code: def yule_walker(X, order=1, method="unbiased", df=None, inv=False):
      """Estimate AR(p) parameters from a sequence X using Yule-Walker equation.
  
      unbiased or maximum-likelihood estimator (mle)
  
      See, for example:
  
      http://en.wikipedia.org/wiki/Autoregressive_moving_average_model
  
      Parameters
      ----------
      X :  ndarray of shape(n)
      order : int, optional
          Order of AR process.
      method : str, optional
          Method can be "unbiased" or "mle" and this determines denominator in
          estimate of autocorrelation function (ACF) at lag k. If "mle", the
          denominator is n=X.shape[0], if "unbiased" the denominator is n-k.
      df : int, optional
          Specifies the degrees of freedom. If df is supplied, then it is assumed
          the X has df degrees of freedom rather than n.
      inv : bool, optional
          Whether to return the inverse of the R matrix (see code)
  
      Returns
      -------
      rho : (`order`,) ndarray
      sigma : int
          standard deviation of the residuals after fit
      R_inv : ndarray
          If `inv` is True, also return the inverse of the R matrix
  
      Notes
      -----
      See also
      http://en.wikipedia.org/wiki/AR_model#Calculation_of_the_AR_parameters
      """
      method = str(method).lower()
      if method not in ["unbiased", "mle"]:
          raise ValueError(
              "ACF estimation method must be 'unbiased or 'MLE'"
          )
      X = np.asarray(X, np.float64)
      if X.ndim != 1:
          raise ValueError(
              "Expecting a vector to estimate AR parameters"
          )
      X -= X.mean(0)
      n = df or X.shape[0]
      if method == "unbiased":
          den = lambda k: n - k
      else:
          den = lambda k: n
      r = np.zeros(order + 1, np.float64)
      r[0] = (X**2).sum() / den(0)
      for k in range(1, order + 1):
          r[k] = (X[0:-k] * X[k:]).sum() / den(k)
      R = spl.toeplitz(r[:-1])
      rho = spl.solve(R, r[1:])
      sigmasq = r[0] - (r[1:] * rho).sum()
      if inv == True:
          return rho, np.sqrt(sigmasq), spl.inv(R)
      return rho, np.sqrt(sigmasq)
  
imports: import warnings
  import numpy
  import numpy.linalg
  import scipy.linalg
  from nibabel.onetime import auto_attr
  from scipy import stats
  from nipy.algorithms.utils.matrices import matrix_rank, pos_recipr
  from .model import LikelihoodModel, LikelihoodModelResults
label_desc: Use image node_id 1 for calling a global function with example usage: yule_walker(X, order, method, df, inv) and returns: rho,  np, rho,  np,  spl