Efficient time-adaptive Expectation Maximization (EM) algorithm for HMM tracking
In ULISSES we are developing data-driven calibration algorithms for autonomous CO2 sensor calibration. This has been done by developing an approach where a Hidden Markov model (HMM) is used to predict on a calibration coefficient to compensate for remaining dependencies of environmental impacts such as temperature on the sensor measurement output. The HMM model is learned from the past sensor data in an unsupervised way using the expectation maximization algorithm. For such an approach, convergence and sample complexity are critical factors.
A common difficulty in gas sensing is to achieve a long-term stable performance due to aging. This unfortunately causes that a previously learned sensor model becomes outdated and needs to be updated regularly. Figure 1 shows degradation of the model.
Figure 1: Calibration model degradation – orange denotes calibrated sensor output and blue denotes measurement of a high-precision reference sensor. Left plot shows 20 day performance immediately after training. The plot to the right show the 20 day performance after 100 days.
A naïve approach would be to re-learn the model from scratch. In this project we have developed and analyzed a computation and data efficient approach where we initialize the algorithm with the prior model. The interval of updates can be decided based on a likelihood difference so that also a convergence rate guarantee can be provided. This approach results in a significant reduction of computations and requires significantly less data and enables to track a sufficiently slowly aging sensor.
Figure 2: The left plot shows the performance that can be achieved after 100 days if the model is monthly updated. The right plot shows the number of iterations needed to update the model using our approach and the traditional approach using the same minimum improvement as stopping criteria. The updates are always done after one month. The traditional approach uses a random initialization. It shows that significantly less data and computations are needed.
Yang You, lead author of the paper and ULISSES researcher with KTH
Y. You. T. J. Oechtering, “Time-adaptive Expectation Maximization Learning Framework for HMM based Data-driven Gas Sensor Calibration,” IEEE Transaction on Industrial Informatics, 2022.