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\begin{document}
\title{Non-Gaussian Bayesian spatiotemporal modelling}

\author{Tha\'{\i}s C.O.~da Fonseca and Mark F.J.~Steel\\
{University of Warwick\\}
\vspace{1.2in}
%Seminar for Dept.~of Statistics, PUC, Chile
}
\maketitle

\overlays{3}{
\begin{slide}{Introduction}

Consider processes that vary continuously in time and space
\[
\{Z(s,t);(s,t)\in D\times T\}
\]
Usual assumption: Gaussian process 
\vspace{0.2in}
\fromSlide{2}{
Convenient, but realistic?
\vspace{0.2in} 

Here we explore non-Gaussian models that accommodate 
}

\fromSlide{3}{


\begin{itemize}

\item regions of increased spatial variance 
\item regions of increased variance in time
\item outliers through fat-tailed nugget effect
\item nonseparable covariance in time and space
%\item Discussion
\end{itemize}}

\end{slide}}






\overlays{2}{
\begin{slide}{Spatiotemporal Modelling}

Start with nonseparable covariance structure:
\vspace{0.1in}

\fromSlide{2}{
$(U,V)$ is bivariate nonnegative random vector \\
$\{Z_1(s;u)\}$ spatial random process with covariance function $C_1(s;u)=\sigma_1 \exp\{-\gamma_1(s)u\}$\\
$\{Z_2(t;v)\}$ temporal random processes with covariance function $C_2(t;v)=\sigma_2 \exp\{-\gamma_2(t)v\}$ \\ 
Define the process
\[
Z(s,t)=Z_1(s;U)Z_2(t;V)
\]
Covariance function of $Z(s,t)$ is valid and generally nonseparable:
\[
C(s,t)=\int C_1(s;u)C_2(t;v)d\mu(u,v)
\]
}
\end{slide}}


\overlays{3}{
\begin{slide}{Spatiotemporal Modelling}

We define $U=X_0+X_1$ and $V=X_0+X_2$\\
\vspace{0.2in}

$X_0$, $X_1$ and $X_2$ independent nonnegative r.v.'s with finite m.g.f.'s $M_{0}$, $M_{1}$ and $M_{2}$\\
\vspace{0.2in}

\fromSlide{2}{
Then the resulting covariance function is given by
\[
C(s,t)=\sigma^2M_0(-\gamma_1(s)-\gamma_2(t))\;M_1(-\gamma_1(s))\;M_2(-\gamma_2(t)),
\]
where $\sigma^2=\sigma_1\sigma_2$
}
\vspace{0.2in}

\fromSlide{3}{
Examples: Cauchy, Mat\'ern, generalized Mat\'ern
}



\end{slide}}

\overlays{3}{\begin{slide}{Scale mixing}

Consider $
\tilde Z(s,t)=\tilde Z_1(s;U)\tilde Z_2(t;V),
$

\fromSlide{2}{
where
\[
\tilde Z_1(s;U) = \sqrt{1-\tau^2}\frac{Z_1(s;U)}{\sqrt{\lambda_1(s)}}+\tau \frac{\epsilon(s)}{\sqrt{h(s)}}
\]
$\{\lambda_1(s)\}$ spatially correlated mixing process \\
$\{\epsilon(s)\}$ uncorrelated Gaussian process (nugget effect)\\ 
$\{h(s)\}$ uncorrelated process in $\Re_+$\\
\vspace{0.1in}

$\lambda_1(s)$ regions in space with larger variances and $h(s)$ outliers\\
\vspace{0.1in}}

\fromSlide{3}{
Also consider temporally correlated mixing process $\{\lambda_2(t)\}$
\[
\tilde Z_2(t;V) = \frac{Z_2(t;V)}{\sqrt{\lambda_2(t)}}
\]
models regions in time with larger variances
}

\end{slide}}

\overlays{3}{\begin{slide}{Scale mixing in space}

Consider smooth process $\lambda_1(s)$ s.t. $\tilde Z_1(s;u)$ is mean square continuous

\vspace{0.2in}

Take $\{\ln(\lambda_1(s))\}$ a Gaussian process with mean $-{\nu}/{2}$ and covariance structure $\nu M_1(-\gamma_1(s))$:\\ $\lambda_{1}(s_i)$ has prior mean 1 and var $\exp(\nu)-1$

\vspace{0.2in}

Induces spatial smoothness (without nugget)
\fromSlide{2}{
\vspace{0.2in}

Consider uncorrelated process $h(s)$ with $\ln( h_i)\sim {\rm N}(-\nu_{h}/2,\nu_{h})$
}
\fromSlide{3}{
\vspace{0.2in}

Kurtosis a (non-monotonous) function of $(\nu,\nu_h)$

\vspace{0.1in}

Limit as $\nu,\nu_h\to 0$: Gaussian}

\end{slide}}


\overlays{3}{\begin{slide}{Scale mixing in time}

Consider smooth process $\lambda_2(t)$ 

\vspace{0.2in}

Take $\{\ln(\lambda_2(t))\}$ a Gaussian process with mean $-{\nu_2}/{2}$ and covariance structure $\nu_2 M_2(-\gamma_2(t))$

\vspace{0.2in}

Induces temporal smoothness
\fromSlide{2}{
\vspace{0.2in}

Allows for periods in time with increased variance

}
\fromSlide{3}{
\vspace{0.1in}

\begin{itemize}
\item Smooth spatial heterogeneity: $\lambda_1$
\item Smooth temporal heterogeneity: $\lambda_2$
\item spatial outliers: $h$
\end{itemize}



Limit as $\nu,\nu_h,\nu_2\to 0$: Gaussian}

\end{slide}}



\overlays{2}{\begin{slide}{Application to temperature data}



\end{slide}}










\begin{slide}{Example: Financial Time Series}

Let $s_0,s_1,\dots,s_T$ be daily observations of a stock price or a stock market index:
Here S\&P 500 data for 1980-87.\\
\vspace{0.2in}
Typically we model the log returns $y_t=\log s_t-\log s_{t-1}$. \\
\vspace{0.2in}

%\centerline{\resizebox{9cm}{5cm}{\includegraphics{data.eps}}}


\end{slide}

\begin{slide}{Example: Financial Time Series}

Such series usually exhibit the following stylized features:
\begin{itemize}
\item Zero or small mean.
\item Volatility clustering.
\item Heavier than normal tails (Sample kurtosis of the returns is 90.3 --
clearly indicating heavy tails).
\end{itemize}
\vspace{0.3in}

\end{slide}

\begin{slide}{Example: Financial Time Series}

We extend existing nonparametric models by allowing the returns distribution to change over time:
\[
\frac{y_t}{\sqrt{h_t}}\sim\N(\mu_t,\sigma^2_t)
\]
\[
\log h_t\sim\N(\delta \log h_{t-1},\sigma^2_v)
\]
\[
(\mu_t,\sigma^2_t)\sim G_t
\]
\[
G_t\sim DPAR(M,H;\lambda)
\]

\vspace{0.2in}

So here we use $G_t$ with Dirichlet marginals, and $H\left(\mu,\sigma^{-2}\right)=\N(\mu\vert 0,0.01\sigma^2)\Ga\left(\left.\sigma^{-2}\right\vert 1,0.1\right)$.

\vspace{0.1in}

Prior distributions are placed on the $\lambda$, $M$, $\delta$ and $\sigma_v^2$.

\end{slide}

\begin{slide}{Results S\&P 500 data}

\begin{figure}[h!]
\begin{center}
\begin{tabular}{ccc}
(a) & (b) & (c)\\
%\psfig{figure=returns_dist_SP.eps,  height=3cm} & \psfig{figure=fpred_SV_SP.eps,  height=3cm} & \psfig{figure=var_SP.eps,  height=3cm}
\end{tabular}
\end{center}
\caption{\small S\&P 500 data:
(a) selected predictive pdf's for observables at times 481, 681 and 1181,
(b) heatmap of the predictive pdf's
 at each time point and (c) variance of the fitted returns over time}
\label{f:dist_SP}
\end{figure}

\end{slide}


\begin{slide}{Example: Income data}
Longitudinal data on GDP of 110 EU regions from 1977 to 1996. Model as:
\[
y_{it}\sim\N(\mu_{it},\sigma^2_{it})
\]
\[
(\mu_{it},\sigma^2_{it})\sim G_t
\]
\[
G_t\sim DPAR(M,H;\lambda)
\]
\[
H\left(\mu,\sigma^{-2}\right)=\N(\mu\vert \mu_0,0.01\sigma^2)\Ga\left(\left.\sigma^{-2}\right\vert 1,0.1\right)
\]

\vspace{0.2in}

So we ignore longitudinal aspect of data (but simply have 110 observations per time period)

\end{slide}



\end{document}
