def water_self_diffusion_coefficient(T=None,
units=None,
warn=True,
err_mult=None):
"""
Temperature-dependent self-diffusion coefficient of water.
Parameters
----------
T : float
Temperature (default: in Kelvin)
units : object (optional)
object with attributes: Kelvin, meter, kilogram
warn : bool (default: True)
Emit UserWarning when outside temperature range.
err_mult : length 2 array_like (default: None)
Perturb paramaters D0 and TS with err_mult[0]*dD0 and
err_mult[1]*dTS respectively, where dD0 and dTS are the
reported uncertainties in the fitted paramters. Useful
for estimating error in diffusion coefficient.
References
----------
Temperature-dependent self-diffusion coefficients of water and six selected
molecular liquids for calibration in accurate 1H NMR PFG measurements
Manfred Holz, Stefan R. Heila, Antonio Saccob;
Phys. Chem. Chem. Phys., 2000,2, 4740-4742
http://pubs.rsc.org/en/Content/ArticleLanding/2000/CP/b005319h
DOI: 10.1039/B005319H
"""
if units is None:
K = 1
m = 1
s = 1
else:
K = units.Kelvin
m = units.meter
s = units.second
if T is None:
T = 298.15 * K
_D0 = D0 * m**2 * s**-1
_TS = TS * K
if err_mult is not None:
_dD0 = dD0 * m**2 * s**-1
_dTS = dTS * K
_D0 += err_mult[0] * _dD0
_TS += err_mult[1] * _dTS
if warn and (_any(T < low_t_bound * K) or _any(T > high_t_bound * K)):
warnings.warn("Temperature is outside range (0-100 degC)")
return _D0 * ((T / _TS) - 1)**gamma
def water_self_diffusion_coefficient(T=None, units=None, warn=True,
err_mult=None):
"""
Temperature-dependent self-diffusion coefficient of water.
Parameters
----------
T: float
Temperature (default: in Kelvin)
units: object (optional)
object with attributes: Kelvin, meter, kilogram
warn: bool (default: True)
Emit UserWarning when outside temperature range.
err_mult: length 2 array_like (default: None)
Perturb paramaters D0 and TS with err_mult[0]*dD0 and
err_mult[1]*dTS respectively, where dD0 and dTS are the
reported uncertainties in the fitted paramters. Useful
for estimating error in diffusion coefficient.
References
----------
Temperature-dependent self-diffusion coefficients of water and six selected
molecular liquids for calibration in accurate 1H NMR PFG measurements
Manfred Holz, Stefan R. Heila, Antonio Saccob;
Phys. Chem. Chem. Phys., 2000,2, 4740-4742
http://pubs.rsc.org/en/Content/ArticleLanding/2000/CP/b005319h
DOI: 10.1039/B005319H
"""
if units is None:
K = 1
m = 1
s = 1
else:
K = units.Kelvin
m = units.meter
s = units.second
if T is None:
T = 298.15*K
_D0 = D0 * m**2 * s**-1
_TS = TS * K
if err_mult is not None:
_dD0 = dD0 * m**2 * s**-1
_dTS = dTS * K
_D0 += err_mult[0]*_dD0
_TS += err_mult[1]*_dTS
if warn and (_any(T < low_t_bound*K) or _any(T > high_t_bound*K)):
warnings.warn("Temperature is outside range (0-100 degC)")
return _D0*((T/_TS) - 1)**gamma
def water_density(T=None,
T0=None,
units=None,
a=None,
just_return_a=False,
warn=True):
"""
Density of water (kg/m3) as function of temperature (K)
according to VSMOW model between 0 and 40 degree Celsius.
Fitted using Thiesen's equation.
Parameters
----------
T : float
Temperature (in Kelvin) (default: 298.15).
T0 : float
Value of T for 0 degree Celsius (default: 273.15).
units : object (optional)
Object with attributes: Kelvin, meter, kilogram.
a : array_like (optional)
5 parameters to the equation.
just_return_a : bool (optional, default: False)
Do not compute rho, just return the parameters ``a``.
warn : bool (default: True)
Emit UserWarning when outside temperature range.
Returns
-------
Density of water (float of kg/m3 if T is float and units is None)
Examples
--------
>>> print('%.2f' % water_density(277.13))
999.97
References
----------
TANAKA M., GIRARD G., DAVIS R., PEUTO A. and BIGNELL N.,
"Recommanded table for the density of water between 0 °C and 40 °C
based on recent experimental reports",
Metrologia, 2001, 38, 301-309.
http://iopscience.iop.org/article/10.1088/0026-1394/38/4/3
doi:10.1088/0026-1394/38/4/3
"""
if units is None:
K = 1
m = 1
kg = 1
else:
K = units.Kelvin
m = units.meter
kg = units.kilogram
if T is None:
T = 298.15 * K
m3 = m**3
if a is None:
a = (
-3.983035 * K, # C
301.797 * K, # C
522528.9 * K * K, # C**2
69.34881 * K, # C
999.974950 * kg / m3)
if just_return_a:
return a
if T0 is None:
T0 = 273.15 * K
t = T - T0
if warn and (_any(t < 0 * K) or _any(t > 40 * K)):
warnings.warn("Temperature is outside range (0-40 degC)")
return a[4] * (1 - ((t + a[0])**2 * (t + a[1])) / (a[2] * (t + a[3])))
def SIC_for_IIR(FO=11,
BO=3,
AR_amp=0.1,
ts_size=5000,
sigma_input_noise=0.,
sigma_output_noise=0.,
order=10,
report_CI=False,
ic_type=None,
ic_max_order=None,
welch_window_size=1000):
"""A function to assess the performance of SIC under additive noise on input, output, both and under
denoising in all the previous scenarios"""
print sigma_input_noise, sigma_output_noise
# a coefficient to reduce the influence of autoregressive coefficients to increase the chance of stability
AR_amp_x = AR_amp_z = AR_amp
# setting the filter coefficients that generates X_t
A_z = append(1., random.randn(FO))
B_z = append(1., random.randn(BO) * AR_amp_z)
#checking whether the first filter has coefficients give rise to a stable IIR filter
while any(abs(roots(A_z)) >= 0.95) or _any(abs(roots(B_z)) >= 0.95):
A_z = append(1., random.randn(FO))
B_z = append(1., random.randn(BO) * AR_amp_z)
# setting the filter coefficients that generates Y_t
A_x = append(1., random.randn(FO))
B_x = append(1., random.randn(BO) * AR_amp_x)
#checking whether the second filter has coefficients give rise to a stable IIR filter
while any(abs(roots(A_x)) >= 0.95) or any(abs(roots(B_x)) >= 0.95):
A_x = append(1., random.randn(FO))
B_x = append(1., random.randn(BO) * AR_amp_x)
#input Z as random noise to a mechanism that generates the cause: X_t
Z = random.randn(ts_size)
X = lfilter(A_z, B_z, Z)
# adding additive noise to the cause
noisy_X = X + random.randn(X.shape[0]) * sigma_input_noise
Y = lfilter(A_x, B_x, X)
# adding additive noise to the effect
noisy_Y = Y + random.randn(Y.shape[0]) * sigma_output_noise
return_SDR_dict = dict()
return_SDR_dict['stochastic_in_out'] = asarray([
stochastic_SDR_estimator(noisy_X,
noisy_Y,
report_CI=report_CI,
ic_max_order=ic_max_order,
ic_type=ic_type),
stochastic_SDR_estimator(noisy_Y,
noisy_X,
report_CI=report_CI,
ic_max_order=ic_max_order,
ic_type=ic_type)
])
if sigma_input_noise > 0:
return_SDR_dict['stochastic_in'] = asarray([
stochastic_SDR_estimator(noisy_X,
Y,
report_CI=report_CI,
ic_max_order=ic_max_order,
ic_type=ic_type),
stochastic_SDR_estimator(Y,
noisy_X,
report_CI=report_CI,
ic_max_order=ic_max_order,
ic_type=ic_type)
])
else:
return_SDR_dict['stochastic_in'] = return_SDR_dict['stochastic_in_out']
if sigma_output_noise > 0:
return_SDR_dict['stochastic_out'] = asarray([
stochastic_SDR_estimator(X,
noisy_Y,
report_CI=report_CI,
ic_max_order=ic_max_order,
ic_type=ic_type),
stochastic_SDR_estimator(noisy_Y,
X,
report_CI=report_CI,
ic_max_order=ic_max_order,
ic_type=ic_type)
])
else:
return_SDR_dict['stochastic_out'] = return_SDR_dict[
'stochastic_in_out']
return_SDR_dict['deterministic_in_out'] = asarray([
deterministic_SDR_estimator(noisy_X,
noisy_Y,
welch_window_size=welch_window_size),
deterministic_SDR_estimator(noisy_Y,
noisy_X,
welch_window_size=welch_window_size)
])
return_SDR_dict['deterministic_in'] = asarray([
deterministic_SDR_estimator(noisy_X,
Y,
welch_window_size=welch_window_size),
deterministic_SDR_estimator(Y,
noisy_X,
welch_window_size=welch_window_size)
])
return_SDR_dict['deterministic_out'] = asarray([
deterministic_SDR_estimator(X,
noisy_Y,
welch_window_size=welch_window_size),
deterministic_SDR_estimator(noisy_Y,
X,
welch_window_size=welch_window_size)
])
return var(X), var(Y), return_SDR_dict
def water_density(T=None, T0=None, units=None, a=None,
just_return_a=False, warn=True):
"""
Density of water (kg/m3) as function of temperature (K)
according to VSMOW model between 0 and 40 degree Celsius.
Fitted using Thiesen's equation.
Parameters
----------
T : float
Temperature (in Kelvin) (default: 298.15).
T0 : float
Value of T for 0 degree Celsius (default: 273.15).
units : object (optional)
Object with attributes: Kelvin, meter, kilogram.
a : array_like (optional)
5 parameters to the equation.
just_return_a : bool (optional, default: False)
Do not compute rho, just return the parameters ``a``.
warn : bool (default: True)
Emit UserWarning when outside temperature range.
Returns
-------
Density of water (float of kg/m3 if T is float and units is None)
Examples
--------
>>> print('%.2f' % water_density(277.13))
999.97
References
----------
TANAKA M., GIRARD G., DAVIS R., PEUTO A. and BIGNELL N.,
"Recommanded table for the density of water between 0 °C and 40 °C
based on recent experimental reports",
Metrologia, 2001, 38, 301-309.
http://iopscience.iop.org/article/10.1088/0026-1394/38/4/3
doi:10.1088/0026-1394/38/4/3
"""
if units is None:
K = 1
m = 1
kg = 1
else:
K = units.Kelvin
m = units.meter
kg = units.kilogram
if T is None:
T = 298.15*K
m3 = m**3
if a is None:
a = (-3.983035*K, # C
301.797*K, # C
522528.9*K*K, # C**2
69.34881*K, # C
999.974950*kg/m3)
if just_return_a:
return a
if T0 is None:
T0 = 273.15*K
t = T - T0
if warn and (_any(t < 0*K) or _any(t > 40*K)):
warnings.warn("Temperature is outside range (0-40 degC)")
return a[4]*(1-((t + a[0])**2*(t + a[1]))/(a[2]*(t + a[3])))