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# The main idea
[Gaussian Processes]( give us _probability distributions_ over **curves and surfaces**.
Unfortunately, these distributions can be hard to visualize.
One approach uses **animations**. Here are the key ideas (my own contributions are in bold):
- Each _frame_ of the animation shows a curve/surface which is _drawn from the distribution_ we want to visualize.
- Consecutive frames show very _similar_ elements (so the animation is _continuous_).
- **Every frame has exactly the same statistical and kinematic properties (there are _no special "keyframes"_).**
- **The motion is smooth and natural (no "kicks").**
For example, here is an uncertain surface. The datapoints have a gap in the middle. The _entire_ surface is animated, but it moves more in the gap because the uncertainty is higher where datapoints are missing.
<img width='100%' src="">
Remarkably, just a single ingredient is needed: the _"Gaussian oscillator"_. This is a particle moving on a continuous path, whose position probability is the standard normal distribution at all times. The right-hand side of the following figure shows independent Gaussian oscillators. The left-hand side visualizes a distribution of curves. Each frame is obtained by multiplying the vector of Gaussian oscillators by the lower-Cholesky decomposition of the covariance matrix (center).
<img width='100%' src="">
My further contribution is to recognize that **Gaussian oscillators are also Gaussian processes**, but in the _time_ domain. This places all future work on Gaussian animations into a familiar and well-studied framework.
# Further reading
- Here is [a poster I presented](
at the [Twelfth World Meeting of ISBA]( (2014).
- **NOTE**: The original version had an error in the implementation! I forgot to divide by `sqrt(N)`. The current version is fixed.
- I gave an invited talk at the [SIAM CSE13]( conference (2013).
Here is [a recording of the talk](,
and the [corresponding slides](
_Paper forthcoming!_