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
Step 1: extrapolate state
and state error from
previous estimates
Step 2: update
extrapolations with new
measurement
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: Dynamics
Prediction of next state:
Particle Filter @work
Generate random xi
from p(xi | xi-1)
Importance
sampling
Original goal …
Compute estimate
of xi at any point
Selection
Compute
importance
weights
Sample new set of
particles based on
importance weights –
filtering
Particle Filter @work
Hidden Markov Models
• Kalman Filter not very accurate
• Particle Filter computationally demanding
• HMMs somewhat in-between
HMMs
• Measurement model: conditional probability
p(zi | xi )
• Dynamic model: limited memory; transition
probabilities

HMMs, more classical application