Dynamic Bayesian Networks¶

In [1]:

import pyAgrum as gum
import pyAgrum.lib.notebook as gnb
import pyAgrum.lib.dynamicBN as gdyn

Building a 2TBN¶

Note the naming convention for a 2TBN : a variable with a name $A$ is present at t=0 with the name $A0$ and at time t as $At$.

In [2]:

# hard coded BN
#twodbn=gum.BayesNet()
#a0,b0,c0,at,bt,ct=[twodbn.add(gum.LabelizedVariable(s,s,6)) 
#                   for s in ["a0","b0","c0","at","bt","ct"]]
#d0,dt=[twodbn.add(gum.LabelizedVariable(s,s,3)) 
#       for s in ["d0","dt"]]

#twodbn.addArc(a0,b0)

#twodbn.addArc(c0,d0)
#twodbn.addArc(c0,at)

#twodbn.addArc(a0,at)
#twodbn.addArc(a0,bt)
#twodbn.addArc(a0,dt)
#twodbn.addArc(b0,bt)
#twodbn.addArc(c0,ct)
#twodbn.addArc(d0,ct)
#twodbn.addArc(d0,dt)

#twodbn.addArc(at,ct)
#twodbn.generateCPTs()

# fast BN version

twodbn=gum.fastBN("d0[3]->ct<-at<-a0->b0->bt<-a0->dt[3]<-d0<-c0->ct;c0->at",6)
twodbn

Out[2]:

2TBN¶

The dbn above actually is a 2TBN and is not correctly shown as a BN. Using the naming convention, it can be shown as a 2TBN.

In [3]:

gdyn.showTimeSlices(twodbn)

unrolling 2TBN¶

A dBN is 'unrolled' using the 2TBN and the time period size. For a couple $a_0$,$a_t$ in the 2TBN, the unrolled dBN will include $a_0, a_1, \cdots, a_{T-1}$

In [4]:

T=5

dbn=gdyn.unroll2TBN(twodbn,T)
gdyn.showTimeSlices(dbn,size="10")

We can infer on bn just as on a normal bn. Following the naming convention in 2TBN, the variables in a dbN are named using the convention $a_i$ where $i$ is the number of their time slice.

In [5]:

gnb.flow.clear()
for i in range(T):
    gnb.flow.add_html(gnb.getPosterior(dbn,target="d{}".format(i),evs={}),"$P(d{})$".format(i))
gnb.flow.display()

$P(d0)$

$P(d1)$

$P(d2)$

$P(d3)$

$P(d4)$

dynamic inference : following variables¶

gdyn.plotFollow directly ask for the 2TBN, unroll it and add evidence evs. Then it shows the dynamic of variable $a$ for instance by plotting $a_0,a_1,\cdots,a_{T-1}$.

In [6]:

import matplotlib.pyplot as plt

plt.rcParams['figure.figsize'] = (10, 2)
gdyn.plotFollow(["a","b","c","d"],twodbn,T=51,evs={'a9':2,'a30':0,'c14':0,'b40':0,'c50':3})  

nsDBN (Non-Stationnary Dynamic Bayesian network)¶

In [7]:

T=15

dbn=gdyn.unroll2TBN(twodbn,T)
gdyn.showTimeSlices(dbn)

Non-stationnaty DBN allows to express that the dBN do not follow the same 2TBN during all steps. A unrolled dbn is a classical BayesNet and then can be changed as you want after unrolling.

In [8]:

# new P(ct|c0) 
pot=gum.Potential().add(twodbn.variableFromName("ct")).add(twodbn.variableFromName("c0"))  
pot.fillWith([1,0,0,0.1]*9).normalizeAsCPT() # 36 valeurs normalized as CPT

Out[8]:

	ct
c0	0	1	2	3	4	5
0	0.4762	0.0000	0.0000	0.0476	0.4762	0.0000
1	0.0000	0.0833	0.8333	0.0000	0.0000	0.0833
2	0.4762	0.0000	0.0000	0.0476	0.4762	0.0000
3	0.0000	0.0833	0.8333	0.0000	0.0000	0.0833
4	0.4762	0.0000	0.0000	0.0476	0.4762	0.0000
5	0.0000	0.0833	0.8333	0.0000	0.0000	0.0833

In [9]:

# from steps 5 to 10, $C_t$ only depends on $C_{t-1}$ and follows this new CPT
for i in range(5,11):
    dbn.eraseArc(f"d{i-1}",f"c{i}")
    dbn.eraseArc(f"a{i}",f"c{i}")
    dbn.cpt(f"c{i}").fillWith(pot,["ct","c0"]) # ct in pot <- first var of cpt, c0 in pot<-second var in cpt
    
gdyn.showTimeSlices(dbn,size="14")

In [10]:

plt.rcParams['figure.figsize'] = (10, 2)
gdyn.plotFollowUnrolled(["a","b","c","d"],dbn,T=15,evs={'a9':2,'c14':0})  