'''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
'''
from __future__ import print_function
from statsmodels.compat.python import range
import numpy as np
from scipy import signal, optimize, linalg
[docs]def arma_generate_sample(ar, ma, nsample, sigma=1, distrvs=np.random.randn,
burnin=0):
"""
Generate a 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
-------
sample : array
sample of ARMA process given by ar, ma of length nsample
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 paramters 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.ARMA(y, (2, 2)).fit(trend='nc', disp=0)
>>> model.params
array([ 0.79044189, -0.23140636, 0.70072904, 0.40608028])
"""
#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
nobs : int
number of terms (lags plus zero lag) to include in returned acovf
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 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 autocorrelation function 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
nobs : int
number of terms (lags plus zero lag) to include in returned acf
Returns
-------
acf : array
autocorrelation 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
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
nobs : int
number of terms (lags plus zero lag) to include in returned pacf
Returns
-------
pacf : array
partial autocorrelation of ARMA process given by ar, ma
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)``
"""
# tested against matlab
return arma_impulse_response(ma, ar, nobs=nobs)
#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)
#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 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 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):
"""
Represent 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.
Parameters
----------
ar : array_like, 1d
Coefficient for autoregressive lag polynomial, including zero lag.
See the notes for some information about the sign.
ma : array_like, 1d
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
-----
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 paramters 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, -ar] # add zero-lag and negate
>>> ma = np.r_[1, ma] # add zero-lag
>>> arma_process = sm.tsa.ArmaProcess(ar, ma)
>>> arma_process.isstationary
True
>>> arma_process.isinvertible
True
>>> y = arma_process.generate_sample(250)
>>> model = sm.tsa.ARMA(y, (2, 2)).fit(trend='nc', disp=0)
>>> model.params
array([ 0.79044189, -0.23140636, 0.70072904, 0.40608028])
"""
# maybe needs special handling for unit roots
def __init__(self, ar, ma, nobs=100):
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=100):
"""
Create ArmaProcess instance from coefficients of the lag-polynomials
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, including zero lag
nobs : int, optional
Length of simulated time series. Used, for example, if a sample
is generated.
"""
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 ARMA estimation results
Parameters
----------
model_results : ARMAResults instance
A fitted model
nobs : int, optional
If None, nobs is taken from the 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:
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__
@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
-------
isstationary : boolean
True if autoregressive roots are outside unit circle
'''
if np.all(np.abs(self.arroots) > 1):
return True
else:
return False
@property
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.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 : 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
'''
#TODO: variable returns like this?
pr = self.ma_roots()
insideroots = np.abs(pr) < 1
if insideroots.any():
pr[np.abs(pr) < 1] = 1./pr[np.abs(pr) < 1]
pnew = np.polynomial.Polynomial.fromroots(pr)
mainv = pnew.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, nsample=100, scale=1., distrvs=None, axis=0,
burnin=0):
'''generate ARMA samples
Parameters
----------
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, then the timeseries is along
axis. All other axis have independent arma samples.
scale : 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
axis : int
See nsample.
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(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(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 = ARIMAProcess(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 = ARIMAProcess(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 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 = ARIMAProcess(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 = ARIMAProcess(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 = ARIMAProcess(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 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))
