Daniel@0: function [pdag, G] = dn_learn_struct_pdag_pc_constrain(adj, cond_indep, n, k, varargin)
Daniel@0: % LEARN_STRUCT_PDAG_PC Learn a partially oriented DAG (pattern) using the PC algorithm
Daniel@0: % Pdag = learn_struct_pdag_pc_constrain(adj, cond_indep, n, k, ...)
Daniel@0: %
Daniel@0: % adj = adjacency matrix learned from dependency network P(i,j) = 1 => i--j; 0 => i  j
Daniel@0: % n is the number of nodes.
Daniel@0: % k is an optional upper bound on the fan-in (default: n)
Daniel@0: % cond_indep is a boolean function that will be called as follows:
Daniel@0: %   feval(cond_indep, x, y, S, ...)
Daniel@0: % where x and y are nodes, and S is a set of nodes (positive integers),
Daniel@0: % and ... are any optional parameters passed to this function.
Daniel@0: %
Daniel@0: %Output
Daniel@0: % pdag  Partially directed graph
Daniel@0: % G     Resulting adjacency graph prior to setting direction arrows
Daniel@0: %
Daniel@0: % The output P is an adjacency matrix, in which
Daniel@0: % P(i,j) = -1 if there is an i->j edge.
Daniel@0: % P(i,j) = P(j,i) = 1 if there is an undirected edge i <-> j
Daniel@0: %
Daniel@0: % The PC algorithm does structure learning assuming all variables are observed.
Daniel@0: % See Spirtes, Glymour and Scheines, "Causation, Prediction and Search", 1993, p117.
Daniel@0: % This algorithm may take O(n^k) time if there are n variables and k is the max fan-in,
Daniel@0: % but this is quicker than the Verma-Pearl IC algorithm, which is always O(n^n).
Daniel@0: % 
Daniel@0: %%  Example
Daniel@0: %%  Given data in a comma separated, filename starting with the variable labels, then the data in rows.
Daniel@0: %%    filename test.txt consists of:
Daniel@0: %%
Daniel@0: %%      Earthquake,Burglar,Radio,Alarm,Call
Daniel@0: %%      1,2,2,2,1
Daniel@0: %%      1,1,2,1,2
Daniel@0: %%      . . .
Daniel@0: %[CovMatrix, obs, varfields] = CovMat('test.txt',5);
Daniel@0: %
Daniel@0: %dn = zeros(5,5); 
Daniel@0: %dn(1,2) = 1;   % This was the known Markov blanket of the system that generated test.txt
Daniel@0: %dn(2,1) = 1;
Daniel@0: %dn(2,4) = 1;
Daniel@0: %dn(4,2) = 1;
Daniel@0: %dn(1,3) = 1;
Daniel@0: %dn(3,1) = 1;
Daniel@0: %dn(1,4) = 1;
Daniel@0: %dn(4,1) = 1;
Daniel@0: %dn(4,5) = 1;
Daniel@0: %dn(5,4) = 1;
Daniel@0: %dn(3,5) = 1; %loop r->c
Daniel@0: %dn(5,3) = 1; %loop c-r
Daniel@0: %dn(3,4) = 1;
Daniel@0: %dn(4,3) = 1;
Daniel@0: %
Daniel@0: %max_fan_in = 4;
Daniel@0: %alpha = 0.05;
Daniel@0: %
Daniel@0: %[pdag G] = learn_struct_pdag_pc_constrain(dn,'cond_indep_fisher_z', 5, max_fan_in, CovMatrix, obs, alpha);
Daniel@0: %%
Daniel@0: %%
Daniel@0: %% Gary Bradski, 7/2002 Modified this to take an adjacency matrix from a dependency network.
Daniel@0: 
Daniel@0:   
Daniel@0: sep = cell(n,n);
Daniel@0: ord = 0;
Daniel@0: done = 0;
Daniel@0: G = ones(n,n);
Daniel@0: G=setdiag(G,0);
Daniel@0: 
Daniel@0: while ~done
Daniel@0:     done = 1;
Daniel@0:     [X,Y] = find(G);
Daniel@0:     for i=1:length(X)
Daniel@0:         x = X(i); y = Y(i);
Daniel@0: %        nbrs = mysetdiff(myunion(neighbors(G, x), neighbors(G,y)), [x y]);%parents, children, but not self
Daniel@0:         nbrs = mysetdiff(myunion(neighbors(adj, x), neighbors(adj,y)), [x y]);%parents, children, but not self
Daniel@0: 
Daniel@0:         if length(nbrs) >= ord & G(x,y) ~= 0
Daniel@0:             done = 0;
Daniel@0:             SS = subsets(nbrs, ord, ord); % all subsets of size ord
Daniel@0:             for si=1:length(SS)
Daniel@0:                 S = SS{si};
Daniel@0:                  %if (feval(dsep,x,y,S,adj)) | (feval(cond_indep, x, y, S, varargin{:}))
Daniel@0:                  if feval(cond_indep, x, y, S, varargin{:})
Daniel@0:                      %if isempty(S)
Daniel@0:                     %  fprintf('%d indep of %d ', x, y);
Daniel@0:                     %else
Daniel@0:                     %  fprintf('%d indep of %d given ', x, y); fprintf('%d ', S);
Daniel@0:                     %end
Daniel@0:                     %fprintf('\n');
Daniel@0:                     
Daniel@0:                     % diagnostic
Daniel@0:                     %[CI, r] = cond_indep_fisher_z(x, y, S, varargin{:});
Daniel@0:                     %fprintf(': r = %6.4f\n', r);
Daniel@0:                     
Daniel@0:                     G(x,y) = 0;
Daniel@0:                     G(y,x) = 0;
Daniel@0:                     adj(x,y) = 0;  %make sure found cond. independencies are marked out
Daniel@0:                     adj(y,x) = 0;
Daniel@0:                     sep{x,y} = myunion(sep{x,y}, S);
Daniel@0:                     sep{y,x} = myunion(sep{y,x}, S);
Daniel@0:                     break; % no need to check any more subsets 
Daniel@0:                 end
Daniel@0:             end
Daniel@0:         end 
Daniel@0:     end
Daniel@0:     ord = ord + 1;
Daniel@0: end
Daniel@0: 
Daniel@0: 
Daniel@0: 
Daniel@0: 
Daniel@0: % Create the minimal pattern,
Daniel@0: % i.e., the only directed edges are V structures.
Daniel@0: 
Daniel@0: pdag = G;
Daniel@0: [X, Y] = find(G);
Daniel@0: % We want to generate all unique triples x,y,z
Daniel@0: % This code generates x,y,z and z,y,x.
Daniel@0: for i=1:length(X)
Daniel@0:   x = X(i);
Daniel@0:   y = Y(i);
Daniel@0:   Z = find(G(y,:));
Daniel@0:   Z = mysetdiff(Z, x);
Daniel@0:   for z=Z(:)'
Daniel@0:     if G(x,z)==0 & ~ismember(y, sep{x,z}) & ~ismember(y, sep{z,x})
Daniel@0:       %fprintf('%d -> %d <- %d\n', x, y, z);
Daniel@0:       pdag(x,y) = -1; pdag(y,x) = 0;
Daniel@0:       pdag(z,y) = -1; pdag(y,z) = 0;
Daniel@0:     end
Daniel@0:   end
Daniel@0: end
Daniel@0: 
Daniel@0: % Convert the minimal pattern to a complete one,
Daniel@0: % i.e., every directed edge in P is compelled
Daniel@0: % (must be directed in all Markov equivalent models),
Daniel@0: % and every undirected edge in P is reversible.
Daniel@0: % We use the rules of Pearl (2000) p51 (derived in Meek (1995))
Daniel@0: 
Daniel@0: old_pdag = zeros(n);
Daniel@0: iter = 0;
Daniel@0: while ~isequal(pdag, old_pdag)
Daniel@0:   iter = iter + 1;
Daniel@0:   old_pdag = pdag;
Daniel@0:   % rule 1
Daniel@0:   [A,B] = find(pdag==-1); % a -> b
Daniel@0:   for i=1:length(A)
Daniel@0:     a = A(i); b = B(i);
Daniel@0:     C = find(pdag(b,:)==1 & G(a,:)==0); % all nodes adj to b but not a
Daniel@0:     if ~isempty(C)
Daniel@0:       pdag(b,C) = -1; pdag(C,b) = 0;
Daniel@0:       %fprintf('rule 1: a=%d->b=%d and b=%d-c=%d implies %d->%d\n', a, b, b, C, b, C);
Daniel@0:     end
Daniel@0:   end
Daniel@0:   % rule 2
Daniel@0:   [A,B] = find(pdag==1); % unoriented a-b edge
Daniel@0:   for i=1:length(A)
Daniel@0:     a = A(i); b = B(i);
Daniel@0:     if any( (pdag(a,:)==-1) & (pdag(:,b)==-1)' );
Daniel@0:       pdag(a,b) = -1; pdag(b,a) = 0;
Daniel@0:       %fprintf('rule 2: %d -> %d\n', a, b);
Daniel@0:     end
Daniel@0:   end
Daniel@0:   % rule 3
Daniel@0:   [A,B] = find(pdag==1); % a-b
Daniel@0:   for i=1:length(A)
Daniel@0:     a = A(i); b = B(i);
Daniel@0:     C = find( (G(a,:)==1) & (pdag(:,b)==-1)' );
Daniel@0:     % C contains nodes c s.t. a-c->ba
Daniel@0:     G2 = setdiag(G(C, C), 1);
Daniel@0:     if any(G2(:)==0) % there are 2 different non adjacent elements of C
Daniel@0:       pdag(a,b) = -1; pdag(b,a) = 0;
Daniel@0:       %fprintf('rule 3: %d -> %d\n', a, b);
Daniel@0:     end
Daniel@0:   end
Daniel@0: end
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