Continuous-Time Bayesian Networks
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In [1]:
import pyAgrum as gum
import pyAgrum.lib.notebook as gnb
import pyAgrum.ctbn as ct
import pyAgrum.ctbn.notebook as ctbnnb
A CTBN(Continuous-Time Bayesian Networks) is a graphical model that allows a Bayesian Network to evolve over continuous time.
(For more details see https://pyagrum.readthedocs.io/en/latest/ctbn.html)
Creating a CTBN
In [2]:
# 1. Creating the ctbn
ctbn = ct.CTBN()
# 2. Adding the random variables
ctbn.add(gum.LabelizedVariable("cloudy?", "cloudy?"))
ctbn.add(gum.LabelizedVariable("rain?", "rain?"))
# 3. Adding the arcs
ctbn.addArc("cloudy?", "rain?")
ctbn.addArc("rain?", "cloudy?")
# 4. Filling the CIMS (Conditional Intensity Matrixes)
ctbn.CIM("cloudy?")[{"rain?": "0"}] = [[-.1, .1],
[1, -1]]
ctbn.CIM("cloudy?")[{"rain?": "1"}] = [[-100, 100],
[.5, -.5]]
ctbn.CIM("rain?")[{"cloudy?": "0"}] = [[-0.5, 0.5],
[1000, -1000]]
ctbn.CIM("rain?")[{"cloudy?": "1"}] = [[-2, 2],
[2, -2]]
# 5. Plotting the CTBN (whith/whithout the CIMs)
ctbnnb.showCtbnGraph(ctbn)
ctbnnb.showCIMs(ctbn)
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| -0.1000 | 0.1000 | |
| 1.0000 | -1.0000 | ||
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| -100.0000 | 100.0000 | |
| 0.5000 | -0.5000 | ||
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|---|---|---|---|
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| -0.5000 | 0.5000 | |
| 1000.0000 | -1000.0000 | ||
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| -2.0000 | 2.0000 | |
| 2.0000 | -2.0000 | ||
cloudy?#i gives the state cloudy? is currently in and cloudy?#j gives its next state.
The CIM object, built around a pyAgrum.Potential is used to store the parameters of the variables, describing the waiting time before switching to another state.
A useful tool for CIMs, mainly used for simple inference, is amalgamation. It allows to merge many CIMs into one.
In [3]:
cimA = ctbn.CIM("cloudy?")
cimB = ctbn.CIM("rain?")
amal = cimA.amalgamate(cimB) # or cimA * cimB
gnb.showPotential(amal.getPotential()) # as a Potential
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|---|---|---|---|---|
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| -0.6000 | 1.0000 | |
| 0.1000 | -3.0000 | |||
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| 0.5000 | 0.0000 | ||
| 0.0000 | 2.0000 | |||
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| 1000.0000 | 0.0000 | |
| 0.0000 | 2.0000 | |||
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| -1100.0000 | 0.5000 | ||
| 100.0000 | -2.5000 | |||
In [4]:
print("\nJoint intensity matrix as numpy array\n")
print(amal.toMatrix()) # as a numpy array
Joint intensity matrix as numpy array
[[-6.0e-01 1.0e-01 5.0e-01 0.0e+00]
[ 1.0e+00 -3.0e+00 0.0e+00 2.0e+00]
[ 1.0e+03 0.0e+00 -1.1e+03 1.0e+02]
[ 0.0e+00 2.0e+00 5.0e-01 -2.5e+00]]
A random variable of a ctbn can be identified by either its name or id.
In any case, most class functions parameters allow for a variable to be identified by either its name or id (type NameOrId).
In [5]:
labels_with_id = ctbn.labels(0)
labels_with_name = ctbn.labels("cloudy?")
print(f"labels using id : {labels_with_id}, using name : {labels_with_name}")
variable_with_id = ctbn.variable(0)
variable_with_name = ctbn.variable("cloudy?")
print(f"The variables 0 and 'cloudy?' are the same ? {variable_with_id==variable_with_name}")
labels using id : ('0', '1'), using name : ('0', '1')
The variables 0 and 'cloudy?' are the same ? True
All functions with NameOrId parameters work the same way
Random CTBN generator
The CtbnGenerator module can be used to quickly generate CTBNs. The parameters value are drawn at random from a value range. The diagonal coefficients are just the negative sum of the ones on the same line (in a CIM). It is also possible to choose the number of variables, the maximum number of parents a variable can have, and the domain size of the variables.
In [6]:
randomCtbn = ct.randomCTBN(valueRange=(1,10), n=8, parMax=2, modal=2)
ctbnnb.showCtbnGraph(randomCtbn)
ctbnnb.showCIMs(randomCtbn)
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| -4.3172 | 4.3172 | |
| 4.8171 | -4.8171 | |||
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| -8.4746 | 8.4746 | ||
| 2.6312 | -2.6312 | |||
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| -5.8418 | 5.8418 | |
| 4.7838 | -4.7838 | |||
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| -6.2430 | 6.2430 | ||
| 2.0341 | -2.0341 | |||
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|---|---|---|---|---|
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| -6.3878 | 6.3878 | |
| 8.9027 | -8.9027 | |||
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| -4.0741 | 4.0741 | ||
| 6.3407 | -6.3407 | |||
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| -9.5064 | 9.5064 | |
| 5.8454 | -5.8454 | |||
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| -8.6840 | 8.6840 | ||
| 6.5747 | -6.5747 | |||
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|---|---|---|---|---|
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| -7.4048 | 7.4048 | |
| 3.9423 | -3.9423 | |||
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| -8.0909 | 8.0909 | ||
| 6.0274 | -6.0274 | |||
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| -6.4239 | 6.4239 | |
| 3.9850 | -3.9850 | |||
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| -8.0607 | 8.0607 | ||
| 1.6419 | -1.6419 | |||
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|---|---|---|---|---|
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| -1.1342 | 1.1342 | |
| 3.7340 | -3.7340 | |||
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| -8.8092 | 8.8092 | ||
| 2.7529 | -2.7529 | |||
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| -9.4600 | 9.4600 | |
| 5.6839 | -5.6839 | |||
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| -6.7917 | 6.7917 | ||
| 9.8443 | -9.8443 | |||
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|---|---|---|---|---|
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| -3.3046 | 3.3046 | |
| 8.3787 | -8.3787 | |||
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| -4.0924 | 4.0924 | ||
| 7.1730 | -7.1730 | |||
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| -5.9062 | 5.9062 | |
| 5.7211 | -5.7211 | |||
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| -1.3430 | 1.3430 | ||
| 1.8059 | -1.8059 | |||
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|---|---|---|---|---|
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| -3.7614 | 3.7614 | |
| 2.4733 | -2.4733 | |||
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| -1.8782 | 1.8782 | ||
| 4.0112 | -4.0112 | |||
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| -7.5913 | 7.5913 | |
| 4.7263 | -4.7263 | |||
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| -8.1558 | 8.1558 | ||
| 7.8676 | -7.8676 | |||
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|---|---|---|
| -8.4816 | 8.4816 | |
| 7.6979 | -7.6979 | |
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|---|---|---|---|---|
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| -5.7592 | 5.7592 | |
| 9.6744 | -9.6744 | |||
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| -8.7535 | 8.7535 | ||
| 8.5555 | -8.5555 | |||
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| -2.2668 | 2.2668 | |
| 2.5436 | -2.5436 | |||
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| -8.9372 | 8.9372 | ||
| 5.2977 | -5.2977 | |||
Inference with CTBNs
Exact inference & Forward Sampling inference
Amalgamation is used to compute exact inference. It consists in merging all the CIMs of a CTBN into one CIM. Then we use the properties of markov process : \(P(X_t) = P(X_0)*exp(Q_Xt)\) for enough time to notice convergence.
Forward sampling works this way:
At time \(t\) :
Draw a “time until next change” \(dt\) value for each variable’s exponential distribution of their waiting time.
Select the variable with smallest transition time : \(next time = t + dt\).
Draw the variable next state uniformly.
loop until \(t = timeHorizon\).
When doing forward sampling, we first iterate this process over a given amount of iterations (called burnIn). We consider the last values as the initial configuration for sampling. The reason is that the initial state of a CTBN is unknown without real data.
In [7]:
# 1. Initialiaze Inference object
ie = ct.SimpleInference(ctbn)
ie_forward = ct.ForwardSamplingInference(ctbn)
gnb.sideBySide(ctbnnb.getCtbnGraph(ctbn), ctbnnb.getCtbnInference(ctbn, ie), ctbnnb.getCtbnInference(ctbn, ie_forward))
In any case, inference can be made using makeInference() function from every Inference object.
In [8]:
import time
# Simple inference
time1 = time.time()
ie.makeInference(t=5000)
pot1 = ie.posterior("cloudy?")
elapsed1 = time.time() - time1
# Sampling inference
time2 = time.time()
ie_forward.makeInference(timeHorizon=5000, burnIn=100)
pot2 = ie_forward.posterior("cloudy?")
elapsed2 = time.time() - time2
gnb.sideBySide(gnb.getPotential(pot1), gnb.getPotential(pot2), captions=[f"Simple inference {round(elapsed1*1000,2)} ms", f"Sampling inference {round(elapsed2*1000, 2)} ms"])
Plotting a sample
Here is an example of a trajectory plot
In [9]:
# 1_1. Find trajectory stored in the inference class
ie_forward.makeInference()
traj = ie_forward.trajectory # a single trajectory List[Tuple[float, str, str]]
# 1_2. Or find it in a csv file
ie_forward.writeTrajectoryCSV('out/trajectory.csv', n=1, timeHorizon=1000, burnIn=100) # to save a trajectory
traj = ct.readTrajectoryCSV('out/trajectory.csv') # a dict of trajectories
# 2. Plot the trajectory
ct.plotTrajectory(ctbn.variable("cloudy?"), traj[0], timeHorizon=40, plotname="plot of cloudy?")
It is also possible to plot the proportions of the states a variable is in over time.
Let’s take the exemple of a variable with domain size=5.
In [10]:
# 1. Create CTBN
new_ctbn = ct.CTBN()
X = gum.LabelizedVariable("X", "X", ["a1", "a2", "a3", "a4", "a5"])
new_ctbn.add(X)
new_ctbn.CIM("X")[:] = [[-10, 2, 4, 0, 4],
[0.5 , -1, 0.5, 0, 0],
[5, 0.5, -10, 3.5, 1],
[50, 0.5, 50, -200.5, 100],
[1, 0, 2, 7, -10]]
# 2. Sampling
ieX = ct.ForwardSamplingInference(new_ctbn)
ieX.writeTrajectoryCSV("out/traj_quinary.csv", n=3, timeHorizon=100)
trajX = ct.readTrajectoryCSV("out/traj_quinary.csv")
# 3. Plotting
ct.plotFollowVar(X, trajX)
Learning a CTBN
The ctbn library has tools to learn the graph of a CTBN as well as the CIMs of its variables
Learning the variables
From a sample, we can find the variables name and their labels. However, if the sample is too short, it is possible that some information may be missed.
In [11]:
newCtbn = ct.CTBNFromData(traj)
print("variables found and their labels")
for name in newCtbn.names():
print(f"variable {name}, labels : {newCtbn.labels(name)}")
variables found and their labels
variable cloudy?, labels : ('0', '1')
variable rain?, labels : ('0', '1')
Learning the graph
In [12]:
# 1. Create a Learner.
learner = ct.Learner('out/trajectory.csv')
# 2. Run learning algorithm
learned_ctbn = learner.learnCTBN()
# 3. Plot the learned CTBN
gnb.show(learned_ctbn)
Let’s note that having a test object is necessary since we want to be able to test independency between variables. For now only 2 tests are available : (Fisher and Chi2) test and Oracle test. Oracle test is only used for benchmarking since it uses the initial ctbn as a reference.
Fitting parameters
It is also possible to approximate the transition time distributions of each variable, i.e the CIMs.
Note that conditioning variables are taken from the ctbn.
In [13]:
cims_before = ctbnnb.getCIMs(ctbn)
learner.fitParameters(ctbn)
cims_after = ctbnnb.getCIMs(ctbn)
gnb.sideBySide(cims_before, cims_after, captions=["original CIMs", "learned CIMs"])
Example from [Nodelman et al, 2003]
This example comes from :
U. Nodelman, C.R. Shelton, and D. Koller (2003). "Learning Continuous Time Bayesian Networks." Proc. Nineteenth Conference on Uncertainty in Artificial Intelligence (UAI) (pp. 451-458).
In [14]:
# 1. Creating the ctbn
drugEffect = ct.CTBN()
drugEffect.add(gum.LabelizedVariable("Eating", ""))
drugEffect.add(gum.LabelizedVariable("Hungry", ""))
drugEffect.add(gum.LabelizedVariable("Full stomach", ""))
drugEffect.add(gum.LabelizedVariable("Uptake", ""))
drugEffect.add(gum.LabelizedVariable("Concentration", "?"))
drugEffect.add(gum.LabelizedVariable("Barometer", ""))
drugEffect.add(gum.LabelizedVariable("Joint pain", ""))
drugEffect.add(gum.LabelizedVariable("Drowsy", ""))
drugEffect.addArc("Hungry","Eating")
drugEffect.addArc("Eating","Full stomach")
drugEffect.addArc("Full stomach","Hungry")
drugEffect.addArc("Full stomach","Concentration")
drugEffect.addArc("Uptake","Concentration")
drugEffect.addArc("Concentration","Drowsy")
drugEffect.addArc("Barometer","Joint pain")
drugEffect.addArc("Concentration","Joint pain")
gnb.show(drugEffect)
In [15]:
from pyAgrum.ctbn.CTBNGenerator import randomCIMs
randomCIMs(drugEffect,(0.1,10))
drugEffect.CIM("Eating")[{"Hungry":0}]=[[-0.1,0.1],
[10,-10]]
drugEffect.CIM("Eating")[{"Hungry":1}]=[[-2,2],
[0.1,-0.1]]
drugEffect.CIM("Full stomach")[{"Eating":0}]=[[-0.1,0.1],
[10,-10]]
drugEffect.CIM("Full stomach")[{"Eating":1}]=[[-2,2],
[0.1,-0.1]]
drugEffect.CIM("Hungry")[{"Full stomach":0}]=[[-0.1,0.1],
[10,-10]]
drugEffect.CIM("Hungry")[{"Full stomach":1}]=[[-2,2],
[0.1,-0.1]]
ieX = ct.ForwardSamplingInference(drugEffect)
ieX.writeTrajectoryCSV("out/drugEffect.csv", n=3, timeHorizon=100)
trajX = ct.readTrajectoryCSV("out/drugEffect.csv")
ct.plotFollowVar(drugEffect.variable("Hungry"), trajX)
ct.plotFollowVar(drugEffect.variable("Full stomach"), trajX)
ct.plotFollowVar(drugEffect.variable("Eating"), trajX)
ct.plotFollowVar(drugEffect.variable("Concentration"), trajX)
Input/output with pickle
In [16]:
import pickle
with open("test.pkl","bw") as f:
pickle.dump(drugEffect,f)
drugEffect
Out[16]:
In [17]:
with open("test.pkl","br") as f:
copyDrug=pickle.load(f)
copyDrug
Out[17]:
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