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classification.html
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<h3 id="classification-of-transition-probability-matrix-tpm-between-health-and-schizophrenia">Classification of Transition Probability Matrix (TPM) between Health and Schizophrenia </h3>
<p>The TPM is a matrix that contains the probabilitys of transition from one of the 16 states to another. These matrices are created from sequences of microstates. For detailed information about the process of obtaining TPM, please refer to the earlier report <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>
<h5 id="1-introduction">1. Introduction </h5>
<p>The foundation of this analysis lies in 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>, 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 chronologically ordered list of microstates. These sequences are categorized into two subsets: <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> for healthy individuals 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> for individuals with schizophrenia. Each subset contains sequences that depict the progression of brain states over time.</p>
<h5 id="2-prototype-probability-matrices">2. Prototype Probability Matrices </h5>
<p>Transition Probability Matrices, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mtext>healthy</mtext></msub></mrow><annotation encoding="application/x-tex">P_{\text{healthy}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">healthy</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mtext>schizo</mtext></msub></mrow><annotation encoding="application/x-tex">P_{\text{schizo}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">schizo</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, are constructed for both groups. These matrices are <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>16</mn><mo>×</mo><mn>16</mn></mrow><annotation encoding="application/x-tex">16 \times 16</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">16</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:0.6444em;"></span><span class="mord">16</span></span></span></span>, with each element <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">p_{ij}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">p</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: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;">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> representing the probability of transitioning from state <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>s</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">s_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">s</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: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">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> to state <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>s</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">s_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></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:0.3117em;"><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:0.05724em;">j</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>, excluding self-transitions <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mrow><mi>i</mi><mi>i</mi></mrow></msub></mrow><annotation encoding="application/x-tex">p_{ii}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</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: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">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>.<br>
The construction of these matrices involves two key steps:</p>
<ol>
<li>
<p><strong>Counting Transitions:</strong> For each group, we count the transitions between states within their respective sequence sets. The count <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>c</mi><mrow><mi>i</mi><mi>j</mi></mrow><mi>G</mi></msubsup></mrow><annotation encoding="application/x-tex">c_{ij}^G</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2361em;vertical-align:-0.3948em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-2.4413em;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;">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 mathnormal mtight">G</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> represents the number of times a transition from state <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>s</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">s_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">s</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: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">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> to state <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>s</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">s_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></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:0.3117em;"><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:0.05724em;">j</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> occurs in group <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</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">G</span></span></span></span>.</p>
</li>
<li>
<p><strong>Normalizing TPMs:</strong> Each TPM is normalized such that the sum of the probabilities in each row equals 1. This normalization ensures that the TPM reflects the probability distribution of transitioning from one state to any other state.</p>
</li>
</ol>
<a href="images/transition/transition_probabilities_no_self.png">
<img src="images/transition/transition_probabilities_no_self.png" alt="transition_probabilities_no_self" style="margin-bottom: 50px; margin-top: 50px; scale: 1.2;">
</a>
<p>Fig.1: Transition Probability Matrices for Healthy and Schizophrenia Groups</p>
<h5 id="31-feature-engineering---intuition">3.1 Feature Engineering - Intuition </h5>
<p>The goal of this analysis is to classify a sequence as either healthy or schizophrenic. To do this, we need to extract features from the TPMs that can be used to train a classifier. Since we are dealing with a small sample size, we want to extract features that are robust and can generalize well to unseen data. Also we want to extract features that are interpretable and can be used to gain insight into the differences between the two groups.</p>
<p>The most strigthforward approach is to use the TPMs as features. However, this approach has one major drawbacks:</p>
<ul>
<li><strong>High Dimensionality:</strong> The TPMs are <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>16</mn><mo>×</mo><mn>16</mn></mrow><annotation encoding="application/x-tex">16 \times 16</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">16</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:0.6444em;"></span><span class="mord">16</span></span></span></span> matrices, which means that each matrix has 256 features. This high dimensionality can lead to overfitting and poor generalization.</li>
</ul>
<p>Since we have the prototype for each group, we can use the prototype to extract features from the TPMs. The idea is to use the prototype as a template to compare the TPMs to. We can then extract features that measure the similarity between the TPMs and the prototype. For every subject we will compare its TPM with the prototype TPMs for both groups. This will result in 2 sets of features for each subject. The features will be the similarity measures between the subject's TPM and the prototype TPMs. By doing this we will be capturing relative position of the subject's TPM with respect to the prototype TPMs, potentially enhancing the classifier's ability to distinguish between the two groups based on these relational features.</p>
<p>To mitigate the high dimensionality of the TPMs and extract meaningful, generalizable features, we employ several similarity measures that offer a comparison between a subject's TPM and the group prototypes. These measures yield robust features that encapsulate the relational positioning of a subject's brain state transitions with respect to the established norms of the healthy and schizophrenic groups.</p>
<h5 id="32-feature-engineering---similarity-measures">3.2 Feature Engineering - Similarity Measures </h5>
<p>We use the following similarity measures to compare the TPMs to the prototypes:</p>
<ol>
<li>
<p><strong>Euclidean Distance</strong>: This measure computes the straight-line distance between two matrices, considering them as vectors in a high-dimensional space. It is used in the function <code>features_map_euclidean</code>, which calculates the Euclidean distance between the flattened prototype matrix and each subject's flattened matrix.</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>d</mi><mo stretchy="false">(</mo><mi>A</mi><mo separator="true">,</mo><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><msqrt><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mo stretchy="false">(</mo><msub><mi>a</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>−</mo><msub><mi>b</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></msqrt></mrow><annotation encoding="application/x-tex">d(A, B) = \sqrt{\sum_{i=1}^{n} \sum_{j=1}^{n} (a_{ij} - b_{ij})^2}</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 mathnormal">d</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</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:3.2929em;vertical-align:-1.4138em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8791em;"><span class="svg-align" style="top:-5.2529em;"><span class="pstrut" style="height:5.2529em;"></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:1.6514em;"><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:-4.3em;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">n</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:1.6514em;"><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:-4.3em;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">n</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">a</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: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;">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:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">b</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: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;">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="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:-3.8391em;"><span class="pstrut" style="height:5.2529em;"></span><span class="hide-tail" style="min-width:0.742em;height:3.3329em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="3.3329em" viewBox="0 0 400000 3332" preserveAspectRatio="xMinYMin slice"><path d="M702 80H40000040
H742v3198l-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
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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>
</li>
<li>
<p><strong>Cosine Similarity</strong>: This metric assesses the cosine of the angle between two non-zero vectors, which in this context are the flattened matrices. The function <code>features_map_cosine</code> calculates this similarity, which is 1 minus the value returned by the <code>scipy.spatial.distance.cosine</code> function, as cosine similarity is typically defined as:</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>cosine</mtext><mo stretchy="false">(</mo><mi>A</mi><mo separator="true">,</mo><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>a</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mi>b</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><mrow><msqrt><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><mi>a</mi><mrow><mi>i</mi><mi>j</mi></mrow><mn>2</mn></msubsup></mrow></msqrt><mo>⋅</mo><msqrt><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><mi>b</mi><mrow><mi>i</mi><mi>j</mi></mrow><mn>2</mn></msubsup></mrow></msqrt></mrow></mfrac></mrow><annotation encoding="application/x-tex">\text{cosine}(A, B) = \frac{\sum_{i=1}^{n} \sum_{j=1}^{n} a_{ij} b_{ij}}{\sqrt{\sum_{i=1}^{n} \sum_{j=1}^{n} a_{ij}^2} \cdot \sqrt{\sum_{i=1}^{n} \sum_{j=1}^{n} b_{ij}^2}}</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">cosine</span></span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</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:3.3601em;vertical-align:-1.73em;"></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:1.6301em;"><span style="top:-2.11em;"><span class="pstrut" style="height:3.1692em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1692em;"><span class="svg-align" style="top:-3.8em;"><span class="pstrut" style="height:3.8em;"></span><span class="mord" style="padding-left:1em;"><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:0.8043em;"><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 mathnormal mtight">n</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:0.8043em;"><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 mathnormal mtight">n</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">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7959em;"><span style="top:-2.4231em;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;">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">2</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 style="top:-3.1292em;"><span class="pstrut" style="height:3.8em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.88em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.88em" viewBox="0 0 400000 1944" preserveAspectRatio="xMinYMin slice"><path d="M983 90
l0 -0
c4,-6.7,10,-10,18,-10 H400000v40
H1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7
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M1001 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.6708em;"><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 sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1692em;"><span class="svg-align" style="top:-3.8em;"><span class="pstrut" style="height:3.8em;"></span><span class="mord" style="padding-left:1em;"><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:0.8043em;"><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 mathnormal mtight">n</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:0.8043em;"><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 mathnormal mtight">n</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">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7959em;"><span style="top:-2.4231em;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;">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">2</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 style="top:-3.1292em;"><span class="pstrut" style="height:3.8em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.88em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.88em" viewBox="0 0 400000 1944" preserveAspectRatio="xMinYMin slice"><path d="M983 90
l0 -0
c4,-6.7,10,-10,18,-10 H400000v40
H1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7
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c-10,12,-21,25,-33,39s-32,39,-32,39c-6,-5.3,-15,-14,-27,-26s25,-30,25,-30
c26.7,-32.7,52,-63,76,-91s52,-60,52,-60s208,722,208,722
c56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5,-658.5
c53.7,-170.3,84.5,-266.8,92.5,-289.5z
M1001 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.6708em;"><span></span></span></span></span></span></span></span><span style="top:-3.3992em;"><span class="pstrut" style="height:3.1692em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.9951em;"><span class="pstrut" style="height:3.1692em;"></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:0.8043em;"><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 mathnormal mtight">n</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:0.8043em;"><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 mathnormal mtight">n</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">a</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: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;">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="mord"><span class="mord mathnormal">b</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: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;">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><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.73em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
</li>
<li>
<p><strong>Correlation Coefficient</strong>: Unlike a single correlation coefficient for the entire dataset, the <code>compute_correlation</code> function calculates the average of row-wise correlation coefficients between the two matrices, reflecting how each individual state's transition probabilities relate between the subject's TPM and the prototype.</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>corr</mtext><mo stretchy="false">(</mo><mi>A</mi><mo separator="true">,</mo><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mi>n</mi></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mtext>corrcoef</mtext><mo stretchy="false">(</mo><msub><mi>A</mi><mi>i</mi></msub><mo separator="true">,</mo><msub><mi>B</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{corr}(A, B) = \frac{1}{n} \sum_{i=1}^{n} \text{corrcoef}(A_i, B_i)</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">corr</span></span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.9291em;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 mathnormal">n</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:0.686em;"><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:1.6514em;"><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:-4.3em;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">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord text"><span class="mord">corrcoef</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">A</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: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">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</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:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="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><msub><mi>A</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">A_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">A</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: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">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>B</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">B_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:0.05017em;">B</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:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> are the <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>-th rows of matrices <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</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">A</span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</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:0.05017em;">B</span></span></span></span>, respectively, and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>corrcoef</mtext><mo stretchy="false">(</mo><msub><mi>A</mi><mi>i</mi></msub><mo separator="true">,</mo><msub><mi>B</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{corrcoef}(A_i, B_i)</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">corrcoef</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">A</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: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">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</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:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> is the correlation coefficient between these rows. The <code>np.corrcoef</code> function is used to calculate the correlation coefficient <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>corrcoef</mtext><mo stretchy="false">(</mo><msub><mi>A</mi><mi>i</mi></msub><mo separator="true">,</mo><msub><mi>B</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{corrcoef}(A_i, B_i)</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">corrcoef</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">A</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: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">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</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:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>.</p>
</li>
</ol>
<p>These similarity measures serve as robust features for machine learning models, designed to capture the relationship between individual brain state transitions and typical patterns seen in healthy or schizophrenic groups. By incorporating the row-wise correlation, the feature set respects the independence of state transitions, which can be crucial for interpreting the underlying neurophysiological processes.</p>
<h5 id="33-feature-engineering---feature-visualization">3.3 Feature Engineering - Feature Visualization </h5>
<p>To gain insight into the features space for each measure, we visualize the features for each group using a 2D scatter plot. The plots are shown below:</p>
<ul>
<li><strong>Euclidean Distance</strong>:</li>
</ul>
<a href="images/classification/feature_scatter_euclid.png">
<img src="images/classification/feature_scatter_euclid.png" alt="feature_scatter_euclid" style="margin-bottom: 50px; margin-top: 50px; scale: 1.0;">
</a>
<p>Fig.2: Scatter plot of the Euclidean distance features for the healthy and schizophrenic groups.</p>
<ul>
<li><strong>Cosine Similarity</strong>:</li>
</ul>
<a href="images/classification/feature_scatter_cosine.png">
<img src="images/classification/feature_scatter_cosine.png" alt="feature_scatter_cosine" style="margin-bottom: 50px; margin-top: 50px; scale: 1.0;">
</a>
<p>Fig.3: Scatter plot of the cosine similarity features for the healthy and schizophrenic groups.</p>
<ul>
<li><strong>Correlation Coefficient</strong>:</li>
</ul>
<a href="images/classification/feature_scatter_corr.png">
<img src="images/classification/feature_scatter_corr.png" alt="feature_scatter_corr" style="margin-bottom: 50px; margin-top: 50px; scale: 1.0;">
</a>
<p>Fig.4: Scatter plot of the correlation coefficient features for the healthy and schizophrenic groups.</p>
<h5 id="4-model-selection">4. Model Selection </h5>
<p>We use the following models to classify the subjects:</p>
<ol>
<li>
<p><strong>Logistic Regression</strong>: This model is used to classify the subjects using the features extracted from the TPMs. The model is trained using the features extracted from the training set and then used to predict the labels for the test set. The model is trained using the <code>sklearn.linear_model.LogisticRegression</code> function.</p>
</li>
<li>
<p><strong>Random Forest</strong>: This model is used to classify the subjects using the features extracted from the TPMs. The model is trained using the features extracted from the training set and then used to predict the labels for the test set. The model is trained using the <code>sklearn.ensemble.RandomForestClassifier</code> function.</p>
</li>
<li>
<p><strong>Support Vector Machine</strong>: This model is used to classify the subjects using the features extracted from the TPMs. The model is trained using the features extracted from the training set and then used to predict the labels for the test set. The model is trained using the <code>sklearn.svm.SVC</code> function.<br>
Kernel used is <code>rbf</code> with default parameters.<br>
For comparison, we also train the model using the <code>linear</code> kernel.</p>
</li>
</ol>
<p>Metrics used to evaluate the models are:</p>
<ul>
<li>
<p><strong>Accuracy</strong>: This metric is used to evaluate the performance of the models. The accuracy is calculated using the <code>sklearn.metrics.accuracy_score</code> function.</p>
</li>
<li>
<p><strong>ROC curve/ AUC</strong>: This metric is used to evaluate the performance of the models. The ROC curve is plotted using the <code>sklearn.metrics.roc_curve</code> function and the AUC is calculated using the <code>sklearn.metrics.auc</code> function.<br>
We have slightly imbalanced dataset so this metric can be used to evaluate the performance of the models in a more precise way than accuracy.</p>
</li>
</ul>
<p>Test set is 20% of the data and the rest is used for training.</p>
<p>Table descriptions:</p>
<ul>
<li><strong>Acc</strong>: Accuracy score for the model. (Here single score is calculated on the test set.)</li>
<li><strong>Roc AUC</strong>: Roc AUC score for the model. (Here single score is calculated on the test set.)</li>
<li><strong>Cv-5</strong>: Cross validation score for the model using 5 folds and accuracy as scoring metric.</li>
<li><strong>Cv-5 AUC</strong>: Cross validation score for the model using 5 folds and AUC as scoring metric.</li>
</ul>
<hr>
<p><strong>4.1</strong> The following table shows the results for <strong><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mi>u</mi><mi>c</mi><mi>l</mi><mi>i</mi><mi>d</mi><mi>e</mi><mi>a</mi><mi>n</mi></mrow><annotation encoding="application/x-tex">Euclidean</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 mathnormal" style="margin-right:0.05764em;">E</span><span class="mord mathnormal">u</span><span class="mord mathnormal">c</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">i</span><span class="mord mathnormal">d</span><span class="mord mathnormal">e</span><span class="mord mathnormal">an</span></span></span></span> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mi>i</mi><mi>s</mi><mi>t</mi><mi>a</mi><mi>n</mi><mi>c</mi><mi>e</mi></mrow><annotation encoding="application/x-tex">Distance</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:0.02778em;">D</span><span class="mord mathnormal">i</span><span class="mord mathnormal">s</span><span class="mord mathnormal">t</span><span class="mord mathnormal">an</span><span class="mord mathnormal">ce</span></span></span></span></strong> features:</p>
<table>
<thead>
<tr>
<th>Model</th>
<th>Acc</th>
<th>Roc AUC</th>
<th>Cv-5</th>
<th>Cv-5 AUC</th>
</tr>
</thead>
<tbody>
<tr>
<td>Logistic Regression</td>
<td>0.53</td>
<td><strong>0.92</strong></td>
<td>0.51</td>
<td>0.61</td>
</tr>
<tr>
<td>SVM (linear)</td>
<td>0.53</td>
<td><strong>0.92</strong></td>
<td>0.52</td>
<td>0.37</td>
</tr>
<tr>
<td>SVM (rbf)</td>
<td><strong>0.76</strong></td>
<td>0.85</td>
<td><strong>0.70</strong></td>
<td><strong>0.79</strong></td>
</tr>
<tr>
<td>Random Forest</td>
<td>0.47</td>
<td>0.60</td>
<td>0.60</td>
<td>0.65</td>
</tr>
</tbody>
</table>
<p>The following figures show the ROC curves for the models (Cv-5 AUC - last column in table):</p>
<ul>
<li><strong>Logistic Regression</strong>:</li>
</ul>
<a href="images/classification/roc_logistic_Euclid.png">
<img src="images/classification/roc_logistic_Euclid.png" alt="roc_logistic_Euclid" style="margin-bottom: 0px; margin-top: 0px; scale: 1.0;">
</a>
<p>Fig.5: ROC curve for the Logistic Regression model.</p>
<ul>
<li><strong>SVM (rbf)</strong>:</li>
</ul>
<a href="images/classification/roc_svm_rbf_Euclid.png">
<img src="images/classification/roc_svm_rbf_Euclid.png" alt="roc_svm_rbf_Euclid" style="margin-bottom: 0px; margin-top: 0px; scale: 1.0;">
</a>
<p>Fig.6: ROC curve for the SVM (rbf) model.</p>
<ul>
<li><strong>SVM (linear)</strong>:</li>
</ul>
<a href="images/classification/roc_svm_linear_Euclid.png">
<img src="images/classification/roc_svm_linear_Euclid.png" alt="roc_svm_linear_Euclid" style="margin-bottom: 0px; margin-top: 0px; scale: 1.0;">
</a>
<p>Fig.7: ROC curve for the SVM (linear) model.</p>
<ul>
<li><strong>Random Forest</strong>:</li>
</ul>
<a href="images/classification/roc_random_forest_Euclid.png">
<img src="images/classification/roc_random_forest_Euclid.png" alt="roc_random_forest_Euclid" style="margin-bottom: 0px; margin-top: 0px; scale: 1.0;">
</a>
<p>Fig.8: ROC curve for the Random Forest model.</p>
<hr>
<p><strong>4.2</strong> The following table shows the results for <strong><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi><mi>o</mi><mi>s</mi><mi>i</mi><mi>n</mi><mi>e</mi></mrow><annotation encoding="application/x-tex">Cosine</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:0.07153em;">C</span><span class="mord mathnormal">os</span><span class="mord mathnormal">in</span><span class="mord mathnormal">e</span></span></span></span> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><mi>i</mi><mi>m</mi><mi>i</mi><mi>l</mi><mi>a</mi><mi>r</mi><mi>i</mi><mi>t</mi><mi>y</mi></mrow><annotation encoding="application/x-tex">Similarity</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 mathnormal" style="margin-right:0.05764em;">S</span><span class="mord mathnormal">imi</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">a</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mord mathnormal">i</span><span class="mord mathnormal">t</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span></strong> features:</p>
<table>
<thead>
<tr>
<th>Model</th>
<th>Acc</th>
<th>Roc AUC</th>
<th>Cv-5</th>
<th>Cv-5 AUC</th>
</tr>
</thead>
<tbody>
<tr>
<td>Logistic Regression</td>
<td>0.53</td>
<td><strong>0.94</strong></td>
<td>0.54</td>
<td>0.67</td>
</tr>
<tr>
<td>SVM (linear)</td>
<td>0.53</td>
<td><strong>0.92</strong></td>
<td>0.54</td>
<td>0.44</td>
</tr>
<tr>
<td>SVM (rbf)</td>
<td>0.71</td>
<td><strong>0.90</strong></td>
<td>0.71</td>
<td>0.79</td>
</tr>
<tr>
<td>Random Forest</td>
<td>0.53</td>
<td>0.60</td>
<td>0.61</td>
<td><strong>0.72</strong></td>
</tr>
</tbody>
</table>
<p>The following figures show the ROC curves for the models (Cv-5 AUC - last column in table):</p>
<ul>
<li><strong>Logistic Regression</strong>:</li>
</ul>
<a href="images/classification/roc_logistic_Euclid.png">
<img src="images/classification/roc_logistic_Euclid.png" alt="roc_logistic_Euclid" style="margin-bottom: 0px; margin-top: 0px; scale: 1.0;">
</a>
<p>Fig.9: ROC curve for the Logistic Regression model.</p>
<ul>
<li><strong>SVM (rbf)</strong>:</li>
</ul>
<a href="images/classification/roc_svm_rbf_Cosine.png">
<img src="images/classification/roc_svm_rbf_Cosine.png" alt="roc_svm_rbf_Cosine" style="margin-bottom: 0px; margin-top: 0px; scale: 1.0;">
</a>
<p>Fig.10: ROC curve for the SVM (rbf) model.</p>
<ul>
<li><strong>SVM (linear)</strong>:</li>
</ul>
<a href="images/classification/roc_svm_linear_Cosine.png">
<img src="images/classification/roc_svm_linear_Cosine.png" alt="roc_svm_linear_Cosine" style="margin-bottom: 0px; margin-top: 0px; scale: 1.0;">
</a>
<p>Fig.11: ROC curve for the SVM (linear) model.</p>
<ul>
<li><strong>Random Forest</strong>:</li>
</ul>
<a href="images/classification/roc_random_forest_Cosine.png">
<img src="images/classification/roc_random_forest_Cosine.png" alt="roc_random_forest_Cosine" style="margin-bottom: 0px; margin-top: 0px; scale: 1.0;">
</a>
<p>Fig.12: ROC curve for the Random Forest model.</p>
<hr>
<p><strong>4.3</strong> The following table shows the results for <strong><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi><mi>o</mi><mi>r</mi><mi>r</mi><mi>e</mi><mi>l</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi></mrow><annotation encoding="application/x-tex">Correlation</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 mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">orre</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">a</span><span class="mord mathnormal">t</span><span class="mord mathnormal">i</span><span class="mord mathnormal">o</span><span class="mord mathnormal">n</span></span></span></span> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi><mi>o</mi><mi>e</mi><mi>f</mi><mi>f</mi><mi>i</mi><mi>c</mi><mi>i</mi><mi>e</mi><mi>n</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">Coefficient</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 mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">oe</span><span class="mord mathnormal" style="margin-right:0.10764em;">ff</span><span class="mord mathnormal">i</span><span class="mord mathnormal">c</span><span class="mord mathnormal">i</span><span class="mord mathnormal">e</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span></span></span></span></strong> features:</p>
<table>
<thead>
<tr>
<th>Model</th>
<th>Acc</th>
<th>Roc AUC</th>
<th>Cv-5</th>
<th>Cv-5 AUC</th>
</tr>
</thead>
<tbody>
<tr>
<td>Logistic Regression</td>
<td>0.65</td>
<td><strong>0.90</strong></td>
<td>0.79</td>
<td>0.85</td>
</tr>
<tr>
<td>SVM (linear)</td>
<td>0.71</td>
<td><strong>0.92</strong></td>
<td>0.73</td>
<td>0.85</td>
</tr>
<tr>
<td>SVM (rbf)</td>
<td>0.82</td>
<td><strong>0.92</strong></td>
<td>0.76</td>
<td>0.86</td>
</tr>
<tr>
<td>Random Forest</td>
<td>0.82</td>
<td><strong>0.90</strong></td>
<td>0.80</td>
<td>0.86</td>
</tr>
</tbody>
</table>
<p>The following figures show the ROC curves for the models (Cv-5 AUC - last column in table):</p>
<ul>
<li><strong>Logistic Regression</strong>:</li>
</ul>
<a href="images/classification/roc_logistic_Correlation.png">
<img src="images/classification/roc_logistic_Correlation.png" alt="roc_logistic_Correlation" style="margin-bottom: 50px; margin-top: 50px; scale: 1.0;">
</a>
<p>Fig.13: ROC curve for the Logistic Regression model.</p>
<ul>
<li><strong>SVM (rbf)</strong>:</li>
</ul>
<a href="images/classification/roc_svm_rbf_Cosine.png">
<img src="images/classification/roc_svm_rbf_Cosine.png" alt="roc_svm_rbf_Cosine" style="margin-bottom: 0px; margin-top: 0px; scale: 1.0;">
</a>
<p>Fig.14: ROC curve for the SVM (rbf) model.</p>
<ul>
<li><strong>SVM (linear)</strong>:</li>
</ul>
<a href="images/classification/roc_svm_linear_Correlation.png">
<img src="images/classification/roc_svm_linear_Correlation.png" alt="roc_svm_linear_Correlation" style="margin-bottom: 0px; margin-top: 0px; scale: 1.0;">
</a>
<p>Fig.15: ROC curve for the SVM (linear) model.</p>
<ul>
<li><strong>Random Forest</strong>:</li>
</ul>
<a href="images/classification/roc_random_forest_Correlation.png">
<img src="images/classification/roc_random_forest_Correlation.png" alt="roc_random_forest_Correlation" style="margin-bottom: 0px; margin-top: 0px; scale: 1.0;">
</a>
<p>Fig.16: ROC curve for the Random Forest model.</p>
<hr>
<h5 id="5-final-thoughts-and-conclusions">5. Final Thoughts and Conclusions </h5>
<p>The classification of subjects into healthy or schizophrenic groups based on Transition Probability Matrices (TPMs) represents a significant results. This study has demonstrated the feasibility of using simple, yet effective, statistical and machine learning techniques to differentiate between the neurophysiological patterns characteristic of these two conditions.</p>
<p>Through the feature engineering process, we have successfully addressed the challenge of high dimensionality inherent in the TPMs by extracting robust and interpretable features. The similarity measures—Euclidean distance, cosine similarity, and correlation coefficient—have been instrumental in transforming the raw TPM data into a format amenable to machine learning algorithms. These features not only reduce the risk of overfitting but also enhance the interpretability of the models, providing valuable insights into the neurodynamics of healthy and schizophrenic subjects.</p>
<p>The visualization of features through scatter plots has offered a preliminary yet insightful look into the distribution and separation capabilities of the features extracted. It has laid the groundwork for understanding how well the features can distinguish between the two groups. In particular, the correlation coefficient features appear to be the most promising, as they show a clear separation between the two groups. And some of the subjects especially in the schizophrenic group are very close to the healthy group. This suggests that the correlation coefficient features can be used to identify subjects with schizophrenia who exhibit brain state transitions similar to those of healthy individuals.</p>
<p>Model selection included Logistic Regression, Random Forest, and Support Vector Machines with linear and radial basis function (rbf) kernels. The choice of these models was guided by their widespread use and proven track record in classification tasks. The evaluation of these models through metrics like accuracy and ROC AUC has provided a comprehensive view of their performance. In particular, the ROC AUC metric has been invaluable for assessing model efficacy in the context of a slightly imbalanced dataset.</p>
<p>The results indicate that the Support Vector Machine with an rbf kernel and features based on the correlation coefficient offers the most promising classification performance. However, the Random Forest model also shows comparable performance, suggesting that an ensemble of decision trees can capture the complex patterns in the data effectively.</p>
<p>It is imperative to note that while the current models perform satisfactorily, there is always room for improvement. Future work may include exploring more complex features and testing additional classifiers. Furthermore, extending the dataset and incorporating longitudinal data could enhance the robustness of the findings. It was also observed that reducing the TPMs by employing dimensionality reduction techniques like Principal Component Analysis (PCA) did not improve the classification performance. It was actually worse than the results obtained using present features. This suggests that the features extracted from the TPMs are already capturing the most important information and reducing the dimensionality further is not helping the models. (The PCA analysis is not included in this report.)</p>
<p>In conclusion, the use of TPMs for classification purposes holds promise not only as a diagnostic tool but also as a means to personalize treatment strategies for individuals with schizophrenia but also for other mental disorders.</p>
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