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

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Presentation transcript:

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

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 )

Particle Filter: Dynamics Prediction of next state:

Particle 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 …

Particle

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

HMMs, more classical application