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""" 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
'count_tied_groups', 'describe', 'f_oneway', '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','skew','skewtest','spearmanr', 'theilslopes','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', ]
_find_repeats, linregress as stats_linregress, theilslopes as stats_theilslopes )
Notes ----- Missing values are considered pair-wise: if a value is missing in x, the corresponding value in y is masked. """
# 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
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
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)
""" 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
"""Find repeats in arr and return a tuple (repeats, repeat_count).
The input is cast to float64. 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.
""" # Make sure we get a copy. ma.compressed promises a "new array", but can # actually return a reference. compr = np.asarray(ma.compressed(arr), dtype=np.float64) try: need_copy = np.may_share_memory(compr, arr) except AttributeError: # numpy < 1.8.2 bug: np.may_share_memory([], []) raises, # while in numpy 1.8.2 and above it just (correctly) returns False. need_copy = False if need_copy: compr = compr.copy() return _find_repeats(compr)
""" Counts the number of tied values.
Parameters ---------- x : sequence Sequence of data on which to counts the ties use_missing : bool, optional 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
"""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 : bool, 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)
""" Returns an array of the modal (most common) value in the passed array.
Parameters ---------- a : array_like n-dimensional array of which to find mode(s). axis : int or None, optional Axis along which to operate. Default is 0. If None, compute over the whole array `a`.
Returns ------- mode : ndarray Array of modal values. count : ndarray Array of counts for each mode.
Notes ----- For more details, see `stats.mode`.
""" 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 ModeResult(*output)
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)
"""Returns the sign of x, or 0 if x is masked.""" return ma.filled(np.sign(x), 0)
""" 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)
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
""" 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 a monotonic 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 ------- correlation : float Spearman correlation coefficient pvalue : float 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)) # need int() here, otherwise numpy defaults to 32 bit # integer on all Windows architectures, causing overflow. # int() will keep it infinite precision. n -= int(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 = sum(v*k*(k**2-1) for (k,v) in iteritems(xties))/12. corr_y = 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 SpearmanrResult(rho, prob)
""" Computes Kendall's rank correlation tau on two variables *x* and *y*.
Parameters ---------- x : sequence First data list (for example, time). y : 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 ------- correlation : float Kendall tau pvalue : 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) # need int() here, otherwise numpy defaults to 32 bit # integer on all Windows architectures, causing overflow. # int() will keep it infinite precision. n -= int(m.sum())
if n < 2: return KendalltauResult(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 -= sum(v*k*(k-1)*(2*k+5)*1. for (k,v) in iteritems(xties)) var_s -= 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 KendalltauResult(tau, prob)
""" 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 = 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 = 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 = 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] = 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
'pvalue'))
"""Calculates a point biserial correlation coefficient and its p-value.
Parameters ---------- x : array_like of bools Input array. y : array_like Input array.
Returns ------- correlation : float R value pvalue : float 2-tailed p-value
Notes ----- Missing values are considered pair-wise: if a value is missing in x, the corresponding value in y is masked.
For more details on `pointbiserialr`, see `stats.pointbiserialr`.
""" 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 PointbiserialrResult(rpb, prob)
'rvalue', 'pvalue', 'stderr'))
""" 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 y is None: x = ma.array(x) if x.shape[0] == 2: x, y = x elif x.shape[1] == 2: x, y = x.T else: msg = ("If only `x` is given as input, it has to be of shape " "(2, N) or (N, 2), provided shape was %s" % str(x.shape)) raise ValueError(msg) else: x = ma.array(x) y = ma.array(y)
x = x.flatten() y = y.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 LinregressResult(slope, intercept, r, prob, sterrest)
r""" Computes the Theil-Sen estimator for a set of points (x, y).
`theilslopes` implements a method for robust linear regression. It computes the slope as the median of all slopes between paired values.
Parameters ---------- y : array_like Dependent variable. x : array_like or None, optional Independent variable. If None, use ``arange(len(y))`` instead. alpha : float, optional Confidence degree between 0 and 1. Default is 95% confidence. Note that `alpha` is symmetric around 0.5, i.e. both 0.1 and 0.9 are interpreted as "find the 90% confidence interval".
Returns ------- medslope : float Theil slope. medintercept : float Intercept of the Theil line, as ``median(y) - medslope*median(x)``. lo_slope : float Lower bound of the confidence interval on `medslope`. up_slope : float Upper bound of the confidence interval on `medslope`.
Notes ----- For more details on `theilslopes`, see `stats.theilslopes`.
""" 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)
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
""" Calculates the T-test for the mean of ONE group of scores.
Parameters ---------- a : array_like sample observation popmean : float or array_like expected value in null hypothesis, if array_like than it must have the same shape as `a` excluding the axis dimension axis : int or None, optional Axis along which to compute test. If None, compute over the whole array `a`.
Returns ------- statistic : float or array t-statistic pvalue : float or array two-tailed p-value
Notes ----- For more details on `ttest_1samp`, see `stats.ttest_1samp`.
""" 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) # force df to be an array for masked division not to throw a warning df = ma.asanyarray(n - 1.0) svar = ((n - 1.0) * v) / df with np.errstate(divide='ignore', invalid='ignore'): t = (x - popmean) / ma.sqrt(svar / n) prob = special.betainc(0.5*df, 0.5, df/(df + t*t))
return Ttest_1sampResult(t, prob)
""" Calculates the T-test for the means of TWO INDEPENDENT samples of scores.
Parameters ---------- a, b : array_like The arrays must have the same shape, except in the dimension corresponding to `axis` (the first, by default). axis : int or None, optional Axis along which to compute test. If None, compute over the whole arrays, `a`, and `b`. equal_var : bool, optional If True, perform a standard independent 2 sample test that assumes equal population variances. If False, perform Welch's t-test, which does not assume equal population variance. .. versionadded:: 0.17.0
Returns ------- statistic : float or array The calculated t-statistic. pvalue : float or array The two-tailed p-value.
Notes ----- For more details on `ttest_ind`, see `stats.ttest_ind`.
""" a, b, axis = _chk2_asarray(a, b, axis)
if a.size == 0 or b.size == 0: return Ttest_indResult(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))
if equal_var: # force df to be an array for masked division not to throw a warning df = ma.asanyarray(n1 + n2 - 2.0) svar = ((n1-1)*v1+(n2-1)*v2) / df denom = ma.sqrt(svar*(1.0/n1 + 1.0/n2)) # n-D computation here! else: vn1 = v1/n1 vn2 = v2/n2 with np.errstate(divide='ignore', invalid='ignore'): df = (vn1 + vn2)**2 / (vn1**2 / (n1 - 1) + vn2**2 / (n2 - 1))
# If df is undefined, variances are zero. # It doesn't matter what df is as long as it is not NaN. df = np.where(np.isnan(df), 1, df) denom = ma.sqrt(vn1 + vn2)
with np.errstate(divide='ignore', invalid='ignore'): t = (x1-x2) / denom probs = special.betainc(0.5*df, 0.5, df/(df + t*t)).reshape(t.shape)
return Ttest_indResult(t, probs.squeeze())
""" Calculates the T-test on TWO RELATED samples of scores, a and b.
Parameters ---------- a, b : array_like The arrays must have the same shape. axis : int or None, optional Axis along which to compute test. If None, compute over the whole arrays, `a`, and `b`.
Returns ------- statistic : float or array t-statistic pvalue : float or array two-tailed p-value
Notes ----- For more details on `ttest_rel`, see `stats.ttest_rel`.
""" 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 Ttest_relResult(np.nan, np.nan)
n = a.count(axis) df = ma.asanyarray(n-1.0) d = (a-b).astype('d') dm = d.mean(axis) v = d.var(axis=axis, ddof=1) denom = ma.sqrt(v / n) with np.errstate(divide='ignore', invalid='ignore'): t = dm / denom
probs = special.betainc(0.5*df, 0.5, df/(df + t*t)).reshape(t.shape).squeeze()
return Ttest_relResult(t, probs)
'pvalue'))
""" 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 ------- statistic : float The Mann-Whitney statistics pvalue : 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 -= 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 MannwhitneyuResult(u, prob)
""" Compute the Kruskal-Wallis H-test for independent samples
Parameters ---------- sample1, sample2, ... : array_like Two or more arrays with the sample measurements can be given as arguments.
Returns ------- statistic : float The Kruskal-Wallis H statistic, corrected for ties pvalue : float The p-value for the test using the assumption that H has a chi square distribution
Notes ----- For more details on `kruskal`, see `stats.kruskal`.
""" 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. - 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 = distributions.chi2.sf(H, df) return KruskalResult(H, prob)
""" 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)
""" 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
""" 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)
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. """
""" Trims an array by masking the data outside some given limits.
Returns a masked version of the input array.
%s
Examples -------- >>> from scipy.stats.mstats import trim >>> z = [ 1, 2, 3, 4, 5, 6, 7, 8, 9,10] >>> print(trim(z,(3,8))) [-- -- 3 4 5 6 7 8 -- --] >>> print(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)
""" 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)
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)
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)
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)
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)
""" 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)
"""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
""" Compute the trimmed mean.
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 (default), then all values are used. Either of the limit values in the tuple can also be None representing a half-open interval. 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). axis : int or None, optional Axis along which to operate. If None, compute over the whole array. Default is None.
Returns ------- tmean : float
Notes ----- For more details on `tmean`, see `stats.tmean`.
""" return trima(a, limits=limits, inclusive=inclusive).mean(axis=axis)
""" 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). axis : int or None, optional Axis along which to operate. If None, compute over the whole array. Default is zero. ddof : int, optional Delta degrees of freedom. Default is 1.
Returns ------- tvar : float Trimmed variance.
Notes ----- For more details on `tvar`, see `stats.tvar`.
""" a = a.astype(float).ravel() if limits is None: n = (~a.mask).sum() # todo: better way to do that? return np.ma.var(a) * n/(n-1.) am = _mask_to_limits(a, limits=limits, inclusive=inclusive)
return np.ma.var(am, axis=axis, ddof=ddof)
""" Compute the trimmed minimum
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 : int or None, optional Axis along which to operate. Default is 0. If None, compute over the whole array `a`. 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, int or ndarray
Notes ----- For more details on `tmin`, see `stats.tmin`.
""" a, axis = _chk_asarray(a, axis) am = trima(a, (lowerlimit, None), (inclusive, False)) return ma.minimum.reduce(am, axis)
""" 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 : int or None, optional Axis along which to operate. Default is 0. If None, compute over the whole array `a`. 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, int or ndarray
Notes ----- For more details on `tmax`, see `stats.tmax`.
""" a, axis = _chk_asarray(a, axis) am = trima(a, (None, upperlimit), (False, inclusive)) return ma.maximum.reduce(am, axis)
""" 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). axis : int or None, optional Axis along which to operate. If None, compute over the whole array. Default is zero. ddof : int, optional Delta degrees of freedom. Default is 1.
Returns ------- tsem : float
Notes ----- For more details on `tsem`, see `stats.tsem`.
""" a = ma.asarray(a).ravel() if limits is None: n = float(a.count()) return a.std(axis=axis, ddof=ddof)/ma.sqrt(n)
am = trima(a.ravel(), limits, inclusive) sd = np.sqrt(am.var(axis=axis, ddof=ddof)) return sd / np.sqrt(am.count())
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 truncated (True) or rounded (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).astype(int) 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).astype(int) 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)
""" Calculates the nth moment about the mean for a sample.
Parameters ---------- a : array_like data moment : int, optional order of central moment that is returned axis : int or None, optional Axis along which the central moment is computed. Default is 0. If None, compute over the whole array `a`.
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.
Notes ----- For more details about `moment`, see `stats.moment`.
""" 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: # Exponentiation by squares: form exponent sequence n_list = [moment] current_n = moment while current_n > 2: if current_n % 2: current_n = (current_n-1)/2 else: current_n /= 2 n_list.append(current_n)
# Starting point for exponentiation by squares a_zero_mean = a - ma.expand_dims(a.mean(axis), axis) if n_list[-1] == 1: s = a_zero_mean.copy() else: s = a_zero_mean**2
# Perform multiplications for n in n_list[-2::-1]: s = s**2 if n % 2: s *= a_zero_mean return s.mean(axis)
""" 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, optional Axis along which to calculate the coefficient of variation. Default is 0. If None, compute over the whole array `a`.
Returns ------- variation : ndarray The calculated variation along the requested axis.
Notes ----- For more details about `variation`, see `stats.variation`.
""" a, axis = _chk_asarray(a, axis) return a.std(axis)/a.mean(axis)
""" Computes the skewness of a data set.
Parameters ---------- a : ndarray data axis : int or None, optional Axis along which skewness is calculated. Default is 0. If None, compute over the whole array `a`. bias : bool, optional If False, then the calculations are corrected for statistical bias.
Returns ------- skewness : ndarray The skewness of values along an axis, returning 0 where all values are equal.
Notes ----- For more details about `skew`, see `stats.skew`.
""" 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
""" Computes the kurtosis (Fisher or Pearson) of a dataset.
Kurtosis is the fourth central moment divided by the square of the variance. If Fisher's definition is used, then 3.0 is subtracted from the result to give 0.0 for a normal distribution.
If bias is False then the kurtosis is calculated using k statistics to eliminate bias coming from biased moment estimators
Use `kurtosistest` to see if result is close enough to normal.
Parameters ---------- a : array data for which the kurtosis is calculated axis : int or None, optional Axis along which the kurtosis is calculated. Default is 0. If None, compute over the whole array `a`. fisher : bool, optional If True, Fisher's definition is used (normal ==> 0.0). If False, Pearson's definition is used (normal ==> 3.0). bias : bool, optional If False, then the calculations are corrected for statistical bias.
Returns ------- kurtosis : array The kurtosis of values along an axis. If all values are equal, return -3 for Fisher's definition and 0 for Pearson's definition.
Notes ----- For more details about `kurtosis`, see `stats.kurtosis`.
""" 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
'variance', 'skewness', 'kurtosis'))
""" Computes several descriptive statistics of the passed array.
Parameters ---------- a : array_like Data array axis : int or None, optional Axis along which to calculate statistics. Default 0. If None, compute over the whole array `a`. ddof : int, optional degree of freedom (default 0); note that default ddof is different from the same routine in stats.describe bias : bool, optional If False, then the skewness and kurtosis calculations are corrected for statistical bias.
Returns ------- nobs : int (size of the data (discarding missing values)
minmax : (int, int) min, max
mean : float arithmetic mean
variance : float unbiased variance
skewness : float biased skewness
kurtosis : float biased kurtosis
Examples -------- >>> from scipy.stats.mstats import describe >>> ma = np.ma.array(range(6), mask=[0, 0, 0, 1, 1, 1]) >>> describe(ma) DescribeResult(nobs=3, minmax=(masked_array(data=0, mask=False, fill_value=999999), masked_array(data=2, mask=False, fill_value=999999)), mean=1.0, variance=0.6666666666666666, skewness=masked_array(data=0., mask=False, fill_value=1e+20), kurtosis=-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, bias=bias) kurt = kurtosis(a, axis, bias=bias)
return DescribeResult(n, mm, m, v, sk, kurt)
"""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)
""" Tests whether the skew is different from the normal distribution.
Parameters ---------- a : array The data to be tested axis : int or None, optional Axis along which statistics are calculated. Default is 0. If None, compute over the whole array `a`.
Returns ------- statistic : float The computed z-score for this test. pvalue : float a 2-sided p-value for the hypothesis test
Notes ----- For more details about `skewtest`, see `stats.skewtest`.
""" 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 SkewtestResult(Z, 2 * distributions.norm.sf(np.abs(Z)))
'pvalue'))
""" Tests whether a dataset has normal kurtosis
Parameters ---------- a : array array of the sample data axis : int or None, optional Axis along which to compute test. Default is 0. If None, compute over the whole array `a`.
Returns ------- statistic : float The computed z-score for this test. pvalue : float The 2-sided p-value for the hypothesis test
Notes ----- For more details about `kurtosistest`, see `stats.kurtosistest`.
""" 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 KurtosistestResult(Z, 2 * distributions.norm.sf(np.abs(Z)))
""" Tests whether a sample differs from a normal distribution.
Parameters ---------- a : array_like The array containing the data to be tested. axis : int or None, optional Axis along which to compute test. Default is 0. If None, compute over the whole array `a`.
Returns ------- statistic : float or array ``s^2 + k^2``, where ``s`` is the z-score returned by `skewtest` and ``k`` is the z-score returned by `kurtosistest`. pvalue : float or array A 2-sided chi squared probability for the hypothesis test.
Notes ----- For more details about `normaltest`, see `stats.normaltest`.
""" a, axis = _chk_asarray(a, axis) s, _ = skewtest(a, axis) k, _ = kurtosistest(a, axis) k2 = s*s + k*k
return NormaltestResult(k2, distributions.chi2.sf(k2, 2))
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, optional 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.]]) >>> print(mquantiles(data, axis=0, limit=(0, 50))) [[19.2 14.6 1.45] [40. 37.5 2.5 ] [42.8 40.05 3.55]]
>>> data[:, 2] = -999. >>> print(mquantiles(data, axis=0, limit=(0, 50))) [[19.200000000000003 14.6 --] [40.0 37.5 --] [42.800000000000004 40.05 --]]
""" 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)
"""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()
""" 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(axis=None)[:n]] = ((np.arange(1, n+1) - alpha) / (n + 1.0 - alpha - beta)) return ma.array(plpos, mask=data._mask)
""" 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
""" 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) >>> print(stats.mstats.sem(a)) [2.8284271247461903 2.8284271247461903 2.8284271247461903 2.8284271247461903]
Find standard error across the whole array, using n degrees of freedom:
>>> print(stats.mstats.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
""" 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 ------- statistic : float The computed F-value of the test. pvalue : float The associated p-value from the F-distribution.
""" # 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_onewayResult(f, prob)
('statistic', 'pvalue'))
"""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 ------- statistic : float the test statistic. pvalue : float the 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 = [find_repeats(row) for row in ranked.T] ties = np.array([y for x, y in repeats if x.size > 0]) 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 FriedmanchisquareResult(chisq, distributions.chi2.sf(chisq, k-1)) |