text:
\section{Introduction} \label{sec:Intro}
In this paper we consider the problem of quickest detection of a change in correlation between
variables in a sequence of high-dimensional vectors. We are also interested in isolating or identifying
the variables whose magnitude correlation with other variables has changed the most. We refer to such variables as \textit{hubs}.
A precise definition will be given below.

The problem of correlation detection and
estimation is an important one in science and engineering. The problem
is particularly difficult when the data involved is high-dimensional; see \cite{pourahmadi2013high},
\cite{fan2016overview}, and \cite{cai2016estimating}
for a survey of the literature. We are interested in the sequential version of the problem where the variables are initially uncorrelated or independent,
and due to some event, the correlation between the variables abruptly changes. This has applications
in problems of fault detection, anomaly detection, detection of changes in time-series data or financial data, etc.
For simplicity, we restrict our attention to multivariate Gaussian random vectors. However, the results in this paper
are application to the broader class of elliptically contoured distributions \cite{bane-firo-hero-isit15}.

If the pre-change parameter are known (means and variances of the uncorrelated variables),
the classical approach to sequential detection of a change in the covariance matrix of a sequence of Gaussian random vectors
is to use the generalized likelihood ratio (GLR) based Cumulative Sum (CUSUM) method of Lorden \cite{lord-amstat-1971}.
In this algorithm one sequentially estimates the covariance of the data and applies classical quickest change detection (QCD)
test using the estimated covariance.
But, this algorithm works only for low dimensional data. One approach to handle high-dimensional data,
is to make assumptions about the structure of the covariance of the matrix, like sparsity, and then estimate the high-dimensional covariance matrix.
Although this approach is appealing, it is generally hard to design sequential tests based on such an approach.
When the pre-change parameters are not known, as is the case in this paper, the problem is even harder. For low dimensional data, one can use a double-GLR approach \cite{lai-jrss-1995},
but optimality of such tests is still under investigation.

In this paper we take a random matrix based approach to solving this problem. We collect the sequence of
high-dimensional vectors in batches
of successive samples from which we form a random matrix.
We then define summary statistics for the
random matrix and obtain approximations for the asymptotic distribution for the summary statistics as the dimension
of the vector goes to infinity. The asymptotic distribution we obtain belongs to a one-parameter exponential family, and
a change in correlation of the random vectors induces a change in the parameter of the random matrix.
We use this fact to detect the change in correlation. This works differs from our effort in \cite{bane-firo-hero-isit15}
in that we also define a family of local summary statistics, one for each variable, and obtain their approximate asymptotic
distributions. We use these local summary statistics to isolate or identify variables that have seen the most amount of change
in correlation with other variables. This fault isolation process is what we call hub discovery.
We obtain asymptotic expressions for delay and false alarm
for our proposed procedure, and show via numerical results that the test is consistent in hub discovery.

\section{Problem Description}\label{sec:Prob}
We have a sequence of samples of independent and identically distributed (i.i.d.)
Gaussian $p$-variate random vectors $\{\mathbf{X}(k)\}$ with densities
\begin{equation}\label{eq:ellipticalDef}
f_{\mathbf{X}(k)}(\mathbf{x}) = \frac{1}{ \sqrt{2\pi}^p |\Sigma_k|^{p/2}} e^{\left(-\frac{1}{2} (\mathbf{x}-\boldsymbol{\mu}_k)^T\boldsymbol{\Sigma}_k^{-1}(\mathbf{x}-\boldsymbol{\mu}_k) \right)}.
\end{equation}
For some time parameter $\gamma$ the samples are assumed to have common covariance
parameter $ \mathbf{\Sigma}_0=D_p$ for $k < \gamma$, where $D_p$ is a digonal matrix,
and common covariance parameter $\mathbf{\Sigma} \neq D_p$ and for $k\geq \gamma$.
$\gamma$ is called the change point and the pre-change and post-change distributions of $\mathbf{X}(k)$
are denoted by $f_{\mathbf{X}}^0$ and  $f_{\mathbf{X}}^1$, respectively.
No assumptions are made about the mean parameter $\boldsymbol{\mu}_k$, and can take different
values for different $k$.
More specifically, the change-point model is described by:
\begin{equation}\label{eq:PreChg_PstChg_Den}
\begin{split}
\mathbf{X}(k) &\sim f_{\mathbf{X}}^{0}(\mathbf{x}) = \frac{1}{ \sqrt{2\pi}^p |\Sigma_0|^{p/2}} e^{-\frac{1}{2}(\mathbf{x}-\boldsymbol{\mu}_k)^T\boldsymbol{\Sigma}_0^{-1}(\mathbf{x}-\boldsymbol{\mu}_k)}, \\
&\; \hspace{7cm} k < \gamma, \\
       &\sim f_{\mathbf{X}}^{1}(\mathbf{x}) = \frac{1}{ \sqrt{2\pi}^p |\Sigma|^{p/2}} e^{-\frac{1}{2}(\mathbf{x}-\boldsymbol{\mu}_k)^T\boldsymbol{\Sigma}^{-1}(\mathbf{x}-\boldsymbol{\mu}_k)}, \\
       &\hspace{7cm} k \geq \gamma
\end{split}
\end{equation}
Thus, in the above change point model, the variables are \textit{uncorrelated before change}, and \textit{correlated after change}.
The objective is to detect this change in correlation as quickly as possible, while avoiding false alarms,
and also correctly isolating the variables that have experienced the most change in their correlation level (to be made precise below).  The mean parameters $\{\boldsymbol{\mu}_k\}$, the diagonal entries in $\boldsymbol{\Sigma}_0$ and the post-change
covariance matrix $\boldsymbol{\Sigma}$, are \textit{unknown}.

A decision-maker sequentially acquires the samples $\{\mathbf{X}(k)\}$.
At each time point $k$ the decision-maker decides to either stop sampling,
declaring that the change has occurred, i.e., $k \geq \gamma$, or to continue sampling.
The decision to stop at time $k$ is only a function of $(\mathbf{X}(1), \cdots, \mathbf{X}(k))$.
Thus, the time at which the decision-maker decides to stop sampling, say $\tau$, is a stopping time
for the vector sequence $\{\mathbf{X}(k)\}$.
At the stopping time $\tau$, the decision maker incurs a delay of $\tau-\gamma$, if $\tau \geq \gamma$.
If $\tau < \gamma$, we say that we have an event of false alarm.
At time $\tau$, using the available data $\left(\mathbf{X}(1), \cdots, \mathbf{X}(\tau)\right)$, the decision maker
also has to identify or isolate variables that have undergone the most amount of change in correlation.
We refer to such variables as hubs. We now formalize the notion of a hub.

For a non-diagonal covariance matrix $\boldsymbol{\Sigma}$ with correlation coefficients $\{\rho_{ki}\}$ define
\begin{equation}\label{eq:V_Sigma}
V_k(\boldsymbol{\Sigma}) = \max_{i \neq k} |\rho_{ki}|, \quad \mbox{for}\; k \in \{1, \cdots, p\},
\end{equation}
to be the maximum magnitude correlation coefficient for the $k$th variable.
Hubs are defined as
\begin{equation}\label{eq:Hub_Hc}
\begin{split}
\mathsf{H} &=\{k: V_k = \max_{1 \leq j \leq p} V_j \}.
\end{split}
\end{equation}
Define the correlation graph $\mathcal{G}(\boldsymbol{\Sigma})$
associated with the post-change correlation matrix $\boldsymbol{\Sigma}$
as an undirected graph with $p$ vertices, each representing a variable in the vector $\mathbf{X}$.
An edge is present between vertices $i$ and $j$ if the magnitude of the correlation coefficient $\rho_{ij}$
between the $i^{th}$ and $j^{th}$ components of the random vector $\mathbf{X}$
is nonzero, i.e., if $|\rho_{ij}| > 0$, $i \neq j$. The correlation graph $\mathcal{G}(\boldsymbol{\Sigma})$ can be treated as a weighted
graph with weight on edge between node $i$ and $j$ equal to $|\rho_{ij}|$, provided the latter is nonzero.
Then, hubs are the nodes with highest maximum weights.

There are also other interesting ways to define
hubs. For example, one can select a threshold $\rho$ and define hubs as those variables for which $V_k > \rho$.
Another way to define hubs is to redefine correlation graphs with edges present only of $|\rho_{ij}| \geq \rho$, and
then define hubs as the nodes with the highest degree. Results on hub discovery for these other type of hubs will be reported elsewhere. In this paper we only discuss hub discovery for hubs defined in \eqref{eq:Hub_Hc}.

At time $\tau$, let
\begin{equation}
\mathcal{D}\left(\mathbf{X}(1), \cdots, \mathbf{X}(\tau)\right) \in 2^{\{1,\cdots,p\}}
\end{equation}
 be the decision function
that selects a subset of the $p$ variables as the hubs.
The decision-maker's objective is to choose a test, i.e. $\{\tau, \mathcal{D}\}$, to
detect this change in correlation of the random vectors, as quickly as possible,
subject to a constraint on the rate of false alarms, and on the rate of false isolation.

A problem formulation that captures the above performance trade-offs is the following.
We seek $\{\tau, \mathcal{D}\}$ to solve
\begin{equation}\label{prob:Pollak}
\begin{split}
 \min_{\{\tau, \mathcal{D}\}}   &  \quad  \sup_{\gamma\geq 1}  \; \mathsf{E}_\gamma[\tau-\gamma| \tau \geq \gamma] \\
 \mbox{subj. to}  &  \quad \mathsf{E}_\infty[\tau] \geq \beta, \\
 \mbox{and }  &  \quad  \sup_\gamma\mathsf{P}_\gamma [\mathcal{D} \neq \mathsf{H} ]\leq \zeta,
\end{split}
\end{equation}
where $\mathsf{E}_\gamma$ is the expectation with respect to the probability measure under which the change occurs at $\gamma$,
$\mathsf{E}_\infty$ is the expectation when the change never occurs, and
$\beta$ and $\zeta$ are user-specified constraints on the mean time to false alarm and the probability
of false isolation, respectively.

This problem formulation is also studied in \cite{tart-sqa-2008} in the context of detecting and isolating
a single affected stream out of a finite number of independent streams of observations, when the exact
pre- and post-change distributions are known.
Even with independent streams, and known distributions, the author in \cite{tart-sqa-2008} obtained a solution for only
a relaxed version of the above problem, by replacing the false isolation probability metric by $\mathsf{P}_1 [\mathcal{D} \neq \mathsf{H} ]$.

In our case there are three major challenges towards developing a test or extending/applying the results from \cite{tart-sqa-2008}:
\begin{enumerate}
\item The means $\{\mu_k\}$, covariance matrices $\boldsymbol{\Sigma}_0$ and $\boldsymbol{\Sigma}$ are unknown
\item The high dimension $p$ that is much larger than the total number of samples we will possibly accumulate.
\item The streams (one for each dimension) in our observation models are not independent: In fact, we are trying to detect a change
in the level of dependence between the streams.
\end{enumerate}

In the sequential analysis literature, for such intractable problems,
an alternative approach often taken is to propose a sub-optimal test,
and provide a detailed performance analysis of the proposed test. The performance analysis is often used to
design the test (to choose thresholds). The effectiveness of the proposed test is then verified by simulations.
See for example \cite{sieg-venk-astat-1995}, \cite{desobry2005online}, and \cite{chen2016sequential}.

In \cite{bane-firo-hero-isit15}, we proposed a test for the non-parametric family of vector elliptically contoured densities, of which the Gaussian case considered here is a special case, with the objective of minimizing
detection delay while avoiding false alarms.
In our test in \cite{bane-firo-hero-isit15}, we used a \textit{global} summary statistic to detect the change in correlation.
However, the global nature of our summary statistics in \cite{bane-firo-hero-isit15} does not allow for classification of where the changes occurred in the correlation matrix, i.e., correlation hub discovery.
In this paper we propose a family of $p$ \textit{local} summary statistics, one for each variable,
and obtain asymptotic densities for these local statistics. We use these and the global
statistic from \cite{bane-firo-hero-isit15} to perform sequential classification of the hubs in addition to detection of the existence of hubs.

\section{A Random Matrix Solution Approach}\label{sec:RandMatModel}
We propose a random matrix based solution to our detection problem.
The decision maker analyzes the $p$-dimensional vector sequence $\{\mathbf{X}(k)\}$ in batches of size $n$, where $p \gg n$.
This leads to a $n \times p$ random matrix sequence $\{\mathbb{X}(m)\}_{m \geq 1}$.
Specifically,
\begin{equation}\label{eq:mathbbXm}
\begin{split}
\mathbb{X}(1) &= [\mathbf{X}(1), \cdots, \mathbf{X}(n)]^T \\
\mathbb{X}(2) &= [\mathbf{X}(n+1), \cdots, \mathbf{X}(2n)]^T, \mbox{ etc.}
\end{split}
\end{equation}
Thus, for each $m$, each of the $n$ rows of the random matrix $\mathbb{X}(m)$ is an i.i.d. samples
of the multivariate Gaussian $p$-variate random vector $\mathbf{X}$ $=[X_1, \cdots, X_p]^T$ \eqref{eq:ellipticalDef}.

A change in the covariance of the random vector sequence $\{\mathbf{X}(k)\}$ as per the description in \eqref{eq:PreChg_PstChg_Den} will
change the covariance of the random matrix sequence $\{\mathbb{X}(m)\}$. We propose to detect the change by using a stopping rule for
the sequence $\{\mathbb{X}(m)\}$.
Note that:
\begin{enumerate}
\item A stopping rule for the sequence $\{\mathbb{X}(m)\}$ is also a stopping rule for the sequence $\{\mathbf{X}(k)\}$, but it takes values in multiples of $n$ (with respect to index $k$).
The precise choice of $n$ will be discussed below.
\item The law of the random matrix $\mathbb{X}$ \textit{at} the change point may consist of random vectors $\mathbf{X}$ from both pre- and post-change distributions. We ignore this issue in the rest of the paper and assume that all the vectors in the matrix either have density $f_{\mathbf{X}}^{0}$ or have density $f_{\mathbf{X}}^{1}$ \eqref{eq:PreChg_PstChg_Den}. However,
see \cite{moustakides2016sequentially} and \cite{bane-chen-garcia-veer-icassp-2014}, where such transient behavior is addressed.
\end{enumerate}

In random matrix theory (RMT), asymptotic distributions are obtained for various functions of the random matrix, e.g.,
a semicircle law for the empirical eigen value distribution of a Wigner matrix, etc \cite{anderson2010introduction}. Such results are in general valid when both the dimensions
of the matrix are taken to infinity. For our random matrix $\mathbb{X}$, however, the batch size $n$ is fixed, and the dimension $p$ is large. This is called the \textit{purely high-dimensional regime} \cite{hero2016foundational}. Thus, traditional RMT theorems cannot be applied in this regime. In this paper, we obtain approximate asymptotic distribution
for a proposed summary statistic (in fact, a family of them) for the random matrix $\mathbb{X}$ in this regime. We then use this approximate distribution(s) to detect a change in the law of the random matrix. We also propose techniques to isolate the hubs.

\section{Local Summary Statistics for the Random Data Matrix}\label{sec:SumStat}
In this section we define a family of local summary statistics to detect hubs \eqref{eq:Hub_Hc}.
The summary statistics are a function
of the sample correlation matrix obtained from the random data matrix $\mathbb{X}$.
We then obtain their asymptotic distribution in the
purely high dimensional regime of  $p\rightarrow \infty$, $n$ fixed. The asymptotic
distributions are members of
a one-parameter exponential family to be specified below.
In Section~\ref{sec:HubDiscAlgo},
we will use these summary statistics and their asymptotic distributions for sequential hub discovery.

For random data matrix $\mathbb{X}$
we write
$$\mathbb{X} = [\mathbf{X}_1, \cdots, \mathbf{X}_p] = [\mathbf{X}^T_{(1)}, \cdots, \mathbf{X}^T_{(n)}]^T,$$
where $\mathbf{X}_i = [X_{1i}, \cdots, X_{ni}]^T$ is the $i^{th}$ column and $\mathbf{X}_{(i)} = [X_{i1}, \cdots, X_{ip}]$
is the $i^{th}$ row.
Define the sample covariance matrix as
$$ \mathbf{S} = \frac{1}{n-1} \sum_{i=1}^n (\mathbf{X}_{(i)} - \bar{\mathbf{X}})^T (\mathbf{X}_{(i)} - \bar{\mathbf{X}}),$$
where $\bar{\mathbf{X}}$ is the sample mean of the $n$ rows of $\mathbb{X}$.
Also define the sample correlation matrix as
$$ \mathbf{R} = \mathbf{D_S}^{-1/2} \mathbf{S} \mathbf{D_S}^{-1/2},  $$
where $\mathbf{D_A}$ denotes the matrix obtained by zeroing out all but the diagonal elements of the matrix $\mathbf{A}$.
Note that, under our assumption that the covariance matrix $\mathbf{\Sigma}$ of the rows of $\mathbb X$ is positive definite, $\mathbf{D_S}$ is invertible with probability one.
Thus $\mathbf{R}_{ij}$, the element in the $i^{th}$ row and the $j^{th}$ column of the matrix $\mathbf{R}$, is the
sample correlation coefficient between the $i^{th}$ and $j^{th}$ columns of $\mathbb X$.

Define the summary statistics (see also \eqref{eq:V_Sigma})
\begin{equation}\label{eq:Sum_Stat_V}
V_k(\mathbb{X}) \; := \; V_k(\mathbf{R})\; := \; \max_{i \neq k}  |\mathbf{R}_{ki} |, \quad k \in \{1, \cdots, p\}.
\end{equation}
This defines a set of $p$ local statistics $\{V_k\}_{k \leq p}$, one for each random variable in the random vector.
We will use these statistics for detection of correlation strength hubs.

Note that the summary statistics are the empirical estimates of the variables $\{V_k\}$ defined for the population correlation matrix $\boldsymbol{\Sigma}$ in \eqref{eq:V_Sigma}. If we could collect enough samples such that $n > p$, then we can rely
on these estimates to check if they are close to zero, because the population version of these quantities are identically zero when the
variables are uncorrelated. But, since $p \gg n$, we may have to collect a large number of samples before we can rely on these
estimates. Such a delay may not be acceptable in practice, or those many samples may not even be available.
In fact, for large $p$, $V_k(\mathbb{X})$ can be arbitrarily close to $1$; see \cite{hero-bala-IT-2012} and Fig.~\ref{fig:fV_histogram_Jeq1} in this paper.

However, as we show below, one can obtain asymptotic law of these estimates as $p \to \infty$. The precise statement is provided
in the theorems below. A change in correlation between
the variables may be reflected in the asymptotic law of these estimates. This phenomena can be utilized for detecting
a change in correlation. The rest of this section is devoted to obtaining the asymptotic laws. The algorithms are
discussed in Section~\ref{sec:HubDiscAlgo}.

We say that a matrix is row sparse of degree $j$ if there are no more than $j$ nonzero entries in any row. We say that
a matrix is block sparse of degree $j$ if the matrix can be reduced to block diagonal form having a single $j \times j$ block,
via row-column permutations. For a threshold parameter $\rho \in [0,1]$ define the correlation graph $\mathcal{G}_\rho(\boldsymbol{\Sigma})$
associated with the post-change correlation matrix $\boldsymbol{\Sigma}$
as an undirected graph with $p$ vertices, each representing a variable in the vector $\mathbf{X}$.
An edge is present between vertices $i$ and $j$ if the magnitude of the correlation coefficient $\rho_{ij}$
between the $i^{th}$ and $j^{th}$ components of the random vector $\mathbf{X}$
is greater than $\rho$, i.e., if $|\rho_{ij}| \geq \rho$, $i \neq j$.
We define $d_k$ to be the degree of vertex $k$ in the graph  $\mathcal{G}_\rho(\boldsymbol{\Sigma})$:
\begin{equation}\label{eq:d_Sigma}
\begin{split}
d_k(\boldsymbol{\Sigma}, \rho) \; &:= \; \# \{i \neq k:  |\mathbf{\rho}_{ki} | \geq \rho \}, \\
&\quad \quad  \quad k \in \{1, \cdots, p\}, \; \rho \in [0,1],
\end{split}
\end{equation}
where $\#A$ is the size of set $A$.
Finally, throughout the index $k$ will be associated with the $k$th variable.

\medskip
\begin{theorem}\label{thm:CorrScr}
Let $\mathbf{\Sigma}$ be the covariance matrix of $\mathbf{X}$ or the rows of $\mathbb{X}$, $\mathbf{\Sigma}$ could be diagonal or non-diagonal depending on whether it is from the pre- or post-change data, respectively. Let $\mathbf{\Sigma}$ be row sparse of degree $j=o(p)$. Also let $p \to \infty$ and $\rho = \rho_p \to 1$ such that
$p(1-\rho^2)^{n/2} \to 0$.
Then,
\begin{equation}
\mathsf{P}(V_k(\mathbb{X}) \leq \rho ) \; \to \; e^{- \Lambda_k },
\end{equation}
where
$$\Lambda_k = \lim_{p\to \infty, \rho \to 1} \Lambda_{k,\rho} ,$$
with
$$\Lambda_{k,\rho} = (p-1)  P_0(\rho) J_k,$$
$$P_0(\rho) = I_{1-\rho^2} ((n-2)/2, 1/2),$$
with $I_x(a,b)$ being the regularized incomplete beta function with parameter $a$ and $b$.
The parameter $J_k$ is a positive real number that is a function of the asymptotic joint density of $\mathbf{X}$.
Further, if the covariance matrix $\mathbf{\Sigma}$ of the p-variate vector $\mathbf{X}$ is block sparse of degree $j$,
then
$$J_k=1 + O((j/p)^{2}).$$
In particular, if $\mathbf{\Sigma}$ is diagonal then $$J_k=1.$$
\end{theorem}
\begin{IEEEproof}
The events $\{V_k(\mathbb{X}) \geq \rho\}$ and $\{d_k > 0\}$ are equivalent. Hence
\begin{equation}\label{eq:V_eq_N}
\mathsf{P}(V_k(\mathbb{X}) \geq \rho) = \mathsf{P}(d_k(\mathbb{X}, \rho) > 0).
\end{equation}
The result now follows from Theorem~\ref{thm:degree} below.
\end{IEEEproof}
\medskip
\begin{theorem}[\cite{firouzi2013local}]\label{thm:degree}
Let $\mathbf{\Sigma}$ be row sparse of degree $j=o(p)$. Also let $p \to \infty$ and $\rho = \rho_p \to 1$ such that
$p(1-\rho^2)^{n/2} \to 0$. Then
 $d_k(\mathbb X,\rho)$ converges in distribution to a Poisson random variable with rate parameter $\Lambda_k$. \end{theorem}
\medskip

Based on the asymptotic distribution obtained in Theorem~\ref{thm:CorrScr},
the  large $p$  distribution of $V_k$ defined in \eqref{eq:Sum_Stat_V} can be approximated by
\begin{equation}\label{eq:CDF_V}
\begin{split}
\mathsf{P}(V_k(\mathbb{X}) \leq \rho) &\approx \exp(-\Lambda_{k,\rho}),\\
&= \exp(-(p-1)  P_0(\rho) J_k) \; \rho \in [0,1].
\end{split}
\end{equation}
where $\Lambda_{k,\rho}$ is as defined in Theorem~\ref{thm:CorrScr}. Using \cite[Thm. 3.1]{firouzi2013local} it can easily be shown that the approximation error associated with (14) decays to zero at least as fast as  $p(1-\rho^2)^{n/2}\Delta$ where $\Delta$ is a dependency coefficient associated with the set of U-scores..

The distribution \eqref{eq:CDF_V} is differentiable everywhere except at $\rho=0$
since $P(V_k(\mathbb{X})=0)>0$. For $\rho>0$ and large $p$,  $V_k$ has density
\begin{equation}\label{eq:PDF_V_1}
f_{V_k}(\rho) \approx - (p-1) P_0'(\rho)  J_k \exp(-(p-1)  P_0(\rho) J_k), \; \rho \in (0,1].
\end{equation}
Note that $f_V$ in \eqref{eq:PDF_V_1} is the density of the Lebesgue continuous component of
the distribution \eqref{eq:CDF_V} and that it integrates to $1-O(e^{-p^2})$ over $\rho\in (0,1]$.

For each $k$, the density $f_{V_k}$ is a member of a one-parameter exponential family with $J_k$ as the unknown parameter.
Then, since $P_0(\rho)' = -\frac{2 (1-\rho^2)^{(n-4)/2}}{B((n-2)/2, 1/2)}$, where $B(a,b)$ is the beta function, we have for $ \rho \in [0,1]$,
the exponential family form of the density $f_V$ with parameter $J_k$:
\begin{equation}\label{eq:PDF_V}
\begin{split}
f_{V_k} &(\rho; J_k)  \\
=&  \; \frac{2 \; (p-1) \; J_k \; (1-\rho^2)^{\frac{n-4}{2}}}{B((n-2)/2, 1/2)}  \exp(-(p-1)  \; J_k   \; P_0(\rho) ).
\end{split}
\end{equation}

Note that the density $f_{V_k}$ corresponds to the $k$th variable, and the whole family $\{f_{V_k}\}$ depends on the index $k$ only through the parameter $J_k$, which can be different for different $k$.
We thus rewrite the family in \eqref{eq:PDF_V} (with complete specification for easy accessibility) as
\medskip
\begin{equation}\label{eq:PDF_V_nok}
\begin{split}
f_{V} &(y; J)  =  J \; C(p, n) \;(1-y^2)^{\frac{n-4}{2}} e^{-(p-1)  \; J   \; P_0(y) },
\end{split}
\end{equation}
\medskip
where
\begin{equation}
\begin{split}
C(p,n) &= \frac{2 \; (p-1)}{B((n-2)/2, 1/2)}, \\
P_0(y) &= I_{1-y^2} ((n-2)/2, 1/2),\\
I_x(a,b) & = \frac{B(x; a,b)}{B(a,b)},\\
B(x, a,b) & = \int_0^x t^{a-1} (1-t)^{b-1} dt,\\
B(a,b) & = B(1,a,b).
\end{split}
\end{equation}

In Fig.~\ref{fig:fV_density} we have plotted the density $f_V$ for various values of $J$ for
$n=10$, and $p=100$. We note that for the chosen values of $n$ and $p$, the density is concentrated close to $1$,
consistent with large values of $\rho$ arising in the purely high dimensional regime assumed in Theorem~\ref{thm:CorrScr} \cite{hero2016foundational}. Note that \textit{the variable $V$ tends to take on higher values as the  parameter $J$ increases}.
This is the fundamental idea used for hub discovery in this paper.
\begin{figure}[htb]
\center
\includegraphics[width=9cm, height=6cm]{fVdiffJ.png
\caption{Plot of density $f_V$ in \eqref{eq:PDF_V_nok} for various values of the parameter $J$ for $n=10, p=100$. }
\label{fig:fV_density
\end{figure}

In Fig.~\ref{fig:fV_histogram_Jeq1} we have plotted the normalized histogram of the samples $V_1(\mathbb{X})$ for multivariate
Gaussian independent random vectors and compare it with the approximation $f_V(\cdot\; ; 1)$. Note that the approximation is quite accurate. Also, note that for this case the population $V_1 = 0$, but the samples are concentrated in the interval $(0.55,0.95)$. This is due to the fact that $p \gg n$.
\begin{figure}[htb]
\center
\includegraphics[width=9cm, height=6cm]{fV_vs_Hist.png
\caption{Comparison of density $f_V$ in \eqref{eq:PDF_V_nok} for $J=1$ for multivariate Gaussian independent random variables with $n=10, p=100$ with the normalized histogram obtained from $10,000$ samples. Note that the population covariance is diagonal, yet the density is concentrated close to $1$.}
\label{fig:fV_histogram_Jeq1
\end{figure}

\section{Global Summary Statistics}\label{sec:globalSumStat}
In \cite{bane-firo-hero-isit15} we proposed a global summary statistic and obtained its asymptotic distribution.
Specifically, we showed that the statistic
\begin{equation}\label{eq:globalU}
U(\mathbb{X}) = \max_k V_k (\mathbb{X}),
\end{equation}
in the limit $p \to \infty$ (taken in a particular sense) and under sparsity assumptions similar to that in Theorem~\ref{thm:CorrScr}, has approximate density in an exponential family given by
\begin{equation}\label{eq:globalgU}
g(u;  \theta) = \frac{D \; \theta }{2} (1-u^2)^{\frac{n-4}{2}} \exp\left(-\frac{D}{2} \theta \;T(u) \right), \; u \in (0,1],
\end{equation}
where
\begin{equation}
\begin{split}
D=D_{p,n}\; &=\; 2p (p-1) B((n-2)/2, 1/2),\\
T(u) &= \int_{u}^1 (1-s^2)^{\frac{n-4}{2}} ds.
\end{split}
\end{equation}
The parameter $\theta$ is a positive parameter and has similar interpretation and nature as that of the parameter $J$
in \eqref{eq:PDF_V_nok}. Specifically, if the rows of $\mathbb{X}$ have diagonal covariance $\mathbf \Sigma$ then $\theta=1$,
otherwise $\theta \neq 1$.
Compared to $V_k (\mathbb{X})$, $U(\mathbb{X})$ has a global view of the covariance matrix. If any of the $V_k$s undergo
a change, this change will be reflected through a change in the distribution of $U$.
However, it is not possible to isolate hubs based only on the statistic $U$.
Sequential tests for detecting a change in correlation using the statistic $U$ is studied in detail in \cite{bane-firo-hero-isit15}. The sequential test from \cite{bane-firo-hero-isit15} will play a fundamental role in what follows.

\section{Multivariate Quickest Hub Discovery}\label{sec:HubDiscAlgo}
We now discuss how to use the results from the previous sections to detect and isolate correlation hubs.

Recall that we have a sequence of $p$-variate Gaussian distributed vectors $\{\mathbf{X}(k)\}$ with a change point model given in \eqref{eq:PreChg_PstChg_Den} and $p \gg 1$. Thus, the vectors are uncorrelated before change and correlated with covariance matrix $\mathbf{\Sigma}$ after change. We collect these vectors in batches of size $n$ and obtain a sequence of $n \times p$ random
matrices $\{\mathbb{X}(m)\}$; see \eqref{eq:mathbbXm}.
For quick detection, we should choose a batch size $n$. Thus, we also have $p \gg n$.

Assuming that the post-change covariance matrix $\mathbf{\Sigma}$ is row sparse, a change in the law of
$\mathbf{X}$ changes the law of the random matrix $\mathbb{X}$, which in turn affects the law of summary statistics
$U(\mathbb{X})$ and $\{V_k(\mathbb{X})\}_{k=1}^p$. Since, the distributions of these statistic each belong to a one-parameter
exponential family \eqref{eq:PDF_V_nok} and \eqref{eq:globalgU}, a change in law within this family is a simple change
in the value of the parameters $\{J_k\}$ and $\theta$. We propose to detect the change in correlation in $\mathbf{X}$ through a change
in the parameters $\{J_k\}$ and $\theta$

From \eqref{eq:PDF_V_nok} and \eqref{eq:globalgU},
it follows that we can choose $n > 4$, because the densities are well defined for these values of $n$.
Note also that here we assume that the change in correlation in $\mathbf{X}$ affects the parameters $\{J_k\}$ and $\theta$, otherwise our test is insensitive to the change in correlation.

\subsection{Correlation Change Detection Using Global Statistic}

Assuming that the post-change covariance matrix $\Sigma$ is row sparse,
we map the random matrix sequence $\{\mathbb{X}(m)\}$ to the summary statistic sequence $\{U(\mathbb{X}(m))\}$.
The random variables $\{U(\mathbb{X}(m))\}$ have approximate
density given by \eqref{eq:globalgU} which is in an exponential family with parameter $\theta$ which is $1$ before the
change and some value $\theta \neq 1$ after change.
For simplicity, we refer to $U(\mathbb{X}(m))$ simply by $U(m)$.
The QCD problem on the density $f_{\mathbf{X}}$, depicted in \eqref{eq:PreChg_PstChg_Den},
is reduced to the following QCD problem
\begin{equation}\label{eq:QCDProb_U}
\begin{split}
U(m) &\sim g(\cdot \; ; 1), \quad\hspace{1.25cm} m < \gamma \\
    &\sim g(\cdot \; ; \theta), \; \quad \theta \neq 1, \; m \geq \gamma.
\end{split}
\end{equation}

Consider the following generalized likelihood ratio (GLR) based QCD tests (see \cite{lord-amstat-1971}, \cite{bane-firo-hero-isit15}) defined by the stopping time:
\begin{equation}\label{eq:tauU}
\begin{split}
&\tau_{\scriptscriptstyle \text{U}} \\
&=\min \left\{m:\max_{1 \leq \ell \leq m} \sup_{\theta: |\theta-1| \geq \epsilon_u} \sum_{i=\ell}^m \log \frac{g(U(i); \theta)}{g(U(i); 1)} > A_u\right\},
\end{split}
\end{equation}
where we assign $\tau_{\scriptscriptstyle \text{U}}=\infty$ if the right hand side of (23) is the empty set. Here $A_u > 0$ and $\epsilon_u>0$ are user-defined parameters, and $\epsilon_u$ represents the minimum magnitude of change, away from $\theta=1$, that the user wishes to detect.

\subsection{Correlation Change Detection Using Local Statistics}
We can also detect a change in correlation using the local sample correlation statistics $\{V_k(\mathbb{X})\}_{k=1}^p$ defined in \eqref{eq:Sum_Stat_V}.
For simplicity, $V_k(\mathbb{X}(m))$ is denoted simply by $V_k(m)$.
For large $p$ and row sparse covariance, the random variables $\{V_k(m)\}$ have approximate
density given by \eqref{eq:PDF_V_nok} which is in an exponential family with parameter $J_k$. This parameter is $1$ before the
change and takes on some value $J_k \neq 1$ after the change.
A natural locally pooled sqequential change detection test is to implement $p$ local sequential change detection tests in parallel, one for each variable $\{V_k(\mathbb{X})\}_{k=1}^p$ (equivalently, each variable in $\mathbf{X}$). An alarm is raised
when \textit{any} of the local tests raises an alarm.

Mathematically, the QCD problem on the density $f_{\mathbf{X}}$, depicted in \eqref{eq:PreChg_PstChg_Den},
is reduced to the following family of QCD problems (one for each $k$):
\begin{equation}\label{eq:QCDProb_1toother}
\begin{split}
V_k(m) &\sim f_V(\cdot \; ; 1), \quad\hspace{1.25cm} m < \gamma \\
    &\sim f_V(\cdot \; ; J_k), \; \quad J_k \neq 1, \; m \geq \gamma.
\end{split}
\end{equation}
Consider the following GLR-based QCD tests defined by the family of local stopping times \cite{lord-amstat-1971}:
\begin{equation}\label{eq:taugk}
\begin{split}
&\tau_{\scriptscriptstyle \text{V}}^{(k)} =\\
&\min  \left\{m: \max_{1 \leq \ell \leq m} \sup_{J_k: |J_k-1| \geq \epsilon_v} \sum_{i=\ell}^m \log \frac{f_{V}(V_k(i);J_k)}{f_{V}(V_k(i); 1)} > A_v\right\},
\end{split}
\end{equation}
where, again we assign $\tau_{\scriptscriptstyle \text{V}}^{(k)}=\infty$ if the right hand side of \eqref{eq:taugk} is the empty set, and
$A_v > 0$ and $\epsilon_v>0$ are user-defined parameters.
This stopping time can be used to detect a change in the parameter $J_k$ for the $k$th variable. Define the overall local stopping time to be
\begin{equation}\label{eq:taug}
\tau_{\scriptscriptstyle \text{V}} = \min_k \tau_{\scriptscriptstyle \text{V}}^{(k)} = \min  \left\{m:  \max_{1 \leq k \leq p} G_k (m) > A_v\right\},
\end{equation}
where
\begin{equation}\label{eq:GLRStat}
G_k (m) := \max_{1 \leq \ell \leq m} \sup_{J_k: |J_k-1| \geq \epsilon_v} \sum_{i=\ell}^m \log \frac{f_{V}(V_k(i);J_k)}{f_{V}(V_k(i); 1)}
\end{equation}
is the GLR statistic.
The test $\tau_{\scriptscriptstyle \text{V}}$ can be used to detect if the parameter of any of the $p$ variables is affected by the change. $\epsilon_v$ represents the minimum magnitude of change, away from $J_k=1$, that the user wishes to detect.

\subsection{Joint Detection and Hub Discovery}
For joint detection and hub discovery, we combine the local and global tests and propose to use the following maximum of the stopping times:
\begin{equation}\label{eq:tauHB}
\tau_{\scriptscriptstyle \text{HB}} = \max\{ \tau_{\scriptscriptstyle \text{V}}, \tau_{\scriptscriptstyle \text{U}}\}.
\end{equation}
Let $D_k(\mathbb{X}(1), \cdots, \mathbb{X}(\tau_{\scriptscriptstyle \text{HB}}))$ be the binary decision variable with value $1$ if variable $k$ is declared a hub. We use the following rule for hub discovery: fix a positive integer $q$ and set
\medskip
\begin{equation}\label{eq:FaultIsoRule}
D_k(\mathbb{X}(1), \cdots, \mathbb{X}(\tau_{\scriptscriptstyle \text{HB}})) = \mathbb{I}_{\left\{    G_k (\tau_{\scriptscriptstyle \text{HB}}) \mbox{ is top $q$ statistic} \right\}}.
\end{equation}
\medskip

To appreciate the above rule, recall that a correlation strength hub is defined as the set of those variables that achieve the maximum in \eqref{eq:globalU}; see \eqref{eq:Hub_Hc}. Thus, there are at least two variables that are correlation strength hubs in the correlation graph. These hub variables in general have value of parameter $J_k$ that is higher than the parameter values for other variables.
However, there may be many variables that have experienced a change in correlation, and their $V_k$ value, and hence
the value of parameter $J_k$, might be close to the values attained by the hubs.
To illustrate this idea, we have plotted in Fig.~\ref{fig:JkPlot} estimated $\{J_k\}$ values from samples of a multivariate Gaussian vector with a row sparse covariance matrix (see Section~\ref{sec:NumericalResults}
on how the covariance matrix is generated). Here the hubs are the two variables with the highest values of $J_k$.
But, as seen in the figure, there are many other variables that have experiences significant changes in correlation, and
hence also have high values of $J_k$.
\begin{figure}[htb]
\center
\includegraphics[width=8cm, height=5cm]{JkPlot_2.png
\caption{Estimated $J_k$ values from a $p \times p$ row-sparse covariance matrix with $p=100$. Here $k$ is the index of the variables in $\mathbf{X}$. Note that the values of parameter $J_k$ for non-hub variables are also significant. Hubs are the two variables with the highest values of the parameter. }
\label{fig:JkPlot
\medskip
\includegraphics[width=8.5cm, height=5.5cm]{GkPlot_2.png
\caption{Values of GLR statistic $G_k$ at the time of stopping $\tau_{\scriptscriptstyle \text{HB}}$ corresponding
to the model used in Fig.~\ref{fig:JkPlot}. Note that the $G_k$ values for the non-hub variables are comparable to those of the hub variables. In practice, one can just plot these $G_k$ values to capture all
the variables that are affected by a change in correlation. A close mathematical equivalent to this idea is the pick-the-top $q$ approach \eqref{eq:FaultIsoRule}.}
\label{fig:GkPlot
\end{figure}
In Fig.~\ref{fig:GkPlot}, we have plotted the values of the GLR statistic $\{G_k\}$ at the time of stopping $\tau_{\scriptscriptstyle \text{HB}}$.
As seen in the figure, in addition to the $G_k$ values of hubs, the values corresponding to other variables
are also significant. Thus, a simple strategy like choosing the variable with the highest $G_k$ value as hub, will not work well.

However, what is clear from Fig.~\ref{fig:JkPlot} and Fig.~\ref{fig:GkPlot} is that high $G_k$ values
at the time of stopping correspond to high $J_k$ values of the summary statistics, and these values together have significantly higher magnitude than the values for the rest of the variables. This motivates
the pick-the-top approach.
In fact, in practice, \textit{one can plot the $G_k$ values this way and identify all the variables that have experienced a change in correlation}.

Picking the variables corresponding to the top $q$ values where $q > 2$ ensures that the hubs are discovered with high probability,
provided a slightly modified definition of hub discovery is used: the variables in the hub $\mathsf{H}$ are considered discovered if
\begin{equation}\label{eq:realDefHubDiscovery}
\mathsf{H} \subset \{k : \mathcal{D}_k= 1\}.
\end{equation}
Indeed, this fact is used in Section~\ref{sec:NumericalResults} for numerical computations,
where it is shown that the proposed rule's probability of false hub discovery goes to zero, as the false alarm rate
goes to zero; see Fig.~\ref{fig:falseIso}.

\medskip
\subsection{Delay and False Alarm Analysis}
The approximate delay and false alarm performance of the stopping rule $\tau_{\scriptscriptstyle \text{HB}}$ is given in the theorem below.
\begin{theorem}\label{thm:PerfTauHB}
Fix any $\epsilon_u > 0$ and $\epsilon_v > 0$. If the densities \eqref{eq:PDF_V_nok} and \eqref{eq:globalgU} are the true distributions of the samples then we have the following:
\begin{enumerate}
\item For the stopping rule $\tau_{\scriptscriptstyle \text{HB}}$, the supremum in \eqref{prob:Pollak} is
achieved at $\gamma=1$, i.e.,
\[
\sup_{\gamma\geq 1} \; \mathsf{E}_\gamma[\tau_{\scriptscriptstyle \text{HB}}-\gamma| \tau_{\scriptscriptstyle \text{HB}} \geq \gamma] = \mathsf{E}_1[\tau_{\scriptscriptstyle \text{HB}}-1].
\]
\item Setting $A_u=A_v=\log \beta$ ensures that as $\beta \to \infty$,
\[
\mathsf{E}_\infty[\tau_{\scriptscriptstyle \text{HB}}] \geq \beta (1+o(1)).
\]
\item For each possible true post-change parameters $J$ and $\theta$, with $|J-1|\geq \epsilon_v$, and $|\theta - 1| \geq \epsilon_u$, as $\beta \to \infty$
\begin{equation}
\begin{split}
\mathsf{E}_1[\tau_{\scriptscriptstyle \text{HB}}] &\leq  \left( \frac{\log \beta}{I(J)}  + \frac{\log \beta}{I(\theta)} \right)(1+o(1)),
\end{split}
\end{equation}
where $I(J)$ is the Kullback-Leibler divergence between the densities $f_V(\cdot; J)$ and $f_V(\cdot; 1)$, and
$I(\theta)$ is the Kullback-Leibler divergence between the densities $g(\cdot; \theta)$ and $g(\cdot; 1)$.
\end{enumerate}
\end{theorem}
\begin{IEEEproof}
The proof follows from standard QCD arguments in the literature (see \cite{veer-bane-elsevierbook-2013}) and because
\[
\tau_{\scriptscriptstyle \text{HB}} \geq \tau_{\scriptscriptstyle \text{U}},
\]
and
\[
\tau_{\scriptscriptstyle \text{HB}} \leq \tau_{\scriptscriptstyle \text{U}} + \tau_{\scriptscriptstyle \text{V}}.
\]
\end{IEEEproof}

\section{Numerical Results}\label{sec:NumericalResults}
In Fig.~\ref{fig:falseIso} we report the hub discovery performance of the proposed algorithm.
We set $\mathbf{\Sigma}_0 = I_p$, the identity matrix.
We assume that the post-change covariance matrix $\mathbf{\Sigma}$ is a row-sparse matrix of degree $j$,
obtained as follows. A $p \times p$ sample from the Wishart distribution is generated and some of
the entries are forced to be zero in such a way that no row has more than $j$ non-zero elements.
Specifically, we retain the top left $j \times j$ block of the matrix, and for each row $k$, $j+1 \leq k \leq (p+j)/2$,
all but the diagonal and the $(p+j+1-k)$th element is forced to zero. Each time an entry $(k,i)$ is set to zero,
the entry $(i,k)$ is also set to zero, to maintain symmetry. Finally, a scaled diagonal matrix is also added to $\mathbf{\Sigma}$
to restore its positive definiteness. We set $n=10$, $p=100$, and $j=5$.

To implement $\tau_{\scriptscriptstyle \text{V}}$ we have chosen $\epsilon_v=1$,
and we use the the maximum likelihood estimator for $J_k$ which,
as a function of $m$ samples $(V_k(1), \cdots, V_k(m))$ from $f_V(\cdot, J)$, is given by
\begin{equation}\label{eq:MLEst_J}
\hat{J}(V_k(1), \cdots, V_k(m)) = \frac{1}{(p-1)\frac{1}{m} \sum_{i=1}^m P_0(V_k(i))}.
\end{equation}
Specifically,
\begin{equation}
\begin{split}
\argmax_{J: J \geq 2} \;  \log \sum_{i=\ell}^m & \frac{f_V(V_k(i);J)}{f_V(V_k(i); 1)} \\
&= \max\{2, \hat{J}(V_k(\ell), \cdots, V_k(m))\}.
\end{split}
\end{equation}
Similar techniques were used to implement $\tau_{\scriptscriptstyle \text{U}}$; see \cite{bane-firo-hero-isit15}.

We consider three scenarios, each with two hubs:
\begin{equation}\label{eq:FaultIsoCases}
\begin{split}
1\;  \mathsf{H} &= \{19, 86\} \mbox{ with } J_{19} = 2.56 \mbox{ and } J_{86}=2.51.\\
2.\;  \mathsf{H} &= \{44, 61\} \mbox{ with } J_{44} = 5.88 \mbox{ and }  J_{61}=5.85.\\
3.\; \mathsf{H} &= \{12, 93\} \mbox{ with } J_{12} = 17 \mbox{ and }  J_{93}=16.
\end{split}
\end{equation}
The parameter $J_k$s are estimated using $1000$ samples. For each case, the total number of false
isolations is calculated using $1000$ sample paths by assuming $\gamma=1$. The parameter $q$ was chosen to be $10$.
The definition \eqref{eq:realDefHubDiscovery} was used as a criterion calculating false isolation or false hub discovery. The results for all the three cases is plotted in Fig.~\ref{fig:falseIso} as a function of the
threshold $A=A_v=A_u$. As shown in Fig.~\ref{fig:falseIso}, the number of false discoveries goes to zero as the threshold is increased. Thus, empirical evidence suggests that
the test is consistent.

\begin{figure}[htb]
\center
\includegraphics[width=9cm, height=6cm]{FaultIso.png
\caption{False Isolation performance for three scenarios in \eqref{eq:FaultIsoCases}}
\label{fig:falseIso
\end{figure}

\medskip
In Fig.~\ref{fig:MFADelay} we plot the delay ($\mathsf{E}_1[\tau_{\scriptscriptstyle \text{HB}}]$) vs the log of mean time to false alarm ($\log \mathsf{E}_\infty[\tau_{\scriptscriptstyle \text{HB}}]$)
for various cases specified in the figure. The values in the figure are obtained
by choosing different values of the threshold $A$ and estimating the delay by choosing the change point $\gamma=1$
and simulating the test for $500$ sample paths.
The mean time to false alarm values are estimated by simulating the test for $1500$ sample paths.
\begin{figure}[htb]
\center
\includegraphics[width=9cm, height=6cm]{MFA_DELAY.png
\caption{MFA-Delay performance for the stopping rule $\tau_{\scriptscriptstyle \text{HB}}$ for various choices of parameter $\{J_k\}$.
The subscripts of $J$ are the hub indices.}
\label{fig:MFADelay
\end{figure}

\section{Conclusions and Future Work}
We have introduced the quickest hub detection and localization problem for high dimensional correlation graphs. Local and global summary statistics were proposed based on purely high dimensional asymptotic theory  for the presence of hub nodes, defined as vertices having either high degree or large correlations. The asymptotic densities of these summary statistics were shown to be in the one dimensional exponential family of densities. These densities were used  to define a hybrid local and global quickest detection test for hubs and the asymptotic performance of this test was evaluated in terms of mean time to detect a change and mean time to false alarm.

\section{Acknowledgments}
This work was partially supported by the Consortium for Verification Technology under Department of Energy National Nuclear Security Administration award number DOE-NA0002534.

\balance
{
\bibliographystyle{ieeetr}

redpajama_set_name: RedPajamaArXiv


text:
Edhird's Blog
Restoring Health: body, mind and spirit
Tag Archives: Christianity
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Check out Praise Moves, a Christian alternative to yoga
Laurette Willis knows what she is talking about, as she was formerly into yoga and the New Age movement for 21 years from the age of 7. Her booklet and webinair are very instructive. I encourage you to send this viral to your friends. Praise Moves is offering a Christian alternative to #yoga, doing healthy stretching and calisthenics without serving two masters.
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Categories: Uncategorized | Tags: calisthenics, Christianity, Laurette Willis, New Age, Praise Moves, stretching, yoga | Permalink.
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Always Winter and Never Christmas
By the Rev. Dr. Ed Hird
This Christmas season, you will not want to miss the 'Chronicles of Narnia' movie 'Voyage of the Dawntreader. Since C. S. Lewis wrote it in 1950, tens of millions of copies of the book in over thirty languages have been sold.
At the heart of Narnia's first book 'The Lion, the Witch, and the Wardrobe' is the abolition of Christmas by the White Witch where it is always winter and never Christmas. C.S. Lewis' alternate title for his book was 'The Hundred Year Winter'. Not once in the past hundred years of Narnia was Christmas ever celebrated.
The White Witch, whose real name is Jadis, punished anyone who wanted the restoration of Christmas, by turning them into stone. The White Witch's most memorable feature was her skin, as white as chalk, or paper, or snow. CS Lewis explains in the Narnia book 'The Magician's Nephew' that the White Witch's skin was made that way by eating an apple from the Emperor's Garden at the beginning of Narnia.
In the midst of this bone-chilling winter, we are told about an ancient prophecy stating that when two sons of Adam and two daughters of Eve filled the four thrones as Kings and Queens of Narnia, the tyranny of the White Witch and her hundred-year winter would end. We are also told that one day the great Lion Aslan will triumphantly return to Narnia: "Wrong will be right, when Aslan comes in sight, At the sound of his roar, sorrows will be no more, When he bares his teeth, winter meets its death, And when he shakes his mane, we shall have spring again." CS Lewis called Aslan a 'supposal' of what might have happened if Christ had come to a world of talking animals and become one of them.
With the remarkable success of the 'Passion of the Christ' Movie and Tolkien's 'Lord of the Rings' Trilogy, many have become more open to spiritually-oriented movies like Voyage of the Dawntreader. Many 'Lord of the Rings' and 'Narnia' buffs may not be aware that it was JRR Tolkien who helped lead his atheist friend CS Lewis to faith in the Lion of the Tribe of Judah (Revelation 5:5).
While teaching at Oxford College, Lewis formed a lasting friendship with JRR Tolkien. Lewis said to Tolkien that tales or myths are 'lies and therefore worthless, even though breathed through silver'. 'No', said Tolkien, 'they are not lies'. Tolkien went on to explain to Lewis that in Jesus Christ, the ancient stories or myths of a dying and rising God entered history and became fact.
Twelve days later, Lewis wrote to another friend Arthur Greeves: "I have just passed on from believing in God to definitely believing in Christ – in Christianity. I will try to explain this another time. My long night talk with Dyson and Tolkien had a good deal to do with it". CS Lewis recalls going by motorcycle with his brother Warren to Whipsnade Zoo, about thirty miles east of Oxford. "When we set out, I did not believe that Jesus Christ is the Son of God, and when we reached the zoo, I did". In his autobiography Surprised by Joy, Lewis commented: "In the Trinity term of 1929 I gave in, and admitted that God was God…perhaps the most dejected and reluctant convert in all England".
This Christmas season, as you take your family and friends to see the Voyage of the Dawntreader, I invite you to discover with CS Lewis that Aslan is the Reason for the Season.
The Rev. Dr. Ed Hird, BSW, MDiv, DMin
-previously published in the North Shore News
-award-winning author of the book Battle for the Soul of Canada
-The sequel book Restoring Health: body, mind and spirit is available online with Amazon.com in both paperback and ebook form. Dr. JI Packer wrote the foreword, saying "I heartily commend what he has written." The book focuses on strengthening a new generation of healthy leaders. Drawing on examples from Titus' healthy leadership in the pirate island of Crete, it shows how we can embrace a holistically healthy life.
In Canada, Amazon.ca has the book available in paperback and ebook. It is also posted on Amazon UK (paperback and ebook), Amazon France (paperback and ebook), and Amazon Germany (paperback and ebook).
Restoring Health is also available online on Barnes and Noble in both paperback and Nook/ebook form. Nook gives a sample of the book to read online.
Indigo also offers the paperback and the Kobo ebook version. You can also obtain it through ITunes as an IBook.
To receive a signed copy within North America, just send a $20 cheque (USD/CAN) to ED HIRD, 102 – 15168 19th Avenue, Surrey, BC, V4A 0A5, Canada.
– In order to obtain a signed copy of the prequel book Battle for the Soul of Canada, please send a $18.50 cheque to ED HIRD, 102 – 15168 19th Avenue, Surrey, BC, V4A 0A5. For mailing the book to the USA, please send $20.00 USD. This can also be done by PAYPAL using the e-mail ed_hird@telus.net . Be sure to list your mailing address. The Battle for the Soul of Canada e-book can be obtained for $4.99 CDN/USD.
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Categories: Uncategorized | Tags: Anglican Coalition in Canada, Arthur Greeves, Aslan, Atheist, Battle for the Soul of Canada, Christ, Christianity, Christmas, CS Lewis, Emperor's Garden, Jadis, Jesus Christ, Kings and Queens of Narnia, Lion of the Tribe of Judah, Narnia Tales, Oxford College, Passion of the Christ, Reason for the Season, Reverend Ed Hird, Son of God, St. Simon's Church North Vancouver, The Hundred Year Winter, The Lion, The Magician's Nephew, the Witch and the Wardrobe, Tolkien Lord of the Rings, Voyage of the Dawn Treader, Whipsnade Zoo, White Witch | Permalink.
Remembering Dear Nana
Nana Allen, my maternal grandmother, died with her prayer book and bible by her bed. I was never successful in talking her out of using the "old-fashioned" Book of Common Prayer, and 'getting with the program'. Nana to me symbolizes the deep Judeo-Christian roots of our beloved Canadian nation. She knew in her heart that I would one day become an Anglican priest, even when I was running from God on the top of Mount Seymour ski hill. Nana, while outwardly a very gentle and proper 'English' lady, was inwardly a prayer warrior who never gave up on her family or her nation. Nana's passionate love for our nation came out most strongly when she watched Hockey Night in Canada, fervently cheering for her favorite team 'The Montreal Canadiens'.
I love Nana very deeply, though she passed away in March 1982, just before my throat operation where I received my voice back. It was hard for Nana to watch me lose my voice, as she was so deeply committed to my calling to the Anglican priesthood. I remember her saying that she wanted to live until I became ordained as a deacon. Then after my first ordination, she decided that she wanted to live until I became priested which she did as well. Within a year of my priesting, she had gone to be with the Lord.
Why am I an Anglican priest today? I believe in my heart of hearts that I am a priest because of my Nana's powerful prayers and personal witness to a biblically faithful Anglicanism. Nana's life embodied to me the heart and soul of genuine Anglican Christianity. Sadly most of the faithful congregations that my Nana attended have since been swallowed by other agendas.
Nana was also a devout Anglophile and royalist. Though she had never been to England, it very much functioned for her as 'the mother country'. Nana was probably more English than the English. My parents finally persuaded her at age 80 to fly to England with them. While she enjoyed the trip tremendously, she felt that England had changed!
As someone who has been an ordained Anglican clergyman for 40 years, I must say similarly to my grandma's comment that the Anglican Church has changed. I value healthy, necessary change, but I grieve when the core values of the Anglican Church are discarded in the relentless search for temporary relevance. I have sadly had to face that we are now often dealing with another gospel, another religion, another faith than the biblical Anglican Christianity that my dear Nana stood for.
I believe that the Anglican Church is the 'canary in the tunnel' for our great nation of Canada. Our Canadian passivity has made us vulnerable to serious cultural meltdown of everything that made Canada great. My grandparents' and parents' generation put everything on the line to defend our great nation in World War I and World War II. How can we do any less in the current battle for the soul of Canada?
We say in the pre-amble to our Constitution that we acknowledge the Supremacy of God. It is time for us as Canadians to turn our words into actions. Our founding forebearers were determined in the words of Psalm 72 that 'he would have dominion for sea to shining sea'. Our original name "The Dominion of Canada" was chosen to deliberately reflect that spiritual commitment as the core of our nation.
My cry is that God would keep our land glorious and free, that God would have mercy upon our rebellious land, that mercy would triumph over judgement. God's heart of love is that we would repent of our turning away from our godly Judeo-Christian heritage and turn back before it is too late.
When I remember my dearly beloved Nana, I am reminded that we have a great heritage as Canadians. Let's not squander it.
-previously published in the Deep Cove Crier/North Shore News
-The sequel book Restoring Health: body, mind and spirit is available online with Amazon.com in both paperback and ebook form. In Canada, Amazon.ca has the book available in paperback and ebook.
It is also posted on Amazon UK (paperback and ebook ), Amazon France (paperback and ebook), and Amazon Germany (paperback and ebook).
To receive a signed copy within North America, just send a $20 cheque (USD/CAN) to ED HIRD,102-15168 19th Avenue, Surrey, BC, Canada V4A 0A5.
– In order to obtain a signed copy of the prequel book Battle for the Soul of Canada, please send a $18.50 cheque to 'Ed Hird', #102-15168 19th Avenue, Surrey, BC, Canada V4A 0A5. For mailing the book to the USA, please send $20.00 USD. This can also be done by PAYPAL using the e-mail ed_hird@telus.net . Be sure to list your mailing address. The Battle for the Soul of Canada e-book can be obtained for $4.99 CDN/USD.
Categories: Uncategorized | Tags: Anglican Mission in the Americas (Canada), Anglican priest, Anglophile, Battle for the Soul of Canada, Bible, book of Common Prayer, canary in the tunnel, Christianity, Constitution, cultural meltdown, deacon, Deep Cove Crier, Dominion of Canada, English, founding forebearers, God keep our land glorious and free, godly, Hockey Night in Canada, Judeo-Christian roots, judgement, Lord Jesus, Montreal Canadians, Montreal Canadiens, mother country, Mount Seymour ski hill, Nana Allen, prayer book, priested, priesting, Psalm 72, rebellion, Royalist, St. Simon's Church North Vancouver, Supremacy of God | Permalink.
by edhird	1 Comment
Chief Joseph Brant: Canadian Hero
My wife and I had the privilege of attending the First Peoples Forgiven Summit in Ottawa. During that time we were able to meet a number of Mohawk believers, including Jonathan Maracle of Broken Walls who led us in remarkable worship music. Canada's most famous Mohawk was Chief Joseph Brant. Recently the Canadian Royal Mint produced a Canadian Loonie with the imprint of Chief Joseph Brant (1742-1807). More Canadians need to hear this story of this Canadian hero. He was described by Mark Jodoin as having the mind of a statesman, the heart of a leader, and the soul of a warrior. Without the military and spiritual support of Chief Brant, Canada would have likely never survived.
Chief Joseph Brant's Mohawk name was Thayendanegea which means "two sticks bound together for strength". Isabel Thompson Kelsay notes that "the most famous (aboriginal) who ever lived, has been for two centuries a virtual unknown." I suspect that he is unknown to most North Americans because he chose the side of Canada in the American revolutionary war. As Canada's premier First Nations leader, Brant had the privilege of meeting both Georges in person: King George III and President George Washington.
Brant learned to speak, read and write English at a New Hampshire school led by Rev Wheelock. Wheelock described Brant as being "of a sprightly genius, a manly and gentle deportment, and of a modest, courteous and benevolent temper." In 1772, Brant was then mentored by Rev John Stuart, being trained in the art of Bible and Prayer Book translation. During that time, Brant developed a deep prayer life, becoming a committed Anglican Christian.
During the American Revolutionary war, Brant was falsely accused of committing atrocities in locations which he was not present, including the tragic Wyoming and Cherry Valley Massacres. Those who knew Brant well testified that he often prevented atrocities through the use of his persuasive leadership. As a devout Anglican Christian, he exhibited compassion and humanity, especially towards women, children, and non-combatants. American Colonel Ichabod Alden commented that he "should much rather fall into the hands of Brant than either of them [Loyalists and Tories]." It was frequently said of Joseph Brant that during the American revolution, he fought with a tomahawk in one hand, a copy of the New Testament in the other.
Joseph Brant's father was one of the sachem/chiefs, known as the Four Indian Kings, who visited Queen Anne in 1710. These chiefs asked 'for missionaries to be sent to the People of the Longhouse to teach them more about Christianity." Queen Anne sent this request to the Anglican Archbishop of Canterbury, promising to build them a chapel. In 1711, Queen Anne's Royal Chapel was built in the Mohawk Valley in New York State. When the Mohawks relocated to Southern Ontario, the Mohawk Royal Chapel was rebuilt there in 1785. Joseph Brant's grave is located right next to the historic Mohawk Chapel, the oldest protestant church in Ontario. Just this past July, Queen Elizabeth, while visiting Ontario, presented the Mohawk Chapel with a set of eight silver hand bells engraved 'The Silver Chain of Friendship 1710-2010'.
On each side of the Mohawk Chapel pulpit are two tablets in the Mohawk language of the Lord's Prayer and the Ten Commandments. Joseph Brant was a brilliant linguist translating the Bible and Anglican Prayer Book into Mohawk (of which there are microfiche copies at Simon Fraser University). He also wrote a concise history of the Bible and a Mohawk language catechism. Brant spoke at least three and possibly all of the Six Nations' languages. When the Chapel was dedicated in 1788, each person was given a Mohawk book containing the Gospel of Mark and the Anglican Prayer Book. At that celebration, sixty five Mohawks were baptized and three couples were married.
When Joseph Brant first visited England in 1775, he was described by a British commander as 'His Majesty's greatest North American subject.', and painted in full aboriginal regalia by George Romney. Receiving a captain's commission, Brant met with the King on two occasions, with a dinner being held in his honour. Brant was honoured by the English leaders in the arts, letters and government, including James Boswell, the famed biographer of Samuel Johnson.
In 1779 Brant was commissioned by the King as 'captain of the Northern Confederate Indians' in recognition of his "astonishing activity and success'. Brant was described as "the perfect soldier, possessed of remarkable stamina, courage under fire, and dedicated to the cause, an able and inspiring leader and a complete gentleman."
Joseph Brant's Six Nations were tragically driven out of their homeland in Central New York. Brant was hurt that in granting their Mohawk homeland in Central New York State to the Americans, England had 'sold the Indians to the US Congress'. Writing to King George III, he reminded the British that "we, the Mohawks, were the first Indian Nation that took you by hand and invited you to live among us, treating you with kindness…" The Six Nations were eventually resettled by Governor Frederick Haldimand in the Grand River area around modern-day Brantford. The British realized that locating the Six Nations in the Grand River area would be a natural protection against any future American invasion. Initially the Mississauga First Nation resisted the concept of having their former foes on their land. One Mississauga Chief Pokquan however persuaded his other chiefs by arguing that other aboriginals would be better neighbours than European settlers, and that Brant's knowledge of the British could prove useful.
The term Brantford comes from Brant's Ford, the shallow part of the Grand River that could be forded. The first years at Brantford were difficult as there was a drought with game being hard to find. Throughout all the challenges, Chief Brant's deep faith sustained him. Chief Brant's sacrificial love for God and nation should inspire all of us. He memorably said: "No person among us desires any other reward for performing a brave and worthwhile action but the consciousness of having served one's nation."
May all of us be willing to learn from the bravery and loyalty of Chief Joseph Brant.
To receive a signed copy within North America, just send a $20 cheque (USD/CAN) to ED HIRD, #102-15168 19th Avenue, Surrey, BC V4A 0A5.
– In order to obtain a signed copy of the prequel book Battle for the Soul of Canada, please send a $18.50 cheque to 'Ed Hird', #102-15168 19th Avenue, Surrey, BC V4A 0A5.
For mailing the book to the USA, please send $20.00 USD. This can also be done by PAYPAL using the e-mail ed_hird@telus.net . Be sure to list your mailing address. The Battle for the Soul of Canada e-book can be obtained for $4.99 CDN/USD.
Categories: Uncategorized | Tags: Aboriginal, America, American Revolutionary War, Anglican Christian, Anglican Coalition in Canada, Anglican Mission in the Americas (Canada), Anglican Prayer Book, Archbishop of Canterbury, Battle for the Soul of Canada, Bible, Brant's Ford, Brantford, bravery, Canada, Canadian Royal Mint, Cherry Valley Massacre, Chief Joseph Brant, Chief Pokquan, Christianity, Colonel Ichabod Alden, First Nations, First Peoples, First Peoples of Canada, Four Indian Kings, George Romney, Governor Frederick Haldimand, Grand River, hand bells, Inuit, Iroquois, Isabel Thompson Kelsay, James Boswell, Jonathan Maracle, Joseph Brant, King George III, linguist, Lord's Prayer, Loyalist, loyalty, Metis, Mississauga First Nation, Mohawk, Mohawk Bible, Mohawk Prayer Book, Mohawk Valley, Nation, New Hampshire, New Testament, New York State, North America, Northern Confederate Indians, Ottawa, People of the Longhouse, President George Washington, protestant, Queen Anne, Queen Anne's Royal Chapel, Queen Elizabeth II, Rev Ed Hird, Rev John Stuart, Rev Wheelcock, Samuel Johnson, Silver Chain of Fellowship, Simon Fraser University, Six Nations, Southern Ontario, St. Simon's Church North Vancouver, Ten Commandments, Thayendanega, Tomahawk, Tories, translatin, United Empire Loyalist, United States of America, US Congress, USA, Wyoming Massacre | Permalink.
Amazing Grace, How Sweet the Sound…
A violent storm was the turning point in John Newton's life. Motherless at age 6 and sent to sea on his 11th birthday, he soon became a teenage rebel. He was press-ganged into the Navy and flogged for desertion. Newton became involved with the African Slave Trade and came close to starvation while living in extreme poverty in Sierra Leone.
You name the corruption … he both saw it and lived it during his wealthy slave trading days. He even ended up for a short time as a slave himself. What he was most enslaved to, however, was a violent temper and a filthy tongue. John was so blasphemous that once even his hardened shipmates threatened to throw him overboard in order to calm a dangerous storm.
Throughout his blasphemy and rebellion, John had family members who never stopped praying for him. Secretly he began to read the "Good Book:' but somehow it never made sense. One night in March 1748, at the age of 23, he was on board a cargo ship which was fighting for its life against heavy seas and rough weather. Worn out with pumping and almost frozen, he called out for God's mercy at the height of the storm, and was amazed to be saved from almost certain death.
Newton's life had many twists and turns. Eventually he renounced his involvement with slave trading, and, at age 39, became an Anglican Priest. He persuaded the young William Wilberforce to stay in politics, and joined him in his fight to abolish the slave trade. After 40 years of prayer and hard work, Wilberforce, through Newton's influence, finally had slavery abolished in 1833.
John Newton is a parable of what the Christian faith is all about: moving from death to life, from slavery to freedom. Newton celebrated his new life in Christ by writing one of the world's most famous songs.
"Amazing Grace, how sweet
That saved a wretch like me.
I once was lost but now
am found,
Was blind but now I see."
May Amazing Grace touch your heart.
The Rev. Dr. Ed Hird
-author of the award-winning book Battle for the Soul of Canada
-previously published in the Deep Cove Crier/North Shore News and the Light Magazine/City Light News
To receive a signed copy within North America, just send a $20 cheque (USD/CAN) to ED HIRD, #102-15168 19th Avenue, Surrey, BC V4A 0A5, Canada.
Categories: Amazing Grace, John Newton, Uncategorized | Tags: Africa, Amazing Grace, Anglican, Anglican Coalition in Canada, Anglican Mission in the Americas (Canada), Bagpipes, Battle for the Soul of Canada, Christianity, England, freedom, John Newton, Reverend Ed Hird, slavery, St. Simon's Church North Vancouver | Permalink.
Sir Sandford Fleming: Inventor of Time
When Sir Sanford Fleming first came to Canada, he was told "Go back to Scotland". The need for engineers was over. Some were convinced that we would only need sixteen miles of railway in Canada. Fortunately for us, Fleming loved a challenge. He was passionate about railways, once driving a bear off the railway tracks with nothing but an umbrella and a loud cry.
Fleming has been described as the outstanding Canadian of the nineteenth century. Prime Minister John A Macdonald appointed him as chief surveyor and engineer of the Canadian Pacific Railway. Fleming knew that he needed to see the route first-hand. With the Rev George Grant, he canoed and portaged across Canada in 1872, creating a best-selling travelogue 'Ocean to Ocean'. The beauty and ruggedness of Canada's wilderness spoke to the depths of his soul.
Our most recognizable Canadian photo is 'The Last Spike', celebrating the completion of our national railway on November 7th 1885. Fleming, our most famous Canadian engineer, was right there at the centre of the photo.
To complete the Canadian Pacific Railway in just ten years was an astronomical task, but Fleming always made time for God in his busyness. Fleming only missed attending church twelve times in his entire life. Sometimes 'church' was simply kneeling by the Rocky mountain railway tracks and giving thanks. On all his surveying trips, no work and travel was done on Sunday if he could help it. He even wrote a worship service that his busy construction crews could use.
After the frustration of his missing an Irish train, Fleming went on to create Meridian Standard Time in 1878. Standard Time replaced the dangerous chaos of 144 different North American time zones. Every city had its own unique time, none of which agreed with any other city. Standard Time went a long way towards keeping locomotives from crashing into each other because of different clocks.
Fleming founded the Canadian Institute which grew into the Royal Society of Canada. He published a dozen books, served for thirty-five years as Chancellor of Queen's University. Canada's very first postage stamp: the three-cent beaver, was the creation of Fleming. Fleming was knighted in 1897 by Queen Victoria for building the world-circling sub-Pacific cable. For the first time in history, the world could communicate instantaneously around the globe. With membership in over seventy international societies, he was Canada's preeminent voice on the world stage. Everyone looked to Sir Sanford Fleming.
Fleming was often snubbed, sidelined, criticized but he never let the nay-sayers stop him from accomplishing his life-goals. Fleming knew that God had put him here on earth to make a difference, to help raise up the nation of Canada from sea to sea. Fleming's strength came from a deep sense that God would never abandon his children.
To receive a signed copy within North America, just send a $20 cheque (USD/CAN) to ED HIRD, 102-15168 19th Avenue, Surrey, BC, Canada V4A 0A5.
Categories: inventor, Sandford Fleming, science, Technology, Time, Uncategorized | Tags: Anglican Coalition in Canada, Anglican Mission in the Americas (Canada), Battle for the Soul of Canada, Canada, Christianity, Church, CPR, railway, Reverend Ed Hird, Scotland, Sir George Fleming, Sir Sandford Fleming, St. Simon's Church North Vancouver, Standard Meridian Time, time, underground cable | Permalink.
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Home / About / News / Engineers Teach Machines to Recognize Tree Species
Engineers Teach Machines to Recognize Tree Species
Engineers from Caltech have developed a method that uses data from satellite and street-level images, such as the ones that you can see in Google maps, to create automatically an inventory of street trees that cities may use to better manage urban forests.
Their work is described in the proceedings of the 2016 IEEE Conference on Computer Vision and Pattern Recognition, which was held in Las Vegas this summer.
"Cities have been surveying their tree populations for decades, but the process is very labor intensive. It usually involves hiring arborists to go out with GPS units to mark the location of each individual tree and identify its species," says senior author Pietro Perona, the Allen E. Puckett Professor of Electrical Engineering in the Division of Engineering and Applied Science. "For this reason, tree surveys are usually only done every 20 to 30 years, and a lot can change in that time."
Perona and his team are not expert arborists. Rather, they are leaders in the field of computer vision: they specialize in creating visual recognition algorithms—computer programs capable of "learning" to recognize objects in images—that can see and understand images much like a human would. These algorithms, by replicating the abilities of experts, can sometimes even understand images better than the average person. As part of an ongoing project called "Visipedia," a collaboration with Serge Belongie (BS '95) of the Joan & Irwin Jacobs Technion–Cornell Institute and Cornell University and the Cornell Lab of Ornithology, the engineers have developed algorithms that can recognize the species of a North American bird from a single picture. (http://merlin.allaboutbirds.org/photo-id)
The team eventually hopes to develop Visipedia's capabilities until it can accurately recognize nearly all living things. But they were inspired to turn their attention toward trees when Perona noticed the effects of the years-long California drought on the trees near the Caltech campus in Pasadena.
"I happened to notice that many people in Pasadena were putting drought-resistant plants in their yards to save water, but when they took out the lawns and stopped watering, many trees started dying, and that seemed like a shame," Perona says. "I realized that computer vision might be able to help. By analyzing automatically satellite and street-level images that are routinely collected, maybe we could carry out an inventory of all the trees and we could see over time how Pasadena is changing, whether the trees that are dying are just a few birch trees, which are not native to California and require frequent watering, or whether it's truly a massive change."
To begin their survey of the Pasadena urban tree population, the team developed a method to automatically "look" at any specific location in the city using aerial and street-level images from Google Maps (Google agreed to let Caltech use the images for research free of charge). They then created an algorithm that detects objects within these images and calculates their geographic location. Although a human could easily look at these photographs, spot an object, and ascertain whether or not that object is a tree, the task is not so simple for a computer.
Perona's research group uses artificial neural networks—algorithms inspired by the brain that allow a computer to "learn" to recognize objects in images. These networks must first receive training from humans. "We train an algorithm the way you would teach a child—by showing it lots of examples," Perona says. "The more examples of trees the algorithm sees, the better it becomes at detecting trees. I must say that a child would learn rather more quickly than our algorithms—right now we need hundreds of examples for each type of tree."
To provide those examples, the team enlisted some human help via a crowdsourcing service called Amazon Mechanical Turk, in which hundreds of workers worldwide can be quickly recruited to complete simple tasks that require human intelligence. In this case, the so-called "turkers" were asked to look at aerial and street-level images of Pasadena and label the trees in each photo. This information was used to train the algorithm to determine which objects were trees.
The engineers next wanted to train the algorithm to identify the species of each tree in the photos—something that the average person cannot do. Fortuitously, the city of Pasadena had partnered in 2013 with a commercial tree management company called Davey Resource Group (DRG) to complete a tree inventory. The survey included species identification, measurements, and the geographical locations of each of the approximately 80,000 trees in the city. Using this information, the engineers trained the algorithm to identify 18 of the more than 200 species of trees in Pasadena.
From Google Maps aerial and street view images, the engineers obtained four different photographs of each tree in Pasadena, taken from different viewpoints and at different distances from the tree. These photos were then analyzed by the algorithm's "brain"—the artificial neural network. The network then produced a list of a few possible tree species and a score of the certainty of each guess. After comparing the algorithm's results with those of the 2013 tree survey, the engineers found that their algorithm could detect and identify a tree's species from Google Maps images with about 80 percent accuracy.
"This was much better than we had expected, and it showed that our method can produce similar results to a tree survey done by humans," says Steve Branson, a postdoctoral scholar in electrical engineering and coauthor on the paper. "A human tree expert can identify species at a higher accuracy than our algorithm, but when these large city tree surveys are done they can't be 100 percent accurate either. You need lots of people to spread out around the city and there will be mistakes."
Eventually, cities could use Perona's computer vision software as part of a long-term technological solution for the management of urban forests. The idea is that the software would continuously collect data about urban street trees from satellite and street level images, which are updated every few months, or from other public images. That information then could be incorporated into software that would help the city understand how its urban forests are evolving, and help in the creation of long-term plans for future street-tree investments.
Although perfecting the algorithm is an ongoing process, Perona says the concept could eventually change the way urban forests are managed.
The study involving Pasadena street trees was published in a paper titled, "Cataloging Public Objects Using Aerial and Street-Level Images—Urban Trees." Results may be browsed on-line at http://vision.caltech.edu/registree/. In addition to Perona and Branson, other coauthors include David Hall from Caltech and Jan Wegner and Konrad Schindler from ETH Zurich. The work in Pasadena was supported by the Office of Naval Research, NASA, and Google.
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Q: Right-continuity of natural filtrations My question: Is the natural filtration of a right-continuous process also right-continuous?
I would say yes, but don't know where to start proving it.
Thanks for your help/ideas!

A: Right continuity fails even for canonical continuous processes.
The natural filtration on $C([0,\infty))$ is not right continuous.
For example, the event $\{\omega: {d^+\over dt}\ \omega_t\mbox{ exists at }t=0\}$ belongs
to ${\cal F}_{0+}$ but not ${\cal F}_0$. In words, you can tell whether
 the function $\omega_t$ has a right derivative at $t=0$ with an infinitesimal peek beyond time 0,
but you cannot tell just from the value of the function $\omega_t$ at time 0.
Right continuous filtrations are nicer to work with, and since it fails for the natural filtration,
we often use the right continuous version instead. Fortunately, many of the nice
properties of right continuous processes carry over even with this enlarged filtration. For example,
Brownian motion is still Markov with respect to ${\cal F}_{t+}$ which leads to interesting results like Blumenthal's zero-one law.

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Most folks will tell you that they love you, but until they prove it don't believe it, fall for, or surrender to it. Because action speaks louder than words.
We know that God loves us by all the things He did to prove it. God clearly proved and demonstrated His unconditional love towards us, in that while we were still sinners, Christ died for us (Romans 5:8). The question is what are you doing, to prove your love for Him?
Don't you think that after all He has done for you, that the least you can do is love Him back, by demonstrating your love for Him in your lifestyle.
When you refuse to love Him, it's proof of your selfishness or your hatred for Him. Aren't you afraid that He might not wake you up one day or prevent that accident you are heading into.
What or who is it that's stopping you from loving your Creator? Are they worth you sacrificing your Heavenly future for unquenchable fire and brimstone? When all you have to do is love God and accept His unspeakable gift of His only begotten Son, Jesus Christ. This is not hard to do.
Romans 10:9 &13 says, "If you confess with your mouth the Lord Jesus, and believe in your heart that God has raised Him from the dead, you will be saved. Then, verse 13 says, "Whoever calls on the Name of the Lord shall be saved." To confirm this, 1 John 1:9 says, "If we confess our sins, He is faithful and just to forgive us our sins and cleanse us from all unrighteousness." We were all born in sin and shaped in iniquity, but His love for us givrs us a chance to change our status from unsaved to saved, unrighteous to righteous, unloved to loved. Why would you refuse this offer?


The 2017 Vermont high school baseball season is upon us. As the season gets underway, the CVU Redhawks baseball program would like to invite any interested students to attend tryouts which takes place next week. If you are interested, please contact Coach Albertson (talbertson@cssu.org).
Varsity Tryouts will take place on mornings before school and on some occasions after school as well. To try out for Varsity, you must be a junior, senior or an invited player. Junior Varsity tryouts will take place after school and must be a freshman or sophomore.
A decision will be made prior to Friday if a JVB team is needed. Cuts will be made after Thursday and/or Friday's tryout session.


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Global recession starting in 2023
Watchman Report / By Marco / 1 minute of reading
This show focuses on the global recession that is starting to hit this year along with how it relates to the upcoming last days and mark of the beast. We are on a path where 42 months after the beast MBS does the deal with Israel will the mark of the beast come to pass and NOBODY will be able to buy or sell without it for the last half of the tribulation period before the second coming. Enjoy and GET READY.
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I am a bible believing son of God by believing the doctrine of justification found in Romans 3:24-26. I believe in Jesus and have faith in his blood fully persuaded. I believe the King James Bible (Pure Cambridge Edition) is the perfect, pure word of God and is ALL AUTHORITY and I read it according to the commandment found in Romans 16:25-27.
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A look ahead into 2023
The Iran Card
Watchman Report's 2022 year end review
Cyrus king of Persia
My trip to the Chicago Oriental Institute


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Green Flag Award pays tribute to HRH The Duke of Edinburgh
The Green Flag Award scheme is saddened to hear of the death of HRH The Duke of Edinburgh. The Duke was a champion of environmental causes including becoming President of Fields in Trust, formerly the National Playing Fields Association (NPFA) in 1948 until 2013. During that period the number of green spaces protected grew to over 2,000.
Early in his royal life, The Duke was said to be appalled to see children playing in the street instead of on green spaces and it became his desire to improve this situation for young people in urban areas. This led to him taking the Presidency of the NPFA, succeeding his uncle, Rear Admiral the Rt. Hon. Earl Mountbatten of Burma.
Green Flag Award Manager Paul Todd said; "Our thoughts today are with Her Majesty the Queen and the Royal Family. He leaves a lasting legacy in the thousands of parks and green spaces that are now protected and will be enjoyed by future generations"
The Green Flag Award scheme works with Fields in Trust to ensure that as many parks and green spaces are protected for future generations and managed to a quality standard set by the Award


WORDLESS WEDNESDAY - "The Girls"
Here is one of the very few photos I have of my mother-in-law, in late summer 1970, taken while on the sidewalk outside our home in Kitsilano, Vancouver BC. I'm holding my second daughter Megan, Reba is holding my first daughter, Pia. Reba died in August 1974 of liver cancer, likely due to a much earlier hepatitis infection [the most common cause of liver cancer].
Following the challenge of writing 52 Ancestors in 52 Weeks, issued by Amy Johnson Crow, here is my 13th installment, for John KUHN, my 5th great-grandfather on my maternal line.
Various researchers have not been certain when the original immigrant John KUHN arrived in British America, with his family. However, it seems to have been by 1732 or 1733, when he is noted living in the Goshenhoppen area, in what is now Berks County, Pennsylvania. Most of the following information has been found in a variety of places including the following: (a) "Eckroth-Eckenroth-Eckenrode Family" Vol. 1 & 2, William Kirkpatrick; published 1987 by Chris Pelikan; (b) FindAGrave Memorial information at Conewago Chapel Basilica Cemetery; (c) Goshenhoppen Church Registers: Births, Marriages, Deaths [I have pdf copies of these]; (d) Early Catholic Migration website; all online.
John KUHN died on 14 Feb 1751, his wife, Anna Barbara died several years later. They are buried - as are many others of my KUHN ancestors - in the Cemetery of the Conewago Chapel Basilica.
KUHN, John, on 14 Feb 1751, carpenter, with whom Rev. Theodore Schneider, the founder of the Goshenhoppen missions, was wont to lodge.
Note that Father Schneider was the first priest arriving in Philadelphia in about 1741, sent to minister to the German parishioners. John Kuhn and his sons and neighbours built a church which stands in Bally, Berks County, PA, as shown in the link in the second paragraph above.
Various early tax registers show the Kuhn surname spelled a variety of ways: Coontz, for example, as well as Koon, Koone, Kuhnz, and other variants.
There have been a few tantalizing clues about precisely where in Wurttemberg the family originally lived, before they emigrated en masse. On several records they are listed as from Wurttemberg, specifically from Schlaitdorf, and there is a very possible birth record in Neckartailfingen, Wurttemberg. Both Schlaitdorf and Neckartailfingen towns are next to each other. A tiny clue, but challenging to research.
If you have information or questions about John KUHN and his family, do leave a comment or contact me via calewis at telus dot net; I would love to hear from you.
Following the challenge of writing 52 Ancestors in 52 Weeks, issued by Amy Johnson Crow, here is my 12th installment, for James FERNALD, my 6th great-grandfather, on my maternal line. James is the son of the Immigrant settler in the Kittery area, John FERNALD.
James' father, John, is listed as one of the early settlers after 1632, on the Kittery Maine town website, under Historical Notes (halfway down the page). I have more research to do in this Fernald line, and see that no mention of John's wife, James' mother, is made on records. Several family trees give the name of Mary Norman, but no document/supports for that statement.
James was born in 1675 in Kittery, now York county, Maine, and married Mary HINCKS (possibly this surname is as a widow) in about 1700-1705. No marriage record is yet found. I suspect that there are other siblings of James, but as yet I have not found clear proof of relationship.
In 1728/9 James' name is listed with a number of others in deeding land designated for a second church, on land between Kittery and Eliot, on "Gower's Hill", belonging to the Fernalds. The link goes to a very attractive 1730 church - scroll down to the lower section where the second church is listed as a Congregational church, first served by Rev. John Eveleth (another important name in my family tree) from 1729-1734. A timeline about Kittery and Eliot may be found here.
James will has been transcribed by others, from Maine Wills, 1640-1760 (pub 1887) p.444. The value of his land and and goods was set at over 4160 pounds. He states in his will a number of times that he is bequeathing "...besides what She has already had out of my Estate." This seems to indicate that he had already given goods or lands to his daughters and one son before his decline.
Two of the three Witnesses on the Will are unknown-to-me Fernalds: Lydia Fernald and Benjamin Fernald. More clues are here for researching the early Fernald history in Kittery and Eliot. As early settlers in the Kittery area in Massachusetts, the Fernald family is likely written up in journals and books/histories of the region. It is clear a trip to our BC Genealogical Society library is needed so I can search in the NEHGS databases online there!
If you have questions or comments, please leave them below, or contact me via calewis at telus dot net.
Following the challenge of writing 52 Ancestors in 52 Weeks, issued by Amy Johnson Crow, here is my 11th installment, for Elizabeth ABREY, wife of the immigrant settler, my 6th great-grandmother on my maternal lines.
I have so little information on Elizabeth, that I'm embarrassed. Which is why I'm writing about her - this tiny bit. I have only her first name; her surname is unsubstantiated by any quality documentation.
As far as is known, the MORGAN family immigrated about 1718, settling in Pennsylvania. Most information on Thomas is based on descriptions of his youngest son, Col. Jacob MORGAN and his exploits.
Elizabeth? She is not mentioned - except for her first name, listing her as his wife [living] at the time of Thomas' will, 6 Dec 1740, probated in Philadelphia, 25 Mar 1741.
The only information I have about her is from a number of family trees who have no documentation attached. Her birth date is estimated only - looking at the birth years of her 4 or 5 sons - to be likely 1668-1675, place unknown but likely in same area as her husband Thomas. Their marriage date is also unknown, but likely to be approximately 1695. Her death date is not listed anywhere that I have found as yet, but is likely to be from 1741-1750.
The Morgan family settled in the Caernarvon area of PA with a large number of Welsh immigrants: Jenkins, Piersol, Morgan, Davis, Evans, Bowen, Hudson, Williams, Allen, Martin, and others. I have an early .pdf map of the area with drawing of land plots, Warrants from early 1700s onwards. Contact me if you wish a copy of this map.
If you have any information or questions about Elizabeth ABREY, wife of Thomas MORGAN, the immigrant settler, do contact me via calewis at telus dot net, and/or comment below.
English Bay near the Sylvia Hotel in Vancouver BC Canada.
I'm in the middle here, aged 18 yrs, just finished high school.
On the left is Marie, one of my Kalifornia Kuhn cousins, about 15 yrs.
And on the right is Wendy, about 16 yrs.
Both these two cousins died of cancer.
As did Marie's father (the elder of my mother's 2 older brothers).
And cancer affected Wendy's mother as well (the eldest of my mother's 2 older sisters).
But in this slightly fuzzy photo, we're 3 teenagers having fun in the sun, August 1961, enjoying the day at the beach.
Following the challenge of writing 52 Ancestors in 52 Weeks, issued by Amy Johnson Crow, here is my 10th installment, for Robert BOLTWOOD, the immigrant settler, my 9th great-grandfather on my maternal lines.
Robert BOLTWOOD, b. abt 1620, possibly in Essex, England, where the name was apparently known from the 1200s. He is listed in several early New England history books as having immigrated first to Connecticut area in about 1643 when he would have been in his early 20s; he is first recorded there, and was a Sergeant. He quickly moved to Massachusetts, and was an early settler in likely Wethersfield MA.
By 1646 he had married Mary GERNOR (or Gernow), who was an early Massachusetts-born child in about 1626 in Northfield. The new Boltwood family apparently moved to Hadley where all - or some of the first-born children were born: one son, and 4 daughters.
Robert had some land in Glastenbury (the eastern part of Wethersfield) - "a grant of a pond and quantity of land adjacent from an Indian chief Peckhaun." [p.409, History of the Town of Northfield, Massachusetts, by J. H. Temple and George Sheldon, published by Joel Munsell, 82 State St., Albany, N.Y., 1875].
He was a first settler of Hadley, Massachusetts, and was made a Freeman on 26 March 1661. He is listed as having land grants, by 1663, in Hadley.
Quite a bit is written about Robert & Mary's only son, Samuel, and some of their children. However, I don't yet have the children of the younger 3 girls who married the 3 Warner men - who are likely brothers. But I haven't researched to determine if it is true that the Warner men were brothers, nor how they might have been related, where they lived, nor who their parents were. Research is never done!
I have a few more clues to research, and I'd love to know more about Robert, one more of our early immigrants from England.
If you have any information or questions about Robert BOLTWOOD or his wife and children, I'd love to hear from you. You can contact me via calewis at telus dot net, and/or leave a comment below. And then, I'll get back to doing more research on Robert and his family.


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