# Dictionary of Mathematical Eponymy: Wahba's Problem

While my thesis has the word ‘topology’ in its title, at heart I’m a vectors-in-3D person. Give me matrices, not manifolds!

So today’s entry in the Dictionary of Mathematical Eponymy is one that brings me joy.

### What is Wahba’s Problem?

The mathematical statement of Wahba’s Problem is as follows:

- Given a list $\mathbf{w}_k$ of $N$ measurements in a reference frame
- … and a list $\mathbf{v}_k$ of $N$ corresponding measurements in the body frame
- … and a list $a_k$ of $N$ corresponding weightings
- Find the special orthogonal matrix $\mathbf{R}$ that minimises:
- $J(\mathbf{R}) = \frac{1}{2}\sum_{k=1}^{N} a_k \left|\left| \mathbf{w}_k - \mathbf{R} \mathbf{v}_k \right|\right|^2$

### Why is it important?

Without knowing what it’s about, I could imagine that formulation looks like gibberish. With some background, it makes a lot more sense!

Wahba’s problem is to figure out how to adjust (for example) attitude measurements from a magnetometer on board a satellite to measurements made from somewhere known.

Those two sets of measurements are likely to disagree – partly due to measurement error, and partly due to systematic error: I don’t know, maybe the magnetometer isn’t pointing the way it should be.

Wahba’s problem is to find the rotation matrix that best corrects for the systematic error so you can adapt any other measurements appropriately.

### Who is Grace Wahba?

Grace Wahba (1934-) studied at Cornell University, University of Maryland and, for her doctoral studies, Stanford, graduating in 1966 before becoming a professor of statistics at the University of Wisconsin-Madison. She retired in 2018.

Her academic forte was finding method to smooth noisy data – as well as posing Wahba’s Problem, she wrote the book (literally) about using splines to model observational data and developed methods applicable to a wide range of fields, and Stanford’s statistics department website ranks her as one of the top statisticians in the USA.

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