# -*- coding: utf-8 -*-
r"""
LB_Keogh
========

This example illustrates the principle of time series envelope and its
relationship to the "LB_Keogh" lower bound [1].

The envelope of a time series consists of two time series such that the
original time series is between the two time series. Denoting the original
time series :math:`X = (X_i)_{1 \leq i \leq n}`, the envelope of this time
series is an ensemble of two time series of same length
:math:`L = (l_i)_{1 \leq i \leq n}` and :math:`U = (u_i)_{1 \leq i \leq n}`
such that for all :math:`i \in \{1, \ldots, n\}`:

.. math::

    u_i = \max(x_{i - r}, \ldots, x_{i + r})

    l_i = \min(x_{i - r}, \ldots, x_{i + r})

where :math:`r` is the radius of the envelope.

The distance between a time series $Q$ and an envelope :math:`(L, U)` is
defined as:

.. math::

    LB_{Keogh}(Q, (L, U)) = \sqrt{\sum_{i=1}^n
    \begin{cases}
          (q_i - u_i)^2 & \text{if $q_i > u_i$}\\
          (q_i - l_i)^2 & \text{if $q_i < l_i$}\\
          0 & \text{otherwise}
    \end{cases}
    }

So it is simply the Euclidean distance between :math:`Q` and the envelope.

[1] E. Keogh and C. A. Ratanamahatana, "Exact indexing of dynamic time
warping". Knowledge and Information Systems, 7(3), 358-386 (2004).
"""

# Author: Romain Tavenard
#         Johann Faouzi
# License: BSD 3 clause
# sphinx_gallery_thumbnail_number = 2

import numpy
import matplotlib.pyplot as plt

from tslearn.generators import random_walks
from tslearn.preprocessing import TimeSeriesScalerMeanVariance
from tslearn import metrics

numpy.random.seed(0)
n_ts, sz, d = 2, 100, 1
dataset = random_walks(n_ts=n_ts, sz=sz, d=d)
scaler = TimeSeriesScalerMeanVariance(mu=0., std=1.)  # Rescale time series
dataset_scaled = scaler.fit_transform(dataset)

plt.figure(figsize=(14, 8))
envelope_down, envelope_up = metrics.lb_envelope(dataset_scaled[0], radius=3)
plt.plot(dataset_scaled[0, :, 0], "r-", label='First time series')
plt.plot(envelope_down[:, 0], "b-", label='Lower envelope')
plt.plot(envelope_up[:, 0], "g-", label='Upper envelope')
plt.legend()
plt.title('Envelope around a time series with radius=3')

plt.figure(figsize=(14, 8))
plt.plot(envelope_down[:, 0], "b-", label='Lower envelope')
plt.plot(envelope_up[:, 0], "g-", label='Upper envelope')
plt.plot(dataset_scaled[1, :, 0], "k-", label='Second time series')
plt.vlines(numpy.arange(sz), dataset_scaled[1, :, 0], numpy.clip(
    dataset_scaled[1, :, 0], envelope_down[:, 0], envelope_up[:, 0]),
           label='Distance', color='orange')
plt.legend()
lb_k_sim = metrics.lb_keogh(dataset_scaled[1],
                            envelope_candidate=(envelope_down, envelope_up))
plt.title('Distance between the second time series and \n'
          'the envelope = {:.4f}'.format(lb_k_sim))

plt.show()