Human-centered Machine Learning

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Human-centered Machine Learning Introduction to Temporal Point Processes (II) Human-centered Machine Learning http://courses.mpi-sws.org/hcml-ws18/

Temporal Point Processes: Basic building blocks Outline of tutorial

Poisson process Intensity of a Poisson process Observations: time Intensity of a Poisson process Observations: Intensity independent of history Uniformly random occurrence Time interval follows exponential distribution

Fitting a Poisson from (historical) timeline Maximum likelihood

Sampling from a Poisson process time We would like to sample: We sample using inversion sampling:

Inhomogeneous Poisson process time Intensity of an inhomogeneous Poisson process Observations: Intensity independent of history

Fitting an inhomogeneous Poisson time Maximum likelihood Design such that max. likelihood is convex (and use CVX)

Nonparametric inhomogeneous Poisson process Positive combination of (Gaussian) RFB kernels:

Sampling from an inhomogeneous Poisson time Thinning procedure (similar to rejection sampling): Sample from Poisson process with intensity Inversion sampling Generate Keep sample with prob. Keep the sample if

Terminating (or survival) process time Intensity of a terminating (or survival) process Observations: Limited number of occurrences

Self-exciting (or Hawkes) process time History, Triggering kernel Intensity of self-exciting (or Hawkes) process: Observations: Clustered (or bursty) occurrence of events Intensity is stochastic and history dependent

Fitting a Hawkes process from a recorded timeline Maximum likelihood The max. likelihood is jointly convex in and (use CVX!)

Sampling from a Hawkes process time Thinning procedure (similar to rejection sampling): Sample from Poisson process with intensity Inversion sampling Generate Keep sample with prob. Keep the sample if

We know how to fit them and how to sample from them Summary Building blocks to represent different dynamic processes: Poisson processes: We know how to fit them and how to sample from them Inhomogeneous Poisson processes: Terminating point processes: Self-exciting point processes:

Temporal Point Processes: Superposition Outline of tutorial

Superposition of processes time Sample each intensity + take minimum = Additive intensity

Mutually exciting process time Bob History Christine time History Clustered occurrence affected by neighbors

Mutually exciting terminating process time Bob Christine time History Clustered occurrence affected by neighbors