from numpy.random import randint from scipy.stats import norm import numpy as np import warnings # Keep python 2/3 compatibility, without using six. At some point, # we may need to add six as a requirement, but right now we can avoid it. try: xrange except NameError: xrange = range class InstabilityWarning(UserWarning): """Issued when results may be unstable.""" pass # On import, make sure that InstabilityWarnings are not filtered out. warnings.simplefilter('always',InstabilityWarning) def ci(data, statfunction=np.average, alpha=0.05, n_samples=10000, method='bca', output='lowhigh', epsilon=0.001, multi=None): """ Given a set of data ``data``, and a statistics function ``statfunction`` that applies to that data, computes the bootstrap confidence interval for ``statfunction`` on that data. Data points are assumed to be delineated by axis 0. Parameters ---------- data: array_like, shape (N, ...) OR tuple of array_like all with shape (N, ...) Input data. Data points are assumed to be delineated by axis 0. Beyond this, the shape doesn't matter, so long as ``statfunction`` can be applied to the array. If a tuple of array_likes is passed, then samples from each array (along axis 0) are passed in order as separate parameters to the statfunction. The type of data (single array or tuple of arrays) can be explicitly specified by the multi parameter. statfunction: function (data, weights=(weights, optional)) -> value This function should accept samples of data from ``data``. It is applied to these samples individually. If using the ABC method, the function _must_ accept a named ``weights`` parameter which will be an array_like with weights for each sample, and must return a _weighted_ result. Otherwise this parameter is not used or required. Note that numpy's np.average accepts this. (default=np.average) alpha: float or iterable, optional The percentiles to use for the confidence interval (default=0.05). If this is a float, the returned values are (alpha/2, 1-alpha/2) percentile confidence intervals. If it is an iterable, alpha is assumed to be an iterable of each desired percentile. n_samples: float, optional The number of bootstrap samples to use (default=10000) method: string, optional The method to use: one of 'pi', 'bca', or 'abc' (default='bca') output: string, optional The format of the output. 'lowhigh' gives low and high confidence interval values. 'errorbar' gives transposed abs(value-confidence interval value) values that are suitable for use with matplotlib's errorbar function. (default='lowhigh') epsilon: float, optional (only for ABC method) The step size for finite difference calculations in the ABC method. Ignored for all other methods. (default=0.001) multi: boolean, optional If False, assume data is a single array. If True, assume data is a tuple/other iterable of arrays of the same length that should be sampled together. If None, decide based on whether the data is an actual tuple. (default=None) Returns ------- confidences: tuple of floats The confidence percentiles specified by alpha Calculation Methods ------------------- 'pi': Percentile Interval (Efron 13.3) The percentile interval method simply returns the 100*alphath bootstrap sample's values for the statistic. This is an extremely simple method of confidence interval calculation. However, it has several disadvantages compared to the bias-corrected accelerated method, which is the default. 'bca': Bias-Corrected Accelerated Non-Parametric (Efron 14.3) (default) This method is much more complex to explain. However, it gives considerably better results, and is generally recommended for normal situations. Note that in cases where the statistic is smooth, and can be expressed with weights, the ABC method will give approximated results much, much faster. 'abc': Approximate Bootstrap Confidence (Efron 14.4, 22.6) This method provides approximated bootstrap confidence intervals without actually taking bootstrap samples. This requires that the statistic be smooth, and allow for weighting of individual points with a weights= parameter (note that np.average allows this). This is _much_ faster than all other methods for situations where it can be used. Examples -------- To calculate the confidence intervals for the mean of some numbers: >> boot.ci( np.randn(100), np.average ) Given some data points in arrays x and y calculate the confidence intervals for all linear regression coefficients simultaneously: >> boot.ci( (x,y), scipy.stats.linregress ) References ---------- Efron, An Introduction to the Bootstrap. Chapman & Hall 1993 """ # Deal with the alpha values if np.iterable(alpha): alphas = np.array(alpha) else: alphas = np.array([alpha/2,1-alpha/2]) if multi == None: if isinstance(data, tuple): multi = True else: multi = False # Ensure that the data is actually an array. This isn't nice to pandas, # but pandas seems much much slower and the indexes become a problem. if multi == False: data = np.array(data) tdata = (data,) else: tdata = tuple( np.array(x) for x in data ) # Deal with ABC *now*, as it doesn't need samples. if method == 'abc': n = tdata[0].shape[0]*1.0 nn = tdata[0].shape[0] I = np.identity(nn) ep = epsilon / n*1.0 p0 = np.repeat(1.0/n,nn) t1 = np.zeros(nn); t2 = np.zeros(nn) try: t0 = statfunction(*tdata,weights=p0) except TypeError as e: raise TypeError("statfunction does not accept correct arguments for ABC ({0})".format(e.message)) # There MUST be a better way to do this! for i in range(0,nn): di = I[i] - p0 tp = statfunction(*tdata,weights=p0+ep*di) tm = statfunction(*tdata,weights=p0-ep*di) t1[i] = (tp-tm)/(2*ep) t2[i] = (tp-2*t0+tm)/ep**2 sighat = np.sqrt(np.sum(t1**2))/n a = (np.sum(t1**3))/(6*n**3*sighat**3) delta = t1/(n**2*sighat) cq = (statfunction(*tdata,weights=p0+ep*delta)-2*t0+statfunction(*tdata,weights=p0-ep*delta))/(2*sighat*ep**2) bhat = np.sum(t2)/(2*n**2) curv = bhat/sighat-cq z0 = norm.ppf(2*norm.cdf(a)*norm.cdf(-curv)) Z = z0+norm.ppf(alphas) za = Z/(1-a*Z)**2 # stan = t0 + sighat * norm.ppf(alphas) abc = np.zeros_like(alphas) for i in range(0,len(alphas)): abc[i] = statfunction(*tdata,weights=p0+za[i]*delta) if output == 'lowhigh': return abc elif output == 'errorbar': return abs(abc-statfunction(tdata))[np.newaxis].T else: raise ValueError("Output option {0} is not supported.".format(output)) # We don't need to generate actual samples; that would take more memory. # Instead, we can generate just the indexes, and then apply the statfun # to those indexes. bootindexes = bootstrap_indexes( tdata[0], n_samples ) stat = np.array([statfunction(*(x[indexes] for x in tdata)) for indexes in bootindexes]) stat.sort(axis=0) # Percentile Interval Method if method == 'pi': avals = alphas # Bias-Corrected Accelerated Method elif method == 'bca': # The value of the statistic function applied just to the actual data. ostat = statfunction(*tdata) # The bias correction value. z0 = norm.ppf( ( 1.0*np.sum(stat < ostat, axis=0) ) / n_samples ) # Statistics of the jackknife distribution jackindexes = jackknife_indexes(tdata[0]) jstat = [statfunction(*(x[indexes] for x in tdata)) for indexes in jackindexes] jmean = np.mean(jstat,axis=0) # Acceleration value a = np.sum( (jmean - jstat)**3, axis=0 ) / ( 6.0 * np.sum( (jmean - jstat)**2, axis=0)**1.5 ) # Raise an exception if the acceleration value is not finite. This will happen if, for example, # every jackknife sample results in the same statistic value. if not np.all(np.isfinite(a)): raise ValueError("BCa acceleration value is not finite, and BCa cannot be used. This \ will happen if, for example, all input rows are identical. Try using the \ percentage interval method instead, though be aware that bootstrapping may \ not be the right option for your data.") zs = z0 + norm.ppf(alphas).reshape(alphas.shape+(1,)*z0.ndim) avals = norm.cdf(z0 + zs/(1-a*zs)) else: raise ValueError("Method {0} is not supported.".format(method)) nvals = np.round((n_samples-1)*avals).astype('int') # commented out pkervell - 16aug15 #if np.any(nvals==0) or np.any(nvals==n_samples-1): # warnings.warn("Some values used extremal samples; results are probably unstable.", InstabilityWarning) # elif np.any(nvals<10) or np.any(nvals>=n_samples-10): # warnings.warn("Some values used top 10 low/high samples; results may be unstable.", InstabilityWarning) if output == 'lowhigh': if nvals.ndim == 1: # All nvals are the same. Simple broadcasting return stat[nvals] else: # Nvals are different for each data point. Not simple broadcasting. # Each set of nvals along axis 0 corresponds to the data at the same # point in other axes. return stat[(nvals, np.indices(nvals.shape)[1:].squeeze())] elif output == 'errorbar': if nvals.ndim == 1: return abs(statfunction(data)-stat[nvals])[np.newaxis].T else: return abs(statfunction(data)-stat[(nvals, np.indices(nvals.shape)[1:])])[np.newaxis].T else: raise ValueError("Output option {0} is not supported.".format(output)) def ci_abc(data, stat=lambda x,y: np.average(x,weights=y) , alpha=0.05, epsilon = 0.001): """ .. note:: Deprecated. This functionality is now rolled into ci. Given a set of data ``data``, and a statistics function ``statfunction`` that applies to that data, computes the non-parametric approximate bootstrap confidence (ABC) interval for ``stat`` on that data. Data points are assumed to be delineated by axis 0. Parameters ---------- data: array_like, shape (N, ...) Input data. Data points are assumed to be delineated by axis 0. Beyond this, the shape doesn't matter, so long as ``statfunction`` can be applied to the array. stat: function (data, weights) -> value The _weighted_ statistic function. This must accept weights, unlike for other methods. alpha: float or iterable, optional The percentiles to use for the confidence interval (default=0.05). If this is a float, the returned values are (alpha/2, 1-alpha/2) percentile confidence intervals. If it is an iterable, alpha is assumed to be an iterable of each desired percentile. epsilon: float The step size for finite difference calculations. (default=0.001) Returns ------- confidences: tuple of floats The confidence percentiles specified by alpha References ---------- Efron, An Introduction to the Bootstrap. Chapman & Hall 1993 bootstrap R package: http://cran.r-project.org/web/packages/bootstrap/ """ return ci(data, statfunction=lambda x,weights: stat(x,weights), alpha=alpha, epsilon=epsilon, method='abc') def bootstrap_indexes(data, n_samples=10000): """ Given data points data, where axis 0 is considered to delineate points, return an generator for sets of bootstrap indexes. This can be used as a list of bootstrap indexes (with list(bootstrap_indexes(data))) as well. """ for _ in range(n_samples): yield randint(data.shape[0], size=(data.shape[0],)) def jackknife_indexes(data): """ Given data points data, where axis 0 is considered to delineate points, return a list of arrays where each array is a set of jackknife indexes. For a given set of data Y, the jackknife sample J[i] is defined as the data set Y with the ith data point deleted. """ base = np.arange(0,len(data)) return (np.delete(base,i) for i in base) def subsample_indexes(data, n_samples=1000, size=0.5): """ Given data points data, where axis 0 is considered to delineate points, return a list of arrays where each array is indexes a subsample of the data of size ``size``. If size is >= 1, then it will be taken to be an absolute size. If size < 1, it will be taken to be a fraction of the data size. If size == -1, it will be taken to mean subsamples the same size as the sample (ie, permuted samples) """ if size == -1: size = len(data) elif (size < 1) and (size > 0): size = round(size*len(data)) elif size > 1: pass else: raise ValueError("size cannot be {0}".format(size)) base = np.tile(np.arange(len(data)),(n_samples,1)) for sample in base: np.random.shuffle(sample) return base[:,0:size]