""" An extension of scipy.stats.stats to support masked arrays :author: Pierre GF Gerard-Marchant :contact: pierregm_at_uga_edu """ #TODO : f_value_wilks_lambda looks botched... what are dfnum & dfden for ? #TODO : ttest_reel looks botched: what are x1,x2,v1,v2 for ? #TODO : reimplement ksonesamp __author__ = "Pierre GF Gerard-Marchant" __docformat__ = "restructuredtext en" __all__ = ['argstoarray', 'betai', 'chisquare','count_tied_groups', 'describe', 'f_oneway','f_value_wilks_lambda','find_repeats','friedmanchisquare', 'gmean', 'hmean', '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 MaskedArray, masked, nomask import itertools import warnings #import scipy.stats as stats import stats import scipy.special as special import scipy.misc as misc #import scipy.stats.futil as futil 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): if axis is None: a = ma.ravel(a) outaxis = 0 else: a = ma.asanyarray(a) outaxis = axis return a, outaxis def _chk2_asarray(a, b, axis): if axis is None: a = ma.ravel(a) b = ma.ravel(b) outaxis = 0 else: a = ma.asanyarray(a) b = ma.asanyarray(b) 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 sequence of sequences. Sequences are filled with missing values to match the length of the longest sequence. Returns ------- output : MaskedArray a (mxn) masked array, where m is the number of arguments and n the length of the longest argument. """ 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 #####-------------------------------------------------------------------------- #---- --- Ranking --- #####-------------------------------------------------------------------------- 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 in x, and returns a dictionary (nb of ties: nb of groups). Parameters ---------- x : sequence Sequence of data on which to counts the ties use_missing : boolean Whether to consider missing values as tied. Examples -------- >>> z = [0, 0, 0, 2, 2, 2, 3, 3, 4, 5, 6] >>> count_tied_groups(z) >>> {2:1, 3:2} >>> # The ties were 0 (3x), 2 (3x) and 3 (2x) >>> z = ma.array([0, 0, 1, 2, 2, 2, 3, 3, 4, 5, 6]) >>> count_tied_groups(z) >>> {2:2, 3:1} >>> # The ties were 0 (2x), 2 (3x) and 3 (2x) >>> z[[1,-1]] = masked >>> count_tied_groups(z, use_missing=True) >>> {2:2, 3:1} >>> # The ties were 2 (3x), 3 (2x) and masked (2x) """ 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) #####-------------------------------------------------------------------------- #---- --- Central tendency --- #####-------------------------------------------------------------------------- def gmean(a, axis=0): a, axis = _chk_asarray(a, axis) log_a = ma.log(a) return ma.exp(log_a.mean(axis=axis)) gmean.__doc__ = stats.gmean.__doc__ def hmean(a, axis=0): a, axis = _chk_asarray(a, axis) if isinstance(a, MaskedArray): size = a.count(axis) else: size = a.shape[axis] return size / (1.0/a).sum(axis) hmean.__doc__ = stats.hmean.__doc__ def mode(a, axis=0): def _mode1D(a): (rep,cnt) = find_repeats(a) if not cnt.ndim: return (0, 0) elif cnt.size: return (rep[cnt.argmax()], cnt.max()) return (a[0], 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__ #####-------------------------------------------------------------------------- #---- --- Probabilities --- #####-------------------------------------------------------------------------- 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__ #####-------------------------------------------------------------------------- #---- --- Correlation --- #####-------------------------------------------------------------------------- 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 : 1D array y : 1D array the same length as x Returns ------- (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 = n*(ma.add.reduce(xm*ym)) r_den = n*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 # t = ma.sqrt(df/((1.0-r)*(1.0+r))) * r if t is masked: prob = 0. else: prob = betai(0.5*df,0.5,df/(df+t*t)) 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 : 1D array y : 1D array the same length as x The lengths of both arrays must be > 2. use_ties : {True, False} optional Whether the correction for ties should be computed. Returns ------- (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 xties.iteritems())/12. corr_y = np.sum(v*k*(k**2-1) for (k,v) in yties.iteritems())/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:]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 xties.iteritems()], dtype=float) corr_y = np.sum([v*k*(k-1) for (k,v) in yties.iteritems()], 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 xties.iteritems()) var_s -= np.sum(v*k*(k-1)*(2*k+5)*1. for (k,v) in yties.iteritems()) v1 = np.sum([v*k*(k-1) for (k, v) in xties.iteritems()], dtype=float) *\ np.sum([v*k*(k-1) for (k, v) in yties.iteritems()], dtype=float) v1 /= 2.*n*(n-1) if n > 2: v2 = np.sum([v*k*(k-1)*(k-2) for (k,v) in xties.iteritems()], dtype=float) * \ np.sum([v*k*(k-1)*(k-2) for (k,v) in yties.iteritems()], 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 extension Kendall's rank correlation tau, designed for seasonal data. Parameters ---------- x: 2D array 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 ties.iteritems()) 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) # cov_jj = ma.empty(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 ties_j.iteritems()) 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] # cov_jj[j] = (nn_p*(2*n_p[j]+5)) # cov_jj[j] -= np.sum(v*k*(k-1)*(2*k+5) for (k,v) in ties_j.iteritems()) # cov_jj[j] /= 18. 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): if len(args) == 1: # more than 1D array? 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: x = ma.array(args[0]).flatten() y = ma.array(args[1]).flatten() m = ma.mask_or(ma.getmask(x), ma.getmask(y)) if m is not nomask: x = ma.array(x,mask=m) y = ma.array(y,mask=m) n = len(x) (xmean, ymean) = (x.mean(), y.mean()) (xm, ym) = (x-xmean, y-ymean) (Sxx, Syy) = (ma.add.reduce(xm*xm), ma.add.reduce(ym*ym)) Sxy = ma.add.reduce(xm*ym) r_den = ma.sqrt(Sxx*Syy) if r_den == 0.0: r = 0.0 else: r = Sxy / r_den if (r > 1.0): r = 1.0 # from numerical error #z = 0.5*log((1.0+r+TINY)/(1.0-r+TINY)) df = n-2 t = r * ma.sqrt(df/(1.0-r*r)) prob = betai(0.5*df,0.5,df/(df+t*t)) slope = Sxy / Sxx intercept = ymean - slope*xmean sterrest = ma.sqrt(1.-r*r) * y.std() return slope, intercept, r, prob, sterrest if stats.linregress.__doc__: linregress.__doc__ = stats.linregress.__doc__ + genmissingvaldoc def theilslopes(y, x=None, alpha=0.05): """Computes the Theil slope over the dataset (x,y), as the median of all slopes between paired values. Parameters ---------- y : sequence Dependent variable. x : {None, sequence} optional Independent variable. If None, use arange(len(y)) instead. alpha : float Confidence degree. 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 """ y = ma.asarray(y).flatten() y[-1] = masked n = len(y) if x is None: x = ma.arange(len(y), dtype=float) else: x = ma.asarray(x).flatten() if len(x) != n: raise ValueError("Incompatible lengths ! (%s<>%s)" % (n,len(x))) m = ma.mask_or(ma.getmask(x), ma.getmask(y)) y._mask = x._mask = m ny = y.count() # slopes = ma.hstack([(y[i+1:]-y[i])/(x[i+1:]-x[i]) for i in range(n-1)]) slopes.sort() medslope = ma.median(slopes) medinter = ma.median(y) - medslope*ma.median(x) # if alpha > 0.5: alpha = 1.-alpha z = stats.distributions.norm.ppf(alpha/2.) # (xties, yties) = (count_tied_groups(x), count_tied_groups(y)) nt = ny*(ny-1)/2. sigsq = (ny*(ny-1)*(2*ny+5)/18.) sigsq -= np.sum(v*k*(k-1)*(2*k+5) for (k,v) in xties.iteritems()) sigsq -= np.sum(v*k*(k-1)*(2*k+5) for (k,v) in yties.iteritems()) sigma = np.sqrt(sigsq) Ru = min(np.round((nt - z*sigma)/2. + 1), len(slopes)-1) Rl = max(np.round((nt + z*sigma)/2.), 0) delta = slopes[[Rl,Ru]] return medslope, medinter, delta[0], delta[1] 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 #####-------------------------------------------------------------------------- #---- --- Inferential statistics --- #####-------------------------------------------------------------------------- def ttest_onesamp(a, popmean): a = ma.asarray(a) x = a.mean(axis=None) v = a.var(axis=None,ddof=1) n = a.count(axis=None) df = n-1 svar = ((n-1)*v) / float(df) t = (x-popmean)/ma.sqrt(svar*(1.0/n)) prob = betai(0.5*df,0.5,df/(df+t*t)) return t,prob ttest_onesamp.__doc__ = stats.ttest_1samp.__doc__ ttest_1samp = ttest_onesamp def ttest_ind(a, b, axis=0): a, b, axis = _chk2_asarray(a, b, axis) (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) / float(df) svar == 0 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,float(df)/(df+t*t)).reshape(t.shape) return t, probs.squeeze() ttest_ind.__doc__ = stats.ttest_ind.__doc__ def ttest_rel(a,b,axis=None): a, b, axis = _chk2_asarray(a, b, axis) if len(a)!=len(b): raise ValueError('unequal length arrays') (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) #zerodivproblem = denom == 0 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__ def chisquare(f_obs, f_exp=None): f_obs = ma.asarray(f_obs) if f_exp is None: f_exp = ma.array([f_obs.mean(axis=0)] * len(f_obs)) f_exp = f_exp.astype(float) chisq = ma.add.reduce((f_obs-f_exp)**2 / f_exp) return chisq, stats.chisqprob(chisq, f_obs.count(0)-1) chisquare.__doc__ = stats.chisquare.__doc__ def mannwhitneyu(x,y, use_continuity=True): """Computes the Mann-Whitney on samples x and y. Missing values in x and/or y are discarded. Parameters ---------- x : sequence y : sequence 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 ties.iteritems())/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() # ssbg = (sumrk**2/ranks.count(-1)).sum() - ranks.sum()**2/ntotal # H = ssbg / (ntotal*(ntotal+1)/12.) 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 ties.iteritems())/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__ _kolmog2 = special.kolmogorov def _kolmog1(x,n): if x <= 0: return 0 if x >= 1: return 1 j = np.arange(np.floor(n*(1-x))+1) return 1 - x * np.sum(np.exp(np.log(misc.comb(n,j)) + (n-j) * np.log(1-x-j/float(n)) + (j-1) * np.log(x+j/float(n)))) def ks_twosamp(data1, data2, alternative="two_sided"): """Computes the Kolmogorov-Smirnov test on two samples. Missing values are discarded. Parameters ---------- data1 : sequence First data set data2 : sequence Second data set alternative : {'two_sided', 'less', 'greater'} optional Indicates the alternative hypothesis. 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 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 : sequence 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 : {(True,True) 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: 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 == 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 == 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 : {(True, True) tuple} optional If relative==False, tuple indicating whether values exactly equal to the absolute limits are allowed. If relative==True, tuple indicating whether the number of data being masked on each side should be rounded (True) or truncated (False). relative : {False, True} optional Whether to consider the limits as absolute values (False) or proportions to cut (True). axis : {None, integer}, 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__: trim.__doc__ = trim.__doc__ % trimdoc def trimboth(data, proportiontocut=0.2, inclusive=(True,True), axis=None): """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 : {0.2, 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. inclusive : {(True, True) tuple} optional Tuple indicating whether the number of data being masked on each side should be rounded (True) or truncated (False). axis : {None, integer}, 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 int(trim*n) values from ONE tail of the data along the given axis, where n is the number of unmasked values. Parameters ---------- data : {ndarray} Data to trim. proportiontocut : {0.2, 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. tail : {'left','right'} optional If left (right), the ``proportiontocut`` lowest (greatest) values will be masked. inclusive : {(True, True) tuple} optional Tuple indicating whether the number of data being masked on each side should be rounded (True) or truncated (False). axis : {None, integer}, optional Axis along which to perform the trimming. If None, the input array is first flattened. """ 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 of the data 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. 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 : {(True, True) tuple} optional Tuple indicating whether the number of data being masked on each side should be rounded (True) or truncated (False). axis : {None, integer}, optional Axis along which to trim. """ #........................ 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 nsize = a.count() 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): shp = a.shape 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)): return trima(a, limits=limits, inclusive=inclusive).var() 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()/ma.sqrt(n) am = trima(a.ravel(), limits, inclusive) sd = np.sqrt(am.var()) return sd / 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 (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. """ 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 gonna 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) #####-------------------------------------------------------------------------- #---- --- Moments --- #####-------------------------------------------------------------------------- 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) vals = ma.where(m2 == 0, 0, m3 / m2**1.5) 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) n = a.count(axis) m2 = moment(a,2,axis) m4 = moment(a,4,axis) vals = ma.where(m2 == 0, 0, m4/ m2**2.0) if not bias: can_correct = (n > 3) & (m2 > 0) if can_correct.any(): 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): """ Computes several descriptive statistics of the passed array. Parameters ---------- a : array axis : int or None 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) 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) #####-------------------------------------------------------------------------- #---- --- Normality Tests --- #####-------------------------------------------------------------------------- def skewtest(a, axis=0): """ Tests whether the skew is different from the normal distribution. This function tests the null hypothesis that the skewness of the population that the sample was drawn from is the same as that of a corresponding normal distribution. Parameters ---------- a : array_like The input array. axis : int or None, optional The axis along which to perform the skew test. Default is 0. If `axis` is None, the array is first flattened. Returns ------- z-score : float The computed z-score for this test. p-value : float a 2-sided p-value for the hypothesis test Notes ----- The sample size must be at least 8. """ 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: warnings.warn( "skewtest only valid for n>=8 ... continuing anyway, n=%i" % 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, (1.0 - stats.zprob(Z))*2 skewtest.__doc__ = stats.skewtest.__doc__ def kurtosistest(a, axis=0): """ Tests whether a dataset has normal kurtosis. This function tests the null hypothesis that the kurtosis of the population from which the sample was drawn is that of the normal distribution: ``kurtosis = 3(n-1)/(n+1)``. Parameters ---------- a : array_like Array of the sample data. axis : int or None, optional The axis to operate along, or None to work on the whole array. The default is 0 (the first axis). Returns ------- z-score : float The computed z-score for this test. p-value : float The 2-sided p-value for the hypothesis test Notes ----- Valid only for n>20. The Z-score is set to 0 for bad entries. """ a, axis = _chk_asarray(a, axis) n = a.count(axis=axis).astype(float) 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)) denom[denom < 0] = masked term2 = ma.power((1-2.0/A)/denom,1/3.0) Z = ( term1 - term2 ) / np.sqrt(2/(9.0*A)) return Z, (1.0-stats.zprob(Z))*2 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__ # Martinez-Iglewicz test # K-S test #####-------------------------------------------------------------------------- #---- --- Percentiles --- #####-------------------------------------------------------------------------- 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-g).x[i] +g.x[i+1]``, where ``x[j]`` is the j-th order statistic, ``i = (floor(n*p+m))``, ``m=alpha+p*(1-alpha-beta)`` and ``g = n*p + m - i``. Typical values of (alpha,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)* : (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 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. alpha : float, optional Plotting positions parameter, default is 0.4. beta : 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 closed interval are ignored. Returns ------- mquantiles : MaskedArray An array containing the calculated quantiles. 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()] # Initialization & checks --------- data = ma.array(a, copy=False) if data.ndim > 2: raise TypeError("Array should be 2D at most !") # if limit: condition = (limit[0] 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 #####-------------------------------------------------------------------------- #---- --- Variability --- #####-------------------------------------------------------------------------- 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): a, axis = _chk_asarray(a, axis) n = a.count(axis=axis) s = a.std(axis=axis,ddof=0) / ma.sqrt(n-1) return s sem.__doc__ = stats.sem.__doc__ zmap = stats.zmap zscore = stats.zscore #####-------------------------------------------------------------------------- #---- --- ANOVA --- #####-------------------------------------------------------------------------- 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 = stats.fprob(dfbg,dfwg,f) return f, prob def f_value_wilks_lambda(ER, EF, dfnum, dfden, a, b): """Calculation of Wilks lambda F-statistic for multivarite 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) #-############################################################################-#