Dynamic Bayesian Networks¶
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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 $At$.
# 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
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.
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 $a_0, a_1, \cdots, a_{T-1}$
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.
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()
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 $a_0,a_1,\cdots,a_{T-1}$.
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)¶
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.
##### new P(ct|c0)
pot=gum.Potential().add(twodbn["ct"]).add(twodbn["c0"])
pot.fillWith([1,0,0,0.1]*9).normalizeAsCPT() # 36 valeurs normalized as CPT
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| 0.4762 | 0.0000 | 0.0000 | 0.0476 | 0.4762 | 0.0000 | |
| 0.0000 | 0.0833 | 0.8333 | 0.0000 | 0.0000 | 0.0833 | |
| 0.4762 | 0.0000 | 0.0000 | 0.0476 | 0.4762 | 0.0000 | |
| 0.0000 | 0.0833 | 0.8333 | 0.0000 | 0.0000 | 0.0833 | |
| 0.4762 | 0.0000 | 0.0000 | 0.0476 | 0.4762 | 0.0000 | |
| 0.0000 | 0.0833 | 0.8333 | 0.0000 | 0.0000 | 0.0833 | |
# 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")
plt.rcParams['figure.figsize'] = (10, 2)
gdyn.plotFollowUnrolled(["a","b","c","d"],dbn,T=15,evs={'a9':2,'c14':0})
