% Written for CVX by Kwangmoo Koh - 12/10/07
%
% The problem of estimating underlying trends in time series data arises in
% a variety of disciplines. The l1 trend filtering method produces trend
% estimates x that are piecewise linear from the time series y.
%
% The l1 trend estimation problem can be formulated as
%
%    minimize    (1/2)*||y-x||^2+lambda*||Dx||_1,
%
% with variable x , and problem data y and lambda, with lambda >0.
% D is the second difference matrix, with rows [0... -1 2 -1 ...0]
%
% CVX is not optimized for the l1 trend filtering problem.
% For large problems, use l1_tf (www.stanford.edu/~boyd/l1_tf/).

% load time series data
y = csvread('snp500.txt'); % log price of snp500
n = length(y);

% form second difference matrix
e = ones(n,1);
D = spdiags([e -2*e e], 0:2, n-2, n);

% set regularization parameter
lambda = 50;

% solve l1 trend filtering problem
cvx_begin
    variable x(n)
    minimize( 0.5*sum_square(y-x)+lambda*norm(D*x,1) )
cvx_end

% plot estimated trend with original signal
figure(1);
plot(1:n,y,'k:','LineWidth',1.0); hold on;
plot(1:n,x,'b-','LineWidth',2.0); hold off;
xlabel('date'); ylabel('log price');

 
Calling Mosek 9.1.9: 5998 variables, 1999 equality constraints
------------------------------------------------------------

MOSEK Version 9.1.9 (Build date: 2019-11-21 11:32:15)
Copyright (c) MOSEK ApS, Denmark. WWW: mosek.com
Platform: MACOSX/64-X86

Problem
  Name                   :                 
  Objective sense        : min             
  Type                   : CONIC (conic optimization problem)
  Constraints            : 1999            
  Cones                  : 1999            
  Scalar variables       : 5998            
  Matrix variables       : 0               
  Integer variables      : 0               

Optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator started.
Freed constraints in eliminator : 0
Eliminator terminated.
Eliminator - tries                  : 1                 time                   : 0.00            
Lin. dep.  - tries                  : 1                 time                   : 0.00            
Lin. dep.  - number                 : 0               
Presolve terminated. Time: 0.00    
Problem
  Name                   :                 
  Objective sense        : min             
  Type                   : CONIC (conic optimization problem)
  Constraints            : 1999            
  Cones                  : 1999            
  Scalar variables       : 5998            
  Matrix variables       : 0               
  Integer variables      : 0               

Optimizer  - threads                : 8               
Optimizer  - solved problem         : the primal      
Optimizer  - Constraints            : 1999
Optimizer  - Cones                  : 1999
Optimizer  - Scalar variables       : 5998              conic                  : 5998            
Optimizer  - Semi-definite variables: 0                 scalarized             : 0               
Factor     - setup time             : 0.00              dense det. time        : 0.00            
Factor     - ML order time          : 0.00              GP order time          : 0.00            
Factor     - nonzeros before factor : 7992              after factor           : 7993            
Factor     - dense dim.             : 2                 flops                  : 6.39e+04        
ITE PFEAS    DFEAS    GFEAS    PRSTATUS   POBJ              DOBJ              MU       TIME  
0   2.3e-02  4.9e+01  1.0e+05  0.00e+00   9.990050000e+04   0.000000000e+00   1.0e+00  0.01  
1   2.7e-05  5.6e-02  7.8e+02  -9.60e-01  5.378312314e+03   -2.260254267e+01  1.2e-03  0.02  
2   6.0e-07  1.3e-03  2.9e+00  8.95e-01   1.285302418e+02   -2.380654240e-01  2.6e-05  0.03  
3   8.9e-08  1.9e-04  1.6e-01  9.97e-01   1.976496798e+01   6.736607045e-01   3.9e-06  0.03  
4   3.9e-08  8.2e-05  4.3e-02  1.00e+00   9.272705213e+00   9.560293535e-01   1.7e-06  0.03  
5   1.9e-08  4.1e-05  1.3e-02  1.00e+00   5.234715125e+00   1.120063862e+00   8.3e-07  0.04  
6   8.1e-09  1.7e-05  3.0e-03  1.00e+00   2.995294554e+00   1.254542023e+00   3.5e-07  0.04  
7   3.2e-09  6.8e-06  6.3e-04  1.00e+00   2.037621544e+00   1.347499165e+00   1.4e-07  0.04  
8   1.3e-09  2.8e-06  1.4e-04  1.00e+00   1.664946352e+00   1.383078762e+00   5.7e-08  0.04  
9   4.6e-10  9.8e-07  2.4e-05  1.00e+00   1.495757307e+00   1.396235461e+00   2.0e-08  0.05  
10  1.4e-10  2.9e-07  3.2e-06  1.00e+00   1.430163148e+00   1.400436699e+00   6.0e-09  0.05  
11  5.7e-11  1.2e-07  8.0e-07  1.00e+00   1.413534193e+00   1.401210314e+00   2.5e-09  0.05  
12  1.6e-11  3.4e-08  1.1e-07  1.00e+00   1.405002838e+00   1.401545731e+00   6.9e-10  0.06  
13  7.0e-12  1.5e-08  3.1e-08  1.00e+00   1.403101534e+00   1.401589316e+00   3.0e-10  0.06  
14  1.8e-12  3.8e-09  3.9e-09  1.00e+00   1.402002257e+00   1.401612236e+00   7.8e-11  0.06  
15  4.5e-13  2.2e-09  4.5e-10  1.00e+00   1.401713452e+00   1.401617081e+00   1.9e-11  0.07  
16  3.5e-14  1.5e-08  1.0e-11  1.00e+00   1.401625686e+00   1.401618075e+00   1.5e-12  0.07  
17  1.4e-13  2.3e-07  1.2e-15  1.00e+00   1.401618171e+00   1.401618153e+00   3.6e-15  0.07  
Optimizer terminated. Time: 0.08    


Interior-point solution summary
  Problem status  : PRIMAL_AND_DUAL_FEASIBLE
  Solution status : OPTIMAL
  Primal.  obj: 1.4016181708e+00    nrm: 1e+00    Viol.  con: 9e-12    var: 0e+00    cones: 0e+00  
  Dual.    obj: 1.4016181528e+00    nrm: 5e+01    Viol.  con: 0e+00    var: 1e-14    cones: 0e+00  
Optimizer summary
  Optimizer                 -                        time: 0.08    
    Interior-point          - iterations : 17        time: 0.07    
      Basis identification  -                        time: 0.00    
        Primal              - iterations : 0         time: 0.00    
        Dual                - iterations : 0         time: 0.00    
        Clean primal        - iterations : 0         time: 0.00    
        Clean dual          - iterations : 0         time: 0.00    
    Simplex                 -                        time: 0.00    
      Primal simplex        - iterations : 0         time: 0.00    
      Dual simplex          - iterations : 0         time: 0.00    
    Mixed integer           - relaxations: 0         time: 0.00    

------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +1.40162