Note: The writing of this vignette is currently in progress. If needed, please, contact the author for assistance. Thanks for your understanding.

Using the PCMFit R-package comprises several steps:

1. Preparing the data, to which we wish to fit phylogenetic comparative model (PCM);
2. Deciding on the type of PCM we wish to infer;
3. Defining the PCM;
4. Calling a PCMFit model inference function on the data and model type we have defined;
5. Evaluating the inferred model.

Before we start, let’s load the packages needed for this tutorial:

library(PCMBase)
library(PCMFit)
library(data.table)
library(ggplot2)
library(ggtree)
library(cowplot)

# make results reproducible
set.seed(4, kind = "Mersenne-Twister", normal.kind = "Inversion")

In the next sections, we go through each of the above steps:

# Step 1. An example dataset

For this tutorial, we will use a simulated non-ultrametric tree of $$N=80$$ tips and simulated $$k=2$$-variate trait values. Both of these are stored in the data-table dtSimulated included with the PCMFit package.

## The tree

tree <- PCMTree(PCMFitDemoObjects$dtSimulated$tree[])
tree
#>
#> Phylogenetic tree with 80 tips and 79 internal nodes.
#>
#> Tip labels:
#>  1, 2, 3, 4, 5, 6, ...
#> Node labels:
#>  81, 82, 83, 84, 85, 86, ...
#>
#> Rooted; includes branch lengths.

## The data

X <- PCMFitDemoObjects$dtSimulated$X[][, seq_len(PCMTreeNumTips(tree))]
dim(X)
#>   2 80

We use the PCMBase::PCMTreePlot() and PCMBase::PCMPlotTraitData2D() functions to generate plots of the tree and the data (note that PCMBase::PCMTreePlot() requires the ggtree R-package which is currently not on CRAN - see the ggtree home page for instructions how to install this package):

plTree <- PCMTreePlot(tree, layout="fan") +
geom_tiplab2(size = 2) +
geom_nodelab(size = 2, color = "black") +
geom_treescale(width = max(PCMTreeNodeTimes(tree)), x = 0, linesize = .25, fontsize = 2, offset = 79)

plX <- PCMPlotTraitData2D(
X[, seq_len(PCMTreeNumTips(tree))],
tree,
scaleSizeWithTime = FALSE,
numTimeFacets = 4) +
geom_text(
aes(x = x, y = y, label = id, color = regime),
size=2,
position = position_jitter(.4, .4)) +
theme_bw() +
theme(legend.position = "bottom")

cowplot::plot_grid(plTree, plX, labels = LETTERS[1:2], nrow=2, rel_heights = c(2,1)) Example phylogenetic comparative data. A: a tree of 80 tips. B: bivariate trait values at the tips of the tree.

# Step 2. Gaussian model types

In this tutorial, we use the six model types $$BM_{A}$$, $$BM_{B}$$, $$OU_{C}$$, $$OU_{D}$$, $$OU_{E}$$, $$OU_{F}$$ from the $$\mathcal{G}_{LInv}$$-family as defined in (Mitov, Bartoszek, and Stadler 2019). These model types are also described in the PCMBase Getting started guide:

# scroll to the right in the following listing to see the aliases for the six
# default model types:
PCMDefaultModelTypes()
#>                                                                                                                                                                                A
#>                                                                                                      "BM__Global_X0__Diagonal_WithNonNegativeDiagonal_Sigma_x__Omitted_Sigmae_x"
#>                                                                                                                                                                                B
#>                                                                                   "BM__Global_X0__UpperTriangularWithDiagonal_WithNonNegativeDiagonal_Sigma_x__Omitted_Sigmae_x"
#>                                                                                                                                                                                C
#>                                       "OU__Global_X0__Schur_Diagonal_WithNonNegativeDiagonal_Transformable_H__Theta__Diagonal_WithNonNegativeDiagonal_Sigma_x__Omitted_Sigmae_x"
#>                                                                                                                                                                                D
#>                    "OU__Global_X0__Schur_Diagonal_WithNonNegativeDiagonal_Transformable_H__Theta__UpperTriangularWithDiagonal_WithNonNegativeDiagonal_Sigma_x__Omitted_Sigmae_x"
#>                                                                                                                                                                                E
#> "OU__Global_X0__Schur_UpperTriangularWithDiagonal_WithNonNegativeDiagonal_Transformable_H__Theta__UpperTriangularWithDiagonal_WithNonNegativeDiagonal_Sigma_x__Omitted_Sigmae_x"
#>                                                                                                                                                                                F
#>                             "OU__Global_X0__Schur_WithNonNegativeDiagonal_Transformable_H__Theta__UpperTriangularWithDiagonal_WithNonNegativeDiagonal_Sigma_x__Omitted_Sigmae_x"

Our purpose for this tutorial will be to select an optimal model for the data. We will compare “global” models in which one model of one of the above model types is fit to the entire tree and data, versus a mixed Gaussian phylogenetic model where different models of the above model types are assigned to different parts of the tree.

# Step 3. Creating model objects

## Global models

### A global $$BM_{B}$$ model

modelBM <- PCM(
PCMDefaultModelTypes()["B"], modelTypes = PCMDefaultModelTypes(), k = 2)

modelBM
#> Brownian motion model
#> S3 class: BM__Global_X0__UpperTriangularWithDiagonal_WithNonNegativeDiagonal_Sigma_x__Omitted_Sigmae_x, BM, GaussianPCM, PCM; k=2; p=5; regimes: 1. Parameters/sub-models:
#> X0 (VectorParameter, _Global, numeric; trait values at the root):
#>  0 0
#> Sigma_x (MatrixParameter, _UpperTriangularWithDiagonal, _WithNonNegativeDiagonal; factor of the unit-time variance rate):
#> , , 1
#>
#>      [,1] [,2]
#> [1,]    0    0
#> [2,]    0    0
#>
#> Sigmae_x (MatrixParameter, _Omitted; factor of the non-heritable variance or the variance of the measurement error):
#>
#> 

### A global $$OU_{F}$$ model

modelOU <- PCM(
PCMDefaultModelTypes()["F"], modelTypes = PCMDefaultModelTypes(), k = 2)

modelOU
#> Ornstein-Uhlenbeck model
#> S3 class: OU__Global_X0__Schur_WithNonNegativeDiagonal_Transformable_H__Theta__UpperTriangularWithDiagonal_WithNonNegativeDiagonal_Sigma_x__Omitted_Sigmae_x, OU, GaussianPCM, PCM, _Transformable; k=2; p=11; regimes: 1. Parameters/sub-models:
#> X0 (VectorParameter, _Global, numeric; trait values at the root):
#>  0 0
#> H (MatrixParameter, _Schur, _WithNonNegativeDiagonal, _Transformable; adaptation rate matrix):
#> , , 1
#>
#>      [,1] [,2]
#> [1,]    0    0
#> [2,]    0    0
#>
#> Theta (VectorParameter, matrix; long-term optimum):
#>      1
#> [1,] 0
#> [2,] 0
#> Sigma_x (MatrixParameter, _UpperTriangularWithDiagonal, _WithNonNegativeDiagonal; factor of the unit-time variance rate):
#> , , 1
#>
#>      [,1] [,2]
#> [1,]    0    0
#> [2,]    0    0
#>
#> Sigmae_x (MatrixParameter, _Omitted; factor of the non-heritable variance or the variance of the measurement error):
#>
#> 

## MGPM models

### A MGPM with the true shift-point configuration and model type mapping

modelTrueTypeMapping <- MixedGaussian(
k = 2,
modelTypes = MGPMDefaultModelTypes(),
mapping = c(4, 3, 2),
X0 = structure(
0, class = c("VectorParameter",
"_Global"), description = "trait values at the root"),
Sigmae_x = structure(
0, class = c("MatrixParameter", "_Omitted",
"_Global"),
description =
"Upper triangular Choleski factor of the non-phylogenetic variance-covariance"))

treeWithTrueShifts <- PCMTree(PCMFitDemoObjects$dtSimulated$tree[])
PCMTreeSetPartRegimes(
treeWithTrueShifts,
part.regime = c(81 = 1, 105 = 2, 125 = 3),
setPartition = TRUE)

### A MGPM with the true shift-point configuration, model type mapping and parameter values

modelTrue <- PCMFitDemoObjects$dtSimulated$model[]
modelTrue
#> Mixed Gaussian model
#> S3 class: MixedGaussian_432, MixedGaussian, GaussianPCM, PCM, _Transformable; k=2; p=18; regimes: 1, 2, 3. Parameters/sub-models:
#> X0 (VectorParameter, _Global, numeric; trait values at the root):
#>   1 -1
#> 1 (OU__Omitted_X0__Schur_Diagonal_WithNonNegativeDiagonal_Transformable_H__Theta__UpperTriangularWithDiagonal_WithNonNegativeDiagonal_Sigma_x__Omitted_Sigmae_x, OU, GaussianPCM, PCM, _Transformable):
#> Ornstein-Uhlenbeck model
#> S3 class: OU__Omitted_X0__Schur_Diagonal_WithNonNegativeDiagonal_Transformable_H__Theta__UpperTriangularWithDiagonal_WithNonNegativeDiagonal_Sigma_x__Omitted_Sigmae_x, OU, GaussianPCM, PCM, _Transformable; k=2; p=7; regimes: 1. Parameters/sub-models:
#> X0 (VectorParameter, _Omitted; trait values at the root):
#>
#> H (MatrixParameter, _Schur, _Diagonal, _WithNonNegativeDiagonal, _Transformable; adaptation rate matrix):
#> , , 1
#>
#>      [,1] [,2]
#> [1,] 1.39 0.00
#> [2,] 0.00 0.59
#>
#> Theta (VectorParameter, matrix; long-term optimum):
#>         1
#> [1,] 5.94
#> [2,] 2.58
#> Sigma_x (MatrixParameter, _UpperTriangularWithDiagonal, _WithNonNegativeDiagonal; Choleski factor of the unit-time variance rate):
#> , , 1
#>
#>      [,1] [,2]
#> [1,] 0.29 0.02
#> [2,] 0.00 0.36
#>
#> Sigmae_x (MatrixParameter, _Omitted; Choleski factor of the non-heritable variance or the variance of the measurement error):
#>
#>
#> 2 (OU__Omitted_X0__Schur_Diagonal_WithNonNegativeDiagonal_Transformable_H__Theta__Diagonal_WithNonNegativeDiagonal_Sigma_x__Omitted_Sigmae_x, OU, GaussianPCM, PCM, _Transformable):
#> Ornstein-Uhlenbeck model
#> S3 class: OU__Omitted_X0__Schur_Diagonal_WithNonNegativeDiagonal_Transformable_H__Theta__Diagonal_WithNonNegativeDiagonal_Sigma_x__Omitted_Sigmae_x, OU, GaussianPCM, PCM, _Transformable; k=2; p=6; regimes: 1. Parameters/sub-models:
#> X0 (VectorParameter, _Omitted; trait values at the root):
#>
#> H (MatrixParameter, _Schur, _Diagonal, _WithNonNegativeDiagonal, _Transformable; adaptation rate matrix):
#> , , 1
#>
#>      [,1] [,2]
#> [1,] 0.37 0.00
#> [2,] 0.00 1.06
#>
#> Theta (VectorParameter, matrix; long-term optimum):
#>         1
#> [1,] 5.41
#> [2,] 2.63
#> Sigma_x (MatrixParameter, _Diagonal, _WithNonNegativeDiagonal; Choleski factor of the unit-time variance rate):
#> , , 1
#>
#>      [,1] [,2]
#> [1,] 0.36  0.0
#> [2,] 0.00  0.4
#>
#> Sigmae_x (MatrixParameter, _Omitted; Choleski factor of the non-heritable variance or the variance of the measurement error):
#>
#>
#> 3 (BM__Omitted_X0__UpperTriangularWithDiagonal_WithNonNegativeDiagonal_Sigma_x__Omitted_Sigmae_x, BM, GaussianPCM, PCM):
#> Brownian motion model
#> S3 class: BM__Omitted_X0__UpperTriangularWithDiagonal_WithNonNegativeDiagonal_Sigma_x__Omitted_Sigmae_x, BM, GaussianPCM, PCM; k=2; p=3; regimes: 1. Parameters/sub-models:
#> X0 (VectorParameter, _Omitted; trait values at the root):
#>
#> Sigma_x (MatrixParameter, _UpperTriangularWithDiagonal, _WithNonNegativeDiagonal; Choleski factor of the unit-time variance rate):
#> , , 1
#>
#>      [,1] [,2]
#> [1,] 0.17 0.14
#> [2,] 0.00 0.50
#>
#> Sigmae_x (MatrixParameter, _Omitted; Choleski factor of the non-heritable variance or the variance of the measurement error):
#>
#>
#> Sigmae_x (MatrixParameter, _Omitted; upper triangular Choleski factor of the non-phylogenetic variance-covariance):
#>
#>

# We specify the tree and trait values for the true model in order to easily
# calculate parameter count likelihood and AIC for it:
attr(modelTrue, "tree") <- treeWithTrueShifts
attr(modelTrue, "X") <- X
attr(modelTrue, "SE") <- X * 0.0

## A view of the data according to the true shift-point configuration


plTree <- PCMTreePlot(treeWithTrueShifts, layout="fan") %<+%
data.table(
node = c(12, 77, 45),
part.model = c(" 1.D ", " 2.C ", " 3.B "),
offset = 5) +
geom_tiplab2(size = 2) +
geom_tiplab2(aes(label = part.model), offset = 16) +
geom_nodelab(size = 2, color = "black") +
geom_treescale(
width = max(PCMTreeNodeTimes(treeWithTrueShifts)), x = 0,
linesize = .25, fontsize = 2, offset = 79)

plX <- PCMPlotTraitData2D(
X[, seq_len(PCMTreeNumTips(treeWithTrueShifts))],
treeWithTrueShifts,
scaleSizeWithTime = FALSE,
numTimeFacets = 4) +
geom_text(
aes(x = x, y = y, label = id, color = regime),
size=2,
position = position_jitter(.4, .4)) +
theme_bw() +
theme(legend.position = "bottom")

cowplot::plot_grid(plTree, plX, labels = LETTERS[1:2], nrow = 2, rel_heights = c(2,1)) Example phylogenetic comparative data. A: a tree of 80 tips partitioned in three evolutionary regimes. Each evolutionary regime is denoted by #.T, where # is the regime identifier and T is the evolutionary model type associated with this regime (a model type among A, …, F). B: bivariate trait values at the tips of the tree.

# Step 4. Model inference

## Specifying parameter limits

An optional but important preliminary step is to explicitly specify the limits for the model parameters. This is needed, because the default settings might not be appropriate for the data in question. Here is an example how the model parameter limits can be set. Note that we specify these limits for the base “BM” and “OU” model types, but they are inherited by their subtypes A, …, F, as specified in the comments.

## File: DefineParameterLimits.R
# lower limits for models A and B
PCMParamLowerLimit.BM <- function(o, k, R, ...) {
o <- NextMethod()
k <- attr(o, "k", exact = TRUE)
R <- length(attr(o, "regimes", exact = TRUE))

if(is.Global(o$Sigma_x)) { if(!is.Diagonal(o$Sigma_x)) {
o$Sigma_x[1, 2] <- -.0 } } else { if(!is.Diagonal(o$Sigma_x)) {
for(r in seq_len(R)) {
o$Sigma_x[1, 2, r] <- -.0 } } } o } # upper limits for models A and B PCMParamUpperLimit.BM <- function(o, k, R, ...) { o <- NextMethod() k <- attr(o, "k", exact = TRUE) R <- length(attr(o, "regimes", exact = TRUE)) if(is.Global(o$Sigma_x)) {
o$Sigma_x[1, 1] <- o$Sigma_x[2, 2] <- 1.0
if(!is.Diagonal(o$Sigma_x)) { o$Sigma_x[1, 2] <- 1.0
}
} else {
for(r in seq_len(R)) {
o$Sigma_x[1, 1, r] <- o$Sigma_x[2, 2, r] <- 1.0
if(!is.Diagonal(o$Sigma_x)) { o$Sigma_x[1, 2, r] <- 1.0
}
}
}
o
}

# lower limits for models C, ..., F.
PCMParamLowerLimit.OU <- function(o, k, R, ...) {
o <- NextMethod()
k <- attr(o, "k", exact = TRUE)
R <- length(attr(o, "regimes", exact = TRUE))

if(is.Global(o$Theta)) { o$Theta <- 0.0
o$Theta <- -1.2 } else { for(r in seq_len(R)) { o$Theta[1, r] <- 0.0
o$Theta[2, r] <- -1.2 } } if(is.Global(o$Sigma_x)) {
if(!is.Diagonal(o$Sigma_x)) { o$Sigma_x[1, 2] <- -.0
}
} else {
if(!is.Diagonal(o$Sigma_x)) { for(r in seq_len(R)) { o$Sigma_x[1, 2, r] <- -.0
}
}
}
o
}

# upper limits for models C, ..., F.
PCMParamUpperLimit.OU <- function(o, k, R, ...) {
o <- NextMethod()
k <- attr(o, "k", exact = TRUE)
R <- length(attr(o, "regimes", exact = TRUE))

if(is.Global(o$Theta)) { o$Theta <- 7.8
o$Theta <- 4.2 } else { for(r in seq_len(R)) { o$Theta[1, r] <- 7.8
o$Theta[2, r] <- 4.2 } } if(is.Global(o$Sigma_x)) {
o$Sigma_x[1, 1] <- o$Sigma_x[2, 2] <- 1.0
if(!is.Diagonal(o$Sigma_x)) { o$Sigma_x[1, 2] <- 1.0
}
} else {
for(r in seq_len(R)) {
o$Sigma_x[1, 1, r] <- o$Sigma_x[2, 2, r] <- 1.0
if(!is.Diagonal(o$Sigma_x)) { o$Sigma_x[1, 2, r] <- 1.0
}
}
}
o
}

## Fitting the global models

fitBM <- PCMFit(model = modelBM, tree = tree, X = X,
metaI = PCMBaseCpp::PCMInfoCpp)
fitOU <- PCMFit(model = modelOU, tree = tree, X = X,
metaI = PCMBaseCpp::PCMInfoCpp)

## Fitting an MGPM model with known shift-point configuration and model type mapping but unknown model parameters

# This can take about 5 minutes to finish
fitMGPMTrueTypeMapping <- PCMFit(
model = modelTrueTypeMapping,
tree = treeWithTrueShifts,
X = X,
metaI = PCMBaseCpp::PCMInfoCpp)

## Fitting an MGPM model with known shift-point configuration and model type mapping starting from the true parameter values

fitMGPMTrueTypeMappingCheat <- PCMFit(
model = modelTrueTypeMapping,
tree = treeWithTrueShifts,
X = X,
matParInit = matrix(PCMParamGetShortVector(modelTrue), 1L),
numRunifInitVecParams = 1000L,
numGuessInitVecParams = 100L,
metaI = PCMBaseCpp::PCMInfoCpp)

# Step 5. Evaluating the model fits

listModels <- list(
RetrieveBestModel(fitBM),
RetrieveBestModel(fitOU),
RetrieveBestModel(fitMGPMTrueTypeMapping),
RetrieveBestModel(fitMGPMTrueTypeMappingCheat),
modelTrue)

dtSummary <- data.table(
model = c(
"Global BM",
"Global OU",
"True MGPM, unknown parameters",
"True MGPM, known true parameters",
"True MGPM, true parameters"),
p = sapply(listModels, PCMParamCount),
logLik = sapply(listModels, logLik),
AIC = sapply(listModels, AIC))
knitr::kable(dtSummary)
model p logLik AIC
Global BM 5 -221.68133 455.3627
Global OU 11 -204.60164 433.2033
True MGPM, unknown parameters 18 -93.65584 233.3117
True MGPM, known true parameters 18 -93.50648 233.0130
True MGPM, true parameters 18 -102.63502 251.2700