# Duration Estimation

In a prior post mentioned that for intra-day variance prediction it made sense to separate variance into 2 processes:

1. intensity process
When is the next event going to occur;  lets call this Tprior + Δt.   This is the more complex process of the two to predict.
2. power process
What is the amplitude of the event at time Tnow + Δt.   The power or amplitude process seems to be fairly well behaved.   An ARMA style process seems like a likely candidate.

Towards this end, I have been exploring models for the intensity process.   Very often this is modeled in terms of duration.   Below is a summary of some results:

ACD Models
ACD processes make overreaching assumptions.  In particular ACD models assume a constant AR decay and innovation contribution across time.   Unfortunately this is not supported by empirical observations.   Here are some results for the best-fitting Wiebull ACD model on HF data:

The R^2 level of 0.0091 does not inspire confidence.

SVR Model
I used an iterative non-parametric machine learning approach (SVR) with a training set of 20 prior observations and a lagged series of the derivatives of the prior 20 durations as the input vector.   Training across the entire series, one gets an in-sample prediction R^2 of 0.9980.   Unfortunately, incremental out of sample does not fair as well:

Distribution of Durations
Here are 2 views on the distribution of durations:

Alternative Models
Some possibilities:

1. markov chain (probabalistic state system)
We model the patterns by categorizing the durations into K separate levels.   To train we observe the chain of states, say {K1, K8, K1,K1,K1,K4} and determine a graph describing the approximate event chains, factorizing and assigning probabilities.
2. ANN
Use a simple feed-forward network, trained with a GA or DE.   This is easy to implement but subject to a variety of problems such as overfitting.

As the ANN is easy to compose, will start there.