text:
\section{Introduction}
\IEEEPARstart{T}{HIS} paper aims to cluster sequences generated by \emph{unknown} continuous distributions into classes so that each class contains all the sequences generated from the same (composite) distribution cluster. By sequence, we mean a set of feature observations generated by an underlying probability distribution. Here each distribution cluster contains a set of distributions
that are close to each other, whereas different clusters are assumed to be far away from each other based on a certain distance metric between distributions. To be more concrete, we assume that the maximum intra-cluster distance (or its upper bound) is smaller than the minimum inter-cluster distance (or its lower bound). This assumption is necessary for the clustering problem to be meaningful. The distance metrics used to characterize the difference of data sequences and corresponding distributions are assumed to have certain properties summarized in Assumption \ref{assump} in Section \ref{sec:sys_model}.

Such unsupervised learning problems have been widely studied  \cite{Bishop06,Barber12}. The problem is typically solved by applying clustering methods, e.g., k-means clustering \cite{Lloyd1982,Stei2000,Kanu2002} and k-medoids clustering\cite{kaufman1987,Laan2003,park2009}, where the data sequences are viewed as multivariate data with Euclidean distance as the distance metric. These clustering algorithms usually require the knowledge of the number of clusters and they differ in how the initial centers are determined. One reasonable way is to choose a data sequence as a center if it has the largest minimum distances to all the existing centers \cite{Katsavounidis1994,Wang:ICASSP2018,Wang:CISS2018}. Alternatively, all the initial centers can be randomly chosen \cite{MORENOSAEZ2014}. With the number of clusters unknown, there are typically two alternative approaches for clustering. One starts with a small number of clusters, e.g., $1$, which is an underestimate of the true number, and proceed to split the existing clusters until convergence \cite{LOPEZ2015,Wang:CISS2018}. The authors in \cite{LOPEZ2015} assumed a maximum number of clusters and the threshold for clustering depended on a pre-determined significance level of the two sample Kolmogorov-Smirnov (KS) test whereas the algorithm proposed in \cite{Wang:CISS2018} did not assume a maximum number of clusters and the threshold for clustering was a function of the intra-cluster and inter-cluster distances. Alternatively, one may start with an overestimated the number of clusters, e.g., every sequence is treated as a cluster, and proceed to merge clusters that are deemed close to each other \cite{Wang:CISS2018}. The algorithms in \cite{Katsavounidis1994,MORENOSAEZ2014,LOPEZ2015} were all validated by simulation results without carrying out an analysis of the error probability.

There are some key differences between the k-means algorithm and the k-medoids algorithm. The k-means algorithm minimizes a sum of squared Euclidean distances. Meanwhile, the k-medoids algorithm assigns data sequences as centers and minimizes a sum of arbitrary distances, which makes it more robust to outliers and noise \cite{Ng2002,kaufman2009}. Moreover, the k-means algorithm requires updating the distances between data sequences and the corresponding centroids in every iteration whereas the k-medoids algorithm only requires the pairwise distances of the data sequences, which can be computed before hand. Thus, the k-medoids algorithm outperforms the k-means algorithm in terms of computational complexity as the number of sequences increases \cite{velmurugan2010}.

Most prior research focused on computational complexity analysis, whereas the error probability and the performance comparison of different clustering algorithms were typically studied through simulations, e.g., \cite{Laan2003,Sheng2006,park2009,velmurugan2010}. This paper attempts to theoretically analyze the error probability for the k-medoids algorithm especially in the asymptotic region. Furthermore, in contrast to previous studies, which frequently used vector norms as the distance metric, e.g., Euclidean distance, our study adopts the distance metrics between distributions for clustering in order to capture the statistical models of data sequences considered in this paper. This formulation based on a distributional distance metric is uniquely suited to the proposed clustering problem, where each data point, i.e., each data sequence, represents a probability distribution and each cluster is a collection of closely related distributions, i.e., composite hypotheses.

Various distance metrics that take the distribution properties into account can be reasonable choices, e.g., KS distance \cite{Wang:CISS2018,Wang:ICASSP2018,LOPEZ2015,MORENOSAEZ2014} and maximum mean discrepancy (MMD) \cite{Gangeh2014}.
Our previous work \cite{Wang:CISS2018,Wang:ICASSP2018} has shown that the KS distance based k-medoids algorithms are exponentially consistent for both known and unknown number of clusters. In this paper, we consider a much more general framework and instead of focusing on a specific distance metric such as KS distance in our prior work \cite{Wang:CISS2018,Wang:ICASSP2018}, we consider any distance metric that satisfies Assumption \ref{assump} for clustering. The rationale is the following: with any distance metric that captures the statistical model of data sequences, one is likely to observe that for a large sample size, 1) sequences generated from the same distribution cluster are close to each other, and 2) sequences generated from different distribution clusters are far away from each other. Thus, if the minimum distance of different distribution clusters is greater than the maximum diameter of all the distribution clusters defined in Section \ref{sec:sys_model}, then it becomes unlikely to make a clustering error as the sample size increases. In this paper, we develop k-mediods distribution clustering algorithms where the distances between distributions are selected to capture the underlying statistical model of the data. We analyze the error probability of the proposed algorithms, which takes the form of exponential decay as the sample size increases under Assumption \ref{assump}.
Furthermore, beyond the KS distance, we also consider another distance metric namely MMD and show that they both satisfy Assumption \ref{assump} so that the error probability of the proposed algorithms decays exponentially fast if the KS distance and MMD are used as the distance metrics.
\par

We note that recent studies \cite{Li2013,Bu2017,Liang16tsp_a,Liang16tsp_b,mai2012} of anomaly detection problems and classification also took into account the statistical model of the data sequences. Our focus here is on the clustering problem, leading to error performance analysis that is substantially different from that in \cite{Li2013,Bu2017,Liang16tsp_a,Liang16tsp_b,mai2012}.

The rest of the paper is organized as follows. In Section \ref{sec:sys_model}, the system model of the clustering problem, the preliminaries of the KS distance and MMD, and notations are introduced. The clustering algorithm given the number of clusters and the corresponding upper bound on the error probability are provided in Section \ref{sec:knownK}, followed by the results of the clustering algorithms with an unknown number of clusters in Section \ref{sec:unknownK}. The simulation results for the KS and MMD based algorithms are provided in Section \ref{sec:numerical_result}.

\section{system model and preliminaries}\label{sec:sys_model}
\subsection{Clustering Problem}
Suppose there are $K$ distribution clusters denoted by $\mathcal{P}_{k}$ for $k=1,\ldots,K$, where $K$ is fixed. Define the intra-cluster distance of $\mathcal{P}_{k}$ and the inter-cluster distance between $\mathcal{P}_{k}$ and $\mathcal{P}_{k^{\prime}}$ for $k\neq k^{\prime}$ respectively as
\begin{equation}\label{eq:cluster_setup1}
\begin{aligned}
d\big(\mathcal{P}_{k}\big)& =\sup_{p_{i},p_{i^{\prime}}\in\mathcal{P}_{k}}d\big(p_{i},p_{i^{\prime}}\big),\\
d\big(\mathcal{P}_{k},\mathcal{P}_{k^{\prime}}\big)& =\inf_{p_{i}\in \mathcal{P}_{k},p_{i^{\prime}}\in\mathcal{P}_{k^{\prime}}}d\big(p_{i},p_{i^{\prime}}\big),
\end{aligned}
\end{equation}
where $d\left(\cdot,\cdot\right)$ is a distance metric between distributions, e.g., the KS distance or MMD defined later in \eqref{eq:defKS} and \eqref{eq:mmdpq} respectively. Then $d\left(\mathcal{P}_{k}\right)$ and $d\left(\mathcal{P}_{k},\mathcal{P}_{k^{\prime}}\right)$ represent the diameter of $\mathcal{P}_{k}$ and the distance between $\mathcal{P}_{k}$ and $\mathcal{P}_{k^{\prime}}$, respectively. Define
\begin{equation}\label{eq:cluster_setup2}
\begin{aligned}
d_{L} &= \max_{k=1,\ldots,K}d\big(\mathcal{P}_{k}\big),\\
d_{H} &= \min_{k \neq k^{\prime}}d\big(\mathcal{P}_{k},\mathcal{P}_{k^{\prime}}\big),\\
\Sigma & = d_{H}+d_{L},\\
\Delta & = d_{H} -d_{L}.
\end{aligned}
\end{equation}
Table \ref{table:notation} summarizes the notations of the generalized form of distances defined in \eqref{eq:cluster_setup1} and \eqref{eq:cluster_setup2} which will be used in the following discussion.
\begin{table}
	\caption{Notations}
	\centering
	\begin{tabular}{ | c | c | c |}
		\hline
		general & KS & MMD \\ \hline
		$d\left(\mathcal{P}_{k}\right)$ & $d_{KS}\left(\mathcal{P}_{k}\right)$ & $\mathbb{MMD}^{2}\left(\mathcal{P}_{k}\right)$\\ \hline
		$d\left(\mathcal{P}_{k},\mathcal{P}_{k^{\prime}}\right)$ & $d_{KS}\left(\mathcal{P}_{k},\mathcal{P}_{k^{\prime}}\right)$ & $\mathbb{MMD}^{2}\left(\mathcal{P}_{k},\mathcal{P}_{k^{\prime}}\right)$\\ \hline
		$d_{L}$ & $d_{L,ks}$ &  $d_{L,mmd}$ \\ \hline
		$d_{H}$ & $d_{H,ks}$ &  $d_{H,mmd}$ \\ \hline
		$\Delta$ &  $\Delta_{ks}$  & $\Delta_{mmd}$ \\ \hline
		$\Sigma$ &  $\Sigma_{ks}$  & $\Sigma_{mmd}$ \\ \hline
	\end{tabular}
	\label{table:notation}
\end{table}
\par
Suppose $M_{k}$ data sequences are generated from the distributions in $\mathcal{P}_{k}$, and hence a total of $\sum_{k=1}^{K}M_{k} = M$ sequences are to be clustered. Without loss of generality, assume that each sequence $\mathbf{x}_{k,j_{k}}$ consists of $n$ independently identically distributed (i.i.d.) samples generated from $p_{k,j_{k}}\in\mathcal{P}_{k}$ for $k=1,\ldots,K$ and $j_{k}\in\{1,\ldots, M_{k}\}$. The $i$-th observation in $\mathbf{x}_{k,j_{k}}$ is denoted by $\mathbf{x}_{k,j_{k}}[i]\in \mathbb{R}^{m}$, where $m<\infty$ and $i\in\{1,\ldots,n\}$. Note that for any fixed $k$, $p_{k,j_{k}}$'s are not necessarily distinct. Namely, for a fixed $k$, some $\mathbf{x}_{k,j_{k}}$'s can be generated from the same distribution. Although the following discussion assumes that all the data sequences have the same length, our analysis can be easily extended to the case with different sequence lengths by replacing $n$ with the minimum sequence length. In order to analyze the error probability of the clustering algorithm, we introduce an assumption relating to the concentration property of the distance metrics:
\begin{assumption}\label{assump}
For any distribution clusters $\{\mathcal{P}_{1},\ldots,\mathcal{P}_{K}\}$ and any sequences $\mathbf{x}_{k,j_{k}}\sim p_{k,j_{k}}$, $\mathbf{x}_{k,j_{k}^{\prime}}\sim p_{k,j_{k}^{\prime}}$ and $\mathbf{x}_{k^{\prime},j_{k^{\prime}}}\sim p_{k^{\prime},j_{k^{\prime}}}$, where $k\neq k^{\prime}$, the following inequalities hold:
\begin{subequations}\label{eq:assump1}
\begin{align}
&d_{L} < d_{H},\label{eq:KSassumptionHT}\\
&P\big(d(\mathbf{x}_{k,j_{k}},\mathbf{x}_{k^{\prime},j_{k^{\prime}}})\leq d_{th}\big) \leq a_{1}e^{-bn}\:\forall d_{th}\in \left(d_{L},d_{H}\right), \label{eq:dist_condition2}\\
&P\big(d(\mathbf{x}_{k,j_{k}},\mathbf{x}_{k,j_{k}^{\prime}})>d_{th}\big) \leq a_{2}e^{-bn}\:\:\forall d_{th}\in
\left(d_{L},d_{H}\right), \label{eq:dist_condition1}\\
&P\big(d(\mathbf{x}_{k,j_{k}},\mathbf{x}_{k,j_{k}^{\prime}})\geq d(\mathbf{x}_{k,j_{k}},\mathbf{x}_{k^{\prime},j_{k^{\prime}}})\big) \leq a_{3}e^{-bn}, \label{eq:dist_condition3}
\end{align}
\end{subequations}
where $a_{i}$'s are some constants independent of distributions, $b$ ($>0$) is a function of $d_{th}$ and $n$ is the sample size.
\hfill\qedsymbol
\end{assumption}
Here \eqref{eq:KSassumptionHT} guarantees that the lower bound of inter-cluster distances is greater than the upper bound of intra-cluster distances. \eqref{eq:dist_condition2} guarantees that the probability that the distance between two sequences generated from different distribution clusters is smaller than $d_{H}$ decays exponentially fast. \eqref{eq:dist_condition1} guarantees that the probability that the distance between two sequences generated from the same distribution cluster is greater than $d_{L}$ decays exponentially fast. \eqref{eq:dist_condition3} guarantees that given two sequences generated from the same cluster and a third sequence generated from another distribution cluster, the probability that the first sequence is actually closer to the third sequence decays exponentially fast.
\par
A clustering output is said to be incorrect if and only if the sequences generated by different distribution clusters are assigned to the same cluster, or
sequences generated by the same distribution cluster are assigned to more than one cluster. Denote by $P_{e}$ the error probability of a clustering algorithm. A clustering algorithm is said to be {\em consistent} if
\begin{equation*}
\lim_{n\rightarrow\infty}P_{e}=0,
\end{equation*}
where $n$ is the sample size. The algorithm is said to be {\em exponentially consistent} if
\begin{equation*}
B = \lim_{n\rightarrow\infty}-\frac{1}{n}\log P_{e}>0.
\end{equation*}
For the case where a clustering algorithm is exponentially consistent, we are also interested in characterizing the error exponent $B$.\par
\subsection{Preliminaries of KS distance}
Suppose $\mathbf{x}$ is generated by the distribution $p$, where $\mathbf{x}[i]\in \mathbb{R}$. Then the empirical cumulative distribution function (c.d.f.) induced by $\mathbf{x}$ is given by
\begin{equation}\label{eq:def-cdf}
F_{\mathbf{x}}\left(a\right)=\frac{1}{n}\sum_{i=1}^{n}1_{[-\infty,a]}\left(\mathbf{x}[i]\right),
\end{equation}
where $1_{[-\infty,x]}\left(\cdot\right)$ is the indicator function. Let the c.d.f. of $p$ evaluated at $a$ be $F_{p}\left(a\right)$. The KS distance is defined as
\begin{equation}\label{eq:defKS}
d_{KS}\left(p,q\right) = \sup_{a\in \mathbb{R}}|F_{p}\left(a\right)- F_{q}\left(a\right)|,
\end{equation}
where $F_{p}\left(a\right)$ and $F_{q}\left(a\right)$ can be either c.d.f's of distributions or empirical c.d.f.'s induced by sequences.
\begin{proposition}\label{proposition1}
	The KS distance satisfies \eqref{eq:dist_condition2} - \eqref{eq:dist_condition3} if $d_{L,ks}<d_{H,ks}$.
\end{proposition}
\begin{proof}
	See Lemmas \ref{lemma:KStest3}, \ref{lemma:KStest2}, and \ref{lemma:KStest4} in Appendix \ref{app:lemmas}.
\end{proof}
\subsection{Preliminaries of MMD}
Suppose $\mathcal{P}$ includes a class of probability distributions, and suppose $\mathcal{H}$ is the reproducing kernel Hilbert space (RKHS) associated with a kernel $g\left(\cdot,\cdot\right)$. Define a mapping from $\mathcal{P}$ to $\mathcal{H}$ such that each distribution $p\in \mathcal{P}$ is mapped into an element in $\mathcal{H}$ as follows
\[\mu_p\left(\cdot\right)=\mathbb{E}_p [g\left(\cdot,x\right)]=\int g\left(\cdot,x\right)dp\left(x\right), \]
where $\mu_p\left(\cdot\right)$ is referred to as the {\em mean embedding} of the distribution $p$ into the Hilbert space $\mathcal{H}$. Due to the reproducing property of $\mathcal{H}$, it is clear that $\mathbb{E}_p[f]=\langle \mu_p,f \rangle_{\mathcal{H}}$ for all $f \in \mathcal{H}$.\par
It has been shown in \cite{Fuku2008,Srip2008,Fuku2009,Srip2010} that for many RKHSs such as those associated with Gaussian and Laplace kernels, the mean embedding is injective, i.e., each $p \in \mathcal{P}$ is mapped to a unique element $\mu_p \in \mathcal{H}$. In this way, many machine learning problems with unknown distributions can be solved by studying mean embeddings of probability distributions without actually estimating the distributions, e.g., \cite{Song2013,Smola2007,Liang16tsp_a,Liang16tsp_b}.  In order to distinguish between two distributions $p$ and $q$, the authors in \cite{Gretton2012} introduced the following notion of MMD based on the mean embeddings $\mu_p$ and $\mu_q$ of $p$ and $q$, respectively:
\begin{equation}\label{eq:mmdpq}
\mathbb{MMD}\left(p,q\right):=\|\mu_p-\mu_q\|_{\mathcal{H}}.
\end{equation}
An unbiased estimator of $\mathbb{MMD}^2\left(p,q\right)$ based on $\mathbf{x}$ and $\mathbf{y}$ which consist of $n$ and $m$ samples, respectively, is given by
\begin{equation}\label{eq:mmdu}
\begin{aligned}
&\quad\:\mathbb{MMD}^2\left(\mathbf{x},\mathbf{y}\right)\\
&=\frac{1}{n\left(n-1\right)}\sum_{i=1}^n\sum_{j\neq i}^n g\left(\mathbf{x}[i],\mathbf{x}[j]\right) +\frac{1}{m\left(m-1\right)}\\
&\qquad \sum_{i=1}^m\sum_{j\neq i}^m g\left(\mathbf{y}[i],\mathbf{y}[j]\right)-\frac{2}{nm}\sum_{i=1}^n\sum_{j=1}^m g\left(\mathbf{x}[i],\mathbf{y}[j]\right).
\end{aligned}
\end{equation}
Assume that the kernel is bounded, i.e., $0 \le g\left(x,y\right) \le \mathbb K$, where $\mathbb{K}$ is finite.
\begin{proposition}\label{proposition2}
	The MMD distance satisfies \eqref{eq:dist_condition2} - \eqref{eq:dist_condition3} if $d_{L,mmd}<d_{H,mmd}$.
\end{proposition}
\begin{proof}
	See Lemmas \ref{lemma:MMD2}, \ref{lemma:MMD1} and \ref{lemma:MMD3} in Appendix \ref{app:lemmas}.
\end{proof}
\subsection{Additional Notations}
The following notations are used in the algorithms and the corresponding proofs. Let $\mathcal{C}_{l}^{t}$ be the $l$-th cluster obtained at the $t$-th cluster update step and let $\mathbf{c}_{l}^{t,a}$, $\mathbf{c}_{l}^{t,e}$ and $\mathbf{c}_{l}^{t,s}$ be the centers of the $l$-th cluster obtained by the center update step, merge step and split step of the $t$-th iteration respectively for $t\geq 1$. Moreover, let $\mathcal{C}_{l}^{0}$ be the $l$-th cluster obtained at the initialization step and $\mathbf{c}_{l}^{0,a}$ be the corresponding center. Let $\hat{K}^{t}$ ($t\geq 1$) be the number of centers before the $t$-th cluster update step and the $t$-th split step. Moreover, use $\hat{K}^{0}$ to denote the number of centers obtained at the center initialization step. For simplicity, all the superscripts are omitted in the following discussion when there is no ambiguity.\par
To further simplify the notation in the algorithms and the proofs, let $\{\mathbf{y}_{i}\}_{i=1}^{M}$ denote the data sequence set $\{\mathbf{x}_{k,j_{k}}\}_{k=1,j_{k}=1}^{K,M_{k}}$. However, the one-to-one mapping from $\{\mathbf{y}_{i}\}_{i=1}^{M}$ onto $\{\mathbf{x}_{k,j_{k}}\}_{k=1,j_{k}=1}^{K,M_{k}}$ is not fixed, i.e., given a fixed $i$, $\mathbf{y}_{i}$ can be any sequence in $\{\mathbf{x}_{k,j_{k}}\}_{k=1,j_{k}=1}^{K,M_{k}}$ unless other constraints are imposed. Denote by $\mathbf{y}_{i}\sim\mathcal{P}_{k}$ if $\mathbf{y}_{i}$ is generated from a distribution $p\in\mathcal{P}_{k}$.
Furthermore, define a set of integers
\begin{equation*}
\begin{aligned}
I_{k_{1}}^{k_{2}} &= \{k_{1},\ldots,k_{2}\},
\end{aligned}
\end{equation*}
where $k_{1},\:k_{2}\in \mathbb{Z}^{+}$ and $k_{1}<k_{2}$.\par

\section{Known number of clusters}\label{sec:knownK}

In this section, we study the clustering algorithm for known $K$, the number of clusters. The method proposed in \cite{Katsavounidis1994} is used for center initialization, as described in Algorithm \ref{KS-Initial-known-C}. The initial $K$ centers are chosen sequentially such that the center of the $k$-th cluster is the sequence that has the largest minimum distance to the previous $k-1$ centers. The clustering algorithm itself is presented in Algorithm \ref{K-means-known-C}. Given the centers, each sequence is assigned to the cluster for which the sequence has the minimum distance to the center. For a given cluster, a sequence is assigned as the center subsequently if the sum of its distances to all the sequences in the cluster is the smallest. The algorithm continues until the clustering result converges. \par
\begin{algorithm}[!bt]
	\caption{Initialization with known $K$}
	\label{KS-Initial-known-C}
	\begin{algorithmic}[1]
		\State \textbf{Input}: Data sequences $\{\mathbf{y}_{i}\}_{i=1}^{M}$, number of clusters $K$.
		\State \textbf{Output}: Partitions $\{\mathcal{C}_{k}\}_{k=1}^{K}$.
		\State \{Center initialization\}
		\State Arbitrarily choose one $\mathbf{y}_{i}$ as $\mathbf{c}_{1}$.
		\For {$k=2\text{ to }K$}
		\State $\mathbf{c}_{k}\leftarrow \text{arg}\max_{\mathbf{y}_{i}}\left(\min_{l\in I_{1}^{k-1}} d\left(\mathbf{y}_{i},\mathbf{c}_{l}\right)\right)$
		\EndFor
		\State \{Cluster initialization\}
		\State Set $\mathcal{C}_{k}\leftarrow \emptyset$ for $1\leq k\leq K$.
		\For {$j=1\text{ to }M$}
		\State $\mathcal{C}_{l}\leftarrow \mathcal{C}_{l}\cup\{\mathbf{y}_{i}\}$, where
		$l=\text{arg}\min_{l\in I_{1}^{K}} d\left(\mathbf{y}_{i},\mathbf{c}_{l}\right)$
		\EndFor
		\State Return $\{\mathcal{C}_{k}\}_{k=1}^{K}$
	\end{algorithmic}
\end{algorithm}

\begin{algorithm}[!bt]
	\caption{Clustering with known $K$}
	\label{K-means-known-C}
	\begin{algorithmic}[1]
		\State \textbf{Input}: Data sequences $\{\mathbf{y}_{i}\}_{i=1}^{M}$, number of clusters $K$.
		\State \textbf{Output}: Partition set $\{\mathcal{C}_{k}\}_{k=1}^{K}$.
		\State Initialize $\{\mathcal{C}_{k}\}_{k=1}^{K}$ by Algorithm \ref{KS-Initial-known-C}.
		\While{not converge}
		\State \{Center update\}
		\For {$k=1\text{ to }K$}
		\State $\mathbf{c}_{k}\leftarrow \arg\min_{\mathbf{y}_{i}\in \mathcal{C}_{k}}\sum_{\mathbf{y}_{j^{\prime}}\in \mathcal{C}_{k}}d\left(\mathbf{y}_{i},\mathbf{y}_{j^{\prime}}\right)$
		\EndFor
		\State $\{\text{Cluster update}\}$
		\For {$j=1\text{ to }M$}
		\If {$\mathbf{y}_{i}\in \mathcal{C}_{k^{\prime}}$ and $d\left(\mathbf{y}_{i},\mathbf{c}_{k}\right)< d\left(\mathbf{y}_{i},\mathbf{c}_{k^{\prime}}\right)$}
		\State $\mathcal{C}_{k}\leftarrow \mathcal{C}_{k}\cup\{\mathbf{y}_{i}\}$ and $\mathcal{C}_{k^{\prime}}\leftarrow \mathcal{C}_{k^{\prime}}\setminus\{\mathbf{y}_{i}\}$.
		\EndIf
		\EndFor
		\EndWhile
		\State Return $\{\mathcal{C}_{k}\}_{k=1}^{K}$
	\end{algorithmic}
\end{algorithm}

The following theorem provides the convergence guarantee for Algorithm \ref{K-means-known-C} via an upper bound on the error probability.
\begin{theorem}\label{theorem:KStest1}
	Algorithm \ref{K-means-known-C} converges after a finite number of iterations. Moreover, under Assumption \ref{assump},
	the error probability of Algorithm \ref{K-means-known-C} after $T$ iterations is upper bounded as follows
	\begin{equation*}
	P_{e}\leq M^{2}\left(a_{1} + a_{2} + \left(T+1\right)a_{3}\right)e^{-bn},
	\end{equation*}
	where $a_{1}$, $a_{2}$, $a_{3}$ and $b$ are as defined in Assumption \ref{assump}.
\end{theorem}
\begin{hproof}
	The idea of proving the upper bound on the error probability is as follows. We first prove that the error probability at the initialization step decays exponentially. Note that the event that an error occurs during the first $T$ iterations is the union of the event that an error occurs at the $t$-th step and the previous $t-1$ iterations are correct for $t=1,\ldots,T$. Thus, if we prove that the error probability at the $t$-th step \emph{given} correct updates from the previous iterations decays exponentially, then so does the error probability of the algorithm by the union bound argument. See Appendix \ref{proof:theorem:KStest1} for details.
\end{hproof}
Theorem \ref{theorem:KStest1} shows that for any given $K$, any distance metric satisfying Assumption \ref{assump} yields an exponentially consistent k-medoids clustering algorithm with the error exponent $b$.
\begin{corollary}\label{coro:knownK}
Suppose the KS distance and MMD are used for Algorithms \ref{KS-Initial-known-C} and \ref{K-means-known-C}, then
	\begin{equation*}
	\begin{aligned}
	&P_{e}^{KS} \leq M^{2}\left(6T+14\right)\exp\left(-\frac{n\Delta_{ks}^{2}}{8}\right),\\
	&P_{e}^{MMD}\leq M^{2}\left(T+3\right)\exp \left(-\frac{n\Delta_{mmd}^2}{256\mathbb{K}^2}\right).
	\end{aligned}
	\end{equation*}
\end{corollary}
\begin{proof}
	By Propositions \ref{proposition1} and \ref{proposition2}, the upper bound on the error probability of Algorithm \ref{K-means-known-C} applies to the KS distance and MMD. Thus, the corollary is obtained by substituting the values specified in Lemmas \ref{lemma:KStest2} - \ref{lemma:MMD3} in the upper bound.
\end{proof}
Corollary \ref{coro:knownK} implies that Algorithm \ref{K-means-known-C} is exponentially consistent under KS and MMD distance metrics with an error exponent no smaller than $\frac{\Delta_{ks}^{2}}{8}$ and $\frac{\Delta_{mmd}^2}{256\mathbb{K}^2}$, respectively.

\section{Unknown number of clusters}\label{sec:unknownK}
In this section, we propose the merge- and split-based algorithms for estimating the number of clusters as well as grouping the sequences.
\subsection{Merge Step}
If a distance metric satisfies \eqref{eq:dist_condition1} and two sequences generated by distributions within the same cluster are assigned as centers, then, with high probability, the distance between the two centers is small. This is the premise of the clustering algorithm based on merging centers that are close to each other.\par
The proposed approach is summarized in Algorithms \ref{KS-Initial-unknown-UC} and \ref{K-means-unknown-UC}. There are two major differences between Algorithms \ref{KS-Initial-unknown-UC} and \ref{K-means-unknown-UC} and Algorithms \ref{KS-Initial-known-C} and \ref{K-means-known-C}. First, the center initialization step of Algorithm \ref{KS-Initial-unknown-UC} keeps generating an increasing number of centers until all the sequences are close to one of the existing centers. Second, an additional Merge Step in Algorithm \ref{K-means-unknown-UC} helps to combine clusters if the corresponding centers have small distances between each other.\par
\begin{algorithm}[!bt]
	\caption{Merge-based initialization with unknown $K$}
	\label{KS-Initial-unknown-UC}
	\begin{algorithmic}[1]
		\State \textbf{Input}: Data sequences $\{\mathbf{y}_{i}\}_{i=1}^{M}$ and threshold $d_{th}$.
		\State \textbf{Output}: Partitions $\{\mathcal{C}_{k}\}_{k=1}^{\hat{K}}$.
		\State $\{\text{Center initialization}\}$
		\State Arbitrarily choose one $\mathbf{y}_{i}$ as $\mathbf{c}_{1}$ and set $\hat{K}=1$.
		\While {$\min_{i\in I_{1}^{M}}\left(\min_{k\in I_{1}^{\hat{K}}} d\left(\mathbf{y}_{i},\mathbf{c}_{k}\right)\right)>d_{th}$}
		\State $\mathbf{c}_{\hat{K}+1}\leftarrow\text{arg}\max_{\mathbf{y}_{i}}\left(\min_{k\in I_{1}^{\hat{K}}} d\left(\mathbf{y}_{i},\mathbf{c}_{k}\right)\right)$
		\State $\hat{K}\leftarrow \hat{K}+1$
		\EndWhile
		\State Clustering initialization specified in Algorithm \ref{KS-Initial-known-C}.
		\State Return $\{\mathcal{C}_{k}\}_{k=1}^{\hat{K}}$
	\end{algorithmic}
\end{algorithm}

\begin{algorithm}[!bt]
	\caption{Merge-based clustering with unknown $K$}
	\label{K-means-unknown-UC}
	\begin{algorithmic}[1]
		\State \textbf{Input}: Data sequences $\{\mathbf{y}_{i}\}_{i=1}^{M}$ and threshold $d_{th}$.
		\State \textbf{Output}: Partition set $\{\mathcal{C}_{k}\}_{k=1}^{\hat{K}}$.
		\State Initialize $\{\mathcal{C}_{k}\}_{k=1}^{\hat{K}}$ by Algorithm \ref{KS-Initial-unknown-UC}.
		\While{not converge}
		\State Center update specified in Algorithm \ref{K-means-known-C}.
		\State $\{\text{Merge Step}\}$
		\For {$k_{1}, k_{2}\in \{1,\ldots,\hat{K}\}$ and $k_{1}\neq k_{2}$}
		\If {$d\left(\mathbf{c}_{k_{1}},\mathbf{c}_{k_{2}}\right)\leq d_{th}$}
		\If {$\sum_{\mathbf{y}_{i}\in \mathcal{C}_{k_{1}}}d\left(\mathbf{c}_{k_{2}},\mathbf{y}_{i}\right) < \sum_{\mathbf{y}_{i}\in \mathcal{C}_{k_{2}}}d\left(\mathbf{c}_{k_{1}},\mathbf{y}_{i}\right)$}
		\State $\mathcal{C}_{k_{2}} \leftarrow \mathcal{C}_{k_{1}}\cup\mathcal{C}_{k_{2}}$ and delete $\mathbf{c}_{k_{1}}$ and $\mathcal{C}_{k_{1}}$.
		\Else
		\State $\mathcal{C}_{k_{1}} \leftarrow \mathcal{C}_{k_{1}}\cup\mathcal{C}_{k_{2}}$ and delete $\mathbf{c}_{k_{2}}$ and $\mathcal{C}_{k_{2}}$.
		\EndIf
		\State $\hat{K}\leftarrow \hat{K} - 1$.
		\EndIf
		\EndFor
		\State Cluster update specified in Algorithm \ref{K-means-known-C}.
		\EndWhile
		\State Return $\{\mathcal{C}_{k}\}_{k=1}^{\hat{K}}$
	\end{algorithmic}
\end{algorithm}
\begin{theorem}\label{theorem:KStest2}
	Algorithm \ref{K-means-unknown-UC} converges after a finite number of iterations. Moreover, under Assumption \ref{assump}, the error probability of Algorithm \ref{K-means-unknown-UC} after $T$ iterations is upper bounded as follows
	\begin{equation*}
	P_{e}\leq M^{2}\left(\left(T+1\right)a_{1} +a_{2} + \left(T+1\right)a_{3}\right)e^{-bn},
	\end{equation*}
	where $a_{1}$, $a_{2}$, $a_{3}$ and $b$ are as defined in Assumption \ref{assump}.
\end{theorem}
\begin{proof}
	The proof shares the same idea as that of Theorem \ref{theorem:KStest1}. See Appendix \ref{proof:theorem:KStest2} for details.
\end{proof}
Theorem \ref{theorem:KStest2} shows that the merge-based algorithm is exponentially consistent under any distance metric satisfying Assumption \ref{assump} with the error exponent $b$.
\begin{corollary}\label{coro:unknownK-merge}
Suppose the KS distance and MMD are used with $d_{th} = \frac{\Sigma_{ks}}{2}$ and $d_{th}=\frac{\Sigma_{mmd}}{2}$. Then the error probability of Algorithm \ref{K-means-unknown-UC} after $T$ iterations is upper bounded as follows
\begin{equation*}
\begin{aligned}
&P_{e}^{KS}\leq M^{2}\left(10T+14\right) \exp{\left(-\frac{n\Delta_{ks}^{2}}{8}\right)},\\
&P_{e}^{MMD}\leq M^{2}\left(2T+3\right) \exp{\left(-\frac{n\Delta_{mmd}^{2}}{256\mathbb{K}^{2}}\right)}.
\end{aligned}
\end{equation*}
\end{corollary}
\begin{proof}
By Propositions \ref{proposition1} and \ref{proposition2}, the upper bound on the error probability of Algorithm \ref{K-means-unknown-UC} in \ref{theorem:KStest2} applies to the KS distance and MMD. Thus, the corollary is obtained by substituting the values specified in Lemmas \ref{lemma:KStest2} - \ref{lemma:MMD3} in the upper bound.
\end{proof}
Corollary \ref{coro:unknownK-merge} implies that Algorithm \ref{K-means-unknown-UC} is exponentially consistent under KS and MMD distance metrics with an error exponent no smaller than $\frac{\Delta_{ks}^{2}}{8}$ and $\frac{\Delta_{mmd}^2}{256\mathbb{K}^2}$, respectively.

\subsection{Split Step}
Suppose a cluster contains sequences generated by different distributions and the center is generated from $p\in\mathcal{P}_{k}$. Then if the distance metric satisfies \eqref{eq:dist_condition2}, the probability that the distances between sequences generated from distribution clusters other than $\mathcal{P}_{k}$ and the center is small decays as the sample size increases. Therefore, it is reasonable to begin with one cluster and then split a cluster if there exists a sequence in the cluster that has a large distance to the center. The corresponding algorithm is summarized in Algorithm \ref{K-means-unknown-spl}.\par
\begin{algorithm}[!bt]
	\caption{Split-based clustering with unknown $K$}
	\label{K-means-unknown-spl}
	\begin{algorithmic}[1]
		\State \textbf{Input}: Data sequences $\{\mathbf{y}_{i}\}_{i=1}^{M}$ and threshold $d_{th}$.
		\State \textbf{Output}: Partition set $\{\mathcal{C}_{k}\}_{k=1}^{\hat{K}}$.
		\State  $\mathcal{C}_{1}=\{\mathbf{y}_{i}\}_{i=1}^{M}$, $\hat{K}=1$ and find $\mathbf{c}_{1}$ by center update specified in Algorithm \ref{K-means-known-C}.
		\While{not converge}
		\State {\{Split Step\}}
		\If {$\max_{k\in I_{1}^{\hat{K}},\: \mathbf{y}_{i}\in \mathcal{C}_{k}}d\left(\mathbf{c}_{k},\mathbf{y}_{i}\right)>d_{th}$}
		\State $\hat{K}\leftarrow \hat{K}+1$.
		\State $k=\arg\max_{k\in I_{1}^{\hat{K}}}\left(\max_{ \mathbf{y}_{i}\in \mathcal{C}_{k}}d\left(\mathbf{c}_{k},\mathbf{y}_{i}\right)\right)$
		\State $\mathbf{c}_{\hat{K}}\leftarrow\arg\max_{\mathbf{y}_{i}\in \mathcal{C}_{k}}d\left(\mathbf{c}_{k},\mathbf{y}_{i}\right)$
		\EndIf
		\State Cluster update specified in Algorithm \ref{K-means-known-C}.
		\EndWhile
		\State Return $\{\mathcal{C}_{k}\}_{k=1}^{\hat{K}}$
	\end{algorithmic}
\end{algorithm}
\begin{definition}\label{def:correct_split}
	Suppose Algorithm \ref{K-means-unknown-spl} obtains $\hat{K}$ clusters at the $t$-th iteration, where $\hat{K}<K$ and $\hat{K}=t$ or $t+1$. Then the correct clustering update result is that each cluster contains all the sequences generated from the distribution cluster that generates the center.
\end{definition}
\begin{theorem}\label{theorem:KStest3}
	Algorithm \ref{K-means-unknown-spl} converges after a finite number of iterations. Moreover, under Assumption \ref{assump}, the error probability of Algorithm \ref{K-means-unknown-spl} after $T$ iterations is upper bounded as follows
	\begin{equation*}
	P_{e}\leq M^{2}T\left(a_{1} + a_{2} + a_{3}\right)e^{-bn},
	\end{equation*}
	where $a_{1}$, $a_{2}$, $a_{3}$ and $b$ are as defined in Assumption \ref{assump}.
\end{theorem}
\begin{hproof}
	An error occurs at the $t$-th iteration if and only if the $\hat{K}$-th center is generated from distribution clusters that generated the previous centers or the clustering result is incorrect. Note that the error event of the first $T$ iterations is the union of the events that an error occurs at the $t$-th iteration while the clustering results in the previous $t-1$ iterations are correct for $t=1,\ldots,T$. Similar to the proof of Theorem \ref{theorem:KStest1}, the error probability is bounded by the union bound. See Appendix \ref{proof:theorem:KStest3} for more details.
\end{hproof}
Theorem \ref{theorem:KStest3} shows that the split-based algorithm is exponentially consistent under any distance metric satisfying Assumption \ref{assump} with the error exponent $b$.
\begin{corollary}\label{coro:unknownK-split}
		Suppose the KS distance and MMD are used with $d_{th} = \frac{\Sigma_{ks}}{2}$ and $d_{th}=\frac{\Sigma_{mmd}}{2}$. Then the error probability of Algorithm \ref{K-means-unknown-spl} after $T$ iterations is upper bounded as follows
		\begin{equation*}
		\begin{aligned}
		&P_{e}^{KS}\leq 14M^{2}T\exp{\left(-\frac{n\Delta_{ks}^{2}}{8}\right)},\\
		&P_{e}^{MMD} \leq 3M^{2}T\exp{\left(-\frac{n\Delta_{mmd}^{2}}{256\mathbb{K}^{2}}\right)}.
		\end{aligned}
		\end{equation*}
\end{corollary}
\begin{proof}
	By Propositions \ref{proposition1} and \ref{proposition2}, the upper bound on the error probability of Algorithm \ref{K-means-unknown-spl} in Theorem \ref{theorem:KStest3} applies to the KS distance and MMD. Thus, the corollary is obtained by substituting the values specified in Lemmas \ref{lemma:KStest2} - \ref{lemma:MMD3} in the upper bound.
\end{proof}
Corollary \ref{coro:unknownK-split} implies that Algorithm \ref{K-means-unknown-spl} is exponentially consistent under KS and MMD with an error exponent no smaller than $\frac{\Delta_{ks}^{2}}{8}$ and $\frac{\Delta_{mmd}^2}{256\mathbb{K}^2}$, respectively.
\section{Numerical Results}\label{sec:numerical_result}
In this section, we provide some simulation results given $K=5$ and $M_{k}=3$ for $k=1,\ldots,5$. Moreover, $\mathbf{x}_{k,j_{k}}[i]\in \mathbb{R}$. Gaussian distributions $\mathcal{N}\left(\mu_{k,j_{k}},\sigma^{2}\right)$ and Gamma distributions $\Gamma\left(a_{k,j_{k}},b\right)$ are used in the simulations. The probability density function (p.d.f.) of $\Gamma(a,b)$ is defined as
\begin{equation*}
\begin{aligned}
f\left(x;\alpha,\beta\right) & = \frac{1}{\beta^{\alpha}\Gamma\left(\alpha\right)}x^{\alpha-1}\exp{\left(-\frac{x}{\beta}\right)}\quad \left(x>0\right),
\end{aligned}
\end{equation*}
where $\alpha>0$, $\beta>0$ and $\Gamma\left(\cdot\right)$ is the Gamma function, respectively. Specifically, the parameters of the distributions are $\mu_{k,j_{k}}\in\{k-\delta,k,k+\delta\}$, $\sigma^{2} = 1$, $\alpha_{k,j_{k}}\in\{2.5k+1-\delta,2.5k+1,2.5k+1+\delta\}$ and $\beta=1$, where $\delta =0$ and $0.1$. Note that when $\delta=0$, all the sequences generated from the same distribution cluster are generated from a single distribution. The exponential kernel function is used in the simulations for the MMD distance, i.e.,
\begin{equation}\label{eq:simu_kernel}
g\left(x,y\right) = e^{-\frac{|x-y|}{2}}.
\end{equation}

\subsection{Known Number of Clusters}
Simulation results for a known number of clusters are shown in Fig.~\ref{fig:algorithm2}. One can observe from the figures that by using both the KS distance and MMD, $\log P_{e}$ is a linear function of the sample size, i.e., $P_{e}$ is exponentially consistent. Moreover, the logarithmic slope of $P_{e}$ with respect to $n$, i.e., the quantity $-\frac{\log P_{e}}{n}$, increases as $\delta$ becomes smaller, which, in the current simulation setting, implies a larger $\Delta$. \par
Furthermore, a good distance metric for Algorithm \ref{K-means-known-C} depends on the underlying distributions. This is because the underlying distributions have different distances under the KS distance and MMD, which results in different values of $\Delta_{ks}$ and $\Delta_{mmd}$.

	\begin{figure}[t]
		\centering

		\subfloat[Gaussian distributions\label{subfig:1-1}]{%
			\includegraphics[width=0.5\textwidth, height=0.15\textheight]{sample_size_vari_mean_composite.pdf}
		}
		\hfill

		\subfloat[Gamma distributions \label{subfig:1-2}]{%
			\includegraphics[width=0.5\textwidth, height=0.15\textheight]{sample_size_gamma_composite.pdf}
		}

		\caption{Performance of Algorithm  2}
		\label{fig:algorithm2}
	\end{figure}

\subsection{Unknown Number of Clusters}
With an unknown number of distribution clusters, the threshold $d_{th}$ specified in Corollaries \ref{coro:unknownK-merge} and \ref{coro:unknownK-split} are used in the simulation. The performance of Algorithms \ref{K-means-unknown-UC} and \ref{K-means-unknown-spl} for the KS distance and MMD are shown in Figs.~\ref{fig:algorithm4_5_KS} and \ref{fig:algorithm4_5_MMD}, respectively. Given the KS distance and MMD, $\log {P_{e}}$'s are linear functions of the sample size when the sample size is large and larger $\Delta$ implies a larger slope of $\log {P_{e}}$. Furthermore, given the same value of $\delta$, Algorithms \ref{K-means-unknown-UC} and \ref{K-means-unknown-spl} have similar performance under the KS distance whereas Algorithm \ref{K-means-unknown-spl} outperforms Algorithm \ref{K-means-unknown-UC} under MMD given $\delta=0$ and $0.1$. Intuitively, smaller $\delta$ implies larger $\Delta$ in the current simulation setting, thereby should result in better clustering performance for a given sample size. However, Fig.~\ref{fig:algorithm4_5_KS}\subref{fig:algorithm4_5_KS-b} indicates that Algorithms \ref{K-means-unknown-UC} and \ref{K-means-unknown-spl} with KS distance performs better with $\delta=0.1$ than that with $\delta=0$ when the sample size is small. This is likely due to the fact that the KS distance between the two sequences is always lower bounded by $\frac{1}{n}$. Thus, with small sample sizes, Algorithms \ref{K-means-unknown-UC} and \ref{K-means-unknown-spl} are likely to overestimate the number of clusters. This can be mitigated by the increased threshold $d_{th}$ to control merging/splitting of cluster centers.

\begin{figure}[!htb]
	\subfloat[Gaussian distributions\label{subfig:2-1}]{%
		\includegraphics[width=0.5\textwidth, height=0.15\textheight]{KS_vari_mean_unknown_K_merge_spl_composite_log_KS.pdf}
	}
	\hfill
	\subfloat[Gamma distributions \label{subfig:2-2}]{%
		\includegraphics[width=0.5\textwidth, height=0.15\textheight]{KS_gamma_unknown_K_merge_spl_composite_log_KS.pdf}
		\label{fig:algorithm4_5_KS-b}
	}
	\caption{Performance of Algorithms \ref{K-means-unknown-UC} and \ref{K-means-unknown-spl} for the KS distance}
	\label{fig:algorithm4_5_KS}
\end{figure}

\begin{figure}[!htb]
	\subfloat[Gaussian distributions\label{subfig:3-1}]{%
		\includegraphics[width=0.5\textwidth, height=0.15\textheight]{KS_vari_mean_unknown_K_merge_spl_composite_log_MMD.pdf}
	}
	\hfill
	\subfloat[Gamma distributions\label{subfig:3-2}]{%
		\includegraphics[width=0.5\textwidth, height=0.15\textheight]{KS_gamma_unknown_K_merge_spl_composite_log_MMD.pdf}
		\label{fig:algorithm4_5_MMD-b}
	}
	\caption{Performance of Algorithms \ref{K-means-unknown-UC} and \ref{K-means-unknown-spl} for MMD}
	\label{fig:algorithm4_5_MMD}
\end{figure}
\subsection{Choice of $d_{th}$}
Note that in general $d_{th}=\omega d_{L} + \left(1-\omega\right)d_{H}$, where $\omega\in\left(0,1\right)$. Theorems \ref{theorem:KStest2} and \ref{theorem:KStest3} only establish the exponential consistency of Algorithms \ref{K-means-unknown-UC} and \ref{K-means-unknown-spl}, respectively. We now investigate the impact on performance given different $\omega$'s. One can observe from Fig.~\ref{fig:algorithm4_5_alpha} that the choice of $d_{th}$ has a significant impact on the performance of Algorithms \ref{K-means-unknown-UC} and \ref{K-means-unknown-spl}. The optimal $d_{th}$ depends on both the value of $\delta$ and the underlying distributions. Moreover, from Fig.~\ref{fig:algorithm4_5_alpha}\subref{fig:algorithm4_5_alpha_KS-a}-\subref{fig:algorithm4_5_alpha_KS-b}, we can see that a smaller $\omega$ which implies larger $d_{th}$ results in better performance for KS distance and the two algorithms always have similar performance. On the other hand, from Fig.~\ref{fig:algorithm4_5_alpha}\subref{fig:algorithm4_5_alpha_MMD-a}-\subref{fig:algorithm4_5_alpha_MMD-b}, we can see that $\omega=0.5$ is a good choice for MMD if $P_{e}$ is of interest and \eqref{eq:simu_kernel} is used as the kernel function. Moreover, when $\omega$ is small, Algorithm \ref{K-means-unknown-spl} outperforms Algorithm \ref{K-means-unknown-UC} under MMD.

\begin{figure}[t]
	\subfloat[KS distance ($n=100$, $\delta=0$)\label{subfig:4-1}]{%
		\includegraphics[width=0.5\textwidth, height=0.15\textheight]{cluster_gamma_unknown_K_merge_spl_composite_KS_alpha_delta0.pdf}
		\label{fig:algorithm4_5_alpha_KS-a}
	}
	\hfill
	\subfloat[KS distance ($n=100$, $\delta=0.1$)\label{subfig:4-2}]{%
		\includegraphics[width=0.5\textwidth, height=0.15\textheight]{cluster_gamma_unknown_K_merge_spl_composite_KS_alpha_delta1.pdf}
		\label{fig:algorithm4_5_alpha_KS-b}
	}
	\hfill
	\subfloat[MMD ($n=80$, $\delta=0$)\label{subfig:4-3}]{%
		\includegraphics[width=0.5\textwidth, height=0.15\textheight]{cluster_gamma_unknown_K_merge_spl_composite_MMD_alpha_delta0.pdf}
		\label{fig:algorithm4_5_alpha_MMD-a}
	}
	\hfill
	\subfloat[MMD ($n=80$, $\delta=0.1$)\label{subfig:4-4}]{%
		\includegraphics[width=0.5\textwidth, height=0.15\textheight]{cluster_gamma_unknown_K_merge_spl_composite_MMD_alpha_delta1.pdf}
		\label{fig:algorithm4_5_alpha_MMD-b}
	}
	\caption{Performance of Algorithms \ref{K-means-unknown-UC} and \ref{K-means-unknown-spl} given different $\omega$}
	\label{fig:algorithm4_5_alpha}
\end{figure}

\section{Conclusion}
This paper studied the k-medoids algorithm for clustering data sequences generated from composite distributions. The convergence of the proposed algorithms and the upper bound on the error probability were analyzed for both known and unknown number of clusters. The error probability of the proposed algorithms were characterized to decay exponentially for distance metrics that satisfied certain properties.
In particular, the KS distance and MMD were shown to satisfy the required condition, and hence the corresponding algorithms were exponentially consistent.\par
One possible generalization of the current work is to investigate the exponential consistency of other clustering algorithms given distributional distance metrics that satisfy the properties similar to that in Assumption \ref{assump}.

redpajama_set_name: RedPajamaArXiv


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before applying css for width, make sure "float:left" is written for label control.
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Andrea Dovizioso attempted to play down his and Ducati's speed on the first day of the MotoGP test at Jerez, stating he did not confirm his suspicions the factory had improved its turning issues with the Desmosedici GP19.
The 32-year old spent Wednesday assessing an early version of the GP19 with two different chassis. Last week at Valencia he felt Ducati had "found something interesting" regarding its well-known turning difficulties. That was not evident here.
And Dovizioso is hopeful of having the chance to test and re-confirm his impressions tomorrow, after a fast crash at turn five brought his day to a premature close. "A small crash," was soon accentuated by his hitting the gravel. Then "I started rolling," he explained.
But overall, it was a positive day. "I think me and Danilo we did a really good pace with the used tyre, apart from the good lap time with the new tyre," he said. "I'm happy about the speed, but we are here to do something different, we want to understand, we have to try to give the best feedback.
"Today we didn't confirm the same better feeling about the new one that we had in Valencia. We want now to understand better why, and maybe we have to try again tomorrow in a different way.
"Because it's always when you are working on small details like this, it's not easy to understand and to feel exactly what happened. So fortunately, the bike worked well, the speed was really good today, but we didn't improve the bike.
"So this is a confirmation that our base is good, but we want to go home with a clear idea, and I hope tomorrow to have no pain to ride, and to make a comparison, because we are here for that, and today the same things didn't happen as at Valencia.
Asked for his impression of the crash, he said, "Nothing, I was pushing, I wanted to improve the lap time. In that moment I had the experimental rear Michelin, already some riders did the lap time with that and it worked well, and we changed the setup, we did a big step in one way, we wanted to understand.
"I tried with a used tyre, and the feeling was good on the front, also if the lap time wasn't so good, and I wanted to push with the new tyre, to really put the bike on the limit, and this is what happened, I wanted to do that. And some positives and negatives came out, that was very important for us.
"But I was a bit too fast in the middle of the corner, that setup didn't work as well as my previous setup in that moment. So I was pushing really hard, and I lost the front. A small crash at Turn 5, but the problem was when I arrived to the gravel, I started rolling.
"So I hit a lot of parts of my body. Fortunately it doesn't look like anything is broken. We did the X-ray, and on the X-ray it's very difficult to see something. It's quite big here [points at base of thumb], but maybe it's just the muscle, so let's see tomorrow when I wake up if I'm able to ride.


text:
Q: Prove $\left(\sqrt{x}+\sqrt{y}\right)^2\ge 2\sqrt{2\left(x+y\right)\sqrt{xy}}$, with $x,\:y\in \mathbb{R}$ and $x\ne 0$ My thinking:
Starting with the basic fact $\left(\sqrt{x}-\sqrt{y}\right)^4\ge 0$ which is trivial.
$\left(\sqrt{x}-\sqrt{y}\right)^4\ge 0\:$
$\rightarrow \:x^2-4\sqrt{yx}\sqrt{x}+6xy-4\sqrt{xy}\sqrt{y}+y^2\ge \:0\:$
$\rightarrow x^2+4\sqrt{yx}\sqrt{x}+6xy+4\sqrt{xy}\sqrt{y}+y^2\ge \:8\sqrt{yx}\sqrt{x}+8\sqrt{xy}\sqrt{y}$
$\rightarrow $ $\left(\sqrt{x}+\sqrt{y}\right)^4\ge \: \:8\sqrt{yx}\sqrt{x}+8\sqrt{xy}\sqrt{y}$
$\rightarrow \frac{\left(\sqrt{x}+\sqrt{y}\right)^4}{8}\ge \sqrt{yx}\sqrt{x}+\sqrt{xy}\sqrt{y}$
$\rightarrow \:\frac{\left(\sqrt{x}+\sqrt{y}\right)^4}{8}\ge x\sqrt{xy}+y\sqrt{xy}$
$\rightarrow \:\frac{\left(\sqrt{x}+\sqrt{y}\right)^4}{4}\ge 2x\sqrt{xy}+2y\sqrt{xy}$
$\rightarrow \:\frac{\left(\sqrt{x}+\sqrt{y}\right)^2}{2}\ge \sqrt{2x\sqrt{xy}+2y\sqrt{xy}}$
$\rightarrow \:\left(\sqrt{x}+\sqrt{y}\right)^2\ge 2\sqrt{2x\sqrt{xy}+2y\sqrt{xy}}$ as needed. QED
I'm not sure if this is correct. If someone can provide some feedback that would be great!

A: Your proof works, but there is a much more direct proof.
First, just write $a=\sqrt x,b=\sqrt y$ and rewrite it as $$(a+b)^2\geq 2\sqrt{(a^2+b^2)(2ab)}$$
Letting $u=a^2+b^2, v=2ab,$ use AM/GM:
$$\frac{u+v}{2}\geq\sqrt{uv}$$

That first step is important from an aesthetic point of view, as much as anything - all those square roots are hard on the eyes, and it is difficult to see what is going on.
If we were to prove AM/GM ourselves, we would, of course, square $u-v=(a-b)^2,$ and you will see quite a lot of your proof is just proving AM/GM for a special case.

A: The given inequality is equivalent to the $t\ge0$ case of
$$(t+1)^4\ge8t(t^2+1)\tag1$$
(Let $t=\sqrt{y/x}$.) Taking the difference in (1),
$$(t+1)^4-8t^3-8t=(t-1)^4,$$
which proves (1) for arbitrary $t.$

redpajama_set_name: RedPajamaStackExchange


SCBWI BI Conference is open for bookings!
You must all know what a HUGE fan of The Society of Children's Book Writers and Illustrators - British Isles (SCBWI - BI) I am.
I've been an active member for over three years and started volunteering at their amazing conference last year.
It is absolutely brilliant and if you're a writer or illustrator of children's books at whatever stage of your journey you will LOVE this conference.


text:
Best Books 2015: Rémi Coignet
Best Books 2015 Best Books 2015 Rémi Coignet 2015 Best Books picks from Rémi Coignet.
Los ultimos dias vistos del rey
By Julián Barón
In 1975, two propaganda books were released in Spain: Los ultimos dias de Franco vistos en TVE and Los primeros dias del rey vistos en TVE, both comprising TV screen shots. On the occasion of the abdication of Juan Carlos in favor of Felipe VI, Julián Barón used a similar format, but turned the propaganda book into a protest book against the scandal-ridden Spanish Monarchy.
This Business of Living
By Daniel Blaufuks
Pierre von Kleist editions
Directly inspired by Cesare Pavese's diary, Daniel Blaufuks' last release is a meditation on the beauty and complexity of life: shadow and light, potential violence (knife, toy gun), the beauty of a woman, the simple presence of lemons or of a glass of water on a Lisbon kitchen table.
By Indra Kumar Jha & Federico Carpani
This book is one of the ugliest, and one of the most beautiful books of the year. One of the ugliest because the images, basic time-stamped snapshots, are devoid of any aesthetic and because it is quite unpleasant to stare death in the face; one of the most beautiful because the photos of the deceased and their families made in Benares by Indra Kumar Jha are brilliantly edited by Federico Carpani and because while it is necessary to stare death in the face, it may not necessarily have to be through any artistic aesthetic strategies.
By JH Engström
Anyone familiar with Engström's work may find the title Tout va bien either ironic or the reflection of some deep shift in his frame of mind. While inscribed in the artist's characteristic style, the book however addresses new issues such as fatherhood, or the place of man in nature. Virtuoso editing and sequencing clearly demonstrate a bookmaker at the top of his art.
By Klara Källström & Thobias Fäldt
The authors, struck by the linguistic gaps between French and English-speakers in Quebec, initiated a reflection about the survival of Amerindian idioms and researched toponyms that had withheld traces of them. They photographed various locations and simply captioned the images with the French and English translations of their names. "Kanata" (Canada) means "Village" in Iroquois.
You Haven't Seen Their Faces
By Daniel Mayrit
Riot Books
Daniel Mayrit spotted the faces of the 100 most influential personalities of London on surveillance video recordings. Annotated in ballpoint pen on each portrait are the scandals or conflicts of interest involving the person. The screw-bound book can be disassembled to become an exhibition or a political tool to be handed out in the streets. A 2015 protest book.
Bottom of the Lake
By Christian Patterson
Koenig Books
How to bounce back after a success such as Redheaded Peckerwood? Christian Patterson does it well with his usual tools (found images, staging, straight photos) but this time through his personal story, by taking hold of the 1973 phonebook of his native city, Fond du Lac, WI.
Belongs to Joe
By Casper Sejersen
Based on the film script of Lars von Trier's Nymphomaniac, this book addresses, through stunning editing, such apparently unrelated issues as botanics, religion and sex. The result is remarkably coherent.
By Carlos Spottorno
PHREE, RM
In black and white, like in a crime novel, Spottorno explores the locations where money and the power it bequeaths are preserved: Switzerland, Luxembourg, or the city of London… The mischievous Spaniard adopts the format of a luxurious promotional booklet for a fictitious private institution coined the WTF Bank.
By Katja Stuke
Fw: Books
On Chinese National Day, Katja Stuke spent several hours filming in Tiananmen Square, one of the world's most monitored sites. From that video, she singled out hundreds of portraits of passers by that she related to news clippings found in Hong Kong dealing with the 1989 massacres: a reflection about the visible and the concealed, the authorized and the banned.
Translated by Frédérique Destribats
Rémi Coignet is editor in chief for The Eyes magazine. In 2014, he published Conversations, a collection of his interviews with distinguished photographers (The Eyes Publishing).
Back to The Best Books of 2015
Labels: Best Books, Best Books 2015, Books

redpajama_set_name: RedPajamaCommonCrawl


Is your iPhone 5S mobile unable to detect the SIM card though it is a valid one? Then the problem is with your SIM card reader. But you have nothing to worry if you avail the services of iPhone Bits Service Center as it specializes in repairing the faulty SIM card readers of iPhone mobiles.
Due to the breaking or burning of an IC or any other important component of a SIM card reader, it would be unable to read even when a valid SIM when inserted. This problem can be rectified at iPhone Bits Service Center in Belfast through their SIM card reader repair service. Customers can either send the device to us through post to our address in Belfast or you can also bring the device to our service center in Belfast if you are in the nearby area. Please make sure that you fill in the required information in the below mentioned Data Repair Sheet and the Service Request sheet when you send us your device to us through post to get it repaired.


The members of the Management Board of UNIQA Insurance Group AG received remuneration of €3.4 million in 2018.
The fixed salary components include remuneration in kind equivalent to €34,788 (2017: €40,656).
The Short-Term Incentive (STI) comprises a variable remuneration component which is paid beginning with the 2017 financial year, partly in the following year and partly after three years (the "deferred component").
The Long-Term Incentive (LTI) corresponds to a share-based remuneration agreement first introduced in 2013, with the beneficiary entitled to receive a cash settlement following a four-year term. Details can be found in the notes to the consolidated financial statements.
In the 2018 financial year, the members of the Management Board of UNIQA Insurance Group AG received variable remuneration and multi-year share-based payments amounting to €1.7 million. Payments (STI) in the amount of €0.4 million are expected to be made in subsequent years for the 2017 financial year. For the 2018 financial year, payments (STI) in the amount of €1.6 million are expected to be made in the years 2019 and 2022. As part of the multi-year share-based payment (LTI), payments of €0.5 million were made to the members of the Management Board of UNIQA Insurance Group AG in 2018. For the subsequent years 2019 to 2022, a payment of €2.1 million is expected for the virtual shares allocated up to 31 December 2018.
The members of the Management Board who are also members of the Management Board of UNIQA Österreich Versicherungen AG received variable remuneration of €0.2 million for their work for UNIQA Österreich Versicherungen AG.
In addition to the above-mentioned employee benefits, the following pension fund contributions were made for the existing pension commitments to the members of the Management Board during the financial year. The compensation payments arise if a member of the Management Board steps down before the age of 65 because pension entitlements are generally funded in full until the age of 65 to avoid over-financing.
The remuneration paid to the members of the Supervisory Board for their work in the 2017 financial year amounted to €481,875. Provisions of €739,375 have been set aside for the remuneration to be paid for work completed in 2018. In 2018, a total of €67,400 was paid to cover attendance fees and out-of-pocket expenses (2017: €61,400). Given the raised requirements and the increasing time necessary for activities in the Audit Committee and in the other committees of the Supervisory Board, the remuneration components for committee functions performed have been increased. This explains the rise from €543,275 in 2017 to €806,775 in 2018.
The Management Board and Supervisory Board intend to propose the remuneration of €739,375 to the 2019 Annual General Meeting for resolution.
Burkhard Gantenbein received Supervisory Board remuneration (including attendance fees) of €18,000 for his activities on the Supervisory Boards of UNIQA Österreich Versicherungen AG and UNIQA International AG in addition to the Supervisory Board remuneration of UNIQA Insurance Group AG. Besides Supervisory Board remuneration (including attendance fees) from UNIQA Insurance Group AG, Jutta Kath also received Supervisory Board remuneration of 19,200 Swiss francs for her work on the Supervisory Board of UNIQA Re AG.
Former members of the Supervisory Board did not receive any remuneration.
The disclosures in accordance with Section 239(1) of the Austrian Commercial Code in conjunction with Section 80b of the Austrian Insurance Supervision Act must be included in the notes to the consolidated financial statements for the financial statements to be in accordance with IFRSs and to release the company from the requirement to prepare financial statements in accordance with the Austrian Commercial Code. The disclosures are defined more broadly for the separate financial statements in accordance with the provisions of the Austrian Commercial Code. The separate financial statements include not only the remuneration for the decision-making functions (Management Board) of UNIQA Insurance Group AG, but also the remuneration paid to the Management Boards of the subsidiaries if such remuneration is based on a contract with UNIQA Insurance Group AG.
International Financial Reporting Standards. Since 2002 the term IFRSs has applied to the overall concept of standards adopted by the International Accounting Standards Board. Standards already adopted beforehand continue to be referred to as International Accounting Standards (IASs).


text:
Home > Opinion > Volkswagen and the death of diesel?
Volkswagen and the death of diesel?
Anumita Roychowdhury22 Oct 2015 9:49 PM GMT
Last month, the US authorities exposed a sinister plot hatched by the Volkswagen Group, one of the world's leading automobile manufacturers that make popular glam diesel car models. Its two-litre turbodiesel engine car brands, often high on shopper's list for their low emissions and fuel economy, were found patched with "defeat devices", which allowed the cars to cheat and pass USA's stringent emissions tests for nitrogen oxides (NOx)—noxious gases liberally spewed out by diesel engines. NOx causes and aggravate lung and heart diseases, and are a key component in the formation of ozone, another deadly gas.
The German carmaker has admitted having installed the defeat device on more than 11 million cars, manufactured under the brand names of Jetta, Beetle, Passat, Golf and even the upmarket Audi 3, which it has fraudulently sold across the US, Europe and perhaps in other parts of the world since 2009.
The corporate fraud caught global attention on September 18 when the US Environmental Protection Agency (USEPA), responsible for enforcing the Clean Air Act and emissions standards, issued "notice of violation" to the conglomerate—Volkswagen AG, Audi AG and Volkswagen Group of America. In its scathing notice, a copy of which is available with Down To Earth, USEPA has laid bare gruesome details of the scam.
The notice of violation states that a defeat device is a cheating software hidden in the electronic control module of the cars. The software is programmed to sense whether the vehicle is being tested in the laboratory, based on parameters such as the position of the steering wheel, vehicle speed, duration of engine operations and barometric pressure. It then switches the car into a clean-running mode, reduces NOx emissions and helps it pass the test. But once on the road, the software switches to run on a separate calibration, which reduces the effectiveness of the NOx control system, and the car begins to emit 10 to 40 times more NOx than the certified level, depending on the driving condition.
Since, in diesel vehicles, it is difficult to balance the NOx control system with cost-effective means of improving energy efficiency, Volkswagen could have altered the software to compromise on emissions.
The impact of this fraud on Volkswagen is immense. Its stock value is falling since the scandal broke—share prices dropped by almost 40 percent within a week; its chief executive, Martin Winterkorn, has lost his job; and worse, its brand value is in tatters, casting a shadow over the much-famed German engineering in the global market.
Volkswagen has reportedly set aside US $7.3 billion to repair the damage. But this is too little. Under the US Clean Air Act, it faces fines of up to $37,500 per defective car. Given that Volkswagen has sold 482,000 defeat device-fitted cars in the US alone, it may have to shell out $18 billion as a civil penalty.
This can wipe out most of its last year's operative profits. Till the magazine went to press, 34 aggrieved customers had filed lawsuits against Volkswagen, claiming that the value of their cars has declined because of the scandal. Volkswagen will have to compensate them directly or through class action lawsuits. Besides, it will cost the earth to recall and fix all the vehicles, not just in the US but elsewhere. Other governments may also impose penalties. Its financial woes could get a lot worse as sales of Volkswagen diesel vehicles have taken a hit worldwide and stopped in the US.
SIX YEARS OF LIES, DECEIT
Exposing Volkswagen's fraud was not easy. In fact, Volkswagen could still be selling lies had it not been for a chance testing.
The NOx standards of the US are the most stringent in the world. In 2007, when the USEPA upgraded its standards for cars and enforced USEPA Tier2-Bin5 standard, Volkswagen could not comply with it and had to pull out of the US market. So in 2009, when it re-entered the crucial market, it came with what now seems a cheating strategy to circumvent the tighter emissions standards. It launched cars with two types of NOx control technologies—lean NOx catalyst and urea-based selective catalytic reducing technology—and convinced American regulators that they had cleaned up diesel car emissions.
Its claims came under suspicion two years ago when the US-based International Council on Clean Transportation (ICCT), along with the West Virginia University, tested a few 2012 and 2013 diesel car models of Volkswagen and BMW to assess if the new cars were meeting the emissions standards in real-world driving conditions. For this, the researchers mounted portable devices to monitor emissions from the vehicles as they moved in cities of California. The results showed that the on-road NOx emissions from cars equipped with lean NOx catalyst exceeded the USEPA Tier2-Bin5 standard by 15 to 35 times; it was five to 20 times for cars equipped with urea-based selective catalytic reducing technology.
ICCT brought the results to the attention of USEPA and California Air Resources Board (CARB), the state's clean-air agency, which ordered Volkswagen to recall fraudulent vehicles and fix the problem. Volkswagen first tried to dismiss the problem of increased emissions as "technical issues or unexpected in-use conditions", but later recalled 482,000 cars and claimed to have rectified the problem. USEPA and CARB conducted confirmatory tests and found little improvement. When challenged, Volkswagen could not give a convincing explanation. "It is only when USEPA and CARB threatened that they would not approve certification of conformity to the Volkswagen 2016 model until the company explained adequately, did Volkswagen admit that it had designed and installed a defeat design," mentions USEPA's notice of violation.
Such frauds can undercut other players in the automobile market, which are investing in advanced technologies to ensure that their products comply with the emissions standards. ICCT and the University of West Virginia had tested a BMW 5X model along with the Volkswagen brands. They found that emissions from BMW were high but more or less within the expected range of variation from the limits: it was close to the US standards for both city and highway driving and higher during uphill and downhill drive. John German of ICCT, who carried out the tests, says, "Without vigilant enforcement, companies that comply with the standards will be placed at a competitive disadvantage. If left unchecked, that could undermine the whole regulatory framework."
The impact of Volkswagen's deceit has spread far beyond the US. Switzerland has taken the stern step by stopping sales of all two-litre diesel Volkswagen cars that meet its Euro V emissions standards; the current Euro VI standards have come into force only recently. The UK, Germany and South Korea have announced investigations into the Volkswagen diesel car models. France, Italy and other EU member states have called for inquiries. In fact, car diesel engines of Volkswagen that are bigger than the tarnished two-litre engines may come under scrutiny.
In India, soon after the Volkswagen controversy came to light, Union Minister of Transport Nitin Gadkari issued a statement that the emission fraud is still not a concern in India, but this is a development that the government is watching. The government has, however, directed the Automotive Research Association of India (ARAI) to carry out tests on Volkswagen cars.
(The author is Executive Director, Centre for Science and Environment. Views expressed are strictly personal)
Anumita Roychowdhury


Two rolls of Catalan flags stand in the window of Gloria Clopés' haberdashery store in the town of Arenys de Munt, 45km from Barcelona.
When a mass rally for Catalan independence was held in Barcelona last month the difference hardly mattered. "People here were desperate to buy either," she says.
Tucked inland away from the ribbon development of the Catalan coast, Arenys de Munt is remarkable for its riera – the sand-strewn main street that doubles up as a flood channel, draining the steep wooded hills around.
Nearly every one of the elegant, modernista facades along the street sports a Catalan flag. This is hardly surprising: for like the water that pours through it in torrents after the autumn rains, Arenys was where the recent Catalan nationalist deluge started.
Here, three years ago, local nationalists held the first of a series of informal, local plebiscites on independence that led inexorably to last month's million-plus rally in Barcelona, an event that has turned Spanish politics on its head.


7 Week Workshop — Sundays, April 28th – June 9th, 2019 – Walk-ins Welcome!
This 7 week session will focus on painting streetscapes.
This workshop runs for 7 weeks: Sunday, April 28th – June 9th. Walk-ins are always welcome!


text:
<resources>

    <!--
        Base application theme, dependent on API level. This theme is replaced
        by AppBaseTheme from res/values-vXX/styles.xml on newer devices.
    -->
    <style name="AppBaseTheme" parent="android:Theme.Light">
        <!--
            Theme customizations available in newer API levels can go in
            res/values-vXX/styles.xml, while customizations related to
            backward-compatibility can go here.
        -->
    </style>

    <!-- Application theme. -->
    <style name="AppTheme" parent="AppBaseTheme">
        <!-- All customizations that are NOT specific to a particular API-level can go here. -->
    </style>

    <style name="MyTheme" parent="android:style/Theme.NoTitleBar.Fullscreen">
        <item name="android:windowAnimationStyle">@style/Animation</item>
    </style>

    <style name="Animation">
        <item name="android:activityOpenEnterAnimation">@null</item>
        <item name="android:activityOpenExitAnimation">@null</item>
        <item name="android:activityCloseEnterAnimation">@null</item>
        <item name="android:activityCloseExitAnimation">@null</item>
        <item name="android:taskOpenEnterAnimation">@null</item>
        <item name="android:taskOpenExitAnimation">@null</item>
        <item name="android:taskCloseEnterAnimation">@null</item>
        <item name="android:taskCloseExitAnimation">@null</item>
        <item name="android:taskToFrontEnterAnimation">@null</item>
        <item name="android:taskToFrontExitAnimation">@null</item>
        <item name="android:taskToBackEnterAnimation">@null</item>
        <item name="android:taskToBackExitAnimation">@null</item>
    </style>

</resources>

redpajama_set_name: RedPajamaGithub


Coulthard, T. J.,Lewin, J. and Macklin, M.G. (2005) Modelling differential catchment response to environmental change. Geomorphology 69, 222-241.
Dennis, I.A., Macklin, M.G., Coulthard, T.J. and Brewer, P.A. (2003) The impact of the October–November 2000 floods on contaminant metal dispersal in the River Swale catchment, North Yorkshire, UK. Hydrological Processes, 17, 1641-1657.
Smith, B.P.G., Naden, P.S., Leeks, G.J.L. and Wass, P.D. (2003) The influence of storm events on fine sediment transport, erosion and deposition within a reach of the River Swale, Yorkshire, UK. Science of the total environment, 314, pp.451-474.
House, W. A. and Warwick, M. S. (1999) Interactions of phosphorus with sediments in the River Swale, Yorkshire, UK. Hydrol. Process., 13: 1103–1115.
Walling, D. E., Owens P.N., and Leeks, G.J.L. (1999) Fingerprinting suspended sediment sources in the catchment of the River Ouse, Yorkshire, UK." Hydrological Processes 13, 955-975.


text:
Home/EDM NEWS/Facebook and UMG Sign Global Licensing Deal
Facebook and UMG Sign Global Licensing Deal
Zach Vaccaro Send an email December 24, 2017
Global Agreement
Universal Music Group (UMG) and Facebook announced that an 'unprecedented global, multi-year agreement' under which UMG licenses both recorded music and publishing catalogs for video and social experiences across Facebook, Instagram, and VR platform Oculus, is in effect.
Overall, the partnership will facilitate deeper engagement between artists and fans. Therefore, empowering users to express themselves through music, share the songs they love, and build communities around music-fueled culture. A variety of features will be enabled across Facebook's platforms. The agreement is intended to serve as a foundation for a strategic partnership that will deliver new music-based experiences online.
Terms of the partnership ensure that users will be able to upload videos that contain licensed music and 'personalize their music experiences' on Facebook, Instagram, and Oculus while sharing videos. Furthermore, functionality will expand in the future by enabling access to a vast library of music across a series of social features.
Executive Vice President of Digital Strategy, Michael Nash, at Universal Music Groups said, "We are creating a dynamic new model for collaboration between music companies and social platforms to advance the interests of recording artists and songwriters while enhancing the social experience of music for their fans."
"This partnership is an important first step demonstrating that innovation and fair compensation for music creators are mutually reinforcing – they thrive together. We look forward to Facebook becoming a significant contributor to a healthy ecosystem for music that will benefit artists, fans, and all those who invest in bringing great music to the world."
Tamara Hrivnak, Head of Music Business Development and Partnerships at Facebook, says, "There is a magnetic relationship between music and community building. We are excited to bring that to life on Facebook, Instagram, Oculus and Messenger in partnership with UMG. Music lovers, artists, and writers will all be right at home as we open up creativity, connection, and innovation through music and video."
The companies claim they will experiment hand-in-hand to introduce new music-based products to these social media platforms. In case, the goal is to develop the next generation of music products that best engage social consumers.
"I'm proud to announce this morning the first partnership between a major music company and Facebook, the largest social media company in the world,"Sir Lucian Grainge, Chairman and CEO of UMG said. "We've entered into a global multi-year agreement which licenses UMG's recorded music and music publishing catalogs for use in video content created by users across Facebook, Instagram, and Oculus."
Read more EDM news on OneEDM here.
Facebook Social Media umg
Zach Vaccaro
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text:
import Vue from 'vue';
import VueRouter from 'vue-router';

// components.
import NewsList from './components/pages/NewsList';
import CatesList from './components/pages/CatesList';
import DeleteList from './components/pages/DeleteList';

import NewsSetting from './components/pages/NewsSetting';
import NewsPinneds from './components/pages/NewsPinneds';

import ManageNews from './components/pages/ManageNews';

// import RecommendList from './components/pages/RecommendList';
// import AddRecommend from './components/pages/AddRecommend';

Vue.use(VueRouter);

const routes = [
    { path: '/', redirect: 'setting' },
    { path: '/newslist', component: NewsList },
    { path: '/deletelist', component: DeleteList },
    { path: '/cates', component: CatesList },
    { path: '/manage', component: ManageNews },
    { path: '/manage/:newsID', component: ManageNews },
    // { path: '/bin', component: BinList },
    // { path: '/rec', component: AddRecommend },
    // { path: '/rec/:rid', component: AddRecommend },
    // { path: '/recommend', component: RecommendList },

    { path: '/setting', component: NewsSetting },
    { path: '/pinneds', component: NewsPinneds }
];
export default new VueRouter({
    routes,
    mode: 'hash',
});


Craving new T-Pain music? The Florida artist is letting his previously-unreleased music go for the low-low price of free in his Soundcloud-released EP "Everything Must Go (Vol. 1)."
Pain will bring his acoustic stylings to six cities in October on a short acoustic tour. Rappers Smino and Felly will accompany as supporting acts.
T-Pain is back today to share a new music video to for the single, which is the perfect extravagant accompaniment for the track. In the clip, Pain has found a match in his voluptuous companion who's the Bonnie to his Clyde.


MonsterMMORPG Autumn Path: Game Map of MonsterMMORPG, Route Id 27, Zone Grass. MonsterMMORPG Autumn Path Game Map: Route Id: 27, Zone: Grass. You can catch 40 different Monsters at this route of MonsterMMORPG game.


text:
Q: Showing prev/next post links doesn't work I have blog which is in sub directory and showing last post on static index page on main site.
main site -> http://example.com
blog -> http://example.com/wp
The blog post is showed correctly on main site but I can't show links for prev and next post/article. This is what I have an what I'm trying
<?php
        define('WP_USE_THEMES', false);
        require('wp/wp-blog-header.php');

        $posts = get_posts('numberposts=1&order=ASC&orderby=post_title');
        foreach ($posts as $post) : setup_postdata( $post ); ?>

<?php the_title(); ?>

<?php the_date(); echo "
";?>

<?php the_content(); ?>

<?php posts_nav_link(); ?>

        <?php endforeach; ?>

So this isn't visible:

<?php posts_nav_link(); ?>

A: I cant comment you because I haven't enough reputations,
First follow https://codex.wordpress.org/Template_Tags/get_posts link
then check the sample code.
and also check the post display setting in the wp-admin panel /wp-admin/options-reading.php settings->reading section.

A: In Layman language it is a wordpress function to fetch posts/custom-posts. Internally it calls WP_Query function. Or you can use WP_Query instead of this. 'posts_nav_link' function does not work with WP_Query & get_posts function. You can try this to get next previous links.
    $posts = get_posts('numberposts=1&order=ASC&orderby=post_title');
    foreach ($posts as $post) : setup_postdata( $post ); ?>

<?php the_title(); ?>

<?php the_date(); echo "
";?>

<?php the_content(); ?>

    <?php endforeach; ?>

<?php previous_post_link(); ?><?php next_post_link(); ?>


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text:
Alexandria Ocasio-Cortez Responds To Claire McCaskill Calling Her "A Bright Shiny New Object"
By Morgan Brinlee
Rick Loomis/Getty Images News/Getty Images
Last week outgoing Democratic Sen. Claire McCaskill celebrated the end of her time in the Senate by taking shots at a number of lawmakers, including Representative-elect Alexandria Ocasio-Cortez. Although a political newcomer, Ocasio-Cortez, a self-described Democratic socialist, has quickly become a well-known name after her surprise defeat of longtime Rep. Joseph Crowley in New York's 14th Congressional District's Democratic primary. Now, Ocasio-Cortez is pushing back on McCaskill for calling her a "thing" and "a bright shiny new object."
"Not sure why fmr Sen. McCaskill keeps going on TV to call me a 'thing' and 'shiny object,' but it's pretty disappointing," Ocasio-Cortez tweeted Saturday.
Last week, McCaskill told CNN that she was "a little confused" as to why Ocasio-Cortez was "the thing." The outgoing Missouri senator went on to say she felt Ocasio-Cortez was "a good example" of the Democratic Party turning its attention from delivering on real issues in ways that would attract independent and white working class voters to focus on "pie-in-the-sky policy ideas." McCaskill went on further to describe Ocasio-Cortez as "a bright shiny new object," saying she "came out of nowhere and surprised people when she beat a very experienced congressman."
On Saturday, Ocasio-Cortez issued her own scathing criticism of McCaskill, who lost her re-election bid to Josh Hawley, Missouri's Republican attorney general, in November. "McCaskill promised she'd '100 percent back Trump up' on his anti-immigrant rhetoric & lost," Ocasio-Cortez tweeted. "In MO, almost all progressive ballot issues won."
The representative-elect then went on to push back on McCaskill's claim that her policy ideas wouldn't resonate with a variety of voters. While visiting Missouri after her primary, Ocasio-Cortez said she knew of "a conservative" who'd embedded at one of her rallies with Ferguson activists and progressive organizers. "You can tell she was inspired + struggled w/ Fox News bc our message resonated."
Earlier in the month, McCaskill told The New York Times podcast "The Daily," that incoming Democrats from "very blue districts" should listen to the Democrats who'd defeated incumbent Republicans and "not [one's who'd] defeated Democrats in primaries, if you know who I mean." McCaskill then brought up Ocasio-Cortez by name, telling "The Daily" that she hoped the newcomer "listens to people who defeated Republicans, because it's the people who defeated Republicans, in this election, that we need to be emulating, not the people who defeated Democrats in primaries."
In a series of tweets published Saturday, Ocasio-Cortez, who is expected to begin her first year as a member of Congress on Jan. 3, also commented on how McCaskill appeared to defend the Republican Party in her interview with CNN last week. "I'm also not sure why McCaskill is covering for the GOP by saying they 'secretly think Trump is nuts,'" she tweeted. "Nobody cares. Trump is melting down our institutions and inciting division between people. At any time GOP could have checked him and choose not to. They're accomplices."
McCaskill had told CNN that the GOP was now "Donald Trump's Republican Party," but alleged that some Republican senators would privately tell you Trump "is nuts." She claimed these Republicans feel the president "doesn't have a grasp of the issues" and was "making rash decisions" and "not listening to people who know the subject matter." The outgoing Democratic senator also claimed that these Republican senators weren't publicly opposing Trump out of a fear of losing the support of their base.
On Sunday, McCaskill appeared to attempt to clarify her remarks in a tweet to Ocasio-Cortez. "In the same interview I said I 'wish you well and hope you hang the moon,'" the outgoing senator wrote. "My point was that results, not rhetoric is what really matters. Most folks are very cynical about all the promises, when they aren't followed by real accomplishments. I look forward to cheering yours!"


Japan Prize winners Yutaka Takahasi, Alain Fischer, and Theodore Friedmann.
TOKYO—The winners of this year's Japan Prizes are grateful for the awards, but also hope the recognition will further their scientific objectives in improving treatments for genetic disorders and reducing the toll from flood disasters.
Theodore Friedmann of the University of California, San Diego, and Alain Fischer of the Imagine Institute in Paris and of the Collège de France will share the prize for "Medical Science and Medicinal Science." Their winning work involved gene therapy, an experimental technique in which genes are inserted into patients to replace mutated genes that are causing disease. Friedmann is credited with originating the concept in the 1970s and furthering basic research. Fischer, in 2000, reported demonstrating the clinical efficacy of gene therapy for the first time, using blood stem cells to treat a fatal genetic disorder called X-linked severe combined immunodeficiency. "Gene therapy has become a reality in recent years with many cases of successful treatments; the field is flourishing," said Hiroshi Komiyama, a chemical engineer and former president of the University of Tokyo who chaired the selection committee.
Yutaka Takahasi, a professor emeritus of civil engineering at the University of Tokyo, convinced engineers and policymakers to look holistically at entire river basins and the hydrological cycle when attempting to control flooding, rather than simply building more dams and higher embankments, Komiyama explained. Takahasi is being recognized in the field of "Resources, Energy and Social Infrastructure."
All three winners, appearing here today, thanked Japan Prize officials and the selection committees. But they all also expressed hope that the awards will highlight ongoing challenges in their respective fields. For gene therapy, "the path from concept to reality has not been straightforward and easy," Friedmann said.
The setbacks have included poor results from initial trials in the 1990s. And in 1999, a patient enrolled in a gene therapy trial in the United States died. It took several years for researchers in the field to address resulting safety and ethical questions. Fischer also suspended his gene therapy trials to solve safety issues. Research picked up again in the later 2000s, particularly with the use of new techniques for safely introducing the genes into the human body.
As for his specialty, Takahasi said that although many lessons have been learned in watershed management, new challenges are emerging as global warming leads to changing rainfall patterns, more powerful typhoons, and other weather anomalies. He said he hopes Japan Prize recognition will get Japan and the world "to recognize the serious threat and come to grips with the new challenges."
The laureates will return to Tokyo in April for the ceremony. at which they will receive a certificate of recognition and a commemorative gold medal. Each prize category carries a cash award of approximately $420,000.
Japan Prize topics are selected annually from within a range of disciplines gathered under two broad areas: physics, chemistry, and engineering; and life science, agriculture, and medicine. The categories for the 2016 prizes are "Materials and Production" and "Biological Production and Biological Environment." The prize is intended to recognize individuals for scientific achievements that also promote peace and prosperity.


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How to Use Expert Advisors on MT4
An expert advisor, also known as an EA, is essentially a program that can automatically place trades for you. This program (or robot) follows a trader's instructions once certain criteria has been met. Thus, it is a form of automated trading, since a trader does not need to do anything for the orders to be entered. The program basically works using if-then statements to create mechanical rules.
Benefits of an Expert Advisor
Apart from placing trades, an expert advisor can also control the position size when entering an order. The position size is influenced by the size of the account. This is because a position size of the same euro value can be a small fraction of an account that has for example €100,000, but may be significant for an account that has €1,000. Therefore, an EA should take into consideration the account equity when determining the position size.
An EA can be easily installed in the MT4 platform. In case you're already following a process or have a system in place, you can create an EA to follow the same process upon a signal. Thus, the EA is essentially replicating whatever you would do. The EA comes in handy for particularly those traders who trade a number of currency pairs as well as for those who are too busy to trade too frequently. Even when you're not at your computer or don't have time to look at the MT4 screen, the EA continues to look for trading opportunities on your behalf.
All you have to do is to leave the computer on, and the EA will place orders and exit trades for you. Moreover, you can study the performance of the EA overtime. This gives you an opportunity to test the instructions or criteria set for the program.
Do keep in mind some of the disadvantages of running EAs. This includes ensuring you have reliable internet connection. Most EAs are set up for a specific market, though some work for multiple trading pairs – just be certain what you are trading. As EAs are math algorithms based on technical analysis they tend to work best in trending markets rather than sideways. Which also means you have to be on wary of sharp fundamental moves.
Since an EA performs the trade automatically, it's critical that the program allows you to set a risk level, taking into consideration your account size.
Issues Associated with EA
There are hundreds of EA programs that can be downloaded from the Internet. However, you'd have to be very careful, because some of these are of lesser quality than others. You can read reviews before deciding on which EA to download. There's also the option of creating the program yourself. It's a little complex, but it may be worth the time. You can also hire a freelancer for creating the program for you.
How to Set Up an Expert Advisor on MT4
Create or download an Expert Advisor. Copy the EA files from the location where you saved it. Now you need to locate the MetaTrader4 folder on your computer. Open this folder and look for the Experts folder. Open this folder and paste the EA files. Your computer may open a pop-up requesting admin permission. Click on the Continue button.
Now open your MT4 platform. On the left bottom on the screen, you will see the Navigator window. Here, you will find a folder named Expert Advisors. Click on the + sign to the left of the Expert Advisors folder. You should be able to see the desired EA in this folder. Click on the desired EA and drag it onto a chart.
A pop-up will open and you can change settings and alerts here. Once done, click OK.
MT4 does not allow two EAs to be attached to the same chart. However, you are allowed to run more than a hundred EAs on the same trading terminal. So, if you want to run two EAs, you can simply open new chart windows, one for each of the EAs you want to run. Simply open as many windows as the number of EAs you wish to run, and attached a different EA to each.
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Apple's iPhone X prices tons of cash.
You'll would like a minimum of $1000/£1000 to urge yourself one – and don't expect costs to vary a lot of in 2018 once the iPhone XI lands.
Nope. Thousand dollar iPhones area unit here to remain, for higher or worse.
You'd be forgiven for thinking that a phone this dearly-won ought to work dead.
But as is commonly the case with technology, things don't perpetually pan out as you'd expect. And Apple's iPhone X is not any exception.
So before you drop a giant 'ol pile of cash on this phone, you would possibly wish to require a glance at a number of the documented problems that have an effect on it.
Please note: not all of those problems (or even most of them) have an effect on the bulk of iPhone X handsets in circulation.
Apple is nice at fixing code problems with updates and has even been acknowledged to require care of hardware problems while not an excessive amount of fuss yet.
All this text is meant to try to to is detail a number of the reportable problems encompassing Apple's iPhone X.
A lot of reviews for a French telephone can glaze over issues, as most don't manifest itself till once many months of usage.
Most reviewers play with a phone for per week and so write a review supported that, that is ok. however it will mean things get uncomprehensible (it's why we tend to perpetually update our reviews).
Multiple folks are whiney on-line – Reddit, for example – that Apple's iPhone X likes to drag hair a small amount an excessive amount of.
According to the myriad of reports, the iPhone X incorporates a habit of housings hair (facial and head-stuff) in between its bezels, and this will cause some painful problems once ending a decision.
The iPhone X is ALL this OLED panel.
But the school is new for Apple, and there's positively a learning curve.
Multiple users have noted that the show can usually become unresponsive to the touch, typically once there's a decision coming back in, and also the solely thanks to fix it's to perform a boot.
Apple's trying into this issue immediately, therefore expect a fix within the terribly close to future.
You get a decision, you wish to answer it. That's what happens commonly.
But with the iPhone X, users area unit coverage delays within the decision registering on the show. The iPhone X rings, however nothing is displayed on the screen for 5-10 seconds. Odd. And annoying.
Apple says FACE ID is super secure.
No one will hack it, however you and your stunning face.
However, multiple users (and YouTube people) have well-tried this is often not precisely the case. This goes double if you have got a twin or associate degree evil doppelgänger.
Apple has additionally warned oldsters to not let their young youngsters use their iPhone X handsets.
Why? as a result of the kid's face continues to be developing and will be confused for its parent's by FACE ID. thence why I like the ol' fingerprint methodology.
This one was noted by KYM's electro-acoustic transducer, AKA the sole iPhone X user on our team.
According to electro-acoustic transducer, the iPhone X battery life is OK, however nothing special. He was apparently expecting a lot of during this regard from Apple's. what is more, once employing a constituent a pair of XL for per week, electro-acoustic transducer was blown away by the distinction.
The constituent a pair of XL lasts FOREVER were his actual words, I believe.
Bus w**nkers aside, paying attention to stuff through your phone's speakers is commonly necessary (unless you're on a bus or public transport), and to try to to this properly you wish tight speakers.
The iPhone X has 2 front-facing stereo speakers. and that they tend to urge the duty done.
But there are lots of problems concerning droning from the speakers at sure volume settings, yet as distortion whenever volume exceeds five hundredth.
Apple's iPhone eight had an identical issue that was fastened with a code update, therefore let's hope Apple will do a similar factor for the iPhone X.