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@ -19,7 +19,7 @@ import scipy.optimize as optimize
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from wafo.misc import meshgrid
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from wafo.misc import meshgrid
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from wafo.wafodata import WafoData
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from wafo.wafodata import WafoData
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from dctpack import dct
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from dctpack import dct, dctn, idctn
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import copy
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import copy
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import numpy as np
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import numpy as np
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@ -63,7 +63,220 @@ def sphere_volume(d, r=1.0):
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Chapman and Hall, pp 105
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Chapman and Hall, pp 105
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"""
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"""
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return (r ** d) * 2.0 * pi ** (d / 2.0) / (d * gamma(d / 2.0))
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return (r ** d) * 2.0 * pi ** (d / 2.0) / (d * gamma(d / 2.0))
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class KDEgauss(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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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(x0, x1,..., xd) : array
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same as kde.eval_grid_fast(x0, x1,..., xd)
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"""
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def __init__(self, data, hs=None, kernel=None, alpha=0.0, xmin=None, xmax=None, inc=128):
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self.dataset = atleast_2d(data)
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self.hs = hs
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self.kernel = kernel if kernel else Kernel('gauss')
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self.alpha = alpha
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self.xmin = xmin
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self.xmax = xmax
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self.inc = inc
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self.initialize()
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def initialize(self):
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self.d, self.n = self.dataset.shape
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self._set_xlimits()
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self._initialize()
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def _initialize(self):
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self._compute_smoothing()
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def _compute_smoothing(self):
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"""Computes the smoothing matrix
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"""
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get_smoothing = self.kernel.get_smoothing
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h = self.hs
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if h is None:
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h = get_smoothing(self.dataset)
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h = np.atleast_1d(h)
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hsiz = h.shape
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if (len(hsiz) == 1) or (self.d == 1):
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if max(hsiz) == 1:
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h = h * np.ones(self.d)
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else:
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h.shape = (self.d,) # make sure it has the correct dimension
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# If h negative calculate automatic values
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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_smoothing(self.dataset[i])
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deth = h.prod()
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self.inv_hs = linalg.diag(1.0 / h)
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else: #fully general smoothing matrix
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deth = linalg.det(h)
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if deth <= 0:
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raise ValueError('bandwidth matrix h must be positive definit!')
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self.inv_hs = linalg.inv(h)
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self.hs = h
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self._norm_factor = deth * self.n
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def _set_xlimits(self):
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amin = self.dataset.min(axis= -1)
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amax = self.dataset.max(axis= -1)
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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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#xyzrange = amax - amin
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#offset = xyzrange / 4.0
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offset = 2 * sigma
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if self.xmin is None:
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self.xmin = amin - offset
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else:
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self.xmin = self.xmin * np.ones((self.d,1))
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if self.xmax is None:
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self.xmax = amax + offset
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else:
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self.xmax = self.xmax * np.ones((self.d,1))
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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 = []
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for i in range(self.d):
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args.append(np.linspace(self.xmin[i], self.xmax[i], self.inc))
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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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X = np.vstack(args)
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d, inc = X.shape
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dx = X[:, 1] - X[:, 0]
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R = X.max(axis=-1)- X.min(axis=-1)
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t_star = (self.hs/R)**2
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I = (np.asfarray(np.arange(0, inc))*pi)**2
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In = []
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for i in range(d):
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In.append(I * t_star[i] * 0.5)
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Inc = meshgrid(*In) if d > 1 else In
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kw = np.zeros((inc,)*d)
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for i in range(d):
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kw += exp(-Inc[i])
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y = kwds.get('y', 1.0)
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d, n = self.dataset.shape
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# Find the binned kernel weights, c.
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c = gridcount(self.dataset, X, y=y)/n
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# Perform the convolution.
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at = dctn(c) * kw
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z = idctn(at)*at.size/np.prod(R)
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return z*(z>0.0)
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#ix = (slice(0, inc),)*d
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#return z[ix] * (z[ix] > 0.0)
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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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else:
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titlestr = 'Kernel density estimate (%s)' % 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 = WafoData(f, args, **kwds2)
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if self.d > 1:
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PL = np.r_[10:90:20, 95, 99, 99.9]
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try:
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ql = qlevels(f, p=PL)
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wdata.clevels = ql
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wdata.plevels = PL
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except:
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pass
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return wdata
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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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if d == 1 and m == 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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else:
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msg = "points have dimension %s, dataset has dimension %s" % (d,
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self.d)
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raise ValueError(msg)
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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_fast
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class _KDE(object):
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class _KDE(object):
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""" Kernel-Density Estimator base class.
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""" Kernel-Density Estimator base class.
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@ -341,7 +554,7 @@ class TKDE(_KDE):
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def __init__(self, data, hs=None, kernel=None, alpha=0.0, xmin=None,
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def __init__(self, data, hs=None, kernel=None, alpha=0.0, xmin=None,
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xmax=None, inc=128, L2=None):
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xmax=None, inc=128, L2=None):
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self.L2 = L2
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self.L2 = L2
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_KDE.__init__(self, data, hs, kernel, alpha, xmin, xmax, inc)
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super(TKDE, self).__init__(data, hs, kernel, alpha, xmin, xmax, inc)
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def _initialize(self):
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def _initialize(self):
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self._check_xmin()
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self._check_xmin()
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@ -574,7 +787,7 @@ class KDE(_KDE):
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"""
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"""
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def __init__(self, data, hs=None, kernel=None, alpha=0.0, xmin=None, xmax=None, inc=128):
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def __init__(self, data, hs=None, kernel=None, alpha=0.0, xmin=None, xmax=None, inc=128):
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_KDE.__init__(self, data, hs, kernel, alpha, xmin, xmax, inc)
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super(KDE, self).__init__(data, hs, kernel, alpha, xmin, xmax, inc)
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def _initialize(self):
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def _initialize(self):
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self._compute_smoothing()
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self._compute_smoothing()
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@ -638,11 +851,14 @@ class KDE(_KDE):
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Xn = np.dot(self.inv_hs, np.vstack(Xnc))
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Xn = np.dot(self.inv_hs, np.vstack(Xnc))
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# Obtain the kernel weights.
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# Obtain the kernel weights.
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kw = self.kernel(Xn)
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norm_fact = (kw.sum()*dx.prod()*self.n)
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norm_fact2 = (self._norm_factor * self.kernel.norm_factor(d, self.n))
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kw = kw/norm_fact
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r = kwds.get('r', 0)
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r = kwds.get('r', 0)
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if r==0:
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if r!=0:
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kw = self.kernel(Xn) / (self._norm_factor * self.kernel.norm_factor(d, self.n))
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kw *= np.vstack(Xnc) ** r
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else:
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kw = np.vstack(Xnc) ** r * self.kernel(Xn) / (self._norm_factor * self.kernel.norm_factor(d, self.n))
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kw.shape = shape0
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kw.shape = shape0
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kw = np.fft.ifftshift(kw)
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kw = np.fft.ifftshift(kw)
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fftn = np.fft.fftn
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fftn = np.fft.fftn
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@ -1434,7 +1650,8 @@ class Kernel(object):
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c = gridcount(A[dim], xa)
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c = gridcount(A[dim], xa)
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N = len(set(A[dim]))
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N = len(set(A[dim]))
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a = dct(c/c.sum(), norm=None)
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#a = dct(c/c.sum(), norm=None)
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a = dct(c/len(A[dim]), norm=None)
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#% now compute the optimal bandwidth^2 using the referenced method
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#% now compute the optimal bandwidth^2 using the referenced method
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I = np.asfarray(np.arange(1, inc))**2
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I = np.asfarray(np.arange(1, inc))**2
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@ -2551,8 +2768,8 @@ def kde_demo1():
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import scipy.stats as st
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import scipy.stats as st
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x = np.linspace(-4, 4, 101)
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x = np.linspace(-4, 4, 101)
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x0 = x / 2.0
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x0 = x / 2.0
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data = np.random.normal(loc=0, scale=1.0, size=7) #rndnorm(0,1,7,1);
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data = np.random.normal(loc=0, scale=1.0, size=7)
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kernel = Kernel('gaus')
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kernel = Kernel('gauss')
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hs = kernel.hns(data)
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hs = kernel.hns(data)
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hVec = [hs / 2, hs, 2 * hs]
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hVec = [hs / 2, hs, 2 * hs]
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@ -2743,6 +2960,58 @@ def kreg_demo1(hs=None, fast=False, fun='hisj'):
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print(kreg.tkde.tkde.inv_hs)
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print(kreg.tkde.tkde.inv_hs)
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print(kreg.tkde.tkde.hs)
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print(kreg.tkde.tkde.hs)
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def kde_gauss_demo(n=50000):
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'''
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KDEDEMO Demonstrate the KDEgauss
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KDEDEMO1 shows the true density (dotted) compared to KDE based on 7
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observations (solid) and their individual kernels (dashed) for 3
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different values of the smoothing parameter, hs.
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'''
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import scipy.stats as st
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#x = np.linspace(-4, 4, 101)
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#data = np.random.normal(loc=0, scale=1.0, size=n)
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#data = np.random.exponential(scale=1.0, size=n)
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# n1 = 128
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# I = (np.arange(n1)*pi)**2 *0.01*0.5
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# kw = exp(-I)
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# pylab.plot(idctn(kw))
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# return
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dist = st.norm
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dist = st.expon
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data = dist.rvs(loc=0, scale=1.0, size=n)
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d, N = np.atleast_2d(data).shape
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if d==1:
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plot_options = [dict(color='red'), dict(color='green'), dict(color='black')]
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else:
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plot_options = [dict(colors='red'), dict(colors='green'), dict(colors='black')]
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pylab.figure(1)
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kde0 = KDE(data, kernel=Kernel('gauss', 'hisj'))
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f0 = kde0.eval_grid_fast(output='plot', ylab='Density')
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f0.plot(**plot_options[0])
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kde1 = TKDE(data, kernel=Kernel('gauss', 'hisj'), L2=0)
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f1 = kde1.eval_grid_fast(output='plot', ylab='Density')
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f1.plot(**plot_options[1])
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kde2 = KDEgauss(data)
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f2 = kde2(output='plot', ylab='Density')
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x = f2.args
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f2.plot(**plot_options[2])
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fmax = dist.pdf(x, 0, 1).max()
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if d==1:
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pylab.plot(x, dist.pdf(x, 0, 1), 'k:')
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pylab.axis([x.min(), x.max(), 0, fmax])
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pylab.show()
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print(fmax/f2.data.max())
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format_ = ''.join(('%g, ')*d)
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format_ = 'hs0=%s hs2=%s' % (format_,format_)
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print(format_ % tuple(kde0.hs.tolist()+kde2.hs.tolist()))
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def test_docstrings():
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def test_docstrings():
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import doctest
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import doctest
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doctest.testmod()
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doctest.testmod()
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@ -2750,4 +3019,5 @@ def test_docstrings():
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if __name__ == '__main__':
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if __name__ == '__main__':
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#test_docstrings()
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#test_docstrings()
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#kde_demo2()
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#kde_demo2()
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kreg_demo1()
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#kreg_demo1(fast=True)
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kde_gauss_demo()
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