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1976 lines
63 KiB
Python
1976 lines
63 KiB
Python
#!/usr/bin/env python
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# -------------------------------------------------------------------------
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# Name: kdetools
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# Purpose:
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#
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# Author: pab
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#
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# Created: 01.11.2008
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# Copyright: (c) pab 2008
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# Licence: LGPL
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# -------------------------------------------------------------------------
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from __future__ import absolute_import, division
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# from abc import ABCMeta, abstractmethod
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import copy
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import warnings
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import numpy as np
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import scipy.stats
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from scipy import interpolate, linalg, special
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from numpy import pi, sqrt, atleast_2d, exp, meshgrid
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from numpy.fft import fftn, ifftn
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from wafo.misc import nextpow2
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from wafo.containers import PlotData
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from wafo.dctpack import dctn, idctn # , dstn, idstn
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from wafo.plotbackend import plotbackend as plt
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from wafo.testing import test_docstrings
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from wafo.kdetools.kernels import iqrange, qlevels, Kernel
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from wafo.kdetools.gridding import gridcount
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import time
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try:
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from wafo import fig
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except ImportError:
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warnings.warn('fig import only supported on Windows')
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__all__ = ['TKDE', 'KDE', 'kde_demo1', 'kde_demo2', 'test_docstrings',
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'KRegression', 'BKRegression']
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def _assert(cond, msg):
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if not cond:
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raise ValueError(msg)
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def _assert_warn(cond, msg):
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if not cond:
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warnings.warn(msg)
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def _invnorm(q):
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return special.ndtri(q)
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class _KDE(object):
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""" Kernel-Density Estimator base class.
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Parameters
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----------
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data : (# of dims, # of data)-array
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datapoints to estimate from
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hs : array-like (optional)
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smooting parameter vector/matrix.
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(default compute from data using kernel.get_smoothing function)
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kernel : kernel function object.
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kernel must have get_smoothing method
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alpha : real scalar (optional)
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sensitivity parameter (default 0 regular KDE)
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A good choice might be alpha = 0.5 ( or 1/D)
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alpha = 0 Regular KDE (hs is constant)
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0 < alpha <= 1 Adaptive KDE (Make hs change)
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Members
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-------
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d : int
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number of dimensions
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n : int
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number of datapoints
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Methods
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-------
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kde.eval_grid_fast(x0, x1,..., xd) : array
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evaluate the estimated pdf on meshgrid(x0, x1,..., xd)
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kde.eval_grid(x0, x1,..., xd) : array
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evaluate the estimated pdf on meshgrid(x0, x1,..., xd)
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kde.eval_points(points) : array
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evaluate the estimated pdf on a provided set of points
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kde(x0, x1,..., xd) : array
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same as kde.eval_grid(x0, x1,..., xd)
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"""
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def __init__(self, data, hs=None, kernel=None, alpha=0.0, xmin=None,
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xmax=None, inc=512):
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self.dataset = atleast_2d(data)
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self.kernel = kernel if kernel else Kernel('gauss')
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self.xmin = xmin
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self.xmax = xmax
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self.hs = hs
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self.inc = inc
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self.alpha = alpha
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self.initialize()
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@property
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def n(self):
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return self.dataset.shape[1]
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@property
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def d(self):
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return self.dataset.shape[0]
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@property
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def sigma(self):
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"""minimum(stdev, 0.75 * interquartile-range)"""
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iqr = iqrange(self.dataset, axis=-1)
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sigma = np.minimum(np.std(self.dataset, axis=-1, ddof=1), iqr / 1.34)
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return sigma
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@property
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def xmin(self):
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return self._xmin
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@xmin.setter
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def xmin(self, xmin):
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if xmin is None:
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self._xmin = self.dataset.min(axis=-1) - 2 * self.sigma
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else:
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self._xmin = xmin * np.ones(self.d)
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@property
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def xmax(self):
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return self._xmax
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@xmax.setter
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def xmax(self, xmax):
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if xmax is None:
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self._xmax = self.dataset.max(axis=-1) + 2 * self.sigma
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else:
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self._xmax = xmax * np.ones(self.d)
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def _replace_negatives_with_default_hs(self, h):
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get_default_hs = self.kernel.get_smoothing
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ind, = np.where(h <= 0)
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for i in ind.tolist():
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h[i] = get_default_hs(self.dataset[i])
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def _check_hs(self, h):
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"""make sure it has the correct dimension and replace negative vals"""
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h = np.atleast_1d(h)
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if (len(h.shape) == 1) or (self.d == 1):
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h = h * np.ones(self.d) if max(h.shape) == 1 else h.reshape(self.d)
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self._replace_negatives_with_default_hs(h)
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return h
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def _invert_hs(self, h):
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if (len(h.shape) == 1) or (self.d == 1):
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determinant = h.prod()
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inv_hs = np.diag(1.0 / h)
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else: # fully general smoothing matrix
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determinant = linalg.det(h)
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_assert(0 < determinant,
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'bandwidth matrix h must be positive definit!')
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inv_hs = linalg.inv(h)
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return inv_hs, determinant
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@property
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def hs(self):
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return self._hs
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@hs.setter
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def hs(self, h):
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if h is None:
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h = self.kernel.get_smoothing(self.dataset)
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h = self._check_hs(h)
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inv_hs, deth = self._invert_hs(h)
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self._norm_factor = deth * self.n
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self._inv_hs = inv_hs
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self._hs = h
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@property
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def inc(self):
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return self._inc
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@inc.setter
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def inc(self, inc):
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if inc is None:
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_tau, tau = self.kernel.effective_support()
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xyzrange = 8 * self.sigma
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L1 = 10
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inc = max(48, (L1 * xyzrange / (tau * self.hs)).max())
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inc = 2 ** nextpow2(inc)
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self._inc = inc
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@property
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def alpha(self):
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return self._alpha
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@alpha.setter
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def alpha(self, alpha):
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self._alpha = alpha
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self._lambda = np.ones(self.n)
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if alpha > 0:
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f = self.eval_points(self.dataset) # pilot estimate
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g = np.exp(np.mean(np.log(f)))
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self._lambda = (f / g) ** (-alpha)
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def initialize(self):
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if self.n > 1:
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self._initialize()
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def _initialize(self):
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pass
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def get_args(self, xmin=None, xmax=None):
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sxmin = self.xmin
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if xmin is not None:
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sxmin = np.minimum(xmin, sxmin)
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sxmax = self.xmax
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if xmax is not None:
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sxmax = np.maximum(xmax, sxmax)
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args = []
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inc = self.inc
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for i in range(self.d):
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args.append(np.linspace(sxmin[i], sxmax[i], inc))
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return args
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def eval_grid_fast(self, *args, **kwds):
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"""Evaluate the estimated pdf on a grid.
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Parameters
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----------
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arg_0,arg_1,... arg_d-1 : vectors
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Alternatively, if no vectors is passed in then
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arg_i = linspace(self.xmin[i], self.xmax[i], self.inc)
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output : string optional
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'value' if value output
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'data' if object output
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Returns
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-------
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values : array-like
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The values evaluated at meshgrid(*args).
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"""
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if len(args) == 0:
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args = self.get_args()
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self.args = args
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return self.eval_grid_fun(self._eval_grid_fast, *args, **kwds)
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def _eval_grid_fast(self, *args, **kwds):
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pass
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def eval_grid(self, *args, **kwds):
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"""Evaluate the estimated pdf on a grid.
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Parameters
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----------
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arg_0,arg_1,... arg_d-1 : vectors
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Alternatively, if no vectors is passed in then
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arg_i = linspace(self.xmin[i], self.xmax[i], self.inc)
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output : string optional
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'value' if value output
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'data' if object output
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Returns
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-------
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values : array-like
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The values evaluated at meshgrid(*args).
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"""
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if len(args) == 0:
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args = self.get_args()
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self.args = args
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return self.eval_grid_fun(self._eval_grid, *args, **kwds)
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def _eval_grid(self, *args, **kwds):
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pass
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def _add_contour_levels(self, wdata):
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p_levels = np.r_[10:90:20, 95, 99, 99.9]
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try:
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c_levels = qlevels(wdata.data, p=p_levels)
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wdata.clevels = c_levels
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wdata.plevels = p_levels
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except Exception as e:
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msg = "Could not calculate contour levels!. ({})".format(str(e))
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warnings.warn(msg)
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def _make_object(self, f, **kwds):
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titlestr = 'Kernel density estimate ({})'.format(self.kernel.name)
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kwds2 = dict(title=titlestr)
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kwds2['plot_kwds'] = dict(plotflag=1)
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kwds2.update(**kwds)
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args = self.args
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if self.d == 1:
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args = args[0]
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wdata = PlotData(f, args, **kwds2)
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if self.d > 1:
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self._add_contour_levels(wdata)
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return wdata
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def eval_grid_fun(self, eval_grd, *args, **kwds):
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output = kwds.pop('output', 'value')
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f = eval_grd(*args, **kwds)
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if output == 'value':
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return f
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return self._make_object(f, **kwds)
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def _check_shape(self, points):
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points = atleast_2d(points)
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d, m = points.shape
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if d != self.d:
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_assert(d == 1 and m == self.d, "points have dimension {}, "
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"dataset has dimension {}".format(d, self.d))
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# points was passed in as a row vector
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points = np.reshape(points, (self.d, 1))
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return points
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def eval_points(self, points, **kwds):
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"""Evaluate the estimated pdf on a set of points.
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Parameters
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----------
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points : (# of dimensions, # of points)-array
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Alternatively, a (# of dimensions,) vector can be passed in and
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treated as a single point.
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Returns
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-------
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values : (# of points,)-array
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The values at each point.
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Raises
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------
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ValueError if the dimensionality of the input points is different than
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the dimensionality of the KDE.
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"""
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points = self._check_shape(points)
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return self._eval_points(points, **kwds)
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def _eval_points(self, points, **kwds):
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pass
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__call__ = eval_grid
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class TKDE(_KDE):
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""" Transformation Kernel-Density Estimator.
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Parameters
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----------
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dataset : (# of dims, # of data)-array
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datapoints to estimate from
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hs : array-like (optional)
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smooting parameter vector/matrix.
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(default compute from data using kernel.get_smoothing function)
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kernel : kernel function object.
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kernel must have get_smoothing method
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alpha : real scalar (optional)
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sensitivity parameter (default 0 regular KDE)
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A good choice might be alpha = 0.5 ( or 1/D)
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alpha = 0 Regular KDE (hs is constant)
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0 < alpha <= 1 Adaptive KDE (Make hs change)
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xmin, xmax : vectors
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specifying the default argument range for the kde.eval_grid methods.
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For the kde.eval_grid_fast methods the values must cover the range of
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the data. (default min(data)-range(data)/4, max(data)-range(data)/4)
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If a single value of xmin or xmax is given then the boundary is the is
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the same for all dimensions.
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inc : scalar integer
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defining the default dimension of the output from kde.eval_grid methods
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(default 512)
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(For kde.eval_grid_fast: A value below 50 is very fast to compute but
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may give some inaccuracies. Values between 100 and 500 give very
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accurate results)
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L2 : array-like
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vector of transformation parameters (default 1 no transformation)
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t(xi;L2) = xi^L2*sign(L2) for L2(i) ~= 0
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t(xi;L2) = log(xi) for L2(i) == 0
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If single value of L2 is given then the transformation is the same in
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all directions.
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Members
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-------
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d : int
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number of dimensions
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n : int
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number of datapoints
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Methods
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-------
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kde.eval_grid_fast(x0, x1,..., xd) : array
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evaluate the estimated pdf on meshgrid(x0, x1,..., xd)
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kde.eval_grid(x0, x1,..., xd) : array
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evaluate the estimated pdf on meshgrid(x0, x1,..., xd)
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kde.eval_points(points) : array
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evaluate the estimated pdf on a provided set of points
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kde(x0, x1,..., xd) : array
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same as kde.eval_grid(x0, x1,..., xd)
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Example
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-------
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N = 20
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data = np.random.rayleigh(1, size=(N,))
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>>> data = np.array([
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... 0.75355792, 0.72779194, 0.94149169, 0.07841119,2.32291887,
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... 1.10419995, 0.77055114, 0.60288273, 1.36883635, 1.74754326,
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... 1.09547561, 1.01671133, 0.73211143, 0.61891719, 0.75903487,
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... 1.8919469 , 0.72433808, 1.92973094, 0.44749838, 1.36508452])
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>>> import wafo.kdetools as wk
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>>> x = np.linspace(0.01, max(data.ravel()) + 1, 10)
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>>> kde = wk.TKDE(data, hs=0.5, L2=0.5)
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>>> f = kde(x)
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>>> f
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array([ 1.03982714, 0.45839018, 0.39514782, 0.32860602, 0.26433318,
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0.20717946, 0.15907684, 0.1201074 , 0.08941027, 0.06574882])
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>>> kde.eval_grid(x)
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array([ 1.03982714, 0.45839018, 0.39514782, 0.32860602, 0.26433318,
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0.20717946, 0.15907684, 0.1201074 , 0.08941027, 0.06574882])
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>>> kde.eval_grid_fast(x)
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array([ 1.04018924, 0.45838973, 0.39514689, 0.32860532, 0.26433301,
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0.20717976, 0.15907697, 0.1201077 , 0.08941129, 0.06574899])
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import pylab as plb
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h1 = plb.plot(x, f) # 1D probability density plot
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t = np.trapz(f, x)
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"""
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def __init__(self, data, hs=None, kernel=None, alpha=0.0,
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xmin=None, xmax=None, inc=512, L2=None):
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self.L2 = L2
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# self.dataset = data
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# self.tkde =
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super(TKDE, self).__init__(data, hs, kernel, alpha, xmin, xmax, inc)
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# @property
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# def dataset(self):
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# return self._dataset
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#
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# @dataset.setter
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# def dataset(self, data):
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# self._dataset = atleast_2d(data)
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# self._tdataset = self._dat2gaus(self._dataset)
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def _initialize(self):
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self._check_xmin()
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tdataset = self._dat2gaus(self.dataset)
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xmin = self.xmin
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if xmin is not None:
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xmin = self._dat2gaus(np.reshape(xmin, (-1, 1)))
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xmax = self.xmax
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if xmax is not None:
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xmax = self._dat2gaus(np.reshape(xmax, (-1, 1)))
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self.tkde = KDE(tdataset, self.hs, self.kernel, self.alpha,
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np.ravel(xmin), np.ravel(xmax),
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self.inc)
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if self.inc is None:
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self.inc = self.tkde.inc
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def _check_xmin(self):
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if self.L2 is not None:
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amin = self.dataset.min(axis=-1)
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# default no transformation
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L2 = np.atleast_1d(self.L2) * np.ones(self.d)
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self.xmin = np.where(L2 != 1,
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np.maximum(self.xmin, amin / 100.0),
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self.xmin)
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def _dat2gaus(self, points):
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if self.L2 is None:
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return points # default no transformation
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# default no transformation
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L2 = np.atleast_1d(self.L2) * np.ones(self.d)
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tpoints = copy.copy(points)
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for i, v2 in enumerate(L2.tolist()):
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tpoints[i] = np.log(points[i]) if v2 == 0 else points[i] ** v2
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return tpoints
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def _gaus2dat(self, tpoints):
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if self.L2 is None:
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return tpoints # default no transformation
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# default no transformation
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L2 = np.atleast_1d(self.L2) * np.ones(self.d)
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points = copy.copy(tpoints)
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for i, v2 in enumerate(L2.tolist()):
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points[i] = np.exp(
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tpoints[i]) if v2 == 0 else tpoints[i] ** (1.0 / v2)
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return points
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def _scale_pdf(self, pdf, points):
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if self.L2 is None:
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return pdf
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# default no transformation
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L2 = np.atleast_1d(self.L2) * np.ones(self.d)
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for i, v2 in enumerate(L2.tolist()):
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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._dat2gaus(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._gaus2dat(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._dat2gaus(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._dat2gaus(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([ 0.21720891, 0.43308789, 0.59017626, 0.55847998, 0.39681482,
|
|
0.23987473, 0.13113066, 0.06062029, 0.02160104, 0.00559028])
|
|
|
|
>>> 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)
|
|
"""
|
|
@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!')
|
|
|
|
# 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): # _KDE):
|
|
|
|
""" 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, *args, **kwds):
|
|
self.method = kwds.pop('method', 'beta')
|
|
self.a = max(kwds.pop('a', 0.5), _TINY)
|
|
self.b = max(kwds.pop('b', 0.5), _TINY)
|
|
self.kreg = KRegression(*args, **kwds)
|
|
# defines bin width (i.e. smoothing) in empirical estimate
|
|
self.hs_e = None
|
|
# self.x = self.kreg.tkde.dataset
|
|
# self.y = self.kreg.y
|
|
|
|
def _set_smoothing(self, hs):
|
|
self.kreg.tkde.hs = hs
|
|
self.kreg.tkde.initialize()
|
|
|
|
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:
|
|
if self.hs_e is None:
|
|
hs1 = self._get_max_smoothing('hste')[0]
|
|
hs2 = self._get_max_smoothing('hos')[0]
|
|
self.hs_e = sqrt(hs1 * hs2)
|
|
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 = self.a
|
|
b = 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
|
|
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.
|
|
|
|
"""
|
|
if self.method.startswith('w'):
|
|
plo, pup = self._wilson_score(n, p, alpha)
|
|
else:
|
|
plo, pup = self._credible_interval(n, p, alpha)
|
|
return plo, pup
|
|
|
|
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(
|
|
hs_e=self.hs_e, alpha=alpha, color=color)
|
|
if hsvec is None:
|
|
hsmax = self._get_max_smoothing(hsfun)[0]
|
|
hsmax = max(hsmax, 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(output='plot', title='Kernel regression', plotflag=1)
|
|
plt.figure(0)
|
|
f.plot(label='p=0')
|
|
|
|
kreg.p = 1
|
|
f1 = kreg(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')
|
|
plt.legend()
|
|
|
|
plt.show()
|
|
|
|
print(kreg.tkde.tkde._inv_hs)
|
|
print(kreg.tkde.tkde.hs)
|
|
|
|
_TINY = np.finfo(float).machar.tiny
|
|
_REALMIN = np.finfo(float).machar.xmin
|
|
_REALMAX = np.finfo(float).machar.xmax
|
|
_EPS = np.finfo(float).eps
|
|
|
|
|
|
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)
|
|
|
|
|
|
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 kreg_demo2(n=100, hs=None, symmetric=False, fun='hisj', plotlog=False):
|
|
x, y, fun1 = _get_data(n, symmetric)
|
|
kreg_demo3(x, y, fun1, hs=None, fun='hisj', plotlog=False)
|
|
|
|
|
|
def kreg_demo3(x, y, fun1, hs=None, fun='hisj', plotlog=False):
|
|
st = scipy.stats
|
|
|
|
alpha = 0.1
|
|
z0 = -_invnorm(alpha / 2)
|
|
|
|
n = x.size
|
|
hopt, hs1, hs2 = _get_regression_smooting(x, y, fun='hos')
|
|
if hs is None:
|
|
hs = hopt
|
|
|
|
forward = _logit
|
|
reverse = _logitinv
|
|
# forward = np.log
|
|
# reverse = np.exp
|
|
|
|
xmin, xmax = x.min(), x.max()
|
|
ni = max(2 * int((xmax - xmin) / hopt) + 3, 5)
|
|
print(ni)
|
|
print(xmin, xmax)
|
|
sml = hopt * 0.1
|
|
xi = np.linspace(xmin - sml, xmax + sml, ni)
|
|
xiii = np.linspace(xmin - sml, xmax + sml, 4 * ni + 1)
|
|
|
|
c = gridcount(x, xi)
|
|
if (y == 1).any():
|
|
c0 = gridcount(x[y == 1], xi)
|
|
else:
|
|
c0 = np.zeros(np.shape(xi))
|
|
yi = np.where(c == 0, 0, c0 / c)
|
|
|
|
kreg = KRegression(x, y, hs=hs, p=0)
|
|
fiii = kreg(xiii)
|
|
yiii = interpolate.interp1d(xi, yi)(xiii)
|
|
fit = fun1(xiii).clip(max=1.0)
|
|
df = np.diff(fiii)
|
|
eerr = np.abs((yiii - fiii)).std() + 0.5 * (df[:-1] * df[1:] < 0).sum() / n
|
|
err = (fiii - fit).std()
|
|
msg = '{0} err={1:1.3f},eerr={2:1.3f}, n={3:d}, hs={4:}, hs1={5:}, '\
|
|
'hs2={6:}'
|
|
title = (msg.format(fun, err, eerr, n, hs, hs1, hs2))
|
|
f = kreg(xiii, output='plotobj', title=title, plotflag=1)
|
|
|
|
# yi[yi==0] = 1.0/(c[c!=0].min()+4)
|
|
# yi[yi==1] = 1-1.0/(c[c!=0].min()+4)
|
|
# yi[yi==0] = fi[yi==0]
|
|
# yi[yi==0] = np.exp(stineman_interp(xi[yi==0], xi[yi>0],np.log(yi[yi>0])))
|
|
# yi[yi==0] = fun1(xi[yi==0])
|
|
try:
|
|
yi[yi == 0] = yi[yi > 0].min() / sqrt(n)
|
|
except:
|
|
yi[yi == 0] = 1. / n
|
|
yi[yi == 1] = 1 - (1 - yi[yi < 1].max()) / sqrt(n)
|
|
|
|
logity = forward(yi)
|
|
|
|
gkreg = KRegression(xi, logity, hs=hs, xmin=xmin - hopt, xmax=xmax + hopt)
|
|
fg = gkreg.eval_grid(
|
|
xi, output='plotobj', title='Kernel regression', plotflag=1)
|
|
sa = (fg.data - logity).std()
|
|
sa2 = iqrange(fg.data - logity) / 1.349
|
|
# print('sa=%g %g' % (sa, sa2))
|
|
sa = min(sa, sa2)
|
|
|
|
# plt.figure(1)
|
|
# plt.plot(xi, slogity-logity,'r.')
|
|
# plt.plot(xi, logity-,'b.')
|
|
# plt.plot(xi, fg.data-logity, 'b.')
|
|
# plt.show()
|
|
# return
|
|
|
|
fg = gkreg.eval_grid(
|
|
xiii, output='plotobj', title='Kernel regression', plotflag=1)
|
|
pi = reverse(fg.data)
|
|
|
|
dx = xi[1] - xi[0]
|
|
ckreg = KDE(x, hs=hs)
|
|
# ci = ckreg.eval_grid_fast(xi)*n*dx
|
|
ciii = ckreg.eval_grid_fast(xiii) * dx * x.size # n*(1+symmetric)
|
|
|
|
# sa1 = np.sqrt(1./(ciii*pi*(1-pi)))
|
|
# plo3 = reverse(fg.data-z0*sa)
|
|
# pup3 = reverse(fg.data+z0*sa)
|
|
fg.data = pi
|
|
pi = f.data
|
|
|
|
# ref Casella and Berger (1990) "Statistical inference" pp444
|
|
# a = 2*pi + z0**2/(ciii+1e-16)
|
|
# b = 2*(1+z0**2/(ciii+1e-16))
|
|
# plo2 = ((a-sqrt(a**2-2*pi**2*b))/b).clip(min=0,max=1)
|
|
# pup2 = ((a+sqrt(a**2-2*pi**2*b))/b).clip(min=0,max=1)
|
|
# Jeffreys intervall a=b=0.5
|
|
# st.beta.isf(alpha/2, x+a, n-x+b)
|
|
ab = 0.07 # 0.055
|
|
pi1 = pi # fun1(xiii)
|
|
pup2 = np.where(pi == 1,
|
|
1,
|
|
st.beta.isf(alpha / 2,
|
|
ciii * pi1 + ab,
|
|
ciii * (1 - pi1) + ab))
|
|
plo2 = np.where(pi == 0,
|
|
0,
|
|
st.beta.isf(1 - alpha / 2,
|
|
ciii * pi1 + ab,
|
|
ciii * (1 - pi1) + ab))
|
|
|
|
averr = np.trapz(pup2 - plo2, xiii) / \
|
|
(xiii[-1] - xiii[0]) + 0.5 * (df[:-1] * df[1:] < 0).sum()
|
|
|
|
# f2 = kreg_demo4(x, y, hs, hopt)
|
|
# Wilson score
|
|
den = 1 + (z0 ** 2. / ciii)
|
|
xc = (pi1 + (z0 ** 2) / (2 * ciii)) / den
|
|
halfwidth = (z0 * sqrt((pi1 * (1 - pi1) / ciii) +
|
|
(z0 ** 2 / (4 * (ciii ** 2))))) / den
|
|
plo = (xc - halfwidth).clip(min=0) # wilson score
|
|
pup = (xc + halfwidth).clip(max=1.0) # wilson score
|
|
# pup = (pi + z0*np.sqrt(pi*(1-pi)/ciii)).clip(min=0,max=1) # dont use
|
|
# plo = (pi - z0*np.sqrt(pi*(1-pi)/ciii)).clip(min=0,max=1)
|
|
|
|
# mi = kreg.eval_grid(x)
|
|
# sigma = (stineman_interp(x, xiii, pup)-stineman_interp(x, xiii, plo))/4
|
|
# aic = np.abs((y-mi)/sigma).std()+ 0.5*(df[:-1]*df[1:]<0).sum()/n
|
|
# aic = np.abs((yiii-fiii)/(pup-plo)).std() + \
|
|
# 0.5*(df[:-1]*df[1:]<0).sum() + \
|
|
# ((yiii-pup).clip(min=0)-(yiii-plo).clip(max=0)).sum()
|
|
|
|
k = (df[:-1] * df[1:] < 0).sum() # numpeaks
|
|
sigmai = (pup - plo)
|
|
aic = (((yiii - fiii) / sigmai) ** 2).sum() + \
|
|
2 * k * (k + 1) / np.maximum(ni - k + 1, 1) + \
|
|
np.abs((yiii - pup).clip(min=0) - (yiii - plo).clip(max=0)).sum()
|
|
|
|
# aic = (((yiii-fiii)/sigmai)**2).sum()+ 2*k*(k+1)/(ni-k+1) + \
|
|
# np.abs((yiii-pup).clip(min=0)-(yiii-plo).clip(max=0)).sum()
|
|
|
|
# aic = averr + ((yiii-pup).clip(min=0)-(yiii-plo).clip(max=0)).sum()
|
|
|
|
fg.plot(label='KReg grid aic={:2.3f}'.format(aic))
|
|
f.plot(label='KReg averr={:2.3f} '.format(averr))
|
|
labtxt = '%d CI' % (int(100 * (1 - alpha)))
|
|
plt.fill_between(xiii, pup, plo, alpha=0.20,
|
|
color='r', linestyle='--', label=labtxt)
|
|
plt.fill_between(xiii, pup2, plo2, alpha=0.20, color='b', linestyle=':',
|
|
label='{:d} CI2'.format(int(100 * (1 - alpha))))
|
|
plt.plot(xiii, fun1(xiii), 'r', label='True model')
|
|
plt.scatter(xi, yi, label='data')
|
|
print('maxp = {}'.format(np.nanmax(f.data)))
|
|
print('hs = {}'.format(kreg.tkde.tkde.hs))
|
|
plt.legend()
|
|
h = plt.gca()
|
|
if plotlog:
|
|
plt.setp(h, yscale='log')
|
|
# plt.show()
|
|
return hs1, hs2
|
|
|
|
|
|
def kreg_demo4(x, y, hs, hopt, alpha=0.05):
|
|
st = scipy.stats
|
|
|
|
n = x.size
|
|
xmin, xmax = x.min(), x.max()
|
|
ni = max(2 * int((xmax - xmin) / hopt) + 3, 5)
|
|
|
|
sml = hopt * 0.1
|
|
xi = np.linspace(xmin - sml, xmax + sml, ni)
|
|
xiii = np.linspace(xmin - sml, xmax + sml, 4 * ni + 1)
|
|
|
|
kreg = KRegression(x, y, hs=hs, p=0)
|
|
|
|
dx = xi[1] - xi[0]
|
|
ciii = kreg.tkde.eval_grid_fast(xiii) * dx * x.size
|
|
# ckreg = KDE(x,hs=hs)
|
|
# ciiii = ckreg.eval_grid_fast(xiii)*dx* x.size #n*(1+symmetric)
|
|
|
|
f = kreg(xiii, output='plotobj') # , plot_kwds=dict(plotflag=7))
|
|
pi = f.data
|
|
|
|
# Jeffreys intervall a=b=0.5
|
|
# st.beta.isf(alpha/2, x+a, n-x+b)
|
|
ab = 0.07 # 0.5
|
|
pi1 = pi
|
|
pup = np.where(pi1 == 1, 1, st.beta.isf(
|
|
alpha / 2, ciii * pi1 + ab, ciii * (1 - pi1) + ab))
|
|
plo = np.where(pi1 == 0, 0, st.beta.isf(
|
|
1 - alpha / 2, ciii * pi1 + ab, ciii * (1 - pi1) + ab))
|
|
|
|
# Wilson score
|
|
# z0 = -_invnorm(alpha/2)
|
|
# den = 1+(z0**2./ciii);
|
|
# xc=(pi1+(z0**2)/(2*ciii))/den;
|
|
# halfwidth=(z0*sqrt((pi1*(1-pi1)/ciii)+(z0**2/(4*(ciii**2)))))/den
|
|
# plo2 = (xc-halfwidth).clip(min=0) # wilson score
|
|
# pup2 = (xc+halfwidth).clip(max=1.0) # wilson score
|
|
# f.dataCI = np.vstack((plo,pup)).T
|
|
f.prediction_error_avg = np.trapz(pup - plo, xiii) / (xiii[-1] - xiii[0])
|
|
fiii = f.data
|
|
|
|
c = gridcount(x, xi)
|
|
if (y == 1).any():
|
|
c0 = gridcount(x[y == 1], xi)
|
|
else:
|
|
c0 = np.zeros(np.shape(xi))
|
|
yi = np.where(c == 0, 0, c0 / c)
|
|
|
|
f.children = [PlotData([plo, pup], xiii, plotmethod='fill_between',
|
|
plot_kwds=dict(alpha=0.2, color='r')),
|
|
PlotData(yi, xi, plotmethod='scatter',
|
|
plot_kwds=dict(color='r', s=5))]
|
|
|
|
yiii = interpolate.interp1d(xi, yi)(xiii)
|
|
df = np.diff(fiii)
|
|
k = (df[:-1] * df[1:] < 0).sum() # numpeaks
|
|
sigmai = (pup - plo)
|
|
aicc = (((yiii - fiii) / sigmai) ** 2).sum() + \
|
|
2 * k * (k + 1) / np.maximum(ni - k + 1, 1) + \
|
|
np.abs((yiii - pup).clip(min=0) - (yiii - plo).clip(max=0)).sum()
|
|
|
|
f.aicc = aicc
|
|
f.labels.title = ('perr={:1.3f},aicc={:1.3f}, n={:d}, '
|
|
'hs={:1.3f}'.format(f.prediction_error_avg, aicc, n, hs))
|
|
|
|
return f
|
|
|
|
|
|
def check_kreg_demo3():
|
|
|
|
plt.ion()
|
|
k = 0
|
|
for n in [50, 100, 300, 600, 4000]:
|
|
x, y, fun1 = _get_data(
|
|
n, symmetric=True, loc1=1.0, scale1=0.6, scale2=1.25)
|
|
k0 = k
|
|
|
|
for fun in ['hste', ]:
|
|
hsmax, _hs1, _hs2 = _get_regression_smooting(x, y, fun=fun)
|
|
for hi in np.linspace(hsmax * 0.25, hsmax, 9):
|
|
plt.figure(k)
|
|
k += 1
|
|
unused = kreg_demo3(x, y, fun1, hs=hi, fun=fun, plotlog=False)
|
|
|
|
# kreg_demo2(n=n,symmetric=True,fun='hste', plotlog=False)
|
|
fig.tile(range(k0, k))
|
|
plt.ioff()
|
|
plt.show()
|
|
|
|
|
|
def check_kreg_demo4():
|
|
plt.ion()
|
|
# test_docstrings()
|
|
# kde_demo2()
|
|
# kreg_demo1(fast=True)
|
|
# kde_gauss_demo()
|
|
# kreg_demo2(n=120,symmetric=True,fun='hste', plotlog=True)
|
|
k = 0
|
|
for _i, n in enumerate([100, 300, 600, 4000]):
|
|
x, y, fun1 = _get_data(
|
|
n, symmetric=True, loc1=0.1, scale1=0.6, scale2=0.75)
|
|
# k0 = k
|
|
hopt1, _h1, _h2 = _get_regression_smooting(x, y, fun='hos')
|
|
hopt2, _h1, _h2 = _get_regression_smooting(x, y, fun='hste')
|
|
hopt = sqrt(hopt1 * hopt2)
|
|
# hopt = _get_regression_smooting(x,y,fun='hos')[0]
|
|
for _j, fun in enumerate(['hste']): # , 'hisj', 'hns', 'hstt'
|
|
hsmax, _hs1, _hs2 = _get_regression_smooting(x, y, fun=fun)
|
|
|
|
fmax = kreg_demo4(x, y, hsmax + 0.1, hopt)
|
|
for hi in np.linspace(hsmax * 0.1, hsmax, 55):
|
|
f = kreg_demo4(x, y, hi, hopt)
|
|
if f.aicc <= fmax.aicc:
|
|
fmax = f
|
|
plt.figure(k)
|
|
k += 1
|
|
fmax.plot()
|
|
plt.plot(x, fun1(x), 'r')
|
|
|
|
# kreg_demo2(n=n,symmetric=True,fun='hste', plotlog=False)
|
|
fig.tile(range(0, k))
|
|
plt.ioff()
|
|
plt.show()
|
|
|
|
|
|
def check_regression_bin():
|
|
plt.ion()
|
|
# test_docstrings()
|
|
# kde_demo2()
|
|
# kreg_demo1(fast=True)
|
|
# kde_gauss_demo()
|
|
# kreg_demo2(n=120,symmetric=True,fun='hste', plotlog=True)
|
|
k = 0
|
|
for _i, n in enumerate([100, 300, 600, 4000]):
|
|
x, y, fun1 = _get_data(
|
|
n, symmetric=True, loc1=0.1, scale1=0.6, scale2=0.75)
|
|
fbest = regressionbin(x, y, alpha=0.05, color='g', label='Transit_D')
|
|
|
|
figk = plt.figure(k)
|
|
ax = figk.gca()
|
|
k += 1
|
|
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()
|
|
|
|
|
|
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')
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|
|
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def _get_regression_smooting(x, y, fun='hste'):
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hs1 = Kernel('gauss', fun=fun).get_smoothing(x)
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# hx = np.median(np.abs(x-np.median(x)))/0.6745*(4.0/(3*n))**0.2
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if (y == 1).any():
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hs2 = Kernel('gauss', fun=fun).get_smoothing(x[y == 1])
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# hy = np.median(np.abs(y-np.mean(y)))/0.6745*(4.0/(3*n))**0.2
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else:
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hs2 = 4 * hs1
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# hy = 4*hx
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|
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# hy2 = Kernel('gauss', fun=fun).get_smoothing(y)
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# kernel = Kernel('gauss',fun=fun)
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# hopt = (hs1+2*hs2)/3
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# hopt = (hs1+4*hs2)/5 #kernel.get_smoothing(x)
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# hopt = hs2
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hopt = sqrt(hs1 * hs2)
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return hopt, hs1, hs2
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|
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def empirical_bin_prb(x, y, hopt, color='r'):
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|
"""Returns empirical binomial probabiltity.
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|
|
|
Parameters
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|
----------
|
|
x : ndarray
|
|
position ve
|
|
y : ndarray
|
|
binomial response variable (zeros and ones)
|
|
|
|
Returns
|
|
-------
|
|
P(x) : PlotData object
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|
empirical probability
|
|
|
|
"""
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|
xmin, xmax = x.min(), x.max()
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ni = max(2 * int((xmax - xmin) / hopt) + 3, 5)
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|
|
|
sml = hopt # *0.1
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xi = np.linspace(xmin - sml, xmax + sml, ni)
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|
|
|
c = gridcount(x, xi)
|
|
if (y == 1).any():
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|
c0 = gridcount(x[y == 1], xi)
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|
else:
|
|
c0 = np.zeros(np.shape(xi))
|
|
yi = np.where(c == 0, 0, c0 / c)
|
|
return PlotData(yi, xi, plotmethod='scatter',
|
|
plot_kwds=dict(color=color, s=5))
|
|
|
|
|
|
def smoothed_bin_prb(x, y, hs, hopt, alpha=0.05, color='r', label='',
|
|
bin_prb=None):
|
|
'''
|
|
Parameters
|
|
----------
|
|
x,y
|
|
hs : smoothing parameter
|
|
hopt : spacing in empirical_bin_prb
|
|
alpha : confidence level
|
|
color : color of plot object
|
|
bin_prb : PlotData object with empirical bin prb
|
|
'''
|
|
if bin_prb is None:
|
|
bin_prb = empirical_bin_prb(x, y, hopt, color)
|
|
|
|
xi = bin_prb.args
|
|
yi = bin_prb.data
|
|
ni = len(xi)
|
|
dxi = xi[1] - xi[0]
|
|
|
|
n = x.size
|
|
|
|
xiii = np.linspace(xi[0], xi[-1], 10 * ni + 1)
|
|
|
|
kreg = KRegression(x, y, hs=hs, p=0)
|
|
# expected number of data in each bin
|
|
ciii = kreg.tkde.eval_grid_fast(xiii) * dxi * n
|
|
|
|
f = kreg(xiii, output='plotobj') # , plot_kwds=dict(plotflag=7))
|
|
pi = f.data
|
|
|
|
st = scipy.stats
|
|
# Jeffreys intervall a=b=0.5
|
|
# st.beta.isf(alpha/2, x+a, n-x+b)
|
|
ab = 0.07 # 0.5
|
|
pi1 = pi
|
|
pup = np.where(pi1 == 1, 1, st.beta.isf(
|
|
alpha / 2, ciii * pi1 + ab, ciii * (1 - pi1) + ab))
|
|
plo = np.where(pi1 == 0, 0, st.beta.isf(
|
|
1 - alpha / 2, ciii * pi1 + ab, ciii * (1 - pi1) + ab))
|
|
|
|
# Wilson score
|
|
# z0 = -_invnorm(alpha/2)
|
|
# den = 1+(z0**2./ciii);
|
|
# xc=(pi1+(z0**2)/(2*ciii))/den;
|
|
# halfwidth=(z0*sqrt((pi1*(1-pi1)/ciii)+(z0**2/(4*(ciii**2)))))/den
|
|
# plo2 = (xc-halfwidth).clip(min=0) # wilson score
|
|
# pup2 = (xc+halfwidth).clip(max=1.0) # wilson score
|
|
# f.dataCI = np.vstack((plo,pup)).T
|
|
f.prediction_error_avg = np.trapz(pup - plo, xiii) / (xiii[-1] - xiii[0])
|
|
fiii = f.data
|
|
|
|
f.plot_kwds['color'] = color
|
|
f.plot_kwds['linewidth'] = 2
|
|
if label:
|
|
f.plot_kwds['label'] = label
|
|
f.children = [PlotData([plo, pup], xiii, plotmethod='fill_between',
|
|
plot_kwds=dict(alpha=0.2, color=color)),
|
|
bin_prb]
|
|
|
|
yiii = interpolate.interp1d(xi, yi)(xiii)
|
|
df = np.diff(fiii)
|
|
k = (df[:-1] * df[1:] < 0).sum() # numpeaks
|
|
sigmai = (pup - plo)
|
|
aicc = (((yiii - fiii) / sigmai) ** 2).sum() + \
|
|
2 * k * (k + 1) / np.maximum(ni - k + 1, 1) + \
|
|
np.abs((yiii - pup).clip(min=0) - (yiii - plo).clip(max=0)).sum()
|
|
|
|
f.aicc = aicc
|
|
f.fun = kreg
|
|
ttl = "perr={0:1.3f}, aicc={1:1.3f}, n={2:d}, hs={3}"
|
|
f.labels.title = ttl.format(f.prediction_error_avg, aicc, n, hs)
|
|
|
|
return f
|
|
|
|
|
|
def regressionbin(x, y, alpha=0.05, color='r', label=''):
|
|
"""Return kernel regression estimate for binomial data.
|
|
|
|
Parameters
|
|
----------
|
|
x : arraylike
|
|
positions
|
|
y : arraylike
|
|
of 0 and 1
|
|
|
|
"""
|
|
|
|
hopt1, _h1, _h2 = _get_regression_smooting(x, y, fun='hos')
|
|
hopt2, _h1, _h2 = _get_regression_smooting(x, y, fun='hste')
|
|
hopt = sqrt(hopt1 * hopt2)
|
|
|
|
fbest = smoothed_bin_prb(x, y, hopt2 + 0.1, hopt, alpha, color, label)
|
|
bin_prb = fbest.children[-1]
|
|
for fun in ['hste']: # , 'hisj', 'hns', 'hstt'
|
|
hsmax, _hs1, _hs2 = _get_regression_smooting(x, y, fun=fun)
|
|
for hi in np.linspace(hsmax * 0.1, hsmax, 55):
|
|
f = smoothed_bin_prb(x, y, hi, hopt, alpha, color, label, bin_prb)
|
|
if f.aicc <= fbest.aicc:
|
|
fbest = f
|
|
# hbest = hi
|
|
return fbest
|
|
|
|
|
|
if __name__ == '__main__':
|
|
if False:
|
|
test_docstrings(__file__)
|
|
else:
|
|
# kde_demo5()
|
|
check_bkregression()
|
|
# check_regression_bin()
|
|
# check_kreg_demo3()
|
|
# check_kreg_demo4()
|
|
|
|
# kreg_demo1(fast=True)
|
|
|
|
# kreg_demo2(n=120,symmetric=True,fun='hste', plotlog=True)
|
|
plt.show('hold')
|