first version of generative model and data

.gitignore

@@ -2,3 +2,10 @@
 *.bbl
 *.bib
 *.pyc
+*.aux
+*.blg
+*.out
+*.fdb_latexmk
+*.fls
+*.log
+

README.txt

@@ -26,7 +26,7 @@ test.solution -- the sample solution on the test data
 Overview
 ========
 
-The model is completely described in ```description.odt``` and this should be
+The model is completely described in ```description.tex``` and this should be
 translated in the Probabilistic Programming Language of your
 choosing. The unlabeled data in ```test.blind``` (and optionally the labeled
 data in ```training.data```) comprises the observations to the model. The
@@ -39,11 +39,11 @@ script ```test.data```. One can also compare the results versus the
 baseline in ```test.solution```.
 
 The files ```generate.py``` and ```solve.py``` have only been provided
-for convenience they shouldn't normally be used. However, if you want to
-check the performance of your model on more than just the provided data
-you may generate more as needed. The sample solver is based loosely on
-the published greedy algorithm, and may be used as a competitive
-baseline.
+for convenience. These files shouldn't normally be used. However, if you
+want to check the performance of your model on more than just the
+provided data you may generate more as needed. The sample solver is
+based loosely on the published greedy algorithm, and may be used as a
+competitive baseline.
 
 Authors
 =======

description.tex

@@ -93,7 +93,7 @@ \subsection{Events}
   e^i_m & \sim \text{Exponential}(\ \cdot\ ;\ \lambda_m, location=2)
 \end{align*}
 
-\subsection{True Arrivals}
+\subsection{True Detections}
 
 The seismic energy from an event travels radially outwards in distinct
 phases, each of which may or may not be detected by a station depending
@@ -103,14 +103,14 @@ \subsection{True Arrivals}
 the surface, refer to \citet{iaspei2011} for a full list. In this work we only
 consider the first arriving phase, the {\em P} phase.
 
-We define a true arrival $\Lambda^{ik}$ as the moment of first arrival
+We define a true detection $\Lambda^{ik}$ as the moment of first arrival
 of the energy from an event $i$ at a seismic station $k$. Various signal
 processing algorithms are applied on the raw waveforms to detect an
 arrival, and then station processing algorithms collect various
 attributes of the
-arrival such as time, azimuth, slowness, and amplitude referenced by
+detection such as time, azimuth, slowness, and amplitude referenced by
 $\Lambda^{ik}_t$, $\Lambda^{ik}_z$, $\Lambda^{ik}_s$, and
-$\Lambda^{ik}_a$ respectively. Time is quite obviously the arrival time
+$\Lambda^{ik}_a$ respectively. Time is quite obviously the detection time
 of the energy, azimuth refers to the geographical direction of the
 incoming seismic waves, and amplitude is the height of the initial
 peak. Slowness is a more peculiar term, it refers to the inverse of the
@@ -139,17 +139,17 @@ \subsubsection{Detection Probability}
 \[\text{logistic}
 (\mu_{d0}^k + \mu_{d1}^k e^i_m + \mu_{d2}^k \Delta_{ik}) \ .\]
 
-\subsubsection{Arrival Time}
+\subsubsection{Detection Time}
 
 The theoretical travel time of a seismic wave at a distance of $\delta$ is
 given by the travel time function,
 \[ I_T(\delta) = -.023 * \delta^2 + 10.7 * \delta\, + 5.\]
 
-The arrival time is a Laplacian centered near the theoretical arrival time,
+The detection time is a Laplacian centered near the theoretical detection time,
 \[ \Lambda_t^{ik} \sim \text{Laplacian}(\ \cdot \ , e^i_t + I_T(\Delta_{ik}) +
 \mu_t^k \  , \ \theta_t^k) . \]
 
-\subsection{Arrival Azimuth}
+\subsection{Detection Azimuth}
 
 The azimuth of location $b=(lon_2, lat_2)$ as observed from location
 $a=(lon_1, lat_1)$ is given by the function $G_z(a, b) \in [0,
@@ -176,13 +176,13 @@ \subsection{Arrival Azimuth}
 \psi'(z_1, z_2) &= (z_2 - z_1) + 360\ \text{mod}\ 360.
 \end{align*}
 
-The difference of the arrival azimuth from the theoretical
+The difference of the detection azimuth from the theoretical
 station-to-event azimuth is distributed as a Laplacian,
 
 \[\psi(G_z(s^k_l, e^i_l), \Lambda_z^{ik}) \sim \text{Laplacian}(\ \cdot
 \ ,\  \mu_z^k, \ \theta_z^k \ ) . \] 
 
-\subsection{Arrival Slowness}
+\subsection{Detection Slowness}
 The slowness at distance $\delta$ given by $I_S(\delta)$ is simply the
 derivative of the travel time. In other words, slowness measures the
 the time that the seismic wave takes to travel between two points very
@@ -191,34 +191,34 @@ \subsection{Arrival Slowness}
 \[ I_S(\delta) = -.046 * \delta + 10.7\, .\]
 Note that $I_S$ is always positive since $\delta \in [0, 180]$.
 
-The arrival slowness is a Laplacian centered near the theoretical
+The detection slowness is a Laplacian centered near the theoretical
 slowness,
 
 \[ \Lambda_s^{ik} \sim \text{Laplacian}(\ \cdot \ , I_S(\Delta_{ik}) +
 \mu_s^k \  , \ \theta_s^k) . \]
 
-\subsection{Arrival Amplitude}
+\subsection{Detection Amplitude}
 
-The log of the arrival amplitude has a Gaussian distribution with a mean
+The log of the detection amplitude has a Gaussian distribution with a mean
 determined by the event magnitude and travel time.
 
 \[\log(\Lambda_a^{ik}) \sim \text{Gaussian}(\ \cdot \ ,\ \mu^k_{a0} 
 + \mu^k_{a1} e^i_m + \mu^k_{a2} I_T(\Delta_{ik})\  ,\ \sigma_a^k \ ) . \]
 
 
-\subsection{False Arrivals}
+\subsection{False Detections}
 
 Each station $k$ has its own time-homogenous Poisson process generating
-false arrivals with rate $\lambda^k_f$. In other words, if $\xi^k$ is
-the set of false arrivals in a time interval $T$,
+false detections with rate $\lambda^k_f$. In other words, if $\xi^k$ is
+the set of false detections in a time interval $T$,
 \begin{align*}
 |\xi^k| & \sim \text{Poisson}(\ \cdot \ , \ \lambda^k_f T),
-\intertext{and the arrival time $\xi^k_t$ is uniformly distributed,}
+\intertext{and the detection time $\xi^k_t$ is uniformly distributed,}
 \xi^k_t & \sim \text{Uniform}(\ \cdot \ , \ 0, \ T) .
 \intertext{The azimuth and slowness are also uniformly distributed
   between their possible values, as follows:}
 \xi^k_z & \sim \text{Uniform}(\ \cdot \ , \ 0, \ 360),  \\
-\xi^k_s & \sim \text{Uniform}(\ \cdot \ , \ I_S(0), \ I_S(180)) .
+\xi^k_s & \sim \text{Uniform}(\ \cdot \ , \ I_S(180), \ I_S(0)) .
 \intertext{However, the log-amplitude is distributed as a Gaussian,}
 \log{\xi^k_a} & \sim \text{Gaussian}(\ \cdot \ , \ \mu^k_f, \sigma^k_f) .
 \end{align*}
@@ -230,20 +230,58 @@ \section{Hyperpriors and Constants}
 R & = 6371 \, km \\
 \lambda_e & \sim \text{Gamma}(\ \cdot \ , \ 6.0, \frac{1}{4 \pi R^2 T}) \\
 \lambda_m & = \log(10) \\
-\mu^k_{d0} & \sim \text{Gaussian}(\ \cdot \ , \ -16.6,\ 4.9) \\
-\mu^k_{d1} & \sim \text{Gaussian}(\ \cdot \ , \ 3.9,\ 0.89) \\
-\mu^k_{d2} & \sim \text{Gaussian}(\ \cdot \ , \ -0.052,\ 0.018) \\
+\left[ 
+  \begin{array}{l}
+    \mu^k_{d0} \\
+    \mu^k_{d1} \\
+    \mu^k_{d2} 
+  \end{array} \right] & = \text{MVarGaussian} \left( \ \cdot \ ; \
+\left[
+  \begin{array}{l}
+    -10.4 \\
+    3.26 \\
+    -.0499
+  \end{array}
+\right]
+ , 
+\left[
+  \begin{array}{lll}
+    13.43 & -2.36 & -.0122 \\
+    -2.36 & .452 & .000112 \\
+    -.0122 & .000112 & .000125 \\
+  \end{array}
+\right]
+\ \right) \\
 \mu_t^k & = 0 \\
-\theta_t^k & \sim \text{InvGamma}(\ \cdot \ , \ 96.9, \ 102.0) \\
+\theta_t^k & \sim \text{InvGamma}(\ \cdot \ , \ 120, \ 118) \\
 \mu_z^k & = 0 \\
-\theta_z^k & \sim \text{InvGamma}(\ \cdot \ , \ 7.0, \ 80.0) \\
+\theta_z^k & \sim \text{InvGamma}(\ \cdot \ , \ 5.2, \ 44) \\
 \mu_s^k & = 0 \\
-\theta_s^k & \sim \text{InvGamma}(\ \cdot \ , \ 7.0, \ 9.0) \\
-\mu^k_{a0} & \sim \text{Gaussian}(\ \cdot \ , \  -7.3, \ 1.1) \\
-\mu^k_{a1} & \sim \text{Gaussian}(\ \cdot \ , \  2, \ 0.21) \\
-\mu^k_{a2} & \sim \text{Gaussian}(\ \cdot \ , \  -0.002, \ 0.00055) \\
+\theta_s^k & \sim \text{InvGamma}(\ \cdot \ , \ 6.7, \ 7.5) \\
+\left[ 
+  \begin{array}{l}
+    \mu^k_{a0} \\
+    \mu^k_{a1} \\
+    \mu^k_{a2} 
+  \end{array} \right] & = \text{MVarGaussian} \left( \ \cdot \ ; \
+\left[
+  \begin{array}{l}
+    -7.3 \\
+    2.03 \\
+    -.00196
+  \end{array}
+\right]
+ , 
+\left[
+  \begin{array}{lll}
+    1.23 & -.227 & -.000175 \\
+    -.227 & .0461 & .0000245 \\
+    -.000175 & .0000245 & .000000302 \\
+  \end{array}
+\right]
+\ \right) \\
 (\sigma^k_a)^2 & \sim \text{InvGamma}(\ \cdot \ , \  21.1, \ 12.6) \\
-\lambda^k_f & \sim \text{Gamma}(\ \cdot \ , \  0.782, \ 0.0018) \\
+\lambda^k_f & \sim \text{Gamma}(\ \cdot \ , \  2.1, \ 0.0013) \\
 \mu^k_f & \sim \text{Gaussian}(\ \cdot \ , \  -0.6, \ 0.6) \\
 (\sigma^k_f)^2 & \sim \text{InvGamma}(\ \cdot \ , \  10.34, \ 9.49) \\
 \end{align*}
@@ -379,6 +417,17 @@ \subsection{Inverse-Gamma}
  \, x^{-\alpha-1} e^{-\frac{\theta}{x}} \, , \] defined over all $x \in
  \mathbb{R}_{>0}$.
 
+\subsection{Multi-variate Gaussian}
+
+The Muti-variate Gaussian distribution with mean vector
+$\mu \in \mathbb{R}^k$, and
+covariance matrix $\Sigma  \in \mathbb{R}^{k \times k}$ has probability density
+
+\[\text{MVarGaussian}(x; \mu, \Sigma) = \frac{1}{\sqrt{(2 \pi)^k | \Sigma
+    |}}
+e^{\left( -\frac{1}{2} (x - \mu)^T \Sigma^{-1} (x - \mu) \right)}
+\]
+defined over all $x \in \mathbb{R}^k$.
 \end{appendices}
 
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