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Python

"""
An extension of scipy.stats.stats to support masked arrays
"""
# Original author (2007): Pierre GF Gerard-Marchant
# TODO : f_value_wilks_lambda looks botched... what are dfnum & dfden for ?
# TODO : ttest_rel looks botched: what are x1,x2,v1,v2 for ?
# TODO : reimplement ksonesamp
from __future__ import division, print_function, absolute_import
__all__ = ['argstoarray',
'betai',
'chisquare','count_tied_groups',
'describe',
'f_oneway','f_value_wilks_lambda','find_repeats','friedmanchisquare',
'kendalltau','kendalltau_seasonal','kruskal','kruskalwallis',
'ks_twosamp','ks_2samp','kurtosis','kurtosistest',
'linregress',
'mannwhitneyu', 'meppf','mode','moment','mquantiles','msign',
'normaltest',
'obrientransform',
'pearsonr','plotting_positions','pointbiserialr',
'rankdata',
'scoreatpercentile','sem',
'sen_seasonal_slopes','signaltonoise','skew','skewtest','spearmanr',
'theilslopes','threshold','tmax','tmean','tmin','trim','trimboth',
'trimtail','trima','trimr','trimmed_mean','trimmed_std',
'trimmed_stde','trimmed_var','tsem','ttest_1samp','ttest_onesamp',
'ttest_ind','ttest_rel','tvar',
'variation',
'winsorize',
'zmap', 'zscore'
]
import numpy as np
from numpy import ndarray
import numpy.ma as ma
from numpy.ma import masked, nomask
from scipy.lib.six import iteritems
import itertools
import warnings
from . import stats
from . import distributions
import scipy.special as special
from . import futil
genmissingvaldoc = """
Notes
-----
Missing values are considered pair-wise: if a value is missing in x,
the corresponding value in y is masked.
"""
def _chk_asarray(a, axis):
# Always returns a masked array, raveled for axis=None
a = ma.asanyarray(a)
if axis is None:
a = ma.ravel(a)
outaxis = 0
else:
outaxis = axis
return a, outaxis
def _chk2_asarray(a, b, axis):
a = ma.asanyarray(a)
b = ma.asanyarray(b)
if axis is None:
a = ma.ravel(a)
b = ma.ravel(b)
outaxis = 0
else:
outaxis = axis
return a, b, outaxis
def _chk_size(a,b):
a = ma.asanyarray(a)
b = ma.asanyarray(b)
(na, nb) = (a.size, b.size)
if na != nb:
raise ValueError("The size of the input array should match!"
" (%s <> %s)" % (na, nb))
return (a, b, na)
def argstoarray(*args):
"""
Constructs a 2D array from a group of sequences.
Sequences are filled with missing values to match the length of the longest
sequence.
Parameters
----------
args : sequences
Group of sequences.
Returns
-------
argstoarray : MaskedArray
A ( `m` x `n` ) masked array, where `m` is the number of arguments and
`n` the length of the longest argument.
Notes
-----
`numpy.ma.row_stack` has identical behavior, but is called with a sequence
of sequences.
"""
if len(args) == 1 and not isinstance(args[0], ndarray):
output = ma.asarray(args[0])
if output.ndim != 2:
raise ValueError("The input should be 2D")
else:
n = len(args)
m = max([len(k) for k in args])
output = ma.array(np.empty((n,m), dtype=float), mask=True)
for (k,v) in enumerate(args):
output[k,:len(v)] = v
output[np.logical_not(np.isfinite(output._data))] = masked
return output
def find_repeats(arr):
"""Find repeats in arr and return a tuple (repeats, repeat_count).
Masked values are discarded.
Parameters
----------
arr : sequence
Input array. The array is flattened if it is not 1D.
Returns
-------
repeats : ndarray
Array of repeated values.
counts : ndarray
Array of counts.
"""
marr = ma.compressed(arr)
if not marr.size:
return (np.array(0), np.array(0))
(v1, v2, n) = futil.dfreps(ma.array(ma.compressed(arr), copy=True))
return (v1[:n], v2[:n])
def count_tied_groups(x, use_missing=False):
"""
Counts the number of tied values.
Parameters
----------
x : sequence
Sequence of data on which to counts the ties
use_missing : boolean
Whether to consider missing values as tied.
Returns
-------
count_tied_groups : dict
Returns a dictionary (nb of ties: nb of groups).
Examples
--------
>>> from scipy.stats import mstats
>>> z = [0, 0, 0, 2, 2, 2, 3, 3, 4, 5, 6]
>>> mstats.count_tied_groups(z)
{2: 1, 3: 2}
In the above example, the ties were 0 (3x), 2 (3x) and 3 (2x).
>>> z = np.ma.array([0, 0, 1, 2, 2, 2, 3, 3, 4, 5, 6])
>>> mstats.count_tied_groups(z)
{2: 2, 3: 1}
>>> z[[1,-1]] = np.ma.masked
>>> mstats.count_tied_groups(z, use_missing=True)
{2: 2, 3: 1}
"""
nmasked = ma.getmask(x).sum()
# We need the copy as find_repeats will overwrite the initial data
data = ma.compressed(x).copy()
(ties, counts) = find_repeats(data)
nties = {}
if len(ties):
nties = dict(zip(np.unique(counts), itertools.repeat(1)))
nties.update(dict(zip(*find_repeats(counts))))
if nmasked and use_missing:
try:
nties[nmasked] += 1
except KeyError:
nties[nmasked] = 1
return nties
def rankdata(data, axis=None, use_missing=False):
"""Returns the rank (also known as order statistics) of each data point
along the given axis.
If some values are tied, their rank is averaged.
If some values are masked, their rank is set to 0 if use_missing is False,
or set to the average rank of the unmasked values if use_missing is True.
Parameters
----------
data : sequence
Input data. The data is transformed to a masked array
axis : {None,int}, optional
Axis along which to perform the ranking.
If None, the array is first flattened. An exception is raised if
the axis is specified for arrays with a dimension larger than 2
use_missing : {boolean}, optional
Whether the masked values have a rank of 0 (False) or equal to the
average rank of the unmasked values (True).
"""
def _rank1d(data, use_missing=False):
n = data.count()
rk = np.empty(data.size, dtype=float)
idx = data.argsort()
rk[idx[:n]] = np.arange(1,n+1)
if use_missing:
rk[idx[n:]] = (n+1)/2.
else:
rk[idx[n:]] = 0
repeats = find_repeats(data.copy())
for r in repeats[0]:
condition = (data == r).filled(False)
rk[condition] = rk[condition].mean()
return rk
data = ma.array(data, copy=False)
if axis is None:
if data.ndim > 1:
return _rank1d(data.ravel(), use_missing).reshape(data.shape)
else:
return _rank1d(data, use_missing)
else:
return ma.apply_along_axis(_rank1d,axis,data,use_missing).view(ndarray)
def mode(a, axis=0):
a, axis = _chk_asarray(a, axis)
def _mode1D(a):
(rep,cnt) = find_repeats(a)
if not cnt.ndim:
return (0, 0)
elif cnt.size:
return (rep[cnt.argmax()], cnt.max())
else:
not_masked_indices = ma.flatnotmasked_edges(a)
first_not_masked_index = not_masked_indices[0]
return (a[first_not_masked_index], 1)
if axis is None:
output = _mode1D(ma.ravel(a))
output = (ma.array(output[0]), ma.array(output[1]))
else:
output = ma.apply_along_axis(_mode1D, axis, a)
newshape = list(a.shape)
newshape[axis] = 1
slices = [slice(None)] * output.ndim
slices[axis] = 0
modes = output[tuple(slices)].reshape(newshape)
slices[axis] = 1
counts = output[tuple(slices)].reshape(newshape)
output = (modes, counts)
return output
mode.__doc__ = stats.mode.__doc__
def betai(a, b, x):
x = np.asanyarray(x)
x = ma.where(x < 1.0, x, 1.0) # if x > 1 then return 1.0
return special.betainc(a, b, x)
betai.__doc__ = stats.betai.__doc__
def msign(x):
"""Returns the sign of x, or 0 if x is masked."""
return ma.filled(np.sign(x), 0)
def pearsonr(x,y):
"""
Calculates a Pearson correlation coefficient and the p-value for testing
non-correlation.
The Pearson correlation coefficient measures the linear relationship
between two datasets. Strictly speaking, Pearson's correlation requires
that each dataset be normally distributed. Like other correlation
coefficients, this one varies between -1 and +1 with 0 implying no
correlation. Correlations of -1 or +1 imply an exact linear
relationship. Positive correlations imply that as `x` increases, so does
`y`. Negative correlations imply that as `x` increases, `y` decreases.
The p-value roughly indicates the probability of an uncorrelated system
producing datasets that have a Pearson correlation at least as extreme
as the one computed from these datasets. The p-values are not entirely
reliable but are probably reasonable for datasets larger than 500 or so.
Parameters
----------
x : 1-D array_like
Input
y : 1-D array_like
Input
Returns
-------
pearsonr : float
Pearson's correlation coefficient, 2-tailed p-value.
References
----------
http://www.statsoft.com/textbook/glosp.html#Pearson%20Correlation
"""
(x, y, n) = _chk_size(x, y)
(x, y) = (x.ravel(), y.ravel())
# Get the common mask and the total nb of unmasked elements
m = ma.mask_or(ma.getmask(x), ma.getmask(y))
n -= m.sum()
df = n-2
if df < 0:
return (masked, masked)
(mx, my) = (x.mean(), y.mean())
(xm, ym) = (x-mx, y-my)
r_num = ma.add.reduce(xm*ym)
r_den = ma.sqrt(ma.dot(xm,xm) * ma.dot(ym,ym))
r = r_num / r_den
# Presumably, if r > 1, then it is only some small artifact of floating
# point arithmetic.
r = min(r, 1.0)
r = max(r, -1.0)
df = n - 2
if r is masked or abs(r) == 1.0:
prob = 0.
else:
t_squared = (df / ((1.0 - r) * (1.0 + r))) * r * r
prob = betai(0.5*df, 0.5, df/(df + t_squared))
return r, prob
def spearmanr(x, y, use_ties=True):
"""
Calculates a Spearman rank-order correlation coefficient and the p-value
to test for non-correlation.
The Spearman correlation is a nonparametric measure of the linear
relationship between two datasets. Unlike the Pearson correlation, the
Spearman correlation does not assume that both datasets are normally
distributed. Like other correlation coefficients, this one varies
between -1 and +1 with 0 implying no correlation. Correlations of -1 or
+1 imply an exact linear relationship. Positive correlations imply that
as `x` increases, so does `y`. Negative correlations imply that as `x`
increases, `y` decreases.
Missing values are discarded pair-wise: if a value is missing in `x`, the
corresponding value in `y` is masked.
The p-value roughly indicates the probability of an uncorrelated system
producing datasets that have a Spearman correlation at least as extreme
as the one computed from these datasets. The p-values are not entirely
reliable but are probably reasonable for datasets larger than 500 or so.
Parameters
----------
x : array_like
The length of `x` must be > 2.
y : array_like
The length of `y` must be > 2.
use_ties : bool, optional
Whether the correction for ties should be computed.
Returns
-------
spearmanr : float
Spearman correlation coefficient, 2-tailed p-value.
References
----------
[CRCProbStat2000] section 14.7
"""
(x, y, n) = _chk_size(x, y)
(x, y) = (x.ravel(), y.ravel())
m = ma.mask_or(ma.getmask(x), ma.getmask(y))
n -= m.sum()
if m is not nomask:
x = ma.array(x, mask=m, copy=True)
y = ma.array(y, mask=m, copy=True)
df = n-2
if df < 0:
raise ValueError("The input must have at least 3 entries!")
# Gets the ranks and rank differences
rankx = rankdata(x)
ranky = rankdata(y)
dsq = np.add.reduce((rankx-ranky)**2)
# Tie correction
if use_ties:
xties = count_tied_groups(x)
yties = count_tied_groups(y)
corr_x = np.sum(v*k*(k**2-1) for (k,v) in iteritems(xties))/12.
corr_y = np.sum(v*k*(k**2-1) for (k,v) in iteritems(yties))/12.
else:
corr_x = corr_y = 0
denom = n*(n**2 - 1)/6.
if corr_x != 0 or corr_y != 0:
rho = denom - dsq - corr_x - corr_y
rho /= ma.sqrt((denom-2*corr_x)*(denom-2*corr_y))
else:
rho = 1. - dsq/denom
t = ma.sqrt(ma.divide(df,(rho+1.0)*(1.0-rho))) * rho
if t is masked:
prob = 0.
else:
prob = betai(0.5*df,0.5,df/(df+t*t))
return rho, prob
def kendalltau(x, y, use_ties=True, use_missing=False):
"""
Computes Kendall's rank correlation tau on two variables *x* and *y*.
Parameters
----------
xdata : sequence
First data list (for example, time).
ydata : sequence
Second data list.
use_ties : {True, False}, optional
Whether ties correction should be performed.
use_missing : {False, True}, optional
Whether missing data should be allocated a rank of 0 (False) or the
average rank (True)
Returns
-------
tau : float
Kendall tau
prob : float
Approximate 2-side p-value.
"""
(x, y, n) = _chk_size(x, y)
(x, y) = (x.flatten(), y.flatten())
m = ma.mask_or(ma.getmask(x), ma.getmask(y))
if m is not nomask:
x = ma.array(x, mask=m, copy=True)
y = ma.array(y, mask=m, copy=True)
n -= m.sum()
if n < 2:
return (np.nan, np.nan)
rx = ma.masked_equal(rankdata(x, use_missing=use_missing), 0)
ry = ma.masked_equal(rankdata(y, use_missing=use_missing), 0)
idx = rx.argsort()
(rx, ry) = (rx[idx], ry[idx])
C = np.sum([((ry[i+1:] > ry[i]) * (rx[i+1:] > rx[i])).filled(0).sum()
for i in range(len(ry)-1)], dtype=float)
D = np.sum([((ry[i+1:] < ry[i])*(rx[i+1:] > rx[i])).filled(0).sum()
for i in range(len(ry)-1)], dtype=float)
if use_ties:
xties = count_tied_groups(x)
yties = count_tied_groups(y)
corr_x = np.sum([v*k*(k-1) for (k,v) in iteritems(xties)], dtype=float)
corr_y = np.sum([v*k*(k-1) for (k,v) in iteritems(yties)], dtype=float)
denom = ma.sqrt((n*(n-1)-corr_x)/2. * (n*(n-1)-corr_y)/2.)
else:
denom = n*(n-1)/2.
tau = (C-D) / denom
var_s = n*(n-1)*(2*n+5)
if use_ties:
var_s -= np.sum(v*k*(k-1)*(2*k+5)*1. for (k,v) in iteritems(xties))
var_s -= np.sum(v*k*(k-1)*(2*k+5)*1. for (k,v) in iteritems(yties))
v1 = np.sum([v*k*(k-1) for (k, v) in iteritems(xties)], dtype=float) *\
np.sum([v*k*(k-1) for (k, v) in iteritems(yties)], dtype=float)
v1 /= 2.*n*(n-1)
if n > 2:
v2 = np.sum([v*k*(k-1)*(k-2) for (k,v) in iteritems(xties)],
dtype=float) * \
np.sum([v*k*(k-1)*(k-2) for (k,v) in iteritems(yties)],
dtype=float)
v2 /= 9.*n*(n-1)*(n-2)
else:
v2 = 0
else:
v1 = v2 = 0
var_s /= 18.
var_s += (v1 + v2)
z = (C-D)/np.sqrt(var_s)
prob = special.erfc(abs(z)/np.sqrt(2))
return (tau, prob)
def kendalltau_seasonal(x):
"""
Computes a multivariate Kendall's rank correlation tau, for seasonal data.
Parameters
----------
x : 2-D ndarray
Array of seasonal data, with seasons in columns.
"""
x = ma.array(x, subok=True, copy=False, ndmin=2)
(n,m) = x.shape
n_p = x.count(0)
S_szn = np.sum(msign(x[i:]-x[i]).sum(0) for i in range(n))
S_tot = S_szn.sum()
n_tot = x.count()
ties = count_tied_groups(x.compressed())
corr_ties = np.sum(v*k*(k-1) for (k,v) in iteritems(ties))
denom_tot = ma.sqrt(1.*n_tot*(n_tot-1)*(n_tot*(n_tot-1)-corr_ties))/2.
R = rankdata(x, axis=0, use_missing=True)
K = ma.empty((m,m), dtype=int)
covmat = ma.empty((m,m), dtype=float)
denom_szn = ma.empty(m, dtype=float)
for j in range(m):
ties_j = count_tied_groups(x[:,j].compressed())
corr_j = np.sum(v*k*(k-1) for (k,v) in iteritems(ties_j))
cmb = n_p[j]*(n_p[j]-1)
for k in range(j,m,1):
K[j,k] = np.sum(msign((x[i:,j]-x[i,j])*(x[i:,k]-x[i,k])).sum()
for i in range(n))
covmat[j,k] = (K[j,k] + 4*(R[:,j]*R[:,k]).sum() -
n*(n_p[j]+1)*(n_p[k]+1))/3.
K[k,j] = K[j,k]
covmat[k,j] = covmat[j,k]
denom_szn[j] = ma.sqrt(cmb*(cmb-corr_j)) / 2.
var_szn = covmat.diagonal()
z_szn = msign(S_szn) * (abs(S_szn)-1) / ma.sqrt(var_szn)
z_tot_ind = msign(S_tot) * (abs(S_tot)-1) / ma.sqrt(var_szn.sum())
z_tot_dep = msign(S_tot) * (abs(S_tot)-1) / ma.sqrt(covmat.sum())
prob_szn = special.erfc(abs(z_szn)/np.sqrt(2))
prob_tot_ind = special.erfc(abs(z_tot_ind)/np.sqrt(2))
prob_tot_dep = special.erfc(abs(z_tot_dep)/np.sqrt(2))
chi2_tot = (z_szn*z_szn).sum()
chi2_trd = m * z_szn.mean()**2
output = {'seasonal tau': S_szn/denom_szn,
'global tau': S_tot/denom_tot,
'global tau (alt)': S_tot/denom_szn.sum(),
'seasonal p-value': prob_szn,
'global p-value (indep)': prob_tot_ind,
'global p-value (dep)': prob_tot_dep,
'chi2 total': chi2_tot,
'chi2 trend': chi2_trd,
}
return output
def pointbiserialr(x, y):
x = ma.fix_invalid(x, copy=True).astype(bool)
y = ma.fix_invalid(y, copy=True).astype(float)
# Get rid of the missing data
m = ma.mask_or(ma.getmask(x), ma.getmask(y))
if m is not nomask:
unmask = np.logical_not(m)
x = x[unmask]
y = y[unmask]
n = len(x)
# phat is the fraction of x values that are True
phat = x.sum() / float(n)
y0 = y[~x] # y-values where x is False
y1 = y[x] # y-values where x is True
y0m = y0.mean()
y1m = y1.mean()
rpb = (y1m - y0m)*np.sqrt(phat * (1-phat)) / y.std()
df = n-2
t = rpb*ma.sqrt(df/(1.0-rpb**2))
prob = betai(0.5*df, 0.5, df/(df+t*t))
return rpb, prob
if stats.pointbiserialr.__doc__:
pointbiserialr.__doc__ = stats.pointbiserialr.__doc__ + genmissingvaldoc
def linregress(*args):
"""
Linear regression calculation
Note that the non-masked version is used, and that this docstring is
replaced by the non-masked docstring + some info on missing data.
"""
if len(args) == 1:
# Input is a single 2-D array containing x and y
args = ma.array(args[0], copy=True)
if len(args) == 2:
x = args[0]
y = args[1]
else:
x = args[:, 0]
y = args[:, 1]
else:
# Input is two 1-D arrays
x = ma.array(args[0]).flatten()
y = ma.array(args[1]).flatten()
m = ma.mask_or(ma.getmask(x), ma.getmask(y), shrink=False)
if m is not nomask:
x = ma.array(x, mask=m)
y = ma.array(y, mask=m)
if np.any(~m):
slope, intercept, r, prob, sterrest = stats.linregress(x.data[~m],
y.data[~m])
else:
# All data is masked
return None, None, None, None, None
else:
slope, intercept, r, prob, sterrest = stats.linregress(x.data, y.data)
return slope, intercept, r, prob, sterrest
if stats.linregress.__doc__:
linregress.__doc__ = stats.linregress.__doc__ + genmissingvaldoc
def theilslopes(y, x=None, alpha=0.95):
y = ma.asarray(y).flatten()
if x is None:
x = ma.arange(len(y), dtype=float)
else:
x = ma.asarray(x).flatten()
if len(x) != len(y):
raise ValueError("Incompatible lengths ! (%s<>%s)" % (len(y),len(x)))
m = ma.mask_or(ma.getmask(x), ma.getmask(y))
y._mask = x._mask = m
# Disregard any masked elements of x or y
y = y.compressed()
x = x.compressed().astype(float)
# We now have unmasked arrays so can use `stats.theilslopes`
return stats.theilslopes(y, x, alpha=alpha)
theilslopes.__doc__ = stats.theilslopes.__doc__
def sen_seasonal_slopes(x):
x = ma.array(x, subok=True, copy=False, ndmin=2)
(n,_) = x.shape
# Get list of slopes per season
szn_slopes = ma.vstack([(x[i+1:]-x[i])/np.arange(1,n-i)[:,None]
for i in range(n)])
szn_medslopes = ma.median(szn_slopes, axis=0)
medslope = ma.median(szn_slopes, axis=None)
return szn_medslopes, medslope
def ttest_1samp(a, popmean, axis=0):
a, axis = _chk_asarray(a, axis)
if a.size == 0:
return (np.nan, np.nan)
x = a.mean(axis=axis)
v = a.var(axis=axis, ddof=1)
n = a.count(axis=axis)
df = n - 1.
svar = ((n - 1) * v) / df
t = (x - popmean) / ma.sqrt(svar / n)
prob = betai(0.5 * df, 0.5, df / (df + t*t))
return t, prob
ttest_1samp.__doc__ = stats.ttest_1samp.__doc__
ttest_onesamp = ttest_1samp
def ttest_ind(a, b, axis=0):
a, b, axis = _chk2_asarray(a, b, axis)
if a.size == 0 or b.size == 0:
return (np.nan, np.nan)
(x1, x2) = (a.mean(axis), b.mean(axis))
(v1, v2) = (a.var(axis=axis, ddof=1), b.var(axis=axis, ddof=1))
(n1, n2) = (a.count(axis), b.count(axis))
df = n1 + n2 - 2.
svar = ((n1-1)*v1+(n2-1)*v2) / df
t = (x1-x2)/ma.sqrt(svar*(1.0/n1 + 1.0/n2)) # n-D computation here!
t = ma.filled(t, 1) # replace NaN t-values with 1.0
probs = betai(0.5 * df, 0.5, df/(df + t*t)).reshape(t.shape)
return t, probs.squeeze()
ttest_ind.__doc__ = stats.ttest_ind.__doc__
def ttest_rel(a, b, axis=0):
a, b, axis = _chk2_asarray(a, b, axis)
if len(a) != len(b):
raise ValueError('unequal length arrays')
if a.size == 0 or b.size == 0:
return (np.nan, np.nan)
(x1, x2) = (a.mean(axis), b.mean(axis))
(v1, v2) = (a.var(axis=axis, ddof=1), b.var(axis=axis, ddof=1))
n = a.count(axis)
df = (n-1.0)
d = (a-b).astype('d')
denom = ma.sqrt((n*ma.add.reduce(d*d,axis) - ma.add.reduce(d,axis)**2) / df)
t = ma.add.reduce(d, axis) / denom
t = ma.filled(t, 1)
probs = betai(0.5*df,0.5,df/(df+t*t)).reshape(t.shape).squeeze()
return t, probs
ttest_rel.__doc__ = stats.ttest_rel.__doc__
# stats.chisquare works with masked arrays, so we don't need to
# implement it here.
# For backwards compatibilty, stats.chisquare is included in
# the stats.mstats namespace.
chisquare = stats.chisquare
def mannwhitneyu(x,y, use_continuity=True):
"""
Computes the Mann-Whitney statistic
Missing values in `x` and/or `y` are discarded.
Parameters
----------
x : sequence
Input
y : sequence
Input
use_continuity : {True, False}, optional
Whether a continuity correction (1/2.) should be taken into account.
Returns
-------
u : float
The Mann-Whitney statistics
prob : float
Approximate p-value assuming a normal distribution.
"""
x = ma.asarray(x).compressed().view(ndarray)
y = ma.asarray(y).compressed().view(ndarray)
ranks = rankdata(np.concatenate([x,y]))
(nx, ny) = (len(x), len(y))
nt = nx + ny
U = ranks[:nx].sum() - nx*(nx+1)/2.
U = max(U, nx*ny - U)
u = nx*ny - U
mu = (nx*ny)/2.
sigsq = (nt**3 - nt)/12.
ties = count_tied_groups(ranks)
sigsq -= np.sum(v*(k**3-k) for (k,v) in iteritems(ties))/12.
sigsq *= nx*ny/float(nt*(nt-1))
if use_continuity:
z = (U - 1/2. - mu) / ma.sqrt(sigsq)
else:
z = (U - mu) / ma.sqrt(sigsq)
prob = special.erfc(abs(z)/np.sqrt(2))
return (u, prob)
def kruskalwallis(*args):
output = argstoarray(*args)
ranks = ma.masked_equal(rankdata(output, use_missing=False), 0)
sumrk = ranks.sum(-1)
ngrp = ranks.count(-1)
ntot = ranks.count()
H = 12./(ntot*(ntot+1)) * (sumrk**2/ngrp).sum() - 3*(ntot+1)
# Tie correction
ties = count_tied_groups(ranks)
T = 1. - np.sum(v*(k**3-k) for (k,v) in iteritems(ties))/float(ntot**3-ntot)
if T == 0:
raise ValueError('All numbers are identical in kruskal')
H /= T
df = len(output) - 1
prob = stats.chisqprob(H,df)
return (H, prob)
kruskal = kruskalwallis
kruskalwallis.__doc__ = stats.kruskal.__doc__
def ks_twosamp(data1, data2, alternative="two-sided"):
"""
Computes the Kolmogorov-Smirnov test on two samples.
Missing values are discarded.
Parameters
----------
data1 : array_like
First data set
data2 : array_like
Second data set
alternative : {'two-sided', 'less', 'greater'}, optional
Indicates the alternative hypothesis. Default is 'two-sided'.
Returns
-------
d : float
Value of the Kolmogorov Smirnov test
p : float
Corresponding p-value.
"""
(data1, data2) = (ma.asarray(data1), ma.asarray(data2))
(n1, n2) = (data1.count(), data2.count())
n = (n1*n2/float(n1+n2))
mix = ma.concatenate((data1.compressed(), data2.compressed()))
mixsort = mix.argsort(kind='mergesort')
csum = np.where(mixsort < n1, 1./n1, -1./n2).cumsum()
# Check for ties
if len(np.unique(mix)) < (n1+n2):
csum = csum[np.r_[np.diff(mix[mixsort]).nonzero()[0],-1]]
alternative = str(alternative).lower()[0]
if alternative == 't':
d = ma.abs(csum).max()
prob = special.kolmogorov(np.sqrt(n)*d)
elif alternative == 'l':
d = -csum.min()
prob = np.exp(-2*n*d**2)
elif alternative == 'g':
d = csum.max()
prob = np.exp(-2*n*d**2)
else:
raise ValueError("Invalid value for the alternative hypothesis: "
"should be in 'two-sided', 'less' or 'greater'")
return (d, prob)
ks_2samp = ks_twosamp
def ks_twosamp_old(data1, data2):
""" Computes the Kolmogorov-Smirnov statistic on 2 samples.
Returns
-------
KS D-value, p-value
"""
(data1, data2) = [ma.asarray(d).compressed() for d in (data1,data2)]
return stats.ks_2samp(data1,data2)
def threshold(a, threshmin=None, threshmax=None, newval=0):
"""
Clip array to a given value.
Similar to numpy.clip(), except that values less than `threshmin` or
greater than `threshmax` are replaced by `newval`, instead of by
`threshmin` and `threshmax` respectively.
Parameters
----------
a : ndarray
Input data
threshmin : {None, float}, optional
Lower threshold. If None, set to the minimum value.
threshmax : {None, float}, optional
Upper threshold. If None, set to the maximum value.
newval : {0, float}, optional
Value outside the thresholds.
Returns
-------
threshold : ndarray
Returns `a`, with values less then `threshmin` and values greater
`threshmax` replaced with `newval`.
"""
a = ma.array(a, copy=True)
mask = np.zeros(a.shape, dtype=bool)
if threshmin is not None:
mask |= (a < threshmin).filled(False)
if threshmax is not None:
mask |= (a > threshmax).filled(False)
a[mask] = newval
return a
def trima(a, limits=None, inclusive=(True,True)):
"""
Trims an array by masking the data outside some given limits.
Returns a masked version of the input array.
Parameters
----------
a : array_like
Input array.
limits : {None, tuple}, optional
Tuple of (lower limit, upper limit) in absolute values.
Values of the input array lower (greater) than the lower (upper) limit
will be masked. A limit is None indicates an open interval.
inclusive : (bool, bool) tuple, optional
Tuple of (lower flag, upper flag), indicating whether values exactly
equal to the lower (upper) limit are allowed.
"""
a = ma.asarray(a)
a.unshare_mask()
if (limits is None) or (limits == (None, None)):
return a
(lower_lim, upper_lim) = limits
(lower_in, upper_in) = inclusive
condition = False
if lower_lim is not None:
if lower_in:
condition |= (a < lower_lim)
else:
condition |= (a <= lower_lim)
if upper_lim is not None:
if upper_in:
condition |= (a > upper_lim)
else:
condition |= (a >= upper_lim)
a[condition.filled(True)] = masked
return a
def trimr(a, limits=None, inclusive=(True, True), axis=None):
"""
Trims an array by masking some proportion of the data on each end.
Returns a masked version of the input array.
Parameters
----------
a : sequence
Input array.
limits : {None, tuple}, optional
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.
Noting n the number of unmasked data before trimming, the
(n*limits[0])th smallest data and the (n*limits[1])th largest data are
masked, and 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.
inclusive : {(True,True) tuple}, optional
Tuple of flags indicating whether the number of data being masked on
the left (right) end should be truncated (True) or rounded (False) to
integers.
axis : {None,int}, optional
Axis along which to trim. If None, the whole array is trimmed, but its
shape is maintained.
"""
def _trimr1D(a, low_limit, up_limit, low_inclusive, up_inclusive):
n = a.count()
idx = a.argsort()
if low_limit:
if low_inclusive:
lowidx = int(low_limit*n)
else:
lowidx = np.round(low_limit*n)
a[idx[:lowidx]] = masked
if up_limit is not None:
if up_inclusive:
upidx = n - int(n*up_limit)
else:
upidx = n - np.round(n*up_limit)
a[idx[upidx:]] = masked
return a
a = ma.asarray(a)
a.unshare_mask()
if limits is None:
return a
# Check the limits
(lolim, uplim) = limits
errmsg = "The proportion to cut from the %s should be between 0. and 1."
if lolim is not None:
if lolim > 1. or lolim < 0:
raise ValueError(errmsg % 'beginning' + "(got %s)" % lolim)
if uplim is not None:
if uplim > 1. or uplim < 0:
raise ValueError(errmsg % 'end' + "(got %s)" % uplim)
(loinc, upinc) = inclusive
if axis is None:
shp = a.shape
return _trimr1D(a.ravel(),lolim,uplim,loinc,upinc).reshape(shp)
else:
return ma.apply_along_axis(_trimr1D, axis, a, lolim,uplim,loinc,upinc)
trimdoc = """
Parameters
----------
a : sequence
Input array
limits : {None, tuple}, optional
If `relative` is False, tuple (lower limit, upper limit) in absolute values.
Values of the input array lower (greater) than the lower (upper) limit are
masked.
If `relative` is True, tuple (lower percentage, upper percentage) to cut
on each side of the array, with respect to the number of unmasked data.
Noting n the number of unmasked data before trimming, the (n*limits[0])th
smallest data and the (n*limits[1])th largest data are masked, and the
total number of unmasked data after trimming is n*(1.-sum(limits))
In each case, the value of one limit can be set to None to indicate an
open interval.
If limits is None, no trimming is performed
inclusive : {(bool, bool) tuple}, optional
If `relative` is False, tuple indicating whether values exactly equal
to the absolute limits are allowed.
If `relative` is True, tuple indicating whether the number of data
being masked on each side should be rounded (True) or truncated
(False).
relative : bool, optional
Whether to consider the limits as absolute values (False) or proportions
to cut (True).
axis : int, optional
Axis along which to trim.
"""
def trim(a, limits=None, inclusive=(True,True), relative=False, axis=None):
"""
Trims an array by masking the data outside some given limits.
Returns a masked version of the input array.
%s
Examples
--------
>>> z = [ 1, 2, 3, 4, 5, 6, 7, 8, 9,10]
>>> trim(z,(3,8))
[--,--, 3, 4, 5, 6, 7, 8,--,--]
>>> trim(z,(0.1,0.2),relative=True)
[--, 2, 3, 4, 5, 6, 7, 8,--,--]
"""
if relative:
return trimr(a, limits=limits, inclusive=inclusive, axis=axis)
else:
return trima(a, limits=limits, inclusive=inclusive)
if trim.__doc__ is not None:
trim.__doc__ = trim.__doc__ % trimdoc
def trimboth(data, proportiontocut=0.2, inclusive=(True,True), axis=None):
"""
Trims the smallest and largest data values.
Trims the `data` by masking the ``int(proportiontocut * n)`` smallest and
``int(proportiontocut * n)`` largest values of data along the given axis,
where n is the number of unmasked values before trimming.
Parameters
----------
data : ndarray
Data to trim.
proportiontocut : float, optional
Percentage of trimming (as a float between 0 and 1).
If n is the number of unmasked values before trimming, the number of
values after trimming is ``(1 - 2*proportiontocut) * n``.
Default is 0.2.
inclusive : {(bool, bool) tuple}, optional
Tuple indicating whether the number of data being masked on each side
should be rounded (True) or truncated (False).
axis : int, optional
Axis along which to perform the trimming.
If None, the input array is first flattened.
"""
return trimr(data, limits=(proportiontocut,proportiontocut),
inclusive=inclusive, axis=axis)
def trimtail(data, proportiontocut=0.2, tail='left', inclusive=(True,True),
axis=None):
"""
Trims the data by masking values from one tail.
Parameters
----------
data : array_like
Data to trim.
proportiontocut : float, optional
Percentage of trimming. If n is the number of unmasked values
before trimming, the number of values after trimming is
``(1 - proportiontocut) * n``. Default is 0.2.
tail : {'left','right'}, optional
If 'left' the `proportiontocut` lowest values will be masked.
If 'right' the `proportiontocut` highest values will be masked.
Default is 'left'.
inclusive : {(bool, bool) tuple}, optional
Tuple indicating whether the number of data being masked on each side
should be rounded (True) or truncated (False). Default is
(True, True).
axis : int, optional
Axis along which to perform the trimming.
If None, the input array is first flattened. Default is None.
Returns
-------
trimtail : ndarray
Returned array of same shape as `data` with masked tail values.
"""
tail = str(tail).lower()[0]
if tail == 'l':
limits = (proportiontocut,None)
elif tail == 'r':
limits = (None, proportiontocut)
else:
raise TypeError("The tail argument should be in ('left','right')")
return trimr(data, limits=limits, axis=axis, inclusive=inclusive)
trim1 = trimtail
def trimmed_mean(a, limits=(0.1,0.1), inclusive=(1,1), relative=True,
axis=None):
"""Returns the trimmed mean of the data along the given axis.
%s
""" % trimdoc
if (not isinstance(limits,tuple)) and isinstance(limits,float):
limits = (limits, limits)
if relative:
return trimr(a,limits=limits,inclusive=inclusive,axis=axis).mean(axis=axis)
else:
return trima(a,limits=limits,inclusive=inclusive).mean(axis=axis)
def trimmed_var(a, limits=(0.1,0.1), inclusive=(1,1), relative=True,
axis=None, ddof=0):
"""Returns the trimmed variance of the data along the given axis.
%s
ddof : {0,integer}, optional
Means Delta Degrees of Freedom. The denominator used during computations
is (n-ddof). DDOF=0 corresponds to a biased estimate, DDOF=1 to an un-
biased estimate of the variance.
""" % trimdoc
if (not isinstance(limits,tuple)) and isinstance(limits,float):
limits = (limits, limits)
if relative:
out = trimr(a,limits=limits, inclusive=inclusive,axis=axis)
else:
out = trima(a,limits=limits,inclusive=inclusive)
return out.var(axis=axis, ddof=ddof)
def trimmed_std(a, limits=(0.1,0.1), inclusive=(1,1), relative=True,
axis=None, ddof=0):
"""Returns the trimmed standard deviation of the data along the given axis.
%s
ddof : {0,integer}, optional
Means Delta Degrees of Freedom. The denominator used during computations
is (n-ddof). DDOF=0 corresponds to a biased estimate, DDOF=1 to an un-
biased estimate of the variance.
""" % trimdoc
if (not isinstance(limits,tuple)) and isinstance(limits,float):
limits = (limits, limits)
if relative:
out = trimr(a,limits=limits,inclusive=inclusive,axis=axis)
else:
out = trima(a,limits=limits,inclusive=inclusive)
return out.std(axis=axis,ddof=ddof)
def trimmed_stde(a, limits=(0.1,0.1), inclusive=(1,1), axis=None):
"""
Returns the standard error of the trimmed mean along the given axis.
Parameters
----------
a : sequence
Input array
limits : {(0.1,0.1), tuple of float}, optional
tuple (lower percentage, upper percentage) to cut on each side of the
array, with respect to the number of unmasked data.
If n is the number of unmasked data before trimming, the values
smaller than ``n * limits[0]`` and the values larger than
``n * `limits[1]`` are masked, and the total number of unmasked
data after trimming is ``n * (1.-sum(limits))``. In each case,
the value of one limit can be set to None to indicate an open interval.
If `limits` is None, no trimming is performed.
inclusive : {(bool, bool) tuple} optional
Tuple indicating whether the number of data being masked on each side
should be rounded (True) or truncated (False).
axis : int, optional
Axis along which to trim.
Returns
-------
trimmed_stde : scalar or ndarray
"""
def _trimmed_stde_1D(a, low_limit, up_limit, low_inclusive, up_inclusive):
"Returns the standard error of the trimmed mean for a 1D input data."
n = a.count()
idx = a.argsort()
if low_limit:
if low_inclusive:
lowidx = int(low_limit*n)
else:
lowidx = np.round(low_limit*n)
a[idx[:lowidx]] = masked
if up_limit is not None:
if up_inclusive:
upidx = n - int(n*up_limit)
else:
upidx = n - np.round(n*up_limit)
a[idx[upidx:]] = masked
a[idx[:lowidx]] = a[idx[lowidx]]
a[idx[upidx:]] = a[idx[upidx-1]]
winstd = a.std(ddof=1)
return winstd / ((1-low_limit-up_limit)*np.sqrt(len(a)))
a = ma.array(a, copy=True, subok=True)
a.unshare_mask()
if limits is None:
return a.std(axis=axis,ddof=1)/ma.sqrt(a.count(axis))
if (not isinstance(limits,tuple)) and isinstance(limits,float):
limits = (limits, limits)
# Check the limits
(lolim, uplim) = limits
errmsg = "The proportion to cut from the %s should be between 0. and 1."
if lolim is not None:
if lolim > 1. or lolim < 0:
raise ValueError(errmsg % 'beginning' + "(got %s)" % lolim)
if uplim is not None:
if uplim > 1. or uplim < 0:
raise ValueError(errmsg % 'end' + "(got %s)" % uplim)
(loinc, upinc) = inclusive
if (axis is None):
return _trimmed_stde_1D(a.ravel(),lolim,uplim,loinc,upinc)
else:
if a.ndim > 2:
raise ValueError("Array 'a' must be at most two dimensional, but got a.ndim = %d" % a.ndim)
return ma.apply_along_axis(_trimmed_stde_1D, axis, a,
lolim,uplim,loinc,upinc)
def tmean(a, limits=None, inclusive=(True,True)):
return trima(a, limits=limits, inclusive=inclusive).mean()
tmean.__doc__ = stats.tmean.__doc__
def tvar(a, limits=None, inclusive=(True,True)):
a = a.astype(float).ravel()
if limits is None:
n = (~a.mask).sum() # todo: better way to do that?
r = trima(a, limits=limits, inclusive=inclusive).var() * (n/(n-1.))
else:
raise ValueError('mstats.tvar() with limits not implemented yet so far')
return r
tvar.__doc__ = stats.tvar.__doc__
def tmin(a, lowerlimit=None, axis=0, inclusive=True):
a, axis = _chk_asarray(a, axis)
am = trima(a, (lowerlimit, None), (inclusive, False))
return ma.minimum.reduce(am, axis)
tmin.__doc__ = stats.tmin.__doc__
def tmax(a, upperlimit, axis=0, inclusive=True):
a, axis = _chk_asarray(a, axis)
am = trima(a, (None, upperlimit), (False, inclusive))
return ma.maximum.reduce(am, axis)
tmax.__doc__ = stats.tmax.__doc__
def tsem(a, limits=None, inclusive=(True,True)):
a = ma.asarray(a).ravel()
if limits is None:
n = float(a.count())
return a.std(ddof=1)/ma.sqrt(n)
am = trima(a.ravel(), limits, inclusive)
sd = np.sqrt(am.var(ddof=1))
return sd / np.sqrt(am.count())
tsem.__doc__ = stats.tsem.__doc__
def winsorize(a, limits=None, inclusive=(True, True), inplace=False,
axis=None):
"""Returns a Winsorized version of the input array.
The (limits[0])th lowest values are set to the (limits[0])th percentile,
and the (limits[1])th highest values are set to the (1 - limits[1])th
percentile.
Masked values are skipped.
Parameters
----------
a : sequence
Input array.
limits : {None, tuple of float}, optional
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.
Noting n the number of unmasked data before trimming, the
(n*limits[0])th smallest data and the (n*limits[1])th largest data are
masked, and 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.
inclusive : {(True, True) tuple}, optional
Tuple indicating whether the number of data being masked on each side
should be rounded (True) or truncated (False).
inplace : {False, True}, optional
Whether to winsorize in place (True) or to use a copy (False)
axis : {None, int}, optional
Axis along which to trim. If None, the whole array is trimmed, but its
shape is maintained.
Notes
-----
This function is applied to reduce the effect of possibly spurious outliers
by limiting the extreme values.
"""
def _winsorize1D(a, low_limit, up_limit, low_include, up_include):
n = a.count()
idx = a.argsort()
if low_limit:
if low_include:
lowidx = int(low_limit * n)
else:
lowidx = np.round(low_limit * n)
a[idx[:lowidx]] = a[idx[lowidx]]
if up_limit is not None:
if up_include:
upidx = n - int(n * up_limit)
else:
upidx = n - np.round(n * up_limit)
a[idx[upidx:]] = a[idx[upidx - 1]]
return a
# We are going to modify a: better make a copy
a = ma.array(a, copy=np.logical_not(inplace))
if limits is None:
return a
if (not isinstance(limits, tuple)) and isinstance(limits, float):
limits = (limits, limits)
# Check the limits
(lolim, uplim) = limits
errmsg = "The proportion to cut from the %s should be between 0. and 1."
if lolim is not None:
if lolim > 1. or lolim < 0:
raise ValueError(errmsg % 'beginning' + "(got %s)" % lolim)
if uplim is not None:
if uplim > 1. or uplim < 0:
raise ValueError(errmsg % 'end' + "(got %s)" % uplim)
(loinc, upinc) = inclusive
if axis is None:
shp = a.shape
return _winsorize1D(a.ravel(), lolim, uplim, loinc, upinc).reshape(shp)
else:
return ma.apply_along_axis(_winsorize1D, axis, a, lolim, uplim, loinc,
upinc)
def moment(a, moment=1, axis=0):
a, axis = _chk_asarray(a, axis)
if moment == 1:
# By definition the first moment about the mean is 0.
shape = list(a.shape)
del shape[axis]
if shape:
# return an actual array of the appropriate shape
return np.zeros(shape, dtype=float)
else:
# the input was 1D, so return a scalar instead of a rank-0 array
return np.float64(0.0)
else:
mn = ma.expand_dims(a.mean(axis=axis), axis)
s = ma.power((a-mn), moment)
return s.mean(axis=axis)
moment.__doc__ = stats.moment.__doc__
def variation(a, axis=0):
a, axis = _chk_asarray(a, axis)
return a.std(axis)/a.mean(axis)
variation.__doc__ = stats.variation.__doc__
def skew(a, axis=0, bias=True):
a, axis = _chk_asarray(a,axis)
n = a.count(axis)
m2 = moment(a, 2, axis)
m3 = moment(a, 3, axis)
olderr = np.seterr(all='ignore')
try:
vals = ma.where(m2 == 0, 0, m3 / m2**1.5)
finally:
np.seterr(**olderr)
if not bias:
can_correct = (n > 2) & (m2 > 0)
if can_correct.any():
m2 = np.extract(can_correct, m2)
m3 = np.extract(can_correct, m3)
nval = ma.sqrt((n-1.0)*n)/(n-2.0)*m3/m2**1.5
np.place(vals, can_correct, nval)
return vals
skew.__doc__ = stats.skew.__doc__
def kurtosis(a, axis=0, fisher=True, bias=True):
a, axis = _chk_asarray(a, axis)
m2 = moment(a, 2, axis)
m4 = moment(a, 4, axis)
olderr = np.seterr(all='ignore')
try:
vals = ma.where(m2 == 0, 0, m4 / m2**2.0)
finally:
np.seterr(**olderr)
if not bias:
n = a.count(axis)
can_correct = (n > 3) & (m2 is not ma.masked and m2 > 0)
if can_correct.any():
n = np.extract(can_correct, n)
m2 = np.extract(can_correct, m2)
m4 = np.extract(can_correct, m4)
nval = 1.0/(n-2)/(n-3)*((n*n-1.0)*m4/m2**2.0-3*(n-1)**2.0)
np.place(vals, can_correct, nval+3.0)
if fisher:
return vals - 3
else:
return vals
kurtosis.__doc__ = stats.kurtosis.__doc__
def describe(a, axis=0,ddof=0):
"""
Computes several descriptive statistics of the passed array.
Parameters
----------
a : array
axis : int or None
ddof : int
degree of freedom (default 0); note that default ddof is different
from the same routine in stats.describe
Returns
-------
n : int
(size of the data (discarding missing values)
mm : (int, int)
min, max
arithmetic mean : float
unbiased variance : float
biased skewness : float
biased kurtosis : float
Examples
--------
>>> ma = np.ma.array(range(6), mask=[0, 0, 0, 1, 1, 1])
>>> describe(ma)
(array(3),
(0, 2),
1.0,
1.0,
masked_array(data = 0.0,
mask = False,
fill_value = 1e+20)
,
-1.5)
"""
a, axis = _chk_asarray(a, axis)
n = a.count(axis)
mm = (ma.minimum.reduce(a), ma.maximum.reduce(a))
m = a.mean(axis)
v = a.var(axis,ddof=ddof)
sk = skew(a, axis)
kurt = kurtosis(a, axis)
return n, mm, m, v, sk, kurt
def stde_median(data, axis=None):
"""Returns the McKean-Schrader estimate of the standard error of the sample
median along the given axis. masked values are discarded.
Parameters
----------
data : ndarray
Data to trim.
axis : {None,int}, optional
Axis along which to perform the trimming.
If None, the input array is first flattened.
"""
def _stdemed_1D(data):
data = np.sort(data.compressed())
n = len(data)
z = 2.5758293035489004
k = int(np.round((n+1)/2. - z * np.sqrt(n/4.),0))
return ((data[n-k] - data[k-1])/(2.*z))
data = ma.array(data, copy=False, subok=True)
if (axis is None):
return _stdemed_1D(data)
else:
if data.ndim > 2:
raise ValueError("Array 'data' must be at most two dimensional, "
"but got data.ndim = %d" % data.ndim)
return ma.apply_along_axis(_stdemed_1D, axis, data)
def skewtest(a, axis=0):
a, axis = _chk_asarray(a, axis)
if axis is None:
a = a.ravel()
axis = 0
b2 = skew(a,axis)
n = a.count(axis)
if np.min(n) < 8:
raise ValueError(
"skewtest is not valid with less than 8 samples; %i samples"
" were given." % np.min(n))
y = b2 * ma.sqrt(((n+1)*(n+3)) / (6.0*(n-2)))
beta2 = (3.0*(n*n+27*n-70)*(n+1)*(n+3)) / ((n-2.0)*(n+5)*(n+7)*(n+9))
W2 = -1 + ma.sqrt(2*(beta2-1))
delta = 1/ma.sqrt(0.5*ma.log(W2))
alpha = ma.sqrt(2.0/(W2-1))
y = ma.where(y == 0, 1, y)
Z = delta*ma.log(y/alpha + ma.sqrt((y/alpha)**2+1))
return Z, 2 * distributions.norm.sf(np.abs(Z))
skewtest.__doc__ = stats.skewtest.__doc__
def kurtosistest(a, axis=0):
a, axis = _chk_asarray(a, axis)
n = a.count(axis=axis)
if np.min(n) < 5:
raise ValueError(
"kurtosistest requires at least 5 observations; %i observations"
" were given." % np.min(n))
if np.min(n) < 20:
warnings.warn(
"kurtosistest only valid for n>=20 ... continuing anyway, n=%i" %
np.min(n))
b2 = kurtosis(a, axis, fisher=False)
E = 3.0*(n-1) / (n+1)
varb2 = 24.0*n*(n-2.)*(n-3) / ((n+1)*(n+1.)*(n+3)*(n+5))
x = (b2-E)/ma.sqrt(varb2)
sqrtbeta1 = 6.0*(n*n-5*n+2)/((n+7)*(n+9)) * np.sqrt((6.0*(n+3)*(n+5)) /
(n*(n-2)*(n-3)))
A = 6.0 + 8.0/sqrtbeta1 * (2.0/sqrtbeta1 + np.sqrt(1+4.0/(sqrtbeta1**2)))
term1 = 1 - 2./(9.0*A)
denom = 1 + x*ma.sqrt(2/(A-4.0))
if np.ma.isMaskedArray(denom):
# For multi-dimensional array input
denom[denom < 0] = masked
elif denom < 0:
denom = masked
term2 = ma.power((1-2.0/A)/denom,1/3.0)
Z = (term1 - term2) / np.sqrt(2/(9.0*A))
return Z, 2 * distributions.norm.sf(np.abs(Z))
kurtosistest.__doc__ = stats.kurtosistest.__doc__
def normaltest(a, axis=0):
a, axis = _chk_asarray(a, axis)
s, _ = skewtest(a, axis)
k, _ = kurtosistest(a, axis)
k2 = s*s + k*k
return k2, stats.chisqprob(k2,2)
normaltest.__doc__ = stats.normaltest.__doc__
def mquantiles(a, prob=list([.25,.5,.75]), alphap=.4, betap=.4, axis=None,
limit=()):
"""
Computes empirical quantiles for a data array.
Samples quantile are defined by ``Q(p) = (1-gamma)*x[j] + gamma*x[j+1]``,
where ``x[j]`` is the j-th order statistic, and gamma is a function of
``j = floor(n*p + m)``, ``m = alphap + p*(1 - alphap - betap)`` and
``g = n*p + m - j``.
Reinterpreting the above equations to compare to **R** lead to the
equation: ``p(k) = (k - alphap)/(n + 1 - alphap - betap)``
Typical values of (alphap,betap) are:
- (0,1) : ``p(k) = k/n`` : linear interpolation of cdf
(**R** type 4)
- (.5,.5) : ``p(k) = (k - 1/2.)/n`` : piecewise linear function
(**R** type 5)
- (0,0) : ``p(k) = k/(n+1)`` :
(**R** type 6)
- (1,1) : ``p(k) = (k-1)/(n-1)``: p(k) = mode[F(x[k])].
(**R** type 7, **R** default)
- (1/3,1/3): ``p(k) = (k-1/3)/(n+1/3)``: Then p(k) ~ median[F(x[k])].
The resulting quantile estimates are approximately median-unbiased
regardless of the distribution of x.
(**R** type 8)
- (3/8,3/8): ``p(k) = (k-3/8)/(n+1/4)``: Blom.
The resulting quantile estimates are approximately unbiased
if x is normally distributed
(**R** type 9)
- (.4,.4) : approximately quantile unbiased (Cunnane)
- (.35,.35): APL, used with PWM
Parameters
----------
a : array_like
Input data, as a sequence or array of dimension at most 2.
prob : array_like, optional
List of quantiles to compute.
alphap : float, optional
Plotting positions parameter, default is 0.4.
betap : float, optional
Plotting positions parameter, default is 0.4.
axis : int, optional
Axis along which to perform the trimming.
If None (default), the input array is first flattened.
limit : tuple
Tuple of (lower, upper) values.
Values of `a` outside this open interval are ignored.
Returns
-------
mquantiles : MaskedArray
An array containing the calculated quantiles.
Notes
-----
This formulation is very similar to **R** except the calculation of
``m`` from ``alphap`` and ``betap``, where in **R** ``m`` is defined
with each type.
References
----------
.. [1] *R* statistical software: http://www.r-project.org/
.. [2] *R* ``quantile`` function:
http://stat.ethz.ch/R-manual/R-devel/library/stats/html/quantile.html
Examples
--------
>>> from scipy.stats.mstats import mquantiles
>>> a = np.array([6., 47., 49., 15., 42., 41., 7., 39., 43., 40., 36.])
>>> mquantiles(a)
array([ 19.2, 40. , 42.8])
Using a 2D array, specifying axis and limit.
>>> data = np.array([[ 6., 7., 1.],
[ 47., 15., 2.],
[ 49., 36., 3.],
[ 15., 39., 4.],
[ 42., 40., -999.],
[ 41., 41., -999.],
[ 7., -999., -999.],
[ 39., -999., -999.],
[ 43., -999., -999.],
[ 40., -999., -999.],
[ 36., -999., -999.]])
>>> mquantiles(data, axis=0, limit=(0, 50))
array([[ 19.2 , 14.6 , 1.45],
[ 40. , 37.5 , 2.5 ],
[ 42.8 , 40.05, 3.55]])
>>> data[:, 2] = -999.
>>> mquantiles(data, axis=0, limit=(0, 50))
masked_array(data =
[[19.2 14.6 --]
[40.0 37.5 --]
[42.8 40.05 --]],
mask =
[[False False True]
[False False True]
[False False True]],
fill_value = 1e+20)
"""
def _quantiles1D(data,m,p):
x = np.sort(data.compressed())
n = len(x)
if n == 0:
return ma.array(np.empty(len(p), dtype=float), mask=True)
elif n == 1:
return ma.array(np.resize(x, p.shape), mask=nomask)
aleph = (n*p + m)
k = np.floor(aleph.clip(1, n-1)).astype(int)
gamma = (aleph-k).clip(0,1)
return (1.-gamma)*x[(k-1).tolist()] + gamma*x[k.tolist()]
data = ma.array(a, copy=False)
if data.ndim > 2:
raise TypeError("Array should be 2D at most !")
if limit:
condition = (limit[0] < data) & (data < limit[1])
data[~condition.filled(True)] = masked
p = np.array(prob, copy=False, ndmin=1)
m = alphap + p*(1.-alphap-betap)
# Computes quantiles along axis (or globally)
if (axis is None):
return _quantiles1D(data, m, p)
return ma.apply_along_axis(_quantiles1D, axis, data, m, p)
def scoreatpercentile(data, per, limit=(), alphap=.4, betap=.4):
"""Calculate the score at the given 'per' percentile of the
sequence a. For example, the score at per=50 is the median.
This function is a shortcut to mquantile
"""
if (per < 0) or (per > 100.):
raise ValueError("The percentile should be between 0. and 100. !"
" (got %s)" % per)
return mquantiles(data, prob=[per/100.], alphap=alphap, betap=betap,
limit=limit, axis=0).squeeze()
def plotting_positions(data, alpha=0.4, beta=0.4):
"""
Returns plotting positions (or empirical percentile points) for the data.
Plotting positions are defined as ``(i-alpha)/(n+1-alpha-beta)``, where:
- i is the rank order statistics
- n is the number of unmasked values along the given axis
- `alpha` and `beta` are two parameters.
Typical values for `alpha` and `beta` are:
- (0,1) : ``p(k) = k/n``, linear interpolation of cdf (R, type 4)
- (.5,.5) : ``p(k) = (k-1/2.)/n``, piecewise linear function
(R, type 5)
- (0,0) : ``p(k) = k/(n+1)``, Weibull (R type 6)
- (1,1) : ``p(k) = (k-1)/(n-1)``, in this case,
``p(k) = mode[F(x[k])]``. That's R default (R type 7)
- (1/3,1/3): ``p(k) = (k-1/3)/(n+1/3)``, then
``p(k) ~ median[F(x[k])]``.
The resulting quantile estimates are approximately median-unbiased
regardless of the distribution of x. (R type 8)
- (3/8,3/8): ``p(k) = (k-3/8)/(n+1/4)``, Blom.
The resulting quantile estimates are approximately unbiased
if x is normally distributed (R type 9)
- (.4,.4) : approximately quantile unbiased (Cunnane)
- (.35,.35): APL, used with PWM
- (.3175, .3175): used in scipy.stats.probplot
Parameters
----------
data : array_like
Input data, as a sequence or array of dimension at most 2.
alpha : float, optional
Plotting positions parameter. Default is 0.4.
beta : float, optional
Plotting positions parameter. Default is 0.4.
Returns
-------
positions : MaskedArray
The calculated plotting positions.
"""
data = ma.array(data, copy=False).reshape(1,-1)
n = data.count()
plpos = np.empty(data.size, dtype=float)
plpos[n:] = 0
plpos[data.argsort()[:n]] = ((np.arange(1, n+1) - alpha) /
(n + 1.0 - alpha - beta))
return ma.array(plpos, mask=data._mask)
meppf = plotting_positions
def obrientransform(*args):
"""
Computes a transform on input data (any number of columns). Used to
test for homogeneity of variance prior to running one-way stats. Each
array in *args is one level of a factor. If an F_oneway() run on the
transformed data and found significant, variances are unequal. From
Maxwell and Delaney, p.112.
Returns: transformed data for use in an ANOVA
"""
data = argstoarray(*args).T
v = data.var(axis=0,ddof=1)
m = data.mean(0)
n = data.count(0).astype(float)
# result = ((N-1.5)*N*(a-m)**2 - 0.5*v*(n-1))/((n-1)*(n-2))
data -= m
data **= 2
data *= (n-1.5)*n
data -= 0.5*v*(n-1)
data /= (n-1.)*(n-2.)
if not ma.allclose(v,data.mean(0)):
raise ValueError("Lack of convergence in obrientransform.")
return data
def signaltonoise(data, axis=0):
"""Calculates the signal-to-noise ratio, as the ratio of the mean over
standard deviation along the given axis.
Parameters
----------
data : sequence
Input data
axis : {0, int}, optional
Axis along which to compute. If None, the computation is performed
on a flat version of the array.
"""
data = ma.array(data, copy=False)
m = data.mean(axis)
sd = data.std(axis, ddof=0)
return m/sd
def sem(a, axis=0, ddof=1):
"""
Calculates the standard error of the mean of the input array.
Also sometimes called standard error of measurement.
Parameters
----------
a : array_like
An array containing the values for which the standard error is
returned.
axis : int or None, optional.
If axis is None, ravel `a` first. If axis is an integer, this will be
the axis over which to operate. Defaults to 0.
ddof : int, optional
Delta degrees-of-freedom. How many degrees of freedom to adjust
for bias in limited samples relative to the population estimate
of variance. Defaults to 1.
Returns
-------
s : ndarray or float
The standard error of the mean in the sample(s), along the input axis.
Notes
-----
The default value for `ddof` changed in scipy 0.15.0 to be consistent with
`stats.sem` as well as with the most common definition used (like in the R
documentation).
Examples
--------
Find standard error along the first axis:
>>> from scipy import stats
>>> a = np.arange(20).reshape(5,4)
>>> stats.sem(a)
array([ 2.8284, 2.8284, 2.8284, 2.8284])
Find standard error across the whole array, using n degrees of freedom:
>>> stats.sem(a, axis=None, ddof=0)
1.2893796958227628
"""
a, axis = _chk_asarray(a, axis)
n = a.count(axis=axis)
s = a.std(axis=axis, ddof=ddof) / ma.sqrt(n)
return s
zmap = stats.zmap
zscore = stats.zscore
def f_oneway(*args):
"""
Performs a 1-way ANOVA, returning an F-value and probability given
any number of groups. From Heiman, pp.394-7.
Usage: ``f_oneway(*args)``, where ``*args`` is 2 or more arrays,
one per treatment group.
Returns: f-value, probability
"""
# Construct a single array of arguments: each row is a group
data = argstoarray(*args)
ngroups = len(data)
ntot = data.count()
sstot = (data**2).sum() - (data.sum())**2/float(ntot)
ssbg = (data.count(-1) * (data.mean(-1)-data.mean())**2).sum()
sswg = sstot-ssbg
dfbg = ngroups-1
dfwg = ntot - ngroups
msb = ssbg/float(dfbg)
msw = sswg/float(dfwg)
f = msb/msw
prob = special.fdtrc(dfbg, dfwg, f) # equivalent to stats.f.sf
return f, prob
def f_value_wilks_lambda(ER, EF, dfnum, dfden, a, b):
"""Calculation of Wilks lambda F-statistic for multivariate data, per
Maxwell & Delaney p.657.
"""
ER = ma.array(ER, copy=False, ndmin=2)
EF = ma.array(EF, copy=False, ndmin=2)
if ma.getmask(ER).any() or ma.getmask(EF).any():
raise NotImplementedError("Not implemented when the inputs "
"have missing data")
lmbda = np.linalg.det(EF) / np.linalg.det(ER)
q = ma.sqrt(((a-1)**2*(b-1)**2 - 2) / ((a-1)**2 + (b-1)**2 - 5))
q = ma.filled(q, 1)
n_um = (1 - lmbda**(1.0/q))*(a-1)*(b-1)
d_en = lmbda**(1.0/q) / (n_um*q - 0.5*(a-1)*(b-1) + 1)
return n_um / d_en
def friedmanchisquare(*args):
"""Friedman Chi-Square is a non-parametric, one-way within-subjects ANOVA.
This function calculates the Friedman Chi-square test for repeated measures
and returns the result, along with the associated probability value.
Each input is considered a given group. Ideally, the number of treatments
among each group should be equal. If this is not the case, only the first
n treatments are taken into account, where n is the number of treatments
of the smallest group.
If a group has some missing values, the corresponding treatments are masked
in the other groups.
The test statistic is corrected for ties.
Masked values in one group are propagated to the other groups.
Returns: chi-square statistic, associated p-value
"""
data = argstoarray(*args).astype(float)
k = len(data)
if k < 3:
raise ValueError("Less than 3 groups (%i): " % k +
"the Friedman test is NOT appropriate.")
ranked = ma.masked_values(rankdata(data, axis=0), 0)
if ranked._mask is not nomask:
ranked = ma.mask_cols(ranked)
ranked = ranked.compressed().reshape(k,-1).view(ndarray)
else:
ranked = ranked._data
(k,n) = ranked.shape
# Ties correction
repeats = np.array([find_repeats(_) for _ in ranked.T], dtype=object)
ties = repeats[repeats.nonzero()].reshape(-1,2)[:,-1].astype(int)
tie_correction = 1 - (ties**3-ties).sum()/float(n*(k**3-k))
ssbg = np.sum((ranked.sum(-1) - n*(k+1)/2.)**2)
chisq = ssbg * 12./(n*k*(k+1)) * 1./tie_correction
return chisq, stats.chisqprob(chisq,k-1)