Chapter 3 - Bayesian Networks in the
Presence of Temporal
Information
Univariate time series
A univariate time series {X(t)} is said to be second order or covariance stationary if the first two
moments, i.e., the mean E(X(t)) and covariance COV(X(t)), are invariant as a function of time
In other words, the first two moments of covariance-stationary time series are invariant over time
…
Multivariate time series
Multivariate time series are commonly modeled as vector auto-regressive (VAR) process
A VAR process is essentially a multivariate extension of an auto-regressive process
…
Lag orders
select suitable lag order via …
- AIC
- BIC
Assumptions in VAR
- multivariate normality
- normality of residuals
- absence of autocorrelation (???)
- heteroscedasticity
Dynamic Bayesian networks
Homogenous dynamic Bayesian networks
- the stochastic process is first-order Markovian
- values at time t of a variable are only dependant on the variables at t -1 (and not t -2, t -3, etc)
- Random values observed at time t are conditionally independent given the
random variables X(t −1) at the previous time t −1
- all information for a variable at a particular time point is in the immediate past
- The temporal profile of any variable Xi cannot be written as a linear combination
of the other profiles
- there are no edges going from one node to another in the same time bin
- The process is homogeneous over time: all arcs in the network and their
directions are invariant over time
- the phenomenon we are modeling is governed by the same set of rules during the whole
experiment
Learning algorithms
- repeated time measurements can be used to perform
linear regression
- only when n >> k
- regularized estimators required in most real world
applications
- LASSO (Least Absolute Shrinkage and Selection Operator)
- James–Stein Shrinkage
- First-Order Conditional Dependencies Approximation
- Modular Networks (SIMoNE)
Non-homogeneous dynamic Bayesian networks
- Homogeneity (Assumption 4 on slide 8) is a strong assumption
which may not be satisfied for real-world data
- ARTIVA (Auto-Regressive TIme VArying) model
Analysis in R
homogenous (non-time varying effects)
- if n >> k
- package::var
- else shrinkage methods
-LASSO
- package::lars
- package::simone (specifically targeted to Bayesian networks)
- other shrinkage methods
- package::GeneNet
- package::G1DBN
non-homogenous (time varying effects)
- package::ARTIVA
