Suppose a snowflake saw a problem that really bothered her. Let's name our snowflake "Greta", just for fun.
Greta is shown a knob she can spin. If she spins it one way, the problem gets worse. If she spins it the other way, the immediate problem improves.
One problem (or consequence) and one action? Simple. In the small table shown above, one spin in the negative direction should reduce the error to zero.
Excel is our friend.
But wait, there were side-effects or unexpected consequences.
|Total error defined as sqrt of sum-of-squares of individual errors. Total error before Greta's fix was 1.0 and now it is 1.29|
Suppose it were snowing outside an Greta had nothing better to do than work on finding the ideal "Correction vector" for this system.*
She is able to beat the error down to 1% of the original value, but at what cost?
It also begs the question, how sensitive is the solution to minor changes in the sensitivity matrix. What if there were minor measurement errors or what if the system becomes increasingly less responsive as larger inputs are jammed into it (insulin resistance, being one example)?
|The two cells that are pink were originally 0.96 and 0.75 respectively. That is, they were each changed by 0.01 which is not a very large amount.|
The first knob must now be turned 531 turns in the negative direction, the second knob must now be turned 384 turns in the positive direction and the third knob must be turned 170 turns in the positive direction.
And the original error of "1.0" can only be reduced to 11% of its original value.
Is this example real?
I believe it is descriptive more often than people realize. If you have the passing thought that you are looking at a "rob Peter to pay Paul" situation, then you are probably looking at this kind of situation.
Consider psychotropic drugs. If they manipulate the same neuro-chemical pathways they will have the same side-effects or consequences. The unwary medical practitioner will find themselves prescribing second drug to counteract the effects of another drug, only to find the therapeutic benefit expected from the first drug is nullified by the second. Dosages increase as the patient teeter-totters from one unhappy state to the other, over-and-over again.
The example of the two psychotropic drugs is an example of a singlular or near-singular system.
And at what cost? The example does not assign a cost (or penalty) to spinning the knob. Greta could be learning to code or lifting weights or reloading ammo in the basement instead of spinning knobs.
*I cheated and used the Excel Solver add-in.