## Monday, 29 May 2017

### Applying the Kriging Models in Structural Reliability

Hi all,

I will then, as promised, talk a bit today about the Kriging surface models.
These models are nothing more than surrogate models that account for a certain level of uncertainty. They are widely used for many fields, but their initial application goes back to geostatistics.

They are an interesting tool that we don't hear much about when learning Engineering. On the other hand, if you talk with a Geologist they will know for sure about what you're talking. I share my office with some people from Geology, and they do. They are all happy when they see me working with it... it's like... "look at this Engineer in trouble with these simple Kriging" haha

Well, as I told before these are nothing more than interpolators. The image below will help you understand (courtesy of Wikipedia):

The idea of the Kriging surrogate model is to approximate a group of points in a N-dimensional space with a curve. Like you would do with a 2nd, 3nd or n degree polynomial. But in this case, we assume that the space between the points we do not know as an error which is Gaussian distributed.
Let's see, you see the red dots, these are the points that we know. If we assume a deterministic interpolation scheme we will have the red line or another line (depending on the order of the approximation) that will in the limit be the same as the trimmed blue line. For such a complex model it's hard to have exactly the blue trimmed line if we use a reasonable amount of points, so we are very likely to be induce in some kind of error in our prediction of the variation of z with x.

Where does the Kriging surface comes into play then? Well, if you assume the Kriging surface for the same set of points you will have the gray area, mixed with the red line. This means that you know that your blue trimmed lined will be, with 95% confidence, inside that area. (!! but it can be out! The Gaussian distribution tails are not bounded). So, let's say it is like a model, that fits infinite curves to a certain group of points.
With one single sample of points for all the domain of x from the Kriging:
If you're lucky you will have the exact same blue curve...well....very very lucky....
If you're not, you will end up with an approximation that is worst than the red line (which is the expected curve). If you take many many "samples of this curve" you will end with the red line, the expected curve.

Can you see the interest now? They are indeed an amazing piece of math. You can tell, well, whats the point? It's all left to the luck? Or, I'll end up with a red curve anyway?

Well, do not forget that so many things in this world follow a Gaussian distribution... and a tool like this one, which is simple and beautiful, can be widely implemented in this world for much more than just approximating curves or a couple of points.

If you have a system's output that is Gaussian distributed and depends on many variables you can use this, like I am doing. If you're not sure about your curve and you want some degrees of uncertainty, here we are :) etc etc...

I am pretty sure that you're amazed, because the first time I saw this I was like: "This is way I am not going anywhere, such a simple and beautiful tool and I couldn't even think remotely on this existing inside my ignorance" :)

It was nice to write to you all.
For those who know me... I know I know...lately it's Kriging for this, Kriging for that... Kriging for beers... Kriging tatoo...I can't avoid it. I love the concept hehe
But I know I know, extra care in the application of them, as good-sense is needed.

See you soon and I hope you find the post interesting,
Rui