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stats.py
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# Copyright (c) Gary Strangman. All rights reserved
#
# Disclaimer
#
# This software is provided "as-is". There are no expressed or implied
# warranties of any kind, including, but not limited to, the warranties
# of merchantability and fitness for a given application. In no event
# shall Gary Strangman be liable for any direct, indirect, incidental,
# special, exemplary or consequential damages (including, but not limited
# to, loss of use, data or profits, or business interruption) however
# caused and on any theory of liability, whether in contract, strict
# liability or tort (including negligence or otherwise) arising in any way
# out of the use of this software, even if advised of the possibility of
# such damage.
#
#
# Heavily adapted for use by SciPy 2002 by Travis Oliphant
"""
stats.py module
#################################################
####### Written by: Gary Strangman ###########
#################################################
A collection of basic statistical functions for python. The function
names appear below.
Some scalar functions defined here are also available in the scipy.special
package where they work on arbitrary sized arrays.
Disclaimers: The function list is obviously incomplete and, worse, the
functions are not optimized. All functions have been tested (some more
so than others), but they are far from bulletproof. Thus, as with any
free software, no warranty or guarantee is expressed or implied. :-) A
few extra functions that don't appear in the list below can be found by
interested treasure-hunters. These functions don't necessarily have
both list and array versions but were deemed useful
CENTRAL TENDENCY: gmean (geometric mean)
hmean (harmonic mean)
medianscore
mode
MOMENTS: moment
variation
skew
kurtosis
normaltest (for arrays only)
MOMENTS HANDLING NAN: nanmean
nanmedian
nanstd
ALTERED VERSIONS: tmean
tvar
tstd
tsem
describe
FREQUENCY STATS: freqtable
itemfreq
scoreatpercentile
percentileofscore
histogram
cumfreq
relfreq
VARIABILITY: obrientransform
signaltonoise (for arrays only)
sem
TRIMMING FCNS: threshold (for arrays only)
trimboth
trim1
around (round all vals to 'n' decimals)
CORRELATION FCNS: paired
pearsonr
fisher_exact
spearmanr
pointbiserialr
kendalltau
linregress
INFERENTIAL STATS: ttest_1samp
ttest_ind
ttest_rel
chisquare
ks_2samp
mannwhitneyu
ranksums
wilcoxon
kruskal
friedmanchisquare
PROBABILITY CALCS: chisqprob
zprob
fprob
betai
## Note that scipy.stats.distributions has many more statistical probability
## functions defined.
ANOVA FUNCTIONS: f_oneway
f_value
SUPPORT FUNCTIONS: ss
square_of_sums
shellsort
rankdata
References
----------
[CRCProbStat2000]_
.. [CRCProbStat2000] Zwillinger, D. and Kokoska, S. (2000). CRC Standard
Probability and Statistics Tables and Formulae. Chapman & Hall: New
York. 2000.
"""
## CHANGE LOG:
## ===========
## since 2001-06-25 ... see scipy SVN changelog
## 05-11-29 ... fixed default axis to be 0 for consistency with scipy;
## cleanup of redundant imports, dead code, {0,1} -> booleans
## 02-02-10 ... require Numeric, eliminate "list-only" functions
## (only 1 set of functions now and no Dispatch class),
## removed all references to aXXXX functions.
## 00-04-13 ... pulled all "global" statements, except from aanova()
## added/fixed lots of documentation, removed io.py dependency
## changed to version 0.5
## 99-11-13 ... added asign() function
## 99-11-01 ... changed version to 0.4 ... enough incremental changes now
## 99-10-25 ... added acovariance and acorrelation functions
## 99-10-10 ... fixed askew/akurtosis to avoid divide-by-zero errors
## added aglm function (crude, but will be improved)
## 99-10-04 ... upgraded acumsum, ass, asummult, asamplevar, var, etc. to
## all handle lists of 'dimension's and keepdims
## REMOVED ar0, ar2, ar3, ar4 and replaced them with around
## reinserted fixes for abetai to avoid math overflows
## 99-09-05 ... rewrote achisqprob/aerfcc/aksprob/afprob/abetacf/abetai to
## handle multi-dimensional arrays (whew!)
## 99-08-30 ... fixed l/amoment, l/askew, l/akurtosis per D'Agostino (1990)
## added anormaltest per same reference
## re-wrote azprob to calc arrays of probs all at once
## 99-08-22 ... edited attest_ind printing section so arrays could be rounded
## 99-08-19 ... fixed amean and aharmonicmean for non-error(!) overflow on
## short/byte arrays (mean of #s btw 100-300 = -150??)
## 99-08-09 ... fixed asum so that the None case works for Byte arrays
## 99-08-08 ... fixed 7/3 'improvement' to handle t-calcs on N-D arrays
## 99-07-03 ... improved attest_ind, attest_rel (zero-division errortrap)
## 99-06-24 ... fixed bug(?) in attest_ind (n1=a.shape[0])
## 04/11/99 ... added asignaltonoise, athreshold functions, changed all
## max/min in array section to maximum/minimum,
## fixed square_of_sums to prevent integer overflow
## 04/10/99 ... !!! Changed function name ... sumsquared ==> square_of_sums
## 03/18/99 ... Added ar0, ar2, ar3 and ar4 rounding functions
## 02/28/99 ... Fixed aobrientransform to return an array rather than a list
## 01/15/99 ... Essentially ceased updating list-versions of functions (!!!)
## 01/13/99 ... CHANGED TO VERSION 0.3
## fixed bug in a/lmannwhitneyu p-value calculation
## 12/31/98 ... fixed variable-name bug in ldescribe
## 12/19/98 ... fixed bug in findwithin (fcns needed pstat. prefix)
## 12/16/98 ... changed amedianscore to return float (not array) for 1 score
## 12/14/98 ... added atmin and atmax functions
## removed umath from import line (not needed)
## l/ageometricmean modified to reduce chance of overflows (take
## nth root first, then multiply)
## 12/07/98 ... added __version__variable (now 0.2)
## removed all 'stats.' from anova() fcn
## 12/06/98 ... changed those functions (except shellsort) that altered
## arguments in-place ... cumsum, ranksort, ...
## updated (and fixed some) doc-strings
## 12/01/98 ... added anova() function (requires NumPy)
## incorporated Dispatch class
## 11/12/98 ... added functionality to amean, aharmonicmean, ageometricmean
## added 'asum' function (added functionality to add.reduce)
## fixed both moment and amoment (two errors)
## changed name of skewness and askewness to skew and askew
## fixed (a)histogram (which sometimes counted points <lowerlimit)
from __future__ import division, print_function, absolute_import
# Standard library imports.
import warnings
import math
from scipy.lib.six.moves import xrange
# friedmanchisquare patch uses python sum
pysum = sum # save it before it gets overwritten
# Scipy imports.
from scipy.lib.six import callable, string_types
from numpy import array, asarray, dot, ma, zeros, sum
import scipy.special as special
import scipy.linalg as linalg
import numpy as np
from . import futil
from . import distributions
# Local imports.
from . import _support
from ._support import _chk_asarray, _chk2_asarray
from ._rank import rankdata, tiecorrect
__all__ = ['find_repeats', 'gmean', 'hmean', 'cmedian', 'mode',
'tmean', 'tvar', 'tmin', 'tmax', 'tstd', 'tsem',
'moment', 'variation', 'skew', 'kurtosis', 'describe',
'skewtest', 'kurtosistest', 'normaltest', 'jarque_bera',
'itemfreq', 'scoreatpercentile', 'percentileofscore',
'histogram', 'histogram2', 'cumfreq', 'relfreq',
'obrientransform', 'signaltonoise', 'sem', 'zmap', 'zscore',
'threshold', 'sigmaclip', 'trimboth', 'trim1', 'trim_mean',
'f_oneway', 'pearsonr', 'fisher_exact',
'spearmanr', 'pointbiserialr', 'kendalltau', 'linregress',
'ttest_1samp', 'ttest_ind', 'ttest_rel',
'kstest', 'chisquare', 'ks_2samp', 'mannwhitneyu',
'tiecorrect', 'ranksums', 'kruskal', 'friedmanchisquare',
'zprob', 'chisqprob', 'ksprob', 'fprob', 'betai',
'glm', 'f_value_wilks_lambda',
'f_value', 'f_value_multivariate',
'ss', 'square_of_sums',
'fastsort', 'rankdata',
'nanmean', 'nanstd', 'nanmedian',
]
def find_repeats(arr):
"""
Find repeats and repeat counts.
Parameters
----------
arr : array_like
Input array
Returns
-------
find_repeats : tuple
Returns a tuple of two 1-D ndarrays. The first ndarray are the repeats
as sorted, unique values that are repeated in `arr`. The second
ndarray are the counts mapped one-to-one of the repeated values
in the first ndarray.
Examples
--------
>>> sp.stats.find_repeats([2, 1, 2, 3, 2, 2, 5])
(array([ 2. ]), array([ 4 ], dtype=int32)
>>> sp.stats.find_repeats([[10, 20, 1, 2], [5, 5, 4, 4]])
(array([ 4., 5.]), array([2, 2], dtype=int32))
"""
v1,v2, n = futil.dfreps(arr)
return v1[:n],v2[:n]
#######
### NAN friendly functions
########
def nanmean(x, axis=0):
"""
Compute the mean over the given axis ignoring nans.
Parameters
----------
x : ndarray
Input array.
axis : int, optional
Axis along which the mean is computed. Default is 0, i.e. the
first axis.
Returns
-------
m : float
The mean of `x`, ignoring nans.
See Also
--------
nanstd, nanmedian
Examples
--------
>>> from scipy import stats
>>> a = np.linspace(0, 4, 3)
>>> a
array([ 0., 2., 4.])
>>> a[-1] = np.nan
>>> stats.nanmean(a)
1.0
"""
x, axis = _chk_asarray(x,axis)
x = x.copy()
Norig = x.shape[axis]
factor = 1.0-np.sum(np.isnan(x),axis)*1.0/Norig
x[np.isnan(x)] = 0
return np.mean(x,axis)/factor
def nanstd(x, axis=0, bias=False):
"""
Compute the standard deviation over the given axis, ignoring nans.
Parameters
----------
x : array_like
Input array.
axis : int or None, optional
Axis along which the standard deviation is computed. Default is 0.
If None, compute over the whole array `x`.
bias : bool, optional
If True, the biased (normalized by N) definition is used. If False
(default), the unbiased definition is used.
Returns
-------
s : float
The standard deviation.
See Also
--------
nanmean, nanmedian
Examples
--------
>>> from scipy import stats
>>> a = np.arange(10, dtype=float)
>>> a[1:3] = np.nan
>>> np.std(a)
nan
>>> stats.nanstd(a)
2.9154759474226504
>>> stats.nanstd(a.reshape(2, 5), axis=1)
array([ 2.0817, 1.5811])
>>> stats.nanstd(a.reshape(2, 5), axis=None)
2.9154759474226504
"""
x, axis = _chk_asarray(x,axis)
x = x.copy()
Norig = x.shape[axis]
Nnan = np.sum(np.isnan(x),axis)*1.0
n = Norig - Nnan
x[np.isnan(x)] = 0.
m1 = np.sum(x,axis)/n
if axis:
d = (x - np.expand_dims(m1, axis))**2.0
else:
d = (x - m1)**2.0
m2 = np.sum(d,axis)-(m1*m1)*Nnan
if bias:
m2c = m2 / n
else:
m2c = m2 / (n - 1.)
return np.sqrt(m2c)
def _nanmedian(arr1d): # This only works on 1d arrays
"""Private function for rank a arrays. Compute the median ignoring Nan.
Parameters
----------
arr1d : ndarray
Input array, of rank 1.
Results
-------
m : float
The median.
"""
cond = 1-np.isnan(arr1d)
x = np.sort(np.compress(cond,arr1d,axis=-1))
if x.size == 0:
return np.nan
return np.median(x)
def nanmedian(x, axis=0):
"""
Compute the median along the given axis ignoring nan values.
Parameters
----------
x : array_like
Input array.
axis : int, optional
Axis along which the median is computed. Default is 0, i.e. the
first axis.
Returns
-------
m : float
The median of `x` along `axis`.
See Also
--------
nanstd, nanmean
Examples
--------
>>> from scipy import stats
>>> a = np.array([0, 3, 1, 5, 5, np.nan])
>>> stats.nanmedian(a)
array(3.0)
>>> b = np.array([0, 3, 1, 5, 5, np.nan, 5])
>>> stats.nanmedian(b)
array(4.0)
Example with axis:
>>> c = np.arange(30.).reshape(5,6)
>>> idx = np.array([False, False, False, True, False] * 6).reshape(5,6)
>>> c[idx] = np.nan
>>> c
array([[ 0., 1., 2., nan, 4., 5.],
[ 6., 7., nan, 9., 10., 11.],
[ 12., nan, 14., 15., 16., 17.],
[ nan, 19., 20., 21., 22., nan],
[ 24., 25., 26., 27., nan, 29.]])
>>> stats.nanmedian(c, axis=1)
array([ 2. , 9. , 15. , 20.5, 26. ])
"""
x, axis = _chk_asarray(x, axis)
if x.ndim == 0:
return float(x.item())
x = x.copy()
x = np.apply_along_axis(_nanmedian, axis, x)
if x.ndim == 0:
x = float(x.item())
return x
#####################################
######## CENTRAL TENDENCY ########
#####################################
def gmean(a, axis=0, dtype=None):
"""
Compute the geometric mean along the specified axis.
Returns the geometric average of the array elements.
That is: n-th root of (x1 * x2 * ... * xn)
Parameters
----------
a : array_like
Input array or object that can be converted to an array.
axis : int, optional, default axis=0
Axis along which the geometric mean is computed.
dtype : dtype, optional
Type of the returned array and of the accumulator in which the
elements are summed. If dtype is not specified, it defaults to the
dtype of a, unless a has an integer dtype with a precision less than
that of the default platform integer. In that case, the default
platform integer is used.
Returns
-------
gmean : ndarray
see dtype parameter above
See Also
--------
numpy.mean : Arithmetic average
numpy.average : Weighted average
hmean : Harmonic mean
Notes
-----
The geometric average is computed over a single dimension of the input
array, axis=0 by default, or all values in the array if axis=None.
float64 intermediate and return values are used for integer inputs.
Use masked arrays to ignore any non-finite values in the input or that
arise in the calculations such as Not a Number and infinity because masked
arrays automatically mask any non-finite values.
"""
if not isinstance(a, np.ndarray): #if not an ndarray object attempt to convert it
log_a=np.log(np.array(a, dtype=dtype))
elif dtype: #Must change the default dtype allowing array type
if isinstance(a,np.ma.MaskedArray):
log_a=np.log(np.ma.asarray(a, dtype=dtype))
else:
log_a=np.log(np.asarray(a, dtype=dtype))
else:
log_a = np.log(a)
return np.exp(log_a.mean(axis=axis))
def hmean(a, axis=0, dtype=None):
"""
Calculates the harmonic mean along the specified axis.
That is: n / (1/x1 + 1/x2 + ... + 1/xn)
Parameters
----------
a : array_like
Input array, masked array or object that can be converted to an array.
axis : int, optional, default axis=0
Axis along which the harmonic mean is computed.
dtype : dtype, optional
Type of the returned array and of the accumulator in which the
elements are summed. If `dtype` is not specified, it defaults to the
dtype of `a`, unless `a` has an integer `dtype` with a precision less
than that of the default platform integer. In that case, the default
platform integer is used.
Returns
-------
hmean : ndarray
see `dtype` parameter above
See Also
--------
numpy.mean : Arithmetic average
numpy.average : Weighted average
gmean : Geometric mean
Notes
-----
The harmonic mean is computed over a single dimension of the input
array, axis=0 by default, or all values in the array if axis=None.
float64 intermediate and return values are used for integer inputs.
Use masked arrays to ignore any non-finite values in the input or that
arise in the calculations such as Not a Number and infinity.
"""
if not isinstance(a, np.ndarray):
a=np.array(a, dtype=dtype)
if np.all(a >0): # Harmonic mean only defined if greater than zero
if isinstance(a, np.ma.MaskedArray):
size = a.count(axis)
else:
if axis == None:
a=a.ravel()
size = a.shape[0]
else:
size = a.shape[axis]
return size / np.sum(1.0/a, axis=axis, dtype=dtype)
else:
raise ValueError("Harmonic mean only defined if all elements greater than zero")
def cmedian(a, numbins=1000):
"""
Returns the computed median value of an array.
All of the values in the input array are used. The input array is first
histogrammed using `numbins` bins. The bin containing the median is
selected by searching for the halfway point in the cumulative histogram.
The median value is then computed by linearly interpolating across that
bin.
Parameters
----------
a : array_like
Input array.
numbins : int
The number of bins used to histogram the data. More bins give greater
accuracy to the approximation of the median.
Returns
-------
cmedian : float
An approximation of the median.
References
----------
[CRCProbStat2000]_ Section 2.2.6
.. [CRCProbStat2000] Zwillinger, D. and Kokoska, S. (2000). CRC Standard
Probability and Statistics Tables and Formulae. Chapman & Hall: New
York. 2000.
"""
# TODO: numpy.median() always seems to be a better choice.
# A better version of this function would take already-histogrammed data
# and compute the median from that.
a = np.ravel(a)
n = float(len(a))
# We will emulate the (fixed!) bounds selection scheme used by
# scipy.stats.histogram(), but use numpy.histogram() since it is faster.
amin = a.min()
amax = a.max()
estbinwidth = (amax - amin)/float(numbins - 1)
binsize = (amax - amin + estbinwidth) / float(numbins)
(hist, bins) = np.histogram(a, numbins,
range=(amin-binsize*0.5, amax+binsize*0.5))
binsize = bins[1] - bins[0]
cumhist = np.cumsum(hist) # make cumulative histogram
cfbin = np.searchsorted(cumhist, n/2.0)
LRL = bins[cfbin] # get lower read limit of that bin
if cfbin == 0:
cfbelow = 0.0
else:
cfbelow = cumhist[cfbin-1] # cum. freq. below bin
freq = hist[cfbin] # frequency IN the 50%ile bin
median = LRL + ((n/2.0-cfbelow)/float(freq))*binsize # MEDIAN
return median
def mode(a, axis=0):
"""
Returns an array of the modal (most common) value in the passed array.
If there is more than one such value, only the first is returned.
The bin-count for the modal bins is also returned.
Parameters
----------
a : array_like
n-dimensional array of which to find mode(s).
axis : int, optional
Axis along which to operate. Default is 0, i.e. the first axis.
Returns
-------
vals : ndarray
Array of modal values.
counts : ndarray
Array of counts for each mode.
Examples
--------
>>> a = np.array([[6, 8, 3, 0],
[3, 2, 1, 7],
[8, 1, 8, 4],
[5, 3, 0, 5],
[4, 7, 5, 9]])
>>> from scipy import stats
>>> stats.mode(a)
(array([[ 3., 1., 0., 0.]]), array([[ 1., 1., 1., 1.]]))
To get mode of whole array, specify axis=None:
>>> stats.mode(a, axis=None)
(array([ 3.]), array([ 3.]))
"""
a, axis = _chk_asarray(a, axis)
scores = np.unique(np.ravel(a)) # get ALL unique values
testshape = list(a.shape)
testshape[axis] = 1
oldmostfreq = np.zeros(testshape)
oldcounts = np.zeros(testshape)
for score in scores:
template = (a == score)
counts = np.expand_dims(np.sum(template, axis),axis)
mostfrequent = np.where(counts > oldcounts, score, oldmostfreq)
oldcounts = np.maximum(counts, oldcounts)
oldmostfreq = mostfrequent
return mostfrequent, oldcounts
def mask_to_limits(a, limits, inclusive):
"""Mask an array for values outside of given limits.
This is primarily a utility function.
Parameters
----------
a : array
limits : (float or None, float or None)
A tuple consisting of the (lower limit, upper limit). Values in the
input array less than the lower limit or greater than the upper limit
will be masked out. None implies no limit.
inclusive : (bool, bool)
A tuple consisting of the (lower flag, upper flag). These flags
determine whether values exactly equal to lower or upper are allowed.
Returns
-------
A MaskedArray.
Raises
------
A ValueError if there are no values within the given limits.
"""
lower_limit, upper_limit = limits
lower_include, upper_include = inclusive
am = ma.MaskedArray(a)
if lower_limit is not None:
if lower_include:
am = ma.masked_less(am, lower_limit)
else:
am = ma.masked_less_equal(am, lower_limit)
if upper_limit is not None:
if upper_include:
am = ma.masked_greater(am, upper_limit)
else:
am = ma.masked_greater_equal(am, upper_limit)
if am.count() == 0:
raise ValueError("No array values within given limits")
return am
def tmean(a, limits=None, inclusive=(True, True)):
"""
Compute the trimmed mean
This function finds the arithmetic mean of given values, ignoring values
outside the given `limits`.
Parameters
----------
a : array_like
array of values
limits : None or (lower limit, upper limit), optional
Values in the input array less than the lower limit or greater than the
upper limit will be ignored. When limits is None, then all values are
used. Either of the limit values in the tuple can also be None
representing a half-open interval. The default value is None.
inclusive : (bool, bool), optional
A tuple consisting of the (lower flag, upper flag). These flags
determine whether values exactly equal to the lower or upper limits
are included. The default value is (True, True).
Returns
-------
tmean : float
"""
a = asarray(a)
# Cast to a float if this is an integer array. If it is already a float
# array, leave it as is to preserve its precision.
if issubclass(a.dtype.type, np.integer):
a = a.astype(float)
# No trimming.
if limits is None:
return np.mean(a,None)
am = mask_to_limits(a.ravel(), limits, inclusive)
return am.mean()
def masked_var(am):
m = am.mean()
s = ma.add.reduce((am - m)**2)
n = am.count() - 1.0
return s / n
def tvar(a, limits=None, inclusive=(True, True)):
"""
Compute the trimmed variance
This function computes the sample variance of an array of values,
while ignoring values which are outside of given `limits`.
Parameters
----------
a : array_like
array of values
limits : None or (lower limit, upper limit), optional
Values in the input array less than the lower limit or greater than the
upper limit will be ignored. When limits is None, then all values are
used. Either of the limit values in the tuple can also be None
representing a half-open interval. The default value is None.
inclusive : (bool, bool), optional
A tuple consisting of the (lower flag, upper flag). These flags
determine whether values exactly equal to the lower or upper limits
are included. The default value is (True, True).
Returns
-------
tvar : float
"""
a = asarray(a)
a = a.astype(float).ravel()
if limits is None:
n = len(a)
return a.var()*(n/(n-1.))
am = mask_to_limits(a, limits, inclusive)
return masked_var(am)
def tmin(a, lowerlimit=None, axis=0, inclusive=True):
"""
Compute the trimmed minimum
This function finds the miminum value of an array `a` along the
specified axis, but only considering values greater than a specified
lower limit.
Parameters
----------
a : array_like
array of values
lowerlimit : None or float, optional
Values in the input array less than the given limit will be ignored.
When lowerlimit is None, then all values are used. The default value
is None.
axis : None or int, optional
Operate along this axis. None means to use the flattened array and
the default is zero
inclusive : {True, False}, optional
This flag determines whether values exactly equal to the lower limit
are included. The default value is True.
Returns
-------
tmin : float
"""
a, axis = _chk_asarray(a, axis)
am = mask_to_limits(a, (lowerlimit, None), (inclusive, False))
return ma.minimum.reduce(am, axis)
def tmax(a, upperlimit, axis=0, inclusive=True):
"""
Compute the trimmed maximum
This function computes the maximum value of an array along a given axis,
while ignoring values larger than a specified upper limit.
Parameters
----------
a : array_like
array of values
upperlimit : None or float, optional
Values in the input array greater than the given limit will be ignored.
When upperlimit is None, then all values are used. The default value
is None.
axis : None or int, optional
Operate along this axis. None means to use the flattened array and
the default is zero.
inclusive : {True, False}, optional
This flag determines whether values exactly equal to the upper limit
are included. The default value is True.
Returns
-------
tmax : float
"""
a, axis = _chk_asarray(a, axis)
am = mask_to_limits(a, (None, upperlimit), (False, inclusive))
return ma.maximum.reduce(am, axis)
def tstd(a, limits=None, inclusive=(True, True)):
"""
Compute the trimmed sample standard deviation
This function finds the sample standard deviation of given values,
ignoring values outside the given `limits`.
Parameters
----------
a : array_like
array of values
limits : None or (lower limit, upper limit), optional
Values in the input array less than the lower limit or greater than the
upper limit will be ignored. When limits is None, then all values are
used. Either of the limit values in the tuple can also be None
representing a half-open interval. The default value is None.
inclusive : (bool, bool), optional
A tuple consisting of the (lower flag, upper flag). These flags
determine whether values exactly equal to the lower or upper limits
are included. The default value is (True, True).
Returns
-------
tstd : float
"""
return np.sqrt(tvar(a, limits, inclusive))
def tsem(a, limits=None, inclusive=(True, True)):
"""
Compute the trimmed standard error of the mean
This function finds the standard error of the mean for given
values, ignoring values outside the given `limits`.
Parameters
----------
a : array_like
array of values
limits : None or (lower limit, upper limit), optional
Values in the input array less than the lower limit or greater than the
upper limit will be ignored. When limits is None, then all values are
used. Either of the limit values in the tuple can also be None
representing a half-open interval. The default value is None.
inclusive : (bool, bool), optional
A tuple consisting of the (lower flag, upper flag). These flags
determine whether values exactly equal to the lower or upper limits
are included. The default value is (True, True).
Returns
-------
tsem : float
"""
a = np.asarray(a).ravel()
if limits is None:
n = float(len(a))
return a.std()/np.sqrt(n)
am = mask_to_limits(a.ravel(), limits, inclusive)
sd = np.sqrt(masked_var(am))
return sd / am.count()
#####################################
############ MOMENTS #############
#####################################
def moment(a, moment=1, axis=0):
"""
Calculates the nth moment about the mean for a sample.
Generally used to calculate coefficients of skewness and
kurtosis.
Parameters
----------
a : array_like
data
moment : int
order of central moment that is returned
axis : int or None
Axis along which the central moment is computed. If None, then the data
array is raveled. The default axis is zero.
Returns
-------
n-th central moment : ndarray or float
The appropriate moment along the given axis or over all values if axis
is None. The denominator for the moment calculation is the number of
observations, no degrees of freedom correction is done.
"""
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 = np.expand_dims(np.mean(a,axis), axis)
s = np.power((a-mn), moment)
return np.mean(s, axis)
def variation(a, axis=0):
"""
Computes the coefficient of variation, the ratio of the biased standard
deviation to the mean.
Parameters
----------
a : array_like
Input array.
axis : int or None
Axis along which to calculate the coefficient of variation.
References
----------
[CRCProbStat2000]_ Section 2.2.20
.. [CRCProbStat2000] Zwillinger, D. and Kokoska, S. (2000). CRC Standard
Probability and Statistics Tables and Formulae. Chapman & Hall: New
York. 2000.
"""
a, axis = _chk_asarray(a, axis)
n = a.shape[axis]
return a.std(axis)/a.mean(axis)
def skew(a, axis=0, bias=True):
"""
Computes the skewness of a data set.
For normally distributed data, the skewness should be about 0. A skewness
value > 0 means that there is more weight in the left tail of the
distribution. The function `skewtest` can be used to determine if the
skewness value is close enough to 0, statistically speaking.
Parameters
----------
a : ndarray
data
axis : int or None