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The following code uses rk4 to simulate the dynamics defined via fe function. However, I encountered the warning Warning: Matrix is singular, close to singular or badly scaled. Results may be inaccurate. RCOND = NaN. .
I felt that the function defined in fe, sometimes the Numerator equals zero, thus creates singularity. But I don't know how to avoid it.
Should I change the function formula (I prefer not to), or should I revise the code, or change the solver?
Thank you in advance for any suggestion.
clear;
global G
tic
n=40;
A = ones(n) - eye(n);
G = graph(A);
num_samples =1;
p.G = A
dim_G = size(p.G);
p.K = 1;
p.N = dim_G(1);
nIters = 2000;
tBegin = 0;
tEnd = 100;
% initial condition
Init = -pi + 2*pi.*rand(p.N,num_samples);
dim_thetaInit = size(Init);
Init = Init(:,1);
[t,sol] = rk4(#fe,tBegin,tEnd,Init,nIters,p);
function td = fe(t,theta,p)
[theta_i,theta_j] = meshgrid(theta);
adj_matrix = p.G;
a = 4;
b = 1;
td= sum(adj_matrix'.*( ( b*cos(theta_i-theta_j) * (2*a^2*cos(theta_i-theta_j)*sin(theta_i-theta_j)-2*b^2*cos(theta_i-theta_j)*sin(theta_i-theta_j)) ) / ( 2*(b^2*(cos(theta_i-theta_j))^2+a^2*(sin(theta_i-theta_j))^2)^(3/2) ) + (b*sin(theta_i-theta_j))/( sqrt( b^2*cos(theta_i-theta_j)^2 + a^2*sin(theta_i-theta_j)^2 ) ) ),1)';
end
function [T,Y] = rk4(f,a,b,ya,m,p)
%---------------------------------------------------------------------------
% RK4 Runge-Kutta solution for y' = f(t,y) with y(a) = ya .
% Sample call
% [T,Y] = rk4('f',a,b,ya,m)
% Inputs
% f name of the function
% a left endpoint of [a,b]
% b right endpoint of [a,b]
% ya initial value
% m number of steps
% p Kuramoto function arguments
% Return
% T solution: vector of abscissas
% Y solution: vector of ordinates
%
% NUMERICAL METHODS: MATLAB Programs, (c) John H . Mathews 1995
% To accompany the text:
% NUMERICAL METHODS for Mathematics, Science and Engineering, 2nd Ed, 1992
% Prentice Hall, Englewood Cliffs, New Jersey, 07632, U . S . A .
% Prentice Hall, Inc .; USA, Canada, Mexico ISBN 0-13-624990-6
% Prentice Hall, International Editions: ISBN 0-13-625047-5
% This free software is compliments of the author .
% E-mail address: in % "mathews#fullerton.edu"
%
% Algorithm 9.4 (Runge-Kutta Method of Order 4) .
% Section 9.5, Runge-Kutta Methods, Page 460
%---------------------------------------------------------------------------
h = (b - a)/m;
T = zeros(1,m+1);
size(ya,1)
Y = zeros(size(ya,1),m+1);
T(1) = a;
Y(:,1) = ya;
for j=1:m
tj = T(j);
yj = Y(:,j);
k1 = h*feval(f,tj,yj,p);
k2 = h*feval(f,tj+h/2,yj+k1/2,p);
k3 = h*feval(f,tj+h/2,yj+k2/2,p);
k4 = h*feval(f,tj+h,yj+k3,p);
Y(:,j+1) = yj + (k1 + 2*k2 + 2*k3 + k4)/6;
T(j+1) = a + h*j;
end
end
I am trying to estimate regression and AR parameters for (loads of) linear regressions with AR error terms. (You could also think of this as a MA process with exogenous variables):
, where
, with lags of length p
I am following the official matlab recommendations and use regArima to set up a number of regressions and extract regression and AR parameters (see reproducible example below).
The problem: regArima is slow! For 5 regressions, matlab needs 14.24sec. And I intend to run a large number of different regression models. Is there any quicker method around?
y = rand(100,1);
r2 = rand(100,1);
r3 = rand(100,1);
r4 = rand(100,1);
r5 = rand(100,1);
exo = [r2 r3 r4 r5];
tic
for p = 0:4
Mdl = regARIMA(3,0,0);
[EstMdl, ~, LogL] = estimate(Mdl,y,'X',exo,'Display','off');
end
toc
Unlike the regArima function which uses Maximum Likelihood, the Cochrane-Orcutt prodecure relies on an iteration of OLS regression. There are a few more particularities when this approach is valid (refer to the link posted). But for the aim of this question, the appraoch is valid, and fast!
I modified James Le Sage's code which covers only AR lags of order 1, to cover lags of order p.
function result = olsc(y,x,arterms)
% PURPOSE: computes Cochrane-Orcutt ols Regression for AR1 errors
%---------------------------------------------------
% USAGE: results = olsc(y,x)
% where: y = dependent variable vector (nobs x 1)
% x = independent variables matrix (nobs x nvar)
%---------------------------------------------------
% RETURNS: a structure
% results.meth = 'olsc'
% results.beta = bhat estimates
% results.rho = rho estimate
% results.tstat = t-stats
% results.trho = t-statistic for rho estimate
% results.yhat = yhat
% results.resid = residuals
% results.sige = e'*e/(n-k)
% results.rsqr = rsquared
% results.rbar = rbar-squared
% results.iter = niter x 3 matrix of [rho converg iteration#]
% results.nobs = nobs
% results.nvar = nvars
% results.y = y data vector
% --------------------------------------------------
% SEE ALSO: prt_reg(results), plt_reg(results)
%---------------------------------------------------
% written by:
% James P. LeSage, Dept of Economics
% University of Toledo
% 2801 W. Bancroft St,
% Toledo, OH 43606
% jpl#jpl.econ.utoledo.edu
% do error checking on inputs
if (nargin ~= 3); error('Wrong # of arguments to olsc'); end;
[nobs nvar] = size(x);
[nobs2 junk] = size(y);
if (nobs ~= nobs2); error('x and y must have same # obs in olsc'); end;
% ----- setup parameters
ITERMAX = 100;
converg = 1.0;
rho = zeros(arterms,1);
iter = 1;
% xtmp = lag(x,1);
% ytmp = lag(y,1);
% truncate 1st observation to feed the lag
% xlag = x(1:nobs-1,:);
% ylag = y(1:nobs-1,1);
yt = y(1+arterms:nobs,1);
xt = x(1+arterms:nobs,:);
xlag = zeros(nobs-arterms,arterms);
for tt = 1 : arterms
xlag(:,nvar*(tt-1)+1:nvar*(tt-1)+nvar) = x(arterms-tt+1:nobs-tt,:);
end
ylag = zeros(nobs-arterms,arterms);
for tt = 1 : arterms
ylag(:,tt) = y(arterms-tt+1:nobs-tt,:);
end
% setup storage for iteration results
iterout = zeros(ITERMAX,3);
while (converg > 0.0001) & (iter < ITERMAX),
% step 1, using intial rho = 0, do OLS to get bhat
ystar = yt - ylag*rho;
xstar = zeros(nobs-arterms,nvar);
for ii = 1 : nvar
tmp = zeros(1,arterms);
for tt = 1:arterms
tmp(1,tt)=ii+nvar*(tt-1);
end
xstar(:,ii) = xt(:,ii) - xlag(:,tmp)*rho;
end
beta = (xstar'*xstar)\xstar' * ystar;
e = y - x*beta;
% truncate 1st observation to account for the lag
et = e(1+arterms:nobs,1);
elagt = zeros(nobs-arterms,arterms);
for tt = 1 : arterms
elagt(:,tt) = e(arterms-tt+1:nobs-tt,:);
end
% step 2, update estimate of rho using residuals
% from step 1
res_rho = (elagt'*elagt)\elagt' * et;
rho_last = rho;
rho = res_rho;
converg = sum(abs(rho - rho_last));
% iterout(iter,1) = rho;
iterout(iter,2) = converg;
iterout(iter,3) = iter;
iter = iter + 1;
end; % end of while loop
if iter == ITERMAX
% error('ols_corc did not converge in 100 iterations');
print('ols_corc did not converge in 100 iterations');
end;
result.iter= iterout(1:iter-1,:);
% after convergence produce a final set of estimates using rho-value
ystar = yt - ylag*rho;
xstar = zeros(nobs-arterms,nvar);
for ii = 1 : nvar
tmp = zeros(1,arterms);
for tt = 1:arterms
tmp(1,tt)=ii+nvar*(tt-1);
end
xstar(:,ii) = xt(:,ii) - xlag(:,tmp)*rho;
end
result.beta = (xstar'*xstar)\xstar' * ystar;
e = y - x*result.beta;
et = e(1+arterms:nobs,1);
elagt = zeros(nobs-arterms,arterms);
for tt = 1 : arterms
elagt(:,tt) = e(arterms-tt+1:nobs-tt,:);
end
u = et - elagt*rho;
result.vare = std(u)^2;
result.meth = 'olsc';
result.rho = rho;
result.iter = iterout(1:iter-1,:);
% % compute t-statistic for rho
% varrho = (1-rho*rho)/(nobs-2);
% result.trho = rho/sqrt(varrho);
(I did not adapt in the last 2 lines the t-test for rho vectors of length p, but this should be straight forward to do..)
I am very new to Scilab, but so far have not been able to find an answer (either here or via google) to my question. I'm sure it's a simple solution, but I'm at a loss. I have a lot of MATLAB scripts I wrote in grad school, but now that I'm out of school, I no longer have access to MATLAB (and can't justify the cost). Scilab looked like the best open alternative. I'm trying to convert my .m files to Scilab compatible versions using mfile2sci, but when running the mfile2sci GUI, I get the error/message shown below. Attached is the original code from the M-file, in case it's relevant.
I Searched Stack Overflow and companion sites, Google, Scilab documentation.
The M-file code follows (it's a super basic MATLAB script as part of an old homework question -- I chose it as it's the shortest, most straightforward M-file I had):
Mmax = 15;
N = 20;
T = 2000;
%define upper limit for sparsity of signal
smax = 15;
mNE = zeros(smax,Mmax);
mESR= zeros(smax,Mmax);
for M = 1:Mmax
aNormErr = zeros(smax,1);
aSz = zeros(smax,1);
ESR = zeros(smax,1);
for s=1:smax % for-loop to loop script smax times
normErr = zeros(1,T);
vESR = zeros(1,T);
sz = zeros(1,T);
for t=1:T %for-loop to carry out 2000 trials per s-value
esr = 0;
A = randn(M,N); % generate random MxN matrix
[M,N] = size(A);
An = zeros(M,N); % initialize normalized matrix
for h = 1:size(A,2) % normalize columns of matrix A
V = A(:,h)/norm(A(:,h));
An(:,h) = V;
end
A = An; % replace A with its column-normalized counterpart
c = randperm(N,s); % create random support vector with s entries
x = zeros(N,1); % initialize vector x
for i = 1:size(c,2)
val = (10-1)*rand + 1;% generate interval [1,10]
neg = mod(randi(10),2); % include [-10,-1]
if neg~=0
val = -1*val;
end
x(c(i)) = val; %replace c(i)th value of x with the nonzero value
end
y = A*x; % generate measurement vector (y)
R = y;
S = []; % initialize array to store selected columns of A
indx = []; % vector to store indices of selected columns
coeff = zeros(1,s); % vector to store coefficients of approx.
stop = 10; % init. stop condition
in = 0; % index variable
esr = 0;
xhat = zeros(N,1); % intialize estimated x signal
while (stop>0.5 && size(S,2)<smax)
%MAX = abs(A(:,1)'*R);
maxV = zeros(1,N);
for i = 1:size(A,2)
maxV(i) = abs(A(:,i)'*R);
end
in = find(maxV == max(maxV));
indx = [indx in];
S = [S A(:,in)];
coeff = [coeff R'*S(:,size(S,2))]; % update coefficient vector
for w=1:size(S,2)
r = y - ((R'*S(:,w))*S(:,w)); % update residuals
if norm(r)<norm(R)
index = w;
end
R = r;
stop = norm(R); % update stop condition
end
for j=1:size(S,2) % place coefficients into xhat at correct indices
xhat(indx(j))=coeff(j);
end
nE = norm(x-xhat)/norm(x); % calculate normalized error for this estimate
%esr = 0;
indx = sort(indx);
c = sort(c);
if isequal(indx,c)
esr = esr+1;
end
end
vESR(t) = esr;
sz(t) = size(S,2);
normErr(t) = nE;
end
%avsz = sum(sz)/T;
aSz(s) = sum(sz)/T;
%aESR = sum(vESR)/T;
ESR(s) = sum(vESR)/T;
%avnormErr = sum(normErr)/T; % produce average normalized error for these run
aNormErr(s) = sum(normErr)/T; % add new avnormErr to vector of all av norm errors
end
% just put this here to view the vector
mNE(:,M) = aNormErr;
mESR(:,M) = ESR;
% had an 'end' placed here, might've been unmatched
mNE%reshape(mNE,[],Mmax)
mESR%reshape(mESR,[],Mmax)]
figure
dimx = [1 Mmax];
dimy = [1 smax];
imagesc(dimx,dimy,mESR)
colormap gray
strESR = sprintf('Average ESR, N=%d',N);
title(strESR);
xlabel('M');
ylabel('s');
strNE = sprintf('Average Normed Error, N=%d',N);
figure
imagesc(dimx,dimy,mNE)
colormap gray
title(strNE)
xlabel('M');
ylabel('s');
The command used (and results) follow:
--> mfile2sci
ans =
[]
****** Beginning of mfile2sci() session ******
File to convert: C:/Users/User/Downloads/WTF_new.m
Result file path: C:/Users/User/DOWNLO~1/
Recursive mode: OFF
Only double values used in M-file: NO
Verbose mode: 3
Generate formatted code: NO
M-file reading...
M-file reading: Done
Syntax modification...
Syntax modification: Done
File contains no instruction, no translation made...
****** End of mfile2sci() session ******
To convert the foo.m file one has to enter
mfile2sci <path>/foo.m
where stands for the path of the directoty where foo.m is. The result is written in /foo.sci
Remove the ```` at the begining of each line, the conversion will proceed normally ?. However, don't expect to obtain a working .sci file as the m2sci converter is (to me) still an experimental tool !
I tried to follow the main tutorial of PlatEMO but I failed to compile it. I tried to modify an already existed function but I've got too many errors.
This is the code I already tried:
classdef counster < PROBLEM
%HELP COUNSTER
methods
%% Initialization
function obj = counster()
obj.Global.M = 2;
if isempty(obj.Global.D)
obj.Global.D = 2;
end
obj.Global.lower =[0,zeros(1,obj.Global.D-1)-2];
obj.Global.upper = [1,zeros(1,obj.Global.D-1)+2];
obj.Global.encoding = 'real';
end
%% Calculate objective values
function PopObj = CalObj(obj,X)
PopObj(:,1) = X(1);
PopObj(:,2) = (1 + X(2))/X(1);
end
%% Calculate constraint violations
function PopCon = CalCon(obj,X)
PopCon(:,1)=-9*X(1)-X(2)+6;
PopCon(:,2)= -9*X(1)-X(2)+1;
end
end
end
and this is an example of built-in function which is correctly working:
classdef CF4 < PROBLEM
% <problem> <CF>
% Constrained benchmark MOP
%------------------------------- Reference --------------------------------
% Q. Zhang, A. Zhou, S. Zhao, P. N. Suganthan, W. Liu, and S. Tiwari,
% Multiobjective optimization test instances for the CEC 2009 special
% session and competition, School of CS & EE, University of Essex, Working
% Report CES-487, 2009.
%------------------------------- Copyright --------------------------------
% Copyright (c) 2018-2019 BIMK Group. You are free to use the PlatEMO for
% research purposes. All publications which use this platform or any code
% in the platform should acknowledge the use of "PlatEMO" and reference "Ye
% Tian, Ran Cheng, Xingyi Zhang, and Yaochu Jin, PlatEMO: A MATLAB platform
% for evolutionary multi-objective optimization [educational forum], IEEE
% Computational Intelligence Magazine, 2017, 12(4): 73-87".
%--------------------------------------------------------------------------
methods
%% Initialization
function obj = CF4()
obj.Global.M = 2;
if isempty(obj.Global.D)
obj.Global.D = 10;
end
obj.Global.lower = [0,zeros(1,obj.Global.D-1)-2];
obj.Global.upper = [1,zeros(1,obj.Global.D-1)+2];
obj.Global.encoding = 'real';
end
%% Calculate objective values
function PopObj = CalObj(obj,X)
D = size(X,2);
J1 = 3 : 2 : D;
J2 = 2 : 2 : D;
Y = X - sin(6*pi*repmat(X(:,1),1,D)+repmat(1:D,size(X,1),1)*pi/D);
h = Y.^2;
temp = Y(:,2) < 3/2*(1-sqrt(1/2));
h(temp,2) = abs(Y(temp,2));
h(~temp,2) = 0.125 + (Y(~temp,2)-1).^2;
PopObj(:,1) = X(:,1) + sum(h(:,J1),2);
PopObj(:,2) = 1-X(:,1) + sum(h(:,J2),2);
end
%% Calculate constraint violations
function PopCon = CalCon(obj,X)
t = X(:,2) - sin(6*pi*X(:,1)+2*pi/size(X,2)) - 0.5*X(:,1) + 0.25;
PopCon = -t./(1+exp(4*abs(t)));
end
%% Sample reference points on Pareto front
function P = PF(obj,N)
P(:,1) = (0:1/(N-1):1)';
P(:,2) = 1 - P(:,1);
temp1 = 0.5<P(:,1) & P(:,1)<=0.75;
temp2 = 0.75<P(:,1);
P(temp1,2) = -0.5*P(temp1,1)+3/4;
P(temp2,2) = 1 - P(temp2,1)+0.125;
end
end
end
>> main('-algorithm',#MOPSO,'-problem',#counster,'-N',200,'-M',10);
It meant to plot a pareto front of counster problem but I have got this:
Index exceeds matrix dimensions.
Error in INDIVIDUAL (line 79)
obj(i).obj = Objs(i,:);
Error in GLOBAL/Initialization (line 151)
Population = INDIVIDUAL(obj.problem.Init(N));
Error in MOPSO (line 23)
Population = Global.Initialization();
Error in GLOBAL/Start (line 120)
obj.algorithm(obj);
Error in main (line 62)
Global.Start();
I got this code for LARS but when I run, it says undefined X. I can't understand what x is. Why is there an error?
function [beta, A, mu, C, c, gamma] = lars(X, Y, option, t, standardize)
% Least Angle Regression (LAR) algorithm.
% Ref: Efron et. al. (2004) Least angle regression. Annals of Statistics.
% option = 'lar' implements the vanilla LAR algorithm (default);
% option = 'lasso' solves the lasso path with a modified LAR algorithm.
% t -- a vector of increasing positive real numbers. If given, LARS
% returns the solution at t.
%
% Output:
% A -- a sequence of indices that indicate the order of variable
% beta: history of estimated LARS coefficients;
% mu -- history of estimated mean vector;
% C -- history of maximal current absolute corrrelations;
% c -- history of current corrrelations;
% gamma: history of LARS step size.
% Note: history is traced by rows. If t is given, beta is just the
% estimated coefficient vector at the constraint ||beta||_1 = t.
%
% Remarks:
% 1. LARS is originally proposed to estimate a sparse coefficient vector
% a noisy over-determined linear system. LARS outputs estimates for all
% shrinkage/constraint parameters (homotopy).
%
% 2. LARS is well suited for Basis Pursuit (BP) purpose in the real
% automatically terminates when the current correlations for inactive
% all zeros. The recovered coefficient vector is the last column of beta
% with the *lasso* option. Hence, this function provides a fast and
% efficient solution for the ell_1 minimization problem.
% Ref: Donoho and Tsaig (2006). Fast solution of ell_1 norm minimization
if nargin < 5, standardize = true; end
if nargin < 4, t = Inf; end
if nargin < 3, option = 'lar'; end
if strcmpi(option, 'lasso'), lasso = 1; else, lasso = 0; end
eps = 1e-10; % Effective zero
[n,p] = size(X);
if standardize,
X = normalize(X);
Y = Y-mean(Y);
end
m = min(p,n-1); % Maximal number of variables in the final active set
T = length(t);
beta = zeros(1,p);
mu = zeros(n,1); % Mean vector
gamma = []; % LARS step lengths
A = [];
Ac = 1:p;
nVars = 0;
signOK = 1;
i = 0;
mu_old = zeros(n,1);
t_prev = 0;
beta_t = zeros(T,p);
ii = 1;
tt = t;
% LARS loop
while nVars < m,
i = i+1;
c = X'*(Y-mu); % Current correlation
C = max(abs(c)); % Maximal current absolute correlation
if C < eps || isempty(t), break; end % Early stopping criteria
if 1 == i, addVar = find(C==abs(c)); end
if signOK,
A = [A,addVar]; % Add one variable to active set
nVars = nVars+1;
end
s_A = sign(c(A));
Ac = setdiff(1:p,A); % Inactive set
nZeros = length(Ac);
X_A = X(:,A);
G_A = X_A'*X_A; % Gram matrix
invG_A = inv(G_A);
L_A = 1/sqrt(s_A'*invG_A*s_A);
w_A = L_A*invG_A*s_A; % Coefficients of equiangular vector u_A
u_A = X_A*w_A; % Equiangular vector
a = X'*u_A; % Angles between x_j and u_A
beta_tmp = zeros(p,1);
gammaTest = zeros(nZeros,2);
if nVars == m,
gamma(i) = C/L_A; % Move to the least squares projection
else
for j = 1:nZeros,
jj = Ac(j);
gammaTest(j,:) = [(C-c(jj))/(L_A-a(jj)), (C+c(jj))/(L_A+a(jj))];
end
[gamma(i) min_i min_j] = minplus(gammaTest);
addVar = unique(Ac(min_i));
end
beta_tmp(A) = beta(i,A)' + gamma(i)*w_A; % Update coefficient estimates
% Check the sign feasibility of lasso
if lasso,
signOK = 1;
gammaTest = -beta(i,A)'./w_A;
[gamma2 min_i min_j] = minplus(gammaTest);
if gamma2 < gamma(i), % The case when sign consistency gets violated
gamma(i) = gamma2;
beta_tmp(A) = beta(i,A)' + gamma(i)*w_A; % Correct the coefficients
beta_tmp(A(unique(min_i))) = 0;
A(unique(min_i)) = []; % Delete the zero-crossing variable (keep the ordering)
nVars = nVars-1;
signOK = 0;
end
end
if Inf ~= t(1),
t_now = norm(beta_tmp(A),1);
if t_prev < t(1) && t_now >= t(1),
beta_t(ii,A) = beta(i,A) + L_A*(t(1)-t_prev)*w_A'; % Compute coefficient estimates corresponding to a specific t
t(1) = [];
ii = ii+1;
end
t_prev = t_now;
end
mu = mu_old + gamma(i)*u_A; % Update mean vector
mu_old = mu;
beta = [beta; beta_tmp'];
end
if 1 < ii,
noCons = (tt > norm(beta_tmp,1));
if 0 < sum(noCons),
beta_t(noCons,:) = repmat(beta_tmp',sum(noCons),1);
end
beta = beta_t;
end
% Normalize columns of X to have mean zero and length one.
function sX = normalize(X)
[n,p] = size(X);
sX = X-repmat(mean(X),n,1);
sX = sX*diag(1./sqrt(ones(1,n)*sX.^2));
% Find the minimum and its index over the (strictly) positive part of X
% matrix
function [m, I, J] = minplus(X)
% Remove complex elements and reset to Inf
[I,J] = find(0~=imag(X));
for i = 1:length(I),
X(I(i),J(i)) = Inf;
end
X(X<=0) = Inf;
m = min(min(X));
[I,J] = find(X==m);
You can have more information in the related paper:
Efron, Bradley; Hastie, Trevor; Johnstone, Iain; Tibshirani, Robert. Least angle regression. Ann. Statist. 32 (2004), no. 2, 407--499. doi:10.1214/009053604000000067.
http://projecteuclid.org/euclid.aos/1083178935.