原始论文

M.T. Rosenstein, J.J. Collins, and C.J. De Luca. A practical method for calculating largest Lyapunov exponents from small data sets. Physica D, 65:117-134, 1993.

下载地址：https://www.physionet.org/content/lyapunov/1.0.0/

python 相关代码

• NonLinear Time Series Analysis（nolitsa）
  
• NOnLinear measures for Dynamical Systems (nolds)
  

混沌系统的常见指标

区分确定性混沌系统与噪声已成为许多不同领域的重要问题。

对于实验产生的时间序列，可以计算这些混沌系统的指标：

• 相关维数（$D_2$），
• Kolmogorov 熵
• Lyapunov 特征指数。

相关维度是对系统复杂程度的估计，熵和特征指数是对混沌程度的估计。

最大李亚普诺夫指数的含义

LLE 描述了相空间中相近的两点（初始间距为$C$）随时间推移指数分离的速率：
$$d(t)=C{e}^{{\lambda }_{1}t}$$其中$d(t)$表示分离距离，$C$表示初始间距，${\lambda }_1$ 为最大李氏指数。

算法流程图

python 代码模块

最近邻

import numpy as np

from scipy import stats
from scipy.spatial import cKDTree as KDTree
from scipy.spatial import distance


def neighbors(y, metric='chebyshev', window=0, maxnum=None):
    """Find nearest neighbors of all points in the given array.

    Finds the nearest neighbors of all points in the given array using
    SciPy's KDTree search.

    Parameters
    ----------
    y : ndarray
        N-dimensional array containing time-delayed vectors.
    metric : string, optional (default = 'chebyshev')
        Metric to use for distance computation.  Must be one of
        "cityblock" (aka the Manhattan metric), "chebyshev" (aka the
        maximum norm metric), or "euclidean".
    window : int, optional (default = 0)
        Minimum temporal separation (Theiler window) that should exist
        between near neighbors.  This is crucial while computing
        Lyapunov exponents and the correlation dimension.
    maxnum : int, optional (default = None (optimum))
        Maximum number of near neighbors that should be found for each
        point.  In rare cases, when there are no neighbors that are at a
        nonzero distance, this will have to be increased (i.e., beyond
        2 * window + 3).

    Returns
    -------
    index : array
        Array containing indices of near neighbors.
    dist : array
        Array containing near neighbor distances.
    """
    if metric == 'cityblock':
        p = 1
    elif metric == 'euclidean':
        p = 2
    elif metric == 'chebyshev':
        p = np.inf
    else:
        raise ValueError('Unknown metric.  Should be one of "cityblock", '
                         '"euclidean", or "chebyshev".')

    tree = KDTree(y)
    n = len(y)

    if not maxnum:
        maxnum = (window + 1) + 1 + (window + 1)
    else:
        maxnum = max(1, maxnum)

    if maxnum >= n:
        raise ValueError('maxnum is bigger than array length.')

    dists = np.empty(n)
    indices = np.empty(n, dtype=int)

    for i, x in enumerate(y):
        for k in range(2, maxnum + 2):
            dist, index = tree.query(x, k=k, p=p)
            valid = (np.abs(index - i) > window) & (dist > 0)

            if np.count_nonzero(valid):
                dists[i] = dist[valid][0]
                indices[i] = index[valid][0]
                break

            if k == (maxnum + 1):
                raise Exception('Could not find any near neighbor with a '
                                'nonzero distance.  Try increasing the '
                                'value of maxnum.')

    return np.squeeze(indices), np.squeeze(dists)

maximum Lyapunov exponent

def mle(y, maxt=500, window=10, metric='euclidean', maxnum=None):
    """Estimate the maximum Lyapunov exponent.

    Estimates the maximum Lyapunov exponent (MLE) from a
    multi-dimensional series using the algorithm described by
    Rosenstein et al. (1993).

    Parameters
    ----------
    y : ndarray
        Multi-dimensional real input array containing points in the
        phase space.
    maxt : int, optional (default = 500)
        Maximum time (iterations) up to which the average divergence
        should be computed.
    window : int, optional (default = 10)
        Minimum temporal separation (Theiler window) that should exist
        between near neighbors (see Notes).
    maxnum : int, optional (default = None (optimum))
        Maximum number of near neighbors that should be found for each
        point.  In rare cases, when there are no neighbors that are at a
        nonzero distance, this will have to be increased (i.e., beyond
        2 * window + 3).

    Returns
    -------
    d : array
        Average divergence for each time up to maxt.

    Notes
    -----
    This function does not directly estimate the MLE.  The MLE should be
    estimated by linearly fitting the average divergence (i.e., the
    average of the logarithms of near-neighbor distances) with time.
    It is also important to choose an appropriate Theiler window so that
    the near neighbors do not lie on the same trajectory, in which case
    the estimated MLE will always be close to zero.
    """
    index, dist = utils.neighbors(y, metric=metric, window=window,
                                  maxnum=maxnum)
    m = len(y)
    maxt = min(m - window - 1, maxt)

    d = np.empty(maxt)
    d[0] = np.mean(np.log(dist))

    for t in range(1, maxt):
        t1 = np.arange(t, m)
        t2 = index[:-t] + t

        # Sometimes the nearest point would be farther than (m - maxt)
        # in time.  Such trajectories needs to be omitted.
        valid = t2 < m
        t1, t2 = t1[valid], t2[valid]

        d[t] = np.mean(np.log(utils.dist(y[t1], y[t2], metric=metric)))

    return d

RANSAC 拟合曲线

需要先安装 sklearn 库

def poly_fit(x, y, degree, fit="RANSAC"):
  # check if we can use RANSAC
  if fit == "RANSAC":
    try:
      # ignore ImportWarnings in sklearn
      with warnings.catch_warnings():
        warnings.simplefilter("ignore", ImportWarning)
        import sklearn.linear_model as sklin
        import sklearn.preprocessing as skpre
    except ImportError:
      warnings.warn(
        "fitting mode 'RANSAC' requires the package sklearn, using"
        + " 'poly' instead",
        RuntimeWarning)
      fit = "poly"

  if fit == "poly":
    return np.polyfit(x, y, degree)
  elif fit == "RANSAC":
    model = sklin.RANSACRegressor(sklin.LinearRegression(fit_intercept=False))
    xdat = np.asarray(x)
    if len(xdat.shape) == 1:
      # interpret 1d-array as list of len(x) samples instead of
      # one sample of length len(x)
      xdat = xdat.reshape(-1, 1)
    polydat = skpre.PolynomialFeatures(degree).fit_transform(xdat)
    try:
      model.fit(polydat, y)
      coef = model.estimator_.coef_[::-1]
    except ValueError:
      warnings.warn(
        "RANSAC did not reach consensus, "
        + "using numpy's polyfit",
        RuntimeWarning)
      coef = np.polyfit(x, y, degree)
    return coef
  else:
    raise ValueError("invalid fitting mode ({})".format(fit))

例子：计算洛伦兹系统的最大李雅普诺夫指数

import warnings
from nolitsa import data, lyapunov
import numpy as np
import matplotlib.pyplot as plt


dt = 0.01
x0 = [0.62225717, -0.08232857, 30.60845379]
x = data.lorenz(length=4000, sample=dt, x0=x0,
               sigma=16.0, beta=4.0, rho=45.92)[1]
plt.plot(range(len(x)),x)
plt.show()

# Choose appropriate Theiler window.
meanperiod = 30
maxt = 250
d = lyapunov.mle(x, maxt=maxt, window=meanperiod)
t = np.arange(maxt) *dt
coefs = poly_fit(t, d, 1)
print('LLE = ', coefs[0])

plt.title('Maximum Lyapunov exponent for the Lorenz system')
plt.xlabel(r'Time $t$')
plt.ylabel(r'Average divergence $\langle d_i(t) \rangle$')
plt.plot(t, d, label='divergence')
plt.plot(t, t * 1.50, '--', label='slope=1.5')
plt.plot(t, coefs[1] +coefs[0]* t, '--', label='RANSAC')
plt.legend()
plt.show()