Sampling and Parameter Fitting with Hawkes Processes
H
UMAN
-
CENTERED
M
ACHINE
L
EARNING
Sampling and parameter fitting with
Hawkes Processes
http://courses.mpi-sws.org/hcml-ws18/
2
Recap: How to fit and why sample?
Raw Data
Infer parameters
(Learning)
Parametrize intensity
Derive Log-likelihood:
Maximum likelihood estimation:
Sampling
(Predicting)
o
Prediction
o
Model Checking
o
Sanity check
o
Gaining Intuition
o
S
i
m
u
l
a
t
o
r
o
Summary statistics
Event times drawn from:
Helps with:
We will first
sample
and then
fit
.
3
Recap: What are Hawkes processes?
time
Intensity of self-exciting
(or Hawkes) process:
Observations:
1.
Clustered (or bursty) occurrence of events
2.
Intensity is stochastic and history dependent
History,
Recap: How to fit a Hawkes process?
time
Maximum
likelihood
The max. likelihood
is
jointly
convex
in and
(use CVX!)
5
Recap: How to sample from a Hawkes process
time
Thinning procedure (similar to rejection sampling):
1.
Sample from Poisson process with intensity
3.
Keep the sample if
Inversion
sampling
2.
Generate
Keep sample with
prob.
6
Coding assignment overview
Sampler
(Assignment 1)
Parameter fitting
(Assignment 2)
Sanity check
(provided)
7
Sampling: Ogata’s algorithm
O
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Drawing 1 sample with intensity
Accepting it with prob
8
Sampling: Achtung I
O
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Sampling: Achtung II
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History up to
but not including
t.
10
Sampling: Achtung III
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is not an actual sample!
11
Sampling: Evaluation
python plot_hawkes.py 1.0 0.5 1.0 10 sampled-sequences.txt output-plot.png
Juxtaposed events
from sequences
Theoretical value
Empirical average
12
Sampling: Evaluation
python plot_hawkes.py 1.0 0.5 1.0 10 sampled-sequences.txt output-plot.png
Juxtaposed events
from sequences
Theoretical value
Empirical average
Poisson (incorrect)
sampler
13
Sampling: Evaluation
python plot_hawkes.py 1.0 0.5 1.0 10 sampled-sequences.txt output-plot.png
Juxtaposed events
from sequences
Theoretical value
Empirical average
Deterministic
(incorrect) sampler
14
Live coding
Show baseline + sanity check
Show Poisson + sanity check
15
Parameter fitting: Problem setting
Parameter fitting
(Assignment 2)
Sampler
(Assignment 1)
samples.txt
samples-test.txt
16
Parameter fitting: Method
Nested sum
Maximum likelihood estimation:
Can we do it faster for
exponential kernels?
17
Parameter fitting: Using
cvxpy
18
Parameter fitting: Using
cvxpy
Change this to maximize the likelihood
19
Live coding
Show baseline + desired output
Evaluation via sampling
20
Happy coding and holidays!
Drop me an e-mail at
utkarshu@mpi-sws.org
Skype:
utkarsh.upadhyay
Q
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s
?
Learn about sampling and parameter fitting with Hawkes processes in the context of human-centered machine learning. Understand the importance of fitting parameters and sampling raw data event times. Explore the characteristics and fitting methods of Hawkes processes, along with coding assignments and sampling algorithms.
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Presentation Transcript
Sampling and parameter fitting with Hawkes Processes HUMAN-CENTERED MACHINE LEARNING http://courses.mpi-sws.org/hcml-ws18/
Recap: How to fit and why sample? Infer parameters (Learning) Sampling (Predicting) Raw Data Event times drawn from: Parametrize intensity Derive Log-likelihood: Helps with: Prediction o Model Checking o Sanity check o Gaining Intuition o Maximum likelihood estimation: Simulator o Summary statistics o We will first sample and then fit. 2
Recap: What are Hawkes processes? time History, Intensity of self-exciting (or Hawkes) process: Observations: 1. Clustered (or bursty) occurrence of events 2. Intensity is stochastic and history dependent 3
Recap: How to fit a Hawkes process? time Maximum likelihood The max. likelihood is jointly convex in and
Recap: How to sample from a Hawkes process time Thinning procedure (similar to rejection sampling): 1. Sample from Poisson process with intensity Inversion sampling 2. Generate Keep sample with prob. 3. Keep the sample if 5
Coding assignment overview Parameter fitting (Assignment 2) Sampler (Assignment 1) Sanity check (provided) 6
Sampling: Ogatas algorithm Ogata s method of thinning Drawing 1 sample with intensity Accepting it with prob 7
Sampling: Achtung I Ogata s method of thinning Careful about exploding intensities! Include a check on i. Ensure 8
Sampling: Achtung II Ogata s method of thinning History up to but not including t. 9
Sampling: Achtung III Ogata s method of thinning is not an actual sample! 10
Sampling: Evaluation python plot_hawkes.py 1.0 0.5 1.0 10 sampled-sequences.txt output-plot.png Theoretical value Empirical average Juxtaposed events from sequences 11
Sampling: Evaluation python plot_hawkes.py 1.0 0.5 1.0 10 sampled-sequences.txt output-plot.png Theoretical value Empirical average Poisson (incorrect) sampler Juxtaposed events from sequences 12
Sampling: Evaluation python plot_hawkes.py 1.0 0.5 1.0 10 sampled-sequences.txt output-plot.png Theoretical value Empirical average Deterministic (incorrect) sampler Juxtaposed events from sequences 13
Live coding Show baseline + sanity check Show Poisson + sanity check 14
Parameter fitting: Problem setting Parameter fitting (Assignment 2) Sampler (Assignment 1) samples.txt samples-test.txt 15
Parameter fitting: Method Maximum likelihood estimation: Nested sum Can we do it faster for exponential kernels? 16
Parameter fitting: Using cvxpy Change this to maximize the likelihood 18
Live coding Show baseline + desired output Evaluation via sampling 19
Happy coding and holidays! Questions? Drop me an e-mail at utkarshu@mpi-sws.org Skype: utkarsh.upadhyay 20
