wolffd@0: function [pdag, G] = learn_struct_pdag_pc(cond_indep, n, k, varargin)
wolffd@0: % LEARN_STRUCT_PDAG_PC Learn a partially oriented DAG (pattern) using the PC algorithm
wolffd@0: % P = learn_struct_pdag_pc(cond_indep, n, k, ...)
wolffd@0: %
wolffd@0: % n is the number of nodes.
wolffd@0: % k is an optional upper bound on the fan-in (default: n)
wolffd@0: % cond_indep is a boolean function that will be called as follows:
wolffd@0: %   feval(cond_indep, x, y, S, ...)
wolffd@0: % where x and y are nodes, and S is a set of nodes (positive integers),
wolffd@0: % and ... are any optional parameters passed to this function.
wolffd@0: %
wolffd@0: % The output P is an adjacency matrix, in which
wolffd@0: % P(i,j) = -1 if there is an i->j edge.
wolffd@0: % P(i,j) = P(j,i) = 1 if there is an undirected edge i <-> j
wolffd@0: %
wolffd@0: % The PC algorithm does structure learning assuming all variables are observed.
wolffd@0: % See Spirtes, Glymour and Scheines, "Causation, Prediction and Search", 1993, p117.
wolffd@0: % This algorithm may take O(n^k) time if there are n variables and k is the max fan-in,
wolffd@0: % but this is quicker than the Verma-Pearl IC algorithm, which is always O(n^n).
wolffd@0: 
wolffd@0:   
wolffd@0: sep = cell(n,n);
wolffd@0: ord = 0;
wolffd@0: done = 0;
wolffd@0: G = ones(n,n);
wolffd@0: G=setdiag(G,0);
wolffd@0: while ~done
wolffd@0:   done = 1;
wolffd@0:   [X,Y] = find(G); 
wolffd@0:   for i=1:length(X)
wolffd@0:     x = X(i); y = Y(i);
wolffd@0:     %nbrs = mysetdiff(myunion(neighbors(G, x), neighbors(G,y)), [x y]);
wolffd@0:     nbrs = mysetdiff(neighbors(G, y), x);  % bug fix by Raanan Yehezkel <raanany@ee.bgu.ac.il> 6/27/04
wolffd@0:     if length(nbrs) >= ord & G(x,y) ~= 0
wolffd@0:       done = 0;
wolffd@0:       %SS = subsets(nbrs, ord, ord); % all subsets of size ord
wolffd@0:       SS = subsets1(nbrs, ord);
wolffd@0:       for si=1:length(SS)
wolffd@0: 	S = SS{si};
wolffd@0: 	if feval(cond_indep, x, y, S, varargin{:})
wolffd@0: 	  %if isempty(S)
wolffd@0:  	  %  fprintf('%d indep of %d ', x, y);
wolffd@0: 	  %else
wolffd@0:  	  %  fprintf('%d indep of %d given ', x, y); fprintf('%d ', S);
wolffd@0:  	  %end
wolffd@0:  	  %fprintf('\n');
wolffd@0: 	  
wolffd@0: 	  % diagnostic
wolffd@0: 	  %[CI, r] = cond_indep_fisher_z(x, y, S, varargin{:});
wolffd@0: 	  %fprintf(': r = %6.4f\n', r);
wolffd@0: 	  
wolffd@0: 	  G(x,y) = 0;
wolffd@0: 	  G(y,x) = 0;
wolffd@0: 	  sep{x,y} = myunion(sep{x,y}, S);
wolffd@0: 	  sep{y,x} = myunion(sep{y,x}, S);
wolffd@0: 	  break; % no need to check any more subsets 
wolffd@0: 	end
wolffd@0:       end
wolffd@0:     end 
wolffd@0:   end
wolffd@0:   ord = ord + 1;
wolffd@0: end
wolffd@0: 
wolffd@0: 
wolffd@0: % Create the minimal pattern,
wolffd@0: % i.e., the only directed edges are V structures.
wolffd@0: pdag = G;
wolffd@0: [X, Y] = find(G);
wolffd@0: % We want to generate all unique triples x,y,z
wolffd@0: % This code generates x,y,z and z,y,x.
wolffd@0: for i=1:length(X)
wolffd@0:   x = X(i);
wolffd@0:   y = Y(i);
wolffd@0:   Z = find(G(y,:));
wolffd@0:   Z = mysetdiff(Z, x);
wolffd@0:   for z=Z(:)'
wolffd@0:     if G(x,z)==0 & ~ismember(y, sep{x,z}) & ~ismember(y, sep{z,x})
wolffd@0:       %fprintf('%d -> %d <- %d\n', x, y, z);
wolffd@0:       pdag(x,y) = -1; pdag(y,x) = 0;
wolffd@0:       pdag(z,y) = -1; pdag(y,z) = 0;
wolffd@0:     end
wolffd@0:   end
wolffd@0: end
wolffd@0: 
wolffd@0: % Convert the minimal pattern to a complete one,
wolffd@0: % i.e., every directed edge in P is compelled
wolffd@0: % (must be directed in all Markov equivalent models),
wolffd@0: % and every undirected edge in P is reversible.
wolffd@0: % We use the rules of Pearl (2000) p51 (derived in Meek (1995))
wolffd@0: 
wolffd@0: old_pdag = zeros(n);
wolffd@0: iter = 0;
wolffd@0: while ~isequal(pdag, old_pdag)
wolffd@0:   iter = iter + 1;
wolffd@0:   old_pdag = pdag;
wolffd@0:   % rule 1
wolffd@0:   [A,B] = find(pdag==-1); % a -> b
wolffd@0:   for i=1:length(A)
wolffd@0:     a = A(i); b = B(i);
wolffd@0:     C = find(pdag(b,:)==1 & G(a,:)==0); % all nodes adj to b but not a
wolffd@0:     if ~isempty(C)
wolffd@0:       pdag(b,C) = -1; pdag(C,b) = 0;
wolffd@0:       %fprintf('rule 1: a=%d->b=%d and b=%d-c=%d implies %d->%d\n', a, b, b, C, b, C);
wolffd@0:     end
wolffd@0:   end
wolffd@0:   % rule 2
wolffd@0:   [A,B] = find(pdag==1); % unoriented a-b edge
wolffd@0:   for i=1:length(A)
wolffd@0:     a = A(i); b = B(i);
wolffd@0:     if any( (pdag(a,:)==-1) & (pdag(:,b)==-1)' );
wolffd@0:       pdag(a,b) = -1; pdag(b,a) = 0;
wolffd@0:       %fprintf('rule 2: %d -> %d\n', a, b);
wolffd@0:     end
wolffd@0:   end
wolffd@0:   % rule 3
wolffd@0:   [A,B] = find(pdag==1); % a-b
wolffd@0:   for i=1:length(A)
wolffd@0:     a = A(i); b = B(i);
wolffd@0:     C = find( (pdag(a,:)==1) & (pdag(:,b)==-1)' );
wolffd@0:     % C contains nodes c s.t. a-c->ba
wolffd@0:     G2 = setdiag(G(C, C), 1);
wolffd@0:     if any(G2(:)==0) % there are 2 different non adjacent elements of C
wolffd@0:       pdag(a,b) = -1; pdag(b,a) = 0;
wolffd@0:       %fprintf('rule 3: %d -> %d\n', a, b);
wolffd@0:     end
wolffd@0:   end
wolffd@0: end
wolffd@0: 
wolffd@0: