Need to explain the variance accounted for better

[moorepants]

Andy had issue with my stating that the nominal joint torques explain 80% of the measured joint torques and the feedback portion of the model explained 5 to 10% more. He thought it was silly to say the nominal joint torques explained anything. I really need to state how much of the variation from nominal is explained by the torque due to the controller.

If K(phi)=0 then we identify mstar which is equivalent to m0 (nominal joint torque). This should simply identify the average joint torques, i.e. the best fit torque given all gait cycles. This is the base trajectories, then the feed back term explains some portion of the variation away from that. If the variation for a given torque at a given phase in the gait cycle is sigma, then K!=0 should explain some percentage of that. I need to report that number. I think it will be something like that it explains 50% of the variation. We'll see.

[moorepants]

From Andy:

I don’t understand your K=0 explaining variance. I think you might be talking
about what part of the motion following a deviated state is just due to the
laws of mechanics. But, because your output is torque, not motions, that should
be zero. It might be passive muscle properties, but those would show up as non-zero
gains in your analysis.

[moorepants]

Given this model for the controller:

m(t) = m0(phi) + K(phi)[x0(phi) - x(t)]

one can rearrange to:

m(t) = m*(phi) - K(phi)x(t)

If K(phi)=0, i.e. no feedback, then m*(phi)=m0(phi).

In this notebook:

http://nbviewer.ipython.org/github/csu-hmc/gait-control-direct-id-paper/blob/master/notebooks/id_m_star_only.ipynb

I show that you can identify m*(phi) with K(phi)=0 which should be no different that computed the mean joint torques.

I exclaimed to Andy and Mont that the model m(t) = m*(phi) - K(phi)x(t), on average, explains 90% (used 80% incorrectly above and in person) of the measured joint torques. Then if k(phi)!=0 the model explains 95% of the measured joint torques.

Because Andy made the assumption that the mean joint torques are the nominal joint torques, m0(phi), I believe he expected me to subtract m0(phi) from the model and simply state that the joint torques from feedback explains some portion of variation in the joint torques, which would be something like 50%.