treec.m

The pattern recognition matlab toolbox

	if isempty(t)
		[f,j,t] = infcrit(a,nlab);
		prwarning(3,'Maxcrit not feasible for decision tree, infcrit is used')
	end
	return

%INFCRIT The information gain and its the best feature split.
% 
% 	[f,j,t] = infcrit(A,nlabels)
% 
% Computes over all features the information gain f for its best 
% threshold from the dataset A and its numeric labels. For f=1: 
% perfect discrimination, f=0: complete mixture. j is the optimum 
% feature, t its threshold. This is a lowlevel routine called for 
% constructing decision trees.

% Copyright: R.P.W. Duin, [e-mail]
% Faculty of Applied Physics, Delft University of Technology
% P.O. Box 5046, 2600 GA Delft, The Netherlands

function [g,j,t] = infcrit(a,nlab)
	prtrace(mfilename);
	[m,k] = size(a);
	c = max(nlab);
	mininfo = ones(k,2);
	% determine feature domains of interest
	[sn,ln] = min(a,[],1); 
	[sx,lx] = max(a,[],1);
	JN = (nlab(:,ones(1,k)) == ones(m,1)*nlab(ln)') * realmax;
	JX = -(nlab(:,ones(1,k)) == ones(m,1)*nlab(lx)') * realmax;
	S = sort([sn; min(a+JN,[],1); max(a+JX,[],1); sx]);
	% S(2,:) to S(3,:) are interesting feature domains
	P = sort(a);
	Q = (P >= ones(m,1)*S(2,:)) & (P <= ones(m,1)*S(3,:));
	% these are the feature values in those domains
	for f=1:k,		% repeat for all features
		af = a(:,f);
		JQ = find(Q(:,f));
		SET = P(JQ,f)';
		if JQ(1) ~= 1
			SET = [P(JQ(1)-1,f), SET];
		end
		n = length(JQ);
		if JQ(n) ~= m
			SET = [SET, P(JQ(n)+1,f)];
		end
		n = length(SET) -1;
		T = (SET(1:n) + SET(2:n+1))/2; % all possible thresholds
		L = zeros(c,n); R = L;     % left and right node object counts per class
		for j = 1:c
			J = find(nlab==j); mj = length(J);
			if mj == 0
				L(j,:) = realmin*ones(1,n); R(j,:) = L(j,:);
			else
				L(j,:) = sum(repmat(af(J),1,n) <= repmat(T,mj,1)) + realmin;
				R(j,:) = sum(repmat(af(J),1,n) > repmat(T,mj,1)) + realmin;
			end
		end
		infomeas =  - (sum(L .* log10(L./(ones(c,1)*sum(L)))) ...
			       + sum(R .* log10(R./(ones(c,1)*sum(R))))) ...
		    ./ (log10(2)*(sum(L)+sum(R))); % criterion value for all thresholds
		[mininfo(f,1),j] = min(infomeas);     % finds the best
		mininfo(f,2) = T(j);     % and its threshold
	end   
	g = 1-mininfo(:,1)';
	[finfo,j] = min(mininfo(:,1));		% best over all features
	t = mininfo(j,2);			% and its threshold
	return

%FISHCRIT Fisher's Criterion and its best feature split 
% 
% 	[f,j,t] = fishcrit(A,nlabels)
% 
% Computes the value of the Fisher's criterion f for all features 
% over the dataset A with given numeric labels. Two classes only. j 
% is the optimum feature, t its threshold. This is a lowlevel 
% routine called for constructing decision trees.

% Copyright R.P.W. Duin, [e-mail]
% Faculty of Applied Physics, Delft University of Technology
% P.O. Box 5046, 2600 GA Delft, The Netherlands

function [f,j,t] = fishcrit(a,nlab)
	prtrace(mfilename);
	[m,k] = size(a);
	c = max(nlab);
	if c > 2
		error('Not more than 2 classes allowed for Fisher Criterion')
	end
	% Get the mean and variances of both the classes:
	J1 = find(nlab==1);
	J2 = find(nlab==2);
	u = (mean(a(J1,:),1) - mean(a(J2,:),1)).^2;
	s = std(a(J1,:),0,1).^2 + std(a(J2,:),0,1).^2 + realmin;
	% The Fisher ratio becomes:
	f = u ./ s;
	% Find then the best feature:
	[ff,j] = max(f);
	% Given the feature, compute the threshold:
	m1 = mean(a(J1,j),1);
	m2 = mean(a(J2,j),1);
	w1 = m1 - m2; w2 = (m1*m1-m2*m2)/2;
	if abs(w1) < eps % the means are equal, so the Fisher
			 % criterion (should) become 0. Let us set the thresold
			 % halfway the domain
			 t = (max(a(J1,j),[],1) + min(a(J2,j),[],1)) / 2;
	else
		t = w2/w1;
	end
	return

%INFSTOP Quinlan's Chi-square test for early stopping
% 
% 	crt = infstop(A,nlabels,j,t)
% 
% Computes the Chi-square test described by Quinlan [1] to be used 
% in maketree for forward pruning (early stopping) using dataset A 
% and its numeric labels. j is the feature used for splitting and t 
% the threshold. 
%
% [1] J.R. Quinlan, Simplifying Decision Trees, 
% Int. J. Man - Machine Studies, vol. 27, 1987, pp. 221-234.
% 
% See maketree, treec, classt, prune 

% Guido te Brake, TWI/SSOR, TU Delft.
% Copyright: R.P.W. Duin, [e-mail]
% Faculty of Applied Physics, Delft University of Technology
% P.O. Box 5046, 2600 GA Delft, The Netherlands

function crt = infstop(a,nlab,j,t)
	prtrace(mfilename);
	[m,k] = size(a);
	c = max(nlab);
	aj = a(:,j);
	ELAB = expandd(nlab); 
	L = sum(ELAB(aj <= t,:),1) + 0.001;
	R = sum(ELAB(aj > t,:),1) + 0.001;
	LL = (L+R) * sum(L) / m;
	RR = (L+R) * sum(R) / m;
	crt = sum(((L-LL).^2)./LL + ((R-RR).^2)./RR);
	return

%PRUNEP Pessimistic pruning of a decision tree
% 
% 	tree = prunep(tree,a,nlab,num)
% 
% Must be called by giving a tree and the training set a. num is the 
% starting node, if omitted pruning starts at the root. Pessimistic 
% pruning is defined by Quinlan.
% 
% See also maketree, treec, mapt 

% Guido te Brake, TWI/SSOR, TU Delft.
% Copyright: R.P.W. Duin, [e-mail]
% Faculty of Applied Physics, Delft University of Technology
% P.O. Box 5046, 2600 GA Delft, The Netherlands

function tree = prunep(tree,a,nlab,num)
	prtrace(mfilename);
	if nargin < 4, num = 1; end;
	[N,k] = size(a);
	c = size(tree,2)-4;
	if tree(num,3) == 0, return, end;
	w = mapping('treec','trained',{tree,num},[1:c]',k,c);
	ttt=tree_map(dataset(a,nlab),w);
	J = testc(ttt)*N;
	EA = J + nleaves(tree,num)./2;   % expected number of errors in tree
	P = sum(expandd(nlab,c),1);     % distribution of classes
					%disp([length(P) c])
					[pm,cm] = max(P);     % most frequent class
					E = N - pm;     % errors if substituted by leave
					SD = sqrt((EA * (N - EA))/N);
					if (E + 0.5) < (EA + SD)	     % clean tree while removing nodes
						[mt,kt] = size(tree);
						nodes = zeros(mt,1); nodes(num) = 1; n = 0;
						while sum(nodes) > n;	     % find all nodes to be removed
							n = sum(nodes);
							J = find(tree(:,3)>0 & nodes==1);
							nodes(tree(J,3)) = ones(length(J),1); 
							nodes(tree(J,4)) = ones(length(J),1); 
						end
						tree(num,:) = [cm 0 0 0 P/N];
						nodes(num) = 0; nc = cumsum(nodes);
						J = find(tree(:,3)>0);% update internal references
						tree(J,[3 4]) = tree(J,[3 4]) - reshape(nc(tree(J,[3 4])),length(J),2);
						tree = tree(~nodes,:);% remove obsolete nodes
					else 
						K1 = find(a(:,tree(num,1)) <= tree(num,2));
						K2 = find(a(:,tree(num,1)) >  tree(num,2));

						tree = prunep(tree,a(K1,:),nlab(K1),tree(num,3));
						tree = prunep(tree,a(K2,:),nlab(K2),tree(num,4));
					end
					return

%PRUNET Prune tree by testset
% 
% 	tree = prunet(tree,a)
% 
% The test set a is used to prune a decision tree. 

% Copyright: R.P.W. Duin, [e-mail]
% Faculty of Applied Physics, Delft University of Technology
% P.O. Box 5046, 2600 GA Delft, The Netherlands

function tree = prunet(tree,a)
	prtrace(mfilename);
	[m,k] = size(a);
	[n,s] = size(tree);
	c = s-4;
	erre = zeros(1,n);
	deln = zeros(1,n);
	w = mapping('treec','trained',{tree,1},[1:c]',k,c);
	[f,lab,nn] = tree_map(a,w);  % bug, this works only if a is dataset, labels ???
	[fmax,cmax] = max(tree(:,[5:4+c]),[],2);
	nngood = nn([1:n]'+(cmax-1)*n);
	errn = sum(nn,2) - nngood;% errors in each node
	sd = 1;
	while sd > 0
		erre = zeros(n,1);
		deln = zeros(1,n);
		endn = find(tree(:,3) == 0)';	% endnodes
		pendl = max(tree(:,3*ones(1,length(endn)))' == endn(ones(n,1),:)');
		pendr = max(tree(:,4*ones(1,length(endn)))' == endn(ones(n,1),:)');
		pend = find(pendl & pendr);		% parents of two endnodes
		erre(pend) = errn(tree(pend,3)) + errn(tree(pend,4));
		deln = pend(find(erre(pend) >= errn(pend))); % nodes to be leaved
		sd = length(deln);
		if sd > 0
			tree(tree(deln,3),:) = -1*ones(sd,s);
			tree(tree(deln,4),:) = -1*ones(sd,s);
			tree(deln,[1,2,3,4]) = [cmax(deln),zeros(sd,3)];
		end
	end
	return

%NLEAVES Computes the number of leaves in a decision tree
% 
% 	number = nleaves(tree,num)
% 
% This procedure counts the number of leaves in a (sub)tree of the 
% tree by using num. If num is omitted, the root is taken (num = 1).
% 
% This is a utility used by maketree. 

% Guido te Brake, TWI/SSOR, TU Delft
% Copyright: R.P.W. Duin, [e-mail]
% Faculty of Applied Physics, Delft University of Technology
% P.O. Box 5046, 2600 GA Delft, The Netherlands

function number = nleaves(tree,num)
	prtrace(mfilename);
	if nargin < 2, num = 1; end
	if tree(num,3) == 0
		number = 1 ;
	else
		number = nleaves(tree,tree(num,3)) + nleaves(tree,tree(num,4));
	end
	return