478 lines
15 KiB
Python
478 lines
15 KiB
Python
"""
|
|
Additional statistics functions with support for masked arrays.
|
|
|
|
"""
|
|
|
|
# Original author (2007): Pierre GF Gerard-Marchant
|
|
|
|
|
|
from __future__ import division, print_function, absolute_import
|
|
|
|
|
|
__all__ = ['compare_medians_ms',
|
|
'hdquantiles', 'hdmedian', 'hdquantiles_sd',
|
|
'idealfourths',
|
|
'median_cihs','mjci','mquantiles_cimj',
|
|
'rsh',
|
|
'trimmed_mean_ci',]
|
|
|
|
|
|
import numpy as np
|
|
from numpy import float_, int_, ndarray
|
|
|
|
import numpy.ma as ma
|
|
from numpy.ma import MaskedArray
|
|
|
|
from . import mstats_basic as mstats
|
|
|
|
from scipy.stats.distributions import norm, beta, t, binom
|
|
|
|
|
|
def hdquantiles(data, prob=list([.25,.5,.75]), axis=None, var=False,):
|
|
"""
|
|
Computes quantile estimates with the Harrell-Davis method.
|
|
|
|
The quantile estimates are calculated as a weighted linear combination
|
|
of order statistics.
|
|
|
|
Parameters
|
|
----------
|
|
data : array_like
|
|
Data array.
|
|
prob : sequence, optional
|
|
Sequence of quantiles to compute.
|
|
axis : int or None, optional
|
|
Axis along which to compute the quantiles. If None, use a flattened
|
|
array.
|
|
var : bool, optional
|
|
Whether to return the variance of the estimate.
|
|
|
|
Returns
|
|
-------
|
|
hdquantiles : MaskedArray
|
|
A (p,) array of quantiles (if `var` is False), or a (2,p) array of
|
|
quantiles and variances (if `var` is True), where ``p`` is the
|
|
number of quantiles.
|
|
|
|
See Also
|
|
--------
|
|
hdquantiles_sd
|
|
|
|
"""
|
|
def _hd_1D(data,prob,var):
|
|
"Computes the HD quantiles for a 1D array. Returns nan for invalid data."
|
|
xsorted = np.squeeze(np.sort(data.compressed().view(ndarray)))
|
|
# Don't use length here, in case we have a numpy scalar
|
|
n = xsorted.size
|
|
|
|
hd = np.empty((2,len(prob)), float_)
|
|
if n < 2:
|
|
hd.flat = np.nan
|
|
if var:
|
|
return hd
|
|
return hd[0]
|
|
|
|
v = np.arange(n+1) / float(n)
|
|
betacdf = beta.cdf
|
|
for (i,p) in enumerate(prob):
|
|
_w = betacdf(v, (n+1)*p, (n+1)*(1-p))
|
|
w = _w[1:] - _w[:-1]
|
|
hd_mean = np.dot(w, xsorted)
|
|
hd[0,i] = hd_mean
|
|
#
|
|
hd[1,i] = np.dot(w, (xsorted-hd_mean)**2)
|
|
#
|
|
hd[0, prob == 0] = xsorted[0]
|
|
hd[0, prob == 1] = xsorted[-1]
|
|
if var:
|
|
hd[1, prob == 0] = hd[1, prob == 1] = np.nan
|
|
return hd
|
|
return hd[0]
|
|
# Initialization & checks
|
|
data = ma.array(data, copy=False, dtype=float_)
|
|
p = np.array(prob, copy=False, ndmin=1)
|
|
# Computes quantiles along axis (or globally)
|
|
if (axis is None) or (data.ndim == 1):
|
|
result = _hd_1D(data, p, var)
|
|
else:
|
|
if data.ndim > 2:
|
|
raise ValueError("Array 'data' must be at most two dimensional, "
|
|
"but got data.ndim = %d" % data.ndim)
|
|
result = ma.apply_along_axis(_hd_1D, axis, data, p, var)
|
|
|
|
return ma.fix_invalid(result, copy=False)
|
|
|
|
|
|
def hdmedian(data, axis=-1, var=False):
|
|
"""
|
|
Returns the Harrell-Davis estimate of the median along the given axis.
|
|
|
|
Parameters
|
|
----------
|
|
data : ndarray
|
|
Data array.
|
|
axis : int, optional
|
|
Axis along which to compute the quantiles. If None, use a flattened
|
|
array.
|
|
var : bool, optional
|
|
Whether to return the variance of the estimate.
|
|
|
|
Returns
|
|
-------
|
|
hdmedian : MaskedArray
|
|
The median values. If ``var=True``, the variance is returned inside
|
|
the masked array. E.g. for a 1-D array the shape change from (1,) to
|
|
(2,).
|
|
|
|
"""
|
|
result = hdquantiles(data,[0.5], axis=axis, var=var)
|
|
return result.squeeze()
|
|
|
|
|
|
def hdquantiles_sd(data, prob=list([.25,.5,.75]), axis=None):
|
|
"""
|
|
The standard error of the Harrell-Davis quantile estimates by jackknife.
|
|
|
|
Parameters
|
|
----------
|
|
data : array_like
|
|
Data array.
|
|
prob : sequence, optional
|
|
Sequence of quantiles to compute.
|
|
axis : int, optional
|
|
Axis along which to compute the quantiles. If None, use a flattened
|
|
array.
|
|
|
|
Returns
|
|
-------
|
|
hdquantiles_sd : MaskedArray
|
|
Standard error of the Harrell-Davis quantile estimates.
|
|
|
|
See Also
|
|
--------
|
|
hdquantiles
|
|
|
|
"""
|
|
def _hdsd_1D(data, prob):
|
|
"Computes the std error for 1D arrays."
|
|
xsorted = np.sort(data.compressed())
|
|
n = len(xsorted)
|
|
|
|
hdsd = np.empty(len(prob), float_)
|
|
if n < 2:
|
|
hdsd.flat = np.nan
|
|
|
|
vv = np.arange(n) / float(n-1)
|
|
betacdf = beta.cdf
|
|
|
|
for (i,p) in enumerate(prob):
|
|
_w = betacdf(vv, (n+1)*p, (n+1)*(1-p))
|
|
w = _w[1:] - _w[:-1]
|
|
mx_ = np.fromiter([np.dot(w,xsorted[np.r_[list(range(0,k)),
|
|
list(range(k+1,n))].astype(int_)])
|
|
for k in range(n)], dtype=float_)
|
|
mx_var = np.array(mx_.var(), copy=False, ndmin=1) * n / float(n-1)
|
|
hdsd[i] = float(n-1) * np.sqrt(np.diag(mx_var).diagonal() / float(n))
|
|
return hdsd
|
|
|
|
# Initialization & checks
|
|
data = ma.array(data, copy=False, dtype=float_)
|
|
p = np.array(prob, copy=False, ndmin=1)
|
|
# Computes quantiles along axis (or globally)
|
|
if (axis is None):
|
|
result = _hdsd_1D(data, p)
|
|
else:
|
|
if data.ndim > 2:
|
|
raise ValueError("Array 'data' must be at most two dimensional, "
|
|
"but got data.ndim = %d" % data.ndim)
|
|
result = ma.apply_along_axis(_hdsd_1D, axis, data, p)
|
|
|
|
return ma.fix_invalid(result, copy=False).ravel()
|
|
|
|
|
|
def trimmed_mean_ci(data, limits=(0.2,0.2), inclusive=(True,True),
|
|
alpha=0.05, axis=None):
|
|
"""
|
|
Selected confidence interval of the trimmed mean along the given axis.
|
|
|
|
Parameters
|
|
----------
|
|
data : array_like
|
|
Input data.
|
|
limits : {None, tuple}, optional
|
|
None or a two item tuple.
|
|
Tuple of the percentages to cut on each side of the array, with respect
|
|
to the number of unmasked data, as floats between 0. and 1. If ``n``
|
|
is the number of unmasked data before trimming, then
|
|
(``n * limits[0]``)th smallest data and (``n * limits[1]``)th
|
|
largest data are masked. The total number of unmasked data after
|
|
trimming is ``n * (1. - sum(limits))``.
|
|
The value of one limit can be set to None to indicate an open interval.
|
|
|
|
Defaults to (0.2, 0.2).
|
|
inclusive : (2,) tuple of boolean, optional
|
|
If relative==False, tuple indicating whether values exactly equal to
|
|
the absolute limits are allowed.
|
|
If relative==True, tuple indicating whether the number of data being
|
|
masked on each side should be rounded (True) or truncated (False).
|
|
|
|
Defaults to (True, True).
|
|
alpha : float, optional
|
|
Confidence level of the intervals.
|
|
|
|
Defaults to 0.05.
|
|
axis : int, optional
|
|
Axis along which to cut. If None, uses a flattened version of `data`.
|
|
|
|
Defaults to None.
|
|
|
|
Returns
|
|
-------
|
|
trimmed_mean_ci : (2,) ndarray
|
|
The lower and upper confidence intervals of the trimmed data.
|
|
|
|
"""
|
|
data = ma.array(data, copy=False)
|
|
trimmed = mstats.trimr(data, limits=limits, inclusive=inclusive, axis=axis)
|
|
tmean = trimmed.mean(axis)
|
|
tstde = mstats.trimmed_stde(data,limits=limits,inclusive=inclusive,axis=axis)
|
|
df = trimmed.count(axis) - 1
|
|
tppf = t.ppf(1-alpha/2.,df)
|
|
return np.array((tmean - tppf*tstde, tmean+tppf*tstde))
|
|
|
|
|
|
def mjci(data, prob=[0.25,0.5,0.75], axis=None):
|
|
"""
|
|
Returns the Maritz-Jarrett estimators of the standard error of selected
|
|
experimental quantiles of the data.
|
|
|
|
Parameters
|
|
----------
|
|
data : ndarray
|
|
Data array.
|
|
prob : sequence, optional
|
|
Sequence of quantiles to compute.
|
|
axis : int or None, optional
|
|
Axis along which to compute the quantiles. If None, use a flattened
|
|
array.
|
|
|
|
"""
|
|
def _mjci_1D(data, p):
|
|
data = np.sort(data.compressed())
|
|
n = data.size
|
|
prob = (np.array(p) * n + 0.5).astype(int_)
|
|
betacdf = beta.cdf
|
|
|
|
mj = np.empty(len(prob), float_)
|
|
x = np.arange(1,n+1, dtype=float_) / n
|
|
y = x - 1./n
|
|
for (i,m) in enumerate(prob):
|
|
W = betacdf(x,m-1,n-m) - betacdf(y,m-1,n-m)
|
|
C1 = np.dot(W,data)
|
|
C2 = np.dot(W,data**2)
|
|
mj[i] = np.sqrt(C2 - C1**2)
|
|
return mj
|
|
|
|
data = ma.array(data, copy=False)
|
|
if data.ndim > 2:
|
|
raise ValueError("Array 'data' must be at most two dimensional, "
|
|
"but got data.ndim = %d" % data.ndim)
|
|
|
|
p = np.array(prob, copy=False, ndmin=1)
|
|
# Computes quantiles along axis (or globally)
|
|
if (axis is None):
|
|
return _mjci_1D(data, p)
|
|
else:
|
|
return ma.apply_along_axis(_mjci_1D, axis, data, p)
|
|
|
|
|
|
def mquantiles_cimj(data, prob=[0.25,0.50,0.75], alpha=0.05, axis=None):
|
|
"""
|
|
Computes the alpha confidence interval for the selected quantiles of the
|
|
data, with Maritz-Jarrett estimators.
|
|
|
|
Parameters
|
|
----------
|
|
data : ndarray
|
|
Data array.
|
|
prob : sequence, optional
|
|
Sequence of quantiles to compute.
|
|
alpha : float, optional
|
|
Confidence level of the intervals.
|
|
axis : int or None, optional
|
|
Axis along which to compute the quantiles.
|
|
If None, use a flattened array.
|
|
|
|
Returns
|
|
-------
|
|
ci_lower : ndarray
|
|
The lower boundaries of the confidence interval. Of the same length as
|
|
`prob`.
|
|
ci_upper : ndarray
|
|
The upper boundaries of the confidence interval. Of the same length as
|
|
`prob`.
|
|
|
|
"""
|
|
alpha = min(alpha, 1 - alpha)
|
|
z = norm.ppf(1 - alpha/2.)
|
|
xq = mstats.mquantiles(data, prob, alphap=0, betap=0, axis=axis)
|
|
smj = mjci(data, prob, axis=axis)
|
|
return (xq - z * smj, xq + z * smj)
|
|
|
|
|
|
def median_cihs(data, alpha=0.05, axis=None):
|
|
"""
|
|
Computes the alpha-level confidence interval for the median of the data.
|
|
|
|
Uses the Hettmasperger-Sheather method.
|
|
|
|
Parameters
|
|
----------
|
|
data : array_like
|
|
Input data. Masked values are discarded. The input should be 1D only,
|
|
or `axis` should be set to None.
|
|
alpha : float, optional
|
|
Confidence level of the intervals.
|
|
axis : int or None, optional
|
|
Axis along which to compute the quantiles. If None, use a flattened
|
|
array.
|
|
|
|
Returns
|
|
-------
|
|
median_cihs
|
|
Alpha level confidence interval.
|
|
|
|
"""
|
|
def _cihs_1D(data, alpha):
|
|
data = np.sort(data.compressed())
|
|
n = len(data)
|
|
alpha = min(alpha, 1-alpha)
|
|
k = int(binom._ppf(alpha/2., n, 0.5))
|
|
gk = binom.cdf(n-k,n,0.5) - binom.cdf(k-1,n,0.5)
|
|
if gk < 1-alpha:
|
|
k -= 1
|
|
gk = binom.cdf(n-k,n,0.5) - binom.cdf(k-1,n,0.5)
|
|
gkk = binom.cdf(n-k-1,n,0.5) - binom.cdf(k,n,0.5)
|
|
I = (gk - 1 + alpha)/(gk - gkk)
|
|
lambd = (n-k) * I / float(k + (n-2*k)*I)
|
|
lims = (lambd*data[k] + (1-lambd)*data[k-1],
|
|
lambd*data[n-k-1] + (1-lambd)*data[n-k])
|
|
return lims
|
|
data = ma.array(data, copy=False)
|
|
# Computes quantiles along axis (or globally)
|
|
if (axis is None):
|
|
result = _cihs_1D(data, alpha)
|
|
else:
|
|
if data.ndim > 2:
|
|
raise ValueError("Array 'data' must be at most two dimensional, "
|
|
"but got data.ndim = %d" % data.ndim)
|
|
result = ma.apply_along_axis(_cihs_1D, axis, data, alpha)
|
|
|
|
return result
|
|
|
|
|
|
def compare_medians_ms(group_1, group_2, axis=None):
|
|
"""
|
|
Compares the medians from two independent groups along the given axis.
|
|
|
|
The comparison is performed using the McKean-Schrader estimate of the
|
|
standard error of the medians.
|
|
|
|
Parameters
|
|
----------
|
|
group_1 : array_like
|
|
First dataset. Has to be of size >=7.
|
|
group_2 : array_like
|
|
Second dataset. Has to be of size >=7.
|
|
axis : int, optional
|
|
Axis along which the medians are estimated. If None, the arrays are
|
|
flattened. If `axis` is not None, then `group_1` and `group_2`
|
|
should have the same shape.
|
|
|
|
Returns
|
|
-------
|
|
compare_medians_ms : {float, ndarray}
|
|
If `axis` is None, then returns a float, otherwise returns a 1-D
|
|
ndarray of floats with a length equal to the length of `group_1`
|
|
along `axis`.
|
|
|
|
"""
|
|
(med_1, med_2) = (ma.median(group_1,axis=axis), ma.median(group_2,axis=axis))
|
|
(std_1, std_2) = (mstats.stde_median(group_1, axis=axis),
|
|
mstats.stde_median(group_2, axis=axis))
|
|
W = np.abs(med_1 - med_2) / ma.sqrt(std_1**2 + std_2**2)
|
|
return 1 - norm.cdf(W)
|
|
|
|
|
|
def idealfourths(data, axis=None):
|
|
"""
|
|
Returns an estimate of the lower and upper quartiles.
|
|
|
|
Uses the ideal fourths algorithm.
|
|
|
|
Parameters
|
|
----------
|
|
data : array_like
|
|
Input array.
|
|
axis : int, optional
|
|
Axis along which the quartiles are estimated. If None, the arrays are
|
|
flattened.
|
|
|
|
Returns
|
|
-------
|
|
idealfourths : {list of floats, masked array}
|
|
Returns the two internal values that divide `data` into four parts
|
|
using the ideal fourths algorithm either along the flattened array
|
|
(if `axis` is None) or along `axis` of `data`.
|
|
|
|
"""
|
|
def _idf(data):
|
|
x = data.compressed()
|
|
n = len(x)
|
|
if n < 3:
|
|
return [np.nan,np.nan]
|
|
(j,h) = divmod(n/4. + 5/12.,1)
|
|
j = int(j)
|
|
qlo = (1-h)*x[j-1] + h*x[j]
|
|
k = n - j
|
|
qup = (1-h)*x[k] + h*x[k-1]
|
|
return [qlo, qup]
|
|
data = ma.sort(data, axis=axis).view(MaskedArray)
|
|
if (axis is None):
|
|
return _idf(data)
|
|
else:
|
|
return ma.apply_along_axis(_idf, axis, data)
|
|
|
|
|
|
def rsh(data, points=None):
|
|
"""
|
|
Evaluates Rosenblatt's shifted histogram estimators for each data point.
|
|
|
|
Rosenblatt's estimator is a centered finite-difference approximation to the
|
|
derivative of the empirical cumulative distribution function.
|
|
|
|
Parameters
|
|
----------
|
|
data : sequence
|
|
Input data, should be 1-D. Masked values are ignored.
|
|
points : sequence or None, optional
|
|
Sequence of points where to evaluate Rosenblatt shifted histogram.
|
|
If None, use the data.
|
|
|
|
"""
|
|
data = ma.array(data, copy=False)
|
|
if points is None:
|
|
points = data
|
|
else:
|
|
points = np.array(points, copy=False, ndmin=1)
|
|
|
|
if data.ndim != 1:
|
|
raise AttributeError("The input array should be 1D only !")
|
|
|
|
n = data.count()
|
|
r = idealfourths(data, axis=None)
|
|
h = 1.2 * (r[-1]-r[0]) / n**(1./5)
|
|
nhi = (data[:,None] <= points[None,:] + h).sum(0)
|
|
nlo = (data[:,None] < points[None,:] - h).sum(0)
|
|
return (nhi-nlo) / (2.*n*h)
|