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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8">
<meta content="width=device-width, initial-scale=1, shrink-to-fit=no" name="viewport">
<title>Markov Chain Visualization</title>
<meta name="description" content="A Markov Chain Visualization and Simulation" />
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<meta content="Markov Chain, Simulation, Visualization, Guide, Educational" name="keywords">
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<link rel="stylesheet" href="css/main.css" />
<link rel="stylesheet" href="css/modal-animation.css" />
</head>
<body>
<div class="container-fluid mb-5">
<div class="row">
<div class="container px-4 py-5" id="custom-cards">
<h1 class="mb-2 mt-3 title">Markov Chains Visualization</h1>
<p class="fs-6 text-secondary">This website is created for the purpose of visualizing Markov Chains.</p>
<p class="fs-6 text-secondary">Developed by: Miguel Robles, Kovie Niño, and Zach Noche</p>
<div class="p-4 border bg-light rounded-3">
<p class="description">
<span class="fw-bold text-secondary">Before we get started,</span> look into how Markov Chain works. If you wish to skip,
you may collapse the division by clicking the title below.
</p>
</div>
<div class="p-4 border bg-light rounded-3 mt-4">
<p class="description">
<span class="fw-bold text-secondary">Want a more detailed explanation?</span> You may find a more detailed explanation of
Markov Chains, complete with the Python code used to derive the simulation results shown below,
by clicking here:
</p>
<div class="d-grid gap-2 d-md-flex justify-content-center">
<a href="https://datalore.jetbrains.com/notebook/BHICuHTWkj8Anq9egAu6Pv/NWwROZYFAb4wT3uGpoB4w9"
class="btn btn-dark " target="_blank">Datalore Jupyter Notebook</a>
</div>
<center>
<strong>(Note that you need a Jetbrains account to access the link)</strong>
</center>
</div>
</div>
<div class="accordion">
<div class="accordion-item">
<h2 class="accordion-header">
<button
class="accordion-button"
type="button"
data-bs-toggle="collapse"
data-bs-target="#collapseOne"
aria-expanded="true"
aria-controls="collapseOne"
>
Introducing Markov Chains
</button>
</h2>
<div id="collapseOne" class="accordion-collapse collapse show" data-bs-parent="#accordionExample">
<div class="accordion-body">
<section class="introduction">
A <strong>Markov Chain</strong> is a random process that undergoes transitions from one state to
another on a state space. The defining characteristic is that the probability of transitioning to any
particular state depends solely on the current state and time elapsed, and not on the sequence of
states that preceded it.
</section>
<hr>
<section class="memorylessness mt-4">
<h5>Memorylessness</h5>
<p>
Memorylessness is a key feature of Markov Chains. If we look at the example, previous states do not
matter as the next state solely relies on the probabilities from the current state.
</p>
</section>
<section class="stochastic-process mt-4">
<h5>Stochastic Process</h5>
<p>
A stochastic process is a collection of random variables:
<span class="math">\{X_t,t \in I \},</span> Where <span class="math">X_t</span> is the set of random
variables at time <span class="math">t</span>, and <span class="math">I</span> is the index set of
the process.
</p>
<p>Markov Chain is a type of Stochastic Process</p>
</section>
<section class="markov-process mt-4">
<h5>Markov Process</h5>
<p>
A sequence of random states S1, S2, ..., with the Markov property, is called a Markov chain. This
process can be entirely defined by the state space and transition matrix.
</p>
<ul>
<li><strong>State Space:</strong> A set of all possible states</li>
<li>
<strong>Transition Matrix:</strong> A matrix wherein each entry on the i-th row and j-th column
(<span class="math">aij</span>) represents the probability of moving from state i to state j.
</li>
</ul>
<strong>Let's create a simple markov chain.</strong>
<div class="text-center">
<img
class="img-fluid"
src="https://deparkes.co.uk/wp-content/uploads/2020/08/GraphView.png"
alt="Markov Transition Graph from Deparkes"
/>
<div>
<small class="text-center text-muted">Markov Transition Graph from Deparkes</small>
</div>
</div>
<table class="weather-table table mt-4">
<thead>
<tr>
<th scope="col"></th>
<th scope="col">Sunny</th>
<th scope="col">Rainy</th>
<th scope="col">Cloudy</th>
</tr>
</thead>
<tbody>
<tr>
<th scope="row">Sunny</th>
<td>0.7</td>
<td>0.2</td>
<td>0.1</td>
</tr>
<tr>
<th scope="row">Rainy</th>
<td>0.1</td>
<td>0.6</td>
<td>0.3</td>
</tr>
<tr>
<th scope="row">Cloudy</th>
<td>0.2</td>
<td>0.5</td>
<td>0.3</td>
</tr>
</tbody>
</table>
<p>
The state space represents the weather conditions. The transition matrix weather chain represents
the probabilities of moving from one weather condition to the other. This matrix is in column form
and each column represents the probabilities of going from one state to another.
</p>
<p>Example:</p>
<div class="sunny-desc">
The 'Sunny' list <strong>[0.7, 0.1, 0.2]</strong> indicates the probabilities of the following
transitions:
<ul>
<li>Transitioning from 'Sunny' to 'Sunny': 0.7</li>
<li>Transitioning from 'Rainy' to 'Sunny': 0.1</li>
<li>Transitioning from 'Cloudy' to 'Sunny': 0.2</li>
</ul>
</div>
<div class="rainy-desc">
The 'Rainy' list <strong>[0.2, 0.6, 0.5]</strong> indicates the probabilities of the following
transitions:
<ul>
<li>Transitioning from 'Sunny' to 'Rainy': 0.2</li>
<li>Transitioning from 'Rainy' to 'Rainy': 0.6</li>
<li>Transitioning from 'Cloudy' to 'Rainy': 0.5</li>
</ul>
</div>
<div class="cloudy-desc">
The 'Cloudy' list <strong>[0.1, 0.3, 0.3]</strong> indicates the probabilities of the following
transitions:
<ul>
<li>Transitioning from 'Sunny' to 'Cloudy': 0.1</li>
<li>Transitioning from 'Rainy' to 'Cloudy': 0.3</li>
<li>Transitioning from 'Cloudy' to 'Cloudy': 0.3</li>
</ul>
</div>
</section>
<hr>
<section class="initial-state-probability mt-4">
<h5>Initial State Probability</h5>
<p>
The initial state probability is typically denoted as a vector, where each entry corresponds to the
probability of the system being in a particular state at the start. The sum of all probabilities in
this vector must equal 1, reflecting the certainty that the system must be in one of the possible
states at the beginning.
</p>
<p>
For example, if we have a Markov chain with states 'Sunny', 'Rainy', and 'Cloudy', we might have an
initial state probability vector like
<strong>[0.5, 0.3, 0.2]</strong>. This would mean that at the start, the system has a 50% chance of
being in the 'Sunny' state, a 30% chance of being in the 'Rainy' state, and a 20% chance of being in
the 'Cloudy' state.
</p>
<p>
When defining Markov Chains, one can either set an initial state or initial state probabilities.
Selecting a sure initial state is simply setting the vector to <strong>[1.0, 0.0, 0.0]</strong>. In this example this
is exactly what we do, making the start state always sunny.
</p>
<p>
<strong> Example Runs from the current set rules: <br /> </strong>
[Initial: Sunny → Sunny → Cloudy → Rainy → Cloudy → Rainy → Rainy → Rainy → Cloudy → Cloudy → Sunny
→ End]
<br />
[Initial: Sunny → Sunny → Cloudy → Cloudy → Cloudy → Sunny → Sunny → Rainy → Rainy → Rainy → Rainy →
End]
<br />
[Initial: Sunny → Sunny → Sunny → Sunny → Sunny → Sunny → Sunny → Sunny → Sunny → Sunny → Sunny →
End]
</p>
</section>
<section class="n-step-transition-matrix mt-4">
<h5>N-Step Transition Matrix</h5>
<p>
What N-Step transition matrix gives us is the probability of reaching a certain future state given
the current state at n steps.
</p>
<p>
A N-Step transition matrix is found by taking the nth power of the original transition matrix. This
is based on the principle that the probability of transitioning from one state to another in n steps
is the sum of the probabilities of all paths from the initial state to the final state that consist
of exactly n steps.
</p>
<p>
If we want to find the 2-step transition matrix, we compute
<span class="math">P^2</span> (i.e., the matrix product of <span class="math">P</span> with itself).
This gives us the probabilities of transitioning from each state to each other state in 2 steps.
</p>
<p>
One important point to remember is that the process of taking powers of the transition matrix
assumes that the Markov chain is time-homogeneous, meaning that the transition probabilities do not
change over time.
</p>
<div class="steps ml-4">
<section class="step-1">
<h6>Get the 2-step transition matrix</h6>
<table class="dot-product table">
<thead>
<tr>
<th scope="col"></th>
<th scope="col">0</th>
<th scope="col">1</th>
<th scope="col">2</th>
</tr>
</thead>
<tbody>
<tr>
<th scope="row">0</th>
<td>0.53</td>
<td>0.31</td>
<td>0.16</td>
</tr>
<tr>
<th scope="row">1</th>
<td>0.19</td>
<td>0.35</td>
<td>0.28</td>
</tr>
<tr>
<th scope="row">2</th>
<td>0.25</td>
<td>0.49</td>
<td>0.26</td>
</tr>
</tbody>
</table>
<p>
It is telling us here that after 2 steps these are the chances to end up in a certain weather
</p>
</section>
<section class="3-step-and-beyond">
<h6>3 Step and Beyond</h6>
<p>
After the two step matrix it gets a little tricky as we need to get three and beyond we need to
get dot product of the previous step and the current transition which can get tedious as more
steps are added. <span class="math">(P^3 = P^2 * P)</span>
</p>
<table class="dot-product table">
<thead>
<tr>
<th scope="col"></th>
<th scope="col">0</th>
<th scope="col">1</th>
<th scope="col">2</th>
</tr>
</thead>
<tbody>
<tr>
<th scope="row">0</th>
<td>0.434</td>
<td>0.372</td>
<td>0.1939</td>
</tr>
<tr>
<th scope="row">1</th>
<td>0.242</td>
<td>0.496</td>
<td>0.2619</td>
</tr>
<tr>
<th scope="row">2</th>
<td>0.2759</td>
<td>0.474</td>
<td>0.2499</td>
</tr>
</tbody>
</table>
</section>
<section class="steady-state">
<h5>Steady State</h5>
<p>
The steady state in a Markov chain is a state where the system's probabilities remain unchanged
even after transitions. That is, the system has reached an equilibrium where the distribution of
states in the future does not depend on the current state.
</p>
<p>
The concept of steady state is central to stationary or time-homogeneous Markov chains, where
the transition probabilities don't change over time.
</p>
<h6>1-Step Transition Matrix</h6>
<table class="dot-product table">
<thead>
<tr>
<th scope="col"></th>
<th scope="col">0</th>
<th scope="col">1</th>
<th scope="col">2</th>
</tr>
</thead>
<tbody>
<tr>
<th scope="row">0</th>
<td>0.7</td>
<td>0.2</td>
<td>0.1</td>
</tr>
<tr>
<th scope="row">1</th>
<td>0.1</td>
<td>0.6</td>
<td>0.3</td>
</tr>
<tr>
<th scope="row">2</th>
<td>0.2</td>
<td>0.5</td>
<td>0.3</td>
</tr>
</tbody>
</table>
<h6>50-Step Transition Matrix</h6>
<table class="dot-product table">
<thead>
<tr>
<th scope="col"></th>
<th scope="col">0</th>
<th scope="col">1</th>
<th scope="col">2</th>
</tr>
</thead>
<tbody>
<tr>
<th scope="row">0</th>
<td>0.3095</td>
<td>0.4523</td>
<td>0.2380</td>
</tr>
<tr>
<th scope="row">1</th>
<td>0.3095</td>
<td>0.4523</td>
<td>0.2380</td>
</tr>
<tr>
<th scope="row">2</th>
<td>0.3095</td>
<td>0.4523</td>
<td>0.2380</td>
</tr>
</tbody>
</table>
<h6>100-Step Transition Matrix</h6>
<table class="dot-product table">
<thead>
<tr>
<th scope="col"></th>
<th scope="col">0</th>
<th scope="col">1</th>
<th scope="col">2</th>
</tr>
</thead>
<tbody>
<tr>
<th scope="row">0</th>
<td>0.3095</td>
<td>0.4523</td>
<td>0.2380</td>
</tr>
<tr>
<th scope="row">1</th>
<td>0.3095</td>
<td>0.4523</td>
<td>0.2380</td>
</tr>
<tr>
<th scope="row">2</th>
<td>0.3095</td>
<td>0.4523</td>
<td>0.2380</td>
</tr>
</tbody>
</table>
<div class="ending">
<h6>What is this all for?</h6>
<p>
From the result of the steady state we can say the long term probability of the day being a
certain weather is dependent on our achieved steady state.
</p>
<p>
<strong>Example</strong>
</p>
<p>
From the results of the weather chain, we can say that the probability of it being sunny is
around <strong>31%</strong>, rainy <strong>45%</strong>, and cloudy <strong>24%</strong>.
</p>
<p>
In a perfect world, where there are no changes due to factors like global warming, and
assuming our matrix is correct, if one gathers enough weather data, then we'd expect these
rough proportions.
</p>
</div>
</section>
<section class="outro">
<h5>Let's try the diagram below!</h5>
<p>You can try to create your own Markov Chain and see how it works.</p>
</section>
</div>
</section>
</div>
</div>
</div>
</div>
<!-- Title -->
<h3 class="mb-1 mt-5 text-center">Markov Chain Diagram</h3>
<small class="text-center text-muted mb-3">
Note: The diagram is interactable, you can drag the nodes around and zoom in and out.
</small>
<div class="d-grid gap-2 d-md-flex justify-content-center">
<button id="simulate-btn" class="btn btn-primary ms-2" type="button">Simulate</button>
</div>
<div class="col-lg-8 col-md-12 mb-3">
<div id="canvas" class="border" style="height: 600px"></div>
</div>
<div class="col mt-2 me-4" >
<div class="card" id="form-tab" style="height: 100%">
<div class="card-body">
<ul class="nav nav-pills nav-fill" id="myTab" role="tablist">
<li class="nav-item" role="presentation">
<button class="nav-link active" id="config-form-tab" data-bs-toggle="tab"
data-bs-target="#config-panel" type="button" role="tab"
aria-controls="config-panel" aria-selected="true">
Markov Chain Configuration
</button>
</li>
<li class="nav-item" role="presentation">
<button class="nav-link" id="nstep-form-tab" data-bs-toggle="tab"
data-bs-target="#nstep-panel" type="button" role="tab" aria-controls="nstep-panel"
aria-selected="false">
N-Step Probability Matrix Calculation
</button>
</li>
<li class="nav-item" role="presentation">
<button class="nav-link" id="timeline-tab" data-bs-toggle="tab"
data-bs-target="#timeline-panel" type="button" role="tab" aria-controls="timeline-panel"
aria-selected="false">
Event Sequence | Timeline
</button>
</li>
</ul>
<div class="tab-content" id="myTabContent">
<div class="tab-pane show active mt-2" id="config-panel" role="tabpanel"
aria-labelledby="config-form-tab">
<hr>
<div class="d-grid gap-2 d-md-flex justify-content-md-center">
<button id="weather-data" class="btn btn-info btn-sm" type="button">Example: Weather</button>
<button id="health-data" class="btn btn-info btn-sm" type="button">Example: Health</button>
<button id="manufacture-data" class="btn btn-info btn-sm" type="button">Example: Manufacture</button>
<button id="sports-data" class="btn btn-info btn-sm" type="button">Example: Sports</button>
</div>
<form id="config-form">
<div class="mb-3">
<label class="form-label">States</label>
<div id="state-inputs"></div>
<div class="d-grid gap-2 d-md-flex justify-content-md-end">
<button id="add-state" class="btn btn-primary mt-2">+</button>
<button id="delete-state" class="btn btn-danger mt-2 ms-2">-</button>
</div>
</div>
<div class="mb-3">
<label class="form-label">Initial State Probabilities</label>
<div id="init-prob-inputs"></div>
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