How to combine quantities of different units to depict a cost function?

Description :

I wish to model the motion of a ball tossed up by a child in the air.

By manually annotating the ball in captured frames, I get a feeling that few things could work in my favour such as constant angle at which the ball travels , constant acceleration of the ball and how well the ball positions depict a parabolic curve (error in fitting parabola).

I would like to make use of these components for formulation of a cost function (for data association). Hence, the cost would look something like the following :

Cost = A + B + C , where A, B, C could be derived from the quantities I mentioned above (for eg. using variance) or could be any other quantities

The objective is to select the point (location of ball) in every frame which yields minimum cost (data association).

Problem :

Since A , B and C are in different units, adding them just like that is not mathematically correct. Also, since the range of values is different for each one of them, some components don't contribute to the cost at all.

One possible solution is to normalize the quantities. But since the resolution of each component is different (minimum value by which they can change) normalization also doesnt help. This is because each component still depicts a different quantity in itself.

For example:

If A , B and C are divided by max(range(A)) , max(range(B)) and max(range(C)) respectively, then all of them can vary from 0 - 1.

But A = 0.5 would not correspond to B = 0.5 and C = 0.5 which is what I want (since I want to give equal weights to all components in my cost equation)

Question:

If I wish to have equal weightage of all components reflected in my cost equation, what is the best way to combine them?

[José-Luis]

Trial and error.

Answer 1

[adi kul]

Just my thoughts, can't you create 3 different functions for each of A,B and C and then give a random array or array [0:0.01:1] of weightage for each and have the conditioned equation such that error across them will be zero.

[Atishay Jain]

I am having difficulty in formulating the required functions for individual components. Will give it a thought and try some experiments in this direction :)

[Justin Dubin]

We're you every able to get this to work? I am in a similar predicament.