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Source code for scikits.statsmodels.tsa.arima_process

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

2009-09-06: copied from try_signal.py
    reparameterized same as signal.lfilter (positive coefficients)


Notes
-----
* pretty fast
* checked with Monte Carlo and cross comparison with statsmodels yule_walker
  for AR numbers are close but not identical to yule_walker
  not compared to other statistics packages, no degrees of freedom correction
* ARMA(2,2) estimation (in Monte Carlo) requires longer time series to estimate parameters
  without large variance. There might be different ARMA parameters
  with similar impulse response function that cannot be well
  distinguished with small samples (e.g. 100 observations)
* good for one time calculations for entire time series, not for recursive
  prediction
* class structure not very clean yet
* many one-liners with scipy.signal, but takes time to figure out usage
* missing result statistics, e.g. t-values, but standard errors in examples
* no criteria for choice of number of lags
* no constant term in ARMA process
* no integration, differencing for ARIMA
* 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)

* two names for lag polynomials ar = rhoy, ma = rhoe ?


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

BigJudge p. 237ff:
If the time series process is a stationary ARMA(p,q), then
minimizing the sum of squares is asymptoticaly (as T-> inf)
equivalent to the exact Maximum Likelihood Estimator

Because Least Squares conditional on the initial information
does not use all information, in small samples exact MLE can
be better.

Without the normality assumption, the least squares estimator
is still consistent under suitable conditions, however not
efficient

Author: josefpktd
License: BSD
'''

import numpy as np
from scipy import signal, optimize, linalg
from scikits.statsmodels.base.model import LikelihoodModel

#this has been copied to new arma_mle.py - keep temporarily for easier lookup
class ARIMA(LikelihoodModel):
    '''currently ARMA only, no differencing used - no I

    parameterized as
         rhoy(L) y_t = rhoe(L) eta_t

    A instance of this class preserves state, so new class instances should
    be created for different examples
    '''
    def __init__(self, endog, exog=None):
        super(ARIMA, self).__init__(endog, exog)
        if endog.ndim == 1:
            endog = endog[:,None]
        elif endog.ndim > 1 and endog.shape[1] != 1:
            raise ValueError("Only the univariate case is implemented")
        self.endog = endog # overwrite endog
        if exog is not None:
            raise ValueError("Exogenous variables are not yet supported.")

    def fit(self, order=(0,0,0), method="ls", rhoy0=None, rhoe0=None):
        '''
        Estimate lag coefficients of an ARIMA process.

        Parameters
        ----------
            order : sequence
                p,d,q where p is the number of AR lags, d is the number of
                differences to induce stationarity, and q is the number of
                MA lags to estimate.
            method : str {"ls", "ssm"}
                Method of estimation.  LS is conditional least squares.
                SSM is state-space model and the Kalman filter is used to
                maximize the exact likelihood.
            rhoy0, rhoe0 : array_like (optional)
                starting values for estimation

        Returns
        -------
            rh, cov_x, infodict, mesg, ier : output of scipy.optimize.leastsq
            rh :
                estimate of lag parameters, concatenated [rhoy, rhoe]
            cov_x :
                unscaled (!) covariance matrix of coefficient estimates
        '''
        if not hasattr(order, '__iter__'):
            raise ValueError("order must be an iterable sequence.  Got type \
%s instead" % type(order))

        p,d,q = order

        if d > 0:
            raise ValueError("Differencing not implemented yet")
            # assume no constant, ie mu = 0
            # unless overwritten then use w_bar for mu
            Y = np.diff(endog, d, axis=0) #TODO: handle lags?

        x = self.endog.squeeze() # remove the squeeze might be needed later
        def errfn( rho):
            #rhoy, rhoe = rho
            rhoy = np.concatenate(([1], rho[:p]))
            rhoe = np.concatenate(([1], rho[p:]))
            etahatr = signal.lfilter(rhoy, rhoe, x)
            #print rho,np.sum(etahatr*etahatr)
            return etahatr

        if rhoy0 is None:
            rhoy0 = 0.5 * np.ones(p)
        if rhoe0 is None:
            rhoe0 = 0.5 * np.ones(q)

        method = method.lower()

        if method == "ls":
            rh, cov_x, infodict, mesg, ier = \
               optimize.leastsq(errfn, np.r_[rhoy0, rhoe0],ftol=1e-10,full_output=True)
#TODO: integrate this into the MLE.fit framework?
        elif method == "ssm":
            pass
        else:
            # fmin_bfgs is slow or doesn't work yet
            errfnsum = lambda rho : np.sum(errfn(rho)**2)
            #xopt, {fopt, gopt, Hopt, func_calls, grad_calls
            rh,fopt, gopt, cov_x, _,_, ier = \
                optimize.fmin_bfgs(errfnsum, np.r_[rhoy0, rhoe0], maxiter=2, full_output=True)
            infodict, mesg = None, None
        self.rh = rh
        self.rhoy = np.concatenate(([1], rh[:p]))
        self.rhoe = np.concatenate(([1], rh[p:])) #rh[-q:])) doesnt work for q=0
        self.error_estimate = errfn(rh)
        return rh, cov_x, infodict, mesg, ier

    def errfn(self, rho=None, p=None, x=None):
        ''' duplicate -> remove one
        '''
        #rhoy, rhoe = rho
        if not rho is None:
            rhoy = np.concatenate(([1],  rho[:p]))
            rhoe = np.concatenate(([1],  rho[p:]))
        else:
            rhoy = self.rhoy
            rhoe = self.rhoe
        etahatr = signal.lfilter(rhoy, rhoe, x)
        #print rho,np.sum(etahatr*etahatr)
        return etahatr

    def predicted(self, rhoy=None, rhoe=None):
        '''past predicted values of time series
        just added, not checked yet
        '''
        if rhoy is None:
            rhoy = self.rhoy
        if rhoe is None:
            rhoe = self.rhoe
        return self.x + self.error_estimate

    def forecast(self, ar=None, ma=None, nperiod=10):
        eta = np.r_[self.error_estimate, np.zeros(nperiod)]
        if ar is None:
            ar = self.rhoy
        if ma is None:
            ma = self.rhoe
        return signal.lfilter(ma, ar, eta)

#TODO: is this needed as a method at all?
    @classmethod
    def generate_sample(cls, ar, ma, nsample, std=1):
        eta = std * np.random.randn(nsample)
        return signal.lfilter(ma, ar, eta)



[docs]def arma_generate_sample(ar, ma, nsample, sigma=1, distrvs=np.random.randn, burnin=0): '''generate an random sample of an ARMA process Parameters ---------- ar : array_like, 1d coefficient for autoregressive lag polynomial, including zero lag ma : array_like, 1d coefficient for moving-average lag polynomial, including zero lag nsample : int length of simulated time series sigma : float standard deviation of noise distrvs : function, random number generator function that generates the random numbers, and takes sample size as argument default: np.random.randn TODO: change to size argument burnin : integer (default: 0) to reduce the effect of initial conditions, burnin observations at the beginning of the sample are dropped Returns ------- acovf : array autocovariance of ARMA process given by ar, ma ''' #TODO: unify with ArmaProcess method eta = sigma * distrvs(nsample+burnin) return signal.lfilter(ma, ar, eta)[burnin:]
[docs]def arma_acovf(ar, ma, nobs=10): '''theoretical autocovariance function of ARMA process Parameters ---------- ar : array_like, 1d coefficient for autoregressive lag polynomial, including zero lag ma : array_like, 1d coefficient for moving-average lag polynomial, including zero lag Returns ------- acovf : array autocovariance of ARMA process given by ar, ma See Also -------- arma_acf acovf Notes ----- Tries to do some crude numerical speed improvements for cases with high persistance. However, this algorithm is slow if the process is highly persistent and only a few autocovariances are desired. ''' #increase length of impulse response for AR closer to 1 #maybe cheap/fast enough to always keep nobs for ir large if np.abs(np.sum(ar)-1) > 0.9: nobs_ir = max(1000, 2* nobs) #no idea right now how large it is needed else: nobs_ir = max(100, 2* nobs) #no idea right now ir = arma_impulse_response(ar, ma, nobs=nobs_ir) #better save than sorry (?), I have no idea about the required precision #only checked for AR(1) while ir[-1] > 5*1e-5: nobs_ir *= 10 ir = arma_impulse_response(ar, ma, nobs=nobs_ir) #again no idea where the speed break points are: if nobs_ir > 50000 and nobs < 1001: acovf = np.array([np.dot(ir[:nobs-t], ir[t:nobs]) for t in range(nobs)]) else: acovf = np.correlate(ir,ir,'full')[len(ir)-1:] return acovf[:nobs]
[docs]def arma_acf(ar, ma, nobs=10): '''theoretical autocovariance function of ARMA process Parameters ---------- ar : array_like, 1d coefficient for autoregressive lag polynomial, including zero lag ma : array_like, 1d coefficient for moving-average lag polynomial, including zero lag Returns ------- acovf : array autocovariance of ARMA process given by ar, ma See Also -------- arma_acovf acf acovf ''' acovf = arma_acovf(ar, ma, nobs) return acovf/acovf[0]
[docs]def arma_pacf(ar, ma, nobs=10): '''partial autocorrelation function of an ARMA process Notes ----- solves yule-walker equation for each lag order up to nobs lags not tested/checked yet ''' apacf = np.zeros(nobs) acov = arma_acf(ar,ma, nobs=nobs+1) apacf[0] = 1. for k in range(2,nobs+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 autoregressive lag-polynomial with leading 1 and lhs sign ma : array_like moving average lag-polynomial with leading 1 worN : {None, int}, optional 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 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 : array frequencies sd : array periodogram, 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.sum(np.isnan(h)) > 0: # this happens with unit root or seasonal unit root' print 'Warning: nan in frequency response h, maybe a unit root' return w, sd
[docs]def arma_impulse_response(ar, ma, nobs=100): '''get the impulse response function (MA representation) for ARMA process Parameters ---------- ma : array_like, 1d moving average lag polynomial ar : array_like, 1d auto regressive lag polynomial nobs : int number of observations to calculate Returns ------- ir : array, 1d 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, nobs=100) ar_representation = arma_impulse_response(ma, ar, nobs=100) fully tested against matlab Examples -------- AR(1) >>> arma_impulse_response([1.0, -0.8], [1.], nobs=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], nobs=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], nobs=10) array([ 1. , 1.3 , 1.24 , 0.992 , 0.7936 , 0.63488 , 0.507904 , 0.4063232 , 0.32505856, 0.26004685]) ''' impulse = np.zeros(nobs) impulse[0] = 1. return signal.lfilter(ma, ar, impulse) #alias, easier to remember
arma2ma = arma_impulse_response #alias, easier to remember
[docs]def arma2ar(ar, ma, nobs=100): '''get the AR representation of an ARMA process Parameters ---------- ar : array_like, 1d auto regressive lag polynomial ma : array_like, 1d moving average lag polynomial nobs : int number of observations to calculate Returns ------- ar : array, 1d coefficients of AR lag polynomial with nobs elements ` Notes ----- This is just an alias for ``ar_representation = arma_impulse_response(ma, ar, nobs=100)`` fully tested against matlab Examples -------- ''' return arma_impulse_response(ma, ar, nobs=100) #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 currently not check whether the MA lagpolynomial of the ARMA process is invertible, neither does it check the roots of the AR lagpolynomial. Parameters ---------- ar_des : array_like coefficients of original AR lag polynomial, including lag zero p, q : int length of desired ARMA lag polynomials n : int number of terms of the impuls_response function to include in the objective function for the approximation mse : string, 'ar' not used yet, Returns ------- ar_app, ma_app : arrays coefficients of the AR and MA lag polynomials of the approximation res : tuple 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 roots outside the unit intervall to ones that are inside. How do we do 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) ## print ar,ma ## print ar_des.shape, ar_approx.shape ## print ar_des ## print ar_approx 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)#, full_output=True) #print res 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
[docs]def lpol2index(ar): '''remove zeros from lagpolynomial, squeezed representation with index Parameters ---------- ar : array_like coefficients of lag polynomial Returns ------- coeffs : array non-zero coefficients of lag polynomial index : array index (lags) of lagpolynomial with non-zero elements ''' ar = np.asarray(ar) index = np.nonzero(ar)[0] coeffs = ar[index] return coeffs, index
[docs]def index2lpol(coeffs, index): '''expand coefficients to lag poly Parameters ---------- coeffs : array non-zero coefficients of lag polynomial index : array index (lags) of lagpolynomial with non-zero elements ar : array_like coefficients of lag polynomial Returns ------- ar : array_like coefficients of lag polynomial ''' n = max(index) ar = np.zeros(n) ar[index] = coeffs return ar #moved from sandbox.tsa.try_fi
[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 : array coefficients of lag polynomial ''' #hide import inside function until we use this heavily from scipy.special import gamma, 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 : array 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 gamma, 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 : array quotient or filtered series rem : array remainder Notes ----- If num is a time series, then this applies the linear filter den^{-1}. If both num and den are both lagpolynomials, 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
[docs]class ArmaProcess(object): '''represents an ARMA process for given lag-polynomials This is a class to bring together properties of the process. It does not do any estimation or statistical analysis. maybe needs special handling for unit roots ''' def __init__(self, ar, ma, nobs=None): self.ar = np.asarray(ar) self.ma = np.asarray(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 @classmethod
[docs] def from_coeffs(cls, arcoefs, macoefs, nobs=None): '''create ArmaProcess instance from coefficients of the lag-polynomials ''' return cls(np.r_[1, -arcoefs], np.r_[1, macoefs], nobs=nobs)
@classmethod
[docs] def from_estimation(cls, model_results, nobs=None): '''create ArmaProcess instance from estimation results ''' arcoefs = model_results.params[:model_results.nar] macoefs = model_results.params[model_results.nar: model_results.nar+model_results.nma] 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: print('other is not a valid type') raise return self.__class__(ar, ma, nobs=self.nobs) def __repr__(self): return 'ArmaProcess(%r, %r, nobs=%d)' % (self.ar.tolist(), self.ma.tolist(), self.nobs) def __str__(self): return 'ArmaProcess\nAR: %r\nMA: %r' % (self.ar.tolist(), self.ma.tolist())
[docs] def acovf(self, nobs=None): nobs = nobs or self.nobs return arma_acovf(self.ar, self.ma, nobs=nobs)
acovf.__doc__ = arma_acovf.__doc__
[docs] def acf(self, nobs=None): nobs = nobs or self.nobs return arma_acf(self.ar, self.ma, nobs=nobs)
acf.__doc__ = arma_acf.__doc__
[docs] def pacf(self, nobs=None): nobs = nobs or self.nobs return arma_pacf(self.ar, self.ma, nobs=nobs)
pacf.__doc__ = arma_pacf.__doc__
[docs] def periodogram(self, nobs=None): nobs = nobs or self.nobs return arma_periodogram(self.ar, self.ma, worN=nobs)
periodogram.__doc__ = arma_periodogram.__doc__
[docs] def impulse_response(self, nobs=None): nobs = nobs or self.nobs return arma_impulse_response(self.ar, self.ma, worN=nobs)
impulse_response.__doc__ = arma_impulse_response.__doc__
[docs] def arma2ma(self, nobs=None): nobs = nobs or self.nobs return arma2ma(self.ar, self.ma, nobs=nobs)
arma2ma.__doc__ = arma2ma.__doc__
[docs] def arma2ar(self, nobs=None): nobs = nobs or self.nobs return arma2ar(self.ar, self.ma, nobs=nobs)
arma2ar.__doc__ = arma2ar.__doc__
[docs] def ar_roots(self): '''roots of autoregressive lag-polynomial ''' return self.arpoly.roots()
[docs] def ma_roots(self): '''roots of moving average lag-polynomial ''' return self.mapoly.roots()
[docs] def isstationary(self): '''Arma process is stationary if AR roots are outside unit circle Returns ------- isstationary : boolean True if autoregressive roots are outside unit circle ''' if np.all(np.abs(self.ar_roots())) > 1: return True else: return False
[docs] def isinvertible(self): '''Arma process is invertible if MA roots are outside unit circle Returns ------- isinvertible : boolean True if moving average roots are outside unit circle ''' if np.all(np.abs(self.ma_roots())) > 1: return True else: return False
[docs] def invertroots(self, retnew=False): '''make MA polynomial invertible by inverting roots inside unit circle Parameters ---------- retnew : boolean If False (default), then return the lag-polynomial as array. If True, then return a new instance with invertible MA-polynomial Returns ------- manew : array new invertible MA lag-polynomial, returned if retnew is false. wasinvertible : boolean 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 ''' pr = self.ma_roots() insideroots = np.abs(pr)<1 if insideroots.any(): pr[np.abs(pr)<1] = 1./pr[np.abs(pr)<1] pnew = poly.Polynomial.fromroots(pr) mainv = pn.coef/pnew.coef[0] wasinvertible = False else: mainv = self.ma wasinvertible = True if retnew: return self.__class__(self.ar, mainv, nobs=self.nobs) else: return mainv, wasinvertible
[docs] def generate_sample(self, size=100, scale=1, distrvs=None, axis=0, burnin=0): '''generate ARMA samples Parameters ---------- size : int or tuple of ints If size is an integer, then this creates a 1d timeseries of length size. If size is a tuple, then the timeseries is along axis. All other axis have independent arma samples. Returns ------- rvs : ndarray random sample(s) of arma process Notes ----- Should work for n-dimensional with time series along axis, but not tested yet. Processes are sampled independently. ''' if distrvs is None: distrvs = np.random.normal if np.ndim(size) == 0: size = [size] if burnin: #handle burin time for nd arrays #maybe there is a better trick in scipy.fft code newsize = list(size) newsize[axis] += burnin newsize = tuple(newsize) fslice = [slice(None)]*len(newsize) fslice[axis] = slice(burnin, None, None) fslice = tuple(fslice) else: newsize = tuple(size) fslice = tuple([slice(None)]*np.ndim(newsize)) eta = scale * distrvs(size=newsize) return signal.lfilter(self.ma, self.ar, eta, axis=axis)[fslice]
__all__ = ['arma_acf', 'arma_acovf', 'arma_generate_sample', 'arma_impulse_response', 'arma2ar', 'arma2ma', 'deconvolve', 'lpol2index', 'index2lpol'] if __name__ == '__main__': # Simulate AR(1) #-------------- # ar * y = ma * eta ar = [1, -0.8] ma = [1.0] # generate AR data eta = 0.1 * np.random.randn(1000) yar1 = signal.lfilter(ar, ma, eta) print "\nExample 0" arest = ARIMA(yar1) rhohat, cov_x, infodict, mesg, ier = arest.fit((1,0,1)) print rhohat print cov_x print "\nExample 1" ar = [1.0, -0.8] ma = [1.0, 0.5] y1 = arest.generate_sample(ar,ma,1000,0.1) arest = ARIMA(y1) rhohat1, cov_x1, infodict, mesg, ier = arest.fit((1,0,1)) print rhohat1 print cov_x1 err1 = arest.errfn(x=y1) print np.var(err1) import scikits.statsmodels.api as sm print sm.regression.yule_walker(y1, order=2, inv=True) print "\nExample 2" nsample = 1000 ar = [1.0, -0.6, -0.1] ma = [1.0, 0.3, 0.2] y2 = ARIMA.generate_sample(ar,ma,nsample,0.1) arest2 = ARIMA(y2) rhohat2, cov_x2, infodict, mesg, ier = arest2.fit((1,0,2)) print rhohat2 print cov_x2 err2 = arest.errfn(x=y2) print np.var(err2) print arest2.rhoy print arest2.rhoe print "true" print ar print ma rhohat2a, cov_x2a, infodict, mesg, ier = arest2.fit((2,0,2)) print rhohat2a print cov_x2a err2a = arest.errfn(x=y2) print np.var(err2a) print arest2.rhoy print arest2.rhoe print "true" print ar print ma print sm.regression.yule_walker(y2, order=2, inv=True) print "\nExample 20" nsample = 1000 ar = [1.0]#, -0.8, -0.4] ma = [1.0, 0.5, 0.2] y3 = ARIMA.generate_sample(ar,ma,nsample,0.01) arest20 = ARIMA(y3) rhohat3, cov_x3, infodict, mesg, ier = arest20.fit((2,0,0)) print rhohat3 print cov_x3 err3 = arest20.errfn(x=y3) print np.var(err3) print np.sqrt(np.dot(err3,err3)/nsample) print arest20.rhoy print arest20.rhoe print "true" print ar print ma rhohat3a, cov_x3a, infodict, mesg, ier = arest20.fit((0,0,2)) print rhohat3a print cov_x3a err3a = arest20.errfn(x=y3) print np.var(err3a) print np.sqrt(np.dot(err3a,err3a)/nsample) print arest20.rhoy print arest20.rhoe print "true" print ar print ma print sm.regression.yule_walker(y3, order=2, inv=True) print "\nExample 02" nsample = 1000 ar = [1.0, -0.8, 0.4] #-0.8, -0.4] ma = [1.0]#, 0.8, 0.4] y4 = ARIMA.generate_sample(ar,ma,nsample) arest02 = ARIMA(y4) rhohat4, cov_x4, infodict, mesg, ier = arest02.fit((2,0,0)) print rhohat4 print cov_x4 err4 = arest02.errfn(x=y4) print np.var(err4) sige = np.sqrt(np.dot(err4,err4)/nsample) print sige print sige * np.sqrt(np.diag(cov_x4)) print np.sqrt(np.diag(cov_x4)) print arest02.rhoy print arest02.rhoe print "true" print ar print ma rhohat4a, cov_x4a, infodict, mesg, ier = arest02.fit((0,0,2)) print rhohat4a print cov_x4a err4a = arest02.errfn(x=y4) print np.var(err4a) sige = np.sqrt(np.dot(err4a,err4a)/nsample) print sige print sige * np.sqrt(np.diag(cov_x4a)) print np.sqrt(np.diag(cov_x4a)) print arest02.rhoy print arest02.rhoe print "true" print ar print ma import scikits.statsmodels.api as sm print sm.regression.yule_walker(y4, order=2, method='mle', inv=True) import matplotlib.pyplot as plt plt.plot(arest2.forecast()[-100:]) #plt.show() ar1, ar2 = ([1, -0.4], [1, 0.5]) ar2 = [1, -1] lagpolyproduct = np.convolve(ar1, ar2) print deconvolve(lagpolyproduct, ar2, n=None) print signal.deconvolve(lagpolyproduct, ar2) print deconvolve(lagpolyproduct, ar2, n=10)