Merge branch 'master' of github.com:robince/partial-info-decomp

calc_pi_Idep_mvn.m

@@ -0,0 +1,91 @@
+function Id = calc_pi_Idep_mvn(C, varsizes);
+% calc_pi_Idep_mvn(C, varsizes)
+% Calculate bivariate PID using Idep approach (James et al. 2017) for
+% Gaussians (Kay & Ince 2018). 
+% C is the full joint covariance matrix with variables in order
+% X0 (first predictor), X1 (second predictor), S (target)
+% varsizes is a length 3 vector containing the dimensionality of each of
+% the above.
+% Returns a length 4 vector containing
+% [Red Unq_1 Unq_2 Syn]
+%
+% James et al. 2017 http://arxiv.org/abs/1709.06653
+% Kay & Ince 2018 
+
+vs = varsizes;
+if length(vs)~=3
+    error('calc_pi_Idep_mvn: only bivariate PID supported')
+end
+if sum(vs)~=size(C,1) || size(C,1)~=size(C,2)
+    error('calc_pi_Idep_mvn: problem with variable specifications')
+end
+
+% variable indexes
+xidx = 1:vs(1);
+yidx = (vs(1)+1):(vs(1)+vs(2));
+zidx = (vs(1)+vs(2)+1):(vs(1)+vs(2)+vs(3));
+
+% extract blockwise covariance components
+Cxx = C(xidx,xidx);
+Cyy = C(yidx,yidx);
+Czz = C(zidx,zidx);
+Cxy = C(xidx,yidx);
+Cxz = C(xidx,zidx);
+Cyz = C(yidx,zidx);
+
+% cholesky square root
+chCxx = chol(Cxx);
+chCyy = chol(Cyy);
+chCzz = chol(Czz);
+
+% Kay & Ince eq. D2
+P = pinv(chCxx)*Cxy*pinv(chCyy);
+Q = pinv(chCxx)*Cxz*pinv(chCzz);
+R = pinv(chCyy)*Cyz*pinv(chCzz);
+
+% standard mutual informations
+mivs = [vs(1)+vs(2)  vs(3)];
+IXY = gauss_mi(C,mivs);
+IY = gauss_mi(C([yidx zidx],[yidx zidx]), vs(2:3));
+Ct = zeros(vs(1)+vs(3));
+tzidx = (vs(1)+1):(vs(1)+vs(3));
+Ct(xidx,xidx) = Cxx;
+Ct(tzidx,tzidx) = Czz;
+Ct(xidx,tzidx) = Cxz;
+Ct(tzidx,xidx) = Cxz';
+IX = gauss_mi(Ct, [vs(1) vs(3)]);
+
+halflog2det = @(X) sum(log2(diag(chol(X))));
+% Dependency lattice edges (Kay & Ince Table 9)
+b = IX;
+
+inum = halflog2det(eye(vs(1)) - R*Q'*Q*R');
+iden = halflog2det(eye(vs(2))-Q'*Q) + halflog2det(eye(vs(2))-R'*R);
+i = inum - iden - IY;
+
+knum = halflog2det(eye(vs(1)) - P'*P);
+Ct = zeros(sum(vs));
+Ct(xidx,xidx) = eye(vs(1));
+Ct(yidx,yidx) = eye(vs(2));
+Ct(zidx,zidx) = eye(vs(3));
+Ct(xidx,yidx) = P;
+Ct(yidx,xidx) = P';
+Ct(xidx,zidx) = Q;
+Ct(zidx,xidx) = Q';
+Ct(yidx,zidx) = R;
+Ct(zidx,yidx) = R';
+kden = halflog2det(Ct);
+k = knum - kden - IY;
+
+% fill out PID
+Xunq = min([b i k]);
+Id = zeros(1,4);
+Id(2) = Xunq;
+Id(1) = IX - Xunq;
+Id(3) = IY - Id(1);
+Id(4) = IXY - sum(Id(1:3));
+
+
+
+
+

gauss_mi.m

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+function I = gauss_mi(C, varsizes)
+
+% C = C + 0.1*eye(size(C));
+vs = varsizes;
+if length(vs)>2
+    error('bivariate MI only')
+end
+if sum(vs) ~= size(C,1)
+    error('variables incorrectly specified')
+end
+
+CX = C(1:vs(1),1:vs(1));
+CY = C((vs(1)+1):end, (vs(1)+1):end);
+
+% use closed form expression
+chX = chol(CX);
+chY = chol(CY);
+chXY = chol(C);
+% normalisations cancel for information
+HX = sum(log(diag(chX))); % + 0.5*Nvarx*log(2*pi*exp(1));
+HY = sum(log(diag(chY))); % + 0.5*Nvary*log(2*pi*exp(1));
+HXY = sum(log(diag(chXY))); % + 0.5*(Nvarx+Nvary)*log(2*pi*exp(1));
+I = (HX + HY - HXY) / log(2);