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circstats.py
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circstats.py
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"""circstats.py -- circular statistics module
This software is licensed under the terms of the MIT License as
follows:
Copyright (c) 2013 Jessica B. Hamrick, Peter W. Battaglia
Permission is hereby granted, free of charge, to any person obtaining
a copy of this software and associated documentation files (the
"Software"), to deal in the Software without restriction, including
without limitation the rights to use, copy, modify, merge, publish,
distribute, sublicense, and/or sell copies of the Software, and to
permit persons to whom the Software is furnished to do so, subject to
the following conditions:
The above copyright notice and this permission notice shall be
included in all copies or substantial portions of the Software.
THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE
LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION
OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION
WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
"""
__all__ = [
'difference',
'wrapdiff',
'normalize',
'csr',
'to_complex',
'to_real',
'mean',
'nanmean',
'resvec',
'nanresvec',
'var',
'nanvar',
'std',
'nanstd',
'confmean',
'nanconfmean',
'corrcc',
'nancorrcc',
'vmpar',
'nanvmpar',
'vmlogpdf',
'vmpdf',
'kappa',
'nankappa'
]
import numpy as np
from scipy.stats import chi2
from scipy.special import iv
def difference(a1, a2, deg=False):
"""Compute the smallest difference between two angle arrays.
Parameters
----------
a1, a2 : np.ndarray
The angle arrays to subtract
deg : bool (default=False)
Whether to compute the difference in degrees or radians
Returns
-------
out : np.ndarray
The difference between a1 and a2
"""
diff = a1 - a2
return wrapdiff(diff, deg=deg)
def wrapdiff(diff, deg=False):
"""Given an array of angle differences, make sure that they lie
between -pi and pi.
Parameters
----------
diff : np.ndarray
The angle difference array
deg : bool (default=False)
Whether the angles are in degrees or radians
Returns
-------
out : np.ndarray
The updated angle differences
"""
if deg:
base = 360
else:
base = np.pi * 2
i = np.abs(diff) > (base / 2.0)
out = diff.copy()
out[i] -= np.sign(diff[i]) * base
return out
def normalize(data, deg=False):
"""Make sure that all values in `data` lie between 0 and 2*pi (or
0 and 360, for degrees).
Parameters
----------
data : number or np.array
The value(s) to normalize
deg : bool (default=False)
Whether the values are in degrees or radians
Returns
-------
out : number np.ndarray
The normalized value(s)
"""
if deg:
base = 360
else:
base = np.pi * 2
data = data % base
return data
def csr(data, axis=None, nanrobust=False, deg=False):
"""Finds the mean cosines, sines, and radii of an array of angles
along the given axis.
Parameters
----------
data : np.ndarray
The angle array
axis : int (default=None)
The axis along which to take the mean computations
nanrobust : bool (default=False)
Ignore nans
deg : bool (default=False)
Whether the values are in degrees or radians
Returns
-------
out : tuple
(mean cosine, mean sine, mean radius)
"""
if deg:
data = np.radians(data)
if nanrobust:
n = np.sum(~np.isnan(data), axis=axis)
C = np.nansum(np.cos(data), axis=axis) / n
S = np.nansum(np.sin(data), axis=axis) / n
else:
C = np.mean(np.cos(data), axis=axis)
S = np.mean(np.sin(data), axis=axis)
R = (C ** 2 + S ** 2) ** 0.5
return (C, S, R)
def to_complex(data, deg=False):
"""Convert an array of angles to complex values, e ** (data * 1j).
Parameters
----------
data : np.ndarray
The angle values
deg : bool (default=False)
Whether the values are in degrees or radians
Returns
-------
out : np.ndarray
An array of complex-valued angles
"""
if deg:
d = np.radians(data)
else:
d = data.copy()
return np.exp(d * 1j)
def to_real(data, deg=False):
"""Convert an array of complex values to angles.
Parameters
----------
data : np.ndarray
The complex values
deg : bool (default=False)
Whether the values are in degrees or radians
Returns
-------
out : np.ndarray
An array of angles
"""
return np.angle(data, deg=deg)
def _mean(alpha, nanrobust, axis=None, w=None, fconf=False):
if w is None:
w = np.ones(alpha.shape)
# weight sum of cos & sin angles
t = w * np.exp(1j * alpha)
if nanrobust:
r = np.nansum(t, axis=axis)
w[np.isnan(t)] = 0
else:
r = np.sum(t, axis=axis)
# obtain mean by
#mu = normalize(np.angle(r))
mu = np.angle(r)
if fconf:
# confidence limits if desired
if nanrobust:
t = nanconfmean(alpha, w=w, axis=axis)
else:
t = confmean(alpha, w=w, axis=axis)
ul = mu + t
ll = mu - t
return (mu, ul, ll)
else:
return mu
def mean(alpha, axis=None, w=None, fconf=False, nancheck=True):
"""Calculate the mean of an array of angles using circular
statistics.
Parameters
----------
alpha : np.ndarray
The array of angles
axis : int (default=None)
The axis along which to take the mean
w : np.ndarray (default=None)
An array of weights, if a weighted mean is desired
fconf : bool (default=False)
Whether or not to return confidence intervals along with the
means
nancheck : bool (default=True)
Whether or not to check for NaNs in the array
Returns
-------
out : np.ndarray OR (np.ndarray, np.ndarray, np.ndarray)
The resulting numpy array, or a tuple of the array, the upper
confidence limit, and the lower confidence limit
References
----------
Berens, P. (2009). CircStat: A MATLAB Toolbox for Circular
Statistics. Journal of Statistical Software, 31(10). Available
from http://www.jstatsoft.org/v31/i10
"""
if nancheck and np.isnan(alpha).any():
raise(ValueError,
"NaN(s) detected in the input array. "
"If you want to ignore this message, "
"please set nancheck to False. If you "
"want to make the mean robust to NaNs, "
"please use the nanmean function instead.")
return _mean(alpha, False, axis=axis, w=w, fconf=fconf)
def nanmean(alpha, axis=None, w=None, fconf=False):
"""Calculate the mean of an array of angles using circular
statistics.
Parameters
----------
alpha : np.ndarray
The array of angles
axis : int (default=None)
The axis along which to take the mean
w : np.ndarray (default=None)
An array of weights, if a weighted mean is desired
fconf : bool (default=False)
Whether or not to return confidence intervals along with the
means
Returns
-------
out : np.ndarray OR (np.ndarray, np.ndarray, np.ndarray)
The resulting numpy array, or a tuple of the array, the upper
confidence limit, and the lower confidence limit
References
----------
Berens, P. (2009). CircStat: A MATLAB Toolbox for Circular
Statistics. Journal of Statistical Software, 31(10). Available
from http://www.jstatsoft.org/v31/i10
"""
return _mean(alpha, True, axis=axis, w=w, fconf=fconf)
def _resvec(alpha, nanrobust, axis=None, w=None, d=0):
if w is None:
w = np.ones(alpha.shape)
# weight sum of cos & sin angles
t = w * np.exp(1j * alpha)
if nanrobust:
r = np.nansum(t, axis=axis)
w[np.isnan(t)] = 0
else:
r = np.sum(t, axis=axis)
# obtain length
r = np.abs(r) / np.sum(w, axis=axis)
# for data with known spacing, apply correction factor to correct
# for bias in the estimation of r (see Zar, p. 601, equ. 26.16)
if d != 0:
r *= d / 2 / np.sin(d / 2)
return r
def resvec(alpha, axis=None, w=None, d=0, nancheck=True):
"""Computes mean resultant vector length for circular data.
Parameters
----------
alpha : np.ndarray
The array of angles
axis : int (default=None)
The axis along which to take the mean
w : np.ndarray (default=None)
An array of weights, if a weighted mean is desired
d : number (default=0)
The bin spacing, if the data is binned
nancheck : bool (default=True)
Whether or not to check for NaNs in the array
Returns
-------
out : np.ndarray
The resulting numpy array of resultant vector lengths
References
----------
Berens, P. (2009). CircStat: A MATLAB Toolbox for Circular
Statistics. Journal of Statistical Software, 31(10). Available
from http://www.jstatsoft.org/v31/i10
"""
if nancheck and np.isnan(alpha).any():
raise(ValueError,
"NaN(s) detected in the input array. If you "
"want to ignore this message, please set nancheck "
"to False. If you want to make the mean robust to "
"NaNs, please use the nanresvec function instead.")
return _resvec(alpha, False, axis=axis, w=w, d=d)
def nanresvec(alpha, axis=None, w=None, d=0):
"""Computes mean resultant vector length for circular data.
Parameters
----------
alpha : np.ndarray
The array of angles
axis : int (default=None)
The axis along which to take the mean
w : np.ndarray (default=None)
An array of weights, if a weighted mean is desired
d : number (default=0)
The bin spacing, if the data is binned
Returns
-------
out : np.ndarray
The resulting numpy array of resultant vector lengths
References
----------
Berens, P. (2009). CircStat: A MATLAB Toolbox for Circular
Statistics. Journal of Statistical Software, 31(10). Available
from http://www.jstatsoft.org/v31/i10
"""
return _resvec(alpha, True, axis=axis, w=w, d=d)
def _var(alpha, nanrobust, axis=None, w=None, d=0, angvar=False):
if w is None:
w = np.ones(alpha.shape)
# compute mean resultant vector length
if nanrobust:
r = nanresvec(alpha, axis=axis, w=w, d=d)
else:
r = resvec(alpha, axis=axis, w=w, d=d)
# apply transformation to var
S = 1 - r # circular variance
if angvar:
var = 2 * S # angular variance
else:
var = S
return var
def var(alpha, axis=None, w=None, d=0, angvar=False, nancheck=True):
"""Computes circular variance for circular data.
Parameters
----------
alpha : np.ndarray
The array of angles
axis : int (default=None)
The axis along which to take the mean
w : np.ndarray (default=None)
An array of weights, if a weighted mean is desired
d : number (default=0)
The bin spacing, if the data is binned
angvar : bool (default=False):
If True, calculate angular variance, otherwise calculate
circular variance
nancheck : bool (default=True)
Whether or not to check for NaNs in the array
Returns
-------
out : tuple (np.ndarray, np.ndarray)
The calculated circular variance and angular variance
References
----------
Berens, P. (2009). CircStat: A MATLAB Toolbox for Circular
Statistics. Journal of Statistical Software, 31(10). Available
from http://www.jstatsoft.org/v31/i10
"""
if nancheck and np.isnan(alpha).any():
raise(ValueError,
"NaN(s) detected in the input array. If you want "
"to ignore this message, please set nancheck to "
"False. If you want to make the mean robust to NaNs, "
"please use the nanvar function instead.")
return _var(alpha, False, axis=axis, w=w, d=d, angvar=angvar)
def nanvar(alpha, axis=None, w=None, d=0, angvar=False):
"""Computes circular variance for circular data.
Parameters
----------
alpha : np.ndarray
The array of angles
axis : int (default=None)
The axis along which to take the mean
w : np.ndarray (default=None)
An array of weights, if a weighted mean is desired
d : number (default=0)
The bin spacing, if the data is binned
angvar : bool (default=False):
If True, calculate angular variance, otherwise calculate
circular variance
Returns
-------
out : tuple (np.ndarray, np.ndarray)
The calculated circular variance and angular variance
References
----------
Berens, P. (2009). CircStat: A MATLAB Toolbox for Circular
Statistics. Journal of Statistical Software, 31(10). Available
from http://www.jstatsoft.org/v31/i10
"""
return _var(alpha, True, axis=axis, w=w, d=d, angvar=angvar)
def _std(data, nanrobust, axis=None, deg=False, varroot=False):
# convert to radians
if deg:
data = np.radians(data)
# compute mean resultant vector length
if nanrobust:
R = nanresvec(data, axis=axis)
else:
R = resvec(data, axis=axis)
# if just compute the square root, compute the variance and then
# take the square root
if varroot:
V = 1 - R
sd = np.sqrt(V)
# otherwise calculate true circular standard deviation
else:
sd = np.sqrt(-2 * np.log(R))
# convert back to degrees, if necessary
if deg:
sd = np.degrees(sd)
return sd
def std(data, axis=None, deg=False, varroot=False, nancheck=True):
"""Computes circular standard deviation for circular data.
Parameters
----------
data : np.ndarray
The array of angles
axis : int (default=None)
The axis along which to take the mean
deg : bool (default=False)
Whether the data is in degrees or radians
varroot : bool (default=False)
If True, compute the square root of the variance. If False,
compute the circlar standard deviation.
nancheck : bool (default=True):
Whether to check for NaNs in the input array
Returns
-------
out : np.ndarray
The calculated circular standard deviation
References
----------
http://en.wikipedia.org/wiki/Directional_statistics#Measures_of_location_and_spread
"""
if nancheck and np.isnan(data).any():
raise (ValueError,
"NaN(s) detected in the input array. If you want "
"to ignore this message, please set nancheck to "
"False. If you want to make the stddev robust to "
"NaNs, please use the nanstd function instead.")
return _std(data, False, axis=axis, deg=deg, varroot=varroot)
def nanstd(data, axis=None, deg=False, varroot=False):
"""Computes circular standard deviation for circular data and is
robust to invalid entries in the array.
Parameters
----------
data : np.ndarray
The array of angles
axis : int (default=None)
The axis along which to take the mean
deg : bool (default=False)
Whether the data is in degrees or radians
varroot : bool (default=False)
If True, compute the square root of the variance. If False,
compute the circlar standard deviation.
Returns
-------
out : np.ndarray
The calculated circular standard deviation
References
----------
http://en.wikipedia.org/wiki/Directional_statistics#Measures_of_location_and_spread
"""
return _std(data, True, axis=axis, deg=deg, varroot=varroot)
def _confmean(alpha, nanrobust, axis=None, ci=0.95, w=None, d=0):
if w is None:
w = np.ones(alpha.shape)
# compute ingredients for conf. lim.
if nanrobust:
r = nanresvec(alpha, axis=axis, w=w, d=d)
else:
r = resvec(alpha, axis=axis, w=w, d=d)
n = np.sum(w, axis=axis)
R = n * r
c2 = chi2.isf(1-ci, 1)
# check for resultant vector length and select appropriate formula
t = np.zeros(r.shape)
tscalar = np.isscalar(r)
if tscalar:
r = np.array([r])
n = np.array([n])
R = np.array([R])
t = np.array([t])
# fill in values
i = (r < 0.9) & (r > np.sqrt(c2 / 2. / n))
t[i] = np.sqrt((2 * n[i] * (2 * R[i]**2 - n[i] * c2)) / (4 * n[i] - c2))
j = r >= 0.9
t[j] = np.sqrt(n[j] ** 2 - (n[j] ** 2 - R[j] ** 2) * np.exp(c2 / n[j]))
t[~(i | j)] = np.nan
# apply final transform
t = np.arccos(t / R)
if tscalar:
t = t[0]
return t
def confmean(alpha, axis=None, ci=0.95, w=None, d=0, nancheck=True):
"""Calculate the confidence limit of the mean of circular data.
Parameters
----------
alpha : np.ndarray
The array of angles
axis : int (default=None)
The axis along which to take the mean
ci : number (default=0.95)
The confidence interval, between 0 and 1
w : np.ndarray (default=None)
An array of weights, if a weighted mean is desired
d : number (default=0)
The bin spacing, if the data is binned
nancheck : bool (default=True)
Whether or not to check for NaNs in the array
Returns
-------
out : np.ndarray
The confidence limit(s) of the input array
References
----------
Berens, P. (2009). CircStat: A MATLAB Toolbox for Circular
Statistics. Journal of Statistical Software, 31(10). Available
from http://www.jstatsoft.org/v31/i10
"""
if nancheck and np.isnan(alpha).any():
raise(ValueError,
"NaN(s) detected in the input array. If you want "
"to ignore this message, please set nancheck to False. "
"If you want to make the mean robust to NaNs, please use "
"the nanconfmean function instead.")
return _confmean(alpha, False, axis=axis, ci=ci, w=w, d=d)
def nanconfmean(alpha, axis=None, ci=0.95, w=None, d=0):
"""Calculate the confidence limit of the mean of circular data.
Parameters
----------
alpha : np.ndarray
The array of angles
axis : int (default=None)
The axis along which to take the mean
ci : number (default=0.95)
The confidence interval, between 0 and 1
w : np.ndarray (default=None)
An array of weights, if a weighted mean is desired
d : number (default=0)
The bin spacing, if the data is binned
Returns
-------
out : np.ndarray
The confidence limit(s) of the input array
References
----------
Berens, P. (2009). CircStat: A MATLAB Toolbox for Circular
Statistics. Journal of Statistical Software, 31(10). Available
from http://www.jstatsoft.org/v31/i10
"""
return _confmean(alpha, True, axis=axis, ci=ci, w=w, d=d)
def _corrcc(alpha1, alpha2, nanrobust, axis=None):
if axis is not None and alpha1.shape[axis] != alpha2.shape[axis]:
raise(ValueError, "shape mismatch")
# compute mean directions
if axis is None:
n = alpha1.size
else:
n = alpha1.shape[axis]
#################################################################
c1 = np.cos(alpha1)
c1_2 = np.cos(2*alpha1)
c2 = np.cos(alpha2)
c2_2 = np.cos(2*alpha2)
s1 = np.sin(alpha1)
s1_2 = np.sin(2*alpha1)
s2 = np.sin(alpha2)
s2_2 = np.sin(2*alpha2)
if nanrobust:
sumfunc = lambda x: np.nansum(x, axis=axis)
else:
sumfunc = lambda x: np.sum(x, axis=axis)
num = 4 * (sumfunc(c1*c2) * sumfunc(s1*s2) -
sumfunc(c1*s2) * sumfunc(s1*c2))
den = np.sqrt((n**2 - sumfunc(c1_2)**2 - sumfunc(s1_2)**2) *
(n**2 - sumfunc(c2_2)**2 - sumfunc(s2_2)**2))
rho = num / den
return rho
def corrcc(alpha1, alpha2, axis=None, nancheck=True):
"""Calculate the correlation coefficient between two sets of
circular data.
Parameters
----------
alpha1 : np.ndarray
The first set of circular data
alpha2 : np.ndarray
The second set of circular data
axis : int (default=None)
The axis along which to take the mean
nancheck : bool (default=True)
Whether or not to check for NaNs in the array
Returns
-------
out : tuple (np.ndarray, np.ndarray)
The correlation coefficient(s) and the p-value(s)
References
----------
Berens, P. (2009). CircStat: A MATLAB Toolbox for Circular
Statistics. Journal of Statistical Software, 31(10). Available
from http://www.jstatsoft.org/v31/i10
Fisher, N. & Lee, A. (1983). A correlation coefficient for circular
data. Biometrika. 70 (2), 327-332.
"""
if nancheck and (np.isnan(alpha1).any() or np.isnan(alpha2).any()):
raise ValueError("NaN(s) detected in the input array. "
"If you want to ignore this message, please set "
"nancheck to False. If you want to make the mean "
"robust to NaNs, please use the nancorrcc function "
"instead.")
return _corrcc(alpha1, alpha2, False, axis=axis)
def nancorrcc(alpha1, alpha2, axis=None):
"""Calculate the correlation coefficient between two sets of
circular data.
Parameters
----------
alpha1 : np.ndarray
The first set of circular data
alpha2 : np.ndarray
The second set of circular data
axis : int (default=None)
The axis along which to take the mean
Returns
-------
out : tuple (np.ndarray, np.ndarray)
The correlation coefficient(s) and the p-value(s)
References
----------
Berens, P. (2009). CircStat: A MATLAB Toolbox for Circular
Statistics. Journal of Statistical Software, 31(10). Available
from http://www.jstatsoft.org/v31/i10
Fisher, N. & Lee, A. (1983). A correlation coefficient for circular
data. Biometrika. 70 (2), 327-332.
"""
return _corrcc(alpha1, alpha2, True, axis=axis)
def _vmpar(alpha, nanrobust, axis=None, w=None, d=0):
if w is None:
w = np.ones(alpha.shape)
if nanrobust:
_kappa = nankappa(alpha, axis=axis, w=w, d=d)
_thetahat = nanmean(alpha, axis=axis, w=w)
else:
_kappa = kappa(alpha, axis=axis, w=w, d=d)
_thetahat = mean(alpha, axis=axis, w=w)
return (_thetahat, _kappa)
def vmpar(alpha, axis=None, w=None, d=0, nancheck=True):
"""Estimate the parameters of a Von Mises distribution.
Parameters
----------
alpha : np.ndarray
The array of angles
axis : int (default=None)
The axis along which to take the mean
w : np.ndarray (default=None)
An array of weights, if a weighted mean is desired
d : number (default=0)
The bin spacing, if the data is binned
nancheck : bool (default=True)
Whether or not to check for NaNs in the array
Returns
-------
out : tuple (np.ndarray, np.ndarray)
The theta and kappa parameters, respectively
References
----------
Berens, P. (2009). CircStat: A MATLAB Toolbox for Circular
Statistics. Journal of Statistical Software, 31(10). Available
from http://www.jstatsoft.org/v31/i10
"""
if nancheck and np.isnan(alpha).any():
raise(ValueError,
"NaN(s) detected in the input array. "
"If you want to ignore this message, please "
"set nancheck to False. If you want to make "
"the mean robust to NaNs, please use the nanvmpar "
"function instead.")
return _vmpar(alpha, False, axis=axis, w=w, d=d)
def nanvmpar(alpha, axis=None, w=None, d=0):
"""Estimate the parameters of a Von Mises distribution.
Parameters
----------
alpha : np.ndarray
The array of angles
axis : int (default=None)
The axis along which to take the mean
w : np.ndarray (default=None)
An array of weights, if a weighted mean is desired
d : number (default=0)
The bin spacing, if the data is binned
Returns
-------
out : tuple (np.ndarray, np.ndarray)
The theta and kappa parameters, respectively
References
----------
Berens, P. (2009). CircStat: A MATLAB Toolbox for Circular
Statistics. Journal of Statistical Software, 31(10). Available
from http://www.jstatsoft.org/v31/i10
"""
return _vmpar(alpha, True, axis=axis, w=w, d=d)
def vmlogpdf(alpha, thetahat=0.0, kappa=1.0):
"""Computes the circular Von Mises log-PDF with preferred
direction thetahat and concentration kappa at each of the angles
in alpha.
The vmpdf is given by f(phi) =
(1/(2pi*I0(kappa))*exp(kappa*cos(phi-thetahat)
Parameters
----------
alpha : np.ndarray
The array of angles
thetahat : number
The preferred direction
kappa : number
Concentration at each of the angles in alpha
Returns
-------
out : np.ndarray
The Von Mises log-PDF for alpha
References
----------
Berens, P. (2009). CircStat: A MATLAB Toolbox for Circular
Statistics. Journal of Statistical Software, 31(10). Available
from http://www.jstatsoft.org/v31/i10
"""
C = -np.log(2 * np.pi * iv(0, kappa))
p = C + (kappa * np.cos(alpha - thetahat))
return p
def vmpdf(alpha, thetahat=0.0, kappa=1.0):
"""Computes the circular Von Mises PDF with preferred direction
thetahat and concentration kappa at each of the angles in alpha.
The vmpdf is given by f(phi) =
(1/(2pi*I0(kappa))*exp(kappa*cos(phi-thetahat)
Parameters
----------
alpha : np.ndarray
The array of angles
thetahat : number
The preferred direction
kappa : number
Concentration at each of the angles in alpha
Returns
-------
out : np.ndarray
The Von Mises PDF for alpha
References
----------
Berens, P. (2009). CircStat: A MATLAB Toolbox for Circular
Statistics. Journal of Statistical Software, 31(10). Available
from http://www.jstatsoft.org/v31/i10
"""
p = np.exp(vmlogpdf(alpha, thetahat=thetahat, kappa=kappa))
return p
def _kappa(alpha, nanrobust, axis=None, w=None, d=0):
if w is None:
w = np.ones(alpha.shape)
if axis is None:
n = alpha.size
else:
n = alpha.shape[axis]
if nanrobust:
R = nanresvec(alpha, axis=axis, w=w, d=d)
else:
R = resvec(alpha, axis=axis, w=w, d=d)
if type(R) is not np.ndarray or len(R.shape) == 0:
R = np.array([R])
i1 = R < 0.53