asr_mk_model {castor} R Documentation

## Ancestral state reconstruction with Mk models and rerooting

### Description

Ancestral state reconstruction of a discrete trait using a fixed-rates continuous-time Markov model (a.k.a. "Mk model"). This function can estimate the (instantaneous) transition matrix using maximum likelihood, or take a specified transition matrix. The function can optionally calculate marginal ancestral state likelihoods for each node in the tree, using the rerooting method by Yang et al. (1995).

### Usage

asr_mk_model( tree,
tip_states,
Nstates               = NULL,
tip_priors            = NULL,
rate_model            = "ER",
transition_matrix     = NULL,
include_ancestral_likelihoods = TRUE,
reroot                = TRUE,
root_prior            = "auto",
Ntrials               = 1,
optim_algorithm       = "nlminb",
optim_max_iterations  = 200,
optim_rel_tol         = 1e-8,
store_exponentials    = TRUE,
check_input           = TRUE,


### Arguments

 tree A rooted tree of class "phylo". The root is assumed to be the unique node with no incoming edge. tip_states An integer vector of size Ntips, specifying the state of each tip in the tree in terms of an integer from 1 to Nstates, where Ntips is the number of tips and Nstates is the number of possible states (see below). Can also be NULL. If tip_states==NULL, then tip_priors must not be NULL (see below). Nstates Either NULL, or an integer specifying the number of possible states of the trait. If Nstates==NULL, then it will be computed based on the maximum value encountered in tip_states or based on the number of columns in tip_priors (whichever is non-NULL). tip_priors A 2D numeric matrix of size Ntips x Nstates, where Nstates is the possible number of states for the character modelled. Hence, tip_priors[i,s] is the likelihood of the observed state of tip i, if the tip's true state was in state s. For example, if you know for certain that a tip is in state k, then set tip_priors[i,s]=1 for s=k and tip_priors[i,s]=0 for all other s. rate_model Rate model to be used for fitting the transition rate matrix. Can be "ER" (all rates equal), "SYM" (transition rate i–>j is equal to transition rate j–>i), "ARD" (all rates can be different), "SUEDE" (only stepwise transitions i–>i+1 and i–>i-1 allowed, all 'up' transitions are equal, all 'down' transitions are equal) or "SRD" (only stepwise transitions i–>i+1 and i–>i-1 allowed, and each rate can be different). Can also be an index matrix that maps entries of the transition matrix to the corresponding independent rate parameter to be fitted. Diagonal entries should map to 0, since diagonal entries are not treated as independent rate parameters but are calculated from the remaining entries in the transition matrix. All other entries that map to 0 represent a transition rate of zero. The format of this index matrix is similar to the format used by the ace function in the ape package. rate_model is only relevant if transition_matrix==NULL. transition_matrix Either a numeric quadratic matrix of size Nstates x Nstates containing fixed transition rates, or NULL. The [r,c]-th entry in this matrix should store the transition rate from state r to state c. Each row in this matrix must have sum zero. If NULL, then the transition rates will be estimated using maximum likelihood, based on the rate_model specified. root_prior Prior probability distribution of the root's states, used to calculate the model's overall likelihood from the root's marginal ancestral state likelihoods. Can be "flat" (all states equal), "empirical" (empirical probability distribution of states across the tree's tips), "stationary" (stationary probability distribution of the transition matrix), "likelihoods" (use the root's state likelihoods as prior) or "max_likelihood" (put all weight onto the state with maximum likelihood). If "stationary" and transition_matrix==NULL, then a transition matrix is first fitted using a flat root prior, and then used to calculate the stationary distribution. root_prior can also be a non-negative numeric vector of size Nstates and with total sum equal to 1. include_ancestral_likelihoods Include the marginal ancestral likelihoods for each node (conditional scaled state likelihoods) in the return values. Note that this may increase the computation time and memory needed, so you may set this to FALSE if you don't need marginal ancestral states. reroot Reroot tree at each node when computing marginal ancestral likelihoods, according to Yang et al. (1995). This is the default and recommended behavior, but leads to increased computation time. If FALSE, ancestral likelihoods at each node are computed solely based on the subtree descending from that node, without rerooting. Ntrials Number of trials (starting points) for fitting the transition matrix. Only relevant if transition_matrix=NULL. A higher number may reduce the risk of landing in a local non-global optimum of the likelihood function, but will increase computation time during fitting. optim_algorithm Either "optim" or "nlminb", specifying which optimization algorithm to use for maximum-likelihood estimation of the transition matrix. Only relevant if transition_matrix==NULL. optim_max_iterations Maximum number of iterations (per fitting trial) allowed for optimizing the likelihood function. optim_rel_tol Relative tolerance (stop criterion) for optimizing the likelihood function. store_exponentials Logical, specifying whether to pre-calculate and store exponentials of the transition matrix during calculation of ancestral likelihoods. This may reduce computation time because each exponential is only calculated once, but requires more memory since all exponentials are stored. Only relevant if include_ancestral_likelihoods==TRUE, otherwise exponentials are never stored. check_input Logical, specifying whether to perform some basic checks on the validity of the input data. If you are certain that your input data are valid, you can set this to FALSE to reduce computation. Nthreads Number of parallel threads to use for running multiple fitting trials simultaneously. This only makes sense if your computer has multiple cores/CPUs and if Ntrials>1, and is only relevant if transition_matrix==NULL. This option is ignored on Windows, because Windows does not support forking.

### Details

For this function, the trait's states must be represented by integers within 1,..,Nstates, where Nstates is the total number of possible states. If the states are originally in some other format (e.g. characters or factors), you should map them to a set of integers 1,..,Nstates. The order of states (if relevant) should be reflected in their integer representation. For example, if your original states are "small", "medium" and "large" and rate_model=="SUEDE", it is advised to represent these states as integers 1,2,3. You can easily map any set of discrete states to integers using the function map_to_state_space.

This function allows the specification of the precise tip states (if these are known) using the vector tip_states. Alternatively, if some tip states are only known in terms of a probability distribution, you can pass these probability distributions using the matrix tip_priors. Note that exactly one of the two arguments, tip_states or tip_priors, must be non-NULL.

Tips must be represented in tip_states or tip_priors in the same order as in tree$tip.label. None of the input vectors or matrixes need include row or column names; if they do, however, they are checked for consistency (if check_input==TRUE). The tree is either assumed to be complete (i.e. include all possible species), or to represent a random subset of species chosen independently of their states. The rerooting method by Yang et al (1995) is used to calculate the marginal ancestral state likelihoods for each node by treating the node as a root and calculating its conditional scaled likelihoods. Note that the re-rooting algorithm is strictly speaking only valid for reversible Mk models, that is, satisfying the criterion π_i Q_{ij}=π_j Q_{ji},\quad\forall i,j, where Q is the transition rate matrix and π is the stationary distribution of the model. The rate models “ER”, 'SYM”, “SUEDE” and “SRD” are reversible. For example, for “SUEDE” or “SRD” choose π_{i+1}=π_{i}Q_{i,i+1}/Q_{i+1,i}. In contrast, “ARD” models are generally not reversible. If tree$edge.length is missing, each edge in the tree is assumed to have length 1. The tree may include multi-furcations (i.e. nodes with more than 2 children) as well as mono-furcations (i.e. nodes with only one child). This function is similar to rerootingMethod in the phytools package (v0.5-64) and similar to ape::ace (v4.1) with options method="ML", type="discrete" and marginal=FALSE, but tends to be much faster than rerootingMethod and ace for large trees.

Internally, this function uses fit_mk to estimate the transition matrix if the latter is not provided. If you only care about estimating the transition matrix but not the ancestral states, consider using the more versatile function fit_mk.

### Value

A list with the following elements:

 success Logical, indicating whether ASR was successful. If FALSE, all other return values may be NULL. Nstates Integer, specifying the number of modeled trait states. transition_matrix A numeric quadratic matrix of size Nstates x Nstates, containing the transition rates of the Markov model. The [r,c]-th entry is the transition rate from state r to state c. Will be the same as the input transition_matrix, if the latter was not NULL. loglikelihood Log-likelihood of the observed tip states under the fitted (or provided) Mk model. If transition_matrix was NULL in the input, then this will be the log-likelihood maximized during fitting. AIC Numeric, the Akaike Information Criterion for the fitted Mk model (if applicable), defined as 2k-2\log(L), where k is the number of independent fitted parameters and L is the maximized likelihood. If the transition matrix was provided as input, then no fitting was performed and hence AIC will be NULL. ancestral_likelihoods Optional, only returned if include_ancestral_likelihoods was TRUE. A 2D numeric matrix, listing the likelihood of each state at each node (marginal ancestral likelihoods). This matrix will have size Nnodes x Nstates, where Nstates was either explicitly provided as an argument, or inferred from tip_states or tip_priors (whichever was non-NULL). The rows in this matrix will be in the order in which nodes are indexed in the tree, i.e. the [n,s]-th entry will be the likelihood of the s-th state at the n-th node. For example, likelihoods[1,3] will store the likelihood of observing the tree's tip states (if reroot=TRUE) or the descending subtree's tip states (if reroot=FALSE), if the first node was in state 3. Note that likelihoods are rescaled (normalized) to sum to 1 for convenience and numerical stability. The marginal likelihoods at a node should not, however, be interpreted as a probability distribution among states.

Stilianos Louca

### References

Z. Yang, S. Kumar and M. Nei (1995). A new method for inference of ancestral nucleotide and amino acid sequences. Genetics. 141:1641-1650.

hsp_mk_model, asr_max_parsimony, asr_squared_change_parsimony, hsp_max_parsimony, map_to_state_space, fit_mk

### Examples

## Not run:
# generate random tree
Ntips = 1000
tree  = generate_random_tree(list(birth_rate_intercept=1),max_tips=Ntips)$tree # create random transition matrix Nstates = 5 Q = get_random_mk_transition_matrix(Nstates, rate_model="ER", max_rate=0.01) cat(sprintf("Simulated ER transition rate=%g\n",Q[1,2])) # simulate the trait's evolution simulation = simulate_mk_model(tree, Q) tip_states = simulation$tip_states
cat(sprintf("Simulated states for last 20 nodes:\n"))
print(tail(simulation$node_states,20)) # reconstruct node states from simulated tip states # at each node, pick state with highest marginal likelihood results = asr_mk_model(tree, tip_states, Nstates, rate_model="ER", Ntrials=2) node_states = max.col(results$ancestral_likelihoods)

# print Mk model fitting summary
cat(sprintf("Mk model: log-likelihood=%g\n",results$loglikelihood)) cat(sprintf("Fitted ER transition rate=%g\n",results$transition_matrix[1,2]))

# print reconstructed node states for last 20 nodes
print(tail(node_states,20))

## End(Not run)

[Package castor version 1.6.8 Index]