Gaussian Mixture and Hidden Markov Model based Pickers

The motivation for this approach was to exploit temporal relation in the data to identify patterns in a given time series that indicate a significant seismic activity has occurred, conventionally called a Pick. One of the graduate students, Mathew Faulkner had already explored Gaussian Mixture model based approach and showed that the technique was effective. Reading his thesis, it struck me that a Gaussian mixture hidden markov model might be worth exploring.

A hidden markov model assumes that system is a markov process with hidden or unobserved states. A markov process in loose sense, says that future state of a system depends only on its present state (memory-less). The hidden markov I used had the structure shown in figure X:

After making the structure, a learning task is required to learn the parameter. The learning task here in HMM is to find, given an output (data stream), the best set of state transition and output probabilities. The task is usually to derive the maximum likelihood estimate of the parameters of the HMM given the set of output sequences. A local maximum likelihood can be derived efficiently using the Baum–Welch algorithm or the Baldi–Chauvin algorithm. The Baum–Welch algorithm is a special case of the expectation-maximization algorithm.

The stream was split into 1 sec buckets and took the RMS value of each bucket to construct a new stream. After this I am moving a sliding window of 20 sec over it.

I constructed the feature set of:

• Short term average (2s)
• Long Term Average(20s)
• Short Term Stdev (2s)
• Long term Stdev (20s)

I started off pick prediction with 2 component Gaussian Hidden Markov Model to utilize the sequential relation in the dataset, assuming No-Pick as state-0 and Pick as state 1. The Expectation Maximization Algorithm was used to iteratively solve for model parameters (state transition matrix), and later calculating probability of Pick in a new sequence for the same sensor.

SciPy again helps here by providing a number of optimized algorithms that can be put together for the above experiment. The figure below shows how effective the model is:

Figure 5. A plot showing comparison of Old Picker and new suggested Hidden markov model picker. The red vertical lines are picks reported by current CSN algorithm (Top) and new suggested model (bottom). Note that new picker almost clears up the false positives.

The plot shows the reduced number of false positives (no seismic event present but picker says there is) after applying the new HMM based picker. The HMM Picker says there is one pick at the time of actual quake. The simulation was run for number of events and stations, and in general reduced false positives were observed.

However, this approach was rejected due to two defects:

1. The transition probabilities between not Quake and Quake are in order of -6 (minus six). Thus in loose sense, 106 data points would be required for proper parameter estimation and system functionality. Not enough data was available at the moment.
2. The HMM uses Expectation Maximization, which converges to local minima. Thus if initial guesses are not good enough, observed states would vary for same input.