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キンラン、ギンラン、ベイジアン・ネットワーク [相鉄沿線]

今年も、こども自然公園にキンランとギンランが咲いていました。












今まで気が付きませんでしたが、追分市民の森にも、キンランとギンランが咲いていました。












キンラン、ギンランとは、関係ないのですがベイジアン・ネットワーク
(キンラン、ギンランも地面の中では菌根菌(きんこんきん、mycorrhizal fungi)のネットワークを作っているらしいです。)

最近ベイジアン・ネットワークの良いテキストを見つけました。

Understanding Bayesian Networks with Examples in R

http://www.bnlearn.com/about/teaching/slides-bnshort.pdf

このスライド中のHands-On ExamplesのCase Studyが、とてもわかりやすいです。


Causal Protein-Signaling Networks Derived from Multiparameter Single-Cell Data (http://science.sciencemag.org/content/308/5721/523)に掲載されているネットワークをRのbnlearnというパッケージを利用して作っていきます。


Causal Protein-Signaling Networks Derived from Multiparameter Single-Cell Data: Fig. 3. Bayesian network inference
results. 22 APRIL 2005 VOL 308 SCIENCE www.sciencemag.org

データは、このページからダウンロードできます。
Bayesian Networks with Examples in R
http://www.bnlearn.com/book-crc/
Sachs' raw observational data.
Sachs' discretized observational data.
Sachs' complete pre-processed data.



図中の上のようなネットワークにしたいのですが、前処理なしでは、下のようになってしまいます。

正しいネットワーク作るのには、どのような前処理が必要かを考えるために、元のデータの分布や関連を調べます。
元のデータの分布や関連を見るのは、Rのpsychというパッケージを使うと便利です。


データの分布がかたよっていたり、データ間の関係が線形相関ではないようなので、データを離散化(Discretisation)したりするようです。
bnlearnには、離散化のためのHartemink’s algorithmのような関数も用意されていています。

Rplot_d_pairs_s.jpeg

離散化後のデータで、ネットワークを作成

graph_02.jpeg


Taking the Interventions into Account
Rplot_pairs_isachs_s.jpeg


The_Final_DAG.jpeg


+ Rのプログラム Bayesian networks Hands-On Examples
# Bayesian networks
# Hands-On Examples
# Case Study: Human Physiology
#
library(bnlearn)

setwd("~/R/bnlearn/crc")
# Exploring the Data
sachs = read.table("sachs.data.txt", header = TRUE)
head(sachs, n = 5)

# A First Try
dag.hiton = si.hiton.pc(sachs, test = "cor", undirected = FALSE)
directed.arcs(dag.hiton)
#undirected.arcs(dag.hiton)

# Compare with the Validated Model
sachs.modelstring =
  paste("[PKC][PKA|PKC][Raf|PKC:PKA][Mek|PKC:PKA:Raf][Erk|Mek:PKA]",
  "[Akt|Erk:PKA][P38|PKC:PKA][Jnk|PKC:PKA][Plcg][PIP3|Plcg]",
  "[PIP2|Plcg:PIP3]")

dag.sachs = model2network(sachs.modelstring)
unlist(compare(dag.sachs, dag.hiton))

par(mfrow=c(2,1))
graphviz.plot(dag.sachs)
graphviz.plot(dag.hiton)


# scatterplots, distributions and correlation
library(psych)
pairs.panels(sachs)

# Discretising Data
# Hartemink’s Information-Preserving Discretisation
dsachs = discretize(sachs, method = "hartemink",
                     breaks = 3, ibreaks = 60, idisc = "quantile")

head(dsachs)

pairs.panels(dsachs)

# However, HITON is still not working...
dag.hiton = si.hiton.pc(dsachs, test = "x2", undirected = FALSE)
unlist(compare(dag.hiton, dag.sachs))
# ... so we switch to a score-based algorithm ...
dag.hc = hc(dsachs, score = "bde", iss = 10, undirected = FALSE)
unlist(compare(dag.hc, dag.sachs))

graphviz.plot(dag.hc)

# ... and frequentist model averaging to remove spurious arcs.
boot = boot.strength(dsachs, R = 500, algorithm = "hc",
  algorithm.args = list(score = "bde", iss = 10))
head(boot[(boot$strength > 0.85) & (boot$direction >= 0.5), ], n = 3)

# Model Averaging from Multiple Searches
nodes = names(dsachs)

# While there is no function in bnlearn that does exactly this, we can
# combine random.graph() and sapply() to generate the random
# starting points and call hc() on each of them.
startlist = random.graph(nodes = nodes, method = "ic-dag",
  num = 500, every = 50)

netlist = lapply(startlist,
  function(net) {
    hc(dsachs, score = "bde", iss = 10, start = net)
  }
)

# Compare Both Approaches with the Validated Network
start = custom.strength(netlist, nodes = nodes)
avg.start = averaged.network(start)
avg.boot = averaged.network(boot)
unlist(compare(avg.start, dag.sachs))
#tp fp fn 
# 3 14  7 
unlist(compare(avg.boot, dag.sachs))
#tp fp fn 
# 6 11  4 
par(mfrow=c(2,1))
graphviz.plot(avg.start)
graphviz.plot(avg.boot)

# Both networks look nothing like the validated network, and in fact fall in
# the same equivalence class.
all.equal(cpdag(avg.boot), cpdag(avg.start))

# The only piece of information we have not taken into account yet are
# the stimulations and the inhibitions, that is, the interventions on the
# variables.
isachs = read.table("sachs.interventional.txt",
  header = TRUE, colClasses = "factor")

# A Naive Approach with Whitelists
wh = matrix(c(rep("INT", 11), names(isachs)[1:11]), ncol = 2)
dag.wh = tabu(isachs, whitelist = wh, score = "bde",
  iss = 10, tabu = 50)

unlist(compare(subgraph(dag.wh, names(isachs)[1:11]), dag.sachs))

graphviz.plot(dag.wh, highlight = list(nodes = "INT",
  arcs = outgoing.arcs(dag.wh, "INT"), col = "darkgrey", fill = "darkgrey"))

# Mixed Observational and Interventional Data
# A more granular way of doing the same thing is to use the mixed observational
# and interventional data posterior score from Cooper & Yoo, which creates an
# implicit intervention binary node for each variable.

INT = sapply(1:11, function(x) which(isachs$INT == x) )
nodes = names(isachs)[1:11]
names(INT) = nodes

# Then we perform model averaging of the resulting causal DAGs, with better
# results.
netlist = lapply(startlist, function(net) {
  tabu(isachs[, 1:11], score = "mbde", exp = INT, iss = 1,
    start = net, tabu = 50)
})
intscore = custom.strength(netlist, nodes = nodes, cpdag = FALSE)
dag.mbde = averaged.network(intscore)
unlist(compare(dag.sachs, dag.mbde))

# The Final DAG
graphviz.plot(dag.mbde, highlight = list(arcs = arcs(dag.sachs)))

####
# Using The Protein Network to Plan Experiments

# First, we need to learn the parameters of the BN given the DAG.
isachs = isachs[, 1:11]
for (i in names(isachs))
  levels(isachs[, i]) = c("LOW", "AVG", "HIGH")
fitted = bn.fit(dag.sachs, isachs, method = "bayes")

# Then we can proceed to perform queries using gRain, on the original BN
library(gRain)
jtree = compile(as.grain(fitted))

# and on a mutilated BN in which we set Erk to LOW with an ideal
# intervention.
jlow = compile(as.grain(mutilated(fitted, evidence = list(Erk = "LOW"))))

par(mfrow=c(2,1))
plot(jtree)
plot(jlow)

# Interventions and Mutilated Graphs

# Variables That are Downstream are Untouched
querygrain(jtree, nodes = "Akt")$Akt
querygrain(jlow, nodes = "Akt")$Akt

# Causal Inference, Posterior Inference
querygrain(jtree, nodes = "PKA")$PKA
querygrain(jlow, nodes = "PKA")$PKA

jlow = setEvidence(jtree, nodes = "Erk", states= "LOW")
querygrain(jlow, nodes = "PKA")$PKA


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