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    \hbox to 6.28in { {\bf EE402 - Discrete Time Systems
	\hfill Spring 2018} }
       \vspace{4mm}
       \hbox to 6.28in { {\Large \hfill Lecture #1 \hfill} }
       \vspace{2mm}
       \hbox to 6.28in { {\it Lecturer: #2 \hfill } }
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\begin{document}

% Lecture Details
\lecture{17}{Asst. Prof. M. Mert Ankarali}

\par

\section*{Discrete-time Luenberger Observer}

In general the state, $x[k]$ of a system
is not accessible and \textit{observers, estimators, filters})
have to be used to extract this information.
The output, $y[k]$, represents the measurements
which is a function of $x[k]$ and $u[k]$.
%
\begin{align*}
  x[k+1] &= G x[k] + H u[k]
  \\
  y[k] &= C x[k] + D u[k]
\end{align*}
%
A Luenberger observers is built using a ``simulated'' model of the
system and the errors caused by the mismatched initial conditions
$x_0 \neq \hat{x}_0$ (or other types of perturbations)
are reduced by introducing output error feedback.

Let's assume that the states of the simulated system
is $\hat{x}[k]$, then the state space equation of this
synthetic system takes the form
%
\begin{align*}
  \hat{x}[k+1] &= \hat{G} x[k] + \hat{H} u[k]
  \\
  \hat{y}[k] &= \hat{C} x[k] + \hat{D} u[k]
\end{align*}
%
Note that since $u[k]$ is the input that is controlled
it is assumed to be known. If $x[0] = \hat{x}[0]$ and
when there is no model mismatch or uncertainty in the system
then we expect that $x[k] = \hat{x}[k]$ and $y[k] = \hat{y}[k]$
for all $k$. When $x[0] = \hat{x}[0]$, then we observe a
difference between the measured and predicted output
$y[k] \neq \hat{y}[k]$. The core idea in Luenberger observers
is feeding the error in the output prediction
$y[k] - \hat{y}[k]$ to the system via a linear feedback gain.
%
\begin{align*}
  \hat{x}[k+1] &= G \hat{x}[k] + H u[k] + L \left( y[k] - \hat{y}[k] \right)
  \\
  \hat{y}[k] &= C x[k] + D u[k]
\end{align*}
%
In order to understand how a Luenberger observer works and
ot choose a proper observer gain $L$, we define an error signal
$e[k] = x[k] - \hat{x}[k]$. The dynamics w.r.t $e[k]$ can be derived
as
%
\begin{align*}
  e[k+1] &= x[k+1] - \hat{x}[k+1]
        \\
     &= \left( G x[k] + H u[k] \right)
  - \left( G \hat{x}[k] + H u[k] + L \left( y[k] - \hat{y}[k] \right)
       \right)
\\
   e[k+1] &= \left( G - L C \right) e[k]
\end{align*}
%
where $e[0] = x[0] - \hat{x}[0]$ denotes the error in the initial
condition.

If the matrix $\left( G - L C \right)$ is stable then the errors
initial condition will diminish eventually. Moreover, in order
to have a good observer/estimator performance the observer
convergence should be sufficiently fast.

\section*{Observer Gain \& Pole Placement}

Similar to the state-feedback gain design,
the fundamental principle of ``pole-placement'' Observer design is that
we first define a desired closed-loop eigenvalue set and
compute associated desired characteristic polynomial.
%
\begin{align*}
 \mathcal{E}^* &= \lbrace \lambda_1^* , \ \cdots, \  \lambda_n^*
                 \rbrace
  \\
  p^*(z) &= \left( z - \lambda_1^* \right) \cdots \left( z - \lambda_n^*
         \right)
  \\
  &= z^n + a_1^* z^{n-1} + \cdots + a_{n-1}^* z + a_n^*
\end{align*}
%
The necessary and sufficient condition on arbitrary observer pole-placement
is that the system should be fully Observable. Then we tune $L$ such
that
%
\begin{align*}
  \mathrm{det} \left( z I - ( G - L C ) \right) = p^*(z)
\end{align*}
%
\subsection*{Direct Design of Observer Gain}

If $n$ is small, the most efficient method could be the direct
design.

\textbf{Example:} Consider the following DT system
%
\begin{align*}
 x[k+1] &= \left[ \begin{array}{cc} 1 & 0 \\ 0 & 2 \end{array} \right] x[k]
    + \left[ \begin{array}{c} 1 \\ 1 \end{array} \right] u[k]
\\
 y[k] &= \left[ \begin{array}{cc} 1 & -1 \end{array} \right] u[k]
\end{align*}
%
Design an observer such that estimater poles are located at
$\lambda_{1,2} = 0$ (Dead-beat Observer)

\textbf{Solution:} Desired characteristic equation can be computed as
%
\begin{align*}
  p^*(z) = z^2
\end{align*}
%
Let $L = \left[ \begin{array}{c} l_2 \\ l_1 \end{array} \right]$, then
the characteristic equation of $(G - L C)$ can be computed as
%
\begin{align*}
  \mathrm{det} \left( z I - ( G - L C ) \right) &=
  \mathrm{det} \left(
  \left[ \begin{array}{cc} z - 1 + l_2 & -l_2 \\ l_1 & z - 2 - l_1 \end{array} \right]
  \right)
\\
&= z^2 + z (l_2 - l_1 - 3) + (l_1 - 2 l_2 + 2)
\end{align*}
%
If we match the equations
%
\begin{align*}
  l_2 - l_1 &= 3
\\
  -l_1 + 2 l_2 &= 2
\\
 l_2 &= -1
\\
 l_1 &= -4
\end{align*}
%
Thus $L = \left[ \begin{array}{c} -1 \\ -4 \end{array} \right]$

\subsection*{Design of Observer Gain Using Reachable Canonical
Form}

Let's assume that the state-space representation is in observable
canonical form
%
\begin{align*}
G &= \left[ \begin{array}{ccccc} 0 & 0 & \cdots & 0 & -a_{n}
              \\ 1 & 0 & \cdots & 0 & -a_{n-1}
\\ \vdots & \vdots & \vdots & \vdots & \vdots
\\ 0 & 0 & \cdots & 0 & -a_2
    \\ 0 & 0 & \cdots & 1 & -a_1 \end{array} \right]
\quad , \quad
H = \left[ \begin{array}{c} (b_n - b_0 a_n)  \\ (b_{n-1} - b_0
             a_{n-1}) \\ \vdots \\ (b_2 - b_0 a_2) \\   (b_1 - b_0
             a_1)
\end{array} \right]
\\ C &= \left[ \begin{array}{ccccc} 0 & 0 & \cdots &  0 & 1 \end{array} \right]
\quad , \quad
D = b_0
\end{align*}
%
Let $L = \left[ \begin{array}{c} l_n \\ \vdots \\ l_1 \end{array} \right]$, then
$( G - L C )$ takes the form
%
\begin{align*}
\left( G - L C \right) &= \left[ \begin{array}{ccccc} 0 & 0 & \cdots & 0 & -a_{n}
              \\ 1 & 0 & \cdots & 0 & -a_{n-1}
\\ \vdots & \vdots & \vdots & \vdots & \vdots
\\ 0 & 0 & \cdots & 0 & -a_2
    \\ 0 & 0 & \cdots & 1 & -a_1 \end{array} \right]
- \left[ \begin{array}{c} l_n \\ \vdots \\ l_1 \end{array} \right]
\left[ \begin{array}{ccccc} 0 & 0 & \cdots &  0 & 1 \end{array}
                                                  \right]
\\
&=
\left[ \begin{array}{ccccc} 0 & 0 & \cdots & 0 & -(a_{n} + l_n)
              \\ 1 & 0 & \cdots & 0 & - (a_{n-1} + l_{n-1})
\\ \vdots & \vdots & \vdots & \vdots & \vdots
\\ 0 & 0 & \cdots & 0 & - (a_2 + l_n)
    \\ 0 & 0 & \cdots & 1 & - (a_1 + l_n) \end{array} \right]
\end{align*}
%
The characterstic equation of $G - LC$ is simply given as
%
\begin{align*}
  p(z) = z^n + ( a_1 - l_1 ) z^{n-1} + \cdots + ( a_{n-1} - l_{n-1} )
  z + ( a_{n} - l_{n} )
\end{align*}
%
Let's assume that desired $p^*(z)$ is equal to
%
\begin{align*}
  p(z) = z^n + a_1^* z^{n-1} + \cdots + a_{n-1}^*
  z + a_{n}^*
\end{align*}
%
Then, the observer gain $L$ is computed as
%
\begin{align*}
L^* = \left[ \begin{array}{c} a_n^* - a_n \\ \vdots \\ a_1^* - a_1 \end{array} \right]
\end{align*}

If the system is not in Observable canonical form, we can find a
transformation that outputs the Observable canonical form
representation
%
The Observability matrix of a state-space representation is given as
%
\begin{align*}
 O = \left[ \begin{array}{c} C \\ C G \\ \vdots \\ C G^{-1} \end{array} \right]
\end{align*}
%
Let's define a transformation matrix $Q$ as follows:
%
\begin{align*}
  Q &= \left( W O \right)^{-1} \quad , \quad x[k] = Q \bar{x}[k]
\\
 \bar{x}[k+1] &= \left[ Q^{-1} G Q \right] \bar{x}[k] + Q^{-1} H u[k]
\\
  y[k] &= C Q \bar{x}[k] + D u[k]
\end{align*}
%
where
%
\begin{align*}
  W = \left[ \begin{array}{ccccc} a_{n-1} & a_{n-2} & \cdots & a_1 & 1
               \\
a_{n-2} & a_{n-3} & \cdots & 1 & 0
\\ \vdots & \vdots & \iddots & & \vdots
\\ a_1 & 1 &  & &
    \\ 1 & 0 & \cdots &  & 0 \end{array} \right]
\end{align*}
%
Then it is given that
%
\begin{align*}
  \bar{G} = Q^{-1} G Q  &=  \left[ \begin{array}{ccccc} 0 & 0 & \cdots & 0 & -a_{n}
              \\ 1 & 0 & \cdots & 0 & -a_{n-1}
\\ \vdots & \vdots & \vdots & \vdots & \vdots
\\ 0 & 0 & \cdots & 0 & -a_2
    \\ 0 & 0 & \cdots & 1 & -a_1 \end{array} \right]
\\
  \bar{C}  = C Q & = \left[ \begin{array}{ccccc} 0 & 0 & \cdots & 0
    & 1 \end{array} \right]
\end{align*}
%
Let's compute $\bar{C} Q^{-1}$
%
\begin{align*}
\bar{C} Q^{-1} &= \bar{C} W O
\\
&= \left[ \begin{array}{ccccc} 0 & 0 & \cdots & 0
    & 1 \end{array} \right]
 \left[ \begin{array}{ccccc} a_{n-1} & a_{n-2} & \cdots & a_1 & 1
               \\
a_{n-2} & a_{n-3} & \cdots & 1 & 0
\\ \vdots & \vdots & \iddots & & \vdots
\\ a_1 & 1 &  & &
    \\ 1 & 0 & \cdots &  & 0 \end{array} \right] O^T
  \\
&= \left[ \begin{array}{ccccc} 1 & 0 & \cdots & 0 & 0 \end{array}
                                                    \right]
\left[ \begin{array}{cccc} C & C G & \cdots & C G^{-1} \end{array}
                                                    \right]
\\
&= C
\end{align*}
%
A similar approach (but longer) can be used to show
that $Q^{-1} G Q = \bar{G}$.

We know how to design a
observer gain $\bar{L}$ for the Observable
canonical form. Given $\bar{L}$, Observer gain
w.r.t. original state-space representation is
computed as
%
\begin{align*}
 L = Q \bar{L}
\end{align*}

\textbf{Example 2:} Consider the following DT system
%
\begin{align*}
 x[k+1] &= \left[ \begin{array}{cc} 1 & 0 \\ 0 & 2 \end{array} \right] x[k]
    + \left[ \begin{array}{c} 1 \\ 1 \end{array} \right] u[k]
\\
 y[k] &= \left[ \begin{array}{cc} 1 & -1 \end{array} \right] u[k]
\end{align*}
%
Design an observer using the Observable canonical form
such that estimater poles are located at $\lambda_{1,2} = 0$ (Dead-beat Observer)

\textbf{Solution:} Characteristic equation of $G$ can be derived as
%
\begin{align*}
 \mathrm{det} \left(  \left[ \begin{array}{cc} z-1 & 0 \\ 0 &
                                                              z-2 \end{array} \right]  \right)
 = z^2 - 3 z + 2
\end{align*}
%
Observability matrix can be computed as
%
\begin{align*}
 O = \left[ \begin{array}{cc} 1 & -1 \\ 1 &
                                                              -2 \end{array} \right]
\end{align*}
%
The matrix $W$ can be computed as
%
\begin{align*}
  W =  \left[ \begin{array}{cc} -3 & 1 \\ 1 & 0 \end{array} \right]
\end{align*}
%
Transformation matrix $Q$ can be computed as
%
\begin{align*}
  Q &= \left( W O^T \right)^{-1}
\\
&= \left(
\left[ \begin{array}{cc} -3 & 1 \\ 1 & 0 \end{array} \right]
\left[ \begin{array}{cc} 1 & -1 \\ 1 & -2 \end{array} \right]
\right)^{-1}
= \left(
\left[ \begin{array}{cc} -2 & 1 \\ 1 & -1 \end{array} \right]
\right)^{-1}
=
\left[ \begin{array}{cc} -1 & -1 \\ -1 & -2 \end{array} \right]
\end{align*}
%
%
Given that desired characteristic polynomial is $p^*(z) = z^2$,
$\bar{L}$ of observable canonical from can be computed as
%
\begin{align*}
  \bar{L} &= \left[ \begin{array}{c} - a_2 \\ -
                                                       a_1 \end{array}
                                                       \right]
\\
&= \left[ \begin{array}{c} -2 \\ 3 \end{array}
                                                       \right]
\end{align*}
%
Finally Observer Gain $L$ can be computed as
%
%
\begin{align*}
 L &= Q \bar{L} =
 \left[ \begin{array}{c} -1 \\ -4 \end{array} \right]
\end{align*}
%
Not surprisingly the result is same with the one found in
first example.

\section*{Closed-Loop Observer \& State-Feedback}

In the state-feedback control policy the input is ideally defined
by the following law
%
\begin{align*}
 u[k] = - K x[k]
\end{align*}
%
However, as mentioned in Observer lecture, in general we don't have
direct access to the all states of the system. In this case, we learnt
how to design an Observer/Estimator of the states. In this respect,
it is natural to assume that in a closed-loop system, the control
policy that define the input should depend on the estimated states
%
\begin{align*}
 u[k] = - K \hat{x}[k]
\end{align*}
%
However the important question how this coupling affect the
closed-loop behavior, and even deeper question can be even
use such a policy. The advantage of LTI systems is that
state-feedback gain, and observer gain can be seperatelly
designed and we guarntee a stable closed-loop performance.
In this section, we will analyze the coupled system
%
Equations of motion for the closed-loop observer \& state-feedback
based control system is given below
%
\begin{align*}
   x[k+1] &= G x[k] + H u[k]
  \\
   y[k] &= C x[k] + D u[k]
  \\
  \hat{x}[k+1] &= G \hat{x}[k] + H u[k] + L \left( y[k] - \hat{y}[k] \right)
  \\
  \hat{y}[k] &= C \hat{x}[k] + D u[k]
  \\
   u[k] &= -K \hat{x}[k]
\end{align*}
%
If we eliminate $u[k]$ and $\hat{y}[k]$ we obtain following
dynamical representation
%
\begin{align*}
   x[k+1] &= G x[k] - H K \hat{x}[k]
  \\
  \hat{x}[k+1] &= G \hat{x}[k] - H K \hat{x}[k] + L C \left( x[k] -
                 \hat{x}[k] \right)
  \\
   y[k] &= C x[k] - D K \hat{x}[k]
\end{align*}
%
Now let's replace $\hat{x}[k]$ with $e[k] = x[k] - \hat{x}[k]$
%
\begin{align*}
   x[k+1] &= ( G - H K ) x[k] + H K e[k]
  \\
  e[k+1] &= ( G - LC ) e[k]
  \\
   y[k] &= (C - D K) x[k] + D K e[k]
\end{align*}
%
Now let's defina a state for the whole system,
$z[k] = \left[ \begin{array}{c} x[k] \\ e[k] \end{array} \right]$
then the state-space representation is given by
%
\begin{align*}
  z[k+1] = \left[ \begin{array}{cc} (G - H K) & H K\\ 0_{n \times n}
                                              & (G - LC) \end{array}
                                                \right] z[k]
\\
 y[k] = \left[ \begin{array}{cc} (C - DK) & D K \end{array}
                                                \right] z[k]
\end{align*}
%
The system matrix is in block diagonal form and the eigenvalues
of this new system matrix is find by taking the union of eigenvalues
of $(G - H K)$ and eigenvalues of $(G - L C)$. Thus a seperate
pole-placement can be performed for the state-feedback controller
and the observer.

% **** This ENDS THE EXAMPLES. DON'T DELETE THE FOLLOWING LINE:
\end{document}
%


\section{Geometry}\label{sec:geometry}

Geometry is the multi-millennium evolution of attempts to measure parts of the earth. Ironically, it may be the main historical justification for the gradual axiomatization of mathematics. Completely abstract results about shapes date at least as early as in Ancient Greece. The important distinction between ancient geometry and modern geometry is the introduction of coordinates in the 17th century.

An axiomatic approach for a theory of plane and, solid figures was developed by Euclid in the third century BC. Later, Hilbert, Tarski and others independently proposed axioms that fit the requirements of modern logic systems. This is known today as \term{synthetic Euclidean geometry} and is mostly of theoretical interest because modern tools are easier to work with.

Descartes' idea of coordinates connects problems of algebra and geometry in such a way that most of today's mathematics seamlessly switches between algebraic and geometric interpretations of the same problem. The study of classical Greek geometry in terms of coordinates is known as \term{analytic geometry}.


\documentclass[a4paper]{article}
\setlength{\oddsidemargin}{-0.7cm}
\setlength{\topmargin}{-1.5cm}
\setlength{\textwidth}{16.5cm}
\setlength{\textheight}{24cm}

\usepackage{amsmath}
\usepackage{graphics}

\newcommand{\ex}[1]{E[ #1 ]}
\newcommand{\xb}{\bar{\bf x}}
\newcommand{\sgm}{\sigma^2({\bf x})}
\newcommand{\ess}{{\cal S}({\bf x})}

\title{Unbiased Estimator for Fourth Moments}
\author{Synge Todo \\
  {\it Department of Physics, University of Tokyo, Tokyo 113-0033, Japan}}

\begin{document}
\maketitle

\section{Hybrids}
Before considering unbiased estimator for the fourth moments, we introduce more hybrids moments in addition to Online Appendix~M of Ref.~\cite{Klements2009}.
\begin{align}
  \begin{split}
    \ex{x_1 x_2 \xb^2} =& \frac{1}{n^2} \ex{x_1 x_2 (\sum_{i=1}^n x_i)^2} \\
    =& \frac{1}{n^2} \ex{x_1 x_2 (x_1^2 + x_2^2 + \sum_{i=3}^n x_i^2 + 2 x_1 x_2 + 2 x_1 (\sum_{i=3}^n x_i) + 2 x_2 (\sum_{i=3}^n x_i) + \sum_{i=3}^n\sum_{j=i+1}^n x_i x_j)} \\
    =& \frac{2}{n^2} \ex{x_1^3 x_2} + \frac{n-2}{n^2} \ex{x_1 x_2 x_3^2} + \frac{2}{n^2} \ex{x_1^2 x_2^2} + \frac{4(n-2)}{n^2} \ex{x_1^2 x_2 x_3} \\ &+ \frac{(n-2)(n-3)}{n^2} \ex{x_1 x_2 x_3 x_4} \\
    =& \frac{2}{n^2} (\ess + 3 \sgm \mu + \mu^3) \mu + \frac{5(n-2)}{n^2} (\sgm + \mu^2) \mu^2 + \frac{2}{n^2} (\sgm^2+\mu^2)^2 \\ &+ \frac{(n-2)(n-3)}{n^2} \mu^4 \\
    =& \frac{2}{n^2} \ess \mu + \frac{5}{n} \sgm \mu^2 + \frac{2}{n^2} \sigma^4 + \mu^4
  \end{split} \\
  \begin{split}
    \ex{x_1^2 x_2 \xb} =& \frac{1}{n} \ex{x_1^2 x_2 (\sum_{i=1}^n x_i)} \\
    =& \frac{1}{n} \ex{x_1^3 x_2 + x_1^2 x_2^2 + (n-2) x_1^2 x_2 x_3} \\
    =& \frac{1}{n} (\ess + 3 \sgm \mu + \mu^3) \mu + \frac{1}{n} (\sgm + \mu^2)^2 + \frac{n-2}{n} (\sgm + \mu^2) \mu^2 \\
    =& \frac{1}{n} \ess \mu + \frac{(n+3)}{n} \sgm \mu^2 + \frac{1}{n} \sigma^4 + \mu^4.
  \end{split}
\end{align}

\section{Square of variance, $\sigma_{\xb}^4({\bf x})$}

Expression for square of variance, $\sigma_{\xb}^4({\bf x})$, given on p.~9 in Online Appendix~M of Ref.~\cite{Klements2009} is not correct.  The correct expression is
\begin{align}
  \begin{split}
    \sigma_{\xb}^4({\bf x}) =& \ex{(x_i-\xb)^2(x_j-\xb)^2} \\
    =& \ex{\frac{1}{n}\sum(x_i-\xb)^2\cdot\frac{1}{n}\sum(x_j-\xb)^2} \\
    =& \frac{1}{n^2} \ex{\sum_i \sum_j (x_i-\xb)^2(x_j-\xb)^2} \\
    =& \frac{1}{n^2} \ex{\sum_i (x_i-\xb)^4} + \frac{1}{n^2} \ex{\sum_i \sum_{j\ne i} (x_i-\xb)^2(x_j-\xb)^2} \\
    =& \frac{1}{n} \ex{(x_1-\xb)^4} + \frac{n-1}{n} \ex{(x_1-\xb)^2(x_2-\xb)^2} \\
    =& \frac{1}{n} \ex{(x_1-\xb)^4} + \frac{n-1}{n} \ex{x_1^2 x_2^2 - 4 x_1^2 x_2 \xb + 2x_1^2 \xb^2 + 4 x_1 x_2 \xb^2 - 4 x_1 \xb^3 + \xb^4} \\
    =& \frac{1}{n} {\cal K_{\xb}(\bf x)}
    + \frac{n-1}{n} (\sgm + \mu^2)^2 \\
    &- \frac{4(n-1)}{n} \Big[ \frac{1}{n} \ess \mu + \frac{(n+3)}{n} \sgm \mu^2 + \frac{1}{n} \sigma^4 + \mu^4 \Big] \\
    &+ \frac{2(n-1)}{n} \Big[ \frac{1}{n^2} {\cal K}({\bf x}) + \frac{2(n+1)}{n^2} \ess \mu + \frac{n+5}{n} \sgm \mu^2 + \frac{n-1}{n^2} \sigma^4 + \mu^4 \Big] \\
    &+ \frac{4(n-1)}{n} \Big[ \frac{2}{n^2} \ess \mu + \frac{5}{n} \sgm \mu^2 + \frac{2}{n^2} \sigma^4 + \mu^4\Big] \\
    &- \frac{4(n-1)}{n} \Big[ \frac{1}{n^3} {\cal K}({\bf x}) + \frac{4}{n^2} \ess \mu + \frac{6}{n} \sgm \mu^2 + \frac{3(n-1)}{n^3} \sigma^4 + \mu^4 \Big] \\
    &+ \frac{n-1}{n} \Big[ \frac{1}{n^3} {\cal K}({\bf x}) + \frac{4}{n^2} \ess \mu + \frac{6}{n} \sgm \mu^2 + \frac{3(n-1)}{n^3} \sigma^4 + \mu^4 \Big] \\
    =& \frac{1}{n} {\cal K_{\xb}(\bf x)} + \frac{(n-1)(2n-3)}{n^4} {\cal K}({\bf x})
    + \frac{(n-1)(n^3-2n^2-3n+9)}{n^4} \sigma^4({\bf x}) \\
    =& \frac{n-1}{n^3} \Big[ (n-1) {\cal K}({\bf x}) + (n^2-2n+3) \sigma^4({\bf x}) \Big].
  \end{split}
  \label{eqn:square_of_variance}
\end{align}
In the last line in Eq.~(\ref{eqn:square_of_variance}), we use the expression for the fourth sample central moment:
\begin{align}
  \begin{split}
    {\cal K_{\xb}(\bf x)} = \ex{(x_i-\xb)^4} = \frac{n-1}{n^3} \Big[ (n^2-3n+3) {\cal K}({\bf x}) + (6n-9)\sigma^4({\bf x}) \Big],
  \end{split}
  \label{eqn:fourth_sample_central_moment}
\end{align}
which is given on p.~8 in Online Appendix~M of Ref.~\cite{Klements2009}.

\section{Unbiased estimator for $\sigma^4({\bf x})$, ${\cal K}({\bf x})$, and fourth cumulant}

By solving Eqs.~(\ref{eqn:square_of_variance}) and (\ref{eqn:fourth_sample_central_moment}), we obtain the unbiased estimators for the square of the variance, the fourth central moment, and the fourth cumulant as
\begin{align}
  \begin{split}
    \sigma^4({\bf x}) = \frac{n}{(n-1)(n-2)(n-3)} \Big[ (n^2-3n+3) \sigma_{\xb}^4({\bf x}) - (n-1) {\cal K_{\xb}(\bf x)} \Big]
  \end{split} \\
  \begin{split}
    {\cal K}({\bf x}) = \frac{n}{(n-1)(n-2)(n-3)} \Big[ (n^2-2n+3) {\cal K_{\xb}(\bf x)} - (6n-9) \sigma_{\xb}^4({\bf x}) \Big]
  \end{split} \\
  \begin{split}
    \kappa_4({\bf x}) = {\cal K}({\bf x}) - 3 \sigma^4({\bf x})
    = \frac{n^2}{(n-1)(n-2)(n-3)} \Big[ (n+1) {\cal K_{\xb}(\bf x)} - 3(n-1) \sigma_{\xb}^4({\bf x}) \Big],
  \end{split}
\end{align}
respectively.

\section{Numerical tests}

We test numerically the above expressions for the Bernoulli distribution ${\cal B}(p=3/4)$ and the normal distribution ${\cal N}(\mu=2,\sigma=3/2)$. We evaluate Eqs.~(1)--(7) for the sample size $n=4,8,16,32,\cdots,1024$. For each sample size, we take the average over 65536 sample sets and estimate the error bar. The results are shown in Figs.~1--7.

\bibliographystyle{naturemag}
\bibliography{fourth-moment}

\clearpage

\begin{figure}
  \begin{center}
    \resizebox{0.47\textwidth}{!}{\includegraphics{plot-eq1-b.pdf}}
    \resizebox{0.47\textwidth}{!}{\includegraphics{plot-eq1-n.pdf}}
  \end{center}
  \caption{Sample-size dependence of $\ex{x_1 x_2 \xb^2}$ for the Bernoulli distribution ${\cal B}(p=3/4)$ (left) and the normal distribution ${\cal N}(\mu=2,\sigma=3/2)$ (right). The green squares denotes the numerical results and the purple line denotes Eq.~(1) calculated by using the exact moments.}
\end{figure}

\begin{figure}
  \begin{center}
    \resizebox{0.47\textwidth}{!}{\includegraphics{plot-eq2-b.pdf}}
    \resizebox{0.47\textwidth}{!}{\includegraphics{plot-eq2-n.pdf}}
  \end{center}
  \caption{Sample-size dependence of $\ex{x_1^2 x_2 \xb}$ for the Bernoulli distribution ${\cal B}(p=3/4)$ (left) and the normal distribution ${\cal N}(\mu=2,\sigma=3/2)$ (right). The green squares denotes the numerical results and the purple line denotes Eq.~(2) calculated by using the exact moments.}
\end{figure}

\begin{figure}
  \begin{center}
    \resizebox{0.47\textwidth}{!}{\includegraphics{plot-eq3-b.pdf}}
    \resizebox{0.47\textwidth}{!}{\includegraphics{plot-eq3-n.pdf}}
  \end{center}
  \caption{Sample-size dependence of $\sigma_{\xb}^4({\bf x})$ for the Bernoulli distribution ${\cal B}(p=3/4)$ (left) and the normal distribution ${\cal N}(\mu=2,\sigma=3/2)$ (right). The green squares denotes the numerical results and the purple line denotes Eq.~(3) calculated by using the exact moments.}
\end{figure}

\begin{figure}
  \begin{center}
    \resizebox{0.47\textwidth}{!}{\includegraphics{plot-eq4-b.pdf}}
    \resizebox{0.47\textwidth}{!}{\includegraphics{plot-eq4-n.pdf}}
  \end{center}
  \caption{Sample-size dependence of ${\cal K_{\xb}(\bf x)}$ for the Bernoulli distribution ${\cal B}(p=3/4)$ (left) and the normal distribution ${\cal N}(\mu=2,\sigma=3/2)$ (right). The green squares denotes the numerical results and the purple line denotes Eq.~(4) calculated by using the exact moments.}
\end{figure}

\begin{figure}
  \begin{center}
    \resizebox{0.47\textwidth}{!}{\includegraphics{plot-eq5-b.pdf}}
    \resizebox{0.47\textwidth}{!}{\includegraphics{plot-eq5-n.pdf}}
  \end{center}
  \caption{Results for the square of the variance, $\sigma^4({\bf x})$, by using the unbiased estimator~(5) for the Bernoulli distribution ${\cal B}(p=3/4)$ (left) and the normal distribution ${\cal N}(\mu=2,\sigma=3/2)$ (right). The green squares denotes the numerical results and the purple line denotes the exact value.}
\end{figure}

\begin{figure}
  \begin{center}
    \resizebox{0.47\textwidth}{!}{\includegraphics{plot-eq6-b.pdf}}
    \resizebox{0.47\textwidth}{!}{\includegraphics{plot-eq6-n.pdf}}
  \end{center}
  \caption{Results for the fourth central moment, ${\cal K}({\bf x})$, by using the unbiased estimator~(6) for the Bernoulli distribution ${\cal B}(p=3/4)$ (left) and the normal distribution ${\cal N}(\mu=2,\sigma=3/2)$ (right). The green squares denotes the numerical results and the purple line denotes the exact value.}
\end{figure}

\begin{figure}
  \begin{center}
    \resizebox{0.47\textwidth}{!}{\includegraphics{plot-eq7-b.pdf}}
    \resizebox{0.47\textwidth}{!}{\includegraphics{plot-eq7-n.pdf}}
  \end{center}
  \caption{Results for the fourth cumulant, $\kappa_4({\bf x})$, by using the unbiased estimator~(6) for the Bernoulli distribution ${\cal B}(p=3/4)$ (left) and the normal distribution ${\cal N}(\mu=2,\sigma=3/2)$ (right). The green squares denotes the numerical results and the purple line denotes the exact value.}
\end{figure}

\end{document}


\label{ch:3}
In this section, we discuss different learning-based approaches that have been explored in recent years. To keep the discussion relevant to the contribution of the thesis, the discussion is focused on bodies work that fall under the banner of reinforcement learning (RL) or inverse reinforcement learning (IRL) based methods. Both RL and IRL have been widely researched in the field of socially-aware navigation. While these methods differ at their core, the problem definition for both the cases is mostly similar but with a key difference. For both RL and IRL setting, the problem at hand is expressed in the form of a Markov decision process (MDP). We define the problem below.
\subsection*{Problem definition:}
A Markov decision process is a discrete-time stochastic control process and can be represented as a 5 tuple
($\mathcal{S}$,$\mathcal{A}$,T,$\gamma$, $\mathcal{R}$), where,
\begin{itemize}
    \item $\mathcal{S}$ is the set of all possible states.
    \item $\mathcal{A}$ is the set of all possible actions.
    \item T is the transition dynamics.
    \item $\gamma$ is the discount factor.
    \item $\mathcal{R}$ is the set of rewards.
\end{itemize}
In reinforcement learning problems, the goal is to find a policy $\pi$: a function that maps a state to an action that maximizes the expected reward obtained.\\
In contrast, in inverse reinforcement learning problems, there is no reward function. Instead, a set of expert demonstrations is provided. This can be denoted by, D = \{$\tau_1$, $\tau_2$, ... \}. The goal is to generate a reward function $\mathcal{R}$, which best explains the expert behavior and a policy that maximizes the reward function.

\section{RL based approaches}
 With their remarkable success in the field of video games and a suite of tasks of similar anatomy, RL has been one of the go-to tools in the field of social navigation in recent years. \\
Chen et al. \cite{chen_socially_2017} address this problem by focusing on what not to do, rather than what to do. They introduce a reward function crafted to induce social norms in the behavior of the agent. The reward function primarily focuses on three different aspects of interaction: overtaking, passing, and crossing as shown in \autoref{fig:chen_socially_crossing}
 \begin{figure}[!htbp]
     \centering
    \includegraphics[width=.6\linewidth]{figures/chapter2_rl_based_approach}
    \caption{Symmetric pairwise collision avoidance in a time efficient manner by the red agent in the three interaction scenarios. The interactions in the top and bottom row are often referred as left and right-handed rules respectively.}
    \label{fig:chen_socially_crossing}
 \end{figure}
%\thcomment{In their paper they write : "This work notes that social norms are one of the many
%	ways to resolve a symmetrical collision avoidance scenario,
%	as illustrated in... "}
They claim that symmetry is one of the driving factors that incorporate social behavior. The reward function they define is given by \autoref{eq:chen_socially_2017_reward_function}.
\begin{align}
\label{eq:chen_socially_2017_reward_function}
\begin{split}
R_{norm}(s^{jn}, a) &=  q_nI(s^{jn} \in S_{norm})\\
s.t. \qquad S_{norm}&=  S_{pass} \cup S_{ovtk} \cup S_{cross}\\
S_{pass} &= \{s^{jn} | d_g > 3, 1 < \tilde{p_x} < 4,
			  -2 < \tilde{p_y} < 0, |\tilde{\phi} - \psi| > 3\pi/4 \} \\
S_{ovtk} &= \{s^{jn} | d_g > 3, 0 < \tilde{p_x} < 3, |v| > |\tilde{v}|
			  0 < \tilde{p_y} < 1, |\tilde{\phi} - \psi| < \pi/4 \} \\
S_{cross} &= \{s^{jn} | d_g > 3, \tilde{d_a} < 2,  \tilde{\phi_{rot}} > 0,
			  -3\pi/4 < \tilde{\phi} - \psi < -\pi/4  \}
\end{split}
\end{align}\\
where, $I$ is an indicator function. $S_{pass}$, $S_{ovtk}$, and $S_{cross}$ are the set of states that fall under the category of passing, overtaking and crossing respectively, $s^{jn}$ is the joint state of the agent and its neighbor, $a$ is the action, $q_{n}$ is a scalar penalty, $d_{g}$ is the distance of the goal from the agent, $d_{a}$ is the distance to the neighboring agent, $\tilde{\phi}$ is the neighboring agent's orientation, $\phi$ is the agent's orientation, $\tilde{\phi_{rot}}$ is the relative rotation angle between the two agents and $\tilde{p_x}$ and $\tilde{p_y}$ are the $x$ and $y$ coordinates of the neighboring agent in the target agent's reference frame. The reward function in \autoref{eq:chen_socially_2017_reward_function} biases the agent towards adhering to the right-hand rules.
\\

A more recent work by the same authors, "Collision avoidance in pedestrian-rich environments with deep reinforcement learning" \cite{everett_collision_2019}, %\thcomment{Yes, it is the title of their paper.}
address drawbacks of their previous work: the inability to tackle a variable number of agents in the environment. They introduce a long short-term memory (LSTM) \cite{hochreiterLongShortTermMemory1997} network architecture as shown in \autoref{fig:everett_lstm_network} that takes as input the information from the nearby pedestrians in a sequence which eliminates the need for specifying a limit on the number of nearby-agents the method can handle.
\begin{figure}[!htbp]
	\centering
	\includegraphics[width=0.6\linewidth]{figures/everett}
	\caption{LSTM based network architecture to account for a variable number of nearby agents.}
	\label{fig:everett_lstm_network}
\end{figure}
 For this work, they select a relatively simple sparse reward function that focuses on collision avoidance and maintaining distance from others as shown in \autoref{eq:everett_reward_function}\\
\begin{align}
\label{eq:everett_reward_function}
R(S^{jn}, \mathbf{u}) =
\begin{cases}
	 1 & \text{if \quad $p=p_g$} \\
	 -0.1+d_{min}/2 & \text{if \quad 0 < $d_{min}$ < 0.2} \\
	 -0.25 & \text{if \quad $d_{min}$ < 0}\\
	 0 & \text{otherwise},
\end{cases}
\end{align}
where $\mathbf{u}$ is the action, and $s^{jn}$ is the joint state representing all the agents in the world, $p$ and $p_{g}$ are the position of the agent and the goal respectively, and $d_{min}$ is the distance to the nearest neighboring agent.
%\textbf{Conclusion}\\
\subsection*{Conclusion}

Reward function is the most succinct definition of the task at hand \cite{abbeel_apprenticeshiplearning_2004}, making it a vital component that determines the final performance of a RL method. We can see similarities between RL methods and other classical approaches. RL method still involve reward engineering, which can get pretty complex even when considering a handful of simple interactions. The reward function by Everett et al. \cite{everett_collision_2019} focuses solely on collision avoidance, and Chen et al. \cite{chen_socially_2017} incorporate an element of naturalness to the mix using symmetry. But it takes into account a very limited number of scenarios: just 3 possible interaction scenarios, which is a small subset of the very many implicit intentions and explicit decisions a human takes while navigating.

Reinforcement learning, as a class of methods, has been widely successful in solving various complex control tasks including, but not limited to, video games making it one of the more preferred choices to tackle the problem of social navigation. \par
Using RL in this particular setting however necessitates the formulation of a reward function that can correctly capture the `social and cultural' characteristics. As noted earlier, coming up with such a set of rules can be difficult and daunting as `social norms' are not always explicit and can vary widely across different societies and even based on a given situation.\\

\section{IRL based approaches}
%\textbf{Learning approaches meet probabilistic road map style path planner.}
Inverse reinforcement learning bypasses the step of engineering the reward function, instead, it aims at recovering the underlying reward function using expert demonstrations, making it an attractive alternative to RL based methods. IRL has been extensively explored to address the problem of social navigation. \\
Vasquez et al. \cite{vasquez_inverse_2014}, present a comprehensive study on the effect of different feature representations and IRL methods on the performance of an agent in the task of social navigation. They examine two IRL methods: max-margin IRL and max-entropy IRL. Max-margin IRL aims at finding a set of weight parameters that maximizes the margin between the feature expectation of the expert demonstrations and that of the trained policy. The goal of max-entropy IRL is to find a set of weight parameters that maximizes the likelihood of the expert demonstrations, where the probability of occurrence of a given roll-out is proportional to the exponential of the reward obtained by it. \\
For the feature representations, they create a pool of measurements garnered from the state of the agent itself and the state of the other entities in the environment and combine these measurements to create 3 feature representations. \\
They test these on a Robot Operating System or, in short, ROS-based pedestrian simulator on 3 scenarios: airport gate, crossing hallway, and intersection. Expert demonstrations for these scenarios are obtained through tele-operation. They find that the performance across the two learning methods are similar. The feature representation, on the other hand, plays a major role in the final performance of the agent hence they come to the conclusion that spending time on modeling the feature representations or working to come up with learning methods that aid in the simplification of building the feature representations might be the way to go.
\\
\par
Kim and Pineau \cite{kim_socially_2016}, present a way to automate the navigation of a wheelchair in a social setting. Their navigation framework comprises of $3$ components, the feature extractor, the IRL module, and the path planner.\\
%\textbf{Feature extractor}
The features are generated from the readings of an RGB-d sensor (3D point cloud) mounted on the robot. The area around the agent is divided into 3D blocks or cells and the features are calculated for each of these spatial cells.
The authors calculate $4$ features namely,
crowd density, speed, velocity, and distance to the goal.
While the calculation of the crowd density and the distance to the goal is straight forward, the calculation of the speed and velocity from 3D point clouds are more involved and the authors use an RGB-d optical flow method based on Farneback RGB optical flow \cite{farneback_optical_flow}.
The result is a $12$-dimensional binary feature vector for each grid cell.
%\textbf{IRL module}
The authors use maximum-a-posteriori Bayesian inverse reinforcement learning (MAP-BIRL) \cite{choi_MAP-BIRL_2011} to calculate the cost function for socially acceptable navigation. The cost of a state is given by
\begin{align}
C(s,a) &=w \cdot \phi(s,a)
\end{align}
where, $\phi(s,a)$ is the feature representation of the state $s$ and $w$ is the weight vector learned from MAP-BIRL.
The weight vector is obtained by obtaining the maximum-a-posteriori (MAP) inference of the following expression:
\begin{align}
L(w) = \sum^M_{m=1} \sum^{H}_{h=1}log[\frac{\exp \mathcal{Q}^{*}_m(s^m_h, a^m_h)}{\sum_{a\in A} \exp(\mathcal{Q}_m^*(s_h^m,a))}]
\end{align}
where, $M$ is the number of trajectories in the expert demonstration, $H$ is the number of states in each trajectory, $s_{h}^{m}$ and  $a_{h}^{m}$ is the state observed and action taken at time step $h^{th}$ of trajectory $m^{th}$ respectively, and $\mathcal{Q}^{*}_m(s^m_h, a^m_h)$ is the cumulative discounted future reward associated with that state-action pair.\\
For path planning, the authors maintain a hierarchical path planner consisting of 3 parts: a global planner, a local planner, and a collision detector.
The global planner chalks out a global path from the starting position of the robot to the goal state. The entire path is broken down into multiple sub-goals and the responsibility of moving from one sub-goal to the next falls on the local planner. During the run, the planner maintains a collision detector based on handcrafted rules.
The authors test their method on different scenarios including one that involves the robot operating in a crowded corridor.\\

Fahad et. al. \cite{fahad_learning_2018}, present a navigation framework to train agents from demonstrations using maximum entropy deep IRL where the objective is to maximize the likelihood of the states visited by the expert.
This is achieved by dividing each demonstration, or trajectory, in this case, into a sum of the individual states encountered by the expert. and minimizing the difference in the state visitation frequency (SVF) of the agent and the expert.\\
The authors present a feature representation consisting of 4 parts:
\begin{enumerate}
    \item Social Affinity Map(SAM): This captures the motion of the pedestrians in the vicinity of the agent. The area near the agent is divided into two concentric circles. The region inside the inner circle is then divided into 4 parts, and the region between the inner and outer circle is divided into 6 parts as shown in \autoref{fig:SAM_local_bins}
    Information from each of these 10 areas (or bins) is then expressed using a 6-dimensional vector that captures the average velocity of the pedestrians in a given bin.
    \begin{figure}
    	\centering
    	\includegraphics[width=0.5\linewidth]{figures/SAM_local_bins.png}
    	\caption{Division of the area around the agent into spatial bins as done in the SAM features. The arrow marks the direction of heading of the agent.}
    	\label{fig:SAM_local_bins}
    \end{figure}
    \item Density feature: Helps provide an idea of how dense the vicinity of the agent is. It is calculated by adding up the number of pedestrians present in the spatial bins (calculated above). Once calculated they are discretized based on some predefined threshold.
    \item Distance feature: Captures the distance between the agent's current location and the goal.
    \item Default Cost feature: Introduced to balance the rest of the other features. This has been proposed by various other works in the past.

\end{enumerate}
The authors train and test their procedure using data collected in the ATC business center in Osaka, Japan.
From these experiments, they show that the method is not only capable of producing a general navigator to negotiate the crowd, but also an agent that show traits of pedestrian behaviors like collision-avoidance and following the leader.

\subsection*{Conclusion}
Both Vasequez et al. \cite{vasquez_inverse_2014} and Fahad et al. \cite{fahad_learning_2018} assume availability of the transition dynamics of the environment which is difficult to obtain in a real-world environment. They also test their models on fairly restrictive scenarios. Vasquez et al. test on 3 specific interaction scenarios: an airport gate, a crossing hallway and an intersection, and, Fahad et al. select a section of a corridor inside a mall. While it represents data taken from actual pedestrians instead of simulation, they lack the richness of human interaction that comes from an open space that puts minimal restriction on the flow of the crowd. Kim et al. \cite{kim_socially_2016} take a different approach and use a layered control architecture, each responsible for a specific part of the navigation. It is unclear as to how much the different control modules interfere in each other's decision and how this affects the overall performance of the method. For example, in an overcrowded environment if the collision detector module keeps over ridding control from the the local planner, then what percentage of the resulting trajectory will be the output of the collision-avoidance module compared to that of the IRL planner.
\par
Given the nature of the task, IRL based methods seem to be a promising avenue to explore. But the problem of social navigation is far from solved and many issues still need addressing. IRL methods are highly dependent on the design of the feature representation used in the algorithm, and a lot of time and effort are needed towards optimizing a set of features that would perform well. Moreover, for most of the existing work, not a lot has been discussed about the generalization of the agent. Most of the experiments are either conducted in a relatively simple environment, like a narrow hallway, or address some specific aspect of crowd navigation like negotiating intersections.
%\begin{comment}
%\textbf{Collection of expert trajectory might be expensive}
%\textbf{The generalization of the method is not that great for now}
%\textbf{The dependence of IRL on feature engineering: IRL methods are highly dependent on the design of the feature representation used in the algorithm, and a lot of time and effort are needed towards optimizing a set of features that would perform well.}
%
%The primary advantage of IRL over the other methods discussed, both model-based and data-driven methods, is that there is no need to specify a handcrafted reward function built to induce certain kinds of behavior within the agent. Most of the current work in this domain primarily focuses on capturing the expert's behavior, thus encapsulating the 'naturalness' in the movements during navigation.
%Disadvantages:
%Comment of IRL and feature engineering.
%Getting expert demonstrations are more expensive compared to RL.
%Hard to generalize.
%\end{comment}


\subsection{Principle of LIGO}

When the gravitational wave reach us, without a doubt, they are only very weak perturbations on our local flat space. Be that as it may, they will provide information about the strong-field regions where they began. it will additionally permit us to decide the wave properties of the gravitational radiation—for ex-sufficient, their spread speed and polarization states \cite{barish1999ligo}. The essential construction of LIGO's interferometers differs a little from the interferometer that Michelson planned more than 125 years prior, however for certain additional highlights. The visible pattern occurring where the coherent waves intersect is simply an "interference" pattern.\cite{collaboration2015advanced}

\subsubsection{Interference Pattern}

In  nature,  the  peaks  and  troughs  of  one  wave cannot absolutely  meet  the  peaks or  troughs  of  another  wave. Regardless  of  how in-sync they are once they merge, the peak of the wave coming out from the interference always equals the sum of the heights of the merging waves on every point wherever they are physically interacting. What  dictates how  well-aligned  the  beams are once  they  merge  is  the path length  they travel before merging. So the core principle of LIGO is interference of light. When the path difference between two light wave is equal to integral multiple of wavelength, then constructive interference occurs where the resultant light will have maximum brightness. And if the path difference is equal to half-integral multiple of wavelength, then destructive interference occurs and resultant light will have minimum brightness. \\

If the beams travel precisely the same distance, their light waves will be absolutely aligned such that they lead to total destructive interference (LIGO is designed to get total destructive interference if no gravitational waves are detected).  But if the lasers don’t travel identical distances, their light waves are not any longer in synchronize as they merge, which implies no light, a bit light, or a light as bright because the original laser beam reaches the photodetector.  And if the arms are changing length over time, a flicker appears as the beams suffer a variety of interference. This time difference is manifested within the interference pattern once the two laser beams superimpose on the path to the photodetector, which can quantify stage movements to ten-billionths of an interference fringe.\cite{barish1999ligo}\\

\begin{figure}[htpb]
    \centering
    \includegraphics[scale=0.585]{images.tex/interference.jpg}
    \caption{Interference. Source :- \href{https://www.tulane.edu/~sanelson/eens211/interference_of_light.htm}{Interference Phenomena by Prof. Stephen A. Nelson}}
\end{figure}

\subsubsection{Differential mode of vibration}

The gravitational waves result in the space to stretch in a direction, at the same time,  compress in a direction perpendicular to it. In LIGO, this results in one arm getting longer whereas the opposite gets shorter, then the other way around, back and forth as long because the wave is passing.  The technical term for this motion is “Differential Arm” motion, or differential displacement, since the arms are at the same time are differing in lengths in opposing ways in which, or deferentially. So because the lengths of the arms differ, thus too will the total path traveled by every laser beam.

\begin{figure}[h]
    \centering
    \includegraphics[scale=0.52]{images.tex/Interferometer.jpg}
    \caption{Michelson interferometer.\; Source :- \href{https://www.sciencenews.org/article/trio-wins-physics-nobel-prize-gravitational-wave-detection}{Sciencenews.org}}
\end{figure}

So because the lengths of the arms change, so too does the space traveled by each beam . A beam travelling in the shorter arm will return to the beam splitter before the beam which is ravelling in the extended arm, then things switches because the arms oscillate between being longer and shorter. Arriving at different times, then the LASER does not meet nicely when recombined at the beam splitter. Instead, they shift in and out of alignment or "phase" as they merge. Unlike optical or radio telescopes, LIGO doesn't see electromagnetic waves. It doesn't need to because gravitational waves aren't a part of the spectrum. In fact, electromagnetic wave is so unimportant to LIGO that its detector components are completely isolated and sheltered from the surface world.

\pagebreak