text:
\section{Introduction}

The concept of networks has been increasingly prevailing for decades, e.g., computer networks, sensor networks or social networks. Regardless of numerous benefits introduced by bridging machines or humans through networks, the interconnect and distributed nature renders networks vulnerable to various kinds of attacks, ranging from physical attacks to internet viruses to groundless rumors through online social networks. This article is concerned with the integrity attacks in sensor networks which are widely embedded in various industrial systems such as smart grid~\cite{MassoudAmin2005} or Supervisory Control And Data Acquisition (SCADA) systems~\cite{boyer2002scada}. During the integrity attack, the adversary can take full control of a subset of sensors and arbitrarily manipulate their measurements. The motivations for launching such an attack in industrial systems may include creating arbitrage opportunities in electricity market, stealing gas or oil without being noticed, posing potential threat to national defense, etc. Since the first SCADA system malware (called Stuxnet) was discovered and extensively investigated~\cite{Chen2010,Fidler2011}, increasing research attention has been paid to resolve the security issues in estimation and control systems~\cite{challengessecurity}.

In this article, we focus on the problem of robust estimation against compromised sensory data in order to mitigate the damage caused by the integrity attack. Robustness for an estimator is urgently needed since quite a number of the commonly used estimators under attack fail to give a reliable estimate and thus lead to poor system performance. For instance, a linear estimator is not robust since one bad measurement is enough to ruin the final estimate. A better estimator may be the geometric median of all measurements~\cite{Lopuhaa1991}. To be concrete, we consider the problem of estimating a vector state $x\in\mathbb R^n$ from measurements collected by $m$ sensors, where the measurements are subject to any random noise. For practical reasons, the spatially distributed sensors cannot be fully guaranteed to be secure. Some of them may be controlled by the attacker and due to the resource limitation the attacker can only attack up to $p<m$ sensors. Without posing any restrictions on the attacker, we assume that the compromised sensory data can be arbitrarily changed.

\emph{Related Work}: A quite similar problem in the context of power systems is bad data detection, which has been studied over the past decades~\cite{Handschin1975,Mili1985}. The method of checking the magnitude of residue is useful for identifying random bad data or outliers but may not work for intentional integrity attacks \cite{henrik2010,Xie2011}. For example, Liu et al.~\cite{liu2009} successfully showed that a stealthy attack changing the state while not being detected is possible. Kim et al.~\cite{Kim2014} studied a so-called framing attack. Under such a attack, the bad data detector is misled to delete those critical measurements, without which the network is unobservable and a convert attack may be launched.

For dynamical systems, detecting malicious components via fault detection and isolation based methods has also been extensively studied,  \cite{fp-ab-fb:09b,Pasqualetti2011,wirelesscontrol,Fawzi2012,chong2015observability}. However, in most of these works, the system is assumed to be noiseless, which greatly favors the failure detector. Pajic et al.~\cite{Pajic2014} improved the work by considering the systems with bounded noise. On the top of sufficient conditions for exact recovery in noiseless case, they showed that the worst error is still bounded even under attack. However, their estimator is based on a combinatorial optimization problem, which in general is computational hard to solve and may not be applicable for large scale systems. In \cite{Mo2010,moscs10security}, the authors use reachability analysis and ellipsoid approximation to characterize all possible biases the adversary can inject to the system.

In the area of statistics, the concept of robust estimators is not new~\cite{Kassam1985,robust2006,robust2009}. The robustness is often measured by breakdown points~\cite{Hampel1971,donoho1983notion} or influence functions~\cite{Hampel1974}. Many existing works studied one or several estimators and discussed the breakdown point properties \cite{Yohai2012,Rousseeuw1992,Hossjer1994,Rousseeuw2012}. However, a unified analysis for most useful estimators is still absent.

Motivated by different behaviors of various estimators under the integrity attacks, we manage to provide a unified robustness analysis framework integrating most commonly used estimators. To reach this goal, we first give a formal definition on the robustness of an estimator. To achieve greater generality, a general convex optimization based estimator is proposed and necessary and sufficient conditions on the robustness of such an estimator is proved. The significance of this work is that the analytical results presented in this manuscript can be used for characterizing and designing a robust estimator in the presence of compromised sensory data.

The rest of the paper is organized as follows. In Section \ref{section:problemsetup} we formulate the robust estimation problem. Our main results on the robustness of a general convex optimization based estimator is presented in Section \ref{section:main}. The concluding remarks are given in Section \ref{section:conlusion}.

\section{Problem Setup}\label{section:problemsetup}
\subsection{System Model}
Assume that $m$ sensors are measuring the state $x$ and the measurement equation for the $i$th sensor is given by
\begin{align}\label{eq:goodsystem}
  z_i = H_i x + w_i,
\end{align}
where $x\in\mathbb R^n$ is the state of interest, $z_i\in \mathbb R^{m_i}$ is the ``true'' measurement collected by the $i$th sensor, and $w_i \in \mathbb R^{m_i}$ is the measurement noise for the $i$th sensor. The measurement matrix $H\triangleq [H_1\T,H_2\T,\ldots,H_i\T]\T\in\mathbb R^{(\sum_i m_i)\times n}$ is assumed to be observable, \ie, $H$ is full column rank. In the presence of attacks, the measurement equation can be written as
\begin{align}\label{eq:badsystem}
  y_i = z_i + a_i = H_i x + w_i + a_i,
\end{align}
where $y_i\in \mathbb R^{m_i}$ is the ``manipulated'' measurement and $a_i\in \mathbb R^{m_i}$ is the attack vector. In other words, the attacker can change the measurement of the $i$th sensor by $a_i$. Denote
\begin{align}
 z&\triangleq[z_1\T,z_2\T,\ldots,z_m\T]\T, &
y&\triangleq[y_1\T,y_2\T,\ldots,y_m\T]\T, \\
w&\triangleq[w_1\T,w_2\T,\ldots,w_m\T]\T,&
 a&\triangleq[a_1\T,a_2\T,\ldots,a_m\T]\T.\nonumber
\end{align}

Denote the index set of all sensors as $\mathcal S\triangleq\{1,2,\ldots,m\}$. For any index set $\mathcal I\subseteq \mathcal S$, define the complement set to be $\mathcal I^c\triangleq \mathcal S\backslash\mathcal I$. In our attack model, we assume that the attacker can only compromise at most $p$ sensors but can arbitrarily choose $a_i$. Formally, a $(p,m)$-sparse attack can be defined as
\begin{definition}[$(p,m)$-sparse attack]
A vector $a$ is called a $(p,m)$-sparse attack if there exists an index set $\mathcal I\subset \mathcal S$, such that:
\begin{enumerate}[(i)]
  \item $\norm{a_i} = 0,~\forall i\in \mathcal I^c ;$
  \item $\card{\mathcal I} \leq p.$
\end{enumerate}
\end{definition}
Define the collection of a possible index set of malicious sensors as
\[
\mathbb C\triangleq\{\mathcal I:\mathcal I\subset\mathcal S,\card{\mathcal I} = p\}.
\]
 The set of all possible $(p,m)$-sparse attacks is denoted as
\[
\mathcal A \triangleq \bigcup_{\mathcal I\in\mathbb C}\{a: \norm{a_i} = 0, i\in \mathcal I^c\}.
\]

The main task of this work is to investigate the generic sufficient and necessary conditions for an estimator to be robust to $(p,m)$-sparse attacks. To this end, we first formally define the robustness of an estimator.

\begin{definition}[Robustness]
An estimator $g:\mathbb R^{\sum_i m_i }\mapsto \mathbb R^n$ which maps the measurements $y$ to a state estimate $\hat x$ is said to be robust to the $(p,m)$-sparse attack if it satisfies the following condition:
\begin{align}
\norm{g(z)-g(z+a)}\leq \mu(z),~\forall a\in\mathcal A,\label{eq:defrobust}
\end{align}
where $\mu:\mathbb R^{\sum_i m_i}\mapsto \mathbb R$ is a real-valued mapping on $z$.
\end{definition}
The robustness implies that the disturbance on the state estimate caused by an arbitrary attack is bounded. A trivial robust estimator is $g(y)=0$ which provides very poor estimate. Therefore, another desirable property for an estimator is translation invariance, which is defined as follows:
\begin{definition}[Translation invariance]
An estimator $g$ is translation invariant if $g(z+Hu)=u+g(z),~\forall u\in\mathbb R^n$.
\end{definition}
\begin{remark}
  Notice that if an estimator is robust and translation invariant, then
  \begin{align*}
    \| g(z) - g(z+a) \|&= \|x + g(w) - x + g(w+a) \| \\
                       &= \|g(w)-g(w+a)\|\leq \mu(w).
  \end{align*}
  Therefore, the maximum bias that can be injected by an adversary is only a function of the noise $w$.
\end{remark}
In the next subsection, we propose a general convex optimization based estimator which is translation invariant.

\subsection{A General Estimator}
A large variety of estimators are developed by the research community to solve the state estimation problem. In order to achieve greater generality, we first propose a general convex optimization based estimator. We then show that many estimators can be rewritten in this general framework.

The estimator that we study in this paper is assumed to have the following form:
\begin{align}
  \hat x = g(y) \triangleq \argmin_{\hat x} \sum_{i\in\mathcal S} f_i(y_i-H_i\hat x),
  \label{eq:general}
\end{align}
where the following properties of function $f_i:\mathbb R^{m_i}\mapsto \mathbb R$ are assumed:
\begin{enumerate}[(i)]
\item $f_i$ is convex.
\item $f_i$ is symmetric, \ie, $f_i(u) = f_i(-u)$.
\item $f_i$ is non-negative and $f_i(0) = 0$.
\end{enumerate}

\begin{remark}\label{remark:1}
  It is easy to check that the estimator $g$ is translation invariant. One can view $y_i-H_i\hat x$ as the residue for the $i$th sensor and $f_i$ as a cost function. The convex constraints on $f_i$ ensures that the minimization problem can be solved in an efficient (possibly also distributed) way. The symmetric assumption on $f_i$ is typically true for many practically used estimator and can actually be relaxed. The last assumption implies that the cost achieves minimum value when the residue is $0$.
\end{remark}
We now investigate several commonly used estimator and show that they can be written as \eqref{eq:general}.
\begin{enumerate}[(a)]
\item Least Square Estimator:
  \begin{align}
    \hat x &=  \argmin_{\hat x} \norm{y-H\hat x}_2^2=  \argmin_{\hat x} \sum_{i\in\mathcal S}\norm{y_i-H_i\hat x}_2^2\nonumber\\
           & = (H\T H)^{-1}H\T y. \label{eq:lsestimator}
  \end{align}

\item Another example is an estimator which minimizes the sum of the $l_1$ norm of the residue, \ie,
  \begin{align}
    \hat x = \argmin_{\hat x} \sum_{i\in\mathcal S}\norm{y_i-H_i\hat x}_1.  \label{eq:medianestimator}
  \end{align}
  In the case that $m_i=n$ and $H_i=\I_n,~\forall i$, the estimate is a vector in which the $i$th entry is the median over the $i$th entries of all measurements $y_i$'s.

\item The following is designed to minimize the sum of the $l_2$ norm of the residue:
  \begin{align}
    \hat x = \argmin_{\hat x} \sum_{i\in\mathcal S}\norm{y_i-H_i\hat x}_2.  \label{eq:geometricmedianestimator}
  \end{align}
  The optimal estimate in the case that $m_i=n$ and $H_i=\I_n,~\forall i$ is the geometric median of all $y_i$'s, which is called an $L_1$ estimator in \cite{Lopuhaa1991}. In other words, $\hat x$ is the point in $\mathbb R^n$ that minimizes the sum of Euclidean distances from $y_i$ to that point.

\item Pajic et al.~\cite{Pajic2014} proposed the following robust estimator in the presence of integrity attack:
  \begin{align*}
    & \mathop{\textit{minimize}}\limits_{\hat x,a,w}&
    & \|w\|^2\\
    &\text{subject to}&
    &y = H\hat x + w + a,\,\|a\|_0\leq q.
  \end{align*}
  However, the minimization problem involves zero-norm, and thus is difficult to solve in general. A commonly adopted approach is to use $L_1$ relaxation to approximate zero-norm, which leads to the following minimization problem:
  \begin{align}
    & \mathop{\textit{minimize}}\limits_{\hat x,a,w}&
    & \|w\|^2+\lambda \|a\|_1\label{eq:optlasso}\\
    &\text{subject to}&
    &y = H\hat x + w + a.\nonumber
  \end{align}
  If we define the following function:
  \begin{align}
    d(u)~\triangleq ~&\mathop{\textit{minimize}}\limits_{a_i}&
    &  \norm{u-a_i}_2^2 + \lambda  \norm{a_i}_1 \label{eq:lasso2}
  \end{align}
  Then one can easily prove that the optimization problem \eqref{eq:optlasso} can be rewritten as
  \begin{align}
    \hat x = \argmin_{\hat x} \sum_{i\in\mathcal S} d(y_i - H_i \hat x).\label{eq:lasso}
  \end{align}
\end{enumerate}

In the next section, we shall present sufficient and necessary conditions for the robustness of the general estimator \eqref{eq:general}. Since \eqref{eq:medianestimator},~\eqref{eq:geometricmedianestimator} and \eqref{eq:lasso} are all special cases of \eqref{eq:general}, we can easily analyze their individual robustness.

\section{Robust Analysis for a General Estimator}\label{section:main}
This section is devoted to the derivation of necessary and sufficient conditions for the robustness of the general estimator.
Denote the compact set $\mathcal U\triangleq\{u\in \mathbb R^n:\norm{u}= 1\}$. Before proceeding to the main results, we need the following lemma.
\begin{lemma}\label{lemma:convex}
  Let $q:\mathbb R\rightarrow \mathbb R$ be a convex function and $q(0) = 0$, then $q(t)/t$ is monotonically non-decreasing on $t\in \mathbb R^+$. Moreover,
  \begin{align}
    q(t+1)-q(t)\geq q(t)/t.
    \label{eq:marginalincrease}
  \end{align}
\end{lemma}
\begin{proof}
  For any $0 < \alpha < 1$, we have
  \begin{align*}
    q(\alpha t) \leq \alpha q(t) + q(0) = \alpha q(t).
  \end{align*}
  Divide both side by $\alpha t$, we can prove that $q(t)/t$ is monotonically non-decreasing. Therefore, $q(t+1)/(t+1)\geq q(t)/t$, which implies \eqref{eq:marginalincrease}.
\end{proof}
As a consequence of Lemma~\ref{lemma:convex}, we know that $f_i(tH_iu)/t$ is monotonically non-decreasing. As a result, there are only two possibilities:
\begin{enumerate}[(i)]
\item $f_i(tH_iu)/t$ is bounded for all $i$ and for all $u$, which implies that the limit $\lim_{t\rightarrow\infty}f_i(tH_iu)/t$ exists.
\item $f_i(tH_iu)/t$ is unbounded for some $i$ and $u$.
\end{enumerate}
The next lemma provides several important properties for the case where $\lim_{t\rightarrow\infty}f_i(tH_iu)/t$ exists, whose proof is reported in the appendix:
\begin{lemma}\label{lemma:1}
  If the following limit is well defined, \ie, finite, for all $u\in \mathbb R^{n}$:
  \begin{align}
    \lim_{t\rightarrow\infty} \frac{f_i(tH_iu)}{t} = C_i(u),
    \label{eq:sublinear}
  \end{align}
  then the following statements are true:
  \begin{enumerate}[(i)]
  \item $C_i(\alpha u)=\card{\alpha} C_i(u)$ and $C_i(u_1+u_2)\leq C_i(u_1)+C_i(u_2)$.
  \item Define the function $h_i(u,v,t):\mathbb R^n\times\mathbb R^{m_i}\times \mathbb R\mapsto\mathbb R$,
    \begin{align}
      h_i(u,v,t) \triangleq \frac{1}{t}\left[f_i(v+tH_iu)-f_i(v)\right].\label{eq:defhfunction}
    \end{align}
    Then the following pointwise limit holds:
    \begin{align}
      \lim_{t\rightarrow\infty}h_i(u,v,t)  = C_i(u).
      \label{eq:deltalimit}
    \end{align}
    Moreover, the convergence is uniform on any compact set of $(u,v)$.
  \item For any $v$ and $u$, we have that
    \begin{align}
      f_i(v+H_iu) - f_i(v) \leq C_i(u).
      \label{eq:deltaw}
    \end{align}
  \end{enumerate}
\end{lemma}
\begin{remark}\label{remark:force}
Intuitively speaking, one can interpret $f_i$ as a potential field and the derivative of $f_i$ as the force generated by sensor $i$ (if it is differentiable). By \eqref{eq:deltaw}, we know that the force from the potential field $f_i$ along the $u$ direction cannot exceed $C_i(u)$ (or $C_i(u)/\|u\|$ to normalize). On the other hand, Equation~\eqref{eq:deltalimit} implies that this bound is achievable.
\end{remark}

We now give the sufficient condition for the robustness of the estimator.
\begin{theorem}[Sufficient condition]
  If the following conditions hold:
  \begin{enumerate}
  \item $C_i(u)$ is well defined for all $u\in \mathbb R^{n}$ and all $i\in\mathcal S$;
  \item the following inequality holds for all non-zero $u$:
    \begin{align}
      \sum_{i\in\mathcal I}C_i(u)<\sum_{i\in\mathcal I^c}C_i(u),~\forall \mathcal I \in\mathbb C,\label{eq:sufficiency}
    \end{align}
  \end{enumerate}
  then the estimator $g$ is robust.
  \label{theorem:sufficient}
\end{theorem}

\begin{proof}
  Our goal is to prove that there exists a $\beta(z)$, such that for any $t\geq \beta(z)$, $\|u\| = 1$, $a\in\mathcal A$, the following inequality holds:
  \begin{align}
    \sum_{i\in\mathcal S}f_i(y_i - H_i\times tu) < \sum_{i\in\mathcal S} f_i(y_i-H_i\times (t+1)u).
    \label{eq:sufficientdiff}
  \end{align}
  As a result, any point $\|\hat x\| \geq \beta(z)+1$ cannot be the solution of the optimization problem since there exists a better point $(\|\hat x\|-1)\hat x/\|\hat x\|$. Therefore, we must have $\|g(y)\|\leq \beta(z) + 1$ and hence the estimator is robust.

  Suppose the set of malicious sensors is $\mathcal I$, to prove \eqref{eq:sufficientdiff}, we will first look at benign sensors. Due to the uniform convergence of $h_i(u,v,t)$ to $C_i(u)$ on $\mathcal U\times\{-z_i\}$ shown in Lemma \ref{lemma:1}, given any $\delta>0$ we can always find a finite constant $N_i$ depending on $\delta$ and $z_i$ such that for all $t \geq N_i(\delta,z_i)$, the following inequality holds:
  \begin{align}
    h_i(-z_i,u,t) = \frac{1}{t}\left[f_i(tH_iu-z_i)-f_i(-z_i)\right]\geq C_i(u) - \delta,
    \label{eq:deltaapprox2}
  \end{align}
  for any $\|u\|=1$. By \eqref{eq:marginalincrease}, we can derive that
 \begin{align}
    f_i((t+1)H_iu-z_i)-f_i(tH_iu-z_i)\geq C_i(u) - \delta.
    \label{eq:deltaapprox3}
  \end{align}
  We define $\beta(z) \triangleq \max_{1\leq i\leq m} N_i(\delta,z_i)$ and fix $\delta$ to be
  \begin{align}
    \delta = \frac{1}{m}\min_{\|u\|=1}\min_{\mathcal I\in\mathbb C}\left(\sum_{i\in\mathcal I^c}C_i(u) - \sum_{i\in\mathcal I}C_i(u)\right).\label{eq:defdelta}
  \end{align}
  Notice that we write $\min_{\|u\|=1}$ instead of $\inf_{\|u\|=1}$ since $C_i(u)$ is continuous and the set $\{u:\|u\|=1\}$ is compact. Hence, the infimum is achievable, which further proves that $\delta > 0$ is strictly positive. Hence, for $i = 1,\dots,m$, if $t > \beta_{\delta}(z)$ we have
  \begin{multline}
    f_i((t+1)H_iu-z_i)-f_i(tH_iu-z_i)\\
    \geq C_i(u) - \delta,\,\forall \|u\|=1.
  \end{multline}
  Since for good sensors, $z_i = y_i$, we know that
  \begin{align}
    \sum_{i\in \mathcal I^c} & \left[f_i((t+1)H_iu-z_i)-f_i(tH_iu-z_i)\right]\nonumber\\
                             &\geq  \sum_{i\in \mathcal I^c} C_i(u) -(m-p) \delta,\,\forall \|u\|=1.
                               \label{eq:suff2}
  \end{align}
  We now consider malicious sensors. By Lemma \ref{lemma:1} (iii), we know that for $i\in\mathcal I$, and any $u$
  \begin{align}
    \sum_{i\in\mathcal I} f_i(y_i - t H_i u)  - \sum_{i\in\mathcal I} f_i(y_i- (t +1)H_i u)\leq \sum_{i\in\mathcal I}C_i(-u).\label{eq:suff1}
  \end{align}
  Hence from \eqref{eq:defdelta},~\eqref{eq:suff1} and~\eqref{eq:suff2}, we know that
  \begin{align*}
    \sum_{i\in\mathcal S} f_i(y_i &- (t+1) H_i u) - \sum_{i\in\mathcal S} f_i(y_i - t H_i u) \\
                                  &\geq \sum_{i\in\mathcal I^c}C_i(u) - \sum_{i\in\mathcal I}C_i(u) - (m-p)\delta > 0,
  \end{align*}
  which proves \eqref{eq:sufficientdiff}.
\end{proof}
\begin{remark}
  Assuming that $y_i$ is a scalar and $w=0$, Fawzi et al.~\cite{Fawzi2012} prove that the state can be exactly recovered under the integrity attack if and only if for all $u\neq 0$, there are at least $2p+1$ non-zero $H_iu$. Notice that if for some $u\neq 0$, there are less than $2p+1$ non-zero $H_iu$, then we can choose $\mathcal I$ to contain the largest $p$ $H_iu$ and thus violate \eqref{eq:sufficiency}. As a result, our sufficient condition is stronger than the ones proposed in \cite{Fawzi2012}. The main reason is that we seek to use convex optimization to solve the state estimation problem, while in \cite{Fawzi2012}, a combinatorial optimization problem is needed to recover the state.
\end{remark}
We next give necessary conditions for the robustness of the estimator.
\begin{theorem}[Necessary Condition I]
  If $ C_i(u)$ is well defined for all $u\in \mathbb R^{n}$ and all $i\in\mathcal S$ but there exist some $\|u_0\|=1,~\mathcal I_0\in\mathbb C$ such that
  \begin{align}
    \sum_{i\in\mathcal I_0}C_i(u_0)>\sum_{i\in\mathcal I_0^c}C_i(u_0),\label{eq:necessity}
  \end{align}
  then the estimator is not robust to the attack.
  \label{theorem:necessity1}
\end{theorem}
\begin{proof}
  The robustness of the estimator is equivalent to that the optimal estimate $\hat x$ satisfies $\norm{\hat x}\leq \mu(z)$ for all $a\in \mathcal A$, where $\mu$ is a real-valued function. To this end, we will prove that for any $r > 0$, there exists a $y$ such that all $\hat x$ that satisfies $\norm{\hat x}\leq r$ cannot be the optimal solution of \eqref{eq:general}.

  We will first look at the compromised sensors. For every $\delta>0$ we can always find a finite constant $N_i(\delta)$ such that for any $\hat x\in\{\hat x:\norm{\hat x}\leq r\}$ and for all $t > N_i$, the following inequality holds:
  \begin{align}
    &f_i(t H_iu_0-H_i\hat x)-f_i(tH_iu_0 - H_i(\hat x+u_0)) \nonumber\\
    \geq &f_i((t+1) H_iu_0-H_i(\hat x+u_0))-f_i(tH_iu_0 - H_i(\hat x+u_0)) \nonumber\\
    \geq & h_i(u_0,-H_i(\hat x+u_0),t)\geq C_i(u_0)-\delta,~\forall i\in \mathcal I_{0}. \label{eq:nece1}
  \end{align}
  The first inequality is derived from \eqref{eq:marginalincrease}. The second inequality is due to the uniform convergence of $h_i(u,v,t)$ to $C_i(u)$ on $\{u_0\}\times \{v:v=-H_ix+u_0,\,\|x\|\leq  r\}$.

  Let us choose
  \begin{align*}
    \delta = \frac{1}{m}\left(\sum_{i\in\mathcal I_{0}}C_i(u_0) - \sum_{i\in \mathcal I_{0}^c}C_i(u_0)\right),
  \end{align*}
  and $t = \max_{i\in\mathcal I_0} N_i(\delta)$ and $y_i = tH_iu_0$ for all $i\in \mathcal I_0$, then we know for any $\|\hat x\|\leq  r$,
  \begin{multline*}
    \sum_{i\in\mathcal I_0}\left[f_i(y_i-H_i\hat x)-f_i(y_i - H_i(\hat x+u_0))\right] \\
    \geq \sum_{i\in \mathcal I_0} C_i(u_0)-p\delta.
  \end{multline*}
  Now let us look at the benign sensors. By Lemma \ref{lemma:1} (iii) we have
  \begin{multline}
    f_i(z_i-H_i(\hat x+u_0))-f_i(z_i-H_i\hat x) \\
    \leq  C_i(u_0),~\forall i\in \mathcal I_0^c.\label{eq:nece2}
  \end{multline}
  From \eqref{eq:nece1} and \eqref{eq:nece2},
  \begin{multline*}
    \sum_{i\in\mathcal S} f_i(y_i-H_i(\hat x+u_0))  - \sum_{i\in\mathcal S} f_i(y_i-H_i\hat x)\\
    \leq \sum_{i\in\mathcal I_0^c}C_i(u_0) - \sum_{i\in\mathcal I_{0}}C_i(u_0) + p\delta < 0.
  \end{multline*}
  Thus for such a $y$ satisfying
  \[ y_i=
    \left\{
      \begin{array}{ll}
        z_i, & \hbox{if } i\in\mathcal I_0^c\\
        tH_iu_0, & \hbox{if } i\in\mathcal I_{0},
      \end{array}
    \right.
  \]
  $\hat x+u_0$ is a better estimate than all $\hat x$ satisfying $\norm{\hat x}\leq  r$. Since $r$ is an arbitrary positive real number, we can conclude that the estimator is not robust.
\end{proof}

Before continuing on, we would like to provide some remarks on the main result. First, it is worth noticing that the existence of a well defined limit of $f_i(tH_iu)/t$ is crucial for the robustness of $g$. For example, the least squares estimator cannot be robust since $f_i$ is in quadratic form. Using the potential field and force analogies in Remark~\ref{remark:force}, one can interpret the results presented in this section as: the estimator $g$ is robust if the force generated by any sensor is bounded and if the combined force of any collection of $p$ sensors is no greater than the combined force of the remaining $m-p$ sensors.

Secondly, one can see that the conditions proved in Theorem~\ref{theorem:sufficient} and \ref{theorem:necessity1} are very tight, with only a trivial gap where the LHS of \eqref{eq:necessity} equals the RHS.

\section{Concluding Remarks}\label{section:conlusion}
We have studied the robust estimation problem where $p$ out of $m$ sensors are under attack. The malicious measurements can be arbitrarily manipulated and thus a robust estimator which can give a reliable estimate is needed. Our interest is not to study any concrete estimator in presence of attacks. Instead, we have considered a general class of estimators which integrate a large number of important estimators as special cases and given sufficient and necessary conditions for the robustness of the estimator. Future works include the robustness analysis for the dynamical state estimation problem.
\section{Appendix}
\subsection*{Proof of Lemma \ref{lemma:1}:}

  \begin{enumerate}[(i)]
  \item If $\alpha = 0$, then clearly $C_i(0)= 0$. On the other hand, if $\alpha\neq 0$, from the definition in \eqref{eq:sublinear}, we have
    \begin{align*}
      C_i(\alpha u) &= \lim_{t\rightarrow\infty} \frac{1}{t}f_i(|\alpha|t H_i u)\\
                    &= \card{\alpha} \lim_{t\rightarrow\infty} \frac{1}{\card{\alpha} t}f_i(|\alpha| t H_i u)= \card{\alpha} C_i(u).
    \end{align*}
    Due to the scaling property of $C_i(u)$ and the convexity of $f_i$, we have
    \begin{align*}
      C_i(u_1+u_2)= 2C_i\left(\frac{u_1+u_2}{2}\right)\leq C_i(u_1) + C_i(u_2).
    \end{align*}
    Therefore, we know that $C_i$ is actually a semi-norm on $\mathbb R^n$
  \item Based on the convexity of $f_i$, we obtain
    \begin{align}
      2f_i(\frac{tH_iu}{2})&\leq f_i(v+tH_iu) + f_i(-v),\label{eq:temp1}\\
      f_i(tH_iu)&\geq 2f_i(\frac{2v+tH_iu}{2}) - f(2v).\label{eq:temp2}
    \end{align}
    Dividing both sides of \eqref{eq:temp1} and \eqref{eq:temp2} by $t$ and taking limit over $t$, we have
    \begin{align}
      C_i(u)&\leq \liminf_{t\rightarrow\infty}\frac{1}{t}f_i(v+tH_iu) +   \lim_{t\rightarrow\infty}\frac{1}{t}f_i(-v),\label{eq:liminf}\\
      C_i(u)&\geq \limsup_{t\rightarrow\infty}\frac{2}{t}f_i(v+\frac{t}{2}H_iu) -   \lim_{t\rightarrow\infty}\frac{1}{t}f_i(2v).\label{eq:limsup}
    \end{align}
    Since $\lim_{t\rightarrow\infty}f_i(-v)/t=\lim_{t\rightarrow\infty}f_i(2v)/t=0$, from \eqref{eq:limsup} and \eqref{eq:liminf} we have the following pointwise limit
    \begin{align*}
      \lim_{t\rightarrow\infty}h_i(u,v,t) = C_i(u).
    \end{align*}
    Notice that for a fixed $(u,v)$, by Lemma~\ref{lemma:convex}, $h(u,v,t)$ is monotonically non-decreasing with respect to $t$. Furthermore, $C_i(u)$ is continuous since it is a semi-norm. Therefore, by Dini's theorem~\cite{rudin1964principles}, $h(u,v,t)$ converges uniformly to $C_i(u)$ on a compact set of $(u,v)$.
  \item By the convexity of $f_i$, we have
    \begin{multline*}
        f_i(v + t H_iu) - f_i(v + (t-1) H_iu)\\
        \leq f_i(v + (t+1)H_iu) - f_i(v + t H_iu),
    \end{multline*}
and
    \begin{multline*}
    f_i(v + (t+1)H_iu) - f_i(v + t H_iu) \\
\leq \frac{1}{t}(f_i(v+tH_iu) - f_i(v)).
\end{multline*}
      Then we can conclude that
    \begin{multline*}
f_i(v + H_iu) - f_i(v)\\
\leq \lim_{t\rightarrow\infty} \frac{1}{t}(f_i(v+tH_iu) - f_i(v)) =C_i(u).
    \end{multline*}
  \end{enumerate}
  \hspace*{\fill}~\QED\par\endtrivlist\unskip

\bibliographystyle{IEEEtran}

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Q: How to calculate running time for a scikit-learn model? How do I compute the time it takes to fit a model in scikit-learn?
It's pretty easy to report performance in accuracy, but is there a way to calculate how much time is spent learning a machine learning model?

A: For general timing advice in Python, good answers already exist (e.g.: Working with time.process_time()). Those should work fine here for simple cases:
from time import process_time

start = process_time()
clf.fit(X, y)         # Assume clf is an instantiated model, and X, y exist
end = process_time()

print(end - start)

In the more general case, we may want to know how long on average certain models take to fit with respect to their hyperparameters. Here, GridSearchCV provides a cv_results_['mean_fit_time'] to help answer this. This tells us that decision trees tend to take longer for higher max_depth.
from sklearn.tree import DecisionTreeClassifier
from sklearn.datasets import make_classification
from sklearn.model_selection import GridSearchCV

X, y = make_classification(n_samples=10_000)

param_grid = {"max_depth": list(range(3, 13))}

clf = GridSearchCV(DecisionTreeClassifier(), param_grid=param_grid)

clf.fit(X, y)
print(clf.cv_results_['mean_fit_time'])
# [0.03812594 0.04834146 0.05935068 0.06945467 0.07974615 0.0890358
#  0.09815149 0.10666809 0.11485486 0.12286725]

Or: cross_validate returns a dictionary with fit_time and score_time:
from sklearn.model_selection import cross_validate

cross_validate(DecisionTreeClassifier(max_depth=4), X, y)
# {
#     'fit_time': array([0.05116796, 0.05008364, 0.04976106, 0.0510118 , 0.04988265]),
#     'score_time': array([0.00071454, 0.00047278, 0.00048947, 0.00048614, 0.00046182]),
#     'test_score': array([0.9375, 0.9255, 0.9325, 0.9375, 0.931 ])
# }

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Henry Ford gründete den Automobilhersteller Ford Motor Company. Er perfektionierte konsequent die Fließbandfertigung im Automobilbau, die allerdings schon Ransom Eli Olds in vereinfachter Form in seiner Automobilfirma Oldsmobile. Henry Ford (* Juli Greenfield Township (Michigan), USA; † 7. April in Dearborn, Michigan) gründete den Automobilhersteller Ford Motor. Henry Ford II (* 4. September in Detroit; † September ebenda), Sohn von Edsel Ford und Enkel von Henry Ford, war der Präsident der Ford Motor. Mit seiner visionären Idee, ein für jedermann erschwingliches Fahrzeug auf den Markt zu bringen, hat Henry Ford das Leben unzähliger Menschen verändert. Er verhalf der US-Autoindustrie zu ihrem großen Durchbruch und ging überaus fair mit seinen Arbeitern um; dennoch ist Henry Ford bis heute.
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By the time he wrote his memoir, he had spoken of the Social Department and the private conditions for profit-sharing in the past tense. He admitted that:.
Welfare work that consists in prying into employees' private concerns is out of date. Men need counsel and men need help, often special help; and all this ought to be rendered for decency's sake.
But the broad workable plan of investment and participation will do more to solidify industry and strengthen organization than will any social work on the outside.
Without changing the principle we have changed the method of payment. In addition to raising his workers' wages, Ford also introduced a new, reduced workweek in The decision was made in , when Ford and Crowther described it as six 8-hour days, giving a hour week, [30] but in it was announced as five 8-hour days, giving a hour week.
On May 1, , the Ford Motor Company's factory workers switched to a five-day hour workweek, with the company's office workers making the transition the following August.
Ford had decided to boost productivity, as workers were expected to put more effort into their work in exchange for more leisure time, and because he believed decent leisure time was good for business since workers would have more time to purchase and consume more goods.
However, charitable concerns also played a role. Ford explained, "It is high time to rid ourselves of the notion that leisure for workmen is either 'lost time' or a class privilege.
Ford was adamantly against labor unions. He explained his views on unions in Chapter 18 of My Life and Work.
Most wanted to restrict productivity as a means to foster employment, but Ford saw this as self-defeating because, in his view, productivity was necessary for economic prosperity to exist.
He believed that productivity gains that obviated certain jobs would nevertheless stimulate the broader economy and grow new jobs elsewhere, whether within the same corporation or in others.
Ford also believed that union leaders had a perverse incentive to foment perpetual socio-economic crises to maintain their power.
Meanwhile, he believed that smart managers had an incentive to do right by their workers, because doing so would maximize their profits.
However, Ford did acknowledge that many managers were basically too bad at managing to understand this fact. But Ford believed that eventually, if good managers such as he could fend off the attacks of misguided people from both left and right i.
Bennett employed various intimidation tactics to quash union organizing. The Dearborn police department and Ford security guards opened fire on workers leading to over sixty injuries and five deaths.
In the late s and early s, Edsel—who was president of the company—thought Ford had to come to a collective bargaining agreement with the unions because the violence, work disruptions, and bitter stalemates could not go on forever.
But Ford, who still had the final veto in the company on a de facto basis even if not an official one, refused to cooperate. For several years, he kept Bennett in charge of talking to the unions trying to organize the Ford Motor Company.
Sorensen's memoir [37] makes clear that Ford's purpose in putting Bennett in charge was to make sure no agreements were ever reached.
Sorensen recounted [38] that a distraught Henry Ford was very close to following through with a threat to break up the company rather than cooperate.
Still, his wife Clara told him she would leave him if he destroyed the family business. In her view, it would not be worth the chaos it would create.
Ford complied with his wife's ultimatum and even agreed with her in retrospect. Overnight, the Ford Motor Company went from the most stubborn holdout among automakers to the one with the most favorable UAW contract terms.
The contract was signed in June Now you're in here and we've given you a union shop and more than you got out of them. That puts you on our side, doesn't it?
We can fight General Motors and Wall Street together, eh? Like other automobile companies, Ford entered the aviation business during World War I , building Liberty engines.
After the war, it returned to auto manufacturing until , when Ford acquired the Stout Metal Airplane Company. Ford's most successful aircraft was the Ford 4AT Trimotor , often called the "Tin Goose" because of its corrugated metal construction.
It used a new alloy called Alclad that combined the corrosion resistance of aluminum with the strength of duralumin. The plane was similar to Fokker 's V.
VII-3m, and some say that Ford's engineers surreptitiously measured the Fokker plane and then copied it. The Trimotor first flew on June 11, , and was the first successful U.
Several variants were also used by the U. The Smithsonian Institution has honored Ford for changing the aviation industry. Ford opposed war, which he viewed as a terrible waste, [40] [41] and supported causes that opposed military intervention.
He and about other prominent peace leaders traveled there. Ford's Episcopalian pastor, Reverend Samuel S.
Marquis, accompanied him on the mission. Marquis headed Ford's Sociology Department from to Ford talked to President Wilson about the mission but had no government support.
His group went to neutral Sweden and the Netherlands to meet with peace activists. A target of much ridicule, Ford left the ship as soon as it reached Sweden.
Ford plants in the United Kingdom produced Fordson tractors to increase the British food supply, as well as trucks and aircraft engines. When the U.
In , with the war on and the League of Nations a growing issue in global politics, President Woodrow Wilson , a Democrat, encouraged Ford to run for a Michigan seat in the U.
Wilson believed that Ford could tip the scales in Congress in favor of Wilson's proposed League. Ford wrote back: "If they want to elect me let them do so, but I won't make a penny's investment.
Ford remained a staunch Wilsonian and supporter of the League. When Wilson made a major speaking tour in the summer of to promote the League, Ford helped fund the attendant publicity.
Ford had opposed the United States entry into World War II [35] [49] and continued to believe that international business could generate the prosperity that would head off wars.
Ford "insisted that war was the product of greedy financiers who sought profit in human destruction"; in , he went so far as to claim that the torpedoing of U.
Like many other businessmen of the Great Depression era, he never liked or entirely trusted the Franklin Roosevelt Administration, and thought Roosevelt was inching the U.
Ford continued to do business with Nazi Germany , including the manufacture of war materiel. Beginning in , with the requisitioning of between and French POWs to work as slave laborers, Ford-Werke contravened Article 31 of the Geneva Convention.
The number of slave laborers grew as the war expanded even though Nazi authorities did not require German companies to use slave laborers.
When Rolls-Royce sought a U. He "lined up behind the war effort" when the U. Before the U. Ford broke ground on Willow Run in the spring of , B component production began in May , and the first complete B came off the line in October At its peak in , the Willow Run plant produced Bs per month, and by Ford was completing each B in eighteen hours, with one rolling off the assembly line every 58 minutes.
When Edsel Ford died of cancer in , aged only 49, Henry Ford nominally resumed control of the company, but a series of strokes in the late s had left him increasingly debilitated, and his mental ability was fading.
Ford was increasingly sidelined, and others made decisions in his name. Ford grew jealous of the publicity Sorensen received and forced Sorensen out in They threatened to sell off their stock, which amounted to three quarters of the company's total shares, if he refused.
Ford was reportedly infuriated, but had no choice but to give in. In the early s, Ford sponsored a weekly newspaper that published strongly antisemitic views.
At the same time, Ford had a reputation as one of the few major corporations actively hiring Black workers. In , Ford's closest aide and private secretary, Ernest G.
Liebold , purchased an obscure weekly newspaper for Ford, The Dearborn Independent. The Independent ran for eight years, from until , with Liebold as editor.
Every Ford franchise nationwide had to carry the paper and distribute it to its customers. During this period, Ford emerged as "a respected spokesman for right-wing extremism and religious prejudice", reaching around , readers through his newspaper.
In Germany, Ford's antisemitic articles from The Dearborn Independent were issued in four volumes, cumulatively titled The International Jew, the World's Foremost Problem published by Theodor Fritsch , founder of several antisemitic parties and a member of the Reichstag.
In a letter written in , Heinrich Himmler described Ford as "one of our most valuable, important, and witty fighters". Ludecke was introduced to Ford by Siegfried Wagner son of the composer Richard Wagner and his wife Winifred , both Nazi sympathizers and antisemites.
Ludecke asked Ford for a contribution to the Nazi cause, but was apparently refused. While these articles explicitly condemned pogroms and violence against Jews, they blamed the Jews themselves for provoking them.
Friends and business associates have said they warned Ford about the contents of the Independent and that he probably never read the articles he claimed he only read the headlines.
A libel lawsuit was brought by San Francisco lawyer and Jewish farm cooperative organizer Aaron Sapiro in response to the antisemitic remarks, and led Ford to close the Independent in December News reports at the time quoted him as saying he was shocked by the content and unaware of its nature.
During the trial, the editor of Ford's "Own Page", William Cameron, testified that Ford had nothing to do with the editorials even though they were under his byline.
Cameron testified at the libel trial that he never discussed the content of the pages or sent them to Ford for his approval. Miller, a former Dearborn Independent employee, swore under oath that Ford had told him he intended to expose Sapiro.
That Cameron would have continued to publish such anti-Semitic material without Ford's explicit instructions seemed unthinkable to those who knew both men.
Stanley Ruddiman, a Ford family intimate, remarked that "I don't think Mr. Cameron ever wrote anything for publication without Mr.
Ford's approval. They formed a coalition of Jewish groups for the same purpose and raised constant objections in the Detroit press.
Before leaving his presidency early in , Woodrow Wilson joined other leading Americans in a statement that rebuked Ford and others for their antisemitic campaign.
A boycott against Ford products by Jews and liberal Christians also had an impact, and Ford shut down the paper in , recanting his views in a public letter to Sigmund Livingston , president of the ADL.
Wallace also found that Ford's apology was likely, or at least partly, motivated by a business that was slumping as a result of his antisemitism, repelling potential buyers of Ford cars.
Ford's apology was well received. According to Pool and Pool , [79] Ford's retraction and apology which were written by others were not even truly signed by him rather, his signature was forged by Harry Bennett , and Ford never privately recanted his antisemitic views, stating in "I hope to republish The International Jew again some time.
In July , before the outbreak of war, the German consul at Cleveland gave Ford, on his 75th birthday, the award of the Grand Cross of the German Eagle , the highest medal Nazi Germany could bestow on a foreigner.
The purpose of the letter was to clarify some general misconceptions that he subscribed or supported directly or indirectly, "any agitation which would promote antagonism toward my Jewish fellow citizens.
Distribution of The International Jew was halted in through legal action by Ford, despite complications from a lack of copyright.
Extremist groups often recycle the material; it still appears on antisemitic and neo-Nazi websites. The decisive anti-Semitic book I was reading and the book that influenced my comrades was I read it and became anti-Semitic.
The book made a great influence on myself and my friends because we saw in Henry Ford the representative of success and also the representative of a progressive social policy.
Robert Lacey wrote in Ford: The Men and the Machines that a close Willow Run associate of Ford reported that when he was shown newsreel footage of the Nazi concentration camps , he "was confronted with the atrocities which finally and unanswerably laid bare the bestiality of the prejudice to which he contributed, he collapsed with a stroke — his last and most serious.
Ford's philosophy was one of economic independence for the United States. His River Rouge Plant became the world's largest industrial complex, pursuing vertical integration to such an extent that it could produce its own steel.
Ford's goal was to produce a vehicle from scratch without reliance on foreign trade. He believed in the global expansion of his company.
He believed that international trade and cooperation led to international peace, and he used the assembly line process and production of the Model T to demonstrate it.
He opened Ford assembly plants in Britain and Canada in , and soon became the biggest automotive producer in those countries.
In , Ford cooperated with Giovanni Agnelli of Fiat to launch the first Italian automotive assembly plants. The first plants in Germany were built in the s with the encouragement of Herbert Hoover and the Commerce Department, which agreed with Ford's theory that international trade was essential to world peace.
In , Ford made an agreement with the Soviets to provide technical aid over nine years in building the first Soviet automobile plant GAZ near Nizhny Novgorod Gorky [87] an additional contract for construction of the plant was signed with The Austin Company on August 23, Ford sent his engineers and technicians to the Soviet Union to help install the equipment and train the working force, while over a hundred Soviet engineers and technicians were stationed at Ford's plants in Detroit and Dearborn "for the purpose of learning the methods and practice of manufacture and assembly in the Company's plants.
All the world is bound to catch some good from it. By , Ford was manufacturing one third of all the world's automobiles. It set up numerous subsidiaries that sold or assembled the Ford cars and trucks:.
Ford's image transfixed Europeans, especially the Germans, arousing the "fear of some, the infatuation of others, and the fascination among all".
They saw the size, tempo, standardization, and philosophy of production demonstrated at the Ford Works as a national service—an "American thing" that represented the culture of the United States.
Both supporters and critics insisted that Fordism epitomized American capitalist development, and that the auto industry was the key to understanding economic and social relations in the United States.
As one German explained, "Automobiles have so completely changed the American's mode of life that today one can hardly imagine being without a car.
It is difficult to remember what life was like before Mr. Ford began preaching his doctrine of salvation". In My Life and Work , Ford predicted that if greed, racism, and short-sightedness could be overcome, then economic and technological development throughout the world would progress to the point that international trade would no longer be based on what today would be called colonial or neocolonial models and would truly benefit all peoples.
Ford maintained an interest in auto racing from to and began his involvement in the sport as both a builder and a driver, later turning the wheel over to hired drivers.
He entered stripped-down Model Ts in races, finishing first although later disqualified in an "ocean-to-ocean" across the United States race in , and setting a one-mile 1.
Ford dropped out of the race and soon thereafter dropped out of racing permanently, citing dissatisfaction with the sport's rules, demands on his time by the booming production of the Model Ts, and his low opinion of racing as a worthwhile activity.
In My Life and Work Ford speaks briefly of racing in a rather dismissive tone, as something that is not at all a good measure of automobiles in general.
He describes himself as someone who raced only because in the s through s, one had to race because prevailing ignorance held that racing was the way to prove the worth of an automobile.
Ford did not agree. But he was determined that as long as this was the definition of success flawed though the definition was , then his cars would be the best that there were at racing.
Nevertheless, Ford did make quite an impact on auto racing during his racing years, and he was inducted into the Motorsports Hall of Fame of America in By this point, Ford, nearing 80 years old, had had several cardiovascular events variously cited as heart attacks or strokes and was mentally inconsistent, suspicious, and generally no longer fit for such immense responsibilities.
Most of the directors did not want to see him as president. But for the previous 20 years, though he had long been without any official executive title, he had always had de facto control over the company; the board and the management had never seriously defied him, and this moment was not different.
The directors elected him, [97] and he served until the end of the war. The administration of President Franklin Roosevelt had been considering a government takeover of the company in order to ensure continued war production, [58] but the idea never progressed.
He died on April 7, , of a cerebral hemorrhage at Fair Lane , his estate in Dearborn, at the age of A public viewing was held at Greenfield Village where up to 5, people per hour filed past the casket.
Funeral services were held in Detroit's Cathedral Church of St. Paul and he was buried in the Ford Cemetery in Detroit. A compendium of short biographies of famous Freemasons , published by a Freemason lodge, lists Ford as a member.
When he received the 33rd degree of the Scottish Rite in , he said, "Masonry is the best balance wheel the United States has.
In , Ford's pastor, and head of his sociology department, Episcopal minister Samuel S. Marquis, claimed that Ford believed, or "once believed," in reincarnation.
Ford published an anti-smoking book, circulated to youth in , called The Case Against the Little White Slaver , which documented many dangers of cigarette smoking attested to by many researchers and luminaries.
Henry Ford long had an interest in materials science and engineering. Ford long had an interest in plastics developed from agricultural products, especially soybeans.
He cultivated a relationship with George Washington Carver for this purpose. This project culminated in , when Ford patented an automobile made almost entirely of plastic , attached to a tubular welded frame.
It ran on grain alcohol ethanol instead of gasoline. The design never caught on. Ford was interested in engineered woods "Better wood can be made than is grown" [] at this time plywood and particle board were little more than experimental ideas ; corn as a fuel source , via both corn oil and ethanol; [] and the potential uses of cotton.
His brother in law, E. Kingsford , used wood scraps from the Ford factory to make the briquets. Ford was a prolific inventor and was awarded U.
Ford had a vacation residence in Fort Myers, Florida next to that of Thomas Edison , which he bought in and used until approximately It is still in existence today and is open as a museum.
He also had a vacation home known today as the "Ford Plantation" in Richmond Hill , Georgia which is still in existence today as a private community.
Ford started buying land in this area and eventually owned 70, acres square miles there. The grand house, made of Savannah-gray brick, had marble steps, air conditioning, and an elevator.
It sat on 55 acres of manicured lawns and flowering gardens. The house became the center of social gatherings with visitations by the Vanderbilts, Rockefellers, and the DuPonts.
It remains the centerpiece of The Ford Plantation today. He contributed substantially to the community, building a chapel and schoolhouse and employing numerous local residents.
Ford had an interest in " Americana ". In the s, Ford began work to turn Sudbury , Massachusetts, into a themed historical village. He moved the schoolhouse supposedly referred to in the nursery rhyme, " Mary Had a Little Lamb ", from Sterling , Massachusetts, and purchased the historic Wayside Inn.
This plan never saw fruition. Ford repeated the concept of collecting historic structures with the creation of Greenfield Village in Dearborn , Michigan.
It may have inspired the creation of Old Sturbridge Village as well. About the same time, he began collecting materials for his museum , which had a theme of practical technology.
It was opened in as the Edison Institute. Although greatly modernized, the museum continues today. From Wikipedia, the free encyclopedia.
This article is about the American industrialist. For other people with the same name, see Henry Ford disambiguation. American businessperson.
Greenfield Township , Michigan , U. Fair Lane , Dearborn , Michigan, U. Republican before Democratic after Clara Jane Bryant. This section does not cite any sources.
Please help improve this section by adding citations to reliable sources. Unsourced material may be challenged and removed.
July Learn how and when to remove this template message. Main article: Willow Run. Main article: Dearborn Independent. Archived from the original on November 19, Random House, Inc.
My Life and Work. Associated Press. September 29, May 26, Edsel Ford, year-old president of the Ford Motor Company, died this morning at his home at Grosse Pointe Shores following an illness of six weeks.
Archived from the original on October 5, Retrieved November 28, Archived from the original on March 25, Retrieved December 1, Moffa Press, New York: St.
Martin's Press. Lake Orion, Mich. April 6, The People's Tycoon. Encyclopedia of American Jewish History.
Abc-clio, , p. Tehabi Books. New York: Public Affairs. The Legend of Henry Ford. Retrieved April 26, Retrieved: August 7, Retrieved June 16, Segal Anti-Semitism in America.
Transaction Publishers. The New York Blueprint. Archived from the original on December 10, Retrieved February 6, See also, Pfal-Traughber, Armin Vienna: Braumüller.
In , Ford was hired as an engineer for the Detroit Edison Company. In , his natural talents earned him a promotion to chief engineer.
All the while, Ford developed his plans for a horseless carriage. In , Ford built his first gasoline-powered buggy, which had a two-cylinder, four-horsepower engine.
In , he constructed his first model car, the Ford Quadricycle. In the same year, he attended a meeting with Edison executives and found himself presenting his automobile plans to Thomas Edison.
The lighting genius encouraged Ford to build a second, better model. By , Ford was awarded with his first patent for a carburetor.
In , with money raised from investors following the development of a third model car, Ford left Edison Illuminating Company to pursue his car-making business full-time.
Ford introduced the Model T , the first car to be affordable for most Americans, in October and continued its construction until For several years, Ford Motor Company posted percent gains.
By , Ford and his son Edsel introduced another successful car, the Model A, and the Ford Motor Company grew into an industrial behemoth.
More than for his profits, Ford became renowned for his revolutionary vision: the manufacture of an inexpensive automobile made by skilled workers who earn steady wages and enjoyed a five-day, hour work week.
Ford was an ardent pacifist and opposed World War I , even funding a peace ship to Europe. Later, in , Ford and his family established the Ford Foundation to provide ongoing grants for research, education and development.
In business, Ford offered profit sharing to select employees who stayed with the company for six months and, most important, who conducted their lives in a respectable manner.
He even went as far as to support a weekly newspaper, The Dearborn Independent , which furthered such views. The Ford company, in turn, said the factory was under the control of the Nazis, not the American corporate headquarters.
Ford was an avid collector of Americana, with a particular interest in technological innovations and the lives of ordinary people: farmers, factory workers, shopkeepers and business people.
He decided to create a place where their lives and interests could be celebrated. Ford's vision for the museum was stated as, "When we are through, we shall have reproduced American life as lived; and that, I think, is the best way of preserving at least a part of our history and tradition.
Fintech Die Robinhood-Anleger und ihre…. Die zuerst nur lokal ausgelegte Stiftung wuchs beträchtlich und hatte bis ihren Fokus auch national und international erweitert. Er selbst errang in seinem Ford 'Sweepstakes' am Jetzt Neu! Örtliche Arbeitskräfte wurden mit sozialen Versprechungen Hotel Sunshine Dortmund, von denen Manipulation Spielautomaten wenig eingehalten Beste Kiste Cs Go, dass es zu Unruhen kam. In: Washington Post. Doch der echte Durchbruch kommt Bayerische Comedy im Jahr Immobilien Die Stolperfallen beim Erbbaurecht. Dieser gehört der Vergangenheit an, bevor wir beschlossen hatten, die höheren Löhne zu zahlen. Auch als Autor trat Ford in Erscheinung. Dabei verlassen Paysafecard Mit Amazon Gutschein Kaufen das Angebot des Drakenmoon. Dort wurden auch Zwangsarbeiter aus Konzentrationslagern eingesetzt, die man für vier Reichsmark pro Tag von der SS auslieh. Mechanik fasziniert ihn: Er repariert Uhren, Italien Schweden Tor alles auseinander, was auf dem Hof an Technik zu finden ist. Einer der App Herunterladen Anleitung ist er auch — fanatisch verehrt von den einen, verhasst bei anderen. Jahrhunderts sonst schwer hatten. Nach seiner Beförderung zum Chefingenieur hatte er nun genügend Zeit und Geld, um sich seinen persönlichen Experimenten mit Verbrennungsmotoren zu widmen. Henry Ford befürwortete, dass die Stelle von Harry Bennett übernommen werden sollte. Auf dieser Seite werden Cookies verwendet. Juni Seine Eltern waren als Farmer tätig. Der Versuch, motorisierte Fahrzeuge zu verkaufen, scheitert zunächst. Klett-Cotta, StuttgartS.
Henry Ford Auf dem Bauernhof
Mit 16 macht sich der Bauernjunge auf ins nahe Detroit und beginnt eine Ausbildung in Beste Spielothek in Lassnitz Murau finden kleinen Metallverarbeitungsbetrieb. Das Buch Der internationale Jude wurde im Iran nachgedruckt. Jahrhunderts noch die meisten Geschichten anfangen: auf einem Bauernhof. Märkte Es geht an die…. Davon wurden in der Zeit von bis über 15 Millionen Stück produziert. Dort wurden auch Zwangsarbeiter aus Konzentrationslagern eingesetzt, die man für vier Reichsmark pro Tag von der SS auslieh.
Henry Ford - Navigationsmenü
Exklusiv NMitarbeiter planen Betriebsratswahlen. Er selbst errang in seinem Ford 'Sweepstakes' am Immobilien Die Stolperfallen beim Erbbaurecht. Ford verbesserte weiterhin die Produktionsgänge, indem die Fertigung von Teilen standardisiert und die Arbeitsteilung noch weiter vorangetrieben wurde. Theodor Fritsch weigerte sich jedoch, die rund
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ICC - International Criminal Court | Institute for War and Peace Reporting
Reporting from DR Congo, Kenya, Sudan, Uganda.
IWPR is at the forefront of media interventions in situations of human suffering and mass abuses around the world. It gives a voice to the vulnerable and delivers crucial analysis on conflict and impunity to both a local and global audience. As the leading news agency reporting on international justice and human rights, IWPR makes an important contribution to the value and effectiveness of local judicial mechanisms as well as international legal institutions such as the International Criminal Court (ICC).
Slow Progress at Rwandan Tribunal
Stephanie Nieuwoudt
At a cost of over a billion dollars, some say the tribunal to try ringleaders of the Rwandan genocide is too expensive and has dragged on for too long.
Ituri: The Congo's Own Rwanda
Francis Mwepu
A savage regional conflict in northeastern Democratic Republic of Congo has claimed more than 60,000 lives and displaced hundreds of thousands of people.
Uganda: Acholi Elders May Hold Key to Peace Talks
Origins of brutal 20-year-old war with the rebel Lord's Resistance Army lie deep in tribal history and culture.
Uganda Peace Talks Bring Hope
Can Sudan's Machar bridge the gap between Kampala and the LRA?
Uganda: Amnesty Offer Blow for Rebel Chief Arrest Plans
With the offer of amnesty for rebel leader Joseph Kony, peace talks suddenly look realistic - but the move worries those who want to see him arrested and brought before the International Criminal Court.
Uganda: Demystifying Kony
LRA chief looks more like a conventional rebel leader than a man described by former followers as a prophet who talked to angels.
Africans at the ICC
Introduction to a series of profiles of prominent Africans working at the International Criminal Court in The Hague.
Uganda: Call for Peace from Rebel Chief's Village
As negotiators gather in Juba, south Sudan, to end 20 years of war in northern Uganda, a plea for peace comes from Joseph Kony's childhood home.
Refugees in Northern Uganda Fear Going Home
Continuing violence and looming land squabbles prompt refugees to reject call to go back to their villages.
World Court Faces Biggest Challenge
New peace initiative for northern Uganda highlights divisions over the International Criminal Court's attempt to arrest rebel leader Joseph Kony.


For 10 short days, award-winning new homebuilder John Wieland Homes and Neighborhoods presents the "Sale of a Lifetime," an unprecented offering of a lifetime structural warranty and up to $100,000 in savings on new homes in Charlotte and Raleigh, N.C. and Charleston, S.C., as well as Atlanta and Nashville, Tenn.
Friday, Jan. 25 through Sunday, Feb. 3, buyers can receive the lifetime structural warranty as well as $10,000 to $100,000 in bonus dollars on every John Wieland home.
This "Sale of a Lifetime," comes from the same builder that recently ranked highest in customer satisfaction among new home builders in Atlanta and Charlotte by J.D. Power and Associates.
Full details on the lifetime warranty are available by visiting any John Wieland neighborhood or www.jwhomes.com/lifetime or by calling 1-800-DRM-HOME(1-800-376-4663).
This entry was posted in Charleston new homes, Charlotte new homes, New Home News, North Carolina, Raleigh new homes, South Carolina on January 15, 2008 by Carol Morgan.


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redpajama_set_name: RedPajamaGithub


When Beneath the Seas 29th annual Ocean Exposition, Adventure and Dive Travel Show convenes at New Jersey's Meadowlands Exposition Center in Secaucus, New Jersey March 18th, 19th and 20th, 2005, Ringmaster for the Film Festival will be the notable photographer, emcee, raconteur, and Master-of-Ceremonies: Ernie Brooks.
If there is anyone who understands photography and the magic that creates an exceptional photograph it is Ernie Brooks. Until recently Ernie was President of the internationally renowned Brooks Institute of Photography. Today in his book, "Silver Seas," he has produced a magnificent collection of images celebrating his view of the underwater world and the creatures that live in it.
At Beneath the Sea, as the master-of-Ceremonies of the Saturday Night Film Festival, Ernie Brooks will bring this sensitive love of the ocean and underwater photography to the people who, in Ernie's words, "are a special people with a common bond ... love of the ocean and the creatures in it."


From Bangkok we flew to a small town in southern Thailand called Krabi. It's approximately 1000 km's south and has a population of only 20,000. It was a refreshing change of pace from Bangkok. Krabi is very laid back and the people are extremely friendly. Many travelers just pass through Krabi on their way to the islands. We found it a worthwhile place to spend some time.
It's very easy to get around Krabi by foot. The locals greeted us with smiles as we walked the streets. During our visit I think the people were pleased to see tourists starting to return after the Tsunami. Although Krabi doesn't sit directly on the ocean the effects of the 2004 Tsunami wiped out the entire ferry dock that sits on the river. It also caused an extreme amount of flooding.
The river is a scenic place to go for a stroll, a boardwalk allows you to take in the views of the dense mangroves lining the river shore. The most notable sight is the massive limestone formations that jet out of the water like breaching whales.
There are some great restaurants to experience Thai food, but if you're craving something more familiar there are a few choices.
A cafe that became our favorite was called May and Marks. It is a family run place that is named after the two oldest children. The husband is Chinese and the wife is Thai, and grandma helps out too. They have many international choices on their menu. They learned how to make the dishes from travelers that have stayed in the guest house above the restaurant. It has become a very popular spot for people who have been on the road for a while. Travelers like to come and get a taste of something from home. They also make homemade breads. The sourdough is a big hit, people come from as far as Phuket to stock up on it.
Thailand's Songkran is in April, it is their New Year and also known as the water festival. From what we heard we were glad to be out of Bangkok during this holiday, it gets pretty crazy. Chiang Mai is supposed to be even worse. It is a time to cleanse the Buddha images with water, but the washing extends to all the people in the streets. There is a version of this holiday celebrated in other Southeast Asia countries such as Laos.
On the first day of Songkran we headed out of our guest house for the day. As soon as we looked into the street we realized we needed to go back in and get rid of our day pack. We placed some money and our compact camera into ziplock bags before placing them into our pockets.
We only made it a few feet out of our hotel before we received our first water blessing. People were everywhere in the streets armed with large water guns, others were using hoses from the front of their shops. Pick up trucks were traveling through the streets with barrels full of water. Anyone driving or walking past these groups of people were fair game to being soaked. Trucks packed with people would circle around the town and stop to have water fights with the street groups and with other trucks.
We stopped by May and Marks to have some lunch. They are a very close family. We saw a great sign of respect while we sat and watched the water activity on the street. A truck pulled up with a bunch of young guys in it. It turned out to be the eldest son and some of his friends.
One guy threatened to wet the daughter, but didn't because she was working. She didn't hesitate to sneak into the back to get a bucket and soak him though. The son did a bow to the mom and put a respectful amount of water on her arms. By this time the grandma had come from the back to check out the scene. The grandson bowed gracefully to her and gently took her hands as he rubbed a small amount of water on them. Grandma was delighted to be included in the action. She seemed to be enjoying sitting in the restaurant and watching the street activity.
As we made our way into the town center it got more intense. The park area around the river had the most activity. Several large barrels of water were set out for the kids to get into the action. The town firetrucks would come by and fill them up occasionally. The entire park had been turned into a community event with food vendors, live music and craft activities set up for the kids.
Another part of the holiday tradition is to smear a paste of baby powder and water onto the sides of people's faces. We were warmly welcomed into the holiday and soon wore the powder on our faces with pride. At first the locals seemed a bit hesitant to get us wet. Once we were marked by the baby powder we became an attractive target. Now they were having fun playing soak the tourists! We had nothing to fight back with other than two small water bottles. They were always kind enough to let us refill the bottles from their water supply. Then they would continue to soak us again!
By late afternoon we were exhausted and had enough for the day. We made our way back to our guest house. We dripped all the way and our wet shoes squeaked with every step. We had nearly dried out as our hotel came into sight, but we were blessed a few more times before we made it to final safety.
The official holiday is three days long. Many shops and places of business close for the period or longer. The water festival only went on for the one day in Krabi. Apparently it goes on for four or five days in Chiang Mai and up to a week or longer in Laos.
It was a great experience to not only witness a different cultural holiday, but to actually participate in it. It's so much fun, I think we should start it at home!


Referring to your story synopsis, have you included a Goal in which all the characters are involved?
If so, describe it below in as much detail as readily comes to mind.
If not, consider your subject matter and the activities in which your characters engage. What singular achievement would affect them all for better or worse?
This entry was posted in Plot Points, Story Development, Write Your Novel Step by Step. Bookmark the permalink.


Impression of a 20 ton London tugboat's hull in clay and sand.
At the end of July, 1997, Max Couper's tugboat was lifted out by harbour crane at Hanover's Lindener Harbour and transported by road into the Sprengel Museum, where it was dropped from five metres into a mound of 100 tons of Hanover clay and sand. It was then removed to leave behind the imprint of its underbelly: the Tug Print.
The tug then returned to London, whilst the Tug Print remained in situ, slowly cracking up.
This was an artwork that had developed in three quite separate ways. First an original accidental imprint had been made some years earlier in a muddy creek of the east coast of England. Then there was the spectacle and performance of making the print in the museum, broadcast on international television. And finally there was the aftermath, in which the print had become a silent part of the museum's collection – begging the question of the visitor of what had made it – obvious only that it must have come from above.


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Minister Launches "Recognition of Child Abuse and Neglect" Book for Social Workers
Roger Sowry
A new resource for social workers launched today by the Minister of Social Welfare, Roger Sowry breaks new ground in New Zealand and internationally in the field of investigating abuse and neglect.
The Recognition of Child Abuse and Neglect book, 'Tirohanga Tukino Tamariki', breaks new ground by encouraging social workers investigating child abuse to look first for the signs of abuse and neglect before categorising the type of abuse that may have occurred.
"Traditionally, child abuse and neglect has been investigated by an abuse type, such as physical or sexual, and then gone on to explore the signs relating to that type of abuse. This had potential to narrowly focus an investigation.
"The book will be issued to every social worker within the Children, Young Persons and their Families Service and will be used as a training resource. It will be an invaluable guide for social workers for all investigations of abuse and neglect of children," says Mr Sowry.
This material will also be published in CYPFS' 'Breaking the Cycle: An interagency Guide to Child Abuse' book for distribution to others in the child protection network such as schools, doctors, refuge workers etc.
The book is the result of nearly two years study and consultation with health professionals, social workers, and community and cultural advisory groups. It is the first of its kind to include reference to cultural practices that can sometimes be confusing to social workers.
"This resource, together with the Breaking the Cycle campaign will not only raise awareness of abuse but will give New Zealanders the confidence to act to eliminate abuse on children in our communities," says Mr Sowry.


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Topic: Arabic language
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Recreating Palestine in Literature
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Anthologies of "world literature" have often used the term to market a largely Western canon. But isn't western literature still implicitly regarded as the measure of all things? And are we not overlooking other literary values out of sheer ignorance? By Fakhri SalehMore
Naguib Mahfouz's influence on Arabic Literature: In the Shadow of the MasterPublication of Arab Literature in English: Change of ParadigmInterview with Hans Magnus Enzensberger: ''Europe Has No Copyright on the Enlightenment''Essay Peter Ripken: Listen to What the Arab World Has to Say
Arabic in Latin Script
Lost in Arabic
Arabic was never easy. But if the language spoken by some 240 million people with its convoluted verb forms and guttural phonology suddenly starts appearing in Latin script, then things get really complicated. And at the same time very straightforward. Details from Mona SarkisMore
Qantara.de Interview with Salah Stétié: "The Arabic Language Is Outmoded" Interview by Saleh Diyab Ireland: The Gaelic Version of the KoranIslamic Studies: On Christian Strophes in the Koran
Interview with Hans Magnus Enzensberger
"Europe Has No Copyright on the Enlightenment"
At the invitation of Sheikh Al Maktoum, the ruler of Dubai, a number of eminent German and Arab intellectuals recently met for a "cultural dialogue". Hans Magnus Enzensberger was among their number, addressing in his speech the question of whether the Arab world needs a new Enlightenment. Loay Mudhoon spoke to the authorMore
Interview with Yasser Hareb: "Culture Can Heal Political Wounds"Goethe in the Emirates: Culture as a Peace-Building Measure


On Saturday, I spent the day with the sun on my back, picking grapes amongst the vines on a south facing hillside in…. Yorkshire! I really could have been anywhere in Europe harvesting the Ortega & Solaris grapes, well other than I had wellies on as it had been very wet the day before and the dew and damp hung heavy on the grass and the grapes!
The day started with a 2 hour drive over to East Yorkshire, we were met with a brew and a chat about the days harvest and introduced to an eclectic mix of fellow volunteers, we swiftly moved on to the first pick a mile down the road at another vineyard aptly named "Paradise" and after a brief lesson with the secateurs we were left to fill the boxes with fantastic sweet, plump Solaris grapes. To grow grapes this far North, (Ryedale Vineyard is the most northerly vineyard in England) does not require any special vines or land other than a disease resistant variety and a south facing slope with good drainage, which is the same the World over.
After a only an hour & a half, it was brew time with homemade cakes, a quick count of how we were doing so as to calculate when lunch would be served back at the lovely farmhouse. We finished the patch of Solaris at Paradise Farm and headed back for a hearty lunch of stew, mash, roast potatoes, cabbage and peas then apple crumble & custard. Apples for the pud from the abundant trees around the vines. Of course the meal was washed down with a glass of dry rose English Wine (should that be Yorkshire Wine?).
Ryedale Vineyard offer the opportunity to volunteer for the grape harvest throughout October with the promise of plenty of Tea & Cakes and a good lunch with wine and the wonderful experience of the harvest. A once in a lifetime experience, get in touch and go along and help, they do B&B so you can make a weekend of it. Thanks to Slow Food North Yorkshire for the heads up about volunteering for grape picking.
This entry was posted in wine, Yorkshire and tagged english, grapes, harvest, Ortega, solaris, vines, vineyard, wine, Yorkshire. Bookmark the permalink.
I'm a researcher for ITV and we're looking for people to take part in the new series of Britain's Best Dish to represent the Northern region by putting forward a dish which they love to cook. This can either be a starter, a main course, or a dessert.
I was wondering if I sent you a poster, if you would be able to upload this to your blog in order to help spread the word.
We're trying to generate as much interest as possible so it would be great if you could help us to let people from your region know about this opportunity.
Please get in touch on danielle.davies@itv.com or 0207 1574719 – If somebody gets in touch with me with an email address, I will be able to send a poster over.


text:
Guaranita yaculica est une espèce d'araignées aranéomorphes de la famille des Pholcidae.

Distribution
Cette espèce est endémique d'Argentine. Elle se rencontre dans les provinces de Salta, de Jujuy et de Corrientes.

Description
Le mâle holotype mesure  et la femelle décrite par Torres, Pardo, González-Reyes, Rodríguez Artigas et Corronca en 2016 mesure .

Étymologie
Son nom d'espèce lui a été donné en référence au lieu de sa découverte, Aguas Blancas-Yaculica.

Publication originale
 Huber, 2000 : New World pholcid spiders (Araneae: Pholcidae): A revision at generic level.Bulletin of the American Museum of Natural History, ,  (texte intégral).

Liens externes

Notes et références

Pholcidae
Espèce d'araignées (nom scientifique)
Faune endémique d'Argentine

redpajama_set_name: RedPajamaWikipedia