""" 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)