How to quantify the skewness of a 2D pressure field?

Question

I'm working with a pressure sensor matrix that measures the weight distribution of a person standing on the sensors. The data comes in as a matrix of pressure values. I'm trying to process this data in some way to quantify how much a person is leaning (or naturally imbalanced) in a general way;

1. The quantity should not depend on the absolute position of the person on the sensors
2. It should not depend on their total mass
3. It should not depend too much on the absolute distance between their feet
4. It should not depend too much on the shape of their feet
5. Should preferably be reliable and robust against noise

I made a first version calculating skewness using central moments but this seems to yield some strange behaviour: The skewness changes direction temporarily and flips back and forth when 2 pressure points pass each other.

I recorded a GIF of a simulation that shows the problem I'm talking about. I'm using 2 simulated weights of different sizes and moving them past each other, then plotting the skewness of the x- and y- marginal distributions as a cursor on a 2D plot.

Skewness measure simulation

My question is if this kind of behavior is expected of skewness and if so, are there any alternatives that might be more suited to this application?

Answer 1

Consider axis $x$. Let both weights be distributed normally. The bigger weight has distribution $N(\mu,1)$ and the smallest weight $N(0,\sigma)$. Let the smaller weight have the fraction $p$ of total mass.

You can think of it as a small weight with size of contact $\sigma$ placed at $x=0$ and larger weight with the size of contact $1$ placed at $x=\mu$.

Then one can show that the non-normalized skewness is: $$ S(\mu)=\mathbb E(X-\mathbb EX)^3=p(1-p)\mu\Big(3-3\sigma^2-(1-2p)\mu^2\Big) $$

If $p<1/2$ (the smaller weight is indeed smaller than the larger) and $\sigma < 1$ (its area of contact is smaller too), the plot of skewness is not monotonous:

Normalization (dividing by $\sigma_0^3$) doesn't change this.

If you need to determine the distribution of weight between the feet, I would go for some clustering algorithms (basically try to find the line that separates the area between feet the best) and then took the integrals of two half planes. What clustering algorithm would do better (robustly) in extreme cases (when 2 feet are close to each other), I can't say. You likely need testing and finding out yourself.