$\ell_1$ trend filtering

A derivative work by Judson Wilson, 5/28/2014.
Adapted from the CVX example of the same name, by Kwangmoo Koh, 12/10/2007

Topic Reference:


The problem of estimating underlying trends in time series data arises in a variety of disciplines. The $\ell_1$ trend filtering method produces trend estimates $x$ that are piecewise linear from the time series $y$.

The $\ell_1$ trend estimation problem can be formulated as \begin{array}{ll} \mbox{minimize} & (1/2)||y-x||_2^2 + \lambda ||Dx||_1, \end{array} with variable $x$ , and problem data $y$ and $\lambda$, with $\lambda >0$. $D$ is the second difference matrix, with rows $$\begin{bmatrix}0 & \cdots & 0 & -1 & 2 & -1 & 0 & \cdots & 0 \end{bmatrix}.$$ CVXPY is not optimized for the $\ell_1$ trend filtering problem. For large problems, use l1_tf (http://www.stanford.edu/~boyd/l1_tf/).

Formulate and solve problem

In [1]:
import numpy as np
import cvxpy as cvx
import scipy as scipy
import cvxopt as cvxopt

# Load time series data: S&P 500 price log.
y = np.loadtxt(open('data/snp500.txt', 'rb'), delimiter=",", skiprows=1)
n = y.size
# Form second difference matrix.
e = np.mat(np.ones((1, n)))
D = scipy.sparse.spdiags(np.vstack((e, -2*e, e)), range(3), n-2, n)
# Convert D to cvxopt sparse format, due to bug in scipy which prevents
# overloading neccessary for CVXPY. Use COOrdinate format as intermediate.
D_coo = D.tocoo()
D = cvxopt.spmatrix(D_coo.data, D_coo.row.tolist(), D_coo.col.tolist())

# Set regularization parameter.
vlambda = 50
# Solve l1 trend filtering problem.
x = cvx.Variable(n)
obj = cvx.Minimize(0.5 * cvx.sum_squares(y - x)
                   + vlambda * cvx.norm(D*x, 1) )
prob = cvx.Problem(obj)
# ECOS and SCS solvers fail to converge before
# the iteration limit. Use CVXOPT instead.

print 'Solver status: ', prob.status
# Check for error.
if prob.status != cvx.OPTIMAL:
    raise Exception("Solver did not converge!")
     pcost       dcost       gap    pres   dres   k/t
 0: -4.6668e-11 -1.0000e+00  1e+05  3e-01  4e-02  1e+00
 1:  5.3971e-02 -6.9550e-03  1e+04  2e-02  3e-03  1e-01
 2:  1.5513e-01  2.2236e-01  3e+03  7e-03  1e-03  1e-01
 3:  1.4714e-01  2.3287e-01  6e+02  1e-03  2e-04  1e-01
 4:  5.3138e-01  5.4713e-01  1e+02  2e-04  3e-05  2e-02
 5:  5.1410e-01  5.3063e-01  1e+02  2e-04  3e-05  2e-02
 6:  7.2454e-01  7.3795e-01  7e+01  1e-04  2e-05  1e-02
 7:  8.0702e-01  8.1346e-01  3e+01  6e-05  1e-05  7e-03
 8:  9.4942e-01  9.5462e-01  2e+01  5e-05  8e-06  6e-03
 9:  1.0232e+00  1.0254e+00  1e+01  2e-05  3e-06  2e-03
10:  1.1802e+00  1.1812e+00  4e+00  8e-06  1e-06  1e-03
11:  1.2596e+00  1.2601e+00  2e+00  4e-06  7e-07  5e-04
12:  1.3502e+00  1.3504e+00  1e+00  2e-06  3e-07  2e-04
13:  1.3794e+00  1.3794e+00  3e-01  6e-07  1e-07  8e-05
14:  1.3956e+00  1.3956e+00  1e-01  3e-07  4e-08  3e-05
15:  1.3993e+00  1.3993e+00  4e-02  8e-08  1e-08  1e-05
16:  1.3994e+00  1.3994e+00  4e-02  8e-08  1e-08  1e-05
17:  1.4009e+00  1.4009e+00  1e-02  2e-08  4e-09  3e-06
18:  1.4012e+00  1.4012e+00  7e-03  1e-08  2e-09  2e-06
19:  1.4014e+00  1.4014e+00  2e-03  4e-09  6e-10  5e-07
20:  1.4014e+00  1.4014e+00  2e-03  3e-09  5e-10  4e-07
21:  1.4014e+00  1.4014e+00  2e-04  3e-10  5e-11  4e-08
22:  1.4014e+00  1.4014e+00  3e-05  6e-11  9e-12  7e-09
23:  1.4014e+00  1.4014e+00  2e-06  3e-12  5e-13  4e-10
24:  1.4014e+00  1.4014e+00  2e-08  4e-14  6e-15  5e-12
Optimal solution found.
Solver status:  optimal

Results plot

In [2]:
import matplotlib.pyplot as plt

# Show plots inline in ipython.
%matplotlib inline

# Plot properties.
plt.rc('text', usetex=True)
plt.rc('font', family='serif')
font = {'family' : 'normal',
        'weight' : 'normal',
        'size'   : 16}
plt.rc('font', **font)

# Plot estimated trend with original signal.
plt.figure(figsize=(6, 6))
plt.plot(np.arange(1,n+1), y, 'k:', linewidth=1.0)
plt.plot(np.arange(1,n+1), np.array(x.value), 'b-', linewidth=2.0)
plt.ylabel('log price')
<matplotlib.text.Text at 0x10e95aad0>