# Simulation and Inference

#### George G. Vega Yon

#### August 18, 2021

This is a simple example of simulation, inference, and prediction with the `aphylo` R package.

## Setup

    library(aphylo)

    ## Loading required package: ape

    # Parameter estimates
    psi  <- c(.05, .025)
    mu_d <- c(.2, .1)
    mu_s <- c(.1, .05)
    Pi   <- .5

## Simulation

    set.seed(223)
    x <- raphylo(n = 200, psi = psi, mu_d = mu_d, mu_s = mu_s, Pi = Pi)
    plot(x)

The simulation function generates an `aphylo` class object which is simply a wrapper containing:

- a 0/1 integer matrix (annotations),
- a `phylo` tree (from the ape package), and
- some information about the tree and annotations.

If needed, we can export the data as follows:

    # Edgelist describing parent->offspring relations
    write.csv(x$tree, file = "tree.tree", row.names = FALSE)

    # Tip annotations
    ann <- with(x, rbind(tip.annotation, node.annotation))
    write.csv(ann, file = "annotations.csv", row.names = FALSE)

    # Event types
    events <- with(x, cbind(c(tip.type*NA, node.type)))
    rownames(events) <- 1:nrow(events)
    write.csv(events, file = "events.csv", row.names = FALSE)

## Inference

To fit the data, we can use MCMC as follows:

    ans <- aphylo_mcmc(x ~ psi + mu_d + mu_s + Pi)

    ## Warning: While using multiple chains, a single initial point has been passed
    ## via `initial`: c(0.1, 0.05, 0.9, 0.5, 0.1, 0.05, 0.5). The values will be
    ## recycled. Ideally you would want to start each chain from different locations.

    ## Convergence has been reached with 10000 steps. Gelman-Rubin's R: 1.0436. (500 final count of samples).

    ans

    ##
    ## ESTIMATION OF ANNOTATED PHYLOGENETIC TREE
    ##
    ##  Call: aphylo_mcmc(model = x ~ psi + mu_d + mu_s + Pi)
    ##  LogLik: -109.0075
    ##  Method used: mcmc (10000 steps)
    ##  # of Leafs: 200
    ##  # of Functions 1
    ##  # of Trees: 1
    ##
    ##          Estimate  Std. Err.
    ##  psi0    0.0744    0.0606
    ##  psi1    0.0569    0.0395
    ##  mu_d0   0.2225    0.1283
    ##  mu_d1   0.1377    0.1007
    ##  mu_s0   0.1312    0.0445
    ##  mu_s1   0.0621    0.0318
    ##  Pi      0.3481    0.2563

For goodness-of-fit analysis, we have a couple of tools. We can compare the predicted values with the observed values:

    plot(ans)

We can also take a look at the surface of the posterior function

    plot_logLik(ans)

And we can also take a look at the prediction scores

    ps <- prediction_score(ans)
    ps       # Printing

    ## Prediction score (H0: Observed = Random)
    ##
    ##  N obs.      : 399
    ##  alpha(0, 1) : 0.41, 0.59
    ##  Observed    : 0.71 ***
    ##  Random      : 0.52
    ##  P(<t)       : 0.0000
    ## --------------------------------------------------------------------------------
    ## Values scaled to range between 0 and 1, 1 being best.
    ##
    ## Significance levels: *** p < .01, ** p < .05, * p < .10
    ## AUC 0.85.
    ## MAE 0.29.

    plot(ps) # and plotting