# 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]:

#twodbn=gum.BayesNet()
#                   for s in ["a0","b0","c0","at","bt","ct"]]
#       for s in ["d0","dt"]]

#twodbn.generateCPTs()
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.fillWith([1,0,0,0.1]*9).normalizeAsCPT() # 36 valeurs normalized as CPT

Out[8]:

ct
c0
0
1
2
3
4
5
0
0.47620.00000.00000.04760.47620.0000
1
0.00000.08330.83330.00000.00000.0833
2
0.47620.00000.00000.04760.47620.0000
3
0.00000.08330.83330.00000.00000.0833
4
0.47620.00000.00000.04760.47620.0000
5
0.00000.08330.83330.00000.00000.0833
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})

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