Markov Modeling of Time-Series Data. via Spectral Analysis

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1 1st International Conference on InfoSymbiotics / DDDAS Session-2: Process Monitoring Nurali Virani Markov Modeling of Time-Series Data Department of Mechanical Engineering The Pennsylvania State University via Spectral Analysis Authors: Devesh K. Jha, Nurali Virani and Asok Ray Combustion instability detection This work has been supported by U.S. Air Force Office of Scientific Research (AFOSR) under Grant No. FA (Dynamic Data-driven Application Systems; PM: Dr. Frederica Darema)

2 Learning Information Representations for Time-Series Data Combustion Instability System health monitoring Flow regime in nuclear plant heat exchangers Challenges? Mathematical characterization and useful representation of data (Data to knowledge) Inference of accurate model structure and efficient estimation of model parameters (Modeling and analysis of hyper-parameters) Motivating Problems: Rotorcraft stability monitoring, structural health monitoring, combustion instability detection, target classification with seismic sensors, two-phase flow regime classification using ultrasonic, and battery state estimation How can we create compact representation of time-series data (using Markov models)?

3 Markov models for information representation from time-series data How to discretize data? 1. Alphabet size? 2. Location of partitions? 1. Uniform 2. Max-Entropy.s 1 s 2 s 1 s 1 s 2 s 1 s 2 s 2 s 1 s 1 s 1.. Markov Modeling? 1. Order Estimation? 2. Parameter Estimation? Symbolic Time-series Analysis Applications Anomaly/Change/Fault Detection Prognostics and Health Monitoring Activity Recognition Sensor Fusion for Event Detection

4 Definition of Probabilistic Finite State Automaton (PFSA) as a generative model Each state is a collection of symbols Collection of memory words once memory (depth) is estimated, e.g., Q={11,12,21,22} with A={1,2} and D=2 Deterministic algebraic structure Sufficient Statistic for the Markov model Advantages 1. Simple to infer model structure 2. Simple to estimate model parameters like symbol emission probabilities (as compared to using Dynamic programming in Hidden Markov Models)

5 Order Estimation for Markov Modeling Definition: Let be the observed symbol sequences where each. Then, the order (or depth) of the process generating is defined as the length such that : Most techniques follow a wrapper like search approach Build Model Find Performance Select the best one. Log-Likelihood based Order Estimation Signal reconstruction-based : Make models in symbolic domain Make predictions in symbolic space Generate predictions in the continuous domain Pick the best one Prediction by a model Euclidean metric

6 Order Estimation for Markov Modeling Definition: Let be the observed symbol sequences where each. Then, the order (or depth) of the process generating is defined as the length such that : Most techniques follow a wrapper like search approach Build Model Find Performance Select the best one. Log-Likelihood based Order Estimation Signal reconstruction-based : Make models in symbolic domain Make predictions in symbolic space Generate predictions in the continuous domain Pick the best one Prediction by a model Entropy rate-based Euclidean metric

7 Order Estimation for Markov Modeling* Approximate estimate *D.K. Jha, A. Srivastav, K. Mukherjee and A. Ray: Depth Estimation in Markov models of Time-Series Data via Spectral Analysis, in American Control Conference, 2015 ** D.K. Jha, A. Srivastav and A. Ray: Depth Estimation in Markov models of Time-Series Data, in preparation

8 Reduced Order Markov Modeling: State Merging with deterministic structure? The deterministic algebraic structure of the finite-state probabilistic graph constraints the state merging process Bigger state space requires more data for convergence, slower performance (?) 1 (?) 1 (?) 22 02; 20 2 (?) 2, 0 (?) 01; , 0 02; 20; 00; Nondeterminism (?) Is that a problem? The targets must be merged together Targets may not be statistically similar; Leads to statistical modeling error?

9 Reduced Order Markov Modeling: State Merging with Nondeterministic Structure Remove the algebraic constraints and allow non-determinism Merge states based on statistical distance Set of symbols s 1 s 2 s N Dynamic Bayesian Network 1. Non-deterministic probabilistic graph 2. Stopping rule (e.g., allowed model distortion or number of states in the final model) 3. Parameter estimation using dynamic factored graphical model 1 1 (for example) 1 Nondeterministic Structure

10 A Primer to Combustion Instability Dynamics Thermo-acoustic Feedback Cycle Low NO x emission regulation Low equivalence ratio combustion Prone to instabilities VERY FAST REACTIVE DYNAMICS (~10 3 s) COMPLEX VELOCITY, THERMAL & ACOUSTIC COUPLING Velocity Fluctuation (Flow Dynamics), u Heat Release Rate Fluctuation (Combustion), q Pressure Fluctuation (Combustor acoustics), p Du Dt + 1 ρ 0 p = 0 NONLINEAR COUPLED DYNAMICS OVERALL MECHANISM OF THERMO-ACOUSTIC INSTABILITY IN COMBUSTOR

11 Schematic of the Combustor Apparatus at Penn State PENN STATE CENTER FOR COMBUSTION, POWER AND PROPULSION Prof. Dominic Santavicca and coworkers Test Apparatus for Methane Gas Combustion STABLE Approximate Empirical Density UNSTABLE

12 Combustion Instability Dynamics STABLE COMBUSTION UNSTABLE COMBUSTION

13 Pressure Data during Lean-Premixed Combustion: Modeling Coarse graining of data results in lots of self-loop Down-sampling is required Find the statistics for the model with D= Different Model structures and parameters in different operating regimes Reflects changes in the temporal model of data

14 Reduced Order Markov Modeling: Comparison and Further Insights Statistically similar states suggest Symbolic Noise States are significantly different Informative Markov chain K-L distance between symbol emissions from different states Stable Combustion Pressure Data Unstable Combustion Pressure Data K-L distance between Information Theoretic Metrics for Comparing Complexity conditional of Markov symbol Chains Maximum Cluster Divergence : emission vs. marginal symbol emission Discrepancy between i.i.d. and Markov Statistics (Information gain by Markov Models)

15 Anomaly Detection during Combustion: Results

16 Behavior in Information Space Combustion Instability Detection: Departure from Stability Detection of departure from stable behavior Stable Case Each point is a row of symbol emission matrix (with A =3) Reduced Model with 3 states Cluster of the stable behavior Receiver Operating Curves (ROC) for different Metrics and Model Parameters Unstable Case Each point is a row of symbol emission matrix (with A =3) Reduced Model with 3 states Unstable behavior sticks to the edges of the simplex

17 Combustion Instability Detection: Detection of Unstable Phase Reduced Model with Two States on Simplex Plane Unstable Behavior Detection of Unstable Behavior Different from detection of departure from stable Train Gaussian Process (GP) to learn the manifold Each point is a row of symbol emission matrix (with A =3) Reduced Model with 2 states Stable Behavior Estimate Likelihoods for Stable and Unstable Instability Detection by the GP model Reduced Order Models perform equally good Reduced Order Model have lesser number of parameters to estimate Faster during test

18 Concluding Remarks Conclusions Compact representation for Markov models of time-series data by state aggregation Spectral analysis provides a computationally efficient technique for memory estimation State-aggregation using agglomerative clustering with symmetric K-L distance Final model is a non-deterministic finite state automata Experimental validation of the approach using pressure time-series from an unstable combustion process in a swirl-stabilized combustor Comparable performance of reduced-order models observed Future Research Use of ideas from Information theory like minimum description length (MDL) for model selection to terminate state aggregation Simultaneous search of the associated hyper-parameters for symbolic dynamics-based Markov modeling (partitions and order)

19 Thank You