# Comparison of Functional Connectivity Analysis Methods in Alzheimer's Disease
## 3 Functional Connectivity Analysis Methods
### Model/Seed Based Methods
#### 3.1.2 Coherence Analysis (CA)  
Despite its wide usage in fMRI analysis, the CCA approach has several drawbacks. The estimation of the correlation at zero lag demanded complex computations when considering all available lags in two brain areas [24]. Some studies have found a strong correlation between regions with no blood flow fluctuations [25]. The correlation is dependent on the characteristics of HRF, which differs from one individual to another from one region to another. A new metric named coherence, which is the spectral representation of correlation in the frequency domain, was introduced [26]. This method demonstrated a more practical technique to assess BOLD time series communication utilizing frequency domain features. For a BOLD fMRI time course \(f_{x}(i)\) with a seed \(f_{y}(i)\) (referenced time course), the coherence can be calculated as:  
\[Coh(\hat{A})=\frac{\big{|}F_{x,y}(\hat{A})\big{|}^{2}}{F_{x,x}(\hat{A}).F_{y,y }(\hat{A})} \tag{4}\]  
Here, \(F_{x,y}(\hat{A})\) is the cross-spectrum calculated by Fourier transformation of cross-covariance using the following equation:  
\[F_{x,y}(\hat{A})=\sum_{\mu}Cov_{x,y}(\mu)\ X\ e^{-j\hat{A}u} \tag{5}\]  
and \(F_{x,x}(\hat{A})\) and \(F_{y,y}(\hat{A})\) are the power spectrum can be calculated by the below equations:  
\[F_{x,x}(\hat{A})=\sum_{\mu}Cov_{x,x}(\mu)\ X\ e^{-j\hat{A}u} \tag{6}\]  
\[F_{y,y}(\hat{A})=\sum_{\mu}Cov_{y,y}(\mu)\ X\ e^{-j\hat{A}u} \tag{7}\]  
Some researchers investigated FC interactions between resting-state networks in AD patients and NCs and found that the interactions between resting-state networks in AD patients and NCs differed [27]. Here we present a visualization of cross-correlation and coherence matrices of AD subjects and NCs (Figure 3). The data utilized in this study were acquired from the Alzheimer's Disease Neuroimaging Initiative (ADNI) database [28]. The ADNI's focus is to explore the pathogenesis and prevention of AD by analyzing a wide variety of medical imaging data. This review analyzed the rs-fMRI data of 29 patients with AD and 30 NC subjects. The demographic and clinical details of the participants have been shown in Table 1.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Group** & **NC (30 Total)** & **AD (29 Total)** \\ \hline \(\text{(Age)}\)1 [FOOTNOTE:1]Footnote 1: \(\text{(Age)  
The fMRI data have been preprocessed using the SPM 12 software package [29, 30]. Using the FC module integrated with the DPARSF toolbox and SPM12, correlation matrices of the FC between white and gray matter have been derived (Figure 3). This analysis was carried out by detecting variations in the correlation.  
In Figure 3, the colored small square boxes represent regions that correlated to rs-fMRI for NC and AD patients. Each box shows the average correlation coefficient. Both CCA and CA methods have a different approach to calculating the correlation between regions, which is why the number of yellow and dark blue boxes are different in both methods. In CCA, the number of yellow blocks is higher than in AD patients, which indicates a strong correlation in NC. Furthermore, the dark-blue and light-blue blocks are more frequent in AD patients, which shows a weakened or missing correlation. Although the yellow and blue blocks seem similar in CA, the intensity of blue is higher in AD patients, which indicates the weak correlation between white and grey matter regions.