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

Let $M$ be a differentiable manifold, $I$ be an interval, and
$g(t)\equiv g_t\equiv g(x,t)$,
$(x,t)\in M\times I$, be
a family of complete riemannian metrics on $M$, which is parameterized
by $t\in I$. Let $\ric_t$ be ricci curvature of $g_t$.
The ricci flow
is a weakly parabolic system of partial differential equations defined by
\begin{equation*}
\pd{t}g_t=-2\ric_t.
\end{equation*}
It was first
introduced by Richard Hamilton in his celebrated 1982
paper\cite{0504.53034} and used to deform riemannian metrics.
 Thereafter, as a powerful tool,
ricci flow helped changing the landscape of differential
geometry greatly. For example, it is a crucial tool to prove Poincar\'e
conjecture and differentiable sphere theorem. See, for example,
 \cite{2003math......3109P} and \cite{MR2449060}.

Ricci flow as a system of partial differential equations, the short time
existence and uniqueness of the solution to initial value problem
are among the most fundamental questions. When underlying manifold
is compact without boundary, they were proved by Hamilton
in the same paper mentioned
above.  Shortly after, DeTurck\cite{MR697987} introduced a
technique now commonly named after himself, to
simplify the proof of existence. Together with the help from
harmonic heat flow, Hamilton used DeTurck's trick to give a new proof to the
uniqueness in \cite{MR1375255}.

For the case of compact manifold with boundary, which we will not discuss
in this paper, interested readers may consult
\cite{2012arXiv1210.0813G} and bibliography  therein.

When the manifold is complete non-compact without boundary, Shi
\cite{MR1001277} first constructed a complete
solution to every initial metric with bounded riemannian curvature. If
the curvature bound is $k_0>0$, then  there
exists a $T=T(n,k_0)$, such that his solution exists on $[0,T]$ and satisfies
\begin{equation*}
  \abs{\nabla^m \riem}^2\leq \frac{c_m}{t^n}
\end{equation*}
for some constant $c_m>0$.
Cabezas-Rivas and Wilking\cite{2011arXiv1107.0606C} improved Shi's result
by removing the curvature upper bound for the initial data, and only
assuming that it is non-collapsing and has nonnegative complex
sectional curvature (see below for definition). Furthermore, their
solution preserves
nonnegativity of complex sectional curvature.

For the uniqueness of solution to classic heat equation,  in contrast,
one cannot
take  the uniqueness of solution for granted. It is Tychonoff who had not only
discovered a non-zero
solution to heat equation on $\mathbb{R}\times [0,\infty)$ with vanishing initial
data, but also proved uniqueness by assuming solutions at most exponential
growth (see e.g. John\cite{MR831655} page 211 and 217). However, in
the case of ricci flow, the situation is revealed to
be subtle by the discovery that, in dimension two and three, the only
solution to ricci flow with
euclidean initial data is euclidean. We will discuss this later.

Nonetheless, following spirit of Tychonoff, Chen and Zhu established a
uniqueness theorem in the
class of metrics with bounded curvature.
\begin{thm}[Chen-Zhu \cite{MR2260930}]
  \label{thm:4}
  Let $(M^n,g)$ be a complete noncompact riemannian manifold of
  dimension $n$ with bounded curvature. Let $g_1(t)$ and $g_2(t)$
  be two solutions to the ricci flow on $M\times [0,T]$ with $g$ as
  their common
  initial data and with bounded curvatures. Then $g_1=g_2$
  for all $(x,t)\in M^n\times [0,T]$.
\end{thm}

Also along this spirit, Kotschwar \cite{2012arXiv1206.3225K} weakened
the condition to  curvature growth rate
being at most quadratic and equivalence of $g(t)$ to $g(0)$.

In lower dimensions, however, the ricci flow is ``better''
than the classic heat equation.
Topping\cite{MR2832165}\cite{2013arXiv1305.1905T}  proved that for
any initial
metric on  two dimensional manifold, which could be noncomplete
and/or with unbounded curvature, ricci flow has one and only one
instantaneously complete solution. Readers may also see
\cite{MR2520796} for related discussion on surface.

In dimension three, Chen\cite{MR2520796} proved that for any
complete noncompact  initial metric, if it has nonnegative and bounded
sectional curvature, then on some time interval depending only on
initial curvature bound, there exists a unique complete ricci flow
solution. As a corollary, the only complete ricci flow solution on $\mathbb{R}^3$
with euclidean initial metric is euclidean. His idea of the proof is first to
prove that the curvature is bounded
over some time interval, then the uniqueness follows from the theorem
of Chen-Zhu.

Another way to weaken the condition is
replacing the bilateral curvature bound by unilateral. We first
consider the case with curvature upper bound.
On riemann surface, Chen and Yan  \cite{MR2721664}
proved that curvature upper bound is sufficient. It is worth noting
that their
theorem does not require curvature lower bound for the initial metric.

In general dimension, when we assume in addition that the initial metric
is with bounded
curvature, one can prove the uniqueness by assuming only
the upper bound of sectional curvature. In fact, this is a
straight forward consequence of Corollary 2.3 in \cite{MR2520796},
which says when scalar curvature is bounded from below initially, it
is preserved along the flow.
So we can apply theorem \ref{thm:4} to obtain uniqueness.

Then we turn to the case with curvature lower bound, which is the main
purpose of this paper.
Before stating our main result, we first remind
readers of the definition of complex sectional curvature.

Let $(M^n,g)$ be a
$n$-dimensional riemannian manifold  and
$T^cM=TM\otimes\mathbb{C}$ be complexified tangent bundle.
Let $\nabla$ be Levi-Civita connection, and $\riem$ be riemannian
curvature defined by
\begin{equation*}
  \riem(u,v,w,z)=g(\nabla_u\nabla_vw-\nabla_v\nabla_uw-\nabla_{[u,v]}w,z)
\end{equation*}
where $u,v,w,z\in T_pM$ for any $p\in M$.  And
\begin{equation*}
  \R:\wedge^2TM\mapsto\wedge^2TM
\end{equation*}
is the riemannian curvature operator induced by $\riem$. Let $\scal$
be the scalar curvature.
We extend both metric $g$ and
curvature tensor $\riem$ to be $\mathbb{C}$-multilinear maps
\begin{equation*}
  g:S^2T^cM\rightarrow \mathbb{C}\quad
  \riem:S^2(\wedge^2T^cM)\rightarrow \mathbb{C}.
\end{equation*}
\begin{definition}
  \label{def:1}
For any $2$-dimensional complex subspace $\sigma\in
T_p^cM$, the complex sectional curvature
is defined by
\begin{equation*}
   \sec^c(\sigma):=\riem(u,v,\bar{v},\bar{u})=g(\R(u\wedge
   v),\overline{u\wedge v}),
\end{equation*}
for any unitary orthogonal vectors $u,v\in \sigma$
i.e. $g(u,\bar{u})=g(v,\bar{v})=1$ and $g(u,\bar{v})=0$, where, e.g.
$\bar{u}$ is the complex conjugate of $u$.
\end{definition}

We can now state the main result.
\begin{thm}
  \label{thm:1}
  Let $(M^n,g)$ be a complete noncompact $n$-dimensional riemannian
  manifold, with complex sectional curvature $\sec^c \geq -1$
  and $\scal\leq s_0$ for some $s_0 > 0$. In addition, the volume of every unit
  ball is bounded from below uniformly by some constant
  $v_0>0$.  Let $g_i(t)$ $i =  1,2$ be
  two solutions to the ricci flow  on
  $M\times [0,T]$ with same initial data $g_i(0) = g$.  If both
  $g_i(t)$ are
  complete riemannian metrics with $\sec^c(g_i(t))\geq -1$, for all $t\in [0,T]$,
  then there exists a constant $C_n>0$ depending only on dimension $n$,
  s.t. $g_1(t) = g_2(t)$ on $ M\times [0,\min\{\epsilon,T\}]$, where $\epsilon=\min\{v_0/2C_n,1/(n-1)\}$.
\end{thm}
We need the volume lower bound for unit ball of initial metric to obtain
injectivity radius lower bound of $g(t)$ when $t>0$. This is crucial to
establish Cheeger-Gromov convergence of ricci flows in our proof. Actually,
the curvature condition alone cannot guarantee the volume lower
bound, which is shown by the following example. Consider a
rotationally
symmetric metric $g=dr^2 + \e{-2r}ds^2$ on a half-cylinder. Its
curvature is $-\frac{(\e{-r})''}{\e{-r}}=-1$. Closing up one end at $r=0$ by
putting on a cap, smoothing if necessary, we get a complete metric
with bounded curvature.  But the volume of unit ball
goes to zero when $r$ increases to infinity. By crossing with $\mathbb{R}^n$ we
get examples for higher dimensions.

With Theorem \ref{thm:4}, above theorem is an immediate
corollary of the following one.
\begin{thm}
  \label{thm:3}
  Let $(M^n,g)$ be as in above theorem. Let $g(t)$ be a ricci flow solution
  on $M^n\times [0,T]$ with $g(0)=g$. If $\sec^c_t\geq -1$
  and $g(t)$ is complete for any  $t\in [0,T]$,
  then $\riem_t$ is uniformly bounded on
  $M\times [0,\min\{\epsilon,T\}]$ for the same $\epsilon$ in above theorem.
\end{thm}
\begin{rmk}
  \label{rmk:5}
  Readers may wonder if the uniqueness is true on
  $[0,T]$. Unfortunately, our method is not strong enough to prove it. Actually,
  Cabezas-Rivas and Wilking had constructed an immortal complete
  ricci flow solution with positive curvature operator (hence with
  positive complex
  sectional curvature) which is bounded if and only if $t\in
  [0,1)$ (See \cite{2011arXiv1107.0606C} Theorem 4 b). And by a
  Toponogov's theorem below (Theorem \ref{thm:2}), it has
  a uniform lower bound for volume of any unit ball in
  initial manifold. So it does satisfy assumptions of
  Theorem \ref{thm:1}. However, our method to prove uniqueness is
  to show that curvature
  is bounded, which is not true general for the whole interval of
  $[0,T]$ as shown by above example. Nonetheless, the question is still open.
\end{rmk}
\begin{rmk}
  \label{rmk:6}
  Authors are in debt to Cabezas-Rivas and
  Wilking \cite{2011arXiv1107.0606C} for
  following their use of complex
  sectional curvature and adapting
  many of their arguments to serve their own purpose.
\end{rmk}

\section{Preliminaries}
\label{sec:preliminaries}

In this section, we list a few facts for later use.

Along ricci flow, the volume form satisfies the following evolution equation.
\begin{equation*}
\pd{t}\,\mathrm{dvol} = -\scal\,\mathrm{dvol}
\end{equation*}
We also write down the rescaling relations for later use. If  $\tilde{g}=k^2g$,
$\widetilde{\sec}^c=k^{-2}\sec^c$.

Remember that  sectional curvature of a tangent plane $\sigma$ spanned by
$u,v\in T_pM$ is defined by
\begin{equation*}
  \sec(\sigma)=\frac{\riem(u,v,v,u)}{\abs{u}^2\abs{v}^2-g(u,v)^2}.
\end{equation*}
We have the following relations between complex sectional curvature and
other curvatures.
\begin{lemma}
  \label{lem:3}
  If $\sec^c>(\geq)C$ for any constant $C$, then sectional
  curvature $\sec >(\geq)C$.
\end{lemma}
\begin{proof}
  Let $u,v\in T_pM\subset T_p^c$ be any pair of orthogonal
  unit vectors and
  $\sigma$ be the complex plane spanned by them. Then sectional
  curvature on the real plane spanned by them is
  $\sec(u,v)=\riem(u,v,v,u)=\sec^c(\sigma)>C$.
\end{proof}
\begin{lemma}
  \label{lem:4}
  If sectional curvature is bounded from below and scalar curvature is
  bounded from above, then the sectional curvature is
  bounded from both sides, as is the curvature operator.
\end{lemma}
\begin{proof}
  Fix a point $p\in M$.
  Suppose for any plane $P\in T_pM$, the sectional curvature $\sec(P)\geq
  c$; and the scalar curvature  $\scal(p)\leq C$  for some fixed
  constants $c\leq C$.

  Then for any o.n. basis $\{e_i\}$ of  $T_pM$,
  \begin{equation*}
    C>\scal=\sum_{i,j}\riem(e_i,e_j,e_j,e_i)=2\sum_{i<j}\riem(e_i,e_j,e_j,e_i)=
    2\sum_{i<j}\sec(P_{ij})
  \end{equation*}
  where $P_{ij}$ is the plane spanned by $\{e_i,e_j\}$. Because
  $\sec$ is bounded from below, by above inequality, $\sec(P_{ij})$ is
  bounded from above by a constant depending on only $c$, $C$, and
  dimension. The above orthonormal basis is
  arbitrary, so sectional curvature at $p$ is bounded from above.

  Any component of curvature operator is a linear combination of
  sectional curvature for various planes in tangent space where the
  coefficients depend only on dimension (see  e.g. Cheeger and Ebin
  \cite{MR2394158} page 14). Thus curvature operator is bounded.
\end{proof}

\section{Proof of Theorem \ref{thm:3}}
\label{sec:proof-theorem}

\begin{proof}[Proof of theorem~\ref{thm:3}]
  \label{pf:1}
  The key to the proof is establishing the following  estimate
  \begin{equation*}
    \scal_t\leq \frac C t.
  \end{equation*}
  Then, using Theorem~\ref{thm:5} in B.L.Chen \cite{MR2520796} (see
  also Simon \cite{MR2439551}), we
  get a uniform bound for $|\riem|$.
  The arguments here are adaptations of those in Cabezas-Rivas and Wilking
  \cite{2011arXiv1107.0606C}.

  \subsection{Estimate Volume Lower Bound}
  \label{sec:estim-volume-lower}

  We first establish uniform volume lower bound for unit balls along
  ricci flow.
  \begin{lemma}
    \label{lem:1}
    Let $(M,g)$ be a complete riemannian $n$-manifold whose unit balls have
    volume lower bound $v_0>0$.  Let $(M,g(t))$ be a solution of
    ricci flow on $t\in [0,T]$ with $g(0)=g$,  and  its
    sectional curvature is
    bounded from below by $-1$ for any $t\in [0,T]$, then there exists
    a constant
    $C_n > 0$ depending only on dimension $n$, s.t. The volume of any ball
    $B_t(p,\mathrm{e})\subset
    (M,g(t))$  bounded from below by $v_0/2$ for $t\in [0,\epsilon]$,
    where $\epsilon=\min\{v_0/2C_n,1/(n-1)\}$.
  \end{lemma}
  \begin{proof}[Proof of Lemma \ref{lem:1}]
    \label{pf:3}
    Because sectional curvature is bounded uniformly from below by $-1$,
    $\ric_t \geq -(n-1)g(t)$, we have the following estimate for
    distance function.
    \begin{lemma}
      \label{lem:5}
      For any fixed points $p,q\in M$,
      \begin{equation}
        \label{eq:2}
        \dist_t(p,q)\leq \dist_0(p,q)\,\e{(n-1)t}.
      \end{equation}
    \end{lemma}
    \begin{proof}
      \label{pf:5}
      Let $\gamma:[0,1]\rightarrow M$ be a length minimizing geodesic with
      respect to $g(0)$ such that $\gamma(0)=p$ and $\gamma(1)=q$. Let
      $\abs{\gamma}_t$ be the length of the curve $\gamma$ with
      respect to $g(t)$. Then
      \begin{equation*}
        \begin{split}
          \td{t}\abs{\gamma}_t
          &=\int_0^1\td{t}\sqrt{g_t(\dot\gamma(s),\dot\gamma(s))}\dd s
          =\int_0^1
          \frac{-\ric(\dot\gamma(s),\dot\gamma(s))}
          {\sqrt{g_t(\dot\gamma(s),\dot\gamma(s))}}\dd s\\
          &\leq\int_0^1 (n-1)\norm{\dot\gamma(s)}_t\dd s=(n-1)\abs{\gamma}_t.
        \end{split}
      \end{equation*}
      By integrating above differential inequality, we get
      \begin{equation*}
        \dist_t(p,q)\leq\abs{\gamma}_t\leq \abs{\gamma}_0\e{(n-1)t}
        =\dist_0(p,q)\e{(n-1)t}.
      \end{equation*}
    \end{proof}
    Therefore $B_0(p,1)\subset B_t(p,\mathrm{e})$ for any $0\leq t\leq 1/(n-1)$ and
    any $p\in M$. Then,
    \begin{equation*}
      \begin{split}
        \td t \vol_t(B_0(p,1))&= -\int_{B_0(p,1)}\scal_t
        \,\mathrm{dvol}_t\\
        &= -\int_{B_t(p,\mathrm{e})}\scal_t\,\mathrm{dvol}_t +
        \int_{B_t(p,\mathrm{e})-B_0(p,1)}\scal_t\,\mathrm{dvol}_t\\
        &\geq -C(n) + \int_{B_t(p,\mathrm{e})}-n(n-1)\,\mathrm{dvol}_t.\\
      \end{split}
    \end{equation*}
    In last step we use $\scal_t\geq -n(n-1)$ and Petrunin's estimate as below.
    \begin{thm}[Petrunin\cite{MR2423998}]
      \label{thm:101}
      Let $M$ be a complete riemannian manifold whose sectional
      curvature is at least $-1$. Then
      \begin{equation*}
        \int_{B(p,1)}\scal \leq C(n)
      \end{equation*}
      for any $p\in M$, where $B(p,1)$ is the unit ball centered  in
      $p$ and $C(n)$ is a constant depending only on dimension $n$.
    \end{thm}
    Then, applying Bishop-Gromov volume comparison,
    \begin{equation*}

        \td t \vol_t(B_0(p,1))

        \geq -C(n) - n(n-1)\vbh \mathrm{e}
        =:-C'(n),

    \end{equation*}
    where $\vbh r$ is the volume of a ball with radius $r$ in
    simply connected hyperbolic space of constant curvature $-1$.
    Integrating above,
    \begin{equation*}
      \vol_{t}(B_{t}(p,\mathrm{e}))\geq
      \vol_{t}(B_0(p,1))\geq \vol_0(B_0(p,1)) - C'(n) t\geq v_0 -C'(n) t.
    \end{equation*}
    Hence, let $\epsilon=\min\{v_0/2C'(n),1/(n-1)\}$,
    \begin{equation*}
      \vol_{t}(B_{t}(p,\mathrm{e})\geq v_0/2\ \text{for any}\ t\in
      [0,\epsilon].
    \end{equation*}
  \end{proof}

  \subsection{Curvature Uniform Bound}
  \label{sec:curv-unif-bound}

  \begin{lemma}
    \label{lem:2}
    Let $(M^n,g(t))$ be a complete solution of ricci flow
    for $M\times [0,\epsilon]$ with $\sec^c\geq -1$. If there
    is a constant $v'>0$ such that
    volume of balls $\vol_t(B_t(p,\mathrm{e})) > v'$, for any $(t,p)\in
    [0,\epsilon]\times M$,  then there exists a
    constant $C>0$ ,  so that
    \begin{equation*}
      \scal_t(p) \leq \frac {C} t
    \end{equation*}
    for any $(p,t)\in M\times (0,\epsilon]$.
  \end{lemma}
  \begin{proof}
    \label{pf:2}
    Prove by contradiction. Suppose there is a
    sequence  ${(p_k,t_k)}\subset M\times (0,\epsilon]$  such that,
    \begin{equation}
      \label{eq:7}
      \scal_k(p_k) > \frac{4^k}{t_k}.
    \end{equation}
    where and hereafter, $B_k=B_{g(t_k)}$, $B_t=B_{g(t)}$, $\scal_t=\scal_{g(t)}$,
    $\scal_k=\scal_{g(t_k)}$, $\dist_t=\dist_{g(t)}$, and  $\dist_k=\dist_{g(t_k)}$.
    We can pick points in space-time to blow up by following trick first
    introduced by Perelman in \cite{2002math.....11159P}.
    \begin{claim}
      \label{claim:2}
      For any sufficiently large integer $k$, there exists
      $\bar{p}_k\in M$ and  $\bar{t}_k\in (0,t_k]$  satisfy
      the following equations
      \begin{equation}
        \label{eq:8}
        \scal_t(x) \leq 2 \scal_{\bar{t}_k}(\bar{p}_k)\quad\text{for all}\
        \begin{cases}
          x \in B_{\bar{t}_k}(\bar{p}_k,
          \frac{k}{\sqrt{\scal_{\bar{t}_k}(\bar{p}_k)}})\ \text{and,}\\
          t \in (\bar{t}_k-\frac{k}{\scal_{\bar{t}_k}(\bar{p}_k)},\bar{t}_k]\\
        \end{cases}
      \end{equation}
      \begin{equation}
        \label{eq:1}
        \scal_{\bar{t}_k}(\bar{p}_k) \geq \frac{4^k}{t_k}.
      \end{equation}
    \end{claim}
    \begin{proof}[Proof of Claim~\ref{claim:2}]
      \label{pf:4}
      We start searching for $(\bar{p}_k,\bar{t}_k)$ from $(p_k,t_k)$
      --- if it satisfies equation \eqref{eq:8},  we
      are done. Otherwise we can find $x_1$ and $\tau_1$
       s.t.
       \begin{equation*}
         \dist_k(x_1,p_k)\leq \frac{k}{\sqrt{\scal_k(p_k)}},\
         \tau_1 \in (t_k-\frac{k}{\scal_k(p_k)},t_k],\ \textrm{and }
         \scal_{\tau_1}(x_1) \geq 2 \scal_k(p_k).
      \end{equation*}
      If $(x_1,\tau_1)$ is not what we are looking for, we can find
      $x_2$ and $\tau_2$, s.t.
      \begin{equation*}
        \dist_{\tau_1}(x_1,x_2)\leq\frac k{\sqrt{\scal_{\tau_1}(x_1)}},\
        \tau_2\in (\tau_1-\frac k{\scal_{\tau_1}(x_1)},\tau_1],
        \ \text{and } \scal_{\tau_2}(x_2) \geq 2 \scal_{\tau_1}(x_1).
      \end{equation*}
      We claim that, along this way only for finite steps, we can find
       desired
      $(\bar{t}_k,\bar{p}_k)$.  Otherwise, we get a sequence of $(x_k,\tau_k)$ and
      $\scal_{\tau_k}(x_k)\rightarrow \infty$ as  $k\rightarrow\infty$. Here readers may
      worry that $\tau_i$ would go below $0$ thus ill-defined. However, by our
      construction, not only this cannot happen, but also, the above
      sequence is well bounded in both space and time, as
      shown in the following discussion. For the convenience, let
      $\tau_0:=t_k$ and $x_0:=p_k$.

      Firstly,
      \begin{equation*}
        \begin{split}
        t_k\geq \tau_{i+1}&\geq
        t_k - \sum_{l=0}^i\frac k {\scal_{\tau_l}(x_l)}\\
        &\geq t_k-t_k\sum_{l=0}^i\frac k {2^l\scal_k(p_k)} \geq
        t_k\left (1- \frac{2k}{\scal_k(p_k)}\right )
        =: \epsilon_k >0.
        \end{split}
      \end{equation*}
      for sufficiently large $k$, where $\epsilon_k$ is a constant
      independent of $i$.

      Secondly, $x_i$ is also
      bounded with respect to a background metric $g_k$. Based on assumption,
      ricci curvature is bounded from below by $-(n-1)$,
      and by definition  $\tau_i < t_k$, then by
      equation~\eqref{eq:2}, for any integer $l\geq 1$
      $\dist_k(x_{l-1},x_l)\leq
      \e{(n-1)t_k}\dist_{\tau_{l-1}}(x_{l-1},x_l)$.
      Thus
      \begin{equation*}
        \begin{split}
          \dist_k(p_k,x_i)&\leq
          \e{(n-1)t_k}\sum_{l=1}^i\dist_{\tau_{l-1}}(x_{l-1},x_l)\\
          &\leq \e{(n-1)t_k} \sum_{l=1}^i\frac k
          {\sqrt{\scal_{\tau_{l-1}}(x_{l-1})}} \\
          &\leq \e{(n-1)t_k} \sum_{l=1}^\infty\frac k
          {(\sqrt{2})^{l-1}\sqrt{\scal_k(p_k)}}\\
          &=:C_k \leq \infty,
        \end{split}
      \end{equation*}
      where $C_k$ is a constant independent of $i$.

      Therefore, $(x_i,\tau_i)$ subconverges to a limit, say
      $(x_\infty,\tau_\infty)$. Then, by continuity of scalar curvature,
      $\scal_{\tau_\infty}(x_\infty)= \infty$, which is absurd.
    \end{proof}

    Now we consider the volume ratios.

    By Bishop-Gromov relative volume comparison theorem, for any
    $0< r\leq \mathrm{e} $,
    \begin{equation*}
      \frac {\vol_{\bar{t}_k}(\bar{p}_k,r)}{r^n}
      =\frac {\vol_{\bar{t}_k}(\bar{p}_k,r)}{\vbh r}\frac{\vbh r}{r^n}
      \geq \frac {\vol_{\bar{t}_k}(\bar{p}_k,\mathrm{e})}{\vbh{\mathrm{e}}}\frac{\vbh r}{r^n}
      \geq \frac{v'}{\vbh{\mathrm{e}}}\frac{\vbh r}{r^n}.
    \end{equation*}
    Using relative volume comparison theorem again for euclidean and
    hyperbolic space, for any $0<r$,
    \begin{equation*}
      c(n) \geq \frac{r^n}{\vbh r
    \end{equation*}
    where $c(n)$ is a constant depending only on dimension $n$. So
    \begin{equation*}
      \frac {\vol_{\bar{t}_k}(\bar{p}_k,r)}{r^n}
      \geq \frac{v'}{\vbh e}\frac 1 {c(n)}=:v''>0
    \end{equation*}
    where $v''=v''(v_0,n)$ is a constant depending only on $v_0$ and
    $n$.

    Let $Q_k=\scal_{\bar{t}_k}(\bar{p}_k)$.
    We rescale metric $g_{\bar{t}_k}$ on
    $B_{\bar{t}_k}(\bar{p}_k,k/\sqrt{Q_k})$ for $k$ large. In this case,
    $k/\sqrt{Q_k} < e$, so above
    lower bound for volume ratio is true for $B_{\bar{t}_k}(\bar{p}_k,r)$ with
    any $r<k/\sqrt{Q_k}$.   We define a new metric by
    parabolic rescaling
    \begin{equation*}
      \tilde{g}_k(x,s):=Q_kg(x,\bar{t}_k + Q_k^{-1}s).
    \end{equation*}
    Hereafter $\widetilde{\scal}_k$ is the scalar curvature of
    $\tilde{g}_k$, so are the other $\tilde{\phantom{g}}$ed quantities.
    Then by definition, $\tilde{g}_k$ is a solution to ricci flow on
    $(-k,0]\times B_{\tilde{g}_k(0)}(\bar{p}_k,k)$.
    Note by equation~\eqref{eq:8}, scalar curvature of $\tilde{g}_k$ is
    bounded from above by 2. And the complex sectional curvature
    \begin{equation}\label{eq:9}
      \widetilde{\sec}^c\geq -\frac 1{Q_k} > -1.
    \end{equation}
    Consequently, by Lemma \ref{lem:3} and  \ref{lem:4}, for some
    constant $c(n)>0$ depending only on dimension,
    \begin{equation}\label{eq:5}
      c(n)|\widetilde{\riem}|\leq \widetilde{\scal}\leq 2.
    \end{equation}
    Because volume ratio is scaling-invariant,
    \begin{equation}\label{eq:6}
      \frac{\vol_{\tilde{g}_k(0)}(B_{\tilde{g}_k(0)}(\bar{p}_k,r))}{r^n}\geq v'' > 0
    \end{equation}
    for any  $0< r\leq k$. Therefore, combining
    equation \eqref{eq:5} and \eqref{eq:6} with injectivity radius
    estimate of Cheeger-Gromov-Taylor\cite{MR658471} (see also
    Cabezas-Rivas and Wilking \cite{2011arXiv1107.0606C} Theorem C.3), we have
    \begin{equation*}
      \inj_{\tilde{g}(0)}(\bar{p}_k) \geq c(v'',n) >0.
    \end{equation*}
    Together with equation \eqref{eq:5},  applying Hamilton's compactness
    theorem (see
    \cite{MR1333936} also \cite{MR2302600}), the sequence
    of pointed ricci flow solution
    $(B_{\tilde{g}_k}(\bar{p}_k,k),\tilde{g}_k(s),\bar{p}_k)$ subconverge
    to a complete ancient solution
    $(M_\infty,g_\infty(s),p_\infty)$ for $s\in (-\infty,0]$ in the
    pointed Cheeger-Gromov sense. From the convergence and
    $\widetilde{\scal}_k(\bar{p}_k)=1$,
    $\scal_{g_\infty(0)}(p_\infty)=1$, hence the
    limit metric is non-flat.  For the
    same reason, its riemannian curvature is
    bounded from equation~\eqref{eq:5} and its complex sectional
    curvature is nonnegative from equation
    \eqref{eq:9}. It is also noncompact for every $s\in
    (-\infty,0]$. We first observe the diameter of $g_\infty(0)$
    is infinity. And we pick a sequence of points $y_i\in M_\infty$
    for $i=1,2,...$,
    s.t. $\dist_{g_\infty(0)}(p_\infty,y_i) = i$. Because ricci
    curvature is nonnegative, using Lemma
    \ref{lem:5} again, we show
    $\dist_{g_\infty(-s)}(p_\infty,y_i)\geq i \rightarrow \infty$.
    With above observations, we can apply Lemma 4.5 in Cabezas-Rivas
    and Wilking \cite{2011arXiv1107.0606C} to this ancient solution,
    and get, for any $s\in (-\infty, 0]$,
    \begin{equation*}
      \lim_{r\rightarrow\infty} \frac {\vol_{g_\infty(s)}(B_{g_\infty}(\cdot,r))}{r^n}=0.
    \end{equation*}
    However, this contradicts to the fact that, for any $r>0$,
    \begin{equation*}
      \frac {\vol_{g_\infty(0)}(B_{g_\infty(0)}(p_\infty,r))}{r^n}>0
    \end{equation*}
     from equation \eqref{eq:6}.
  \end{proof}
  Now we are ready to use following theorem to conclude the proof.
  \begin{thm}[B.L.Chen\cite{MR2520796} or M.Simon\cite{MR2439551}]
    \label{thm:5}
    There is a constant $C=C(n)$ with the following property. Suppose
    we have a smooth solution to the ricci flow on $M^n\times [0,T]$
    such that $B_t(x_0,r_0)$, $0\leq t\leq T$, is compactly contained
    in $M$ and
    \begin{enumerate}
    \item $|\riem|\leq r_0^{-2}$ on $B_0(x_0,r_0)$ at $t=0$;
    \item
      \begin{equation*}
        |\riem|_t(x)\leq \frac K t
      \end{equation*}
      where $K\geq 1$, $\dist_t(x_0,x)<r_0$, whenever $0\leq t\leq T$.
    \end{enumerate}
    Then we have
    \begin{equation*}
      |\riem|_t(x)\leq \e{CK}(r_0-\dist_t(x_0,x))^{-2}
    \end{equation*}
    whenever $0\leq t \leq T$, $\dist_t(x_0,x)<r_0$.
  \end{thm}
  Let $x_0$ be any point in $M$. By our assumption, the ricci flow is
  complete for any $t$ hence any
  ball in $M$ is compactly contained. Again by assumption,
  $|\riem_0|_0$ is bounded.
  Thus we may choose $r_0$ small enough so that condition 1 is
  satisfied. Finally, lemma \ref{lem:2} gives condition 2, i.e.
  \begin{equation*}
   c(n)|\riem|_t(x)\leq \scal_t\leq \frac C t.
  \end{equation*}
  Consequently, there exists a $C$ depending on only dimension $n$,
  such that
  \begin{equation*}
    |\riem_t(x)|\leq C
  \end{equation*}
  whenever $0\leq t\leq \epsilon$ and $\dist_t(x_0,x)\leq
  \frac{C'}{r_0^2}$. For $x_0\in M$ is arbitrary, we have $\riem$ is
  uniformly bounded on $M\times [0,\epsilon]$.
\end{proof}

\section{Corollaries}
\label{sec:corollaries}

We conclude the paper by a few corollaries of the main theorem.

\subsection{Solutions with  Nonnegative Curvature}
\label{sec:case-nonn-curv}

If we assume further, that $(M^n,g)$ has nonnegative complex sectional
curvature, together with upper bound for scalar curvature,
it has nonnegative sectional curvature by Lemma \ref{lem:3} and
\ref{lem:4} stated in the next section.
Then $g$ has a positive lower bound for injectivity radius
by using Toponogov's injectivity radius estimate as follows.
\begin{thm}[Toponogov\cite{MR0262974} also Maeda\cite{MR0365411}]
  \label{thm:2}
  If $M$ is a complete noncompact riemannian manifold and for all
  tangent two-plane $\sigma$ its sectional curvature $K_\sigma$
  satisfies the inequality $0\leq K(\sigma)\leq \lambda$ then there
  exists a constant $i_0>0$ such that for all $p\in M$the injectivity
  radius $i(p)$ satisfies
  \begin{equation*}
    i(p)\geq i_0.
  \end{equation*}
  Further, if $0< K(\sigma)\leq \lambda$ for all $\sigma$, then for all
  $p\in M$,
  \begin{equation*}
    i(p)\geq \frac \pi {\sqrt{\lambda}}.
  \end{equation*}
\end{thm}
\begin{rmk}
  \label{rmk:2}
  Beware the lower bound $i_0$ in the
  first part,
  depends not only $\lambda$ but also on $g$ in a general way. In fact,
  a sequence of
  ``thiner'' and  ``thiner'' flat tori whose injetivity radii approach
   $0$ is an example.  We only use this part of the theorem in this paper.
\end{rmk}
Because the sectional
curvature is bounded from above, say by a positive constant $C_0$, by
volume comparison, the volume of any ball with fixed radius
$i_0$ a lower bound. Rescaling the metric
if necessary, by Theorem~\ref{thm:3}, we get the
following corollary.
\begin{cor}
  \label{cor:2}
  Let $(M^n,g)$ be a complete noncompact $n$-dimensional  riemannian
  manifold, with complex sectional curvature $\sec^c \geq 0$
  and $\scal\leq s_0$ for some $s_0 > 0$. Let $g_i(t)$ $i=1,2$ be
  two ricci flow solutions on
  $M\times [0,T]$, with same initial data $g_i(0) = g$. If both of them are
  complete riemannian metrics and
  $\sec^c(g_i(t))\geq 0$ for every $t\in [0,T]$,
  then there exists a constant $\epsilon>0$, s.t.
  $g_1(t) = g_2(t)$ on $M\times [0,\min\{\epsilon,T\}]$.
\end{cor}

\subsection{Solutions on Three Manifolds}
\label{sec:case-three-manifold}

Next let us consider the three manifold. In this case sectional curvature,
complex sectional curvature, and curvature operator
bounded from below are all equivalent. Then we have the following corollary.
\begin{cor}
  \label{cor:3}
    Let $(M,g)$ be a smooth complete riemannian three manifold with bounded sectional
  curvature. And the volume of unit balls in $(M,g)$ is uniformly
  bounded by $v_0>0$. Let $g_i(t)$, $i=1,2$
  $t\in[0,T]$ be two complete solutions of ricci flow with $g_i(0)=g$. If their
  sectional curvatures are both bounded from below for any
  $t\in [0,T]$, then there exists a constant $C_n$ depending on dimension $n$, s.t.
  $g_1(t) = g_2(t)$ on $M\times [0,\min\{\epsilon,T\}]$, where
  $\epsilon=\min\{v_0/2C_n,1/(n-1)\}$.
\end{cor}
\begin{rmk}
  \label{rmk:8}
  Readers may notice that above corollary together with the
  fact that nonnegativity of sectional curvature is preserved by
  $3$-dimensional along ricci flow(Corollary 2.3 in
  \cite{MR2520796}), imply B-L Chen's strong uniqueness
  theorem (Theorem 1.1 in same paper above). However, this in
  essential is not a different proof because two crucial
  components in it are from B-L Chen's paper. One is the
  preservation of nonnegative sectional curvature, the other one is
  Theorem~\ref{thm:5} which we use to prove our main theorem.
\end{rmk}

\subsection{Warped Product Solution}
\label{sec:case-warped-product}

If we consider the ricci flow solutions in only the
class of rotationally symmetric metrics, the curvature condition can
be weaken to ricci bounded from below. Actually we can slightly
enlarge the class. For convenience of stating our corollary, we need the
following definition.
\begin{definition}
  \label{def:3}
  We call $g$ a warped product metric on a manifold $M$ if they
  satisfy the following conditions:
  \begin{enumerate}
  \item $M$ is diffeomorphic to $I\times N$ for some interval $I$ and
    manifold $N$.
  \item There exist a smooth function $f:I\mapsto\mathbb{R}^+$ and a
    riemannian metric $h$ on $N$ s.t. $g=\dd r^2+f^2(r)h$ on coordinate
    system $(r,p)\in I\times N$.
  \end{enumerate}
  We call $(N,h)$ the warped component and $\nabla r$ the radial direction.
\end{definition}
\begin{cor}
  \label{cor:4}
  Let $(M,g)$ be a smooth complete noncompact riemannian manifold
  satisfying the following conditions:
  \begin{enumerate}
  \item $(M,g)$ has bounded sectional curvature.
  \item Volume of any unit ball in $(M,g)$ is bounded from below uniformly
    by a positive constant.
  \item $g$ is a warped product metric and its warped component has nonnegative
    curvature operator.
  \end{enumerate}
  Let $g_i(t)$, $i=1,2$ $t\in[0,T]$ be two complete
  solutions of ricci flow with $g_i(0)=g$. In addition, for any fixed
  $t\in [0,T]$, they satisfy the following conditions:
  \begin{enumerate}
  \item $g_i(t)$ is a warped product metric whose warped
    component is complete and has nonnegative curvature operator.
  \item  Ricci curvature of $g_i$ is bounded from below in
    radial direction by a uniform constant independent of $t$.
  \end{enumerate}
  Then there exists an $\epsilon>0$ such that $g_1(t)=g_2(t)$
  for any $t\in [0,\min\{\epsilon,T\}]$.
\end{cor}

Above corollary is a consequence of facts below.
For convenience, we introduce $\phi=\log f$. Then
$f''/f=\phi''+(\phi')^2$ and $f'/f=\phi'$. Because $f>0$ is a smooth
function, $\phi$ is smooth wherever $f$ is defined.
Then the curvatures (see e.g. Petersen\cite{MR2243772})
\begin{equation*}
  \R(\nabla r\wedge X)=-(\phi''+(\phi)^2)\nabla r\wedge X.
\end{equation*}
where $X$ is any vector field on $N$.
Let $\{E_a\}$, $1\leq a \leq n(n-1)/2$ be a o.n. frame of
$\wedge^2T_xN$, which diagonalize the curvature operator $\R^h$,
with correspondent eigenvalues $\lambda_a$. It is easy to see
\begin{equation*}
  \R(E_a)=(\lambda_a\e{-2\phi}-(\phi')^2)E_a.
\end{equation*}
Let $e_\alpha$ $0\leq\alpha\leq n+1$ be an o.n. frame for $g$ and
$e_0=\nabla r$. Hence $e_i$ $i=1,2,\cdots,n$ are orthogonal frame on $(N,h)$.
We can choose $e_i$ further s.t. they diagonalize $\ric^h$, then for
any $i\neq j$,
\begin{equation*}
  \ric(e_0,e_0)=-n(\phi''+(\phi')^2)\quad \ric(e_0,e_i)=0=\ric(e_i,e_j).
\end{equation*}

Then let us prove the following lemma.
\begin{lemma}
  \label{lem:6}
  Let $a:[0,+\infty)\rightarrow \mathbb{R}$ be a smooth function s.t. $a'+a^2\leq C^2$
  for some constant $C\geq 0$. Then, when $C>0$,
  $-C\leq a\leq \max\{a(0),C\}$, when $C=0$, $0\leq a\leq a(0)$.
\end{lemma}
\begin{proof
  The case $C>0$. Consider the correspondent ODE
  \begin{equation*}
    A'+A^2=C^2.
  \end{equation*}
  When $A(0)=\pm C$, $A\equiv \pm C$.
  When $A(0)\neq \pm C$, its solution is
  \begin{equation*}
    A=C+\frac{2C}{K\e{2Cr}-1}\quad
    \text{where $K$ is a constant satisfying}\ A(0)=C(1+\frac 2 {K-1}).
  \end{equation*}
  Then let us has a closer look of $A$ with different initial values
  other than $\pm C$.
  \begin{enumerate}
  \item   When $A(0)> C$. Then $K > 1$. Hence  $A(r)$ is
    decreasing and $A\rightarrow C$ as $r\rightarrow\infty$. So $A(0)\geq A\geq C$ as
    long as $r\geq 0$.
  \item   When $-C<A(0)<C$. Then $K<0$. So $A(r)$ is increasing and $A\rightarrow C$
    as $r\rightarrow \infty$. So   $C\geq A\geq -C$ as long as $r\geq 0$.
  \item When $A(0)< -C$.  Then $0<K<1$. So $A(r)$ is decreasing and $A\rightarrow
    -\infty$ as $r\rightarrow r_0$ for some $0<r_0< \infty$.
  \end{enumerate}

  Let $A(0)=a(0)$, by comparison principle of ordinary differential
  equations, $a(r)\leq A(r)$ as long as $A(r)$ exists. So,
  when $a(0)<-C$, $a(r)$ becomes discontinuous at some finite $r$,
  which contradicts to our assumption on $a$. Consequently, $a(0)\geq
  -C$,
  thus $a(r)\leq A(r)\leq \max\{a(0),C\}$. Not only so, we can
  further conclude that $a(r)\geq -C$. Otherwise, there exists some
  $\tilde{r}>0$ s.t. $a(\tilde{r})< -C$. Due to the fact that the
  differential inequality
  satisfied by $a$ is translation invariant, it becomes discontinuous
  for the same reason mention as above.

  Conclusion for  $C=0$ follows a similar argument.
\end{proof}

We have the following fact for warped product metrics.
\begin{prop}
  \label{prop:2}
     Let $(N,h)$ be a complete riemannian $n$-manifold with nonnegative
   curvature operator. Let $g=\dd r^2 + f^2(r)h$ be a complete
   noncompact metric on $\mathbb{R}\times N$ or $\mathbb{R}^+\times N$. If the ricci
   curvature of $g$ $\ric(\nabla r, \nabla r)\geq -C^2$ for some constant $C\geq
   0$, then the  curvature operator $\R\geq -C'\id$ for constant
   $C'$.
\end{prop}
\begin{rmk}
  \label{rmk:7}
  When $f(r)$ is defined on  $\mathbb{R}^+\times N$, $g$ is complete if and
  only if
  $f(r)$ is with certain restrictions and  $(N,h)$ is
  a standard sphere (up to scaling). Details can be found in
  Petersen\cite{Petersen-warped-product-e}.
\end{rmk}
\begin{proof}
  Without loss of generality, assume $f>0$ is a smooth function on
  $[0,+\infty)$.
  Since $\ric(\nabla r,\nabla r)\geq -C^2$,
  \begin{equation*}
    \phi''+(\phi')^2\leq \frac{C^2}{n}.
  \end{equation*}
  By Lemma \ref{lem:6}, $\phi'$ is bounded.
  And by assumption $\R^h$ is nonnegative, $\R(E_a)\geq
  -(\phi')^2E_a$ is bounded from below.
\end{proof}
\begin{proof}[Proof of Corollary~\ref{cor:4} ]
  By lemma above, we get $g_i(t)$ are both with curvature
  operators bounded from below,  hence with complex sectional
  curvature bounded from below. Then the
  corollary follows from Theorem~\ref{thm:1}.
\end{proof}

\noindent\textsc{acknowledgement: } Both authors thank professor Xiu-xiong
Chen for his valuable suggestions, constant motivating, and
encouragement. The first author thanks professor Xiu-xiong Chen and
Mathematics Department of Stony Brook University for their hospitality
during his visit and preparation of this paper.

redpajama_set_name: RedPajamaArXiv


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President-elect Joe Biden's Secretary of State Pick Antony J. Blinken Discusses Multilateral Diplomacy and U.S. Global Leadership with Walter Russell Mead in 2016 Discussion, Reposted this Week by Council on Foreign Relations
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redpajama_set_name: RedPajamaCommonCrawl


text:
Govt to clarify taxation of PDPs
The Government announced today that it will clarify the law to ensure that bonus issues of shares distributed under profit distribution plans (PDPs) are taxed in the same way as shares issued under other dividend reinvestment plans. For more information see the media statement.
Hon Bill English
Deputy Prime Minister
Minister of Finance
Government to clarify taxation of profit distribution plans
The Government will clarify the law to ensure that bonus issues of shares distributed under profit distribution plans (PDPs) are taxed in the same way as shares issued under other dividend reinvestment plans, Finance Minister Bill English and Revenue Minister Peter Dunne announced today.
"We are concerned about a possible lack of consistency in the tax treatment of the two types of share issues," Mr English and Mr Dunne said.
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text:
solution "AnimatProjectGen"
	configurations { "Debug", "Release" }
	platforms {"x32"}

	project "AnimatProjectGen"
		language "C#"
		kind     "ConsoleApp"
		files  { "../*.cs"}
		links  { "System" }

		configuration {"../Templates/**.lua" }
			buildaction "Embed"

		configuration { "Debug" }
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			flags   { "Symbols" }

		configuration { "Release" }
			 flags { "Optimize" }

redpajama_set_name: RedPajamaGithub


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The story unfolds in another 'possible' modern age. The main characters are youngsters called "Kildren", who are destined to live eternally in their adolescence. The Kildren are conscious that every day could be the last, because they fight a "war as entertainment" organized and operated by adults. But as they embrace the reality they are faced wit… more The story unfolds in another 'possible' modern age. The main characters are youngsters called "Kildren", who are destined to live eternally in their adolescence. The Kildren are conscious that every day could be the last, because they fight a "war as entertainment" organized and operated by adults. But as they embrace the reality they are faced with, they live their day-to-day lives to the full. less
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text:
\section*{Author Summary}

Infectious disease mapping provides a powerful synthesis of evidence in an effective, visually condensed form. With the advent of new web-based data sources and systematic data collection in the form of cross-sectional surveys and health facility reporting, there is high demand for accurate methods to predict spatial maps. The primary technique used in spatial mapping is known as Gaussian process regression (GPR). GPR is a flexible stochastic model that not only allows the modelling of disease driving factors such as the environment but can also capture unknown residual spatial correlations in the data.  We introduce a method that blends state-of-the-art machine learning methods with GPR to produce a model that substantially out-performs all other methods commonly used in disease mapping. The utility of this new approach also extends far beyond just mapping, and can be used for general machine learning applications across computational biology, including Bayesian optimisation and mechanistic modelling.

\section*{Introduction}

Infectious disease mapping with model-based geostatistics\cite{Diggle2007Model-basedGeostatistics} can provide a powerful synthesis of the available evidence base to assist surveillance systems and support progress towards global health targets, revealing the geographical bounds of disease occurrence and the spatial patterns of transmission intensity and clinical burden. A recent review found that out of 174 infectious diseases with a strong rational for mapping only 7 (4\%) have thus far been comprehensively mapped\cite{Hay2013GlobalDisease.b}, with a lack of accurate, population representative, geopositioned data having been the primary factor impeding progress. In recent years this has begun to change as increasing volumes of spatially referenced data are collected from both cross-sectional household surveys and web-based data sources (e.g. Health Map\cite{Freifeld2008HealthMap:Reports.}), bringing new opportunities for scaling up the global mapping of diseases. Alongside this surge in new data, novel statistical methods are needed that can generalise to new data accurately while remaining computationally tractable on large datasets.  In this paper we will introduce one such method designed with these aims in mind.

Owing to both a long history of published research in the field and a widespread appreciation amongst endemic countries for the value of cross-sectional household surveys as guides to intervention planning, malaria is an example of a disease that \textit{has} been comprehensively mapped. Over the past decade, volumes of publicly-available malaria prevalence data---defined as the proportion of parasite positive individuals in a sample---have reached sufficiency to allow for detailed spatio-temporal mapping \cite{Bhatt2015The2015.}. From a statistical perspective, the methodological mainstay of these malaria prevalence mapping efforts has been Gaussian process regression\cite{Rasmussen2006GaussianLearning,Bishop2006PatternLearning, Gething2011A2010,Hay2009A2007.}.  Gaussian processes are a flexible semi-parametric regression technique defined entirely through a mean function, $\mu(\cdot)$, and a covariance function, $k(\cdot,\cdot)$. The mean function models an underlying trend, such as the effect of environmental/socio-economic factors, while the covariance function applies a Bayesian shrinkage to residual variation from the mean such that points close to each other in space and time tend towards similar values. The resulting ability of Gaussian processes to strike a parsimonious balance in the weighting of explained and unexplained spatio-temporal variation has led to their near exclusive use in contemporary studies of the geography of malaria prevalence \cite{Hay2009A2007,Gething2011A2010,Diggle2007Model-basedGeostatistics, Gosoniu2012SpatiallyData.,Adigun2015MalariaData.,Bhatt2015The2015.}.

Outside of disease mapping, Gaussian processes have been used for numerous applications in machine learning, including regression\cite{Rasmussen2006GaussianLearning,Bishop2006PatternLearning, Diggle2007Model-basedGeostatistics}, classification\cite{Rasmussen2006GaussianLearning}, and optimisation\cite{Snoek2012PracticalAlgorithms}; their popularity leading to the development of efficient computational techniques and statistical parametrisations. A key challenge for the implementation of Gaussian process models arises in statistical learning (or inference) of any underlying parameters controlling the chosen mean and covariance functions. Learning is typically performed using Markov Chain Monte Carlo (MCMC) or by maximizing the marginal likelihood \cite{Rasmussen2006GaussianLearning}, both of which are made computationally demanding by the need to compute large matrix inverses returned by the covariance function. The complexity of this inverse operation is $\mathcal{O}(n^3)$ in computation and $\mathcal{O}(n^2)$ in storage in the naive case \cite{Rasmussen2006GaussianLearning}, which imposes practical limits on data sizes\cite{Rasmussen2006GaussianLearning}. MCMC techniques may be further confounded by mixing problems in the Markov chains. These challenges have necessitated the use of highly efficient MCMC methods, such as Hamiltonian MCMC\cite{homan2014no}, or posterior approximation approaches, such as the integrated nested Laplace approximation \cite{Rue2009ApproximateApproximations}, expectation propagation\cite{vanhatalo2010approximate,Minka2001ExpectationInference,Rasmussen2006GaussianLearning}, and variational inference\cite{hensman2013gaussian,Opper2009TheRevisited.}. Many of these methods  adopt finite dimensional representations of the covariance function yielding sparse precision matrices, either by specifying a fully independent training conditional (FITC) structure\cite{quinonero2005unifying} or by identifying a Gaussian Markov Random Field (GMRF) in approximation to the continuous process \cite{Lindgren2011AnApproachb}.

Alongside these improved methods for inference, recent research has focussed on model development to increase the flexibility and diversity of parametrisations for the covariance function, with new techniques utilising solutions to stochastic partial differential equations (allowing for easy extensions to non-stationary and anisotropic forms\cite{Lindgren2011AnApproachb}), the combination of kernels additively and multiplicatively\cite{duvenaud2013structure}, and various spectral representations\cite{wilson2013gaussian}.

One aspect of Gaussian processes that has remained largely neglected is the mean function which is often---and indeed with justification in some settings---simply set to zero and ignored. However, in the context of disease mapping, where the biological phenomena are driven by a complex interplay of environmental and socioeconomic factors, the mean plays a central role in improving the predictive performance of Gaussian process models. Furthermore, it has also been shown that using a well-defined mean function can allow for simpler covariance functions (and hence simpler, scalable inference techniques)  \cite{Fuglstad2015DoesFields}.

The steady growth of remotely-sensed data with incredible spatio-temporal richness \cite{Weiss2015Re-examiningApproach.} combined with well-developed biological models\cite{Weiss2014AirPrediction.} has meant that there is a rich suite of environmental and socio-economic covariates currently available. In previous malaria mapping efforts these covariates have been modelled as simple linear predictors \cite{Gething2011A2010,Hay2009A2007,Gosoniu2012SpatiallyData.} that fail to capture complex non-linearities and interactions, leading to a reduced overall predictive performance. Extensive covariate engineering can be performed by introducing large sets of non-linear and interacting transforms of the covariates, but this brute force combinatorial problem quickly becomes computationally inefficient\cite{Weiss2015Re-examiningApproach.,Bhatt2015The2015.}.

In the field of machine learning and data science there has been great success with algorithmic approaches that neglect the covariance and focus on learning from the covariates alone\cite{Hastie2009TheLearning,caruana2006empiricalb}. These include tree based algorithms such as boosting \cite{Friedman2001GreedyMachine} and random forests \cite{BreimanRandomForests}, generalized additive spline models \cite{HastieGam,Wood2006GeneralizedR}, multivariate adaptive regression splines \cite{Friedman1991MultivariateSplines}, and regularized regression models \cite{Zou2005RegularizationNet}. The success of these methods is grounded in their ability to manipulate the bias-variance tradeoff\cite{Geman1992NeuralDilemma}, capture interacting non-linear effects, and perform automatic covariate selection. The technical challenges of hierarchically embedding these algorithmic methods within the Gaussian process framework are forbidding, and many of the approximation methods that make Gaussian process models computationally tractable would struggle with their inclusion. Furthermore, it is unclear which of these approaches would best characterize the mean function when applied across different diseases and settings. In this paper we propose a simplified embedding method based on stacked generalisation \cite{Wolpert1992StackedGeneralization,BreimanStackedRegressions} that focuses on improving the mean function of a Gaussian process, thereby allowing for substantial improvements in the predictive accuracy beyond what has been achieved in the past.

\section*{Results}

\begin{figure} \centering \subfigure[]{\includegraphics[width=0.75\textwidth]{Zone1.pdf}\label{fig:fig2a}} \subfigure[]{\includegraphics[width=0.75\textwidth]{Zone1.pdf}\label{fig:fig2b}} \subfigure[]{\includegraphics[width=0.75\textwidth]{Zone3.pdf}\label{fig:fig2c}} \subfigure[]{\includegraphics[width=0.75\textwidth]{Zone4.pdf}\label{fig:fig2d}} \caption{Comparisons of cross-validation MSE versus MAE versus correlation. Plot (a) is Eastern African, (b) Southern African, (c) North Eastern Africa and (d) Western Africa.  Legend abbreviations: (1) sGP - stacked Gaussian process, (2) CWM - stacked constrained weighted mean, (3) GP - standard Gaussian process, (4) GBM - Gradient boosted trees, (5) GAS - Generalised additive splines, (6) FR - Random forests, (7) MARS - Multivariate adaptive regression splines and (8) LIN - Elastic net regularised linear regression.}\label{fig:fig2} \end{figure}

The results of our analysis are summarised in figure \ref{fig:fig2} where pairwise comparisons of MSE versus MAE versus correlation are shown. We found a consistent ranking pattern in the generalisation performance of the methods across the Eastern, Southern and Western African regions (Figures \ref{fig:fig2a},\ref{fig:fig2b} and \ref{fig:fig2d}), with the stacked Gaussian process approach presented in this paper outperforming all other methods. The constrained weighted mean stacked approach was the next best method followed by the standard Gaussian process (with a linear mean) and Gradient boosted trees. Random forests, multivariate adaptive regression splines and generalised additive splines all had similar performance and the worst performing method was the elastic net regularised regression. For the North Eastern region (Figure \ref{fig:fig2c}), again the stacked Gaussian process approach was the best performing method, but the standard Gaussian process performed better than the constrained weighted mean stacked approach, though only in terms of MAE and MSE.

One average, across all regions, the stacked Gaussian process approach reduced the MAE and MSE by $9\% \; [1\% - 13\%]$ (values in square brackets are the min and max across all regions)  and $16\% \; [2\% - 24\%]$ respectively and increased the correlation by $3\% \; [1\% - 5\%]$ over the next best constrained weighted mean approach thereby empirically reinforcing the theoretical bounds derived in Appendix \ref{proof}. When compared the the widely used elastic net linear regression the relative performance increase of the Gaussian process stacked approach is stark, with reduced MAE and MSE by  $25\% \; [12\% - 33\%]$ and $25\% \; [19\% - 30\%]$ respectively and increase in correlation by $39\% \; [20\% - 50\%]$.

Compared to the standard Gaussian process previously used in malaria mapping the stacked Gaussian process approach reduced MAE and MSE by $10\% \; [3\% - 14\%]$  and $18\% \; [9\% - 26\%]$ respectively and increased the correlation by $6\% \; [3\% - 7\%]$.

Consistently across all regions the best non-stacked method was the standard Gaussian process with a linear mean function. Of the level zero generalisers gradient boosted trees were the best performing method, with performance close to that of  the standard Gaussian process. The standard Gaussian process only had a modest improvement over Gradient boosted trees with average reductions in MAE and MSE of $4\% \; [1\% - 8\%]$  and $7\% \; [1\% - 13\%]$ respectively and increases in correlation of $3\% \; [1\% - 7\%]$.

\section*{Discussion}

All the level zero generalisation methods used in this paper have been previously applied to a diverse set of machine learning problems and have track records of good generalisability \cite{Hastie2009TheLearning}. For example, in closely related ecological applications, these level zero methods have been shown to far surpass classical learning approaches\cite{Elith2008ATrees}. However, as introduced by Wolpert\cite{Wolpert1992StackedGeneralization}, rather than picking one level zero method, an ensemble via a second generaliser has the ability to improve prediction beyond that achievable by the constituent parts\cite{Krogh95neuralnetwork}. Indeed, in all previous applications \cite{BreimanStackedRegressions,Wolpert1992StackedGeneralization,vanderLaan2007SuperLearner,Sill2009Feature-WeightedStacking} ensembling by stacking has consistently produced the best predictive models across a wide range of regression and classification techniques. The most popular level one generaliser is the constrained weighted mean with convex combinations. The key attraction of this level one generaliser is the ease of implementation and theoretical properties \cite{Krogh95neuralnetwork,vanderLaan2007SuperLearner}. In this paper we show that, for disease mapping, stacking using Gaussian processes is more predictive and generalises better than both any single method in isolation, and the more common stacking approach using a constrained weighted mean.

The key benefit of stacking is summarised in equation \ref{eq:biasvariance} where the total error of an ensemble model can be reduced by using multiple, very different, but highly predictive models. However, stacking using a constrained weighted mean only ensures that the predictive power of the covariates are fully utilised, and does not exploit the predictive power that could be gained from characterising any residual covariance structure. The standard Gaussian process suffers from the inverse situation where the covariates are underexploited, and predictive power is instead gained from leveraging residual spatio-temporal covariance. In a standard Gaussian process the mean function is usually paramaterised through simple linear basis functions\cite{Rasmussen2004GaussianLearning} that are often unable to model the complex non linear interactions needed to correctly capture the true underlying mean. This inadequacy is best highlighted by the poor generalisation performance of the elastic net regularised regression method across all regions. The trade off between the variance explained by the covariates versus that explained by the covariance function will undoubtedly vary from setting to setting. For example in the Eastern, Southern and Western African regions, the constrained weighted mean stacking approach performs better than the standard Gaussian process, and the level 0 gradient boosted trees generaliser performs almost as well as the standard Gaussian process. For these regions, this shows a strong influence of the covariates on the underlying process. In contrast, for the North Eastern African region, the standard Gaussian process does better than both the constrained weighted mean approach (in terms of error not correlation) and all of the Level 0 generalisers, suggesting a weak influence of the covariates. However, for all zones, the stacked Gaussian process approach is consistently the best approach across all predictive metrics. By combining both the power of Gaussian processes to characterise a complex covariance structure, and multiple algorithmic approaches to fully exploit the covariates, the stacked Gaussian process approach combines the best of both worlds, and predicts well in all settings.

This paper introduces one way of stacking that is tailored for spatio-temporal data. However the same principles are applicable to purely spatial or purely temporal data, settings in which Gaussian process models excel. Additionally, there is no constraint on the types of level 0 generalisers than can be used; dynamical models of disease transmission e.g.~\cite{Smith2004StaticsMosquitoes.} \cite{Griffin2014EstimatesAfrica} can be fitted to data and used as the mean function within the stacked framework. Using dynamical models in this way can constrain the mean to include known biological mechanisms that can potentially improve generalisability, allow for forecast predictions and help restrict the model to only plausible functions when data is sparse. Finally multiple different stacking schemes can be designed (see Appendix \ref{alternative} for details), and relaxations on linear combinations can be implemented (e.g. \cite{Sill2009Feature-WeightedStacking}).

Gaussian processes are increasingly being used for expensive optimisation problems \cite{OHagan1991BayesHermiteQuadrature} and Bayesian quadrature \cite{Hennig2015ProbabilisticComputations}. In current implementations both of these applications are limited to low dimensional problems typically with $n<10$. Future work will explore the potential for stacking to extend these approaches to high dimensional settings. The intuition is that the level 0 generalisers can accurately and automatically learn much of the latent structure in the data, including complex features like non-stationarity, which are a challenge for Gaussian processes. Learning this underlying structure through the mean can leave a much simpler residual structure\cite{Fuglstad2015DoesFields} to be modelled by the level one Gaussian process.

In this paper we have focused primarily on prediction, that is neglecting any causal inference and only searching for models with the lowest generalisation error. Determining causality from the complex relationships fitted through the stacked algorithmic approaches is difficult, but empirical methods such as partial dependence  \cite{Friedman2001GreedyMachine} or individual conditional expectation\cite{Goldstein2015PeekingExpectation} plots can be used to approximate the marginal relationships from the various covariates. Similar statistical techniques can also be used to determine covariate importance.

Increasing volumes of data and computational capacity afford unprecedented opportunities to scale up infectious disease mapping for public health uses\cite{Pigott2015PrioritisingMapping.}.  Maps of diseases and socio-economic indicators are increasingly being used to inform policy \cite{Bhatt2013TheDengue.,Bhatt2015The2015.}, creating demand for methods to produce accurate estimates at high spatial resolutions. Many of these maps can subsequently be used in other models but, in the first instance, creating these maps requires continuous covariates, the bulk of which, come from remotely sensed sources. For many indicators, such as HIV or Tuberculosis, these remotely sensed covariates serve as proxies for complex phenomenon, and as such, the simple mean functions in standard Gaussian processes are insufficient to predict with accuracy and low generalisation error. The stacked Gaussian process approach introduced in this paper provides an intuitive, easy to implement method that predicts accurately through exploiting information in both the covariates and covariance structure.

\section*{Methods}
\subsection*{Gaussian process regression}

We define our response, $\mathbf{y}_{s,t} = \{y_{(s,t)[1]},...,y_{(s,t)[n]}\}$, as a vector of $n$ empirical logit transformed malaria prevalence surveys at location-time pairs, $(s,t)[i]$, with $\mathbf{X}_{s,t}=\{(\mathbf{x}_{1:m})[1],\ldots,(\mathbf{x}_{1:m})[n]\}$ denoting a corresponding $n\times m$ design matrix of $m$ covariates (see Section \ref{covs}). The likelihood of the observed response is $\mathbb{P}(\mathbf{y}_{s,t}|\mathbf{f}_{s,t},\mathbf{X}_{s,t},\theta)$, which we will write simply as $\mathbb{P}(y|f(s,t),\theta)$, suppressing the spatio-temporal indices for ease of notation.  Naturally, $f(s,t)$ $[=\mathbf{f}_{s,t}]$ is the realisation of a Gaussian process with mean function, $\mu_\theta(\cdot)$, and covariance function, $k_\theta(\cdot,\cdot)$, controlled by elements of a low-dimensional vector of hyperparameters, $\theta$.  Formally, the Gaussian process is defined as an $(s,t)$-indexed stochastic process for which the joint distribution over any finite collection of points, $(s,t)[i]$, is multivariate Gaussian with mean vector, $\mu_i = m((s,t)[i]|\theta)$, and covariance matrix, $\Sigma_{i,j}=k((s,t)[i],(s,t)[j]|\theta)$. The Bayesian hierarchy is completed by defining a prior distribution for $\theta$, which may potentially include hyperparameters for the likelihood (e.g., over-dispersion in a beta-binomial) in addition to those on parametrising the mean and covariance functions.  In hierarchical notation, supposing for clarity an iid Normal likelihood with variance, $\sigma_e^2$:
\begin{eqnarray} \label{eq:eq1}
\theta & \sim & \pi(\theta)\nonumber\\
f(s,t)|\mathbf{X}_{s,t},\theta & \sim & GP(\mu_{\theta},k_{\theta})\nonumber \\
y|f(s,t),\mathbf{X}_{s,t},\theta & \sim & N(f(s,t),\mathds{1}\sigma_e^2)
\end{eqnarray}
Following Bayes theorem the posterior distribution resulting from this hierarchy becomes:
\begin{equation}\label{eq:posterior}
\mathbb{P}(\theta,f(s,t)|y) = \frac{\mathbb{P}(y|f(s,t),\theta) \mathbb{P}(f(s,t)|\theta)\mathbb{P}(\theta)}{\int \int \mathbb{P}(y|f(s,t),\theta) \{d\mathbb{P}(f(s,t)|\theta)\} \{d\mathbb{P}(\theta)\}},
\end{equation}
where the denominator in Equation \ref{eq:posterior} is the marginal likelihood, $\mathbb{P}(y)$.

Given the hierarchical structure in Equation \ref{eq:eq1} and the conditional properties of Gaussian distributions, the conditional predictive distribution for the mean of observations, $z$ $[=\mathbf{z}_{s^\prime,t^\prime}]$, at location-time pairs, $(s^\prime,t^\prime)[j]$, for a given $\theta$ is also Gaussian with form:
\begin{eqnarray}
z|y,\theta &\sim& N(\mu^\ast,\Sigma^\ast) \\
\label{eq:eq2a} \mu^\ast &=&  \mu_{(s^\prime,t^\prime)|\theta}+\Sigma_{(s^\prime,t^\prime),(s,t)|\theta}\Sigma_{y|(s,t),\theta}^{-1}\left(y-\mu_{(s,t)|\theta}\right)\\
 \label{eq:eq2b} \Sigma^\ast &=&  \Sigma_{(s^\prime,t^\prime)|\theta} - \Sigma_{(s^\prime,t^\prime),(s,t)|\theta}\Sigma_{y|(s,t),\theta}^{-1}\Sigma_{(s,t),(s^\prime,t^\prime)|\theta}
\end{eqnarray}
where $\Sigma_{y|(s,t),\theta}= \left ( \Sigma_{\theta} +\mathds{1}\sigma_e^2\right)$. For specific details on the parametrisation of $Sigma$ see Appendix

When examining the conditional expectation in equation \ref{eq:eq2b} and splitting the summation into terms $\mu_{(s^\prime,t^\prime)|\theta}$ and $\Sigma_{(s^\prime,t^\prime),(s,t)|\theta}\Sigma_{y|(s,t),\theta}^{-1}\left(y-\mu_{(s,t)|\theta}\right)$,  it is clear that the first specifies a global underlying mean while the second augments the residuals from that mean by the covariance function. Clearly if the mean function fits the data perfectly the covariance in the second term of the expectation would drop out and conversely if the mean function is zero, then only the covariance function would model the data. This expectation therefore represents a balance between the underlying trend and the residual correlated noise.

In most applications of Gaussian process regression a linear mean function ($\mu_{\theta}=\mathbf{X}_{s,t}\mathbf{\beta}$) is used, where $\mathbf{\beta}$ is a vector of $m$ coefficients. However, when a rich suite of covariates is available this linear mean may be sub-optimal, limiting the generalisation accuracy of the model. To improve on the linear mean, covariate basis terms can be expanded to include parametric nonlinear transforms and interactions, but finding the optimal set of bases is computationally demanding and often leaves the researcher open to data snooping\cite{Abu-Mostafa2012LearningData}. In this paper we propose using an alternative two stage statistical procedure to first obtain a set of candidate non-linear mean functions using multiple different algorithmic methods fit without reference to the assumed spatial covariance structure and then include those means in the Gaussian process via stacked generalisation.

\subsection*{Stacked generalisation}\label{stackedgen}

Stacked generalisation \cite{Wolpert1992StackedGeneralization}, also called stacked regression \cite{BreimanStackedRegressions}, is a general ensemble approach to combining different models. In brief stacked generalisers combine different models together to produce a meta-model with equal or better predictive performance than the constituent parts \cite{vanderLaan2007SuperLearner}. In the context of malaria mapping our goal is to fuse multiple algorithmic methods with Gaussian process regression to both fully exploit the information contained in the covariates and model spatio-temporal correlations.

To present stacked generalisation we begin by introducing standard ensemble methods and show that stacked generalisation is simply a special case of this powerful technique. To simplify notation we suppress the spatio-temporal index and dependence on $\theta$. Consider $\mathcal{L}$ models, with outputs $\tilde{y}_i(x),i={1,...,\mathcal{L}}$. The choice of these models is described in appendix \ref{genmethods}. We denote the true target function as $f(x)$ and can therefore write the regression equation as $y_i(x)=f(x) + \epsilon_i(x)$. The average sum-of-squares error for model $i$ is defined as $E_i=\mathbb{E}[(\tilde{y}_i(x)-f(x))^2]$. Our goal is to estimate an ensemble model across all $\mathcal{L}$ models, denoted as $M(\tilde{y}_1,..,\tilde{y}_{\mathcal{L}})$. The simplest choice for $C$ is an average across all models $M(\tilde{y}_1,..,\tilde{y}_{\mathcal{L}})=\tilde{y}_\mathrm{avg}(x)=\frac{1}{\mathcal{L}}\sum_{i=1}^{\mathcal{L}} \tilde{y}_i(x)$. However this average assumes that the error of all models are the same, and that all models perform equally well. The assumption of equal performance may hold when using variants of a single model (i.e. bagging) but is unsuitable when very different models are used. Therefore a simple extension would be to use a weighted mean across models $M(\tilde{y}_1,..,\tilde{y}_{\mathcal{L}})=\tilde{y}_\mathrm{wavg}(x)=\sum_{i=1}^{\mathcal{L}} \beta_i \tilde{y}_i(x)$ subject to constraints $\beta>0 \; \forall i,\sum_{i=1}^{\mathcal{L}}\beta=1$ (convex combinations). These constraints prevent extreme predictions in well predicting models and impose the sensible inequality $\tilde{y}_\mathrm{min}(x)\leq \tilde{y}_\mathrm{wavg}(x)\leq \tilde{y}_\mathrm{max}$ \cite{BreimanStackedRegressions}. The optimal $\beta$s can be found by quadratic programming or by Bayesian linear regression with a Dirichlet/categorical prior on the coefficients. One particularly interesting result of combining models using this constrained weighted mean is the resulting decomposition of error into two terms \cite{Krogh95neuralnetwork}
\begin{equation}
\label{eq:biasvariance}
\begin{split}
\mathbb{E}[(\tilde{y}_\mathrm{wavg}(x)-f(x))^2] &=\sum_{i=1}^n \beta_i \mathbb{E}[(\tilde{y}_i(x)-f(x))^2] \\ & - \sum_{i=1}^n \beta_i \mathbb{E}[(\tilde{y}_i(x)-\tilde{y}_\mathrm{wavg}(x))^2]
\end{split}
\end{equation} Equation \ref{eq:biasvariance} is a reformulating of the standard bias-variance decomposition\cite{Geman1992NeuralDilemma} where first term describes the average error of all models, and the second (termed the ambiguity) is the spread of each member of the ensemble around the weighted mean, and measures the disagreement among models. Equation \ref{eq:biasvariance} shows that combining multiple models with low error but with large disagreements produces a lower overall error. It should be noted that equation \ref{eq:biasvariance} makes the assumption that $y(x)=f(x)$.

Combination of models in an ensemble as described above can potentially lead to reductions in errors. However the ensemble models introduced so far are based only on training data and therefore neglect the issue of model complexity and tell us nothing about the ability to generalise to new data. To state this differently, the constrained weighted mean model will always allocate the highest weight to the model that most over fits the data. The standard method of addressing this issue is to use cross validation as a measure of the generalisation error and select the best performing of the $\mathcal{L}$ models. Stacked generalisation provides a technique to combine the power of ensembles described above, but also produces models that can generalise well to new data. The principle idea behind stacked generalisation is to train $\mathcal{L}$ models (termed level 0 generalisers) and generalise their combined behaviour via a second model (termed the level 1 generaliser). Practically this is done by specifying a $K$-fold cross validation set, training all $\mathcal{L}$ level 0  models on these sets and using the cross validation predictions to train a level 1 generaliser. This calibrates the level 1 model based on the generalisation ability of the level 0 models. After this level 1 calibration, all level 0 models are refitted using the full data set and these predictions are used in the level 1 model without refitting (This procedure is more fully described in algorithm \ref{alg:stgen} and the schematic design shown in Appendix figure \ref{fig:fig3}). The combination of ensemble modelling with the ability to generalise well has made stacking one of the best methods to achieve state-of-the art predictive accuracy \cite{Puurula2014KaggleSolution,BreimanStackedRegressions, vanderLaan2007SuperLearner}.

Defining the most appropriate level 1 generaliser based on a rigorous optimality criteria is still an open problem, with most applications using the constrained weighted mean specified above \cite{BreimanStackedRegressions,vanderLaan2007SuperLearner}. Using the weighted average approach can be seen as a general case of cross validation, where standard cross validation would select a single model by specifying a single $\beta_i$ as 1 and all other $\beta_i$s as zero. Additionally, it has been shown that using the constrained weighted mean method will perform asymptotically as well as the best possible choice among the family of weight combinations \cite{vanderLaan2007SuperLearner}.

Here we suggest using Gaussian process regression as the level 1 generaliser. Revisiting equation \ref{eq:eq2a} we can replace $\mu_{(s^\prime,t^\prime)|\theta}$ with a linear stacked function $\mu_{(s^\prime,t^\prime)|\theta}=\sum_{i=1}^{\mathcal{L}} \beta_i \tilde{y}_{i}(s^\prime,t^\prime)$ across $\mathcal{L}$ level 0 generalisers, where the subscript denotes predictions from the $i$th Level 0 generaliser (see algorithm \ref{alg:stgen}. We also impose inequality constraints on $\beta_i$ such that $\beta>0 \; \forall i,\sum_{i=1}^{\mathcal{L}}\beta=1$. This constraint allows the $\beta$s to approximately sum to one and helps computational tractability. It should be noted that empirical analysis suggests that simply imposing $\beta>0$ is normally sufficient \cite{BreimanStackedRegressions}.

The intuition in this extended approach is that the stacked mean of the Gaussian process uses multiple different methods to exploit as much predictive capacity from the covariates and then leaves the spatio-temporal residuals to be captured through the Gaussian process covariance function. In Appendix \ref{proof} we show that this approach yields all the benefits of using constrained weighted mean (equation \ref{eq:biasvariance}) but allows for a further reduction in overall error from the covariance function of the Gaussian process.

\subsection*{Data, Covariates and Experimental Design} \label{covs} The hierarchical structure most commonly used in infectious disease mapping is that shown in Equation  \ref{eq:eq1}. In malaria studies our response data are often discrete random variables representing the number of individuals testing positive for the \emph{Plasmodium falciparum}  (\emph{Pf}) malaria parasite, $N^{+}$, out of the total number tested, $N$, at a given location.  If the response is aggregated (e.g.) from the individual household level to a cluster or enumeration area level usually the centroid of the component sites is used as the spatial position datum.  The ratio of $N^{+}$ to $N$ is defined as the parasite rate or prevalence. To aid computation, the response data transformed to the empirical logit of observed prevalence \cite{Bhatt2015The2015.,Diggle2007Model-basedGeostatistics}, which for $N \gtrsim 20$ can be well approximated by a Gaussian likelihood. Pre-modelling standardisation of the available prevalence data for age and diagnostic type has also been performed on the sample used here, as described in depth in \cite{Bhatt2015The2015.,Gething2011A2010}. Our analysis is performed over Sub-Saharan Africa with the study area and dataset partitioned into 4 epidemiologically-distinct regions \cite{Gething2011A2010}---Eastern Africa, Western Africa, North Eastern Africa, and Southern Africa---each of which was modelled separately (see Figure \ref{fig:fig1}.

\begin{figure}[] \centering  \includegraphics[width=0.7\textwidth]{Fig1.png} \caption{Study area of stable malaria transmission in Sub-Saharan Africa. Our analysis was performed on 4 zones - Western Africa, North eastern Africa, Eastern Africa and Southern Africa} \label{fig:fig1} \end{figure}

The covariates (i.e., independent variables) used in this research consist of raster layers spanning the entire continent at a 2.5 arc-minute (5 km x 5 km) spatial resolution. The majority of these raster covariates were derived from high temporal resolution satellite images that were first gap filled \cite{Weiss2014AnTime-series.} to eliminate missing data (resulting primarily from persistent cloud cover over equatorial forests) and then aggregated to create a dynamic (i.e. temporally varying) dataset for every month throughout the study period (2000-2015). The list of covariates is presented in Table \ref{table1} and detailed information on individual covariates can be found here \cite{Weiss2015Re-examiningApproach.,Weiss2014AirPrediction., Weiss2014AnTime-series.}. The set of monthly dynamic covariates was further expanded to include lagged versions of the covariate at 2 month, 4 month and 6 month lags. The main objective of this study was to judge the predictive performance of the various generalisation methods and therefore no variable selection or thinning of the covariate set was performed. It should be noted however that many of the level 0 generalisers performed variable selection automatically (e.g. elastic net regression).

The level 0 generalisers used were gradient boosted trees \cite{Friedman2001GreedyMachine,Friedman2002StochasticBoosting}, random forests\cite{breiman2001random}, elastic net regularised regression\cite{Zou2005RegularizationNet}, generalised additive splines \cite{Wood2006GeneralizedR,Hastie2009TheLearning} and multivariate adaptive regression splines\cite{Friedman1991MultivariateSplines}. The level 1 generalisers used were stacking using a constrained weighted mean and stacking using Gaussian process regression. We also fitted a standard Gaussian process for benchmark comparisons with the level 0 and 1 generalisers. Stacked fitting was performed following algorithm \ref{alg:stgen}. Full analysis and $K$-fold Cross validation was performed 5 times and then averaged to reduce any bias from the choices of cross validation set. The averaged cross validation results were used to estimate the generalisation error by calculating the mean squared error (MSE), mean absolute error (MAE) and the correlation.

\pagebreak

\section*{Tables and Algorithms}

\begin{algorithm} \caption{Stacked Generalisation Algorithm: The algorithm proceeds as follows. In line 2 to 4 the covariates,response and number of cross validation folds is defined. Lines 6 to 9 fits all level 0 generalisers to the full data set. Lines 10 to 16 fits all level 0 generalisers to cross validation data sets. Line 17 to 18 fits a level 1 generaliser to the cross validation predictions and Line 19 returns the final output by using the level 1 generaliser to predict on the full predictions}\label{alg:stgen} \begin{algorithmic}[1] \Procedure{Stack}{}\Comment{covariate and response input} \State \textbf{Input} $X$ as a $n\times m$ design matrix \State \textbf{Input} $y$ as a $n$ vector of responses \State \textbf{Input} $v$ cross validation folds \State \textbf{choose} $l$, $ \mathcal{L}(y,X)$ models\Comment{level zero generalisers} \State \textbf{define} $n\times l$ matrix $P$ \Comment{matrix of predictions} \For{$i\gets 1, l$} \State \textbf{fit} $\mathcal{L}_i(y,X)$ \State \textbf{predict} $P_{\cdot,i} = \mathcal{L}_i(y,X)$ \EndFor

\State \textbf{split} $X,y$ into $\{g_1,...g_v\}$  groups $\{X_{g_1},..,X_{g_v}\}$ and $\{y_{g_1},..,y_{g_v}\}$ \Comment {training set} \State \textbf{add} remaining samples to $\{X_{/g_1},..,X_{/g_v}\}$ and $\{y_{/g_1},..,y_{/g_v}\}$ \Comment {testing set} \State \textbf{define} $n\times l$ matrix $H$ \Comment{matrix cross validation of predictions} \For{$i\gets 1, l$} \For{$j\gets 1, v$} \State \textbf{fit} $\mathcal{L}_i(y_{g_j},X_{g_j})$ \State \textbf{predict} $H_{/g_j,i} = \mathcal{L}_i(y_{/g_j},X_{/g_j})$ \EndFor \EndFor \State \textbf{choose} $\mathcal{L}^*(y,H)$ model \Comment{level one generaliser} \State \textbf{fit} $\mathcal{L}^*(y,H)$  \\ \Return {$\mathcal{L}^*(y,P)$} \Comment{final prediction output} \EndProcedure \end{algorithmic} \end{algorithm}

\begin{table}[H] \caption{List of Environmental, Socio-demographic and Land type covariates used.} \label{table1} \begin{tabular}{|>{\centering}p{4cm}|>{\centering}p{4cm}|>{\centering}p{ 2cm}|>{\centering}p{4cm}|} \hline \multicolumn{1}{|>{\centering}p{4cm}|}{Variable Class} & \multicolumn{3}{c|}{Variable(s)SourceType}\tabularnewline \hline \hline Temperature & Land Surface Temperature (day, night, and diurnal-flux) & MODIS Product & Dynamic Monthly\tabularnewline \hline Temperature Suitability & Temperature Suitability for Plasmodium falciparum & Modeled Product  & Dynamic Monthly\tabularnewline \hline Precipitation & Mean Annual Precipitation & WorldClim  & Synoptic\tabularnewline \hline Vegetation Vigor & Enhanced Vegetation Index & MODIS Derivative & Dynamic Monthly\tabularnewline \hline Surface Wetness & Tasseled Cap Wetness & MODIS Derivative & Dynamic Monthly\tabularnewline \hline Surface Brightness  & Tasseled Cap Brightness  & MODIS Derivative & Dynamic Monthly\tabularnewline \hline IGBP Landcover & Fractional Landcover  & MODIS Product & Dynamic Annual\tabularnewline \hline IGBP Landcover Pattern & Landcover Patterns  & MODIS Derivative & Dynamic Annual\tabularnewline \hline Terrain Steepness & SRTM Derivatives & MODIS Product & Static\tabularnewline \hline Flow \& Topographic Wetness & Topographically Redistributed Water & SRTM Derivatives & Static\tabularnewline \hline Elevation & Digital Elevation Model & SRTM & Static\tabularnewline \hline Human Population & AfriPop  & Modeled Products & Dynamic Annual\tabularnewline \hline Infrastructural Development & Accessibility to Urban Centers and Nighttime Lights & Modeled Product and VIIRS  & Static\tabularnewline \hline Moisture Metrics & Aridity and Potential Evapotranspiration & Modeled Products & Synoptic\tabularnewline \hline \end{tabular} \end{table}

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text:
Dan + Shay
2019 ACM Awards Behind The Scenes: Dan + ShayGo behind the scenes at the 54th annual ACM Awards to hear from 2019 Duo of the Year, Dan + Shay.
Country Duo Featuring Wexford Native Picks Up First GrammyCountry duo Dan + Shay, featuring Wexford native Dan Smyers, picked up their first Grammy on Sunday.
Florida Georgia Line Coming To KeyBank Pavilion, Dan + Shay To Be Opening ActFlorida Georgia Line is coming back to western Pennsylvania for a concert.
Pittsburgh Native Wins At CMT AwardsPittsburgh native Dan Smyers and his country music partner Shay Mooney, collectively known as Dan and Shay, won the CMT Music Award for Best Duo Video of the Year.
Nick Jonas Comes Back to Texas, Pairs with Dan + Shay on 'Jealous' & 'Chains' at ACM AwardsHe performed the song with country duo Dan + Shay.
Interview: Dan + Shay's 'Magical' Journey to the ACM Awards"We were just sharing our stories about meeting girls in the summer and that summer love that you had," Shay said.


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To introduce travel to Taiwan, Taiwan Tourism Bureau present a promotion program for [Time for Taiwan!] on July 10th afternoon in Manila. This event will be at the meeting room 7 of SMX convention center at 2nd Floor. During the program Taiwan Tourism will introduce to the local Philippines people the best tour package program to Taiwan, a free lucky draw of free flight ticket, hotel accommodation voucher and several Taiwan special gifts are prepared for the attendees, it is recommended to local Philippines people must come to event and grab the big prize!
This program activity is at the same time of the Travel Madness Fair in Manila, Taiwan Tourism specially invite the local famous celebrity, and famous blogger to be the program promoter for this event, by introducing the tasty food, shopping, romantic love scenery, flowers seasons, amusement parks in Taiwan, the event will share the Taiwan's most fun, premium travel package, target to invite the Philippines people to make their travel plan for a tour to Taiwan.
The culinary culture of the Chinese people goes back a very long time; and while Chinese food can be enjoyed in every large city in the world today, true gourmets know that only in Taiwan is it possible to enjoy fine authentic cuisine from all the different regions of China.
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From the north to the south, there are many large shopping centers and chain department stores in Taiwan. For example the north has Taipei 101 shopping complex, Ximending and Zhongxiao-Dunhua Shopping Area in Taipei; at Central Taiwan there are Fengjia night market, Jingming 1st Street shopping area in Taichung; South Taiwan has New Juejiang Commercial Area, Dream Mall in Kaoshiung. The easy transportation of Taiwan high speed train, Taipei MRT, Kaoshiung express train are very convenient for travelers to enjoy premium travel experience and full of fun any time.
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In the mid to late 1970's, there was a TV show that circled around a father/son duo that collected and sold scrap metal. The father was played by the famous comedian Redd Foxx, and the hilarity ensued when various difficult situations came up that the scrap metal recyclers were not quite sure how to handle. However, the show does illustrate some very important things about the business in which Sanford and Son were involved--scrap metal recycling. The pair did it "old school" and lived on the profits they made. Here is how you can do scrap metal pickup and recycling old school and earn a living too.
As the 1970s show clearly illustrated, the business is operated out of your own home. This cuts operation costs dramatically. You may have to clear it with the city in which you reside unless you live on the outskirts of town where it is relatively acceptable to collect your scrap metal and deposit it on the lawn.
You will need a truck, and the more it can hold, the better. While you could advertise that you pick up scrap metal, you can also drive around town on garbage collection days and look for any metal scrap that has been set curbside. Since it is clearly being tossed to the curb and no longer wanted, it is free to take, although it is generally considered polite to ask first. If no one is available to ask, then you can take it and toss it in your truck. You can do this just about any day of the week that you do not have a scheduled scrap metal pickup appointment.
Even though you are attempting to collect, sort, recycle and sell scrap metal, you still want it to be reasonably organized. When you bring a truckload back home to your property, try to place similar objects in their own piles (e.g., old bike frames together on one pile, old car parts together on another pile, etc.). That way, any customer looking to buy a specific piece of scrap metal knows exactly which pile to search and you can easily point that pile out to the customer. As for everything you intend to sell to a scrap metal recycling foundry, place that far off to the side, maybe in a dumpster, until you can have it transported to the foundry.


Today the University of Virginia Center for Politics released its latest book, Pendulum Swing, drawing on the collective wisdom of nearly two dozen of the nation's top political analysts, journalists, and academics for comprehensive analysis of the 2010 midterm elections and the subsequent policy implications. As President Barack Obama makes his first address to the 112th Congress, this timely publication by Center for Politics Director Larry Sabato and a national team of contributing experts delves into the overlooked details of the 2010 elections. Looking back at the titanic political shift in Washington, DC and around the nation, the authors explore how that shift could affect the 2012 presidential election. The book is the newest in an ongoing annual election series developed by the Center for Politics and published by Pearson/Longman publishers (www.pearsonhighered.com).


Industrial Growth Partners has sold Tucson, Arizona-based IOTA Engineering, a maker of lighting products and power equipment, to Acuity Brands Inc. No financial terms were disclosed.
San Francisco, CA – Industrial Growth Partners is pleased to announce the sale of IOTA Engineering, L.L.C. to Acuity Brands, Inc. (NYSE: AYI) in a transaction that closed on May 1, 2018.
IOTA, based in Tucson, Arizona, is a leading manufacturer of highly-engineered emergency lighting products and power equipment for commercial and institutional applications, both in the U.S. and internationally.
IGP, together with members of the IOTA management team and certain outside investors, purchased the Company in February 2017. Since that time, the Company has continued to execute on its strategic plan to strengthen its leading market position providing critical power supply equipment. During IGP's ownership, the Company invested heavily to expand engineering capabilities, research and development facilities, and sales infrastructure.
For more information, please contact contact Dan Delaney or Micah Wickersheim at 415-882-4550 or go to igpequity.com.