def arma_estimate(X, P, Q, lag):
"""Autoregressive and moving average estimators.
This function provides an estimate of the autoregressive
parameters, the moving average parameters, and the driving
white noise variance of an ARMA(P,Q) for a complex or real data sequence.
The parameters are estimated using three steps:
* Estimate the AR parameters from the original data based on a least
squares modified Yule-Walker technique,
* Produce a residual time sequence by filtering the original data
with a filter based on the AR parameters,
* Estimate the MA parameters from the residual time sequence.
:param array X: Array of data samples (length N)
:param int P: Desired number of AR parameters
:param int Q: Desired number of MA parameters
:param int lag: Maximum lag to use for autocorrelation estimates
:return:
* A - Array of complex P AR parameter estimates
* B - Array of complex Q MA parameter estimates
* RHO - White noise variance estimate
.. note::
* lag must be >= Q (MA order)
**dependencies**:
* :meth:`spectrum.correlation.CORRELATION`
* :meth:`spectrum.covar.arcovar`
* :meth:`spectrum.arma.ma`
.. plot::
:width: 80%
:include-source:
from spectrum import arma_estimate, arma2psd, marple_data
import pylab
a,b, rho = arma_estimate(marple_data, 15, 15, 30)
psd = arma2psd(A=a, B=b, rho=rho, sides='centerdc', norm=True)
pylab.plot(10 * pylab.log10(psd))
pylab.ylim([-50,0])
:reference: [Marple]_
"""
R = CORRELATION(X, maxlags=lag, norm='unbiased')
R0 = R[0]
#C Estimate the AR parameters (no error weighting is used).
#C Number of equation errors is M-Q .
MPQ = lag - Q + P
N = len(X)
Y = np.zeros(N-P, dtype=complex)
for K in range(0, MPQ):
KPQ = K + Q - P+1
if KPQ < 0:
Y[K] = R[-KPQ].conjugate()
if KPQ == 0:
Y[K] = R0
if KPQ > 0:
Y[K] = R[KPQ]
# The resize is very important for the normalissation.
Y.resize(lag)
if P <= 4:
res = arcovar_marple(Y.copy(), P) #! Eq. (10.12)
ar_params = res[0]
else:
res = arcovar(Y.copy(), P) #! Eq. (10.12)
ar_params = res[0]
# the .copy is used to prevent a reference somewhere. this is a bug
# to be tracked down.
Y.resize(N-P)
#C Filter the original time series
for k in range(P, N):
SUM = X[k]
#SUM += sum([ar_params[j]*X[k-j-1] for j in range(0,P)])
for j in range(0, P):
SUM = SUM + ar_params[j] * X[k-j-1] #! Eq. (10.17)
Y[k-P] = SUM
# Estimate the MA parameters (a "long" AR of order at least 2*IQ
#C is suggested)
#Y.resize(N-P)
ma_params, rho = ma(Y, Q, 2*Q) #! Eq. (10.3)
return ar_params, ma_params, rho
def arma_estimate(X, P, Q, lag):
"""Autoregressive and moving average estimators.
This function provides an estimate of the autoregressive
parameters, the moving average parameters, and the driving
white noise variance of an ARMA(P,Q) for a complex or real data sequence.
The parameters are estimated using three steps:
* Estimate the AR parameters from the original data based on a least
squares modified Yule-Walker technique,
* Produce a residual time sequence by filtering the original data
with a filter based on the AR parameters,
* Estimate the MA parameters from the residual time sequence.
:param array X: Array of data samples (length N)
:param int P: Desired number of AR parameters
:param int Q: Desired number of MA parameters
:param int lag: Maximum lag to use for autocorrelation estimates
:return:
* A - Array of complex P AR parameter estimates
* B - Array of complex Q MA parameter estimates
* RHO - White noise variance estimate
.. note::
* lag must be >= Q (MA order)
**dependencies**:
* :meth:`spectrum.correlation.CORRELATION`
* :meth:`spectrum.covar.arcovar`
* :meth:`spectrum.arma.ma`
.. plot::
:width: 80%
:include-source:
from spectrum import *
from pylab import *
a,b, rho = arma_estimate(marple_data, 15, 15, 30)
psd = arma2psd(A=a, B=b, rho=rho, sides='centerdc', norm=True)
plot(10 * log10(psd))
ylim([-50,0])
:reference: [Marple]_
"""
R = CORRELATION(X, maxlags=lag, norm='unbiased')
R0 = R[0]
# C Estimate the AR parameters (no error weighting is used).
# C Number of equation errors is M-Q .
MPQ = lag - Q + P
N = len(X)
Y = zeros(N - P, dtype=complex)
for K in range(0, MPQ):
KPQ = K + Q - P + 1
if KPQ < 0:
Y[K] = R[-KPQ].conjugate()
if KPQ == 0:
Y[K] = R0
if KPQ > 0:
Y[K] = R[KPQ]
# The resize is very important for the normalissation.
Y.resize(lag)
if P <= 4:
res = arcovar_marple(Y.copy(), P) # ! Eq. (10.12)
ar_params = res[0]
else:
res = arcovar(Y.copy(), P) # ! Eq. (10.12)
ar_params = res[0]
# the .copy is used to prevent a reference somewhere. this is a bug
# to be tracked down.
Y.resize(N - P)
# C Filter the original time series
for k in range(P, N):
SUM = X[k]
# SUM += sum([ar_params[j]*X[k-j-1] for j in range(0,P)])
for j in range(0, P):
SUM = SUM + ar_params[j] * X[k - j - 1] # ! Eq. (10.17)
Y[k - P] = SUM
# Estimate the MA parameters (a "long" AR of order at least 2*IQ
# C is suggested)
# Y.resize(N-P)
ma_params, rho = ma(Y, Q, 2 * Q) # ! Eq. (10.3)
return ar_params, ma_params, rho