code: def acf(
      x,
      adjusted=False,
      nlags=None,
      qstat=False,
      fft=True,
      alpha=None,
      bartlett_confint=True,
      missing="none",
  ):
      """
      Calculate the autocorrelation function.
  
      Parameters
      ----------
      x : array_like
         The time series data.
      adjusted : bool, default False
         If True, then denominators for autocovariance are n-k, otherwise n.
      nlags : int, optional
          Number of lags to return autocorrelation for. If not provided,
          uses min(10 * np.log10(nobs), nobs - 1). The returned value
          includes lag 0 (ie., 1) so size of the acf vector is (nlags + 1,).
      qstat : bool, default False
          If True, returns the Ljung-Box q statistic for each autocorrelation
          coefficient.  See q_stat for more information.
      fft : bool, default True
          If True, computes the ACF via FFT.
      alpha : scalar, default None
          If a number is given, the confidence intervals for the given level are
          returned. For instance if alpha=.05, 95 % confidence intervals are
          returned where the standard deviation is computed according to
          Bartlett"s formula.
      bartlett_confint : bool, default True
          Confidence intervals for ACF values are generally placed at 2
          standard errors around r_k. The formula used for standard error
          depends upon the situation. If the autocorrelations are being used
          to test for randomness of residuals as part of the ARIMA routine,
          the standard errors are determined assuming the residuals are white
          noise. The approximate formula for any lag is that standard error
          of each r_k = 1/sqrt(N). See section 9.4 of [2] for more details on
          the 1/sqrt(N) result. For more elementary discussion, see section 5.3.2
          in [3].
          For the ACF of raw data, the standard error at a lag k is
          found as if the right model was an MA(k-1). This allows the possible
          interpretation that if all autocorrelations past a certain lag are
          within the limits, the model might be an MA of order defined by the
          last significant autocorrelation. In this case, a moving average
          model is assumed for the data and the standard errors for the
          confidence intervals should be generated using Bartlett's formula.
          For more details on Bartlett formula result, see section 7.2 in [2].
      missing : str, default "none"
          A string in ["none", "raise", "conservative", "drop"] specifying how
          the NaNs are to be treated. "none" performs no checks. "raise" raises
          an exception if NaN values are found. "drop" removes the missing
          observations and then estimates the autocovariances treating the
          non-missing as contiguous. "conservative" computes the autocovariance
          using nan-ops so that nans are removed when computing the mean
          and cross-products that are used to estimate the autocovariance.
          When using "conservative", n is set to the number of non-missing
          observations.
  
      Returns
      -------
      acf : ndarray
          The autocorrelation function for lags 0, 1, ..., nlags. Shape
          (nlags+1,).
      confint : ndarray, optional
          Confidence intervals for the ACF at lags 0, 1, ..., nlags. Shape
          (nlags + 1, 2). Returned if alpha is not None.
      qstat : ndarray, optional
          The Ljung-Box Q-Statistic for lags 1, 2, ..., nlags (excludes lag
          zero). Returned if q_stat is True.
      pvalues : ndarray, optional
          The p-values associated with the Q-statistics for lags 1, 2, ...,
          nlags (excludes lag zero). Returned if q_stat is True.
  
      Notes
      -----
      The acf at lag 0 (ie., 1) is returned.
  
      For very long time series it is recommended to use fft convolution instead.
      When fft is False uses a simple, direct estimator of the autocovariances
      that only computes the first nlag + 1 values. This can be much faster when
      the time series is long and only a small number of autocovariances are
      needed.
  
      If adjusted is true, the denominator for the autocovariance is adjusted
      for the loss of data.
  
      References
      ----------
      .. [1] Parzen, E., 1963. On spectral analysis with missing observations
         and amplitude modulation. Sankhya: The Indian Journal of
         Statistics, Series A, pp.383-392.
      .. [2] Brockwell and Davis, 1987. Time Series Theory and Methods
      .. [3] Brockwell and Davis, 2010. Introduction to Time Series and
         Forecasting, 2nd edition.
      """
      adjusted = bool_like(adjusted, "adjusted")
      nlags = int_like(nlags, "nlags", optional=True)
      qstat = bool_like(qstat, "qstat")
      fft = bool_like(fft, "fft", optional=False)
      alpha = float_like(alpha, "alpha", optional=True)
      missing = string_like(
          missing,
          "missing",
          options=("none", "raise", "conservative", "drop"),
      )
      x = array_like(x, "x")
      # TODO: should this shrink for missing="drop" and NaNs in x?
      nobs = x.shape[0]
      if nlags is None:
          nlags = min(int(10 * np.log10(nobs)), nobs - 1)
  
      avf = acovf(
          x, adjusted=adjusted, demean=True, fft=fft, missing=missing
      )
      acf = avf[: nlags + 1] / avf[0]
      if not (qstat or alpha):
          return acf
      _alpha = alpha if alpha is not None else 0.05
      if bartlett_confint:
          varacf = np.ones_like(acf) / nobs
          varacf[0] = 0
          varacf[1] = 1.0 / nobs
          varacf[2:] *= 1 + 2 * np.cumsum(acf[1:-1] ** 2)
      else:
          varacf = 1.0 / len(x)
      interval = stats.norm.ppf(1 - _alpha / 2.0) * np.sqrt(varacf)
      confint = np.array(lzip(acf - interval, acf + interval))
      if not qstat:
          return acf, confint
      qstat, pvalue = q_stat(acf[1:], nobs=nobs)  # drop lag 0
      if alpha is not None:
          return acf, confint, qstat, pvalue
      else:
          return acf, qstat, pvalue
  
imports: from __future__ import annotations
  from statsmodels.compat.numpy import lstsq
  from statsmodels.compat.pandas import deprecate_kwarg
  from statsmodels.compat.python import Literal, lzip
  from statsmodels.compat.scipy import _next_regular
  from typing import Union
  import warnings
  import numpy
  from numpy.linalg import LinAlgError
  import pandas
  from scipy import stats
  from scipy.interpolate import interp1d
  from scipy.signal import correlate
  from statsmodels.regression.linear_model import OLS, yule_walker
  from statsmodels.tools.sm_exceptions import CollinearityWarning, InfeasibleTestError, InterpolationWarning, MissingDataError, ValueWarning
  from statsmodels.tools.tools import Bunch, add_constant
  from statsmodels.tools.validation import array_like, bool_like, dict_like, float_like, int_like, string_like
  from statsmodels.tsa._bds import bds
  from statsmodels.tsa._innovations import innovations_algo, innovations_filter
  from statsmodels.tsa.adfvalues import mackinnoncrit, mackinnonp
  from statsmodels.tsa.tsatools import add_trend, lagmat, lagmat2ds
label_desc: Use image node_id 5 for calling a global function with example usage: acf(x, adjusted, nlags, qstat, fft, alpha, bartlett_confint, missing) and returns: acf, acf,  confint, acf,  confint,  qstat,  pvalue, acf,  qstat,  pvalue