# Summary Statistics

    library(MethEvolSIM)

# Introduction

Here we present summary statistics that can be used, e.g. in an Approximate Bayesian Computation method (Csillery, Francois, and Blum 2012), to estimate parameters of the methylation dynamics model of MethEvolSIM. The statistics include mean frequencies, standard deviations, neighbor correlations of methylation states in island and non-island regions. Additionally, for analyses comparing different samples (comparison of tree tips), some of the summary statistics are based on the mean per-site frequency of methylation changes and the minimum number of changes per islands leading to different global methylation states, calculated with the Fitch algorithm (Fitch 1971).

The provided set of functions allows the computation of summary statistics for methylation data in a genomic region with structural categorization into two types of structures. In this vignette, we will refer to these structures as islands and non-island structures.

The `data` argument must be in one of the following formats, depending on the analysis:

1. For analyses without tip comparison: A list where each element represents the methylation states in a given structure (`data[[structure]]`):

    # Example: a single sample with 3 genomic structures
    # (1) island with 10 partially-methylated sites
    # (2) non-island with 5 methylated sites
    # (3) island with 15 unmethylated sites

    data <- list(rep(0.5, 10),  # Partially methylated
                 rep(1,5),      # Methylated
                 rep(0,15))     # Unmethylated
    data
    #> [[1]]
    #>  [1] 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5
    #>
    #> [[2]]
    #> [1] 1 1 1 1 1
    #>
    #> [[3]]
    #>  [1] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2. For analyses comparing multiple tips in a phylogenetic tree: A nested list where the first level represents the tips, and the second level represents the structures within each tip (`data[[tip]][[structure]]`).

    # Example: data from 3 tips of a tree,
    # each with 3 genomic structures
    data <- list(
        # Tip 1
        list(c(rep(0.5,5), rep(0,5)),  # 5 partially methylated, 5 unmethylated
             c(rep(1,4), 0.5),         # 4 methylated, 1 unmethylated
             c(rep(0,7), rep(0.5,8))), # 7 unmethylated, 8 partially methylated
        # Tip 2
        list(c(rep(0.5,9), 1), # 9 partially methylated, 1 methylated
             c(rep(0.5,4), 1), # 4 partially methylated, 1 methylated
             c(rep(0,8), rep(0.5,7))), # 8 unmethylated, 7 partially-methylated
        # Tip 3
        list(c(1, rep(0,8), 1), # first and last methylated, rest unmethylated
             c(rep(0.5,3), rep(1,2)), # 3 methylated, 1 unmethylated
             c(rep(0.5,15)))) # all partially methylated

In this case, `data` must be pre-filtered to include only sites present in all tips, ensuring a valid comparison between species or samples.

Regardless of the format, methylation values should be represented as follows:

- `0` for unmethylated sites,

- `0.5` for partially methylated sites,

- `1` for methylated sites.

Intermediate values (e.g., obtained from pooled empirical data) should be categorized before analysis with `categorize_siteMethSt`.

    non_categorized_data <- list(
      # Tip 1
        list(c(0.1, 0.7, 0.9), # First structure
             c(0.3, 0.5, 0.9)), # Second structure
      # Tip 2
        list(c(0.2, 0.8, 0.6), # First structure
             c(0.9, 0.4, 0.7)) # Second structure
      )

    # Transform the data with custom thresholds
    categorized_data <- categorize_siteMethSt(data, u_threshold = 0.15, m_threshold = 0.85)

    categorized_data
    #> [[1]]
    #> [[1]][[1]]
    #>  [1] 0.5 0.5 0.5 0.5 0.5 0.0 0.0 0.0 0.0 0.0
    #>
    #> [[1]][[2]]
    #> [1] 1.0 1.0 1.0 1.0 0.5
    #>
    #> [[1]][[3]]
    #>  [1] 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5
    #>
    #>
    #> [[2]]
    #> [[2]][[1]]
    #>  [1] 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 1.0
    #>
    #> [[2]][[2]]
    #> [1] 0.5 0.5 0.5 0.5 1.0
    #>
    #> [[2]][[3]]
    #>  [1] 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.5 0.5 0.5 0.5 0.5 0.5 0.5
    #>
    #>
    #> [[3]]
    #> [[3]][[1]]
    #>  [1] 1 0 0 0 0 0 0 0 0 1
    #>
    #> [[3]][[2]]
    #> [1] 0.5 0.5 0.5 1.0 1.0
    #>
    #> [[3]][[3]]
    #>  [1] 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

Finally, when phylogenetic data is provided, the order of tips in `data` must match the tip order in the Newick tree format (left to right) or the order of `tree$tip.label()` when the tree is provided as a `phylo` object from the `ape` package.

    # Example tree in Newick format for the above data
    newick_tree <- "((tip1:1, tip2:1):1, tip3:2);"

    # Example tree as a phylo object from the ape package
    library(ape)
    phylo_tree <- read.tree(text = newick_tree)
    phylo_tree$tip.label
    #> [1] "tip1" "tip2" "tip3"

# Summary Statistics

## Frequency of Methylation States per Genomic Structure Type

### Global Mean

- Partially-methylated sites: Let \(x_{i,t}\) and \(y_{j,t}\) be the frequency of methylation state \(p\) in island \(i \in\{1, \cdots, I\}\) and non-island \(j \in\{1, \cdots, J\}\) at tip \(t\), respectively. We define the mean global frequency of \(x\) and \(y\) as:

\[ \overline{x}=\frac1I\sum_{i=1}^{I}\frac1T\sum_{t=1}^{T}x_{i,t} \\ \overline{y}=\frac1J\sum_{j=1}^{J}\frac1T\sum_{t=1}^{T}y_{j,t} \]

where \(x_{i,t}\) and \(y_{j,t}\) represent the respective frequencies at each tip.

- Methylated sites: Similarly, let \(w_{i,t}\) and \(z_{j,t}\) be the frequency of methylation state \(m\) in islands and non-islands. The mean global frequencies are:

\[ \overline{w}=\frac1I\sum_{i=1}^{I}\frac1T\sum_{t=1}^{T}w_{i,t} \\ \overline{z}=\frac1J\sum_{j=1}^{J}\frac1T\sum_{t=1}^{T}z_{j,t} \]

These values can be computed using the following functions:

    # 1 tip / sample / replicate
    sample_n <- 1
    index_islands <- c(1, 3)
    index_nonislands <- c(2, 4)
    data <- list(c(.5, .5, 0, 0, 0, .5), c(.5, 0, 0, .5), c(.5, .5, 0), c(0, 1, .5)) # tip 1
    get_islandMeanFreqP(index_islands, data, categorized_data = T, sample_n)
    #> [1] 0.5833333
    get_nonislandMeanFreqP(index_nonislands, data, categorized_data = T, sample_n)
    #> [1] 0.4166667
    get_islandMeanFreqM(index_islands, data, categorized_data = T, sample_n)
    #> [1] 0
    get_nonislandMeanFreqM(index_nonislands, data, categorized_data = T, sample_n)
    #> [1] 0.1666667

    # 2 tip / sample / replicate
    sample_n <- 2
    index_islands <- c(1, 3)
    index_nonislands <- c(2, 4)
    data <- list(
      list(c(.5, .5, 0, 0, 0, .5), c(.5, 0, 0, .5), c(.5, .5, 0), c(0, 0, .5)), # tip 1
      list(c(0, .5, .5, 1, 1, .5), c(1, .5, 1, .5), c(0, .5, .5), c(1, .5, 1)) # tip 2
      )
    get_islandMeanFreqP(index_islands, data, categorized_data = T, sample_n)
    #> [1] 0.5833333
    get_nonislandMeanFreqP(index_nonislands, data, categorized_data = T, sample_n)
    #> [1] 0.4166667
    get_islandMeanFreqM(index_islands, data, categorized_data = T, sample_n)
    #> [1] 0.08333333
    get_nonislandMeanFreqM(index_nonislands, data, categorized_data = T, sample_n)
    #> [1] 0.2916667

Note that in the calculation of the means the relative frequencies \(x_{i, t}\), \(y_{j, t}\), \(w_{i, t}\) and \(z_{j, t}\) are not weighted by the lengths of island \(i\) or non-island \(j\).

### Mean SD of Observed Frequencies per Structure Type at Tree Tips

- Partially-methylated sites: We define the standard deviation of \(x\) and \(y\) at tip \(t\) as:

\[ \sigma_{t}(x)=\sqrt{\frac{1}{I-1}\sum_{i=1}^{I}(x_{i,t}-\overline{x_{.,t}})^2} \\ \sigma_{t}(y)=\sqrt{\frac{1}{J-1}\sum_{j=1}^{J}(y_{j,t}-\overline{y_{.,t}})^2} \]

where \(\overline{x_{.,t}}\) and \(\overline{y_{.,t}}\) represent mean frequencies at tip \(t\). The mean standard deviation across tips is:

\[ \hat{\sigma}(x)=\frac{1}{T}\sum_{t=1}^{T}\sigma_t(x) \\ \hat{\sigma}(y)=\frac{1}{T}\sum_{t=1}^{T}\sigma_t(y) \]

- Methylated sites: Similarly, for \(w\) and \(z\):

\[ \hat{\sigma}(w)=\frac{1}{T}\sum_{t=1}^{T}\sigma_t(w) \\ \hat{\sigma}(z)=\frac{1}{T}\sum_{t=1}^{T}\sigma_t(z) \]

These values can be computed using:

    # 1 tip / sample / replicate
    sample_n <- 1
    index_islands <- c(1, 3)
    index_nonislands <- c(2, 4)
    data <- list(c(.5, .5, 0, 1, 1, .5), c(.5, 0, 1, .5), c(.5, .5, 0), c(0, 0, .5))
    get_islandSDFreqP(index_islands, data, categorized_data = T, sample_n)
    #> [1] 0.1178511
    get_nonislandSDFreqP(index_nonislands, data, categorized_data = T, sample_n)
    #> [1] 0.1178511
    get_islandSDFreqM(index_islands, data, categorized_data = T, sample_n)
    #> [1] 0.2357023
    get_nonislandSDFreqM(index_nonislands, data, categorized_data = T, sample_n)
    #> [1] 0.1767767

    # 2 tip / sample / replicate
    sample_n <- 2
    index_islands <- c(1, 3)
    index_nonislands <- c(2, 4)
    data <- list(
      list(c(.5, .5, 0, 0, 0, 1), c(.5, 0, 0, .5), c(1, .5, 0), c(0, 0, .5)), # tip 1
      list(c(0, .5, .5, 1, 1, .5), c(1, .5, 1, .5), c(0, .5, .5), c(1, .5, 1)) # tip 2
      )
    get_islandSDFreqP(index_islands, data, categorized_data = T, sample_n)
    #> [1] 0.05892557
    get_nonislandSDFreqP(index_nonislands, data, categorized_data = T, sample_n)
    #> [1] 0.1178511
    get_islandSDFreqM(index_islands, data, categorized_data = T, sample_n)
    #> [1] 0.1767767
    get_nonislandSDFreqM(index_nonislands, data, categorized_data = T, sample_n)
    #> [1] 0.05892557

Note that we average across tips as there can be correlations between the mean standard deviation of tips (e.g. under the case of an event affecting the states at two tips).

## Mean Correlation of a Central Segment of Methylation States per Genomic Structure Type

Let \(s\) be the length of two adjacent segments of central positions to consider. Let \(l\) be the number of positions excluded at the start and end of each structure. Define:

\[ \overline{Cor(x)_{s,l}}=\frac1I\frac1T\sum_{i=1}^{I}\sum_{t=1}^{T}Cor(x_{i,t,[p:p'-1]}, x_{i,t,[p+1:p']}) \\ \overline{Cor(y)_{s,l}}=\frac1J\frac1T\sum_{j=1}^{J}\sum_{t=1}^{T}Cor(y_{j,t,[p:p'-1]}, y_{j,t,[p+1:p']}) \]

Correlations are computed only for segment pairs where standard deviation is non-zero:

\[ \sigma(x_{i,t,[p:p'-1]}) \neq 0 \text{ and } \sigma(x_{i,t,[p+1:p']})\neq 0 \\ \sigma(y_{j,t,[p:p'-1]}) \neq 0 \text{ and } \sigma(y_{j,t,[p+1:p']})\neq 0 \]

In this formulation:

- Only structures with a minimum length of \(n = s + 2 \cdot l\) are considered.

- If \(l = 0\), all the positions are considered.

These values can be computed with the functions `compute_meanCor_i` for island structures and `compute_meanCor_ni` for non-island structures as in the following examples:

    # 1 tip / sample / replicate
    sample_n <- 1
    index_islands <- c(1, 3)
    index_nonislands <- c(2, 4)
    data <- list(c(.5, 0, 0, 0, .5, .5, .5, .5, .5, 1, .5, 0, 0, 0, .5, .5, .5, .5,
                   .5, 1, .5, 0, 0, 0, .5, .5, .5, .5, .5, 1), # 30 sites
                   c(.5, 1, 1, 1, .5, .5, 1, 1, 1, .5, .5, 1, 1, 1, .5, .5, 1, 1, 1,
                     .5, .5, 1, 1, 1, .5), # 25 sites
                   c(.5, 0, 0, .5, 1, .5, 0, 0, .5, 1, .5, 0, 0, .5, 1, .5, 0, 0,
                     .5, 1, .5, 0, 0, .5, 1, .5, 0, 0, .5, 1, .5, 0, 0, .5, 1, .5,
                     0, 0, .5, 1), # 40 sites
                   c(1, 1, 1, .5, .5, .5, 0, 0, 0, .5, 1, 1, 1, .5, .5, .5, 0, 0, 0,
                     .5, 1, 1, 1, .5, .5, .5, 0, 0, 0, .5,
                     .5, 0, 0, 0, .5)) # 35 sites
    compute_meanCor_i(index_islands, minN_CpG = 10,
                      shore_length = 5, data, sample_n = 1, categorized_data = T)
    #> [1] 0.3626687
    compute_meanCor_ni(index_nonislands, minN_CpG = 10,
                       shore_length = 5, data, sample_n = 1, categorized_data = T)
    #> [1] 0.3623932

    # 2 tip / sample / replicate
    sample_n <- 2
    index_islands <- c(1, 3)
    index_nonislands <- c(2, 4)
    data <- list(
      # tip 1
        list(c(.5, 0, 0, 0, .5, .5, .5, .5, .5, 1, .5, 0, 0, 0, .5, .5, .5, .5, .5,
               1, .5, 0, 0, 0, .5, .5, .5, .5, .5, 1), # 30 sites
             c(.5, 1, 1, 1, .5, .5, 1, 1, 1, .5, .5, 1, 1, 1, .5, .5, 1, 1, 1, .5,
               .5, 1, 1, 1, .5), # 25 sites
             c(.5, 0, 0, .5, 1, .5, 0, 0, .5, 1, .5, 0, 0, .5, 1, .5, 0, 0, .5, 1,
               .5, 0, 0, .5, 1, .5, 0, 0, .5, 1, .5, 0, 0, .5, 1, .5, 0, 0,
               .5, 1), # 40 sites
             c(1, 1, 1, .5, .5, .5, 0, 0, 0, .5, 1, 1, 1, .5, .5, .5, 0, 0, 0, .5,
               1, 1, 1, .5, .5, .5, 0, 0, 0, .5, .5, 0, 0, 0, .5)), # 35 sites
      # tip 2
        list(c(.5, 0, 0, .5, .5, .5, 0, 0, .5, 1, .5, 0, 0, 0, 0, .5, .5, 1, 1, 1,
               .5, 0, 0, 0, .5, .5, 1, 1, 1, 1), # 30 sites
             c(.5, .5, 1, 1, .5, .5, 1, 1, 1, .5, .5, 0, 0, 0, .5, .5, 1, 1, 1, .5,
               .5, 1, 1, 1, .5), # 25 sites
             c(.5, 0, 0, .5, 1, .5, 0, 0, .5, 1, .5, 0, 0, .5, .5, .5, 0, 0, .5, 1,
               1, 1, 1, .5, 1, .5, 0, 0, .5, 1, .5, 0, 0, .5, 1, .5, 0, 0,
               .5, 1), # 40 sites
             c(1, 1, 1, .5, .5, .5, 0, 0, 0, .5, 1, 1, 1, 1, .5, .5, 0, 0, 0, .5, 1,
               1, 1, .5, .5, .5, .5, .5, 0, .5, .5, .5, .5, 0, .5)) # 35 sites
      )
    compute_meanCor_i(index_islands, minN_CpG = 10,
                      shore_length = 5, data, sample_n = 2, categorized_data = T)
    #> [1] 0.4371918
    compute_meanCor_ni(index_nonislands, minN_CpG = 10,
                       shore_length = 5, data, sample_n = 2, categorized_data = T)
    #> [1] 0.5021673

Note that correlation measures how much the state of each site says about the next. So, when there is no variation (e.g. in middle segment state u of a site is followed by state u of the next), there is no correlation.

## Comparison of tree tips

### Mean Frequency of Methylation State Changes at Single Sites per Structure Type

Let \(S_k\) be the set of sites in the genomic structure indexed by \(k\). For each site \(s \in S_k\), let the methylation state at a given tip \(t\) be denoted as \(m_{k,s,t}\), where \(m_{k,s,t} \in \{0, 0.5, 1\}\) represents the unmethylated, partially-methylated, or methylated state, respectively.

For a given cherry \((t_1, t_2)\), that is two tips \(t_1\) and \(t_2\) that are direct offspring of the same internal node, we compute the proportion of sites with differing methylation states between the two tips.

Define the indicator variable: \[ \delta_{k,s,t_1,t_2} = \begin{cases} 1, & \text{if } m_{k,s,t_1} \neq m_{k,s,t_2}, \\ 0, & \text{if } m_{k,s,t_1} = m_{k,s,t_2}. \end{cases} \]

Then, the proportion of sites with different methylation states in structure \(k\) for the cherry \((t_1, t_2)\) is given by:

\[ F_k(t_1, t_2) = \frac{1}{|S_k|} \sum_{s \in S_k} \delta_{k,s,t_1,t_2} \]

where \(|S_k|\) is the total number of sites in structure \(k\), and the sum counts the number of sites where the methylation state differs between tips \(t_1\) and \(t_2\).

This statistic quantifies the proportion of differing methylation states for a given genomic structure in a cherry.

To compute the weighted mean frequency of methylation changes across all structures in a cherry, we separate island and non-island structures.

For a given cherry \((t_1, t_2)\), let \(\mathcal{I}\) be the set of indices for island structures and \(\mathcal{N}\) be the set of indices for non-island structures. Each structure \(k\) has a total number of sites \(|S_k|\), which we use as weights.

The weighted mean frequency of methylation changes for island structures is:

\[ \bar{F}_{\text{island}}(t_1, t_2) = \frac{\sum_{k \in \mathcal{I}} |S_k| F_k(t_1, t_2)}{\sum_{k \in \mathcal{I}} |S_k|} \]

Similarly, the weighted mean frequency for non-island structures is:

\[ \bar{F}_{\text{non-island}}(t_1, t_2) = \frac{\sum_{k \in \mathcal{N}} |S_k| F_k(t_1, t_2)}{\sum_{k \in \mathcal{N}} |S_k|} \]

These expressions compute the mean frequency of methylation state differences across all island and non-island structures, weighted by the number of sites in each structure.

Functions:

- `get_cherryDist(tree)` to get the distances between the tips of each cherry. It identifies each cherry by the tip names and the tip indices. The tip indices correspond to (a) the index from left to right on the newick string, (b) the order of the tip label in the `phylo_object$tip.label`, and (c) the index in the methylation data list `data[[tip]][[structure]]` as obtained with the function `simulate_evolData()` when the given tree has several tips.

- `countSites_cherryMethDiff(cherryDist, data)` to get for each cherry the counts of sites with half-methylation and full-methylation changes per genomic structure (island or non-island).

- `freqSites_cherryMethDiff(tree,data)`. The function first validates the tree structure and extracts pairwise distances between cherry tips with `get_cherryDist`. It then validates the input data and counts half and full methylation differences for each cherry at each structure using `countSites_cherryMethDiff`. Finally, it normalizes these (half and full) counts by the number of sites per structure to compute frequencies.

- `get_siteFChange_cherry(tree,data)` uses `freqSites_cherryMethDiff(tree,data)` and then computes the frequency of sites with any change of methylation state (full or half) for each cherry at each genomic structure.

- `MeanSiteFChange_cherry(tree, data, index_islands, index_nonislands)`. The function computes the per-cherry frequency of sites with different methylation states at each structure (islands and non-islands) using `get_siteFChange_cherry`. Then it calculates the weighted mean site frequency of methylation changes for each cherry separately for islands and non-islands.

    # Set example tree and methylation data
    tree <- "((a:1.5,b:1.5):2,(c:2,d:2):1.5);"
    data <- list(
        list(rep(1,10), rep(0,5), rep(1,8)), # tip a
        list(rep(1,10), rep(0.5,5), rep(0,8)), # tip b
        list(rep(1,10), rep(0.5,5), rep(0,8)), # tip c
        list(c(rep(0,5), rep(0.5, 5)), c(0, 0, 1, 1, 1), c(0.5, 1, rep(0, 6)))) # d

    # Set the index for islands and non-island structures
    index_islands <- c(2)
    index_nonislands <- c(1, 3)

    MeanSiteFChange_cherry(data = data,
                           categorized_data = T,
                           tree = tree,
                           index_islands = index_islands,
                           index_nonislands = index_nonislands)
    #>   tip_names tip_indices dist nonisland_meanFChange island_meanFChange
    #> 1       a-b         1-2    3             0.4444444                  1
    #> 2       c-d         3-4    4             0.6666667                  1

### Fitch Parsimony Estimate for Minimum Global Methylation State Changes at CpG Islands

Let \(I\) be the set of CpG islands in a given genomic region For each island \(i \in I\), the mean methylation level across sites is computed with `get_meanMeth_islands(index_islands, data)` and categorized into one of three global methylation states—unmethylated (\(u\)), partially methylated (\(p\)), or methylated (\(m\))—based on user-defined thresholds \(u_{\text{thresh}}\) and \(m_{\text{thresh}}\). This categorization is given by the function:

\[ \operatorname{categorize\_islandGlbSt}(\mu_i, u_{\text{thresh}}, m_{\text{thresh}}) = \begin{cases} u, & \text{if } \mu_i \leq u_{\text{thresh}}, \\ m, & \text{if } \mu_i \geq m_{\text{thresh}}, \\ p, & \text{otherwise}. \end{cases} \]

Given a phylogenetic tree \(T\) with tips corresponding to sampled species or individuals, we aim to estimate the minimum number of state changes required to explain the observed distribution of island methylation states across the tips. This is achieved using the Fitch algorithm (Fitch 1971) as implemented in `compute_fitch(meth, tree)`.

The function `computeFitch_islandGlbSt(T, M)`, where \(M\) is the matrix of categorized methylation states across tips, returns a vector \(C\) where each entry \(C_i\) represents the minimum number of changes required for island \(i\) under the Fitch parsimony criterion.

    # Example with data from a single island structure
    # and three tips

    tree <- "((bla:1,bah:1):2,booh:2);"

    data <- list(
      #Tip 1
      list(c(rep(1,9), rep(0,1))), # m
      #Tip 2
      list(c(rep(0.5,10))), # p
      #Tip 3
      list(c(rep(0.5,9), rep(0.5,1)))) # p

    index_islands <- c(1)

    computeFitch_islandGlbSt(index_islands, data, tree,
                             u_threshold = 0.1, m_threshold = 0.9)
    #> [1] 1

    # Example: data from a genomic region consisting on 3 structures with 10 sites each
    # one island, one non-island, one island
    # and a tree with 8 tips
    tree <- "(((a:1,b:1):1,(c:1,d:1):1):1,((e:1,f:1):1,(g:1,h:1):1):1);"
    data <- list(
      #Tip 1
      list(c(rep(1,5), rep(0,5)), # p
           c(rep(0,9), 1),
            c(rep(1,8), rep(0.5,2))), # m
      #Tip 2
      list(c(rep(0.5,9), rep(0.5,1)), # p
            c(rep(0.5,9), 1),
           c(rep(0,8), rep(0.5,2))), # u
      #Tip 3
      list(c(rep(1,9), rep(0.5,1)), # m
           c(rep(0.5,9), 1),
           c(rep(0.5,8), rep(0.5,2))), # p
      #Tip 4
      list(c(rep(1,9), rep(0.5,1)), # m
           c(rep(1,9), 0),
           c(rep(0.5,8), rep(0.5,2))), # p
      #Tip 5
      list(c(rep(0,5), rep(0,5)), # u
           c(rep(0,9), 1),
           c(rep(0.5,8), rep(0.5,2))), # p
      #Tip 6
      list(c(rep(0,9), rep(0.5,1)), # u
           c(rep(0.5,9), 1),
           c(rep(1,8), rep(0.5,2))), # m
      #Tip 7
      list(c(rep(0,9), rep(0.5,1)), # u
           c(rep(0.5,9), 1),
           c(rep(0,8), rep(0.5,2))), # u
      #Tip 8
      list(c(rep(0,9), rep(0.5,1)), # u
           c(rep(1,9), 0),
           c(rep(0,9), rep(0.5,1)))) # u

    index_islands <- c(1,3)

    computeFitch_islandGlbSt(index_islands, data, tree,
                             u_threshold = 0.1, m_threshold = 0.9)
    #> [1] 2 4

For genomic regions containing multiple CpG islands, we can summarize the per-island estimates using the mean to obtain an overall measure of the minimum state changes.

    mean(computeFitch_islandGlbSt(index_islands, data, tree,
                                  u_threshold = 0.1, m_threshold = 0.9))
    #> [1] 3

### Mean Number of Significant Methylation Frequency Changes per Island and Cherry

A cherry \(c\) is defined as a pair of direct offspring (tips \(t_1\) and \(t_2\)) of the same internal node in the phylogenetic tree. For a given cherry \(c\), let \(\mathcal{I}\) be the set of indices for island structures. For each island structure \(i \in \mathcal{I}\) at each cherry tip \(t_c \in \{t_{c,1}, t_{c,2}\}\) in each phylogenetic tree, the number of sites in each methylation state (unmethylated \(0\), partially-methylated \(0.5\), methylated \(1\)) is counted. This count is represented as:

\[ \text{count}_{\text{UPM}}(c,t_c, i) = \left( n_0, n_{0.5}, n_1 \right) \]

where \(n_0\), \(n_{0.5}\), and \(n_1\) are the counts of unmethylated, partially-methylated, and methylated sites at tip \(t_c\), respectively.

The distribution of methylation states at each island between the two tips \(t_1\) and \(t_2\) of a given cherry is compared using a chi-squared test. The null hypothesis of the test is that the frequency distributions of states in the two islands follow the same multinomial distribution. Note that not only IWEs can lead to deviations from this null hypothesis but also neighbor-dependent SSEs can lead to slight deviations from the assumption of multinomial distributions. The test statistic is calculated using the contingency table \(T_{c,i}\) for tips \(t_{c,1}\) and \(t_{c,2}\):

\[ T_{c,i} = \begin{pmatrix} n_0^{t_{c,1}} & n_{0.5}^{t_{c,1}} & n_1^{t_{c,1}} \\ n_0^{t_{c,2}} & n_{0.5}^{t_{c,2}} & n_1^{t_{c,2}} \end{pmatrix} \] For each cherry \(c\) and each island \(i\), the methylation frequencies at each cherry tip are compared using the chi-squared test. The p-value for the chi-squared test is calculated via Monte Carlo simulation to improve reliability, even when the expected frequencies do not meet the assumptions of the chi-squared approximation (i.e., expected counts of at least 5 in each category). The obtained p-values are stored in a dataframe for each cherry and for each island.

The significance of the methylation frequency changes is determined based on a user-defined threshold \(p_{\text{threshold}}\). If the p-value is smaller than the threshold, the change is considered significant. This is done for each cherry and each island \(i \in \mathcal{I}\) by comparing each p-value to the threshold:

\[ \text{Significant Change}_{c,i} = \begin{cases} 1, & \text{if } p_{\text{value}}(c,i) \lt p_{\text{threshold}} \\ 0, & \text{if } p_{\text{value}}(c,i) \ge p_{\text{threshold}} \end{cases} \]

For each cherry , the mean number of significant changes across all islands \(i \in \mathcal{I}\) is computed as the average of the significant changes for each island:

\[ \text{Mean Number of Significant Changes per Island}_{c} = \frac{1}{|\mathcal{I}|} \sum_{i \in \mathcal{I}} \text{Significant Change}_{c,i} \]

where \(|\mathcal{I}|\) is the total number of islands.

Functions:

- `compare_CherryFreqs(tip1, tip2)` to perform a chi-squared test to compare the distribution of methylation states between two cherry tips.

- `pValue_CherryFreqsChange_i(data, index_islands, tree)` uses `compare_CherryFreqs(tip1, tip2)` to get for each cherry and each island the chi-square p-value of the comparison of the distribution of methylation states between two cherry tips.

- `mean_CherryFreqsChange_i(data, categorized_data = FALSE, index_islands, tree, pValue_threshold)`. The function uses `pValue_CherryFreqsChange_i(data, index_islands, tree)` and counts the mean number of significant changes per island at each cherry.

    # Set example tree and methylation data
      tree <- "((a:1,b:1):2,c:2);"
      data <- list(
        #Tip a
        list(c(rep(1,9), rep(0,1)), # Structure 1: island
             c(rep(0,9), 1), # Structure 2: non-island
             c(rep(0,9), rep(0.5,1))),  # Structure 3: island
        #Tip b
        list(c(rep(0,9), rep(0.5,1)),  # Structure 1: island
             c(rep(0.5,9), 1), # Structure 2: non-island
             c(rep(0,9), rep(0,1))), # Structure 3: island
        #Tip c
        list(c(rep(1,9), rep(0.5,1)),  # Structure 1: island
             c(rep(0.5,9), 1), # Structure 2: non-island
             c(rep(0,9), rep(0.5,1)))) # Structure 3: island

      index_islands <- c(1,3)

      mean_CherryFreqsChange_i(data, categorized_data = T,
                               index_islands, tree,
                               pValue_threshold = 0.05)
    #>   first_tip_name second_tip_name first_tip_index second_tip_index dist island_1
    #> 1              a               b               1                2    2     TRUE
    #>   island_3 FreqsChange
    #> 1    FALSE         0.5

### Mean Number of Significant Methylation Frequency Changes per Island Across All Tree Tips

For a given phylogenetic tree with a number of tips \(N\), let \(t_n \in \{t_1, t_2, \cdots, t_N\) represent the tree tips. For a given genomic region, let \(\mathcal{I}\) be the set of indices for island structures. For each island structure \(i \in \mathcal{I}\) at each tip \(t_n\) in the phylogenetic tree, the number of sites in each methylation state (unmethylated \(0\), partially-methylated \(0.5\), methylated \(1\)) is counted. This count is represented as:

\[ \text{count}_{\text{UPM}}(t_n, i) = \left( n_0, n_{0.5}, n_1 \right) \]

where \(n_0\), \(n_{0.5}\), and \(n_1\) are the counts of unmethylated, partially-methylated, and methylated sites at tip \(t_n\), respectively.

The distribution of methylation states at each island across all tips is compared using a chi-squared test. The null hypothesis of the test is that the frequency distributions of states in the two islands follow the same multinomial distribution. Note that not only IWEs can lead to deviations from this null hypothesis but also neighbor-dependent SSEs can lead to slight deviations from the assumption of multinomial distributions. The test statistic is calculated using the contingency table \(T_{i}\) for all tree tips:

\[ T_{i} = \begin{pmatrix} n_0^{t_{1}} & n_{0.5}^{t_{1}} & n_1^{t_{1}} \\ n_0^{t_{2}} & n_{0.5}^{t_{2}} & n_1^{t_{2}} \\ \cdots & \cdots & \cdots \\ n_0^{t_{N}} & n_{0.5}^{t_{N}} & n_1^{t_{N}} \end{pmatrix} \]

For each island \(i\), the methylation frequencies are compared across tips using the chi-squared test. The p-value for the chi-squared test is calculated via Monte Carlo simulation to improve reliability, even when the expected frequencies do not meet the assumptions of the chi-squared approximation (i.e., expected counts of at least 5 in each category).

The significance of the methylation frequency changes for each island \(i \in \mathcal{I}\) across tips is determined based on a user-defined threshold \(p_{\text{threshold}}\). If the p-value is smaller than the threshold, the change is considered significant.

\[ \text{Significant Change}_{i} = \begin{cases} 1, & \text{if } p_{\text{value}}(i) < p_{\text{threshold}} \\ 0, & \text{if } p_{\text{value}}(i) \ge p_{\text{threshold}} \end{cases} \]

For the given tree, the mean number of significant changes per island across all tips is computed as:

\[ \text{Mean Number of Significant Changes per Island} = \frac{1}{|\mathcal{I}|} \sum_{i \in \mathcal{I}} \text{Significant Change}_{i} \]

where \(|\mathcal{I}|\) is the total number of islands.

Function:

- `mean_TreeFreqsChange_i(tree, data, categorized_data = FALSE, index_islands, pValue_threshold)`.

    # Set example tree and methylation data
    tree <- "((a:1,b:1):2,(c:2,d:2):1);"
      data <- list(
        #Tip a
        list(c(rep(1,9), rep(0,1)), # Structure 1: island
             c(rep(0,9), 1), # Structure 2: non-island
             c(rep(0,9), rep(0,1))), # Structure 3: island
        #Tip b
        list(c(rep(0,9), rep(0.5,1)), # Structure 1: island
             c(rep(0.5,9), 1), # Structure 2: non-island
             c(rep(0,9), rep(0,1))),# Structure 3: island
        #Tip c
        list(c(rep(0,9), rep(0.5,1)), # Structure 1: island
             c(rep(0.5,9), 1), # Structure 2: non-island
             c(rep(1,9), rep(0,1))),# Structure 3: island
        #Tip d
        list(c(rep(0,9), rep(0.5,1)), # Structure 1: island
             c(rep(0.5,9), 1), # Structure 2: non-island
             c(rep(1,8), rep(0.5,2)))) # Structure 3: island

      index_islands <- c(1,3)

      mean_TreeFreqsChange_i(tree, data, categorized_data = T,
                               index_islands,
                               pValue_threshold = 0.05)
    #> [1] 1

# References

Csillery, Katalin, Olivier Francois, and Michael G. B. Blum. 2012. “Abc: An r Package for Approximate Bayesian Computation (ABC).” Methods in Ecology and Evolution. https://doi.org/10.1111/j.2041-210X.2011.00179.x.
Fitch, Walter M. 1971. “Toward Defining the Course of Evolution: Minimum Change for a Specific Tree Topology.” Systematic Biology 20 (4): 406–16.