Sampling: series

Previous:

• sampling points in box shaped clusters

In an earlier post, we used the following design goal for a good sampling:

"generate a set of random points with a lot of features, using minimal coding effort"

An approach based on generating a random (non-complete) binary tree of boxes gave nice results:

From previous post

We can plot the boxes for clarity:

Fractally generated boxes.

Fractally generated boxes, with random sampled points.

Now I had the idea that this could be used to generate interesting time series. We can put the current value on the x-axis, and sample the next value from the profile distribution. Illustration:

Profile / conditional distribution of y(t) when y(t-1)=0.4. Notice that box density = conditional density. 

So let's generate some time series with this. We start at y(0) = 0, and then update recursively using the conditional distribution induced by the boxes. 

Sampling directly from the conditional distribution. 

Well, that is a bit disappointing. The generated time series just looks kind of random. It is quite stateless. We can try residual sampling instead. Then we change the update equation, from:

to:

B is the box conditional distribution, and lambda is a small coefficient. This makes it more like a derivative. An other way of seeing it, in terms of the direct sampling, is that is is as if we have forced all boxes to be around the y(t)=y(t-1) line. It also gives more interesting time series:

Residual sampling - we only make small updates. Time series has more "features". However, it only explores a small part of the state space. 

This method looks more promising, but clearly we want to get better at generating conditional distributions that will make the sampling explore a larger portion of the state space.