-
Notifications
You must be signed in to change notification settings - Fork 0
/
transition-probability-matrix.html
107 lines (88 loc) · 82.4 KB
/
transition-probability-matrix.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
<!DOCTYPE html><html><head>
<title>transition-probability-matrix</title>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/[email protected]/dist/katex.min.css">
<style>
code[class*=language-],pre[class*=language-]{color:#333;background:0 0;font-family:Consolas,"Liberation Mono",Menlo,Courier,monospace;text-align:left;white-space:pre;word-spacing:normal;word-break:normal;word-wrap:normal;line-height:1.4;-moz-tab-size:8;-o-tab-size:8;tab-size:8;-webkit-hyphens:none;-moz-hyphens:none;-ms-hyphens:none;hyphens:none}pre[class*=language-]{padding:.8em;overflow:auto;border-radius:3px;background:#f5f5f5}:not(pre)>code[class*=language-]{padding:.1em;border-radius:.3em;white-space:normal;background:#f5f5f5}.token.blockquote,.token.comment{color:#969896}.token.cdata{color:#183691}.token.doctype,.token.macro.property,.token.punctuation,.token.variable{color:#333}.token.builtin,.token.important,.token.keyword,.token.operator,.token.rule{color:#a71d5d}.token.attr-value,.token.regex,.token.string,.token.url{color:#183691}.token.atrule,.token.boolean,.token.code,.token.command,.token.constant,.token.entity,.token.number,.token.property,.token.symbol{color:#0086b3}.token.prolog,.token.selector,.token.tag{color:#63a35c}.token.attr-name,.token.class,.token.class-name,.token.function,.token.id,.token.namespace,.token.pseudo-class,.token.pseudo-element,.token.url-reference .token.variable{color:#795da3}.token.entity{cursor:help}.token.title,.token.title .token.punctuation{font-weight:700;color:#1d3e81}.token.list{color:#ed6a43}.token.inserted{background-color:#eaffea;color:#55a532}.token.deleted{background-color:#ffecec;color:#bd2c00}.token.bold{font-weight:700}.token.italic{font-style:italic}.language-json .token.property{color:#183691}.language-markup .token.tag .token.punctuation{color:#333}.language-css .token.function,code.language-css{color:#0086b3}.language-yaml .token.atrule{color:#63a35c}code.language-yaml{color:#183691}.language-ruby .token.function{color:#333}.language-markdown .token.url{color:#795da3}.language-makefile .token.symbol{color:#795da3}.language-makefile .token.variable{color:#183691}.language-makefile .token.builtin{color:#0086b3}.language-bash .token.keyword{color:#0086b3}pre[data-line]{position:relative;padding:1em 0 1em 3em}pre[data-line] .line-highlight-wrapper{position:absolute;top:0;left:0;background-color:transparent;display:block;width:100%}pre[data-line] .line-highlight{position:absolute;left:0;right:0;padding:inherit 0;margin-top:1em;background:hsla(24,20%,50%,.08);background:linear-gradient(to right,hsla(24,20%,50%,.1) 70%,hsla(24,20%,50%,0));pointer-events:none;line-height:inherit;white-space:pre}pre[data-line] .line-highlight:before,pre[data-line] .line-highlight[data-end]:after{content:attr(data-start);position:absolute;top:.4em;left:.6em;min-width:1em;padding:0 .5em;background-color:hsla(24,20%,50%,.4);color:#f4f1ef;font:bold 65%/1.5 sans-serif;text-align:center;vertical-align:.3em;border-radius:999px;text-shadow:none;box-shadow:0 1px #fff}pre[data-line] .line-highlight[data-end]:after{content:attr(data-end);top:auto;bottom:.4em}html body{font-family:'Helvetica Neue',Helvetica,'Segoe UI',Arial,freesans,sans-serif;font-size:16px;line-height:1.6;color:#333;background-color:#fff;overflow:initial;box-sizing:border-box;word-wrap:break-word}html body>:first-child{margin-top:0}html body h1,html body h2,html body h3,html body h4,html body h5,html body h6{line-height:1.2;margin-top:1em;margin-bottom:16px;color:#000}html body h1{font-size:2.25em;font-weight:300;padding-bottom:.3em}html body h2{font-size:1.75em;font-weight:400;padding-bottom:.3em}html body h3{font-size:1.5em;font-weight:500}html body h4{font-size:1.25em;font-weight:600}html body h5{font-size:1.1em;font-weight:600}html body h6{font-size:1em;font-weight:600}html body h1,html body h2,html body h3,html body h4,html body h5{font-weight:600}html body h5{font-size:1em}html body h6{color:#5c5c5c}html body strong{color:#000}html body del{color:#5c5c5c}html body a:not([href]){color:inherit;text-decoration:none}html body a{color:#08c;text-decoration:none}html body a:hover{color:#00a3f5;text-decoration:none}html body img{max-width:100%}html body>p{margin-top:0;margin-bottom:16px;word-wrap:break-word}html body>ol,html body>ul{margin-bottom:16px}html body ol,html body ul{padding-left:2em}html body ol.no-list,html body ul.no-list{padding:0;list-style-type:none}html body ol ol,html body ol ul,html body ul ol,html body ul ul{margin-top:0;margin-bottom:0}html body li{margin-bottom:0}html body li.task-list-item{list-style:none}html body li>p{margin-top:0;margin-bottom:0}html body .task-list-item-checkbox{margin:0 .2em .25em -1.8em;vertical-align:middle}html body .task-list-item-checkbox:hover{cursor:pointer}html body blockquote{margin:16px 0;font-size:inherit;padding:0 15px;color:#5c5c5c;background-color:#f0f0f0;border-left:4px solid #d6d6d6}html body blockquote>:first-child{margin-top:0}html body blockquote>:last-child{margin-bottom:0}html body hr{height:4px;margin:32px 0;background-color:#d6d6d6;border:0 none}html body table{margin:10px 0 15px 0;border-collapse:collapse;border-spacing:0;display:block;width:100%;overflow:auto;word-break:normal;word-break:keep-all}html body table th{font-weight:700;color:#000}html body table td,html body table th{border:1px solid #d6d6d6;padding:6px 13px}html body dl{padding:0}html body dl dt{padding:0;margin-top:16px;font-size:1em;font-style:italic;font-weight:700}html body dl dd{padding:0 16px;margin-bottom:16px}html body code{font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:.85em;color:#000;background-color:#f0f0f0;border-radius:3px;padding:.2em 0}html body code::after,html body code::before{letter-spacing:-.2em;content:'\00a0'}html body pre>code{padding:0;margin:0;word-break:normal;white-space:pre;background:0 0;border:0}html body .highlight{margin-bottom:16px}html body .highlight pre,html body pre{padding:1em;overflow:auto;line-height:1.45;border:#d6d6d6;border-radius:3px}html body .highlight pre{margin-bottom:0;word-break:normal}html body pre code,html body pre tt{display:inline;max-width:initial;padding:0;margin:0;overflow:initial;line-height:inherit;word-wrap:normal;background-color:transparent;border:0}html body pre code:after,html body pre code:before,html body pre tt:after,html body pre tt:before{content:normal}html body blockquote,html body dl,html body ol,html body p,html body pre,html body ul{margin-top:0;margin-bottom:16px}html body kbd{color:#000;border:1px solid #d6d6d6;border-bottom:2px solid #c7c7c7;padding:2px 4px;background-color:#f0f0f0;border-radius:3px}@media print{html body{background-color:#fff}html body h1,html body h2,html body h3,html body h4,html body h5,html body h6{color:#000;page-break-after:avoid}html body blockquote{color:#5c5c5c}html body pre{page-break-inside:avoid}html body table{display:table}html body img{display:block;max-width:100%;max-height:100%}html body code,html body pre{word-wrap:break-word;white-space:pre}}.markdown-preview{width:100%;height:100%;box-sizing:border-box}.markdown-preview ul{list-style:disc}.markdown-preview ul ul{list-style:circle}.markdown-preview ul ul ul{list-style:square}.markdown-preview ol{list-style:decimal}.markdown-preview ol ol,.markdown-preview ul ol{list-style-type:lower-roman}.markdown-preview ol ol ol,.markdown-preview ol ul ol,.markdown-preview ul ol ol,.markdown-preview ul ul ol{list-style-type:lower-alpha}.markdown-preview .newpage,.markdown-preview .pagebreak{page-break-before:always}.markdown-preview pre.line-numbers{position:relative;padding-left:3.8em;counter-reset:linenumber}.markdown-preview pre.line-numbers>code{position:relative}.markdown-preview pre.line-numbers .line-numbers-rows{position:absolute;pointer-events:none;top:1em;font-size:100%;left:0;width:3em;letter-spacing:-1px;border-right:1px solid #999;-webkit-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none}.markdown-preview pre.line-numbers .line-numbers-rows>span{pointer-events:none;display:block;counter-increment:linenumber}.markdown-preview pre.line-numbers .line-numbers-rows>span:before{content:counter(linenumber);color:#999;display:block;padding-right:.8em;text-align:right}.markdown-preview .mathjax-exps .MathJax_Display{text-align:center!important}.markdown-preview:not([data-for=preview]) .code-chunk .code-chunk-btn-group{display:none}.markdown-preview:not([data-for=preview]) .code-chunk .status{display:none}.markdown-preview:not([data-for=preview]) .code-chunk .output-div{margin-bottom:16px}.markdown-preview .md-toc{padding:0}.markdown-preview .md-toc .md-toc-link-wrapper .md-toc-link{display:inline;padding:.25rem 0}.markdown-preview .md-toc .md-toc-link-wrapper .md-toc-link div,.markdown-preview .md-toc .md-toc-link-wrapper .md-toc-link p{display:inline}.markdown-preview .md-toc .md-toc-link-wrapper.highlighted .md-toc-link{font-weight:800}.scrollbar-style::-webkit-scrollbar{width:8px}.scrollbar-style::-webkit-scrollbar-track{border-radius:10px;background-color:transparent}.scrollbar-style::-webkit-scrollbar-thumb{border-radius:5px;background-color:rgba(150,150,150,.66);border:4px solid rgba(150,150,150,.66);background-clip:content-box}html body[for=html-export]:not([data-presentation-mode]){position:relative;width:100%;height:100%;top:0;left:0;margin:0;padding:0;overflow:auto}html body[for=html-export]:not([data-presentation-mode]) .markdown-preview{position:relative;top:0;min-height:100vh}@media screen and (min-width:914px){html body[for=html-export]:not([data-presentation-mode]) .markdown-preview{padding:2em calc(50% - 457px + 2em)}}@media screen and (max-width:914px){html body[for=html-export]:not([data-presentation-mode]) .markdown-preview{padding:2em}}@media screen and (max-width:450px){html body[for=html-export]:not([data-presentation-mode]) .markdown-preview{font-size:14px!important;padding:1em}}@media print{html body[for=html-export]:not([data-presentation-mode]) #sidebar-toc-btn{display:none}}html body[for=html-export]:not([data-presentation-mode]) #sidebar-toc-btn{position:fixed;bottom:8px;left:8px;font-size:28px;cursor:pointer;color:inherit;z-index:99;width:32px;text-align:center;opacity:.4}html body[for=html-export]:not([data-presentation-mode])[html-show-sidebar-toc] #sidebar-toc-btn{opacity:1}html body[for=html-export]:not([data-presentation-mode])[html-show-sidebar-toc] .md-sidebar-toc{position:fixed;top:0;left:0;width:300px;height:100%;padding:32px 0 48px 0;font-size:14px;box-shadow:0 0 4px rgba(150,150,150,.33);box-sizing:border-box;overflow:auto;background-color:inherit}html body[for=html-export]:not([data-presentation-mode])[html-show-sidebar-toc] .md-sidebar-toc::-webkit-scrollbar{width:8px}html body[for=html-export]:not([data-presentation-mode])[html-show-sidebar-toc] .md-sidebar-toc::-webkit-scrollbar-track{border-radius:10px;background-color:transparent}html body[for=html-export]:not([data-presentation-mode])[html-show-sidebar-toc] .md-sidebar-toc::-webkit-scrollbar-thumb{border-radius:5px;background-color:rgba(150,150,150,.66);border:4px solid rgba(150,150,150,.66);background-clip:content-box}html body[for=html-export]:not([data-presentation-mode])[html-show-sidebar-toc] .md-sidebar-toc a{text-decoration:none}html body[for=html-export]:not([data-presentation-mode])[html-show-sidebar-toc] .md-sidebar-toc .md-toc{padding:0 16px}html body[for=html-export]:not([data-presentation-mode])[html-show-sidebar-toc] .md-sidebar-toc .md-toc .md-toc-link-wrapper .md-toc-link{display:inline;padding:.25rem 0}html body[for=html-export]:not([data-presentation-mode])[html-show-sidebar-toc] .md-sidebar-toc .md-toc .md-toc-link-wrapper .md-toc-link div,html body[for=html-export]:not([data-presentation-mode])[html-show-sidebar-toc] .md-sidebar-toc .md-toc .md-toc-link-wrapper .md-toc-link p{display:inline}html body[for=html-export]:not([data-presentation-mode])[html-show-sidebar-toc] .md-sidebar-toc .md-toc .md-toc-link-wrapper.highlighted .md-toc-link{font-weight:800}html body[for=html-export]:not([data-presentation-mode])[html-show-sidebar-toc] .markdown-preview{left:300px;width:calc(100% - 300px);padding:2em calc(50% - 457px - 300px / 2);margin:0;box-sizing:border-box}@media screen and (max-width:1274px){html body[for=html-export]:not([data-presentation-mode])[html-show-sidebar-toc] .markdown-preview{padding:2em}}@media screen and (max-width:450px){html body[for=html-export]:not([data-presentation-mode])[html-show-sidebar-toc] .markdown-preview{width:100%}}html body[for=html-export]:not([data-presentation-mode]):not([html-show-sidebar-toc]) .markdown-preview{left:50%;transform:translateX(-50%)}html body[for=html-export]:not([data-presentation-mode]):not([html-show-sidebar-toc]) .md-sidebar-toc{display:none}
/* Please visit the URL below for more information: */
/* https://shd101wyy.github.io/markdown-preview-enhanced/#/customize-css */
</style>
<!-- The content below will be included at the end of the <head> element. --><script type="text/javascript">
document.addEventListener("DOMContentLoaded", function () {
// your code here
});
</script></head><body for="html-export">
<div class="crossnote markdown-preview ">
<div class="home-btn">
<a href="index.html">
<img src="images/home.png" alt="home" style="width: 50px; height: 50px;">
</a>
<h3 id="eeg-microstate-sequences--transition-probability-matrix-tpm">EEG Microstate Sequences <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">→</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">→</span></span></span></span> Transition Probability Matrix (TPM) </h3>
<h4 id="transition-probability-matrix-p">Transition Probability Matrix <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span> </h4>
<p>To gain a more nuanced understanding of the transition dynamics, we transform the count-based transition matrices into probability matrices. This transformation provides a clearer picture of the likelihood of transitioning from one state to another.</p>
<p>For both the healthy and schizophrenia groups, we normalize the transition matrices by dividing each entry by the sum of its respective row. This process converts the raw counts into probabilities, where each row of the matrix sums to 1. The transformed matrices are denoted as <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mtext>healthy</mtext></msub></mrow><annotation encoding="application/x-tex">P_{\text{healthy}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">healthy</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mtext>schizo</mtext></msub></mrow><annotation encoding="application/x-tex">P_{\text{schizo}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">schizo</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> for the healthy and schizophrenia groups, respectively.</p>
<p>The transformation process is defined mathematically for each state <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> and transition <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> as follows:</p>
<p>For the <strong>healthy group</strong>:<br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msubsup><mi>P</mi><mrow><mi>i</mi><mi>j</mi></mrow><mtext>healthy</mtext></msubsup><mo>=</mo><mfrac><msubsup><mi>T</mi><mrow><mi>i</mi><mi>j</mi></mrow><mtext>healthy</mtext></msubsup><mrow><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi mathvariant="normal">∣</mi><mi>U</mi><mi mathvariant="normal">∣</mi></mrow></munderover><msubsup><mi>T</mi><mrow><mi>i</mi><mi>k</mi></mrow><mtext>healthy</mtext></msubsup></mrow></mfrac></mrow><annotation encoding="application/x-tex">P_{ij}^{\text{healthy}} = \frac{T_{ij}^{\text{healthy}}}{\sum_{k=1}^{|U|} T_{ik}^{\text{healthy}}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.38em;vertical-align:-0.413em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.967em;"><span style="top:-2.4231em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">healthy</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.413em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.0061em;vertical-align:-1.2361em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.77em;"><span style="top:-2.11em;"><span class="pstrut" style="height:3.0448em;"></span><span class="mord"><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∣</span><span class="mord mathnormal mtight" style="margin-right:0.10903em;">U</span><span class="mord mtight">∣</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.967em;"><span style="top:-2.3987em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">ik</span></span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">healthy</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.2748em;"><span class="pstrut" style="height:3.0448em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.8478em;"><span class="pstrut" style="height:3.0448em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.967em;"><span style="top:-2.4231em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">healthy</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.413em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.2361em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
<p>For the <strong>schizophrenia group</strong>:<br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msubsup><mi>P</mi><mrow><mi>i</mi><mi>j</mi></mrow><mtext>schizo</mtext></msubsup><mo>=</mo><mfrac><msubsup><mi>T</mi><mrow><mi>i</mi><mi>j</mi></mrow><mtext>schizo</mtext></msubsup><mrow><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi mathvariant="normal">∣</mi><mi>U</mi><mi mathvariant="normal">∣</mi></mrow></munderover><msubsup><mi>T</mi><mrow><mi>i</mi><mi>k</mi></mrow><mtext>schizo</mtext></msubsup></mrow></mfrac></mrow><annotation encoding="application/x-tex">P_{ij}^{\text{schizo}} = \frac{T_{ij}^{\text{schizo}}}{\sum_{k=1}^{|U|} T_{ik}^{\text{schizo}}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2822em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8991em;"><span style="top:-2.453em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">schizo</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.87em;vertical-align:-1.2361em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6339em;"><span style="top:-2.11em;"><span class="pstrut" style="height:3.0448em;"></span><span class="mord"><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∣</span><span class="mord mathnormal mtight" style="margin-right:0.10903em;">U</span><span class="mord mtight">∣</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8309em;"><span style="top:-2.3987em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">ik</span></span></span></span><span style="top:-3.0448em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">schizo</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.2748em;"><span class="pstrut" style="height:3.0448em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.8296em;"><span class="pstrut" style="height:3.0448em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-2.4413em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">schizo</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3948em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.2361em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
<p>where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>U</mi><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">|U|</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mord">∣</span></span></span></span> is the number of unique states, and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>T</mi><mrow><mi>i</mi><mi>j</mi></mrow><mtext>healthy</mtext></msubsup></mrow><annotation encoding="application/x-tex">T_{ij}^{\text{healthy}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.38em;vertical-align:-0.413em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.967em;"><span style="top:-2.4231em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">healthy</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.413em;"><span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>T</mi><mrow><mi>i</mi><mi>j</mi></mrow><mtext>schizo</mtext></msubsup></mrow><annotation encoding="application/x-tex">T_{ij}^{\text{schizo}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2439em;vertical-align:-0.3948em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-2.4413em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">schizo</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3948em;"><span></span></span></span></span></span></span></span></span></span> are the original transition counts from state <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> to <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> in the healthy and schizophrenia groups, respectively.</p>
<p>This probability-based approach allows for a more refined and accurate comparison between the two groups, highlighting differences in the likelihood of state transitions rather than merely their frequency.</p>
<p>In pursuit of uncovering the similarities and dissimilarities in state transition dynamics between healthy individuals and those with schizophrenia, advanced analytical methods are employed, such as correlation analysis and Distance calculations.</p>
<h4 id="correlation-analysis-of-transition-dynamics">Correlation Analysis of Transition Dynamics </h4>
<p>Correlation analysis is a statistical method that measures the strength and direction of a linear relationship between two variables. In the context of transition matrices, it's used to compare how similarly two groups transition between states. This is done by computing the Pearson correlation coefficient for each corresponding pair of states between the two groups and then averaging these coefficients to obtain an overall measure of similarity.</p>
<p>The correlation coefficient <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span> for states <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> in the healthy group <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>and the schizophrenia group <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span></span></span></span> is calculated as follows:</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>r</mi><mi>i</mi></msub><mo>=</mo><mtext>corr</mtext><mo stretchy="false">(</mo><msub><mi>P</mi><mrow><mi>i</mi><mo separator="true">,</mo><mi>H</mi></mrow></msub><mo separator="true">,</mo><msub><mi>P</mi><mrow><mi>i</mi><mo separator="true">,</mo><mi>S</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">r_i = \text{corr}(P_{i, H}, P_{i, S})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord text"><span class="mord">corr</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.08125em;">H</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">S</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></p>
<p>where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>corr</mtext></mrow><annotation encoding="application/x-tex">\text{corr}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord text"><span class="mord">corr</span></span></span></span></span> denotes the Pearson correlation function, and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mrow><mi>i</mi><mo separator="true">,</mo><mi>H</mi></mrow></msub></mrow><annotation encoding="application/x-tex">P_{i, H}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.08125em;">H</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mrow><mi>i</mi><mo separator="true">,</mo><mi>S</mi></mrow></msub></mrow><annotation encoding="application/x-tex">P_{i, S}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">S</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> represent the probability distributions of transitioning from state <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> to all other states for the healthy and schizophrenia groups, respectively.</p>
<p>The overall correlation across all states is then given by:</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>r</mi><mo>=</mo><mfrac><mn>1</mn><mrow><mi mathvariant="normal">∣</mi><mi>U</mi><mi mathvariant="normal">∣</mi></mrow></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi mathvariant="normal">∣</mi><mi>U</mi><mi mathvariant="normal">∣</mi></mrow></munderover><msub><mi>r</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">r = \frac{1}{|U|} \sum_{i=1}^{|U|} r_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.2387em;vertical-align:-1.2777em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mord">∣</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.961em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.386em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∣</span><span class="mord mathnormal mtight" style="margin-right:0.10903em;">U</span><span class="mord mtight">∣</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></p>
<p>where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>U</mi><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">|U|</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mord">∣</span></span></span></span> is the total number of unique states.</p>
<p>A correlation coefficient of 1 indicates a perfect positive correlation, while a value of -1 signifies a perfect negative correlation. A coefficient of 0 indicates no correlation between the two groups.</p>
<h4 id="distance-for-comparative-analysis">Distance for Comparative Analysis </h4>
<p>The Distance is a measure used to quantify the difference between two matrices. It's particularly useful in this context as it provides a single value that captures the overall disparity in transition probabilities between the two groups, accounting for the entirety of the transition matrix.</p>
<p>For matrices <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mtext>healthy</mtext></msub></mrow><annotation encoding="application/x-tex">P_{\text{healthy}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">healthy</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mtext>schizo</mtext></msub></mrow><annotation encoding="application/x-tex">P_{\text{schizo}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">schizo</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, representing the transition probabilities of the healthy and schizophrenia groups respectively, the Distance is defined as:</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>Distance</mtext><mo>=</mo><msqrt><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi mathvariant="normal">∣</mi><mi>U</mi><mi mathvariant="normal">∣</mi></mrow></munderover><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi mathvariant="normal">∣</mi><mi>U</mi><mi mathvariant="normal">∣</mi></mrow></munderover><mo stretchy="false">(</mo><msubsup><mi>P</mi><mrow><mi>i</mi><mi>j</mi></mrow><mtext>schizo</mtext></msubsup><mo>−</mo><msubsup><mi>P</mi><mrow><mi>i</mi><mi>j</mi></mrow><mtext>healthy</mtext></msubsup><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></msqrt></mrow><annotation encoding="application/x-tex">\text{Distance} = \sqrt{\sum_{i=1}^{|U|} \sum_{j=1}^{|U|} (P_{ij}^{\text{schizo}} - P_{ij}^{\text{healthy}})^2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord">Distance</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.6025em;vertical-align:-1.4138em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.1888em;"><span class="svg-align" style="top:-5.5625em;"><span class="pstrut" style="height:5.5625em;"></span><span class="mord" style="padding-left:1.056em;"><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.961em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.386em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∣</span><span class="mord mathnormal mtight" style="margin-right:0.10903em;">U</span><span class="mord mtight">∣</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.961em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.386em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∣</span><span class="mord mathnormal mtight" style="margin-right:0.10903em;">U</span><span class="mord mtight">∣</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.4138em;"><span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8309em;"><span style="top:-2.4231em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span><span style="top:-3.0448em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">schizo</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.413em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.967em;"><span style="top:-2.4231em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">healthy</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.413em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-4.1488em;"><span class="pstrut" style="height:5.5625em;"></span><span class="hide-tail" style="min-width:0.742em;height:3.6425em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="3.6425em" viewBox="0 0 400000 3642" preserveAspectRatio="xMinYMin slice"><path d="M702 80H40000040
H742v3508l-4 4-4 4c-.667.7 -2 1.5-4 2.5s-4.167 1.833-6.5 2.5-5.5 1-9.5 1
h-12l-28-84c-16.667-52-96.667 -294.333-240-727l-212 -643 -85 170
c-4-3.333-8.333-7.667-13 -13l-13-13l77-155 77-156c66 199.333 139 419.667
219 661 l218 661zM702 80H400000v40H742z"></path></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.4138em;"><span></span></span></span></span></span></span></span></span></span></p>
<p>where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>U</mi><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">|U|</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mord">∣</span></span></span></span> is the total number of unique states.</p>
<p>A smaller Distance indicates a closer resemblance between the transition behaviors of the two groups, while a larger value signifies greater differences.</p>
<p>By calculating both the overall Correlation and the Distance, we can gain a deeper understanding of the transition dynamics characteristic of each group, potentially unveiling unique patterns that could serve as biomarkers for schizophrenia.</p>
<h4 id="transition-probability-matrix---including-self-transitions">Transition Probability Matrix - including self-transitions </h4>
<p>Transition probability matrices are constructed by normalizing the transition matrices by dividing each entry by the sum of its respective row. This process converts the raw counts into probabilities, where each row of the matrix sums to 1. The transformed matrices are denoted as <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mtext>healthy</mtext></msub></mrow><annotation encoding="application/x-tex">P_{\text{healthy}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">healthy</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mtext>schizo</mtext></msub></mrow><annotation encoding="application/x-tex">P_{\text{schizo}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">schizo</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> for the healthy and schizophrenia groups, respectively.</p>
<p>The Distance and correlation coefficient are calculated for the transition probability matrices of the healthy and schizophrenia groups. The results are as follows:</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>Distance</mtext><mo>=</mo><mn>0.038</mn></mrow><annotation encoding="application/x-tex">\text{Distance} = 0.038</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord">Distance</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0.038</span></span></span></span></span></p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>Correlation Coefficient</mtext><mo>=</mo><mn>0.99</mn></mrow><annotation encoding="application/x-tex">\text{Correlation Coefficient} = 0.99</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord">Correlation Coefficient</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0.99</span></span></span></span></span></p>
<a href="images/transition/transition_probabilities_self.png">
<img src="images/transition/transition_probabilities_self.png" alt="transition_probabilities_self" style="margin-bottom: 50px; margin-top: 50px; scale: 1.2;">
</a>
<p>Fig.9 We can see that due to the high number of self-transitions, the transition probability matrices are dominated by the diagonal entries. This is reflected in the high correlation coefficient and low Distance, which indicate a high degree of similarity between the two groups. However, this is misleading, as the self-transitions are not indicative of the actual transition dynamics between states but rather the tendency of each group to remain in the same state. And this is of course informative as we saw on count distributions of self-transitions. But it is not what we are looking for. We are interested in the transitions between states. So let's remove the self-transitions from the transition matrices and see what happens.</p>
<h4 id="transition-probability-matrix---excluding-self-transitions">Transition Probability Matrix - excluding self-transitions </h4>
<p>Obtained <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mtext>healthy</mtext></msub></mrow><annotation encoding="application/x-tex">P_{\text{healthy}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">healthy</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mtext>schizo</mtext></msub></mrow><annotation encoding="application/x-tex">P_{\text{schizo}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">schizo</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> for the healthy and schizophrenia groups, respectively without self-transitions.</p>
<p>The Distance and correlation coefficient are calculated for the transition probability matrices of the healthy and schizophrenia groups. The results are as follows:</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>Distance</mtext><mo>=</mo><mn>0.287</mn></mrow><annotation encoding="application/x-tex">\text{Distance} = 0.287</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord">Distance</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0.287</span></span></span></span></span></p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>Correlation Coefficient</mtext><mo>=</mo><mn>0.75</mn></mrow><annotation encoding="application/x-tex">\text{Correlation Coefficient} = 0.75</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord">Correlation Coefficient</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0.75</span></span></span></span></span></p>
<a href="images/transition/transition_probabilities_no_self.png">
<img src="images/transition/transition_probabilities_no_self.png" alt="transition_probabilities_no_self" style="margin-bottom: 50px; margin-top: 50px; scale: 1.2;">
</a>
<p>Fig.10 We can see that the Distance is much higher and the correlation coefficient is much lower. This is because the self-transitions are not influencing the analysis. The transition probability matrices are now dominated by the off-diagonal entries, which reflect the actual transition dynamics between states. This is what we are looking for.</p>
<h4 id="direct-graph">Direct Graph </h4>
<p>The transition from matrix representations of state sequences to visual graph structures offers an intuitive understanding of the complex dynamics within each group. By pruning and normalizing the matrices, we can construct directed graphs that reveal the most probable paths and highlight the interdependencies between states.</p>
<p>For the <strong>healthy group</strong>, the directed graph <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>G</mi><mtext>healthy</mtext></msub></mrow><annotation encoding="application/x-tex">G_{\text{healthy}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">G</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">healthy</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> is formed by connecting state <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> to state <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> if the probability <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>P</mi><mrow><mi>i</mi><mi>j</mi></mrow><mtext>healthy</mtext></msubsup></mrow><annotation encoding="application/x-tex">P_{ij}^{\text{healthy}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.38em;vertical-align:-0.413em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.967em;"><span style="top:-2.4231em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">healthy</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.413em;"><span></span></span></span></span></span></span></span></span></span> exceeds a predetermined threshold. This threshold is set to capture the top 10% of transition probabilities, ensuring that only the most significant connections are visualized.</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>G</mi><mtext>healthy</mtext></msub><mo>=</mo><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mi>i</mi><mo separator="true">,</mo><mi>j</mi><mo stretchy="false">)</mo><mtext> </mtext><mi mathvariant="normal">∣</mi><mtext> </mtext><msubsup><mi>P</mi><mrow><mi>i</mi><mi>j</mi></mrow><mtext>healthy</mtext></msubsup><mo>≥</mo><mtext>threshold</mtext><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">G_{\text{healthy}} = \{ (i, j) \ | \ P_{ij}^{\text{healthy}} \geq \text{threshold} \}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">G</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">healthy</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.38em;vertical-align:-0.413em;"></span><span class="mopen">{(</span><span class="mord mathnormal">i</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mclose">)</span><span class="mspace"> </span><span class="mord">∣</span><span class="mspace"> </span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.967em;"><span style="top:-2.4231em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">healthy</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.413em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">threshold</span></span><span class="mclose">}</span></span></span></span></span></p>
<p>Similarly, for the <strong>schizophrenia group</strong>, the directed graph <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>G</mi><mtext>schizo</mtext></msub></mrow><annotation encoding="application/x-tex">G_{\text{schizo}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">G</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">schizo</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is established using the same principle.</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>G</mi><mtext>schizo</mtext></msub><mo>=</mo><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mi>i</mi><mo separator="true">,</mo><mi>j</mi><mo stretchy="false">)</mo><mtext> </mtext><mi mathvariant="normal">∣</mi><mtext> </mtext><msubsup><mi>P</mi><mrow><mi>i</mi><mi>j</mi></mrow><mtext>schizo</mtext></msubsup><mo>≥</mo><mtext>threshold</mtext><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">G_{\text{schizo}} = \{ (i, j) \ | \ P_{ij}^{\text{schizo}} \geq \text{threshold} \}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">G</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">schizo</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2822em;vertical-align:-0.3831em;"></span><span class="mopen">{(</span><span class="mord mathnormal">i</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mclose">)</span><span class="mspace"> </span><span class="mord">∣</span><span class="mspace"> </span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8991em;"><span style="top:-2.453em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">schizo</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">threshold</span></span><span class="mclose">}</span></span></span></span></span></p>
<p>The vertices of these graphs correspond to the unique states, while the edges depict the transitions with probabilities surpassing the pruning threshold. These graphs provide a visual representation of the most likely transitions, as well as insights into the overall structure and connectivity of the state dynamics.</p>
<p>The Distance of the difference between the two graphs weight adjacency matrices is:</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>Distance</mtext><mo>=</mo><mn>0.61</mn></mrow><annotation encoding="application/x-tex">\text{Distance} = 0.61</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord">Distance</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0.61</span></span></span></span></span></p>
<a href="images/transition/graphs_prob.png">
<img src="images/transition/graphs_prob.png" alt=" Directed Graph" style="margin-bottom: 50px; margin-top: 50px; scale: 1.0;">
</a>
<p>Fig.11 and Fig.12 illustrate the directed graphs for the healthy and schizophrenia groups, respectively. The vertices represent the unique states, while the edges depict the transitions with probabilities surpassing the pruning threshold. These graphs provide a visual representation of the most likely transitions, as well as insights into the overall structure and connectivity of the state dynamics. The position and color of the vertices is the same for both graphs as well labels for the vertices. The thickness of the edges is proportional to the probability of the transition. The size of the node it its degree, i.e., the number of edges connected to it.</p>
<p>The analysis of these graphs can elucidate characteristics such as resilience, adaptability, and potential biomarkers within the EEG microstate sequences for each group. The directed graphs not only emphasize the significant transitions but also pave the way for identifying key differences in the neurological patterns associated with healthy and schizophrenic brains. We see now that performing analysis with self-transitions included in the transition matrices can lead to misleading results.</p>
<hr>
<h6 id="author-łukasz-furmancracernetgmailcom">Author: <a href="[email protected]">Łukasz Furman</a> </h6>
</div>
</body></html>