the dural membrane.
Transforaminal epidural steroid injections involve the insertion of a needle into the epidural space above a nerve root, targeting just one side of the spine. This type of injection may require more imaging guidance, but it is regarded as having a lower risk of dural puncture.
Caudal epidural steroid injections are applied into the epidural space of the sacral region, which is at the end of the spine. Larger amounts of medication can be delivered with this type of injection. Additionally, caudal injections have a relatively low risk of dural puncture. However, caudal epidural steroid injections may be less accurate than interlaminar or transforaminal injections.
Each type of epidural steroid injection is best suited for certain back pain conditions. For instance, caudal injections may be indicated for people with degenerative lumbar spinal stenosis. Transforaminal injections might be a good choice for those with sciatica, potentially even working well enough to provide an alternative to surgery. For some conditions, such as unilateral lumbosacral radicular pain (ULRP), there may be no long-term difference in pain relief between transforaminal and interlaminar injections. A trained physician or pain specialist will be able to assess each case individually and decide which type of epidural spinal injection is the best choice.
There are a few other types of steroid injections that can be used for back pain. These include facet joint injections and sacroiliac joint injections.
This method is very similar to the epidural injection technique. The main difference is that the needle is inserted into the facet joint, which is the moveable piece of the spine that connects one vertebra to another.
This joint is located between the sacrum, the bone that is located at the base of the spine, and the ilium, the largest, uppermost part of the pelvis commonly known as the hip bone. The sacroiliac joint acts as a shock absorber for the upper body and transmits force throughout the hips and legs. Pain in the legs, hips, and lower back can all be caused by inflammation or arthritis in this joint.
Many conditions that present with chronic pain can be treated using traditional oral pain medication rather than steroid injections. Furthermore, your doctor will perform an in-depth history and analysis of current symptoms to identify the characteristics that can affect the success of this kind of treatment. It is standard practice to rule out other pain conditions before moving to epidural steroid injections for back pain.
Here are a few examples of back pain conditions steroid injections are commonly recommended for.
This condition occurs when the jelly-like interior of the intervertebral disc ruptures through the outer layer through a weakened point. This leads to the irritation of nearby spinal nerves that can lead to back pain, numbness, and weakness in the extremities.
Back pain can be caused by certain infections of the spine. There is a much higher probability of this occurring when a patient is recovering from spinal surgery or has a history of abusing drugs. Infection should always be considered as a possibility for back pain when a fever is present.
This is a common injury that occurs in the neck region when the tissues in the cervical area are damaged by forceful movements well outside the normal range of motion. Motor vehicle collisions and other events such as extreme impacts in football, riding a roller coaster, and falling off of a bicycle are the major causes of this disorder. Symptoms for whiplash include headaches, shoulder, neck, and back pain, fatigue, dizziness, and visual disturbances.
This condition occurs when the spinal canal narrows which can lead to compression of the spinal cord and nearby nerves. In some instances, no symptoms appear immediately and usually develop and worsen over time, but the most common symptoms include neck pain and back pain.
Epidural steroid injections for back pain are usually only considered after medication and physical therapy have been exhausted, but before major surgery is considered. The main factor that sets injections apart from oral medication is that the medicine is injected directly into the problem area, targeting the specific muscle and nerves that are in pain. They tend to vary greatly between patients: some provide instantaneous results that can last up to a whole year and others may only experience a brief reprieve from the pain for about one week.
Epidural steroid injections can be administered up to three times per year, but there is no consensus in the medical community about the frequency. Some doctors space injections out evenly over the year. Others advocate for all three injections to be given over a series, spaced at two to four week intervals. Most agree, however, that if no pain relief is felt at the first injection, others may not be beneficial.
Epidural steroid injections can take place in a clinic, as an outpatient in a hospital setting, or at a doctor's office. In any setting the procedure is the same, and generally takes between 15 and 30 minutes.
You'll change into a gown so that the doctor will have access to the injection site along the spine.
You'll lie down on an X-ray table, placing a small pillow under the abdomen to slightly raise your spine. If this position is painful, you may be given the option to lie curled on your side or to sit up.
Your back will be cleaned, sterilized, and numbed with a local anesthetic.
Using fluoroscopy (X-ray) as a guide, your doctor will insert a thin needle into the epidural space. Correct placement of the needle is crucial to properly deliver medication and to prevent side effects. Fluoroscopy helps the doctor to be precise and accurate.
After the needle is in place, contrast dye is injected to ensure proper placement and medication is administered, ranging in amounts from three to ten mL. While not painful, some patients report feeling pressure as the medication is injected.
When the procedure is complete, you will be monitored for 15 minutes or so before being discharged.
You may experience some soreness and swelling at the injection site, but this can generally be addressed with an ice pack. Rest is usually prescribed for the remainder of the day, but you can expect to resume normal activity the day after a epidural steroid injection. You may notice a slight increase in back pain for a couple days, but with continued movement that pain should decrease exponentially as the medication works on the affected area.
Physicians have lots of tools at their disposal to make certain that these injections are as safe as possible. Imaging techniques, such as fluoroscopy, are used to provide visualization of the area; contrast dye might be utilized, as well. Proper visualization allows the physician to make sure the needle is placed correctly. Still, it is possible for mistakes to occur, such as a dural membrane puncture.
The medication in an epidural steroid injection can potentially cause some side effects. Most are relatively mild, like headaches or hot flashes. Nausea or diarrhea are possible, as well. Also, it's possible for a temporary increase in blood sugar to occur. This may make epidural steroid injections a less attractive pain management option for diabetic people. It's also possible that epidural steroid injections might affect bone density if used too often. This can lead to issues like bone loss or an increased risk of fracture.
Have you had any experiences with epidural steroid injections for back pain? Did you find any pain relief from the procedure? If you're interested in learning more about these procedures, click the button below to talk to one of our pain doctors about this treatment option.
Thanks for helping me understand that epidural steroid injections can relieve back pain because it's a combination of anti-inflammatory and also a local anesthetic. I've been suffering from chronic back pain and inflammation for the past months, and I think epidural steroid injection would help. I'll look for a good clinic tomorrow to know more about interventional injections services. Thanks for the advice!


On, S. L. W. and Podivinsky, E. and Thom, K. (2007) New, portable and cost-effective AFLP genotyping system for the Campylobacteraceae. In: 14th International Workshop on Campylobacter, Helicobacter and Related Organisms, 2-5 September, 2007, Rotterdam, Holland.
This list was generated on Thu Apr 25 20:56:21 2019 NZST.


The included remote is ultra slim and totally controls the Wave Radio IV from the volume to the pre-set stations.
While combining simplicity and affordability, the Wave offers a sleek design and daily color customization options. The Movo Wave app is available for free download on the App Store and Google Play.
Wave is becoming a major player in the electric bike market.
The Balticus Watches include the Big Wave and the Grey Seal. The Big Wave takes inspiration from the Great Wave Off Kanagawa, the iconic Japanese piece of art.
The Powerbreather Wave edition comes with different types of tips so you can even use the device in open, choppy water.


10-Round Detachable rotary magazine. Extended magazine release. Push-button manual safety. Combination scope base adapter. Hammer-forged barrel. Polymer trigger housing. Aluminum receiver. Sights. Contoured buttpad and barrel band.


Try your first scuba dive on the outer Great Barrier Reef onboard Cairn's newest, fastest and most luxurious vessel for the best value introductory dive package in Queensland.
That's right, if you have never dived before, and have always wished to experience first-hand the beauty of the underwater world you can arrange an Introductory dive with one of our friendly and professional Instructors.
Through the PADI Discover Scuba Diving program you will be taught all the basic skills required to dive down to a depth of up to 12 metres while accompanied by one of our experienced PADI qualified instructors.
Divers are broken up into small groups of 4 people, are always guided and the experience lasts up to 20 minutes giving you time to feel comfortable and begin to enjoy your first dive on the Great Barrier Reef.
Evolution's massive back deck, combined with specially designed fold out 'Wings' which to provide more space, comfort and safety for our guests as they enter the water.
Lycra suits are included free of charge. Wet Suits are available for hire at $10 per person.


text:
Museum could become landmark
by John Ruch • December 9, 2011 • 0 Comments
LMA—The Isabella Stewart Gardner Museum could become an official Boston landmark just in time for its grand reopening in January. That would be exactly four years after Mission Hill activists sought the landmark status to block the museum's massive expansion.
The art museum at 280 The Fenway is closed until Jan. 19, when it will open to unveil its new, modern wing. The plans for the expansion were once controversial because of little public notice and the demolition of a historic carriage house to make way for the wing. The addition also required moving artwork, which was forbidden by the will of Gardner, who founded the museum in what was once her home. The unadvertised art move was discovered by the Gazette.
In 2008, Friends of Historic Mission Hill filed a petition to declare the entire museum complex a city historic landmark. The status would require city approval of exterior building changes. The Boston Landmarks Commission (BLC) accepted the petition, but tabled a vote for years. Instead, the BLC used the petition as leverage to get ongoing review of the expansion's design.
The museum opposed the landmarking, with Director Anne Hawley once saying it would "jeopardize" the museum's existence. But the museum participated in the design review process.
Now that the design is done, the BLC is considering an actual vote on the landmarking. It considered the issue at its Nov. 8 meeting, but requested more staff reports, according to the BLC. The museum landmarking likely will return to the BLC agenda next month.
A museum spokesperson did not respond to Gazette questions for this article.
The century-old museum was designed to look like a Venetian palace and still contains Gardner's furnishings. The glassy new building will house modern art, performance space and other features the older building lacked.
The museum is known for its eccentric yet high-quality collection. But it lost several masterpieces, including works by Vermeer and Rembrandt, in an infamous 1990 robbery. The works remain missing.
← Senior housing project nets grant
Health careers school to move here →

redpajama_set_name: RedPajamaCommonCrawl


text:
Misbah to lead Pakistan in Australia A home series by AFP
Player: Misbah-ul-Haq
Event: Australia A in Pakistan 2005/06
DateLine: 5th September 2005
Test discard Misbah-ul-Haq will lead a strong Pakistan A team against Australia A who arrive in Pakistan this week for a series of one-day and four-day matches, cricket officials said Monday.
"We hope that a tough Australia A series will harden our players for the forthcoming series against England," Pakistan Cricket Board's chief selector Wasim Bari told AFP.
Brad Haddin, understudy to Australian wicket-keeper Adam Gilchrist, will lead a 14-man Australia A squad for the series of three one-day and two four-day matches.
The Australia A due in Pakistan on Thursday will open the tour with the first four-day match at Rawalpindi from September 11-14.
The second four-day match will also be played at Rawalpindi September 17-20 while the three one-dayers are scheduled for September 23, 25 and 27 - all in Lahore.
The Australia A team will include four members currently with the Australian Ashes squad in England -- Haddin, batsman Brad Hodges, pacemen Shaun Tait and Stuart Clark.
Cricket Australia (CA) have named four reserves should any of the four be required to play in the fifth Ashes Test at The Oval starting from September 8.
Bari said the Australia A team consisted of quality players and would test the Pakistani youngsters to the hilt. "We are taking Australia A team very seriously and the series would give us a chance to judge our fringe players who could be played in the tougher assignments of the national team," he said.
Pakistan will host England and then India in their hectic international schedule spread from November to February next year.
Both England and India will play three Tests and five one-day matches each.
Fit-again pacers Mohammad Sami and Umar Gul are part of the 15-man Pakistan A squad while injury-prone pacer Shoaib Ahktar is likely to be tested in the second four-day match for which a final squad will be announced later.
Pakistan A squad: Misbah-ul-Haq (captain), Yasir Hameed, Salman Butt, Shoaib Malik, Bazid Khan, Asim Kamal, Zulqarnain Haider, Faisal Iqbal, Yasir Arafat, Mohammad Sami, Mohammad Asif, Umar Gul, Ashar Zaidi, Mohammad Irshad, Imran Tahir
(Article: Copyright © 2005 AFP)


text:
Jesús de Polanco (* 7. November 1929 in Madrid, Spanien; † 21. Juli 2007 ebenda) war ein spanischer Medienunternehmer. Im Jahr 2005 wurde er auf Platz 210 der reichsten Menschen weltweit und 2006 auf Platz 258 geführt.

Leben
Polanco stammt aus kleinen Verhältnissen und finanzierte sein Jurastudium als Buchverkäufer. Ende der 1950er Jahre gründete den Buchverlag Santillana, durch Revinvestitionen war dies der Nucleus der Prisa-Gruppe, an der er zuletzt mit 64 Prozent beteiligt war.

In Spanien gründete er kurz nach dem Tod des Diktators Francos im Mai 1976 die seither größte und international bekannteste Tageszeitung Spaniens, El País, die traditionell der sozialistischen Partei PSOE nahestand. Aufgrund seines politischen Einflusses trug er den Spitznamen Jesús del Gran Poder, und seine Dominanz in den spanischen Medien wurde von politisch rechter Seite teilweise heftig kritisiert. Seine Gegner beschuldigten ihn, ein Monopol zu nutzen, um tendenziöse Berichterstattung zu betreiben. Der Prisa wurde unter anderem im Jahr 2004 vorgeworfen, mittels ihrer Berichterstattung über die Madrider Zuganschläge die öffentliche Meinung einseitig informiert und so die Spanische Parlamentswahl 2004 zuungunsten des Partido Popular beeinflusst zu haben.
Von nicht nationalistischer Seite und international wurde jedoch insbesondere Polancos Einfluss bei der Festigung der spanischen Demokratie hervorgehoben.

Unter Polancos Führung expandierte Prisa auf den lateinamerikanischen Markt, engagierte sich unter seiner Leitung aber auch in Portugal und bei der Tageszeitung Le Monde in Frankreich.

Polanco starb Mitte Juli 2007 an Komplikationen in Verbindung mit einer Arthritis.

Weblinks
  Nachruf bei REUTERS Deutschland
 Prisa-Gruppe (englisch)

Einzelnachweise

Medienunternehmer
Verleger (Spanien)
Spanier
Geboren 1929
Gestorben 2007
Mann

redpajama_set_name: RedPajamaWikipedia


On Thursday 5th December 2013, more than 300 delegates met in Stratford upon Avon for the National Insulation Association's (NIA) Annual Conference.
The conference was opened by the NIA's Chair Tony Hardiman MBE, and an introductory presentation was delivered by Neil Marshall, Chief Executive of the NIA. Essex Insulation's Sales & Marketing Director, Mike Easdon, who chairs the NIA Market/Business Development Group, set out NIA's plans for 2014 to support its members in maximising business opportunities.
On behalf of the NIA, Mike is actively involved in a DECC review panel and has delivered a number of Green Deal and ECO presentations at conferences throughout the UK. Essex Insulation is a leading participant within a number of trade bodies and regularly liaises with Government and other key stakeholders on current energy efficiency legislation and schemes.
The proposed changes to the ECO scheme were the hot topic of discussion at this year's conference. The day also included a wide range of high profile external speakers including Dave Sowden, CEO of the Micropower Council, Claire Curtiss-Thomas, CEO of the BBA, Paul King CEO of the UKGBC, Lawrence Slade, Chief Operating Officer of Energy UK and Richard Lambert, CEO of the National Landlords Association.


text:
import {Component, View, bootstrap, Attribute} from 'angular2/angular2';
import {RouteConfig, ROUTER_DIRECTIVES, ROUTER_BINDINGS} from 'angular2/router';
import {Http, HTTP_BINDINGS} from 'angular2/http';

/* Custom Angular2 components */
import {Home}         from './components/home/home';
import {UserDetails}  from './components/user_details/user_details';
import {About}        from './components/about/about';
import {F1Drivers}    from './components/F1Drivers/F1Drivers';
import {Details}      from './components/F1Drivers/Details/details';
import {EventsDemo}   from './components/Events/events';
import {GitHubAPI}    from './services/api/github_api';
import {LoginComp}    from './components/forms/forms';
import UserRepo       from './services/repositories/user_repo'; //here, {} aren't used when export default ...( #http://stackoverflow.com/questions/21117160/what-is-export-default-in-javascript)
/*
*Create a whole component. Requires a @View.
 *  */
@Component({
  selector: 'app',
  /*
  * Specifies a list of objects that will created using dependency injection
  * and will be visible to the components and its view children.
  * NOTE :
  * `bindings` : are available to a component and its children,
  * `viewBindings` : are only available to a particular view,
  * `lifecycle` : refers to different stages through which a component might go (create, destroy, change detection etc.)
   so you can think of those as callbacks that are executed when "something happens" to a given component / directive
   * */
  viewBindings: [UserRepo, GitHubAPI]
})
@RouteConfig([
    { path: '/', component: Home, as: 'home',  loader: function(obj) { console.log('loader: ' : obj )} },
    { path: '/user_details/:username', component: UserDetails, as: 'user_details' },
    { path: '/about', component: About, as: 'about' },
    { path: '/drivers', component: F1Drivers, as: 'drivers' },
    { path: '/details/:name', component: Details, as: 'details' },
    { path: '/events', component: EventsDemo, as: 'events' },
    //{ path: '/form', component: LoginComp, as: 'form' }
])
@View({
  templateUrl: './app.html?v=<%= VERSION %>',
  directives: [ROUTER_DIRECTIVES]
})
class App {

}

bootstrap(App, [ROUTER_BINDINGS, HTTP_BINDINGS]);
//import {RouteConfig, RouterOutlet, RouterLink, routerInjectables} from 'angular2/router';

redpajama_set_name: RedPajamaGithub


A printable list of past NBA Finals Championship winners and results throughout NBA history. The NBA playoffs close with the NBA Finals which was first played in 1947 and has been held every year. The NBA Finals is played with the Champion of the Eastern Conference and the Champion of the Western Conference in a best-of-seven series. Today, the NBA Finals Champions would receive the Larry O'Brien Champion Trophy.