#!/usr/bin/env python # ------------------------------------------------------------------------- # Name: kdetools # Purpose: # # Author: pab # # Created: 01.11.2008 # Copyright: (c) pab 2008 # Licence: LGPL # ------------------------------------------------------------------------- from __future__ import absolute_import, division # from abc import ABCMeta, abstractmethod import copy import warnings import numpy as np import scipy.stats from scipy import interpolate, linalg, special from numpy import sqrt, atleast_2d, meshgrid from numpy.fft import fftn, ifftn from wafo.misc import nextpow2 from wafo.containers import PlotData from wafo.plotbackend import plotbackend as plt from wafo.testing import test_docstrings from wafo.kdetools.kernels import iqrange, qlevels, Kernel from wafo.kdetools.gridding import gridcount try: from wafo import fig except ImportError: warnings.warn('fig import only supported on Windows') __all__ = ['TKDE', 'KDE', 'kde_demo1', 'kde_demo2', 'test_docstrings', 'KRegression', 'BKRegression'] _TINY = np.finfo(float).machar.tiny # _REALMIN = np.finfo(float).machar.xmin _REALMAX = np.finfo(float).machar.xmax _EPS = np.finfo(float).eps def _assert(cond, msg): if not cond: raise ValueError(msg) def _assert_warn(cond, msg): if not cond: warnings.warn(msg) def _invnorm(q): return special.ndtri(q) def _logit(p): pc = p.clip(min=0, max=1) return (np.log(pc) - np.log1p(-pc)).clip(min=-40, max=40) # def _logitinv(x): # return 1.0 / (np.exp(-x) + 1) class _KDE(object): def __init__(self, data, kernel=None, xmin=None, xmax=None): self.dataset = data self.xmin = xmin self.xmax = xmax self.kernel = kernel if kernel else Kernel('gauss') @property def dataset(self): return self._dataset @dataset.setter def dataset(self, data): self._dataset = atleast_2d(data) @property def n(self): return self.dataset.shape[1] @property def d(self): return self.dataset.shape[0] @property def sigma(self): """minimum(stdev, 0.75 * interquartile-range)""" iqr = iqrange(self.dataset, axis=-1) sigma = np.minimum(np.std(self.dataset, axis=-1, ddof=1), iqr / 1.34) return sigma @property def xmin(self): return self._xmin @xmin.setter def xmin(self, xmin): if xmin is None: xmin = self.dataset.min(axis=-1) - 2 * self.sigma self._xmin = self._check_xmin(xmin*np.ones(self.d)) def _check_xmin(self, xmin): return xmin @property def xmax(self): return self._xmax @xmax.setter def xmax(self, xmax): if xmax is None: xmax = self.dataset.max(axis=-1) + 2 * self.sigma self._xmax = self._check_xmax(xmax * np.ones(self.d)) def _check_xmax(self, xmax): return xmax def get_args(self, xmin=None, xmax=None): sxmin = self.xmin if xmin is not None: sxmin = np.minimum(xmin, sxmin) sxmax = self.xmax if xmax is not None: sxmax = np.maximum(xmax, sxmax) args = [] inc = self.inc for i in range(self.d): args.append(np.linspace(sxmin[i], sxmax[i], inc)) return args def eval_grid_fast(self, *args, **kwds): """Evaluate the estimated pdf on a grid using fft. Parameters ---------- arg_0,arg_1,... arg_d-1 : vectors Alternatively, if no vectors is passed in then arg_i = linspace(self.xmin[i], self.xmax[i], self.inc) output : string optional 'value' if value output 'data' if object output Returns ------- values : array-like The values evaluated at meshgrid(*args). """ return self.eval_grid_fun(self._eval_grid_fast, *args, **kwds) def _eval_grid_fast(self, *args, **kwds): pass def eval_grid(self, *args, **kwds): """Evaluate the estimated pdf on a grid. Parameters ---------- arg_0,arg_1,... arg_d-1 : vectors Alternatively, if no vectors is passed in then arg_i = linspace(self.xmin[i], self.xmax[i], self.inc) output : string optional 'value' if value output 'data' if object output Returns ------- values : array-like The values evaluated at meshgrid(*args). """ return self.eval_grid_fun(self._eval_grid, *args, **kwds) def _eval_grid(self, *args, **kwds): pass def _add_contour_levels(self, wdata): p_levels = np.r_[10:90:20, 95, 99, 99.9] try: c_levels = qlevels(wdata.data, p=p_levels) wdata.clevels = c_levels wdata.plevels = p_levels except Exception as e: msg = "Could not calculate contour levels!. ({})".format(str(e)) warnings.warn(msg) def _make_object(self, f, **kwds): titlestr = 'Kernel density estimate ({})'.format(self.kernel.name) kwds2 = dict(title=titlestr) kwds2['plot_kwds'] = dict(plotflag=1) kwds2.update(**kwds) args = self.args if self.d == 1: args = args[0] wdata = PlotData(f, args, **kwds2) if self.d > 1: self._add_contour_levels(wdata) return wdata def eval_grid_fun(self, eval_grd, *args, **kwds): if len(args) == 0: args = self.get_args() self.args = args output = kwds.pop('output', 'value') f = eval_grd(*args, **kwds) if output == 'value': return f return self._make_object(f, **kwds) def _check_shape(self, points): points = atleast_2d(points) d, m = points.shape if d != self.d: _assert(d == 1 and m == self.d, "points have dimension {}, " "dataset has dimension {}".format(d, self.d)) # points was passed in as a row vector points = np.reshape(points, (self.d, 1)) return points def eval_points(self, points, **kwds): """Evaluate the estimated pdf on a set of points. Parameters ---------- points : (# of dimensions, # of points)-array Alternatively, a (# of dimensions,) vector can be passed in and treated as a single point. Returns ------- values : (# of points,)-array The values at each point. Raises ------ ValueError if the dimensionality of the input points is different than the dimensionality of the KDE. """ points = self._check_shape(points) return self._eval_points(points, **kwds) def _eval_points(self, points, **kwds): pass __call__ = eval_grid class TKDE(_KDE): """ Transformation Kernel-Density Estimator. Parameters ---------- dataset : (# of dims, # of data)-array datapoints to estimate from hs : array-like (optional) smooting parameter vector/matrix. (default compute from data using kernel.get_smoothing function) kernel : kernel function object. kernel must have get_smoothing method alpha : real scalar (optional) sensitivity parameter (default 0 regular KDE) A good choice might be alpha = 0.5 ( or 1/D) alpha = 0 Regular KDE (hs is constant) 0 < alpha <= 1 Adaptive KDE (Make hs change) xmin, xmax : vectors specifying the default argument range for the kde.eval_grid methods. For the kde.eval_grid_fast methods the values must cover the range of the data. (default min(data)-range(data)/4, max(data)-range(data)/4) If a single value of xmin or xmax is given then the boundary is the is the same for all dimensions. inc : scalar integer defining the default dimension of the output from kde.eval_grid methods (default 512) (For kde.eval_grid_fast: A value below 50 is very fast to compute but may give some inaccuracies. Values between 100 and 500 give very accurate results) L2 : array-like vector of transformation parameters (default 1 no transformation) t(xi;L2) = xi^L2*sign(L2) for L2(i) ~= 0 t(xi;L2) = log(xi) for L2(i) == 0 If single value of L2 is given then the transformation is the same in all directions. Members ------- d : int number of dimensions n : int number of datapoints Methods ------- kde.eval_grid_fast(x0, x1,..., xd) : array evaluate the estimated pdf on meshgrid(x0, x1,..., xd) kde.eval_grid(x0, x1,..., xd) : array evaluate the estimated pdf on meshgrid(x0, x1,..., xd) kde.eval_points(points) : array evaluate the estimated pdf on a provided set of points kde(x0, x1,..., xd) : array same as kde.eval_grid(x0, x1,..., xd) Example ------- N = 20 data = np.random.rayleigh(1, size=(N,)) >>> data = np.array([ ... 0.75355792, 0.72779194, 0.94149169, 0.07841119,2.32291887, ... 1.10419995, 0.77055114, 0.60288273, 1.36883635, 1.74754326, ... 1.09547561, 1.01671133, 0.73211143, 0.61891719, 0.75903487, ... 1.8919469 , 0.72433808, 1.92973094, 0.44749838, 1.36508452]) >>> import wafo.kdetools as wk >>> x = np.linspace(0.01, max(data.ravel()) + 1, 10) >>> kde = wk.TKDE(data, hs=0.5, L2=0.5) >>> f = kde(x) >>> f array([ 1.03982714, 0.45839018, 0.39514782, 0.32860602, 0.26433318, 0.20717946, 0.15907684, 0.1201074 , 0.08941027, 0.06574882]) >>> kde.eval_grid(x) array([ 1.03982714, 0.45839018, 0.39514782, 0.32860602, 0.26433318, 0.20717946, 0.15907684, 0.1201074 , 0.08941027, 0.06574882]) >>> kde.eval_grid_fast(x) array([ 1.04018924, 0.45838973, 0.39514689, 0.32860532, 0.26433301, 0.20717976, 0.15907697, 0.1201077 , 0.08941129, 0.06574899]) import pylab as plb h1 = plb.plot(x, f) # 1D probability density plot t = np.trapz(f, x) """ def __init__(self, data, hs=None, kernel=None, alpha=0.0, xmin=None, xmax=None, inc=512, L2=None): self.L2 = L2 super(TKDE, self).__init__(data, kernel, xmin, xmax) tdataset = self._transform(self.dataset) txmin = np.ravel(self._transform(np.reshape(self.xmin, (-1, 1)))) txmax = np.ravel(self._transform(np.reshape(self.xmax, (-1, 1)))) self.tkde = KDE(tdataset, hs, self.kernel, alpha, txmin, txmax, inc) def _check_xmin(self, xmin): if self.L2 is not None: amin = self.dataset.min(axis=-1) L2 = np.atleast_1d(self.L2) * np.ones(self.d) xmin = np.where(L2 != 1, np.maximum(xmin, amin / 100.0), xmin) return xmin @property def inc(self): return self.tkde.inc @inc.setter def inc(self, inc): self.tkde.inc = inc @property def hs(self): return self.tkde.hs @hs.setter def hs(self, hs): self.tkde.hs = hs def _transform(self, points): if self.L2 is None: return points # default no transformation L2 = np.atleast_1d(self.L2) * np.ones(self.d) tpoints = copy.copy(points) for i, v2 in enumerate(L2.tolist()): tpoints[i] = np.log(points[i]) if v2 == 0 else points[i] ** v2 return tpoints def _inverse_transform(self, tpoints): if self.L2 is None: return tpoints # default no transformation L2 = np.atleast_1d(self.L2) * np.ones(self.d) points = copy.copy(tpoints) for i, v2 in enumerate(L2.tolist()): points[i] = np.exp( tpoints[i]) if v2 == 0 else tpoints[i] ** (1.0 / v2) return points def _scale_pdf(self, pdf, points): if self.L2 is None: return pdf # default no transformation L2 = np.atleast_1d(self.L2) * np.ones(self.d) for i, v2 in enumerate(L2.tolist()): factor = v2 * np.sign(v2) if v2 else 1 pdf *= np.where(v2 == 1, 1, points[i] ** (v2 - 1) * factor) if (np.abs(np.diff(pdf)).max() > 10).any(): msg = ''' Numerical problems may have occured due to the power transformation. Check the KDE for spurious spikes''' warnings.warn(msg) return pdf def eval_grid_fast2(self, *args, **kwds): """Evaluate the estimated pdf on a grid. Parameters ---------- arg_0,arg_1,... arg_d-1 : vectors Alternatively, if no vectors is passed in then arg_i = gauss2dat(linspace(dat2gauss(self.xmin[i]), dat2gauss(self.xmax[i]), self.inc)) output : string optional 'value' if value output 'data' if object output Returns ------- values : array-like The values evaluated at meshgrid(*args). """ return self.eval_grid_fun(self._eval_grid_fast, *args, **kwds) def _interpolate(self, points, f, *args, **kwds): ipoints = meshgrid(*args) # if self.d > 1 else args for i in range(self.d): points[i].shape = -1, points = np.asarray(points).T fi = interpolate.griddata(points, np.ravel(f), tuple(ipoints), method='linear', fill_value=0.0) self.args = args r = kwds.get('r', 0) if r == 0: return fi * (fi > 0) return fi def _get_targs(self, args): targs = [] if len(args): targs0 = self._transform(list(args)) xmin = [min(t) for t in targs0] xmax = [max(t) for t in targs0] targs = self.tkde.get_args(xmin, xmax) return targs def _eval_grid_fast(self, *args, **kwds): if self.L2 is None: f = self.tkde.eval_grid_fast(*args, **kwds) self.args = self.tkde.args return f targs = self._get_targs(args) tf = self.tkde.eval_grid_fast(*targs) self.args = self._inverse_transform(list(self.tkde.args)) points = meshgrid(*self.args) f = self._scale_pdf(tf, points) if len(args): return self._interpolate(points, f, *args, **kwds) return f def _eval_grid(self, *args, **kwds): if self.L2 is None: return self.tkde.eval_grid(*args, **kwds) targs = self._transform(list(args)) tf = self.tkde.eval_grid(*targs, **kwds) points = meshgrid(*args) f = self._scale_pdf(tf, points) return f def _eval_points(self, points): """Evaluate the estimated pdf on a set of points. Parameters ---------- points : (# of dimensions, # of points)-array Alternatively, a (# of dimensions,) vector can be passed in and treated as a single point. Returns ------- values : (# of points,)-array The values at each point. Raises ------ ValueError if the dimensionality of the input points is different than the dimensionality of the KDE. """ if self.L2 is None: return self.tkde.eval_points(points) tpoints = self._transform(points) tf = self.tkde.eval_points(tpoints) f = self._scale_pdf(tf, points) return f class KDE(_KDE): """ Kernel-Density Estimator. Parameters ---------- data : (# of dims, # of data)-array datapoints to estimate from hs : array-like (optional) smooting parameter vector/matrix. (default compute from data using kernel.get_smoothing function) kernel : kernel function object. kernel must have get_smoothing method alpha : real scalar (optional) sensitivity parameter (default 0 regular KDE) A good choice might be alpha = 0.5 ( or 1/D) alpha = 0 Regular KDE (hs is constant) 0 < alpha <= 1 Adaptive KDE (Make hs change) xmin, xmax : vectors specifying the default argument range for the kde.eval_grid methods. For the kde.eval_grid_fast methods the values must cover the range of the data. (default min(data)-range(data)/4, max(data)-range(data)/4) If a single value of xmin or xmax is given then the boundary is the is the same for all dimensions. inc : scalar integer (default 512) defining the default dimension of the output from kde.eval_grid methods (For kde.eval_grid_fast: A value below 50 is very fast to compute but may give some inaccuracies. Values between 100 and 500 give very accurate results) Members ------- d : int number of dimensions n : int number of datapoints Methods ------- kde.eval_grid_fast(x0, x1,..., xd) : array evaluate the estimated pdf on meshgrid(x0, x1,..., xd) kde.eval_grid(x0, x1,..., xd) : array evaluate the estimated pdf on meshgrid(x0, x1,..., xd) kde.eval_points(points) : array evaluate the estimated pdf on a provided set of points kde(x0, x1,..., xd) : array same as kde.eval_grid(x0, x1,..., xd) Example ------- N = 20 data = np.random.rayleigh(1, size=(N,)) >>> data = np.array([ ... 0.75355792, 0.72779194, 0.94149169, 0.07841119, 2.32291887, ... 1.10419995, 0.77055114, 0.60288273, 1.36883635, 1.74754326, ... 1.09547561, 1.01671133, 0.73211143, 0.61891719, 0.75903487, ... 1.8919469 , 0.72433808, 1.92973094, 0.44749838, 1.36508452]) >>> x = np.linspace(0, max(data.ravel()) + 1, 10) >>> import wafo.kdetools as wk >>> kde = wk.KDE(data, hs=0.5, alpha=0.5) >>> f = kde(x) >>> f array([ 0.17252055, 0.41014271, 0.61349072, 0.57023834, 0.37198073, 0.21409279, 0.12738463, 0.07460326, 0.03956191, 0.01887164]) >>> kde.eval_grid(x) array([ 0.17252055, 0.41014271, 0.61349072, 0.57023834, 0.37198073, 0.21409279, 0.12738463, 0.07460326, 0.03956191, 0.01887164]) >>> kde.eval_grid_fast(x) array([ 0.20729484, 0.39865044, 0.53716945, 0.5169322 , 0.39060223, 0.26441126, 0.16388801, 0.08388527, 0.03227164, 0.00883579]) >>> kde0 = wk.KDE(data, hs=0.5, alpha=0.0) >>> kde0.eval_points(x) array([ 0.2039735 , 0.40252503, 0.54595078, 0.52219649, 0.3906213 , 0.26381501, 0.16407362, 0.08270612, 0.02991145, 0.00720821]) >>> kde0.eval_grid(x) array([ 0.2039735 , 0.40252503, 0.54595078, 0.52219649, 0.3906213 , 0.26381501, 0.16407362, 0.08270612, 0.02991145, 0.00720821]) >>> f = kde0.eval_grid(x, output='plotobj') >>> f.data array([ 0.2039735 , 0.40252503, 0.54595078, 0.52219649, 0.3906213 , 0.26381501, 0.16407362, 0.08270612, 0.02991145, 0.00720821]) >>> f = kde0.eval_grid_fast() >>> np.allclose(np.interp(x, kde0.args[0], f), ... [ 0.20398034, 0.40252166, 0.54593292, 0.52218993, 0.39062245, ... 0.26381651, 0.16407487, 0.08270847, 0.02991439, 0.00882095]) True >>> f1 = kde0.eval_grid_fast(output='plot') >>> np.allclose(np.interp(x, f1.args, f1.data), ... [ 0.20398034, 0.40252166, 0.54593292, 0.52218993, 0.39062245, ... 0.26381651, 0.16407487, 0.08270847, 0.02991439, 0.00882095]) True h = f1.plot() import pylab as plb h1 = plb.plot(x, f) # 1D probability density plot t = np.trapz(f, x) """ def __init__(self, data, hs=None, kernel=None, alpha=0.0, xmin=None, xmax=None, inc=512): super(KDE, self).__init__(data, kernel, xmin, xmax) self.hs = hs self.inc = inc self.alpha = alpha def _replace_negatives_with_default_hs(self, h): get_default_hs = self.kernel.get_smoothing ind, = np.where(h <= 0) for i in ind.tolist(): h[i] = get_default_hs(self.dataset[i]) def _check_hs(self, h): """make sure it has the correct dimension and replace negative vals""" h = np.atleast_1d(h) if (len(h.shape) == 1) or (self.d == 1): h = h * np.ones(self.d) if max(h.shape) == 1 else h.reshape(self.d) self._replace_negatives_with_default_hs(h) return h def _invert_hs(self, h): if (len(h.shape) == 1) or (self.d == 1): determinant = h.prod() inv_hs = np.diag(1.0 / h) else: # fully general smoothing matrix determinant = linalg.det(h) _assert(0 < determinant, 'bandwidth matrix h must be positive definit!') inv_hs = linalg.inv(h) return inv_hs, determinant @property def hs(self): return self._hs @hs.setter def hs(self, h): if h is None: h = self.kernel.get_smoothing(self.dataset) h = self._check_hs(h) self._inv_hs, deth = self._invert_hs(h) self._norm_factor = deth * self.n self._hs = h @property def inc(self): return self._inc @inc.setter def inc(self, inc): if inc is None: _tau, tau = self.kernel.effective_support() xyzrange = 8 * self.sigma L1 = 10 inc = max(48, (L1 * xyzrange / (tau * self.hs)).max()) inc = 2 ** nextpow2(inc) self._inc = inc @property def alpha(self): return self._alpha @alpha.setter def alpha(self, alpha): self._alpha = alpha self._lambda = np.ones(self.n) if alpha > 0: f = self.eval_points(self.dataset) # pilot estimate g = np.exp(np.mean(np.log(f))) self._lambda = (f / g) ** (-alpha) @staticmethod def _make_flat_grid(dx, d, inc): Xn = [] x0 = np.linspace(-inc, inc, 2 * inc + 1) for i in range(d): Xn.append(x0[:-1] * dx[i]) Xnc = meshgrid(*Xn) for i in range(d): Xnc[i].shape = (-1,) return np.vstack(Xnc) def _kernel_weights(self, Xn, dx, d, inc): kw = self.kernel(Xn) norm_fact0 = (kw.sum() * dx.prod() * self.n) norm_fact = (self._norm_factor * self.kernel.norm_factor(d, self.n)) if np.abs(norm_fact0 - norm_fact) > 0.05 * norm_fact: warnings.warn( 'Numerical inaccuracy due to too low discretization. ' + 'Increase the discretization of the evaluation grid ' + '(inc={})!'.format(inc)) norm_fact = norm_fact0 kw = kw / norm_fact return kw def _eval_grid_fast(self, *args, **kwds): X = np.vstack(args) d, inc = X.shape dx = X[:, 1] - X[:, 0] Xnc = self._make_flat_grid(dx, d, inc) Xn = np.dot(self._inv_hs, Xnc) kw = self._kernel_weights(Xn, dx, d, inc) r = kwds.get('r', 0) if r != 0: fun = self._moment_fun(r) kw *= fun(np.vstack(Xnc)) kw.shape = (2 * inc, ) * d kw = np.fft.ifftshift(kw) y = kwds.get('y', 1.0) if self.alpha > 0: warnings.warn('alpha parameter is not used for binned kde!') # Find the binned kernel weights, c. c = gridcount(self.dataset, X, y=y) # Perform the convolution. z = np.real(ifftn(fftn(c, s=kw.shape) * fftn(kw))) ix = (slice(0, inc),) * d if r == 0: return z[ix] * (z[ix] > 0.0) return z[ix] def _eval_grid(self, *args, **kwds): grd = meshgrid(*args) shape0 = grd[0].shape d = len(grd) for i in range(d): grd[i] = grd[i].ravel() f = self.eval_points(np.vstack(grd), **kwds) return f.reshape(shape0) def _moment_fun(self, r): if r == 0: return lambda x: 1 return lambda x: (x ** r).sum(axis=0) @property def norm_factor(self): return self._norm_factor * self.kernel.norm_factor(self.d, self.n) def _loop_over_data(self, data, points, y, r): fun = self._moment_fun(r) d, m = points.shape inv_hs, lambda_ = self._inv_hs, self._lambda kernel = self.kernel y_d_lambda = y / lambda_ ** d result = np.zeros((m,)) for i in range(self.n): dxi = points - data[:, i, np.newaxis] tdiff = np.dot(inv_hs / lambda_[i], dxi) result += fun(dxi) * kernel(tdiff) * y_d_lambda[i] return result / self.norm_factor def _loop_over_points(self, data, points, y, r): fun = self._moment_fun(r) d, m = points.shape inv_hs, lambda_ = self._inv_hs, self._lambda kernel = self.kernel y_d_lambda = y / lambda_ ** d result = np.zeros((m,)) for i in range(m): dxi = points[:, i, np.newaxis] - data tdiff = np.dot(inv_hs, dxi / lambda_[np.newaxis, :]) result[i] = np.sum(fun(dxi) * kernel(tdiff) * y_d_lambda, axis=-1) return result / self.norm_factor def _eval_points(self, points, **kwds): """Evaluate the estimated pdf on a set of points. Parameters ---------- points : (# of dimensions, # of points)-array Alternatively, a (# of dimensions,) vector can be passed in and treated as a single point. Returns ------- values : (# of points,)-array The values at each point. Raises ------ ValueError if the dimensionality of the input points is different than the dimensionality of the KDE. """ d, m = points.shape _assert(d == self.d, "d={} expected, got {}".format(self.d, d)) y = kwds.get('y', 1) r = kwds.get('r', 0) more_points_than_data = m >= self.n if more_points_than_data: return self._loop_over_data(self.dataset, points, y, r) return self._loop_over_points(self.dataset, points, y, r) class KRegression(object): """ Kernel-Regression Parameters ---------- data : (# of dims, # of data)-array datapoints to estimate from y : # of data - array response variable p : scalar integer (0 or 1) Nadaraya-Watson estimator if p=0, local linear estimator if p=1. hs : array-like (optional) smooting parameter vector/matrix. (default compute from data using kernel.get_smoothing function) kernel : kernel function object. kernel must have get_smoothing method alpha : real scalar (optional) sensitivity parameter (default 0 regular KDE) A good choice might be alpha = 0.5 ( or 1/D) alpha = 0 Regular KDE (hs is constant) 0 < alpha <= 1 Adaptive KDE (Make hs change) xmin, xmax : vectors specifying the default argument range for the kde.eval_grid methods. For the kde.eval_grid_fast methods the values must cover the range of the data. (default min(data)-range(data)/4, max(data)-range(data)/4) If a single value of xmin or xmax is given then the boundary is the is the same for all dimensions. inc : scalar integer (default 128) defining the default dimension of the output from kde.eval_grid methods (For kde.eval_grid_fast: A value below 50 is very fast to compute but may give some inaccuracies. Values between 100 and 500 give very accurate results) Members ------- d : int number of dimensions n : int number of datapoints Methods ------- kde.eval_grid_fast(x0, x1,..., xd) : array evaluate the estimated pdf on meshgrid(x0, x1,..., xd) kde.eval_grid(x0, x1,..., xd) : array evaluate the estimated pdf on meshgrid(x0, x1,..., xd) kde.eval_points(points) : array evaluate the estimated pdf on a provided set of points kde(x0, x1,..., xd) : array same as kde.eval_grid(x0, x1,..., xd) Example ------- >>> import wafo.kdetools as wk >>> N = 100 >>> x = np.linspace(0, 1, N) >>> ei = np.random.normal(loc=0, scale=0.075, size=(N,)) >>> ei = np.sqrt(0.075) * np.sin(100*x) >>> y = 2*np.exp(-x**2/(2*0.3**2))+3*np.exp(-(x-1)**2/(2*0.7**2)) + ei >>> kreg = wk.KRegression(x, y) >>> f = kreg(output='plotobj', title='Kernel regression', plotflag=1) >>> np.allclose(f.data[:5], ... [ 3.18670593, 3.18678088, 3.18682196, 3.18682932, 3.18680337]) True h = f.plot(label='p=0') """ def __init__(self, data, y, p=0, hs=None, kernel=None, alpha=0.0, xmin=None, xmax=None, inc=128, L2=None): self.tkde = TKDE(data, hs=hs, kernel=kernel, alpha=alpha, xmin=xmin, xmax=xmax, inc=inc, L2=L2) self.y = np.atleast_1d(y) self.p = p def eval_grid_fast(self, *args, **kwds): self._grdfun = self.tkde.eval_grid_fast return self.tkde.eval_grid_fun(self._eval_gridfun, *args, **kwds) def eval_grid(self, *args, **kwds): self._grdfun = self.tkde.eval_grid return self.tkde.eval_grid_fun(self._eval_gridfun, *args, **kwds) def _eval_gridfun(self, *args, **kwds): grdfun = self._grdfun s0 = grdfun(*args, r=0) t0 = grdfun(*args, r=0, y=self.y) if self.p == 0: return (t0 / (s0 + _TINY)).clip(min=-_REALMAX, max=_REALMAX) elif self.p == 1: s1 = grdfun(*args, r=1) s2 = grdfun(*args, r=2) t1 = grdfun(*args, r=1, y=self.y) return ((s2 * t0 - s1 * t1) / (s2 * s0 - s1 ** 2)).clip(min=-_REALMAX, max=_REALMAX) __call__ = eval_grid_fast class BKRegression(object): ''' Kernel-Regression on binomial data method : {'beta', 'wilson'} method is one of the following 'beta', return Bayesian Credible interval using beta-distribution. 'wilson', return Wilson score interval a, b : scalars parameters of the beta distribution defining the apriori distribution of p, i.e., the Bayes estimator for p: p = (y+a)/(n+a+b). Setting a=b=0.5 gives Jeffreys interval. ''' def __init__(self, data, y, method='beta', a=0.05, b=0.05, p=0, hs_e=None, hs=None, kernel=None, alpha=0.0, xmin=None, xmax=None, inc=128, L2=None): self.method = method self.a = max(a, _TINY) self.b = max(b, _TINY) self.kreg = KRegression(data, y, p=p, hs=hs, kernel=kernel, alpha=alpha, xmin=xmin, xmax=xmax, inc=inc, L2=L2) # defines bin width (i.e. smoothing) in empirical estimate self.hs_e = hs_e @property def hs_e(self): return self._hs_e @hs_e.setter def hs_e(self, hs_e): if hs_e is None: hs1 = self._get_max_smoothing('hste')[0] hs2 = self._get_max_smoothing('hos')[0] hs_e = sqrt(hs1 * hs2) self._hs_e = hs_e def _set_smoothing(self, hs): self.kreg.tkde.hs = hs x = property(fget=lambda cls: cls.kreg.tkde.dataset.squeeze()) y = property(fget=lambda cls: cls.kreg.y) kernel = property(fget=lambda cls: cls.kreg.tkde.kernel) hs = property(fset=_set_smoothing, fget=lambda cls: cls.kreg.tkde.hs) def _get_max_smoothing(self, fun=None): """Return maximum value for smoothing parameter.""" x = self.x y = self.y if fun is None: get_smoothing = self.kernel.get_smoothing else: get_smoothing = getattr(self.kernel, fun) hs1 = get_smoothing(x) # hx = np.median(np.abs(x-np.median(x)))/0.6745*(4.0/(3*n))**0.2 if (y == 1).any(): hs2 = get_smoothing(x[y == 1]) # hy = np.median(np.abs(y-np.mean(y)))/0.6745*(4.0/(3*n))**0.2 else: hs2 = 4 * hs1 # hy = 4*hx hopt = sqrt(hs1 * hs2) return hopt, hs1, hs2 def get_grid(self, hs_e=None): if hs_e is None: hs_e = self.hs_e x = self.x xmin, xmax = x.min(), x.max() ni = max(2 * int((xmax - xmin) / hs_e) + 3, 5) sml = hs_e # *0.1 xi = np.linspace(xmin - sml, xmax + sml, ni) return xi def _wilson_score(self, n, p, alpha): # Wilson score z0 = -_invnorm(alpha / 2) den = 1 + (z0 ** 2. / n) xc = (p + (z0 ** 2) / (2 * n)) / den halfwidth = (z0 * sqrt((p * (1 - p) / n) + (z0 ** 2 / (4 * (n ** 2))))) / den plo = xc - halfwidth.clip(min=0) # wilson score pup = xc + halfwidth.clip(max=1.0) # wilson score return plo, pup def _credible_interval(self, n, p, alpha): # Jeffreys intervall a=b=0.5 # st.beta.isf(alpha/2, y+a, n-y+b) y = n*p, n-y = n*(1-p) a, b = self.a, self.b st = scipy.stats pup = np.where(p == 1, 1, st.beta.isf(alpha / 2, n * p + a, n * (1 - p) + b)) plo = np.where(p == 0, 0, st.beta.isf(1 - alpha / 2, n * p + a, n * (1 - p) + b)) return plo, pup def prb_ci(self, n, p, alpha=0.05): """Return Confidence Interval for the binomial probability p. Parameters ---------- n : array-like number of Bernoulli trials p : array-like estimated probability of success in each trial alpha : scalar confidence level """ if self.method.startswith('w'): return self._wilson_score(n, p, alpha) return self._credible_interval(n, p, alpha) def prb_empirical(self, xi=None, hs_e=None, alpha=0.05, color='r', **kwds): """Returns empirical binomial probabiltity. Parameters ---------- x : ndarray position vector y : ndarray binomial response variable (zeros and ones) alpha : scalar confidence level color: used in plot Returns ------- P(x) : PlotData object empirical probability """ if xi is None: xi = self.get_grid(hs_e) x = self.x y = self.y c = gridcount(x, xi) # + self.a + self.b # count data if np.any(y == 1): c0 = gridcount(x[y == 1], xi) # + self.a # count success else: c0 = np.zeros(np.shape(xi)) prb = np.where(c == 0, 0, c0 / (c + _TINY)) # assume prb==0 for c==0 CI = np.vstack(self.prb_ci(c, prb, alpha)) prb_e = PlotData(prb, xi, plotmethod='plot', plot_args=['.'], plot_kwds=dict(markersize=6, color=color, picker=5)) prb_e.dataCI = CI.T prb_e.count = c return prb_e def prb_smoothed(self, prb_e, hs, alpha=0.05, color='r', label=''): """Return smoothed binomial probability. Parameters ---------- prb_e : PlotData object with empirical binomial probabilites hs : smoothing parameter alpha : confidence level color : color of plot object label : label for plot object """ x_e = prb_e.args n_e = len(x_e) dx_e = x_e[1] - x_e[0] n = self.x.size x_s = np.linspace(x_e[0], x_e[-1], 10 * n_e + 1) self.hs = hs prb_s = self.kreg(x_s, output='plotobj', title='', plot_kwds=dict( color=color, linewidth=2)) # dict(plotflag=7)) m_nan = np.isnan(prb_s.data) if m_nan.any(): # assume 0/0 division prb_s.data[m_nan] = 0.0 # prb_s.data[np.isnan(prb_s.data)] = 0 # expected number of data in each bin c_s = self.kreg.tkde.eval_grid_fast(x_s) * dx_e * n plo, pup = self.prb_ci(c_s, prb_s.data, alpha) prb_s.dataCI = np.vstack((plo, pup)).T prb_s.prediction_error_avg = (np.trapz(pup - plo, x_s) / (x_s[-1] - x_s[0])) if label: prb_s.plot_kwds['label'] = label prb_s.children = [PlotData([plo, pup], x_s, plotmethod='fill_between', plot_kwds=dict(alpha=0.2, color=color)), prb_e] p_e = prb_e.eval_points(x_s) p_s = prb_s.data dp_s = np.sign(np.diff(p_s)) k = (dp_s[:-1] != dp_s[1:]).sum() # numpeaks sigmai = _logit(pup) - _logit(plo) + _EPS aicc = ((((_logit(p_e) - _logit(p_s)) / sigmai) ** 2).sum() + 2 * k * (k + 1) / np.maximum(n_e - k + 1, 1) + np.abs((p_e - pup).clip(min=0) - (p_e - plo).clip(max=0)).sum()) prb_s.aicc = aicc return prb_s def prb_search_best(self, prb_e=None, hsvec=None, hsfun='hste', alpha=0.05, color='r', label=''): """Return best smoothed binomial probability. Parameters ---------- prb_e : PlotData object with empirical binomial probabilites hsvec : arraylike (default np.linspace(hsmax*0.1,hsmax,55)) vector smoothing parameters hsfun : method for calculating hsmax """ if prb_e is None: prb_e = self.prb_empirical(alpha=alpha, color=color) if hsvec is None: hsmax = max(self._get_max_smoothing(hsfun)[0], self.hs_e) hsvec = np.linspace(hsmax * 0.2, hsmax, 55) hs_best = hsvec[-1] + 0.1 prb_best = self.prb_smoothed(prb_e, hs_best, alpha, color, label) aicc = np.zeros(np.size(hsvec)) for i, hi in enumerate(hsvec): f = self.prb_smoothed(prb_e, hi, alpha, color, label) aicc[i] = f.aicc if f.aicc <= prb_best.aicc: prb_best = f hs_best = hi prb_best.score = PlotData(aicc, hsvec) prb_best.hs = hs_best self._set_smoothing(hs_best) return prb_best def kde_demo1(): """KDEDEMO1 Demonstrate the smoothing parameter impact on KDE. KDEDEMO1 shows the true density (dotted) compared to KDE based on 7 observations (solid) and their individual kernels (dashed) for 3 different values of the smoothing parameter, hs. """ st = scipy.stats x = np.linspace(-4, 4, 101) x0 = x / 2.0 data = np.random.normal(loc=0, scale=1.0, size=7) kernel = Kernel('gauss') hs = kernel.hns(data) hVec = [hs / 2, hs, 2 * hs] for ix, h in enumerate(hVec): plt.figure(ix) kde = KDE(data, hs=h, kernel=kernel) f2 = kde(x, output='plot', title='h_s = {0:2.2f}'.format(float(h)), ylab='Density') f2.plot('k-') plt.plot(x, st.norm.pdf(x, 0, 1), 'k:') n = len(data) plt.plot(data, np.zeros(data.shape), 'bx') y = kernel(x0) / (n * h * kernel.norm_factor(d=1, n=n)) for i in range(n): plt.plot(data[i] + x0 * h, y, 'b--') plt.plot([data[i], data[i]], [0, np.max(y)], 'b') plt.axis([min(x), max(x), 0, 0.5]) def kde_demo2(): '''Demonstrate the difference between transformation- and ordinary-KDE. KDEDEMO2 shows that the transformation KDE is a better estimate for Rayleigh distributed data around 0 than the ordinary KDE. ''' st = scipy.stats data = st.rayleigh.rvs(scale=1, size=300) x = np.linspace(1.5e-2, 5, 55) kde = KDE(data) f = kde(output='plot', title='Ordinary KDE (hs={0:})'.format(kde.hs)) plt.figure(0) f.plot() plt.plot(x, st.rayleigh.pdf(x, scale=1), ':') # plotnorm((data).^(L2)) # gives a straight line => L2 = 0.5 reasonable hs = Kernel('gauss').get_smoothing(data**0.5) tkde = TKDE(data, hs=hs, L2=0.5) ft = tkde(x, output='plot', title='Transformation KDE (hs={0:})'.format(tkde.tkde.hs)) plt.figure(1) ft.plot() plt.plot(x, st.rayleigh.pdf(x, scale=1), ':') plt.figure(0) def kde_demo3(): '''Demonstrate the difference between transformation and ordinary-KDE in 2D KDEDEMO3 shows that the transformation KDE is a better estimate for Rayleigh distributed data around 0 than the ordinary KDE. ''' st = scipy.stats data = st.rayleigh.rvs(scale=1, size=(2, 300)) # x = np.linspace(1.5e-3, 5, 55) kde = KDE(data) f = kde(output='plot', title='Ordinary KDE', plotflag=1) plt.figure(0) f.plot() plt.plot(data[0], data[1], '.') # plotnorm((data).^(L2)) % gives a straight line => L2 = 0.5 reasonable hs = Kernel('gauss').get_smoothing(data**0.5) tkde = TKDE(data, hs=hs, L2=0.5) ft = tkde.eval_grid_fast( output='plot', title='Transformation KDE', plotflag=1) plt.figure(1) ft.plot() plt.plot(data[0], data[1], '.') plt.figure(0) def kde_demo4(N=50): '''Demonstrate that the improved Sheather-Jones plug-in (hisj) is superior for 1D multimodal distributions KDEDEMO4 shows that the improved Sheather-Jones plug-in smoothing is a better compared to normal reference rules (in this case the hns) ''' st = scipy.stats data = np.hstack((st.norm.rvs(loc=5, scale=1, size=(N,)), st.norm.rvs(loc=-5, scale=1, size=(N,)))) # x = np.linspace(1.5e-3, 5, 55) kde = KDE(data, kernel=Kernel('gauss', 'hns')) f = kde(output='plot', title='Ordinary KDE', plotflag=1) kde1 = KDE(data, kernel=Kernel('gauss', 'hisj')) f1 = kde1(output='plot', label='Ordinary KDE', plotflag=1) plt.figure(0) f.plot('r', label='hns={0}'.format(kde.hs)) # plt.figure(2) f1.plot('b', label='hisj={0}'.format(kde1.hs)) x = np.linspace(-9, 9) plt.plot(x, (st.norm.pdf(x, loc=-5, scale=1) + st.norm.pdf(x, loc=5, scale=1)) / 2, 'k:', label='True density') plt.legend() def kde_demo5(N=500): '''Demonstrate that the improved Sheather-Jones plug-in (hisj) is superior for 2D multimodal distributions KDEDEMO5 shows that the improved Sheather-Jones plug-in smoothing is better compared to normal reference rules (in this case the hns) ''' st = scipy.stats data = np.hstack((st.norm.rvs(loc=5, scale=1, size=(2, N,)), st.norm.rvs(loc=-5, scale=1, size=(2, N,)))) kde = KDE(data, kernel=Kernel('gauss', 'hns')) f = kde(output='plot', plotflag=1, title='Ordinary KDE, hns={0:s}'.format(str(list(kde.hs)))) kde1 = KDE(data, kernel=Kernel('gauss', 'hisj')) f1 = kde1(output='plot', plotflag=1, title='Ordinary KDE, hisj={0:s}'.format(str(list(kde1.hs)))) plt.figure(0) plt.clf() f.plot() plt.plot(data[0], data[1], '.') plt.figure(1) plt.clf() f1.plot() plt.plot(data[0], data[1], '.') def kreg_demo1(hs=None, fast=False, fun='hisj'): """""" N = 100 # ei = np.random.normal(loc=0, scale=0.075, size=(N,)) ei = np.array([ -0.08508516, 0.10462496, 0.07694448, -0.03080661, 0.05777525, 0.06096313, -0.16572389, 0.01838912, -0.06251845, -0.09186784, -0.04304887, -0.13365788, -0.0185279, -0.07289167, 0.02319097, 0.06887854, -0.08938374, -0.15181813, 0.03307712, 0.08523183, -0.0378058, -0.06312874, 0.01485772, 0.06307944, -0.0632959, 0.18963205, 0.0369126, -0.01485447, 0.04037722, 0.0085057, -0.06912903, 0.02073998, 0.1174351, 0.17599277, -0.06842139, 0.12587608, 0.07698113, -0.0032394, -0.12045792, -0.03132877, 0.05047314, 0.02013453, 0.04080741, 0.00158392, 0.10237899, -0.09069682, 0.09242174, -0.15445323, 0.09190278, 0.07138498, 0.03002497, 0.02495252, 0.01286942, 0.06449978, 0.03031802, 0.11754861, -0.02322272, 0.00455867, -0.02132251, 0.09119446, -0.03210086, -0.06509545, 0.07306443, 0.04330647, 0.078111, -0.04146907, 0.05705476, 0.02492201, -0.03200572, -0.02859788, -0.05893749, 0.00089538, 0.0432551, 0.04001474, 0.04888828, -0.17708392, 0.16478644, 0.1171006, 0.11664846, 0.01410477, -0.12458953, -0.11692081, 0.0413047, -0.09292439, -0.07042327, 0.14119701, -0.05114335, 0.04994696, -0.09520663, 0.04829406, -0.01603065, -0.1933216, 0.19352763, 0.11819496, 0.04567619, -0.08348306, 0.00812816, -0.00908206, 0.14528945, 0.02901065]) x = np.linspace(0, 1, N) y0 = 2 * np.exp(-x ** 2 / (2 * 0.3 ** 2)) + \ 3 * np.exp(-(x - 1) ** 2 / (2 * 0.7 ** 2)) y = y0 + ei kernel = Kernel('gauss', fun=fun) hopt = kernel.hisj(x) kreg = KRegression( x, y, p=0, hs=hs, kernel=kernel, xmin=-2 * hopt, xmax=1 + 2 * hopt) if fast: kreg.__call__ = kreg.eval_grid_fast f = kreg(x, output='plot', title='Kernel regression', plotflag=1) plt.figure(0) f.plot(label='p=0') kreg.p = 1 f1 = kreg(x, output='plot', title='Kernel regression', plotflag=1) f1.plot(label='p=1') # print(f1.data) plt.plot(x, y, '.', label='data') plt.plot(x, y0, 'k', label='True model') from statsmodels.nonparametric.kernel_regression import KernelReg kreg2 = KernelReg(y, x, ('c')) y2 = kreg2.fit(x) plt.plot(x, y2[0], 'm', label='statsmodel') plt.legend() plt.show() print(kreg.tkde.tkde._inv_hs) print(kreg.tkde.tkde.hs) def _get_data(n=100, symmetric=False, loc1=1.1, scale1=0.6, scale2=1.0): st = scipy.stats dist = st.norm norm1 = scale2 * (dist.pdf(-loc1, loc=-loc1, scale=scale1) + dist.pdf(-loc1, loc=loc1, scale=scale1)) def fun1(x): return ((dist.pdf(x, loc=-loc1, scale=scale1) + dist.pdf(x, loc=loc1, scale=scale1)) / norm1).clip(max=1.0) x = np.sort(6 * np.random.rand(n, 1) - 3, axis=0) y = (fun1(x) > np.random.rand(n, 1)).ravel() # y = (np.cos(x)>2*np.random.rand(n, 1)-1).ravel() x = x.ravel() if symmetric: xi = np.hstack((x.ravel(), -x.ravel())) yi = np.hstack((y, y)) i = np.argsort(xi) x = xi[i] y = yi[i] return x, y, fun1 def check_bkregression(): plt.ion() k = 0 for _i, n in enumerate([50, 100, 300, 600]): x, y, fun1 = _get_data(n, symmetric=True, loc1=0.1, scale1=0.6, scale2=0.75) bkreg = BKRegression(x, y, a=0.05, b=0.05) fbest = bkreg.prb_search_best( hsfun='hste', alpha=0.05, color='g', label='Transit_D') figk = plt.figure(k) ax = figk.gca() k += 1 # fbest.score.plot(axis=ax) # axsize = ax.axis() # ax.vlines(fbest.hs,axsize[2]+1,axsize[3]) # ax.set(yscale='log') fbest.labels.title = 'N = {:d}'.format(n) fbest.plot(axis=ax) ax.plot(x, fun1(x), 'r') ax.legend(frameon=False, markerscale=4) # ax = plt.gca() ax.set_yticklabels(ax.get_yticks() * 100.0) ax.grid(True) fig.tile(range(0, k)) plt.ioff() plt.show('hold') if __name__ == '__main__': if False: test_docstrings(__file__) else: # kde_demo5() # check_bkregression() kreg_demo1(hs=0.04, fast=True) plt.show('hold')