-
Notifications
You must be signed in to change notification settings - Fork 0
/
correlation-matrices.html
145 lines (126 loc) · 116 KB
/
correlation-matrices.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
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
<!DOCTYPE html><html><head>
<title>correlation-matrices</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="correlation-matrices-of-eeg-microstate-sequences-in-healthy-and-schizophrenic-individuals">Correlation Matrices of EEG Microstate Sequences in Healthy and Schizophrenic Individuals </h3>
<p>Continuing our deep dive into the analysis of EEG microstate sequences, we now turn our attention to understanding the intricate correlations within and between healthy and schizophrenia groups. This approach involves creating and examining correlation matrices, providing valuable insights into the interconnectedness of different brain states. The histograms and statistical tests employed further enhance our grasp of the nuanced relationships in these groups.</p>
<h4 id="creating-correlation-matrices">Creating Correlation Matrices </h4>
<p>The correlation matrices are generated by calculating the Pearson correlation coefficients between all pairs of sequences within each group. The process can be mathematically described 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><msub><mtext>Corr</mtext><mtext>healthy</mtext></msub><mo>=</mo><mtext>corrcoef</mtext><mo stretchy="false">(</mo><mtext>healthy_seqs</mtext><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{Corr}_{\text{healthy}} = \text{corrcoef}(\text{healthy\_seqs})</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 text"><span class="mord">Corr</span></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-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:1.06em;vertical-align:-0.31em;"></span><span class="mord text"><span class="mord">corrcoef</span></span><span class="mopen">(</span><span class="mord text"><span class="mord">healthy_seqs</span></span><span class="mclose">)</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><msub><mtext>Corr</mtext><mtext>schizo</mtext></msub><mo>=</mo><mtext>corrcoef</mtext><mo stretchy="false">(</mo><mtext>schizo_seqs</mtext><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{Corr}_{\text{schizo}} = \text{corrcoef}(\text{schizo\_seqs})</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 text"><span class="mord">Corr</span></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-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:1.06em;vertical-align:-0.31em;"></span><span class="mord text"><span class="mord">corrcoef</span></span><span class="mopen">(</span><span class="mord text"><span class="mord">schizo_seqs</span></span><span class="mclose">)</span></span></span></span></span></p>
<p>where <code>corrcoef</code> is the numpy function that computes the correlation coefficients.</p>
<p>Since the diagonal elements of a correlation matrix always equal 1 (as a sequence is perfectly correlated with itself), these are set to 0 to focus on the relationships between different sequences.</p>
<p>The matrices are visualized as heatmaps, allowing us to observe patterns of correlation across the groups.</p>
<a href="images/transition/correlation_matrices.png">
<img src="images/transition/correlation_matrices.png" alt="Correlation Matrices" style="margin-left: 0px; scale: 0.9">
</a>
<p>Fig.1 and Fig.2 Correlation matrices for healthy and schizophrenia groups, respectively. Entrie <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi><mo separator="true">,</mo><mi>j</mi></mrow><annotation encoding="application/x-tex">i,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">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></span></span> represents the correlation between subject sequence <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 subject sequence <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> within the group.</p>
<h4 id="histograms-of-correlations">Histograms of Correlations </h4>
<p>To further dissect these relationships, histograms of the flattened correlation matrices are plotted. These histograms display the distribution of correlation coefficients, providing insights into the prevalence of various degrees of correlation within each group.</p>
<p>The histograms for each group 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>Histogram</mtext><mo stretchy="false">(</mo><msub><mtext>Corr</mtext><mtext>healthy</mtext></msub><mi mathvariant="normal">.</mi><mtext>flatten</mtext><mo stretchy="false">(</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{Histogram}(\text{Corr}_{\text{healthy}}.\text{flatten}())</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord text"><span class="mord">Histogram</span></span><span class="mopen">(</span><span class="mord"><span class="mord text"><span class="mord">Corr</span></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-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="mord">.</span><span class="mord text"><span class="mord">flatten</span></span><span class="mopen">(</span><span class="mclose">))</span></span></span></span></span><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><mtext>Histogram</mtext><mo stretchy="false">(</mo><msub><mtext>Corr</mtext><mtext>schizo</mtext></msub><mi mathvariant="normal">.</mi><mtext>flatten</mtext><mo stretchy="false">(</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{Histogram}(\text{Corr}_{\text{schizo}}.\text{flatten}())</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 text"><span class="mord">Histogram</span></span><span class="mopen">(</span><span class="mord"><span class="mord text"><span class="mord">Corr</span></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-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="mord">.</span><span class="mord text"><span class="mord">flatten</span></span><span class="mopen">(</span><span class="mclose">))</span></span></span></span></span></p>
<p>Combined histogram plots the distributions of both groups together, allowing for a direct comparison</p>
<a href="images/transition/histogram_correlations.png">
<img src="images/transition/histogram_correlations.png" alt="Histogram of Correlations" style="margin-left: 0px; scale: 0.9">
</a>
<p>Fig.3 Histograms of correlations for healthy and schizophrenia groups.</p>
<h4 id="statistical-significance-analysis">Statistical Significance Analysis </h4>
<p>To assess the statistical significance of the observed differences in correlations between the two groups, a bootstrap method is employed. This method involves repeatedly resampling the flattened correlation matrices and calculating the mean difference in correlations for each sample. The distribution of these differences is then plotted as a histogram.</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>Bootstrap Difference</mtext><mo>=</mo><mfrac><mn>1</mn><msub><mi>n</mi><mtext>bootstrap</mtext></msub></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><msub><mi>n</mi><mtext>bootstrap</mtext></msub></munderover><mo stretchy="false">(</mo><mtext>Mean</mtext><mo stretchy="false">(</mo><mtext>Resample</mtext><mo stretchy="false">(</mo><msub><mtext>Corr</mtext><mtext>healthy</mtext></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>−</mo><mtext>Mean</mtext><mo stretchy="false">(</mo><mtext>Resample</mtext><mo stretchy="false">(</mo><msub><mtext>Corr</mtext><mtext>schizo</mtext></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{Bootstrap Difference} = \frac{1}{n_{\text{bootstrap}}} \sum_{i=1}^{n_{\text{bootstrap}}} (\text{Mean}(\text{Resample}(\text{Corr}_{\text{healthy}})) - \text{Mean}(\text{Resample}(\text{Corr}_{\text{schizo}})))</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord">Bootstrap Difference</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:3.0431em;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 class="mord mathnormal">n</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">bootstrap</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 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:0.9721em;"><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:1.7655em;"><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:-4.4141em;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 class="mord mathnormal mtight">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3488em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">bootstrap</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2901em;"><span></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:1.2777em;"><span></span></span></span></span></span><span class="mopen">(</span><span class="mord text"><span class="mord">Mean</span></span><span class="mopen">(</span><span class="mord text"><span class="mord">Resample</span></span><span class="mopen">(</span><span class="mord"><span class="mord text"><span class="mord">Corr</span></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-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="mclose">))</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">Mean</span></span><span class="mopen">(</span><span class="mord text"><span class="mord">Resample</span></span><span class="mopen">(</span><span class="mord"><span class="mord text"><span class="mord">Corr</span></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-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="mclose">)))</span></span></span></span></span></p>
<p>The p-value, derived from the bootstrap analysis, indicates the likelihood of observing such a difference under the null hypothesis (no significant difference between the groups).</p>
<p>The bootstrap histogram is as follows: <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>Histogram</mtext><mo stretchy="false">(</mo><mtext>Bootstrap Difference</mtext><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{Histogram}(\text{Bootstrap Difference})</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 text"><span class="mord">Histogram</span></span><span class="mopen">(</span><span class="mord text"><span class="mord">Bootstrap Difference</span></span><span class="mclose">)</span></span></span></span></span> <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>p-value</mtext><mo>=</mo><mn>0.037</mn></mrow><annotation encoding="application/x-tex">\text{p-value} = 0.037</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord">p-value</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">0.037</span></span></span></span></span></p>
<a href="images/transition/bootstrap_histogram.png">
<img src="images/transition/bootstrap_histogram.png" alt="Bootstrap Histogram" style="margin-left: 0px; scale: 0.9">
</a>
<p>Fig.4 Bootstrap histogram showcasing the differences in correlations.</p>
<p>Now, we can construct transition matrices for both the healthy and schizophrenia groups, providing a quantitative framework to examine the state transitions.</p>
<p>Let's assume that we have a set of sequences <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> of states, where each sequence <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">s \in S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">s</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:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span></span></span></span> is a list of states visited in order. <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>S</mi><mtext>healthy</mtext></msub></mrow><annotation encoding="application/x-tex">S_{\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:0.05764em;">S</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:-0.0576em;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>S</mi><mtext>schizo</mtext></msub></mrow><annotation encoding="application/x-tex">S_{\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:0.05764em;">S</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:-0.0576em;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> represent the sequences for the healthy and schizophrenia groups, respectively.</p>
<h4 id="transition-matrix-t">Transition Matrix <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</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;">T</span></span></span></span> </h4>
<p>The transition matrix <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</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;">T</span></span></span></span> is a square matrix where the entry <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>T</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">T_{ij}</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;">T</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:-0.1389em;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><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> represents the number of transitions 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 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>. Given the set of sequences <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>, the transition matrix is constructed as follows:</p>
<ol>
<li>Identify the set of unique states <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>U</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:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span></span></span></span> across all sequences in <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>.</li>
<li>Initialize a matrix <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</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;">T</span></span></span></span> with dimensions <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><mo>×</mo><mi mathvariant="normal">∣</mi><mi>U</mi><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">|U| \times |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 class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><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>, 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, with all entries set to zero.</li>
<li>For each sequence <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">s \in S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">s</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:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span></span></span></span>, and for each consecutive pair of states <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>s</mi><mi>k</mi></msub><mo separator="true">,</mo><msub><mi>s</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(s_k, s_{k+1})</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="mopen">(</span><span class="mord"><span class="mord mathnormal">s</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 mathnormal mtight" style="margin-right:0.03148em;">k</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="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">s</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 mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> in <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:0.4306em;"></span><span class="mord mathnormal">s</span></span></span></span>, increment the matrix entry <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>T</mi><mrow><msub><mi>s</mi><mi>k</mi></msub><mo separator="true">,</mo><msub><mi>s</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></msub></mrow><annotation encoding="application/x-tex">T_{s_k, s_{k+1}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9809em;vertical-align:-0.2975em;"></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:0.1514em;"><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 mtight"><span class="mord mathnormal mtight">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3488em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.1512em;"><span></span></span></span></span></span></span><span class="mpunct mtight">,</span><span class="mord mtight"><span class="mord mathnormal mtight">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3488em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2107em;"><span></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:0.2975em;"><span></span></span></span></span></span></span></span></span></span> by 1.</li>
</ol>
<p>Mathematically, the construction can be written 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><msub><mi>T</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><munder><mo>∑</mo><mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></munder><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi mathvariant="normal">∣</mi><mi>s</mi><mi mathvariant="normal">∣</mi><mo>−</mo><mn>1</mn></mrow></munderover><mo stretchy="false">[</mo><msub><mi>s</mi><mi>k</mi></msub><mo>=</mo><mi>i</mi><mo>∧</mo><msub><mi>s</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mi>j</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">T_{ij} = \sum_{s \in S} \sum_{k=1}^{|s|-1} [s_k = i \land s_{k+1} = j]</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;">T</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:-0.1389em;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><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:3.2827em;vertical-align:-1.3217em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.05em;"><span style="top:-1.8557em;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">s</span><span class="mrel mtight">∈</span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">S</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><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.3217em;"><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:-1.8479em;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:0.03148em;">k</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">s</span><span class="mord mtight">∣</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.3021em;"><span></span></span></span></span></span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal">s</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 mathnormal mtight" style="margin-right:0.03148em;">k</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:0.6595em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">s</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 mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><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 mathnormal" style="margin-right:0.05724em;">j</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><mo stretchy="false">[</mo><mo>⋅</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[ \cdot ]</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="mopen">[</span><span class="mord">⋅</span><span class="mclose">]</span></span></span></span> is the Iverson bracket, which is 1 if the condition is true, and 0 otherwise.</p>
<p>Since the transition matrix is a square matrix, it can be visualized as a heatmap, where the rows represent the source states, and the columns represent the destination states.</p>
<a href="images/healthy_matrix.png">
<img src="images/healthy_matrix.png" alt="Healthy Matrix" style="margin-left: 40px; scale: 0.9;">
</a>
<p>Fig.5 Transition matrix <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>T</mi><mrow><mi>h</mi><mi>e</mi><mi>a</mi><mi>l</mi><mi>t</mi><mi>h</mi><mi>y</mi></mrow></msub></mrow><annotation encoding="application/x-tex">T_{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;">T</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 mathnormal mtight">h</span><span class="mord mathnormal mtight">e</span><span class="mord mathnormal mtight">a</span><span class="mord mathnormal mtight">lt</span><span class="mord mathnormal mtight">h</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</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> for the healthy group. The rows represent the source states, and the columns represent the destination states. The entry <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>T</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">T_{ij}</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;">T</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:-0.1389em;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><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> represents the number of transitions 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 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>. The diagonal entries are set to <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>, for visualization purposes.</p>
<a href="images/schizo_matrix.png">
<img src="images/schizo_matrix.png" alt="Schizo Matrix" style="margin-left: 40px; scale: 0.9">
</a>
<p>Fig.6 Transition matrix <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>T</mi><mrow><mi>s</mi><mi>c</mi><mi>h</mi><mi>i</mi><mi>z</mi><mi>o</mi></mrow></msub></mrow><annotation encoding="application/x-tex">T_{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;">T</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 mathnormal mtight">sc</span><span class="mord mathnormal mtight">hi</span><span class="mord mathnormal mtight">zo</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 schizophrenia group. The rows represent the source states, and the columns represent the destination states. The entry <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>T</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">T_{ij}</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;">T</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:-0.1389em;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><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> represents the number of transitions 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 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>. The diagonal entries are set to <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>, for visualization purposes.</p>
<p>The analysis of transition matrices for the healthy and schizophrenia groups provides a comprehensive understanding of state transition dynamics in these populations. The matrices, visualized in Figures 1 and 2, offer a macroscopic view of how frequently each state transitions to another. Before we transform the transition matrices into a probability matrix lets delve deeper into the characteristics of these transitions.</p>
<h4 id="comparative-analysis-of-transition-matrices">Comparative Analysis of Transition Matrices </h4>
<p>A notable aspect of the transition matrices is the ratio of self-transitions (transitions from a state to itself) to all transitions. The self-loops can be conceptualized 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><msub><mi>T</mi><mrow><mi>i</mi><mi>i</mi></mrow></msub><mo>=</mo><munder><mo>∑</mo><mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></munder><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi mathvariant="normal">∣</mi><mi>s</mi><mi mathvariant="normal">∣</mi><mo>−</mo><mn>1</mn></mrow></munderover><mo stretchy="false">[</mo><msub><mi>s</mi><mi>k</mi></msub><mo>=</mo><msub><mi>s</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mi>i</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">T_{ii} = \sum_{s \in S} \sum_{k=1}^{|s|-1} [s_k = s_{k+1} = i]</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;">T</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:-0.1389em;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">ii</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:3.2827em;vertical-align:-1.3217em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.05em;"><span style="top:-1.8557em;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">s</span><span class="mrel mtight">∈</span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">S</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><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.3217em;"><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:-1.8479em;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:0.03148em;">k</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">s</span><span class="mord mtight">∣</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.3021em;"><span></span></span></span></span></span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal">s</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 mathnormal mtight" style="margin-right:0.03148em;">k</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:0.6389em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">s</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 mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><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 mathnormal">i</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><mo stretchy="false">[</mo><mo>⋅</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[ \cdot ]</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="mopen">[</span><span class="mord">⋅</span><span class="mclose">]</span></span></span></span> is the Iverson bracket, which is 1 if the condition is true, and 0 otherwise.<br>
And the ratio between the number of self-loops and the total number of transitions 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><mfrac><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><msub><mi>T</mi><mrow><mi>i</mi><mi>i</mi></mrow></msub></mrow><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><msub><mi>T</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\sum_{i=1}^{|U|} T_{ii}}{\sum_{i=1}^{|U|} \sum_{j=1}^{|U|} T_{ij}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0713em;vertical-align:-1.3537em;"></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:1.7176em;"><span style="top:-2.11em;"><span class="pstrut" style="height:3.0279em;"></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:1.0279em;"><span style="top:-2.4003em;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">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;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:0.2997em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><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:1.0279em;"><span style="top:-2.4003em;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:0.05724em;">j</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;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:0.4358em;"><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:0.3117em;"><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" style="margin-right:0.05724em;">ij</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 style="top:-3.2579em;"><span class="pstrut" style="height:3.0279em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.7176em;"><span class="pstrut" style="height:3.0279em;"></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:1.0279em;"><span style="top:-2.4003em;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">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;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:0.2997em;"><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:0.3117em;"><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">ii</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><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.3537em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
<p>This ratio reflects the tendency of each group to remain in the same state rather than transition to different states. The computed ratios are as follows:</p>
<ul>
<li><strong>Healthy Group Self-Transition Ratio</strong>: 0.8887</li>
<li><strong>Schizophrenia Group Self-Transition Ratio</strong>: 0.8892</li>
</ul>
<p>These ratios indicate a high prevalence of self-transitions in both groups, suggesting a tendency towards state stability or persistence in both healthy and schizophrenia subjects.</p>
<p>To compare the number of transition between the healthy and schizophrenia groups, the Frobenius norm (we can think of it as a distance between two matrices) of the difference between their transition matrices is calculated 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>T</mi><mrow><mi>i</mi><mi>j</mi></mrow><mtext>schizo</mtext></msubsup><mo>−</mo><msubsup><mi>T</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|} (T_{ij}^{\text{schizo}} - T_{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;">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:-2.4231em;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;">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 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>This norm quantifies the overall dissimilarity between the two matrices.</p>
<hr>
<h4 id="distribution-of-self-transitions">Distribution of Self-Transitions </h4>
<p>Self-transitions represent the counts of a state transitioning to itself, reflecting a sort of persistence or stability within that state.</p>
<p>For the <strong>healthy group</strong>, we define the self-transition 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> as <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>SelfTrans</mtext><mrow><mi>i</mi><mo separator="true">,</mo><mtext>healthy</mtext></mrow></msub></mrow><annotation encoding="application/x-tex">\text{SelfTrans}_{i, \text{healthy}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord text"><span class="mord">SelfTrans</span></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-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 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>, which is extracted from the diagonal of the transition matrix <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>T</mi><mtext>healthy</mtext></msub></mrow><annotation encoding="application/x-tex">T_{\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;">T</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>:</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><mtext>SelfTrans</mtext><mrow><mi>i</mi><mo separator="true">,</mo><mtext>healthy</mtext></mrow></msub><mo>=</mo><msubsup><mi>T</mi><mrow><mi>i</mi><mi>i</mi></mrow><mtext>healthy</mtext></msubsup></mrow><annotation encoding="application/x-tex">\text{SelfTrans}_{i, \text{healthy}} = T_{ii}^{\text{healthy}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord text"><span class="mord">SelfTrans</span></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-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 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:1.2439em;vertical-align:-0.2769em;"></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">ii</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:0.2769em;"><span></span></span></span></span></span></span></span></span></span></span></p>
<p>Similarly, for the <strong>schizophrenia group</strong>, the self-transition 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> are denoted as <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>SelfTrans</mtext><mrow><mi>i</mi><mo separator="true">,</mo><mtext>schizo</mtext></mrow></msub></mrow><annotation encoding="application/x-tex">\text{SelfTrans}_{i, \text{schizo}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord text"><span class="mord">SelfTrans</span></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-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 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:0.2861em;"><span></span></span></span></span></span></span></span></span></span>, obtained from the diagonal of the transition matrix <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>T</mi><mtext>schizo</mtext></msub></mrow><annotation encoding="application/x-tex">T_{\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;">T</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>:</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><mtext>SelfTrans</mtext><mrow><mi>i</mi><mo separator="true">,</mo><mtext>schizo</mtext></mrow></msub><mo>=</mo><msubsup><mi>T</mi><mrow><mi>i</mi><mi>i</mi></mrow><mtext>schizo</mtext></msubsup></mrow><annotation encoding="application/x-tex">\text{SelfTrans}_{i, \text{schizo}} = T_{ii}^{\text{schizo}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord text"><span class="mord">SelfTrans</span></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-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 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:0.2861em;"><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:1.1461em;vertical-align:-0.247em;"></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:0.8991em;"><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">ii</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:0.247em;"><span></span></span></span></span></span></span></span></span></span></span></p>
<p>These occurrences are essential for understanding the intrinsic dynamics of each group. A high self-transition counts suggests a tendency for the system to remain in the same state over time, which could be indicative of a stable or persistent pattern of brain activity. Conversely, lower self-transition counts imply a greater likelihood of transitioning to different states, suggesting more dynamic or variable brain activity.</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 with Self-Transitions</mtext><mo>=</mo><mn>22220</mn></mrow><annotation encoding="application/x-tex">\text{Distance with Self-Transitions} = 22220</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">Distance with Self-Transitions</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">22220</span></span></span></span></span></p>
<a href="images/transition/self_transition_counts.png">
<img src="images/transition/self_transition_counts.png" alt="Self-Transitions-counts" style="margin-left: 0px; scale: 0.9">
</a>
<p>Fig.7 Self-transition ocurrences in the healthy and schizophrenia groups.</p>
<hr>
<h4 id="distribution-of-transitions">Distribution of Transitions </h4>
<p>After examining self-transitions, we now explore the distribution of transitions between distinct states in both the healthy and schizophrenia groups. This excludes self-transitions and provides insight into the dynamic interplay of different states.</p>
<p>For the <strong>healthy group</strong>, let <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>T</mi><mtext>healthy</mtext></msub></mrow><annotation encoding="application/x-tex">T_{\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;">T</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> be the transition matrix, where <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> represents the 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 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>. The sum of transition counts 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>, excluding self-transitions, 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><mtext>SumTrans</mtext><mrow><mi>i</mi><mo separator="true">,</mo><mtext>healthy</mtext></mrow></msub><mo>=</mo><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mi>j</mi><mo mathvariant="normal">≠</mo><mi>i</mi></mrow><mrow><mi mathvariant="normal">∣</mi><mi>U</mi><mi mathvariant="normal">∣</mi></mrow></munderover><msubsup><mi>T</mi><mrow><mi>i</mi><mi>j</mi></mrow><mtext>healthy</mtext></msubsup></mrow><annotation encoding="application/x-tex">\text{SumTrans}_{i, \text{healthy}} = \sum_{j=1, j \neq i}^{|U|} T_{ij}^{\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 text"><span class="mord">SumTrans</span></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-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 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:3.3992em;vertical-align:-1.4382em;"></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:-1.8479em;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 class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight"><span class="mrel mtight"><span class="mord vbox mtight"><span class="thinbox mtight"><span class="rlap mtight"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord mtight"><span class="mrel mtight"></span></span></span><span class="fix"></span></span></span></span></span><span class="mrel mtight">=</span></span><span class="mord mathnormal mtight">i</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:1.4382em;"><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:-2.4231em;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></p>
<p>Similarly, for the <strong>schizophrenia group</strong>, let <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>T</mi><mtext>schizo</mtext></msub></mrow><annotation encoding="application/x-tex">T_{\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;">T</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> be the transition matrix, where <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> indicates the counts 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 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>. The sum of transition counts 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>, excluding self-transitions, is 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><msub><mtext>SumTrans</mtext><mrow><mi>i</mi><mo separator="true">,</mo><mtext>schizo</mtext></mrow></msub><mo>=</mo><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mi>j</mi><mo mathvariant="normal">≠</mo><mi>i</mi></mrow><mrow><mi mathvariant="normal">∣</mi><mi>U</mi><mi mathvariant="normal">∣</mi></mrow></munderover><msubsup><mi>T</mi><mrow><mi>i</mi><mi>j</mi></mrow><mtext>schizo</mtext></msubsup></mrow><annotation encoding="application/x-tex">\text{SumTrans}_{i, \text{schizo}} = \sum_{j=1, j \neq i}^{|U|} T_{ij}^{\text{schizo}}</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 text"><span class="mord">SumTrans</span></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-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 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:0.2861em;"><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:3.3992em;vertical-align:-1.4382em;"></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:-1.8479em;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 class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight"><span class="mrel mtight"><span class="mord vbox mtight"><span class="thinbox mtight"><span class="rlap mtight"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord mtight"><span class="mrel mtight"></span></span></span><span class="fix"></span></span></span></span></span><span class="mrel mtight">=</span></span><span class="mord mathnormal mtight">i</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:1.4382em;"><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:0.8991em;"><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></span></span></span></p>
<p>Here, <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> denotes the total number of unique states. The results of these calculations yield a vector for each group. Each element of these vectors corresponds to the cumulative counts of a state transitioning to all other states. This analysis offers a comprehensive view of how states are interconnected and interact with one another within each group.</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 without Self-Transitions</mtext><mo>=</mo><mn>922</mn></mrow><annotation encoding="application/x-tex">\text{Distance without Self-Transitions} = 922</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">Distance without Self-Transitions</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">922</span></span></span></span></span></p>
<a href="images/transition/transition_counts.png">
<img src="images/transition/transition_counts.png" alt="Transitions-counts" style="margin-left: 0px; scale: 0.9">
</a>
<p>Fig.8 Transition ocurrences in the healthy and schizophrenia groups.</p>
<p>Dealing with self-loops in transition matrices, especially when analyzing sequences of states in contexts like EEG microstate analysis, requires careful consideration. Self-loops, representing transitions from a state to itself, can disproportionately influence the analysis. That from now we separate the self-loops from the transition matrices and analyze them separately. But first, let's transform the transition matrices into probability matrices. → <a href="transition-probability-matrix.html">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)</a></p>
<hr>
<h6 id="author-łukasz-furmancracernetgmailcom">Author: <a href="[email protected]">Łukasz Furman</a> </h6>
</div>
</body></html>