Goal-Equivalent Manifold (GEM) decomposition for a scalar goal

The scalar-goal special case of UCM (Cusumano & Cesari 2006): given the gradient of a scalar goal function, split the execution variability into goal-equivalent (along the level set – does not change the goal) and non-goal-equivalent (along the gradient – changes it), and report the motor-equivalent ratio.

Usage

goalEquivalentManifold(execution, goal_gradient = NULL, goal = NULL)

Arguments

execution

An N x m matrix of execution/body variables across trials.

goal_gradient

Length-m gradient of the goal function at the mean (the direction in which the goal changes fastest). Provide this, or goal.

goal

Optional scalar goal function R^m -> R; its gradient is computed numerically at the mean when goal_gradient is not given.

Value

a gem_result list: gev (goal-equivalent variance, along the GEM), ngev (non-goal-equivalent variance, along the gradient), me_ratio (gev / ngev, > 1 = variability channelled into the goal-irrelevant direction), log_me_ratio.

References

Cusumano JP, Cesari P (2006) Biol Cybern 94:367-379.

See also

uncontrolledManifold()

Examples

set.seed(2)
x1 <- rnorm(200, 10, 3); x2 <- 20 - x1 + rnorm(200, 0, 0.4)  # x1+x2 ~ const
goalEquivalentManifold(cbind(x1, x2), goal_gradient = c(1, 1))$me_ratio
#> [1] 268.2205