Conditional Linear Gaussian models
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In [1]:
import pyAgrum as gum
import pyAgrum.lib.notebook as gnb
import pyAgrum.lib.bn_vs_bn as gcm
import pyAgrum.clg as gclg
import pyAgrum.clg.notebook as gclgnb
Build a CLG model
From scratch
Suppose we want to build a CLG with these specifications \(A={\cal N}(5,1)\), \(B={\cal N}(4,3)\) and \(C=2.A+3.B+{\cal N}(3,2)\)
In [2]:
model=gclg.CLG()
model.add(gclg.GaussianVariable("A",5,1))
model.add(gclg.GaussianVariable("C",3,2))
model.add(gclg.GaussianVariable("B",4,3))
model.addArc("A","C",2)
model.addArc("B","C",3)
model
Out[2]:
From SEM (Structural Equation Model)
We can create a Conditional Linear Gaussian Bayesian networ(CLG model) using a SEM-like syntax.
A = 4.5 [0.3] means that the mean of the distribution for Gaussian random variable A is 4.5 and ist standard deviation is 0.3.
B = 3 + 0.8F [0.3] means that the mean of the distribution for the Gaussian random variable B is 3 and the standard deviation is 0.3.
pyAgrum.CLG.SEM is a set of static methods to manipulate this kind of SEM.
In [3]:
sem2="""
A=4.5 [0.3] # comments are allowed
F=7 [0.5]
B=3 + 1.2F [0.3]
C=9 + 2A + 1.5B [0.6]
D=9 + C + F[0.7]
E=9 + D [0.9]
"""
model2 = gclg.SEM.toclg(sem2)
In [4]:
gnb.show(model2)
One can of course build the SEM from a CLG using pyAgrum.CLG.SEM.tosem :
In [5]:
gnb.flow.row(model,"<pre><div align='left'>"+gclg.SEM.tosem(model)+"</div></pre>",
captions=["the first CLG model","the SEM from the CLG"])
B=4[3] A=5[1] C=3+2A+3B[2]
And this SEM allows of course input/output format for CLG
In [6]:
gclg.SEM.saveCLG(model2,"out/model2.sem")
print("=== file content ===")
with open("out/model2.sem","r") as file:
for line in file.readlines():
print(line,end="")
print("====================")
=== file content ===
F=7.0[0.5]
B=3.0+1.2F[0.3]
A=4.5[0.3]
C=9.0+2.0A+1.5B[0.6]
D=9.0+F+C[0.7]
E=9.0+D[0.9]
====================
In [7]:
model3=gclg.SEM.loadCLG("out/model2.sem")
gnb.sideBySide(model2,model3,captions=["saved model","loaded model"])
input/output with pickle
In [8]:
import pickle
with open("test.pkl","bw") as f:
pickle.dump(model3,f)
model3
Out[8]:
In [9]:
with open("test.pkl","br") as f:
copyModel3=pickle.load(f)
copyModel3
Out[9]:
Exact or approximated Inference
Exact inference : Variable Elimination
Compute some posterior using difference exact inference
In [10]:
ie=gclg.CLGVariableElimination(model2)
ie.updateEvidence({"D":3})
print(ie.posterior("A"))
print(ie.posterior("B"))
print(ie.posterior("C"))
print(ie.posterior("D"))
print(ie.posterior("E"))
print(ie.posterior("F"))
v=ie.posterior("E")
print(v)
print(f" - mean(E|D=3)={v.mu()}")
print(f" - stdev(E|D=3)={v.sigma()}")
A:1.9327650111193468[0.28353638852446156]
B:-2.5058561897702[0.41002992170553515]
C:3.9722757598220895[0.5657771474513671]
D:3[0]
E:12.0[0.9]
F:-2.9836916234247597[0.32358490464094586]
E:12.0[0.9]
- mean(E|D=3)=12.0
- stdev(E|D=3)=0.9
In [11]:
gnb.sideBySide(model2,gclgnb.getInference(model2,evs={"D":3},size="3!"),gclgnb.getInference(model2,evs={"D":3,"F":1}),
captions=["The CLG","First inference","Second inference"])
Approximated inference : MonteCarlo Sampling
When the model is too complex for exact infernece, we can use forward sampling to generate 5000 samples from the original CLG model.
In [12]:
fs = gclg.ForwardSampling(model2)
fs.makeSample(5000).tocsv("./out/model2.csv")
We will use the generated database to do learning. But before, we can also compute posterior but without evidence :
In [13]:
ie=gclg.CLGVariableElimination(model2)
print("| 'Exact' inference | Results from sampling |")
print("|------------------------------------------|------------------------------------------|")
for i in model2.names():
print(f"| {str(ie.posterior(i)):40} | {str(gclg.GaussianVariable(i,fs.mean_sample(i),fs.stddev_sample(i))):40} |")
| 'Exact' inference | Results from sampling |
|------------------------------------------|------------------------------------------|
| A:4.499999999999998[0.3] | A:4.502363024169802[0.3036745541350688] |
| F:7.000000000000008[0.5000000000000002] | F:6.996457547190566[0.5047526239929775] |
| B:11.399999999999999[0.6708203932499367] | B:11.395468084089094[0.6767485530521845] |
| C:35.099999999999994[1.3162446581088183] | C:35.11191960486344[1.3237822638375893] |
| D:51.10000000000002[1.8364367672206963] | D:51.10560364380086[1.8559870614893375] |
| E:60.100000000000016[2.0451161336217565] | E:60.095007796503175[2.0400481268440025] |
Now with the generated database and the original model, we can calculate the log-likelihood of the model.
In [14]:
print("log-likelihood w.r.t orignal model : ", model2.logLikelihood("./out/model2.csv"))
log-likelihood w.r.t orignal model : -22179.640208993103
Learning a CLG from data
Use the generated database to do our RAvel Learning. This part needs some time to run.
In [15]:
# RAveL learning
learner = gclg.CLGLearner("./out/model2.csv")
We can get the learned_clg model with function learn_clg() which contains structure learning and parameter estimation.
In [16]:
learned_clg = learner.learnCLG()
gnb.sideBySide(model2,learned_clg,
captions=['original CLG','learned CLG'])
Compare the learned model’s structure with that of the original model’.
In [17]:
cmp=gcm.GraphicalBNComparator(model2,learned_clg)
print(f"F-score(original_clg,learned_clg) : {cmp.scores()['fscore']}")
F-score(original_clg,learned_clg) : 0.6666666666666666
Get the learned model’s parameters and compare them with the original model’s parameters using the SEM syntax.
In [18]:
gnb.flow.row("<pre><div align='left'>"+gclg.SEM.tosem(model2)+"</div></pre>",
"<pre><div align='left'>"+gclg.SEM.tosem(learned_clg)+"</div></pre>",
captions=["original sem","learned sem"])
F=7.0[0.5] B=3.0+1.2F[0.3] A=4.5[0.3] C=9.0+2.0A+1.5B[0.6] D=9.0+F+C[0.7] E=9.0+D[0.9]
E=60.095[2.04] F=6.996[0.505] B=2.966+1.2F[0.297] D=5.248+1.02F+0.64E[0.72] A=4.502[0.304] C=2.125+1.19A+0.64B+0.4D[0.461]
We can algo do parameter estimation only with function fitParameters() if we already have the structure of the model.
In [19]:
# We can copy the original CLG
copy_original = gclg.CLG(model2)
# RAveL learning again
RAveL_l = gclg.CLGLearner("./out/model2.csv")
# Fit the parameters of the copy clg
RAveL_l.fitParameters(copy_original)
copy_original
Out[19]:
Compare two CLG models
We first create two CLG from two SEMs.
In [20]:
# TWO DIFFERENT CLGs
# FIRST CLG
clg1=gclg.SEM.toclg("""
# hyper parameters
A=4[1]
B=3[5]
C=-2[5]
#equations
D=A[.2] # D is a noisy version of A
E=1+D+2B [2]
F=E+C+B+E [0.001]
""")
# SECOND CLG
clg2=gclg.SEM.toclg("""
# hyper parameters
A=4[1]
B=3+A[5]
C=-2+2B+A[5]
#equations
D=A[.2] # D is a noisy version of A
E=1+D+2B [2]
F=E+C [0.001]
""")
This cell shows how to have a quick view of the differences
In [21]:
print(gum.config)
[core]
default_maxnumberofthreads = 24
[notebook]
potential_visible_digits = 4
potential_with_colors = True
potential_color_0 = #FF7F64
potential_color_1 = #7FFF64
potential_with_fraction = False
potential_fraction_limit = 50
potential_fraction_round_error = 1e-6
potential_fraction_with_latex = True
histogram_horizontal_visible_digits = 2
histogram_vertical_visible_digits = 2
histogram_horizontal_threshold = 8
histogram_line_threshold = 40
histogram_color = darkseagreen
histogram_edge_color = darkgreen
histogram_use_percent = True
histogram_discretized_visualisation = histogram
histogram_discretized_scale = 1.0
histogram_mode = compact
histogram_epsilon = 1e-8
potential_parent_values = merge
figure_facecolor = #E0E0E0
flow_background_color = transparent
flow_border_color = transparent
flow_border_width = 0
graph_format = svg
show_inference_time = True
default_arc_color = #4A4A4A
default_node_bgcolor = #404040
default_node_fgcolor = white
evidence_bgcolor = sandybrown
evidence_fgcolor = black
default_node_cmap = Pastel1
default_arc_cmap = BuGn
default_edge_cmap = BuGn
default_graph_size = 5
default_graph_inference_size = 8
graph_rankdir = TB
graph_layout = dot
default_markovrandomfield_view = factorgraph
junctiontree_graph_size = 10
junctiontree_with_names = True
junctiontree_separator_bgcolor = palegreen
junctiontree_separator_fgcolor = black
junctiontree_separator_fontsize = 8
junctiontree_clique_bgcolor = burlywood
junctiontree_clique_fgcolor = black
junctiontree_clique_fontsize = 10
junctiontree_map_cliquescale = 0.3
junctiontree_map_sepscale = 0.1
junctiontree_map_edgelen = 1
junctiontree_map_size = 10
graphdiff_missing_style = dashed
graphdiff_missing_color = red
graphdiff_overflow_style = dashed
graphdiff_overflow_color = purple
graphdiff_reversed_style = solid
graphdiff_reversed_color = purple
graphdiff_correct_style = solid
graphdiff_correct_color = grey
[BN]
allow_modification_when_saving = False
[factorgraph]
default_node_bgcolor = coral
default_node_fgcolor = black
default_factor_bgcolor = burlywood
edge_length = 0.7
edge_length_inference = 0.9
graph_layout = neato
[dynamicBN]
default_graph_size = 6
[influenceDiagram]
default_graph_size = 6
default_chance_bgcolor = #808080
default_chance_fgcolor = white
default_utility_bgcolor = #50508A
default_utility_fgcolor = white
default_decision_bgcolor = #9A5050
default_decision_fgcolor = white
chance_shape = ellipse
utility_shape = hexagon
decision_shape = box
decision_arc_style = tapered, bold, dotted
utility_arc_style = dashed
default_id_size = 6
default_id_inference_size = 6
utility_visible_digits = 2
utility_show_stdev = True
utility_show_loss = False
[credalnet]
default_node_bgcolor = #404040
default_node_fgcolor = white
histo_max_color = #BBFFAA
[causal]
show_latent_names = False
latex_do_prefix = \text{do}(
latex_do_suffix = )
default_graph_size = 2.5
default_node_bgcolor = #404040
default_node_fgcolor = white
default_latent_bgcolor = #A08080
default_latent_fgcolor = black
[ROC]
draw_color = #008800
fill_color = #AAEEAA
[ctbn]
show_latent_names = False
default_graph_size = 2.5
default_node_bgcolor = #404040
default_node_fgcolor = white
default_latent_bgcolor = #A08080
default_latent_fgcolor = black
[bnmixture]
default_graph_size = 5
default_line_size = 1.0
default_head_size = 0.25
default_arrow_type = normal
default_arc_cmap = Greens
default_arc_color = #4A4A4A
default_arc_style = solid
default_node_bgcolor = #404040
default_node_fgcolor = white
default_layout = fdp
default_overlap = 0
default_bar_capsize = 1.5
default_bar_height = 0.8
default_boot_histo_scale = 2.0
default_histo_scale = 1.0
correct_arc_style = solid
correct_arc_color = green
incorrect_arc_style = dashed
incorrect_arc_color = green
left_quantile = 0.2
right_quantile = 0.8
[Pickle]
add_version = True
add_date = True
In [22]:
gnb.flow.row(clg1,clg2,gcm.graphDiff(clg1,clg2),
gcm.graphDiffLegend(),
gcm.graphDiff(clg2,clg1))
We compare the CLG models.
In [23]:
# We use the F-score to compare the two CLGs
cmp=gcm.GraphicalBNComparator(clg1,clg1)
print(f"F-score(clg1,clg1) : {cmp.scores()['fscore']}")
cmp=gcm.GraphicalBNComparator(clg1,clg2)
print(f"F-score(clg1,clg2) : {cmp.scores()['fscore']}")
F-score(clg1,clg1) : 1.0
F-score(clg1,clg2) : 0.7142857142857143
In [24]:
# The complete list of structural scores is :
print("score(clg1,clg2) :")
for score,val in cmp.scores().items():
print(f" - {score} : {val}")
score(clg1,clg2) :
- count : {'tp': 5, 'tn': 21, 'fp': 3, 'fn': 1}
- recall : 0.8333333333333334
- precision : 0.625
- fscore : 0.7142857142857143
- dist2opt : 0.41036907507483766
Forward Sampling
In [25]:
# We create a simple CLG with 3 variables
clg = gclg.CLG()
# prog=« sigma=2;X=N(5);Y=N(3);Z=X+Y »
A = gclg.GaussianVariable(mu=2, sigma=1, name='A')
B = gclg.GaussianVariable(mu=1, sigma=2, name='B')
C = gclg.GaussianVariable(mu=2, sigma=3, name='C')
idA = clg.add(A)
idB = clg.add(B)
idC = clg.add(C)
clg.addArc(idA, idB, 1.5)
clg.addArc(idB, idC, 0.75)
# We can show it as a graph
original_clg = gclgnb.CLG2dot(clg)
original_clg
Out[25]:
In [26]:
fs = gclg.ForwardSampling(clg)
fs.makeSample(10)
Out[26]:
<pyAgrum.clg.ForwardSampling.ForwardSampling at 0x118cfd1f0>
In [27]:
print("A's sample_variance: ", fs.variance_sample(0))
print("B's sample_variance: ", fs.variance_sample('B'))
print("C's sample_variance: ", fs.variance_sample(2))
A's sample_variance: 0.42342322842206004
B's sample_variance: 6.498064948578582
C's sample_variance: 11.149870887296805
In [28]:
print("A's sample_mean: ", fs.mean_sample('A'))
print("B's sample_mean: ", fs.mean_sample('B'))
print("C's sample_mean: ", fs.mean_sample('C'))
A's sample_mean: 1.9929620169987534
B's sample_mean: 4.503100699918417
C's sample_mean: 5.706727657782962
In [29]:
fs.toarray()
Out[29]:
array([[ 1.23883751, 1.17686467, 1.12420003],
[ 2.48183731, 4.81084019, 6.99692555],
[ 1.18582355, 5.66586213, 11.28495059],
[ 1.59743112, 0.69401504, 2.28371226],
[ 1.85195856, 5.39300381, 5.74532108],
[ 1.98001931, 3.45440182, 3.51062802],
[ 3.06546191, 7.66348816, 11.52314793],
[ 2.89189006, 9.32051452, 6.69346805],
[ 2.35629654, 2.89995015, 4.28207469],
[ 1.2800643 , 3.95206649, 3.62284839]])
In [30]:
# export to dataframe
fs.topandas()
Out[30]:
| A | B | C | |
|---|---|---|---|
| 0 | 1.238838 | 1.176865 | 1.124200 |
| 1 | 2.481837 | 4.810840 | 6.996926 |
| 2 | 1.185824 | 5.665862 | 11.284951 |
| 3 | 1.597431 | 0.694015 | 2.283712 |
| 4 | 1.851959 | 5.393004 | 5.745321 |
| 5 | 1.980019 | 3.454402 | 3.510628 |
| 6 | 3.065462 | 7.663488 | 11.523148 |
| 7 | 2.891890 | 9.320515 | 6.693468 |
| 8 | 2.356297 | 2.899950 | 4.282075 |
| 9 | 1.280064 | 3.952066 | 3.622848 |
In [31]:
# export to csv
fs.makeSample(10000)
fs.tocsv('./out/samples.csv')
PC-algorithm & Parameter Estimation
The module allows to investigale more deeply into the learning algorithm.
We first create a random CLG model with 5 variables.
In [32]:
# Create a new random CLG
clg = gclg.randomCLG(nb_variables=5, names="ABCDE")
# Display the CLG
print(clg)
CLG{nodes: 5, arcs: 5, parameters: 15}
We then do the Forward Sampling and CLGLearner.
In [33]:
n = 20 # n is the selected values of MC number n in n-MCERA
K = 10000 # K is the list of selected values of number of samples
Delta = 0.05 # Delta is the FWER we want to control
# Sample generation
fs = gclg.ForwardSampling(clg)
fs.makeSample(K).tocsv('./out/clg.csv')
# Learning
RAveL_l = gclg.CLGLearner('./out/clg.csv',n_sample=n,fwer_delta=Delta)
We use the PC algorithme to learn the structure of the model.
In [34]:
# Use the PC algorithm to get the skeleton
C = RAveL_l.PC_algorithm(order=clg.nodes(), verbose=False)
print("The final skeleton is:\n", C)
The final skeleton is:
{0: {2}, 1: {2, 3}, 2: set(), 3: set(), 4: set()}
In [35]:
# Create a Mixedgraph to display the skeleton
RAveL_MixGraph = gum.MixedGraph()
# Add variables
for i in range(len(clg.names())):
RAveL_MixGraph.addNodeWithId(i)
# Add arcs and edges
for father, kids in C.items():
for kid in kids:
if father in C[kid]:
RAveL_MixGraph.addEdge(father, kid)
else:
RAveL_MixGraph.addArc(father, kid)
RAveL_MixGraph
Out[35]:
In [36]:
# Create a BN with the same structure as the CLG
bn = gum.BayesNet()
# add variables
for name in clg.names():
new_variable = gum.LabelizedVariable(name,'a labelized variable',2)
bn.add(new_variable)
# add arcs
for arc in clg.arcs():
bn.addArc(arc[0], arc[1])
# Compare the result above with the EssentialGraph
Real_EssentialGraph = gum.EssentialGraph(bn)
Real_EssentialGraph
Out[36]:
In [37]:
# create a CLG from the skeleton of PC algorithm
clg_PC = gclg.CLG()
for node in clg.nodes():
clg_PC.add(clg.variable(node))
for father,kids in C.items():
for kid in kids:
clg_PC.addArc(father, kid)
# Compare the structure of the created CLG and the original CLG
print(f"F-score : {clg.CompareStructure(clg_PC)}")
F-score : 0.5
We can also do the parameter learning.
In [38]:
id2mu, id2sigma, arc2coef = RAveL_l.estimate_parameters(C)
for node in clg.nodes():
print(f"Real Value: node {node} : mu = {clg.variable(node)._mu}, sigma = {clg.variable(node)._sigma}")
print(f"Estimation: node {node} : mu = {id2mu[node]}, sigma = {id2sigma[node]}")
for arc in clg.arcs():
print(f"Real Value: arc {arc} : coef = {clg.coefArc(*arc)}")
print(f"Estimation: arc {arc} : coef = {(arc2coef[arc] if arc in arc2coef else '-')}")
Real Value: node 0 : mu = -1.466848572985544, sigma = 1.9514642651358898
Estimation: node 0 : mu = -1.477800164365639, sigma = 1.9452271398161107
Real Value: node 1 : mu = 2.9199869887532524, sigma = 1.0618293117736646
Estimation: node 1 : mu = -71.75993337308283, sigma = 80.92168605727542
Real Value: node 2 : mu = 2.9355816369315493, sigma = 4.496492712039123
Estimation: node 2 : mu = 0.5365222585893274, sigma = 0.19221279933838867
Real Value: node 3 : mu = 2.4592840967453977, sigma = 7.410140817711454
Estimation: node 3 : mu = 10.966766311109211, sigma = 22.012372950956188
Real Value: node 4 : mu = -1.5686837870467354, sigma = 3.935343144759555
Estimation: node 4 : mu = -1.5848581255944825, sigma = 3.9769747243490663
Real Value: arc (0, 1) : coef = 7.681396654035673
Estimation: arc (0, 1) : coef = -
Real Value: arc (2, 1) : coef = -5.520416583970681
Estimation: arc (2, 1) : coef = -
Real Value: arc (4, 3) : coef = -5.216928735069795
Estimation: arc (4, 3) : coef = -
Real Value: arc (0, 2) : coef = -5.782205786820264
Estimation: arc (0, 2) : coef = 1.379828617950918
Real Value: arc (1, 3) : coef = 7.196680713870754
Estimation: arc (1, 3) : coef = 7.199986238446008
In [ ]: