# Signal recovery by 1D total variation¶

In this example, we generate a signal that is piecewise constant. We then observe some random and corrupted measurements from that signal and then try to recover that signal using L1 and 1D total variation (TV1D) penalties.

Given a ground truth vectors, the signal that we observe is given by

y = sign(X ground_truth + noise)

where X is a random matrix. We obtain the vector ground_truth by solving an optimization problem using lightning’s `lightning.classification.FistaClassifier`.

The 1D total variation is also known as fused lasso. Python source code: `plot_1d_total_variation.py`

```# Author: Fabian Pedregosa <f@bianp.net>

import numpy as np
import matplotlib.pyplot as plt
from lightning.classification import FistaClassifier
from sklearn.grid_search import GridSearchCV

# generate some synthetic data
n_samples = 200
ground_truth = np.concatenate((
np.ones(20), - np.ones(20), np.zeros(40)))
n_features = ground_truth.size
np.random.seed(0)  # for reproducibility
X = np.random.rand(n_samples, n_features)
# generate y as a linear model, y = sign(X w + noise)
y = np.sign(X.dot(ground_truth) + 0.5 * np.random.randn(n_samples)).astype(np.int)

for penalty in ('l1', 'tv1d'):
clf = FistaClassifier(penalty=penalty)
gs = GridSearchCV(clf, {'alpha': np.logspace(-3, 3, 10)})
gs.fit(X, y)
coefs = gs.best_estimator_.coef_
plt.plot(coefs.ravel(), label='%s penalty' % penalty, lw=3)

plt.plot(ground_truth, lw=3, marker='^', markevery=5, markersize=10, label="ground truth")
plt.grid()
plt.legend()
plt.ylim((-1.5, 1.5))
plt.show()
```

Total running time of the example: 0.83 seconds ( 0 minutes 0.83 seconds)