3432 lines
62 KiB
Plaintext
3432 lines
62 KiB
Plaintext
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\begin_layout Title
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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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EEEM030
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January 2021
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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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\begin_layout Abstract
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abstract
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Andy Pack / 6420013
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January 2021
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\end_layout
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\begin_layout Section
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Introduction
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\end_layout
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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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The hidden Markov model extends the standard Markov chain by modeling a
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system where the states are not directly observable but are hidden
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\begin_inset CommandInset citation
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LatexCommand cite
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key "standford-hmm-book"
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literal "false"
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\end_inset
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, instead the states are inferred from the visible observations.
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status open
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Described markov chains
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\end_layout
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\begin_inset Flex TODO Note (inline)
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status open
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Difference with hidden markov models
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\end_layout
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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 Flex TODO Note (inline)
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status open
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\begin_layout Plain Layout
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Application to machine learning and speech in particular
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\end_layout
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\end_inset
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\end_layout
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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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\end_layout
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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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\end_inset
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),
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\end_layout
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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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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\[
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\pi_{j}=\left\{ 0.44,0.56\right\}
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\]
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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 Formula
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\[
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a_{ij}=\left[\begin{array}{cc}
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0.92 & 0.06\\
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0.04 & 0.93
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\end{array}\right]
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\]
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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 Formula
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\[
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\eta_{i}=\left\{ 0.02,0.03\right\}
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\]
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\end_inset
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\end_layout
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\begin_layout Standard
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These, in combination, describe a state topology than can be seen graphically
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in figure
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "fig:State-topology"
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plural "false"
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caps "false"
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noprefix "false"
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\end_inset
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.
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\end_layout
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\begin_layout Standard
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As previously mentioned, the states are described by 1D continuous probability
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density functions (PDF) of a Gaussian profile.
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Each has an associated mean and variance as seen below,
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\end_layout
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State,
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\begin_inset Formula $i$
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\end_inset
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\end_inset
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1
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2
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</row>
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<row>
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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\begin_inset Text
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\begin_layout Plain Layout
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Mean,
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\begin_inset Formula $\mu_{i}$
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\end_inset
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\end_layout
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\end_inset
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1.00
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4.00
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</cell>
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<row>
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<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
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\begin_inset Text
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\begin_layout Plain Layout
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Variance,
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\begin_inset Formula $\varSigma_{i}$
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\end_inset
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\end_layout
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\end_inset
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</cell>
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<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
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1.44
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\end_inset
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</cell>
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0.49
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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 Standard
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Finally, a set of observations from the above model are provided below,
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\end_layout
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\begin_layout Standard
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\noindent
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\align center
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\begin_inset Formula
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\[
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\mathcal{O}=\left\{ 3.8,4.2,3.4,-0.4,1.9,3.0,1.6,1.9,5.0\right\}
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\]
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\end_inset
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\end_layout
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\begin_layout Standard
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These are used as the basis for training the model using the Baum-Welch
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equations
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\begin_inset Flex TODO Note (Margin)
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status open
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Expectation-maximisation
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status open
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Cite
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.
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\end_layout
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\begin_layout Section
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Implementation
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\begin_inset Float figure
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wide false
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sideways false
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\noindent
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filename StateTopology.png
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lyxscale 50
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width 65col%
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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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State topology for the specified Markov Model
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\begin_inset CommandInset label
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LatexCommand label
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name "fig:State-topology"
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\end_inset
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\end_layout
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\end_inset
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\begin_layout Plain Layout
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\begin_layout Standard
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The work was completed using Python
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\begin_inset CommandInset citation
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LatexCommand cite
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key "python"
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literal "false"
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and the Jupyter notebook
|
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\begin_inset CommandInset citation
|
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LatexCommand cite
|
|
key "jupyter"
|
|
literal "false"
|
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\end_inset
|
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|
|
.
|
|
Apart from the standard library, NumPy
|
|
\begin_inset CommandInset citation
|
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LatexCommand cite
|
|
key "numpy"
|
|
literal "false"
|
|
|
|
\end_inset
|
|
|
|
was used for a n-dimensional array implementation.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
An implementation of the Gaussian function was written in order to derive
|
|
output probability densities for each state given an observation, see listing
|
|
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "maths-listing"
|
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plural "false"
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caps "false"
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noprefix "false"
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\end_inset
|
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|
.
|
|
This was used to plot each of the PDFs and throughout the likelihood calculatio
|
|
ns.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
The forward and backward procedure are outlined by
|
|
\begin_inset CommandInset citation
|
|
LatexCommand cite
|
|
key "standford-hmm-book"
|
|
literal "false"
|
|
|
|
\end_inset
|
|
|
|
, the equations for both forward and backward likelihood can be seen below,
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Flex TODO Note (inline)
|
|
status open
|
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|
|
\begin_layout Plain Layout
|
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Finish?
|
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\end_layout
|
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|
|
\end_inset
|
|
|
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|
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\end_layout
|
|
|
|
\begin_layout Section
|
|
Results
|
|
\end_layout
|
|
|
|
\begin_layout Subsection
|
|
Output Probability Density Functions
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
The Gaussian probability functions described by the initial parameters can
|
|
be seen in figure
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "fig:PDFs"
|
|
plural "false"
|
|
caps "false"
|
|
noprefix "false"
|
|
|
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\end_inset
|
|
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.
|
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\end_layout
|
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\begin_layout Standard
|
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\begin_inset Float figure
|
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wide false
|
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sideways false
|
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status open
|
|
|
|
\begin_layout Plain Layout
|
|
\noindent
|
|
\align center
|
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\begin_inset Graphics
|
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filename res/pdfs.png
|
|
lyxscale 50
|
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width 70col%
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\end_inset
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\end_layout
|
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\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"
|
|
|
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\end_inset
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\end_layout
|
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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
|
|
|
|
\begin_layout Subsection
|
|
Observation Probability Densities
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
The output probability densities for each observation in each state can
|
|
be seen in table
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "tab:obs-prob-dens"
|
|
plural "false"
|
|
caps "false"
|
|
noprefix "false"
|
|
|
|
\end_inset
|
|
|
|
.
|
|
These observations can be seen overlaid on the original PDFs in figure
|
|
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "fig:PDFs-w-obs"
|
|
plural "false"
|
|
caps "false"
|
|
noprefix "false"
|
|
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
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|
\begin_layout Standard
|
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\begin_inset Float table
|
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wide false
|
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sideways false
|
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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 Tabular
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<lyxtabular version="3" rows="11" columns="4">
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<features tabularvalignment="middle">
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<column alignment="center" valignment="top">
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<column alignment="center" valignment="top">
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<column alignment="center" valignment="top" width="10col%">
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<column alignment="center" valignment="top" width="10col%">
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<row>
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<cell multirow="3" alignment="center" valignment="middle" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
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\begin_inset Text
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\begin_layout Plain Layout
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t
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\end_layout
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\end_inset
|
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</cell>
|
|
<cell multirow="3" alignment="center" valignment="middle" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
|
|
\begin_inset Text
|
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\begin_layout Plain Layout
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\begin_inset Formula $o_{t}$
|
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\end_inset
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\end_layout
|
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\end_inset
|
|
</cell>
|
|
<cell multicolumn="1" alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
\begin_inset Text
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|
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\begin_layout Plain Layout
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\begin_inset Formula $b_{i}\left(o_{t}\right)$
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\end_inset
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\end_layout
|
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\end_inset
|
|
</cell>
|
|
<cell multicolumn="2" alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
\begin_inset Text
|
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\begin_layout Plain Layout
|
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\end_layout
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\end_inset
|
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</cell>
|
|
</row>
|
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<row>
|
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<cell multirow="4" alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
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\begin_layout Plain Layout
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\end_layout
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\end_inset
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</cell>
|
|
<cell multirow="4" alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
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\begin_layout Plain Layout
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\end_layout
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\end_inset
|
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</cell>
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
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|
\begin_layout Plain Layout
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|
\begin_inset Formula $i=1$
|
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\end_inset
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|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
|
|
\begin_inset Text
|
|
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|
\begin_layout Plain Layout
|
|
\begin_inset Formula $i=2$
|
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\end_inset
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|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
</row>
|
|
<row>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
\begin_inset Text
|
|
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|
\begin_layout Plain Layout
|
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1
|
|
\end_layout
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\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
\begin_inset Text
|
|
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|
\begin_layout Plain Layout
|
|
3.8
|
|
\end_layout
|
|
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|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
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0.022
|
|
\end_layout
|
|
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|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
\begin_inset Text
|
|
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|
\begin_layout Plain Layout
|
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0.55
|
|
\end_layout
|
|
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|
\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
|
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2
|
|
\end_layout
|
|
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|
\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
|
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3
|
|
\end_layout
|
|
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|
\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
|
|
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\begin_layout Plain Layout
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6
|
|
\end_layout
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|
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</cell>
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|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
\begin_inset Text
|
|
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\begin_layout Plain Layout
|
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3.0
|
|
\end_layout
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\end_inset
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</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
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\begin_layout Plain Layout
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0.083
|
|
\end_layout
|
|
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|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
\begin_inset Text
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|
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\begin_layout Plain Layout
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0.21
|
|
\end_layout
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\end_inset
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</cell>
|
|
</row>
|
|
<row>
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
\begin_inset Text
|
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7
|
|
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</cell>
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|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
\begin_inset Text
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1.6
|
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\end_inset
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</cell>
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|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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|
\begin_inset Text
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0.29
|
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\end_layout
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\end_inset
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</cell>
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
\begin_inset Text
|
|
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\begin_layout Plain Layout
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0.0016
|
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</cell>
|
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</row>
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<row>
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
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8
|
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</cell>
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
\begin_inset Text
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\begin_layout Plain Layout
|
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1.9
|
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|
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\end_inset
|
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</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
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0.25
|
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|
|
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\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
\begin_inset Text
|
|
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\begin_layout Plain Layout
|
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0.0063
|
|
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|
|
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\end_inset
|
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</cell>
|
|
</row>
|
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<row>
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<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
|
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9
|
|
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</cell>
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
|
|
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5.0
|
|
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|
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\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
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|
\begin_layout Plain Layout
|
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0.0013
|
|
\end_layout
|
|
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|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
|
|
\begin_inset Text
|
|
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|
\begin_layout Plain Layout
|
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0.21
|
|
\end_layout
|
|
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|
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|
|
</cell>
|
|
</row>
|
|
</lyxtabular>
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Caption Standard
|
|
|
|
\begin_layout Plain Layout
|
|
Probability densities for each observation from each state to (2 s.f)
|
|
\begin_inset CommandInset label
|
|
LatexCommand label
|
|
name "tab:obs-prob-dens"
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\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-w-obs.png
|
|
lyxscale 50
|
|
width 70col%
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Caption Standard
|
|
|
|
\begin_layout Plain Layout
|
|
Previous PDFs (figure
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "fig:PDFs"
|
|
plural "false"
|
|
caps "false"
|
|
noprefix "false"
|
|
|
|
\end_inset
|
|
|
|
) with observations highlighted
|
|
\begin_inset CommandInset label
|
|
LatexCommand label
|
|
name "fig:PDFs-w-obs"
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Subsection
|
|
Forward Procedure and Likelihoods
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
The forward likelihoods calculated as part of the forward procedure can
|
|
be seen in figure
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "tab:Forward-likelihoods"
|
|
plural "false"
|
|
caps "false"
|
|
noprefix "false"
|
|
|
|
\end_inset
|
|
|
|
.
|
|
The natural logarithm of these results can be seen graphically in figure
|
|
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "fig:forward-log-like"
|
|
plural "false"
|
|
caps "false"
|
|
noprefix "false"
|
|
|
|
\end_inset
|
|
|
|
.
|
|
The log-likelihoods were used here as subsequent values can decrease by
|
|
orders of magnitude making it hard to display, as a monotonic function,
|
|
the relationship between values remains relevant when presenting the log
|
|
of each value
|
|
\begin_inset Flex TODO Note (Margin)
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
Reword?
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
.
|
|
For state 1, the forward likelihood generally decreases over time, this
|
|
pattern is not followed by state 2.
|
|
State 2 begins decreasing before rapidly decreasing and increasing.
|
|
\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 $\alpha_{t}\left(i\right)$
|
|
\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
|
|
|
|
\end_inset
|
|
|
|
|
|
\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/forward-logline.png
|
|
lyxscale 50
|
|
width 50col%
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Caption Standard
|
|
|
|
\begin_layout Plain Layout
|
|
Forward log-likelihood for each state over all observations
|
|
\begin_inset CommandInset label
|
|
LatexCommand label
|
|
name "fig:forward-log-like"
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
Using the forward procedure, the observation likelihood,
|
|
\begin_inset Formula $P\left(\mathcal{O}|\lambda\right)$
|
|
\end_inset
|
|
|
|
, was calculated as follows
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Formula
|
|
\begin{equation}
|
|
P\left(\mathcal{O}|\lambda\right)=\eta_{1}\alpha_{9}\left(1\right)+\eta_{2}\alpha_{9}\left(2\right)\label{eq:forward-symbolic}
|
|
\end{equation}
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Formula
|
|
\begin{equation}
|
|
P\left(\mathcal{O}|\lambda\right)=\left(0.02\cdot5.63\times10^{-10}\right)+\left(0.03\cdot6.02\times10^{-9}\right)=1.92\times10^{-10}\label{eq:forward}
|
|
\end{equation}
|
|
|
|
\end_inset
|
|
|
|
to 2 decimal places.
|
|
\end_layout
|
|
|
|
\begin_layout Subsection
|
|
Backward Procedure and Likelihoods
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
The backward likelihoods were calculated as part of the backwards procedure
|
|
and can be seen in table
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "tab:Backward-likelihoods"
|
|
plural "false"
|
|
caps "false"
|
|
noprefix "false"
|
|
|
|
\end_inset
|
|
|
|
, the log-likelihoods can be seen in figure
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "fig:backward-log-like"
|
|
plural "false"
|
|
caps "false"
|
|
noprefix "false"
|
|
|
|
\end_inset
|
|
|
|
.
|
|
Reading backwards in time (from bottom to top), for both states the backwards
|
|
likelihood can be seen to decrease to a minimum at
|
|
\begin_inset Formula $t=1$
|
|
\end_inset
|
|
|
|
.
|
|
\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 Standard
|
|
\begin_inset Float figure
|
|
wide false
|
|
sideways false
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
\noindent
|
|
\align center
|
|
\begin_inset Graphics
|
|
filename res/backward-logline.png
|
|
lyxscale 50
|
|
width 50col%
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Caption Standard
|
|
|
|
\begin_layout Plain Layout
|
|
Backward log-likelihood for each state over all observations
|
|
\begin_inset CommandInset label
|
|
LatexCommand label
|
|
name "fig:backward-log-like"
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
Finalising the backward procedure can also produce
|
|
\begin_inset Formula $P\left(\mathcal{O}|\lambda\right)$
|
|
\end_inset
|
|
|
|
,
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Formula
|
|
\begin{equation}
|
|
P\left(\mathcal{O}|\lambda\right)=\pi_{1}b_{1}\left(o_{1}\right)\beta_{1}\left(1\right)+\pi_{2}b_{2}\left(o_{1}\right)\beta_{1}\left(2\right)\label{eq:backward-symbolic}
|
|
\end{equation}
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Formula
|
|
\begin{equation}
|
|
P\left(\mathcal{O}|\lambda\right)=\left(0.44\cdot0.022\cdot6.58\times10^{-11}\right)+\left(0.56\cdot0.55\cdot6.25\times10^{-10}\right)=1.92\times10^{-10}\label{eq:backward}
|
|
\end{equation}
|
|
|
|
\end_inset
|
|
|
|
to 2 decimal places.
|
|
Looking back to equation
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "eq:forward"
|
|
plural "false"
|
|
caps "false"
|
|
noprefix "false"
|
|
|
|
\end_inset
|
|
|
|
, these can be seen to be the same, as expected
|
|
\begin_inset Flex TODO Note (Margin)
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
Maybe move to discussion?
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\begin_layout Subsection
|
|
Occupation Likelihoods
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
The above forward and backward likelihoods were used to calculate the occupation
|
|
likelihoods of each state at each time step, the results can be seen in
|
|
table
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "tab:Occupation-likelihoods"
|
|
plural "false"
|
|
caps "false"
|
|
noprefix "false"
|
|
|
|
\end_inset
|
|
|
|
and are presented graphically in figure
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "fig:occupation-likelihood-bars"
|
|
plural "false"
|
|
caps "false"
|
|
noprefix "false"
|
|
|
|
\end_inset
|
|
|
|
.
|
|
\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>
|
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<row>
|
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
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|
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|
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|
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1
|
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|
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|
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\end_inset
|
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</cell>
|
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
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|
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|
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0.003295
|
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|
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|
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|
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</cell>
|
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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|
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|
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0.9967
|
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\end_layout
|
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|
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</cell>
|
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</row>
|
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<row>
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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2
|
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|
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</cell>
|
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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0.003055
|
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\end_layout
|
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|
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</cell>
|
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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\begin_inset Text
|
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|
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0.9969
|
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\end_layout
|
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</cell>
|
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</row>
|
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<row>
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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3
|
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|
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|
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|
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</cell>
|
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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0.1040
|
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|
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\end_inset
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</cell>
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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\begin_inset Text
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|
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0.8960
|
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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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<row>
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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4
|
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\end_layout
|
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|
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</cell>
|
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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0.9999...
|
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|
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|
|
</cell>
|
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
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\begin_inset Text
|
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|
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\begin_inset Formula $8.818\times10^{-9}$
|
|
\end_inset
|
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|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
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</row>
|
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<row>
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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5
|
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\end_layout
|
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|
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\end_inset
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</cell>
|
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
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\begin_inset Text
|
|
|
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\begin_layout Plain Layout
|
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0.9997
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
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\begin_inset Text
|
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\begin_layout Plain Layout
|
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0.0002579
|
|
\end_layout
|
|
|
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\end_inset
|
|
</cell>
|
|
</row>
|
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<row>
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
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\begin_inset Text
|
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|
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\begin_layout Plain Layout
|
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6
|
|
\end_layout
|
|
|
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\end_inset
|
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</cell>
|
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
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\begin_layout Plain Layout
|
|
0.9925
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
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\begin_inset Text
|
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\begin_layout Plain Layout
|
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0.007517
|
|
\end_layout
|
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|
</cell>
|
|
</row>
|
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<row>
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
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\begin_inset Text
|
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\begin_layout Plain Layout
|
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7
|
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\end_layout
|
|
|
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</cell>
|
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
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\begin_inset Text
|
|
|
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0.9995
|
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\end_layout
|
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|
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\end_inset
|
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</cell>
|
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
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\begin_inset Text
|
|
|
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\begin_layout Plain Layout
|
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0.0005095
|
|
\end_layout
|
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|
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|
|
</cell>
|
|
</row>
|
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<row>
|
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
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\begin_inset Text
|
|
|
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\begin_layout Plain Layout
|
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8
|
|
\end_layout
|
|
|
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\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
|
\begin_layout Plain Layout
|
|
0.9761
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
|
|
\begin_inset Text
|
|
|
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\begin_layout Plain Layout
|
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0.02386
|
|
\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
|
|
0.05868
|
|
\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
|
|
0.9413
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
</row>
|
|
</lyxtabular>
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Caption Standard
|
|
|
|
\begin_layout Plain Layout
|
|
Occupation likelihoods for each state at each time step (4 s.f.)
|
|
\begin_inset CommandInset label
|
|
LatexCommand label
|
|
name "tab:Occupation-likelihoods"
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\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/occupation-line.png
|
|
lyxscale 50
|
|
width 70col%
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Caption Standard
|
|
|
|
\begin_layout Plain Layout
|
|
Occupation likelihoods for each state for each observation
|
|
\begin_inset CommandInset label
|
|
LatexCommand label
|
|
name "fig:occupation-likelihood-bars"
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Subsection
|
|
Baum-Welch Re-estimations
|
|
\end_layout
|
|
|
|
\begin_layout Subsubsection
|
|
Output Parameters
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
Following one iteration of Baum-Welch training, the output Gaussian parameters
|
|
were estimated as follows (2 d.p.),
|
|
\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
|
|
|
|
|
|
\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
|
|
\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
|
|
1.67
|
|
\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
|
|
4.09
|
|
\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
|
|
1.85
|
|
\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
|
|
0.37
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
</row>
|
|
</lyxtabular>
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
This represents the following deltas from the original parameters (2 s.f.),
|
|
\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
|
|
|
|
|
|
\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
|
|
\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
|
|
+0.67
|
|
\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
|
|
+0.089
|
|
\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
|
|
+0.41
|
|
\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
|
|
-0.11
|
|
\end_layout
|
|
|
|
\end_inset
|
|
</cell>
|
|
</row>
|
|
</lyxtabular>
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
Displayed graphically, these two sets of Gaussian functions can be seen
|
|
in figure
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "fig:Re-estimated-PDFs"
|
|
plural "false"
|
|
caps "false"
|
|
noprefix "false"
|
|
|
|
\end_inset
|
|
|
|
.
|
|
Compared to the original function, increasing the standard deviation has
|
|
widened and shortened the new function for state 1.
|
|
The slight increase in mean has also shifted the function to the right.
|
|
For state 2, the new function has a tighter, taller distribution as a result
|
|
of the decreased variance; the mean has, again, been slightly shifted to
|
|
the right.
|
|
\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/re-est-pdfs.png
|
|
lyxscale 50
|
|
width 70col%
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Caption Standard
|
|
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\begin_layout Plain Layout
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Probability density functions following re-estimation of parameters
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\begin_inset CommandInset label
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LatexCommand label
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name "fig:Re-estimated-PDFs"
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\end_inset
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\end_layout
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\end_inset
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\end_layout
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\end_inset
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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
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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
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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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\end_layout
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\begin_layout Section
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\start_of_appendix
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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
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filename "../constants.py"
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lstparams "caption={Constants file defining initial state transition probabilities and PDF parameters},label={constants-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 CommandInset include
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LatexCommand lstinputlisting
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filename "../maths.py"
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lstparams "caption={Maths utility file with definition for a gaussian},label={maths-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
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\begin_inset CommandInset include
|
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LatexCommand lstinputlisting
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filename "../markov.py"
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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
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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
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The development of the model behind this report was completed using Jupyter
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Notebook.
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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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\begin_inset listings
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lstparams "language=Python,basicstyle={\ttfamily}"
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inline true
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status open
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\begin_layout Plain Layout
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\noindent
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%%
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\end_layout
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\end_inset
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= cell delimiter).
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\end_layout
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\begin_layout Standard
|
|
\begin_inset CommandInset include
|
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LatexCommand lstinputlisting
|
|
filename "../notebook.py"
|
|
lstparams "commentstyle={\\large\\bfseries\\color{commentgreen}},caption={Plain output of Jupyter Notebook used for development and debugging},label={notebook-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
|
|
\end_inset
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\end_layout
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|
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\begin_layout Section
|
|
Extensions
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
A few extra calculations were done with the model, figure
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "fig:5-training-iterations"
|
|
plural "false"
|
|
caps "false"
|
|
noprefix "false"
|
|
|
|
\end_inset
|
|
|
|
shows how the output probability density functions move during 5 iterations
|
|
of training.
|
|
State 1's output function can be seen to spread out (increasing variance)
|
|
and shift slightly to the right (increasing mean).
|
|
State 2's mean begins by increasing before decreasing below the initial
|
|
value.
|
|
The variance also decreases, tightening the function.
|
|
\end_layout
|
|
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|
\begin_layout Standard
|
|
\begin_inset Float figure
|
|
placement h
|
|
wide false
|
|
sideways false
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
\noindent
|
|
\align center
|
|
\begin_inset Graphics
|
|
filename res/iterated-pdfs.png
|
|
lyxscale 50
|
|
width 70col%
|
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|
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\end_inset
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\end_layout
|
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\begin_layout Plain Layout
|
|
\begin_inset Caption Standard
|
|
|
|
\begin_layout Plain Layout
|
|
5 iterations of training completed using the Baum-Welch equations
|
|
\begin_inset CommandInset label
|
|
LatexCommand label
|
|
name "fig:5-training-iterations"
|
|
|
|
\end_inset
|
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|
|
\end_layout
|
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\end_inset
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\end_layout
|
|
|
|
\end_inset
|
|
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|
|
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\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Formula
|
|
\[
|
|
\hat{a}_{ij}=\left[\begin{array}{cc}
|
|
0.80 & 0.19\\
|
|
0.26 & 0.50
|
|
\end{array}\right]
|
|
\]
|
|
|
|
\end_inset
|
|
|
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|
|
\end_layout
|
|
|
|
\end_body
|
|
\end_document
|