Processing Sequential Sensor Data

The “John Krumm perspective”Thomas Plötz

November 29th, 2011

Sequential Data?

Sequential Data!

Sequential Data Analysis – Challenges

• Segmentation vs. Classification“chicken and egg” problem

• Noise, noise, and noise …• … more noise

• [Evaluation – “Ground Truth”?]

Noise …

filtering trivial (technically)

- lag

- no higher level variables (speed)

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

Kalman Filter

state and observations:

Explicit consideration of noise:

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:

Kalman Filter – Parameters

Kalman Filter @work

• Two-step procedure for every zi

• Result: mean and covariance of xi

Generalization: Particle Filter

• No linearity assumption, no Gaussian noise• Sequence of unknown state vectors xi, and

measurement vectors zi

• Probabilistic model for measurements, e.g. (!):

• … and for dynamics:

PF samples from it, i.e., generates xi subject to p(xi | xi-1)

Particle Filter: DynamicsPrediction of next state:

Particle Filter @workGenerate random xi from p(xi | xi-1)

Sample new set of particles based on importance weights – filtering

Original goal …

Particle Filter @work

Hidden Markov Models

• Kalman Filter not very accurate• Particle Filter computationally demanding• HMMs somewhat in-between

HMMs

• Measurement model: conditional probability

• Dynamic model: limited memory; transition probabilities

€

p(zi | xi )

HMMs, more classical application