2020-12-10 21:43:15 +00:00
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\pdf_title "Hidden Markov Models"
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\pdf_author "Andy Pack"
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\begin_body
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\begin_layout Title
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\size giant
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Hidden Markov Model Analysis
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\end_layout
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\begin_layout Author
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Andy Pack
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\end_layout
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EEEM030
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2020-12-10 21:43:15 +00:00
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January 2021
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\size large
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Department of Electrical and Electronic Engineering
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Faculty of Engineering and Physical Sciences
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University of Surrey
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2020-12-10 22:59:55 +00:00
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\begin_layout Abstract
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abstract
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Andy Pack / 6420013
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\begin_layout Left Footer
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January 2021
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\backslash
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pagenumbering{arabic}
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\begin_layout Section
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Introduction
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\end_layout
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2020-12-31 19:30:39 +00:00
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\begin_layout Standard
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The Markov model is a stochastic model for analysing random processes.
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The model describes a system of multiple states and the transitions between
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them; a key feature is the
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\emph on
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Markov property
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\emph default
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which states that the future state of the system depends only on the current
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state
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\begin_inset CommandInset citation
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LatexCommand cite
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key "towards-data-science-markov-intro"
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literal "false"
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\end_inset
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.
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This is also described as the system being
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\emph on
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memoryless
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\emph default
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.
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status open
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Described markov chains
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status open
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Difference with hidden markov models
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status open
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Application to machine learning and speech in particular
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\begin_layout Subsection
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Brief
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\end_layout
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\begin_layout Standard
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This work analyses a continuous-density hidden Markov model of 2 emitting
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states describing a 1D observation space.
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Various statistics and calculations will be conducted on the model including
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forward, backward, occupation and transition likelihoods.
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Using these, an iteration of training will be completed using the Baum-Welch
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equations resulting in a new set of transition probabilities and re-estimated
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output probabilities.
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\begin_layout Standard
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The model is specified as follows; the transition probabilities are described
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by entry (
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\begin_inset Formula $\pi$
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\end_inset
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) and exit (
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\begin_inset Formula $\eta$
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\end_inset
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) probabilities for each state along with a matrix of state-to-state transitions
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(
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\begin_inset Formula $a$
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),
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\begin_layout Standard
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\begin_inset Formula
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\[
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A=\left\{ \pi_{j},a_{ij},\eta_{i}\right\}
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\]
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\end_inset
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\begin_layout Standard
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|
|
|
\begin_inset Formula
|
|
|
|
|
\[
|
|
|
|
|
\pi_{j}=\left\{ 0.44,0.56\right\}
|
|
|
|
|
\]
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
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|
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|
|
|
|
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|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
\begin_inset Formula
|
|
|
|
|
\[
|
|
|
|
|
a_{ij}=\left[\begin{array}{cc}
|
|
|
|
|
0.92 & 0.06\\
|
|
|
|
|
0.04 & 0.93
|
|
|
|
|
\end{array}\right]
|
|
|
|
|
\]
|
|
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|
|
|
|
|
\end_inset
|
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|
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|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
\begin_inset Formula
|
|
|
|
|
\[
|
|
|
|
|
\eta_{i}=\left\{ 0.02,0.03\right\}
|
|
|
|
|
\]
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
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|
|
|
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|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
These, in combination, describe a state topology than can be seen graphically
|
|
|
|
|
in figure
|
|
|
|
|
\begin_inset CommandInset ref
|
|
|
|
|
LatexCommand ref
|
|
|
|
|
reference "fig:State-topology"
|
|
|
|
|
plural "false"
|
|
|
|
|
caps "false"
|
|
|
|
|
noprefix "false"
|
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|
|
|
\end_inset
|
|
|
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|
|
|
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|
|
.
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
As previously mentioned, the states are described 1D continuous probability
|
|
|
|
|
density functions (PDF) of a Gaussian profile.
|
|
|
|
|
Each has an associated mean and variance as seen below,
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
\noindent
|
|
|
|
|
\align center
|
|
|
|
|
\begin_inset Tabular
|
|
|
|
|
<lyxtabular version="3" rows="3" columns="3">
|
|
|
|
|
<features tabularvalignment="middle">
|
|
|
|
|
<column alignment="center" valignment="top">
|
|
|
|
|
<column alignment="center" valignment="top">
|
|
|
|
|
<column alignment="center" valignment="top">
|
|
|
|
|
<row>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
State,
|
|
|
|
|
\begin_inset Formula $i$
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
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|
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|
|
\end_layout
|
|
|
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|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
1
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
2
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
</row>
|
|
|
|
|
<row>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
Mean,
|
|
|
|
|
\begin_inset Formula $\mu_{i}$
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
1.00
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
4.00
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
</row>
|
|
|
|
|
<row>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
Variance,
|
|
|
|
|
\begin_inset Formula $\varSigma_{i}$
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
1.44
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
0.49
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
</row>
|
|
|
|
|
</lyxtabular>
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
Finally, a set of observations from the above model are provided below,
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
\noindent
|
|
|
|
|
\align center
|
|
|
|
|
\begin_inset Formula
|
|
|
|
|
\[
|
|
|
|
|
\mathcal{O}=\left\{ 3.8,4.2,3.4,-0.4,1.9,3.0,1.6,1.9,5.0\right\}
|
|
|
|
|
\]
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
These are used as the basis for training the model using the Baum-Welch
|
|
|
|
|
equations.
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Section
|
|
|
|
|
Implementation
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
\begin_inset Float figure
|
|
|
|
|
wide false
|
|
|
|
|
sideways false
|
|
|
|
|
status open
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\noindent
|
|
|
|
|
\align center
|
|
|
|
|
\begin_inset Graphics
|
|
|
|
|
filename StateTopology.png
|
|
|
|
|
lyxscale 50
|
|
|
|
|
width 65col%
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Caption Standard
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
State topology for the specified Markov Model
|
|
|
|
|
\begin_inset CommandInset label
|
|
|
|
|
LatexCommand label
|
|
|
|
|
name "fig:State-topology"
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Section
|
|
|
|
|
Results
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Subsection
|
|
|
|
|
Output Probability Density Functions
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
\begin_inset Float figure
|
|
|
|
|
wide false
|
|
|
|
|
sideways false
|
|
|
|
|
status open
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\noindent
|
|
|
|
|
\align center
|
|
|
|
|
\begin_inset Graphics
|
|
|
|
|
filename res/pdfs.png
|
|
|
|
|
width 50col%
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Caption Standard
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
Probability density functions for both states across the observation space
|
|
|
|
|
\begin_inset CommandInset label
|
|
|
|
|
LatexCommand label
|
|
|
|
|
name "fig:PDFs"
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Subsection
|
|
|
|
|
Observation Probability Densities
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
\begin_inset Float table
|
|
|
|
|
wide false
|
|
|
|
|
sideways false
|
|
|
|
|
status open
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\noindent
|
|
|
|
|
\align center
|
|
|
|
|
\begin_inset Tabular
|
|
|
|
|
<lyxtabular version="3" rows="11" columns="4">
|
|
|
|
|
<features tabularvalignment="middle">
|
|
|
|
|
<column alignment="center" valignment="top">
|
|
|
|
|
<column alignment="center" valignment="top">
|
|
|
|
|
<column alignment="center" valignment="top" width="10col%">
|
|
|
|
|
<column alignment="center" valignment="top" width="10col%">
|
|
|
|
|
<row>
|
|
|
|
|
<cell multirow="3" alignment="center" valignment="middle" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
t
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell multirow="3" alignment="center" valignment="middle" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Formula $o_{t}$
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell multicolumn="1" alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Formula $b_{i}\left(o_{t}\right)$
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell multicolumn="2" alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
</row>
|
|
|
|
|
<row>
|
|
|
|
|
<cell multirow="4" alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell multirow="4" alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Formula $i=1$
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Formula $i=2$
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
</row>
|
|
|
|
|
<row>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
1
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
3.8
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
0.022
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
0.55
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
</row>
|
|
|
|
|
<row>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
2
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
4.2
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
0.0095
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
0.55
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
</row>
|
|
|
|
|
<row>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
3
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
3.4
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
0.045
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
0.39
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
</row>
|
|
|
|
|
<row>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
4
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
-0.4
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
0.17
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
1.5
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
</row>
|
|
|
|
|
<row>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
5
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
1.9
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
0.25
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
0.0063
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
</row>
|
|
|
|
|
<row>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
6
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
3.0
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
0.083
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
0.21
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
</row>
|
|
|
|
|
<row>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
7
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
1.6
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
0.29
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
0.0016
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
</row>
|
|
|
|
|
<row>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
8
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
1.9
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
0.25
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
0.0063
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
</row>
|
|
|
|
|
<row>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
9
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
5.0
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
0.0013
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
0.21
|
|
|
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\end_layout
|
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|
\end_inset
|
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|
</cell>
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</row>
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|
</lyxtabular>
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|
|
\end_inset
|
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|
\end_layout
|
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|
\begin_layout Plain Layout
|
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|
|
\begin_inset Caption Standard
|
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|
|
\begin_layout Plain Layout
|
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|
Probability densities for each observation from each state to (2 s.f)
|
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|
|
\begin_inset CommandInset label
|
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|
LatexCommand label
|
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|
|
name "tab:obs-prob-dens"
|
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\end_inset
|
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\end_layout
|
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|
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\end_inset
|
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|
\end_layout
|
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|
\begin_layout Plain Layout
|
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|
\end_layout
|
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\end_inset
|
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|
\end_layout
|
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|
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|
|
\begin_layout Standard
|
|
|
|
|
\begin_inset Float figure
|
|
|
|
|
wide false
|
|
|
|
|
sideways false
|
|
|
|
|
status open
|
|
|
|
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|
|
\begin_layout Plain Layout
|
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|
|
\noindent
|
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|
|
\align center
|
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|
|
\begin_inset Graphics
|
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|
|
filename res/pdfs-w-obs.png
|
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|
|
|
width 50col%
|
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|
\end_inset
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\end_layout
|
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|
\begin_layout Plain Layout
|
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|
|
\begin_inset Caption Standard
|
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|
|
|
|
\begin_layout Plain Layout
|
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|
|
Previous PDFs (figure
|
|
|
|
|
\begin_inset CommandInset ref
|
|
|
|
|
LatexCommand ref
|
|
|
|
|
reference "fig:PDFs"
|
|
|
|
|
plural "false"
|
|
|
|
|
caps "false"
|
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|
|
noprefix "false"
|
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|
|
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|
|
\end_inset
|
|
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|
|
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|
|
) with observations highlighted
|
|
|
|
|
\begin_inset CommandInset label
|
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|
|
|
LatexCommand label
|
|
|
|
|
name "fig:PDFs-w-obs"
|
|
|
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|
|
\end_inset
|
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|
\end_layout
|
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|
|
\end_inset
|
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|
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|
|
\end_layout
|
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|
|
\end_inset
|
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|
|
\end_layout
|
|
|
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|
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|
|
\begin_layout Subsection
|
|
|
|
|
Forward Procedure and Likelihoods
|
|
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|
|
\end_layout
|
|
|
|
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|
|
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|
|
\begin_layout Standard
|
|
|
|
|
\begin_inset Float table
|
|
|
|
|
wide false
|
|
|
|
|
sideways false
|
|
|
|
|
status open
|
|
|
|
|
|
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|
|
\begin_layout Plain Layout
|
|
|
|
|
\noindent
|
|
|
|
|
\align center
|
|
|
|
|
\begin_inset Tabular
|
|
|
|
|
<lyxtabular version="3" rows="11" columns="3">
|
|
|
|
|
<features tabularvalignment="middle">
|
|
|
|
|
<column alignment="center" valignment="top">
|
|
|
|
|
<column alignment="center" valignment="top" width="15col%">
|
|
|
|
|
<column alignment="center" valignment="top" width="15col%">
|
|
|
|
|
<row>
|
|
|
|
|
<cell multirow="3" alignment="center" valignment="middle" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
t
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell multicolumn="1" alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Formula $\alpha_{t}\left(i\right)$
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell multicolumn="2" alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
</row>
|
|
|
|
|
<row>
|
|
|
|
|
<cell multirow="4" alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Formula $i=1$
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Formula $i=2$
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
</row>
|
|
|
|
|
<row>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
1
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Formula $9.61\times10^{-3}$
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
0.306
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
</row>
|
|
|
|
|
<row>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
2
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Formula $2.00\times10^{-4}$
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
0.156
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
</row>
|
|
|
|
|
<row>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
3
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Formula $2.89\times10^{-4}$
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
0.0573
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
</row>
|
|
|
|
|
<row>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
4
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Formula $4.31\times10^{-4}$
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Formula $8.01\times10^{-11}$
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
</row>
|
|
|
|
|
<row>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
5
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Formula $9.95\times10^{-5}$
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Formula $1.64\times10^{-7}$
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
</row>
|
|
|
|
|
<row>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
6
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Formula $7.59\times10^{-6}$
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Formula $1.26\times10^{-6}$
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
</row>
|
|
|
|
|
<row>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
7
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Formula $2.06\times10^{-6}$
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Formula $2.59\times10^{-9}$
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
</row>
|
|
|
|
|
<row>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
8
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Formula $4.76\times10^{-7}$
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Formula $7.99\times10^{-10}$
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
</row>
|
|
|
|
|
<row>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
9
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Formula $5.63\times10^{-10}$
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Formula $6.02\times10^{-9}$
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
</row>
|
|
|
|
|
</lyxtabular>
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Caption Standard
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
Forward likelihoods for each state at each time step
|
|
|
|
|
\begin_inset CommandInset label
|
|
|
|
|
LatexCommand label
|
|
|
|
|
name "tab:Forward-likelihoods"
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Subsection
|
|
|
|
|
Backward Procedure and Likelihoods
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
\begin_inset Float table
|
|
|
|
|
wide false
|
|
|
|
|
sideways false
|
|
|
|
|
status open
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\noindent
|
|
|
|
|
\align center
|
|
|
|
|
\begin_inset Tabular
|
|
|
|
|
<lyxtabular version="3" rows="11" columns="3">
|
|
|
|
|
<features tabularvalignment="middle">
|
|
|
|
|
<column alignment="center" valignment="top">
|
|
|
|
|
<column alignment="center" valignment="top" width="15col%">
|
|
|
|
|
<column alignment="center" valignment="top" width="15col%">
|
|
|
|
|
<row>
|
|
|
|
|
<cell multirow="3" alignment="center" valignment="middle" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
t
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell multicolumn="1" alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Formula $\beta_{t}\left(i\right)$
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell multicolumn="2" alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
</row>
|
|
|
|
|
<row>
|
|
|
|
|
<cell multirow="4" alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Formula $i=1$
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Formula $i=2$
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
</row>
|
|
|
|
|
<row>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
1
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Formula $6.58\times10^{-11}$
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Formula $6.25\times10^{-10}$
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
</row>
|
|
|
|
|
<row>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
2
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Formula $2.93\times10^{-9}$
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Formula $1.23\times10^{-9}$
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
</row>
|
|
|
|
|
<row>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
3
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Formula $6.90\times10^{-8}$
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Formula $3.00\times10^{-9}$
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
</row>
|
|
|
|
|
<row>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
4
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Formula $4.45\times10^{-7}$
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Formula $2.11\times10^{-8}$
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
</row>
|
|
|
|
|
<row>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
5
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Formula $1.93\times10^{-6}$
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Formula $3.02\times10^{-7}$
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
</row>
|
|
|
|
|
<row>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
6
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Formula $2.51\times10^{-5}$
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Formula $1.15\times10^{-6}$
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
</row>
|
|
|
|
|
<row>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
7
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Formula $9.30\times10^{-5}$
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Formula $3.77\times10^{-5}$
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
</row>
|
|
|
|
|
<row>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
8
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Formula $3.93\times10^{-4}$
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Formula $5.73\times10^{-3}$
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
</row>
|
|
|
|
|
<row>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
9
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
0.02
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
0.03
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
</row>
|
|
|
|
|
</lyxtabular>
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Caption Standard
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
Backward likelihoods for each state at each time step
|
|
|
|
|
\begin_inset CommandInset label
|
|
|
|
|
LatexCommand label
|
|
|
|
|
name "tab:Backward-likelihoods"
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Subsection
|
|
|
|
|
Observations Likelihood
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Subsection
|
|
|
|
|
Occupation Likelihoods
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
\begin_inset Float table
|
|
|
|
|
wide false
|
|
|
|
|
sideways false
|
|
|
|
|
status open
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\noindent
|
|
|
|
|
\align center
|
|
|
|
|
\begin_inset Tabular
|
|
|
|
|
<lyxtabular version="3" rows="11" columns="3">
|
|
|
|
|
<features tabularvalignment="middle">
|
|
|
|
|
<column alignment="center" valignment="top">
|
|
|
|
|
<column alignment="center" valignment="top" width="15col%">
|
|
|
|
|
<column alignment="center" valignment="top" width="15col%">
|
|
|
|
|
<row>
|
|
|
|
|
<cell multirow="3" alignment="center" valignment="middle" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
t
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell multicolumn="1" alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Formula $\gamma_{t}\left(i\right)$
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell multicolumn="2" alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
</row>
|
|
|
|
|
<row>
|
|
|
|
|
<cell multirow="4" alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Formula $i=1$
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Formula $i=2$
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
</row>
|
|
|
|
|
<row>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
1
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
0.003295
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
0.9967
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
</row>
|
|
|
|
|
<row>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
2
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
0.003055
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
0.9969
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
</row>
|
|
|
|
|
<row>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
3
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
0.1040
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
|
|
|
|
</cell>
|
|
|
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
|
|
|
\begin_inset Text
|
|
|
|
|
|
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
0.8960
|
|
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4
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0.9999...
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0.9997
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\begin_inset Text
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0.0002579
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0.9925
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\begin_inset Text
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0.9995
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\begin_inset Text
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\end_layout
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</lyxtabular>
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\begin_inset Caption Standard
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\begin_layout Plain Layout
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Occupation likelihoods for each state at each time step (4 s.f.)
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\begin_inset CommandInset label
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LatexCommand label
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name "tab:Occupation-likelihoods"
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\end_inset
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\end_layout
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|
2020-12-10 22:59:55 +00:00
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\begin_layout Subsection
|
2020-12-31 19:30:39 +00:00
|
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Baum-Welch Re-estimations
|
2020-12-10 22:59:55 +00:00
|
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\end_layout
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|
2020-12-31 19:30:39 +00:00
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\begin_layout Subsubsection
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Output Parameters
|
2020-12-10 22:59:55 +00:00
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\end_layout
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|
2020-12-31 19:30:39 +00:00
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\begin_inset Float figure
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sideways false
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\noindent
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\align center
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\begin_inset Graphics
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filename res/re-est-pdfs.png
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width 50col%
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\end_inset
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Probability density functions following re-estimation of parameters
|
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|
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LatexCommand label
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name "fig:Re-estimated-PDFs"
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\begin_layout Subsubsection
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State Transitions
|
2020-12-10 22:59:55 +00:00
|
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\end_layout
|
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\begin_layout Section
|
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Discussion
|
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\end_layout
|
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\begin_layout Section
|
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Conclusion
|
2020-12-10 21:43:15 +00:00
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\end_layout
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\begin_layout Standard
|
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|
\begin_inset CommandInset label
|
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LatexCommand label
|
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|
name "sec:bibliography"
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\end_inset
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\begin_inset CommandInset bibtex
|
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LatexCommand bibtex
|
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|
btprint "btPrintCited"
|
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|
bibfiles "references"
|
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options "bibtotoc"
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\end_inset
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2020-12-31 19:30:39 +00:00
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2020-12-10 21:43:15 +00:00
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\end_layout
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\begin_layout Section
|
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|
\start_of_appendix
|
2020-12-10 22:59:55 +00:00
|
|
|
Intermediate Results
|
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|
\end_layout
|
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\begin_layout Section
|
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Source Code
|
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\end_layout
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\begin_layout Standard
|
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|
\begin_inset CommandInset include
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|
|
LatexCommand lstinputlisting
|
|
|
|
|
filename "../constants.py"
|
2020-12-31 19:30:39 +00:00
|
|
|
lstparams "caption={Constants file defining initial state transition probabilities and PDF parameters},label={constants-listing}"
|
2020-12-10 22:59:55 +00:00
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\end_inset
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|
2020-12-24 14:58:45 +00:00
|
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\end_layout
|
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|
\begin_layout Standard
|
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|
\begin_inset CommandInset include
|
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|
|
LatexCommand lstinputlisting
|
|
|
|
|
filename "../maths.py"
|
2020-12-31 19:30:39 +00:00
|
|
|
lstparams "caption={Maths utility file with definition for a gaussian},label={maths-listing}"
|
2020-12-24 14:58:45 +00:00
|
|
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|
\end_inset
|
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|
2020-12-23 20:12:08 +00:00
|
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\end_layout
|
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|
\begin_layout Standard
|
|
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|
|
\begin_inset CommandInset include
|
|
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|
|
LatexCommand lstinputlisting
|
|
|
|
|
filename "../markov.py"
|
2020-12-31 19:30:39 +00:00
|
|
|
lstparams "caption={Markov model object defining forward and backward procedure, occupation likelihoods, Baum-Welch equations},label={markov-listing}"
|
|
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|
\end_inset
|
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\end_layout
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\begin_layout Standard
|
|
|
|
|
\begin_inset CommandInset include
|
|
|
|
|
LatexCommand lstinputlisting
|
|
|
|
|
filename "../markovlog.py"
|
|
|
|
|
lstparams "caption={Child Markov model re-implementing calculations using log probability and likelihood},label={log-markov-listing}"
|
|
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\end_inset
|
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\end_layout
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\begin_layout Standard
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\begin_inset Newpage newpage
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\end_inset
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\end_layout
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\begin_layout Standard
|
|
|
|
|
The development of the model behind this report was completed using Jupyter
|
|
|
|
|
Notebook.
|
|
|
|
|
The used notebook can be seen formatted in plain text below, the relevant
|
|
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mark scheme elements are referenced in brackets.
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\end_layout
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\begin_layout Standard
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\begin_inset CommandInset include
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LatexCommand lstinputlisting
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filename "../notebook.py"
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lstparams "caption={Plain output of Jupyter Notebook used for development and debugging},label={notebook-listing}"
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2020-12-23 20:12:08 +00:00
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\end_inset
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2020-12-10 21:43:15 +00:00
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\end_layout
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\end_body
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\end_document
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