1. Processing Sequential Sensor Data The “John Krumm perspective” Thomas Plötz November 29 th, 2011

2. Sequential Data?

3. Sequential Data!

5. Sequential Data Analysis – Challenges Segmentation vs. Classification “chicken and egg” problem Noise, noise, and noise … … more noise  [Evaluation – “Ground Truth”?]

6. Noise … filtering trivial (technically) - lag - no higher level variables (speed)

7. States vs. Direct Observations Idea: Assume (internal) state of the “system” Approach: Infer this very state by exploiting measurements / observations Examples: – Kalman Filter – Particle Filter – Hidden Markov Models

8. Kalman Filter state and observations: Explicit consideration of noise:

9. Kalman Filter – Linear Dynamics State at time i: linear function of state at time i-1 plus noise: System matrix describes linear relationship between i and i-1:

10. Kalman Filter – Parameters

11. Kalman Filter @work Two-step procedure for every z i Result: mean and covariance of x i Step 1: extrapolate state and state error from previous estimates Step 2: update extrapolations with new measurement

13. Generalization: Particle Filter No linearity assumption, no Gaussian noise Sequence of unknown state vectors x i, and measurement vectors z i Probabilistic model for measurements, e.g. (!): … and for dynamics:  PF samples from it, i.e., generates x i subject to p(x i | x i-1 )

14. Particle Filter: Dynamics Prediction of next state:

15. Particle Filter @work Importance sampling Compute importance weights Selection Compute estimate of xi at any point Generate random x i from p(x i | x i-1 ) Sample new set of particles based on importance weights – filtering Original goal …

16. Particle Filter @work

18. Hidden Markov Models Kalman Filter not very accurate Particle Filter computationally demanding HMMs somewhat in-between

19. HMMs Measurement model: conditional probability Dynamic model: limited memory; transition probabilities

21. HMMs, more classical application