learn_struct_pdag_pc_constrain.m

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function [pdag, G] = dn_learn_struct_pdag_pc_constrain(adj, cond_indep, n, k, varargin)
% LEARN_STRUCT_PDAG_PC Learn a partially oriented DAG (pattern) using the PC algorithm
% Pdag = learn_struct_pdag_pc_constrain(adj, cond_indep, n, k, ...)
%
% adj = adjacency matrix learned from dependency network P(i,j) = 1 => i--j; 0 => i  j
% n is the number of nodes.
% k is an optional upper bound on the fan-in (default: n)
% cond_indep is a boolean function that will be called as follows:
%   feval(cond_indep, x, y, S, ...)
% where x and y are nodes, and S is a set of nodes (positive integers),
% and ... are any optional parameters passed to this function.
%
%Output
% pdag  Partially directed graph
% G     Resulting adjacency graph prior to setting direction arrows
%
% The output P is an adjacency matrix, in which
% P(i,j) = -1 if there is an i->j edge.
% P(i,j) = P(j,i) = 1 if there is an undirected edge i <-> j
%
% The PC algorithm does structure learning assuming all variables are observed.
% See Spirtes, Glymour and Scheines, "Causation, Prediction and Search", 1993, p117.
% This algorithm may take O(n^k) time if there are n variables and k is the max fan-in,
% but this is quicker than the Verma-Pearl IC algorithm, which is always O(n^n).
% 
%%  Example
%%  Given data in a comma separated, filename starting with the variable labels, then the data in rows.
%%    filename test.txt consists of:
%%
%%      Earthquake,Burglar,Radio,Alarm,Call
%%      1,2,2,2,1
%%      1,1,2,1,2
%%      . . .
%[CovMatrix, obs, varfields] = CovMat('test.txt',5);
%
%dn = zeros(5,5); 
%dn(1,2) = 1;   % This was the known Markov blanket of the system that generated test.txt
%dn(2,1) = 1;
%dn(2,4) = 1;
%dn(4,2) = 1;
%dn(1,3) = 1;
%dn(3,1) = 1;
%dn(1,4) = 1;
%dn(4,1) = 1;
%dn(4,5) = 1;
%dn(5,4) = 1;
%dn(3,5) = 1; %loop r->c
%dn(5,3) = 1; %loop c-r
%dn(3,4) = 1;
%dn(4,3) = 1;
%
%max_fan_in = 4;
%alpha = 0.05;
%
%[pdag G] = learn_struct_pdag_pc_constrain(dn,'cond_indep_fisher_z', 5, max_fan_in, CovMatrix, obs, alpha);
%%
%%
%% Gary Bradski, 7/2002 Modified this to take an adjacency matrix from a dependency network.

  
sep = cell(n,n);
ord = 0;
done = 0;
G = ones(n,n);
G=setdiag(G,0);

while ~done
    done = 1;
    [X,Y] = find(G);
    for i=1:length(X)
        x = X(i); y = Y(i);
%        nbrs = mysetdiff(myunion(neighbors(G, x), neighbors(G,y)), [x y]);%parents, children, but not self
        nbrs = mysetdiff(myunion(neighbors(adj, x), neighbors(adj,y)), [x y]);%parents, children, but not self

        if length(nbrs) >= ord & G(x,y) ~= 0
            done = 0;
            SS = subsets(nbrs, ord, ord); % all subsets of size ord
            for si=1:length(SS)
                S = SS{si};
                 %if (feval(dsep,x,y,S,adj)) | (feval(cond_indep, x, y, S, varargin{:}))
                 if feval(cond_indep, x, y, S, varargin{:})
                     %if isempty(S)
                    %  fprintf('%d indep of %d ', x, y);
                    %else
                    %  fprintf('%d indep of %d given ', x, y); fprintf('%d ', S);
                    %end
                    %fprintf('\n');
                    
                    % diagnostic
                    %[CI, r] = cond_indep_fisher_z(x, y, S, varargin{:});
                    %fprintf(': r = %6.4f\n', r);
                    
                    G(x,y) = 0;
                    G(y,x) = 0;
                    adj(x,y) = 0;  %make sure found cond. independencies are marked out
                    adj(y,x) = 0;
                    sep{x,y} = myunion(sep{x,y}, S);
                    sep{y,x} = myunion(sep{y,x}, S);
                    break; % no need to check any more subsets 
                end
            end
        end 
    end
    ord = ord + 1;
end




% Create the minimal pattern,
% i.e., the only directed edges are V structures.

pdag = G;
[X, Y] = find(G);
% We want to generate all unique triples x,y,z
% This code generates x,y,z and z,y,x.
for i=1:length(X)
  x = X(i);
  y = Y(i);
  Z = find(G(y,:));
  Z = mysetdiff(Z, x);
  for z=Z(:)'
    if G(x,z)==0 & ~ismember(y, sep{x,z}) & ~ismember(y, sep{z,x})
      %fprintf('%d -> %d <- %d\n', x, y, z);
      pdag(x,y) = -1; pdag(y,x) = 0;
      pdag(z,y) = -1; pdag(y,z) = 0;
    end
  end
end

% Convert the minimal pattern to a complete one,
% i.e., every directed edge in P is compelled
% (must be directed in all Markov equivalent models),
% and every undirected edge in P is reversible.
% We use the rules of Pearl (2000) p51 (derived in Meek (1995))

old_pdag = zeros(n);
iter = 0;
while ~isequal(pdag, old_pdag)
  iter = iter + 1;
  old_pdag = pdag;
  % rule 1
  [A,B] = find(pdag==-1); % a -> b
  for i=1:length(A)
    a = A(i); b = B(i);
    C = find(pdag(b,:)==1 & G(a,:)==0); % all nodes adj to b but not a
    if ~isempty(C)
      pdag(b,C) = -1; pdag(C,b) = 0;
      %fprintf('rule 1: a=%d->b=%d and b=%d-c=%d implies %d->%d\n', a, b, b, C, b, C);
    end
  end
  % rule 2
  [A,B] = find(pdag==1); % unoriented a-b edge
  for i=1:length(A)
    a = A(i); b = B(i);
    if any( (pdag(a,:)==-1) & (pdag(:,b)==-1)' );
      pdag(a,b) = -1; pdag(b,a) = 0;
      %fprintf('rule 2: %d -> %d\n', a, b);
    end
  end
  % rule 3
  [A,B] = find(pdag==1); % a-b
  for i=1:length(A)
    a = A(i); b = B(i);
    C = find( (G(a,:)==1) & (pdag(:,b)==-1)' );
    % C contains nodes c s.t. a-c->ba
    G2 = setdiag(G(C, C), 1);
    if any(G2(:)==0) % there are 2 different non adjacent elements of C
      pdag(a,b) = -1; pdag(b,a) = 0;
      %fprintf('rule 3: %d -> %d\n', a, b);
    end
  end
end