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function beta = larsen(X, y, lambda2, stop, trace)
% LARSEN The LARSEN algorithm for elastic net regression.
% BETA = LARSEN(X, Y) performs elastic net regression on the variables
% in X to approximate the response Y. Variables X are assumed to be
% normalized (zero mean, unit length), the response Y is assumed to be
% centered. The ridge term coefficient, lambda2, has a default value of
% 1e-6. This keeps the ridge influence low while making p > n possible.
% BETA = LARSEN(X, Y, LAMBDA2) adds a user-specified LAMBDA2. LAMBDA2 =
% 0 produces the lasso solution.
% BETA = LARSEN(X, Y, LAMBDA2, STOP) with nonzero STOP will perform
% elastic net regression with early stopping. If STOP is negative, its
% absolute value corresponds to the desired number of variables. If STOP
% is positive, it corresponds to an upper bound on the L1-norm of the
% BETA coefficients.
% BETA = LARSEN(X, Y, LAMBDA2, STOP, TRACE) with nonzero TRACE will
% print the adding and subtracting of variables as all elastic net
% solutions are found.
% Returns BETA where each row contains the predictor coefficients of
% one iteration. A suitable row is chosen using e.g. cross-validation,
% possibly including interpolation to achieve sub-iteration accuracy.
%
% Author: Karl Skoglund, IMM, DTU, kas@imm.dtu.dk
% Reference: 'Regularization and Variable Selection via the Elastic Net' by
% Hui Zou and Trevor Hastie, 2005.
%% Input checking
if nargin < 5
trace = 0;
end
if nargin < 4
stop = 0;
end
if nargin < 3
lambda2 = 1e-6;
end
%% Elastic net variable setups
[n p] = size(X);
maxk = 8*(n+p); % Maximum number of iterations
if lambda2 < eps
nvars = min(n-1,p); %Pure LASSO
else
nvars = p; % Elastic net
end
if stop > 0,
stop = stop/sqrt(1 + lambda2);
end
if stop == 0
beta = zeros(2*nvars, p);
elseif stop < 0
beta = zeros(2*round(-stop), p);
else
beta = zeros(100, p);
end
mu = zeros(n, 1); % current "position" as LARS-EN travels towards lsq solution
I = 1:p; % inactive set
A = []; % active set
R = []; % Cholesky factorization R'R = X'X where R is upper triangular
lassocond = 0; % Set to 1 if LASSO condition is met
stopcond = 0; % Set to 1 if early stopping condition is met
k = 0; % Algorithm step count
vars = 0; % Current number of variables
d1 = sqrt(lambda2); % Convenience variables d1 and d2
d2 = 1/sqrt(1 + lambda2);
if trace
disp(sprintf('Step\tAdded\tDropped\t\tActive set size'));
end
%% Elastic net main loop
while vars < nvars && ~stopcond && k < maxk
k = k + 1;
c = X'*(y - mu)*d2;
[C j] = max(abs(c(I)));
j = I(j);
if ~lassocond % if a variable has been dropped, do one iteration with this configuration (don't add new one right away)
R = cholinsert(R,X(:,j),X(:,A),lambda2);
A = [A j];
I(I == j) = [];
vars = vars + 1;
if trace
disp(sprintf('%d\t\t%d\t\t\t\t\t%d', k, j, vars));
end
end
s = sign(c(A)); % get the signs of the correlations
GA1 = R\(R'\s);
AA = 1/sqrt(sum(GA1.*s));
w = AA*GA1;
u1 = X(:,A)*w*d2; % equiangular direction (unit vector) part 1
u2 = zeros(p, 1); u2(A) = d1*d2*w; % part 2
if vars == nvars % if all variables active, go all the way to the lsq solution
gamma = C/AA;
else
a = (X'*u1 + d1*u2)*d2; % correlation between each variable and eqiangular vector
temp = [(C - c(I))./(AA - a(I)); (C + c(I))./(AA + a(I))];
gamma = min([temp(temp > 0); C/AA]);
end
% LASSO modification
lassocond = 0;
temp = -beta(k,A)./w';
[gamma_tilde] = min([temp(temp > 0) gamma]);
j = find(temp == gamma_tilde);
if gamma_tilde < gamma,
gamma = gamma_tilde;
lassocond = 1;
end
mu = mu + gamma*u1;
if size(beta,1) < k+1
beta = [beta; zeros(size(beta,1), p)];
end
beta(k+1,A) = beta(k,A) + gamma*w';
% Early stopping at specified bound on L1 norm of beta
if stop > 0
t2 = sum(abs(beta(k+1,:)));
if t2 >= stop
t1 = sum(abs(beta(k,:)));
s = (stop - t1)/(t2 - t1); % interpolation factor 0 < s < 1
beta(k+1,:) = beta(k,:) + s*(beta(k+1,:) - beta(k,:));
stopcond = 1;
end
end
% If LASSO condition satisfied, drop variable from active set
if lassocond == 1
R = choldelete(R,j);
I = [I A(j)];
A(j) = [];
vars = vars - 1;
if trace
disp(sprintf('%d\t\t\t\t%d\t\t\t%d', k, j, vars));
end
end
% Early stopping at specified number of variables
if stop < 0
stopcond = vars >= -stop;
end
end
% trim beta
if size(beta,1) > k+1
beta(k+2:end, :) = [];
end
% divide by d2 to avoid double shrinkage
beta = beta/d2;
if k == maxk
disp('LARS-EN warning: Forced exit. Maximum number of iteration reached.');
end
%% Fast Cholesky insert and remove functions
% Updates R in a Cholesky factorization R'R = X'X of a data matrix X. R is
% the current R matrix to be updated. x is a column vector representing the
% variable to be added and X is the data matrix containing the currently
% active variables (not including x).
function R = cholinsert(R, x, X, lambda)
diag_k = (x'*x + lambda)/(1 + lambda); % diagonal element k in X'X matrix
if isempty(R)
R = sqrt(diag_k);
else
col_k = x'*X/(1 + lambda); % elements of column k in X'X matrix
R_k = R'\col_k'; % R'R_k = (X'X)_k, solve for R_k
R_kk = sqrt(diag_k - R_k'*R_k); % norm(x'x) = norm(R'*R), find last element by exclusion
R = [R R_k; [zeros(1,size(R,2)) R_kk]]; % update R
end
% Deletes a variable from the X'X matrix in a Cholesky factorisation R'R =
% X'X. Returns the downdated R. This function is just a stripped version of
% Matlab's qrdelete.
function R = choldelete(R,j)
R(:,j) = []; % remove column j
n = size(R,2);
for k = j:n
p = k:k+1;
[G,R(p,k)] = planerot(R(p,k)); % remove extra element in column
if k < n
R(p,k+1:n) = G*R(p,k+1:n); % adjust rest of row
end
end
R(end,:) = []; % remove zero'ed out row |
methode elastic net regression
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