mean_abs_change_scalar_ou: Compute the expected absolute change of an Ornstein-Uhlenbeck...

In castor: Efficient Phylogenetics on Large Trees

Compute the expected absolute change of an Ornstein-Uhlenbeck process.

Description

Given a scalar Ornstein-Uhlenbeck process at stationarity, compute the expected absolute net change over a specific time interval. In other words, if X(t) is the process at time t, compute the conditional expectation of |X(t)-X(0)| given that X(0) is randomly drawn from the stationary distribution. This quantity may be used as a measure for the speed at which a continuous trait evolves over time.

Usage

mean_abs_change_scalar_ou(stationary_mean,
                          stationary_std,
                          decay_rate,
                          delta,
                          rel_error = 0.001,
                          Nsamples = NULL)

Arguments

stationary_mean	Numeric, the stationary mean of the OU process, i.e., its equilibrium (\mu).
stationary_std	Positive numeric, the stationary standard deviation of the OU process. Note that this is \sigma/\sqrt{2\lambda}, where \sigma is the volatility.
decay_rate	Positive numeric, the decay rate or "rubber band" parameter of the OU process (\lambda).
delta	Positive numeric, the time step for which to compute the expected absolute change.
rel_error	Positive numeric, the relative tolerable standard estimation error (relative to the true mean absolute displacement).
Nsamples	Integer, number of Monte Carlo samples to use for estimation. If NULL, this is determined automatically based on the desired accuracy (rel_error).

Details

The scalar OU process is a continuous-time stochastic process that satisfies the following stochastic differential equation:

dX = \lambda\cdot(\mu-X)\ dt + \sigma\ dW,

where W is a Wiener process (aka. "standard Brownian Motion"), \mu is the equilibrium, \sigma is the volatility and \lambda is the decay rate. The OU process is commonly used to model the evolution of a continuous trait over time. The decay rate \lambda alone is not a proper measure for how fast a trait changes over time (despite being erroneously used for this purpose in some sudies), as it only measures how fast the trait tends to revert to \mu when it is far away from \mu. Similarly, the volatility \sigma alone is not a proper measure of evolutionary rate, because it only describes how fast a trait changes when it is very close to the equilibrium \mu, where the tendency to revert is negligible and the process behaves approximately as a Brownian Motion.

This function uses Monte Carlo integration to estimate the expected absolute change, by repeatedly sampling start values X(0) from the OU's stationary distribution, computing the conditional expected absolute change given the sampled start value, and then averaging those conditional expectations.

Value

A non-negative numeric, specifying the expected absolute change of the OU process.

Author(s)

Stilianos Louca

See Also

simulate_ou_model

Examples

# compute the expected absolute change of an OU process after 10 time units
expected_abs_change = mean_abs_change_scalar_ou(stationary_mean=5, 
                                                stationary_std=1,
                                                decay_rate=0.1,
                                                delta=10)

castor documentation built on Aug. 18, 2023, 1:07 a.m.