Source code for statsmodels.tsa.arima_process

"""ARMA process and estimation with scipy.signal.lfilter

Notes
-----
* written without textbook, works but not sure about everything
  briefly checked and it looks to be standard least squares, see below

* theoretical autocorrelation function of general ARMA
  Done, relatively easy to guess solution, time consuming to get
  theoretical test cases, example file contains explicit formulas for
  acovf of MA(1), MA(2) and ARMA(1,1)

Properties:
Judge, ... (1985): The Theory and Practise of Econometrics

Author: josefpktd
License: BSD
"""
import numpy as np
from scipy import signal, optimize, linalg

from statsmodels.compat.pandas import Appender
from statsmodels.tools.docstring import remove_parameters, Docstring
from statsmodels.tools.validation import array_like

__all__ = ['arma_acf', 'arma_acovf', 'arma_generate_sample',
           'arma_impulse_response', 'arma2ar', 'arma2ma', 'deconvolve',
           'lpol2index', 'index2lpol']


NONSTATIONARY_ERROR = """\
The model's autoregressive parameters (ar) indicate that the process
 is non-stationary. arma_acovf can only be used with stationary processes.
"""


[docs]def arma_generate_sample(ar, ma, nsample, scale=1, distrvs=None, axis=0, burnin=0): """ Simulate data from an ARMA. Parameters ---------- ar : array_like The coefficient for autoregressive lag polynomial, including zero lag. ma : array_like The coefficient for moving-average lag polynomial, including zero lag. nsample : int or tuple of ints If nsample is an integer, then this creates a 1d timeseries of length size. If nsample is a tuple, creates a len(nsample) dimensional time series where time is indexed along the input variable ``axis``. All series are unless ``distrvs`` generates dependent data. scale : float The standard deviation of noise. distrvs : function, random number generator A function that generates the random numbers, and takes ``size`` as argument. The default is np.random.standard_normal. axis : int See nsample for details. burnin : int Number of observation at the beginning of the sample to drop. Used to reduce dependence on initial values. Returns ------- ndarray Random sample(s) from an ARMA process. Notes ----- As mentioned above, both the AR and MA components should include the coefficient on the zero-lag. This is typically 1. Further, due to the conventions used in signal processing used in signal.lfilter vs. conventions in statistics for ARMA processes, the AR parameters should have the opposite sign of what you might expect. See the examples below. Examples -------- >>> import numpy as np >>> np.random.seed(12345) >>> arparams = np.array([.75, -.25]) >>> maparams = np.array([.65, .35]) >>> ar = np.r_[1, -arparams] # add zero-lag and negate >>> ma = np.r_[1, maparams] # add zero-lag >>> y = sm.tsa.arma_generate_sample(ar, ma, 250) >>> model = sm.tsa.ARIMA(y, (2, 0, 2), trend='n').fit(disp=0) >>> model.params array([ 0.79044189, -0.23140636, 0.70072904, 0.40608028]) """ distrvs = np.random.standard_normal if distrvs is None else distrvs if np.ndim(nsample) == 0: nsample = [nsample] if burnin: # handle burin time for nd arrays # maybe there is a better trick in scipy.fft code newsize = list(nsample) newsize[axis] += burnin newsize = tuple(newsize) fslice = [slice(None)] * len(newsize) fslice[axis] = slice(burnin, None, None) fslice = tuple(fslice) else: newsize = tuple(nsample) fslice = tuple([slice(None)] * np.ndim(newsize)) eta = scale * distrvs(size=newsize) return signal.lfilter(ma, ar, eta, axis=axis)[fslice]
[docs]def arma_acovf(ar, ma, nobs=10, sigma2=1, dtype=None): """ Theoretical autocovariances of stationary ARMA processes Parameters ---------- ar : array_like, 1d The coefficients for autoregressive lag polynomial, including zero lag. ma : array_like, 1d The coefficients for moving-average lag polynomial, including zero lag. nobs : int The number of terms (lags plus zero lag) to include in returned acovf. sigma2 : float Variance of the innovation term. Returns ------- ndarray The autocovariance of ARMA process given by ar, ma. See Also -------- arma_acf : Autocorrelation function for ARMA processes. acovf : Sample autocovariance estimation. References ---------- .. [*] Brockwell, Peter J., and Richard A. Davis. 2009. Time Series: Theory and Methods. 2nd ed. 1991. New York, NY: Springer. """ if dtype is None: dtype = np.common_type(np.array(ar), np.array(ma), np.array(sigma2)) p = len(ar) - 1 q = len(ma) - 1 m = max(p, q) + 1 if sigma2.real < 0: raise ValueError('Must have positive innovation variance.') # Short-circuit for trivial corner-case if p == q == 0: out = np.zeros(nobs, dtype=dtype) out[0] = sigma2 return out elif p > 0 and np.max(np.abs(np.roots(ar))) >= 1: raise ValueError(NONSTATIONARY_ERROR) # Get the moving average representation coefficients that we need ma_coeffs = arma2ma(ar, ma, lags=m) # Solve for the first m autocovariances via the linear system # described by (BD, eq. 3.3.8) A = np.zeros((m, m), dtype=dtype) b = np.zeros((m, 1), dtype=dtype) # We need a zero-right-padded version of ar params tmp_ar = np.zeros(m, dtype=dtype) tmp_ar[:p + 1] = ar for k in range(m): A[k, :(k + 1)] = tmp_ar[:(k + 1)][::-1] A[k, 1:m - k] += tmp_ar[(k + 1):m] b[k] = sigma2 * np.dot(ma[k:q + 1], ma_coeffs[:max((q + 1 - k), 0)]) acovf = np.zeros(max(nobs, m), dtype=dtype) try: acovf[:m] = np.linalg.solve(A, b)[:, 0] except np.linalg.LinAlgError: raise ValueError(NONSTATIONARY_ERROR) # Iteratively apply (BD, eq. 3.3.9) to solve for remaining autocovariances if nobs > m: zi = signal.lfiltic([1], ar, acovf[:m:][::-1]) acovf[m:] = signal.lfilter([1], ar, np.zeros(nobs - m, dtype=dtype), zi=zi)[0] return acovf[:nobs]
[docs]def arma_acf(ar, ma, lags=10): """ Theoretical autocorrelation function of an ARMA process. Parameters ---------- ar : array_like Coefficients for autoregressive lag polynomial, including zero lag. ma : array_like Coefficients for moving-average lag polynomial, including zero lag. lags : int The number of terms (lags plus zero lag) to include in returned acf. Returns ------- ndarray The autocorrelations of ARMA process given by ar and ma. See Also -------- arma_acovf : Autocovariances from ARMA processes. acf : Sample autocorrelation function estimation. acovf : Sample autocovariance function estimation. """ acovf = arma_acovf(ar, ma, lags) return acovf / acovf[0]
[docs]def arma_pacf(ar, ma, lags=10): """ Theoretical partial autocorrelation function of an ARMA process. Parameters ---------- ar : array_like, 1d The coefficients for autoregressive lag polynomial, including zero lag. ma : array_like, 1d The coefficients for moving-average lag polynomial, including zero lag. lags : int The number of terms (lags plus zero lag) to include in returned pacf. Returns ------- ndarrray The partial autocorrelation of ARMA process given by ar and ma. Notes ----- Solves yule-walker equation for each lag order up to nobs lags. not tested/checked yet """ # TODO: Should use rank 1 inverse update apacf = np.zeros(lags) acov = arma_acf(ar, ma, lags=lags + 1) apacf[0] = 1. for k in range(2, lags + 1): r = acov[:k] apacf[k - 1] = linalg.solve(linalg.toeplitz(r[:-1]), r[1:])[-1] return apacf
[docs]def arma_periodogram(ar, ma, worN=None, whole=0): """ Periodogram for ARMA process given by lag-polynomials ar and ma. Parameters ---------- ar : array_like The autoregressive lag-polynomial with leading 1 and lhs sign. ma : array_like The moving average lag-polynomial with leading 1. worN : {None, int}, optional An option for scipy.signal.freqz (read "w or N"). If None, then compute at 512 frequencies around the unit circle. If a single integer, the compute at that many frequencies. Otherwise, compute the response at frequencies given in worN. whole : {0,1}, optional An options for scipy.signal.freqz/ Normally, frequencies are computed from 0 to pi (upper-half of unit-circle. If whole is non-zero compute frequencies from 0 to 2*pi. Returns ------- w : ndarray The frequencies. sd : ndarray The periodogram, also known as the spectral density. Notes ----- Normalization ? This uses signal.freqz, which does not use fft. There is a fft version somewhere. """ w, h = signal.freqz(ma, ar, worN=worN, whole=whole) sd = np.abs(h) ** 2 / np.sqrt(2 * np.pi) if np.any(np.isnan(h)): # this happens with unit root or seasonal unit root' import warnings warnings.warn('Warning: nan in frequency response h, maybe a unit ' 'root', RuntimeWarning) return w, sd
[docs]def arma_impulse_response(ar, ma, leads=100): """ Compute the impulse response function (MA representation) for ARMA process. Parameters ---------- ar : array_like, 1d The auto regressive lag polynomial. ma : array_like, 1d The moving average lag polynomial. leads : int The number of observations to calculate. Returns ------- ndarray The impulse response function with nobs elements. Notes ----- This is the same as finding the MA representation of an ARMA(p,q). By reversing the role of ar and ma in the function arguments, the returned result is the AR representation of an ARMA(p,q), i.e ma_representation = arma_impulse_response(ar, ma, leads=100) ar_representation = arma_impulse_response(ma, ar, leads=100) Fully tested against matlab Examples -------- AR(1) >>> arma_impulse_response([1.0, -0.8], [1.], leads=10) array([ 1. , 0.8 , 0.64 , 0.512 , 0.4096 , 0.32768 , 0.262144 , 0.2097152 , 0.16777216, 0.13421773]) this is the same as >>> 0.8**np.arange(10) array([ 1. , 0.8 , 0.64 , 0.512 , 0.4096 , 0.32768 , 0.262144 , 0.2097152 , 0.16777216, 0.13421773]) MA(2) >>> arma_impulse_response([1.0], [1., 0.5, 0.2], leads=10) array([ 1. , 0.5, 0.2, 0. , 0. , 0. , 0. , 0. , 0. , 0. ]) ARMA(1,2) >>> arma_impulse_response([1.0, -0.8], [1., 0.5, 0.2], leads=10) array([ 1. , 1.3 , 1.24 , 0.992 , 0.7936 , 0.63488 , 0.507904 , 0.4063232 , 0.32505856, 0.26004685]) """ impulse = np.zeros(leads) impulse[0] = 1. return signal.lfilter(ma, ar, impulse)
[docs]def arma2ma(ar, ma, lags=100): """ A finite-lag approximate MA representation of an ARMA process. Parameters ---------- ar : ndarray The auto regressive lag polynomial. ma : ndarray The moving average lag polynomial. lags : int The number of coefficients to calculate. Returns ------- ndarray The coefficients of AR lag polynomial with nobs elements. Notes ----- Equivalent to ``arma_impulse_response(ma, ar, leads=100)`` """ return arma_impulse_response(ar, ma, leads=lags)
[docs]def arma2ar(ar, ma, lags=100): """ A finite-lag AR approximation of an ARMA process. Parameters ---------- ar : array_like The auto regressive lag polynomial. ma : array_like The moving average lag polynomial. lags : int The number of coefficients to calculate. Returns ------- ndarray The coefficients of AR lag polynomial with nobs elements. Notes ----- Equivalent to ``arma_impulse_response(ma, ar, leads=100)`` """ return arma_impulse_response(ma, ar, leads=lags)
# moved from sandbox.tsa.try_fi
[docs]def ar2arma(ar_des, p, q, n=20, mse='ar', start=None): """ Find arma approximation to ar process. This finds the ARMA(p,q) coefficients that minimize the integrated squared difference between the impulse_response functions (MA representation) of the AR and the ARMA process. This does not check whether the MA lag polynomial of the ARMA process is invertible, neither does it check the roots of the AR lag polynomial. Parameters ---------- ar_des : array_like The coefficients of original AR lag polynomial, including lag zero. p : int The length of desired AR lag polynomials. q : int The length of desired MA lag polynomials. n : int The number of terms of the impulse_response function to include in the objective function for the approximation. mse : str, 'ar' Not used. start : ndarray Initial values to use when finding the approximation. Returns ------- ar_app : ndarray The coefficients of the AR lag polynomials of the approximation. ma_app : ndarray The coefficients of the MA lag polynomials of the approximation. res : tuple The result of optimize.leastsq. Notes ----- Extension is possible if we want to match autocovariance instead of impulse response function. """ # TODO: convert MA lag polynomial, ma_app, to be invertible, by mirroring # TODO: roots outside the unit interval to ones that are inside. How to do # TODO: this? # p,q = pq def msear_err(arma, ar_des): ar, ma = np.r_[1, arma[:p - 1]], np.r_[1, arma[p - 1:]] ar_approx = arma_impulse_response(ma, ar, n) return (ar_des - ar_approx) # ((ar - ar_approx)**2).sum() if start is None: arma0 = np.r_[-0.9 * np.ones(p - 1), np.zeros(q - 1)] else: arma0 = start res = optimize.leastsq(msear_err, arma0, ar_des, maxfev=5000) arma_app = np.atleast_1d(res[0]) ar_app = np.r_[1, arma_app[:p - 1]], ma_app = np.r_[1, arma_app[p - 1:]] return ar_app, ma_app, res
_arma_docs = {'ar': arma2ar.__doc__, 'ma': arma2ma.__doc__}
[docs]def lpol2index(ar): """ Remove zeros from lag polynomial Parameters ---------- ar : array_like coefficients of lag polynomial Returns ------- coeffs : ndarray non-zero coefficients of lag polynomial index : ndarray index (lags) of lag polynomial with non-zero elements """ ar = array_like(ar, 'ar') index = np.nonzero(ar)[0] coeffs = ar[index] return coeffs, index
[docs]def index2lpol(coeffs, index): """ Expand coefficients to lag poly Parameters ---------- coeffs : ndarray non-zero coefficients of lag polynomial index : ndarray index (lags) of lag polynomial with non-zero elements Returns ------- ar : array_like coefficients of lag polynomial """ n = max(index) ar = np.zeros(n + 1) ar[index] = coeffs return ar
[docs]def lpol_fima(d, n=20): """MA representation of fractional integration .. math:: (1-L)^{-d} for |d|<0.5 or |d|<1 (?) Parameters ---------- d : float fractional power n : int number of terms to calculate, including lag zero Returns ------- ma : ndarray coefficients of lag polynomial """ # hide import inside function until we use this heavily from scipy.special import gammaln j = np.arange(n) return np.exp(gammaln(d + j) - gammaln(j + 1) - gammaln(d))
# moved from sandbox.tsa.try_fi
[docs]def lpol_fiar(d, n=20): """AR representation of fractional integration .. math:: (1-L)^{d} for |d|<0.5 or |d|<1 (?) Parameters ---------- d : float fractional power n : int number of terms to calculate, including lag zero Returns ------- ar : ndarray coefficients of lag polynomial Notes: first coefficient is 1, negative signs except for first term, ar(L)*x_t """ # hide import inside function until we use this heavily from scipy.special import gammaln j = np.arange(n) ar = - np.exp(gammaln(-d + j) - gammaln(j + 1) - gammaln(-d)) ar[0] = 1 return ar
# moved from sandbox.tsa.try_fi
[docs]def lpol_sdiff(s): """return coefficients for seasonal difference (1-L^s) just a trivial convenience function Parameters ---------- s : int number of periods in season Returns ------- sdiff : list, length s+1 """ return [1] + [0] * (s - 1) + [-1]
[docs]def deconvolve(num, den, n=None): """Deconvolves divisor out of signal, division of polynomials for n terms calculates den^{-1} * num Parameters ---------- num : array_like signal or lag polynomial denom : array_like coefficients of lag polynomial (linear filter) n : None or int number of terms of quotient Returns ------- quot : ndarray quotient or filtered series rem : ndarray remainder Notes ----- If num is a time series, then this applies the linear filter den^{-1}. If both num and den are both lag polynomials, then this calculates the quotient polynomial for n terms and also returns the remainder. This is copied from scipy.signal.signaltools and added n as optional parameter. """ num = np.atleast_1d(num) den = np.atleast_1d(den) N = len(num) D = len(den) if D > N and n is None: quot = [] rem = num else: if n is None: n = N - D + 1 input = np.zeros(n, float) input[0] = 1 quot = signal.lfilter(num, den, input) num_approx = signal.convolve(den, quot, mode='full') if len(num) < len(num_approx): # 1d only ? num = np.concatenate((num, np.zeros(len(num_approx) - len(num)))) rem = num - num_approx return quot, rem
_generate_sample_doc = Docstring(arma_generate_sample.__doc__) _generate_sample_doc.remove_parameters(['ar', 'ma']) _generate_sample_doc.replace_block('Notes', []) _generate_sample_doc.replace_block('Examples', [])
[docs]class ArmaProcess(object): r""" Theoretical properties of an ARMA process for specified lag-polynomials. Parameters ---------- ar : array_like Coefficient for autoregressive lag polynomial, including zero lag. Must be entered using the signs from the lag polynomial representation. See the notes for more information about the sign. ma : array_like Coefficient for moving-average lag polynomial, including zero lag. nobs : int, optional Length of simulated time series. Used, for example, if a sample is generated. See example. Notes ----- Both the AR and MA components must include the coefficient on the zero-lag. In almost all cases these values should be 1. Further, due to using the lag-polynomial representation, the AR parameters should have the opposite sign of what one would write in the ARMA representation. See the examples below. The ARMA(p,q) process is described by .. math:: y_{t}=\phi_{1}y_{t-1}+\ldots+\phi_{p}y_{t-p}+\theta_{1}\epsilon_{t-1} +\ldots+\theta_{q}\epsilon_{t-q}+\epsilon_{t} and the parameterization used in this function uses the lag-polynomial representation, .. math:: \left(1-\phi_{1}L-\ldots-\phi_{p}L^{p}\right)y_{t} = \left(1+\theta_{1}L+\ldots+\theta_{q}L^{q}\right)\epsilon_{t} Examples -------- ARMA(2,2) with AR coefficients 0.75 and -0.25, and MA coefficients 0.65 and 0.35 >>> import statsmodels.api as sm >>> import numpy as np >>> np.random.seed(12345) >>> arparams = np.array([.75, -.25]) >>> maparams = np.array([.65, .35]) >>> ar = np.r_[1, -arparams] # add zero-lag and negate >>> ma = np.r_[1, maparams] # add zero-lag >>> arma_process = sm.tsa.ArmaProcess(ar, ma) >>> arma_process.isstationary True >>> arma_process.isinvertible True >>> arma_process.arroots array([1.5-1.32287566j, 1.5+1.32287566j]) >>> y = arma_process.generate_sample(250) >>> model = sm.tsa.ARIMA(y, (2, 0, 2), trend='n').fit(disp=0) >>> model.params array([ 0.79044189, -0.23140636, 0.70072904, 0.40608028]) The same ARMA(2,2) Using the from_coeffs class method >>> arma_process = sm.tsa.ArmaProcess.from_coeffs(arparams, maparams) >>> arma_process.arroots array([1.5-1.32287566j, 1.5+1.32287566j]) """ # TODO: Check unit root behavior def __init__(self, ar=None, ma=None, nobs=100): if ar is None: ar = np.array([1.]) if ma is None: ma = np.array([1.]) self.ar = array_like(ar, 'ar') self.ma = array_like(ma, 'ma') self.arcoefs = -self.ar[1:] self.macoefs = self.ma[1:] self.arpoly = np.polynomial.Polynomial(self.ar) self.mapoly = np.polynomial.Polynomial(self.ma) self.nobs = nobs
[docs] @classmethod def from_coeffs(cls, arcoefs=None, macoefs=None, nobs=100): """ Create ArmaProcess from an ARMA representation. Parameters ---------- arcoefs : array_like Coefficient for autoregressive lag polynomial, not including zero lag. The sign is inverted to conform to the usual time series representation of an ARMA process in statistics. See the class docstring for more information. macoefs : array_like Coefficient for moving-average lag polynomial, excluding zero lag. nobs : int, optional Length of simulated time series. Used, for example, if a sample is generated. Returns ------- ArmaProcess Class instance initialized with arcoefs and macoefs. Examples -------- >>> arparams = [.75, -.25] >>> maparams = [.65, .35] >>> arma_process = sm.tsa.ArmaProcess.from_coeffs(ar, ma) >>> arma_process.isstationary True >>> arma_process.isinvertible True """ arcoefs = [] if arcoefs is None else arcoefs macoefs = [] if macoefs is None else macoefs return cls(np.r_[1, -np.asarray(arcoefs)], np.r_[1, np.asarray(macoefs)], nobs=nobs)
[docs] @classmethod def from_estimation(cls, model_results, nobs=None): """ Create an ArmaProcess from the results of an ARMA estimation. Parameters ---------- model_results : ARMAResults instance A fitted model. nobs : int, optional If None, nobs is taken from the results. Returns ------- ArmaProcess Class instance initialized from model_results. """ arcoefs = model_results.arparams macoefs = model_results.maparams nobs = nobs or model_results.nobs return cls(np.r_[1, -arcoefs], np.r_[1, macoefs], nobs=nobs)
def __mul__(self, oth): if isinstance(oth, self.__class__): ar = (self.arpoly * oth.arpoly).coef ma = (self.mapoly * oth.mapoly).coef else: try: aroth, maoth = oth arpolyoth = np.polynomial.Polynomial(aroth) mapolyoth = np.polynomial.Polynomial(maoth) ar = (self.arpoly * arpolyoth).coef ma = (self.mapoly * mapolyoth).coef except: raise TypeError('Other type is not a valid type') return self.__class__(ar, ma, nobs=self.nobs) def __repr__(self): msg = 'ArmaProcess({0}, {1}, nobs={2}) at {3}' return msg.format(self.ar.tolist(), self.ma.tolist(), self.nobs, hex(id(self))) def __str__(self): return 'ArmaProcess\nAR: {0}\nMA: {1}'.format(self.ar.tolist(), self.ma.tolist())
[docs] @Appender(remove_parameters(arma_acovf.__doc__, ['ar', 'ma', 'sigma2'])) def acovf(self, nobs=None): nobs = nobs or self.nobs return arma_acovf(self.ar, self.ma, nobs=nobs)
[docs] @Appender(remove_parameters(arma_acf.__doc__, ['ar', 'ma'])) def acf(self, lags=None): lags = lags or self.nobs return arma_acf(self.ar, self.ma, lags=lags)
[docs] @Appender(remove_parameters(arma_pacf.__doc__, ['ar', 'ma'])) def pacf(self, lags=None): lags = lags or self.nobs return arma_pacf(self.ar, self.ma, lags=lags)
[docs] @Appender(remove_parameters(arma_periodogram.__doc__, ['ar', 'ma', 'worN', 'whole'])) def periodogram(self, nobs=None): nobs = nobs or self.nobs return arma_periodogram(self.ar, self.ma, worN=nobs)
[docs] @Appender(remove_parameters(arma_impulse_response.__doc__, ['ar', 'ma'])) def impulse_response(self, leads=None): leads = leads or self.nobs return arma_impulse_response(self.ar, self.ma, leads=leads)
[docs] @Appender(remove_parameters(arma2ma.__doc__, ['ar', 'ma'])) def arma2ma(self, lags=None): lags = lags or self.lags return arma2ma(self.ar, self.ma, lags=lags)
[docs] @Appender(remove_parameters(arma2ar.__doc__, ['ar', 'ma'])) def arma2ar(self, lags=None): lags = lags or self.lags return arma2ar(self.ar, self.ma, lags=lags)
@property def arroots(self): """Roots of autoregressive lag-polynomial""" return self.arpoly.roots() @property def maroots(self): """Roots of moving average lag-polynomial""" return self.mapoly.roots() @property def isstationary(self): """ Arma process is stationary if AR roots are outside unit circle. Returns ------- bool True if autoregressive roots are outside unit circle. """ if np.all(np.abs(self.arroots) > 1.0): return True else: return False @property def isinvertible(self): """ Arma process is invertible if MA roots are outside unit circle. Returns ------- bool True if moving average roots are outside unit circle. """ if np.all(np.abs(self.maroots) > 1): return True else: return False
[docs] def invertroots(self, retnew=False): """ Make MA polynomial invertible by inverting roots inside unit circle. Parameters ---------- retnew : bool If False (default), then return the lag-polynomial as array. If True, then return a new instance with invertible MA-polynomial. Returns ------- manew : ndarray A new invertible MA lag-polynomial, returned if retnew is false. wasinvertible : bool True if the MA lag-polynomial was already invertible, returned if retnew is false. armaprocess : new instance of class If retnew is true, then return a new instance with invertible MA-polynomial. """ # TODO: variable returns like this? pr = self.maroots mainv = self.ma invertible = self.isinvertible if not invertible: pr[np.abs(pr) < 1] = 1. / pr[np.abs(pr) < 1] pnew = np.polynomial.Polynomial.fromroots(pr) mainv = pnew.coef / pnew.coef[0] if retnew: return self.__class__(self.ar, mainv, nobs=self.nobs) else: return mainv, invertible
[docs] @Appender(str(_generate_sample_doc)) def generate_sample(self, nsample=100, scale=1., distrvs=None, axis=0, burnin=0): return arma_generate_sample(self.ar, self.ma, nsample, scale, distrvs, axis=axis, burnin=burnin)