def ar_nitime(x, order=1, center=False):
"""Derive a model of the noise present in the functional timeseries for
the calculation of the standardized DVARS.
- Borrowed from nipy.algorithms.AR_est_YW. aka "from nitime import
algorithms as alg".
:type x: Nibabel data
:param x: The vector of one voxel's timeseries.
:type order: int
:param order: (default: 1) Which lag of the autocorrelation of the
timeseries to use in the calculation.
:type center: bool
:param center: (default: False) Whether to center the timeseries (to
demean it).
:rtype: float
:return: The modeled noise value for the current voxel's timeseries.
"""
from nitime.lazy import scipy_linalg as linalg
import nitime.utils as utils
if center:
x = x.copy()
x = x - x.mean()
r_m = utils.autocorr(x)[:order + 1]
Tm = linalg.toeplitz(r_m[:order])
y = r_m[1:]
ak = linalg.solve(Tm, y)
return ak[0]
def ar_nitime(x, order=1, center=False):
"""Derive a model of the noise present in the functional timeseries for
the calculation of the standardized DVARS.
- Borrowed from nipy.algorithms.AR_est_YW. aka "from nitime import
algorithms as alg".
:type x: Nibabel data
:param x: The vector of one voxel's timeseries.
:type order: int
:param order: (default: 1) Which lag of the autocorrelation of the
timeseries to use in the calculation.
:type center: bool
:param center: (default: False) Whether to center the timeseries (to
demean it).
:rtype: float
:return: The modeled noise value for the current voxel's timeseries.
"""
from nitime.lazy import scipy_linalg as linalg
import nitime.utils as utils
if center:
x = x.copy()
x = x - x.mean()
r_m = utils.autocorr(x)[:order + 1]
Tm = linalg.toeplitz(r_m[:order])
y = r_m[1:]
ak = linalg.solve(Tm, y)
return ak[0]
def AR_est_YW(x, order, rxx=None):
r"""Determine the autoregressive (AR) model of a random process x using
the Yule Walker equations. The AR model takes this convention:
.. math::
x(n) = a(1)x(n-1) + a(2)x(n-2) + \dots + a(p)x(n-p) + e(n)
where e(n) is a zero-mean white noise process with variance sig_sq,
and p is the order of the AR model. This method returns the a_i and
sigma
The orthogonality property of minimum mean square error estimates
states that
.. math::
E\{e(n)x^{*}(n-k)\} = 0 \quad 1\leq k\leq p
Inserting the definition of the error signal into the equations above
yields the Yule Walker system of equations:
.. math::
R_{xx}(k) = \sum_{i=1}^{p}a(i)R_{xx}(k-i) \quad1\leq k\leq p
Similarly, the variance of the error process is
.. math::
E\{e(n)e^{*}(n)\} = E\{e(n)x^{*}(n)\} = R_{xx}(0)-\sum_{i=1}^{p}a(i)R^{*}(i)
Parameters
----------
x : ndarray
The sampled autoregressive random process
order : int
The order p of the AR system
rxx : ndarray (optional)
An optional, possibly unbiased estimate of the autocorrelation of x
Returns
-------
ak, sig_sq : The estimated AR coefficients and innovations variance
"""
if rxx is not None and type(rxx) == np.ndarray:
r_m = rxx[:order + 1]
else:
r_m = utils.autocorr(x)[:order + 1]
Tm = linalg.toeplitz(r_m[:order])
y = r_m[1:]
ak = linalg.solve(Tm, y)
sigma_v = r_m[0].real - np.dot(r_m[1:].conj(), ak).real
return ak, sigma_v
def AR_est_YW(x, order, rxx=None):
r"""Determine the autoregressive (AR) model of a random process x using
the Yule Walker equations. The AR model takes this convention:
.. math::
x(n) = a(1)x(n-1) + a(2)x(n-2) + \dots + a(p)x(n-p) + e(n)
where e(n) is a zero-mean white noise process with variance sig_sq,
and p is the order of the AR model. This method returns the a_i and
sigma
The orthogonality property of minimum mean square error estimates
states that
.. math::
E\{e(n)x^{*}(n-k)\} = 0 \quad 1\leq k\leq p
Inserting the definition of the error signal into the equations above
yields the Yule Walker system of equations:
.. math::
R_{xx}(k) = \sum_{i=1}^{p}a(i)R_{xx}(k-i) \quad1\leq k\leq p
Similarly, the variance of the error process is
.. math::
E\{e(n)e^{*}(n)\} = E\{e(n)x^{*}(n)\} = R_{xx}(0)-\sum_{i=1}^{p}a(i)R^{*}(i)
Parameters
----------
x : ndarray
The sampled autoregressive random process
order : int
The order p of the AR system
rxx : ndarray (optional)
An optional, possibly unbiased estimate of the autocorrelation of x
Returns
-------
ak, sig_sq : The estimated AR coefficients and innovations variance
"""
if rxx is not None and type(rxx) == np.ndarray:
r_m = rxx[:order + 1]
else:
r_m = utils.autocorr(x)[:order + 1]
Tm = linalg.toeplitz(r_m[:order])
y = r_m[1:]
ak = linalg.solve(Tm, y)
sigma_v = r_m[0].real - np.dot(r_m[1:].conj(), ak).real
return ak, sigma_v
def ar_nitime(x, order=1, center=False):
"""
Borrowed from nipy.algorithms.AR_est_YW.
aka from nitime import algorithms as alg.
We could speed this up by having the autocorr only compute lag1.
"""
from nitime.lazy import scipy_linalg as linalg
import nitime.utils as utils
if center:
x = x.copy()
x = x - x.mean()
r_m = utils.autocorr(x)[:order + 1]
Tm = linalg.toeplitz(r_m[:order])
y = r_m[1:]
ak = linalg.solve(Tm, y)
return ak[0]
def ar_nitime(x, order=1, center=False):
"""
Borrowed from nipy.algorithms.AR_est_YW.
aka from nitime import algorithms as alg.
We could speed this up by having the autocorr only compute lag1.
"""
from nitime.lazy import scipy_linalg as linalg
import nitime.utils as utils
if center:
x = x.copy()
x = x - x.mean()
r_m = utils.autocorr(x)[:order + 1]
Tm = linalg.toeplitz(r_m[:order])
y = r_m[1:]
ak = linalg.solve(Tm, y)
return ak[0]
