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One wonderful festival that takes place in Japan in July is the Star Festival, also known as Tanabata. Tanabata means "evening of the seventh"… the Star Festival takes place at night on July 7th every year. Tanabata is originally based on a Chinese legend.
Tanabata celebrates the meeting of a husband and wife. They are Orihime (the star Vega) and Hikoboshi (the star Altair)… to meet they must cross the river Amanogawa (the Milky Way). They can only meet once a year on this night… it's the one time they can cross the river. The story is that a flock of magpies help Orihime cross the river on this day only. If it's a cloudy night, that means the river was too high and Orihame and Hikoboshi couldn't meet. So in Japan everyone wishes for a clear night full of stars on July 7th.
If you'd like to share any photos, songs or memories of this festival, feel free to in the comments below or email me at lisa@mamalisa.com .
Many thanks to Ayako Egawa for letting us know about this festival and for sharing the wonderful photo!
This article was posted on Thursday, July 30th, 2009 at 7:15 pm	and is filed under Children's Songs, Countries & Cultures, Customs and Traditions, Holiday Songs, Holidays Around the World, Japan, Japanese, Japanese Kids Songs, Languages, Mama Lisa, Sasa no ha sara-sara, Star Festival, Star Festival Songs, Tanabata, YouTube. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.
I stumbled across this page while doing research for my blogsite http://HowToWriteInJapanese.Blogspot.com.
The Japanese really do know how to celebrate festivals in a simple yet beautiful way. It would be fantastic to get my kids to celebrate a day when they 'write their wishes on strips of fancy paper and put them on displays made of the branches of bamboo trees'.
I've bumped into your site whilst procrastinating and it is helping me in this endeavour greatly. I love the festival above, thank you.
I will now commence day-dreaming of my future trip to Japan.


text:
\section{Introduction}
\label{sec:intro}

Given pairs of observations $(x_{i},y_{i}) \in \mathbb{R}^{p} \times \mathbb{R}^{q}$ for $i=1,\ldots,n$, assume they are independently identically distributed as $F_{\ensuremath{X} \ensuremath{Y}}$. Two random variables are independent if and only if the joint distribution equals the product of the marginal distributions. The statistical hypothesis for testing independence is
\begin{align*}
& H_{0}: F_{\ensuremath{X} \ensuremath{Y}}=F_{\ensuremath{X}}F_{\ensuremath{Y}},\\
& H_{A}: F_{\ensuremath{X} \ensuremath{Y}} \neq F_{\ensuremath{X}}F_{\ensuremath{Y}}.
\end{align*}
Detecting the potential relationships underlying the sample data has long been a fundamental question in theoretical and applied research. The traditional Pearson correlation \citep{Pearson1895} has been a valuable tool in quantifying the linear association and applied in many branches of statistics and machine learning. To detect all types of dependence structures, a number of universally consistent methods have been proposed recently, such as the distance correlation \citep{SzekelyRizzoBakirov2007, SzekelyRizzo2009}, the Hilbert-Schmidt independence criterion \citep{GrettonEtAl2005,GrettonGyorfi2010}, the Heller-Heller-Gorfine statistics \citep{HellerGorfine2013,heller2016consistent}, the multiscale graph correlation \citep{mgc1, mgc2}, among many others. The Hilbert-Schmidt independence criterion (HSIC) can be thought of as a kernel version of distance correlation and vice versa \citep{SejdinovicEtAl2013,exact2}, and the multiscale graph correlation is an optimal local version of distance correlation. These universally consistent dependence measures have been applied to two-sample testing \citep{RizzoSzekely2016, exact1}, conditional and partial testing \citep{gretton2007,SzekelyRizzo2014,Wang2015}, feature screening \citep{LiZhongZhu2012, bal2013, Zhong2015,mgc4}, clustering \citep{Szekely2005,Rizzo2010}, time-series \citep{Zhou2012,Pitsillou2018,mgc5}, graph testing \citep{mgc2,mgc6}, etc.

To populate these methods to big data analysis, a big hurdle is the time complexity. Computing the distance correlation typically requires $\mc{O}(n^2)$; and to compute its p-value for testing, the standard approach is to estimate the null distribution of distance correlation via permutation, which requires $\mc{O}(r n^2)$ where $r$ is the number of random permutations and typically at least $100$ or more. In comparison, for one-dimensional ($p=q=1$) data the Pearson correlation can be computed in $\mc{O}(n)$, and a Pearson correlation t-test is readily available to compute the p-value in constant time complexity $\mc{O}(1)$. The computational advantage makes Pearson extremely fast and attractive in practice. Recent works have successfully expedited the distance correlation computation into $\mc{O}(n \log(n))$ under one-dimensional data and Euclidean distance \citep{Huo2016,Hu2018}. The testing part, however, remains difficult and less well-understood, because the null distribution of distance correlation has no fixed nor known density in general. Some notable works include the distance correlation t-test \citep{SzekelyRizzo2013a}, HSIC Gamma test \citep{GrettonGyorfi2010}, and the subsampling approach \citep{zhang2018}, which provided special treatments and very valuable insights into the problem.

In this paper, we propose a chi-square test for the bias-corrected distance correlation. It is simple and straightforward to use, has a constant time complexity without the need to permute nor subsampling nor parameter estimation, and does not rely on any distributional assumption on data. In particular, the bias-corrected distance correlation can be computed in $\mc{O}(n \log n)$ under one-dimensional data and Euclidean distance, which renders the method comparable in speed to the Pearson correlation t-test and scalable to billions of observations. The test is applicable to any multivariate data, any strong negative type metric or characteristic kernel, and can be utilized for K-sample and partial testing.

Theory-wise, we prove the chi-square test is universally consistent and valid for sufficiently large $n$ and sufficiently small type 1 error level $\alpha$ (generally $n \geq 20$ and $\alpha \leq 0.05$ suffice), has a similar testing power as the permutation test, and is the most powerful among all valid tests of distance correlation using known distributions (i.e., any test that is based on a fixed distribution, such as using t-test or F-test). In particular, we prove that the underlying chi-square distribution can well-approximate and dominate the limiting null distribution of the bias-corrected distance correlation in upper tail, and establish a testing power inequality among the chi-square test, the distance correlation t-test from \citet{SzekelyRizzo2013a}, and the standard permutation test.

The advantages of the chi-square test are supported by simulations and real data experiments. The code are openly available on Github and implemented in Matlab\footnote{\url{https://github.com/neurodata/mgc-matlab}} and Python\footnote{\url{https://github.com/neurodata/mgc}}. The Appendix includes detailed background information, all theorem proofs and intermediate results, and detailed simulation functions.

\section{Method}
\label{sec:main}
\subsection{The Distance Correlation Chi-Square Test}

The proposed chi-square test is stated in Algorithm~\ref{alg1}: given paired sample data $(\mathbf{X},\mathbf{Y}) \in \mathbb{R}^{n \times (p+q)}$, first compute the bias-corrected sample distance correlation $C=\sct{Dcor}_{n}(\mathbf{X},\mathbf{Y})$, then take the p-value as $p=Prob(nC < \chi^{2}_{1} -1)$. Then at any pre-specified type 1 error level $\alpha$, the independence hypothesis is rejected if and only if $p < \alpha$. Unless mentioned otherwise, in this paper distance correlation always means the bias-corrected sample distance correlation (see Appendix Section~\ref{sec:bg} for the algebraic expression).

As the chi-square distribution is standard in every software package, the p-value computation takes $\mc{O}(1)$ regardless of sample size, which is much faster than the standard permutation approach (see Appendix Algorithm~\ref{alg2}) requiring $\mc{O}(r n^2)$. The statistic computation and thus Algorithm~\ref{alg1} on its whole require $\mc{O}(n^2)$ in general.

\begin{algorithm}
\caption{The Distance Correlation Chi-Square Test for Independence}
\label{alg1}
\begin{algorithmic
\Require Paired sample data $(\mathbf{X},\mathbf{Y})=\{(x_i,y_i) \in \mathbb{R}^{p+q} \mbox{ for }  i \in [n]\}$.
\Ensure The bias-corrected distance correlation $C$ and its p-value $p$.
\Function{FastTest}{$\mathbf{X},\mathbf{Y}$}
\State $C=\sct{Dcor}_{n}(\mathbf{X},\mathbf{Y})$; \Comment{the bias-corrected distance correlation}
\State $p=1-F_{\chi^{2}_{1}-1}(n \cdot C)$; \Comment{reject the null when $p<\alpha$}
\EndFunction
\end{algorithmic}
\end{algorithm}

The chi-square test is well behaved from the following theorem (which follows directly from Theorem~\ref{cor1} in Section~\ref{sec:main1}):
\begin{theorem}
\label{thm7}
The distance correlation chi-square test that rejects independence if and only if
\begin{align*}
n \sct{Dcor}_{n}(\mathbf{X},\mathbf{Y}) \geq F_{\chi^{2}_{1}-1}^{-1}(1-\alpha)
\end{align*}
is a valid and universally consistent test for sufficiently large $n$ and sufficiently small type 1 error level $\alpha$.
\end{theorem}
In practice, $n \geq 20$ suffices, and the validity empirically holds for any $\alpha \leq 0.05$.

\subsection{Fast Statistic Computation}
\label{sec:ext}
In the special case of $p=q=1$ and Euclidean distance, Algorithm~\ref{alg1} can run significantly faster:

\begin{theorem}
\label{thm8}
Suppose $p=q=1$ and we use the Euclidean distance for the bias-corrected distance correlation. Then Algorithm~\ref{alg1} can be implemented with a computational complexity of $\mc{O}(n \log n)$.
\end{theorem}

Essentially the bias-corrected sample distance correlation can be computed in $\mc{O}(n \log n)$ based on the results from \citet{Huo2016} and \citet{Hu2018}. This makes one-dimensional testing comparable in speed to the Pearson correlation t-test. On a standard Windows 10 machine using MATLAB, we are able to test independence between a million pairs of observations ($p=q=1$) within $10$ seconds with space requirement of $\mc{O}(n)$. Previously, the statistic computation and the permutation test need a space complexity of $\mc{O}(n^2)$ and a time complexity of $\mc{O}(r n^2)$, which would have required external disk storage and days of computation to finish testing the same amount of data. The implementation details can be found in Appendix and Github Matlab code.

\subsection{Chi-Square Test for K-Sample}
The chi-square test is readily applicable to any inference task using distance correlation, or any statistic involving a similar trace operation with bias-corrected matrix modification. Two immediate extensions are the K-sample test and partial test. Previously, the permutation test has been the standard approach for both of them.

Given $K$ sets of sample data $\mathbf{U}^{k}=[u_{1}^{k}|u_{2}^{k}|\ldots|u_{n_{k}}^{k}] \in \mathbb{R}^{p \times n_k}$ for $k=1,\ldots,K$, denote $\sum_{k=1}^{K} n_{k}=n$. The K-sample testing problem assumes $u_i^{k}$ are independently and identically distributed as $F_{U^{k}}$ for each $i$ and $k$, and aim to test
\begin{align*}
&H_{0} : F_{U^{1}} = F_{U^{2}} = \cdots = F_{U^{K}}, \\
&H_{A} : \mbox{ there exists at least one $F_{U^{k}}$ that is different from other distributions.}
\end{align*}
Algorithm~\ref{alg3} shows the K-sample variant of chi-square test.

\begin{algorithm}
\caption{The Distance Correlation Chi-Square Test for K-sample}
\label{alg3}
\begin{algorithmic
\Require Sample data $\{\mathbf{U}^{k} \in \mathbb{R}^{p \times n_k} \mbox{ for }  k=1,\ldots,K\}$.
\Ensure The distance correlation $C$ and its p-value $p$.
\Function{FastKSample}{$\{\mathbf{U}^{k}\}$}
\Statex{\textbf{(1)} Data Transformation:}
\State $\mathbf{X}=[\ \mathbf{U}^{1} \ |\ \mathbf{U}^{2} \ | \cdots | \ \mathbf{U}^{K} \ ]$; \Comment{concatenate Data}
\State $\mathbf{Y}=zeros(K,N)$; \Comment{a zero matrix}
\For{$k=1,\ldots,K$}
\For{$i=1,\ldots,n_k$}
\State $\mathbf{Y}(k,\sum_{j=1}^{k-1}n_j + i)=1$;
\EndFor
\EndFor

\Statex{\textbf{(2)} Test:}
\State $C=\sct{Dcor}_{n}(\mathbf{X},\mathbf{Y})$;
\State $p=1-F_{\chi^{2}_{1}-1}(n \cdot C)$; \Comment{reject the null when $p<\alpha$}
\EndFunction
\end{algorithmic}
\end{algorithm}

It is shown in \citet{exact1} that by concatenating the sample data into $\mathbf{X}$ and forming an indicator matrix $\mathbf{Y}$ via one-hot encoding, $\sct{Dcor}_{n}(\mathbf{X}, \mathbf{Y}) \rightarrow 0$ if and only if the null hypothesis is true. Then the next corollary follows from Theorem~\ref{thm7}.

\begin{corollary}
\label{cor3}
For sufficiently large $n$ and sufficiently small type 1 error level $\alpha$ and any $K \geq 2$, Algorithm~\ref{alg3} is valid and universally consistent for testing $F_{U^{1}} = F_{U^{2}} = \cdots = F_{U^{K}}$.
\end{corollary}

\subsection{Chi-Square Test for Partial}
Another application is to test whether the partial distance correlation equals $0$ or not. Given three sample data $\mathbf{X},\mathbf{Y},\mathbf{Z}$ of same sample size $n$, the partial distance correlation $\sct{PDcor}_{n}(\mathbf{X},\mathbf{Y};\mathbf{Z})$ and its population version $\sct{PDcor}(X,Y;Z)$ are defined in \citet{SzekelyRizzo2014}. Algorithm~\ref{alg4} shows the chi-square test for fast partial testing,

\begin{algorithm}
\caption{The Distance Correlation Chi-Square Test for Partial}
\label{alg4}
\begin{algorithmic
\Require Paired sample data $(\mathbf{X},\mathbf{Y}, \mathbf{Z})=\{(x_i,y_i,z_i) \in \mathbb{R}^{p+q+s} \mbox{ for }  i \in [n]\}$.
\Ensure The partial distance correlation $C$ and its p-value $p$.
\Function{FastPartial}{$\mathbf{X},\mathbf{Y}, \mathbf{Z}$}
\State $C=\sct{PDcor}_{n}(\mathbf{X},\mathbf{Y};\mathbf{Z})$; \Comment{implementation details in \cite{SzekelyRizzo2014} or Github Matlab code}
\State $p=1-F_{\chi^{2}_{1}-1}(n \cdot C)$;\Comment{reject the null when $p<\alpha$}
\EndFunction
\end{algorithmic}
\end{algorithm}

The partial statistic shares the same trace formulation as bias-corrected distance correlation and operates on the same bias-corrected matrix modification, except using projected distance matrices rather than Euclidean distance matrices. Therefore we have the following result:
\begin{corollary}
\label{cor4}
For sufficiently large $n$ and sufficiently small type 1 error level $\alpha$, Algorithm~\ref{alg4} is valid and consistent for testing $\sct{PDcor}(X,Y;Z)=0$.
\end{corollary}

\section{Supporting Theory}
\label{sec:main1}

In this section we show the theory behind the chi-square test. Note that the results hold for any strong negative type metric or any characteristic kernel for $X$ and $Y$, e.g., Euclidean distance, Gaussian kernel, Laplacian kernel, etc., which means the chi-square test is also applicable to the Hilbert-Schmidt Independence Criterion. Some results (like Theorem~\ref{thm1} and Theorem~\ref{cor1}) are stated in limit or for sufficiently large sample size, for which $n \geq 20$ generally suffices for the limiting null distribution to be almost the same as the actual null.

\subsection{The Limiting Null Distribution and the Centered Chi-Square Distribution}

\begin{theorem}
\label{thm1}
The limiting null distribution of the bias-corrected distance correlation satisfies
\begin{align*}
n\sct{Dcor}_{n}(\mathbf{X},\mathbf{Y}) &\stackrel{D}{\rightarrow} \sum\limits_{i,j=1}^{\infty} w_{ij} (\mc{N}_{ij}^{2}-1),
\end{align*}
where the weights satisfy $w_{ij} \in [0,1]$ and $\sum\limits_{i,j=1}^{\infty} w_{ij}^{2} = 1$, and $\mc{N}_{ij}$ are independent standard normal distribution.
\end{theorem}

This theorem follows from \citet{zhang2018}. For different choice of metric and different marginal distributions, the weights $\{w_{ij}\}$ and the limiting null distribution are different, which is the main obstacle to design a test using known distribution. Nevertheless, from Theorem~\ref{thm1} we observe the mean and variance of $n\sct{Dcor}_{n}(\mathbf{X},\mathbf{Y})$ are always fixed and equal the mean and variance of $\chi^{2}_{1}-1$, which we shall call the centered chi-square distribution and denote by $U$ from now on.

\subsection{Upper Tail Dominance and Distribution Bounds}

We aim to show that $U \sim \chi^{2}_{1}-1$ dominates the limiting null distribution at some upper tail probability $\alpha$. We denote $F_{V}(x)$ as the cumulative distribution function of random variable $V$ at argument $x$, $F^{-1}_{V}(1-\alpha)$ as the inverse cumulative distribution function of random variable $V$ at probability $1-\alpha$, and formally define upper tail stochastic dominance as follows:
\begin{definition}
Given two random variables $U$ and $V$, we say $U$ dominates $V$ in upper tail at probability level $\alpha$ if and only if
\begin{align*}
F_{V}(x) \geq F_{U}(x)
\end{align*}
for all $x \geq F_{U}^{-1}(1-\alpha)$. This is denoted by
\begin{align*}
V \preceq_{\alpha} U.
\end{align*}
\end{definition}

The next theorem plays a key role in proving the remaining theorems:
\begin{theorem}
\label{thm3}
Assume $U, U_1, U_2, \ldots, U_m$ are independently and identically distributed as $\chi_{1}^{2}-1$, and the weights $\{w_i \in [0,1], i=1,\ldots,m\}$ are decreasingly ordered and satisfies $\sum\limits_{i=1}^{m} w_{i}^{2}=1$. The summation density satisfies
\begin{align*}
f_{\sum\limits_{i=1}^{m}w_{i} U_i}(x) = O(e^{-x c/2})
\end{align*}
where the constant $c=\frac{1}{w_{1}} \in [1, \sqrt{m}]$.
\end{theorem}

Namely, the summation density is determined by the largest weight, which equals $U$ if and only if $w_1=1$. We can then bound the upper tail distribution of bias-corrected distance correlation as follows:
\begin{theorem}
\label{cor1}
For sufficiently large $n$, there exists $\alpha>0$ such that
\begin{align*}
\mc{N}(0,2) \preceq_{\alpha} n\sct{Dcor}_{n}(\mathbf{X},\mathbf{Y}) \preceq_{\alpha} U
\end{align*}
regardless of the metric choice or marginal distributions.
\end{theorem}
Therefore, despite the null distribution is subject to change without a fixed nor known density form, the centered chi-square distribution is always a valid approximation choice. Theorem~\ref{thm7} in Section~\ref{sec:main} (the validity and consistency of the chi-square test) follows trivially from Theorem~\ref{cor1}.

\subsection{Validity Level and Testing Power}
\label{sec:example}
The limiting null distribution have closed-form densities in some special cases, for which the largest validity level $\alpha$ (i.e., the largest type 1 error level so the chi-square test is still valid) can be exactly determined, e.g., $0.05$ in Theorem~\ref{thm4} and at most $0.0875$ in Theorem~\ref{thm6}.
\begin{theorem}
\label{thm4}
Given $m \geq 1$, assume the weights in Theorem~\ref{thm1} satisfy $w_i=\frac{1}{\sqrt{m}}$ for all $i =1,\ldots,m$ and zero otherwise. It follows that \begin{align*}
n \sct{Dcor}_{n}(\mathbf{X},\mathbf{Y}) \stackrel{D}{\rightarrow} \frac{\chi^{2}_{m}-m}{\sqrt{m}} \preceq_{0.05} U,
\end{align*}
where $\chi^{2}_{m}$ is the chi-square distribution of degree $m$.
\end{theorem}

\begin{corollary}
Under the same assumption of Theorem~\ref{thm3}. Let $m_{1} = \left \lfloor{1/w_{1}^{2}}\right \rfloor$ and $m_{2} = \left \lceil{1/w_{1}^{2}}\right \rceil$, it always holds that
\begin{align*}
\frac{\chi^{2}_{m_{2}}-m_{2}}{\sqrt{m_{2}}} \preceq_{\alpha} \sum\limits_{i=1}^{m}w_{i} U_i \preceq_{\alpha} \frac{\chi^{2}_{m_{1}}-m_{1}}{\sqrt{m_{1}}} \preceq_{\alpha}U.
\end{align*}
\end{corollary}

Next, we offer two random variable examples that achieve the known densities. Theorem~\ref{thm5} shows that the limiting null distribution can equal the centered chi-square distribution. Theorem~\ref{thm6} establishes the normal distribution (which is also used in the t-test in \citet{SzekelyRizzo2013a}) with a simplified condition and a simplified proof, i.e., only one random variable is required to have infinite dimension.
\begin{theorem}
\label{thm5}
When $X$ and $Y$ are two independent binary random variables, $n \sct{Dcor}_{n}(\mathbf{X},\mathbf{Y}) \stackrel{D}{\rightarrow} U$.
\end{theorem}

\begin{theorem}
\label{thm6}
Assume $X$ is independent of $Y$, $X$ is continuous, and each dimension of $X$ is exchangeable with positive finite variance. As $n,p \rightarrow \infty$, it holds that
\begin{align*}
&n \sct{Dcor}_{n}(\mathbf{X},\mathbf{Y}) \stackrel{D}{\rightarrow} \mc{N}(0,2) \preceq_{0.0875} U.
\end{align*}
\end{theorem}

As the distance correlation t-test essentially uses a t-transformation of $\mc{N}(0,2)$, we can view the t-test and the chi-square test as two-sides of the permutation test. In particular, the distance correlation t-test is an invalid test that slightly inflates the type $1$ error level, while the chi-square test is valid but always conservative in power:
\begin{corollary}
\label{cor0}
At any type 1 error level $\alpha \leq 0.05$, denote the testing power of distance correlation chi-square test, distance correlation t-test, and the permutation test as $\beta_{\alpha}^{\chi}, \beta_{\alpha}^{t}, \beta_{\alpha}$ respectively. At any $n$ and $\alpha$ such that Theorem~\ref{cor1} holds, there exists $\alpha_{1} \in (0,\alpha]$ and $\alpha_{2} \in (\alpha,0.0875]$ such that
\begin{align*}
\beta_{\alpha}^{\chi} = \beta_{\alpha_{1}} \leq \beta_{\alpha} < \beta_{\alpha_{2}} = \beta_{\alpha}^{t}.
\end{align*}
The actual $\alpha_{1}$ and $\alpha_{2}$ depend on the metric choice and marginal distributions.
\end{corollary}

Finally, there is no other valid test of distance correlation that is as fast and as powerful as the chi-square test:
\begin{corollary}
\label{cor2}
At any $n$ and $\alpha$ such that Theorem~\ref{cor1} holds, the chi-square test is the most powerful test among all valid tests of distance correlation using known distributions. Namely, for any valid test $z$ of distance correlation using a fixed distribution, it always holds that $\beta_{\alpha}^{\chi} \geq \beta_{\alpha}^{z}$.
\end{corollary}

\section{Numerical Experiments}
\label{sec:simu}

This section evaluates the numerical advantages of the chi-square test. First, we show that the centered chi-square distribution approximates the true null distribution of bias-corrected distance correlation in simulation. Second, the chi-square test is compared to existing tests on 1D data. Then the comparison is carried out on multivariate and high-dimensional data. And finally, we use a real data application to demonstrate the practical usage and superior performance of the chi-square test. Simulation function details are in Appendix~\ref{app_sim}.

\subsection{Null Distribution Approximation}
\label{simu0}

The top row of Figure~\ref{fig1} visualizes the centered chi-square distribution $U$ and the normal distribution $N(0,2)$ (divide each by $n$), and compare them with the actual null distribution of $\sct{Dcor}_{n}(\mathbf{X},\mathbf{Y})$ in varying dimensions. The centered chi-square distribution and the normal distribution are plotted by a solid line and an dashed line, respectively. We set sample size at $n=100$, generate independent $\mathbf{X}$ and $\mathbf{Y}$ for $r=10$,$000$ replicates and different $p,q$, and plot the null distribution in dotted line. The left panel shows the null distributions at $p=q=1$, the center panel is for $p=q=10$, and the right panel is for $p=q=100$. As expected from the theorems, the upper tail of the null distribution for $\alpha \leq 0.05$ (equivalently y-axis greater than $0.95$) always lies between and gradually shifts from $U$ to $N(0,2)$ as dimension increases.

The bottom row of Figure~\ref{fig1} shows the weights used in the corresponding limiting null distribution, associated with the ordered eigenvalue list $\{\lambda_i, i=1,\ldots,10\}$ normalized by $-(\sum\limits_{i=1}^{n}\lambda_i^2)^{0.5}$ (see proof of Theorem~\ref{thm1}). When $p=1$, the leading weight plays a dominating role such that $n \sct{Dcor}_{n}(\mathbf{X},\mathbf{Y}) \stackrel{D}{\approx} U$. As $p,q$ increase, all weights becoming similar such that $n \sct{Dcor}_{n}(\mathbf{X},\mathbf{Y}) \stackrel{D}{\rightarrow} N(0,2)$. Therefore, the most conservative the chi-square test can be is to approximate a normal distribution by a chi-square distribution of same mean and variance.

\begin{figure}
\includegraphics[width=1.0\textwidth,trim={1cm 0.5cm 1cm 0.5cm},clip]{FastFig1}
\caption{The top row compares the centered chi-square distribution, the normal distribution, and the actual null distribution of distance correlation in case of varying dimensions. The bottom row shows the weights used in the limiting null distribution in each case.}
\label{fig1}
\end{figure}

\subsection{One-Dimensional Random Variables}
\label{simu1}

Here we evaluate distance correlation chi-square test, permutation test, distance correlation t-test, HSIC Gamma test (only applicable to biased HSIC), and the subsampling method for four different 1D simulations: linear, quadratic, spiral, and independent. We extensively evaluated the tests on many different dependency types and decided to illustrate four representative simulations only, as the phenomenon is qualitatively similar throughout. All simulations are one-dimensional in this section, that is, $p=q=1$. In each simulation, we sample $n=20,40,\ldots, 200$ points, generate sample data $1$,$000$ times, run each test and reject at $\alpha=0.05$ level, and compute how often the test is correctly rejected (the testing power), which is then plotted against the sample size.

The top row of Figure~\ref{fig2} shows the power of distance correlation under Euclidean distance, the middle row shows the power of Hilbert-Schmidt independence criterion (equivalently the distance correlation using Gaussian kernel), and the bottom row shows the running time for the top row in log scale. The performance is the same throughout all dependency types and sample size and metric choice: the permutation test is the benchmark for testing power, but significantly slower than the fast alternatives; the distance correlation chi-square test has almost the same testing power as the benchmark permutation test, and the fastest method; the distance correlation t-test is equally fast, but consistently inflates the power in most cases, e.g., it has a power of $0.07$ for independence vs about $0.05$ of the permutation test; the subsampling method always yields degraded power.

Note that the HSIC Gamma test requires some parameter estimation using mean, variance, and kernel bandwidth, thus only applicable to the middle row for the biased HSIC. Its testing power is almost the same as the chi-square and permutation tests. This does not mean the chi-square test approximates the Gamma test in distribution. Rather, it implies that the Gamma test approximates the null distribution of biased HSIC well; the chi-square test approximates the bias-corrected null distribution well; then the permutation test has almost the same testing power with or without bias-correction.

\begin{figure}
\includegraphics[width=1.0\textwidth,trim={2cm 0cm 2cm 0cm},clip]{FastFig2}
\caption{Evaluate distance correlation using different tests for linear, quadratic, spiral, and independent simulations. The top row shows the power using the Euclidean distance, the center row shows the power using the Gaussian kernel, and the bottom row shows the running time (in log scale) for each method in the top row.}
\label{fig2}
\end{figure}

\subsection{Increasing-Dimensional Random Variables}
\label{simu2}
We consider four multivariate settings: equal variance (each dimension is exchangeable with same variance), minimal variance (first few dimensions has same variance while remaining dimensions have very small variance), dependent coordinates (consecutive dimensions are dependent), and varying marginals (the marginal distribution of each dimension is different). We fix the sample size and $q=1$, increase $p$ accordingly in each simulation, and compute the testing power at $\alpha=0.05$ based on $1000$ Monte-Carlo replicates.

The testing power and the running time are plotted against dimension in Figure~\ref{fig3}, offering almost the same interpretation as Figure~\ref{fig2}. In particular, the equal variance simulation is the only setting here satisfying the assumption of Theorem~\ref{thm6}, in which case the t-test only minimally inflate the testing power and the chi-square test exhibits a slightly more conservative testing power vs the permutation test. In the other three high-dimensional settings, the dimensions are no longer exchangeable, and the chi-square test has almost the same power as the permutation test. In terms of running time, the chi-square test and the t-test are the best, which do not increase much as dimension increases.

Finally, we evaluate the testing power for the multivariate simulation in Figure~\ref{fig1}, and present the results in Figure~\ref{fig4}. Regardless of $p,q,n$, the chi-square test is always similar in power as the permutation test and slightly conservative.

\begin{figure}
\centering
\includegraphics[width=1.0\textwidth,trim={2cm 0cm 2cm 0cm},clip]{FastFig3}
\caption{Evaluate distance correlation using different tests for four increasing-dimensional simulations using Euclidean distance. The first row shows the testing power in each simulation, and the second row shows the running time in log scale for each method in the respective first row.}
\label{fig3}
\end{figure}

\begin{figure}
\centering
\includegraphics[width=0.7\textwidth,trim={1cm 0.5cm 1cm 0.5cm},clip]{FastFig4}
\caption{The Testing Power for the simulation in Figure~\ref{fig1}. The left panel fix $n=200$, and let $p,q$ increases; the right panel set $p=q=50$, and let $n$ increases. }
\label{fig4}
\end{figure}

\subsection{Real Data Application}

Here we apply the tests to do feature selection on a proteomics data \citep{mgc1,PMID21248225}. The data consist of $318$ proteolytic peptides measurements, derived from the blood samples of $98$ individuals harboring pancreatic ($n=10$), ovarian ($n=24$), colorectal cancer ($n=28$), and healthy controls  ($n=33$). We would like to identify potential biomarkers for the pancreatic cancer, because it is lethal and no clinically useful biomarkers are currently available \citep{Bhat2012}.

For each biomarker, we first apply the distance correlation permutation test (using $500$ replicates) between the blood measurement of each peptide and the class label ($1$ for pancreatic cancer, $0$ otherwise). This yields $318$ p-values, and we take all peptides with p-value less than $0.05$ as the positive ones (in total 19 of them), and the remaining 299 peptides as the negative ones. Then we apply the chi-square test to each peptide vs the class label, and compute the true positive (i.e., when a peptide has p-value $<0.05$ from the chi-square test, whether the peptide also has p-value $<0.05$ from the permutation test) and true negative. Repeat for the t-test and subsampling method.

The result is summarized in Table~\ref{table1}. The chi-square test is fast and almost perfect for both true positive and true negative. The t-test is as fast, but inflates one of two errors. The subsampling is actually non-performing (all peptides have p-value $>0.1$ so every peptide is a negative) --- this is because due to the imbalanced label vector, most subsampled labels are a constant $0$, so the subsampled statistics end up being $0$ most of the time.

\begin{table*}[!ht]
\centering
\caption{Report the running time (in seconds), true positive and true negative for each method. Permutation test is the benchmark for determining the positive and negative peptides. Among all the approaches, the chi-square test is the fastest with almost perfect result. }
\label{table1}%
\begin{tabular}{l|l l l }
\toprule
Method & Running Time & True Positive & True Negative \\
\midrule
Permutation Test 	& 149.8  & 100\% & 100\% \\
\midrule
Chi-Square Test 	& \textbf{0.32}  & \textbf{92\%} & \textbf{98\%} \\
T Test   & 0.34 & \textbf{92\%} & 68\% \\
Subsampling & 0.48 & NaN & 94\% \\
\bottomrule
\end{tabular}
\end{table*}

\section{Discussion}
\label{sec:dis}
This paper proposes a new chi-square test for testing independence. It is very computationally efficient and scalable to big data, valid and universally consistent, applicable to general metric and kernel choices and other tasks like K-sample and partial, achieves similar power as the permutation test, and compares very favorably against all existing tests in simulations and real data. We expect the chi-square test to be widely used in practice due to its computational and performance advantages, and we plan to further investigate the stochastic dominance in theory and its potential applications in other inference tasks.

In particular, by Corollary~\ref{cor0} the chi-square test is always more conservative than the permutation test. But the power loss is negligible in 1D data and minimal in high-dimensional data as shown in Figure~\ref{fig2} and Figure~\ref{fig3}. In comparison, the t-test always inflates the testing power. As shown in Figure~\ref{fig1}, at the type 1 error level $0.05$ the t-test inflates the error to $(0.05,0.07]$, while the chi-square test has a conservative error within $[0.02,0.05]$. Viewed in another way, the t-test at $\alpha=0.05$ is equivalent in power to a permutation test at $\alpha \in (0.05,0.07]$, while the chi-square test at $\alpha=0.05$ is equivalent in power to a permutation test at $\alpha \in [0.02,0.05]$. Inflating the type 1 error slightly may not be a big issue in a single test, but can cause larger deviation in multiple comparison, as evidenced tin the real data application; whereas the chi-square test is not susceptible to this issue.

\section*{Acknowledgment}
This work was supported by the National Science Foundation award DMS-1921310, the National Institute of Health award R01MH120482, and DARPA L2M program FA8650-18-2-7834.
The authors thank Dr. Austin Brockmeier, Mr. Ronak Mehta, Dr. Yuexiao Dong, Dr. Carey Priebe for helpful discussions and suggestions. We also thank the editors and reviewers for the valuable suggestions that greatly improved the paper in organization, expositions, and experiments.

\bibliographystyle{apalike}

redpajama_set_name: RedPajamaArXiv


text:
Pilar Sanders is Suing Deion Sanders and His Daughter for $200 Million
By Valaney March 3, 2012
NFL Hall of Famer Deion Sanders is being sued by his wife for $200 million dollars. Friday morning on The Ben and Skin show on ESPN Dallas 103.3 FM, Deion defended himself against the lawsuit and the allegations made against him and his daughter.
"Where are the going to get the money from?" is the question he asked the host of the radio show. Deion then went on to say "That's just like me suing a homeless guy for some burger and fries."
Pilar Sanders filed the law suit last week at the district court in Collin County. The reason for the lawsuit? Slanderous comments made on twitter by Sanders and his daughter Deiondra Sanders.
The daughter of the Hall of Famer tweeted that Pilar is "The number one gold digger of the year" and many other comments. Deion is accused of co signing with his daughter tweeting "tired of all [Pilar's] lies and foolishness."
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It is a shame that all this is happening, could anything else happen to this NFL Hall of Famer?
Valaney
I am a passionate person. Every task that I pick I do it with passion. I am artist that loves all things artsy. I like to create things that many people can interpret differently. I am passionate about the youth and spend time mentoring, so maybe I can make a difference in their lives and the community as a whole.
Oprah Lands First Interview With Bobbi Kristina and the Houston Family
Is Our Food Making Us Sick? The Toxicity of American Food
New Single Alert: LUMiN's "The Worst Thing"
Why buying a property in Pune is a good deal for NRIs?
Aerosmith and Run DMC reunite for explosive Grammys performance
Gambling Revenue in Pennsylvania Skyrockets
Wiz Khalifa, Ty Dolla $ign, Lil Yachty & Sueco the Child release 'Speed Me Up' for the Sonic Movie
No Coming Back to the Unhealthy Life: Relapse Prevention Strategies
A Short Guide to the Different Types of Exercise
5 Tips To Help Children Be More Comfortable At The Dentist
Here Comes the Sun: 7 of the Hottest Updates in Solar Tech
Warm Wintry Vibes: "I Still Love You" by The Jayplayers
Why you Need to Try Virtual Healthcare
Weekend Wanderlust: How to Spend an Epic 3 Days In New Orleans
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text:
Arab-Andalusian Music
with Haig Yazdjian and Dimitris Koukoulitakis
The personal style with the characteristic Arabic and Armenian sound colors of oud virtuoso Haig Yazdjian encounters the Mediterranean sound of flamenco guitar of Dimitris Koukoulitakis in a fusion of Arab-Andalusian music with unexpected musical colors and paths.
Haig Yazdjian, Oud, vocals
Haig Yazdjian was born by Armenian parents in Syria, and has been living and working in Greecesince 1980. There, he discovered the enchanting elements of Eastern music, and took up the oud as the means to express an inner undiscovered domain. Soon, he became extremely popular as an oud soloist, and one of the most characteristic vocalists. After four solo albums and numerous participations in other artists' discographies, in 2007 comes into circulation the first studio album of Haig Yazdjian, entitled "Amalur". Amalur in Basque language means "Mother Earth". Haig, a citizen of the world feels that the sound of the word Amalur "beats" in pace with the music. Haig Yazdjian has collaborated with many important Greek composers and singers and soloists with international prestige. Among these are Nikos Xydakis, Eleftheria Arvanitaki, Maria Farantouri, Dimitra Galani, Ellie Paspala, Eleni Vitali, Loreenna McKennitt, David Lynch, Dave Spilain, Omar Faruk Tekbilek, Djivan Gasparyan, Theodossii Spassov and Glen Velez.
[Biography updated on: 31/10/2012]
Dimitris Koukoulitakis, Guitar
Performances: 2 | Duration: 180'
Saturday, 3 November, 2012 @ 21:00
Thursday, 20 December, 2012 @ 21:00


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WASHINGTON, D.C. -- The Biosimilars Forum and Medicines for Europe released a joint report today exploring how European countries have successfully instituted and sustained a multi-source, competitive biosimilars market over the last decade. The report also lays out a roadmap for United States (U.S.) policymakers to ensure that patients have access to the treatments they need and that the healthcare system at-large achieves the significant potential for cost-savings associated with a vibrant biosimilars market.
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Proactively encouraging and incentivizing switching to biosimilars by physicians in order to increase access and affordability and assessing shared savings models that can foster this.
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text:
PlayStation Store releases for 12th April 2016
Dark Souls III and Ratchet & Clank leads this week's releases on the PlayStation Store .
Dark Souls III and Ratchet & Clank leads this week's releases on the PlayStation Store.
Alongside the return of these two big franchises this week also sees the release of roguelike platformer Risk of Rain and the narratively playful, Stories: The Path of Destinies.
And if Samurai Warriors 1 through 3 weren't enough, it's your lucky week…
Here's the full line-up.
PlayStation Store releases for 12th April 2016.
Risk of Rain (Cross Buy)
Samurai Warriors 4 Empires
Stories: The Path of Destinies
Buy PlayStation Store Wallet Top-Up cards on Amazon.
Related Topics:PlayStation 3PlayStation 4PlayStation StorePlayStation Vita
New PlayStation Store releases (January 18-22, 2021)
PlayStation Plus free games for January 2021 revealed
Save up to 75% in the PlayStation Store January Sale
New PlayStation Store releases (December 21-25, 2020)


text:
Víctor Hugo Amatti (born 20 June 1965) is a former Argentine footballer who played for clubs of Chile.

External links

 Víctor Hugo Amatti at playmakerstats.com (English version of ceroacero.es)

1965 births
Living people
Argentine footballers
Argentine people of Italian descent
Argentine expatriate footballers
Deportes Iquique footballers
Lota Schwager footballers
Club Deportivo Palestino footballers
Coquimbo Unido footballers
C.D. Arturo Fernández Vial footballers
Santiago Wanderers footballers
Primera B de Chile players
Chilean Primera División players
Expatriate footballers in Chile
Argentine expatriate sportspeople in Chile
Naturalized citizens of Chile
Association football forwards

redpajama_set_name: RedPajamaWikipedia


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text:
19TH HOLE: THE READERS TAKE OVER
Gay Flood
ROSE AND RUTHVEN
Sir:
Regarding your May 28 cover subject, Pete Rose, phooey! Boo! How about Ray Knight, Rose's replacement at third base in Cincinnati? Knight is doing a remarkable job in the field and at the plate and is one of the chief contributors to the rise in the NL West of McNamara's Band. Yea, Ray!
WILLIAM F. O'BRIEN
I realize your production deadline was closing in and a cover had to be chosen, but since when is a 24-12 baseball team in May more important than the Stanley Cup finals?
STUART AGLER
Jackson Heights, N.Y.
My congratulations to Bruce Newman for an informative and delightful cover story on Pete Rose (He's the Phillie Fillip). Great job. Great subject.
CARSON EVANS
Lexington, Ky.
For those critical fans who feel that Pete Rose is being paid too much for what he does, it's a shame they don't have an opportunity to watch him play.
MARC ANDREWS
Wilmington, Del.
I just read your article on the Phillies and Dick Ruthven's comments about pitching in Atlanta. Durn! How could Ruthven or anyone else suffer from "terminal boredom" in Atlanta? The sun is always shining, the girls are pretty, the fans are friendly, the players are friendly. It's a great place to be, win or lose.
I was going to write a scathing letter disputing Ruthven's claim that he pitched before "800 fans," but, admittedly, he did pitch before 970 one time. However, the average crowd that saw Ruthven pitch at Atlanta Stadium during his two-plus seasons (1976-78) here was 13,360, and he was 14-17 for those games, which was better than his won-lost record on the road. He also lost the game that 970 attended.
A million people saw a last-place team play last year, and if a million come when we're in last, lots more will come when we're in first. And the Braves will win someday soon—without the help of Dick Ruthven.
CHARLIE'S A's (CONT.)
I applaud Ron Fimrite for his exposè" of the Charlie O. A's (They're Just Mad About Charlie, May 21). The potential was there to build one of the best franchises in major league baseball. In addition, given Finley's records with other professional teams, the current situation in Oakland doesn't seem that surprising.
His other major venture in the Bay Area still has many in National Hockey League circles scratching their heads. He outfitted the Seals in those tacky Kelly-green and California-gold uniforms, complete with green and gold skates. He then had the audacity to nickname them the Golden Seals, as if we couldn't tell that by looking at them. He began his policy of a tight pocketbook in 1972, when he let half his team jump to the World Hockey Association. And he traded other members of his squad to NHL franchises for unproven talent and that old reliable: cash.
In less than four years he put a loser on Oakland ice, a team far worse and much less competitive than the one he had bought in 1970. Finley asked the NHL to take the disaster off his hands, and the league did, later peddling the team off to an eventual death in Cleveland (at the time, no place was considered a worse hockey market than Oakland, but the NHL found one).
One admirable quality about Finley is his consistency. But the constant deterioration of his franchises and his knack for alienating both his players and his teams' fans aren't admirable. If the right people are doing the selling, one can sell anything to anyone, anywhere, anytime.
I hope that the other owners and the American League won't attempt to bail Finley out of his current mess. Maybe the embarrassing situation that he faces with the Coliseum will teach him a lesson.
LAURENCE CONWAY
I believe that the other American League owners must do their part to oust Charles Finley, otherwise the A's franchise will never again amount to anything. Baseball's brass hats never act unless alerted by a sudden weightless feeling in their pocketbooks. A few more red-ink-stained trips out here, however, should convince them that it is in their interest to buy the club from Finley and sell it to someone ready, willing and able to run it correctly. As long as Finley owns the club, the A's will continue to be an embarrassment not only to the Bay Area, but also to the league—despite the efforts of the young, hustling Oakland players, who deserve better.
GARY SMITH
Berkeley, Calif.
In your article you said, "Once inside [the Oakland Coliseum], the odds on catching a foul ball are much better than anywhere else." That is so true. Last year I caught five foul balls during one game. That's the only reason I go to A's games—for foul balls. As for the A's, who cares? I'm a Giants fan.
TOM WITTENBERG
Danville, Calif.
I appreciated your article on decathlon historian and figure filbert Frank Zarnowski (He's Every Inch a Decathnut, May 7). I am an ardent fan of the decathlon, and my roommate, David Lee Steen, is Canada's and the University of California's premier competitor in this grueling competition.
Zarnowski cites the Mulkeys—Phil and Phil Jr.—of the U.S. as the world's father-son decathlon champions, with a combined total of 14,548 points. He further claims that the Jewlews, senior and junior, of the U.S.S.R., are second with an aggregate of 14,451 points. After your article was published, my roommate scored 7,647 points in the Pac-10 decathlon championships in Tempe, Ariz. on May 16-17. Add this to the 6,860 points Dave's father, Donald Steen, scored in Eugene, Ore. on May 4-5, 1957, and the Steens, with a total of 14,507, surely outpoint the second-place Soviets.
Additionally, Donald Steen's total was based on the 1950 scoring tables. If you convert his marks to today's tables, I believe that the Steens have the world father-son record.
JAY PARDINI
FAST-PITCHER
Thanks for your article on Ty Stofflet (This Guy Can Rise It, Drop It and Pop It at 104 mph. May 28). As a dabbler in the art of fast-pitch softball hurling, I can appreciate his sacrifices and admire his accomplishments.
It was also encouraging to see coverage of the fast-pitch game, which requires much more skill and dedication than does its more popular counterpart, slow-pitch.
Perhaps, in time, fast-pitch will regain popularity. Until then, we fast-pitch players can take solace in the fact that our game is a sport, while slow-pitch is just a recreation.
BILL LOHMANN
In some states, fast-pitch Softball is classified according to a pitcher's ability. Those who have competed in Pennsylvania or who have seen Ty Stofflet throw know that these classifications are C, B, A, major and Ty Stofflet. Thanks to your article, everyone will now know that Stofflet is in a class above the rest in all other ways, too. And thanks to Jack McCallum's great article, I now know of another restaurant in Dutch country where I can "eat 'til my stomach ouches" me. McCallum touched the hearts of fast-pitch enthusiasts across the country.
BOB STONEBRAKER
Piscataway, N.J.
I played some fast-pitch softball while growing up in Ocala, Fla. I loved the game and held all pitchers in awe. How they could throw that hard underhand and still have control was beyond me. Thanks for giving the game in general and Ty Stofflet in particular the recognition both deserve. Now you need to write an article on the man who catches for Stofflet—he must be a super athlete!
THE REV. DOUG MOORE
Crestwood, Ky.
•For the past five seasons, Stofflet's catcher has been Carl Solarek, a three-time Amateur Softball Association All-America. His manager, Rocco Santilli, calls Solarek "the best defensive catcher in the country."—ED.
With such impressive pitching statistics, it isn't surprising that Ty Stofflet prefers the tempo of fast-pitch to that of slow-pitch. Why shouldn't he—he's involved in every play. But what about the fielders behind him who might legitimately wonder if the ball will ever be hit in their direction? Some fun to take your position in the field and watch an endless line of "would-be hitters" either strike out or meekly ground out!
In slow-pitch, being able to hit the ball is fun, but that's only half of the game. The other half is when you're in the field and you know that every batter is going to make "contact" with the ball, which means there is a chance for running, fielding and throwing with every at bat.
With a good fast-pitch pitcher like Stofflet, a team could hide three lead gloves in the field and never be hurt. In slow-pitch, one fielder with cement hands is soon found and exploited.
As you might guess, I am a strong advocate of slow-pitch. I'm still trying to get the cobwebs out of my glove from my fast-pitch days.
DENNIS L. GREENHAW
Independence, Kans.
Slow-pitch softball is growing by leaps and bounds because it's fun to watch something more than a great pitcher and catcher dominate a game. Stofflet can have his form of softball, but tell him not to forget to wake up the fans at the end of the game so they can go home.
BRYANT C. TAYLOR
Glendora, Calif.
COUSINEAU (CONT.)
Although I enjoyed your article on Tom Cousineau ("You Made a Wise Choice," May 21) and thought that Douglas S. Looney did an excellent job of writing, there appears to be a mistake. Looney states that, since the NFL draft was initiated in 1936, Ohio State's Cousineau is only the second linebacker to be drafted No. 1, the first having been Texas' Tommy Nobis, by Atlanta in 1966. However, according to us Philadelphia Eagle fans and a book by Jack McCallum and Chuck Bednarik called Bednarik: Last of the Sixty-Minute Men, Penn's Bednarik was the first linebacker to be drafted No. 1 by an NFL team (the Eagles in 1949). This would make Cousineau the third linebacker so chosen.
RANDY KIENZLE
Paoli, Pa.
•Officially, Bednarik was drafted as a center, not a linebacker, but he played both ways. At Penn, Bednarik averaged 58 minutes a game. And during his 14-year pro career he continued to play both ways, although he was chiefly renowned as a linebacker, a position at which he was a seven-time All Pro.—ED.
SEALS AND IRONMEN
Your article Ironman (May 14) was indeed a tribute to the men, and to Lyn Lemaire, who participated in the Hawaiian Iron Man Triathlon, perhaps the ultimate test of an individual's motivation, fortitude and endurance. I read the story with personal pride and interest because it was my privilege to have served with second-place finisher John Dunbar during his tour in the navy on Seal Team One. He was an exemplary team man then, and it doesn't surprise me to see him continuing to successfully test himself. Although your brief description of Seal training is exaggerated, it does give the reader a flavor of the rigors that Seals must endure. We attempt to push a man well past what he thinks is his breaking point to show him that he always has something left in reserve. Certainly the participants in the Iron Man Triathlon have experienced the satisfaction of reaching way down and never coming up empty-handed.
THOMAS N. LAWSON
Lieutenant Commander, USN
Seal Team Two
Norfolk, Va.
Address editorial mail to SPORTS ILLUSTRATED, Time & Life Building, Rockefeller Center, New York, New York, 10020.


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text:
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Ford Mustang fans are invited to sign the electronic birthday card marking 50 years of their favorite car
Ford Buyer's Guide
DEARBORN, MI--April 10, 2014; Fans can help set a new GUINNESS WORLD RECORDS® title for most contributions on an electronic card with more than 50,000 signatures and messages
50 Years of Mustang will be celebrated at a variety of events in several locations
Ford Mustang is turning 50 next week and fans everywhere are invited to sign an electronic birthday card. Everyone who wants to help commemorate 50 years of Mustang can customize and add their own birthday message for their favorite car at Mustang-50.
Over the course of five decades Mustang has become one of the most popular cars in both the offline and online worlds, with sales of more than 9 million units and more than 6.2 million fans on Facebook. Mustang fans can participate in a GUINNESS WORLD RECORDS official attempt by signing the card, leaving a message and sharing the link on social media. 50,000 signatures and messages are needed to set new GUINNESS WORLD RECORDS title.
"The spirit of Mustang infuses everything we do at Ford," said Steve Ling, car marketing manager. "Mustang inspires a feeling of freedom and independence for drivers and fans, and it helps to inspire everyone at Ford to create the best products we can for all of our customers in every segment."
April 17 marks 50 years since Mustang officially went on sale, and the car's global legions of fans will honor the occasion with events including a new 2015 Mustang convertible on the observation deck of the Empire State Building, dual celebrations organized by the Mustang Club of America, convoys of Mustangs road tripping across the country and an employee celebration at Ford Motor Company headquarters.
2015 Ford Mustang The all-new Mustang is the most advanced version yet of the iconic pony car. The sleek new design of Mustang fastback and convertible is backed up by world-class performance from a range of available engines, including a new turbocharged EcoBoost® 2.3-liter and an upgraded 5.0-liter V8 with more than 420 horsepower.
In addition to state-of-the-art connectivity systems, Mustang features available advanced driver-assist systems, track apps, launch control and more.
Drivers in China and around the world will have an opportunity to experience the freedom and excitement of Mustang firsthand when the car becomes available there for the first time in 2015.
Search, Explore, and Discover!
Copyright 1996– The Auto Channel.


The San Diego Padres have acquired Marc Rzepczynski from the Cleveland Indians in exchange for Abraham Almonte, according to Keith Law. The deal has been confirmed by the teams.
The 29-year-old Rzepcynski has posted an ERA of 4.43 in 20 1/3 innings this season. The lefty has walked 4.4 batters per nine innings, which has been one of the main causes of his struggles this season.
Almonte, 26, is hitting a slash of .204/.271/.259 in 62 plate appearances this season. The switch-hitter has seen more time in Triple-A, hitting .275/.361/.414 with four homers and 35 RBIs in 282 plate appearances.


Over the course of Summer and Fall 2018, the Eppley Institute was tasked with assisting the Montgomery County Parks (Maryland) Volunteer Services Office with developing and delivering required training to over 375 people who were awarded deer hunting permits in parks across the district. The outcome of the course was to help participants understand the goals of the program, follow the proper procedures, and participate safely in managed deer hunts.
To deliver the training, the Eppley Institute developed a custom training portal for Montgomery Parks. Accounts were created for participants and participants were provided a link to access the training. Once the participant completed the training and successfully passed an assessment, they were provided a certificate of completion that could be presented at check-in on the day of the deer hunt.
The custom training portal also provided administrators of this program with the ability to verify course completions, track participation levels, gauge levels of engagement with the training materials, monitor assessment results, and survey participants on their reaction to the training.
As a result of this project, Montgomery County was able to make the required training more easily accessible and better verify that participants had completed the training, all while reducing the cost of training delivery.


text:
FBI, This Week: Wildlife Trafficking
The FBI uses its investigative expertise in transnational criminal enterprise cases to help address illicit wildlife trafficking networks.
Mollie Halpern: The FBI uses its investigative expertise in transnational criminal enterprise cases to help address illicit wildlife trafficking networks.
In that way, the FBI can be a valuable partner to the U.S. Fish and Wildlife Service and the Department of Homeland Security.
Maxwell Marker worked and oversaw these types of investigations as an FBI section chief.
Maxwell Marker: Typically, organized crime groups are all about money and profit, so they do a lot of different crimes. One piece of that criminal activity may be wildlife trafficking. And that's typically how the FBI gets involved, because of our organized crime expertise and the individuals that work organized crime within the Bureau.
Halpern: African, Asian, Middle Eastern, and other organized crime groups traffic wildlife products like ivory and rhinoceros horns from Africa into the United States for profit.
Endangered fish from Mexico have also been trafficked into the U.S.
The wildlife is used for traditional medicine, food delicacies, and as status symbols.
Marker: It's shocking to me the volume of wildlife that are trafficked in this day and age, because you have a huge international awareness now of the fact that a lot of this wildlife isn't going to be around anymore, 10, 15, 20, 50 years from now. But it doesn't seem to be slowing people down.
Halpern: The FBI is also a training resource for law enforcement in countries like Kenya, where the wildlife are sourced.
The Bureau provides its international law enforcement partners with techniques to enhance their investigative skills beyond targeting poachers and into dismantling criminal networks.
With FBI, This Week, I'm Mollie Halpern of the Bureau.
01.21.2020 — Esta Semana en el FBI: Ley de Protección de las Víctimas de la Trata de Personas
01.21.2020 — FBI, This Week: 2019 Preliminary Semiannual Uniform Crime Report Released
01.16.2020 — Se Busca por el FBI: Jose Arturo Navarrete, Jr.
01.16.2020 — Gotcha: Brink's Robbery Anniversary
01.16.2020 — FBI, This Week: Illicit Mining and Human Rights Abuses


Still judging people by their skin color? Which cave did you just crawl out of — and will you go crawl back in it?
The comment was not addressed to the author of the piece, but to it's readers, and supports in every way the point you are making in the piece. Feel free not to be carried off by knee jerk paranoia next time.
Reply in commentary to general piece–won't copy & paste to individual comment reply.


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Kuta is one of Bali's most famous destinations, a brash, lively seaside resort promising fun in the sun by day and non-stop partying by night. What was once a quiet fishing village nestled around a sweeping expanse of sand is today a pumping holiday destination where surfing, shopping and beach fun give way to bars and nightclubs galore as the sun goes down. The centre and south of Kuta draw crowds of younger backpackers looking for carefree entertainment. It's here that the lion's share of budget restaurants, accommodation and surf shops can be found. Further north, Legian caters to families and couples and upmarket Seminyak appeals to those looking for a dash more luxury.
Kuta centres on the spectacular stretch of Kuta Beach, a lengthy strip met by some of the country's best surfing waves. A lack of dangerous rocks or coral reefs, plus clean breaks and warm waters, make the conditions perfect for experienced and beginner surfers alike, and it was this that first drew tourists to what was then just a sleepy fishing village back in the 1970s. Today it's a surfing haven, with surf shops lining the main street and a true 'surfer dude' ambience. Water sports opportunities abound, abundant dive shops offer exhilarating scuba experiences, and traditional Indonesian massages are brought straight to your sun lounger. Indeed, it's almost impossible to visit Kuta without enjoying the ultimate relaxing pampering session. There are dozens of spas offering a range of wellness experiences at some of the cheapest prices in all of Indonesia.
Away from the beach, one of Kuta's most popular activities is shopping. There is an eclectic range of shops to explore, be it the rows of small stalls selling wacky T-shirts and kitsch souvenirs around Kuta Square or the big air-conditioned shopping malls. Pick up the very latest beachwear by international brands in the surfing and sports shops, get a traditionally woven hammock to take home, or enjoy some friendly bartering with the hawkers on the beach. Shopping malls are where the big brand names can be found, amid modern architecture, good eateries and lots of entertainment options.
Kuta's nightlife is legendary and the streets teem with revellers until the small hours. Start the evening with sunset cocktails in a laid-back beachfront bar and end in one of the booming discos that line the strip. It's worth taking a moment during your trip, however, to pay a visit to the Monument of Human Tragedy, built to commemorate the 2002 and 2005 bombings in Bali nightclubs by Islamic extremists.
None would describe Kuta as Indonesia at its most traditional, and temples and historical sights certainly aren't why most people visit. But for something more authentically Balinese, get up early (or stay up late) and you might glimpse locals throwing offerings into the Indian Ocean near the temple on the beach. North Kuta is home to the Tanah Lot, one of Bali's most important Hindu temples, which is balanced precariously on an offshore rocky outcrop.


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Infections of the abdomen, blood, or bones.
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These symptoms appear till your body gets adjusted to the medication.
What Are The Contraindications Of Netilmicin?