\section*{Abstract}
A novel approach towards construction of absolutely continuous distributions over the unit interval is proposed. Considering two absolutely continuous random variables with positive support, this method conditions on their convolution to generate a new random variable in the unit interval. This approach is demonstrated using some popular choices of the positive random variables such as the exponential, Lindley, gamma. Some existing distributions like the uniform and the beta are formulated with this method. Several new structures of density functions having potential for future application in real life problems are also provided. One of the new distributions having one parameter is considered for parameter estimation and real life modelling application and shown to provide better fit than the popular one parameter Topp-Leone model.\\

\noindent\textbf{\large{Keywords:}} Topp-Leone, Uniform, Beta, Exponential, Gamma, Lindley, Convolution, Ferlie-Gumble- Morgenstern copula.\\

\noindent\textbf{\large{MSC 2010:}} 60E05. \\

\section{Introduction}
Consider a departmental store where customers arrive independently of each other. Suppose the store opens at 10:00 a.m. and the second customer arrives at 11:00 a.m. Arrival time of the first customer was not observed. Then the store manager may be interested in finding the probability of the first customer arriving within half an hour from opening the store. If the customers arrive according to a Poisson process, then the probability of the first customer arriving within 10:30 a.m. is $0.5$. In fact, for given arrival time of the second customer, arrival time of the first customer is conditionally uniform over the duration that is uniform over 10:00 a.m to 11:00 a.m. This follows from the assumption that the inter-arrival times of the customers are independently and identically distributed (iid) exponential random variates. However, the conditional distribution does not remain uniform when any one of the above  assumptions is relaxed. Still the resulting conditional distribution has support $(0,1)$, the unit interval. This simple yet previously unaddressed approach clearly opens up a scope for constructing new distributions in the unit interval.\\

Importance of distributions with support $(0,1)$ is well established for modelling proportions, scores, rates, indices etc. Responses in the unit interval are often focus of investigation in many fields of application including finance (Gómez-Déniz et al. 2014, Biswas et al. 2020), public health (Mazucheli et al. 2019, Biswas and Chakraborty 2019), demography (Andreopoulos et al. 2019). The first model that comes to the mind of practitioners is obviously the well known Beta distribution (Johnson et al. 1995). Besides this, the Kumaraswamy's distribution (Kumaraswamy 1980, Jones 2009) and the Topp-Leone distribution (Topp and Leone 1955) also received attention. In recent times, a number of new models have been proposed as alternatives to the Beta and Kumaraswamy's distributions to enhance modelling flexibility as par context (Grassia 1977, Tadikamalla and Johnson 1982, Barndorff-Nielsen and Jorgensen 1991, Lemonte et al. 2013, Pourdarvish et al. 2015, Mazucheli et al. 2018, 2019, 2020). This recent surge in the number of research papers devoted to proposing new distribution in the unit interval clearly exhibits their growing relevance.\\

Most of the above distributions are generated through a suitable variable transformation of a baseline distribution with positive support. For $X$ being a positive valued random variable, the transformations $U=X/(1+X)$, $U=1/(1+X)$, $U=\exp(-X)$ are popularly used to derive a distribution with support $(0,1)$. Besides this, a general approach based on the cumulative distribution function-quantile function by Smithson and Shou (2017), Rodrigues et al. (2019) and Cancho et al. (2020). The objective of this paper is to describe a general procedure for construction of distributions with support $(0,1)$ from conditional convolution  approach.\\

Rest of the article is organized as follows. The new method of deriving distributions on $(0,1)$ is presented in the next section. Section 3 is devoted for illustrating the proposed methodology. Section 4 provides some detail on a new one parameter distribution along with real life application. This article is concluded with a potential generalization and further scopes.
\section{Method}
Consider a two component absolutely continuous random vector $(X,Y)$ with joint cumulative distribution function (cdf) $F_{X,Y}$ and joint probability density function (pdf) $f_{X,Y}$. Let $F_X$ and $f_X$ are the marginal cdf and the marginal pdf of $X$, respectively. Similarly, $F_Y$ and $f_Y$ denote the marginal cdf and the marginal pdf of $Y$. The conditional cdf and pdf of $X$ given the realization on $Y$ are $F_{X|Y}$ and $f_{X|Y}$, respectively. \\

We assume that both $X$ and $Y$ have positive support, that is $P(X>0)=P(Y>0)=1$. For $Z=X+Y$, the cdf of $Z$ can be expressed as

\begin{equation}\label{eq:Fz}
    F_Z(z)=\int_0^z F_{X|Y}(z-y) f_Y(y)\,dy.
\end{equation}
The corresponding pdf is

\begin{equation}\label{eq:fz}
    f_Z(z)=\int_0^z f_{X,Y}(z-y,y)\,dy=\int_0^z f_{X|Y}(z-y) f_Y(y)\, dy.
\end{equation}
In particular, when $X$ and $Y$ are independent, the cdf and the pdf of $Z$ can be obtained by replacing $F_{X|Y}(z-y)$ by $F_X(z-y)$ in (\ref{eq:Fz}) and $f_{X|Y}(z-y)$ by $f_X(z-y)$ in (\ref{eq:fz}), respectively. \\

We are interested in the conditional distribution of $X$ given $Z=1$. The corresponding random variable be $U$ with support $(0,1)$, the unit interval. Now $P(U\leq u)=P(X\leq u|Z=1)=P(X\leq u , X+Y=1)/f_Z(1)$. $P(X\leq u , X+Y=1)$ can be seen as $\int_0^u f_{X,Y}(v,1-v)\, dv$ and hence the pdf of $U$ is

\begin{equation}\label{eq:fu}
    f_U(u)=\frac{f_{X,Y}(u,1-u)}{f_Z(1)}, \,\, \textrm{for}\,\, 0<u<1.
\end{equation}
If $X$ and $Y$ are independently distributed, then $f_{X,Y}(u,1-u)$ can be replaced by $f_X(u)f_Y(1-u)$ in (\ref{eq:fu}).

It may be noted that, bivariate distributions on $(0,\infty)\times (0,\infty)$ are seldom naturally available. The common techniques to construct bivariate distributions from given marginals involve marginal transformation method, trivariate reduction and bivariate copula (Balakrishnan and Lai 2009). To demonstrate the proposed method for constructing distributions on the unit interval, some bivariate distributions will be considered in Section 3. The bivariate density we use here is derived using Ferlie-Gumble-Morgenstern (FGM) copula (Gumble 1960) when the marginal densities $f_X$ and $f_Y$ are available. The joint density function of $(X,Y)$ with additional parameter $\alpha\in [-1,1]$ using FGM copula is

\begin{equation}\label{eq:fgm}
    f_{X,Y}(x,y)=f_X(x)f_Y(y)[1+\alpha(2F_X(x)-1)(2F_Y(y)-1)].
\end{equation}

\section{Main results}
In this section we apply the proposed method to first derive distributions in $(0, 1)$ which are known followed by some new ones which to the best of our knowledge are not yet reported in the literature.

\subsection{Known distributions}
As outlined in Section 1, let $X$ and $Y$ follow exponential distribution independently with same parameter $\theta$. Then $f_X(x|\theta)=\theta\exp(-\theta x)$ and $f_Y(y|\theta)=\theta\exp(-\theta y)$. Putting the the expressions of $f_X(x|\theta)$ and $f_Y(y|\theta)$ in (\ref{eq:fz}), we get $f_Z(1|\theta)=\theta^2\exp(-\theta)$. Now, putting the expressions of $f_X(x|\theta)$, $f_Y(y|\theta)$ and $f_Z(1|\theta)$ in (\ref{eq:fu}) we get $f_U(u)=1$ for $0<u<1$ which is pdf of the uniform uniform distribution over unit interval. \\

Let $G(\alpha,\beta)$ denote the gamma distribution with density function $\alpha^\beta x^{\beta-1}\exp(-\alpha x)/\Gamma(\beta)$ for $\alpha,\beta >0$. Consider $X\sim$G$(\alpha,\beta_1)$ and $Y\sim$G$(\alpha,\beta_2)$ independently. Using (\ref{eq:fz}) it is easy to see that $Z\sim G(\alpha,\beta_1+\beta_2)$. Now, putting $f_X(x|\alpha,\beta_1)$, $f_Y(y|\alpha,\beta_2)$ and $f_Z(1)=\alpha^{\beta_1+\beta_2}\exp(-\alpha)/\Gamma(\beta_1+\beta_2)$ in (\ref{eq:fu}) we get $f_U(u|\beta_1,\beta_2)=\Gamma(\beta_1+\beta_2)u^{\beta_1-1}(1-u)^{\beta_2-1}/\Gamma(\beta_1)\Gamma(\beta_2)$ which is the pdf of beta distribution with parameters $\beta_1$ and $\beta_2$.\\

While the uniform and beta cases are easily seen, generating the other distributions in the unit interval as mentioned in Section 1 using the proposed method remain a challenge. However, we do not claim that the distributions can (or cannot) be generated using the proposed method.

\subsection{New distributions}
Here we consider different distributions for $(X, Y)$ to construct a number of new distributions on the unit interval.\\

\noindent\textbf{Distribution 1:} Let $X$ and $Y$ follow exponential distribution independently with parameter $\theta_1$ and $\theta_2$, respectively. Putting $f_X(x|\theta_1)=\theta_1\exp(-\theta_1x)$ and $f_Y(y|\theta_2)=\theta_2\exp(-\theta_2y)$ in (\ref{eq:fz}) we get
\begin{equation}
    f_Z(1)=\begin{cases}
    \frac{\theta_1\theta_2(e^{-\theta_1}-e^{-\theta_2})}{\theta_1-\theta_2}\quad \textrm{for}\quad \theta_1\neq\theta_2\\
    \theta^2e^{-\theta} \quad \textrm{for}\quad \theta_1=\theta_2=\theta.
    \end{cases}
\end{equation}
Now, putting $f_X(x|\theta_1)$, $f_Y(y|\theta_2)$ and $f_Z(1)$ in (\ref{eq:fu}) and replacing $\theta_1-\theta_2$ by $\delta$ we obtain
\begin{equation} \label{eq:d1}
f_U(u)=\begin{cases}
\frac{\delta e^{-\delta u}}{1-e^{-\delta}}\quad \textrm{for} \quad \delta\neq 0\\
1\quad \textrm{for} \quad \delta=0.
\end{cases}
\end{equation}
The density function in (\ref{eq:d1}) reduces to that of truncated exponential distribution with region of truncation $(1,\infty)$ when we restrict the parameter space to $\delta>0$. However, the new genesis does not require the parameter $\delta$ to be positive. Thus the density in (\ref{eq:d1}) provides more flexibility with $\delta\in\mathcal{R}$, the real line despite having the same mathematical structure as the said truncated exponential distribution. From Figure 1 it is clear that the density can be both increasing and decreasing depending on $\delta$. For $\delta<0$, it is increasing with mode at $1$ and for $\delta>0$, it is decreasing with mode at $0$.\\

\begin{figure}[H]
\begin{center}
\includegraphics[height=3in,width=4in,angle=0]{figure1.pdf}
\caption{Plot of the density function of Distribution 1.}
\end{center}
\end{figure}

\noindent\textbf{Distribution 2:} Let $X$ and $Y$ both marginally follow exponential distribution with parameters $\theta$. Putting $f_X(x|\theta)=\theta\exp(-\theta x)$, $F_X(x|\theta)=1-\exp(-\theta x)$, $f_Y(y|\theta)=\theta\exp(-\theta y)$ and $F_Y(y|\theta)=1-\exp(-\theta y)$ in (\ref{eq:fgm}) we obtain the following bivariate exponential distribution with parameters $\theta>0$ and $-1\leq\alpha\leq 1$.
\begin{equation}\label{eq:fgmd2}
    f_{X,Y}(x,y|\theta,\alpha)=\theta^2 e^{-\theta(x+y)}\left[1+\alpha(1-2e^{-\theta x})(1-2e^{-\theta y})\right]
\end{equation}
Using (\ref{eq:fgmd2}) in (\ref{eq:fz}) we obtain
\begin{equation}
    f_Z(1)=\theta^2 e^{-\theta}\left[1+\alpha\left\{1+4e^{-\theta}-\frac{4}{\theta}+4\frac{e^{-\theta}}{\theta}\right\}\right].
\end{equation}
Putting (\ref{eq:fgmd2}) and (\ref{eq:fz}) in (\ref{eq:fu}) we obtain the following density function.
\begin{equation}\label{eq:d2}
    f_U(u)=\frac{1+\alpha(1-2e^{-\theta u})(1-2e^{-\theta(1-u)})}{\left[1+\alpha\left\{1+4e^{-\theta}-\frac{4}{\theta}+4\frac{e^{-\theta}}{\theta}\right\}\right]}
\end{equation}
This distribution has two parameters and may prove to be useful in modelling applications. It is clear that for $\alpha=0$, $X$ and $Y$ are independent and consequently (\ref{eq:d2}) reduces to uniform distribution. From Figure 2, it can be seen that the distribution is symmetric about $0.5$ and it possesses both bath-tub shaped and bell-shaped curves.\\

\begin{figure}[H]
\begin{center}
\includegraphics[height=3in,width=6in,angle=0]{figure2.jpg}
\caption{Plot of the density function of Distribution 2.}
\end{center}
\end{figure}

\noindent\textbf{Distribution 3:} Let $X$ and $Y$ follow Lindley distribution independently and identically with parameter $\theta>0$. Putting $f_X(x|\theta)=\theta^2(1+x)\exp(-\theta x)/(1+\theta)$ and $f_Y(y|\theta)=\theta^2(1+y)\exp(-\theta y)/(1+\theta)$ in (\ref{eq:fz}) we get
\begin{equation}\label{eq:LLZ}
    f_Z(1)=\frac{13}{6}\frac{e^{-\theta}\theta^4}{(1+\theta)^2}.
\end{equation}
As in previous cases, using the density functions of $X$ and $Y$ along with (\ref{eq:LLZ}) in (\ref{eq:fu}) we obtain
\begin{equation}\label{eq:LLU}
    f_U(u)=\frac{6}{13}(2-u)(1+u).
\end{equation}
The resulting distribution is free from any parameter and thus not flexible as clear from Figure 3. Clearly (\ref{eq:LLU}) is symmetric about $0.5$.\\

\begin{figure}[H]
\begin{center}
\includegraphics[height=3in,width=4in,angle=0]{figure3.pdf}
\caption{Plot of the density function of Distribution 3.}
\end{center}
\end{figure}

\noindent\textbf{Distribution 4:} Let $X$ and $Y$ follow Lindley distribution independently with parameters $\theta_1$ and $\theta_2$, respectively. Here $f_X(x|\theta_1)=\theta_1^2(1+x)\exp(-\theta_1 x)/(1+\theta_1)$ and $f_Y(y|\theta_2)=\theta_2^2(1+y)\exp(-\theta_2 y)/(1+\theta_2)$. The density function of $Z=X+Y$ is evaluated using (\ref{eq:fz}) but we do not report it here due to its messy structure. However, the density function of $U$ obtained using $f_X(x|\theta_1)$, $f_Y(y|\theta_2)$ and $f_Z(1)$ in (\ref{eq:fu}) is
\begin{equation}\label{eq:LLU2}
    f_U(u)=\begin{cases}
    \frac{\delta^3 e^{-\delta u} (u-2)(1+u)}{(2-\delta-2\delta^2)+e^{-\delta}(2\delta^2-2-\delta)}\quad \textrm{for} \quad \delta\neq0\\
    \frac{6}{13}(2-u)(1+u)\quad \textrm{for} \quad \delta=0
    \end{cases}
\end{equation}
for $\delta=\theta_1-\theta_2\in \mathcal{R}$. Obviously the density in (\ref{eq:LLU2}) having one parameter is an extension of (\ref{eq:LLU}). From Figure 4 and Figure 1, it can be seen that the Distribution 4 and Distribution 1 behave quite similarly.\\

\begin{figure}[H]
\begin{center}
\includegraphics[height=3in,width=4in,angle=0]{figure4.pdf}
\caption{Plot of the density function of Distribution 4.}
\end{center}
\end{figure}

\noindent\textbf{Distribution 5:} Let $X$ follow exponential distribution with parameter $\theta$ and $Y$ follow Lindley distribution with parameter $\theta$ independently. Here $f_X(x|\theta)=\theta\exp(-\theta x)$ and $f_Y(y|\theta)=\theta^2(1+y)\exp(-\theta y)/(1+\theta)$. From (\ref{eq:fu}) we get
\begin{equation}\label{eq:EL1}
    f_U(u)=\frac{2}{3}(2-u).
\end{equation}
This distribution has no parameter and thus not flexible for modelling purposes . The density in (\ref{eq:EL1}) is decreasing and it decreases from $4/3$ at $u=0$ to $2/3$ at $u=1$ linearly as given in Figure 5.\\

\begin{figure}[H]
\begin{center}
\includegraphics[height=3in,width=4in,angle=0]{figure5.pdf}
\caption{Plot of the density function of Distribution 5.}
\end{center}
\end{figure}

\noindent \textbf{Distribution 6:} Let $X$ follow exponential distribution with parameter $\theta_1$ and $Y$ follow Lindley distribution with parameter $\theta_2$ independently. Here $f_X(x|\theta_1)=\theta_1\exp(-\theta_1 x)$ and $f_Y(y|\theta_2)=\theta_2^2(1+y)\exp(-\theta_2 y)/(1+\theta_2)$. Using the density functions first in (\ref{eq:fz}) and then in (\ref{eq:fu}) we get
\begin{equation}\label{EL2}
    f_U(u)=\begin{cases}
    \frac{\delta^2 e^{\delta(1-u)}(2-u)}{1-\delta+(2\delta-1)e^\delta}\quad \textrm{for}\quad \delta\neq 0\\
    \frac{2}{3}(2-u)\quad \textrm{for}\quad \delta= 0.
    \end{cases}
\end{equation}
Here $\delta=\theta_1-\theta_2\in \mathcal{R}$. A few representative plots of the density function are provided in Figure 6.\\

\begin{figure}[h]
\begin{center}
\includegraphics[height=3in,width=4in,angle=0]{figure6.pdf}
\caption{Plot of the density function of Distribution 6.}
\end{center}
\end{figure}

\noindent \textbf{Distribution 7:} Consider $X$ following exponential distribution with parameter $\theta$ and $Y$ following gamma distribution with parameters $\alpha$ and $\beta$. Putting $f_X(x|\theta)=\theta\exp(-\theta x)$ and $f_Y(y|\alpha,\beta)=\alpha^\beta y^{\beta-1}\exp(-\alpha x)/\Gamma(\beta)$ in (\ref{eq:fz}) we get
\begin{equation}\label{eq:egz}
    f_Z(1)=\begin{cases}
    \frac{\theta \alpha^\beta e^{-\theta}\{\Gamma(\beta)-\Gamma(\alpha-\theta)\}}{\Gamma(\beta) (\alpha-\theta)^\beta}\quad \textrm{for} \quad \alpha\neq \theta\\
    \frac{\alpha^{\beta+1}\exp(-\alpha)}{\Gamma(\beta+1)} \quad \textrm{for}\quad \alpha=\theta.
    \end{cases}
\end{equation}
Here $\Gamma(s,t)$ denotes the incomplete gamma function $\int_t^\infty v^{s-1} \exp(-v)\, dv$. Now using the density functions of $X$ and $Y$ along with (\ref{eq:egz}) in (\ref{eq:fu}) we obtain
\begin{equation}\label{eq:egu}
    f_U(u)=\begin{cases}
    \frac{\delta^\beta}{\Gamma(\beta)-\Gamma(\beta,\delta)}e^{\delta(1-u)}(1-u)^{\beta-1}\quad \textrm{for}\quad \beta>0,\delta\neq 0\\
    \beta (1-u)^{\beta-1}\quad \textrm{for} \quad \alpha>0,\delta=0
    \end{cases}
\end{equation}
where $\delta=\alpha-\theta\in \mathcal{R}$. The density in (\ref{eq:egu}) has two parameters and may prove to be a competitor of the popular beta distribution as evident from Figure 7.\\

\begin{figure}[H]
\begin{center}
\includegraphics[height=2in,width=6in,angle=0]{figure7.jpg}
\caption{Plot of the density function of Distribution 7.}
\end{center}
\end{figure}

\noindent \textbf{Distribution 8:} Let $X$ be a Lindley random variable with parameter $\theta$ and $Y$ be a gamma random variable with parameters $\theta$ and $\beta$. Here $f_X(x|\theta)=\theta^2(1+x)\exp(-\theta x)/(1+\theta)$ and $f_Y(y|\theta,\beta)=\theta^\beta y^{\beta-1}\exp(-\theta x)/\Gamma(\beta)$. Putting the densities of $x$ and $Y$ first in (\ref{eq:fz}) and then using $f_Z(1)$ along with the density functions in (\ref{eq:fu}) we obtain
\begin{equation}\label{eq:flg}
    f_U(u)=\frac{\beta(1+\beta)}{2+\beta} (1+u) (1-u)^{\beta-1}\quad \textrm{for} \quad \beta>0.
\end{equation}
The parameter $\theta$ labelling the distributions of both $X$ and $Y$ surprisingly plays no role in (\ref{eq:flg}). Despite being a one-parameter distribution, it is quite flexible for modelling purposes since the corresponding density function has a wide range of shapes as can be seen from Figure 8. The derived distribution can also be seen as a mixture of two beta distributions where the mixing proportions are also functions of the parameter $\beta$ .

\begin{figure}[H]
\begin{center}
\includegraphics[height=3in,width=4in,angle=0]{figure8.pdf}
\caption{Plot of the density function of Distribution 8.}
\end{center}
\end{figure}

\section{LCG: A new one parameter distribution}
Distribution 8 is derived from the distribution of Lindley random variable conditioned on the convolution of the same with gamma random variable. Thus a working abbreviation for the distribution with pdf in (\ref{eq:flg}) is taken as $LCG(\beta)$ and we write $U\sim LCG(\beta)$. The cumulative distribution function (cdf) of $LCG(\beta)$ is
\begin{equation}\label{eq:Flg}
    F_U(u)=1-\frac{(1-u)^\beta(2+\beta+u\beta)}{2+\beta},\quad  0<u<1.
\end{equation}
Figure 9 exhibits the functional form of $F_U$ for different choices of $\beta$. With increasing $\beta$, the cdf becomes more and more convex in shape.
\begin{figure}[H]
\begin{center}
\includegraphics[height=3in,width=4in,angle=0]{figure10.pdf}
\caption{Plot of the cumulative distribution function of LCG distribution.}
\end{center}
\end{figure}
The corresponding hazard rate is
\begin{equation}\label{eq:hlg}
    h(u)=\frac{\beta(1+\beta)}{2+\beta+u\beta}\frac{1+u}{1-u},\quad   0<u<1.
\end{equation}
The LCG family has increasing failure rate as shown in Figure 10.
\begin{figure}[H]
\begin{center}
\includegraphics[height=3in,width=4in,angle=0]{figure9.pdf}
\caption{Plot of the hazard rate of LCG distribution.}
\end{center}
\end{figure}
\noindent The lower order moments are very simple, the mean and the variance of $LCG(\beta)$ is given below.
\begin{equation}
    E(U)=\frac{4+\beta}{(2+\beta)^2}\quad \textrm{and} \quad V(U)=\frac{\beta(16+9\beta+\beta^2)}{(2+\beta)^4(3+\beta)}
\end{equation}

\subsection{Generating random observations}
To generate random observations from $LCG(\beta)$ it is convenient to use the cdf in (\ref{eq:Flg}). We employ cdf inversion technique as the structure is not too complex. For a random observation $v$ from uniform distribution over $(0,1)$ we equate $F_U(u)=v$ and find solution of the equation in $u$. This $u$ is a random observation from $LCG(\beta)$. This technique produces the following nonlinear equation in $u$
\begin{equation}\label{eq:Flgv}
    (1-u)^\beta(2+\beta+u\beta)-(1-v)(2+\beta)=0.
\end{equation}
We solve (\ref{eq:Flgv}) by \textit{uniroot} function available on basic \textbf{R}.
\subsection{Estimation}
Since $E(U)$ is structurally simple, a method of moment estimator can easily be implemented. For a given dataset $\bm{u}=(u_1, u_2, ..., u_n)$, we calculate the sample mean $\Bar{u}=(1/n)\sum_{i=1}^n u_i$ and form the following equation.
\begin{equation}\label{eq:mme1}
    E(U)=\frac{4+\beta}{(2+\beta)^2}=\Bar{u}
\end{equation}
The equation in (\ref{eq:mme1}) can be rewritten as
\begin{equation}\label{eq:mme2}
    \Bar{u}\beta^2+(4\Bar{u}-1)\beta+4(\Bar{u}-1)=0.
\end{equation}
The method of moment estimator $\hat{\beta}_{MM}$ is the solution of (\ref{eq:mme2}) in $\beta$. \\

The log-likelihood function of $\beta$ given the dataset $\bm{u}$ is
\begin{equation}\label{mle1}
    l(\beta|\bm{u})=n\log\left[\frac{\beta(1+\beta)}{2+\beta}\right]+\sum_{i=1}^n \log(1+u_i)+(\beta-1)\sum_{i=1}^n \log(1-u_i).
\end{equation}
Differentiating $l(\beta|\bm{u})$ with respect to $\beta$ and equating it with $0$ yields the following.
\begin{equation}\label{eq:mle2}
    T(\bm{u})\beta^2+(3T(\bm{u})-1)\beta+(2T(\bm{u})-4)=0
\end{equation}
Here $T(\bm{u})=-(1/n)\sum_{i=1}^n \log (1-u_i)$.  To solve both (\ref{eq:mme2}) and (\ref{eq:mle2}) in $\beta$, we use \textit{polyroot} function available in basic \textbf{R}. The solution of (\ref{eq:mle2}) is the maximum likelihood estimate namely $\hat{\beta}_{ML}$.
\begin{center}
\begin{table}[h!]
\centering
\vspace{1cm}
	\caption{Bias and mean-squared error (MSE) of $\hat{\beta}_{MM}$ and $\hat{\beta}_{ML}$.}
	\label{tab:simul}
	\begin{tabular}{cccccc}
		\hline\\
		$n$ & $\beta$ & Bias($\hat{\beta}_{MM}$) & MSE($\hat{\beta}_{MM}$) & Bias($\hat{\beta}_{ML}$) & MSE($\hat{\beta}_{ML}$)  \\
		\\\hline
		\multirow{3}{*}{20} & 2.30 & 0.07792 &0.27779  & -0.21959 &0.33440   \\
		& 7.80 &0.34196   &3.74806  &0.21788  &3.72484\\
		&  15.00&0.52377  &12.36922  &0.46433  &12.36009  \\ \hline
		\multirow{3}{*}{40} & 2.30 &0.01869  &0.12236  &-0.29453  &0.21342   \\
		& 7.80 &0.13403   &1.48331  &-0.00307  &1.48377\\
		&15.00 &0.38477   &5.71176  &0.30913  &5.72911  \\ \hline
				\multirow{3}{*}{60} &2.30  &0.01719  &0.08165  &-0.29909  &0.17421  \\
		& 7.80 &0.12145  &0.96437  &-0.02729  &0.96883 \\
		&15.00 & 0.16190  &3.51346  &0.08157&3.51912 \\ \hline

				\multirow{3}{*}{80} & 2.30 &0.00449 &0.06157 &-0.31263 &0.16230  \\
		&  7.80 &0.12302 &0.70574 &-0.03383 &0.70355  \\
		& 15.00 & 0.14265 & 2.69120 &0.05492  &2.69554  \\ \hline

				\multirow{3}{*}{100} & 2.30  & 0.01120 &0.04923  &-0.30954  &0.14639   \\
		& 7.80 &0.06890   &0.55969  &-0.08804  &0.57761\\
		& 15.00 & 0.17952 &2.35752  &0.08494  &2.35630  \\ \hline

	\end{tabular}%
\end{table}
\end{center}
A simulation experiment is performed to study the relative performance of the estimators. We generate 1000 samples of different sizes ($n$) using the method in Section 4.1 and compute the average bias and mean squared error of the estimators. From Table 1, it is seen that the performance of both the estimators are quite similar. With increasing $\beta$, the mean squared error of both the estimators increase. As desired the mean squared error and absolute bias of both the estimators decrease with increase in sample size.
\subsection{Application}
Consider the following datasets from Caramanis et al. (1983) providing computation time of two different algorithms namely SC16 and P3. \\

\noindent SC16:= (0.853, 0.759, 0.866, 0.809, 0.717, 0.544, 0.492, 0.403, 0.344, 0.213, 0.116,
0.116, 0.092, 0.070, 0.059, 0.048, 0.036, 0.029, 0.021, 0.014, 0.011, 0.008, 0.006)\\

\noindent P3:= (0.853, 0.759, 0.874, 0.800, 0.716, 0.557, 0.503, 0.399, 0.334, 0.207, 0.118,
0.118, 0.097, 0.078, 0.067, 0.056, 0.044, 0.036, 0.026, .019, 0.014, 0.010)\\

\noindent Mazumdar and Gaver (1984) comparison of the mentioned algorithms for estimating the unit capacity factor. For a population level comparison one may refer to Genc (2013) where the authors fitted Topp-Leone model to both SC16 and P3 datasets for further analysis. It is to be noted that to improve the population level comparison, one need to select a better suited model. One parameter distributions on the unit interval is rare and the Topp-Leone distribution is a benchmark in this category. The introduced LCG distribution is a new addition to this category and hence a real life data driven comparison with Topp-Leone is sufficient to justify its utility. Topp-Leone model fits the datasets well as can be seen in Genc (2013). However, we compare fitting of the Topp-Leone model with that of the LCG in Table 2  and find that the proposed LCG model is far better as indicated by the corresponding AIC values.
\begin{center}
\begin{table}[h!]
\centering
\vspace{1cm}
	\caption{Goodness of fit.}
	\label{tab:simul}
	\begin{tabular}{ccccc}
		\hline\\
		Dataset & Distribution & MLE & Log-likelihood & AIC  \\
		\\\hline
		\multirow{2}{*}{SC16}  & Topp-Leone & 0.5943 & -11.3660 & 25.7837   \\
		 & LCG & 1.9876 & 2.9459 & -3.8981 \\ \hline
		\multirow{2}{*}{P3} & Topp-Leone & 0.6778 & -10.9016 & 23.8032    \\

		  & LCG & 1.8646 &2.3009 &-2.6098\\ \hline
	\end{tabular}%
\end{table}
\end{center}

\section{Discussion}
A number of new distributions with simple yet flexible structures are derived in the current work. Throughout the article we discussed about the distribution of $U=X|X+Y=1$. It is obvious that in many situations the distribution of $V=Y|X+Y=1$ is different from $U$ and hence may be of interest fom new distribution perspective. However, we refrain from detailing such density function since the same can be done by simply finding density function of $V=1-U$. We envisage that the proposed method and the derived distributions will be a useful addition in the existing literature.  In fact the LCG is shown to be a better one parameter model than the benchmark Topp-Leone distribution. Since the LCG model possesses simple structure of the mean, modelling unit responses with covariates using LCG is of interest. Moreover Section 4.2 hints a productive study on the stress-strength reliability of the LCG distribution. These problems are under consideration and will be reported as independent works. Detailed investigation on the other derived distributions are required and further research on this is warranted. This article is intended to motivate researchers to delve into the possibilities of exploring new distributions with different choices of the joint distributions of $(X,Y)$. The proposed method can naturally be extended. One may consider the random vector $(X_1, X_2, ..., X_n)$ with $P(X_i>0)=$ for $i=1, 2, ..., n$. For $k<n$, $U=X_1+X_2+...+X_k|X_1+X_2+...+X_n=1$  is a random variable with support $(0,1)$. In our case $n=2$ and $k=1$.

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\section*{Appendix: R-codes}
\subsubsection*{C1: Random sample generation from LCG distribution.}
\begin{lstlisting}[language=R]
rGL=function(n,beta)
{
sample=0
for(i in 1:n)
{
v=runif(1)
f=function(u)
{
(1-u)^beta*(2+beta+u*beta)-(1-v)*(2+beta)
}
proc=uniroot(f,c(0,1))
sample[i]=proc$root
}
return(sample)
}
\end{lstlisting}

\subsubsection*{C2: Method of moment estimation for LCG distribution.}
\begin{lstlisting}[language=R]
MME=function(data)
{
ubar=mean(data)
z=c(4*(ubar-1),4*ubar-1,ubar)
return(max(Re(polyroot(z))))
}
\end{lstlisting}

\subsubsection*{C2: Maximum likelihood estimation for LCG distribution.}
\begin{lstlisting}[language=R]
MLE=function(data)
{
Tu=-mean(log(1-data))
z=c(2*Tu-4,3*Tu-1,Tu)
return(max(Re(polyroot(z))))
}
\end{lstlisting}
\end{document}


\section{Introduction} \label{Sec1}

Every sports tournament has to provide appropriate incentives for the contestants to exert effort \citep{Szymanski2003}. In particular, the ranking method should not reward teams for poor performance \citep{VaziriDabadghaoYihMorin2018}. However, there are a number of reasons why a team may consider \emph{tanking} (deliberately losing) in a competition: \citet{KendallLenten2017} identify several examples when the misaligned rules had such unforeseen consequences.

Unsurprisingly, academic scholars have studied various theoretical models of sports contests in view of \emph{incentive incompatibility}.
\citet{Pauly2014} derives an impossibility theorem for championships consisting of two qualifying tournaments with disjoint sets of participants. \citet{Vong2017} proves that, if more than one contestants advance to the next round, some players can benefit from shirking to qualify with a lower rank.
\citet{DagaevSonin2018} investigate tournament systems, composed of multiple round-robin and knockout tournaments with the same set of participants when the sets of winners of noncumulative prizes have a nonempty intersection. \citet{Csato2020f} considers group-based qualification systems where teams from different groups are compared, which can create incentives for both teams to play a draw instead of winning \citep{Csato2020d}. \citet{Csato2021a} presents how the ignorance of these theoretical findings has lead to problems in European football.
\citet{KrumerMegidishSela2020b} show that strategic considerations may motivate a contestant to lose in a round-robin tournament because this can result in a higher expected payoff.

Although the round-robin format in which each team meets all the others is one of the most common sports tournaments, it requires a lot of time. On the other hand, if the competitors can play only against a limited number of opponents, the set of matches should be chosen carefully. This can be achieved through \emph{seeding}, by ordering the entrants based on playing history and/or the judgement of experts to pair them according to their ranks.
The problem of seeding in knockout tournaments has been thoroughly explored, see e.g.\ \citet{Hwang1982, HorenRiezman1985, Schwenk2000, GrohMoldovanuSelaSunde2012, DagaevSuzdaltsev2018, ArlegiDimitrov2020, DellaCroceDragottoScatamacchia2020}. The seeding rules of the most prominent football competition, the FIFA World Cup have also got serious attention \citep{ScarfYusof2011, Guyon2015a, LalienaLopez2019, CeaDuranGuajardoSureSiebertZamorano2020}. Similarly, several statistical papers have analysed the effect of seeding on tournament outcome \citep{MonksHusch2009, CoronaForrestTenaWiper2019, DagaevRudyak2019, EngistMerkusSchafmeister2021}.

However, the previous literature has scarcely addressed the incentive compatibility of the seeding rules except for \citet{Csato2020c}, a paper revealing a unique shortcoming in the UEFA Champions League group stage draw that emerged only in the 2015/16 season due to a misaligned way of filling vacant slots.
Our main contribution resides in a more universal result: traditional seeding systems based on exogenous measures of teams' strengths are generically incentive incompatible, but they can easily be made strategyproof.

Our roadmap is as follows.
Section~\ref{Sec2} presents two real-world cases to highlight the issue. A mathematical model is given in Section~\ref{Sec3}. We provide a strategyproof seeding mechanism and summarise policy implications in Section~\ref{Sec4}. Finally, Section~\ref{Sec6} concludes.

\section{Case studies from the real world} \label{Sec2}

Let us see two motivating examples.

\begin{example} \label{Examp1}
Assume the following hypothetical modifications to real-world results in the \href{https://en.wikipedia.org/wiki/2018_FIFA_World_Cup_qualification}{2018 FIFA World Cup qualification}:
\begin{itemize}
\item
Wales vs.\ Republic of Ireland was 1-1 (instead of 0-1) on 9 October 2017 in \href{https://en.wikipedia.org/wiki/2018_FIFA_World_Cup_qualification_\%E2\%80\%93_UEFA_Group_D}{UEFA Group D}. Consequently, Wales would have been 18 and the Republic of Ireland would have been 17 points in that group, thus Wales would have advanced to the \href{https://en.wikipedia.org/wiki/2018_FIFA_World_Cup_qualification_\%E2\%80\%93_UEFA_Second_Round}{UEFA Second Round}. There Wales would have been in Pot 1 rather than Denmark, therefore the tie Wales vs.\ Denmark would have been possible (in fact, Denmark played against the Republic of Ireland). Suppose that Wales qualified for the World Cup instead of Denmark.
\item
The first leg of Sweden vs.\ Italy was 1-1 (instead of 1-0) on 10 November 2017 in the UEFA Second Round, hence Italy qualified for the World Cup.
\end{itemize}
In the \href{https://en.wikipedia.org/wiki/2018_FIFA_World_Cup_seeding}{draw for the 2018 FIFA World Cup}, the composition of the pots depended on the \href{https://www.fifa.com/fifa-world-ranking/ranking-table/men/rank/id11976/}{October 2017 FIFA World Ranking}. The only exception was the automatic assignment of the host---Russia---to Pot 1 besides the seven highest-ranked qualified teams.
Hence Uruguay ($17$th in the relevant FIFA ranking) would have been drawn from Pot 3 as among the best $16$ teams, only Chile would have not qualified in the above scenario (Wales was the $14$th and Italy was the $15$th in the FIFA World Ranking of October 2017).

\begin{table}[t!]
  \centering
  \caption{Pot composition in the hypothetical 2018 FIFA World Cup}
  \label{Table1}
\rowcolors{3}{}{gray!20}
\begin{threeparttable}
    \begin{tabularx}{\textwidth}{LLLL} \toprule
    Pot 1 & Pot 2 & Pot 3 & Pot 4 \\ \bottomrule
    Russia (65) & Spain (8) & \textbf{Uruguay (17)} & Serbia (38) \\
    Germany (1) & \textbf{Peru (10)} & Iceland (21) & Nigeria (41) \\
    Brazil (2) & Switzerland (11) & Costa Rica (22) & Australia (43) \\
    Portugal (3) & England (12) & Sweden (25) & Japan (44) \\
    Argentina (4) & Colombia (13) & Tunisia (28) & Morocco (48) \\
    Belgium (5) & \emph{Wales (14)} & Egypt (30) & Panama (49) \\
    Poland (6) & \emph{Italy (15)} & Senegal (32) & South Korea (62) \\
    France (7) & Mexico (16) & Iran (34) & Saudi Arabia (63) \\ \bottomrule
    \end{tabularx}

\begin{tablenotes} \footnotesize
	\item The pots are determined by the FIFA World Ranking of October 2017, see the numbers in parenthesis.
	\item Russia is the top seed as host.
	\item Teams written in \emph{italics} qualified only in the hypothetical but feasible scenario of Example~\ref{Examp1}.
	\item Uruguay (17)---the top team in Pot 3---would have been drawn from Pot 2 due to losing against Paraguay (34) in the South American qualifiers of the 2018 FIFA World Cup since then either Paraguay (34) or New Zealand (122) would have qualified for the World Cup instead of Peru (10). The national teams affected by this tanking are written in \textbf{bold}.
\end{tablenotes}
\end{threeparttable}
\end{table}

The allocation of the teams in the above scenario is given in Table~\ref{Table1}.
Consider what would have happened if the result of the match Paraguay ($34$) vs.\ Uruguay, played on 5 September 2017 in the \href{https://en.wikipedia.org/wiki/2018_FIFA_World_Cup_qualification_(CONMEBOL)}{South American qualifiers}, would have been 2-1 instead of 1-2. Then Uruguay would have remained second and Paraguay would have been fifth in this qualifying competition. Paraguay would have played against New Zealand ($122$) in the \href{https://en.wikipedia.org/wiki/2018_FIFA_World_Cup_qualification_(OFC\%E2\%80\%93CONMEBOL_play-off)}{OFC--CONMEBOL qualification play-off}, thus Peru ($10$) could not have qualified for the World Cup. Therefore, Uruguay would have been drawn from the stronger Pot 2 instead of Pot 3 merely due to its loss against Paraguay.
It probably means a substantial advantage: in the 2018 FIFA World Cup, seven and two teams advanced to the knockout stage from Pots 2 and 3, respectively.
\end{example}

Example~\ref{Examp1} contains a small sloppiness since we have not checked whether the October 2017 FIFA World Ranking would have been modified if the result of Paraguay vs.\ Uruguay would have changed. However, this minor issue does not affect the potential case of incentive incompatibility.

\begin{example} \label{Examp2}
In the \href{https://en.wikipedia.org/wiki/2020\%E2\%80\%9321_UEFA_Europa_League_group_stage#Draw}{draw for the 2020/21 UEFA Europa League group stage}, the composition of the pots was determined by the 2020 UEFA club coefficients, available at \url{https://kassiesa.net/uefa/data/method5/trank2020.html}.
Assume the following hypothetical modifications to real-world results in the \href{https://en.wikipedia.org/wiki/2020\%E2\%80\%9321_UEFA_Europa_League_qualifying_phase_and_play-off_round\#Play-off_round}{play-off round of the qualifying phase}, with the first favourite team advancing to the group stage in place of the second unseeded underdog (the UEFA club coefficients are given in parenthesis):
\begin{itemize}
\item
Viktoria Plze{\v n} ($34.0$) against Hapoel Be'er Sheva ($14.0$);
\item
Basel ($58.5$) against CSKA Sofia ($4.0$);
\item
Sporting CP ($50.0$) against LASK ($14.0$);
\item
Copenhagen ($42.0$) against Rijeka ($11.0$);
\item
VfL Wolfsburg ($36.0$) against AEK Athens ($16.5$).
\end{itemize}
There were $48$ teams in the group stage. Leicester City ($22.0$) was ranked $20$th because Rapid Wien ($22.0$) had the same 2020 UEFA club coefficient but the tie-breaking criterion---coefficient in the next most recent season in which they are not equal \citep[Annex~D.8]{UEFA2020b}---preferred the latter club. Due to the above changes, five teams with a higher coefficient than Leicester City would have qualified instead of five teams with a lower coefficient. Hence, Leicester City would have been only the $25$th highest-ranked, namely, the best club in Pot 3 as each of the four pot contain $12$ clubs.
In addition, suppose that Leicester defeated Norwich City at home by 2-1 (instead of 0-0) in the \href{https://en.wikipedia.org/wiki/2019\%E2\%80\%9320_Premier_League}{2019/20 English Premier League}. Then Leicester City would have remained fifth at the end of the season with $64$ points.

\begin{table}[t!]
  \centering
  \caption{Pot composition in the hypothetical 2020/21 UEFA Europa League}
  \label{Table2}
\rowcolors{3}{}{gray!20}
    \begin{tabularx}{1\textwidth}{LL} \toprule
    Pot 1 & Pot 2 \\ \bottomrule
    Arsenal (91.0) & Gent (39.5) \\
    \textbf{Tottenham Hotspur (85.0)} & PSV Eindhoven (37.0) \\
    Roma (80.0) & \emph{VfL Wolfsburg (36.0)} \\
    Napoli (77.0) & Celtic (34.0) \\
    Benfica (70.0) & \emph{Viktoria Plze{\v n} (34.0)} \\
    Bayer Leverkusen (61.0) & Dinamo Zagreb (33.5) \\
    \emph{Basel (58.5)} & Sparta Prague (30.5) \\
    Villarreal (56.0) & Slavia Prague (27.5) \\
    \emph{Sporting CP (50.0)} & Ludogorets Razgrad (26.0) \\
    CSKA Moscow (44.0) & Young Boys (25.5) \\
    \emph{Copenhagen (42.0)} & Crvena Zvezda (22.75) \\
    Braga (41.0) & Rapid Wien (22.0) \\ \bottomrule
    \end{tabularx}

\vspace{0.25cm}
\begin{threeparttable}
\rowcolors{3}{}{gray!20}
    \begin{tabularx}{1\textwidth}{LL} \toprule
    Pot 3 & Pot 4 \\ \bottomrule
    \textbf{Leicester City (22.0)} & 1899 Hoffenheim (14.956) \\
    PAOK (21.0) & CFR Cluj (12.5) \\
    Qaraba{\u g} (21.0) & Zorya Luhansk (12.5) \\
    Standard Li{\` e}ge (20.5) & Nice (11.849) \\
    Real Sociedad (20.476) & Lille (11.849) \\
    Granada (20.456) & Dundalk (8.5) \\
    Milan (19.0) & Slovan Liberec (8.0) \\
    AZ Alkmaar (18.5) & Antwerp (7.58) \\
    Feyenoord (17.0) & Lech Poznan (7.0) \\
    Maccabi Tel Aviv (16.5) & Sivasspor (6.72) \\
    Rangers (16.25) & Wolfsberger AC (6.585) \\
    Molde (15.0) & Omonia (5.35) \\ \bottomrule
    \end{tabularx}

\begin{tablenotes} \footnotesize
	\item The pots are determined by the 2020 UEFA club coefficients, shown in parenthesis.
	\item Teams written in \emph{italics} qualified only in the hypothetical but feasible scenario of Example~\ref{Examp2}.
	\item Leicester City (22.0)---the top team in Pot 3---could have been drawn from Pot 2 due to losing against Wolverhampton (18.092) in the 2019/20 English Premier League since then the latter team could have qualified for the UEFA Europa League group stage instead of Tottenham Hotspur (85.0). The clubs affected by this tanking are written in \textbf{bold}.
\end{tablenotes}
\end{threeparttable}
\end{table}

The allocation of the clubs in the above scenario is presented in Table~\ref{Table2}.
Consider what would have happened if the outcome of the match Wolverhampton Wanderers ($18.092$) vs.\ Leicester City, played on 14 February 2020 in the 2019/20 Premier League, would have been 1-0 instead of 0-0. Leicester would have remained fifth with $62$ points, while Wolverhampton would have been sixth with $61$ points rather than Tottenham Hotspur ($85.0$), which scored $59$ points. Consequently, Wolverhampton would have entered the Europa League qualification in the second qualifying round, and it could have qualified for the group stage in the place of Tottenham. Then Leicester would have been drawn from the stronger Pot 2 merely due to its loss against Wolverhampton.
This probably means an advantage, although in the 2020/21 Europa League, six teams advanced to the knockout stage from both Pots 2 and 3, respectively.
\end{example}

\section{Theoretical background} \label{Sec3}

Consider a round-robin qualifying tournament with a set of teams $T$, where each team $t \in T$ has a coefficient $\xi_t$. The teams ranked between the $p$th and $q$th ($p \leq q$) qualify for the second stage. A qualified team $t$ has a seeding value $\Psi_t$.
Any team prefers if more teams play in the second round with a lower seeding value.
In other words, it is a common belief that these measures positively correlate with true abilities. All coefficients used in practice are constructed along this line. For instance, the \href{https://en.wikipedia.org/wiki/FIFA_World_Rankings}{FIFA World Ranking} and the \href{https://en.wikipedia.org/wiki/UEFA_coefficient}{UEFA coefficients} for national teams, countries, and clubs alike award more points for wins (draws) than for draws (losses), thus better achievements in the past translate into a higher value.

Therefore, the first goal for every team is to qualify and the second goal is to qualify together with teams having a lower coefficient.
Any team $t \in T$ may tank to improve the second objective without deteriorating the first. Denote the (strict) rankings of the qualifying tournament by $\succ$ and $\succ'$, as well as the sets of teams qualified by
\[
Q = \{ s \in T: p-1 \leq |r \in T: r \succ s| \leq q-1 \} \text{ and}
\]
\[
Q' = \{ s \in T: p-1 \leq |r \in T: r \succ' s| \leq q-1 \}
\]
before and after tanking, respectively.

\begin{definition} \label{Def1}
\emph{Incentive incompatibility with respect to seeding}:
A round-robin qualifying tournament is said to be \emph{incentive incompatible with respect to seeding} if there exists a team $t \in T$ with a tanking strategy such that:
\begin{itemize}
\item
team $t$ qualifies for the second stage both before and after tanking, that is, $t \in Q$ and $t \in Q'$;
\item
team $t$ has a better seeding position after tanking than before tanking, namely, $|s \in Q': \Psi_s > \Psi_t| < |s \in Q: \Psi_s > \Psi_t|$.
\end{itemize}
Otherwise, the qualifying tournament is called \emph{incentive compatible with respect to seeding}.
\end{definition}

As Section~\ref{Sec2} reveals, a qualifying tournament is incentive incompatible if $\Psi_t = \xi_t$ for all $t \in Q$ and $2 \leq |Q| < |T|$. In particular, a situation may exist where team $i$ has already secured qualification, while teams $j$ and $k$ compete for another slot such that $\xi_k > \xi_i > \xi_j$. Then team $i$ may consider losing against team $j$ in order to push it to the next stage at the expense of team $k$ as team $i$ can get a better seeding pot by taking team $j$ to the second round instead of team $k$.

The result below provides sufficient conditions to prevent a strategic manipulation of this type. For the sake of simplicity, we assume that ties in the seeding values of qualified teams are broken in favour of the team ranked higher in the qualifying tournament.

\begin{proposition} \label{Prop1}
A round-robin qualifying tournament is incentive incompatible with respect to seeding if at least one of the following conditions hold:
\begin{itemize}
\item
Only one team is allowed to qualify: $p = q$;
\item
All teams qualify: $p = 1$ and $q = |T|$;
\item
The seeding value of each qualified team $t \in Q$ in the second stage is equal to the maximal coefficient of the teams that are ranked lower than team $t$ in the qualifying round-robin tournament: $\Psi_t = \max \{ \xi_s: t \succ s \}$ for all $t \in Q$.
\end{itemize}
\end{proposition}

\begin{proof}
If $p = 1$, then $t \in Q$ leads to $|s \in Q: \Psi_s > \Psi_t| = 0$. Consequently, no tanking strategy can satisfy Definition~\ref{Def1}. \\
$p = 1$ and $q = |T|$ result in $Q = T$, thus $|s \in Q': \Psi_s > \Psi_t| = |s \in Q: \Psi_s > \Psi_t|$, which excludes the existence of a tanking strategy satisfying Definition~\ref{Def1}. \\
$\Psi_t = \max \{ \xi_s: t \succ s \}$ for all $t \in Q$ implies that
\[
s \in Q \text{ and } \Psi_s > \Psi_t \iff s \in Q \text{ and } s \succ t.
\]
Since team $t$ cannot be ranked higher in the round-robin qualifying tournament after tanking,
\[
|s \in Q': \Psi_s > \Psi_t| = |s \in Q': s \succ t| \geq |s \in Q: s \succ t| = |s \in Q: \Psi_s > \Psi_t|
\]
holds. Hence, the qualifying tournament is incentive compatible with respect to seeding.
\end{proof}

\section{Discussion} \label{Sec4}

Now we present a general procedure to guarantee our requirement, incentive compatibility with respect to seeding, on the basis of the theoretical model. Some alternative ideas are also outlined shortly.

According to Proposition~\ref{Prop1}, there are three ways to achieve strategyproofness in a round-robin qualifying tournament. However, the first two conditions---when exactly one team qualifies or all teams qualify for the next round---could not offer a universal rule. Nonetheless, they can be exploited in certain cases, for example, only one team advanced from the \href{https://en.wikipedia.org/wiki/2018_FIFA_World_Cup_qualification_(OFC)}{Oceanian (OFC) section of the 2018 FIFA World Cup qualification}.
Fortunately, there is a third opportunity, that is, to calculate the seeding value of any qualified team $t \in Q$ as $\Psi_t = \max \{ \xi_s: t \succ s \}$. In other words, team $t$ is seeded in the second stage based on the maximum of coefficients $\xi_s$ of all teams $s$ ranked lower than team $t$ in its round-robin qualifying competition. This is a reasonable rule: if team $i$ finishes ahead of team $j$ in a league, why is it judged worse for the draw in the next round?
Our proposal is called \emph{strategyproof seeding}.

\begin{sidewaystable
  \centering
  \caption{Alternative rules for the draw of the UEFA Champions League group stage in the 2020/21 season}
  \label{Table3}
  \renewcommand\arraystretch{0.8}
\begin{threeparttable}
\rowcolors{1}{gray!20}{}
    \begin{tabularx}{\textwidth}{lccCCcCCc} \toprule \hiderowcolors
    Club  & Country & Position & \multicolumn{2}{c}{Coefficient}      & Inherited from & \multicolumn{3}{c}{Pot allocation}  \\
          &       &       & Official & Proposed &       & Official & Proposed & Change \\ \bottomrule \showrowcolors
    Bayern Munich & \multicolumn{2}{c}{CL TH (Germany 1st)} & 136   & 136   & ---   & 1 (1) & 1 (1) & --- \\
    Sevilla & \multicolumn{2}{c}{EL TH (Spain 4th)} & 102   & 102   & ---   & 1 (1) & 1 (1) & --- \\
    Real Madrid & Spain & 1st   & 134   & 134   & ---   & 1 (1) & 1 (1) & --- \\
    Liverpool & England & 1st   & 99    & 116   & Manchester City (2nd) & 1 (1) & 1 (1) & --- \\
    Juventus & Italy & 1st   & 117   & 117   & ---   & 1 (1) & 1 (1) & --- \\
    Paris Saint-Germain & France & 1st   & 113   & 113   & ---   & 1 (1) & 1 (1) & --- \\
    Zenit Saint Petersburg & Russia & 1st   & 64    & 64    & ---   & 1 (1) & 1 (1) & --- \\
    Porto & Portugal & 1st   & 75    & 75    & ---   & 1 (2) & 1 (3) & --- \\
    Barcelona & Spain & 2nd   & 128   & 128   & ---   & 2 (2) & 2 (2) & --- \\
    Atl\'etico Madrid & Spain & 3rd   & 127   & 127   & ---   & 2 (2) & 2 (2) & --- \\
    Manchester City & England & 2nd   & 116   & 116   & ---   & 2 (2) & 2 (2) & --- \\
    Manchester United & England & 3rd   & 100   & 100   & ---   & 2 (2) & 2 (2) & --- \\
    Shakhtar Donetsk & Ukraine & 1st   & 85    & 85    & ---   & 2 (2) & 2 (2) & --- \\
    Borussia Dortmund & Germany & 2nd   & 85    & 85    & ---   & 2 (1) & 2 (1) & --- \\
    Chelsea & England & 4th   & 83    & 91    & Arsenal (8th) & 2 (2) & 2 (2) & --- \\
    Ajax  & Netherlands & 1st   & 69.5  & 69.5  & ---   & 2 (2) & 3 (3) & \textcolor{BrickRed}{\rotatebox[origin=c]{270}{\ding{212}}} \\
    Dynamo Kyiv & Ukraine & 2nd   & 55    & 55    & ---   & 3 (3) & 3 (3) & --- \\
    Red Bull Salzburg & Austria & 1st   & 53.5  & 53.5  & ---   & 3 (3) & 4 (4) & \textcolor{BrickRed}{\rotatebox[origin=c]{270}{\ding{212}}} \\
    RB Leipzig & Germany & 3rd   & 49    & 61    & Bayer Leverkusen (5th) & 3 (3) & 3 (3) & --- \\
    Inter Milan & Italy & 2nd   & 44    & 80    & Roma (5th) & 3 (3) & 3 (3) & --- \\
    Olympiacos & Greece & 1st   & 43    & 43    & ---   & 3 (3) & 4 (4) & \textcolor{BrickRed}{\rotatebox[origin=c]{270}{\ding{212}}} \\
    Lazio & Italy & 4th   & 41    & 80    & Roma (5th) & 3 (3) & 3 (3) & --- \\
    Krasnodar & Russia & 3rd   & 35.5  & 44    & CSKA Moscow (4th) & 3 (3) & 4 (4) & \textcolor{BrickRed}{\rotatebox[origin=c]{270}{\ding{212}}} \\
    Atalanta & Italy & 3rd   & 33.5  & 80    & Roma (5th) & 3 (3) & 3 (3) & --- \\
    Lokomotiv Moscow & Russia & 2nd   & 33    & 44    & CSKA Moscow (4th) & 4 (4) & 4 (4) & --- \\
    Marseille & France & 2nd   & 31    & 83    & Lyon (7th) & 4 (4) & 2 (2) & \textcolor{PineGreen}{\rotatebox[origin=c]{90}{\ding{212}}} \textcolor{PineGreen}{\rotatebox[origin=c]{90}{\ding{212}}} \\
    Club Brugge & Belgium & 1st   & 28.5  & 39.5  & Gent (2nd) & 4 (4) & 4 (4) & --- \\
    Borussia M\"onchengladbach & Germany & 4th   & 26    & 61    & Bayer Leverkusen (5th) & 4 (4) & 3 (3) & \textcolor{PineGreen}{\rotatebox[origin=c]{90}{\ding{212}}} \\
    Istanbul Ba{\c s}ak{\c s}ehir & Turkey & 1st   & 21.5  & 54    & Be{\c s}ikta{\c s} (3rd) & 4 (4) & 4 (4) & --- \\
    Midtjylland & Denmark & 1st   & 14.5  & 42    & Copenhagen (2nd) & 4 (4) & 4 (4) & --- \\
    Rennes & France & 3rd   & 14    & 83    & Lyon (7th) & 4 (4) & 3 (2) & \textcolor{PineGreen}{\rotatebox[origin=c]{90}{\ding{212}}} \\
    Ferencv\'aros & Hungary & 1st   & 9     & 10.5  & Feh\'erv\'ar (2nd) & 4 (4) & 4 (4) & --- \\
    \bottomrule
    \end{tabularx}

\begin{tablenotes} \footnotesize
	\item CL (EL) TH stands for the UEFA Champions League (Europa League) titleholder.
	\item The column ``Inherited from'' shows the club of the domestic league whose UEFA club coefficient is taken over.
	\item Proposed pot is the pot that contains the club if the current seeding policy applies to Pot 1. Since this rule is incentive compatible \citep{Csato2020c}, the pot according to the amendment suggested by \citet[Section~5]{Csato2020c} is reported in parenthesis for both the official and the strategyproof seeding systems.
	\item The column ``Change'' shows the movements of clubs between the pots due to the strategyproof seeding if the current seeding regime applies to Pot 1.
\end{tablenotes}
\end{threeparttable}
\end{sidewaystable}

Table~\ref{Table3} applies strategyproof seeding for the 2020/21 Champions League group stage. Even though the club coefficients of $15$ teams, including the $11$ lowest-ranked, are increased, it has only a moderated effect on the composition of pots as one German and two French teams benefit at the expense of four teams from the Netherlands, Austria, Greece, and Russia. That amendment usually favours the highest-ranked associations, where some clubs emerging without a robust European record (recall the unlikely triumph of Leicester City in the 2015/16 English Premier League \citep{BBC2016}) can ``obtain'' the performances of clubs with considerable achievements at the international level. Thus strategyproof seeding contributes to the success of underdogs in the European cups, which may be advantageous for the long-run competitive balance in the top leagues. In addition, it probably better reflects the true abilities of the teams since playing more matches reduces the role of luck in sports tournaments \citep{McGarrySchutz1997, ScarfYusofBilbao2009, LasekGagolewski2018, Csato2021b, CsatoBiroSziklai2021}. Consequently, it is more difficult to perform better in a round-robin league than in the Champions League or Europa League.

From the 2018/19 season onwards, UEFA club coefficients are determined either as the sum of all points won in the previous five years or as the association coefficient over the same period, \emph{whichever is the higher} \citep{Kassies2021, UEFA2021}. While this rule was not effective in the 2020/21 Champions League, the lower bound applied in the case of some Spanish, German, and French teams in the 2020/21 Europa League.
A somewhat similar policy is used in the UEFA Champions League and Europa League qualification, too, if a later round is drawn before the identity of the teams is known: ``\emph{If, for any reason, any of the participants in such rounds are not known at the time of the draw, the coefficient of the club with the higher coefficient of the two clubs involved in an undecided tie is used for the purposes of the draw.}'' \citep[Article~13.03]{UEFA2020a}.
Therefore, the principle of strategyproof seeding is not unknown in UEFA club competitions, which can support its implementation.

Table~\ref{Table3} reinforces that the strategyproof seeding system may result in more ties than the current definition. If some teams inherit their seeding values from the same lower-ranked team, then these remain identical, and the tie should be broken by drawing of lots \citep[Annex~D.8]{UEFA2020b}. Although tie-breaking does not affect incentive compatibility, it is reasonable to prefer the teams ranked higher in the domestic league. If clubs from other associations also have the same coefficient (which has a much lower probability), they can be assigned arbitrarily in this equivalence class. Alternatively, the original club coefficients can be used for tie-breaking.

Our incentive compatible mechanism has further favourable implications. UEFA has modified the pot allocation policy in the Champions League from the 2015/16 season, probably inspired by the previous year when Manchester City, the English champion, was drawn from the second pot but Arsenal, the fourth-placed team in England, was drawn from the first pot. This decision---intended to strengthen the position of domestic titleholders \citep{UEFA2015e}---has considerable sporting effects \citep{CoronaForrestTenaWiper2019, DagaevRudyak2019}, especially since the poor way of filling vacancies leads to incentive incompatibility \citep{Csato2020c}. On the other hand, the proposed seeding rule guarantees that a national champion has at least the same seeding value as any team ranked lower in its domestic league.

Naturally, other strategyproof seeding policies can be devised. One example is the system of the 2020 UEFA European Championship: the ranking of all entrants on the basis of their results in the qualification. However, that principle is not appropriate if the achievements in the qualifying tournament(s) cannot be compared.
Another solution might be to associate seeding positions not with the coefficients but with the path of qualification. For instance, a club can be identified in the UEFA Champions League as the Spanish runner-up rather than by its name. The results of these ``labels'' can be measured by the achievements of the corresponding teams \citep{Guyon2015b}. Nonetheless, this principle seems to be difficult to apply for the UEFA European Championship.

To conclude, the recommended strategyproof seeding mechanism provides incentive compatibility in any setting. While other rules are also able to eliminate perverse incentives, they are unlikely to be independent of the particular characteristics of the tournament.

\section{Conclusions} \label{Sec6}

The present work has analysed a mathematical model of seeding for sports tournaments where the teams qualify from round-robin competitions. Several contests are designed this way, including the most prestigious football tournaments (FIFA World Cup, UEFA European Championship, UEFA Champions League). The sufficient conditions of incentive incompatibility have turned out to be quite restrictive: if each competitor is considered with its own coefficient (usually a measure of its past performance), only one or all of them should qualify from every round-robin contest.

Similarly to the main findings of \citet{Vong2017} and \citet{KrumerMegidishSela2020b}, our result has the flavour of an impossibility theorem at first glance. However, here we can achieve strategyproofness by giving to each qualified competitor the highest coefficient of all competitors that are ranked lower in its round-robin qualifying tournament for seeding purposes.

The central message of our paper for decision makers is consonant with the conclusion of \citet{HaugenKrumer2021}, that is, tournament design should be included into the family of traditional topics discussed by sports management. In particular, administrators are strongly encouraged to follow our recommendation in order to prevent the occurrence of costly scandals in the future.

\section*{Acknowledgements}
\addcontentsline{toc}{section}{Acknowledgements}
\noindent
Three anonymous reviewers provided valuable comments on an earlier draft. \\
We are indebted to the \href{https://en.wikipedia.org/wiki/Wikipedia_community}{Wikipedia community} for collecting and structuring valuable information on the sports tournaments discussed. \\
The research was supported by the MTA Premium Postdoctoral Research Program grant PPD2019-9/2019.

\bibliographystyle{apalike}


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

With the rapid development of speech recognition and natural language processing, voice user interface (VUI) has become a fundamental use case for today's smart devices (e.g., smartphone, smartwatch, laptop, smart speaker, and smart appliance). However, voice interaction suffers from several limitations that severely hinder its usage in daily life. First, audible speech is not suitable in some scenarios, such as in a meeting or when someone is sleeping. Second, environmental noise, like traffic noise, industrial machinery noise, and speech from bystanders, can make speech recognition challenging or even impossible. Third, people are unlikely to use voice input in public areas due to its risk of privacy leakage.

Recent advances in silent speech recognition have opened up new possibilities to counterbalance the above limitations. Some methods are based on computer vision technology \cite{zhou2014review} to capture the visual features of lip movements. The adoption of deep learning substantially boosts the precision of vision-based speech recognition \cite{assael2016lipnet,chung2017lip}. However, these methods are highly sensitive to lighting conditions, which means they cannot work in dark environments. Some other works exploit a variety of face-worn sensors for speech sensing, such as EMG electrodes \cite{wand2016deep,wand2018domain,kapur2018alterego}, RFID tags \cite{wang2019rfid}, bone-conduction vibration sensors \cite{maruri2018v}. While a significant drawback of these works is that the skin-attached sensors are very invasive to users. Besides, the reattachment of the sensors may cause changes in the recorded signal, which will significantly degrade their performance \cite{wand2018domain}.

\begin{figure}{}
	\centering
	\includegraphics[width=3in]{lipreading2.pdf}
	\caption{Sensing lip movements with acoustic signals generated by smart devices.}
	\label{lipreading2}
\end{figure}
{}
Instead of the aforementioned approaches, another trend of works utilizes ultrasound. The use of high-frequency ultrasound (MHz level) has a long history in medical voice interface research \cite{Sonies1981}, aiming to provide an alternative to electrolarynx for some patients who have lost their voicebox. In recent research, most of the works use ultrasound to build real-time 2D tongue images \cite{denby2004,hueber2011,Hakoun2016}. However, the method using high-frequency ultrasound requires a special ultrasonic-imaging device that is not convenient for daily uses. Therefore, researchers also developed some applications based on low-frequency (LF) ultrasound. Inspired by gesture recognition \cite{gupta2012soundwave,ruan2016audiogest}, they employed LF ultrasound to detect lip movements \cite{zhang2017hearing,tan2017silenttalk,lu2019lip}, instead of creating tongue images. In these works, acoustic sensing systems for simple lip-reading mainly use the Doppler shift of the received signal. Nevertheless, due to limited frequency precision, the Doppler shift can only provide coarse-grained estimation\cite{wang2016device}, which is not suitable to capture subtle lip movements.

In this paper, we put forward a non-invasive silent speech interface, using LF ultrasound with some critical modifications. Our contributions focus on the followings:

(1). Propose an end-to-end silent speech interface for continuous recognition using acoustic signals, which is completely non-invasive and needs no extra hardware modification except existing smart devices.

(2). Leverage the phase information of the received signals, instead of Doppler shift, to obtain the fine-grained estimation of lip movements, and carefully design the signal preprocessing pipeline.

(3). Employ CNNs to extract representative features, and use the attention-based encoder-decoder network to enable end-to-end recognition as well as learn the underlying language model.

Our method can be deployed on existing smart devices, exploiting speaker and microphone for lip-reading. People don't need to wear any sensors, but only need to move the devices near their mouths. As Figure~\ref{lipreading2} depicts, when people speak to the devices, our system leverages the speaker to emit the inaudible signal and the microphone to listen to the signal reflected by moving lips. Then the system analyses the reflected signal to recognize speech.

\begin{figure*}
		\centering
		\subfloat[The Doppler shift]{\includegraphics[width=2.3in]{doppler}}
		\hspace{10pt}
		\subfloat[Phase]{\includegraphics[width=2.1in]{pham}}
		\hspace{10pt}
		\subfloat[Phase delta]{\includegraphics[width=2.1in]{signal_diff}}
		\caption{Different acoustic signals of a lip movement from 1.2s to 2.5s when speaking the word ``WiFi". (a) shows the STFT result of the Doppler shift, where we can hardly observe signals after 1.8s. (b) and (c) shows the phase and phase delta signals, respectively. We demonstrate two frequency channels with $k=0,6$. It is clear that the phase delta performs the best.}
		\label{doppler_pham}
\end{figure*}

\section{Signal Measurement and Preprocessing}

\subsection{Limitations of Doppler Shift}
Most existing smart devices can emit and record sound waves with frequency up to 23 kHz, and researchers showed that sound waves higher than 17 kHz are usually inaudible to most people\cite{rodriguez2014extended}. Therefore, the speaker and microphone of the devices can act as an active sonar to sense surroundings. Many researchers used the Doppler shift calculated by STFT to estimate movements. However, the resolution of STFT is limited by the fundamental constraints of time-frequency analysis. Figure~\ref{doppler_pham}(a) shows the STFT results of a moving lip from 1.2s to 2.5s when speaking the word ``WiFi". The frequency of the emitted signal is 17.35 kHz. We remove the inference of the Line-of-Sight (LOS) signal by calculating the difference between two successive samples\cite{lu2019lip} in the frequency-domain. Since small frequency variations are buried in the wide frequency band around 17.35 kHz, we can hardly observe the Doppler shift signals after 1.8s.

\subsection{Phase Measurement}

To overcome the limitations of the Doppler shift, we leverage phase information of received signal to profile fine-grained lip movements. The wavelength of sound waves up to 17 kHz is less than 2 cm, meaning that a small movement of frequency will significantly change the phase of the received sound wave. Therefore, the signal phase is susceptible to subtle changes of propagation distance.

The phase signals can be calculated through the coherent detector, as Figure~\ref{lipreading2} depicts. Firstly, the inaudible signals reflected by moving lips are collected by the microphone of smartphone. Secondly, these signals are fed into low-pass filters (LPF) to get In-phase ($I$) component and Quadrature ($Q$) component respectively. Thirdly, these two components are combined together to get the phase features. Specificly, the speaker of smartphone emit the Continuous Wave (CW) signal of $Acos(2\pi ft)$, where $A$ is the amplitude, $t$ is the sampling index on time axis, and $f$ is the frequency of the sound, which is higher than 17 kHz. The sampling rate is 48 kHz. Without loss of generality, we assume there is only one propagation path $d_p(t)$, here $p$ denote propagation. And thus the received signal of reflection can be denoted as:
\begin{equation}
\begin{aligned}
R_p &= A_p cos(2\pi f(t-d_p(t)/c) - \theta_p)
\\
&= A_p cos(2\pi ft-\phi_p),
\end{aligned}
\end{equation}
where $c$ is the speed of the sound, and $\theta_p$ is the phase shift brought by the hardware. $A_p$ and $\phi_p$ are amplitude and phase, respectively. The received signal will be multiplied by $cos(2\pi ft)$ :
\begin{equation}
\begin{aligned}
R_p\times cos(2\pi ft) &= A_p cos(2\pi ft-\phi_p)\times cos(2\pi ft)
\\
&= \frac{A_p}{2} cos(4\pi ft-\phi_p) + \frac{A_p}{2} cos(\phi_p).
\end{aligned}
\end{equation}

The first term in the equation has a high frequency of $2f$ and thus can be removed by a LPF. Then, we can get the $I$ component of the base-band signal as $I_p = \frac{A_p}{2} cos(\phi_p)$. To reduce computational complexity, the $I$ component will pass through a moving average filter with a window size of 200 and an overlap of 0.5. This makes the sampling rate decreased from 48 kHz to 480 Hz.  Similarly, we can get the $Q$ component as  $Q_p = \frac{A_p}{2} sin(\phi_p)$. Then, these two components are combined as the real and imaginary parts of a complex signal:
\begin{equation}
B_p=\frac{A_p}{2}e^{-j\phi_p}.
\label{base-band}
\end{equation}

We can easily get the phase signals $\phi_p(t)$ from Equation~\ref{base-band}. Figure~\ref{doppler_pham}(b) shows the phase profile obtained from the same sound record that produces the spectrogram in Figure~\ref{doppler_pham}(a). We can clearly observe patterns caused by lip movements. The profiles exhibit significant fluctuations from 1.2s to 2.5s.

\subsection{Signal Preprocessing}

\textbf{Multi-frequency acoustic signals.} Wireless signals with different frequencies will experience different multipath fading when propagating in the air \cite{tse2005fundamentals}. Therefore, we simultaneously transmit sound waves at multiple frequencies to mitigate frequency selective fading as well as enhance the capability to profile multipath environments. In particular, we generate signal $A\sum_k cos[2\pi (f+k\delta f)t]$, which is the superposition of multi-frequency sound waves. $k$ depicts the $k$th frequency channel, and $\delta f$ is the frequency interval between adjacent channels. In the receiver, we get the phase values for each frequency using the corresponding coherent detector. All the frequencies fall into the band of 17$\sim$23 kHz. Considering the signal energy for each frequency and limited bandwidth, we set the number of channels $k$ to 8 and $\delta f$ to 700 Hz.

\textbf{Multipath elimination.} The received signals are the mixtures of multipath signals. Besides dynamic signals caused by moving lips, there exist static signals including the LOS signal (i.e., the signal directly propagated from the speaker to microphone) and surrounding reflections (from face and body), which are usually much stronger than dynamic signals. Moreover, static signals may also change slowly with the movements of the face or body. Figure~\ref{doppler_pham}(b) shows that the phase still increases slowly after 2.5s (the end time of lip movements). Static signals are irrelevant or even harmful to lip-reading. To eliminate this extraneous information, we calculate the first order difference of phase between two consecutive samples at time $t-1$ and $t$, and denote it as \textit{phase delta}:
\begin{equation}
\Delta \phi_p(t) = \phi_p(t) - \phi_p(t-1).
\end{equation}

Figure~\ref{doppler_pham}(c) shows the phase delta signals of the same sound record. It can be observed that the phase delta signals are approximately zero in the absence of lip movements, which confirms that static signals are almost completely eliminated. In addition, signals with different frequencies fluctuate differently, which shows they experience multipath fading. We also follow the idea in \cite{Kumar2011} and calculate \textit{phase double-delta}, which stands for the second-order difference of phase signals:
\begin{equation}
\Delta \Delta \phi_p(t) = \Delta \phi_p(t) - \Delta \phi_p(t-1).
\end{equation}

All of these phase features can be candidates for our input, and we also try various combinations of these features in our experiments. The details are shown in Table~\ref{input}. Here, the operator $[\cdot,\cdot]$ in the last row represents concatenation operation.

\begin{table}[th]
	\caption{different choices of phase features $x(t)$.}
	\begin{center}
		\begin{tabular}{c|c}
			\hline
			feature name & phase features $x(t)$  \\
			\hline
			\hline
			phase & $\phi_p(t)$  \\
			\hline
			phase delta & $\Delta \phi_p(t)$ \\
			\hline
			phase delta + double-delta & $[\Delta \phi_p(t), \Delta \Delta\phi_p(t))]$  \\
			\hline
		\end{tabular}
		\label{input}
	\end{center}
\end{table}

\textbf{Data augmentation.} In addition, we try to implement data augmentation on the input features $x(t)$. Deep networks usually have a huge demand for the training data. However, it is hard for us to collect sufficient data for lip-reading in practice. In this paper, considering that people speak lip commands at different speeds, we borrow the time-warping technique used in speech recognition \cite{ko2015audio} to enrich training data. Specifically, given a phase signal $x(t)$, we expand or contract the raw signal in the time axis by a factor $\alpha$, thereby generating a new profile $x(\alpha t)$. The data augmentation mechanism can effectively avoid overfitting and improve the robustness against different speeds.

\section {End-to-end Lip-Reading}
Figure~\ref{overview} illustrates the overall architecture of our framework. The continuous signal stream is firstly segmented into a series of overlapping clips. We employ CNNs to extract features for each clip. As the phase features are temporal signals with different carrier frequencies, we apply convolutions over time and frequency to extract patterns. In particular, assuming there are $N$ clips in total, for the $n$th clip, the input data matrix is $x_{n}^{(T\times{D})}=[x(\tau),x(\tau+1),x(\tau+2),...,x(\tau+T-1)]$, where $D$ is the feature dimension, $\tau$ is the beginning time of this clip and $T$ is the time length in each clip. We input this data matrix into a 3-layer CNN. Pooling and batch normalization (BN) are applied to each layer. After the convolutional layers, fully-connected layers are employed to get the representation vector for this clip.

After feature extraction and processing, we exploit an attention-based encoder-decoder network to achieve end-to-end lip-reading. We break down the sequence learning task into two phases. In the encoding phase, the CNN outputs are projected into a latent space in the form of a fixed size vector, which is later used in the decoding phase to generate sentence labels.

\textbf{Encoder.} In the encoder phase, we implement a 3-layer CNN on $x_n$, and feed the results $f_n$ into a two-layer LSTM, noted as $LSTM_{enc}(\cdot)$, to model temporal changes and output the hidden state $o_n$:
\begin{equation}
f_n = CNN_{\times 3}(x_n),
\end{equation}
\begin{equation}
o_n = LSTM_{enc}(f_n,o_{n+1}).
\end{equation}

Note that the LSTM ingests the inputs in reverse time order, which can shorten long-term dependencies between the beginning of the signal stream and sentence labels, as shown in \cite{sutskever2014sequence}. We denote the final output matrix as $O=[o_N, o_{N-1}, ... ,o_1]$, in which the last column vector $o_1$ corresponds to the latent embedding of the input sequence, and pass it to the decoder.

\begin{figure}
	\centering
	\includegraphics[width=3in]{overview.pdf}
	\caption{An overview of our end-to-end framework.}
	\label{overview}
\end{figure}

\textbf{Decoder.} The decoder is also based on a two-layer LSTM, noted as $LSTM_{dec}(\cdot)$. Besides, we utilize the attention mechanism to force the model to learn to focus on specific parts of the input sequence when decoding \cite{bahdanau2014neural}. The key idea of the attention mechanism is to assign a weight $\alpha_u$ for each encoder output $o_n$ at step $u$ of the decoder and generate a context vector $c_u$:
\begin{equation}
\alpha_u = Attention(h_{u-1}, O),
\end{equation}
\begin{equation}
c_u = O \cdot \alpha_u.
\end{equation}

Then, the hidden state $h_u$ of the decoder at step $u$ can be updated as:
\begin{equation}
y_u, h_u = LSTM_{dec}(h_{u-1}, c_u, g_{u-1}),
\label{yu_cal}
\end{equation}
where $y_u$ is the predicted word label, and $g_{u-1}$ is the word embedding of $y_{u-1}$. The initial $h_0$ is the latent vector $o_1$, and $y_0$ is a special label $<BOS>$ indicating the start of a sentence.  The decoder phase will end when predicting a label $<EOS>$, which indicates the end of the sequence.

In the training phase, the probability of label $y_u$ at step $u$ can be calculated based on Equation~\ref{yu_cal}. Thus given the signal stream X, the conditional probability of the target sentence $Y$ is:
\begin{equation}
p(Y|X) = \prod_{u} p(y_u).
\end{equation}

We minimize the corresponding cross-entropy loss to update all of the network parameters. In the decoder phase, given the posterior probability distributions of labels at each step, we employ the beam search algorithm to generate the final sequence.

\section{Implementation and Evaluation}
\label{sec:eval}

\begin{figure*}[t]
	\centering
	\subfloat[The WER across volunteers in domain-independent test]{\includegraphics[width=2.2in]{sentence_wer2}}
	\hspace{5pt}
	\subfloat[Top-10 WER in unseen sentence test]{\includegraphics[width=2.2in]{unseensentence}}
	\hspace{5pt}
	\subfloat[Comparison with CTC in different test settings]{\includegraphics[width=2.1in]{CTC2}}
	\caption{Experimental results.}
	\label{results}
\end{figure*}

\subsection{Data Collection and Training}
We use one Samsung smartphone to collect lip sentence datasets. All the sentences are Standard Chinese. 10 volunteers are recruited in the data collection. Volunteers perform the experiments at 8 locations (i.e., a laboratory, a meeting room, and a bedroom). Each volunteer performs 5 sessions, and they repeat each sentence 5 times at each session. Different sessions are on different days to ensure data diversity. People can change the distance between the lip and the smartphone according to their habits. We carefully select 54 sentences which are frequently used in voice interfaces of smartphones, e.g., ``What's the weather like tomorrow'' and ``Take a picture on a wide angle'' . A total of 29 words are included in these sentences. In total, we collect 13500 samples. In the training phase, we perform data augmentation for each lip commands with 10 different scaling factor $\alpha$, meaning that the number of samples increases 10 times the original one. The range of $\alpha$ is 0.5$\sim$2.

\subsection{Impact of Signal Processing}

The signal processing pipeline plays a key role in the recognition task. In this paper, we leverage the phase information instead of the coarse-grained Doppler shift to capture lip movements. For all evaluations, we employ the word error rate (WER) as the criterion. Table~\ref{result of sp} lists the WER of different feature choices in the domain-independent test setting.

\begin{table}[h]
	\caption{WER under different signal preprocessing mechanisms}
	\begin{center}
		\begin{tabular}{c|c}
			\hline
			Mechanisms & WER(\%)  \\
			\hline
			\hline
			Doppler shift & 29.4  \\
			\hline
			phase & 32.1  \\
			\hline
			phase delta & 15.6  \\
			\hline
			phase delta + double-delta & 11.2  \\
			\hline
			\textbf{phase delta + double-delta + augmentation} & \textbf{8.4}  \\
			\hline
		\end{tabular}
		\label{result of sp}
	\end{center}
\end{table}

In particular, \textit{Doppler shift, phase, phase delta,} and \textit{phase delta + double-delta} are four types of input features calculated under multiple frequencies. In our experiments, \textit{phase delta + double-delta} gets the lowest WER, outperforming any other features. By the way, we also test \textit{phase delta + double-delta} with only one frequency, but the result (WER 38.2\%) is much worse than using 8 frequency channels. Therefore, the multi-frequency mechanism improves the accuracy by a large margin. Then, we implement data augmentation on \textit{phase delta + double-delta + augmentation} and get the best result in our experiments. We can clearly see that each component in our pipeline boosts the recognition performance to some extent.

\subsection{Evaluation and Performance}
\label{eval_perf}
We evaluate our method by three evaluation strategies:

\textbf{Domain-dependent test:} Domains in this paper refer to users and environments, both of which have an impact on the phase profiles. Domain-dependent means we ignore the impact of different domains, and randomly divide all the data into training, validation, and testing sets. In the domain-dependent test, we randomly select 70\% of the dataset as training data, 10\% as validation data and 20\% as testing data. The WER of testing data here is \textbf{2.6\%}.

\textbf{Domain-independent test:} We perform leave-one-domain-out cross-validation to validate the capacity of our method to deal with domain diversity. The model hyperparameters are fixed in each test, which are tuned on the validation data of domain-dependent test. We make sure that the training data and the testing data are collected from different users and positions. For the domain-independent test, we present the WERs across volunteers in Figure~\ref{results}(a). From the figure, we get the WERs ranging from 3.5\% to 12.2\%, and the average is \textbf{8.4\%}. The results show that the fine-grained phase profiles as well as the deep learning networks are capable of capturing the key characteristics of these lip sentences, thus generalizing very well across different domains.

\textbf{Unseen sentences test:} We also evaluate the performance in translating unseen sentences (sentences not in the training set). As there is no public and large-scale dataset for acoustic-based lip-reading, the recognition ability for unseen sentences can eliminate the burden to collect all possible sentences. For the unseen sentences test, we perform leave-one-sentence-out validation. We list Top-10 WER in Figure~\ref{results}(b) to show the worst cases. The highest WER is 18.2\%, and the average for all 54 sentences is \textbf{8.1\%}. This is a very impressive result considering that the testing sentences are not included in the training set.

\subsection{Comparison with CTC}
In this paper, we follow the WAS network \cite{chung2017lip} to use the attention-based encoder-decoder framework for sequence modeling. Another popular method used in speech recognition and lip-reading is Connectionist Temporal Classification (CTC) \cite{graves2006connectionist,graves2014towards,assael2016lipnet}. The advantage of the attention model is that it explicitly uses the history of the target label, while CTC assumes the output labels are not conditioned on each other. We make a comparison with CTC by replacing the decoder network with CTC. Figure~\ref{results}(c) presents the result under three evaluation mechanisms. From the figure, we can see that the attention model outperforms CTC notably, especially in the domain-independent conditions. This is mainly due to the ability of the attention to learn internal language models, which is very helpful in the decoding phase.

\section{Conclusion}
In this paper, we propose a non-invasive silent speech recognition method, which uses the inaudible acoustic signals generated by smart devices for lip-reading. We leverage the phase information of the received signals to characterize fine-grained lip movements. And we propose an end-to-end recognition framework which combines the CNN and attention-based encoder-decoder network. We show that the combination of phase delta and double-delta features can get high accuracy on continuous silent speech recognition based on the dataset we collect. The WER under domain-dependent, domain-independent, and unseen sentence tests are 2.6\%, 8.4\%, and 8.1\%, respectively, demonstrating the feasibility and effectiveness of our method. As our method can be seamlessly applied to existing voice-controlled smart devices without any modifications, we believe it can significantly contribute to the advancement of silent voice recognition. Future works include exploring various sequence learning architectures like CTC-attention joint model. We are also interested to combine traditional acoustic signals and the silent signals in speech recognition and voice activity detection tasks.

\section{Acknowledgement}
This paper is supported by National Key Research and Development Program of China under Grant No.2017YFB1401202, No.2018YFB0204400 and No.2018YFB1003500.

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