To gain a more nuanced understanding of the transition dynamics, we transform the count-based transition matrices into probability matrices. This transformation provides a clearer picture of the likelihood of transitioning from one state to another.
For both the healthy and schizophrenia groups, we normalize the transition matrices by dividing each entry by the sum of its respective row. This process converts the raw counts into probabilities, where each row of the matrix sums to 1. The transformed matrices are denoted as ( P_{\text{healthy}} ) and ( P_{\text{schizo}} ) for the healthy and schizophrenia groups, respectively.
The transformation process is defined mathematically for each state ( i ) and transition ( j ) as follows:
For the healthy group: [ P_{ij}^{\text{healthy}} = \frac{T_{ij}^{\text{healthy}}}{\sum_{k=1}^{|U|} T_{ik}^{\text{healthy}}} ]
For the schizophrenia group: [ P_{ij}^{\text{schizo}} = \frac{T_{ij}^{\text{schizo}}}{\sum_{k=1}^{|U|} T_{ik}^{\text{schizo}}} ]
where ( |U| ) is the number of unique states, and ( T_{ij}^{\text{healthy}} ) and ( T_{ij}^{\text{schizo}} ) are the original transition counts from state ( i ) to ( j ) in the healthy and schizophrenia groups, respectively.
This probability-based approach allows for a more refined and accurate comparison between the two groups, highlighting differences in the likelihood of state transitions rather than merely their frequency.
In pursuit of uncovering the similarities and dissimilarities in state transition dynamics between healthy individuals and those with schizophrenia, advanced analytical methods are employed, such as correlation analysis and Distance calculations.
Correlation analysis is a statistical method that measures the strength and direction of a linear relationship between two variables. In the context of transition matrices, it's used to compare how similarly two groups transition between states. This is done by computing the Pearson correlation coefficient for each corresponding pair of states between the two groups and then averaging these coefficients to obtain an overall measure of similarity.
The correlation coefficient ( r ) for states ( i ) in the healthy group ( H )and the schizophrenia group ( S ) is calculated as follows:
[ r_i = \text{corr}(P_{i, H}, P_{i, S}) ]
where ( \text{corr} ) denotes the Pearson correlation function, and ( P_{i, H} ) and ( P_{i, S} ) represent the probability distributions of transitioning from state ( i ) to all other states for the healthy and schizophrenia groups, respectively.
The overall correlation across all states is then given by:
[ r = \frac{1}{|U|} \sum_{i=1}^{|U|} r_i ]
where ( |U| ) is the total number of unique states.
A correlation coefficient of 1 indicates a perfect positive correlation, while a value of -1 signifies a perfect negative correlation. A coefficient of 0 indicates no correlation between the two groups.
The Distance is a measure used to quantify the difference between two matrices. It's particularly useful in this context as it provides a single value that captures the overall disparity in transition probabilities between the two groups, accounting for the entirety of the transition matrix.
For matrices ( P_{\text{healthy}} ) and ( P_{\text{schizo}} ), representing the transition probabilities of the healthy and schizophrenia groups respectively, the Distance is defined as:
[ \text{Distance} = \sqrt{\sum_{i=1}^{|U|} \sum_{j=1}^{|U|} (P_{ij}^{\text{schizo}} - P_{ij}^{\text{healthy}})^2} ]
where ( |U| ) is the total number of unique states.
A smaller Distance indicates a closer resemblance between the transition behaviors of the two groups, while a larger value signifies greater differences.
By calculating both the overall Correlation and the Distance, we can gain a deeper understanding of the transition dynamics characteristic of each group, potentially unveiling unique patterns that could serve as biomarkers for schizophrenia.
Transition probability matrices are constructed by normalizing the transition matrices by dividing each entry by the sum of its respective row. This process converts the raw counts into probabilities, where each row of the matrix sums to 1. The transformed matrices are denoted as ( P_{\text{healthy}} ) and ( P_{\text{schizo}} ) for the healthy and schizophrenia groups, respectively.
The Distance and correlation coefficient are calculated for the transition probability matrices of the healthy and schizophrenia groups. The results are as follows:
Fig.9 We can see that due to the high number of self-transitions, the transition probability matrices are dominated by the diagonal entries. This is reflected in the high correlation coefficient and low Distance, which indicate a high degree of similarity between the two groups. However, this is misleading, as the self-transitions are not indicative of the actual transition dynamics between states but rather the tendency of each group to remain in the same state. And this is of course informative as we saw on count distributions of self-transitions. But it is not what we are looking for. We are interested in the transitions between states. So let's remove the self-transitions from the transition matrices and see what happens.
Obtained ( P_{\text{healthy}} ) and ( P_{\text{schizo}} ) for the healthy and schizophrenia groups, respectively without self-transitions.
The Distance and correlation coefficient are calculated for the transition probability matrices of the healthy and schizophrenia groups. The results are as follows:
Fig.10 We can see that the Distance is much higher and the correlation coefficient is much lower. This is because the self-transitions are not influencing the analysis. The transition probability matrices are now dominated by the off-diagonal entries, which reflect the actual transition dynamics between states. This is what we are looking for.
The transition from matrix representations of state sequences to visual graph structures offers an intuitive understanding of the complex dynamics within each group. By pruning and normalizing the matrices, we can construct directed graphs that reveal the most probable paths and highlight the interdependencies between states.
For the healthy group, the directed graph ( G_{\text{healthy}} ) is formed by connecting state ( i ) to state ( j ) if the probability ( P_{ij}^{\text{healthy}} ) exceeds a predetermined threshold. This threshold is set to capture the top 10% of transition probabilities, ensuring that only the most significant connections are visualized.
[ G_{\text{healthy}} = { (i, j) \ | \ P_{ij}^{\text{healthy}} \geq \text{threshold} } ]
Similarly, for the schizophrenia group, the directed graph ( G_{\text{schizo}} ) is established using the same principle.
[ G_{\text{schizo}} = { (i, j) \ | \ P_{ij}^{\text{schizo}} \geq \text{threshold} } ]
The vertices of these graphs correspond to the unique states, while the edges depict the transitions with probabilities surpassing the pruning threshold. These graphs provide a visual representation of the most likely transitions, as well as insights into the overall structure and connectivity of the state dynamics.
The Distance of the difference between the two graphs weight adjacency matrices is:
Fig.11 and Fig.12 illustrate the directed graphs for the healthy and schizophrenia groups, respectively. The vertices represent the unique states, while the edges depict the transitions with probabilities surpassing the pruning threshold. These graphs provide a visual representation of the most likely transitions, as well as insights into the overall structure and connectivity of the state dynamics. The position and color of the vertices is the same for both graphs as well labels for the vertices. The thickness of the edges is proportional to the probability of the transition. The size of the node it its degree, i.e., the number of edges connected to it.
The analysis of these graphs can elucidate characteristics such as resilience, adaptability, and potential biomarkers within the EEG microstate sequences for each group. The directed graphs not only emphasize the significant transitions but also pave the way for identifying key differences in the neurological patterns associated with healthy and schizophrenic brains. We see now that performing analysis with self-transitions included in the transition matrices can lead to misleading results.