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@ -1,71 +1,20 @@
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from __future__ import division
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import numpy as np
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#from math import pow
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#from numpy import zeros,dot
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from numpy import abs, size, convolve, linalg, concatenate # @UnresolvedImport
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# from math import pow
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# from numpy import zeros,dot
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from numpy import (pi, abs, size, convolve, linalg, concatenate, sqrt)
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from scipy.sparse import spdiags
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from scipy.sparse.linalg import spsolve, expm
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from scipy.signal import medfilt
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from wafo.dctpack import dctn, idctn
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import scipy.optimize as optimize
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from scipy.signal import _savitzky_golay
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from scipy.ndimage import convolve1d
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from scipy.ndimage.morphology import distance_transform_edt
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import warnings
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from wafo.plotbackend import plotbackend as plt
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__all__ = ['calc_coeff', 'smooth', 'smooth_last',
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'SavitzkyGolay', 'Kalman', 'HodrickPrescott']
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def calc_coeff(n, degree, diff_order=0):
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""" calculates filter coefficients for symmetric savitzky-golay filter.
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see: http://www.nrbook.com/a/bookcpdf/c14-8.pdf
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n means that 2*n+1 values contribute to the smoother.
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degree is degree of fitting polynomial
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diff_order is degree of implicit differentiation.
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0 means that filter results in smoothing of function
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1 means that filter results in smoothing the first
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derivative of function.
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and so on ...
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"""
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order_range = np.arange(degree + 1)
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k_range = np.arange(-n, n + 1, dtype=float).reshape(-1, 1)
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b = np.mat(k_range ** order_range)
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#b = np.mat([[float(k)**i for i in order_range] for k in range(-n,n+1)])
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coeff = linalg.pinv(b).A[diff_order]
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return coeff
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def smooth_last(signal, coeff, k=0):
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n = size(coeff - 1) // 2
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y = np.squeeze(signal)
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if y.ndim > 1:
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coeff.shape = (-1, 1)
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first_vals = y[0] - abs(y[n:0:-1] - y[0])
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last_vals = y[-1] + abs(y[-2:-n - 2:-1] - y[-1])
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y = concatenate((first_vals, y, last_vals))
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return (y[-2 * n - 1 - k:-k] * coeff).sum(axis=0)
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def smooth(signal, coeff, pad=True):
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"""applies coefficients calculated by calc_coeff() to signal."""
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n = size(coeff - 1) // 2
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y = np.squeeze(signal)
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if n == 0:
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return y
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if pad:
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first_vals = y[0] - abs(y[n:0:-1] - y[0])
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last_vals = y[-1] + abs(y[-2:-n - 2:-1] - y[-1])
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y = concatenate((first_vals, y, last_vals))
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n *= 2
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d = y.ndim
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if d > 1:
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y1 = y.reshape(y.shape[0], -1)
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res = []
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for i in range(y1.shape[1]):
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res.append(convolve(y1[:, i], coeff)[n:-n])
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res = np.asarray(res).T
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else:
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res = convolve(y, coeff)[n:-n]
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return res
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__all__ = ['SavitzkyGolay', 'Kalman', 'HodrickPrescott']
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class SavitzkyGolay(object):
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@ -81,21 +30,62 @@ class SavitzkyGolay(object):
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n : int
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the size of the smoothing window is 2*n+1.
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degree : int
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the order of the polynomial used in the filtering.
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the degree of the polynomial used in the filtering.
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Must be less than `window_size` - 1, i.e, less than 2*n.
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diff_order : int
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order of the derivative to compute (default = 0 means only smoothing)
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0 means that filter results in smoothing of function
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1 means that filter results in smoothing the first derivative of the
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function and so on ...
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delta : float, optional
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The spacing of the samples to which the filter will be applied.
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This is only used if deriv > 0. Default is 1.0.
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axis : int, optional
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The axis of the array `x` along which the filter is to be applied.
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Default is -1.
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mode : str, optional
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Must be 'mirror', 'constant', 'nearest', 'wrap' or 'interp'. This
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determines the type of extension to use for the padded signal to
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which the filter is applied. When `mode` is 'constant', the padding
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value is given by `cval`. See the Notes for more details on 'mirror',
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'constant', 'wrap', and 'nearest'.
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When the 'interp' mode is selected (the default), no extension
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is used. Instead, a degree `polyorder` polynomial is fit to the
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last `window_length` values of the edges, and this polynomial is
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used to evaluate the last `window_length // 2` output values.
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cval : scalar, optional
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Value to fill past the edges of the input if `mode` is 'constant'.
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Default is 0.0.
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Notes
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-----
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The Savitzky-Golay is a type of low-pass filter, particularly
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suited for smoothing noisy data. The main idea behind this
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approach is to make for each point a least-square fit with a
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polynomial of high order over a odd-sized window centered at
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the point.
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The Savitzky-Golay is a type of low-pass filter, particularly suited for
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smoothing noisy data. The main idea behind this approach is to make for
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each point a least-square fit with a polynomial of high order over a
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odd-sized window centered at the point.
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Details on the `mode` options:
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'mirror':
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Repeats the values at the edges in reverse order. The value
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closest to the edge is not included.
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'nearest':
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The extension contains the nearest input value.
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'constant':
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The extension contains the value given by the `cval` argument.
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'wrap':
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The extension contains the values from the other end of the array.
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For example, if the input is [1, 2, 3, 4, 5, 6, 7, 8], and
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`window_length` is 7, the following shows the extended data for
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the various `mode` options (assuming `cval` is 0)::
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mode | Ext | Input | Ext
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-----------+---------+------------------------+---------
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'mirror' | 4 3 2 | 1 2 3 4 5 6 7 8 | 7 6 5
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'nearest' | 1 1 1 | 1 2 3 4 5 6 7 8 | 8 8 8
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'constant' | 0 0 0 | 1 2 3 4 5 6 7 8 | 0 0 0
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'wrap' | 6 7 8 | 1 2 3 4 5 6 7 8 | 1 2 3
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Examples
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--------
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@ -120,21 +110,19 @@ class SavitzkyGolay(object):
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Cambridge University Press ISBN-13: 9780521880688
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"""
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def __init__(self, n, degree=1, diff_order=0):
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def __init__(self, n, degree=1, diff_order=0, delta=1.0, axis=-1,
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mode='interp', cval=0.0):
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self.n = n
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self.degree = degree
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self.diff_order = diff_order
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self.calc_coeff()
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def calc_coeff(self):
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""" calculates filter coefficients for symmetric savitzky-golay filter.
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"""
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n = self.n
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order_range = np.arange(self.degree + 1)
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k_range = np.arange(-n, n + 1, dtype=float).reshape(-1, 1)
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b = np.mat(k_range ** order_range)
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#b =np.mat([[float(k)**i for i in order_range] for k in range(-n,n+1)])
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self._coeff = linalg.pinv(b).A[self.diff_order]
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self.mode = mode
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self.cval = cval
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self.axis = axis
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self.delta = delta
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window_length = 2 * n + 1
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self._coeff = _savitzky_golay.savgol_coeffs(window_length,
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degree, deriv=diff_order,
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delta=delta)
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def smooth_last(self, signal, k=0):
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coeff = self._coeff
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@ -152,7 +140,24 @@ class SavitzkyGolay(object):
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def __call__(self, signal):
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return self.smooth(signal)
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def smooth(self, signal, pad=True):
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def smooth(self, signal):
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x = np.asarray(signal)
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if x.dtype != np.float64 and x.dtype != np.float32:
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x = x.astype(np.float64)
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coeffs = self._coeff
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mode, axis = self.mode, self.axis
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if mode == "interp":
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window_length, polyorder = self.n * 2 + 1, self.degree
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deriv, delta = self.diff_order, self.delta
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y = convolve1d(x, coeffs, axis=axis, mode="constant")
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_savitzky_golay._fit_edges_polyfit(x, window_length, polyorder,
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deriv, delta, axis, y)
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else:
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y = convolve1d(x, coeffs, axis=axis, mode=mode, cval=self.cval)
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return y
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def _smooth(self, signal, pad=True):
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"""
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Returns smoothed signal (or it's n-th derivative).
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@ -190,6 +195,490 @@ class SavitzkyGolay(object):
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return res
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def evar(y):
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"""Noise variance estimation. Assuming that the deterministic function Y
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has additive Gaussian noise, EVAR(Y) returns an estimated variance of this
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noise.
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Note:
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----
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A thin-plate smoothing spline model is used to smooth Y. It is assumed
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that the model whose generalized cross-validation score is minimum can
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provide the variance of the additive noise. A few tests showed that
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EVAR works very well with "not too irregular" functions.
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Examples:
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--------
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1D signal
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>>> n = 1e6
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>>> x = np.linspace(0,100,n);
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>>> y = np.cos(x/10)+(x/50)
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>>> var0 = 0.02 # noise variance
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>>> yn = y + sqrt(var0)*np.random.randn(*y.shape)
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>>> s = evar(yn) # estimated variance
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>>> np.abs(s-var0)/var0 < 3.5/np.sqrt(n)
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True
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2D function
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>>> xp = np.linspace(0,1,50)
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>>> x, y = np.meshgrid(xp,xp)
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>>> f = np.exp(x+y) + np.sin((x-2*y)*3)
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>>> var0 = 0.04 # noise variance
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>>> fn = f + sqrt(var0)*np.random.randn(*f.shape)
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>>> s = evar(fn) # estimated variance
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>>> np.abs(s-var0)/var0 < 3.5/np.sqrt(50)
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True
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3D function
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>>> yp = np.linspace(-2,2,50)
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>>> [x,y,z] = meshgrid(yp,yp,yp, sparse=True)
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>>> f = x*exp(-x**2-y**2-z**2)
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>>> var0 = 0.5 # noise variance
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>>> fn = f + sqrt(var0)*np.random.randn(*f.shape)
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>>> s = evar(fn) # estimated variance
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>>> np.abs(s-var0)/var0 < 3.5/np.sqrt(50)
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True
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Other example
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-------------
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http://www.biomecardio.com/matlab/evar.html
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Note:
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----
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EVAR is only adapted to evenly-gridded 1-D to N-D data.
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See also
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--------
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VAR, STD, SMOOTHN
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"""
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# Damien Garcia -- 2008/04, revised 2009/10
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y = np.atleast_1d(y)
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d = y.ndim
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sh0 = y.shape
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S = np.zeros(sh0)
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sh1 = np.ones((d,))
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cos = np.cos
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pi = np.pi
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for i in range(d):
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ni = sh0[i]
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sh1[i] = ni
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t = np.arange(ni).reshape(sh1) / ni
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S += cos(pi * t)
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sh1[i] = 1
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S2 = 2 * (d - S).ravel()
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# N-D Discrete Cosine Transform of Y
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dcty2 = dctn(y).ravel() ** 2
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def score_fun(L, S2, dcty2):
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# Generalized cross validation score
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M = 1 - 1. / (1 + 10 ** L * S2)
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noisevar = (dcty2 * M ** 2).mean()
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return noisevar / M.mean() ** 2
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# fun = lambda x : score_fun(x, S2, dcty2)
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Lopt = optimize.fminbound(score_fun, -38, 38, args=(S2, dcty2))
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M = 1.0 - 1.0 / (1 + 10 ** Lopt * S2)
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noisevar = (dcty2 * M ** 2).mean()
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return noisevar
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def smoothn(data, s=None, weight=None, robust=False, z0=None, tolz=1e-3,
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maxiter=100, fulloutput=False):
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'''
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SMOOTHN fast and robust spline smoothing for 1-D to N-D data.
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Parameters
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----------
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data : array like
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uniformly-sampled data array to smooth. Non finite values (NaN or Inf)
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are treated as missing values.
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s : real positive scalar
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smooting parameter. The larger S is, the smoother the output will be.
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Default value is automatically determined using the generalized
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cross-validation (GCV) method.
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weight : string or array weights
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weighting array of real positive values, that must have the same size
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as DATA. Note that a zero weight corresponds to a missing value.
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robust : bool
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If true carry out a robust smoothing that minimizes the influence of
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outlying data.
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tolz : real positive scalar
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Termination tolerance on Z (default = 1e-3)
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maxiter : scalar integer
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Maximum number of iterations allowed (default = 100)
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z0 : array-like
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Initial value for the iterative process (default = original data)
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Returns
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-------
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z : array like
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smoothed data
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To be made
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----------
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Estimate the confidence bands (see Wahba 1983, Nychka 1988).
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Reference
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---------
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Garcia D, Robust smoothing of gridded data in one and higher dimensions
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with missing values. Computational Statistics & Data Analysis, 2010.
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http://www.biomecardio.com/pageshtm/publi/csda10.pdf
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Examples:
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--------
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1-D example
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>>> import matplotlib.pyplot as plt
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>>> x = np.linspace(0,100,2**8)
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>>> y = np.cos(x/10)+(x/50)**2 + np.random.randn(*x.shape)/10
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>>> y[np.r_[70, 75, 80]] = np.array([5.5, 5, 6])
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>>> z = smoothn(y) # Regular smoothing
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>>> zr = smoothn(y,robust=True) # Robust smoothing
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>>> h=plt.subplot(121),
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>>> h = plt.plot(x,y,'r.',x,z,'k',linewidth=2)
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>>> h=plt.title('Regular smoothing')
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>>> h=plt.subplot(122)
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>>> h=plt.plot(x,y,'r.',x,zr,'k',linewidth=2)
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>>> h=plt.title('Robust smoothing')
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2-D example
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>>> xp = np.r_[0:1:.02]
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>>> [x,y] = np.meshgrid(xp,xp)
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>>> f = np.exp(x+y) + np.sin((x-2*y)*3);
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>>> fn = f + np.random.randn(*f.shape)*0.5;
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>>> fs = smoothn(fn);
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>>> h=plt.subplot(121),
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>>> h=plt.contourf(xp,xp,fn)
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>>> h=plt.subplot(122)
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>>> h=plt.contourf(xp,xp,fs)
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2-D example with missing data
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n = 256;
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y0 = peaks(n);
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y = y0 + rand(size(y0))*2;
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I = randperm(n^2);
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y(I(1:n^2*0.5)) = NaN; lose 1/2 of data
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y(40:90,140:190) = NaN; create a hole
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z = smoothn(y); smooth data
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subplot(2,2,1:2), imagesc(y), axis equal off
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title('Noisy corrupt data')
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subplot(223), imagesc(z), axis equal off
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title('Recovered data ...')
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subplot(224), imagesc(y0), axis equal off
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title('... compared with original data')
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3-D example
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[x,y,z] = meshgrid(-2:.2:2);
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xslice = [-0.8,1]; yslice = 2; zslice = [-2,0];
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vn = x.*exp(-x.^2-y.^2-z.^2) + randn(size(x))*0.06;
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subplot(121), slice(x,y,z,vn,xslice,yslice,zslice,'cubic')
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title('Noisy data')
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v = smoothn(vn);
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subplot(122), slice(x,y,z,v,xslice,yslice,zslice,'cubic')
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title('Smoothed data')
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Cardioid
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t = linspace(0,2*pi,1000);
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x = 2*cos(t).*(1-cos(t)) + randn(size(t))*0.1;
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y = 2*sin(t).*(1-cos(t)) + randn(size(t))*0.1;
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z = smoothn(complex(x,y));
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plot(x,y,'r.',real(z),imag(z),'k','linewidth',2)
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axis equal tight
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Cellular vortical flow
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[x,y] = meshgrid(linspace(0,1,24));
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Vx = cos(2*pi*x+pi/2).*cos(2*pi*y);
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Vy = sin(2*pi*x+pi/2).*sin(2*pi*y);
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Vx = Vx + sqrt(0.05)*randn(24,24); adding Gaussian noise
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Vy = Vy + sqrt(0.05)*randn(24,24); adding Gaussian noise
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I = randperm(numel(Vx));
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Vx(I(1:30)) = (rand(30,1)-0.5)*5; adding outliers
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Vy(I(1:30)) = (rand(30,1)-0.5)*5; adding outliers
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Vx(I(31:60)) = NaN; missing values
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Vy(I(31:60)) = NaN; missing values
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Vs = smoothn(complex(Vx,Vy),'robust'); automatic smoothing
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subplot(121), quiver(x,y,Vx,Vy,2.5), axis square
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title('Noisy velocity field')
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subplot(122), quiver(x,y,real(Vs),imag(Vs)), axis square
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title('Smoothed velocity field')
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See also SMOOTH, SMOOTH3, DCTN, IDCTN.
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-- Damien Garcia -- 2009/03, revised 2010/11
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Visit
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http://www.biomecardio.com/matlab/smoothn.html
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for more details about SMOOTHN
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'''
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y = np.atleast_1d(data)
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sizy = y.shape
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noe = y.size
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if noe < 2:
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return data
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weightstr = 'bisquare'
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W = np.ones(sizy)
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# Smoothness parameter and weights
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if weight is None:
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pass
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elif isinstance(weight, str):
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weightstr = weight.lower()
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else:
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W = weight
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# Weights. Zero weights are assigned to not finite values (Inf or NaN),
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# (Inf/NaN values = missing data).
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IsFinite = np.isfinite(y)
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nof = IsFinite.sum() # number of finite elements
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W = W * IsFinite
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if (W < 0).any():
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raise ValueError('Weights must all be >=0')
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else:
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W = W / W.max()
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isweighted = (W < 1).any() # Weighted or missing data?
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isauto = s is None # Automatic smoothing?
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# Creation of the Lambda tensor
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# Lambda contains the eingenvalues of the difference matrix used in this
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# penalized least squares process.
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d = y.ndim
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Lambda = np.zeros(sizy)
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siz0 = [1, ] * d
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for i in range(d):
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siz0[i] = sizy[i]
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Lambda = Lambda + \
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np.cos(pi * np.arange(sizy[i]) / sizy[i]).reshape(siz0)
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siz0[i] = 1
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Lambda = -2 * (d - Lambda)
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if not isauto:
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Gamma = 1. / (1 + s * Lambda ** 2)
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# Upper and lower bound for the smoothness parameter
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# The average leverage (h) is by definition in [0 1]. Weak smoothing occurs
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# if h is close to 1, while over-smoothing appears when h is near 0. Upper
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# and lower bounds for h are given to avoid under- or over-smoothing. See
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# equation relating h to the smoothness parameter (Equation #12 in the
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# referenced CSDA paper).
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N = (np.array(sizy) != 1).sum() # tensor rank of the y-array
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hMin = 1e-6
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hMax = 0.99
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sMinBnd = (((1 + sqrt(1 + 8 * hMax ** (2. / N))) / 4. /
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hMax ** (2. / N)) ** 2 - 1) / 16
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sMaxBnd = (((1 + sqrt(1 + 8 * hMin ** (2. / N))) / 4. /
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hMin ** (2. / N)) ** 2 - 1) / 16
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# Initialize before iterating
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Wtot = W
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# Initial conditions for z
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if isweighted:
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# With weighted/missing data
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# An initial guess is provided to ensure faster convergence. For that
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# purpose, a nearest neighbor interpolation followed by a coarse
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# smoothing are performed.
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if z0 is None:
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z = InitialGuess(y, IsFinite)
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else:
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# an initial guess (z0) has been provided
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z = z0
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else:
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z = np.zeros(sizy)
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z0 = z
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y[~IsFinite] = 0 # arbitrary values for missing y-data
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tol = 1
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RobustIterativeProcess = True
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RobustStep = 1
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# Error on p. Smoothness parameter s = 10^p
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errp = 0.1
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# Relaxation factor RF: to speedup convergence
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RF = 1.75 if isweighted else 1.0
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norm = linalg.norm
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# Main iterative process
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while RobustIterativeProcess:
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# "amount" of weights (see the function GCVscore)
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aow = Wtot.sum() / noe # 0 < aow <= 1
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exitflag = True
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for nit in range(1, maxiter + 1):
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DCTy = dctn(Wtot * (y - z) + z)
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if isauto and not np.remainder(np.log2(nit), 1):
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# The generalized cross-validation (GCV) method is used.
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# We seek the smoothing parameter s that minimizes the GCV
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# score i.e. s = Argmin(GCVscore).
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# Because this process is time-consuming, it is performed from
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# time to time (when nit is a power of 2)
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log10s = optimize.fminbound(
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gcv, np.log10(sMinBnd), np.log10(sMaxBnd),
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args=(aow, Lambda, DCTy, y, Wtot, IsFinite, nof, noe),
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xtol=errp, full_output=False, disp=False)
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s = 10 ** log10s
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Gamma = 1.0 / (1 + s * Lambda ** 2)
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z = RF * idctn(Gamma * DCTy) + (1 - RF) * z
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# if no weighted/missing data => tol=0 (no iteration)
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tol = norm(z0.ravel() - z.ravel()) / norm(
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z.ravel()) if isweighted else 0.0
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if tol <= tolz:
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break
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z0 = z # re-initialization
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else:
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exitflag = False # nit<MaxIter;
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if robust:
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# -- Robust Smoothing: iteratively re-weighted process
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# --- average leverage
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h = sqrt(1 + 16 * s)
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h = sqrt(1 + h) / sqrt(2) / h
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h = h ** N
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# take robust weights into account
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Wtot = W * RobustWeights(y - z, IsFinite, h, weightstr)
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# re-initialize for another iterative weighted process
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isweighted = True
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tol = 1
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RobustStep = RobustStep + 1
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# 3 robust steps are enough.
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RobustIterativeProcess = RobustStep < 4
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else:
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RobustIterativeProcess = False # stop the whole process
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# Warning messages
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if isauto:
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if abs(np.log10(s) - np.log10(sMinBnd)) < errp:
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warnings.warn('''s = %g: the lower bound for s has been reached.
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Put s as an input variable if required.''' % s)
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elif abs(np.log10(s) - np.log10(sMaxBnd)) < errp:
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warnings.warn('''s = %g: the Upper bound for s has been reached.
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Put s as an input variable if required.''' % s)
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if not exitflag:
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warnings.warn('''Maximum number of iterations (%d) has been exceeded.
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Increase MaxIter option or decrease TolZ value.''' % (maxiter))
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if fulloutput:
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return z, s
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else:
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return z
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def gcv(p, aow, Lambda, DCTy, y, Wtot, IsFinite, nof, noe):
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# Search the smoothing parameter s that minimizes the GCV score
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s = 10 ** p
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Gamma = 1.0 / (1 + s * Lambda ** 2)
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# RSS = Residual sum-of-squares
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if aow > 0.9: # aow = 1 means that all of the data are equally weighted
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# very much faster: does not require any inverse DCT
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RSS = linalg.norm(DCTy.ravel() * (Gamma.ravel() - 1)) ** 2
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else:
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# take account of the weights to calculate RSS:
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yhat = idctn(Gamma * DCTy)
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RSS = linalg.norm(sqrt(Wtot[IsFinite]) *
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(y[IsFinite] - yhat[IsFinite])) ** 2
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TrH = Gamma.sum()
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GCVscore = RSS / nof / (1.0 - TrH / noe) ** 2
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return GCVscore
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# Robust weights
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def RobustWeights(r, I, h, wstr):
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# weights for robust smoothing.
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MAD = np.median(abs(r[I] - np.median(r[I]))) # median absolute deviation
|
|
|
|
|
u = abs(r / (1.4826 * MAD) / sqrt(1 - h)) # studentized residuals
|
|
|
|
|
if wstr == 'cauchy':
|
|
|
|
|
c = 2.385
|
|
|
|
|
W = 1. / (1 + (u / c) ** 2) # Cauchy weights
|
|
|
|
|
elif wstr == 'talworth':
|
|
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|
|
c = 2.795
|
|
|
|
|
W = u < c # Talworth weights
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|
|
|
|
else: # bisquare weights
|
|
|
|
|
c = 4.685
|
|
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|
|
W = (1 - (u / c) ** 2) ** 2 * ((u / c) < 1)
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|
|
|
|
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|
|
W[np.isnan(W)] = 0
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|
|
|
return W
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|
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|
|
def InitialGuess(y, I):
|
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|
|
|
# Initial Guess with weighted/missing data
|
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|
|
|
# nearest neighbor interpolation (in case of missing values)
|
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|
z = y
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|
|
if (1 - I).any():
|
|
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|
|
notI = ~I
|
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|
|
z, L = distance_transform_edt(notI, return_indices=True)
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|
z[notI] = y[L.flat[notI]]
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|
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|
|
# coarse fast smoothing using one-tenth of the DCT coefficients
|
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|
siz = z.shape
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|
d = z.ndim
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|
z = dctn(z)
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|
|
for k in range(d):
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|
z[int((siz[k] + 0.5) / 10) + 1::, ...] = 0
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|
z = z.reshape(np.roll(siz, -k))
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z = z.transpose(np.roll(range(z.ndim), -1))
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# z = shiftdim(z,1);
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|
z = idctn(z)
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return z
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def test_smoothn_1d():
|
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|
|
x = np.linspace(0, 100, 2 ** 8)
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|
y = np.cos(x / 10) + (x / 50) ** 2 + np.random.randn(x.size) / 10
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|
y[np.r_[70, 75, 80]] = np.array([5.5, 5, 6])
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|
|
z = smoothn(y) # Regular smoothing
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|
zr = smoothn(y, robust=True) # Robust smoothing
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|
|
plt.subplot(121),
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|
|
unused_h = plt.plot(x, y, 'r.', x, z, 'k', linewidth=2)
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|
|
|
|
plt.title('Regular smoothing')
|
|
|
|
|
plt.subplot(122)
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|
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|
|
plt.plot(x, y, 'r.', x, zr, 'k', linewidth=2)
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|
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|
|
plt.title('Robust smoothing')
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|
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|
|
plt.show('hold')
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|
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|
|
def test_smoothn_2d():
|
|
|
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|
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|
|
|
# import mayavi.mlab as plt
|
|
|
|
|
xp = np.r_[0:1:.02]
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|
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|
|
[x, y] = np.meshgrid(xp, xp)
|
|
|
|
|
f = np.exp(x + y) + np.sin((x - 2 * y) * 3)
|
|
|
|
|
fn = f + np.random.randn(*f.shape) * 0.5
|
|
|
|
|
fs, s = smoothn(fn, fulloutput=True) # @UnusedVariable
|
|
|
|
|
fs2 = smoothn(fn, s=2 * s)
|
|
|
|
|
plt.subplot(131),
|
|
|
|
|
plt.contourf(xp, xp, fn)
|
|
|
|
|
plt.subplot(132),
|
|
|
|
|
plt.contourf(xp, xp, fs2)
|
|
|
|
|
plt.subplot(133),
|
|
|
|
|
plt.contourf(xp, xp, f)
|
|
|
|
|
plt.show('hold')
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
def test_smoothn_cardioid():
|
|
|
|
|
t = np.linspace(0, 2 * pi, 1000)
|
|
|
|
|
cos = np.cos
|
|
|
|
|
sin = np.sin
|
|
|
|
|
randn = np.random.randn
|
|
|
|
|
x0 = 2 * cos(t) * (1 - cos(t))
|
|
|
|
|
x = x0 + randn(t.size) * 0.1
|
|
|
|
|
y0 = 2 * sin(t) * (1 - cos(t))
|
|
|
|
|
y = y0 + randn(t.size) * 0.1
|
|
|
|
|
z = smoothn(x + 1j * y, robust=False)
|
|
|
|
|
plt.plot(x0, y0, 'y',
|
|
|
|
|
x, y, 'r.',
|
|
|
|
|
z.real, z.imag, 'k', linewidth=2)
|
|
|
|
|
plt.show('hold')
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
class HodrickPrescott(object):
|
|
|
|
|
|
|
|
|
|
'''Smooth data with a Hodrick-Prescott filter.
|
|
|
|
@ -445,7 +934,7 @@ class Kalman(object):
|
|
|
|
|
H = self.H
|
|
|
|
|
PHT = np.dot(P, H.T)
|
|
|
|
|
innovation_covariance = np.dot(H, PHT) + self.R
|
|
|
|
|
#return np.linalg.solve(PHT, innovation_covariance)
|
|
|
|
|
# return np.linalg.solve(PHT, innovation_covariance)
|
|
|
|
|
return np.dot(PHT, np.linalg.inv(innovation_covariance))
|
|
|
|
|
|
|
|
|
|
def _update_state_from_observation(self, x, z, K):
|
|
|
|
@ -491,7 +980,6 @@ def test_kalman():
|
|
|
|
|
for i, zi in enumerate(z):
|
|
|
|
|
x[i] = filt(zi, u) # perform a Kalman filter iteration
|
|
|
|
|
|
|
|
|
|
import matplotlib.pyplot as plt
|
|
|
|
|
_hz = plt.plot(z, 'r.', label='observations')
|
|
|
|
|
# a-posteriori state estimates:
|
|
|
|
|
_hx = plt.plot(x, 'b-', label='Kalman output')
|
|
|
|
@ -502,8 +990,7 @@ def test_kalman():
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
def lti_disc(F, L=None, Q=None, dt=1):
|
|
|
|
|
'''
|
|
|
|
|
LTI_DISC Discretize LTI ODE with Gaussian Noise
|
|
|
|
|
"""LTI_DISC Discretize LTI ODE with Gaussian Noise.
|
|
|
|
|
|
|
|
|
|
Syntax:
|
|
|
|
|
[A,Q] = lti_disc(F,L,Qc,dt)
|
|
|
|
@ -531,7 +1018,8 @@ def lti_disc(F, L=None, Q=None, dt=1):
|
|
|
|
|
Which can be used for integrating the model
|
|
|
|
|
exactly over time steps, which are multiples
|
|
|
|
|
of dt.
|
|
|
|
|
'''
|
|
|
|
|
|
|
|
|
|
"""
|
|
|
|
|
n = np.shape(F)[0]
|
|
|
|
|
if L is None:
|
|
|
|
|
L = np.eye(n)
|
|
|
|
@ -547,13 +1035,13 @@ def lti_disc(F, L=None, Q=None, dt=1):
|
|
|
|
|
Phi = np.vstack((np.hstack((F, np.dot(np.dot(L, Q), L.T))),
|
|
|
|
|
np.hstack((np.zeros((n, n)), -F.T))))
|
|
|
|
|
AB = np.dot(expm(Phi * dt), np.vstack((np.zeros((n, n)), np.eye(n))))
|
|
|
|
|
#Q = AB[:n, :] / AB[n:(2 * n), :]
|
|
|
|
|
# Q = AB[:n, :] / AB[n:(2 * n), :]
|
|
|
|
|
Q = np.linalg.solve(AB[n:(2 * n), :].T, AB[:n, :].T)
|
|
|
|
|
return A, Q
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
def test_kalman_sine():
|
|
|
|
|
'''Kalman Filter demonstration with sine signal.'''
|
|
|
|
|
"""Kalman Filter demonstration with sine signal."""
|
|
|
|
|
sd = 1.
|
|
|
|
|
dt = 0.1
|
|
|
|
|
w = 1
|
|
|
|
@ -578,17 +1066,17 @@ def test_kalman_sine():
|
|
|
|
|
|
|
|
|
|
# Track and animate
|
|
|
|
|
m = M.shape[0]
|
|
|
|
|
MM = np.zeros((m, n))
|
|
|
|
|
PP = np.zeros((m, m, n))
|
|
|
|
|
_MM = np.zeros((m, n))
|
|
|
|
|
_PP = np.zeros((m, m, n))
|
|
|
|
|
'''In this demonstration we estimate a stationary sine signal from noisy
|
|
|
|
|
measurements by using the classical Kalman filter.'
|
|
|
|
|
'''
|
|
|
|
|
filt = Kalman(R=R, x=M, P=P, A=A, Q=Q, H=H, B=0)
|
|
|
|
|
|
|
|
|
|
# Generate random voltages and watch the filter operate.
|
|
|
|
|
#n = 50
|
|
|
|
|
#truth = np.random.randn(n) * np.sqrt(q) + V0
|
|
|
|
|
#z = truth + np.random.randn(n) * np.sqrt(r) # measurement
|
|
|
|
|
# n = 50
|
|
|
|
|
# truth = np.random.randn(n) * np.sqrt(q) + V0
|
|
|
|
|
# z = truth + np.random.randn(n) * np.sqrt(r) # measurement
|
|
|
|
|
truth = X
|
|
|
|
|
z = Y
|
|
|
|
|
x = np.zeros((n, m))
|
|
|
|
@ -596,7 +1084,6 @@ def test_kalman_sine():
|
|
|
|
|
for i, zi in enumerate(z):
|
|
|
|
|
x[i] = filt(zi, u=0).ravel()
|
|
|
|
|
|
|
|
|
|
import matplotlib.pyplot as plt
|
|
|
|
|
_hz = plt.plot(z, 'r.', label='observations')
|
|
|
|
|
# a-posteriori state estimates:
|
|
|
|
|
_hx = plt.plot(x[:, 0], 'b-', label='Kalman output')
|
|
|
|
@ -620,7 +1107,8 @@ def test_kalman_sine():
|
|
|
|
|
# T,Y,'ro',...
|
|
|
|
|
# T(k),M(1),'k*',...
|
|
|
|
|
# T(1:k),MM(1,1:k),'k-');
|
|
|
|
|
# legend('Real signal','Measurements','Latest estimate','Filtered estimate')
|
|
|
|
|
# legend('Real signal','Measurements','Latest estimate',
|
|
|
|
|
# 'Filtered estimate')
|
|
|
|
|
# title('Estimating a noisy sine signal with Kalman filter.');
|
|
|
|
|
# drawnow;
|
|
|
|
|
#
|
|
|
|
@ -629,9 +1117,11 @@ def test_kalman_sine():
|
|
|
|
|
# end
|
|
|
|
|
#
|
|
|
|
|
# clc;
|
|
|
|
|
# disp('In this demonstration we estimate a stationary sine signal from noisy measurements by using the classical Kalman filter.');
|
|
|
|
|
# disp('In this demonstration we estimate a stationary sine signal '
|
|
|
|
|
# 'from noisy measurements by using the classical Kalman filter.');
|
|
|
|
|
# disp(' ');
|
|
|
|
|
# disp('The filtering results are now displayed sequantially for 10 time step at a time.');
|
|
|
|
|
# disp('The filtering results are now displayed sequantially for 10 time '
|
|
|
|
|
# 'step at a time.');
|
|
|
|
|
# disp(' ');
|
|
|
|
|
# disp('<push any key to see the filtered and smoothed results together>')
|
|
|
|
|
# pause;
|
|
|
|
@ -646,7 +1136,8 @@ def test_kalman_sine():
|
|
|
|
|
# title('Filtered and smoothed estimate of the original signal');
|
|
|
|
|
#
|
|
|
|
|
# clc;
|
|
|
|
|
# disp('The filtered and smoothed estimates of the signal are now displayed.')
|
|
|
|
|
# disp('The filtered and smoothed estimates of the signal are now '
|
|
|
|
|
# 'displayed.')
|
|
|
|
|
# disp(' ');
|
|
|
|
|
# disp('RMS errors:');
|
|
|
|
|
# %
|
|
|
|
@ -658,8 +1149,7 @@ def test_kalman_sine():
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
class HampelFilter(object):
|
|
|
|
|
'''
|
|
|
|
|
Hampel Filter.
|
|
|
|
|
"""Hampel Filter.
|
|
|
|
|
|
|
|
|
|
HAMPEL(X,Y,DX,T,varargin) returns the Hampel filtered values of the
|
|
|
|
|
elements in Y. It was developed to detect outliers in a time series,
|
|
|
|
@ -718,7 +1208,8 @@ class HampelFilter(object):
|
|
|
|
|
window filters. Please visit his blog at:
|
|
|
|
|
http://exploringdatablog.blogspot.com/2012/01/moving-window-filters-and
|
|
|
|
|
-pracma.html
|
|
|
|
|
'''
|
|
|
|
|
|
|
|
|
|
"""
|
|
|
|
|
def __init__(self, dx=None, t=3, adaptive=None, fulloutput=False):
|
|
|
|
|
self.dx = dx
|
|
|
|
|
self.t = t
|
|
|
|
@ -790,7 +1281,7 @@ class HampelFilter(object):
|
|
|
|
|
Y0 = smgauss(X, Y0, DX)
|
|
|
|
|
|
|
|
|
|
T = self.t
|
|
|
|
|
## Prepare Output
|
|
|
|
|
# Prepare Output
|
|
|
|
|
self.UB = Y0 + T * S0
|
|
|
|
|
self.LB = Y0 - T * S0
|
|
|
|
|
outliers = np.abs(Y - Y0) > T * S0 # possible outliers
|
|
|
|
@ -806,7 +1297,6 @@ class HampelFilter(object):
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
def test_hampel():
|
|
|
|
|
import matplotlib.pyplot as plt
|
|
|
|
|
randint = np.random.randint
|
|
|
|
|
Y = 5000 + np.random.randn(1000)
|
|
|
|
|
outliers = randint(0, 1000, size=(10,))
|
|
|
|
@ -827,7 +1317,6 @@ def test_hampel():
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
def plot_hampel(Y, YY, res):
|
|
|
|
|
import matplotlib.pyplot as plt
|
|
|
|
|
X = np.arange(len(YY))
|
|
|
|
|
plt.plot(X, Y, 'b.') # Original Data
|
|
|
|
|
plt.plot(X, YY, 'r') # Hampel Filtered Data
|
|
|
|
@ -836,14 +1325,14 @@ def plot_hampel(Y, YY, res):
|
|
|
|
|
plt.plot(X, res['UB'], 'r--') # Upper Bounds on Hampel Filter
|
|
|
|
|
i = res['outliers']
|
|
|
|
|
plt.plot(X[i], Y[i], 'ks') # Identified Outliers
|
|
|
|
|
#plt.show('hold')
|
|
|
|
|
# plt.show('hold')
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
def test_tide_filter():
|
|
|
|
|
# import statsmodels.api as sa
|
|
|
|
|
# import statsmodels.api as sa
|
|
|
|
|
import wafo.spectrum.models as sm
|
|
|
|
|
sd = 10
|
|
|
|
|
Sj = sm.Jonswap(Hm0=4.* sd)
|
|
|
|
|
Sj = sm.Jonswap(Hm0=4.*sd)
|
|
|
|
|
S = Sj.tospecdata()
|
|
|
|
|
|
|
|
|
|
q = (0.1 * sd) ** 2 # variance of process noise s the car operates
|
|
|
|
@ -851,7 +1340,7 @@ def test_tide_filter():
|
|
|
|
|
b = 0 # no system input
|
|
|
|
|
u = 0 # no system input
|
|
|
|
|
|
|
|
|
|
from scipy.signal import butter, lfilter, filtfilt, lfilter_zi
|
|
|
|
|
from scipy.signal import butter, filtfilt, lfilter_zi # lfilter,
|
|
|
|
|
freq_tide = 1. / (12 * 60 * 60)
|
|
|
|
|
freq_wave = 1. / 10
|
|
|
|
|
freq_filt = freq_wave / 10
|
|
|
|
@ -859,7 +1348,7 @@ def test_tide_filter():
|
|
|
|
|
freq = 1. / dt
|
|
|
|
|
fn = (freq / 2)
|
|
|
|
|
|
|
|
|
|
P = 10* np.diag([1, 0.01])
|
|
|
|
|
P = 10 * np.diag([1, 0.01])
|
|
|
|
|
R = r
|
|
|
|
|
H = np.atleast_2d([1, 0])
|
|
|
|
|
|
|
|
|
@ -876,19 +1365,21 @@ def test_tide_filter():
|
|
|
|
|
|
|
|
|
|
filt = Kalman(R=R, x=np.array([[tide[0]], [0]]), P=P, A=A, Q=Q, H=H, B=b)
|
|
|
|
|
filt2 = Kalman(R=R, x=np.array([[tide[0]], [0]]), P=P, A=A, Q=Q, H=H, B=b)
|
|
|
|
|
#y = tide + 0.5 * np.sin(freq_wave * w * t)
|
|
|
|
|
# y = tide + 0.5 * np.sin(freq_wave * w * t)
|
|
|
|
|
# Butterworth filter
|
|
|
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|
b, a = butter(9, (freq_filt / fn), btype='low')
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#y2 = [lowess(y[max(i-60,0):i + 1], t[max(i-60,0):i + 1], frac=.3)[-1,1] for i in range(len(y))]
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#y2 = [lfilter(b, a, y[:i + 1])[i] for i in range(len(y))]
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#y3 = filtfilt(b, a, y[:16]).tolist() + [filtfilt(b, a, y[:i + 1])[i] for i in range(16, len(y))]
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#y0 = medfilt(y, 41)
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zi = lfilter_zi(b, a)
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#y2 = lfilter(b, a, y)#, zi=y[0]*zi) # standard filter
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# y2 = [lowess(y[max(i-60,0):i + 1], t[max(i-60,0):i + 1], frac=.3)[-1,1]
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# for i in range(len(y))]
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# y2 = [lfilter(b, a, y[:i + 1])[i] for i in range(len(y))]
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# y3 = filtfilt(b, a, y[:16]).tolist() + [filtfilt(b, a, y[:i + 1])[i]
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# for i in range(16, len(y))]
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# y0 = medfilt(y, 41)
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_zi = lfilter_zi(b, a)
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# y2 = lfilter(b, a, y)#, zi=y[0]*zi) # standard filter
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y3 = filtfilt(b, a, y) # filter with phase shift correction
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y4 =[]
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y4 = []
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y5 = []
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for i, j in enumerate(y):
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for _i, j in enumerate(y):
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tmp = filt(j, u=u).ravel()
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tmp = filt2(tmp[0], u=u).ravel()
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# if i==0:
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@ -896,16 +1387,16 @@ def test_tide_filter():
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# print(filt2.x)
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y4.append(tmp[0])
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y5.append(tmp[1])
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y0 = medfilt(y4, 41)
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_y0 = medfilt(y4, 41)
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print(filt.P)
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# plot
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import matplotlib.pyplot as plt
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plt.plot(t, y, 'r.-', linewidth=2, label='raw data')
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#plt.plot(t, y2, 'b.-', linewidth=2, label='lowess @ %g Hz' % freq_filt)
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#plt.plot(t, y2, 'b.-', linewidth=2, label='filter @ %g Hz' % freq_filt)
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# plt.plot(t, y2, 'b.-', linewidth=2, label='lowess @ %g Hz' % freq_filt)
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# plt.plot(t, y2, 'b.-', linewidth=2, label='filter @ %g Hz' % freq_filt)
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plt.plot(t, y3, 'g.-', linewidth=2, label='filtfilt @ %g Hz' % freq_filt)
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plt.plot(t, y4, 'k.-', linewidth=2, label='kalman')
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#plt.plot(t, y5, 'k.', linewidth=2, label='kalman2')
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# plt.plot(t, y5, 'k.', linewidth=2, label='kalman2')
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plt.plot(t, tide, 'y-', linewidth=2, label='True tide')
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plt.legend(frameon=False, fontsize=14)
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plt.xlabel("Time [s]")
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@ -914,14 +1405,31 @@ def test_tide_filter():
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def test_smooth():
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import matplotlib.pyplot as plt
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t = np.linspace(-4, 4, 500)
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y = np.exp(-t ** 2) + np.random.normal(0, 0.05, t.shape)
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coeff = calc_coeff(n=0, degree=0, diff_order=0)
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ysg = smooth(y, coeff, pad=True)
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n = 11
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ysg = SavitzkyGolay(n, degree=1, diff_order=0)(y)
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plt.plot(t, y, t, ysg, '--')
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plt.show()
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plt.show('hold')
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def test_hodrick_cardioid():
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t = np.linspace(0, 2 * np.pi, 1000)
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cos = np.cos
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sin = np.sin
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randn = np.random.randn
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x0 = 2 * cos(t) * (1 - cos(t))
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x = x0 + randn(t.size) * 0.1
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y0 = 2 * sin(t) * (1 - cos(t))
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y = y0 + randn(t.size) * 0.1
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smooth = HodrickPrescott(w=20000)
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|
# smooth = HampelFilter(adaptive=50)
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|
z = smooth(x) + 1j * smooth(y)
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|
plt.plot(x0, y0, 'y',
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|
|
x, y, 'r.',
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|
z.real, z.imag, 'k', linewidth=2)
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|
plt.show('hold')
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def test_docstrings():
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|
@ -930,9 +1438,12 @@ def test_docstrings():
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|
doctest.testmod(optionflags=doctest.NORMALIZE_WHITESPACE)
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|
|
if __name__ == '__main__':
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|
|
#test_kalman_sine()
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|
test_tide_filter()
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|
#test_docstrings()
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|
#test_hampel()
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|
#test_kalman()
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|
test_docstrings()
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|
# test_kalman_sine()
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|
# test_tide_filter()
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|
# test_hampel()
|
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|
|
# test_kalman()
|
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|
|
# test_smooth()
|
|
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|
|
# test_hodrick_cardioid()
|
|
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|
|
test_smoothn_1d()
|
|
|
|
|
# test_smoothn_cardioid()
|
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|