Added SavitzkyGolay class to sg_filter.py
Refined confidence intervals in kreg_demo2 in kdetools.pymaster
Per.Andreas.Brodtkorb
13 years ago
parent
1cdcb0d632
commit
d292a14330
@ -1,84 +1,423 @@
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#from math import *
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import numpy as np
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from numpy import zeros, convolve, dot, linalg, size #@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 abs, size, convolve, linalg, concatenate #@UnresolvedImport
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all = ['calc_coeff','smooth']
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__all__ = ['calc_coeff', 'smooth', 'smooth_last']
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def _resub(D, rhs):
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""" solves D D^T = rhs by resubstituion.
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D is lower triangle-matrix from cholesky-decomposition """
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M = D.shape[0]
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def calc_coeff(n, degree, diff_order=0):
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x1= zeros((M,),float)
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""" calculates filter coefficients for symmetric savitzky-golay filter.
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x2= zeros((M,),float)
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see: http://www.nrbook.com/a/bookcpdf/c14-8.pdf
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# resub step 1
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n means that 2*n+1 values contribute to the
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for l in range(M):
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smoother.
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sum = rhs[l]
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for n in range(l):
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sum -= D[l,n]*x1[n]
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x1[l] = sum/D[l,l]
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# resub step 2
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degree is degree of fitting polynomial
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for l in range(M-1,-1,-1):
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sum = x1[l]
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for n in range(l+1,M):
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sum -= D[n,l]*x2[n]
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x2[l] = sum/D[l,l]
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return x2
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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 calc_coeff(num_points, pol_degree, diff_order=0):
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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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"""
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""" applies coefficients calculated by calc_coeff()
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Calculates filter coefficients for symmetric savitzky-golay filter.
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to signal """
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see: http://www.nrbook.com/a/bookcpdf/c14-8.pdf
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n = size(coeff - 1) // 2
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y = np.squeeze(signal)
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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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class SavitzkyGolay(object):
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r"""Smooth (and optionally differentiate) data with a Savitzky-Golay filter.
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The Savitzky-Golay filter removes high frequency noise from data.
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It has the advantage of preserving the original shape and
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features of the signal better than other types of filtering
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approaches, such as moving averages techniques.
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Parameters
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Parameters
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----------
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----------
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num_points : scalar, integer
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n : int
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means that 2*num_points+1 values contribute to the smoother.
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the size of the smoothing window is 2*n+1.
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pol_degree : scalar, integer
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degree : int
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is degree of fitting polynomial
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the order of the polynomial used in the filtering.
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diff_order : scalar, integer
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Must be less than `window_size` - 1, i.e, less than 2*n.
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is degree of implicit differentiation.
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diff_order : int
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the 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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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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1 means that filter results in smoothing the first derivative of function.
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derivative of function.
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and so on ...
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and so on ...
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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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Examples
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--------
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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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>>> ysg = SavitzkyGolay(n=15, degree=4).smooth(y)
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>>> import matplotlib.pyplot as plt
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>>> plt.plot(t, y, label='Noisy signal')
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>>> plt.plot(t, np.exp(-t**2), 'k', lw=1.5, label='Original signal')
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>>> plt.plot(t, ysg, 'r', label='Filtered signal')
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>>> plt.legend()
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>>> plt.show()
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References
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----------
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.. [1] A. Savitzky, M. J. E. Golay, Smoothing and Differentiation of
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Data by Simplified Least Squares Procedures. Analytical
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Chemistry, 1964, 36 (8), pp 1627-1639.
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.. [2] Numerical Recipes 3rd Edition: The Art of Scientific Computing
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W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery
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Cambridge University Press ISBN-13: 9780521880688
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"""
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"""
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def __init__(self, n, degree=1, diff_order=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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def smooth_last(self, signal, k=0):
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coeff = self._coeff
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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(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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Parameters
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----------
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y : array_like, shape (N,)
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the values of the time history of the signal.
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pad : bool
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pad first and last values to lessen the end effects.
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Returns
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-------
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ys : ndarray, shape (N)
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the smoothed signal (or it's n-th derivative).
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"""
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coeff = self._coeff
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n = size(coeff - 1) // 2
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y = np.squeeze(signal)
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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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# setup normal matrix
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class Kalman(object):
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A = zeros((2*num_points+1, pol_degree+1), float)
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'''
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for i in range(2*num_points+1):
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Kalman filter object - updates a system state vector estimate based upon an
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for j in range(pol_degree+1):
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observation, using a discrete Kalman filter.
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A[i,j] = pow(i-num_points, j)
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The Kalman filter is "optimal" under a variety of
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# calculate diff_order-th row of inv(A^T A)
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circumstances. An excellent paper on Kalman filtering at
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ATA = dot(A.transpose(), A)
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the introductory level, without detailing the mathematical
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rhs = zeros((pol_degree+1,), float)
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underpinnings, is:
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rhs[diff_order] = 1
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D = linalg.cholesky(ATA)
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"An Introduction to the Kalman Filter"
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wvec = _resub(D, rhs)
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Greg Welch and Gary Bishop, University of North Carolina
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http://www.cs.unc.edu/~welch/kalman/kalmanIntro.html
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# calculate filter-coefficients
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coeff = zeros((2*num_points+1,), float)
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PURPOSE:
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for n in range(-num_points, num_points+1):
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The purpose of each iteration of a Kalman filter is to update
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x = 0.0
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the estimate of the state vector of a system (and the covariance
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for m in range(pol_degree+1):
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of that vector) based upon the information in a new observation.
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x += wvec[m]*pow(n, m)
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The version of the Kalman filter in this function assumes that
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coeff[n+num_points] = x
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observations occur at fixed discrete time intervals. Also, this
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return coeff
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function assumes a linear system, meaning that the time evolution
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of the state vector can be calculated by means of a state transition
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matrix.
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USAGE:
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filt = Kalman(R, x, P, A, B=0, u=0, Q, H)
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x = filt(z)
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filt is a "system" object containing various fields used as input
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and output. The state estimate "x" and its covariance "P" are
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updated by the function. The other fields describe the mechanics
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of the system and are left unchanged. A calling routine may change
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these other fields as needed if state dynamics are time-dependent;
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otherwise, they should be left alone after initial values are set.
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The exceptions are the observation vector "z" and the input control
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(or forcing function) "u." If there is an input function, then
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"u" should be set to some nonzero value by the calling routine.
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System dynamics
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---------------
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The system evolves according to the following difference equations,
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where quantities are further defined below:
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x = Ax + Bu + w meaning the state vector x evolves during one time
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step by premultiplying by the "state transition
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matrix" A. There is optionally (if nonzero) an input
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vector u which affects the state linearly, and this
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linear effect on the state is represented by
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premultiplying by the "input matrix" B. There is also
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gaussian process noise w.
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z = Hx + v meaning the observation vector z is a linear function
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of the state vector, and this linear relationship is
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represented by premultiplication by "observation
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matrix" H. There is also gaussian measurement
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noise v.
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where w ~ N(0,Q) meaning w is gaussian noise with covariance Q
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v ~ N(0,R) meaning v is gaussian noise with covariance R
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VECTOR VARIABLES:
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s.x = state vector estimate. In the input struct, this is the
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"a priori" state estimate (prior to the addition of the
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information from the new observation). In the output struct,
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this is the "a posteriori" state estimate (after the new
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measurement information is included).
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s.z = observation vector
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s.u = input control vector, optional (defaults to zero).
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MATRIX VARIABLES:
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s.A = state transition matrix (defaults to identity).
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s.P = covariance of the state vector estimate. In the input struct,
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this is "a priori," and in the output it is "a posteriori."
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(required unless autoinitializing as described below).
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s.B = input matrix, optional (defaults to zero).
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s.Q = process noise covariance (defaults to zero).
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s.R = measurement noise covariance (required).
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s.H = observation matrix (defaults to identity).
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NORMAL OPERATION:
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(1) define all state definition fields: A,B,H,Q,R
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(2) define intial state estimate: x,P
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(3) obtain observation and control vectors: z,u
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(4) call the filter to obtain updated state estimate: x,P
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(5) return to step (3) and repeat
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INITIALIZATION:
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If an initial state estimate is unavailable, it can be obtained
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from the first observation as follows, provided that there are the
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same number of observable variables as state variables. This "auto-
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intitialization" is done automatically if s.x is absent or NaN.
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x = inv(H)*z
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P = inv(H)*R*inv(H')
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This is mathematically equivalent to setting the initial state estimate
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covariance to infinity.
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Example (Automobile Voltimeter):
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-------
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# Define the system as a constant of 12 volts:
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>>> V0 = 12
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>>> h = 1 # voltimeter measure the voltage itself
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>>> q = 1e-5 # variance of process noise s the car operates
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>>> r = 0.1**2 # variance of measurement error
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>>> b = 0 # no system input
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>>> u = 0 # no system input
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>>> filt = Kalman(R=r, A=1, Q=q, H=h, B=b, u=u)
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# Generate random voltages and watch the filter operate.
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>>> n = 50
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>>> truth = np.random.randn(n)*np.sqrt(q) + V0
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>>> z = truth + np.random.randn(n)*np.sqrt(r) # measurement
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>>> x = np.zeros(n)
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>>> for i, zi in enumerate(z):
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... x[i] = filt(zi) # perform a Kalman filter iteration
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>>> import matplotlib.pyplot as plt
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>>> hz = plt.plot(z,'r.', label='observations')
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>>> hx = plt.plot(x,'b-', label='Kalman output') # a-posteriori state estimates:
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>>> ht = plt.plot(truth,'g-', label='true voltage')
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>>> plt.legend()
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>>> plt.title('Automobile Voltimeter Example')
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'''
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def smooth(signal, coeff):
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def __init__(self, R, x=None, P=None, A=None, B=0, u=0, Q=None, H=None):
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"""
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self.R = R
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applies coefficients calculated by calc_coeff()
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self.x = x
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to signal
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self.P = P
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"""
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self.u = u
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self.A = A
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self.B = B
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self.Q = Q
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self.H = H
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self.reset()
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def reset(self):
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self._filter = self._filter_first
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def _filter_first(self, z):
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self._filter = self._filter_main
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auto_init = self.x is None
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if auto_init:
|
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|
n = np.size(z)
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|
else:
|
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|
n = np.size(self.x)
|
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|
if self.A is None:
|
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|
self.A = np.eye(n, n)
|
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|
self.A = np.atleast_2d(self.A)
|
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|
if self.Q is None:
|
||||||
|
self.Q = np.zeros((n, n))
|
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|
self.Q = np.atleast_2d(self.Q)
|
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|
if self.H is None:
|
||||||
|
self.H = np.eye(n, n)
|
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|
self.H = np.atleast_2d(self.H)
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|
# if np.diff(np.shape(self.H)):
|
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|
# raise ValueError('Observation matrix must be square and invertible for state autointialization.')
|
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|
HI = np.linalg.inv(self.H)
|
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|
if self.P is None:
|
||||||
|
self.P = np.dot(np.dot(HI, self.R), HI.T)
|
||||||
|
self.P = np.atleast_2d(self.P)
|
||||||
|
if auto_init:
|
||||||
|
#initialize state estimate from first observation
|
||||||
|
self.x = np.dot(HI, z)
|
||||||
|
return self.x
|
||||||
|
else:
|
||||||
|
return self._filter_main(z)
|
||||||
|
|
||||||
|
|
||||||
|
def _filter_main(self, z):
|
||||||
|
''' This is the code which implements the discrete Kalman filter:
|
||||||
|
'''
|
||||||
|
A = self.A
|
||||||
|
H = self.H
|
||||||
|
P = self.P
|
||||||
|
|
||||||
|
# Prediction for state vector and covariance:
|
||||||
|
x = np.dot(A, self.x) + np.dot(self.B, self.u)
|
||||||
|
P = np.dot(np.dot(A, P), A.T) + self.Q
|
||||||
|
|
||||||
|
# Compute Kalman gain factor:
|
||||||
|
PHT = np.dot(P, H.T)
|
||||||
|
K = np.dot(PHT, np.linalg.inv(np.dot(H, PHT) + self.R))
|
||||||
|
|
||||||
|
# Correction based on observation:
|
||||||
|
self.x = x + np.dot(K, z - np.dot(H, x))
|
||||||
|
self.P = P - np.dot(K, np.dot(H, P))
|
||||||
|
|
||||||
|
# Note that the desired result, which is an improved estimate
|
||||||
|
# of the system state vector x and its covariance P, was obtained
|
||||||
|
# in only five lines of code, once the system was defined. (That's
|
||||||
|
# how simple the discrete Kalman filter is to use.) Later,
|
||||||
|
# we'll discuss how to deal with nonlinear systems.
|
||||||
|
|
||||||
|
|
||||||
|
return self.x
|
||||||
|
def __call__(self, z):
|
||||||
|
return self._filter(z)
|
||||||
|
|
||||||
|
def test_kalman():
|
||||||
|
V0 = 12
|
||||||
|
h = np.atleast_2d(1) # voltimeter measure the voltage itself
|
||||||
|
q = 1e-9 # variance of process noise as the car operates
|
||||||
|
r = 0.05 ** 2 # variance of measurement error
|
||||||
|
b = 0 # no system input
|
||||||
|
u = 0 # no system input
|
||||||
|
filt = Kalman(R=r, A=1, Q=q, H=h, B=b, u=u)
|
||||||
|
|
||||||
|
# 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
|
||||||
|
x = np.zeros(n)
|
||||||
|
|
||||||
|
for i, zi in enumerate(z):
|
||||||
|
x[i] = filt(zi) # perform a Kalman filter iteration
|
||||||
|
|
||||||
|
import matplotlib.pyplot as plt
|
||||||
|
hz = plt.plot(z, 'r.', label='observations')
|
||||||
|
hx = plt.plot(x, 'b-', label='Kalman output') # a-posteriori state estimates:
|
||||||
|
ht = plt.plot(truth, 'g-', label='true voltage')
|
||||||
|
plt.legend()
|
||||||
|
plt.title('Automobile Voltimeter Example')
|
||||||
|
plt.show()
|
||||||
|
|
||||||
N = size(coeff-1)/2
|
|
||||||
res = convolve(signal, coeff)
|
|
||||||
return res[N:-N]
|
|
||||||
|
|
||||||
|
def test_smooth():
|
||||||
|
import matplotlib.pyplot as plt
|
||||||
|
t = np.linspace(-4, 4, 500)
|
||||||
|
y = np.exp(-t ** 2) + np.random.normal(0, 0.05, t.shape)
|
||||||
|
coeff = calc_coeff(num_points=3, degree=2, diff_order=0)
|
||||||
|
ysg = smooth(y, coeff, pad=True)
|
||||||
|
|
||||||
|
plt.plot(t, y, t, ysg, '--')
|
||||||
|
plt.show()
|
||||||
|
if __name__ == '__main__':
|
||||||
|
test_kalman()
|
||||||
|
#test_smooth()
|
||||||
|
|
||||||
|
|||||||
Loading…
Reference in New Issue