# Substitution models¶

BEAST models discrete data as evolving on a phylogenetic tree according to a Continuous Time Markov Chain.

When configuring a BEASTling analysis, for each model you must choose a substitution model. Different substitution models are more or less appropriate for different kinds of data. BEASTling supports two models for binary data (commonly used to model the evolution of cognacy relationships for lexical data) and two models for multistate data (appropriate for structural/typological data). The substitution models are:

## Binary CTMC¶

(set `model=binaryctmc`

in config file)

The binary CTMC model is defined for binary datasets, i.e. sets where every datapoint is either a 0 or a 1. It is the simplest possible model for this kind of data. Binary models are most typically used for cognate data, but can be used for binary structural data also.

When this model is used, if the specified datafile contains multistate data, BEASTling will automatically translate this into the appropriate number of binary features. This approach means that you can have a single data file which can be used to generate binary and multistate analyses, and also lets BEASTling share mutation rates across binary features corresponding to a single multistate feature.

If the datafile only contains features with two values, you should explicitly tell BEASTling what kind of data it is. If you have pre-binarised multistate data such as cognate class assignments, you should include `binarised=True`

(or `binarized=True`

) in your config’s `[model]`

section. If your data is genuinely binary in nature (e.g. absence/presence structural data) you should set `binarised=False`

instead. This lets BEASTling perform ascertainment correction correctly (even if constant features are retained, ascertainment correction must be performed for the impossibility of all zero features when multistate data has been recoded into binary form). See Ascertainment correction for more explanation of ascertainment correction in BEASTling.

## Binary Covarion¶

(set `model=covarion`

in config file)

The binary Covarion model is defined for binary datasets, i.e. sets where every datapoint is either a 0 or a 1. This model introduces a latent “fast” or “slow” state, which controls the rate of transitions between 0 and 1 (transitions in either direction are always equally probable). This model is more sophisticated than the binary CTMC model and usually represents a better fit to data.

All of the comments regarding the binarisation of multistate data and ascertainment correction for this process in the description of the binary CTMC model apply identically to the binary Covarion model.

This model estimates two parameters, a switching factor which governs how frequently the latent state switches between “fast” and “slow”, and a parameter denoted “alpha” which controls the difference in speed between the two states. By default, these parameters are shared across all features in the dataset, i.e. BEAST will estimate 2 parameters for n features. By setting the boolean parameter `share_params`

to `False`

, you can change this behaviour and give each feature its own pair of parameters, i.e. have BEAST estimate 2n parameters for n features. This will slow your analysis down but for some datasets may enable a much better fit to the data.

## Mk¶

(set `model=mk`

in config file)

The Lewis Mk model is the simplest generic substitution model available in BEASTling. It is a generalisation of the classic JC69 model from genetics to a statespace of arbitrary size. Transitions are possible from any state to any other state, and every transition is equally probable. No parameters are estimated, increasing analysis speed. This model could be used with any dataset, but the assumptions are not a good match for cognate data.

If using this model, you may set `frequencies=approx`

for a slight performance improvement (smaller XML, possibly faster and less memory-intensive MCMC). This will use “approximately empirical” equilibrium distributions where possible, by rounding to a single decimal point, e.g. the empirical distributions (0.82, 0.18) and (0.76, 0.24) would both be replaced by (0.8, 0.2). For large datasets and small state spaces, this means that many distinct features will have identical equilibrium frequencies, allowing them to share the same SubstitutionModel object. This cuts down on repeated calculations.

## BSVS¶

(set `model=bsvs`

in config file)

The Bayesian Stochastic Variable Selection (BSVS) model is a rich model suitable for structural data. Compared to the Lewis Mk model, it permits non-equal transition probabilities between different states, and also tries to set a number of probabilities to zero, i.e. transitions from some states to others will be disallowed. This model is suitable for attempting to uncover preferential directions of change in the evolution of particular linguistic features. Note that this model is very parameter intensive and analyses will be much slower than Mk analyses for the same data.

A `bsvs`

model accepts two additional parameters, `symmetric`

and `svsprior`

.
They change the behaviour as follows.

A symmetric model (`symmetric=True`

, which is the default value) assumes that transition rates between states are symmetric, i.e. for two states A and B, transitions from A to B occur at the same rate as transitions from B to A. An asymmetric model (`symmetric=False`

) has double the number of parameters, because the rates A→B and B→A are estimated separately.

The `svsprior`

property specifies the shape of the prior distribution which is placed over the number of non-zero rate. Possible choices are `poisson'' and ``exponential`

, with `poisson`

being the default.

The size of the statespace for a particular feature determines a maximum possible number of non-zero rates (the entire matrix), and also a minimum possible number (to ensure that the Markov chain is ergodic). The non-zero rate prior is defined over this range, so both the Poisson and exponential priors have an offset, rather than beginning their support at zero.

The default Poisson prior is the more conservative choice. BEASTling will set the mean of the Poisson distribution equal to the midpoint between the minimum and maximum possible number of non-zero rates. In this way, the model has no strong preference for sparse matrices over dense matrices or vice versa (as the mean of the Poisson is usually approximately equal to the median), while still encouraging the setting of rates to zero if the data supports it.

The exponential prior, on the other hand, is biased toward sparse transition matrices. BEASTling will set the mean of the exponential distribution such that 99% of the probability density lies between the minimum and maximum possible number of non-zero rates. In this way, matrices with the majority of rates set to zero are much more probable a priori than matrices with the majority of rates being non-zero. This prior is the better choice when you want to fit a model that permits the minimum number of transitions required to explain the data.