Matlab NaN and Inf issue - matlab

Question

So, I'm implementing the EM algorithm in Matlab, but my matrices quickly end up contaminated by NaN and Inf values. I think it might be caused by matrix inversions, but I'm not sure that's the only reason.
Here is the code:
function [F, Q, R, x_T, P_T] = em_algo(y, G)
% y_t = G_t'*x_t + v_t 1*1 = 1*p p*1
% x_t = F*x_t-1 + w_t p*1 = p*p p*1
% G is T*p
p = size(G,2); % p = nb assets ; G = T*p
q = size(y,2); % q = nb observations ; y = T*q
T = size(y,1); % y is T*1
F = eye(p); % = Transition matrix p*p
Q = eye(p); % innovation (v) covariance matrix p*p
R = eye(q); % noise (w) covariance matrix q x q
x_T_old = zeros(p,T);
mu0 = zeros(p,1);
Sigma = eye(p); % Initial state covariance matrix p*p
converged = 0;
i = 0;
max_iter = 60; % only for testing purposes
while ~converged
if i > max_iter
break;
end
% E step = smoothing
fprintf('Iteration %d\n',i);
[x_T,P_T,P_Tm2] = smoother(G,F,Q,R,mu0,Sigma,y);
%x_T
% M step
A = zeros(p,p);
B = zeros(p,p);
C = zeros(p,p);
R = eye(q);
for t = 2:T % eq (9) in EM paper
A = A + (P_T(:,:,t-1) + (x_T(:,t-1)*x_T(:,t-1)'));
end
for t = 2:T % eq (10)
%B = B + (P_Tm2(:,:,t-1) + (x_T(:,t)*x_T(:,t-1)'));
B = B + (P_Tm2(:,:,t) + (x_T(:,t)*x_T(:,t-1)'));
end
for t = 1:T %eq (11)
C = C + (P_T(:,:,t) + (x_T(:,t)*x_T(:,t)'));
end
F = B*inv(A); %eq (12)
Q = (1/T)*(C - (B*inv(A)*B')); % eq (13) pxp
for t = 1:T
bias = y(t) - (G(t,:)*x_T(:,t));
R = R + ((bias*bias') + (G(t,:)*P_T(:,:,t)*G(t,:)'));
end
R = (1/T)*R;
if i>1
err = norm(x_T-x_T_old)/norm(x_T_old);
if err < 1e-4
converged = 1;
end
end
x_T_old = x_T;
i = i+1;
end
fprintf('EM algorithm iterated %d times\n',i);
end
This iterates until convergence (which never happens due to my issue) and calls smoother.m at each iteration:
function [x_T, P_T, P_Tm2] = smoother(G,F,Q,R,mu0,Sigma,y)
% G is T*p
p = size(mu0,1); % mu0 is p*1
T = size(y,1); % y is T*1
J = zeros(p,p,T);
K = zeros(p,T); % gain matrix
x = zeros(p,T);
x(:,1) = mu0;
x_m1 = zeros(p,T);
x_T = zeros(p,T); % x values when we know all the data
% Notation : x = xt given t ; x_m1 = xt given t-1 (m1 stands for minus
% one)
P = zeros(p,p,T);% array of cov(xt|y1...yt), eq (6) in Shumway & Stoffer 1982
P(:,:,1) = Sigma;
P_m1 = zeros(p,p,T); % Same notation ; = cov(xt, xt-1|y1...yt) , eq (7)
P_T = zeros(p,p,T);
P_Tm2 = zeros(p,p,T); % cov(xT, xT-1|y1...yT)
for t = 2:T %starts at t = 2 because at each time t we need info about t-1
x_m1(:,t) = F*x(:,t-1); % eq A3 ; pxp * px1 = px1
P_m1(:,:,t) = (F*P(:,:,t-1)*F') + Q; % A4 ; pxp * pxp = pxp
if nnz(isnan(P_m1(:,:,t)))
error('NaNs in P_m1 at time t = %d',t);
end
if nnz(isinf(P_m1(:,:,t)))
error('Infs in P_m1 at time t = %d',t);
end
K(:,t) = P_m1(:,:,t)*G(t,:)'*pinv((G(t,:)*P_m1(:,:,t)*G(t,:)') + R); %A5 ; pxp * px1 * 1*1 = p*1
%K(:,t) = P_m1(:,:,t)*G(t,:)'/((G(t,:)*P_m1(:,:,t)*G(t,:)') + R); %A5 ; pxp * px1 * 1*1 = p*1
% The matrix inversion seems to generate NaN values which quickly
% contaminate all the other matrices. There is no warning about
% (close to) singular matrices or whatever. The use of pinv()
% instead of inv() seems to solve the problem... but I don't think
% it's the appropriate way to deal with it, there must be something
% wrong elsewhere
if nnz(isnan(K(:,t)))
error('NaNs in K at time t = %d',t);
end
x(:,t) = x_m1(:,t) + (K(:,t)*(y(t)-(G(t,:)*x_m1(:,t)))); %A6
P(:,:,t) = P_m1(:,:,t) - (K(:,t)*G(t,:)*P_m1(:,:,t)); %A7
end
x_T(:,T) = x(:,T);
P_T(:,:,T) = P(:,:,T);
for t = T:-1:2 % we stop at 2 since we need to use t-1
%P_m1 seem to get really huge (x10^22...), might lead to "Inf"
%values which in turn might screw pinv()
%% inv() caused NaN value to appear, pinv seems to solve the issue
J(:,:,t-1) = P(:,:,t-1)*F'*pinv(P_m1(:,:,t)); % A8 pxp * pxp * pxp
%J(:,:,t-1) = P(:,:,t-1)*F'/(P_m1(:,:,t)); % A8 pxp * pxp * pxp
x_T(:,t-1) = x(:,t-1) + J(:,:,t-1)*(x_T(:,t)-(F*x(:,t-1))); %A9 % Becomes NaN during 8th iteration!
P_T(:,:,t-1) = P(:,:,t-1) + J(:,:,t-1)*(P_T(:,:,t)-P_m1(:,:,t))*J(:,:,t-1)'; %A10
nans = [nnz(isnan(J)) nnz(isnan(P_m1)) nnz(isnan(F)) nnz(isnan(x_T)) nnz(isnan(x_m1))];
if nnz(nans)
error('NaN invasion at time t = %d',t);
end
end
P_Tm2(:,:,T) = (eye(p) - K(:,T)*G(T,:))*F*P(:,:,T-1); % %A12
for t = T:-1:3 % stop at 3 because use of t-2
P_Tm2(:,:,t-1) = P_m1(:,:,t-1)*J(:,:,t-2)' + J(:,:,t-1)*(P_Tm2(:,:,t)-F*P(:,:,t-1))*J(:,:,t-2)'; % A11
end
end
The NaNs and Infs start popping around the ~8th iteration.
I guess in there somewhere I'm doing something unholy with my matrices, but I really have no clue about what's wrong. I trust your expertise.
Thanks in advance for the help.
Rody :
Here is how I generate the data (it's not "real world" data yet, just some test data generated to check that nothing goes wront) :
T = 500;
nbassets = 3;
G = .1 + randn(T,nbassets); % random walk trajectories
y = (1:T).';
y = 1.01.^y; % 1 * T % Exponential 1% returns curve
Dan :
You're right. I indeed lack the math background to really understand how the formulas are derived. I know it doesn't help, but I'm not sure I can remedy that for the time being. :/
Rody : Yes indeed, I arrived at the same conclusion. But I really have no clue what makes it go wrong like that.
Here is a link to the paper :
http://www.stat.pitt.edu/stoffer/em.pdf
The formulas for the smoother are all at the very end, in the appendix. Thanks for your time so far.

Answer 1

As the user appears to have inserted the answer into his question I will post it here:
As mentioned by #Rody the cause of the problem was that the use of inv created NaN or Inf values.
The user 'solved' this by using pinv instead.